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+# UniMRSeg: Unified Modality-Relax Segmentation via Hierarchical Self-Supervised Compensation
+
+Xiaoqi Zhao<sup>1</sup> Youwei Pang<sup>2</sup> Chenyang Yu<sup>3</sup>  
+Lihe Zhang<sup>3</sup> Huchuan Lu<sup>3</sup> Shijian Lu<sup>2</sup> Georges El Fakhri<sup>1</sup>  
+<sup>1</sup Yale University, USA  
+<sup>2</sup Nanyang Technological University, Singapore  
+<sup>3</sup Dalian University of Technology, China  
+xiaoqi.zhao@yale.edu
+
+# Abstract
+
+Multi-modal image segmentation faces real-world deployment challenges from incomplete/corrupted modalities degrading performance. While existing methods address training-inference modality gaps via specialized per-combination models, they introduce high deployment costs by requiring exhaustive model subsets and model-modality matching. In this work, we propose a unified modality-relax segmentation network (UniMRSeg) through hierarchical self-supervised compensation (HSSC). Our approach hierarchically bridges representation gaps between complete and incomplete modalities across input, feature and output levels. First, we adopt modality reconstruction with the hybrid shuffled-masking augmentation, encouraging the model to learn the intrinsic modality characteristics and generate meaningful representations for missing modalities through cross-modal fusion. Next, modality-invariant contrastive learning implicitly compensates the feature space distance among incomplete-complete modality pairs. Furthermore, the proposed lightweight reverse attention adapter explicitly compensates for the weak perceptual semantics in the frozen encoder. Last, UniMRSeg is fine-tuned under the hybrid consistency constraint to ensure stable prediction under all modality combinations without large performance fluctuations. Without bells and whistles, UniMRSeg significantly outperforms the state-of-the-art methods under diverse missing modality scenarios on MRI-based brain tumor segmentation, RGB-D semantic segmentation, RGB-D/T salient object segmentation. The code will be released at https://github.com/Xiaqi-Zhao-DLUT/UniMRSeg.
+
+# 1 Introduction
+
+Visual multi-modal image segmentation has become a cornerstone in critical applications such as autonomous driving [9], medical diagnostics [23], and robotics [67], where complementary visual cues (e.g., RGB-D, MRI sequences) improve scene understanding. Advanced hybrid CNN-Transformer [14, 46, 39, 79, 68, 40], global-local attention [64, 66, 81, 82, 84, 78, 75, 41], multiscale [38, 36, 37, 83], dynamic convolution [35, 77, 76] techniques have achieved remarkable success under idealized settings with complete modalities. However, real-world scenarios often suffer from incomplete modality inputs due to sensor failures, low-quality data and clinical constraints. For example, although it is ideal to use four complementary MRI modalities—fluid-attenuated inversion recovery (Flair), contrast-enhanced T1-weighted (T1ce), T1-weighted (T1), and T2-weighted (T2)—for brain tumor diagnosis, variations in scanning protocols and patient conditions may limit the ability to obtain all MRI scans.
+
+![](images/9877d25aa9ee774c74766aba94f3610782f723b6e6ae491fad14eb4115c112e7.jpg)  
+Figure 1: Top: UniMRSeg has a unified framework and parameters within each segmentation task to handle 21 modality combinations (RGB-Depth: 3, RGB-Thermal: 3, MRI: 15). Bottom: Box plots compare UniMRSeg with existing methods across four benchmarks, displaying average performance (red dots) and standard deviation (error bars).
+
+Recent research on addressing missing modalities falls into two main challenges. Firstly, most methods [71, 58, 72, 80] focus on designing adaptable cross-modal interaction to amalgamate multi-modal features while preserving generalizable architectures for single-modal scenarios. However, during inference, diverse modality combinations require separate model parameters [71, 58] or independent encoder parameters [69, 74], which not only increase resource consumption in practical deployment but also necessitate additional manual or automatic modality classification as a prerequisite. Although some works [25, 8] leverage the knowledge distillation from complete to incomplete modalities, they still demand multiple models for each modality subset, complicating clinical deployment. Secondly, modality reconstruction-based methods [80, 69, 25, 85] aim to predict missing modality inputs to align features during training and inference. Since segmentation requires precise spatial features and boundary information, the pre-trained reconstruction model prioritizes global feature compression, resulting in insufficient feature representation. Therefore, it is difficult to directly inherit the ability to reduce the modality gap obtained by input-level [80, 25] or feature-level [69, 85] reconstruction for downstream tasks (i.e., image segmentation). In particular, cascading low-quality reconstruction predictions [25] as input to the segmentation network will increase error propagation and degrade performance.
+
+In this paper, we propose a unified modality-relax segmentation framework (UniMRSeg). As shown in Fig. 1, UniMRSeg shares  $100\%$  of its parameters across all possible modality input combinations in a given segmentation task. Since complete modality inputs typically yield the best prediction, our goal is to ensure that UniMRSeg, after training, can approach complete modality representation quality during inference with
+
+![](images/71cf1e9975a7f4390c89c7fc1d9cf563257132bc1dfa903c00bf132c3d363563.jpg)  
+Figure 2: Illustration of the hierarchical compensation.
+
+arbitrary modality inputs. To achieve this, we fully exploit the power of self-supervision [1, 16, 4, 57] in representation learning and propose a hierarchical compensation mechanism that operates at the input level, feature level, and output level, as shown in Fig. 2. First, we adopt cross-modal reconstruction as a pretext task. Unlike previous methods [80, 2] that either discard complete modalities or employ partial masking strategies, our approach simultaneously applies global and local masking mechanisms. This dual masking design enables the model to capture both fine-grained local patterns and holistic semantic representations from intra-modality and cross-modality interactions.
+
+Furthermore, we introduce a channel-wise modality shuffling operation that deliberately breaks the correspondence between input and reconstructed modalities. This operation implicitly formulates a modality classification task, compelling the model to disentangle modality-specific characteristics while eliminating dependence on modality category priors during inference. Next, we leverage contrastive learning to finish the feature-level compensation. Specifically, we construct complete and randomly missing modalities from the same sample as positive pairs, and those from other samples as negative pairs. To enhance the inheritance of representation for downstream segmentation tasks [33, 7], we jointly optimize the spatial distance metric and object segmentation at this stage to guide feature clustering in a direction that is beneficial for the results of the segmentation. Considering the inevitable prediction errors in the aforementioned pixel-level reconstruction and feature contrast learning, they may still limit the boundaries of the compensation mechanism. Inspired by adapter-based methods [26, 5, 61], we design a lightweight reverse attention adapter to explicitly compensate for weak perceptual semantics in the frozen encoder, while embedding a 3D Swin Transformer [28, 52] to capture high-response mutual attention patterns across modalities. By adding feature-level consistency constraints, we ensure that the adapter is aware of the partial representation defects inherent in any missing modal combination. Last, UniMRSeg is fine-tuned by enforcing segmentation consistency constraints, and the knowledge presented by the complete modality at the output will be distilled to all missing modality combinations as supervision information. Through the hierarchical three-level compensation strategy, UniMRSeg achieves optimal average performance, the highest full-modality performance, and minimal performance variance, as shown in Fig. 1.
+
+Our main contributions can be summarized as follows:
+
+- We propose a unified framework, i.e., UniMRSeg, with one set of parameters adapting to varying modality-missing scenarios for general multi-modal segmentation.  
+- We integrate pixel-level modality reconstruction, feature-level contrastive learning, and prediction-level label distillation to construct a hierarchical annotation-free compensation mechanism, breaking through the previous isolated self-supervised research paradigm.  
+- Benefiting from the reverse attention adapter, UniMRSeg can explicitly obtain the compensation about difficult regions with weak perception of the complete modality, aligning missing and complete modality representations.  
+- Extensive comparisons conducted on brain tumor, salient object and semantic segmentation tasks in MRI (15 combinations), RGB-D (3 combinations), and RGB-T (3 combinations) modalities across 2D and 3D images within medical and natural scenes, show that our method consistently achieves the best performance in all individual modality combinations while attaining superior average accuracy with minimal standard deviation.
+
+# 2 Related Works
+
+# 2.1 Incomplete Multi-modal Image Segmentation
+
+There are three popular research patterns to handle missing modalities situations. I) Generalizable Architectures. Some methods [58, 71, 73] aim to fully integrate multi-modal features while minimizing structural modifications in single-modal scenarios. CMX [71] introduces a cross-modal interaction attention mechanism that combines both the CNN and Transformer. Tokenfusion [58] focuses on efficient token-based fusion, pruning multiple single-modal Transformer and repurposing the pruned units for multi-modal fusion. II) Projection-based Methods. Hetero [11] and HeMIS [15] perform arithmetic operations (e.g., averaging) in the projected space to obtain the final segmentation result. SFusion [30] proposes a self-attention-based fusion block, where extracted features from available modalities are projected as tokens and processed through a self-attention layer to capture cross-modal relationships. III) Reconstruction Strategies. M3AE [25] refines the segmentation network by reusing the reconstructed modality images as inputs for fine-tuning. MaskMentor [85] and SSLSOD [80] treat modality reconstruction as a pre-training representation task, where missing and complete modalities share weights for joint training. Zeng et al. [69] enforce consistency between missing modality feature reconstruction and complete modality features at the token level. Different from them, we aim to achieve a simple yet efficient architecture while completing the comprehensive compensation of multi-modal and multi-combinatorial representations with unified parameters by constructing multi-granularity self-supervised tasks that are not limited to pixel/token-level reconstruction.
+
+# 2.2 Self-supervised Learning
+
+Self-supervised learning (SSL) has emerged as a powerful paradigm for learning representations without relying on manual annotations. Three principal SSL mechanisms dominate current research: I) Mask-based SSL involves training models to predict missing or masked parts of an input. Early image inpainting works [42, 70] evolved into modern masked autoencoders [16, 60] using high-ratio masking and asymmetric encoder-decoder architectures to various visual tasks, including image segmentation, depth estimation, and low-level image restoration. II) Contrastive learning encourages models to bring similar data points closer while distancing dissimilar ones. SimCLR [4] establishes data augmentation principles with momentum encoders, while MoCo series [17, 6] address negative sample scarcity through dynamic queues. CLIP [44] leverages cross-modal contrastive learning to embed images and text into a unified semantic space, enabling zero-shot transfer capabilities and providing a universal representation foundation for multi-modal tasks like medical image retrieval and caption generation. III) Knowledge distillation [19] transfers supervision via teacher-student paradigms. In image segmentation, Chen et al. [49] propose to normalize the activation map of each channel to obtain a soft probability map. For cross-modal scenarios, Gupta et al. [13] transfer supervision from labeled RGB images to unlabeled depth and optical flow images. Existing studies usually develop these paradigms in isolation. We hope to take the incomplete multi-modal image segmentation task as an opportunity to effectively combine these three different levels of self-supervision techniques and demonstrate their synergy.
+
+# 3 Approach
+
+Preliminaries. In this section, we adopt the MRI-based brain tumor segmentation with the T1, T1ce, T2 and Flair modalities as the targeted task to describe the proposed multi-stage learning framework. Let  $I \in \mathbb{R}^{1 \times H \times W \times T}$  be the input modality-specific sequence with  $T$  slices for the model, where 1,  $H$ , and  $W$  are the channel, height, and width of the slice. The final model generates the segmentation tensor  $P \in \mathbb{R}^{N \times H \times W \times T}$ , where each channel corresponds to a specific class and indicates the probability of each spatial-temporal location being assigned to that class.
+
+Multi-stage Learning Framework. Our framework aims to bridge the performance gap between complete and incomplete multi-modal inputs through three-stage progressive learning, specifically designed to empower flexible handling of diverse missing modality scenarios while maintaining segmentation robustness. The overall pipeline is shown in Fig. 3. The basic models in all stages follow a unified 3D U-Net-style [45] encoder-decoder structure and embed a 3D ASPP [3] with dilation rates of [1, 6, 12, 18] into the high-level feature. The following sections elaborate on the technical specifics of each learning stage.
+
+# 3.1 Multi-granular Modality Reconstruction
+
+This stage emphasizes enhancing the representation capabilities under diverse potential input-side missing modality scenarios. Existing work [16] has demonstrated that effective data perturbations facilitate learning more robust representations. To encourage the model to mine implicit contextual relationships across modalities, this stage integrates three strategies, including modality dropout, modality shuffle and spatial masking, to enable multi-granular information reconstruction.
+
+Data Perturbation. For the complete multi-modal input, we first apply the random modality dropout strategy to randomly discard some modalities with a  $50\%$  probability and generate the output. And it also preserves at least one modality, thus retaining the fundamental information for overall reconstruction. Furthermore, we randomly shuffle the order of remaining modalities, which mitigates the model's reliance on fixed modality order and further decouples modality-agnostic representations from their inherent sequential dependencies in existing paradigms [85, 25]. Additionally, the following spatial masking strategy randomly masks a portion of the input data, thus simulating missing effects within the sequences from available modalities.
+
+Data Reconstruction. We input these remaining perturbed samples into the 3D U-Net-based reconstruction network, and obtain the output through a ReLU function. The normalized slices from the original complete modalities are used as reconstruction objectives based on the combination loss of L1 and SSIM [59], thus overall achieving self-supervised pre-training in the first stage.
+
+![](images/99ab8efe11a8d219fa44920d61eb8ef23f34ff2bfb327a672737ee0b664fd860.jpg)  
+Figure 3: Multi-stage learning framework built on our encoder-decoder network. Stage 1: complete modality reconstruction based on multi-granular random perturbations. Stage 2: modality-invariant contrastive learning for enhancing incomplete-modality representation. Stage 3: incomplete modality adaptive fine-tuning via hybrid consistency constraints.
+
+# 3.2 Modality-invariant Contrastive Learning
+
+The core idea behind this stage is to implicitly compensate for the feature biases introduced by missing modalities through contrastive learning.
+
+Representation Construction. Without loss of generality, we consider a batch containing two input samples with complete modalities, i.e.,  $I_{1}$  and  $I_{2}$ . By applying random modality dropout to each of them while ensuring at least one and at most three modalities remain active, we can obtain two extended samples  $(\hat{I}_1$  and  $\hat{I}_2)$  with missing modalities. And then, we initialize the segmentation model by load the weights of the first stage except for the last output layer.  $\{I_1, \hat{I}_1, I_2, \hat{I}_2\}$  are then fed into the segmentation model to extract multi-level encoder features and generate the final predictions  $\{P_1, \hat{P}_1, P_2, \hat{P}_2\}$ , respectively. After the global average pooling, those features are converted into four vector sets  $\{f_1^i\}_{i=1}^5$ ,  $\{\hat{f}_1\}_{i=1}^5$ ,  $\{f_2\}_{i=1}^5$ , and  $\{\hat{f}_2\}_{i=1}^5$ , respectively. They are utilized for the following representation contrastive learning.
+
+Contrastive Learning. The aforementioned representation vectors are utilized to construct positive-negative sample relationships. And we introduce the NT-Xent loss [4] in each feature level to minimize distances between positive pairs and maximize separation of negative pairs. Specifically, for the  $i^{th}$  level, positive pairs are from the same input sample and its augmented variant, i.e.,  $I_k^i$  and  $\hat{I}_k$  ( $k \in \{1,2\}$ ). Negative pairs are from distinct input sources, e.g.,  $(I_1, I_2)$  and  $(I_1, \hat{I}_2)$ . Considering the vector set  $\mathbf{f}^i = \{f_1^i, \hat{f}_1^i, f_2^i, \hat{f}_2^i\}$ , we can calculate the NT-Xent loss as follows:
+
+$$
+l ^ {i} (u, v) = - \log \frac {\exp \left(\operatorname {s i m} \left(\mathbf {f} _ {u} ^ {i} , \mathbf {f} _ {v} ^ {i}\right) / \tau\right)}{\sum_ {k = 1} ^ {2 B} \mathbb {I} _ {[ k \neq i ]} \exp \left(\operatorname {s i m} \left(\mathbf {f} _ {u} ^ {i} , \mathbf {f} _ {k} ^ {i}\right) / \tau\right)} \tag {1}
+$$
+
+$$
+L _ {\mathrm {N T - X e n t}} = \sum_ {i = 1} ^ {5} \sum_ {k = 1} ^ {B} \frac {l ^ {i} (2 k - 1 , 2 k) + l ^ {i} (2 k , 2 k - 1)}{2 B \times 5} \tag {2}
+$$
+
+where  $B = 2$  and sim are the batch size and the cosine similarity.  $\mathbb{I}_{[k\neq i]}$  is an indicator function that results in 1 when  $k\neq i$  otherwise 0. Such a contrastive learning encourages the model to learn modality-invariant representations, focusing on the underlying target semantic features rather than being biased by modality-specific characteristics.
+
+Segmentation Constraint. The common Dice loss  $l_{\mathrm{Dice}}$  is used to supervise the segmentation predictions as follows:
+
+$$
+L _ {\mathrm {D i c e}} = \frac {1}{B} \sum_ {k = 1} ^ {B} \sum_ {P \in \left\{P _ {k}, \hat {P} _ {k} \right\}} l _ {\mathrm {D i c e}} \left(P, G _ {k}\right) \tag {3}
+$$
+
+It is worth noting that despite applying the contrastive learning, the segmentation constraint is also necessary to guide the encoder's representation learning. This ensures the learned features are optimized and aligned for our ultimate objective, i.e., the segmentation task.
+
+# 3.3 Incomplete Modality Adaptive Fine-tuning
+
+Due to potential errors in the reconstruction from the first stage and the positive-negative sample distance control in the second stage, we introduce the following adaptive fine-tuning guided by the complete modalities to dynamically compensate for these representation errors. As shown in Fig. 3, our framework implements parallel pipelines to handle complete modality samples and their all potential incomplete counterparts (14 valid variants for 4 MRI modalities). The model directly generates the ideal intermediate features  $\{F^i\}_{i=1}^5$  and segmentation prediction  $P$  as the reference from complete modality samples via the forward propagation. And its prediction  $P$  is supervised by the GT mask using the Dice loss. The incomplete samples additionally require targeted optimization through lightweight adapters  $\{\mathbf{A}^i\}_{i=1}^5$ , which progressively aligns their features  $\{\hat{F}^i\}_{i=1}^5$  and prediction  $\hat{P}$  with those references. To maintain the representation capabilities pre-trained from the first two stages, we freeze the entire encoder during this stage and only fine-tune the decoder and adapters.
+
+Reverse Attention Adapter. Taking the  $i^{th}$  encoder stage as an example, we attach the adapter component to the frozen encoder stage  $\mathbf{E}^i$ . In the feature propagation path for incomplete multi-modal samples, the feature  $\hat{F}^{i - 1}$  from the previous encoder layer is first processed by  $\mathbf{E}^i$  to obtain the base features  $\hat{F}_{cp}^i$ . Meanwhile,  $\hat{F}^{i - 1}$  also undergoes 3D convolutions and output the initial adaptive feature  $\hat{F}_{ada - in}^i$ , which is then integrated with  $\hat{F}_{cp}^i$  via element-wise addition and sequential 3D convolutions. The generated feature  $\hat{F}_h^i$  is fed into a 3D Swin Transformer [29] block to establish global contextual correlations between the  $\hat{F}_{cp}^i$  and  $\hat{F}_{ada - in}^i$ . And then the global average pooling is applied across channel and sequence dimensions, followed by the sigmoid to generate the mutual attention. We hope to capture the difficult semantic parts that cannot be perceived by the first two stages and then compensate  $\hat{F}_{cp}^i$ . Therefore, we apply the reverse operation to the mutual attention map and generate the reverse attention map. And it is multiplied to  $\hat{F}_h^i$  and generate the adapted feature  $\hat{F}_{ada}^i$ , which highlights differential information between modality-complete and -incomplete representations. Finally, the sum  $\hat{F}^i$  of  $\hat{F}_{ada}^i$  and  $\hat{F}_{cp}^i$  replaces original  $\hat{F}_{cp}^i$  as the real input to the subsequent process. The roles played by the above features need to be activated with the help of feature-level consistency constraints.
+
+Hybrid Consistency Constraints. The consistency constraint process depicted in Fig. 3 involves the levels of encoder features and final predictions. For the feature-level consistency, we compare the intermediate features  $\{F^i\}_{i=1}^5$  from modality-complete samples and  $\{\hat{F}^i\}_{i=1}^5$  compensated by the adapter from modality-incomplete samples:
+
+$$
+L _ {f c} = \frac {1}{B} \sum_ {k = 1} ^ {B} \sum_ {m} ^ {M} \frac {1}{5} \sum_ {i = 1} ^ {5} \| F _ {k} ^ {i} - \hat {F} _ {k, m} ^ {i} \| _ {1} \tag {4}
+$$
+
+where  $M$  denotes the number of potential valid modality-incomplete samples and it is 14 in our MRI experiments. Besides, the prediction-level consistency between the predicted segmentation maps from modality-complete and -incomplete samples can be formulated by:
+
+$$
+L _ {p c} = \frac {1}{B} \sum_ {k = 1} ^ {B} \sum_ {m = 1} ^ {M} l _ {\text {D i c e}} \left(P _ {k} ^ {i}, \hat {P} _ {k, m} ^ {i}\right) \tag {5}
+$$
+
+By accumulating these consistency constraints, the parameters of the decoder and adapters are jointly optimized to minimize the difference between the representations from modality-complete and -incomplete samples.
+
+Table 1: Quantitative comparison of brain tumor segmentation on BraTS2020 [32].  $\uparrow$  and  $\downarrow$  indicate that the larger scores and the smaller ones are better, respectively. Note: Official implementations are used when available. All methods are evaluated under a unified missing modality setting when possible. Some methods use different training data and lack released code, making exact re-training infeasible. Extended results under various settings are in the Appendix.
+
+<table><tr><td rowspan="2" colspan="2">Modality</td><td colspan="14">Dice score(%)</td><td></td></tr><tr><td colspan="5">Whole Tumor (Whole)</td><td colspan="4">Tumor Core (Core)</td><td colspan="5">Enhancing Tumor (Enhancing)</td><td></td></tr><tr><td colspan="2">Flair T1 T1ce T2</td><td>NestedFormer [63]</td><td>SFusion [30]</td><td>ShaSpec [55]</td><td>PASSION [48]</td><td>UniMRSeg [48]</td><td>NestedFormer [63]</td><td>SFusion [30]</td><td>ShaSpec [55]</td><td>PASSION [48]</td><td>NestedFormer [63]</td><td>SFusion [30]</td><td>ShaSpec [55]</td><td>PASSION [48]</td><td>UniMRSeg [48]</td><td></td></tr><tr><td>○</td><td>○</td><td>23.22</td><td>58.54</td><td>63.10</td><td>64.83</td><td>75.14</td><td>8.26</td><td>37.20</td><td>42.83</td><td>50.36</td><td>56.31</td><td>2.31</td><td>16.34</td><td>22.15</td><td>30.23</td><td>35.19</td></tr><tr><td>○</td><td>○</td><td>24.61</td><td>60.82</td><td>57.31</td><td>60.92</td><td>68.87</td><td>27.40</td><td>45.19</td><td>54.59</td><td>54.78</td><td>79.20</td><td>35.85</td><td>44.34</td><td>64.93</td><td>69.12</td><td>79.20</td></tr><tr><td>○</td><td>○</td><td>8.15</td><td>55.69</td><td>55.40</td><td>54.10</td><td>60.41</td><td>6.77</td><td>29.84</td><td>34.26</td><td>39.93</td><td>45.70</td><td>3.55</td><td>14.13</td><td>18.23</td><td>22.13</td><td>28.70</td></tr><tr><td>○</td><td>○</td><td>61.18</td><td>74.25</td><td>79.54</td><td>77.35</td><td>85.33</td><td>34.01</td><td>43.23</td><td>52.74</td><td>50.27</td><td>61.81</td><td>29.84</td><td>30.92</td><td>37.40</td><td>39.60</td><td>45.61</td></tr><tr><td>○</td><td>○</td><td>47.22</td><td>72.91</td><td>70.60</td><td>75.82</td><td>81.64</td><td>40.02</td><td>71.82</td><td>75.93</td><td>80.35</td><td>82.01</td><td>49.13</td><td>66.18</td><td>74.23</td><td>79.03</td><td>79.58</td></tr><tr><td>○</td><td>○</td><td>40.87</td><td>60.30</td><td>67.58</td><td>67.90</td><td>72.98</td><td>52.34</td><td>75.62</td><td>78.92</td><td>79.85</td><td>84.60</td><td>56.87</td><td>66.09</td><td>77.13</td><td>78.32</td><td>80.23</td></tr><tr><td>○</td><td>○</td><td>70.54</td><td>84.40</td><td>80.23</td><td>82.39</td><td>86.22</td><td>42.84</td><td>64.31</td><td>60.20</td><td>60.55</td><td>67.90</td><td>33.61</td><td>36.53</td><td>48.08</td><td>46.05</td><td>51.14</td></tr><tr><td>○</td><td>○</td><td>22.77</td><td>67.35</td><td>72.40</td><td>77.71</td><td>77.98</td><td>13.16</td><td>42.92</td><td>52.80</td><td>51.60</td><td>59.71</td><td>7.26</td><td>32.29</td><td>36.02</td><td>33.42</td><td>41.49</td></tr><tr><td>○</td><td>○</td><td>59.18</td><td>83.71</td><td>85.31</td><td>86.64</td><td>87.78</td><td>32.42</td><td>55.62</td><td>63.53</td><td>52.87</td><td>70.00</td><td>29.40</td><td>32.09</td><td>43.52</td><td>43.34</td><td>53.12</td></tr><tr><td>○</td><td>○</td><td>58.57</td><td>84.90</td><td>82.80</td><td>85.74</td><td>87.96</td><td>43.91</td><td>73.30</td><td>76.52</td><td>78.94</td><td>85.71</td><td>54.05</td><td>69.48</td><td>77.28</td><td>79.08</td><td>81.59</td></tr><tr><td>○</td><td>○</td><td>76.67</td><td>80.52</td><td>84.12</td><td>85.42</td><td>86.91</td><td>63.12</td><td>77.30</td><td>79.34</td><td>80.38</td><td>85.92</td><td>69.73</td><td>74.36</td><td>76.02</td><td>78.12</td><td>81.28</td></tr><tr><td>○</td><td>○</td><td>73.00</td><td>78.90</td><td>81.13</td><td>81.31</td><td>85.50</td><td>42.25</td><td>63.62</td><td>61.93</td><td>63.95</td><td>67.43</td><td>32.44</td><td>30.03</td><td>39.89</td><td>43.60</td><td>49.57</td></tr><tr><td>○</td><td>○</td><td>74.30</td><td>80.65</td><td>83.20</td><td>83.52</td><td>87.51</td><td>60.04</td><td>76.28</td><td>78.93</td><td>79.92</td><td>82.78</td><td>64.47</td><td>70.02</td><td>74.68</td><td>79.09</td><td>80.50</td></tr><tr><td>○</td><td>○</td><td>58.77</td><td>70.52</td><td>74.36</td><td>76.31</td><td>76.57</td><td>60.32</td><td>78.48</td><td>80.65</td><td>82.36</td><td>84.92</td><td>69.46</td><td>67.18</td><td>73.39</td><td>76.04</td><td>77.15</td></tr><tr><td>○</td><td>○</td><td>81.07</td><td>84.92</td><td>85.02</td><td>85.90</td><td>88.74</td><td>67.02</td><td>78.84</td><td>84.26</td><td>84.75</td><td>86.01</td><td>73.78</td><td>72.09</td><td>75.59</td><td>80.72</td><td>82.18</td></tr><tr><td colspan="2">Average↑</td><td>52.01</td><td>73.23</td><td>74.81</td><td>76.39</td><td>80.64</td><td>39.59</td><td>60.90</td><td>65.16</td><td>66.06</td><td>73.33</td><td>40.78</td><td>48.14</td><td>55.90</td><td>58.53</td><td>63.10</td></tr><tr><td colspan="2">Std Dev↓</td><td>23.09</td><td>10.47</td><td>10.08</td><td>10.07</td><td>8.43</td><td>19.53</td><td>17.07</td><td>15.45</td><td>15.34</td><td>13.04</td><td>24.20</td><td>21.80</td><td>21.62</td><td>21.84</td><td>19.86</td></tr></table>
+
+Table 2: Quantitative comparison of segmentation performance across RGB, Depth and Thermal modalities.  
+
+<table><tr><td colspan="2">Modality</td><td colspan="5">RGB-D Salient Object Segmentation (STERE [34])</td><td colspan="5">RGB-T Salient Object Segmentation (VT1000 [54])</td><td colspan="5">RGB-D Semantic Segmentation (SUN-RGBD [50])</td></tr><tr><td colspan="2">RGB Depth/Thermal</td><td>SSLSOD CAVER [80]</td><td>CAYER [39]</td><td>PopNet [62]</td><td>GateNet [82]</td><td>UniMRSeg [82]</td><td>SSLSOD [80]</td><td>LSNet [86]</td><td>CAYER [39]</td><td>CONTRINET [51]</td><td>UniMRSeg [51]</td><td>TokenFusion [58]</td><td>CMX CMXNeXt [71]</td><td>MaskMentor [72]</td><td>MultiMentor [85]</td><td>MultiMentor [87]</td></tr><tr><td colspan="2">○</td><td>.846</td><td>.825</td><td>.916</td><td>.844</td><td>.918</td><td>.784</td><td>.698</td><td>.720</td><td>.735</td><td>.847</td><td>48.1</td><td>49.6</td><td>48.6</td><td>49.8</td><td>51.7</td></tr><tr><td>○</td><td>•</td><td>.732</td><td>.614</td><td>.645</td><td>.653</td><td>.839</td><td>.894</td><td>.837</td><td>.861</td><td>.854</td><td>.911</td><td>40.6</td><td>42.9</td><td>41.0</td><td>41.2</td><td>47.3</td></tr><tr><td>•</td><td>•</td><td>.885</td><td>.917</td><td>.917</td><td>.919</td><td>.923</td><td>.930</td><td>.924</td><td>.936</td><td>.929</td><td>.938</td><td>51.0</td><td>52.4</td><td>51.9</td><td>53.0</td><td>53.2</td></tr><tr><td colspan="2">Average ↑</td><td>.821</td><td>.785</td><td>.826</td><td>.805</td><td>.893</td><td>.869</td><td>.820</td><td>.839</td><td>.839</td><td>.899</td><td>46.6</td><td>48.3</td><td>47.2</td><td>48.0</td><td>50.7</td></tr><tr><td colspan="2">Std Dev ↓</td><td>.080</td><td>.155</td><td>.157</td><td>.137</td><td>.047</td><td>.076</td><td>.114</td><td>.110</td><td>.098</td><td>.047</td><td>5.4</td><td>4.9</td><td>5.6</td><td>6.1</td><td>3.1</td></tr></table>
+
+# 4 Experiments
+
+# 4.1 Datasets and Metrics
+
+In this work, we conduct experimental comparisons on four popular visual multi-modal image segmentation tasks to show the generalizability of the proposed method. I) Brain Tumor Segmentation. We follow most brain tumor segmentation methods [22, 63, 10, 30] use the BraTS2020 dataset [32], which contains 369 images with four modality scans: T1ce, T1, T2, and Flair, along with three annotated regions: enhancing tumor, tumor core, and whole tumor, which are mutually inclusive. The dataset is split into training (315), validation (17), and test (37) sets. II) RGB-D Salient Object Segmentation. We adopt the same training set as most methods [80, 82, 39], i.e., 1,485 samples from the NJUD [20] and 700 samples from the NLPR [43]. The test dataset is STERE [34], which contains 1,000 RGB and depth image pairs with complex scenes. III) RGB-T Salient Object Segmentation. We follow the setting of recent works [39, 86, 51], the training set only contains the 2,500 samples from VT5000 [53] and adopt the VT1000 [54] as the test set which contains 1,000 pairs of RGB-T images including more than 400 kinds of common objects collected in 10 types of scenes under different illumination conditions. IV) RGB-D Semantic Segmentation. SUN-RGBD [50] is the popular indoor scene benchmark with 37 classes. It contains 10,335 pairs of RGB-D images, with 5,285 pairs allocated for training and 5,050 for testing. We introduce some widely used metrics in each field for fair evaluation, including Dice for brain tumor segmentation, S-measure [12] ( $S_{m}$ ) for salient object segmentation and IoU for semantic segmentation.
+
+# 4.2 Implementation Details
+
+All experiments are conducted on one NVIDIA A800 GPU. We adopt basic image augmentation techniques to avoid overfitting, including random flipping, rotating and border clipping. We train the model for 300 epochs based on the AdamW optimizer [31] with a warmup schedule, an initial learning rate of 0.0001, and a weight decay of 0.00001. For RGB-D and RGB-T tasks, we concatenate the inputs into a 4-channel tensor to maintain the single-stream architecture. For fair comparison, we separately adopt ResNet-50 [18] and ConvNext-B [27] as the backbone for RGB-D/T salient object segmentation and RGB-D semantic segmentation, which are widely used in their respective fields.
+
+![](images/349a51437694fd12cefb41fce54c9999e59dadbddcb007390f59132d076d5267.jpg)
+
+![](images/872fb0ef7324cfd7ca00a3a3261ad94875e4ea75f4eef2b02e921783ce32be35.jpg)
+
+![](images/aa1374add408f5841bc5f2bd95464602a81de2d4ee68dfd9091295c411b30d7f.jpg)  
+(a) MRI-based Brain Tumor Segmentation  
+(b) RGB-D Semantic Segmentation  
+Figure 4: Qualitative comparisons of predictions from different methods across different modality combinations. Best viewed on screen.
+
+![](images/9846a098060b389cf35101fe400978ec496fac02174bd9e719703bac4969bea7.jpg)  
+(c) RGB-D Salient Object Segmentation
+
+![](images/766066ccc632a1254cfa6d5e820d8c3ac6b8e8e6daef4b532127d03e8103edd5.jpg)  
+(d) RGB-T Salient Object Segmentation
+
+# 4.3 Evaluation
+
+Quantitative Results. We conduct thorough comparisons on all the four tasks, as shown in Tab. 1 and Tab. 2. All these models are either directly tested for performance or retrained based on their publicly released code. To simulate missing modalities in practice and keep consistent with training, we fill missing modalities in MRI with zero pixel values and directly copy other existing modalities to complete the dual input of any missing modality in RGB-D and RGB-T tasks. In addition, we follow some methods with depth estimation functions [80, 62, 85] to cascade the intrinsic depth predictor as the input of the models for RGB-D salient object segmentation and RGB-D semantic segmentation tasks. It can be seen that our proposed UniMRSeg achieves excellent performance on all tasks with the best average performance and lowest standard deviation. These high accuracy, strong robustness and generalization capabilities across multiple modalities show its potential as a unified, reliable, and efficient segmentation framework.
+
+Qualitative Results. In Fig. 4a, PASSION [48] exhibits significant prediction fluctuations for enhancing tumor (blue) and tumor core (red) under missing modalities, while UniMRSeg maintains highly consistent predictions aligned with ground truth. T1 and Flair modalities notably degrade PASSION's boundary discrimination for whole tumor (green), revealing mutually exclusive fusion deficiencies. In Fig. 4b, UniMRSeg achieves precise door segmentation (red) by fusing color and geometric features across RGB/RGB-D modes, whereas CMX [71] fails completely with depth-only inputs. Leveraging spatial topology of chairs, UniMRSeg robustly segments blurred wall-adjacent boxes using pure depth data, demonstrating superior shape reasoning. Fig. 4c shows PopNet [62] produces identical spherical predictions in RGB/RGB-D modes via embedded depth prediction, but degenerates to depth map binarization without RGB. In contrast, UniMRSeg reconstructs RGB semantics from depth and fuses cross-modal features to accurately segment humans and spheres. Fig. 4d verifies thermal modality's advantage in resolving RGB's color-similarity-induced sticking issues. Remarkably, UniMRSeg achieves complete target segmentation using thermal data alone.
+
+# 4.4 Ablation Study
+
+In this section, we show the effectiveness of each component on the brain tumor segmentation task with the most complex modality combination. The baseline model is 3D-UNet without any pre-training.
+
+Each Components. As shown in Tab. 3, the three data perturbations in stage 1 demonstrate the effectiveness of learning intra- and inter-modal features for multi-granular reconstruction. Their accumulation during pre-training leads to over  $14.7\%$  performance gain over the baseline in subsequent segmentation. Stage 2 highlights the importance of jointly training the segmentation decoder and the encoder guided by spatial distance clustering, which promote each other. Stage 3 further optimizes cross-modal alignment through reverse attention adapters and prediction-level consistency
+
+Table 3: Ablation study of each component.  
+
+<table><tr><td rowspan="2">Models</td><td colspan="3">Average Dice score(%)</td></tr><tr><td>Whole</td><td>Core</td><td>Enhancing</td></tr><tr><td>Baseline</td><td>63.31</td><td>51.60</td><td>38.40</td></tr><tr><td colspan="4">Stage 1: Multi-granular Modality Reconstruction (Sec. 3.1)</td></tr><tr><td>+ Random Modality Dropout</td><td>66.98</td><td>55.47</td><td>42.25</td></tr><tr><td>+ Random Modality Shuffle</td><td>67.78</td><td>56.85</td><td>44.17</td></tr><tr><td>+ Random Spatial Masking</td><td>69.35</td><td>59.89</td><td>47.12</td></tr><tr><td colspan="4">Stage 2: Modality-invariant Contrastive Learning (Sec. 3.2)</td></tr><tr><td>+ Contrastive Learning (Encoder)</td><td>72.45</td><td>64.02</td><td>51.45</td></tr><tr><td>+ Segmentation Constraint (Decoder)</td><td>74.53</td><td>65.25</td><td>53.97</td></tr><tr><td colspan="4">Stage 3: Incomplete Modality Adaptive Fine-tuning (Sec. 3.3)</td></tr><tr><td>+ Feature-Level Consistency (Adapter)</td><td>78.12</td><td>69.38</td><td>59.25</td></tr><tr><td>+ Prediction-Level Consistency (Segmentation)</td><td>80.64</td><td>73.33</td><td>63.10</td></tr><tr><td colspan="4">Three-Stage Design vs. Unified Single Stage</td></tr><tr><td>Three-Stage Design</td><td>80.64</td><td>73.33</td><td>63.10</td></tr><tr><td>Unified Single Stage</td><td>20.32</td><td>13.67</td><td>10.03</td></tr></table>
+
+Table 4: Ablation study of the reverse attention adapter.  
+
+<table><tr><td rowspan="2">Models</td><td colspan="3">Average Dice score(%)</td></tr><tr><td>Whole</td><td>Core</td><td>Enhancing</td></tr><tr><td>UniMRSeg</td><td>80.64</td><td>73.33</td><td>63.10</td></tr><tr><td>w/o Rerseve Attention</td><td>78.92</td><td>69.26</td><td>60.45</td></tr><tr><td>w/o Rerseve Attention+Mutual Attention</td><td>78.12</td><td>68.98</td><td>60.23</td></tr><tr><td>w/o 3D Swim Transformer+Rerseve Attention+Mutual Attention</td><td>77.44</td><td>68.74</td><td>59.24</td></tr><tr><td>Fine-tune Encoder in Stage 3</td><td>77.05</td><td>68.15</td><td>58.20</td></tr></table>
+
+Table 5: Evaluation of different compensation levels.  
+
+<table><tr><td rowspan="2">Input Level (Stage 1)</td><td rowspan="2">Feature Level (Stage 2 and Adapter)</td><td rowspan="2">Output Level (Segmentation Consistency)</td><td colspan="3">Average Dice score(%)</td></tr><tr><td>Whole</td><td>Core</td><td>Enhancing</td></tr><tr><td>✓</td><td>-</td><td>-</td><td>69.35</td><td>59.89</td><td>47.12</td></tr><tr><td>-</td><td>✓</td><td>-</td><td>72.46</td><td>62.30</td><td>50.92</td></tr><tr><td>-</td><td>-</td><td>✓</td><td>69.47</td><td>57.25</td><td>48.31</td></tr><tr><td>✓</td><td>✓</td><td>-</td><td>78.12</td><td>69.38</td><td>59.25</td></tr><tr><td>✓</td><td>-</td><td>✓</td><td>74.45</td><td>66.40</td><td>54.38</td></tr><tr><td>-</td><td>✓</td><td>✓</td><td>75.65</td><td>67.48</td><td>54.95</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>80.64</td><td>73.33</td><td>63.10</td></tr><tr><td>Baseline</td><td></td><td></td><td>63.31</td><td>51.60</td><td>38.40</td></tr></table>
+
+![](images/2527cd439c12e1ad2472d0dbd464ec42bc0574721059d9a03ac6e1454d4decd4.jpg)  
+(a) Baseline
+
+![](images/09d7d0f6b745d12fba3c4790f04a8b3ad87bd43e60e5e8589b8ced9a2edf167c.jpg)  
+(b) Stage 1  
+Figure 5: Average response value of feature maps of each encoding layer under different modal combinations at different stages.
+
+![](images/af1a093be615a132c40b912a1e46fad494d80c518c39968721304ee565a5a691.jpg)  
+(c) Stage 2
+
+![](images/3df38383ee1fdc2c06d50ae1e8bb66e3a7ff917c18c2c312779f672f8e792af7.jpg)  
+(d) Stage 3
+
+constraints. The latter alone surpasses the stage 2 decoder by  $12.0\%$ , showing the importance of enforcing prediction consistency across modalities. Gains in each stage are consistently and gradually improved, with the final model outperforming the baseline by over  $44.6\%$  on average.
+
+Three-Stage Design vs. Unified Single Stage. UniMRSeg is built upon a self-supervised pretraining framework, with training initialized from scratch starting at Stage 1. For each task (e.g., BraTS 2020), self-supervised training is conducted solely on the task's own training set, rather than relying on externally labeled datasets such as ImageNet-pretrained weights for initialization. Our three-stage design includes: 1) Pretrain both the encoder and decoder through arbitrary modality reconstruction. 2) Pretrain the encoder with contrastive learning. The segmentation task here is only used to guide the contrastive objective, not as a final goal. 3) Perform the downstream segmentation task. In this way, our three-stage design explicitly aims to reduce the representation gap between complete and incomplete modalities in the encoder-decoder space during the final segmentation stage. If all these tasks are trained jointly in a single-stage model, it would fall into the scope of multi-task learning, rather than self-supervised pretraining.
+
+In multi-task learning, the goal is to leverage multiple complementary clues for collaborative learning. These clues can come from the data level [21, 56, 77], or from structural supervision types [47, 24, 65]. Most approaches often incorporate deliberate designs for task-sharing and task-specific components to enable effective joint prediction across tasks. If the three stages are forcibly merged into a single training stage, two major issues will arise: 1) The model is trained with inputs that involve Random Modality Dropout, Random Modality Shuffle, and Random Spatial Masking. The encoder is supervised using the NT-Xent contrastive loss, an adapter is incorporated, and the model is simultaneously tasked with both segmentation and reconstruction. Such a fully entangled training setup lacks a clear task hierarchy, and there is no explicit coordination among the input, encoder, decoder, and output. 2) When all tasks are treated equally without ordering, joint optimization becomes highly difficult. In the unified single-stage training attempt, the total loss involves six parts. As shown in Tab. 3, we compare the three-stage model with the unified single-stage model. We observed two phenomena during the experiments: 1) The single stage model failed to converge, with loss plateauing early. 2) The optimization process was highly unstable, with different losses fluctuating in turn and failing to decrease consistently together. These phenomena and performance clearly indicate that unified single stage training is unable to achieve effective coordination among the various designs and supervision signals. The lack of a clear training order, combined with competing optimization objectives, leads to mutual interference.
+
+Reverse Attention Adapter. Tab. 4 shows ablations on the reverse attention adapter. There are three key findings: 1) Removing reverse attention alone leads to a  $4.2\%$  average drop, confirming its role in semantic compensation for missing modalities. 2) Subsequent elimination of mutual attention yields negligible performance variation, indicating that only emphasizing high-response regions between adapter layers and frozen encoding layers cannot achieve meaningful feature compensation. 3) Replacing the 3D Swin Transformer with a 3D convolution of similar parameters significantly degrades performance, validating the Transformer's advantage in cross-modal correlation modeling. Additionally, we compare freezing vs. fine-tuning the encoder during stage 3. Fine-tuning causes over  $6.4\%$  performance drops, indicating that it destroys the adapter-encoder synergy and the compensation mechanism relies on stable encoder representations rather than task-specific tuning.
+
+To further explain this observation, we provide two perspectives: the inheritance between Stage 2 and Stage 3, and the rationale behind the lightweight reverse attention adapter (RAA). 1) Unlike general SSL methods [4, 17, 16] that focus on learning generic representations without targeting specific downstream tasks, our contrastive learning in Stage 2 is task-aware. We co-train the segmentation head to guide the encoder toward modality-invariant features that are directly beneficial for segmentation, rather than general-purpose representations. This task-guided design ensures that the learned contrastive space aligns with the downstream segmentation objective. Therefore, fine-tuning the encoder in Stage 3 would undermine the task-guided contrastive representations that were carefully established. 2) The RAA itself is designed as a residual correction bridging the encoder representation gap between incomplete and complete modality inputs. Formally:
+
+$$
+f _ {\text {i n c}} + \mathcal {A} \left(f _ {\text {i n c}}\right) \approx f _ {\text {c o m}}, \tag {6}
+$$
+
+where  $f_{\mathrm{inc}}$  denotes encoder features from incomplete modalities,  $f_{\mathrm{com}}$  represents encoder features from complete modalities, and  $\mathcal{A}(\cdot)$  is the learnable adapter. During Stage 3, both  $f_{\mathrm{inc}}$  and  $f_{\mathrm{com}}$  are frozen, while only  $\mathcal{A}$  is trained. This constrained setup ensures that the adapter focuses purely on compensating missing information, enabling stable and efficient optimization. Once the encoder is unfrozen, however, all three components become variables, making their roles unclear and the optimization unstable. In conclusion, freezing the encoder in Stage 3 is a deliberate and necessary design choice to preserve the task-guided contrastive representations learned earlier. The RAA thus works in synergy with a stable encoder to effectively compensate for missing modalities and maintain robust segmentation performance.
+
+Hierarchical Compensation Mechanism. As shown in Tab. 5, single-level compensations each surpass the baseline. Cross-level combinations reveal nonlinear synergy. Notably, merely integrating input- and feature-level compensations outperforms existing methods in Tab. 1. The fully compensated model achieves  $36.2\%$  average gain over baseline across categories, demonstrating consistent collaborative synergy among all three levels without mutual exclusion. To quantify modality representation gaps, Fig. 5 visualizes the average feature activation values across different encoding layers under various modality combinations. Fig. 5a displays the baseline visual distributions, where single-modality, dual-modality, and triple-modality combinations exhibit pronounced representation gaps compared to the full-modality setup. Fig. 5b shows results after stage 1 pre-training, where the compensation capability for single-modality inputs is significantly enhanced. Fig. 5c illustrates the results under the joint constraints of contrastive learning and segmentation tasks, where the representations of all the modalities are further aligned. In Fig. 5d, the reverse attention adapter in stage 3 effectively compensates for cumulative errors from modality reconstruction in stage 1 and spatial discrepancy differentiation in stage 2, using only a small number of parameters.
+
+# 5 Conclusion
+
+In this work, we propose a novel modality-relax segmentation framework based on a hierarchical self-supervised compensation strategy. Through multi-granularity data perturbation, segmentation task-guided feature distance constraints, and the design of a reverse attention adapter, we simultaneously integrate three different self-supervised techniques into the proposed UniMRSeg model and complete the representation compensation of the missing modality from the input level, feature level, and output level. UniMRSeg is simple yet effective, achieving dominant performance in four different multimodal image segmentation tasks. We hope that this research paradigm focusing on representation-level compensation can inspire more visual tasks that require modality-relax conditions in the future.
+
+Acknowledgements: This work was supported by the National Natural Science Foundation of China under Grant 62431004, and by Dalian Science and Technology Innovation Foundation under Grant 2023JJ12GX015.
+
+# References
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+
+# NeurIPS Paper Checklist
+
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+
+# 3. Theory assumptions and proofs
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+Table 6: Quantitative comparison of brain tumor segmentation on BraTS2020.  $\uparrow$  and  $\downarrow$  indicate that higher and lower scores are better, respectively. Note: The reported results of RFNet [10] and mmFormer [74] are taken from the PASSION paper [48], where they are evaluated using the same training and test splits of BraTS2020.  
+
+<table><tr><td rowspan="2" colspan="3">Modality</td><td colspan="10">Dice score(%)</td></tr><tr><td colspan="3">Whole Tumor (Whole)</td><td colspan="3">Tumor Core (Core)</td><td colspan="4">Enhancing Tumor (Enhancing)</td></tr><tr><td colspan="3">Flair T1 T1ce T2</td><td>RFNet mmFormer PASSION-RFNet UniMRSeg [10]</td><td>[74]</td><td>[48]</td><td>RFNet mmFormer PASSION-RFNet UniMRSeg [10]</td><td>[74]</td><td>[48]</td><td>RFNet mmFormer PASSION-RFNet UniMRSeg [10]</td><td>[74]</td><td>[48]</td><td>RFNet mmFormer PASSION-RFNet UniMRSeg [10]</td></tr><tr><td>○</td><td>○</td><td>○●</td><td>82.25</td><td>83.33</td><td>81.35</td><td>84.54</td><td>66.77</td><td>63.74</td><td>53.69</td><td>68.94</td><td>40.08</td><td>38.20</td></tr><tr><td>○</td><td>○</td><td>●○</td><td>69.27</td><td>69.87</td><td>75.02</td><td>80.49</td><td>77.24</td><td>76.40</td><td>77.25</td><td>78.33</td><td>65.81</td><td>67.25</td></tr><tr><td>○</td><td>●</td><td>○○</td><td>68.53</td><td>69.60</td><td>72.20</td><td>79.72</td><td>59.53</td><td>56.00</td><td>57.20</td><td>63.32</td><td>31.99</td><td>28.66</td></tr><tr><td>●</td><td>○</td><td>○○</td><td>82.28</td><td>83.34</td><td>83.95</td><td>84.45</td><td>64.30</td><td>61.74</td><td>58.60</td><td>66.69</td><td>36.67</td><td>33.76</td></tr><tr><td>○</td><td>○</td><td>●●</td><td>83.94</td><td>85.47</td><td>82.60</td><td>85.66</td><td>81.90</td><td>81.36</td><td>78.16</td><td>82.43</td><td>69.18</td><td>69.83</td></tr><tr><td>○</td><td>●</td><td>●○</td><td>74.07</td><td>74.74</td><td>76.45</td><td>81.46</td><td>81.45</td><td>80.50</td><td>79.83</td><td>82.06</td><td>68.56</td><td>70.70</td></tr><tr><td>●</td><td>●</td><td>○○</td><td>85.51</td><td>86.60</td><td>86.78</td><td>86.27</td><td>70.28</td><td>67.07</td><td>65.57</td><td>72.00</td><td>41.19</td><td>38.51</td></tr><tr><td>○</td><td>●</td><td>○●</td><td>84.90</td><td>85.81</td><td>82.30</td><td>85.97</td><td>70.39</td><td>66.61</td><td>61.70</td><td>72.09</td><td>43.61</td><td>41.02</td></tr><tr><td>●</td><td>○</td><td>○●</td><td>86.42</td><td>87.73</td><td>86.85</td><td>86.76</td><td>70.70</td><td>68.66</td><td>61.54</td><td>72.34</td><td>43.62</td><td>42.17</td></tr><tr><td>○</td><td>●</td><td>○●○</td><td>85.18</td><td>87.39</td><td>85.57</td><td>86.54</td><td>80.16</td><td>79.67</td><td>78.38</td><td>81.91</td><td>68.38</td><td>68.11</td></tr><tr><td>●</td><td>●</td><td>○○</td><td>86.61</td><td>87.99</td><td>87.46</td><td>88.81</td><td>81.67</td><td>80.71</td><td>79.54</td><td>82.41</td><td>69.47</td><td>70.86</td></tr><tr><td>●</td><td>●</td><td>○●</td><td>87.63</td><td>88.60</td><td>87.91</td><td>89.87</td><td>72.83</td><td>69.89</td><td>66.50</td><td>80.69</td><td>45.21</td><td>43.22</td></tr><tr><td>○</td><td>●</td><td>○●●</td><td>87.36</td><td>88.84</td><td>87.45</td><td>89.46</td><td>81.97</td><td>81.25</td><td>78.88</td><td>83.68</td><td>68.76</td><td>69.91</td></tr><tr><td>○</td><td>●</td><td>●●●</td><td>85.40</td><td>86.40</td><td>83.04</td><td>89.04</td><td>83.28</td><td>82.23</td><td>80.00</td><td>84.64</td><td>71.04</td><td>70.82</td></tr><tr><td>●</td><td>●</td><td>●●</td><td>88.30</td><td>89.27</td><td>88.20</td><td>90.45</td><td>82.80</td><td>81.90</td><td>80.79</td><td>85.26</td><td>69.64</td><td>70.62</td></tr><tr><td colspan="3">Average ↑</td><td>82.51</td><td>83.67</td><td>83.15</td><td>85.97</td><td>75.02</td><td>73.18</td><td>70.51</td><td>77.12</td><td>55.55</td><td>54.91</td></tr><tr><td colspan="3">Std Dev ↓</td><td>6.27</td><td>6.45</td><td>4.86</td><td>3.26</td><td>7.45</td><td>8.48</td><td>9.66</td><td>6.95</td><td>14.56</td><td>16.24</td></tr></table>
+
+Table 7: Quantitative comparison of brain tumor segmentation on BraTS2018.  $\uparrow$  and  $\downarrow$  indicate that higher and lower scores are better, respectively. Note: The reported results of mmFormer [74], M3AE [25] and M3FeCon [69] are taken from the M3FeCon [69], where they are evaluated using the same training and test splits of BraTS2018.  
+
+<table><tr><td rowspan="2" colspan="3">Modality</td><td colspan="11">Dice score(%)</td><td></td></tr><tr><td colspan="3">Whole Tumor (Whole)</td><td colspan="3">Tumor Core (Core)</td><td colspan="5">Enhancing Tumor (Enhancing)</td><td></td></tr><tr><td rowspan="2" colspan="3">Flair T1 T1ce T2</td><td>mmFormer M3AE</td><td>M3FeCon</td><td>UniMRSeg</td><td>mmFormer M3AE</td><td>M3FeCon</td><td>UniMRSeg</td><td>mmFormer M3AE</td><td>M3FeCon</td><td>UniMRSeg</td><td></td><td></td><td></td></tr><tr><td>[74]</td><td>[25]</td><td>[69]</td><td>[74]</td><td>[25]</td><td>[69]</td><td>[74]</td><td>[25]</td><td>[69]</td><td></td><td></td><td></td></tr><tr><td>○</td><td>○</td><td>○●</td><td>81.43</td><td>84.22</td><td>85.13</td><td>87.10</td><td>64.61</td><td>69.14</td><td>72.48</td><td>76.45</td><td>41.92</td><td>46.93</td><td>49.32</td><td>54.20</td></tr><tr><td>○</td><td>○</td><td>●</td><td>72.62</td><td>75.16</td><td>75.26</td><td>82.22</td><td>75.93</td><td>82.53</td><td>82.31</td><td>82.25</td><td>71.37</td><td>73.04</td><td>75.88</td><td>75.46</td></tr><tr><td>○</td><td>●</td><td>○</td><td>67.92</td><td>73.83</td><td>75.24</td><td>82.29</td><td>56.96</td><td>65.77</td><td>65.72</td><td>69.70</td><td>31.38</td><td>36.54</td><td>44.35</td><td>49.20</td></tr><tr><td>●</td><td>○</td><td>○</td><td>86.37</td><td>88.04</td><td>89.05</td><td>89.85</td><td>61.61</td><td>66.02</td><td>69.42</td><td>73.52</td><td>37.98</td><td>34.96</td><td>46.59</td><td>51.35</td></tr><tr><td>○</td><td>○</td><td>●</td><td>83.25</td><td>85.58</td><td>86.61</td><td>88.57</td><td>79.07</td><td>83.85</td><td>84.75</td><td>84.85</td><td>73.13</td><td>74.39</td><td>76.55</td><td>76.75</td></tr><tr><td>○</td><td>●</td><td>○</td><td>74.75</td><td>76.50</td><td>79.16</td><td>83.25</td><td>79.01</td><td>83.11</td><td>82.88</td><td>83.97</td><td>72.77</td><td>74.58</td><td>76.78</td><td>78.41</td></tr><tr><td>●</td><td>●</td><td>○</td><td>87.46</td><td>88.49</td><td>90.17</td><td>91.45</td><td>66.47</td><td>70.53</td><td>72.82</td><td>76.72</td><td>41.75</td><td>48.16</td><td>48.47</td><td>53.20</td></tr><tr><td>○</td><td>●</td><td>○●</td><td>82.46</td><td>86.34</td><td>86.03</td><td>88.15</td><td>69.89</td><td>71.46</td><td>72.01</td><td>77.43</td><td>43.85</td><td>44.73</td><td>50.17</td><td>55.12</td></tr><tr><td>●</td><td>○</td><td>○●</td><td>87.99</td><td>89.31</td><td>90.46</td><td>91.78</td><td>70.18</td><td>70.59</td><td>72.81</td><td>76.95</td><td>46.47</td><td>40.57</td><td>51.12</td><td>56.09</td></tr><tr><td>●</td><td>○●</td><td>○</td><td>87.61</td><td>88.85</td><td>90.35</td><td>91.98</td><td>78.34</td><td>84.03</td><td>84.14</td><td>85.20</td><td>73.93</td><td>74.08</td><td>76.23</td><td>78.43</td></tr><tr><td>●</td><td>●</td><td>○●</td><td>87.77</td><td>88.07</td><td>89.62</td><td>89.99</td><td>80.22</td><td>83.72</td><td>84.81</td><td>85.41</td><td>74.31</td><td>73.42</td><td>76.74</td><td>78.72</td></tr><tr><td>●</td><td>●</td><td>○●</td><td>88.08</td><td>89.24</td><td>90.24</td><td>91.89</td><td>71.97</td><td>72.41</td><td>75.98</td><td>80.26</td><td>46.51</td><td>44.15</td><td>52.63</td><td>60.12</td></tr><tr><td>●</td><td>○</td><td>●●</td><td>88.47</td><td>89.46</td><td>90.07</td><td>92.01</td><td>79.94</td><td>84.25</td><td>84.64</td><td>85.47</td><td>74.53</td><td>74.68</td><td>77.64</td><td>79.02</td></tr><tr><td>○</td><td>●</td><td>●●</td><td>83.08</td><td>85.06</td><td>86.79</td><td>90.48</td><td>80.89</td><td>84.13</td><td>84.39</td><td>85.49</td><td>73.49</td><td>73.37</td><td>77.33</td><td>79.36</td></tr><tr><td>●</td><td>●</td><td>●●</td><td>89.93</td><td>89.56</td><td>90.69</td><td>92.45</td><td>86.23</td><td>84.24</td><td>85.03</td><td>86.44</td><td>76.36</td><td>74.85</td><td>77.81</td><td>79.87</td></tr><tr><td colspan="3">Average ↑</td><td>83.28</td><td>85.18</td><td>86.32</td><td>88.90</td><td>73.42</td><td>77.05</td><td>78.28</td><td>80.67</td><td>58.65</td><td>59.23</td><td>63.84</td><td>67.02</td></tr><tr><td colspan="3">Std Dev ↓</td><td>6.37</td><td>5.29</td><td>5.25</td><td>3.50</td><td>8.02</td><td>7.34</td><td>6.60</td><td>5.05</td><td>16.51</td><td>16.17</td><td>14.05</td><td>12.25</td></tr></table>
+
+# Appendix
+
+# A Performance Comparison on BraTS2020 and BraTS2018
+
+To enable fair comparisons with a broader range of methods, we adopt the same data split settings as used in the PASSION paper [48] for BraTS2020, where the dataset is divided into 219 cases for training, 50 for validation, and 100 for testing. For BraTS2018, we follow the data split protocol from M3FeCon [69], using 200 cases for training and 85 for testing. As shown in Tab. 6 and Tab. 7, our method consistently achieves the best average performance and lowest standard deviation.
+
+# B Necessity of the Three-Stage Training Strategy
+
+The proposed three-stage training pipeline, comprising Multi-granular Modality Reconstruction, Modality-Invariant Contrastive Learning, and Incomplete Modality Adaptive Fine-tuning, is designed to address the unique challenges of missing modality segmentation, which cannot be effectively tackled using standard end-to-end training.
+
+Unlike single-modality RGB image tasks that benefit from large-scale pre-trained models, many other modalities such as depth, infrared, and medical imaging still lack general-purpose pre-trained encoders. As a result, most models initialized randomly suffer from suboptimal feature representations and unstable performance. Moreover, simple end-to-end optimization struggles to bridge the representation gap between complete and incomplete modality inputs, leading to poor generalization to unseen modality combinations.
+
+To this end, each stage in our training strategy plays a complementary role in strengthening the model's robustness and adaptability:
+
+- Stage 1 (Modality Reconstruction): Enables the model to learn cross-modality structural priors and recover semantically meaningful representations from incomplete inputs.  
+- Stage 2 (Contrastive Learning): Encourages the network to align features between complete and missing modality inputs, enhancing modality-invariant representations.  
+- Stage 3 (Unified Fine-tuning): Refines prediction consistency across different modality combinations and ensures deployment under a unified model.  
+- Each stage serves a distinct and complementary role (input-level structure recovery, feature-level modality alignment, and output-level consistency). Extensive ablation studies (Sec. 4.4) validate that each stage contributes distinct improvements in average accuracy. Without any of the stages, performance consistently degrade.
+
+# C Discussion on Training Complexity and Efficiency
+
+Although UniMRSeg adopts a three-stage training strategy, it is deliberately designed for efficiency and simplicity. We analyze the training complexity from two perspectives: concise design and inference simplicity.
+
+**Concise Design:** All three stages in UniMRSeg share a unified 3D U-Net-style encoder-decoder backbone, integrated with a lightweight 3D ASPP module. No additional complex or stacked modules are introduced. This ensures both architectural clarity and training efficiency. Stage 1 utilizes a masked autoencoding-based reconstruction head. This head operates on top of the shared encoder and does not require extra encoder-decoder branches, keeping the design compact. Stage 2 applies a contrastive loss to already-computed latent features, without introducing new network components. Stage 3 focuses on refining segmentation via a lightweight reverse attention adapter and decoder. The encoder is frozen, making this phase computationally efficient and requiring minimal fine-tuning.
+
+# Single-Stage Inference with Unified Weights:
+
+Despite the multi-stage training, inference remains a single-stage process. The final model trained at stage 3 merges all learned representations into a unified network with shared parameters across all modality combinations. Unlike other methods that require ensemble inference [25, 8] or modality-specific branches [69, 74], UniMRSeg performs fast, unified
+
+Table 8: Efficiency comparison.  
+
+<table><tr><td>Metrics</td><td colspan="4">RFNet [10] mmFormer [74] M3AE [25] UniMRSeg</td></tr><tr><td>Parameters (MB)↓</td><td>34</td><td>106</td><td>167</td><td>87</td></tr><tr><td>FLOPs (G)↓</td><td>148</td><td>748</td><td>248</td><td>202</td></tr></table>
+
+inference without requiring modality recognition or dynamic model selection. As shown in Tab. 8, UniMRSeg achieves comparable computational efficiency to state-of-the-art methods.
+
+# D Modality-Agnostic Robustness via Random Modality Shuffle
+
+Previous multi-modal segmentation methods [69, 74, 10, 25] often rely on a fixed modality-to-channel correspondence, where each input modality (e.g., T1, T2, T1ce, Flari) is assigned to a specific encoder or channel during both training and inference. This design imposes a strong prior assumption: the modality type of each input channel must be known in advance and must strictly align with the model's expected input structure during inference. However, such assumptions significantly limit the scalability and automation of AI-driven medical image analysis pipelines, especially in real-world applications where modality labels may be missing, inconsistent, or ambiguous.
+
+To alleviate this constraint, we introduce a random modality shuffle strategy in stage 1, where the input modalities are randomly permuted at each training iteration. This forces the model to learn modality-invariant representations and reduces its reliance on fixed input orders.
+
+As shown in Tab. 9, even after randomly shuffling the modality-channel order five times during inference, the segmentation performance remains stable with minimal variation. These results demonstrate that the random shuffle not only improves performance (see Tab. 3) but also enhances practical robustness and scalability. This aligns with the broader goal of building fully automated and generalizable medical AI systems.
+
+Table 9: Ablation study on the impact of different modality-channel orders during inference. Flair-T1-T1ce-T2 denotes the best-performing fixed input order, while 5-Ave represents the average result of five inference runs using random modality shuffle.
+
+<table><tr><td rowspan="2">Models</td><td colspan="3">Average Dice score(%)</td></tr><tr><td>Whole</td><td>Core</td><td>Enhancing</td></tr><tr><td>Flair-T1-T1ce-T2</td><td>80.64</td><td>73.33</td><td>63.10</td></tr><tr><td>5-Ave</td><td>80.54</td><td>73.27</td><td>63.05</td></tr></table>
+
+# E Limitations and Future Work
+
+While the three-stage training strategy significantly improves model generalizability and segmentation accuracy under missing modalities, it inevitably increases the training pipeline complexity. This additional training overhead may pose challenges for practitioners in time-constrained or resource-limited environments. We acknowledge this trade-off and consider streamlining or accelerating the training process as a critical future direction. Several promising avenues to achieve this include: I) Curriculum-based Optimization. Introducing a curriculum learning schedule that gradually transitions from reconstruction to segmentation may allow for progressive learning in a single stage. II) Modality-aware Parameter Sharing. Parameter-efficient fine-tuning (e.g., adapters or LoRA) across stages may allow most model weights to be reused, reducing memory and time overhead. III) Cross-task Shared Training Objectives. Reformulating the three tasks under a shared objective (e.g., information bottleneck or consistency maximization) may unify their gradients and reduce stage-specific training routines. These directions hold the potential to preserve the benefits of multi-stage training while simplifying the optimization process, making the proposed method more accessible to broader applications in multi-modal image analysis.
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+# UniMotion: A Unified Motion Framework for Simulation, Prediction and Planning
+
+Nan Song<sup>1</sup> Junzhe Jiang<sup>1</sup> Jingyu Li<sup>1,2</sup> Xiatian Zhu<sup>3</sup> Li Zhang<sup>1,2*</sup>
+
+$^{1}$ School of Data Science, Fudan University
+
+$^{2}$ Shanghai Innovation Institute  $^{3}$ University of Surrey
+
+https://github.com/LogosRoboticsGroup/UniMotion
+
+# Abstract
+
+Motion simulation, prediction and planning are foundational tasks in autonomous driving, each essential for modeling and reasoning about dynamic traffic scenarios. While often addressed in isolation due to their differing objectives, such as generating diverse motion states or estimating optimal trajectories, these tasks inherently depend on shared capabilities: understanding multi-agent interactions, modeling motion behaviors, and reasoning over temporal and spatial dynamics. Despite this underlying commonality, existing approaches typically adopt specialized model designs, which hinders cross-task generalization and system scalability. More critically, this separation overlooks the potential mutual benefits among tasks. Motivated by these observations, we propose UniMotion, a unified motion framework that captures shared structures across motion tasks while accommodating their individual requirements. Built on a decoder-only Transformer architecture, UniMotion employs dedicated interaction modes and tailored training strategies to simultaneously support these motion tasks. This unified design not only enables joint optimization and representation sharing but also allows for targeted fine-tuning to specialize in individual tasks when needed. Extensive experiments on the Waymo Open Motion Dataset demonstrate that joint training leads to robust generalization and effective task integration. With further fine-tuning, UniMotion achieves state-of-the-art performance across a range of motion tasks, establishing it as a versatile and scalable solution for autonomous driving.
+
+# 1 Introduction
+
+Motion understanding is a cornerstone of autonomous driving, supporting essential capabilities such as interactive behavior modeling, motion analysis, and decision-making for the ego vehicle. Core motion tasks include motion simulation, trajectory prediction, and ego planning. Among them, trajectory prediction [1, 2, 3] and ego planning [4] are critical intermediaries between perception modules and control systems in the driving pipeline, while motion simulation [5] generates rich and diverse agent behaviors, serving both development and evaluation for motion models. The complexity of real-world driving, characterized by uncertain behaviors and densely interactive environments, makes these tasks challenging.
+
+Accordingly, recent research has produced a wide range of task-specific models to optimize performance for its designated objective. Trajectory prediction and planning models have primarily focused on comprehensive representation learning [6, 7, 8, 9], along with a growing emphasis on precise waypoint estimation [10, 11, 12, 13]. In contrast, motion simulation has leaned heavily on
+
+![](images/31017e98cc176fddb756143f7d12c38469053ae9e2f6b0438c2b6d7e03e63af4.jpg)  
+Figure 1: Overview of our UniMotion pipeline. (a) We train jointly our model with combined generative and forecasting supervision, resulting in a multi-task model that can simultaneously address all kinds of motion tasks in autonomous driving. (b) We further adopt dedicated fine-tuning strategies for each task, producing task-specific models to promote specialization.
+
+![](images/b73e4585398c1992dc228ae5c732f6eb71911509a5a4175c96d148d5a1ecab26.jpg)
+
+generative modeling. Inspired by the success of Large Language Models (LLMs), recent GPT-style architectures [14, 15] have been applied to simulate agent behaviors. Furthermore, this paradigm has also been introduced into prediction [16, 17] and planning [13] tasks to achieve progressive and more accurate regression.
+
+While these task-customized models have proven effective for different motion tasks, it is challenging to transfer models and learned knowledge across tasks. Considering that these tasks are essentially similar due to their reliance on interaction modeling and motion reasoning, our goal is to build a general model that encourages knowledge sharing among motion tasks and enables cross-task generalization. To achieve this aim, we revisit the formulation of motion tasks and propose a unified motion framework for autonomous driving. We begin by abstracting existing motion tasks into two fundamental categories: diverse motion generation and long-range trajectory forecasting, which serve as a conceptual foundation for unification. From this, we develop UniMotion, a unified framework based on a decoder-only Transformer, chosen for its simplicity, scalability, and strong generative capacity across tasks. By incorporating task-aware interaction modes and training strategies (see Figure 1(a)), UniMotion enables joint training across simulation, prediction and planning, fostering shared representations and efficient multi-task learning. To further enhance task proficiency, we also introduce dedicated fine-tuning strategies (see Figure 1(b)) that adapt the unified model to specific objectives, improving performance and practical deployment.
+
+Our contributions are summarized as follows: (i) We abstract motion tasks into two core categories and propose UniMotion, a unified Transformer-based framework that jointly models simulation, prediction and planning to promote generality and cross-task knowledge sharing. (ii) We design effective fine-tuning strategies that specialize the jointly trained model for individual tasks, allowing the model to master task-specific expertise. (iii) Extensive experiments on the Waymo Open Motion Dataset (WOMD) show that UniMotion achieves competitive performance under joint training and sets new state-of-the-art results when fine-tuned for specific tasks.
+
+# 2 Related work
+
+# 2.1 Motion tasks in autonomous driving
+
+Existing motion tasks in autonomous driving can be roughly categorized into three types according to their objectives: motion simulation, trajectory prediction, and ego planning. We provide a detailed discussion for each type in the following.
+
+Motion simulation focuses on simulating diverse and complex motion patterns, augmenting training samples and offering off-board assessment for autonomous driving algorithms. To effectively measure simulation quality, Sim Agents [5] first proposes a benchmark with comprehensive metrics, expecting models to progressively evolve the motion states of all traffic agents. Motivated by the performance of LLMs, existing methods [14, 15] tokenize the agent trajectories and map elements, and employ a scalable decoder-only model to perform auto-regressive trajectory generation. Besides, CATK [18] utilizes closed-loop supervised finetuning with Closest Among Top-K rollouts, which mitigates covariate shift in open-loop behavior cloning and further improves these GPT-style methods. In contrast, UniMM [19] introduces a unified mixture model with both continuous mixture models and discrete GPT-style models to address simulation.
+
+Trajectory prediction anticipates models to rapidly estimate the future trajectories of traffic participants, providing prior knowledge for subsequent decision-making. As a foundation task in autonomous driving, trajectory prediction has been widely explored [6, 7, 8, 11] with Transformer frameworks. Recent methods [20, 21, 22] uncover detailed temporal information and achieve more accurate prediction. Moreover, there are methods utilizing particular strategies for impressive performance improvements, such as model pre-training [23, 24] and trajectory post-refinement [10, 12].
+
+Ego planning is responsible for giving final driving trajectories for ego vehicle according to the driving environment and upstream results. Although rule-based models [25, 26] still hold a crucial position, learning-based methods have emerged and exceeded. PlanTF [27] and PLUTO [28] effectively alleviate the limit of imitation-based planners through designed model architecture and training strategies. Besides, BeTopNet [9] further improves planning performance by explicitly representing behavioral topology among multi-agent future states.
+
+# 2.2 GPT-style motion models
+
+GPT-style LLMs have demonstrated strong capabilities in language tasks, which stimulates the emergence of similar architectures in autonomous driving [29, 30]. Given that both trajectories and text share a sequential structure, it is natural to introduce analogous modules and designs into motion models. In particular, the aforementioned simulation methods directly utilize decoder-only models with next-token prediction for simulation tasks, generating diverse multi-agent trajectory sets. There are also some prediction and planning methods adopting modified auto-regressive decoding. For example, MotionLM [17] and Trajeglish [16] introduce a causal decoder for iterative trajectory prediction, and similarly, CarPlanner [13] adopts consistent auto-regressive planning with reinforcement learning. In contrast, we unify all motion tasks under a GPT-style model, fully exploiting the capability of this framework.
+
+# 3 Methodology
+
+# 3.1 Preliminary
+
+Task definition. Motion understanding in autonomous driving focuses on three tasks: simulation, prediction and planning. All these tasks share the same traffic scenario input, including historical agent states  $\mathcal{A}_h$  and a high-definition (HD) map  $\mathcal{M}$  of driving scenes. Motion simulation is typically designed to support other motion tasks. Given  $\mathcal{A}_h$  as initial motion states, it aims to iteratively and concurrently simulate multi-agent states  $\{\mathcal{A}_s^i\}_{i=1}^K$ , where  $K$  represents the number of generated rollouts. Simulation aims to exhibit high short-term generation diversity. In contrast, trajectory prediction provides distant future trajectories  $\mathcal{A}_f$  of traffic agents over a span of time  $T_f$ . In real-world scenarios, we expect the predictors to generate the results directly and efficiently, rather than relying on segment-by-segment auto-regressive generation [16, 17]. Accordingly, this task evaluates the ability of models to perform accurate long-range predictions. Ego planning aims to determine feasible trajectories for the ego vehicle by referencing surrounding agents. It can be viewed as first predicting the behaviors of other traffic participants, followed by the progressive generation of the ego vehicle's trajectory. In light of the above observations, we summarize motion tasks into two fundamental components: diverse motion generation and long-range trajectory forecasting. As motivated earlier, we strive to integrate these tasks into a unified framework and jointly develop both capabilities in a single model, thereby enhancing overall motion understanding and fostering mutual complementarity between tasks.
+
+Input representation. To ensure the generalization of representations across tasks, we standardize the input format via tokenization following [14, 15]. Concretely, the trajectories of all agents are segmented at fixed time intervals and normalized to form a trajectory set, which is then clustered to construct the agent token vocabulary  $S_{a}$ . Similarly, map tokens  $S_{m}$  are generated by dividing original map polylines into fixed-length segments followed by clustering. The corresponding agent token embeddings  $E_{sa} \in \mathbb{R}^{K_a \times C}$  and map token embeddings  $E_{sm} \in \mathbb{R}^{K_m \times C}$  obtained by applying MLP modules to  $S_{a}$  and  $S_{m}$ , respectively, where  $K_{(\cdot)}$  and  $C$  are the number of tokens and the dim of feature embeddings. Subsequently, the agent states  $\mathcal{A}$  and a HD map  $\mathcal{M}$  are tokenized based on geometric similarity of segments, forming the input agent embeddings  $E_{a} \in \mathbb{R}^{N_{a} \times T \times C}$  and map
+
+![](images/af350afd9c1d30e9a56b86f8a8f97da463017c313d6dbc16fc7ec27b41532b0b.jpg)  
+Figure 2: Overview of UniMotion architecture. It takes as input tokenized agent trajectories and map polylines, and adopts a decoder-only structure equipped with an additional map encoder to yield motion states. By employing task-specific attention masks and training objectives, UniMotion is empowered to flexibly and effectively address multiple motion tasks simultaneously.
+
+embeddings  $E_{m} \in \mathbb{R}^{N_{m} \times C}$ , where  $N_{a}$  and  $N_{m}$  denote the number of agents and map segments, and  $T$  denotes the time frames.
+
+# 3.2 UniMotion for multiple motion tasks
+
+As depicted in Figure 2, the overall design of our UniMotion follows a decoder-only architecture for agent trajectory generation and prediction, with an additional map context encoder. For each specific motion task, we introduce dedicated interactive patterns and training objectives for specialization, but enable weight sharing to learn general representations.
+
+# 3.2.1 Scene context encoding and motion decoding
+
+To ensure the robustness and generalizability of the model, we employ the Transformer architecture with relative positional embedding following [8, 14, 15], which remains invariant under coordinate transformations. Specifically, we first stack several self-attention modules to encode map embeddings, yielding map features  $F_{m}$ . Then, agent embeddings  $E_{a}$ , along with  $F_{m}$ , are further delivered to a agent Transformer decoder constructed with factorized attentions, which are composed of temporal self-attention, agent-map cross-attention and agent-agent self-attention. This can be formulated as:
+
+$$
+F _ {a} = \mathrm {M H S A} _ {t} (\mathrm {Q} = F _ {a}, \mathrm {K} = \mathrm {V} = F _ {a} + \mathcal {R} _ {t} (P _ {a})),
+$$
+
+$$
+F _ {a} = \mathrm {M H C A} _ {a - m} (\mathrm {Q} = F _ {a}, \mathrm {K} = \mathrm {V} = F _ {m} + \mathcal {R} _ {a - m} \left(P _ {a}, P _ {m}\right)), \tag {1}
+$$
+
+$$
+F _ {a} = \mathrm {M H S A} _ {a - a} (\mathrm {Q} = F _ {a}, \mathrm {K} = \mathrm {V} = F _ {a} + \mathcal {R} _ {a - a} (P _ {a})),
+$$
+
+where  $\mathcal{R}$  is the relative positional embedding for spatial and temporal position information  $P_{(\cdot)}$ , and the initial  $F_{a}$  is directly derived from  $E_{a}$ . This decoder module enables the thorough interaction among agents and between agents and road elements, finally giving step-wise motion states.
+
+# 3.2.2 Joint training across motion tasks
+
+Following the designs of multi-task language models, we apply customized interactive attention masks for different motion tasks and introduce tailored training objectives to improve both stepwise generation and long-range prediction. At inference time, we adaptively leverage the model's capabilities according to the specific task.
+
+Motion simulation. The motion simulation task closely resembles language generation, which aims to iteratively select suitable tokens from motion vocabulary. Accordingly, we adopt a causal attention mask and employ Next-Token Prediction (NTP) as the learning strategy, modeling the generation of
+
+![](images/de7c377a769003988d1a37b50166a559c9a0e39cf6885a27a37eb84706e0027f.jpg)  
+(a) RL-based fine-tuning
+
+![](images/fe4ed5c1f183c7e74c6f0c38095be931705c5d7f2ed0b3ee47a94f380e162029.jpg)  
+(b) Multi-modal fine-tuning
+
+![](images/dfdd2faf90e8942867b654ff9baef810e7c8fbafd456d468c629e9b3efffa9c7.jpg)  
+(c) P2G fine-tuning  
+Figure 3: Illustration of task-specific fine-tuning strategies. We adopt (a) RL-based fine-tuning to improve simulation likelihood while maintaining closed-loop consistency. (b) Multi-modal fine-tuning is introduced to refine the multi-modal behavior of predictions. With reference to the planning inference, we utilize (c) Pred2Gen fine-tuning to address distribution mismatch.
+
+the next agent token  $\mathcal{A}_t$  at step  $t$  from the conditional probability  $P(\mathcal{A}_t\mid \mathcal{A}_{< t})$ . During inference, motion processes for all agents are simultaneously simulated through auto-regressive generation.
+
+Trajectory prediction. We regard trajectory prediction as a sequence-to-sequence task, which maps historical segments to future trajectories. Tokens within the historical segments are allowed to attend only to each other, while future tokens can attend both to preceding tokens within the future segment and to all tokens in the historical segments. Moreover, we propose Long-range Future Regression (LFR) to meet the strict requirements of prediction task on accuracy. Specifically, we replace token classification with normalized trajectory regression, and it predicts a complete long-range future trajectory instead of short segments with token-level length, in which case future trajectories  $\mathcal{A}_f$  can be derived from:
+
+$$
+\mathcal {A} _ {f} = \operatorname {M L P} \left(F _ {a}\right), \mathcal {A} _ {f} \in \mathbb {R} ^ {N _ {a} \times T \times T _ {f} \times 2}. \tag {2}
+$$
+
+Notably, all these tokens are utilized for dense supervision in the training phase, while we only need to perform single-pass reasoning at current step for inference.
+
+Ego planning. As mentioned in Section 3.1, we consider planning task to be the combination of generation and prediction. Hence, the training objectives for simulation and prediction naturally benefit the planning task, without requiring any additional training strategies. After training, we adopt a two-stage inference approach for ego trajectory generation. Concretely, we first jointly predict future trajectories for all traffic participants, and then tokenize the future parts of these predictions to form the future tokens. Given the predicted future tokens of surrounding agents, we then progressively generate the planning trajectory of ego vehicle.
+
+# 3.2.3 Task-specific fine-tuning strategies
+
+Despite the capability of solving multiple tasks simultaneously, models are typically expected to exhibit higher specialization for individual tasks in real-world applications. To this end, we introduce task-specific fine-tuning strategies to further enhance the specialization of UniMotion. Overall, we prefer not to introduce additional parameters during fine-tuning stage. According to this principle, we utilize reinforcement learning-based fine-tuning for simulation and two-stage pred2gen finetuning for planning. Additionally, trajectory prediction requires to produce multi-modal outputs, which will significantly increase the computational burden under dense supervision. To alleviate this, we employ a multi-modal decoder to support the prediction process as a compromise.
+
+In line with the purpose of Reinforcement Learning from Human Feedback (RLHF) in LLMs, we enhance the model to generate trajectories that align with human driving behavior and traffic rules, which cannot be constrained through direct supervision alone. Hence, we introduce RL-based fine-tuning for simulation. As shown in Figure 3(a), we first generate  $n$  rollouts of driving scenes as a group following GRPO [31], with activating the gradient for only one rollout. Subsequently,
+
+rewards  $R$  are computed using the same algorithms as the metrics in Sim Agents [5], which consist of kinematic reward and collision reward for simplicity. We formulate this as:
+
+$$
+R = \exp \left(\frac {1}{T} \sum_ {i = 1} ^ {T} \log q ^ {\mathrm {g t}}\right) + \mathbf {1} \{\text {n o t c o l l i s i o n} \}, \tag {3}
+$$
+
+where the kinematic reward is defined as the exponentiated log-likelihood of ground truth samples under the distribution induced by corresponding generated trajectories, whereas the collision reward is an indicator function. Then, the advantage  $A$  is derived from the normalized rewards within the group of each agent. Notably, we only supervise one rollout to reduce computational overhead. More details are provided in Section 3.3 and supplementary materials.
+
+To improve multi-modality of predicted trajectories, we introduce a lightweight transformer decoder similar to [11, 32] as illustrated in Figure 3(b). At this stage, we only focus on the agents of interest that are utilized for evaluation, and perform prediction within perspective local system. To this end, the coordinate-invariant features are transformed into the local agent features  $F_{a}^{*}$  and map features  $F_{m}^{*}$  through adding the local normalized position content encoded by MLP layer. For each interested agent, the estimated multi-modal trajectories  $a_{f}^{*} \in \mathbb{R}^{N_{mo} \times T_{f} \times 2}$  for  $N_{mo}$  modes can be derived by:
+
+$$
+a _ {f} ^ {*} = \operatorname {T r a n s f o r m e r} \left(\mathrm {Q} = F _ {m o}, \mathrm {K} = \mathrm {V} = \left\{F _ {a} ^ {*}, F _ {m} ^ {*} \right\}\right), \tag {4}
+$$
+
+where  $F_{mo}$  denotes the mode features encoded from intention points, and  $F_{a}^{*}$  and  $F_{m}^{*}$  are with respect to current interested agent. Benefiting from distinct intention points, our model can provide more diverse trajectory choices after fine-tuning.
+
+During planning inference, the predicted trajectory tokens for surrounding agents will participate in and significantly affect ego generation, inconsistent with the independent training process. To alleviate potential distribution mismatch, we further fine-tune ego generation using the predicted results. As shown in Figure 3(c), the incorrectly predicted trajectories are first filtered out and replaced by the ground truth. Then, we tokenize all predictions and feed them into our network for ego generation, which is fine-tuned with end-to-end supervision.
+
+# 3.3 Model training
+
+During the joint training process, we mainly adopt NTP classification loss  $\mathcal{L}_{\mathrm{ntp}}$  between next tokens  $\mathcal{A}_s$  and corresponding ground truth  $\hat{\mathcal{A}}_s$  for generation supervision, and LFR regression loss  $\mathcal{L}_{\mathrm{lfr}}$  between  $\mathcal{A}_f$  and ground truth  $\hat{\mathcal{A}}_f$  for trajectory prediction supervision. We combine these two individual losses with equal weights to form the overall loss  $\mathcal{L}$ , formulated as follows:
+
+$$
+\mathcal {L} = \mathcal {L} _ {\mathrm {n t p}} \left(\mathcal {A} _ {s}, \hat {\mathcal {A}} _ {s}\right) + \mathcal {L} _ {\mathrm {l f r}} \left(\mathcal {A} _ {f}, \hat {\mathcal {A}} _ {f}\right), \tag {5}
+$$
+
+where we employ the cross-entropy loss and the smooth-L1 loss for  $\mathcal{L}_{\mathrm{ntp}}$  and  $\mathcal{L}_{\mathrm{lfr}}$ , respectively.
+
+During the fine-tuning stage, we employ different supervision strategies for each motion task. For simulation fine-tuning, we follow [33] to simplify the GRPO but still preserve the consistency constraint through closed-loop supervision on the gradient-activated rollout  $\mathcal{A}_r\in \mathbb{R}^{N_a\times T}$ . Besides, in light of efficiency concerns, we only use  $\mathcal{A}_r$  rather than all rollouts to perform a single policy update per iteration. The simulation fine-tuning loss is:
+
+$$
+\mathcal {L} _ {\mathrm {f t - s i m}} = \mathcal {L} _ {\mathrm {c e}} \left(\mathcal {A} _ {r}, \hat {\mathcal {A}} _ {r}\right) + w _ {\mathrm {s}} \mathcal {L} _ {\mathrm {p}} \left(\mathcal {A} _ {r}, A\right), \tag {6}
+$$
+
+where  $\mathcal{L}_{\mathrm{ce}}$  is the cross-entropy loss between  $\mathcal{A}_r$  and ground truth  $\hat{\mathcal{A}}_r$ ,  $\mathcal{L}_{\mathrm{p}}$  is the policy utilized in GRPO with taking  $\mathcal{A}_r$  and advantage  $A$  as inputs, and  $w_{s}$  is a balancing factor. The prediction loss consists of Gaussian NLL loss  $\mathcal{L}_{\mathrm{nll}}$  to supervise multi-modal trajectories  $a_{f}^{*}$  and cross-entropy loss for mode classification. Meanwhile, the model also learns to regress the local future trajectories  $\mathcal{A}_f^*$  for other agents as auxiliary supervision. The loss can be formulated as:
+
+$$
+\mathcal {L} _ {\mathrm {f t - p r e d}} = w _ {\mathrm {p r}} \mathcal {L} _ {\mathrm {l f r}} \left(\mathcal {A} _ {f} ^ {*}, \hat {\mathcal {A}} _ {f} ^ {*}\right) + \mathcal {L} _ {\mathrm {n l l}} \left(a _ {f} ^ {*}, \hat {a} _ {f} ^ {*}\right) + \mathcal {L} _ {\mathrm {c e}}. \tag {7}
+$$
+
+In addition, we simply supervise the generation  $a_{\mathrm{ego}}$  for ego vehicle and the predictions  $\mathcal{A}_f^{\mathrm{surr}}$  for surrounding agents in the planning fine-tuning stage as:
+
+$$
+\mathcal {L} _ {\mathrm {f t - p l a n}} = w _ {\mathrm {p l}} \mathcal {L} _ {\mathrm {l f r}} \left(\mathcal {A} _ {f} ^ {\text {s u r r}}, \hat {\mathcal {A}} _ {f} ^ {\text {s u r r}}\right) + \mathcal {L} _ {\mathrm {n t p}} \left(a _ {\mathrm {e g o}}, \hat {a} _ {\mathrm {e g o}}\right). \tag {8}
+$$
+
+Table 1: Performance comparison on 2025 Waymo Open Sim Agents Challenge.  
+
+<table><tr><td>Method</td><td>Realism Meta Metric ↑</td><td>Kinematic Metrics ↑</td><td>Interactive Metrics ↑</td><td>Map-based Metrics ↑</td><td>minADE ↓</td></tr><tr><td>UniTAM</td><td>0.7698</td><td>0.4869</td><td>0.7876</td><td>0.9085</td><td>1.3940</td></tr><tr><td>UniTFormer</td><td>0.7776</td><td>0.4892</td><td>0.7997</td><td>0.9140</td><td>1.3592</td></tr><tr><td>LLM2AD</td><td>0.7779</td><td>0.4846</td><td>0.8048</td><td>0.9109</td><td>1.2827</td></tr><tr><td>UniMM [34]</td><td>0.7829</td><td>0.4914</td><td>0.8089</td><td>0.9161</td><td>1.2949</td></tr><tr><td>CATK [18]</td><td>0.7846</td><td>0.4931</td><td>0.8106</td><td>0.9177</td><td>1.3065</td></tr><tr><td>UniMotion</td><td>0.7851</td><td>0.4943</td><td>0.8105</td><td>0.9187</td><td>1.3036</td></tr></table>
+
+Table 2: Performance comparison on WOMD Prediction Leaderboard. For each metric, the best and second best results are highlighted in bold and underline, respectively.  
+
+<table><tr><td>Method</td><td>minADE ↓</td><td>minFDE ↓</td><td>MR ↓</td><td>mAP ↑</td><td>Soft mAP ↑</td></tr><tr><td>HDGT [35]</td><td>0.5933</td><td>1.2055</td><td>0.1854</td><td>0.3577</td><td>0.3709</td></tr><tr><td>MTR [11]</td><td>0.6050</td><td>1.2207</td><td>0.1351</td><td>0.4129</td><td>0.4216</td></tr><tr><td>MTR++ [36]</td><td>0.5906</td><td>1.1939</td><td>0.1298</td><td>0.4329</td><td>0.4414</td></tr><tr><td>MGTR [37]</td><td>0.5918</td><td>1.2135</td><td>0.1298</td><td>0.4505</td><td>0.4599</td></tr><tr><td>EDA [32]</td><td>0.5718</td><td>1.1702</td><td>0.1169</td><td>0.4487</td><td>0.4596</td></tr><tr><td>ControlMTR [38]</td><td>0.5897</td><td>1.1916</td><td>0.1282</td><td>0.4414</td><td>0.4572</td></tr><tr><td>RMP-YOLO [39]</td><td>0.5737</td><td>1.1697</td><td>0.1160</td><td>0.4523</td><td>0.4673</td></tr><tr><td>UniMotion</td><td>0.5718</td><td>1.1643</td><td>0.1162</td><td>0.4534</td><td>0.4642</td></tr></table>
+
+# 4 Experiments
+
+# 4.1 Experimental settings
+
+Datasets and metrics. To evaluate the performance of our method, we conduct extensive experiments on the Waymo Open Motion Dataset, comprising 486,995 training scenarios, 44,097 validation scenarios and 44,920 testing scenarios. Each scenario has an observation window of 9.1 seconds, consisting of 91 frames sampled at  $10\mathrm{Hz}$ . Given 11-frame historical agent states and a HD map, the motion models are tasked with generating or predicting the future states of 80 frames. Additionally, prediction and planning tasks only perform one-step inference at the current time, while simulation task requires 32 simulations per scenario.
+
+We employ respective benchmark metrics to assess our model. For simulation, the core metric is the Realism Meta Metric, which is utilized to measure the similarity between the simulations and the real-world distribution. It is calculated by the weighted sum of Kinematic Metrics for motion features, Interactive Metrics to capture the interaction among agents and Map-based Metrics to test the agent behavior on the map. For prediction, 6 predicted trajectories for each interested agent are expected for evaluation, with metrics including minADE (minimum Average Displacement Error), minFDE (minimum Final Displacement Error), Miss Rate, mAP and Soft MAP. In addition, due to the absence of an official benchmark for planning, we adopt the metrics proposed by [40] as measures.
+
+Implementation details. We tokenize the trajectories and map elements following [15, 18], with 2,048 tokens for agents and 1,024 tokens for map. During the training phase, We train our models using the AdamW [41] optimizer with a total batch size of 48 on 8 NVIDIA A6000 GPUs for 30 epochs, with the weight decay set to 0.01. We initialize the learning rate of  $5 \times 10^{-4}$ , which is then decayed following a cosine annealing schedule. Regarding the fine-tuning stage, we employ the same learning rate for prediction, while a lower learning rate of  $5 \times 10^{-5}$  for simulation and planning. Beside, we also fine-tune these tasks with the same epochs.
+
+Each agent token covers 0.5 seconds, and each map token covers about 0.5 meters. After tokenization, the original 91-step trajectories in Waymo are compressed into 18 frames. For the training phase, the training time for full data is approximately 3.5 days, while using  $20\%$  of the data takes around 1 day in our experimental settings. In addition, we set the balance factor  $w_{\mathrm{s}}$ ,  $w_{\mathrm{pr}}$  and  $w_{\mathrm{pl}}$  to 0.1, 0.1 and 0.5, respectively.
+
+Table 3: Performance comparison for Waymo planning.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Collision ↓</td><td rowspan="2">Red light ↓</td><td rowspan="2">Off route ↓</td><td colspan="3">Planning error ↓</td></tr><tr><td>@1s</td><td>@3s</td><td>@5s</td></tr><tr><td>IL</td><td>5.469</td><td>2.772</td><td>4.816</td><td>0.175</td><td>1.416</td><td>4.194</td></tr><tr><td>IL+Prediction</td><td>3.930</td><td>1.670</td><td>4.542</td><td>0.127</td><td>0.892</td><td>2.901</td></tr><tr><td>Sep. Plan+Pred.</td><td>1.813</td><td>1.327</td><td>0.527</td><td>0.238</td><td>1.251</td><td>3.466</td></tr><tr><td>DIPP [40]</td><td>1.802</td><td>1.235</td><td>0.506</td><td>0.227</td><td>1.187</td><td>3.335</td></tr><tr><td>UniMotion</td><td>1.565</td><td>1.309</td><td>0.477</td><td>0.083</td><td>0.591</td><td>2.246</td></tr></table>
+
+# 4.2 Comparison with state of the art
+
+We compare the performance of our UniMotion with top-ranked competitors on Sim Agents [5], WOMD Prediction and Waymo Planning [40] benchmarks. The leaderboards for Sim Agents and WOMD Prediction are presented in Table 1 and Table 2, respectively. For simulation, our method achieve the best score of 78.51 on Realism Meta Metric. Benefiting from the kinematic reward of fine-tuning, the model is encouraged to generate trajectories with higher motion similarity to real-world situations, driving better Kinematic Metrics than other methods. However, the minADE metric exhibits relatively poorer performance, which can be attributed to the tokenization-based design's stronger emphasis on generation diversity at the cost of slightly reduced prediction accuracy. The leaderboard of the prediction task demonstrate that our method achieves promising results across almost all metrics, except for Soft mAP. Regarding the planning task, UniMotion has far outperformed most of previous approaches as depicted in Table 3, showing that our method stands distinctly ahead of others in terms of planning error. Moreover, to ensure generality and consistency, we supervise model training solely using logged trajectories. This mechanism leads to suboptimal performance in reacting to red lights, which can be mitigated by incorporating traffic rule constraints, as in [40].
+
+# 4.3 General ablation study
+
+We conduct ablation studies on the WOMD validation split to examine the effectiveness of each component in UniMotion. For efficiency, we use only  $20\%$  of the training data for ablation studies, while adopting the default experimental settings following Section 4.1 for all other aspects. We focus on the core modules here, while leaving task-specific ablation to the supplementary materials.
+
+Effects of NTP and LFR. As shown in Table 4, we simply assess the effectiveness of NTP and LFR in our network on simulation and prediction tasks. The first and second rows demonstrate the results of training UniMotion using only NTP or LFR, respectively. Despite the absence of dedicated supervision for another one of tasks, we report all relevant metrics to compare with joint training and showcase the model's ability to handle multiple tasks. It can be observed that the model benefits from both NTP and LFR in generation and prediction, while it performs poorly on another task. In the third row, we implement a joint training strategy with these two types of supervision, simultaneously endowing the model with diverse generative and long-range predictive capabilities. Besides, we find that joint training further improves prediction accuracy compared to using LFR alone, which we attribute to the diverse generation directions serving as targets to guide the prediction process.
+
+Table 4: Ablation on joint training with NTP and LFR. "Kin."/“Inter.”: Kinematic/Interactive metrics.  
+
+<table><tr><td rowspan="2">NTP</td><td rowspan="2">LFR</td><td colspan="4">Simulation</td><td colspan="3">Prediction</td></tr><tr><td>Kin.</td><td>Inter.</td><td>Map</td><td>minADE</td><td>minADE</td><td>minFDE</td><td>mAP</td></tr><tr><td>✓</td><td></td><td>0.4884</td><td>0.7961</td><td>0.9148</td><td>1.4074</td><td>0.7697</td><td>1.5547</td><td>0.2629</td></tr><tr><td></td><td>✓</td><td>0.4401</td><td>0.7742</td><td>0.8949</td><td>1.2965</td><td>0.6668</td><td>1.3613</td><td>0.2935</td></tr><tr><td>✓</td><td>✓</td><td>0.4892</td><td>0.7968</td><td>0.9144</td><td>1.4011</td><td>0.6508</td><td>1.3296</td><td>0.3147</td></tr></table>
+
+Effects of task-specific fine-tuning. After training a preliminary model with a joint strategy, we further tailor it with designs specific to the target task. As demonstrated in Table 5, fine-tuning methods can intuitively bring consistent improvements across all motion tasks. For simulation, all metrics exhibit varying degrees of gains after fine-tuning, especially in the reward items. The 3rd and
+
+4th rows of prediction task show that the mAP metric has been remarkably enhanced by introducing an extra anchor-based decoder, which can cover more potential future trajectories. In addition, noticeable improvements are also observed on the minADE and MR metrics, achieving the expected goals of fine-tuning. Regrading the planning task, the 5th row shows limited performance due to the distribution mismatch between predicted tokens and original trajectory tokens. After fine-tuning, the model is adapted to the two-stage process, and there are considerable improvements for all metrics.
+
+Table 5: Ablation on task-specific fine-tuning for simulation, prediction and planning. Notably, we selected a set of metrics with substantial variation to facilitate comparison. Additionally, we fine-tune planning model with the data following [40] instead of  $20\%$  data.  
+
+<table><tr><td>Task</td><td>ft.</td><td colspan="4">Metrics</td></tr><tr><td rowspan="3">Simulation</td><td>✘</td><td>Kinematic ↑</td><td>Interactive ↑</td><td>Map-based ↑</td><td>minADE ↓</td></tr><tr><td rowspan="2">✓</td><td>0.4892</td><td>0.7968</td><td>0.9144</td><td>1.4011</td></tr><tr><td>0.4939</td><td>0.8032</td><td>0.9159</td><td>1.3904</td></tr><tr><td rowspan="3">Prediction</td><td>✘</td><td>minADE ↓</td><td>minFDE ↓</td><td>MR ↓</td><td>mAP ↑</td></tr><tr><td rowspan="2">✓</td><td>0.6508</td><td>1.3296</td><td>0.1626</td><td>0.3147</td></tr><tr><td>0.6413</td><td>1.3368</td><td>0.1493</td><td>0.3856</td></tr><tr><td rowspan="3">Planning</td><td>✘</td><td>Collision ↓</td><td>Err @1s ↓</td><td>Err @3s ↓</td><td>Err @5s ↓</td></tr><tr><td rowspan="2">✓</td><td>1.7347</td><td>0.1326</td><td>0.7943</td><td>2.7809</td></tr><tr><td>1.5650</td><td>0.0834</td><td>0.5908</td><td>2.2455</td></tr></table>
+
+# 4.4 Task-specific ablation study
+
+In this section, we adopt the same experimental settings and conduct task-specific ablation studies on the simulation and prediction tasks to investigate the effects of different module designs.
+
+Effects of RL-based fine-tuning for simulation. We present the effects of different constraints in Table 6. As shown in the 2nd and 3rd rows, closed-loop supervised fine-tuning leads to notable improvements, while relying solely on RL-based supervision, like DAPO [33], fails to produce satisfactory results. We attribute this to two primary factors: insufficient reward signals and limited model capacity.
+
+Table 6: Ablation on RL-based fine-tuning. "con:": consistency constraint. "Kin."/“Inter.”: Kinematic/Interactive metrics.  
+
+<table><tr><td>con.</td><td>rl</td><td>Kin.</td><td>Inter.</td><td>Map</td><td>minADE</td></tr><tr><td></td><td></td><td>0.4892</td><td>0.7968</td><td>0.9144</td><td>1.4011</td></tr><tr><td>✓</td><td></td><td>0.4921</td><td>0.8019</td><td>0.9156</td><td>1.3933</td></tr><tr><td></td><td>✓</td><td>0.4913</td><td>0.7974</td><td>0.9120</td><td>1.3967</td></tr><tr><td>✓</td><td>✓</td><td>0.4939</td><td>0.8032</td><td>0.9159</td><td>1.3904</td></tr></table>
+
+Effects of fine-tuning for prediction. We present the effects of different fine-tuning strategies in Table 7. The comparison between the 2nd and 3rd rows demonstrates that although the end-to-end fine-tuning strategy (without intention anchors) can significantly reduce prediction errors, it severely compromises prediction diversity and confidence, leading to degraded mAP performance. Hence, we retain the intention anchors as part of the fine-tuning process.
+
+Table 7: Ablation on prediction fine-tuning.  
+
+<table><tr><td>Strategy</td><td>minADE</td><td>minFDE</td><td>MR</td><td>mAP</td></tr><tr><td>w/o ft.</td><td>0.6508</td><td>1.3296</td><td>0.1626</td><td>0.3147</td></tr><tr><td>ft. w/o intention</td><td>0.6174</td><td>1.2639</td><td>0.1313</td><td>0.3389</td></tr><tr><td>ft. w/ intention</td><td>0.6413</td><td>1.3368</td><td>0.1493</td><td>0.3856</td></tr></table>
+
+![](images/a5a30c5386cd09e2f5c86d76d3fd860cce95bf7bd776f949a069b4964a13cbcf.jpg)  
+Figure 4: Qualitative results on WOMD validation split. The top and bottom rows present two rollouts of simulation and the results of prediction and planning, respectively. The red and green arrowed curves are ground truth and predicted trajectories.
+
+# 4.5 Qualitative results
+
+In Figure 4, we present qualitative results of our network on three motion tasks. The first two panels illustrate the diverse simulation results of all traffic agents at 0s, 4s, and 8s, respectively. The bottom-left figure depicts the multi-modal predictions for interested agents, while the bottom-right figure presents the ego trajectory generation process in a token-by-token manner. These qualitative results demonstrate our network's capability in handling multiple motion tasks.
+
+# 5 Conclusion
+
+In this work, we abstract mainstream motion tasks into two core types and aim to address all motion tasks within a unified framework, thereby promoting knowledge sharing and enhancing model generalization. Achieving this, we present UniMotion, an integrated model designed particularly to support joint training for these tasks. The critical components of our framework are distinct interactive modes and training strategies. Besides, we further perform task-specific fine-tuning for each task, fulfilling the requirement for expert-level models in real-world applications. Our extensive experiments on several task benchmarks comprehensively demonstrate that UniMotion is capable of unifying motion tasks, and the fine-tuned versions surpass the current state-of-the-art performance in the respective tasks.
+
+Limitations and future work. Although we employ a decode-only framework to ensure flexibility, the coupled spatial and temporal interaction and relative position embedding rely on detailed positional information of traffic elements, constraining the model from leveraging more LLM-related technologies. Moreover, after applying tokenization and conducting joint multi-task training, it is worthwhile to explore cross-dataset learning, which could lead to substantial improvements in joint training performance. Limited by computational resources and training time, we have not explored this technique in the present work.
+
+# Acknowledgments
+
+This work was supported in part by National Natural Science Foundation of China (Grant No. 62376060).
+
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+[38] Jiawei Sun, Chengran Yuan, Shuo Sun, Shanze Wang, Yuhang Han, Shuailei Ma, Zefan Huang, Anthony Wong, Keng Peng Tee, and Marcelo H Ang. Controlmtr: Control-guided motion transformer with scene-compliant intention points for feasible motion prediction. In IEEE ITSC, 2024. 7  
+[39] Jiawei Sun, Jiahui Li, Tingchen Liu, Chengran Yuan, Shuo Sun, Zefan Huang, Anthony Wong, Keng Peng Tee, and Marcelo H Ang Jr. Rmp-yolo: A robust motion predictor for partially observable scenarios even if you only look once. arXiv preprint, 2024. 7  
+[40] Zhiyu Huang, Haochen Liu, Jingda Wu, and Chen Lv. Differentiable integrated motion prediction and planning with learnable cost function for autonomous driving. TNNLS, 2023. 7, 8, 9  
+[41] Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. In  $ICLR$ , 2018. 7
+
+# A Details for simulation fine-tuning
+
+After obtaining the generated rollout  $\mathcal{A}_r$  and advantage  $A$ , we update policy gradient using simplified objective to ensure stable training of lightweight models, the gradient of which for token  $t$  can be estimated as:
+
+$$
+\nabla_ {\theta} \log \mathcal {F} _ {\theta} \left(\mathcal {A} _ {r} [ t ]\right) \cdot A _ {\left(\mathcal {A} _ {r}\right)} [ t ], \tag {9}
+$$
+
+where  $\mathcal{F}_{\theta}$  denotes our trainable models. Besides, considering the relatively uniform rewards and the large number of iterations, we perform policy update for only one generation per iteration to promote efficient training.
+
+# B Discussions
+
+Why we employ decoder-only Transformers as the backbone? We opt for a decoder-only Transformer structure because it is better suited for multi-step generative tasks like simulation, where input and output lengths can vary. Conversely, the encoder-decoder structure, while excellent for sequence-to-sequence prediction tasks, cannot generalize as well across the diverse demands of our multi-task framework. Adopting a decoder-only architecture during joint training ensures stronger generalization capabilities across all our tasks.
+
+Why the joint training in UniMotion works well? As detailed in Section 3.1, our primary objectives are NTP and LFR. Intuitively, this joint training paradigm fosters the learning of shared, rich intermediate representations across multiple tasks. This enables tokens within the model to attend simultaneously to both their immediate surroundings and more distant scene contexts, leading to a more holistic understanding of the environment.
+
+Further, despite their apparent differences, the two task objectives possess crucial commonalities. Specifically, while LFR provides direct supervision on long-range logged trajectories, the initial motion direction embedded within these trajectories concurrently serves as the maximization objective for NTP. This underlying consistency and synergistic relationship between LFR and NTP are fundamental to ensuring stable joint training and preventing mode collapse, allowing the model to learn effectively from both detailed local cues and broad long-range dependencies.
+
+# C More qualitative results
+
+We provide more qualitative results of our framework in Figure 5 and Figure 6 on the validation set of Waymo Open Motion Dataset [1, 5].
+
+# D Failure cases
+
+Although our UniMotion achieves strong performance across multiple motion tasks, it still encounters certain failure cases. We provide qualitative results and analysis to offer a more comprehensive understanding, which we hope will support future efforts toward developing more robust and capable algorithms, as shown in Figure 7 and Figure 8.
+
+In simulation task, there is still a possibility of collisions in complex scenario as illustrated in Figure 7 (a). Besides, we have also observed some unrealistic simulations in intersection scenarios as shown in Figure 7 (b), where agents remain stationary despite the green light. We provide the failure cases of prediction task in Figure 8 (a), which demonstrate that it is challenging to predict complex motion patterns, especially sudden turns. Similar issues are also observed in planning, particularly when dealing with sudden U-turns. Additionally, the model might yield reasonable trajectories that fail to reflect the driver's subjective intention.
+
+![](images/c1e155950225367f58eb6ace0f57c27409b630e0819337a0cc12ff886a42862c.jpg)  
+(a) Parking lot scene
+
+![](images/3af2f2cf8e89f502010ea36ae7472ff1c19211946d97aa572a616045352e0503.jpg)  
+(b) Intersection scene  
+Figure 5: Qualitative results on the Sim Agents. We present several complex scenarios in autonomous driving, each accompanied by two rollouts. The circled areas highlight the key differences between the two rollouts.
+
+![](images/464b3de393969e77be0c906819e8b5c734227b2d2609b2fda14e97dc270e254b.jpg)
+
+![](images/3b2cea10085fcec49b9b35a7ebc2e184e91b64a2ee442614feacae5d21fa1b1c.jpg)  
+(a) Prediction
+
+![](images/c341d400f3896e75a1902246cbd6547ac8c60c67534d9eccb543a78d0520e2ad.jpg)
+
+![](images/f1fc90bf5cf4dbf5d4b409e85e1a3ce15dc4208fa26b3b1a4e8539dfd3422fa6.jpg)  
+Figure 6: Qualitative results on the WOMD for prediction and planning task.
+
+![](images/3dbafd2535fcec3bb8cf6bf8e03a6cd79aa44c1c61d0aaa8640280c9037c12c7.jpg)  
+(b) Planning
+
+![](images/52044002deef6a6a8e7a5fd28fd5cd7deda2b02007523db2241a535062d1c936.jpg)
+
+![](images/6b2442d69d2c064b2bfc6f8c8f855f52a07f780f8862621aed946eebbf4a8b87.jpg)
+
+![](images/a79e08c56aa0329a1d740003e3a4bdb50e82997d3616d28310b203db8e7c609b.jpg)  
+(a) Collision
+
+![](images/397a240dbc498ebd52842402df319931f504da7414862b4605ceab2aa43606f7.jpg)
+
+![](images/90634ae22338e63ba6f388d215607deb2c8d22576bd6575650a272316c747967.jpg)
+
+![](images/b3a2b36ac4a04dca51f45b4b849774df38408b7b769cb0b576e14944f7d38843.jpg)  
+(b) Non-red-light stop
+
+![](images/c42f9d7596831378c681b927560631fbe4155dc4eac4a9e2a719b93bd55f31ce.jpg)  
+Figure 7: Failure cases in simulation. In the first row, the model causes a collision in a crowded parking lot. In the second row, it incorrectly incorrectly stop without a red light at an intersection.
+
+![](images/762bc99a1db32c0903ba7aa0520bb9e6908cdf1bfcdb28037babdabdb2644e3e.jpg)
+
+![](images/63e3d0f7b758ac2933d71652ecfe76064d148d73084f635b8a5382358501d0d4.jpg)  
+(a) Prediction
+
+![](images/32dbf075717e172bfd3d2685604f7959372d3e558327f86c19e7d4774eadb5f0.jpg)
+
+![](images/5c0d198c5a627bdd0eb65c95ce3e8f4e5e26021c9127690adb1ba21a62f7e0ae.jpg)  
+Figure 8: Failure cases in prediction and planning. The model fails to accurately predict the trajectories of vehicles and pedestrians. In terms of planning, it struggles with scenarios where the targets involve drivers' subjective intentions.
+
+![](images/4a0edb8666a904405c9dbaeeb11f02666bcf4577403c213f424797076932e1bb.jpg)  
+(b) Planning
+
+![](images/e809d9d7a134c7f70f424e3acd18bf92115d948961f50429a36b50dbe1be6bec.jpg)
+
+# NeurIPS Paper Checklist
+
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+
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+
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+
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+
+# 3. Theory assumptions and proofs
+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
+# 9. Code of ethics
+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
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+
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+
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+
+# 13. New assets
+
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+
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+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
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+
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+
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+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+# Answer: [NA]
+
+Justification: This paper does not involve crowdsourcing nor research with human subjects.
+
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+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
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+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+# Answer: [NA]
+
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+
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+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
\ No newline at end of file
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+# UniPixel: Unified Object Referring and Segmentation for Pixel-Level Visual Reasoning
+
+Ye Liu $^{1,2}$ , Zongyang Ma $^{2,3}$ , Junfu Pu $^{2}$ , Zhongang Qi $^{4}$ , Yang Wu $^{5}$ , Ying Shan $^{2}$ , Chang Wen Chen $^{1*}$
+
+<sup>1</sup> The Hong Kong Polytechnic University <sup>2</sup> ARC Lab, Tencent PCG
+
+$^{3}$  Institute of Automation, Chinese Academy of Sciences  $^{4}$  vivo Mobile Communication Co.
+
+$^{5}$  MindWingman Technology (Shenzhen) Co., Ltd.
+
+coco.ye.liu@connect.polyu.hk
+
+https://polyu-chenlab.github.io/unipixel/
+
+![](images/d27254d971bc18a15e10eed7ec6fbe751fa1820d87232a03d82cbc2db6fe3d1e.jpg)  
+Figure 1: UniPixel flexibly supports a large variety of fine-grained image and video understanding tasks, including referring/reasoning/interactive segmentation, motion-grounded video reasoning, and referred video description & question answering. It can also handle a novel PixelQA task that jointly requires object-centric referring, segmentation, and question answering in videos.
+
+# Abstract
+
+Recent advances in Large Multi-modal Models (LMMs) have demonstrated their remarkable success as general-purpose multi-modal assistants, with particular focuses on holistic image- and video-language understanding. Conversely, less attention has been given to scaling fine-grained pixel-level understanding capabilities, where the models are expected to realize pixel-level alignment between visual
+
+signals and language semantics. Some previous studies have applied LMMs to related tasks such as region-level captioning and referring expression segmentation. However, these models are limited to performing either referring or segmentation tasks independently and fail to integrate these fine-grained perception capabilities into visual reasoning. To bridge this gap, we propose UniPixel, a large multi-modal model capable of flexibly comprehending visual prompt inputs and generating mask-grounded responses. Our model distinguishes itself by seamlessly integrating pixel-level perception with general visual understanding capabilities. Specifically, UniPixel processes visual prompts and generates relevant masks on demand, and performs subsequent reasoning conditioning on these intermediate pointers during inference, thereby enabling fine-grained pixel-level reasoning. The effectiveness of our approach has been verified on 10 benchmarks across a diverse set of tasks, including pixel-level referring/segmentation and object-centric understanding in images/videos. A novel PixelQA task that jointly requires referring, segmentation, and question answering is also designed to verify the flexibility of our method.
+
+# 1 Introduction
+
+Large Multi-modal Models (LMMs) have been the de facto standard for developing general-purpose assistants. By effectively aligning multi-modalities with language, their significance has been demonstrated across various applications, including multi-modal analysis [59, 19, 107, 108, 48], autonomous driving (AD) [16, 79, 106, 11], and Embodied AI [111, 22, 30, 99].
+
+In the field of visual-language understanding, efforts have been dedicated to developing holistic understanding models, where simple projection layers between visual encoders and LLMs are utilized to bridge vision and language modalities. Supported by large-scale alignment pre-training and visual instruction tuning, such a straightforward paradigm achieves strong performance in holistic understanding tasks such as captioning [40, 6, 114] and general question answering [36, 24, 54, 49]. However, these models exhibit two fundamental limitations in fine-grained scenarios. First, their interactions with users are limited to text format, lacking support for more intuitive forms of communication such as drawing points/boxes as references or grounding model responses with key regions represented by masks. Second, the internal reasoning process of these models predominantly operates at a coarse level, directly perceiving the entire content rather than reasoning over specific objects/regions, making them hard to understand fine-grained details. Some previous studies have explored the application of LMMs to related tasks such as region-level captioning [12, 102, 103], referring expression segmentation [29, 55, 41, 21, 71, 62], and reasoning segmentation [32, 27, 96, 4, 112]. Nevertheless, their models are limited to performing either referring or segmentation tasks independently via rigidly defined input/output templates (e.g., "It's <SEG>." in LISA [32]), lacking the flexibility to comprehend user-referred concepts and generate mask-grounded responses simultaneously. More importantly, these methods cannot integrate such fine-grained perception capabilities with their original human-like [90, 89, 88, 92, 91] multi-modal reasoning abilities, resulting in degraded performance on general visual understanding benchmarks [100, 93, 28].
+
+In this work, we seek to bridge this gap by introducing UniPixel, a large multi-modal model that can flexibly comprehend visual prompt inputs (i.e., points, boxes, and masks) and generate mask-grounded responses. Our model significantly differentiates itself from existing ones by unifying the internal representations of referred and segmented objects via a novel object memory bank, which is a hashmap storing the spatial-temporal information of object-of-interests. During inference, UniPixel initializes the object memory bank and updates it on demand by adding object-centric information according to the context. The model responses are then generated conditioning on the fine-grained object memory. Benefits from such unification, UniPixel is able to perform not only basic referring/segmentation tasks, but also flexible pixel-level reasoning tasks that require simultaneous visual prompt comprehension and mask prediction. As illustrated in Fig. 1 (the last row), given a video $^2$ , a question, and optionally a visual prompt (e.g., a point specified by a click on an object in any frame), UniPixel can (1) infer the mask for the referred object in the corresponding frame, (2) propagate it to all video frames containing the same instance, (3) extract the mask-grounded object features, and finally (4) answer the question conditioning on both the video-level and object-centric
+
+![](images/b7fe5e3c9d0788a68c97a5e9702bf1dc9ba843de1d6f4bd2ad91c1374c1e67a6.jpg)  
+Figure 2: Schematic comparison between UniPixel and its counterparts. To the best of our knowledge, UniPixel is the first unified method supporting simultaneous object referring and segmentation.
+
+![](images/f021bae1a962adc3de63fa30181af01a5542a40fcb4340549d8a12f7344e9446.jpg)  
+How does the behavior of [3] differ from [1]?
+
+information. All these operations are seamlessly conducted within a single model, eliminating the need for external frame samplers [96], mask generators [102, 103], or object trackers [4].
+
+We evaluate the effectiveness of UniPixel from two aspects, i.e., basic referring/segmentation capabilities and flexible pixel-level reasoning capabilities. For the first aspect, we conduct extensive experiments on 10 public benchmarks across 9 image/video referring/segmentation tasks. Our method achieves state-of-the-art performance in diverse scenarios. Notably, on the challenging video reasoning segmentation and referred video QA tasks, our 3B model obtains  $62.1\mathcal{J}\& \mathcal{F}$  on ReVOS [96] and  $72.8\%$  Acc on VideoRefer-Bench[103], surpassing strong counterparts with  $7\mathrm{B}\sim 13\mathrm{B}$  parameters. Further ablation studies also demonstrate the mutual reinforcement effect of referring and segmentation. For the second aspect, we introduce a novel PixelQA task that jointly requires object-centric referring, segmentation, and QA in videos, which cannot be handled by existing methods. UniPixel establishes a strong baseline for this novel setting. Our contributions are summarized below:
+
+1. We propose UniPixel, a unified large multi-modal model that supports flexible object referring and segmentation in images and videos, via a novel object memory bank design.  
+2. Our model achieves state-of-the-art performance on 10 public benchmarks across 9 referring/segmentation tasks, verifying the mutual reinforcement effect of such unification.  
+3. We also introduce a novel PixelQA task that jointly requires object-centric referring, segmentation, and QA in videos, where UniPixel establishes a strong baseline for this setting.
+
+# 2 Related Work
+
+Large Multi-modal Models The remarkable success of large multi-modal models (LMMs) has shifted the paradigm of visual-language understanding from close-ended experts to open-ended task solvers. Early attempts [43, 42, 17, 115] involve an MLP projector or Q-Former [34] to align visual encoders to LLMs, enabling open-ended tasks such as visual question answering. With advanced designs such as dynamic resolution and data augmentation, open-source models, e.g., Qwen-VL [2, 77, 3] and InternVL [14, 74, 13] series, have narrowed the gap with advanced proprietary models like the GPT [58, 59] and Gemini families [67, 18]. Recent studies [60, 25, 51, 37, 48] also explore test-time scaling on visual-language understanding. However, these methods are spatially coarse-grained. UniPixel can also be regarded as an object-centric test-time scaling approach, where key objects are first segmented then encoded to facilitate the subsequent reasoning process.
+
+Visual Referring and Segmentation To meet the growing demand for fine-grained visual understanding [50, 46, 47, 45, 84], recent efforts have focused on enhancing LMMs with object referring and segmentation capabilities, as compared in Fig. 2. LISA [32] is a representative model that enables LMM-based segmentation by integrating SAM [31] as its decoder. They also introduced a novel reasoning segmentation task, requiring models to perform segmentation based on implicit queries. Other works in this direction [104, 69, 110, 65, 27] have explored advanced mask decoders, more flexible tasks, and larger-scale datasets. Recent studies have also extended these capabilities to videos [4, 96, 101]. Additionally, some research has examined regional understanding through boxes [12] and masks [102, 103]. While recent approaches attempt to unify these two capabilities, they either support only images [65] or rely on sub-optimal, tool-based pipelines [23]. To the best of knowledge, UniPixel is the first end-to-end method unifying object referring and mask prediction.
+
+![](images/a3622479aa81da759e67d33e39c140ef21c4ea85fc48660ab611aa7858c6911d.jpg)  
+Figure 3: The architecture of UniPixel. Given a video, a question, and visual prompts, the model encodes them into tokens via the visual encoder, prompt encoder, and tokenizer, respectively, then predicts a spatial-temporal mask for each visual prompt via the mask decoder. The masks are updated into the object memory bank, and subsequently injected into the prompt for pixel-level reasoning.
+
+# 3 Method
+
+Problem Formulation We provide a unified definition for pixel-level reasoning tasks. Formally, the inputs are an image or a video  $\mathcal{X}$ , a text prompt  $\mathcal{T}$ , and  $N$  optional visual prompts  $\{\mathcal{P}_i\}_{i=1}^N$  where each  $\mathcal{P}_i$  could be a point, box, or mask on a specific frame. The outputs are textual responses to the prompt with  $K$  grounded spatial-temporal masks  $\{\mathcal{M}_i\}_{i=1}^K$ . Here, both  $N$  and  $K$  could be zero (degenerating to a normal visual understanding task) and  $K$  is not necessarily equal to  $N$ , as the model may segment extra objects/regions that are not specified by the visual prompts.
+
+Overview Fig. 3 presents an overview of UniPixel. It is built upon the Qwen2.5-VL [3] framework, consisting of an LLM backbone and a ViT-based visual encoder that supports dynamic resolution inputs. Given a video and a text prompt, the model first tokenizes them via the visual encoder and text tokenizer, then sends them into the LLM for response generation. To boost this framework from holistic-level to pixel-level, we introduce (1) a prompt encoder (Sec.3.1) supporting three types of visual prompts, (2) an object memory bank (Sec.3.2) for storing object information and injecting it into the response generation process, and (3) a mask decoder (Sec.3.3) for generating spatial-temporal masks. We also extend the LLM's vocabulary by adding <REF>, <MEM>, and <SEG> tokens. The former two serve as placeholders in the input prompt that would be replaced by visual prompt and memory tokens, respectively, while the <SEG> token is utilized to trigger and guide the mask decoding process. Detailed designs and interactions among these components are illustrated as follows.
+
+# 3.1 Prompt Encoder
+
+This module aims to effectively encode each visual prompt into a single token that can be processed by the LLM. We denote a point prompt as a tuple  $(x,y,t)$  containing its spatial coordinates  $(x,y)$  and the corresponding frame index  $t$ . For box prompts, it is extended to  $(x_{1},y_{1},x_{2},y_{2},t)$  containing the positions of top-left and bottom-right corners. A mask prompt is densely represented by a 2D binary mask  $\mathbf{m}_{ij}\in \{0,1\}$  with the same shape as the encoded target frame.
+
+![](images/4595ce7c0e8fb1f75dc95338dbd36d46848192dac365420ae42cdf8977bb57fa.jpg)  
+Figure 4: Joint positional & temporal encoding for point  $(\mathrm{X}_1\mathrm{Y}_1\mathrm{T})$  and box  $(\mathrm{X}_1\mathrm{Y}_1\mathrm{X}_2\mathrm{Y}_2\mathrm{T})$  prompts.
+
+For sparse prompts (points and boxes), as shown in Fig. 4, we encode each position  $(x_{i},y_{i})$  as the sum of a 2D Fourier embedding [73] and a learnable type embedding (indicating whether it is a single point, top-left corner, or bottom-right corner). For box prompts, we merge the two positional embeddings
+
+by concatenating them along the channel dimension and linearly projecting them back to the original size. Frame indices are also encoded similarly with 1D Fourier embeddings. The resulting positional and temporal embeddings are concatenated again, and then projected to the LLM's embedding space via a GELU  $\rightarrow$  Linear block, such that the sparse coordinates in a point/box are encoded into a compact high-dimensional token. This design is inspired by [31, 66] with two key differences: (1) the spatial-only embeddings are extended to include temporal information, and (2) the negative points are discarded. For dense prompts (masks), we directly resize the binary masks and apply masked pooling on the outputs of the visual encoder. An M  $\rightarrow$  L projector (Linear  $\rightarrow$  GELU  $\rightarrow$  Linear) is leveraged to project the pooled visual features to the LLM's embedding space.
+
+# 3.2 Object Memory Bank
+
+Although sparse prompts contain rich positional and temporal information indicating the objects that users are referring to, it is still hard for the model to focus on these important regions. Previous studies [12, 102, 103] also confirm that direct region cropping can generally provide better object awareness compared to positional pointers. To seamlessly integrate such a mechanism while preserving the flexibility of visual prompts (e.g., allow pointing on a single frame instead of drawing complete masks on all frames), we propose an object memory bank to bridge sparse visual prompts and dense object masks. This is a hashmap where the keys are object IDs and the values are the corresponding spatial-temporal masks. It is initialized as an empty storage for every chat session, and is dynamically updated on demand. We define two operations for the object memory bank, namely memory pre-filling and memory injection. Below is an example of memory-enhanced multi-round conversation.
+
+Prompt 1: How does the behavior of [1] <REF> differ from [2] <REF> and [3] <REF?>
+
+<REF> detected, enhancing the prompt with object memory.
+
+# Memory Pre-filling Response:
+
+The relevant regions for this question are [1]  $<\mathrm{SEG}>$  [2]  $<\mathrm{SEG}>$  [3]  $<\mathrm{SEG}>$  [4]  $<\mathrm{SEG}>$ . ← 4 objects saved into the memory
+
+# Memory Injected Prompt:
+
+Here is a video with 4 frames denoted as  $<1>$  to  $<4>$ . The highlighted regions are as follows:
+
+[1]: <1> <MEM> <2> <MEM> <3> <MEM> ← This object cannot be seen in the last frame  
+[2]:<1><MEM><2><MEM><3><MEM><4><MEM>  
+[3]:<1><MEM><2><MEM><3><MEM><4><MEM>  
+[4]:<1><MEM><2><MEM><3><MEM><4><MEM>  
+How does the behavior of [1] differ from [2] and [3]?
+
+Response 1: [1] appears disinterested and focuses on nibbling on the ground, while [2] is engaging with [4], who is offering some food to [2] and [3].
+
+Prompt 2: What food is [4] offering?  $\leftarrow$  Users can directly refer to objects in the memory
+
+Response 2: [4] is offering carrots.
+
+Memory Pre-filling This operation is triggered upon the detection of <REF> tokens in the input prompt, aiming to thoroughly analyze the referred objects and predict their corresponding masks. In this stage, the model responds with object IDs and <SEG> tokens for the relevant objects according to the context, and predicts their spatial-temporal masks accordingly. These object-mask pairs are then saved into the object memory bank.
+
+Memory Injection We inject the features of the saved objects into the prompt to enhance object-awareness. Similar to the mask prompt encoder described in Sec. 3.1, each frame-level object mask is downsampled to match the resolution of visual tokens. We then apply masked pooling to aggregate object-centric features. Each frame-level mask is condensed into a single feature token, projected through the mask projector, and subsequently utilized to replace the corresponding <MEM> token in the memory-injected prompt. Through this pre-filling and injection mechanism, object-centric information is effectively integrated into the model inference process.
+
+Why using object memory bank? An alternative is directly appending a <SEG> token to each <REF> token, followed by masked pooled features obtained during inference. However, we do not adopt this approach for two reasons: (1) During mask prediction, the <SEG> tokens, due to the unidirectional nature of causal self-attention, are unable to aggregate the full context of the prompt, thereby compromising the quality of predicted masks. (2) By utilizing the object memory bank, we can effectively decouple regional understanding and mask prediction, allowing each to benefit from referring and segmentation data during training, thus enhancing both capabilities.
+
+Table 1: Comparison with state-of-the-art methods on ReVOS [96] val split. The best and second-best results are marked bold and underlined, respectively.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Size</td><td colspan="3">Referring</td><td colspan="3">Reasoning</td><td colspan="3">Overall</td><td rowspan="2">R</td></tr><tr><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td></tr><tr><td colspan="12">Non-LLM-based Specialists</td></tr><tr><td>MTTR [5]</td><td>-</td><td>29.8</td><td>30.2</td><td>30.0</td><td>20.4</td><td>21.5</td><td>21.0</td><td>25.1</td><td>25.9</td><td>25.5</td><td>5.6</td></tr><tr><td>LMPM [21]</td><td>-</td><td>29.0</td><td>39.1</td><td>34.1</td><td>13.3</td><td>24.3</td><td>18.8</td><td>21.2</td><td>31.7</td><td>26.4</td><td>3.2</td></tr><tr><td>ReferFormer [85]</td><td>-</td><td>31.2</td><td>34.3</td><td>32.7</td><td>21.3</td><td>25.6</td><td>23.4</td><td>26.2</td><td>29.9</td><td>28.1</td><td>8.8</td></tr><tr><td colspan="12">LLM-based Generalists</td></tr><tr><td>LISA [32]</td><td>13B</td><td>45.2</td><td>47.9</td><td>46.6</td><td>34.3</td><td>39.1</td><td>36.7</td><td>39.8</td><td>43.5</td><td>41.6</td><td>8.6</td></tr><tr><td>TrackGPT [72]</td><td>13B</td><td>48.3</td><td>50.6</td><td>49.5</td><td>38.1</td><td>42.9</td><td>40.5</td><td>43.2</td><td>46.8</td><td>45.0</td><td>12.8</td></tr><tr><td>VISA [96]</td><td>13B</td><td>55.6</td><td>59.1</td><td>57.4</td><td>42.0</td><td>46.7</td><td>44.3</td><td>48.8</td><td>52.9</td><td>50.9</td><td>14.5</td></tr><tr><td>HyperSeg [81]</td><td>3B</td><td>56.0</td><td>60.9</td><td>58.5</td><td>50.2</td><td>55.8</td><td>53.0</td><td>53.1</td><td>58.4</td><td>55.7</td><td>-</td></tr><tr><td>InstructSeg [82]</td><td>3B</td><td>54.8</td><td>59.2</td><td>57.0</td><td>49.2</td><td>54.7</td><td>51.9</td><td>52.0</td><td>56.9</td><td>54.5</td><td>-</td></tr><tr><td>GLUS [39]</td><td>7B</td><td>56.0</td><td>60.7</td><td>58.3</td><td>48.8</td><td>53.9</td><td>51.4</td><td>52.4</td><td>57.3</td><td>54.9</td><td>17.9</td></tr><tr><td>ViLLa [112]</td><td>6B</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>54.9</td><td>59.1</td><td>57.0</td><td>-</td></tr><tr><td>Sa2VA [101]</td><td>4B</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>53.2</td><td>-</td></tr><tr><td>UniPixel (Ours)</td><td>3B</td><td>62.3</td><td>66.7</td><td>64.5</td><td>57.1</td><td>62.1</td><td>59.6</td><td>59.7</td><td>64.4</td><td>62.1</td><td>19.0</td></tr><tr><td>UniPixel (Ours)</td><td>7B</td><td>63.9</td><td>67.8</td><td>65.8</td><td>59.4</td><td>63.7</td><td>61.5</td><td>61.7</td><td>65.7</td><td>63.7</td><td>19.4</td></tr></table>
+
+# 3.3 Mask Decoder
+
+We adopt SAM 2.1 [66] as the mask decoder to disentangle the discrete language modeling and continuous mask prediction capabilities. For each <SEG> token, we extract its last-layer hidden states, downsample them via an  $\mathrm{L} \rightarrow \mathrm{M}$  projector (architecturally identical to the  $\mathrm{M} \rightarrow \mathrm{L}$  projector), and reshape them into two tokens. Using two tokens ensures better preservation of object information when downsampling from high- to low-dimensional channel space. These tokens prompt the mask decoder to predict the mask on the first frame, which is then propagated to the other frames.
+
+# 3.4 Model Training
+
+The training loss for UniPixel is a linear combination of language modeling loss and mask decoding losses [66], including a focal loss and dice loss for mask prediction, a mean-absolute-error (MAE) loss for IoU prediction, and a cross-entropy loss for objectness prediction. The loss weights are set to 1, 100, 5, 5, and 5, respectively. We train the model through a three-stage progressive alignment recipe. The datasets are listed in Tab. 12. In the first stage, we pre-train the sparse prompt encoder using  $851\mathrm{K}$  regional captioning data. Then, we align the LLM and mask decoder by training the  $\mathrm{L} \rightarrow \mathrm{M}$  projector on  $87\mathrm{K}$  referring segmentation data. In the last stage, we further unfreeze the  $\mathrm{M} \rightarrow \mathrm{L}$  projector and mask decoder, and apply LoRA [26] on the visual encoder and LLM. The model is jointly trained on a large-scale corpus with around 1M samples for diverse tasks.
+
+# 4 Experiments
+
+We evaluate the effectiveness of UniPixel by conducting extensive experiments across a diverse set of benchmarks. Specifically, we study the following research questions.
+
+Q1. Whether UniPixel is flexible and effective on basic image/video referring and segmentation tasks compared to the corresponding representative methods?  
+Q2. How does it perform on the more challenging PixelQA task, which requires joint referring, segmentation, and question answering in videos?  
+Q3. What effects does each architectural design contribute? More importantly, does the unified modeling of referring and segmentation lead to a mutual reinforcement effect?
+
+Detailed information about the benchmarks, evaluation metrics, implementation details, and more experimental results can be found in the appendix.
+
+# 4.1 Q1: Comparison with State-of-the-Arts on Referring and Segmentation Tasks
+
+Reasoning Video Object Segmentation We begin with the most challenging ReVOS [96] dataset, which requires models to predict masks based on implicit text queries demanding complex reasoning abilities based on world knowledge. The results are shown in Tab. 1. Our 3B variant outperforms all
+
+Table 2: Comparison with state-of-the-art methods on referring video object segmentation (RVOS) and motion-grounded video reasoning datasets, including MeViS [21] (val1), Ref-YouTube-VOS [71] (val1), Ref-DAVIS17 [62] (val1), and GroundMoRe [20] (test). The best and second-best results are marked bold and underlined, respectively.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Size</td><td colspan="3">MeViS</td><td colspan="3">Ref-YouTube-VOS</td><td colspan="3">Ref-DAVIS17</td><td colspan="3">GroundMoRe</td></tr><tr><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td></tr><tr><td colspan="14">Non-LLM-based Specialists</td></tr><tr><td>ReferFormer [85]</td><td>-</td><td>29.8</td><td>32.2</td><td>31.0</td><td>61.3</td><td>64.6</td><td>62.9</td><td>58.1</td><td>64.1</td><td>61.1</td><td>11.2</td><td>14.3</td><td>12.7</td></tr><tr><td>LMPM [21]</td><td>-</td><td>34.2</td><td>40.2</td><td>37.2</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>12.7</td><td>14.0</td><td>13.3</td></tr><tr><td>OnlineRefer [83]</td><td>-</td><td>-</td><td>-</td><td>-</td><td>61.6</td><td>65.5</td><td>63.5</td><td>61.6</td><td>67.7</td><td>64.8</td><td>-</td><td>-</td><td>-</td></tr><tr><td colspan="14">LLM-based Generalists</td></tr><tr><td>PixelLM [69]</td><td>7B</td><td>36.3</td><td>41.1</td><td>38.7</td><td>54.3</td><td>55.7</td><td>55.0</td><td>63.4</td><td>70.0</td><td>66.7</td><td>9.9</td><td>10.0</td><td>10.0</td></tr><tr><td>LISA [32]</td><td>13B</td><td>35.8</td><td>40.0</td><td>37.9</td><td>54.0</td><td>54.8</td><td>54.4</td><td>63.2</td><td>68.8</td><td>66.0</td><td>6.3</td><td>6.7</td><td>6.5</td></tr><tr><td>VISA [96]</td><td>13B</td><td>41.8</td><td>47.1</td><td>44.5</td><td>61.4</td><td>64.7</td><td>63.0</td><td>67.0</td><td>73.8</td><td>70.4</td><td>5.3</td><td>4.7</td><td>5.9</td></tr><tr><td>VideoLISA [4]</td><td>3.8B</td><td>41.3</td><td>47.6</td><td>44.4</td><td>61.7</td><td>65.7</td><td>63.7</td><td>64.9</td><td>72.7</td><td>68.8</td><td>-</td><td>-</td><td>-</td></tr><tr><td>VideoGLaMM [56]</td><td>3.8B</td><td>42.1</td><td>48.2</td><td>45.2</td><td>65.4</td><td>68.2</td><td>66.8</td><td>73.3</td><td>65.6</td><td>69.5</td><td>-</td><td>-</td><td>-</td></tr><tr><td>ViLLa [112]</td><td>6B</td><td>46.5</td><td>52.3</td><td>49.4</td><td>64.6</td><td>70.4</td><td>67.5</td><td>70.6</td><td>78.0</td><td>74.3</td><td>-</td><td>-</td><td>-</td></tr><tr><td>GLUS [39]</td><td>7B</td><td>48.5</td><td>54.2</td><td>51.3</td><td>65.5</td><td>69.0</td><td>67.3</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Sa2VA [101]</td><td>4B</td><td>-</td><td>-</td><td>46.2</td><td>-</td><td>-</td><td>70.0</td><td>-</td><td>-</td><td>73.8</td><td>-</td><td>-</td><td>-</td></tr><tr><td>MoRA [20]</td><td>7B</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>27.4</td><td>26.9</td><td>27.2</td></tr><tr><td>UniPixel (Ours)</td><td>3B</td><td>50.4</td><td>55.7</td><td>53.1</td><td>68.6</td><td>72.3</td><td>70.5</td><td>70.7</td><td>77.8</td><td>74.2</td><td>36.0</td><td>38.7</td><td>37.4</td></tr><tr><td>UniPixel (Ours)</td><td>7B</td><td>53.2</td><td>58.3</td><td>55.8</td><td>69.5</td><td>72.4</td><td>71.0</td><td>72.7</td><td>80.1</td><td>76.4</td><td>36.5</td><td>39.1</td><td>37.8</td></tr></table>
+
+Table 3: Comparison with state-of-the-art methods on image referring expression segmentation (RES) and reasoning segmentation datasets, including RefCOCO+/g [29, 55] and ReasonSeg [32] (val). The best and second-best results are marked bold and underlined, respectively.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Size</td><td colspan="3">RefCOCO</td><td colspan="3">RefCOCO+</td><td colspan="2">RefCOCOg</td><td colspan="2">ReasonSeg</td></tr><tr><td>val</td><td>testA</td><td>testB</td><td>val</td><td>testA</td><td>testB</td><td>val(U)</td><td>test(U)</td><td>gIoU</td><td>cIoU</td></tr><tr><td colspan="12">Non-LLM-based Specialists</td></tr><tr><td>ReLU [41]</td><td>-</td><td>73.8</td><td>76.5</td><td>70.2</td><td>66.0</td><td>71.0</td><td>57.7</td><td>65.0</td><td>66.0</td><td>-</td><td>-</td></tr><tr><td>X-Decoder [116]</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>64.6</td><td>-</td><td>22.6</td><td>17.9</td></tr><tr><td>SEEM [117]</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>65.7</td><td>-</td><td>25.5</td><td>21.2</td></tr><tr><td colspan="12">LLM-based Image Generalists</td></tr><tr><td>NExT-Chat [104]</td><td>7B</td><td>74.7</td><td>78.9</td><td>69.5</td><td>65.1</td><td>71.9</td><td>56.7</td><td>67.0</td><td>67.0</td><td>-</td><td>-</td></tr><tr><td>PixelLM [69]</td><td>7B</td><td>73.0</td><td>76.5</td><td>68.2</td><td>66.3</td><td>71.7</td><td>58.3</td><td>69.3</td><td>70.5</td><td>-</td><td>-</td></tr><tr><td>LISA [32]</td><td>7B</td><td>74.9</td><td>79.1</td><td>72.3</td><td>65.1</td><td>70.8</td><td>58.1</td><td>67.9</td><td>70.6</td><td>61.3</td><td>62.9</td></tr><tr><td>Groundhog [110]</td><td>7B</td><td>78.5</td><td>79.9</td><td>75.7</td><td>70.5</td><td>75.0</td><td>64.9</td><td>74.1</td><td>74.6</td><td>56.2</td><td>-</td></tr><tr><td>LaSagnA [80]</td><td>7B</td><td>76.8</td><td>78.7</td><td>73.8</td><td>66.4</td><td>70.6</td><td>60.1</td><td>70.6</td><td>71.9</td><td>48.8</td><td>47.2</td></tr><tr><td>M2SA [27]</td><td>13B</td><td>74.6</td><td>77.6</td><td>71.0</td><td>64.0</td><td>68.1</td><td>57.6</td><td>69.0</td><td>69.3</td><td>-</td><td>-</td></tr><tr><td colspan="12">LLM-based Video Generalists</td></tr><tr><td>VideoLISA [4]</td><td>3.8B</td><td>73.8</td><td>76.6</td><td>68.8</td><td>63.4</td><td>68.8</td><td>56.2</td><td>68.3</td><td>68.8</td><td>61.4</td><td>67.1</td></tr><tr><td>VISA [96]</td><td>7B</td><td>72.4</td><td>75.5</td><td>68.1</td><td>59.8</td><td>64.8</td><td>53.1</td><td>65.5</td><td>66.4</td><td>52.7</td><td>57.8</td></tr><tr><td>Vitron [23]</td><td>7B</td><td>75.5</td><td>79.5</td><td>72.2</td><td>66.7</td><td>72.5</td><td>58.0</td><td>67.9</td><td>68.9</td><td>-</td><td>-</td></tr><tr><td>Sa2VA [101]</td><td>4B</td><td>78.9</td><td>-</td><td>-</td><td>71.7</td><td>-</td><td>-</td><td>74.1</td><td>-</td><td>-</td><td>-</td></tr><tr><td>UniPixel (Ours)</td><td>3B</td><td>80.5</td><td>82.6</td><td>76.9</td><td>74.3</td><td>78.9</td><td>68.4</td><td>76.3</td><td>77.0</td><td>64.0</td><td>56.2</td></tr><tr><td>UniPixel (Ours)</td><td>7B</td><td>80.8</td><td>83.0</td><td>77.4</td><td>75.3</td><td>80.1</td><td>70.0</td><td>76.4</td><td>77.1</td><td>60.5</td><td>58.7</td></tr></table>
+
+existing methods with larger LLMs (including Sa2VA-4B [101] also with SAM 2 decoder), achieving 62.1 overall  $\mathcal{J}\&\mathcal{F}$ . The 7B model further boosts the performance to 64.0  $\mathcal{J}\&\mathcal{F}$  - an improvement of  $12\%$  over the previous state-of-the-art - demonstrating that UniPixel can effectively understand implicit queries based on its world knowledge, and accurately generate masks as responses.
+
+Referring Video Object Segmentation The performance comparisons on MeViS [21], Ref-YouTube-VOS [71], and Ref-DAVIS17 [62] datasets are presented in Tab. 2. UniPixel consistently achieves the best performance among its counterparts. Its advantage is particularly evident on the more challenging MeViS dataset, where our 3B model outperforms GLUS-7B [39] by around  $3.5\%$ , as well as the similarly sized VideoGLaMM-3.8B [56] by  $17\%$ . More experimental results on MeViS [21] val $^u$  set and Ref-SAV [101] val set are provided in Tab. 4 and Tab. 5, respectively. Ref-SAV features long referring descriptions, large object motion, large camera motion, and heavy occlusion compared with existing datasets. Given these complex descriptions and video content, our method consistently performs better than counterparts, including those fine-tuned on the target dataset.
+
+Motion-Grounded Video Reasoning We also evaluate our method on GroundMoRe [20] dataset (results shown in Tab. 2), which highlights visual answer generation that requires joint spatial and
+
+Table 4: Experimental results on MeViS [21] valu set. Post means applying post optimization.  
+
+<table><tr><td>Method</td><td>Size</td><td>FT</td><td>J</td><td>F</td><td>J&amp;F</td></tr><tr><td>LMPM [21]</td><td>-</td><td>X</td><td>36.5</td><td>43.9</td><td>40.2</td></tr><tr><td>LISA [32]</td><td>7B</td><td>X</td><td>39.9</td><td>46.5</td><td>43.2</td></tr><tr><td>LISA [32] + XMem [15]</td><td>7B</td><td>X</td><td>41.9</td><td>49.3</td><td>45.6</td></tr><tr><td>VideoLISA [4]</td><td>7B</td><td>X</td><td>48.4</td><td>54.9</td><td>51.7</td></tr><tr><td>VideoLISA [4] + Post</td><td>7B</td><td>X</td><td>50.9</td><td>58.1</td><td>54.5</td></tr><tr><td>Sa2VA [101]</td><td>4B</td><td>X</td><td>-</td><td>-</td><td>52.1</td></tr><tr><td>Sa2VA [101]</td><td>8B</td><td>X</td><td>-</td><td>-</td><td>57.0</td></tr><tr><td>UniPixel (Ours)</td><td>3B</td><td>X</td><td>56.1</td><td>63.2</td><td>59.7</td></tr><tr><td>UniPixel (Ours)</td><td>7B</td><td>X</td><td>58.4</td><td>65.0</td><td>61.7</td></tr></table>
+
+Table 5: Comparison on Ref-SAV [101] val set. FT means fine-tuning after pre-/co-training.  
+
+<table><tr><td>Method</td><td>Size</td><td>FT</td><td>J</td><td>F</td><td>J&amp;F</td></tr><tr><td>UniRef++ [86]</td><td>-</td><td>X</td><td>11.6</td><td>9.5</td><td>10.5</td></tr><tr><td>UNINEXT [95]</td><td>-</td><td>X</td><td>8.8</td><td>6.4</td><td>7.6</td></tr><tr><td>LMPM [21]</td><td>-</td><td>X</td><td>12.2</td><td>9.8</td><td>10.3</td></tr><tr><td>VISA [96]</td><td>7B</td><td>X</td><td>13.2</td><td>11.3</td><td>11.8</td></tr><tr><td>Sa2VA [101]</td><td>8B</td><td>X</td><td>39.6</td><td>43.0</td><td>41.3</td></tr><tr><td>UniRef++ [86]</td><td>-</td><td>✓</td><td>15.8</td><td>13.4</td><td>14.6</td></tr><tr><td>Sa2VA [101]</td><td>8B</td><td>✓</td><td>48.3</td><td>51.7</td><td>50.0</td></tr><tr><td>UniPixel (Ours)</td><td>3B</td><td>X</td><td>66.9</td><td>67.6</td><td>67.2</td></tr><tr><td>UniPixel (Ours)</td><td>7B</td><td>X</td><td>68.5</td><td>69.6</td><td>69.0</td></tr></table>
+
+Table 6: Fine-tuned performance on referring expression segmentation (RES) datasets, including Ref-COCO+/+g [29, 55]. The best and second-best results are marked **bold** and **underlined**, respectively.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Size</td><td colspan="3">RefCOCO</td><td colspan="3">RefCOCO+</td><td colspan="2">RefCOCOg</td></tr><tr><td>val</td><td>testA</td><td>testB</td><td>val</td><td>testA</td><td>testB</td><td>val(U)</td><td>test(U)</td></tr><tr><td>LISA [32]</td><td>7B</td><td>74.9</td><td>79.1</td><td>72.3</td><td>65.1</td><td>70.8</td><td>58.1</td><td>67.9</td><td>70.6</td></tr><tr><td>GSVA [87]</td><td>7B</td><td>77.2</td><td>78.9</td><td>73.5</td><td>65.9</td><td>69.6</td><td>59.8</td><td>72.7</td><td>73.3</td></tr><tr><td>OMG-LLaVA [109]</td><td>7B</td><td>78.0</td><td>80.3</td><td>74.1</td><td>69.1</td><td>73.1</td><td>63.0</td><td>72.9</td><td>72.9</td></tr><tr><td>GLaMM [65]</td><td>7B</td><td>79.5</td><td>83.2</td><td>76.9</td><td>72.6</td><td>78.7</td><td>64.6</td><td>74.2</td><td>74.9</td></tr><tr><td>Sa2VA [101]</td><td>4B</td><td>80.4</td><td>-</td><td>-</td><td>74.3</td><td>-</td><td>-</td><td>75.7</td><td>-</td></tr><tr><td>UniPixel (Ours)</td><td>3B</td><td>81.9</td><td>83.5</td><td>78.6</td><td>75.3</td><td>80.3</td><td>70.6</td><td>77.2</td><td>78.5</td></tr><tr><td>UniPixel (Ours)</td><td>7B</td><td>83.0</td><td>84.9</td><td>80.4</td><td>77.8</td><td>82.3</td><td>72.7</td><td>78.7</td><td>79.7</td></tr></table>
+
+Table 7: Experimental results on referring expression comprehension (REC) datasets, including Ref-COCO/+/g [29, 55]. The best and second-best results are marked bold and underlined, respectively.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Size</td><td colspan="3">RefCOCO</td><td colspan="3">RefCOCO+</td><td colspan="2">RefCOCOg</td></tr><tr><td>val</td><td>testA</td><td>testB</td><td>val</td><td>testA</td><td>testB</td><td>val(U)</td><td>test(U)</td></tr><tr><td>OFA [78]</td><td>-</td><td>80.0</td><td>83.7</td><td>76.4</td><td>68.3</td><td>76.0</td><td>61.8</td><td>67.6</td><td>67.6</td></tr><tr><td>Shikra [9]</td><td>7B</td><td>87.0</td><td>90.6</td><td>80.2</td><td>81.6</td><td>87.4</td><td>72.1</td><td>82.3</td><td>82.2</td></tr><tr><td>MiniGPT-v2 [8]</td><td>7B</td><td>88.7</td><td>91.6</td><td>85.3</td><td>79.9</td><td>85.1</td><td>74.4</td><td>84.4</td><td>84.6</td></tr><tr><td>Vitron [23]</td><td>7B</td><td>90.9</td><td>93.2</td><td>89.3</td><td>83.7</td><td>89.1</td><td>76.9</td><td>86.4</td><td>87.0</td></tr><tr><td>UniPixel (Ours)</td><td>3B</td><td>91.8</td><td>93.8</td><td>87.5</td><td>86.3</td><td>90.8</td><td>80.3</td><td>88.0</td><td>88.2</td></tr><tr><td>UniPixel (Ours)</td><td>7B</td><td>92.0</td><td>94.4</td><td>88.1</td><td>87.2</td><td>91.9</td><td>82.1</td><td>88.6</td><td>88.7</td></tr></table>
+
+temporal grounding. Note that we mainly compare the results with MoRA [20], which is fine-tuned on GroundMoRe while other methods are evaluated under the zero-shot setting. Benefit from the strong pixel-level reasoning capability, UniPixel significantly performs better than the baseline.
+
+Referring Expression Segmentation and Reasoning Segmentation Tab. 3 compares the image segmentation capabilities using explicit and implicit queries. We evaluate our co-trained model on RefCOCO+/g [29, 55] and ReasonSeg [32]. While state-of-the-art performance has been achieved on RES datasets, we observe that the reasoning segmentation data (239 samples) can be easily overwhelmed by the other samples during training due to its limited size. Tab. 6 presents the RES performance after fine-tuning. We follow the common practice that jointly fine-tunes the model on RefCOCO+/g datasets [29, 55], and then evaluate on them separately. These results demonstrate the generalizability of UniPixel when facing both explicit and implicit queries.
+
+Referring Expression Comprehension Our method also supports referring expression comprehension by inferring the bounding boxes from predicted masks. Its performance (accuracy with IoU  $\geqslant 0.5$ ) is compared with representative methods in Tab. 7. Benefiting from the high-quality mask prediction, UniPixel can also achieve very competitive performance on this simpler task.
+
+Referred Video Description and Question Answering We study UniPixel's regional understanding capabilities on VideoRefer-Bench [103], which contains two subsets for description and question answering tasks. The comparisons are in Tab. 8 and Tab. 9. BQ, SQ, RQ, CQ, and FP denote basic questions, sequential questions, relational questions, reasoning questions, and future predictions, respectively. Both tasks leverage mask prompts as inputs, where single-frame and multi-frame modes denote applying the masks only on a specific frame and on all frames, respectively. UniPixel can
+
+Table 8: Comparison with state-of-the-art methods on VideoRefer-BenchD [103]. The best and second-best results are marked bold and underlined, respectively.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Size</td><td colspan="5">Single-Frame</td><td colspan="5">Multi-Frame</td></tr><tr><td>SC</td><td>AD</td><td>TD</td><td>HD</td><td>Avg.</td><td>SC</td><td>AD</td><td>TD</td><td>HD</td><td>Avg.</td></tr><tr><td colspan="12">General LMMs</td></tr><tr><td>LLaVA-OV [33]</td><td>7B</td><td>2.62</td><td>1.58</td><td>2.19</td><td>2.07</td><td>2.12</td><td>3.09</td><td>1.94</td><td>2.50</td><td>2.41</td><td>2.48</td></tr><tr><td>Qwen2-VL [77]</td><td>7B</td><td>2.97</td><td>2.24</td><td>2.03</td><td>2.31</td><td>2.39</td><td>3.30</td><td>2.54</td><td>2.22</td><td>2.12</td><td>2.55</td></tr><tr><td>InternVL2 [74]</td><td>26B</td><td>3.55</td><td>2.99</td><td>2.57</td><td>2.25</td><td>2.84</td><td>4.08</td><td>3.35</td><td>3.08</td><td>2.28</td><td>3.20</td></tr><tr><td>GPT-4o-mini [59]</td><td>-</td><td>3.56</td><td>2.85</td><td>2.87</td><td>2.38</td><td>2.92</td><td>3.89</td><td>3.18</td><td>2.62</td><td>2.50</td><td>3.05</td></tr><tr><td>GPT-4o [59]</td><td>-</td><td>3.34</td><td>2.96</td><td>3.01</td><td>2.50</td><td>2.95</td><td>4.15</td><td>3.31</td><td>3.11</td><td>2.43</td><td>3.25</td></tr><tr><td colspan="12">Image Referring LMMs</td></tr><tr><td>Ferret [98]</td><td>7B</td><td>3.08</td><td>2.01</td><td>1.54</td><td>2.14</td><td>2.19</td><td>3.20</td><td>2.38</td><td>1.97</td><td>1.38</td><td>2.23</td></tr><tr><td>Osprey [102]</td><td>7B</td><td>3.19</td><td>2.16</td><td>1.54</td><td>2.45</td><td>2.34</td><td>3.30</td><td>2.66</td><td>2.10</td><td>1.58</td><td>2.41</td></tr><tr><td colspan="12">Video Referring LMMs</td></tr><tr><td>Elysium [75]</td><td>7B</td><td>2.35</td><td>0.30</td><td>0.02</td><td>3.59</td><td>1.57</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Artemis [63]</td><td>7B</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>3.42</td><td>1.34</td><td>1.39</td><td>2.90</td><td>2.26</td></tr><tr><td>VideoRefer [103]</td><td>7B</td><td>4.41</td><td>3.27</td><td>3.03</td><td>2.97</td><td>3.42</td><td>4.44</td><td>3.27</td><td>3.10</td><td>3.04</td><td>3.46</td></tr><tr><td>UniPixel (Ours)</td><td>3B</td><td>4.04</td><td>3.15</td><td>3.10</td><td>3.37</td><td>3.42</td><td>4.08</td><td>3.13</td><td>3.13</td><td>3.42</td><td>3.44</td></tr><tr><td>UniPixel (Ours)</td><td>7B</td><td>3.83</td><td>3.07</td><td>2.96</td><td>3.62</td><td>3.37</td><td>3.82</td><td>3.05</td><td>3.01</td><td>3.57</td><td>3.36</td></tr></table>
+
+Table 9: Comparison with state-of-the-art methods on VideoRefer-Bench $^{\text{Q}}$  [103] (mask prompts). MF denotes multi-frame mode. Full question types are in Sec. 4.1.  
+
+<table><tr><td>Method</td><td>Size</td><td>MF</td><td>BQ</td><td>SQ</td><td>RQ</td><td>CQ</td><td>FP</td><td>Avg.</td></tr><tr><td colspan="9">General LMMs</td></tr><tr><td>LLaVA-OV [33]</td><td>7B</td><td>X</td><td>58.7</td><td>62.9</td><td>64.7</td><td>87.4</td><td>76.3</td><td>67.4</td></tr><tr><td>Qwen2-VL [77]</td><td>7B</td><td>X</td><td>62.0</td><td>69.6</td><td>54.9</td><td>87.3</td><td>74.6</td><td>66.0</td></tr><tr><td>InternVL2 [74]</td><td>26B</td><td>X</td><td>58.5</td><td>63.5</td><td>53.4</td><td>88.0</td><td>78.9</td><td>65.0</td></tr><tr><td>GPT-4o-mini [59]</td><td>-</td><td>X</td><td>57.6</td><td>67.1</td><td>56.5</td><td>85.9</td><td>75.4</td><td>65.8</td></tr><tr><td>GPT-4o [59]</td><td>-</td><td>X</td><td>62.3</td><td>74.5</td><td>66.0</td><td>88.0</td><td>73.7</td><td>71.3</td></tr><tr><td colspan="9">Image Referring LMMs</td></tr><tr><td>Ferret [98]</td><td>7B</td><td>X</td><td>35.2</td><td>44.7</td><td>41.9</td><td>70.4</td><td>74.6</td><td>48.8</td></tr><tr><td>Osprey [102]</td><td>7B</td><td>X</td><td>45.9</td><td>47.1</td><td>30.0</td><td>48.6</td><td>23.7</td><td>39.9</td></tr><tr><td colspan="9">Video Referring LMMs</td></tr><tr><td>VideoRefer [103]</td><td>7B</td><td>X</td><td>75.4</td><td>68.6</td><td>59.3</td><td>89.4</td><td>78.1</td><td>71.9</td></tr><tr><td>UniPixel (Ours)</td><td>3B</td><td>X</td><td>73.6</td><td>70.3</td><td>60.7</td><td>88.8</td><td>78.0</td><td>72.2</td></tr><tr><td>UniPixel (Ours)</td><td>7B</td><td>X</td><td>68.9</td><td>73.1</td><td>64.7</td><td>88.8</td><td>83.3</td><td>73.4</td></tr><tr><td>VideoRefer [103]</td><td>7B</td><td>✓</td><td>-</td><td>70.6</td><td>60.5</td><td>-</td><td>-</td><td>72.1</td></tr><tr><td>UniPixel (Ours)</td><td>3B</td><td>✓</td><td>75.3</td><td>70.7</td><td>62.3</td><td>87.4</td><td>77.2</td><td>72.8</td></tr><tr><td>UniPixel (Ours)</td><td>7B</td><td>✓</td><td>70.6</td><td>74.6</td><td>64.7</td><td>88.8</td><td>82.5</td><td>74.1</td></tr></table>
+
+Table 10: Evaluation results on our newly introduced PixelQA task. All the visual prompts are applied in a single frame. See Sec. 4.2 for detailed settings.  
+
+<table><tr><td>Method</td><td>Size</td><td>J</td><td>F</td><td>J&amp;F</td><td>Acc</td></tr><tr><td colspan="6">Point Prompts</td></tr><tr><td>InternVL2 [74]</td><td>26B</td><td>-</td><td>-</td><td>-</td><td>60.8</td></tr><tr><td>Qwen2-VL [77]</td><td>72B</td><td>-</td><td>-</td><td>-</td><td>69.3</td></tr><tr><td>UniPixel (Ours)</td><td>3B</td><td>57.3</td><td>64.4</td><td>60.9</td><td>71.1</td></tr><tr><td>UniPixel (Ours)</td><td>7B</td><td>42.1</td><td>47.1</td><td>44.6</td><td>71.4</td></tr><tr><td colspan="6">Box Prompts</td></tr><tr><td>InternVL2 [74]</td><td>26B</td><td>-</td><td>-</td><td>-</td><td>61.3</td></tr><tr><td>Qwen2-VL [77]</td><td>72B</td><td>-</td><td>-</td><td>-</td><td>69.0</td></tr><tr><td>UniPixel (Ours)</td><td>3B</td><td>57.8</td><td>64.7</td><td>61.3</td><td>70.3</td></tr><tr><td>UniPixel (Ours)</td><td>7B</td><td>41.1</td><td>46.4</td><td>43.8</td><td>71.4</td></tr><tr><td colspan="6">Mixed (50% Points + 50% Boxes)</td></tr><tr><td>InternVL2 [74]</td><td>26B</td><td>-</td><td>-</td><td>-</td><td>60.9</td></tr><tr><td>Qwen2-VL [77]</td><td>72B</td><td>-</td><td>-</td><td>-</td><td>69.1</td></tr><tr><td>UniPixel (Ours)</td><td>3B</td><td>57.2</td><td>64.1</td><td>60.6</td><td>70.8</td></tr><tr><td>UniPixel (Ours)</td><td>7B</td><td>42.3</td><td>47.5</td><td>44.9</td><td>71.4</td></tr></table>
+
+effectively comprehend both types of prompts, and accurately respond with object-centric descriptions or answers, surpassing strong models including GPT-4o [59] and VideoRefer [103].
+
+# 4.2 Q2: Pixel-Level Video Question Answering (PixelQA)
+
+We design the new PixelQA task based on VideoRefer-Bench $^{\text{Q}}$  [103], where the original mask prompts are replaced with more challenging point or box prompts. Given these ambiguous visual cues, models are expected to correctly identify the target object according to the question and the visual prompt, then respond with both the textual answer and the corresponding object masks. We report the mask prediction  $\mathcal{I} \& \mathcal{F}$  and MCQ accuracy in Tab. 10. Note that none of the existing methods supports this scenario. Thus, we apply set-of-mark prompts [97] directly on video frames, and evaluate the QA accuracies of two strong LMMs [77, 74] as our baselines. Aside from point- or box-only prompts, we also explore a more flexible setting that randomly chooses different prompts for different objects. The results verify that our memory pre-filling & injection paradigm effectively enhances the model's reasoning capabilities. Visualizations of this task are shown in Fig. 5.
+
+# 4.3 Q3: Key Ablation Studies
+
+Effect of Task Unification We study the effect of task unification in Tab. 11 (a). Unifying referring and segmentation capabilities into a single model and training them jointly leads to better results
+
+![](images/fed3eced9fec65b6fd71b568917fd0ee46ca0ee6d269bb3a5443313b32dabbf2.jpg)  
+If [1] continues moving forward, what is a likely future event?  
+What is a likely future event with [2]?
+
+![](images/73e95afe7568bdf8dd2614f4b9617cd70b34f6693e90d2d475d30221d7027ba4.jpg)  
+(A) The bear will encounter other animals (B) The bear will find a place to rest (C) The bear will start running (D) The bear will climb the stone wall
+
+![](images/6ec2ab668ff445083af57c5bd462b1a8d5f92ad2f56459b772e334d4f675d8fd.jpg)
+
+![](images/baf7ce051f86c86dbb8fc63ddfcccef0d37b3c61e1669c4bd941255ffcdb0b56.jpg)
+
+![](images/3b0e4beb650b1d3d2cf284d02a21ab1f0efab1ea3c283aa8407f5c99c3b6dea8.jpg)
+
+![](images/8d75b14646973ebe0609edf5811f0752e8596c9b4291f18ee63137af8c0dcb00.jpg)
+
+![](images/972e151b3fde7ec906816bc35ce14faf2a47b6df4501e342dd4bcec51c29801a.jpg)
+
+![](images/ed7853322e16a00bbcebd81d5e19c44167d72f57d68314776589387258d1eb99.jpg)  
+(A) He will stop and rest (B) He will start walking slowly (C) He will continue to navigate through more obstacles (D) He will sit down and take a break  
+How is [4] related to [2]?
+
+![](images/185ff2ad27d7981ac68f73eb48e743e952794bdc011c10efd89dafccccee8dd3.jpg)
+
+![](images/d5fd0922016cefa5eb9d19545911732fa1cf17d23b7406d980685cfbca06bb86.jpg)
+
+![](images/965da8e0e5b991e5761cedfd868bc0292feee6ad5cd46edc176d941d060ab3cd.jpg)
+
+![](images/ae4c619c3c28f15c06e8d35ba31ed94a92a5d9b2c1651284545ee89618604b77.jpg)
+
+![](images/d70e4c3e7923efdf7c844a3e784aed126e290ec4f80f5ec0010769511647630c.jpg)
+
+![](images/0542bb8dd20fa5a5ee76fce094f38c1fc0e4998555a56e927ed31462da272834.jpg)
+
+![](images/5d7584af293026973f36e228731dc8a1d32b793edae547f2a3c62fac464434cc.jpg)  
+(A) [4] is holding [2] (B) [4] is controlling [2] with a leash (C) [4] is walking away from [2] (D) [4] is ignoring [2]  
+Figure 5: Visualization of the outputs from UniPixel on PixelQA task. Star marks and boxes refer to point and box prompts, respectively. The boxed frames denote where the visual prompts are applied. Given different types of visual prompts on a single frame, our method can flexibly infer the relevant object, track it across the entire video, and involve its features in reasoning.
+
+![](images/1a9c0ee843a28cbcd22bf4364768a887e4b9ce9a966e31e9e25ae1cad53d05ea.jpg)
+
+![](images/4e1d033b71672015bbd823120519d86a81978775382716e3607eb2d541cfb94a.jpg)
+
+![](images/ebfc9f11c2927927cfc476a00fcad3f85affa1b428b9be8552c727494299ec02.jpg)
+
+![](images/a921225ff8d1f3f06f3f52e1ee0b7e2e5cfdccd565f4ce1c20c769885716c099.jpg)
+
+![](images/c37b4b2cab3daf0cc0da86e8a837fea42bc94721a1430519d976fb53340864a6.jpg)
+
+![](images/144b7c7e25f74706a7fcc9af73acf4e6bbc40eaec1bd9f6b18925b97e92c1be4.jpg)
+
+Table 11: Key ablation studies with UniPixel-3B on PixelQA (mixed). See Sec. 4.3 for explanations.  
+(a) Task Unification  
+
+<table><tr><td>Refer</td><td>Segment</td><td>Memory</td><td>J&amp;F</td><td>Acc</td></tr><tr><td>✓</td><td></td><td></td><td>-</td><td>64.6</td></tr><tr><td></td><td>✓</td><td></td><td>47.5</td><td>-</td></tr><tr><td>✓</td><td>✓</td><td></td><td>48.2</td><td>67.4</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>49.0</td><td>68.5</td></tr></table>
+
+(b) Object Memory Bank  
+
+<table><tr><td>Referring Method</td><td>J&amp;F</td><td>Acc</td></tr><tr><td>①&lt;REF&gt;</td><td>46.8</td><td>64.5</td></tr><tr><td>②&lt;REF&gt;&lt;SEG&gt;</td><td>47.8</td><td>64.9</td></tr><tr><td>③&lt;REF&gt;&lt;SEG&gt;+Pooling</td><td>47.5</td><td>66.3</td></tr><tr><td>④ Object Memory Bank</td><td>49.0</td><td>68.5</td></tr></table>
+
+(c) Prompt Encoder & Mask Decoder  
+
+<table><tr><td>Encoder</td><td>Decoder</td><td>J&amp;F</td><td>Acc</td></tr><tr><td>w/o Time</td><td>-</td><td>44.3</td><td>63.7</td></tr><tr><td>w/ Time</td><td>-</td><td>49.0</td><td>68.5</td></tr><tr><td>-</td><td>Independent</td><td>46.1</td><td>66.2</td></tr><tr><td>-</td><td>Propagation</td><td>49.0</td><td>68.5</td></tr></table>
+
+on both tasks (first three rows), demonstrating the mutual reinforcement effect of such unification. Incorporating memory pre-filling as an auxiliary task (last row) brings extra improvements.
+
+Effect of Object Memory Bank Tab. 11 (b) verifies the effectiveness of object memory bank.  $①$  means using a single token for each referred object.  $②$  means adding an extra segmentation token to segment it as an auxiliary task.  $③$  further appends masked-pooled visual tokens after it. The results show that (1) both adding auxiliary segmentation task and masked-pooled features help regional understanding, and (2) decoupling them via object memory bank can further boost the performance.
+
+Design Space of Prompt Encoder & Mask Decoder We compare different prompt encoder and mask decoder designs in Tab. 11 (c). The performance significantly drops when the temporal encoding in the prompt encoder is removed (first two rows). For the mask decoder (last two rows), we explore an alternative strategy that treats video frames independently (as batched images), which could largely accelerate inference but lead to sub-optimal accuracies. We hypothesize that this is because the LLM-generated <SEG> token cannot well-capture the object information in all frames, thus disentangling the segmentation and tracking capabilities to an external module is reasonable.
+
+# 5 Conclusion
+
+In this work, we proposed UniPixel, a large multi-modal model that supports flexible pixel-level visual reasoning. It unifies the internal representations of referred and segmented objects through a novel object memory bank. We observe that by such unification, the performance of object referring and segmentation can be jointly enhanced. Extensive experiments on diverse pixel-level understanding tasks, including the PixelQA task, demonstrate the significance of the proposed method. We hope this work inspires future advancements in pixel-level visual understanding.
+
+# Acknowledgements
+
+This study was supported by The Hong Kong RGC Grant (15229423) and a financial support from ARC Lab, Tencent PCG (ZGG9). We also acknowledge The University Research Facility in Big Data Analytics (UBDA) at The Hong Kong Polytechnic University for providing computing resources that have contributed to the research results reported within this paper.
+
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+
+# Appendix
+
+In this appendix, we provide more details about the training data, model implementation, and experimental settings to complement the main paper. Additional analysis, ablation studies, visualizations, and discussions are also incorporated. Below is the table of contents.
+
+A. Model
+
+1. Implementation Details  
+2. Training Recipe
+
+B. Experiments
+
+1. Tasks and Benchmarks  
+2. Evaluation Metrics  
+3. More Experimental Results  
+4. Ablation Studies  
+5. Qualitative Results
+
+C. Discussions
+
+1. Limitations & Future Work  
+2. Potential Societal Impacts
+
+D. Licenses
+
+# A Model
+
+# A.1 Implementation Details
+
+We instantiate our base models with 3B and 7B versions of Qwen2.5-VL [3]. Both variants employ pre-trained SAM 2.1 [66] with Hiera Base+ [70] backbone as the mask decoder. The  $\mathrm{M}\rightarrow \mathrm{L}$  projector is initialized with the weights from the  $\mathrm{V}\rightarrow \mathrm{L}$  projector of Qwen2.5-VL. The hidden size inside the prompt encoder is 256. To reduce GPU memory and accelerate training, we randomly sample 8 frames per video, with each frame resized to  $316^{2} \sim 448^{2}$  pixels ( $128 \sim 256$  tokens per frame). The frame sampling strategies follow the specifications of each benchmark during inference. The mask decoder has a fixed resolution of  $768 \times 768$ . For each segmentation sample, up to 5 objects are randomly selected to compute the mask prediction losses. During training, LoRA adapters [26] with rank=128 and alpha=256 are applied to all QKV0 layers in the visual encoder and LLM. The input sequences are restricted to 4K tokens. We train the model with 8 RTX A6000 Ada (48G) GPUs, with a global batch size of 256 for stages 1 and 2, and 32 for stage 3. In the first two stages, the learning rates are set to 1e-3. In the last stage, it is set to 5e-6 for the mask decoder and 2e-5 for all the other parameters, respectively. A linear warmup in the first  $3\%$  steps followed by cosine decay is adopted in all stages. The configurations of datasets are introduced in the following section.
+
+# A.2 Training Recipe
+
+The detailed distribution of training datasets for UniPixel is shown in Tab. 12. Within the three-stage training recipe, we first pre-train the sparse prompt encoder using short caption samples from Inst-IT [61] and VideoRefer [103]. For each sample, we randomly select a point inside the ground truth mask (50%) or generate an augmented box from it (50%). This stage aims to enable the model with simple visual prompt comprehension and regional captioning capabilities on images and videos. In the second stage, we align the LLM and mask decoder using referring object segmentation datasets [29, 55, 71]. We use short caption/query samples for the first two stages to focus on alignment rather than knowledge learning. For the last stage, we collect a large-scale, high-quality corpus called UniPixel-SFT-1M $^3$  to jointly train the model on diverse pixel-level tasks. The original annotations have been rewritten using task-specific templates to incorporate instructions. All the repurposed datasets and pre-processing pipelines will be publicly available to facilitate future research.
+
+Table 12: The distribution of training datasets for UniPixel. We use different background colors to denote object referring, object segmentation, regional understanding, memory pre-filling, and general video understanding data, respectively.  
+
+<table><tr><td rowspan="2">Stage</td><td rowspan="2">Dataset</td><td colspan="6">Inputs</td><td colspan="2">Outputs</td><td rowspan="2">#Samples</td><td rowspan="2">#Repeat</td><td rowspan="2">Ratio</td></tr><tr><td>Text</td><td>Image</td><td>Video</td><td>Point</td><td>Box</td><td>Mask</td><td>Text</td><td>Mask</td></tr><tr><td rowspan="2">1</td><td>Inst-IT-Image-Short-Caption [61]</td><td>✓</td><td>✓</td><td></td><td>✓</td><td>✓</td><td></td><td>✓</td><td></td><td>351K</td><td>1</td><td>41.2%</td></tr><tr><td>VideoRefer-Short-Caption [103]</td><td>✓</td><td></td><td>✓</td><td>✓</td><td>✓</td><td></td><td>✓</td><td></td><td>500K</td><td>1</td><td>58.8%</td></tr><tr><td rowspan="5">2</td><td>RefCOCO [29]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>17K</td><td>5</td><td>20.8%</td></tr><tr><td>RefCOCO+ [29]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>17K</td><td>5</td><td>20.8%</td></tr><tr><td>RefCOCOg [55]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>22K</td><td>5</td><td>26.8%</td></tr><tr><td>RefClef [29]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>18K</td><td>5</td><td>22.0%</td></tr><tr><td>Ref-Youtube-VOS [71]</td><td>✓</td><td></td><td>✓</td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>13K</td><td>3</td><td>9.5%</td></tr><tr><td rowspan="28">3</td><td>Osprey-Conversation [102]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td>✓</td><td>✓</td><td></td><td>1.4K</td><td>5</td><td>0.1%</td></tr><tr><td>Osprey-Detail-Description [102]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td>✓</td><td>✓</td><td></td><td>29K</td><td>5</td><td>2.5%</td></tr><tr><td>Osprey-Pos-Neg [102]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td>✓</td><td>✓</td><td></td><td>20K</td><td>5</td><td>1.7%</td></tr><tr><td>VideoRefer-Detailed-Caption [103]</td><td>✓</td><td></td><td>✓</td><td></td><td></td><td>✓</td><td>✓</td><td></td><td>120K</td><td>5</td><td>10.1%</td></tr><tr><td>VideoRefer-QA [103]</td><td>✓</td><td></td><td>✓</td><td></td><td></td><td>✓</td><td>✓</td><td></td><td>69K</td><td>5</td><td>5.8%</td></tr><tr><td>Inst-IT-Video-QA [61]</td><td>✓</td><td></td><td>✓</td><td></td><td></td><td>✓</td><td>✓</td><td></td><td>159K</td><td>5</td><td>13.4%</td></tr><tr><td>VideoRefer-QA-Memory [103]</td><td>✓</td><td></td><td>✓</td><td>✓</td><td>✓</td><td></td><td>✓</td><td>✓</td><td>69K</td><td>3</td><td>3.5%</td></tr><tr><td>Inst-IT-QA-Memory [61]</td><td>✓</td><td></td><td>✓</td><td>✓</td><td>✓</td><td></td><td>✓</td><td>✓</td><td>158K</td><td>3</td><td>8.0%</td></tr><tr><td>RefCOCO [29]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>17K</td><td>10</td><td>2.9%</td></tr><tr><td>RefCOCO+ [29]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>17K</td><td>10</td><td>2.9%</td></tr><tr><td>RefCOCOg [55]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>22K</td><td>10</td><td>3.7%</td></tr><tr><td>RefClef [29]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>18K</td><td>10</td><td>3.0%</td></tr><tr><td>ReasonSeg [32]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>1.6K</td><td>10</td><td>0.3%</td></tr><tr><td>ADE20K [113]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>20K</td><td>3</td><td>1.0%</td></tr><tr><td>COCOSTuff [7]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>118K</td><td>3</td><td>6.0%</td></tr><tr><td>Mapillary Vistas [57]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>18K</td><td>3</td><td>0.9%</td></tr><tr><td>PACO-LVIS [64]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>46K</td><td>3</td><td>2.3%</td></tr><tr><td>PASCAL-Part [10]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>4.4K</td><td>3</td><td>0.2%</td></tr><tr><td>Ref-/YouTube-VOS [71]</td><td>✓</td><td></td><td>✓</td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>13K</td><td>5</td><td>1.1%</td></tr><tr><td>Ref-DAVIS17 [62]</td><td>✓</td><td></td><td>✓</td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>0.6K</td><td>10</td><td>0.1%</td></tr><tr><td>Ref-SAV [101]</td><td>✓</td><td></td><td>✓</td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>56K</td><td>3</td><td>2.8%</td></tr><tr><td>MeViS [21]</td><td>✓</td><td></td><td>✓</td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>23K</td><td>5</td><td>1.9%</td></tr><tr><td>LV-VIS [76]</td><td>✓</td><td></td><td>✓</td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>11K</td><td>3</td><td>0.6%</td></tr><tr><td>ViCaS [1]</td><td>✓</td><td></td><td>✓</td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>41K</td><td>3</td><td>2.1%</td></tr><tr><td>ReVOS [96]</td><td>✓</td><td></td><td>✓</td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>29K</td><td>5</td><td>2.5%</td></tr><tr><td>GroundMoRe [20]</td><td>✓</td><td></td><td>✓</td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>5.6K</td><td>3</td><td>0.3%</td></tr><tr><td>LLaVA-1.5-Mix-665K [42]</td><td>✓</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td></td><td>647K</td><td>1</td><td>10.9%</td></tr><tr><td>VideoGPT+ Instruct [52]</td><td>✓</td><td></td><td>✓</td><td></td><td></td><td></td><td>✓</td><td></td><td>573K</td><td>1</td><td>9.7%</td></tr></table>
+
+# B Experiments
+
+# B.1 Tasks and Benchmarks
+
+Our method is extensively evaluated across 9 fine-grained image/video understanding tasks. The benchmark(s) used for each task are listed as follows:
+
+1. Reasoning Video Object Segmentation: ReVOS [96]  
+2. Referring Video Object Segmentation: MeViS [21], Ref-YouTube-VOS [71], Ref-DAVIS17 [62], Ref-SAV [101]  
+3. Motion-Grounded Video Reasoning: GroundMoRe [20]  
+4. Referring Expression Segmentation: RefCOCO [29], RefCOCO+ [29], RefCOCOg [55]  
+5. Reasoning Segmentation: ReasonSeg [32]  
+6. Referring Expression Comprehension: RefCOCO [29], RefCOCO+ [29], RefCOCOg [55]  
+7. Referred Video Description: VideoRefer-Bench [103]  
+8. Referred Video Question Answering: VideoRefer-Bench [103]  
+9. Flexible Pixel-Level Understanding: PixelQA (Ours)
+
+# B.2 Evaluation Metrics
+
+For video segmentation tasks, we adopt  $\mathcal{I}\&\mathcal{F}$  as the main metric to jointly consider region similarity  $\mathcal{I}$  and contour accuracy  $\mathcal{F}$ . Image segmentation is evaluated using cIoU (the cumulative intersection over the cumulative union) and gIoU (the average of all per-image IoUs) following existing work. For referred video description and question answering tasks, we follow the official evaluation protocols to report GPT-4o [59] scores and MCQ accuracy, respectively. For referring expression comprehension,
+
+Table 13: Performance comparison on general video question answering (VideoQA) on MVBench [36]. Note that UniPixel is the only model supporting pixel-level referring & segmentation.  
+
+<table><tr><td>Model</td><td>Size</td><td>AS</td><td>AP</td><td>AA</td><td>FA</td><td>UA</td><td>OE</td><td>OI</td><td>OS</td><td>MD</td><td>AL</td><td>ST</td><td>AC</td><td>MC</td><td>MA</td><td>SC</td><td>FP</td><td>CO</td><td>EN</td><td>ER</td><td>CI</td><td>Avg.</td></tr><tr><td>GPT-4V [58]</td><td>-</td><td>55.5</td><td>63.5</td><td>72.0</td><td>46.5</td><td>73.5</td><td>18.5</td><td>59.0</td><td>29.5</td><td>12.0</td><td>40.5</td><td>83.5</td><td>39.0</td><td>12.0</td><td>22.5</td><td>45.0</td><td>47.5</td><td>52.0</td><td>31.0</td><td>59.0</td><td>11.0</td><td>43.5</td></tr><tr><td>Video-ChatGPT [53]</td><td>7B</td><td>23.5</td><td>26.0</td><td>62.0</td><td>22.5</td><td>26.5</td><td>54.0</td><td>28.0</td><td>40.0</td><td>23.0</td><td>20.0</td><td>31.0</td><td>30.5</td><td>25.5</td><td>39.5</td><td>48.5</td><td>29.0</td><td>33.0</td><td>29.5</td><td>26.0</td><td>35.5</td><td>32.7</td></tr><tr><td>Video-LLaMA [105]</td><td>7B</td><td>27.5</td><td>25.5</td><td>51.0</td><td>29.0</td><td>39.0</td><td>48.0</td><td>40.5</td><td>38.0</td><td>22.5</td><td>22.5</td><td>43.0</td><td>34.0</td><td>22.5</td><td>32.5</td><td>45.5</td><td>32.5</td><td>40.0</td><td>30.0</td><td>21.0</td><td>37.0</td><td>34.1</td></tr><tr><td>VideoChat [35]</td><td>7B</td><td>33.5</td><td>26.5</td><td>56.0</td><td>33.5</td><td>40.5</td><td>53.0</td><td>40.5</td><td>30.0</td><td>25.5</td><td>27.0</td><td>48.5</td><td>35.0</td><td>20.5</td><td>42.5</td><td>46.0</td><td>26.5</td><td>41.0</td><td>23.5</td><td>23.5</td><td>36.0</td><td>35.5</td></tr><tr><td>Video-LLaVA [38]</td><td>7B</td><td>46.0</td><td>42.5</td><td>56.5</td><td>39.0</td><td>53.5</td><td>53.0</td><td>48.0</td><td>41.0</td><td>29.0</td><td>31.5</td><td>82.5</td><td>45.0</td><td>26.0</td><td>53.0</td><td>41.5</td><td>33.5</td><td>41.5</td><td>27.5</td><td>38.5</td><td>31.5</td><td>43.0</td></tr><tr><td>TimeChat [68]</td><td>7B</td><td>40.5</td><td>36.0</td><td>61.0</td><td>32.5</td><td>53.0</td><td>53.5</td><td>41.5</td><td>29.0</td><td>19.5</td><td>26.5</td><td>66.5</td><td>34.0</td><td>20.0</td><td>43.5</td><td>42.0</td><td>36.5</td><td>36.0</td><td>29.0</td><td>35.0</td><td>35.0</td><td>38.5</td></tr><tr><td>PLLAVA [94]</td><td>7B</td><td>58.0</td><td>49.0</td><td>55.5</td><td>41.0</td><td>61.0</td><td>56.0</td><td>61.0</td><td>36.0</td><td>23.5</td><td>26.0</td><td>82.0</td><td>39.5</td><td>42.0</td><td>52.0</td><td>45.0</td><td>42.0</td><td>53.5</td><td>30.5</td><td>48.0</td><td>31.0</td><td>46.6</td></tr><tr><td>ST-LLM [44]</td><td>7B</td><td>66.0</td><td>53.5</td><td>84.0</td><td>44.0</td><td>58.5</td><td>80.5</td><td>73.5</td><td>38.5</td><td>42.5</td><td>31.0</td><td>86.5</td><td>36.5</td><td>56.5</td><td>78.5</td><td>43.0</td><td>44.5</td><td>46.5</td><td>34.5</td><td>41.5</td><td>58.5</td><td>54.9</td></tr><tr><td>VideoGPT+ [52]</td><td>4B</td><td>69.0</td><td>60.0</td><td>83.0</td><td>48.5</td><td>66.5</td><td>85.5</td><td>75.5</td><td>36.0</td><td>44.0</td><td>34.0</td><td>89.5</td><td>39.5</td><td>71.0</td><td>90.5</td><td>45.0</td><td>53.0</td><td>50.0</td><td>29.5</td><td>44.0</td><td>60.0</td><td>58.7</td></tr><tr><td>VideoChat2 [36]</td><td>7B</td><td>75.5</td><td>58.0</td><td>83.5</td><td>50.5</td><td>60.5</td><td>87.5</td><td>74.5</td><td>45.0</td><td>47.5</td><td>44.0</td><td>82.5</td><td>37.0</td><td>64.5</td><td>87.5</td><td>51.0</td><td>66.5</td><td>47.0</td><td>35.0</td><td>37.0</td><td>72.5</td><td>60.4</td></tr><tr><td>UniPixel (Ours)</td><td>3B</td><td>69.5</td><td>62.5</td><td>83.0</td><td>48.5</td><td>76.5</td><td>86.5</td><td>66.5</td><td>38.0</td><td>49.0</td><td>40.5</td><td>87.0</td><td>49.0</td><td>74.0</td><td>95.0</td><td>49.0</td><td>45.0</td><td>63.5</td><td>34.5</td><td>58.0</td><td>73.5</td><td>62.5</td></tr><tr><td>UniPixel (Ours)</td><td>7B</td><td>71.0</td><td>68.0</td><td>84.0</td><td>45.0</td><td>78.0</td><td>91.5</td><td>66.5</td><td>35.5</td><td>57.5</td><td>43.0</td><td>91.5</td><td>47.0</td><td>73.5</td><td>92.5</td><td>58.0</td><td>53.0</td><td>74.0</td><td>37.5</td><td>49.0</td><td>69.0</td><td>64.3</td></tr></table>
+
+Table 14: Effectiveness justification of multi-stage training. The best and second-best results are marked bold and underlined, respectively. The three-stage recipe leads to optimal performance.  
+
+<table><tr><td rowspan="2">Stage 1</td><td rowspan="2">Stage 2</td><td rowspan="2">Stage 3</td><td colspan="3">ReVOS</td><td colspan="3">MeViS (valu)</td><td colspan="2">VideoRefer-BenchQ</td></tr><tr><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>Single-Frame</td><td>Multi-Frame</td></tr><tr><td rowspan="3">✓</td><td></td><td>✓</td><td>58.3</td><td>63.6</td><td>61.0</td><td>54.8</td><td>61.9</td><td>58.4</td><td>71.1</td><td>71.5</td></tr><tr><td></td><td>✓</td><td>59.0</td><td>63.4</td><td>61.2</td><td>55.2</td><td>62.1</td><td>58.7</td><td>71.8</td><td>72.3</td></tr><tr><td>✓</td><td>✓</td><td>59.6</td><td>63.5</td><td>61.6</td><td>55.7</td><td>62.5</td><td>59.1</td><td>71.2</td><td>71.6</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>59.7</td><td>64.4</td><td>62.1</td><td>56.1</td><td>63.2</td><td>59.7</td><td>72.2</td><td>72.8</td></tr></table>
+
+we leverage mean accuracies, where a predicted bounding box is considered correct when it has the intersection over union (IoU) with the ground truth no less than 0.5.
+
+# B.3 More Experimental Results
+
+General Video Question Answering We also evaluate UniPixel on MVBench [36] to compare its general video understanding capabilities with existing methods. The results are illustrated in Tab. 13. Note that our method is the only one in the table that supports referring and segmentation. By jointly training on holistic-level and pixel-level data, UniPixel can effectively balance the capabilities under both scenarios, demonstrated by the strong performance compared with holistic-level models.
+
+# B.4 Ablation Studies
+
+Effect of Multi-stage Training We investigate the effectiveness of multi-stage training in Tab. 14. As shown in the first line, directly training the model using large-scale data only leads to sub-optimal performance, due to the unaligned representations among prompt encoder, LLM, and mask decoder. We observe that pre-training either the sparse prompt encoder or the  $\mathrm{L}\rightarrow \mathrm{M}$  projector (the second and third lines) brings performance gains on both tasks (referring and segmentation). We hypothesize that this is because pre-aligning either of them can alleviate the burden of joint-task learning in stage 3. The last row verifies that the performance can be further boosted by pre-aligning both of them.
+
+Number of Hidden Tokens for Mask Decoder As mentioned in the main paper, there is a huge gap between the feature dimensions of the LLM and the mask decoder, thus splitting the <SEG> token into more hidden tokens can better preserve the object information from the LLM. We ablate this mechanism in Tab. 15. According to the results, using only 1 hidden token cannot fully preserve the object information, as the mask prediction performance is sub-optimal. However, we also observe that using more than 2 hidden tokens (e.g., 4 or 8) only brings negligible performance gain. Therefore, we choose 2 hidden tokens per object in our final model.
+
+Training Strategy for the  $\mathbf{M}\rightarrow \mathbf{L}$  projector The  $\mathrm{M}\to \mathrm{L}$  projector aims to project the masked-pooled object-centric features to the LLM's embedding space. Since the object features originate from the visual encoder, it is possible to re-use the pre-trained weights of the original  $\mathrm{V}\rightarrow \mathrm{L}$  projector in Qwen2.5-VL. Its effects are studied in Tab. 16. We investigated two strategies: 1) re-using the weights and 2) adding an extra pre-training stage for better alignment. The comparison shows that directly re-using weights without extra pre-training can achieve the best results.
+
+Table 15: Ablation study on the number of hidden tokens for each  $<\mathrm{SEG}>$ . Performance gains are negligible with more than 2 tokens/object.  
+
+<table><tr><td rowspan="2">#Tokens</td><td colspan="3">ReVOS</td><td colspan="3">MeViS (valu)</td></tr><tr><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td></tr><tr><td>1</td><td>59.6</td><td>63.5</td><td>61.6</td><td>55.8</td><td>62.5</td><td>59.2</td></tr><tr><td>2</td><td>59.7</td><td>64.4</td><td>62.1</td><td>56.1</td><td>63.2</td><td>59.7</td></tr><tr><td>4</td><td>59.8</td><td>63.9</td><td>61.9</td><td>56.8</td><td>63.1</td><td>59.9</td></tr><tr><td>8</td><td>59.5</td><td>64.0</td><td>61.8</td><td>56.4</td><td>62.8</td><td>59.6</td></tr></table>
+
+Table 16: Ablation study on  $\mathrm{M}\rightarrow \mathrm{L}$  projector. Init and PT denote weight initialization from  $\mathrm{V}\rightarrow \mathrm{L}$  projector and extra pre-training, respectively.  
+
+<table><tr><td rowspan="2">Init</td><td rowspan="2">PT</td><td colspan="2">VideoRefer-BenchQ</td><td>PixelQA</td></tr><tr><td>Single-Frame</td><td>Multi-Frame</td><td>Mixed Acc</td></tr><tr><td></td><td></td><td>71.4</td><td>71.9</td><td>67.7</td></tr><tr><td></td><td>✓</td><td>71.5</td><td>71.7</td><td>67.4</td></tr><tr><td>✓</td><td></td><td>72.4</td><td>72.6</td><td>68.2</td></tr><tr><td>✓</td><td>✓</td><td>72.2</td><td>72.8</td><td>68.5</td></tr></table>
+
+Table 17: Ablation study on training data used in stage 3. The best and second-best results are marked bold and underlined, respectively. Gradually adding more pixel-level data brings performance gains.  
+
+<table><tr><td rowspan="2">Regional</td><td rowspan="2">Segmentation</td><td rowspan="2">Memory</td><td rowspan="2">General</td><td colspan="3">ReVOS</td><td colspan="3">MeViS (valu)</td><td colspan="2">VideoRefer-BenchQ</td></tr><tr><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>Single-Frame</td><td>Multi-Frame</td></tr><tr><td>✓</td><td></td><td></td><td></td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>72.1</td><td>72.0</td></tr><tr><td></td><td>✓</td><td></td><td></td><td>58.9</td><td>63.8</td><td>61.4</td><td>56.0</td><td>63.2</td><td>59.6</td><td>-</td><td>-</td></tr><tr><td>✓</td><td>✓</td><td></td><td></td><td>59.2</td><td>63.7</td><td>61.5</td><td>55.8</td><td>63.1</td><td>59.5</td><td>72.3</td><td>72.6</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td></td><td>59.6</td><td>64.5</td><td>62.1</td><td>56.3</td><td>63.5</td><td>59.9</td><td>72.4</td><td>72.5</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>59.7</td><td>64.4</td><td>62.1</td><td>56.1</td><td>63.2</td><td>59.7</td><td>72.2</td><td>72.8</td></tr></table>
+
+Combination of Training Data Tab. 17 studies the effect of the combination of multi-task cotraining data in stage 3. Compared with training only on the regional or segmentation data, leveraging both of them leads to considerable performance on both tasks. Incorporating memory pre-filling data (requiring both referring and segmentation) can further boost the performance. We also mix some general holistic-level video understanding data to preserve the original capabilities of the pre-trained model, while it slightly affects the performance on pixel-level tasks.
+
+# B.5 Qualitative Results
+
+Fig.  $6\sim 11$  present more visualizations of outputs from UniPixel on different pixel-level understanding tasks. Our method can effectively handle flexible visual prompts [103], implicit queries [32, 96], long queries [101], and motion-grounded questions [20].
+
+# C Discussion
+
+# C.1 Limitations & Future Work
+
+Due to the limited computing resources, we did not further scale up the training data to incorporate more pixel-level tasks such as grounded caption generation (GCG) on images [65] or videos [56], which are interesting scenarios and their data may bring more performance gains. Besides, the mask decoder currently predicts the first mask on the first frame and propagates it to the following frames, while it potentially supports predicting on the best frame (defined as the frame with the best view of the target) and propagates it to both sides of the video. We will focus in our future work to explore more pixel-level understanding tasks and more flexible mechanisms for the mask decoder.
+
+# C.2 Potential Societal Impacts
+
+This work introduces a new framework for pixel-level visual-language understanding, which could potentially be used in education, surveillance, and healthcare industries, where flexible interactions with the users and fine-grained understanding of images & videos are required. In other scenarios requiring multi-modal assistants, our method can also serve as a more advanced alternative. To the best of our knowledge, there are no potential negative societal impacts to declare.
+
+# D Licenses
+
+Our model is built based on the pre-trained Qwen2.5-VL [3] and SAM 2.1 [66] models. They are both licensed under the Apache License 2.0 (https://www.apache.org/licenses/License-2.0).
+
+What action does [1] perform that involves [3]?
+
+(A) [1] extends an arm across [3]'s chest (B) [1] hands something to [3] (C) [1] talks to [3] (D) [1] ignores [3]
+
+![](images/598df0f2b0dea42cd854ae336e479cfe4930f1cf19d5369c13525a52b0578570.jpg)
+
+![](images/fbbfe0508cddde48d6551515850e775c6a3aa1120b639e03cada6c9444363eed.jpg)
+
+What is [1] wearing?
+
+(A) Blue sweatshirt and black jeans (B) Red sweatshirt and light blue jeans (C) Green t-shirt and white pants (D) Yellow hoodie and dark blue jeans
+
+![](images/44764a6c5298a4df5ec3f963c3e7b0aa05ebd0e11ccdbd726a5d2ac0ab9cbf6b.jpg)
+
+![](images/2210f0b3e7ed7afc58ad67a3728358f99c16443814e70057b928083c62bfd93d.jpg)
+
+If [1] continues to move forward, what is a likely future event involving [2]?
+
+(A) [2] will run away (B) [2] will sit down and stop moving (C) [2] will start barking (D) [2] will continue walking by the wheelchair
+
+![](images/46aa4f64858b873762eb199dc8744130253ddeeeded02e656b5f505a74f4bbc1.jpg)
+
+![](images/bc3e5ba9e52d6d6d629bb8860e6393b7d9dd0a870e371d82f55c35b2fc87e089.jpg)
+
+If  $\langle \text{object1} \rangle \langle \text{region} \rangle$  continues riding the bike, what is a likely future event?
+
+(A) [1] will stop (B) [1] will start running (C) [1] will change a different outfit (D) [1] will continue to challenge different high difficulty movements
+
+![](images/2039555bbb778e2c7776f34cdb40cc5096188a72319a4373fd11b43b32cf5a9c.jpg)  
+Figure 6: Visualization of the predictions from UniPixel on PixelQA.
+
+![](images/9eade8ef4199f7bbc4f975f6b4b86e98038e8835138245b6fc314f8d41e4d33d.jpg)
+
+Please segment the cow that is the furthest from the camera in this video.
+
+![](images/c214033751d38905dd53df3606348f23ff2301e657a08e1e47eb04929cf297a7.jpg)
+
+Which goldfish is on the left side of the screen at the beginning of the video? Please provide the segmentation mask.
+
+![](images/3ce3350c1f61acd244f41163aaa207d7a018a8fb2aa9fa8ee27b8589650de0fd.jpg)
+
+Where is the instrument that serves to shield from the sun or protect from rain and snow?
+
+![](images/44ad87fdce59cb8335e6047f6b709b33f90f16a2068586690f26b0fefb6ba3c0.jpg)
+
+Can you find the skunk that has black fur all over its body and a tuft of white fur on its head and the tip of its tail?
+
+![](images/05a42575bb3e82296a39c10c0030a1a8b9d4967558b440059f99f885db398711.jpg)
+
+Which ferret(s) is/are being licked by a cat consistently? Please provide the segmentation mask.
+
+![](images/517856bc4719087d43991a44799ff91699e032fef005bf6b0e28ed52c64cb531.jpg)
+
+Can you segment the insect(s) belonging to the superfamily Papilionoidea of the Lepidoptera order in this video?
+
+![](images/6a572394a4fe7ef40fe99490bcdb98739b8070393b3c48f858ff8366a7c96c59.jpg)
+
+Please segment the zebra which is younger in this video.
+
+![](images/a16866fe16cef25e55fc26e160f796f502197ab4f55beec88e97035befe83004.jpg)  
+Figure 7: Visualization of the predictions from UniPixel on ReVOS [96].
+
+Please segment the black swan in this video.
+
+![](images/b18dd21c283f847ff03520e00de7f112f7028ca4b556f69f11ba8dc39cee9573.jpg)
+
+Where is the man wearing a cap and shorts in this video? Respond with the segmentation mask.
+
+![](images/3dd8dcde89a917814d2fd09c0d3250614af1ac38254ba6149e9abfbb17798a78.jpg)
+
+Can you find the blue wooden car in the frames?
+
+![](images/5ab5a05406e57c228de16efbf4a16d7df1ec3fc6757170e799b7a57a5def20b7.jpg)
+
+Segment and track the green motorbike in this video.
+
+![](images/6f7b61c69a90ac7d98d40ffbd2061116963244110ae70765aeadd3e045530021.jpg)
+
+Where is the rope? Give me the segmentation results directly.
+
+![](images/de36923c7f9c43f52e68511d359106093146c6ab8661bdcec91809fd3135d5da.jpg)  
+Figure 8: Visualization of the predictions from UniPixel on Ref-DAVIS17 [62].
+
+Q: Who might not open the cooler if not for feeding the walrus a fish?
+
+A: The woman.
+
+![](images/c3d7e1d045bbb891401f6bddbd5042659634e0a3f43382e2371335112e2d23d1.jpg)
+
+Q: What might not be given to the woman by the man if he did not eat by himself?
+
+A: The bag.
+
+![](images/03f1a4fe83475b01817f93b97a2bf668c833a5cbeacbdc26367aa1372628ad0a.jpg)
+
+Q: Who opens the ziploc bag to transfer the crushed Oreo cookies into the bowl?
+
+A: The girl.
+
+![](images/5287d0e9d83bf03456aae5d1f056fd50f43d8fdf6bea88e6bd34103c84e5d94d.jpg)
+
+Q: Who dribbles the ball before he shoots it?
+
+A: The man in the black shorts.
+
+![](images/7821481530431fee4813afeb5e10ac5521fd711da8d6659860f855d0fd75ed93.jpg)
+
+Q: Who kicks the ball into the goal?
+
+A:The boy.
+
+![](images/19b1868cc0551ea64cbf84d2de556bc138ae5fb5d4a7529e0251be6e3c4b39c4.jpg)
+
+Q: Who asked if the little girl could carry the box before she picked it up?
+
+A: The man.
+
+![](images/f8e74bf86772e7cf56fbcbbf0d4e1d3dcab6b5289cd97b1f606498943171d274.jpg)  
+Figure 9: Visualization of the predictions from UniPixel on GroundMoRe [20].
+
+Find the object according to the description: The object is a dark-colored backpack with light-colored accents, featuring multiple compartments and pockets, securely fastened to an individual's back. The person is dressed in dark clothing and ascending an escalator in a public setting, likely a mall or transportation hub. The backpack has adjustable straps and a top handle, appearing functional for carrying various items. The individual moves steadily up the escalator, indicating a purposeful journey.
+
+![](images/74d1d97f9bbbe80b333cb6e1601249436fbbeb06f6d9c96e6093ac2d8ae3153b.jpg)
+
+Analyze the following sentences and provide the corresponding segmentation mask: The object is a dark-colored sedan, likely blue or black, parked on an unpaved surface, possibly a dirt road or an area with loose soil. It has four doors, a visible rear spoiler on the trunk, silver wheels, and tinted windows. The car is slightly tilted, suggesting it might be parked on uneven ground or experiencing some form of imbalance. Throughout the video, the sedan remains stationary, with no indication of movement or actions being performed by the vehicle.
+
+![](images/55b381ee1aba3e09095c7412de0038320d09d7a19741d1e3a39c4086ea102e8a.jpg)
+
+Please segment the object according to the description: The object is a person with long dark hair, wearing a dark top and a patterned skirt with geometric designs. This individual is stationary or moving very slowly in the background of a retail store, possibly a furniture or home goods store. The person remains in close proximity to another shopper pushing a shopping cart, suggesting they might be together or interacting. The scene captures a typical shopping experience.
+
+![](images/4288e01e1cbf92f6706eb7eee1f6aa3c66da48b212fa935c479a5da60c6d15ba.jpg)  
+Figure 10: Visualization of the predictions from UniPixel on Ref-SAV [101].
+
+Find the lens that is more suitable for photographing nearby objects.
+
+![](images/7bc544627a0d047acbde230cdc4eab06733cec39b4346a2c682daddae7e0ff7a.jpg)
+
+Which part of the vehicle must be used to display identifying information as required by law? Segment the target directly.
+
+Where is the goat nearest to the bottom stone? Give me the segmentation mask.
+
+![](images/1c1bf68fe49e0128a380f0ec05f75914ddd044b2b3cb9d575a8578c7e24a6224.jpg)
+
+What item in the picture can provide information to help guide travelers through this rugged terrain that can be challenging to navigate?
+
+Please localize the place where piano players should sit in this image.
+
+![](images/4a8a3675f84c796ea29c65c90a0973a23efbaa6ec718f3b6afb3ce7f2e358843.jpg)
+
+Where is the place where the garbage should be put? Please respond with the segmentation mask.
+
+Segment the place where the patient lies down to receive examination in this image.
+
+![](images/faece438adbdb6b56e091376cb3904d759254803e1945e4247dc891675d19cab.jpg)
+
+In some rural areas, horse-drawn carts are still used for transportation and carrying goods. What is the main source of power that drives the cart in the picture?
+
+![](images/a6b9702d94558c585fa340a1a7b4f9dd73ec3f60ed647b9071d3ea74ee6a44b5.jpg)  
+Figure 11: Visualization of the predictions from UniPixel on ReasonSeg [32].
+
+![](images/35946298a76436b7b9c5c7e1873b6ab961c62c1738c3598f9a8c4ffa0b5cbff8.jpg)
+
+![](images/5a9895d72bb1a789bc2693ce212fe68882011f827d402342e0fafe274b151b15.jpg)
+
+![](images/e63ea9814b67ca423427bf5e0056365a2da17ec6b74c177030f10f293d84c61c.jpg)
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: To the best of our knowledge, all the claims made in the abstract and introduction are verified either by referring to previous studies or by conducting experiments.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: The limitations of this work are carefully discussed in the appendix.
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [NA]
+
+Justification: The paper does not include theoretical results. All the claims shall be verified through ablation studies.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+# Answer: [Yes]
+
+Justification: We clearly present the list of datasets used for training, the implementation details, and the evaluation metrics we used to produce the experimental results in the appendix.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.  
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.  
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.  
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example  
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: We open-source all the code, checkpoints, data, and training logs to ensure full reproducibility.
+
+Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).  
+- The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.  
+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.  
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: All the detailed settings, hyperparameters, and other necessary information are provided in the appendix.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [No]
+
+Justification: We are not able to fully report the error bars given the limited computing resources.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
+
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).  
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: This information is included in the implementation details in the appendix.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: We confirm that the research conforms to the NeurIPS Code of Ethics.
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: The potential societal impacts are discussed in the appendix.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.
+
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: The paper poses no such risks.
+
+# Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: We provided citations to the original papers for all the used datasets and base models.
+
+# Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
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+# 13. New assets
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+
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+
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\ No newline at end of file
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+# UniRelight: Learning Joint Decomposition and Synthesis for Video Relighting
+
+Kai He $^{1,2,3}$  Ruofan Liang $^{1,2,3}$  Jacob Munkberg $^{1}$  Jon Hasselgren $^{1}$
+
+Nandita Vijaykumar $^{2,3}$  Alexander Keller $^{1}$  Sanja Fidler $^{1,2,3}$
+
+Igor Gilitschenski $^{2,3\dagger}$  Zan Gojcic $^{1\dagger}$  Zian Wang $^{1,2,3\dagger}$
+
+$^{1}$ NVIDIA  $^{2}$ University of Toronto  $^{3}$ Vector Institute
+
+# Abstract
+
+We address the challenge of relighting a single image or video, a task that demands precise scene intrinsic understanding and high-quality light transport synthesis. Existing end-to-end relighting models are often limited by the scarcity of paired multi-illumination data, restricting their ability to generalize across diverse scenes. Conversely, two-stage pipelines that combine inverse and forward rendering can mitigate data requirements but are susceptible to error accumulation and often fail to produce realistic outputs under complex lighting conditions or with sophisticated materials. In this work, we introduce a general-purpose approach that jointly estimates albedo and synthesizes relit outputs in a single pass, harnessing the generative capabilities of video diffusion models. This joint formulation enhances implicit scene comprehension and facilitates the creation of realistic lighting effects and intricate material interactions, such as shadows, reflections, and transparency. Trained on synthetic multi-illumination data and extensive automatically labeled real-world videos, our model demonstrates strong generalization across diverse domains and surpasses previous methods in both visual fidelity and temporal consistency. Our project page is https://research.nvidia.com/labs/toronto-ai/UniRelight/.
+
+# 1 Introduction
+
+Lighting is crucial in defining a scene's visual appearance, influencing both aesthetics and the perception of objects and materials. Modifying illumination while preserving scene content enables a wide range of applications, including creative editing, simulation, and robust vision systems. Realistic relighting necessitates an accurate understanding of scene geometry and materials, as well as faithful light transport modeling. Recent methods aim to learn these complex relationships from data, but a significant challenge is the scarcity of multi-illumination datasets capturing the same scene under varied lighting conditions. Acquiring such data in the real world requires controlled environments, calibrated equipment, and repeated captures, making it costly and impractical. As a result, most existing approaches rely on synthetic data or focus on narrow domains like portraiture, limiting their ability to generalize to real-world scenes.
+
+To address the lack of multi-illumination data, recent works [69, 39] have decomposed the relighting task into two stages: an inverse rendering step that estimates scene attributes such as albedo, normal, and depth (i.e., G-buffers), followed by a forward rendering step that synthesizes relit images conditioned on these estimates. This pipeline avoids the need for multi-illumination supervision, allowing each stage to be trained separately using single-illumination data. However, inverse and forward rendering are inherently coupled, and modeling them separately with discrete intermediate representations introduces significant limitations. The forward renderer is highly sensitive to errors in
+
+![](images/fc6c5a544ffd02a281c2660ab72e81aace09886fc5493d96b09e463ca8bad158.jpg)  
+Input & Albedo Pred.
+
+![](images/208641accb7554c730b31a50036855db1dfb948487f5bbfd84475c68633a62d5.jpg)
+
+![](images/1bf5ee94f42ab67f801152baf0a4aec309ee6c6596935e8f84766762909d39e9.jpg)  
+Figure 1: Given an input image (top left) or video, our method jointly estimates albedo (bottom left) and synthesizes relit videos with novel lighting conditions using provided HDR probes. Notably, our estimated albedo maps effectively demodulate shadows and specular highlights, while the relit images exhibit plausible shadows and specular highlights.
+
+![](images/a6351b86d69858856971b3826a01d5e9ff1294acbe1db816c0a24dbbde7f8793.jpg)  
+Relighting I
+
+![](images/a8dcd2ae8d563c77cb82017138edb85690bc1577d2fda081955e422cbdc389ee.jpg)
+
+![](images/5726e91013a41af0ac8b4723820ce445c69344582baa72a6995d2f0f0588eab8.jpg)  
+Relighting II
+
+![](images/381e3df3994b237b726c96534a8afda236e2a7f7a49404489e9e67a8bcb5034f.jpg)
+
+the estimated G-buffers and struggles to capture complex material properties not encoded by them, such as transparency and subsurface scattering.
+
+Learning an end-to-end relighting model in low data regime is challenging. While prior methods for intrinsic estimation (e.g., geometry [28, 19] and albedo [32, 39]) trained on synthetic data often generalize well to real-world scenes, we observe that directly training relighting models as video-to-video translation leads to poor generalization to unseen domains. We argue that robust relighting requires an explicit understanding of illumination and scene properties.
+
+Hence, we propose a relighting framework that jointly models the distribution of scene intrinsics and illumination. Inspired by VideoJAM [12], we train a video generative model that jointly denoises latent space for relighting and albedo demodulation in a single pass. Practically, we concatenate both latent representations and treat them as a single video clip. This design is motivated by the hypothesis that demodulation provides a strong prior for the relighting task, such as removing shadows. This joint formulation encourages the model to learn an internal representation of scene structure, leading to improved generalization across diverse and unseen domains. Our approach contrasts with two-stage pipelines that rely on explicit G-buffer estimation. Instead, we implicitly reason about intrinsic scene representation, enabling better representation learning, reducing error accumulation, and modeling of complex visual effects such as specular highlights, transparency, and subsurface scattering.
+
+Our formulation also offers flexibility to learn from diverse sources of supervision. We leverage high-quality synthetic data for full supervision and complement it with real-world single illumination videos that can be auto-labeled at scale. This hybrid training strategy enables the model to handle complex lighting effects while significantly improving generalization to unseen domains, see Figure 1. UNIRELIGHT enables high-quality relighting and intrinsic decomposition from a single input image or video, producing temporally consistent shadows, reflections, and transparency, and outperforms state-of-the-art methods.
+
+# 2 Related Work
+
+Inverse rendering estimates intrinsic scene properties such as geometry, materials, and lighting from input images. Traditional approaches use hand-crafted priors within an optimization framework [34, 3, 21, 11, 77, 2] to handle low-order effects like diffuse shading and reflectance. Recently, the field has been revitalized by applying supervised and self-supervised learning [4, 33, 35, 57, 66, 36, 37, 10, 61, 60, 63, 6] to inverse rendering tasks. These methods typically require large, domain-specific datasets, and struggle to generalize outside the training domain.
+
+![](images/df509f8a6ff6b060ee5d27a3a188f946bd93ddd2920df4bfdd2649b992f30aed.jpg)  
+Figure 2: Method overview. Given an input video  $\mathbf{I}$  and a target lighting configuration  $(\mathbf{E}_{\mathrm{ldr}},\mathbf{E}_{\mathrm{log}},\mathbf{E}_{\mathrm{dir}})$ , our method jointly predicts a relit video  $\hat{\mathbf{l}}_E$  and its corresponding albedo  $\hat{\mathbf{a}}$ . We use a pretrained VAE encoder-decoder pair  $(\mathcal{E},\mathcal{D})$  to map input and output videos to a latent space. The latents for the target relit video and albedo are concatenated along the temporal (frame) dimension with the encoded input video. Lighting features  $\mathbf{h}^{\mathbf{E}}$ , derived from the environment maps, are concatenated along the channel dimension with the relit video latent. A finetuned DiT video model denoises the joint latent according to Equation 1, enabling consistent generation of both relit appearance and intrinsic decomposition.
+
+Most related to our approach are methods that leverage large generative image and video models for inverse rendering [18, 32, 54, 40, 30, 14, 47, 41]. Notably, RGB  $\leftrightarrow$  X [69] demonstrates both intrinsic decomposition and neural rendering from intrinsics using finetuned image diffusion models. The DiffusionRenderer [39] extends RGB  $\leftrightarrow$  X to video and also supports relighting.
+
+Relighting is the task of modifying the lighting conditions in images or videos. Many methods first reconstruct 3D scenes from multi-view images [46, 22, 13, 9, 71, 76, 70, 56, 62, 38, 64, 42, 58, 25]. Previously, surface reflectance has been captured using calibrated light stages [15]. Now, relighting is supported by material properties recovered from inverse rendering. These methods typically require per-scene optimization and are limited to static, object-centric scenes with single illumination. Training across multiple scenes has led to the exploration of latent feature learning [78, 43, 72, 7], often incorporating neural rendering modules with intrinsic buffers as priors [53, 20, 50, 31, 65].
+
+Recent approaches [55, 31, 68, 26, 39, 5, 75, 45] leverage diffusion models for relighting tasks. These methods are often domain-specific, such as portraits, single objects, and outdoor scenes. Despite promising results, a significant challenge remains the need for multi-illumination datasets [48], which are difficult to capture at scale.
+
+Joint generative modeling approaches enable diffusion models to predict multiple modalities. Matrix3D [44] predicts pose estimation, depth, and novel view synthesis using a single DiT [52] model. VideoJAM [12] extends this by predicting both generated pixels and their corresponding motion from a single DiT. We leverage this approach to jointly predict a relit image and albedo.
+
+# 3 Preliminaries: Video Diffusion Models
+
+Diffusion models approximate a data distribution  $p_{\mathrm{data}}(\mathbf{I})$  by learning to iteratively denoise samples corrupted by Gaussian noise [59, 23, 17]. For efficiency, most video diffusion models (VDMs) operate in a lower-dimensional latent space [8, 1]. Given an RGB video  $\mathbf{I} \in \mathbb{R}^{L \times H \times W \times 3}$  consisting of  $L$  frames at resolution  $H \times W$ , a pre-trained VAE encoder  $\mathcal{E}$  encodes the video into a latent tensor  $\mathbf{z} = \mathcal{E}(\mathbf{I}) \in \mathbb{R}^{l \times h \times w \times C}$ . Then, the final video  $\hat{\mathbf{I}}$  is reconstructed by decoding  $\mathbf{z}$  with a pre-trained VAE decoder  $\mathcal{D}$ . Both the training and inference stages of the VDM are conducted in this latent space.
+
+In this work, we fine-tune a recent Diffusion Transformer (DiT) video model, Cosmos-Predict1 [1]. Encoding and decoding to and from latent space are performed by the pre-trained VAE Cosmos-1.0-Tokenizer-CV8x8x8, which compresses the video by a factor of eight along the spatial and temporal dimensions:  $l = \frac{L}{8}$ ,  $C = 16$ ,  $h = \frac{H}{8}$ , and  $w = \frac{W}{8}$ . The base model supports text- and image-guided video generation at a resolution of  $704 \times 1280$  pixels.
+
+To train the VDM, noisy versions  $\mathbf{z}_{\tau} = \alpha_{\tau}\mathbf{z}_0 + \sigma_{\tau}\epsilon$  are constructed by adding Gaussian noise  $\epsilon$ , with the noise schedule provided by  $\alpha_{\tau}$  and  $\sigma_{\tau}$  following the EDM [27]. The diffusion model parameters  $\theta$  of the denoising function  $\mathbf{f}_{\theta}$  are optimized using the denoising score matching objective [27]. Once trained, iteratively applying  $\mathbf{f}_{\theta}$  to a sample of Gaussian noise will produce a sample of  $p_{\mathrm{data}}(\mathbf{I})$ .
+
+# 4 Method
+
+We propose a generative relighting framework by fine-tuning a DiT-based video diffusion model [52, 1], which jointly predicts the albedo and relit appearance from an input image or video under a target lighting condition.
+
+At the core of our method is a joint denoising architecture: the latent representations for albedo and relit video are concatenated at the token level and denoised in a single pass using the DiT model. This formulation enables cross-modal interaction via self-attention, allowing the model to capture shared scene structure and improve generalization and temporal consistency.
+
+We leverage a combination of synthetic datasets, multi-illumination data, and auto-labeled real-world data to train our model, ensuring robust generalization and high-quality output. In the following sections, we detail the model architecture, data strategy, and training objectives.
+
+# 4.1 Model Design
+
+Given an input video  $\mathbf{I}$  and a temporally-varying target lighting condition  $\mathbf{E} \in \mathbb{R}^{L \times H \times W \times 3}$  of the same dimensions, our goal is to train a model  $\mathbf{f}_{\theta}$  that jointly denoises the albedo  $\mathbf{a}$  of the input video and a relit video  $\mathbf{I}_{\mathbf{E}}$  under the target illumination  $\mathbf{E}$ .
+
+As illustrated in Figure 2, the model comprises a VAE encoder-decoder pair, denoted  $(\mathcal{E},\mathcal{D})$ , and a transformer-based denoising function,  $\mathbf{f}_{\theta}$ . We use the VAE encoder  $\mathcal{E}$  to separately encode the input video  $\mathbf{I}$ , the albedo  $\mathbf{a}$ , and the relit video  $\mathbf{I_E}$ , producing the corresponding latent tensors  $(\mathbf{z^I},\mathbf{z^a},\mathbf{z^E})$ .
+
+Unlike VideoJAM [12], which compresses a pair of latents  $(\mathbf{z}^x, \mathbf{z}^y)$  using a linear layer, we find this approach does not yield acceptable quality for our applications. We adopt a simple yet effective strategy: concatenating the latents  $\mathbf{z}^{\mathbf{l}}$ ,  $\mathbf{z}^{\mathbf{a}}$ , and  $\mathbf{z}^{\mathbf{E}}$  along the temporal (frame) dimension. This formulation enables the DiT model to apply full self-attention across input, albedo, and relit frames, facilitating cross-modal information exchange.
+
+Token embeddings. To distinguish between the three modalities in the concatenated sequence, we combine standard RoPE positional embeddings with dedicated type embeddings. We first apply the same positional embedding to each of the three video clips, encoding the spatial-temporal position of tokens within each clip. To indicate modality, we introduce a learnable type embedding  $\mathbf{c}_{\mathrm{emb}} \in \mathbb{R}^{K_{\mathrm{emb}} \times C_{\mathrm{emb}}}$ , where  $K_{\mathrm{emb}} = 3$  denotes the number of video types. Each type embedding is broadcast across spatial-temporal dimensions and concatenated along the channel dimension with its corresponding latent representation:  $\mathbf{c}_{\mathrm{emb}}^{0}$  is appended to  $\mathbf{z}^{\mathbf{E}}$ ,  $\mathbf{c}_{\mathrm{emb}}^{1}$  to  $\mathbf{z}^{\mathbf{a}}$ , and  $\mathbf{c}_{\mathrm{emb}}^{2}$  to  $\mathbf{z}^{\mathbf{I}}$ .
+
+Condition mask. We attach a binary mask to each frame indicating whether it should be treated as a condition (e.g., input source video) or a denoising target (e.g., relit video). During training, we randomly vary which modalities are used as inputs, allowing the model to generalize under partial supervision. For instance, we may randomly treat a as either a conditioning input or a prediction target (see Section 4.3 for details).
+
+Encoding HDR lighting. Environment maps contain high dynamic range (HDR) values that often exceed the intensity range of standard images, while the VAEs used in latent diffusion models are trained on low dynamic range (LDR) inputs and cannot directly encode these high-intensity signals.
+
+To address this, we follow prior work [26, 39] and represent the lighting condition using three complementary buffers: (1) an LDR panorama  $\mathbf{E}_{\mathrm{ldr}}$  obtained via standard Reinhard tonemapping; (2) a normalized log-intensity map  $\mathbf{E}_{\mathrm{log}} = \log (\mathbf{E} + 1) / E_{\mathrm{max}}$ , where  $E_{\mathrm{max}}$  is the maximum log intensity; and (3) a directional encoding  $\mathbf{E}_{\mathrm{dir}} \in \mathbb{R}^{L \times H \times W \times 3}$ , where each pixel stores a unit vector indicating its direction in the camera coordinate system. These representations are passed through the VAE encoder, yielding  $\mathbf{h}^{\mathbf{E}} = (\mathcal{E}(\mathbf{E}_{\mathrm{ldr}}), \mathcal{E}(\mathbf{E}_{\mathrm{log}}), \mathcal{E}(\mathbf{E}_{\mathrm{dir}}))$  which is concatenated along the channel dimension with  $\mathbf{z}^{\mathbf{E}}$ . We also append a binary condition mask to indicate whether lighting features are present. For input video and albedo tokens, we use zero-padded placeholders to maintain shape consistency.
+
+# 4.2 Data Strategy
+
+Similar to prior work [39], we combine a large-scale synthetic dataset with smaller real-world datasets. The synthetic dataset offers a large volume of high-quality data, which, when combined with
+
+powerful diffusion models, shows impressive generalization to unseen domains [28, 19]. However, using synthetic target images biases the model output, resulting in a rendered look. Therefore, we also utilize smaller and lower-quality datasets of automatically labeled real-world data.
+
+Synthetic data curation. Our synthetic dataset consists of rendered video clips that are tuples  $(\mathbf{I},\mathbf{I}_{\mathbf{E}},\mathbf{a},\mathbf{E})$  of input video, relit result, albedo, and environment map. To create a large dataset with complex and diverse scenes and lighting conditions, we follow the methodology of the Diffusion-Renderer [39] and procedurally generate simple scenes with randomized environment map lighting. We curate a collection of 36,500 3D objects from the Objaverse LVIS subset, 4,260 PBR materials, and 766 HDRI environment maps, gathered from publicly available resources. Each scene contains a ground plane with a randomly selected material, and up to three randomly placed 3D objects. We perform collision detection to avoid intersecting objects. Additionally, we place up to three primitive objects (cube, sphere, and cylinder) with randomized shapes and materials, to increase variety. A randomly selected HDR environment map illuminates each scene. We generate videos with random motions including camera orbits, camera oscillation, lighting rotation, object rotation and translation. Each scene is rendered with two different illuminations under the same motion.
+
+We rendered the dataset using a custom path tracer based on OptiX [51] at high sample counts with path length three to capture global illumination effects, producing a total of 108k videos with 57 frames per video at a resolution of  $704 \times 1280$  pixels.
+
+MIT multi-illumination labeling. The multi-illumination dataset of Murmann et al. [48] consists of 985 real-world indoor scenes for training and 30 scenes for testing, each lit under 25 different conditions. Each image contains a reflective chrome sphere that has been isolated and exported as an HDR image, which can be easily remapped to the latitude-longitude representation expected by our model. We create a dataset by randomly selecting pairs from the 25 lighting conditions and extracting  $(\mathbf{I},\mathbf{E},\mathbf{I}_{\mathbf{E}})$ .
+
+To obtain pseudo-groundtruth albedo labels a, we re-implement the inverse renderer from Diffusion-Renderer [39] based on the Cosmos-1.0-Diffusion-7BVideo2World model [49], fine-tuned on our synthetic dataset. The resulting model produces high-quality albedo estimates across diverse images and videos. For each scene, we estimate albedo under all 25 lighting conditions and average the results to obtain a stable albedo map.
+
+Real-world auto-labeling. While multi-illumination datasets provide effective supervision for relighting, they are costly to acquire and limited in scale. In contrast, the Internet offers an abundance of high-quality videos captured under single illumination. To leverage this resource, we curate 150k real-world video clips, each with 57 frames at a resolution of  $704 \times 1280$ , and automatically annotate them with albedo maps to create paired data (I, a) using the reproduced inverse rendering model. These auto-labeled RGB-albedo pairs significantly increase the diversity and realism of our training set, enabling improved generalization to real-world scenes.
+
+# 4.3 Training
+
+Our model is trained on a combination of the synthetic video dataset, the MIT multi-illumination dataset, and real-world auto-labeled data. For image datasets, we treat images as single-frame videos. For both the synthetic video dataset and the MIT multi-illumination dataset, each data sample consists of an input video  $\mathbf{I}$ , its corresponding albedo  $\mathbf{a}$ , a new environment map  $\mathbf{E}$ , and the target relit video  $\mathbf{I}_{\mathbf{E}}$  under this new illumination. The target latent variable for these datasets is constructed by concatenating the latent of the relit video,  $\mathbf{z}_0^{\mathbf{E}}$ , and the corresponding albedo,  $\mathbf{z}_0^{\mathbf{a}}$ , along the temporal dimension. Noise is introduced independently to both  $\mathbf{z}_0^{\mathbf{E}}$  and  $\mathbf{z}_0^{\mathbf{a}}$  to produce  $\mathbf{z}_{\tau}^{\mathbf{E}}$  and  $\mathbf{z}_{\tau}^{\mathbf{a}}$ . The model parameters are optimized by minimizing the objective function:
+
+$$
+\hat {\mathbf {z}} ^ {\mathbf {E}} (\theta), \hat {\mathbf {z}} ^ {\mathbf {a}} (\theta) = \mathbf {f} _ {\theta} \left(\left[ \mathbf {z} _ {\tau} ^ {\mathbf {E}} + \mathbf {h} ^ {\mathbf {E}}, \mathbf {z} _ {\tau} ^ {\mathbf {a}}, \mathbf {z} ^ {\mathbf {I}} \right]; \mathbf {c} _ {\text {e m b}}, \tau\right) \tag {1}
+$$
+
+$$
+\mathcal {L} (\theta) = \mathbb {E} _ {\left(\mathbf {z} _ {0} ^ {\mathbf {E}}, \mathbf {z} _ {0} ^ {\mathbf {a}}\right) \sim p _ {\mathrm {d a t a}, \epsilon} \sim \mathcal {N} (0, \sigma^ {2} I)} \left[ \left\| \hat {\mathbf {z}} ^ {\mathbf {E}} (\theta) - \mathbf {z} _ {0} ^ {\mathbf {E}} \right\| _ {2} ^ {2} + \lambda_ {\mathbf {a}} \| \hat {\mathbf {z}} ^ {\mathbf {a}} (\theta) - \mathbf {z} _ {0} ^ {\mathbf {a}} \| _ {2} ^ {2} \right], \tag {2}
+$$
+
+where  $[\cdot]$  denotes concatenation in the temporal dimension, and  $\lambda_{\mathbf{a}} = 0.1$  is a scalar weight for albedo.
+
+Training strategy. Our training process incorporates specific conditioning strategies tailored to the data source. For synthetic data and the MIT multi-illumination dataset, we apply three different
+
+![](images/60f918d223dffd23eddb93620c43a0b545d8da99ef0317adce294d8de8b584e7.jpg)  
+Figure 3: Qualitative comparison on the synthetic dataset and MIT multi-illumination dataset. Our method produces high-quality inter-reflections and shadows in synthetic scenes (top rows). Crucially, on the MIT multi-illumination dataset (bottom rows), it delivers relighting results with higher accuracy than baselines, which fail when faced with complex materials.
+
+conditioning strategies. In  $12\%$  of training steps, the input video is dropped (i.e.,  $\mathbf{z}^{\mathbf{I}} = \mathbf{0}$ ), and the ground truth albedo serves as the sole conditioning signal. The denoising function is then  $\mathbf{f}_{\theta}([\mathbf{z}_{\tau}^{\mathbf{E}} + \mathbf{h}^{\mathbf{E}}, \mathbf{z}_0^{\mathbf{a}}, \mathbf{0}]; \mathbf{c}_{\mathrm{emb}}, \tau)$ . In another  $18\%$  of the training steps, the model is conditioned on both the input video  $\mathbf{z}^{\mathbf{I}}$  and the ground truth albedo, resulting in the function  $\mathbf{f}_{\theta}([\mathbf{z}_{\tau}^{\mathbf{E}} + \mathbf{h}^{\mathbf{E}}, \mathbf{z}_0^{\mathbf{a}}, \mathbf{z}^{\mathbf{I}}]; \mathbf{c}_{\mathrm{emb}}, \tau)$ . When using these two specific conditions, the conditioning latents of the input video and albedo have their corresponding condition masks set to 1. For the remaining  $70\%$  of the training steps, we use the default denoising function where we take input video tokens as conditions and predict the denoised relit video and albedo.
+
+Our real-world auto-labeled data consists of RGB videos and corresponding albedos but lacks pairwise RGB videos and environment maps. The provided albedo  $\mathbf{a}$  is used as a condition, and the RGB video is treated as the target relit video  $\mathbf{I}_{\mathbf{E}}$ . Since the original input video and the environment map information are unavailable for this dataset, their respective representations are set to  $\mathbf{0}$ : The environment map embedding  $\mathbf{h}^{\mathbf{E}}$  within the primary input array becomes  $\mathbf{0}$ , and the conditional input video representation  $\mathbf{z}^{\mathbf{I}}$  is dropped. The denoising function is thus  $\mathbf{f}_{\theta}([ \mathbf{z}_{\tau}^{\mathbf{E}} + \mathbf{0}, \mathbf{z}_0^{\mathbf{a}}, \mathbf{0} ]; \mathbf{c}_{\mathrm{emb}}, \tau)$ . We adopt a 10% probability of dropping conditions to enable classifier-free guidance [24].
+
+# 5 Results
+
+Before evaluating our method, we introduce baselines, metrics, and datasets. Then we conduct ablation studies on the principle of joint modeling and real-world auto-labeling of data. We refer to model implementation details in the Supplement.
+
+Baselines. For the relighting task, we compare with the 2D methods DiLightNet [68], NeuralGaffer [26], and DiffusionRenderer [39]. As the original DiffusionRenderer is built on Stable Video Diffusion [8], to isolate the effect of the base model and enable a fair algorithmic comparison, we re-implement DiffusionRenderer using the same Cosmos [49] backbone as our method, and adopt a data curation strategy closely following the original paper. We refer to this enhanced variant as DiffusionRenderer (Cosmos). For albedo estimation, we compare with IntrinsicImageDiffusion [32] and DiffusionRenderer [39].
+
+Metrics. We apply the standard image metrics PSNR, SSIM, and LPIPS [74] to evaluate relighting and albedo estimation results. For relighting, we also conduct a user study to evaluate the perceptual quality of the produced results. Following prior works [32, 26, 39], we counteract scale ambiguity in both relighting and albedo estimation by a three-channel scaling factor using least-squares error minimization before computing the metrics.
+
+![](images/ed9e3bed342f3b27c5f63df9c1799ad325baffcbcbf8f3f37edf91b9251ae4f3.jpg)  
+Figure 4: Qualitative comparison on in-the-wild data. Our method generates more plausible results than the baselines, with higher quality and more realistic appearance.
+
+Table 1: Quantitative evaluation of relighting, including a user study, where "Ours preferred" indicates the preference over the baselines. A preference over the  $>50\%$  indicates Ours outperforming baselines.  
+
+<table><tr><td rowspan="2"></td><td colspan="3">SyntheticScenes</td><td colspan="4">MIT multi-illumination</td></tr><tr><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>Ours preferred</td></tr><tr><td>DiLightNet [68]</td><td>19.13</td><td>0.584</td><td>0.319</td><td>15.79</td><td>0.539</td><td>0.368</td><td>92% ± 8%</td></tr><tr><td>Neural Gaffer [26]</td><td>19.17</td><td>0.638</td><td>0.263</td><td>17.87</td><td>0.683</td><td>0.241</td><td>84% ± 2%</td></tr><tr><td>DiffusionRenderer [39]</td><td>24.17</td><td>0.768</td><td>0.217</td><td>16.99</td><td>0.582</td><td>0.342</td><td>96% ± 4%</td></tr><tr><td>DiffusionRenderer (Cosmos)</td><td>26.61</td><td>0.841</td><td>0.222</td><td>17.29</td><td>0.622</td><td>0.355</td><td>88% ± 8%</td></tr><tr><td>Ours</td><td>26.97</td><td>0.847</td><td>0.190</td><td>20.76</td><td>0.749</td><td>0.251</td><td>n/a</td></tr></table>
+
+Datasets. We have curated a high-quality synthetic dataset named SyntheticScenes for quantitative and qualitative evaluation of both relighting and albedo estimation tasks. This dataset includes 3D assets from PolyHaven [67] and Objaverse [16], ensuring that no assets were used in the training of our method or any of the baseline methods. SyntheticScenes features 40 scenes, each constructed with a plane textured with random physically-based materials. Each scene is rendered into a 57-frame video sequence under four different lighting conditions, incorporating orbiting camera motions and rotating environment lighting. We cycle through the lighting conditions, selecting one video as the input, and the following as the relighting target, resulting in four different relighting tasks per scene. Additionally, we utilize the test set of MIT multi-illumination benchmark [48], which includes 30 scenes under 25 different illuminations. In this dataset, images are captured sequentially under adjacent flashlight illuminations. For our relighting tasks, the image captured under the  $i$ -th lighting condition serves as input, while the image captured under the  $(i + 12)$ -th lighting condition is utilized as the target illumination.
+
+# 5.1 Evaluation
+
+Qualitative comparison. In Figure 3 and Figure 4, we present a comparative analysis of our method against recent state-of-the-art relighting techniques: DiLightNet [68], NeuralGaffer [26], and DiffusionRenderer [39]. Our approach demonstrates superior performance, particularly in handling intricate shadows and inter-reflections. As illustrated in Figure 3, while DiffusionRenderer performs well on the synthetic dataset, it struggles to accurately represent complex materials—such as anisotropic surfaces, glass, and transparent objects—when utilizing G-buffers as an intermediate state, leading to suboptimal results. NeuralGaffer and DiLightNet do not produce accurate shadows and reflections, leading to poor relighting effects. Specifically, NeuralGaffer often makes very subtle, or no changes to the input video.
+
+Figure 4 compares relighting results on in-the-wild data. For the first two object-centric scenes, we provided DiLightNet and NeuralGaffer with masks (we used DiLightNet's auto-masking, which is based on Segment Anything [29]) to produce more reasonable results. Notably, DiffusionRenderer struggles with fur, human skin, and car windows in the first three cases, as these are hard to represent
+
+![](images/b275fbeeafd15a4d9815273b380719173c1f9edb9aa0de0c3a23dd83ad06c11c.jpg)  
+Input
+
+![](images/0e55a66ac171cdf9020339e006934cc33e4af8591873f43f019e287b957fedfb.jpg)  
+Ours (w/o joint modeling)
+
+![](images/ab9db577e0650f0d55a76b509e3f52ca5cd206c4e0f02056a84d5d673f2026ed.jpg)  
+Shadow baking artifact
+
+![](images/0f07bff43fb710240c918706bdeed9d742b97215ba3b04557b20467fc28d0e4b.jpg)  
+Ours
+
+![](images/7d2117759fe1989dfb9c0819505b15ef9dbf84c33da59b8b62d61c02ad5d78d7.jpg)  
+Env.
+
+![](images/d93f1f37d387e85919442ffe1723fcc485080d8758bd9c523824ea2a533a9d90.jpg)  
+Figure 5: Ablation on joint modeling. Relighting results on urban street scenes. The orange and green crops highlight regions where the pure relighting model (w/o joint modeling) clearly bakes shadows from the input image into the relit result. Our joint model correctly demodulates the shadows.
+
+![](images/e2a2b120511c019b88a902a607b920f71210021352e0813725cb09314c5c4eae.jpg)
+
+![](images/8d840c7fd572607d10c730a07669fc27582f62339a4a945cbc30c1b1426d131f.jpg)
+
+![](images/b34008aeb35fe237d25dff041db16693a2638d99de53962a288d20de1e488bc9.jpg)
+
+![](images/21f19722a40889d777fad554f083ba7fb24f3e7e3be106c4ee27b0ec71131b1c.jpg)
+
+with simple G-buffers. It also misestimates the material of the plastic basin, leading to metal-like appearances. Overall, our relighting model is more general-purpose and can be used more effectively in diverse scenes, yielding comparatively more accurate and high-fidelity relighting results.
+
+Quantitative comparison. In Table 1, we present a comparative analysis of our method against all baselines on both SyntheticScenes and the MIT multi-illumination benchmark [48]. DiffusionRenderer works well on SyntheticScenes, and the Cosmos version—a stronger base model—leads to improved fidelity. However, on the MIT multi-illumination dataset, DiffusionRenderer fails to model the more challenging materials using only the G-buffers, and even the Cosmos version cannot bridge this gap.
+
+An outlier is observed in the LPIPS score for NeuralGaffer on the MIT multi-illumination dataset. As mentioned before, NeuralGaffer makes minimal changes to the input and performs poorly, but still achieves the lowest LPIPS score. We speculate that LPIPS considers the images sufficiently similar due to the relatively small deviation in light direction for this dataset.
+
+User study. We conducted a user study to evaluate the perceptual quality of our relighting method compared to the baseline methods on the MIT multi-illumination dataset. Participants were shown a ground-truth relit image alongside two relighting results—one generated by our method and the other by a baseline model, with the order randomly shuffled. They were asked to select the result that more closely resembled the ground truth, considering aspects such as transparency, shadows, and reflections. For each sample pair, 11 users made a binary selection, and majority voting determined the preferred method for each comparison. We repeated the study three times, involving a total of 33 users, and report the average percentage of samples where our method is preferred over baselines in Table 1, along with the standard deviation for the three experiments. Our results are consistently preferred, strongly outperforming the other baselines.
+
+Albedo estimation. While our primary focus is on relighting, we also evaluate the quality of our albedo estimates compared to the baselines in Table 2. While the Cosmos version of DiffusionRenderer has a slight edge, our method performs or on par with previous work in all metrics.
+
+Table 2: Quantitative evaluation of albedo estimation on SyntheticScenes.  
+
+<table><tr><td></td><td colspan="3">SyntheticScenes</td></tr><tr><td></td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td></tr><tr><td>IntrinsicImageDiffusion [32]</td><td>16.41</td><td>0.543</td><td>0.395</td></tr><tr><td>DiffusionRenderer [39]</td><td>26.22</td><td>0.837</td><td>0.166</td></tr><tr><td>DiffusionRenderer (Cosmos)</td><td>28.56</td><td>0.911</td><td>0.131</td></tr><tr><td>Ours</td><td>28.07</td><td>0.877</td><td>0.167</td></tr></table>
+
+# 5.2 Ablation Study
+
+Ablation on joint modeling. We ablate our joint modeling approach with quantitative and qualitative results in Table 3 and Figure 5. We compare our model to a pure relighting model (without joint modeling), whose denoising function is defined as  $\mathbf{f}_{\theta}([{\bf z}_{\tau}^{E} + {\bf h}^{{\bf E}}];{\bf z}^{{\bf I}},\tau)$ . Both models are trained exclusively on synthetic data. The quantitative results in Table 3 reveal that our joint model slightly improves relighting quality compared to the ablated model. The table also shows that providing ground truth albedo as input to our joint model further enhances quality, highlighting the model's capacity to leverage albedo information effectively. For a qualitative assessment, Figure 5 compares the results on urban street scenes outside the training distribution, where urban street scenes typically exhibit strong shadows under sunlight. The joint modeling approach demonstrates superior generalization, characterized by reduced false shadowing, as clearly shown in the figures, whereas the model without joint modeling largely bakes the shadow from the input.
+
+![](images/9c38f207394e7eba94b61b31623f9174af5ae5f79a1e5c224763f0dd5a692f56.jpg)  
+Input
+
+![](images/fb8ffc0768c5be31acc81960f2c72cc0fdc5e50f8be0571a998446f10ff336c4.jpg)  
+w/o auto-labeled data
+
+![](images/22fe44786b4fdbb163a4cb0551fd392a3641a7afa31ad17d444ec7955de383b8.jpg)  
+Ours  
+Env.
+
+![](images/e2cf22c2c3cb7bde8321b214274b0bcac7066151201d6e205a46b46be2c34c91.jpg)  
+Figure 6: Ablation on real-world auto-labeled data. Although the dataset is sparsely labeled, it helps the model generalize to natural images.
+
+![](images/f917291a9639286bbd2af6d17d1cfdd6bb467b15aa58e4d5fd7cbf75af1cfd51.jpg)
+
+![](images/8c5cb56900c83d9d094594c27de01463692745223e21a9d2d15d898983c62d66.jpg)
+
+![](images/4ab4baa0c41e27587f2e40b580c998a710472dbf153717bcd275ff07b245e34d.jpg)
+
+![](images/7a92f8a7e6860de00548fdc3c8e381c71340efed2e59f67d23dae29d9aa2ad9a.jpg)  
+Input
+
+![](images/53c51084608832fef02eaadda02ad40a214e2b903caf3bc1ba88a2afeddabd5e.jpg)  
+Random Samples
+
+![](images/dd44e68064fb2daf4346310f50e0c78dc4e071cb7e5448c9f385c1b158c32082.jpg)  
+Figure 7: Application for data augmentation. The top left image with green outline is the input image.
+
+![](images/e02b9445883d58acdd36a1dc0bef6d331ce2372509d41ea8e2d289cd0e57266a.jpg)
+
+Table 3: Ablations on joint modeling designs evaluated on the SyntheticScenes dataset.  
+
+<table><tr><td>Ablated Versions</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td></tr><tr><td>Ours (w/o joint modeling)</td><td>26.42</td><td>0.842</td><td>0.191</td></tr><tr><td>Ours</td><td>26.97</td><td>0.847</td><td>0.190</td></tr><tr><td>Ours (w/ GT albedo)</td><td>27.15</td><td>0.857</td><td>0.181</td></tr></table>
+
+Table 4: User study of relighting on StreetScenes. Each row compares an ablated variant against the base version (Ours w/o auto-labeled data), reporting percentage of samples where users prefer the base version.  
+
+<table><tr><td>Ablated Versions</td><td>Ours (w/o auto-labeled data) preferred</td></tr><tr><td>Ours (w/o auto-labeled data, w/o joint modeling)</td><td>68% ± 14%</td></tr><tr><td>Ours (w/ auto-labeled data)</td><td>45% ± 8%</td></tr><tr><td>Ours (w/o auto-labeled data, w/ inference-time albedo)</td><td>54% ± 13%</td></tr></table>
+
+Ablation on real-world automatically labeled data. We ablate the usefulness of our automatically labeled real-world data in Figure 6. The model trained on solely the synthetic and MIT multi-illumination datasets (center column) exhibits noticeable artifacts for out-of-focus regions and outdoor scenes. This is expected, as both datasets have very limited depth of field and primarily feature indoor scenes or lack sky-like backgrounds. As shown in the figure, even though the real-world auto-labeled dataset contains only a subset (I, a) of the video labels, it significantly helps the model generalize to out-of-domain data, thereby improving image quality.
+
+User study. We conducted a user study to evaluate the perceptual impact of our design choices on relighting quality, using 19 urban street scenes. For each scene, participants were shown a reference image and two relit videos produced by ablated versions of our method (randomized order). They were asked to select the video with more realistic lighting, focusing on aspects such as shadows and reflections. As before, each comparison was evaluated by 11 participants through binary selection, and the preferred result was determined by majority vote. We repeated the study three times with a total of 33 participants and report the average preference rate and the standard deviation in Table 4.
+
+When trained only on multi-illumination data (w/o auto-labeled data), the joint modeling variant is strongly preferred over direct relighting. Adding real-world auto-labeled data further improves perceptual quality, consistent with our qualitative and quantitative findings. When comparing models with and without estimated albedo at inference time, the larger standard deviation in user preference suggests the perceptual quality is comparable. This indicates our method does not strictly rely on estimated albedo at test time.
+
+# 5.3 Application: Illumination Augmentation
+
+Our model's strong generalization capability enables effective data augmentation for scenarios such as driving and robotics scenes. We test our model under conditions without environment maps and with different random seeds. Figure 7 shows five random illuminations of one input scene. Our model generates diverse data, including nighttime and dusk scenes, demonstrating that it accurately models the illumination distribution and can sample realistic relighting results under varying lighting conditions.
+
+# 6 Discussion
+
+To overcome data scarcity and the limitations of decoupled two-stage methods in realistic relighting, we present a novel framework UNIRELIGHT, that jointly models scene intrinsics and illumination. This approach enhances generalization, reduces error accumulation, and more effectively captures complex visual effects by implicitly reasoning about scene properties rather than relying on explicit G-buffers. Furthermore, when trained on real-world auto-labeled data, our model achieves state-of-the-art, high-quality, and realistic relighting results.
+
+Limitations and future work. While our method performs well on diverse data, our model cannot handle emitting objects, such as toggling lights within scenes. Our design, which is focused on environmental lighting, does not extend to emittance effects, which remains an area of future research. Moreover, incorporating text-based relighting could significantly enhance usability. Future work could investigate the conditioning of lighting with natural language by incorporating text crossattention in our DiT model, a feature currently not implemented. While our method can generate reasonable shadows, integrating our generative approach with explicit 3D information—such as traced shadow buffers—represents a promising direction for improving physical accuracy. Besides, in UNIRELIGHT, we incorporate the albedo to improve the model's generalization, and predicting additional properties could further enhance the model's understanding of scene structure and materials. However, memory requirements scale with the number of properties, making this approach expensive for high-resolution videos; it is worth exploring this direction further.
+
+# Acknowledgments and Disclosure of Funding
+
+The authors thank Tianshi Cao and Huan Ling for their insightful discussions that contributed to this project. This work was conducted at and supported by NVIDIA.
+
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+
+[37] Zhengqin Li, Ting-Wei Yu, Shen Sang, Sarah Wang, Sai Bi, Zexiang Xu, Hong-Xing Yu, Kalyan Sunkavalli, Miloš Hasan, Ravi Ramamoorthi, et al. OpenRooms: an end-to-end open framework for photorealistic indoor scene datasets. arXiv preprint arXiv:2007.12868, 2020. 2  
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+[44] Yuanxun Lu, Jingyang Zhang, Tian Fang, Jean-Daniel Nahmias, Yanghai Tsin, Long Quan, Xun Cao, Yao Yao, and Shiwei Li. Matrix3D: Large Photogrammetry Model All-in-One. arXiv preprint arXiv:2502.07685, 2025. 3  
+[45] Yiqun Mei, Mingming He, Li Ma, Julien Philip, Wenqi Xian, David M George, Xueming Yu, Gabriel Dedic, Ahmet Levent Tasel, Ning Yu, et al. Lux post facto: Learning portrait performance relighting with conditional video diffusion and a hybrid dataset. In CVPR, 2025. 3  
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+[47] Jacob Munkberg, Zian Wang, Ruofan Liang, Tianchang Shen, and Jon Hasselgren. VideoMat: Extracting PBR Materials from Video Diffusion Models. In EGSR - CGF Track, 2025. 3  
+[48] Lukas Murmann, Michael Gharbi, Miika Aittala, and Fredo Durand. A multi-illumination dataset of indoor object appearance. In ICCV, 2019. 3, 5, 7, 8  
+[49] NVIDIA. Cosmos world foundation model platform for physical AI. arXiv preprint arXiv:2501.03575, 2025. 5, 6, 21  
+[50] Rohit Pandey, Sergio Orts-Escalano, Chloe LeGendre, Christian Haene, Sofien Bouaziz, Christoph Rhemann, Paul Debevec, and Sean Fanello. Total relighting: Learning to relight portraits for background replacement. ACM ToG, 2021. 3  
+[51] Steven G. Parker, James Bigler, Andreas Dietrich, Heiko Friedrich, Jared Hoberock, David Luebke, David McAllister, Morgan McGuire, Keith Morley, Austin Robison, and Martin Stich. OptiX: a general purpose ray tracing engine. ACM ToG, 2010. 5  
+[52] William Peebles and Saining Xie. Scalable diffusion models with transformers. arXiv preprint arXiv:2212.09748, 2022.3, 4  
+[53] Julien Philip, Michael Gharbi, Tinghui Zhou, Alexei A Efros, and George Drettakis. Multi-view relighting using a geometry-aware network. ACM ToG, 2019. 3  
+[54] Pakkapon Phongthawee, Worameth Chinchuthakun, Nontaphat Sinsunthithet, Amit Raj, Varun Jampani, Pramook Khungurn, and Supasorn Suwajanakorn. DiffusionLight: light probes for free by painting a chrome ball. arXiv preprint arxiv:2312.09168, 2023. 3  
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+
+# NeurIPS Paper Checklist
+
+The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit.
+
+Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:
+
+- You should answer [Yes], [No], or [NA].  
+- [NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available.  
+- Please provide a short (1–2 sentence) justification right after your answer (even for NA).
+
+The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper.
+
+The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes]" is generally preferable to "[No]", it is perfectly acceptable to answer "[No]" provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No]" or "[NA]" is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found.
+
+IMPORTANT, please:
+
+- Delete this instruction block, but keep the section heading "NeurIPS Paper Checklist",  
+- Keep the checklist subsection headings, questions/answers and guidelines below.  
+- Do not modify the questions and only use the provided macros for your answers.
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: The abstract and introduction clearly state our contributions, which are fully supported by the experiments and analysis presented.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: We discuss limitations in Section 6.
+
+# Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [NA]
+
+Justification: The paper does not contain theoretical results or formal proofs.
+
+Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: We include all necessary details for reproduction in the main paper and supplementary materials.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.  
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.  
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.  
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example  
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+# Answer: [No]
+
+Justification: We plan to release the code and data upon acceptance. The internal guidelines of our institution prevent us from releasing code at this stage.
+
+# Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).  
+- The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.  
+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.
+
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: All relevant details are provided in the paper and supplementary materials.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [No]
+
+Justification: Due to the high computational cost of training large-scale video diffusion models, we did not perform multiple runs to report error bars.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).  
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We include sufficient information in supplementary materials.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.
+
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: Our work adheres to the NeurIPS Code of Ethics.
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: We discussed societal impacts of our work in the paper.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: Our work does not involve models or data with high risk of misuse.
+
+# Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: Third-party assets are properly credited, and their licenses are respected.
+
+# Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: Our paper does not release new assets.
+
+# Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+# Answer: [Yes]
+
+Justification: We include the details of all crowdsourcing experiments in supplementary materials.
+
+# Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+# Answer: [Yes]
+
+Justification: An approval is obtained for all of our experiments.
+
+# Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+# Answer: [NA]
+
+Justification: LLMs were not used in the core methods of this research.
+
+# Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+In this Appendix, we first discuss the broader impact of our project (Appendix A). We then provide additional implementation details of our model and experiments (Appendix B), followed by further results and analysis (Appendix C). Please refer to the accompanying video for additional qualitative results and comparisons.
+
+# A Broader Impact
+
+We present UniRelight, a generative framework that jointly estimates albedo and synthesizes relit videos from a single input, enabling diverse lighting manipulation across both synthetic and real-world scenes. This capability can support a range of applications, including creative content generation, visual effects, virtual production, and potentially data augmentation for training more robust computer vision models in domains such as robotics and autonomous driving.
+
+As with all generative video models, UniRelight may reflect biases present in its training data. Such biases could lead to relighting results that fail to generalize to underrepresented scene types or lighting conditions. Furthermore, tools for lighting manipulation carry the risk of misuse, such as altering or misrepresenting visual content in sensitive contexts like surveillance or media.
+
+We discourage the use of UniRelight in applications where relighting may contribute to misinformation, misattribution, or privacy violations. In human-centric use cases, we recommend careful dataset curation to ensure fair representation across skin tones, races, and gender identities. Practitioners are encouraged to critically assess and de-bias training data to mitigate unintended harms where appropriate.
+
+# B Experimental Details
+
+# B.1 Implementation Details
+
+We fine-tune our models based on Cosmos-Predict1-7B-Video2World [49], a pre-trained DiT video diffusion model.
+
+The encoded latents  $\mathbf{z}^{\mathrm{I}},\mathbf{z}^{\mathrm{a}},\mathbf{z}^{\mathrm{E}},\mathcal{E}(\mathbf{E}_{\mathrm{ldr}}),\mathcal{E}(\mathbf{E}_{\mathrm{log}})$  and  $\mathcal{E}(\mathbf{E}_{\mathrm{dir}})$  are all in  $\mathbb{R}^{l\times h\times w\times C}$ , where  $C = 16$ . We use  $C_{\mathrm{emb}} = 3$  as the dimension of the type embedding. Thus, the concatenated tokens have a channel dimension of  $16 + 1 + (16 + 1)\times 3 + 3 = 71$ , where each latent has an associated binary condition mask (added as an extra channel), and the lighting features  $(\mathbf{E}_{\mathrm{ldr}},\mathbf{E}_{\mathrm{log}},\mathbf{E}_{\mathrm{dir}})$  each include a condition mask to indicate whether they are provided.
+
+We adopt an image-video co-training strategy and train the model in two stages. Firstly, we train with only synthetic data by mixing the image data (sampling one frame from the video data) and the video data in a ratio of  $1:1$  for 15,000 iterations. Then we train with all data with random sampling, including synthetic video data, synthetic image data, real-world auto-labeled data, and MIT multi-illumination data, in a ratio of  $8:1:3:2$  for 12,000 iterations. We augment the real-world auto-labeled data with random flipping.
+
+All training is done with a batch size of 64, using the AdamW optimizer with a learning rate of  $2 \times 10^{-5}$ , with mixed-precision (BF16) training at a resolution of  $480 \times 848$  pixels. The AdamW optimizer was employed with a weight decay of 0.1. The exponential decay rates for the moment estimates  $\beta$  are set to 0.9 for the first moment and 0.99 for the second moment, with  $\epsilon$  at  $1 \times 10^{-10}$ . The total training of two stages takes around 4 days on 32 A100 GPUs.
+
+During inference, we use 35 denoising steps. We do not apply classifier-free guidance (CFG), as we empirically found that inference without CFG yields more accurate and visually consistent results.
+
+Baseline configurations. Since DiLightNet [68] requires a text prompt per example, we use meta/llama-3.2-90b-vision-instruct to generate a short prompt for each example in the datasets based on the first image in each clip with the instruction "What is in this image? Describe the materials. Be concise and produce an answer with a few sentences, no more than 50 words."
+
+As each of the baselines generates videos in different resolutions, for UNet-based baselines, we run inference on the model with the video first resized to  $486 \times 864$  and then center-cropped to a
+
+![](images/9ce02258916a677a0f5cfd62f5a3ab405a39e7a37ef86c88932ba9b1c4275e3e.jpg)  
+Figure 8: Synthetic data visualization. Randomly sampled example images are shown from our synthetic rendering data. The top-most images show albedo maps, while the bottom two rows display rendered scenes under two different illuminations with the corresponding environment maps.
+
+Table 5: Quantitative comparison with ICLight.  
+
+<table><tr><td></td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td></tr><tr><td>IC-Light [73]</td><td>18.08</td><td>0.834</td><td>0.096</td></tr><tr><td>Ours</td><td>23.19</td><td>0.901</td><td>0.079</td></tr></table>
+
+Table 6: Evaluation of inference runtime cost.  
+
+<table><tr><td></td><td>Runtime cost (seconds) ↓</td></tr><tr><td>DiffusionRenderer [39]</td><td>566.6</td></tr><tr><td>DiffusionRenderer (Cosmos)</td><td>780.0</td></tr><tr><td>Ours</td><td>445.5</td></tr></table>
+
+resolution of  $448 \times 832$ ; for our DiT-based model, we run inference on the model with the resized video with resolution of  $486 \times 864$  and then center-cropped to  $448 \times 832$  to align the results.
+
+Quantitative comparison configurations. For quantitative evaluation, we apply background masks to the synthetic dataset to focus on foreground appearance. For the MIT multi-illumination dataset, we follow the dataset protocol and mask out light probes in all outputs before computing metrics.
+
+# B.2 User Study Details
+
+We conducted two user studies on Amazon Mechanical Turk to evaluate the perceptual quality of relighting results.
+
+MIT multi-illumination dataset is a public benchmark with ground truth relighting. Participants were shown three images: a ground truth relit image and two relighting results—one generated by our method and one by a baseline model. Their task was to choose the result that more closely resembled the ground truth, considering attributes such as transparency, shadows, and reflections.
+
+The exact instructions shown to participants were as follows:
+
+Carefully compare Image A, the Reference Image, and Image B. Your task is to determine which image (A or B) is more similar to the Reference Image.
+
+To make an informed decision, you may zoom in to examine the details. Pay close attention to aspects such as lighting, reflections, and shadows, as these can affect how natural the image appears.
+
+Once you have compared the images, select the one that best matches the Reference Image.
+
+□ Image A  
+□ Image B
+
+We evaluated 30 scenes from the test set, comparing our method against four baselines. The study was repeated three times with 11 unique participants in each run. In total, this resulted in  $30 \times 4 \times 11 \times 3 = 3960$  individual comparisons.
+
+StreetScenes Dataset. This dataset contains 19 urban street scenes without ground-truth relighting. Participants were shown a reference image along with two relit videos generated by different ablated versions of our method.
+
+The instructions presented to participants were as follows:
+
+In this study, you will be shown a Reference Image and two videos - Video A and Video B - that changes the lighting of the scene. Your task is to watch both
+
+![](images/22e14b5ee2ca2f2d480a7832ced8a3c4e4a233f903be0e23a5f1db2b81c9b951.jpg)  
+Figure 9: Qualitative comparison with IC-Light [73]. We provide the environmental background used for IC-Light conditioning, with the reference environment ball on the left. Our method produces higher-quality and more accurate relighting results.
+
+videos and choose which one (A or B) you think has more realistic shadows and reflections. To make an informed decision, you may zoom in to examine the details. Pay close attention to aspects such as lighting, reflections, and shadows, as these can affect how natural the image appears.
+
+Once you have compared the videos, select the one that has more realistic lighting effects.
+
+Video A  
+Video B
+
+We evaluated 19 scenes, comparing a base version against two ablated versions of our method. The study was repeated three times with 11 unique participants in each run. In total, this resulted in  $19 \times 2 \times 11 \times 3 = 1254$  individual comparisons.
+
+# B.3 Synthetic data visualization
+
+We show a synthetic data visualization in Figure 8. Each scene contains albedo videos, two environment maps, and pairwise videos rendered under the environment maps.
+
+# C Additional Results
+
+Runtime cost. We evaluate the inference runtime of our model on a 57-frame video at a resolution of  $480 \times 848$ . The overall inference time for performing 35 denoising steps, including VAE encoding and decoding, is 445.5 seconds, measured on a single A100 GPU.
+
+To contextualize this cost, we compare our method against two baselines: DiffusionRenderer [39] and its Cosmos-based variant. All methods use 35 denoising steps for consistency. DiffusionRenderer is run at  $448 \times 832$  resolution (slightly smaller than ours but divisible by 32 to fit its architecture) while the Cosmos variant is run at our native resolution.
+
+For baseline methods, the total runtime is the sum of the inverse rendering and forward rendering durations. Notably, DiffusionRenderer requires five separate inverse rendering passes and one forward rendering pass per video, resulting in significantly higher computational cost. In contrast, our approach performs joint relighting and albedo estimation in a single pass and is correspondingly faster. Full timing results are shown in Table 6.
+
+Comparison with IC-Light [73]. We compare our method with the single-image relighting approach IC-Light [73] on object-centric synthetic data, as shown in Table 5 and Figure 9. Note that the
+
+![](images/5ef610476996498659046fe9406297a39e4bcc26a10738e80c34a6fa9eade30c.jpg)  
+Figure 10: Additional qualitative comparison on MIT multi-illumination dataset. Our method consistently achieves more accurate relighting results than all baselines on the MIT multi-illumination dataset, demonstrating strong capability in relighting complex materials.
+
+two methods follow different relighting formulations: IC-Light is designed for object relighting using background context as the primary cue—without access to an explicit environment map, while our method is conditioned on full HDR illumination, but is not specifically tuned for object-centric data.
+
+Our method shows improved quantitative and qualitative performance. Since IC-Light relies on background appearance as its primary cue and has less information about the surrounding lighting, it may retain input-specific effects in its outputs—such as specular highlights and shadows, which can limit accuracy under novel lighting conditions. In contrast, our method produces more faithful relighting results, with sharper specular highlights, more realistic shadows, and improved visual fidelity.
+
+Additional qualitative comparison on the MIT multi-illumination dataset. We provide additional qualitative comparisons on the MIT Multi-Illumination dataset in Figure 10. To ensure a fair comparison, we include results from our re-implemented version of DiffusionRenderer using the Cosmos backbone, which achieves higher visual fidelity than the original implementation. Our method
+
+![](images/094d1f27dc9d7884a16c5531a435f6ef1d7fe8780f4a39d021588d80ea7290f8.jpg)  
+Input
+
+![](images/98a0ef12b94f44d7909272da13f4c265a7743ede491361bec84e98885cec6ce7.jpg)  
+Relighting I
+
+![](images/38e9c4a189161c7890d6397565762c7979f7126258d918243c9d7a8871f9de56.jpg)  
+Relighting II
+
+![](images/c82780ac0a7ac2357295c5cdbdb3372132ceec2043856b79a613e8c2766b670d.jpg)  
+Relighting III
+
+![](images/6b14c2e21fe6192909ea033619165b1d9c63a153e986276f0ef845a95d9f33f0.jpg)  
+Figure 11: Additional qualitative results under point-light illumination. The bottom right of each column indicates the target lighting conditions. Our results show strong robustness of our method under point-light illumination.
+
+![](images/fbe1745a8515694f5d92b706aed1f5fa1219a142552e6dabcb9553f5766aab67.jpg)
+
+![](images/38c0fee9e46db2735997b5a061509d6cb6811a852b14cf644942b12283e2d167.jpg)
+
+![](images/10f6d7ab2c1b0b6a9de30f7e651e7b3330b679189f8cd5e6f23f16c83ae67083.jpg)
+
+![](images/220e2ba589f6e7edd8d6ad8ad5de1ca9b319103b113d542d5a16f8b80e7123ee.jpg)  
+Input
+
+![](images/ca53d2bf3780936001aa4c1b6969dd2e0ad730dc33a604e3914312f8a29e21a9.jpg)  
+Estimated Albedo
+
+![](images/1e4e1d8483c7ba1fe25f48bc16f1e09c4a940a504c2f712aff934c2042984a33.jpg)  
+Relighting
+
+![](images/73bbc93b4ccada7addf8ff1bc586d5e4ca42b11a491c0d066a41e5697456c9df.jpg)  
+Env.
+
+![](images/a531ee4980ceb85bb47c2f3fb58c6cf0c50606c1cb9a3bd5ddf6c9595b8967c4.jpg)
+
+![](images/682b4fc1085511af53bf7db102175b59cb43aa3c1ee01822897fb8cd22a27059.jpg)
+
+![](images/749b66c6e1a7cf8261dd81ef0e8bde1a9bf4910a236fad5dc0ffb5b6dce70dec.jpg)
+
+![](images/9e64057bd8d0ef3bc3f19a8e95363663a405b76dcdb52217370b0af495472552.jpg)
+
+![](images/240b732a2ad8a1452f7dbd2291253f7a81d679cd1cdeef82398f76af1f773fb0.jpg)
+
+![](images/b59b26713c7f40161596a8cbdfdd8a1e68f2a665f94d5c8885e5978e17a919fb.jpg)
+
+![](images/e53ea253220df3469d1f0919fb27378f57f2e00a80fbe463edf4bbfa4ccdff0b.jpg)
+
+![](images/ebea8c79ce652e666731f09c8604a2d50e123d45dd19f21f62d54e992e88dbac.jpg)
+
+![](images/53ad06874b2aa6c8e9fb69316c20aa86fe2032ce09b429dc9452a5f5968aac1b.jpg)
+
+![](images/04889c02731a642e4c4fe02cc36c3d812b81db96d89fffd0cf0f9d1bfc6f924e.jpg)
+
+![](images/af4c4bfd8a7954b314823ebf8af4114cf990a8935ffb7fc7cd8d676dc1a7a0fd.jpg)
+
+![](images/5244bca55478b1ba41bfea80776cf7d00198e0a4843880ca4bc406b4659c2427.jpg)
+
+![](images/87b1f57a00913401fb5f66381ffeaec93cf15e02e62134faed864346f66ff455.jpg)  
+Figure 12: Additional qualitative results on real scenes. Our method provides high-quality albedo estimation and realistic relighting results.
+
+![](images/59c0564e06a3eeab8ad0d5b92b4e5b350dedb4a98ead898a058c82f4f97c9757.jpg)
+
+![](images/cf280cd73fddecc54bc4868ec512092ba56e0ddde0ddf2b7688355f9417f0d3c.jpg)
+
+![](images/fc9e8d5c3d3554bbb188f1259214ed27705f5b5332725f07f511db33086fb160.jpg)
+
+consistently produces more accurate transparency, specular highlights, and shadows across scenes, demonstrating strong capability in handling complex materials and outperforming all baselines in visual quality.
+
+Additional qualitative results under point-light illumination. We further evaluate the robustness of our method in extreme cases, such as point-light illuminations, which do not exist in our training data. As shown in Figure 11, our method produces high-quality relighting results, demonstrating the strong robustness and generalization capability of our method.
+
+Additional qualitative results on real scenes. We present additional results on real scenes in Figure 12. Our method produces high-quality albedo and relighting results with realistic specular highlights and shadows under target lighting conditions.
\ No newline at end of file
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+# UniSite: The First Cross-Structure Dataset and Learning Framework for End-to-End Ligand Binding Site Detection
+
+Jigang Fan $^{1,*}$  Quanlin Wu $^{1,*}$  Shengjie Luo $^{2}$  Liwei Wang $^{1,2,3,\dagger}$
+
+<sup>1</sup>Center for Data Science, Peking University
+
+$^{2}$ State Key Laboratory of General Artificial Intelligence, Peking University
+
+<sup>3</sup>Center for Machine Learning Research, Peking University
+
+{jigangfan,luosj}  $@$ stu.pku.edu.cn，{quanlin,wanglw}@pku.edu.cn
+
+# Abstract
+
+The detection of ligand binding sites for proteins is a fundamental step in Structure-Based Drug Design. Despite notable advances in recent years, existing methods, datasets, and evaluation metrics are confronted with several key challenges: (1) current datasets and methods are centered on individual protein-ligand complexes and neglect that diverse binding sites may exist across multiple complexes of the same protein, introducing significant statistical bias; (2) ligand binding site detection is typically modeled as a discontinuous workflow, employing binary segmentation and subsequent clustering algorithms; (3) traditional evaluation metrics do not adequately reflect the actual performance of different binding site prediction methods. To address these issues, we first introduce UniSite-DS, the first UniProt (Unique Protein)-centric ligand binding site dataset, which contains 4.81 times more multi-site data and 2.08 times more overall data compared to the previously most widely used datasets. We then propose UniSite, the first end-to-end ligand binding site detection framework supervised by set prediction loss with bijective matching. In addition, we introduce Average Precision based on Intersection over Union (IoU) as a more accurate evaluation metric for ligand binding site prediction. Extensive experiments on UniSite-DS and several representative benchmark datasets demonstrate that IoU-based Average Precision provides a more accurate reflection of prediction quality, and that UniSite outperforms current state-of-the-art methods in ligand binding site detection. The dataset and codes will be made publicly available at https://github.com/quanlin-wu/unisite.
+
+# 1 Introduction
+
+The detection of ligand binding sites on target proteins is one of the most critical steps in modern drug discovery strategies [1, 2, 3]. Structure-based drug design approaches begin with the three-dimensional structure of the target protein, from which deep, druggable cavities are identified. These regions, referred to as binding sites or binding pockets, are composed of sets of protein residues. Once the protein's sites are recognized, virtual screening of a molecular library can be performed using methods such as protein-ligand docking and protein-ligand affinity prediction [4, 5]. Alternatively, de novo molecular design [6, 7] can be conducted based on the local structure of the binding sites to identify potential candidate compounds. As a fundamental step, the accurate identification of protein binding sites can significantly facilitate and influence subsequent steps in drug discovery.
+
+Over the past several decades, some endeavours have been made to detect protein ligand binding sites. These methods have evolved from traditional techniques based on geometry [8], template searching [9], and energy probes [10], to machine learning methods based on surface features [11], and further to deep learning methods utilizing convolutional neural networks (CNNs) [12] and graph neural networks (GNNs) [13, 14, 15]. Concurrently, a series of protein-ligand datasets have also been established progressively, including scPDB [16] and PDBbind [17] datasets for protein-ligand complex structures, as well as benchmark datasets such as Holo4K [18] and COACH420 [19] for evaluating binding site detection methods.
+
+Although the above efforts have significantly advanced the field of ligand binding site detection, current methods, datasets, and evaluation metrics are confronted with substantial challenges:
+
+Issue 1. All previous methods and datasets are PDB (Protein Data Bank file)-centric, specifically focusing on individual protein-ligand structures, which introduces considerable statistical bias. Due to experimental constraints, only a limited number of binding sites in the protein are typically observed in one single protein-ligand structure where ligands are bound. However, one protein can be associated with numerous distinct protein-ligand structures, which exhibit high structural similarity in their protein components yet considerable variation in their binding site regions [20, 21, 22] (Figure 1). But existing datasets and methods only regard these structures as individual data entries, focusing on limited binding sites in single PDB structure. Training and evaluating on PDB-centric datasets introduces significant statistical bias, as the annotation paradigm of individual PDB structures overlooks many other ground truth binding sites.
+
+Issue 2. Existing methods employ discontinuous workflows for binding site detection. Most approaches [8, 11, 13, 14] first perform semantic segmentation to generate binary masks of potential binding residues/atoms, then cluster them into discrete binding sites. Alternative implementations only predict binding site centers [15], and the associated residues need to be extracted using external methods. These fragmented pipelines highly rely on the post-processing methods (e.g. clustering algorithms), inherently limit end-to-end optimization and struggle with overlapping binding sites.
+
+Issue 3. Traditional evaluation metrics inadequately reflect the actual performance of binding site detection. The most widely used evaluation metrics are DCC and DCA [11]. DCC represents the distance between the predicted binding site center and the ground truth binding site center. DCA denotes the shortest distance between the predicted binding site center and any heavy atom of the ligand. These metrics suffer from two fundamental limitations (Figure 4): (1) they completely disregard the structural properties such as shape, size, and residue composition of binding sites, which are crucial for downstream tasks (Appendix A), and (2) the absence of proper matching criteria between predictions and ground truth may lead to double-counting of predictions.
+
+To address the issues mentioned above, this paper makes the following contributions:
+
+1) We introduce UniSite-DS, a manually curated, UniProt (Unique Protein)-centric dataset of protein ligand binding sites. Leveraging the unique identifiers assigned to protein sequences in UniProt [23], we systematically integrated all ligand binding sites associated with given unique protein across multiple PDB structures. To the best of our knowledge, it is the first UniProt-centric dataset. Notably, UniSite-DS includes 4.81 times more multi-site proteins than existing datasets [16, 17], and the overall size of the dataset is 2.08 times larger. The Uniprot-centric dataset corrects the statistical bias of previous PDB-centric datasets, thereby resolving Issue 1 and significantly broadening the available data.  
+2) We propose UniSite-1D and UniSite-3D, two end-to-end methods for protein ligand binding site detection. Both models utilize a transformer encoder-decoder architecture, supervised by a set prediction loss with bijective matching. UniSite 1D/3D directly predict  $N$  potentially overlapping binding sites without requiring post-processing clustering steps, thus completely resolves Issue 2. The UniSite-1D variant operates exclusively on 1D protein sequence inputs, providing structure-free binding site detection capability. For enhanced performance, the UniSite-3D variant incorporates 3D structural information while maintaining the same end-to-end prediction framework.  
+3) To overcome the limitations inherent in traditional evaluation methods outlined in Issue 3, we introduce an Average Precision (AP) metric based on Intersection over Union (IoU) for fair and comprehensive binding site assessment. Extensive experiments have demonstrated that the IoU-based AP maintains strong concordance with method rankings under traditional metrics while overcoming their key limitations, providing a more accurate reflection of prediction quality.
+
+4) Extensive experiments on UniSite-DS and classical datasets have demonstrated that our methods outperform the current state-of-the-art methods in protein ligand binding site detection. These results indicate that the end-to-end detection framework, which operates without the need for specialized feature engineering, is already capable of exhibiting strong performance for binding site detection.
+
+![](images/39a515f12d98d5abd993a6fe70107d4999e8139f68262ac262e5cb78ff937e84.jpg)  
+Figure 1: Comparison between UniSite-DS and previous datasets. (Top left) In PDBbind2020, only one ligand binding site and one structure are recorded for UniProt ID Q8WS26. (Top right) In contrast, UniSite-DS integrates distinct binding sites across all available structures (highly similar, mean TM-Score=0.99), identifying 17 unique ligand binding sites derived from 13 representative PDB entries. (Bottom left and center) Comparison of UniSite-DS with other widely used datasets in terms of multi-site entries and the number of unique proteins. For HOLO4K and COACH420, the most widely used mlig subsets were selected, where each entry corresponds to a PDB structure, while in UniSite-DS, each entry corresponds to a UniProt ID. (Bottom right) Distribution of the number of unique proteins in UniSite-DS with respect to the number of distinct binding sites they contain.
+
+# 2 UniSite-DS: The First Uniprot-centric Dataset
+
+A key challenge in detecting protein binding sites is how to identify all potential binding sites [20, 21]. Most proteins contain an inherently conserved binding site, commonly referred to as the active site. The active site is shared among members of the same protein family, which means that molecules targeting this site will simultaneously target all other proteins within the family, which is highly likely to lead to off-target effects and side effects [24]. Identifying other binding sites within the protein that can be targeted is a crucial strategy. These sites are often located in regions topologically distant from the active site and can modulate the protein's function through allosteric effects [25, 26, 27].
+
+As illustrated in Figure 1, one single protein can correspond to a large number of different ligand-bound structures. While the overall protein structure tends to be highly conserved, the ligand binding site regions vary considerably across these structures. The motivation behind constructing UniSite-DS lies in the recognition that identifying all potential binding sites of a protein requires a comprehensive examination of all its ligand-bound structures—an important consideration that has been overlooked by previous methods and datasets.
+
+To construct UniSite-DS, we performed the following search and processing steps: (1) We utilized AHoJ [28] to systematically search for all protein-ligand interactions in the PDB database [29]; (2) To ensure dataset quality, we excluded entries with a resolution greater than  $2.5\AA$  or those determined by non-crystallographic methods; (3) Following P2Rank's filtering criteria [11], we removed solvent molecules and ligands composed of fewer than five atoms, resulting in a total
+
+of 143,197 protein-ligand interaction entries; (4) For each interaction, binding site residues were identified within a  $4.5\AA$  radius of the ligand; (5) We discarded entries with three or fewer binding site residues to eliminate "floating" ligands; (6) Leveraging UniProt's unique protein sequence identifiers [23], we mapped binding site residues from all protein-ligand interactions of each UniProt entry to their corresponding sequences via SIFTS annotations [30], integrating all ligand binding sites across different PDB structures; (7) To eliminate data redundancy among ligand binding sites, we applied Non-Maximum Suppression (NMS) with an Intersection over Minimum (IoM) threshold of 0.7 and an Intersection over Union (IoU) threshold of 0.5, excluding highly overlapping sites. This process resulted in 13,464 distinct UniProt IDs, of which 4,846 contained multiple ligand-binding sites; (8) Based on the criteria from Proteina [31], we set the sequence length threshold to 800; (9) We manually inspected all UniProt IDs with more than ten ligand binding sites, as well as those where a single protein-ligand complex structure contributed three or more binding sites. As a result, we identified 11,510 valid UniProt IDs, including 3,670 with multiple ligand binding sites. The distribution of ligand binding sites is shown in Figure 1. More details about the UniSite-DS curation workflow and manual inspection process are provided in Appendix B.
+
+As the first UniProt-centric dataset, UniSite-DS encompasses 4.81 times more multi-site entries than previous datasets, and covers 2.96 times more UniProt entries than the widely used PDBbind dataset [17], as well as 2.08 times more than sc-PDB [16] (Figure 1). UniSite-DS eliminates the statistical biases inherent in earlier datasets and significantly expands the available data on multi-site ligand binding sites. Notably, case studies conducted using UniSite-DS (Appendix E) highlighted the limitations of current binding site prediction methods in handling multi-site proteins. This observation motivated us to develop a novel end-to-end method for protein-ligand binding site detection.
+
+# 3 The Proposed Methodology
+
+In this paper, we formulate protein ligand binding site detection as a set prediction task: given a protein  $P$  with an amino acid sequence  $S$  of length  $L$ , the goal of binding site detection is to identify a set of binding sites  $\{m_i^{gt}\}_{i=1}^{N_{gt}}$ , where each binding site is represented by a binary mask  $m_i^{gt} \in \{0,1\}^L$ . Here,  $m_{ij}^{gt} = 1$  indicates that the  $j$ -th residue is part of the  $i$ -th site, while  $m_{ij}^{gt} = 0$  means it is not. Currently, most learning-based binding site detection methods adopt a discontinuous workflow: first predicting a score for each amino acid residue or heavy atom, and then clustering them into distinct binding sites. To streamline this process, we propose UniSite, the first UniProt-centric and direct set prediction approach that adheres to the end-to-end paradigm (Figure 2). Two components are essential for direct set prediction in this context: (1) a set prediction loss based on bijective match-
+
+![](images/281f8d99c9f05898997dc33a1d943bf766f476f61c8059e904f55cf26463c0e9.jpg)  
+Figure 2: Comparison of detection approaches. (Top) Conventional learning-based binding site detection methods typically employ a discontinuous workflow: first predicting binary masks for residues/atoms, then clustering these masks into distinct binding sites. (Bottom) In contrast, our method directly outputs a set of  $N$  potentially overlapping binding sites in a single step.
+
+ing between predicted and ground truth binding sites; and (2) an architecture capable of predicting a set of sites in a single forward pass. The architecture of UniSite is shown in detail in Figure 3.
+
+# 3.1 Set prediction loss for binding site detection
+
+UniSite infers a fixed-size set of  $N$  predictions  $z = \{(p_i, m_i) | m_i \in \{0, 1\}^L\}_{i=1}^N$  in a single forward pass, where  $m_i$  represents the predicted binding site, and  $p_i$  denotes the probability of binding and  $\emptyset$  (non-binding) category. Since the ground truth set  $|z^{gt}| = N^{gt}$  and the prediction set  $|z| = N$  typically have unequal sizes, we assume  $N \geq N^{gt}$  and pad the ground truth set with  $\emptyset$  (non-binding) tokens. The padded ground truth set is defined as  $z_{pad}^{gt} = \{(c_i^{gt}, m_i^{gt}) | c_i^{gt} \in \{1, \emptyset\}, m_i^{gt} \in \{0, 1\}^L\}_{i=1}^N$ , where  $c_i^{gt} = 1$  indicates a true binding site and  $c_i^{gt} = \emptyset$  corresponds to the padding.
+
+![](images/1f2a3062ea41d4cfed0f3eb0d0ad3068615abe4e3c83e9ff4e5c7c57d6c83ccd.jpg)  
+Figure 3: The architecture of UniSite. Our models employ an encoder to extract the residue-level features. Then a decoder module is used to generate embeddings of the  $N$  predicted binding sites. Finally, the segmentation module outputs  $N$  potentially overlapping binding site predictions. The encoder comprises dual pathways: a sequence encoder and an optional structural encoder, allowing UniSite to operate with either sequence-only input or combined sequence-structure information.
+
+To train a set prediction model, we require a bijective matching  $\sigma$  between the predicted set  $z$  and the padded ground truth set  $z_{pad}^{gt}$ . This matching is obtained by minimizing matching cost  $\mathcal{L}_{\mathrm{match}}$ :
+
+$$
+\hat {\sigma} = \arg \min  \sum_ {i} ^ {N} \mathcal {L} _ {\text {m a t c h}} \left(z _ {i} ^ {g t}, z _ {\sigma (i)}\right) \tag {1}
+$$
+
+where  $\sigma$  is a permutation of  $N$  elements and  $\mathcal{L}_{\mathrm{match}}$  quantifies the pairwise matching cost between ground truth site  $z_{i}^{gt}$  and the prediction with index  $\sigma(i)$ . Following prior work [32, 33], we employ the Hungarian algorithm to compute the optimal matching and use the matching cost defined as:
+
+$$
+\mathcal {L} _ {\text {m a t c h}} \left(z _ {i} ^ {g t}, z _ {\sigma (i)}\right) = - \mathbf {1} _ {\left\{c _ {i} ^ {g t} \neq \emptyset \right\}} \log p _ {\sigma (i)} \left(c _ {i} ^ {g t}\right) + \mathbf {1} _ {\left\{c _ {i} ^ {g t} \neq \emptyset \right\}} \mathcal {L} _ {\text {m a s k}} \left(m _ {i} ^ {g t}, m _ {\sigma (i)}\right) \tag {2}
+$$
+
+where  $\mathcal{L}_{\mathrm{mask}} = \lambda_{\mathrm{bce}}\mathcal{L}_{\mathrm{bce}} + \lambda_{\mathrm{dice}}\mathcal{L}_{\mathrm{dice}}$  is a combination of BCE loss and dice loss [34],  $p_{\sigma (i)}$  and  $m_{\sigma (i)}$  denote the predicted probability and the binding site for the  $\sigma (i)$ -th prediction, respectively. This matching cost considers both the class prediction and the similarity of the predicted and ground truth binding sites. Given the optimal matching  $\hat{\sigma}$ , we compose a cross-entropy classification loss and the binary mask loss  $\mathcal{L}_{\mathrm{mask}}$  for each predicted site to train model parameters:
+
+$$
+\mathcal {L} _ {\text {m a s k} \& \operatorname {c l s}} \left(z ^ {g t}, z\right) = \lambda_ {\operatorname {c l s}} \sum_ {i} ^ {N} - \log p _ {\hat {\sigma} (i)} \left(c _ {i} ^ {g t}\right) + \mathbf {1} _ {\left\{c _ {i} ^ {g t} \neq \emptyset \right\}} \mathcal {L} _ {\text {m a s k}} \left(m _ {i} ^ {g t}, m _ {\hat {\sigma} (i)}\right) \tag {3}
+$$
+
+# 3.2 UniSite architecture
+
+As illustrated in Figure 3, UniSite comprises three main components: (1) an encoder module that extracts residue-level representations  $\mathcal{F} \in \mathbb{R}^{L \times d_{\mathrm{model}}}$  of the protein; (2) a decoder module consisting of multiple Transformer decoder layers to generate embeddings of the  $N$  predicted binding sites, and (3) a segmentation module which combines the residue-level representations and the decoder embeddings to produce the final predictions  $\{(p_i, m_i) | m_i \in \{0, 1\}^L\}_{i=1}^N$ . This architecture maintains conciseness while demonstrating strong compatibility with existing protein representation methods. In our implementation, we construct two variants: UniSite-1D using only sequence encoding, and UniSite-3D incorporating both sequence and structural encoders.
+
+Sequence encoder. The amino acid sequence encodes the primary information of a protein and serves as the fundamental input for the UniProt-centric binding site detection. First, an input embedding module receives three inputs: (1) learnable embeddings for the 21 amino acid types (20 standard amino acids plus an "unknown" category); (2) sinusoidal positional embedding [35] for residue indices; (3) pre-trained ESM-2 [36] protein embeddings. These three components are
+
+concatenated along the feature dimension, and subsequently processed by a 3-layer multilayer perceptron (MLP) to generate the initial per-residue features. Then a stack of Transformer encoder layers process the combined features to capture residue-residue interactions and global sequence patterns.
+
+Structural encoder. Protein structure serves as the most critical input for structure-based tasks, including ligand binding site detection. Recent advances have proposed various structural feature extraction approaches, including hand-crafted algorithms [11], CNN-based methods [12, 37] and graph neural network approaches [13, 15]. To demonstrate the generality and effectiveness of our approach, we utilize GearNet-Edge [38], a standard E(3)-invariant GNN model, without introducing any custom architecture or specialized feature engineering.
+
+Given a protein  $P$ , the protein structure is represented as a residue-level relational graph  $\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{R})$ , where  $\mathcal{V}$  and  $\mathcal{E}$  represent the set of nodes and edges respectively, and  $\mathcal{R}$  is the set of edge types. Each node in the protein graph represents the alpha carbon of a residue, while sequential edges, radius edges and  $K$ -nearest neighbor edges are considered in the graph. Based on the defined protein graph, the node features are updated through the relational graph convolution layers [39] as follows:
+
+$$
+u _ {i} ^ {(l)} = \operatorname {R e L U} \left(\mathrm {B N} \left(\sum_ {r \in \mathcal {R}} W _ {r} \sum_ {j \in \mathcal {N} _ {r} (i)} h _ {j} ^ {(l - 1)}\right)\right), h _ {i} ^ {(l)} = h _ {i} ^ {(l - 1)} + u _ {i} ^ {(l)} \tag {4}
+$$
+
+where  $h_i^{(l)}$  represents the feature of node  $i$  at the  $l$ -th layer,  $\mathcal{N}_r(i) = \{j \in \mathcal{V} | (j, i, r) \in \mathcal{E}\}$  denotes the neighborhood of node  $i$  with the edge type  $r$ , and  $W_r$  is the convolutional kernel matrix shared within the edge type  $r$ . Specifically, BN represents a batch normalization layer and ReLU denotes the ReLU activation function.
+
+Unlike conventional binding site detection methods that rely on structural input, our framework allows the structural encoder to be optionally included. When incorporated, the structural features are concatenated with sequence features and projected via a linear layer to match the decoder's channels.
+
+Transformer decoder. The decoder comprises multiple Transformer decoder layers [35] that simultaneously process  $N$  embeddings of dimension  $d_{\mathrm{model}}$ ,  $\mathcal{Q} \in \mathbb{R}^{N \times d_{\mathrm{model}}}$ , through multi-head self-attention and cross-attention mechanisms. Following established practices in [32, 40], these input embeddings are learnable positional embeddings which we refer to as site queries. The attention mechanisms enable the decoder to perform global reasoning over all potential binding sites while incorporating contextual information from the residue-level protein features output by the encoder.
+
+Segmentation module. We process the  $N$  site queries through a linear classifier followed by the softmax activation to generate class probabilities  $\{p_i = (p_i^{site}, p_i^\emptyset) | p_i^{site}, p_i^\emptyset \in [0,1]\}_{i=1}^N$ . Here, the classifier predicts an additional  $\emptyset$  (non-binding) category to indicate when a query does not correspond to any actual binding site. For mask prediction, the site queries  $\mathcal{Q} \in \mathbb{R}^{N \times d_{\mathrm{model}}}$  are converted to  $N$  mask embeddings  $\mathcal{E}_{\mathrm{mask}} \in \mathbb{R}^{N \times d_{\mathrm{model}}}$  by a MLP. Finally, the binary mask prediction for each query is computed via dot-production between the  $i$ -th mask embedding and the residue-level protein features  $\mathcal{F} \in \mathbb{R}^{L \times d_{\mathrm{model}}}$ , followed by a sigmoid activation:
+
+$$
+m _ {i} [ j ] = \operatorname {s i g m o i d} \left(\mathcal {E} _ {\text {m a s k}} [ i,: ] \cdot \mathcal {F} [ j,: ] ^ {T}\right) \tag {5}
+$$
+
+# 4 Rethinking the Evaluation Metrics for Binding Site Detection
+
+DCC (Distance between the predicted binding site center and the true binding site center) and DCA (Shortest distance between the predicted binding site center and any heavy atom of the ligand) are the two most widely-used metrics for binding site detection. A binding site prediction is considered successful when its DCC or DCA value is below a predetermined threshold. Previous works [11, 37, 41, 14] quantify prediction performance via the Success Rate, defined as the ratio of successful predictions to the total number of ground truth sites:
+
+$$
+\text {S u c c e s s} \left(\mathrm {D C C} \text {o r} \mathrm {D C A}\right) = \frac {\mid \{\text {P r e d i c t e d s i t e s} \mid \mathrm {D C C} \text {o r D C A} <   \text {t h r e s h o l d} \} \mid}{\mid \{\text {G r o u n d t r u t h s i t e s} \mid} \tag {6}
+$$
+
+However, these metrics suffer from two critical limitations: (Limitation 1) They disregard the prediction scores or ranks, and predictions may be double-counted due to the absence of proper matching criteria (Figure 4 A and Table S1). (Limitation 2) They only evaluate the center of
+
+![](images/6beb0725168f9a4e8d055dd7c3491ca3190b6412e320ee3cf6a0c3867ef678a9.jpg)  
+Figure 4: DCC or DCA failure cases. (A) Repeated counting of the same predicted site since absence of matching. (B) Different ligands bound to the same site lead to deviations in DCC or DCA calculations. (C-D) Failed predictions classified as successful by DCC or DCA but below the IoU threshold.
+
+![](images/cb0590c39d38364d465a59d2d5aecca6c9a0b361350e1dc47af61be2c76f0ad1.jpg)
+
+![](images/8376a08f4741f91c0a646b85cdbc71b11b227d14e0b0a9f319e3536cf40f84df.jpg)
+
+![](images/d81de26640c81b2d657d52ee450baabae965e56e74fa501b3b3e91a19d949e5b.jpg)
+
+binding sites and are ligand-dependent (DCC typically considers the ligand center as site center). Since different ligands can bound to one binding site, relying solely on ligand-centered evaluation causes these metrics to completely miss key structural properties such as the shape, size, and residue composition of the binding site. It leads to evaluation failures in certain scenarios (Figure 4 B-D), disregard the crucial information required for downstream tasks (Appendix A).
+
+The quantitative analysis of the DCC and DCA metrics is provided in Appendix G to further substantiate their evaluative flaws. The analysis reveals that approximately  $20\%$  of proteins are subject to double-counting during evaluation. Furthermore, the measured mean ground truth DCC  $(2.15\mathring{\mathrm{A}},92.65\% < 4\mathring{\mathrm{A}})$  and DCA  $(1.57\mathring{\mathrm{A}},98.88\% < 4\mathring{\mathrm{A}})$  exhibit a significant deviation from the ideal value of 0, revealing a systematic bias inherent to these metrics. These results directly correspond to the previously discussed limitations and confirm that DCC and DCA metrics significantly distort model performance assessment.
+
+Previous works have recognized these limitations. To address Limitation 1, a common approach is to calculate the DCC or DCA for either the top- $n$  or top- $(n+2)$  predicted binding sites [11, 13, 14], where  $n$  is the number of ground truth sites. For Limitation 2, DeepSurf [41] and Utgés et al. [42] propose computing the IoU between the predicted and ground truth binding site residues. Given two binding sites  $m_A, m_B \in \{0,1\}^L$ , where  $L$  is the protein sequence length, the IoU of two sites is defined as:
+
+$$
+\operatorname {I o U} \left(m _ {A}, m _ {B}\right) = \frac {\operatorname {s u m} \left(m _ {A} \& m _ {B}\right)}{\operatorname {s u m} \left(m _ {A} \mid m _ {B}\right)}, \text {w h e r e} m _ {A}, m _ {B} \in \{0, 1 \} ^ {L} \tag {7}
+$$
+
+However, these methods fail to address the core issues, as they still lack proper matching between predicted and ground truth sites, and the top- $n$  or top- $(n+2)$  metrics introduce information leakage.
+
+To overcome these limitations, we propose to calculate the Average Precision (AP) metric based on the residue-level IoU as a fair metric for method evaluation. We calculate AP as follows: First, we sort all predictions by confidence scores. Then, we match each ground truth site to the predicted site with the highest score and residue-level IoU above a predetermined threshold, enforcing a one-to-one assignment constraint. Finally, we compute AP as the area under the interpolated precision-recall curve following COCO evaluation protocols [43], which is widely used in object detection. The pseudo-code for AP calculation is provided in Appendix K. The AP metric offers two significant advantages: (1) the residue-level IoU enables accurate shape and size comparison between binding sites; (2) the one-to-one matching scheme inherently prevents double-counting of predictions.
+
+# 5 Experiments
+
+# 5.1 Settings
+
+Dataset. UniSite-DS is used for training and validation. We employ MMSeq2 [44] to ensure that no test UniProt sequence has similarity above 0.9 to any sequence in the training set. For each UniProt sequence, we select the PDB structure with the highest sequence identity as the representative structure. Additionally, we compare UniSite with baseline methods on widely-used binding site benchmark datasets, HLO4K [11] and COACH420 [11]. Following DeepSurf [41] and EquiPocket [13], we use the mlig subsets of HLO4K and COACH420 for evaluation. Since
+
+our models are trained under a UniProt-centric schema, we only consider single-chain structures, denoting the test datasets as HLO04K-sc and COACH420 (all structures in COACH420 are originally single-chain). All test UniProt entries are strictly excluded from the training set. More details are provided in Appendix F.
+
+Implementation details. We set  $d_{\mathrm{model}} = 256$  by default. The transformer encoder consists of 6 standard Transformer encoder layers with a feed-forward dimension 1024 and the dropout rate of 0.1. We employ 6 Transformer decoder layers following the architecture of DETR [32]. By default, We use 32 site queries, where each query is associated with a learnable positional encoding and a zero-initialized query embedding. The multi-layer perceptron in the segmentation module consists of 2 hidden layers with 256 channels. For mask prediction, we use a combination of BCE loss and dice loss [34]:
+
+$$
+\mathcal {L} _ {\text {m a s k}} = \lambda_ {\text {b c e}} \mathcal {L} _ {\text {b c e}} + \lambda_ {\text {d i c e}} \mathcal {L} _ {\text {d i c e}} \tag {8}
+$$
+
+where  $\lambda_{\mathrm{bce}} = \lambda_{\mathrm{dice}} = 5.0$ . The classification loss weight  $\lambda_{\mathrm{cls}}$  is set to 2.0, and we downweight the classification loss by a factor of 10 when  $c_i^{gt} = \emptyset$  to mitigate class imbalance. Following DETR [32], we apply segmentation modules which share the same weights after each decoder layer, and supervise their predictions by the set prediction loss. We optimize the model using the AdamW optimizer [45] with a learning rate of  $1.0 \times 10^{-4}$  and a weight decay factor of 0.05. For the structural encoder, we implement the GearNet-Edge network following the origin paper [38] without specialized feature engineering. Notably, we train the GearNet-Edge network from scratch rather than loading pre-trained weights. All models are trained on 8 NVIDIA RTX 4090 GPUs.
+
+Our method predicts the associated residues along with a confidence score for each binding site. For applications requiring binding site centers, we compute these as the centroids of the convex hull encompassing all atoms within each predicted binding site [14, 46].
+
+Table 1: Results on UniSite-DS. We highlight the top two performing methods for each metric in bold. a Fpocket-rescore denotes sites initially predicted by Fpocket and subsequently rescored by P2Rank. b VN-EGNN only outputs centers of predicted sites. For each center, We include the residues within a  $9\AA$  radius, which has the best AP performance (Appendix H).  
+
+<table><tr><td>Method</td><td>Type</td><td>Input</td><td>AP0.3↑</td><td>AP0.5↑</td></tr><tr><td>Fpocket [8]</td><td>Geometry-based</td><td>structure</td><td>0.1836</td><td>0.1017</td></tr><tr><td>Fpocket-rescorea</td><td rowspan="2">Machine-learning</td><td>structure + Fpocket result</td><td>0.5075</td><td>0.2349</td></tr><tr><td>P2Rank [11]</td><td>structure</td><td>0.5056</td><td>0.2157</td></tr><tr><td>DeepPocket [12]</td><td>CNN-based</td><td>structure + Fpocket result</td><td>0.4273</td><td>0.2334</td></tr><tr><td>GrASP [14]</td><td rowspan="2">GNN-based</td><td>structure</td><td>0.4469</td><td>0.2848</td></tr><tr><td>VN-EGNNb [15]</td><td>structure</td><td>0.1621</td><td>0.0705</td></tr><tr><td>UniSite-1D</td><td rowspan="2">Ours</td><td>sequence</td><td>0.5121</td><td>0.3033</td></tr><tr><td>UniSite-3D</td><td>structure</td><td>0.5603</td><td>0.3835</td></tr></table>
+
+Evaluation metrics. For comprehensive evaluation, we compare IoU-based AP and traditional DCC or DCA metrics in HLO04K-sc and COACH420. Since a Uniprot-centric data entry can contain ligands from multiply PDB structures, coordinate-dependent metrics like DCC and DCA become unsuitable due to potential inconsistencies in ligand spatial arrangements. Consequently, we merely use IoU-based AP metric on UniSite-DS. Following EquiPocket [13], we set the DCC or DCA threshold to  $4\mathring{\mathrm{A}}$  , and compute the DCC or DCA success rate of top- $n$  predictions. We calculate AP using IoU thresholds of 0.3 and 0.5.
+
+# 5.2 Results on UniSite-DS
+
+The results on UniSite-DS are shown in Table 1. The geometry-based method Fpocket [8] exhibits inferior performance since it merely considers the geometry and electronegativity. P2Rank [11] achieves better results by extracting the protein surface features with Random Forest. DeepPocket [12] utilizes 3D-CNN to rescore and refine Fpocket predictions, improving the preformance in  $\mathrm{AP}_{0.5}$ . Notably, Fpocket-rescore, which combines Fpocket's initial predictions with P2Rank's re-ranking, surpasses both P2Rank and DeepPocket, highlighting the importance of proper scoring in binding site detection as captured by our AP metric. For graph models, GrASP [14] achieves further improvements in  $\mathrm{AP}_{0.5}$  by employing graph attention networks. However, VN-EGNN [15] performs poorly under
+
+AP metrics, because it only outputs predicted binding site centers, discarding structural properties (shape and size) or residue identification. For evaluation purposes, we include the residues within a  $9\AA$  radius of each predicted center, which has the best AP performance (Appendix H).
+
+Trained on the UniProt-centric dataset, UniSite-1D outperforms all baseline methods without protein structure, demonstrating remarkable capability for structure-free binding site detection, particularly valuable for site-aware protein-ligand docking (Appendix A). UniSite-3D further improves the performance remarkably by incorporating structure information. The above observations not only validate the effectiveness of our methods, but also reveal the significant statistical biases inherent in previous PDB-centric datasets, which limit the performance of prior methods.
+
+# 5.3 Results on HLO4K-sc and COACH420
+
+The results on HLO04K-sc and COACH420 are shown in Table 2. Fpocket [8], P2Rank [11] and DeePocket [12] exhibit a consistent performance ranking across different datasets and metrics. The improvement achieved by Fpocket-rescore is also consistently evident. These indicate the concordance between our proposed IoU-based AP and traditional DCC or DCA metrics. Besides, the AP metric demonstrates superior discriminative power. On HLO04K-sc, while DeepPocket and GrASP [14] show almost identical performance in  $\mathrm{DCA}_{\mathrm{top - n}}(< 0.01)$ , they diverge substantially ( $>0.10$ ) in  $\mathrm{AP}_{0.3}$ . Similarly, on COACH420, performance differences among Fpocket-rescore, P2Rank, and GrASP are more pronounced under AP evaluation than with DCC metrics. As an exception, VN-EGNN [15] performs well in DCC or DCA while performing poorly under AP, as it merely predicts the binding site centers, discarding structural properties and residue identification. Notably, it is problematic since both the structural properties and the residue identification of binding sites are critical for downstream tasks (Appendix A). Since UniSite-DS is a UniProt-centric dataset while HLO04K-sc and COACH420 are both PDB-centric, there is a training-test gap for UniSite-1D/3D. Even so, both UniSite-1D and UniSite-3D maintain strong performance on the two PDB-centric benchmarks across all evaluation metrics, demonstrating the effectiveness of our framework. More results are provided in Appendix J.
+
+Table 2: Results on HLO4K-sc and COACH420. We highlight the top two performing methods for each metric in bold. a Fpocket-rescore denotes sites initially predicted by Fpocket and subsequently rescored by P2Rank. b VN-EGNN only outputs centers of predicted sites. For each center, We include the residues within a  $9\AA$  radius, which has the best AP performance (Appendix H).  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">HOL04K-sc</td><td colspan="3">COACH420</td></tr><tr><td>AP0.3↑</td><td>DCCtop-n↑</td><td>DCAtop-n↑</td><td>AP0.3↑</td><td>DCCtop-n↑</td><td>DCAtop-n↑</td></tr><tr><td>Fpocket [8]</td><td>0.2711</td><td>0.3076</td><td>0.4382</td><td>0.2106</td><td>0.2708</td><td>0.4107</td></tr><tr><td>Fpocket-rescorea</td><td>0.5899</td><td>0.5183</td><td>0.7654</td><td>0.5602</td><td>0.4405</td><td>0.7113</td></tr><tr><td>P2Rank [11]</td><td>0.6011</td><td>0.5300</td><td>0.8188</td><td>0.6188</td><td>0.4643</td><td>0.7411</td></tr><tr><td>DeepPocket [12]</td><td>0.5415</td><td>0.4925</td><td>0.7369</td><td>0.5184</td><td>0.3958</td><td>0.6756</td></tr><tr><td>GrASP [14]</td><td>0.6668</td><td>0.5131</td><td>0.7416</td><td>0.7150</td><td>0.4851</td><td>0.7620</td></tr><tr><td>VN-EGNNb[15]</td><td>0.2606</td><td>0.5861</td><td>0.6999</td><td>0.2637</td><td>0.5446</td><td>0.7530</td></tr><tr><td>UniSite-1D (ours)</td><td>0.6867</td><td>0.5538</td><td>0.7692</td><td>0.5921</td><td>0.4554</td><td>0.7351</td></tr><tr><td>UniSite-3D (ours)</td><td>0.7091</td><td>0.5716</td><td>0.7879</td><td>0.7196</td><td>0.4702</td><td>0.7381</td></tr></table>
+
+Table 3: Effect of sequence similarity. The second column indicates the sequence identity between training sets and test sets.  
+
+<table><tr><td>Method</td><td>Similarity</td><td>AP0.3↑</td><td>AP0.5↑</td></tr><tr><td>UniSite-1D</td><td></td><td>0.5121</td><td>0.3033</td></tr><tr><td>UniSite-3D</td><td>&lt;0.9</td><td>0.5603</td><td>0.3835</td></tr><tr><td>UniSite-1D</td><td></td><td>0.5056</td><td>0.2945</td></tr><tr><td>UniSite-3D</td><td>&lt;0.7</td><td>0.5579</td><td>0.3734</td></tr><tr><td>UniSite-1D</td><td></td><td>0.4338</td><td>0.2243</td></tr><tr><td>UniSite-3D</td><td>&lt;0.5</td><td>0.4677</td><td>0.2801</td></tr></table>
+
+Table 4: Effect of site queries. This table shows results of UniSite-3D trained with a varying number of site queries.  
+
+<table><tr><td># of queries</td><td>AP0.3↑</td><td>AP0.5↑</td></tr><tr><td>16</td><td>0.5515</td><td>0.3795</td></tr><tr><td>32</td><td>0.5603</td><td>0.3835</td></tr><tr><td>48</td><td>0.5562</td><td>0.3861</td></tr><tr><td>64</td><td>0.5615</td><td>0.3867</td></tr></table>
+
+# 5.4 Ablation study
+
+Sequence similarity. Both the structure and function of proteins diverge with the decreasing of sequence similarity. It is necessary to evaluate our protein ligand binding site detection method across varying similarity thresholds. We employ MMSeqs2 [44] to partition UnSite-DS with three similarity thresholds: 0.5, 0.7 and 0.9, ensuring that no test protein exceeds the corresponding similarity to any training protein. Compared to threshold 0.9, UnSite-1D/3D exhibit only a slight decrease under threshold 0.7 (Table 3), indicating that our methods possess generalization ability. As proteins with sequence similarity below 0.5 typically belong to evolutionarily distant families and exhibit markedly different structural folds, we observe significant AP performance degradation for UniSite-1D and UniSite-3D under threshold 0.5.
+
+Number of site queries. As shown in Table 4, UniSite-3D exhibits stable performance across varying numbers of site queries. we select 32 queries as our default configuration, considering both the computation cost and the coverage of ground truth binding sites (99.5% proteins in UniSite-DS have sites less than 20).
+
+# 6 Concluding Remarks and Future Perspectives
+
+Key Contributions. In this paper, we introduce UniSite-DS, the first UniProt-centric dataset of protein ligand binding sites, which systematically integrates all ligand binding sites across multiple PDB structures for each unique protein. UniSite-DS corrects the statistical bias in previously available PDB-centric datasets and methods, while significantly broadening the available data. To amend the discontinuous workflows in existing binding site detection methods, we proposed UniSite-1D/3D, two end-to-end methods supervised by set prediction loss with bijective matching. In addition, we introduce IoU-based AP as a more accurate evaluation metric. Extensive experiments on UniSite-DS and several benchmark datasets demonstrate that our frameworks achieve superior performance, and the IoU-based AP metric can provide a more accurate reflection of binding site prediction quality.
+
+Limitations and Future Work. The current version of UniSite-DS involves manual curation to remove unreasonable entries. A promising direction for future work is to develop automated methods for repairing and reintegrating excluded data to further enhance the dataset's coverage and quality. Additionally, our current model design aims to demonstrate the effectiveness of the end-to-end ligand binding site learning framework, without incorporating specialized feature engineering. Future investigations could explore the inclusion of specialized feature engineering to further improve model performance and generalization ability.
+
+# Acknowledgements
+
+We thank the reviewers for their constructive feedback, and Hannes Stärk, Chenyu Wang, Zuobai Zhang, Zaixi Zhang, and Bozitao Zhong for helpful discussions. Liwei Wang is supported by National Science and Technology Major Project (2022ZD0114902) and National Science Foundation of China (NSFC92470123, NSFC62276005).
+
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+
+# A Impact of Binding Site Accuracy on Downstream Tasks
+
+The accuracy of protein-ligand binding site detection is critical for downstream tasks. Taking molecular docking as an example (Figure S1), a comparison of different methods leads to the following conclusions: (1) The definition of binding site residues can strongly impact the docking performance, highlighting the importance of our proposed IoU-based AP metric. (2) Docking methods that do not specify binding sites (blind docking) show significant performance improvements when provided with binding site information, emphasizing the importance of accurate binding site identification.
+
+![](images/2d3155674f1a7a5ac16ab39b8f504de194bd6462cd8e0708431295b12e1f19b4.jpg)  
+Figure S1: The significant impact of binding site detection on molecular docking. (A) Gold [47] defines the binding site using a sphere. (B) AutoDock Vina [48] defines the binding site using a cube. (C) DeepDock [49] and (D) Uni-Mol [50] identify the binding site by applying a fixed radius around the ligand. (E) Docking success rates on the PoseBusters dataset under different binding site configurations. Docking success rate is defined as the proportion of predictions with an RMSD less than  $2\AA$ . Data sourced from [51, 52, 53, 54].
+
+![](images/0cd822d47020d118d1c45f3056584cf927d280f76de3479ad5e64f13756e61dc.jpg)
+
+# B UniSite-DS Curation Workflow and Representative Manual Inspection Case Studies
+
+As shown in Figure S2, the UniSite-DS workflow comprises three main components: (I) dataset curation, (II) quality control, and (III) manual inspection. Below, we describe two representative types of cases encountered during manual inspection that required special consideration:
+
+1. Case 1. Supramolecular assemblies across multiple protein subunits. The Photosystem I-LHCI Supercomplex is a core component of the photosynthetic machinery, consisting of multiple subunits such as UniProt ID: P05310 (Photosystem I P700 chlorophyll a apoprotein A1, PDB: 7dkz_A), Q41038 (Chlorophyll a-b binding protein, 7dkz_B), and Q32904 (Chlorophyll a-b binding protein 3, 7dkz_C). Although each of these proteins binds a large number of Chlorophyll a molecules, they function as part of a tightly integrated supramolecular assembly. Therefore, they are not suitable to be included as independent entries in the UniSite-DS database and have been excluded in the current version.  
+2. Case 2. Large composite cavities jointly formed by multiple ligands. Trypanothione reductase (UniProt ID: Q389T8) is a key enzyme in Trypanosoma brucei that specifically catalyzes the reduction of trypanothione, functionally analogous to glutathione reductase in mammals. The ligands from structures PDB: 5s9x_A, 5s9t_A, and 2wov_C together form a large binding cavity. These ligand binding sites could potentially be merged. However, since each ligand actually occupies only a portion of the cavity rather than the entire cavity, whether such a composite cavity formed by different ligands should be considered a unified binding site often depends on system-specific definitions in the literature. As a result, entries that require further literature-based validation were excluded from the current version of the UniSite-DS database.
+
+# I) Curation Process
+
+Systematic retrieval of all protein-ligand interactions from the PDB database  
+Identification of binding site residues within 4.5 Å of each ligand  
+- Integration of all ligand binding sites across different PDB structures using UniProt identifiers and SIFTS annotations  
+Redundancy removal using NMS with IoM  $\geq$  0.7 and IoU  $\geq 0.5$  to exclude highly overlapping binding sites  
+Additional quality control and manual inspection to ensure high-quality dataset
+
+# II) Quality Control
+
+- Exclude structures with resolution  $>2.5\AA$  or determined by non-crystallographic methods  
+- Remove solvent molecules
+
+- Discard entries with  $\leq 3$  binding site residues to eliminate floating ligands
+
+# III) Manual Inspection
+
+- Manual inspection of entries with  $>10$  ligand binding sites or with  $\geq 3$  sites contributed by a single protein-ligand complex  
+Entries involving supramolecular assemblies across multiple protein subunits  
+- Entries with large composite cavities jointly formed by multiple ligands, each occupying only part of the cavity
+
+# IV) Manual Inspection Case Studies
+
+Case 1) Entries involving supramolecular assemblies across multiple protein subunits
+
+![](images/449e5baeccf25fcf7e145fd8e98ae2b96473b2ae7ab8987c5e8b1cc0f1464555.jpg)
+
+Case 2) Entries with large composite cavities jointly formed by multiple ligands, each occupying only part of the cavity
+
+![](images/096990baca25a7ba8f43ba74d96aac082f21744bdf40ad7e8a2b0eeaf7f32c66.jpg)  
+Figure S2: Overview of the UniSite-DS workflow. Workflow of (I) dataset curation, (II) quality control, and (III) manual inspection for UniSite-DS, together with (IV) representative manual inspection case studies.
+
+# C Related Work
+
+Over the past several decades, numerous methods have been developed for detecting protein-ligand binding sites, accompanied by advances in techniques leveraging the geometric, physical and chemical features of proteins.
+
+Early methods relied on traditional computational algorithms. Since most binding sites show up as cavities in protein 3D structures, geometry-based methods (Fpocket [8], LigSite [55]) identify and rank these hollow cavities through hand-crafted features like alpha spheres [56]. Template-based methods (FINDSITE [9] and LIBRA [57]) predict ligand binding sites by comparing the query protein with templates from known protein structure database. These methods typically generate a large number of predicted sites while performing poorly in ranking them.
+
+Subsequent approaches like PRANK [58] and P2Rank [11] employ traditional machine learning methods, particularly Random Forest. Based on the predictions of Fpocket [8], PRANK assigns "ligandibility" scores, which denotes ligand binding potential, to candidate sites. P2Rank is a widely used method which integrates the geometric features of the protein surface with Random Forest Algorithm.
+
+In recent years, deep learning methods have emerged for protein-ligand binding site detection. CNN-based approaches [12, 37, 41] treat protein structures as 3D images, applying 3D convolutional neural networks similar to those used in computer vision. Alternatively, GNN-based methods [15, 14, 13] utilize graph neural networks by constructing graphs incorporating both geometric and chemical features of proteins. Despite their improved performance, these methods typically adopt discontinuous workflows: they first perform semantic segmentation to generate binary masks of potential binding residues/atoms, then cluster these masks into discrete binding sites. This fragmented pipeline heavily depends on post-processing (e.g., clustering algorithms), inherently limiting end-to-end optimization and struggling with overlapping binding sites. DETR-based architectures have also been adapted for other protein tasks, such as ProtDETR [59], which performs enzyme function classification rather than binding site detection.
+
+# D Baseline Justification
+
+We selected representative baseline methods that span the major methodological paradigms in ligand binding site detection. Fpocket [8] represents traditional geometry-based computational algorithms. P2Rank [11] is the most widely used machine learning approach. Among deep learning approaches, DeepPocket [12], GrASP [14], and VN-EGNN [15] were included as they represent different neural network architectures: DeepPocket employs 3D CNNs, while GrASP and VN-EGNN utilize GNNs.
+
+For a fair and consistent comparison, we extracted ligand binding site residues from each baseline's output files according to their respective standard formats:
+
+- Fpocket: For each predicted pocket, Fpocket outputs a file named pocket{index}_atm.pdb, which contains the atomic coordinates of the predicted binding site in PDB format. These atoms were directly used to identify the corresponding binding residues.  
+- P2Rank: P2Rank generates a {name}.pdb_prediction.csv file for each input structure. The residue_ids column specifies the chain IDs and residue IDs of residues constituting each predicted binding site. These identifiers were parsed to obtain the binding site residues.  
+- DeepPocket: DeepPocket refines and re-scores the pockets predicted by Fpocket, while maintaining the same output format.  
+- GrASP: GrASP outputs a {name}_probs.pdb file, where the bfactor column encodes the predicted binding probabilities of heavy atoms. Following the original publication, we filtered atoms based on their predicted scores and clustered them spatially into distinct binding sites.  
+- VN-EGNN: VN-EGNN produces a prediction.csv file containing four columns (x, y, z, rank) that specify the 3D coordinates and ranking of predicted pocket centers. As the method does not directly output residue-level predictions, binding site residues were obtained by including all residues within a defined radius, as described in Appendix H.
+
+# E Case Study of Classical Methods
+
+As discussed in Section 1, 2, previous approaches are developed based on PDB-centric datasets, which introduce substantial statistical biases. We conduct a case study of these methods, and the results (Figure S3) indicate that, for multi-site proteins, existing approaches are weak in distinguishing between different ligand binding sites. This limitation motivated us to investigate the problems in existing approaches and to develop a new methodology.
+
+![](images/a8593d6268c43e506a64cab8e00f018698a2ff7b21c8ce2a746b44a461c90ab3.jpg)  
+A Ground Truth Sites
+
+![](images/5bc8080a3ab29790d11fcc6ec437da9b30c831b417165f703ac92da0033ad8a3.jpg)  
+C GrASP
+
+![](images/246270c574791a4c734489482ea8bc86ef4a1484a50ee484a96c49719be4d04a.jpg)  
+Figure S3: Case study of classical methods. For UniProt ID Q7YYQ9, the two ground truth binding sites are represented by dark blue and orange meshes, respectively. All predicted binding sites are shown as surfaces. (A) The two ground truth binding sites are colored in dark blue and orange. (B) The two binding sites predicted by our UniSite-3D method are colored in cyan and purple. (C) The single binding site predicted by the GNN-based method GrASP [14] is colored in green. (D) The single binding site predicted by the classical machine learning method P2Rank [11] is colored in red. (E) The single binding site predicted by the geometry-based method Fpocket [8] is colored in cyan.
+
+![](images/0dbfcdcdf1084e029e11d18ca83429959df1caad7e2ae1c0bc72173f717341d1.jpg)  
+E Fpocket
+
+![](images/4280c28fbec93143a478c54aea3f6043064676274ee6ac1f74beb3cb159801a6.jpg)
+
+# F External Datasets
+
+scPDB [16] is a famous dataset for protein-ligand binding site detection, commonly employed for training and validation in recent studies ( [15, 14, 13]). scPDB provides both protein and ligand structures, accompanied by the structures of binding site extracted via VolSite [60]. Notably, only one binding site and one corresponding ligand are annotated for each data entry. In this work, we use the 2017 release of scPDB, which contains 17,594 structures and 5,550 unique proteins. (Source: http://bioinfo-pharma.u-strasbg.fr/scPDB/)
+
+PDBBind [17] is a widely used dataset to study protein-ligand interaction, especially for protein-ligand docking [51, 61]. Similar to scPDB, PDBBind annotates one ligand structure and one binding site structure in each data entry. In this paper, we use the general set of v2020, the latest academic-free edition, which comprises 19,443 structures and 3,888 unique proteins. (Source: http://www.pdbbind.org.cn/download/)
+
+HOL04K and COACH420 are two benchmark datasets utilized for protein-ligand binding site detection. Follow VN-EGNN [15], EquiPocket [13] and GrASP [14], we employ the mlig subsets of these two dataset, which contain explicitly specified relevant ligands. HOL04K-mlig comprises 3,204 structures and 1,259 unique proteins, while COACH420-mlig covers 284 structures and 265 unique proteins. (Source: https://github.com/rdk/p2rank-datasets)
+
+# G Quantitative Analysis of Evaluation Flaws in DCC and DCA Metrics
+
+In Section 4, we discussed two critical limitations in the design of the DCC and DCA metrics. Here, we conduct a quantitative analysis on the HOLO4K-sc benchmark to demonstrate the flawed evaluation introduced by these metrics.
+
+**Limitation 1.** The absence of proper matching criteria may lead to double-counting of predictions (Figure 4 A). We quantified the proportion of proteins affected by double-counting during evaluation on the HOLO4K-sc benchmark (Table S1). The results reveal that DCC and DCA metrics suffer from widespread double counting artifacts, which significantly distort model performance assessment.
+
+Table S1: Double counting (DC) rate of DCC or DCA metrics on HOL04K-sc. We highlight the top two performing methods for each metric in bold. a Fpocket sites rescored by P2Rank. b Residues within 9 Å of each VN-EGNN predicted center, which has the best AP performance (Appendix H).  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">HOL04K-sc</td></tr><tr><td>DC of DCCtop-n</td><td>DC of DCAtop-n</td></tr><tr><td>Fpocket [8]</td><td>18.80%</td><td>18.31%</td></tr><tr><td>Fpocket-rescorea</td><td>13.23%</td><td>13.66%</td></tr><tr><td>P2Rank [11]</td><td>18.98%</td><td>18.31%</td></tr><tr><td>DeepPocket [12]</td><td>13.53%</td><td>14.21%</td></tr><tr><td>GrASP [14]</td><td>16.29%</td><td>14.88%</td></tr><tr><td>VN-EGNNb[15]</td><td>12.80%</td><td>11.70%</td></tr><tr><td>UniSite-1D (ours)</td><td>8.88%</td><td>8.94%</td></tr><tr><td>UniSite-3D (ours)</td><td>9.86%</td><td>10.04%</td></tr></table>
+
+**Limitation 2.** DCC or DCA only evaluate the center of binding sites and are ligand-dependent, which leads to evaluation failures in certain scenarios (Figure 4 B-D). For HLO4K-sc, we calculated these metrics of the centroid of ground truth binding residues for each protein. The results indicate that the mean ground truth DCC is  $2.15\,\text{\AA}$  ( $92.65\% < 4\,\text{\AA}$ ), and the mean ground truth DCA is  $1.57\,\text{\AA}$  ( $98.88\% < 4\,\text{\AA}$ ). However, in principle, both DCC and DCA should ideally be 0 when evaluated using ground truth binding residues, indicating these metrics inherently contain systematic bias.
+
+To mitigate some of DCC's inherent limitations, we defined a corrected metric, DCC-residue, which uses the center of ground truth binding residues rather than the ligand center for calculation. This modification resolves failure cases caused by ligand diversity in traditional DCC evaluation. As shown in Table S2, the corrected DCC-residue metric exhibits improved consistency with IoU-based AP in ranking the performance of different methods.
+
+Table S2: IoU-based AP, DCC-residue and DCC results on HOL04K-sc. We highlight the top two performing methods for each metric in bold. a Fpocket sites rescored by P2Rank. b Residues within 9Å of each VN-EGNN predicted center, which has the best AP performance (Appendix H).  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">HOL04K-sc</td></tr><tr><td>AP0.3↑</td><td>DCC-residue top-n↑</td><td>DCCtop-n↑</td></tr><tr><td>Fpocket [8]</td><td>0.2711</td><td>0.2982</td><td>0.3076</td></tr><tr><td>Fpocket-rescorea</td><td>0.5899</td><td>0.5005</td><td>0.5183</td></tr><tr><td>P2Rank [11]</td><td>0.6011</td><td>0.4972</td><td>0.5300</td></tr><tr><td>DeepPocket [12]</td><td>0.5415</td><td>0.4902</td><td>0.4925</td></tr><tr><td>GrASP [14]</td><td>0.6668</td><td>0.5379</td><td>0.5131</td></tr><tr><td>VN-EGNNb [15]</td><td>0.2606</td><td>0.5997</td><td>0.5861</td></tr><tr><td>UniSite-1D (ours)</td><td>0.6867</td><td>0.6400</td><td>0.5538</td></tr><tr><td>UniSite-3D (ours)</td><td>0.7091</td><td>0.6264</td><td>0.5716</td></tr></table>
+
+# H The AP Evaluation of VN-EGNN
+
+Since VN-EGNN [15] outputs only the centers of predicted binding sites, it is non-trivial to identify the binding residues. In order to evaluate the IoU-based AP, we include the residues with a fixed radius for each predicted center (Figure S4). As shown in Table S3, the radius of  $9\AA$  has the best performance across all datasets.
+
+![](images/46a700269f5cc657b492212589e06ef1096469460780c0a70625c75d616f5798.jpg)  
+Figure S4: Binding sites derived from the predicted center by VN-EGNN using different radii. Binding site residues are visualized using surface representation: (A) residues within a  $4.5\AA$  radius; (B) residues within a  $6\AA$  radius; (C) residues within a  $9\AA$  radius; (D) residues within a  $12\AA$  radius.
+
+![](images/8db0b25f46b95fd2d1434540158615320b9bd9d88ba12a95dbde8497ed4f4d20.jpg)
+
+![](images/e3029b179c1f29a3b763a09721330fd4ff565a6d19139466b831dfe218961d9e.jpg)
+
+![](images/bb8ff317621c952e429185a594dd7a162d9c60e2942625ccd560afc6bce67b7f.jpg)
+
+Table S3: AP evaluation results of VN-EGNN using different radii.  
+
+<table><tr><td rowspan="2">Radius</td><td colspan="2">UniSite-DS</td><td colspan="2">HOLO4K-sc</td><td colspan="2">COACH420</td></tr><tr><td>AP0.3↑</td><td>AP0.5↑</td><td>AP0.3↑</td><td>AP0.5↑</td><td>AP0.3↑</td><td>AP0.5↑</td></tr><tr><td>4.5Å</td><td>0.0054</td><td>0.0004</td><td>0.0049</td><td>0.0001</td><td>0.0072</td><td>0.0007</td></tr><tr><td>6Å</td><td>0.0894</td><td>0.0088</td><td>0.1290</td><td>0.0172</td><td>0.1814</td><td>0.0189</td></tr><tr><td>9Å</td><td>0.1621</td><td>0.0705</td><td>0.2606</td><td>0.1346</td><td>0.2637</td><td>0.1138</td></tr><tr><td>12Å</td><td>0.0087</td><td>0.0010</td><td>0.1411</td><td>0.0014</td><td>0.1444</td><td>0.0013</td></tr></table>
+
+# I The Recall Evaluation of Different Methods
+
+The AP metric provides a comprehensive evaluation of binding site detection performance by considering both precision and recall. However, it is also important to evaluate how many true binding sites can be recovered under different IoU thresholds, since the goal of binding site detection is to identify novel and biologically meaningful sites/pockets. We computed the Recall of different methods across multiple IoU thresholds on the HLO4K-sc dataset, and the results are summarized in Table S4. UniSite-3D achieves state-of-the-art Recall across all IoU thresholds.
+
+Notably, the high recall obtained by Fpocket is due to its tendency to output nearly all potential cavities in a protein, rather than accurately identifying biologically relevant pockets. In fact, Fpocket tends to assign lower scores to true binding sites, making it challenging for biologists to distinguish meaningful pockets. This limitation has motivated many studies to develop rescoring strategies for Fpocket. While Recall reflects the potential of a method to detect biologically significant pockets, it does not account for the confidence score of each prediction. This limitation is addressed by the AP metric, which is why we adopt it as a fair and balanced criterion for evaluating different methods.
+
+Table S4: The AP and Recall results on H0LO4K-sc. We highlight the top two performing methods for each metric in bold. a Fpocket sites rescored by P2Rank. b Residues within 9 Å of each VN-EGNN predicted center, which has the best AP performance (Appendix H).  
+
+<table><tr><td rowspan="2">Method</td><td colspan="6">HOL04K-sc</td></tr><tr><td>AP0.3</td><td>AP0.5</td><td>Recall0.3</td><td>Recall0.5</td><td>Recall0.7</td><td>Recall0.9</td></tr><tr><td>Fpocket [8]</td><td>0.2711</td><td>0.1488</td><td>0.8361</td><td>0.5922</td><td>0.2130</td><td>0.0253</td></tr><tr><td>Fpocket-rescorea</td><td>0.5899</td><td>0.2847</td><td>0.8361</td><td>0.5922</td><td>0.2130</td><td>0.0253</td></tr><tr><td>P2Rank [11]</td><td>0.6011</td><td>0.2625</td><td>0.7814</td><td>0.5337</td><td>0.1868</td><td>0.0089</td></tr><tr><td>DeepPocket [12]</td><td>0.5415</td><td>0.2891</td><td>0.7514</td><td>0.5824</td><td>0.2584</td><td>0.022</td></tr><tr><td>GrASP [14]</td><td>0.6668</td><td>0.4126</td><td>0.7186</td><td>0.5374</td><td>0.2537</td><td>0.0159</td></tr><tr><td>VN-EGNNb[15]</td><td>0.2606</td><td>0.1346</td><td>0.7289</td><td>0.4874</td><td>0.0566</td><td>0</td></tr><tr><td>UniSite-1D (ours)</td><td>0.6867</td><td>0.4595</td><td>0.8212</td><td>0.6199</td><td>0.3535</td><td>0.0824</td></tr><tr><td>UniSite-3D (ours)</td><td>0.7091</td><td>0.5446</td><td>0.8469</td><td>0.6901</td><td>0.4106</td><td>0.1039</td></tr></table>
+
+# J Full Results on HLO04K-sc and COACH420
+
+Table S5: Full results on HLO4K-sc. We highlight the top two performing methods for each metric in bold. a Fpocket sites rescored by P2Rank. b Residues within 9Å of each VN-EGNN predicted center, which has the best AP performance (Appendix H).  
+
+<table><tr><td rowspan="2">Method</td><td colspan="6">HOL04K-sc</td></tr><tr><td>AP0.3</td><td>AP0.5</td><td>DCCtop-n</td><td>DCCtop-n+2</td><td>DCAtop-n</td><td>DCAtop-n+2</td></tr><tr><td>Fpocket [8]</td><td>0.2711</td><td>0.1488</td><td>0.3076</td><td>0.4181</td><td>0.4382</td><td>0.5941</td></tr><tr><td>Fpocket-rescorea</td><td>0.5899</td><td>0.2847</td><td>0.5183</td><td>0.5941</td><td>0.7654</td><td>0.8577</td></tr><tr><td>P2Rank [11]</td><td>0.6011</td><td>0.2625</td><td>0.5300</td><td>0.5623</td><td>0.8188</td><td>0.8652</td></tr><tr><td>DeepPocket [12]</td><td>0.5415</td><td>0.2891</td><td>0.4925</td><td>0.5478</td><td>0.7369</td><td>0.7851</td></tr><tr><td>GrASP [14]</td><td>0.6668</td><td>0.4126</td><td>0.5131</td><td>0.5267</td><td>0.7416</td><td>0.7612</td></tr><tr><td>VN-EGNNb[15]</td><td>0.2606</td><td>0.1346</td><td>0.5861</td><td>0.6339</td><td>0.6999</td><td>0.7500</td></tr><tr><td>UniSite-1D (ours)</td><td>0.6867</td><td>0.4595</td><td>0.5538</td><td>0.6400</td><td>0.7692</td><td>0.8305</td></tr><tr><td>UniSite-3D (ours)</td><td>0.7091</td><td>0.5446</td><td>0.5716</td><td>0.6470</td><td>0.7879</td><td>0.8422</td></tr></table>
+
+Table S6: Full results on COACH420. We highlight the top two performing methods for each metric in bold. a Fpocket sites rescored by P2Rank. b Residues within 9Å of each VN-EGNN predicted center, which has the best AP performance (Appendix H).  
+
+<table><tr><td rowspan="2">Method</td><td colspan="6">COACH420</td></tr><tr><td>AP0.3</td><td>AP0.5</td><td>DCCtop-n</td><td>DCCtop-n+2</td><td>DCAtop-n</td><td>DCAtop-n+2</td></tr><tr><td>Fpocket [8]</td><td>0.2106</td><td>0.1219</td><td>0.2708</td><td>0.3750</td><td>0.4107</td><td>0.5714</td></tr><tr><td>Fpocket-rescorea</td><td>0.5602</td><td>0.2905</td><td>0.4405</td><td>0.5179</td><td>0.7113</td><td>0.8333</td></tr><tr><td>P2Rank [11]</td><td>0.6188</td><td>0.2618</td><td>0.4643</td><td>0.5000</td><td>0.7411</td><td>0.8034</td></tr><tr><td>DeepPocket [12]</td><td>0.5184</td><td>0.2512</td><td>0.3958</td><td>0.4821</td><td>0.6756</td><td>0.7560</td></tr><tr><td>GrASP [14]</td><td>0.7150</td><td>0.4914</td><td>0.4851</td><td>0.4970</td><td>0.7620</td><td>0.7917</td></tr><tr><td>VN-EGNNb[15]</td><td>0.2637</td><td>0.1138</td><td>0.5446</td><td>0.6071</td><td>0.7530</td><td>0.7768</td></tr><tr><td>UniSite-1D (ours)</td><td>0.5921</td><td>0.2998</td><td>0.4554</td><td>0.5238</td><td>0.7351</td><td>0.8006</td></tr><tr><td>UniSite-3D (ours)</td><td>0.7196</td><td>0.3977</td><td>0.4702</td><td>0.5387</td><td>0.7381</td><td>0.8095</td></tr></table>
+
+# K Pseudo-code for Average Precision Calculation
+
+Algorithm 1 Average Precision Calculation  
+Input: prediction_list: A list, where each element represents the predictions for one protein. Each element is of the form  $\{(s_i, m_i) | s_i \in \mathbb{R}, m_i \in \{0, 1\}^L\}_{i=1}^N$ , where  $m_i$  represents the  $i$ -th predicted binding site as a binary mask of length  $L$ , and  $s_i$  denotes the  $i$ -th confidence score. groundtruth_list: A list, where each element represents the ground truth (gt) binding sites for one protein. Each element is of the form  $\{m_i^{gt} | m_i^{gt} \in \{0, 1\}^L\}_{i=1}^{N_{gt}}$ , where  $m_i^{gt}$  represents the  $i$ -th ground truth binding site as a binary mask of length  $L$ . iou_threshold: IoU threshold for considering a prediction as True Positive. Output: AP: Average Precision under the given IoU threshold.
+
+Step 1: Determine TP (True Positive) or FP (False Positive) for each prediction  
+1: for predictions_per_protein, ground truths_per_protein in ZIP(prediction_list, groundtruths_list) do  
+2: Set all ground truths in ground_truths_per_protein as unused  
+3: Sort predictions_per_protein = {  $(s_i, m_i) | s_i \in \mathbb{R}, m_i \in \{0, 1\}^L\}_{i=1}^N$  by decreasing confidence score  $s_i$   
+4: for  $i = 1$  to  $N$  do  
+5: Find ground Truth_j that has the max residue-level IoU( $m_j^{gt}$ ,  $m_i$ ) with prediction_i  
+6: if IoU( $m_j^{gt}$ ,  $m_i$ ) > iou_threshold and groundtruth_j is unused then  
+7: Mark prediction_i as TP  
+8: Set ground Truth_j as used  
+9: else  
+10: Mark prediction_i as FP  
+11: end if  
+12: end for  
+13: end for
+
+Step 2: Sort all predictions across all proteins by decreasing confidence scores  
+14: allpredictions  $\leftarrow$  FLATTEN_ALL(prediction_list)   
+15: all-ground_truths  $\leftarrow$  FLATTEN_ALL(groundtruthsList)   
+16: Sort allpredictions by decreasing confidence scores
+
+Step 3: Calculate the precision-recall curve  
+Step 4: Calculate average precision  
+27: Calculate AP as the area under the precision-recall curve  
+28: return AP  
+17: Set cum_TP, cum_BP as 0, precision_list and recall_list as empty  
+18: for prediction_i in allpredictions do  
+19: if prediction_i is marked as TP then  
+20: cum_TP  $\leftarrow$  cum_TP + 1  
+21: else  
+22: cum_BP  $\leftarrow$  cum_BP + 1  
+23: end if  
+24: precision_list.append(cum_TP / (cum_TP + cum_BP))  
+25: recall_list.append(cum_TP / LEN(all-groundtruths))  
+26: end for
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: The claims reflect our contribution about dataset, methodology and evaluation metrics, which are illustrated in Section 2, Section 3 and Section 4 respectively.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: We discussed the limitations in Section 6.
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [NA]
+
+Justification: This paper does not include theoretical results.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+# Answer: [Yes]
+
+Justification: We described the details of our model architecture, evaluation metrics, and experiment settings in Section 3-5.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.  
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.  
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.  
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example  
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+# Answer: [Yes]
+
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+
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+
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\ No newline at end of file
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+# UniTok: A Unified Tokenizer for Visual Generation and Understanding
+
+Chuofan Ma $^{1,2}$  Yi Jiang $^{2\dagger}$  Junfeng Wu $^{2,3}$  Jihan Yang $^{1}$   
+Xin Yu $^{1}$  Zehuan Yuan $^{2*}$  Bingyue Peng $^{2}$  Xiaojuan Qi $^{1\dagger*}$
+
+<sup>1</sup>The University of Hong Kong <sup>2</sup>ByteDance Inc.
+
+$^{3}$ Huazhong University of Science and Technology
+
+# Abstract
+
+Visual generative and understanding models typically rely on distinct tokenizers to process images, presenting a key challenge for unifying them within a single framework. Recent studies attempt to address this by connecting the training of VQVAE (for autoregressive generation) and CLIP (for understanding) to build a unified tokenizer. However, directly combining these training objectives has been observed to cause severe loss conflicts. In this paper, we show that reconstruction and semantic supervision do not inherently conflict. Instead, the underlying bottleneck stems from limited representational capacity of discrete token space. Building on these insights, we introduce UniTok, a unified tokenizer featuring a novel multi-codebook quantization mechanism that effectively scales up the vocabulary size and bottleneck dimension. In terms of final performance, UniTok sets a new record of 0.38 rFID and  $78.6\%$  zero-shot accuracy on ImageNet. Besides, UniTok can be seamlessly integrated into MLLMs to unlock native visual generation capability, without compromising the understanding performance. Additionally, we show that UniTok favors CFG-free generation, reducing gFID from 14.6 to 2.5 on ImageNet  $256 \times 256$  benchmark. GitHub: https://github.com/FoundationVision/UniTok.
+
+![](images/24d3288e99c12cc8d9b4b983ea4ccdf5415dd803d97462bfb369723bc48afe47.jpg)  
+Figure 1: The major challenge in unified tokenizer training. CLIP supervision cannot be easily incorporated into VQVAE training – This provides only marginal improvements in understanding performance, while drastically degrading reconstruction FID.
+
+![](images/e2e32779d8eda7fc9c15646b84602c23d59a4d20c99b46ad5155e54f524a924b.jpg)
+
+![](images/f357035c3ab7ca29285d01e54d5090b031ebef4e9ab9937423f9f04a566ee8c3.jpg)
+
+![](images/93709fddb0d411ee90ed97270d77c5226ce45906cd17ac8bfa9e70f79556b345.jpg)
+
+# 1 Introduction
+
+The advent of GPT-4o [33] highlights the immense potential of Multimodal Large Language Models (MLLMs) with native visual generation capabilities [12, 46, 62, 73, 58]. These unified models offer precise control in multimodal interactions, enabling exceptional fluency in tasks such as multi-turn image editing and visual in-context learning. However, a fundamental dilemma remains in the choice of visual tokenizers for unified MLLMs – e.g., the CLIP [38, 71] tokenizer excels in multimodal understanding but complicates generative modeling due to its high-dimensional, continuous feature
+
+space; Conversely, the discrete VQVAE [8] tokenizer fits autoregressive generation but struggles to capture essential semantics for understanding [62].
+
+In this work, we aim to design a unified visual tokenizer to bridge the gap in multimodal generation and understanding. Intuitively, this can be achieved by integrating CLIP supervision into VQVAE training, resulting in a discrete tokenizer capturing both fine-grained details and high-level semantics. However, we empirically find this training recipe confronts severe convergence issues [61] and largely falls behind the CLIP baseline in multimodal understanding (Figure 1). While prior studies commonly attribute these challenges to conflicts between semantic and pixel-level feature learning [61, 37, 58], recent progress in visual generation suggests the opposite, showing semantic regularization could benefit tokenizers in reconstruction-oriented training [63, 4, 19]. Such disparity motivates us question: Do reconstruction and semantic losses truly conflict in tokenizer training?
+
+To study the problem, we conduct a comprehensive ablation on the unified tokenizer training paradigm (Figure 3), which yields several intriguing findings: First, we show that removing reconstruction supervision, which leads to a vector-quantized CLIP model, does not improve understanding performance compared to the unified tokenizer. This observation indicates that the performance gap between unified and CLIP tokenizers mainly arises from vector quantization, rather than conflicts between learning objectives; Further analysis reveals that this gap is driven by two key factors: token factorization, which projects tokens into a lower-dimensional space for code index lookup [65], and discretization. These operations are essential for vector quantization but inevitably compromise the expressiveness of visual tokens. We thus argue that the primary bottleneck of unified tokenizers lies in the limited representational capacity of discrete token space.
+
+In light of the issue, we consider expanding the vocabulary size and latent code dimension, which allows for a closer approximation of the continuous feature space. However, extensive studies have shown that doing so could result in low codebook utilization [74, 65] and diminishing performance gains [67]. To address this, we introduce multi-codebook quantization to partition the visual token into several chunks, each discretized using a small, separate sub-codebook, akin to the multi-head attention mechanism [54]. This design exponentially scales the vocabulary size with the number of sub-codebooks, while avoiding the optimization problems of large monolithic codebooks. Besides, we replace traditional linear projection layers with adapted attention modules for token factorization, which is observed to consistently improve training stability and understanding performance.
+
+Building upon these techniques, we train a unified tokenizer called UniTok to bridge visual generation and understanding. Through extensive experiments, we demonstrate that UniTok achieves comparable or even better performance to domain-specific tokenizers: On ImageNet evaluation, UniTok records an impressive 0.38 reconstruction FID and  $78.6\%$  zero-shot accuracy at  $256 \times 256$  resolution; In building unified MLLMs, UniTok enables the MLLM with native visual generation capabilities while maintaining decent understanding performance. It outperforms the Liquid [59] baseline with a VQGAN tokenizer by  $5.5\%$  on VQAv2 [13],  $9.2\%$  on TextVQA [42], and 339 points on MME [64]; In addition, we demonstrate that semantic supervision leads to improved latent space structure for autoregressive generation, i.e., for class-conditional image generation on ImageNet  $256 \times 256$ , UniTok significantly reduces generation FID without classifier-free guidance from 14.6 to 2.5 under the LlamaGen [43] framework, which aligns with recent findings in diffusion modeling [19, 4, 63].
+
+# 2 Related Work
+
+Image Tokenization for Generation. In the domain of visual generation, image tokenization plays an important role in encoding raw pixels into compact latent features for generative modeling [53, 39]. Among a variety of tokenizers, the vector-quantized tokenizer [53] is favored for its discrete latent space and compatibility with autoregressive or masked generative models [48, 43, 3, 66]. The pioneering work VQVAE [53] initially introduced the concept of discretizing continuous tokens by mapping them to the nearest neighbors in a learnable codebook. Built on this, VQGAN [8] added perceptual loss [72] and discriminator loss [16] to improve the reconstruction quality. ViT-VQGAN [65] subsequently advanced the framework with the transformer architecture. In recent literature, considerable efforts have been devoted to developing better quantization methods such as residual quantization [18] and lookup-free quantization [67], which also constitute a focal point of this paper.
+
+Image Tokenization for Understanding. The unprecedented success of large language models (LLMs) [57, 1, 51, 47] has catalyzed the development of multimodal large language models (MLLMs)
+
+[28, 25, 31]. As a critical component of MLLMs, the selection of an effective vision tokenizer has been the subject of extensive study [55, 49]. A common choice of the vision tokenizer is the pretrained CLIP model [38], which undergoes alignment with language during its pretraining phase. While self-supervised learning models, such as DINOv2 [34], are shown to be advantageous at region-level tasks [30]. However, these tokenizers predominantly encode images into a continuous feature space, presenting challenges for uniformly modeling both vision and text tokens. To address this, some works have explored discretizing CLIP tokens [10] or employing VQVAE encoders [27, 62]. Yet, these methods have been observed to substantially impair understanding performance of MLLMs.
+
+Unified Vision-Language Models. The rise of MLLMs is not limited to the realm of visual understanding. Recent advancements have witnessed an increasing focus on unifying visual generation and understanding within one MLLM [7, 60, 46, 73, 62, 50, 21]. Specifically, a line of works employs continuous visual tokenizers for image encoding, and leverages pretrained diffusion models for image synthesis [7, 11, 44]. This approach inevitably increases model complexity and disconnects the visual sampling process from the MLLM. In contrast, another stream of research adopts VQVAE models to encode images into discrete tokens [46, 56, 62, 61, 59]. These tokens are subsequently modeled using the same cross-entropy loss that is applied to text tokens, facilitating a unified approach to multimodal learning. However, as reconstruction-oriented VQVAE does not naturally align with the LLM token space, these models typically suffer from degraded visual comprehension capabilities. Our research aligns with the second approach, with a particular focus on the tokenizer design that is suitable for both generation and understanding tasks.
+
+# 3 Method
+
+In this section, we introduce UniTok, a unified tokenizer well-suited for both visual generation and understanding tasks. We start with a unified training recipe that integrates reconstruction (VQVAE) and semantic (CLIP) supervisions (Section 3.1). However, we find that simply combining both training objectives leads to severe performance degradation, which can be mainly attributed to limited representational capacity of discrete tokens (Section 3.2). To this end, we propose multi-codebook quantization and attention projection to enhance the latent feature space and derive unified visual representations (Section 3.3). An overview of the framework is presented in Figure 2.
+
+![](images/e4a3740cba98d9b66dfe4c6a152f661c36e3ced96cc08deb4ac416eb0852f389.jpg)  
+Figure 2: An overview of UniTok. The tokenizer is trained to reconstruct the input image while aligning its discrete latent features with the text caption. For vector quantization, each visual token is split into multiple chunks, which then undergo code index lookup on corresponding sub-codebooks.
+
+# 3.1 Unified Supervision
+
+Visual generative and understanding models typically impose distinct demands on the visual tokenizers. For instance, generation emphasizes precise encoding of the visual signals, whereas understanding prioritizes capturing high-level semantics. To accommodate both requirements, we jointly train the tokenizer with  $(i)$  a VQVAE-based reconstruction loss to preserve low-level information, and  $(ii)$  an image-text contrastive loss that enhances high-level semantics of the features.
+
+To be specific, the VQVAE-based loss term  $\mathcal{L}_{\mathrm{recon}}$  consists of a pixel-level reconstruction loss  $\mathcal{L}_{\mathrm{R}}$ , a perceptual loss  $\mathcal{L}_{\mathrm{P}}$  based on the LPIPS metric [72], a discriminator loss  $\mathcal{L}_{\mathrm{G}}$  to enhance reconstruction
+
+fidelity [16], and a vector quantization loss  $\mathcal{L}_{\mathrm{VQ}}$  to minimize distance between the encoder output and its nearest code entry. It is denoted as:
+
+$$
+\mathcal {L} _ {\text {r e c o n}} = \mathcal {L} _ {\mathrm {R}} + \lambda_ {\mathrm {V Q}} \mathcal {L} _ {\mathrm {V Q}} + \lambda_ {\mathrm {P}} \mathcal {L} _ {\mathrm {P}} + \lambda_ {\mathrm {G}} \mathcal {L} _ {\mathrm {G}}, \tag {1}
+$$
+
+where  $\lambda$  is the weight factor for the corresponding loss term. The image-text contrastive loss term  $\mathcal{L}_{\mathrm{contra}}$  is basically the same as in CLIP [38]. Therefore, the final loss term can be written as:
+
+$$
+\mathcal {L} = \mathcal {L} _ {\text {r e c o n}} + \lambda_ {\text {c o n t r a}} \mathcal {L} _ {\text {c o n t r a}}. \tag {2}
+$$
+
+We simply choose  $\lambda_{\mathrm{contra}} = 1$  in this paper.
+
+# 3.2 Quantization Bottleneck
+
+Despite being augmented with CLIP supervision, we find that the unified tokenizer exhibits unsatisfactory performance in visual understanding tasks, significantly lagging behind the commonly used CLIP tokenizer. To figure out the underlying cause of this underperformance, we break down the key components involved in training a unified tokenizer, as illustrated in Figure 3. Starting with the CLIP baseline, we provide a step-by-step walk-through of all changes in following paragraphs.
+
+![](images/cc182200a28d3a44eac10b00a8abb6ca35b9a889c5a6cd3e5e5066fd82fdfcc8.jpg)  
+Figure 3: Roadmap from CLIP to UniTok. It is observed that major degradation in understanding performance comes from token factorization and discretization, rather than reconstruction supervision. The proposed multi-codebook quantization and attention projection effectively address this by scaling up the vocabulary size and bottleneck dimension. The VQA score is measured using the average score across the VQAv2, GQA, TextVQA, and POPE benchmarks. All tokenizers are trained from scratch on  $512\mathrm{m}$  image-text pairs from DataComp.
+
+Factorization. Modern VQ-tokenizers typically project continuous tokens to a lower-dimensional latent space for code index lookup (e.g. from 768-d to 8-d), known as token factorization [65]. This increases the relative density of codes by compressing the latent code space, thereby reducing quantization error. To evaluate the impact of factorization in CLIP training, we add two linear projection layers on top of the CLIP vision encoder (right before average pooling), which transforms tokens from 768-d to 16-d and then back to 768-d. Notably, vector quantization and reconstruction supervision are not included at this stage. Surprisingly, it turns out that this channel compression operation significantly compromises the expressiveness of tokens, leading to severe performance degradation in downstream VQA tasks.
+
+Discretization. Based on the implementation described above, we further introduce vector quantization to CLIP training, which maps factorized tokens to their nearest code entries. Compared to language tokenizers with vocabularies exceeding 200k entries, the vocabulary size of modern VQ-tokenizers is markedly smaller (i.e., typically ranging from 4k to 16k). Mapping continuous tokens to such a small codebook results in considerable information loss. This is validated in our experiment, which demonstrates that discretizing the factorized tokens with a 16k codebook causes an average accuracy drop of 2.1 in VQA tasks.
+
+Reconstruction Supervision. Finally, we integrate reconstruction losses into the training process to build a unified tokenizer, as outlined in Section 3.1. Previous literature suggests that loss conflict between VQVAE and CLIP is a major cause of performance degradation in joint training [61]. We observe a similar phenomenon where joint training results in sub-optimal ImageNet zero-shot classification accuracy and reconstruction FID compared to specialized training. However, surprisingly, we find that this degradation has negligible impacts on downstream understanding performance.
+
+Moreover, the degradation in classification accuracy and reconstruction FID diminishes after we improve the quantization methods (detailed in the next section). Based on these observations, we speculate that the perceived loss conflict is only a superficial issue, and the primary cause of the underperformance lies in the limited representational capacity of discrete tokens.
+
+# 3.3 UniTok
+
+A straightforward solution to breaking the quantization bottleneck could be increasing the codebook size and the latent code dimension. However, current studies on VQVAE tokenizers suggest that there is diminishing gain in scaling and the performance saturates after the codebook size reaches 16k [67, 43]. Continuing expansion results in a substantial portion of codes being rarely used or becoming 'dead' during training, which negatively impacts downstream task performance [65]. To address this, we propose multi-codebook quantization and attention projection in the following paragraphs.
+
+Multi-codebook quantization (MCQ) discretizes the latent tokens with a set of independent codebooks. Specifically, the latent vector  $f \in \mathbb{R}^d$  is first evenly split into  $n$  chunks  $\{f_1, f_2, \dots, f_n\}$ , where  $f_i \in \mathbb{R}^{\frac{d}{n}}$ . The subsequent quantization process is denoted as:
+
+$$
+\hat {f} = \operatorname {C o n c a t} \left(\mathcal {Q} \left(Z _ {1}, f _ {1}\right), \mathcal {Q} \left(Z _ {2}, f _ {2}\right), \dots , \mathcal {Q} \left(Z _ {n}, f _ {n}\right)\right) \tag {3}
+$$
+
+where  $\hat{f}$  is the discretized latent vector,  $\mathcal{Q}$  is the code index lookup operation, and  $Z_{i}$  is  $i$ -th subcodebook. Compared to conventional quantization methods, the proposed MCQ effectively scales up the vocabulary size. For instance, by increasing the number of sub-codebooks from 1 to 4, and suppose each sub-codebook contains 16k code entries, the theoretical vocabulary size exponentially increases from  $2^{14}$  to  $2^{56}$  (i.e., there are up to  $2^{14\times 4}$  possible combinations of codes for each token). As the size of each individual codebook remains constant, it circumvents the optimization problem associated with large codebooks. Besides, the dimensionality of the latent codes also scales proportionally with the number of codebooks (i.e., increasing from 16-d to 64-d in this case), which further enhances the representational capacity of discrete representations.
+
+Discussions. MCQ shares a similar concept with residual quantization (RQ) [18] in using multiple codes to quantize a token, but differs fundamentally in design philosophy: RQ follows a coarse-to-fine quantization order, whereas MCQ adopts a divide-and-conquer strategy. This distinction gives MCQ unique advantages when operating in high-dimensional latent spaces, where codes tend to become increasingly sparse. For instance, with a latent dimension of 64-d, we observe that MCQ's quantization loss is 15 to 45 times lower than that of RQ. This is because MCQ partitions the original latent space into multiple low-dimensional subspaces for quantization. Our ablation study in Table 7 further confirms the superiority of MCQ in unified tokenizer training.
+
+Attention projection. Existing VQ methods usually employ linear or convolutional projection layers for token factorization. But as shown in Figure 3, this over-simplified design fails to preserve rich semantics when compressing the feature dimensions, leading to degraded understanding performance. To alleviate this problem, we suggest adapting the multi-head attention modules for factorization. Specifically, instead of concatenating features from multiple heads after the attention calculation, we replace the concatenation operation with average pooling to realize channel compression. Figure 6 provides a detailed illustration of the adaptation. Despite its simplicity, we find this design effectively strengthens the representational power of factorized tokens and stabilizes training.
+
+# 3.4 Unified MLLM
+
+We proceed to develop a unified multimodal model with UniTok. Particularly, we leverage the unified framework introduced in Liquid [59], which models (discrete-valued) vision and language sequences with a universal next-token prediction loss. But instead of learning the visual codebook from scratch, we reuse code embeddings of UniTok by projecting them to the MLLM token space with an MLP projector. Notably, despite UniTok encodes an image into  $H \times W \times K$  codes (where  $K$  represents the number of sub-codebooks), we simplify this for MLLM input by merging every  $K$  consecutive codes into a single visual token. Similarly, when it comes to visual token prediction, we make each token autoregressively predict the next  $K$  codes, using a depth transformer head as implemented in RQ-Transformer [18] and VILA-U [61]. This design maintains efficiency for visual generation in the context of multi-codebooks.
+
+# 4 Experiments
+
+# 4.1 Implementation Details
+
+Tokenizer Setup. Leading VQVAE tokenizers predominantly adopt the CNN architecture, while ViT is preferred in CLIP training for its scalability. To take advantage of both, we choose a hybrid architecture, ViTamin-L/16 [5], to instantiate UniTok. We configure UniTok with eight sub-codebooks, each containing 4,096 code entries and a latent dimension set to 8-d (the global latent dimension is thus 64-d). The discriminator is initialized with pretrained DINOv2-S [34]. We train the tokenizer for one epoch on the public dataset DataComp-1B [9] consisting of 1.28B image-text pairs, with all images resized to  $256 \times 256$  resolution and a global batch size of 16k. The learning rate is set to 1e-3 for the tokenizer and 2e-4 for the discriminator. Besides, we prepare two settings for evaluation: one with pretrained CLIP weight initialization and one with random initialization (the default setting).
+
+MLLM Setup. We instantiate a unified MLLM described in Section 3.4 with the Llama-2-7B base model [52]. Following Liquid, we first pretrain the model on a mix of multimodal data, which is composed of 10M language data from DCLM [22], 30M internal MidJourney-style synthetic data, and 30M re-captioned image-text pairs from COYO [32] and Laion [41]. Subsequently, we finetune the model on 1.5M text-to-image data and 1.5M multimodal instruction tuning data introduced in Mini-Gemini [23]. Specifically, the learning rate is set to 5e-5 in the pretraining stage and 2e-5 in the finetuning stage. For visual understanding evaluation, we report results on standard VQA benchmarks including VQAv2 [13], GQA [14], TextVQA [42], POPE [24], MME [64], and MM-Vet [70]. For visual generation evaluation, we report results on GenAI-Bench [26] and MJHQ-30K [20].
+
+# 4.2 Tokenizer Comparison
+
+We benchmark UniTok on ImageNet using two primary metrics: Fréchet Inception Distance (FID) to evaluate reconstruction quality, and top-1 zero-shot accuracy to assess image-text alignment. The results are presented in Table 1. To provide a fair comparison with tokenizers trained on small datasets, we also train a version of UniTok on OpenImages [17] solely with reconstruction supervision. It can be seen that UniTok excels in reconstruction quality compared to both unified and domain-specific tokenizers, recording an impressive  $0.38\mathrm{rFID}$  on ImageNet with  $16\times$  downsampling ratio. As a discrete tokenizer, UniTok even surpasses the continuous VAE tokenizer from Stable Diffusion v2.1 [40], showcasing the superiority of the proposed multi-codebook quantization. For the perception performance, we observe that randomly initialized UniTok demonstrates suboptimal zero-shot classification accuracy. This is expected as current training schedule (i.e., one epoch on 1.28B samples) is insufficient for CLIP training to fully converge. It can be seen that initializing the model with pretrained CLIP weights largely alleviates the problem, boosting the zero-shot accuracy from  $70.8\%$  to  $78.6\%$ . In complement to quantitative results, we provide examples of reconstructed images in Figure 4.
+
+Table 1: Comparison on ImageNet reconstruction FID and zero-shot classification accuracy. rFID is measured at  $256\times 256$  resolution with  $16\times$  downsample ratio.  $\dagger$  indicates model using pretrained CLIP weights for initialization. \* indicates model trained on OpenImages.
+
+<table><tr><td>Method</td><td>#Tokens</td><td>rFID ↓</td><td>Accuracy</td></tr><tr><td colspan="4">VQVAE Model</td></tr><tr><td>VQ-GAN* [8]</td><td>256</td><td>4.98</td><td>-</td></tr><tr><td>RQ-VAE [18]</td><td>256</td><td>1.30</td><td>-</td></tr><tr><td>VAR* [48]</td><td>680</td><td>0.90</td><td>-</td></tr><tr><td>UniTok*</td><td>256</td><td>0.33</td><td>-</td></tr><tr><td colspan="4">CLIP Model</td></tr><tr><td>CLIP [38]</td><td>256</td><td>-</td><td>76.2</td></tr><tr><td>SigLIP [71]</td><td>256</td><td>-</td><td>80.5</td></tr><tr><td>ViTamin [5]</td><td>256</td><td>-</td><td>81.2</td></tr><tr><td colspan="4">Unified Model</td></tr><tr><td>TokenFlow†[37]</td><td>680</td><td>1.37</td><td>-</td></tr><tr><td>VILA-U†[61]</td><td>256</td><td>1.80</td><td>73.3</td></tr><tr><td>UniTok</td><td>256</td><td>0.41</td><td>70.8</td></tr><tr><td>UniTok†</td><td>256</td><td>0.38</td><td>78.6</td></tr></table>
+
+# 4.3 Class-Conditional Image Generation
+
+Recent studies on diffusion models indicate that injecting semantics into VAE training leads to a better-structured latent space, considerably enhancing guidance-free generation performance [63, 4, 19]. To evaluate whether UniTok possesses similar properties, we test it within the LlamaGen framework for class-conditional image generation. As shown in Table 2, UniTok reduces the FID by 12.11 compared to the VQGAN baseline in CFG-free generation, under the same generator setup. This implies that UniTok learns a more structured code distribution, benefiting autoregressive modeling.
+
+![](images/73210aef10fb2b885d793ed79c8965106e0e16728a202b9fd0102279b2d71574.jpg)
+
+![](images/5323887858f10c132d21133378866105238b6f77266eb6d93c71305f42e0c479.jpg)
+
+![](images/5ccecd61b1b0444eb5f362dcbaec910c171d28b57d68939f02c703ec04064d38.jpg)  
+Original Images
+
+![](images/2c9622fd6ad934e2de058f49c3cf584e96a048c8fb753e14e2fb02a94c22792a.jpg)
+
+![](images/bcebc10c5f3469aafe8ef8fa5b077d81f4389dbacacd1c0e3c16107c6aa41118.jpg)
+
+![](images/9f8f9dff8b814187481c22c2c81b80aa4ff6ad5cf18a94aee4e5913b9accdb30.jpg)  
+Figure 4: Qualitative results on image reconstruction in a resolution of  $256 \times 256$ .
+
+![](images/e7be833009d85fd3bff50e3f18e5edc98d7cd5795a12587558328a73c714be69.jpg)  
+Reconstructed Images
+
+![](images/481f5f1c989f47709c98e94d98ead8a016e3281f54e4c07e284facaa2780fe40.jpg)
+
+![](images/3bd9335edf05aa99d97d9ebb3ed3e8eb27c1b6d3a88c8d9dd12f5cf49c56321a.jpg)
+
+![](images/6443277a0e6980cb5ff5b4d58eb0a508cb9adc1ed891a0e7d116ab318053459c.jpg)
+
+Table 2: Class-conditional image generation results on ImageNet  ${256} \times  {256}$  . †: VQGAN tokenizer from LlamaGen. ‡: Images are generated at  ${384} \times  {384}$  resolution and then resized to  ${256} \times  {256}$  for evaluation. 'Pre.': precision; 'Rec.': recall; 'CFG': classifier-free-guidance.  
+
+<table><tr><td rowspan="2">Tokenizer</td><td rowspan="2">rFID</td><td rowspan="2">Generator</td><td rowspan="2">#Params.</td><td colspan="4">Generation w/o CFG</td><td colspan="4">Generation w/ CFG</td></tr><tr><td>gFID↓</td><td>IS↑</td><td>Pre.</td><td>Rec.</td><td>gFID↓</td><td>IS↑</td><td>Pre.</td><td>Rec.</td></tr><tr><td colspan="12">Diffusion Models</td></tr><tr><td>SD-VAE [40]</td><td>0.61</td><td>DiT [35]</td><td>675M</td><td>9.62</td><td>121.5</td><td>0.67</td><td>0.67</td><td>2.27</td><td>278.2</td><td>0.83</td><td>0.57</td></tr><tr><td>VAVAE [63]</td><td>0.28</td><td>LightningDiT [63]</td><td>675M</td><td>2.17</td><td>205.6</td><td>0.77</td><td>0.65</td><td>1.35</td><td>295.3</td><td>0.79</td><td>0.65</td></tr><tr><td colspan="12">Masked Generative Models</td></tr><tr><td>LFQ [67]</td><td>0.9</td><td>MAGVIT-v2 [67]</td><td>307M</td><td>3.07</td><td>213.1</td><td>-</td><td>-</td><td>1.91</td><td>324.3</td><td>-</td><td>-</td></tr><tr><td>TiTok-L [69]</td><td>2.21</td><td>MaskGIT [3]</td><td>177M</td><td>3.15</td><td>173.0</td><td>-</td><td>-</td><td>2.77</td><td>199.8</td><td>-</td><td>-</td></tr><tr><td colspan="12">Autoregressive Models</td></tr><tr><td>VQGAN†</td><td>2.19</td><td>LlamaGen‡[43]</td><td>1.4B</td><td>14.65</td><td>86.3</td><td>0.63</td><td>0.68</td><td>2.34</td><td>253.9</td><td>0.81</td><td>0.60</td></tr><tr><td>UniTok (Ours)</td><td>0.41</td><td>LlamaGen [43]</td><td>1.4B</td><td>2.51</td><td>216.7</td><td>0.82</td><td>0.57</td><td>2.77</td><td>227.5</td><td>0.81</td><td>0.57</td></tr></table>
+
+# 4.4 Unified Understanding and Generation
+
+Understanding Performance. We evaluate the understanding performance of UniTok on diverse VQA benchmarks in Table 3. Our unified MLLM showcases clear advantages when compared to other unified models that also utilize a discrete visual tokenizer. Specifically, UniTok significantly outperforms the Chameleon model, which relies on a traditional VQVAE tokenizer, by  $7.2\%$  higher accuracy on VQAv2. Additionally, it surpasses VILA-U, another model with a unified tokenizer, by  $3.3\%$  in accuracy on the TextVQA benchmark and by a notable margin of 112 points on the MME-Perception scores. Furthermore, we can see that UniTok largely narrows the performance gap with MLLMs that incorporate continuous visual tokenizers. These strong results confirm the candidacy of UniTok as a unified visual tokenizer for multimodal models.
+
+Generation Performance. Table 4 presents the text-to-image generation performance of our unified MLLM on the GenEval benchmark. We show that UniTok not only outperforms most of the unified MLLMs, but also demonstrates competitive performance against domain experts (diffusion models) trained on billions of images. Besides, UniTok achieves non-trivial improvements over Liquid while using exactly the same set of text-to-image training data, highlighting the importance of a unified tokenizer. We also provide results on GenAI-Bench in Table 10 and Table 11 in Appendix.
+
+We further evaluate the quality of images generated by our model on the MJHQ-30K benchmark, details of which are presented in Table 5. Notably, as this benchmark primarily relies on the FID score
+
+Table 3: Comparison with unified multi-modal large language models on VQA benchmarks.  
+
+<table><tr><td>Method</td><td>LLM</td><td>Token Type</td><td>Res.</td><td>VQAv2</td><td>GQA</td><td>TextVQA</td><td>POPE</td><td>MME</td><td>MM-Vet</td></tr><tr><td>Emu [45]</td><td>Llama-13B</td><td>Continuous</td><td>224</td><td>52.0</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>LaVIT [15]</td><td>Llama-7B</td><td>Continuous</td><td>224</td><td>66.0</td><td>46.8</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DreamLLM [7]</td><td>Vicuna-7B</td><td>Continuous</td><td>224</td><td>72.9</td><td>-</td><td>41.8</td><td>-</td><td>-</td><td>26.6</td></tr><tr><td>Unified-IO 2 [29]</td><td>6.8B from scratch</td><td>Continuous</td><td>384</td><td>79.4</td><td>-</td><td>-</td><td>87.7</td><td>-</td><td>-</td></tr><tr><td>Janus [58]</td><td>DeepSeek-1.3B</td><td>Continuous</td><td>384</td><td>77.3</td><td>59.1</td><td>-</td><td>87.0</td><td>1338</td><td>34.3</td></tr><tr><td>CM3Leon [68]</td><td>7B from scratch</td><td>Discrete</td><td>256</td><td>47.6</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>LWM [27]</td><td>Llama-2-7B</td><td>Discrete</td><td>256</td><td>55.8</td><td>44.8</td><td>18.8</td><td>75.2</td><td>-</td><td>-</td></tr><tr><td>Show-o [62]</td><td>Phi-1.5-1.3B</td><td>Discrete</td><td>256</td><td>59.3</td><td>48.7</td><td>-</td><td>73.8</td><td>948</td><td>-</td></tr><tr><td>Chameleon [46]</td><td>34B from scratch</td><td>Discrete</td><td>512</td><td>69.6</td><td>-</td><td>-</td><td>-</td><td>-</td><td></td></tr><tr><td>Liquid [59]</td><td>Gemma-7B</td><td>Discrete</td><td>512</td><td>71.3</td><td>58.4</td><td>42.4</td><td>81.1</td><td>1119</td><td>-</td></tr><tr><td>VILA-U [61]</td><td>Llama-2-7B</td><td>Discrete</td><td>256</td><td>75.3</td><td>58.3</td><td>48.3</td><td>83.9</td><td>1336</td><td>27.7</td></tr><tr><td>UniTok</td><td>Llama-2-7B</td><td>Discrete</td><td>256</td><td>76.8</td><td>61.1</td><td>51.6</td><td>83.2</td><td>1448</td><td>33.9</td></tr></table>
+
+Table 4: Comparison with other visual generation methods on the GenEval benchmark.  
+
+<table><tr><td>Method</td><td>Type</td><td>#Data</td><td>Single Obj.</td><td>Two Obj.</td><td>Counting</td><td>Colors</td><td>Position</td><td>Color Attri.</td><td>Overall↑</td></tr><tr><td>SD v2.1 [39]</td><td>Diffusion</td><td>2000M</td><td>0.98</td><td>0.51</td><td>0.44</td><td>0.85</td><td>0.07</td><td>0.17</td><td>0.50</td></tr><tr><td>SD-XL [36]</td><td>Diffusion</td><td>2000M</td><td>0.98</td><td>0.74</td><td>0.39</td><td>0.85</td><td>0.15</td><td>0.23</td><td>0.55</td></tr><tr><td>DALL-E 3 [2]</td><td>Diffusion</td><td>-</td><td>0.96</td><td>0.87</td><td>0.47</td><td>0.83</td><td>0.43</td><td>0.45</td><td>0.67</td></tr><tr><td rowspan="2">Show-o [62]</td><td rowspan="2">Discrete Diff.</td><td>36M</td><td>0.95</td><td>0.52</td><td>0.49</td><td>0.82</td><td>0.11</td><td>0.28</td><td>0.53</td></tr><tr><td>2.0B</td><td>0.98</td><td>0.80</td><td>0.66</td><td>0.84</td><td>0.31</td><td>0.50</td><td>0.68</td></tr><tr><td>LWM [27]</td><td>Autoregressive</td><td>-</td><td>0.93</td><td>0.41</td><td>0.46</td><td>0.79</td><td>0.09</td><td>0.15</td><td>0.47</td></tr><tr><td>Janus [58]</td><td>Autoregressive</td><td>-</td><td>0.97</td><td>0.68</td><td>0.30</td><td>0.84</td><td>0.46</td><td>0.42</td><td>0.61</td></tr><tr><td>Liquid [59]</td><td>Autoregressive</td><td>30M</td><td>0.98</td><td>0.73</td><td>0.32</td><td>0.76</td><td>0.17</td><td>0.37</td><td>0.55</td></tr><tr><td>UniTok</td><td>Autoregressive</td><td>30M</td><td>0.99</td><td>0.71</td><td>0.36</td><td>0.79</td><td>0.26</td><td>0.45</td><td>0.59</td></tr></table>
+
+for evaluation, high-resolution images are preferred because they potentially capture more fine-grained details. Despite this makes FID across different resolutions less comparable, we show that our model achieves impressive performance even at the smallest resolution, showcasing its ability to generate high-quality, detail-rich images.
+
+We present some examples of the images generated by our model in Figure 5, using text prompts sampled from MJHQ-30K. The visualization results demonstrate our model is capable of synthesizing photorealistic and visually appealing images. Moreover, the model is able to comprehend a wide spectrum
+
+of concepts, such as 'Vincent van Gogh painting style' and 'bitcoin', and flexibly combine these concepts to synthesize creative images.
+
+Table 5: Results on MJHQ-30K.  
+
+<table><tr><td>Method</td><td>Type</td><td>Res.</td><td>FID↓</td></tr><tr><td>SD-XL [36]</td><td>Diffusion</td><td>1024</td><td>9.55</td></tr><tr><td>PixArt [6]</td><td>Diffusion</td><td>1024</td><td>6.14</td></tr><tr><td>Playground [20]</td><td>Diffusion</td><td>1024</td><td>4.48</td></tr><tr><td>Liquid [59]</td><td>Autoregressive</td><td>512</td><td>5.47</td></tr><tr><td>Janus [58]</td><td>Autoregressive</td><td>384</td><td>10.10</td></tr><tr><td>LWM [27]</td><td>Autoregressive</td><td>256</td><td>17.77</td></tr><tr><td>Show-o [62]</td><td>Discrete Diff.</td><td>256</td><td>15.18</td></tr><tr><td>VILA-U [61]</td><td>Autoregressive</td><td>256</td><td>12.81</td></tr><tr><td>UniTok</td><td>Autoregressive</td><td>256</td><td>7.46</td></tr></table>
+
+# 4.5 Ablation Studies
+
+Impact of Supervision Types. To ablate the impact of contrastive and reconstruction losses in UniTok training, we conduct experiments on tokenizers trained with different supervision types, as shown in Table 6. It is worth noting that all the tokenizers are vector-quantized even though some do not have reconstruction supervision. First, we show that reconstruction-oriented tokenizer significantly lags behind tokenizers with contrastive supervision in visual understanding performance. This observation evidences the limitations of traditional VQVAE. Second, we demonstrate that reconstruction and contrastive training objectives do not inherently conflict, or can be addressed by enhancing discrete feature space. With multi-codebook quantization, the jointly trained tokenizer not only exhibits understanding performance on par with the tokenizer trained solely with contrastive loss, but also slightly improves generation performance over the reconstruction-oriented tokenizer.
+
+MCQ v.s. RQ. Following discussions in Section 3.3, we provide an apple-to/apple comparison between multi-codebook quantization and residual quantization in Table 7. For fair comparisons, we
+
+![](images/9b74248716873f198241c048072f2a0cd3390a148e305cde73b90e1a78220dca.jpg)  
+Figure 5: Images generated in a resolution of  $256 \times 256$  with our unified MLLM.
+
+Table 6: Impact of different supervision types on downstream generation and understanding performance. The rFID and gFID are measured on the ImageNet  $(256\times 256)$  validation set. LlamaGen-L [43] is adopted as the generator for gFID evaluation.  
+
+<table><tr><td rowspan="2">Supervision</td><td colspan="2">Generation</td><td colspan="6">Understanding</td></tr><tr><td>rFID ↓</td><td>gFID ↓</td><td>VQAv2</td><td>GQA</td><td>SciQA</td><td>TextVQA</td><td>POPE</td><td>MME</td></tr><tr><td>Contrastive</td><td>-</td><td>-</td><td>68.95</td><td>56.89</td><td>65.64</td><td>49.89</td><td>82.34</td><td>1373</td></tr><tr><td>Reconstruction</td><td>0.82</td><td>3.59</td><td>56.33</td><td>47.53</td><td>63.26</td><td>43.65</td><td>77.09</td><td>902</td></tr><tr><td>Recon. + Contra.</td><td>0.72</td><td>3.26</td><td>69.14</td><td>56.06</td><td>65.25</td><td>49.22</td><td>81.42</td><td>1333</td></tr></table>
+
+directly implement RQ on our codebase, keeping all the training settings the same as UniTok. Both tokenizers are trained on a 512M subset of DataComp-1B. Notably, unlike MCQ, RQ by default uses a shared large codebook. To keep the global codebook size the same as UniTok, we set the codebook size of RQ to 32768. It can be seen that RQ demonstrates inferior reconstruction performance and lower classification accuracy compared to MCQ under the high bottleneck dimension (64-d) setting.
+
+Number of Sub-Codebooks. To gain deeper insights into multi-codebook quantization, we evaluate how tokenizer performance changes with the number of sub-codebooks in Table 8. Specifically, the size of a codebook is denoted as  $A \times B$ , where  $A$  is the number of sub-codebook and  $B$  is the size of sub-codebook. For rFID evaluation, we train the tokenizer solely with reconstruction loss on OpenImages [17], and evaluated it on ImageNet (256 × 256) validation set. While for ImageNet zero-shot accuracy evaluation, the tokenizer is trained on DataComp-1B 128m subset using only contrastive loss. Given a constant global codebook size, we see that increasing the number of subcodebooks consistently improves reconstruction FID and classification accuracy. This indicates that MCQ generally benefits vector-quantized models, independent of the training objectives.
+
+Table 7: MCQ v.s. RQ.  
+
+<table><tr><td>Method</td><td>Code Shape</td><td>Code Dim.</td><td>rFID↓</td><td>Accuracy</td></tr><tr><td>RQ</td><td>16×16×8</td><td>64</td><td>3.46</td><td>58.8</td></tr><tr><td>MCQ</td><td>16×16×8</td><td>64</td><td>0.55</td><td>63.7</td></tr><tr><td colspan="5">Codebook / Vocabulary</td></tr><tr><td colspan="5">1×16384 / 214</td></tr><tr><td colspan="5">2×8192 / 226</td></tr><tr><td colspan="5">4×4096 / 248</td></tr><tr><td colspan="5">8×2048 / 288</td></tr><tr><td colspan="5">Accuracy</td></tr><tr><td colspan="5">41.0%</td></tr><tr><td colspan="5">43.9%</td></tr><tr><td colspan="5">44.7%</td></tr><tr><td colspan="5">46.1%</td></tr></table>
+
+Table 8: Ablation on number of sub-codebooks.
+
+CLIP Weight Initialization. We notice that higher ImageNet accuracy does not guarantee superior downstream performance. In Table 9, we ablate the impact of CLIP weight initialization on visual understanding performance. Specifically, we adopt the classic LLaVA framework for evaluation,
+
+replacing the original CLIP tokenizer with UniTok while keeping all other the training settings unchanged. One tokenizer is initialized with the pretrained ViTamin-L-256 [5] weights, while the other is randomly initialized. To our surprise, UniTok that is trained from scratch surpasses the one initialized with pretrained CLIP weights, despite the latter actually achieves better zero-shot classification accuracy. This suggests downstream VQA performance may not be highly correlated with ImageNet classification accuracy. More importantly, it also implies that CLIP weight initialization may serve as a negative prior for unified tokenizers, as the unified visual feature space could drastically differ from CLIP feature space.
+
+Table 9: Comparison of different initialization methods under the LLaVA framework.  $\dagger$  indicates the model uses CLIP weights for initialization. We highlight the default setting of UniTok in gray.  
+
+<table><tr><td>Tokenizer</td><td>VQAv2</td><td>GQA</td><td>TextVQA</td><td>POPE</td><td>MME</td></tr><tr><td>UniTok†</td><td>69.9</td><td>56.2</td><td>49.3</td><td>81.2</td><td>1331</td></tr><tr><td>UniTok</td><td>72.4</td><td>58.2</td><td>51.6</td><td>82.4</td><td>1392</td></tr></table>
+
+# 5 Limitations and Conclusion
+
+This paper studies unified visual tokenization for generation and understanding, which serves as the cornerstone of unified multimodal large language models. We investigate the training paradigm of unified tokenizers and identify that the current challenge in unification mainly arises from the limited representational power of discrete tokens. To address this limitation, we introduce multi-codebook quantization and attention projection to build a unified tokenizer called UniTok. We show that UniTok excels in downstream visual generation and understanding tasks. The ablation study further reveals that discriminative and generative representation learning does not inherently conflict. We hope our findings could inspire future research in this domain.
+
+However, due to limited computational resources, UniTok is only trained for one epoch, which is not sufficient for CLIP-based semantic representation learning. We believe extending the training schedule could further benefit the tokenizer, especially in understanding performance.
+
+Acknowledgments This work has been supported by the National Key R&D Program of China (Grant No. 2022YFB3608300), Hong Kong Research Grant Council - Early Career Scheme (Grant No. 27209621), General Research Fund Scheme (Grant No. 17202422, 17212923, 17215025) Themebased Research (Grant No. T45-701/22-R) and Shenzhen Science and Technology Innovation Commission (SGDX20220530111405040). We sincerely thank Seng Ye for the insightful discussions and valuable contributions to this project.
+
+# References
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+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes].
+
+Justification: We have included the motivation, empirical findings, technical innovations, and major achievements of our work in the abstract and introduction.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: Please see 'Limitations and Conclusion' section.
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [NA]
+
+Justification: This paper does not include theoretical results.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
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+
+# 4. Experimental result reproducibility
+
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+
+# Answer: [Yes]
+
+Justification: We have included the implementation details of our tokenizer and MLLM (such as training data, hyperparameter configurations, testing framework, etc) in the 'Experiments' part.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
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+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: Most code and data for this paper are open-source, except the unified MLLM training data, which includes some internal, non-public data. However, our main focus - tokenizer training - uses only publicly available data.
+
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+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: Please see our code for the full set of hyperparameter configurations.
+
+Guidelines:
+
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+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [No]
+
+Justification: We mainly evaluate our methods using ImageNet FID and zero-shot accuracy, metrics known for low variance. Following prior work, we omit error bars in our experiments.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
+
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: It takes roughly 50 hours with the equivalent computing power of 256 A100 GPUs to train the tokenizer.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
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+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: Please see Appendix.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.
+
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+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [No]
+
+Justification: We will add safeguards in our future release.
+
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+
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+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: We strictly followed the licenses to use other assets (like DataComp) with credits.
+
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+
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+
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+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
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+
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+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
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+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: This paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
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+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components.
+
+Guidelines:
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+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Attention Projection Modules
+
+Figure 6 illustrates the adaptations we made to the tradition MHA module, which enables channel compression and expansion in token factorization.
+
+![](images/b328deb3bbde08dc3d61670755469c8ab73cb3e926e54a645e099e9b951f739f.jpg)  
+Channel Compression Block
+
+![](images/19e71bb401230a41d2981c126a0ee85491ba0778e220f5f22159dfce56886cd5.jpg)  
+Channel Expansion Block  
+Figure 6: Modified attention blocks for factorization. Modules in yellow indicate a change in the number of channels.  $C$  and  $c$  stand for the channel dimension,  $h$  is the number of heads in the multi-head attention module.  $C = h \times c$ .
+
+# B More Generation Results
+
+We provide the results on GenAI-Bench in Table 10 and Table 11. UniTok consistently delivers superior generation performance on this benchmark.
+
+Table 10: Comparison with other visual generation methods on GenAI-Bench (basic prompts).  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Type</td><td rowspan="2">#Training Images</td><td rowspan="2">Attribute↑</td><td rowspan="2">Scene↑</td><td colspan="3">Relation↑</td><td rowspan="2">Overall↑</td></tr><tr><td>Spatial</td><td>Action</td><td>Part</td></tr><tr><td>SD v2.1 [39]</td><td>Diffusion</td><td>2000M</td><td>0.80</td><td>0.79</td><td>0.76</td><td>0.77</td><td>0.80</td><td>0.78</td></tr><tr><td>SD-XL [36]</td><td>Diffusion</td><td>2000M</td><td>0.84</td><td>0.84</td><td>0.82</td><td>0.83</td><td>0.89</td><td>0.83</td></tr><tr><td>Midjourney v6</td><td>Diffusion</td><td>-</td><td>0.88</td><td>0.87</td><td>0.87</td><td>0.87</td><td>0.91</td><td>0.87</td></tr><tr><td>DALL-E 3 [2]</td><td>Diffusion</td><td>-</td><td>0.91</td><td>0.90</td><td>0.92</td><td>0.89</td><td>0.91</td><td>0.90</td></tr><tr><td>Show-o [62]</td><td>Discrete Diff.</td><td>36M</td><td>0.72</td><td>0.72</td><td>0.70</td><td>0.70</td><td>0.75</td><td>0.70</td></tr><tr><td>LWM [27]</td><td>Autoregressive</td><td>-</td><td>0.63</td><td>0.62</td><td>0.65</td><td>0.63</td><td>0.70</td><td>0.63</td></tr><tr><td>VILA-U [61]</td><td>Autoregressive</td><td>15M</td><td>0.78</td><td>0.78</td><td>0.77</td><td>0.78</td><td>0.79</td><td>0.76</td></tr><tr><td>Liquid [59]</td><td>Autoregressive</td><td>30M</td><td>0.84</td><td>0.86</td><td>0.81</td><td>0.83</td><td>0.91</td><td>0.83</td></tr><tr><td>UniTok</td><td>Autoregressive</td><td>30M</td><td>0.85</td><td>0.87</td><td>0.86</td><td>0.86</td><td>0.89</td><td>0.85</td></tr></table>
+
+Table 11: Comparison with other visual generation methods on GenAI-Bench (advanced prompts).  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Type</td><td rowspan="2">#Training Images</td><td rowspan="2">Count↑</td><td rowspan="2">Differ↑</td><td rowspan="2">Compare↑</td><td colspan="2">Logical↑</td><td rowspan="2">Overall↑</td></tr><tr><td>Negate</td><td>Universal</td></tr><tr><td>SD v2.1 [39]</td><td>Diffusion</td><td>2000M</td><td>0.68</td><td>0.70</td><td>0.68</td><td>0.54</td><td>0.64</td><td>0.62</td></tr><tr><td>SD-XL [36]</td><td>Diffusion</td><td>2000M</td><td>0.71</td><td>0.73</td><td>0.69</td><td>0.50</td><td>0.66</td><td>0.63</td></tr><tr><td>Midjourney v6</td><td>Diffusion</td><td>-</td><td>0.78</td><td>0.78</td><td>0.79</td><td>0.50</td><td>0.76</td><td>0.69</td></tr><tr><td>DALL-E 3 [2]</td><td>Diffusion</td><td>-</td><td>0.82</td><td>0.78</td><td>0.82</td><td>0.48</td><td>0.80</td><td>0.70</td></tr><tr><td>Show-o [62]</td><td>Discrete Diff.</td><td>36M</td><td>0.70</td><td>0.62</td><td>0.71</td><td>0.51</td><td>0.65</td><td>0.60</td></tr><tr><td>LWM [27]</td><td>Autoregressive</td><td>-</td><td>0.59</td><td>0.58</td><td>0.54</td><td>0.49</td><td>0.52</td><td>0.53</td></tr><tr><td>VILA-U [61]</td><td>Autoregressive</td><td>15M</td><td>0.70</td><td>0.71</td><td>0.74</td><td>0.53</td><td>0.66</td><td>0.64</td></tr><tr><td>Liquid [59]</td><td>Autoregressive</td><td>30M</td><td>0.76</td><td>0.73</td><td>0.74</td><td>0.46</td><td>0.74</td><td>0.65</td></tr><tr><td>UniTok</td><td>Autoregressive</td><td>30M</td><td>0.76</td><td>0.76</td><td>0.79</td><td>0.46</td><td>0.73</td><td>0.67</td></tr></table>
+
+# C Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
\ No newline at end of file
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+# UniTraj: Learning a Universal Trajectory Foundation Model from Billion-Scale Worldwide Traces
+
+Yuanshao Zhu $^{1,2,3,*}$ , James Jianqiao Yu $^{4,\dagger}$ , Xiangyu Zhao $^{2,\dagger}$ , Xun Zhou $^{4}$ , Liang Han $^{4}$ , Xuetao Wei $^{1}$ , Yuxuan Liang $^{3,\dagger}$
+
+$^{1}$  Southern University of Science and Technology,  $^{2}$  City University of Hong Kong
+
+<sup>3</sup> The Hong Kong University of Science and Technology (Guangzhou)
+
+$^{4}$  Harbin Institute of Technology, Shenzhen
+
+yuanshao@ieee.org, jqyu@ieee.org, xianzhao@cityu.edu.hk
+
+zhouxun2023@hit.edu.cn, han.liang@hit.edu.cn
+
+weixt@sustech.edu.cn, yuxliang@outlook.com
+
+# Abstract
+
+Building a universal trajectory foundation model is a promising solution to address the limitations of existing trajectory modeling approaches, such as task specificity, regional dependency, and data sensitivity. Despite its potential, data preparation, pre-training strategy development, and architectural design present significant challenges in constructing this model. Therefore, we introduce UniTraj, a Universal Trajectory foundation model that aims to address these limitations through three key innovations. First, we construct WorldTrace, an unprecedented dataset of 2.45 million trajectories with billions of GPS points spanning 70 countries, providing the diverse geographic coverage essential for region-independent modeling. Second, we develop novel pre-training strategies—Adaptive Trajectory Resampling and Self-supervised Trajectory Masking—that enable robust learning from heterogeneous trajectory data with varying sampling rates and quality. Finally, we tailor a flexible model architecture to accommodate a variety of trajectory tasks, effectively capturing complex movement patterns to support broad applicability. Extensive experiments across multiple tasks and real-world datasets demonstrate that UniTraj consistently outperforms existing methods, exhibiting superior scalability, adaptability, and generalization, with WorldTrace serving as an ideal yet non-exclusive training resource. The implementation codes and full dataset are available at https://github.com/Yasoz/UniTraj.
+
+# 1 Introduction
+
+Trajectory data, as the digital footprints of human movement, is becoming a fundamental data source for understanding mobility patterns and transforming urban intelligence [3]. These spatiotemporal sequences unlock critical insights across diverse applications: from optimizing transportation networks that alleviate congestion in megacities, enhancing location-based services that personalize user experiences [18, 2], to powering logistics systems that determine the efficiency of global supply chains [12, 24, 36]. Despite their significance, extracting meaningful patterns (from statistical methods to deep learning [25]) of trajectory data presents profound challenges due to their inherent complexity, varying lengths, irregular sampling rates, and region-specific characteristics.
+
+As trajectory data continues to expand exponentially, three critical limitations in current approaches have become increasingly apparent: (1) Task Specificity: Current approaches are typically designed
+
+![](images/fe0e294753ec471d040ffa8e1f650a1f65d696110dfbd59e59407c7063636aca.jpg)  
+Figure 1: Overview of this work, we propose a trajectory foundation model and also collect a worldwide trajectory dataset. The pre-trained UniTraj can be used as a backbone while adapters are trained for different regions and tasks.
+
+for single-purpose applications, limiting their generalizability and requiring substantial re-engineering for new tasks. (2) Regional Dependency: Many models are developed and trained on data from specific geographic regions, making them ineffective when applied to different locations with distinct mobility patterns and infrastructure. (3) Data Sensitivity: Real-world trajectory data often contains noise, irregular sampling, or missing entries, making models highly sensitive to data quality and necessitating extensive preprocessing, which reduces robustness. These limitations point to a fundamental gap: the absence of a universal foundation model capable of operating across diverse tasks, geographic regions, and data quality levels. While foundation models have revolutionized NLP [1, 7] and CV [9, 13] by providing versatile, pre-trained architectures that generalize across domains, trajectory analysis has not yet benefited from this paradigm shift. Creating this model would transform trajectory intelligence from its current fragmented state to a unified approach with significantly enhanced generalization capabilities [27, 39].
+
+However, building such a model presents key challenges: (1) Data preparation: The first challenge is to prepare a sufficiently diverse trajectory dataset that spans different geographic regions and appropriate sampling rates. Existing datasets lack sufficient geographic diversity and scale, also limited by proprietary restrictions and collection costs. This data scarcity severely hampers model generalizability and cross-regional research efforts on a global scale. (2) Pre-training Strategy: Developing robust and scalable pre-training strategies is another challenge. Real-world trajectory data exhibits heterogeneous quality with noising, varying sampling rates, and missing points. Effective pre-training must accommodate these inconsistencies while learning robust representations that transfer across diverse contexts. (3) Model Design: The last challenge involves selecting and tailoring an effective model architecture. A universal foundation model requires an architecture that balances adaptability across tasks with computational efficiency, capturing complex spatio-temporal dependencies without overfitting to specific regional information or trajectory patterns.
+
+To address these challenges, we introduce Universal Trajectory foundation model (UniTraj) supported by three key innovations. As shown in Figure 1, we firstly construct WorldTrace, the first trajectory dataset with large-scale, high-quality, and global distribution, which provides the essential foundation for region-agnostic modeling. Then, we design several novel pre-training strategies—adaptive resampling and self-supervised masking—that enable robust learning from heterogeneous trajectory data with varying sampling rates and quality, bridging the gap between regional variations and inconsistent data. Finally, we design a flexible model architecture that captures complex spatio-temporal dependencies while adapting to diverse trajectory tasks, creating a versatile backbone for trajectory modeling. Collectively, UniTraj achieves task-adaptive, region-independent, and data quality resilience, delivering a scalable and efficient solution for trajectory analysis applications. In summary, our research makes the following key contributions:
+
+- We introduce WorldTrace, a pioneering trajectory dataset spanning 70 countries with 2.45 million trajectories and billions of GPS points. Its unprecedented global diversity and quality overcome the limitations of existing region-specific datasets, offering a comprehensive and open groundwork for facilitating trajectory modeling research.  
+- We propose UniTraj, trained on WorldTrace and equipped with novel pre-training and masking strategies that effectively capture complex spatio-temporal dependencies. This model significantly
+
+enhances generalizability across tasks and geographical contexts, adapts to the heterogeneity of data, and provides a scalable and efficient solution for a wide range of trajectory analysis applications.
+
+- We demonstrated the effectiveness of UniTraj through comprehensive experiments on multiple trajectory analysis tasks. The results show significantly improved performance of zero-shot and fine-tuning settings, confirming its potential as a versatile backbone for diverse trajectory modeling tasks, performing optimally when trained on diverse and high-quality datasets like WorldTrace.
+
+# 2 Related Work
+
+Trajectory Datasets. Trajectory datasets are foundational for advancing mobility research, yet existing collections vary (geographic coverage, data quality, and granularity) considerably in their utility and limitations. Well-known datasets, such as GeoLife [46], collected over five years by 182 users, has contributed significantly to fields like travel mode detection [5] and traffic flow analysis [19]. However, its limited geographic coverage and participant diversity restrict its generalizability. Ehicle-focused datasets such as Porto [28], T-drive [42], and Electric Vehicle Data [34] provide valuable mobility insights but frequently exhibit low or inconsistent sampling rates that complicate analysis. Synthetic alternatives like SynMob [49] offer uniform sampling but lack the regional diversity and quality variations essential for robust model development. Proprietary collections including GAIA [8] and Grab-Posisi [15] contain high-quality data but remain largely inaccessible due to regulatory and commercial constraints. These limitations—geographic constraints, sampling irregularities, and access restrictions—collectively impede the development of universal trajectory models. The community urgently needs comprehensive, openly accessible datasets with global coverage to advance trajectory modeling research and enable effective model generalization.
+
+Foundation Models. The success of foundation models in natural language processing and computer vision, exemplified by BERT [7], GPT-3 [1], and Vision Transformers [9], has demonstrated how large-scale pretraining can yield highly generalizable representations across diverse tasks. This paradigm has recently extended to time series and spatio-temporal domains, with models like TST [44], TimeFM [6], and Moirai [38] leveraging Transformer architectures to capture temporal dependencies. In spatio-temporal prediction specifically, approaches such as UniST [43], Opencity [20], and ClimaX [29] have shown promise in traffic flow and climate modeling, respectively. However, these models often remain tailored to specific tasks or regions, limiting their broader applicability. Trajectory-specific models like TrajGDM [4], BigCity [41], and TrajFM [23] address certain tasks but lack the scalability and robustness needed for cross-task or cross-region applications. While unsupervised learning approaches like MAE [13] and TimeFM [6] have proven effective for images and time series, trajectory modeling presents unique challenges that demand greater flexibility to accommodate diverse mobility patterns, geographic contexts, and sampling characteristics without extensive task-specific modifications. To summary, there remains a pressing need for trajectory foundation models that unify multiple tasks within a single framework, providing robust, transferable representations that generalize across tasks and handle data variability while maintaining computational efficiency.
+
+# 3 Preliminary
+
+Definition 1: (Trajectory). A trajectory represents the sequential record of movement through space over time. Formally, a trajectory  $\pmb{\tau}$  of length  $n$  is expressed as a sequence of continuously sampled GPS points:  $\pmb{\tau} = \{p_1, p_2, \dots, p_n\}$ , where each point  $p_i = \langle \mathrm{lng}_i, \mathrm{lat}_i, t_i \rangle$  denotes the spatial coordinates (longitude and latitude) at timestamp  $t_i$ . The sampling interval between consecutive points is defined as  $\Delta t_i = t_i - t_{i-1}$ , for  $i = 2, \dots, n$ . These intervals may be uniform within or across trajectories, or vary significantly based on data collection methods and environmental factors.
+
+Definition 2: (Trajectory Dataset). A trajectory dataset comprises multiple trajectories, each capturing the movement of an object over time. Formally, it is given by  $\mathcal{D} = \{\tau_1,\tau_2,\dots ,\tau_{|\mathcal{D}|}\}$ , where  $|\mathcal{D}|$  denotes the total number of trajectories in the dataset. These collections may vary in geographic coverage, sampling rates, and quality depending on their source and application scenario.
+
+Problem Statement: (Universal Trajectory Modeling). Building upon the above definitions, this study aims to develop a universal foundation model for trajectory data that can adapt to diverse tasks and geographic contexts while accommodating heterogeneous data sources. Formally, consider a set
+
+of trajectories  $\mathcal{D} = \{\pmb{\tau}_i\}_{i=1}^{|\mathcal{D}|}$ , where each  $\pmb{\tau}_i$  is defined as in Definition 1. The goal is:
+
+$$
+F: \boldsymbol {\tau} \mapsto \mathbf {h} \in \mathbb {R} ^ {d}, \tag {1}
+$$
+
+which projects a raw trajectory  $\tau$  into a d-dimensional representation  $\mathbf{h}$ . This function  $F(\cdot)$  must capture intrinsic spatio-temporal patterns within trajectories while demonstrating three key capabilities: (1) task adaptability across various applications including classification, prediction, and anomaly detection; (2) region independence, enabling zero-shot generalization to different geographic contexts; and (3) resilience to data quality variations, effectively handling inconsistent sampling rates, varying trajectory lengths, and noise without extensive preprocessing or task-specific re-engineering.
+
+# 4 Methodology
+
+In this section, we describe the methodology for developing UniTraj, addressing the key challenges outlined in the introduction. Our approach is structured around answering three fundamental questions: (1) How to construct a diverse and high-quality trajectory dataset that enables cross-regional generalization? (2) How to develop robust and scalable pre-training strategies that accommodate heterogeneous trajectory data? and (3) How to design an effective model architecture that adapts across diverse trajectory tasks?
+
+# 4.1 WorldTrace Dataset Construction
+
+To address the data preparation challenge, we introduce WorldTrace, a large-scale, globally distributed trajectory dataset specifically designed to support universal trajectory modeling. Below, we introduce our data acquisition process, preprocessing pipeline, and key dataset statistics, demonstrating WorldTrace's suitability as a foundation for developing robust and generalizable trajectory foundation models. Detailed information on processing, analysis, and copyright can be found in Appendix A. The full dataset is available on the Hugging Face<sup>3</sup> and ModelScope<sup>4</sup> platforms.
+
+Data Acquisition. We sourced raw trajectory data from OpenStreetMap (OSM) GPS traces [30], focusing on contributions uploaded between 2021-2023 and tagged for motorized movement to ensure data currency and relevance. This approach minimizes device heterogeneity and outdated data impacts. All collected data is stored in the standardized GPX format (an XML schema), containing latitude, longitude, timestamps, and optional metadata, providing a uniform structure that simplifies parsing and preprocessing. During acquisition, we implemented preliminary filtering to exclude trajectories with obvious anomalies such as coordinates outside valid ranges or duplicate entries.
+
+Table 1: Summary statistics of WorldTrace.  
+
+<table><tr><td>Statistic</td><td>Value</td></tr><tr><td>Number of Trajectories</td><td>2.45 Million</td></tr><tr><td>Total Raw Points</td><td>8.8 Billion</td></tr><tr><td>Geographical Covered</td><td>70 Countries</td></tr><tr><td>Sampling Interval</td><td>1 sec (normalized)</td></tr><tr><td>Time Span</td><td>08/2021 – 12/2023</td></tr><tr><td>Avg. Duration</td><td>6 min</td></tr><tr><td>Avg. Distance</td><td>5.73 km</td></tr><tr><td>Avg. Speed</td><td>48.0 km/h</td></tr></table>
+
+Data Preprocessing. Our preprocessing pipeline balances preserving authentic movement patterns with removing noise and inconsistencies, which includes the following steps:
+
+1. Normalization: The original data had a high sampling frequency of up to  $10\mathrm{Hz}$ , causing redundancy and increased storage demands. We therefore resampled trajectories to a uniform rate of one point per second (1 Hz), preserving essential motion details while reducing data size. In addition, by standardizing trajectories to 1s/point, we can perform better resampling during subsequent model training to accommodate frequency inconsistent issues.  
+2. Filtering: We discarded trajectories with fewer than 32 points or covering distances below 100 meters, as such short trips often lack meaningful patterns and introduce noise. Following established practices [5], we also removed trajectories containing implausible speeds (e.g., exceeding  $120\mathrm{km / h}$ ), typically caused by GPS errors or anomalies. We also apply distance- and loop-based outlier detection to identify and remove trajectories that deviate markedly from the expected path.
+
+3. Calibration: Given that GPS signals can suffer from errors due to building obstructions, multipath effects, and receiver noise [14], we applied map-matching techniques [40] to align raw GPS points with underlying road networks. This calibration step is common practice in trajectory data processing and is widely used in data collection and related research to correct positioning errors [8, 37], improve spatial accuracy, and make trajectory analysis more reliable.
+
+Data Analysis and Statistics. After acquiring and preprocessing the raw trajectory data, we conducted an in-depth analysis to examine the characteristics and quality of the WorldTrace dataset. Table 1 summarizes key statistics of WorldTrace. Overall, the dataset contains approximately 2.45 million trajectories and 8.8 billion raw GPS points, covering 70 countries across all inhabited continents. The data spans August 2021 to December 2023, with an average trajectory duration of about six minutes (with normalized to a 1-second sampling interval), an average distance of  $5.73\mathrm{km}$  and an average speed of  $48.0~\mathrm{km / h}$ . The number of points per trajectory ranges from 32 to more than 600, averaging around 358 points. Collectively, these attributes confirm WorldTrace's suitability for developing universal trajectory models that can address varied spatiotemporal patterns and broad geographical contexts.
+
+# 4.2 Pre-Training Strategies
+
+Having established a diverse trajectory dataset, we develop robust pre-training strategies to learn robust and transferable spatio-temporal representations. Rather than relying on task-specific supervision, we leverage unannotated trajectory data to capture both local and global movement patterns. To address the heterogeneous data quality challenges (varying sampling rates, differing lengths, and missing points) posed by real-world trajectory, we propose two strategies tailored specifically for trajectory: Adaptive Trajectory Resampling and Self-supervised Trajectory Masking. Due to space limitations, more details and analysis about pre-training strategies can be found in Appendix B.
+
+Adaptive Trajectory Resampling (ATR). Real-world trajectory data often exhibits inconsistent sampling intervals and lengths due to diverse collection standards, device capabilities, and user behaviors. Such discrepancies challenge model generalization, as features learned under one sampling regime may not transfer to another. Inspired by common practice of multi-scale representation learning, ATR strategy addresses these issues through two complementary components:
+
+- Dynamic Multi-Scale Resampling. This approach dynamically adjusts sampling frequency based on trajectory length, ensuring shorter trajectories retain fine-grained detail while longer ones are efficiently compressed. Specifically, we design a logarithmic resampling function  $R(n)$  to implement this strategy:
+
+$$
+R (n) = R _ {\min } + \left(1 - R _ {\min }\right) \cdot \frac {\ln \left(n - n _ {\min } + 1\right)}{\ln \left(n _ {\max } - n _ {\min } + 1\right)}, \tag {2}
+$$
+
+where  $n_{\mathrm{min}}$  and  $n_{\mathrm{max}}$  define thresholds for trajectory lengths considered "short" or "long", and  $R_{\mathrm{min}}$  is the minimum sampling ratio. This logarithmic function creates a smooth transition in sampling density ( $n_{\mathrm{min}} < n < n_{\mathrm{max}}$ ), providing three key benefits: (1) preserving critical motion patterns across trajectory lengths, (2) reducing overfitting by limiting redundancy in densely sampled data, and (3) exposing the model to diverse temporal resolutions during training.
+
+- Interval Consistent Resampling. This component focuses on the sampling rate, imposing a uniform time interval  $\Delta t$  between consecutive points within each track:
+
+$$
+\boldsymbol {\tau} ^ {\prime} = \left\{p _ {k _ {j}} \mid k _ {j} = 1 + (j - 1) \Delta t, j = 1, 2, \dots , m \right\}. \tag {3}
+$$
+
+By ensuring consistent spacing, this approach simplifies downstream modeling by creating regular temporal structures that make time-dependent patterns easier to learn, while mitigating complications from missing data or irregular sampling.
+
+Combining these approaches, ATR enables models to learn representations that generalize across varying sampling rates and trajectory lengths (analysis presented in Appendix B.1), which is a critical capability for universal trajectory modeling.
+
+Self-supervised Trajectory Masking (STM). Trajectory data is often incomplete or irregular due to device limitations, communication failures, and environmental factors. Motivated by masked auto-encoding methods from visual and language models, we introduce a tailored self-supervised
+
+trajectory masking strategy, in which part of the input trajectory is hidden, forcing the model to infer local and global dependencies. Given a resampled trajectory  $\tau' = \{p_1, p_2, \ldots, p_n\}$ , we define a masking function  $\mathcal{M}(\tau', r)$  that replaces a fraction  $r$  of points with a [MASK] tokens:
+
+$$
+\tilde {\tau} = \mathcal {M} \left(\tau^ {\prime}, r\right) = \left\{p _ {1}, \dots , \left[ \mathrm {M A S K} \right] _ {i \in \mathbf {I}}, \dots , p _ {n} \right\}, \tag {4}
+$$
+
+where  $\mathbf{I} \subseteq \{1, 2, \dots, n\}$  and  $r = |\mathbf{I}| / n$ . To comprehensively address different data incompleteness scenarios, (see Appendix B.2 for details) we employ four complementary masking strategies:
+
+- Random Masking: Uniformly samples points to mask  $(\mathbf{I}_{\mathrm{rand}} \sim \mathrm{Uniform}(\{1,2,\dots,n\}))$ , forcing the model to infer both short-range and long-range dependencies. By forcing the reconstruction of randomly omitted points, the approach enhances the model's ability to generalize to diverse gaps.  
+- Block Masking: Conceals consecutive points  $(\mathbf{I}_{\mathrm{block}} = \{k, k + 1, \dots, k + b - 1\})$  to simulate sensor failures, encouraging reconstruction of continuous segments. This approach prompts the model to utilize surrounding context for reconstructing entire missing segments, encouraging it to capture longer-range dependencies.  
+- Key Points Masking: Identifies and masks critical turning points using the Ramer-Douglas-Peucker algorithm [10]:  $\mathbf{I}_{\mathrm{key}} = \{p_k \mid d_{\max}(p_k, \overline{p_1 p_n}) > \epsilon\}$  ( $d_{\max}(\cdot)$  is the maximum perpendicular distance between point  $p_k$  and line  $\overline{p_1 p_n}$ ,  $\epsilon$  is the threshold). This focuses learning on structurally significant points (sharp turns or notable speed changes) that define the trajectory's shape.  
+- Last N Masking: Masks final trajectory points  $(\mathbf{I}_{\mathrm{last}} = \{n - N + 1, n - N + 2, \dots, n\})$ . This setting emulates real-world forecasting tasks where future data is unavailable and must be inferred from historical observations, making it particularly effective for prediction scenarios.
+
+# 4.3 Universal Trajectory Modeling
+
+To effectively leverage the diverse trajectory data and robust pre-training strategies described above, we need to design a model architecture that can capture local and global patterns while freeing itself from regional and task-specific constraints. Our motivation for adopting this structure design is as follows: (1) We need an architecture that can be generalized to a wide range of tasks without extensive restructuring. Therefore, we adopted minimal trajectory data information (latitude, longitude, and timestamp) and ignored other region-bound information such as POI and geographical context. (2) This structure uses the reconstruction of missing points in partial observations as a proxy task and can inherit the masking strategy introduced earlier. (3) The separation of encoding and decoding enables flexible application to various downstream tasks through transfer learning or fine-tuning. More details about the architecture and parameters can be found in Appendix C.
+
+Building Trajectory Embedding. Effective trajectory modeling requires transforming raw spatial and temporal data into structured embeddings that capture both local and global movement patterns. To ensure the generality of the model, we only use the latitude, longitude, and time information of the trajectory, and embed the spatial and temporal components separately to form a unified representation. For the spatial component, we normalize trajectory and map them into a  $d$ -dimensional space using a 1D convolutional, yielding a spatial embedding  $h_i^s$ . Similarly, the temporal component, based on the time intervals  $\Delta t_i$ , is embedded into the same  $d$ -dimensional space via a linear layer, resulting in a temporal embedding  $h_i^t$ . This decoupled design enables the model to effectively learn relative movement and temporal dependencies, and also cope with situations where one component may be absent. Beyond point-wise embedding, modeling the relationships between trajectory points is critical for understanding movement patterns. We adopt Rotary Position Encoding (RoPE) [32], which applies rotational transformations in the embedding space. The advantage of RoPE is its ability to preserve relative positional relationships while allowing for flexible encoding of spatial-temporal patterns across varying trajectory scales.
+
+Adaptive Representation Learning. Based on the trajectory embeddings, we use a encoder-decoder architecture with RoPE-enhanced attention mechanism to adaptively learn a general representation of trajectories. The encoder processes the visible points in a trajectory those that are unmasked during training. Given a masked trajectory  $\tilde{\tau} = \{p_1,\dots ,[\mathrm{MASK}]_{i\in \mathbf{I}},\dots ,p_n\}$ , we first extract the embedding representations of the unmasked points  $\mathbf{H} = \{h_1,h_2,\dots ,h_m\}$  (where  $m\leq n$  and  $i\notin \mathbf{I}$ ) through the embedding steps. The encoder, denoted as  $\mathbf{E}_{\theta}$ , processes these visible embeddings to generate latent representations:  $z_{\mathrm{enc}} = \mathbf{E}_{\theta}\left(\mathbf{H}\right)$ . The decoder reconstructs masked trajectory points based on the latent embeddings produced by the encoder. It receives the visible embeddings and mask
+
+Table 2: Performance comparison of UniTraj with trajectory recovery tasks. The results are reported in MAE and RMSE with meters. Bold denotes the best results and underline denotes the second-best results.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">WorldTrace</td><td colspan="2">Chengdu</td><td colspan="2">Xi&#x27;an</td><td colspan="2">GeoLife</td><td colspan="2">Grab-Posisi</td><td colspan="2">Porto</td></tr><tr><td>MAE</td><td>RMSE</td><td>MAE</td><td>RMSE</td><td>MAE</td><td>RMSE</td><td>MAE</td><td>RMSE</td><td>MAE</td><td>RMSE</td><td>MAE</td><td>RMSE</td></tr><tr><td>Linear</td><td>427.68</td><td>516.15</td><td>205.74</td><td>258.52</td><td>176.49</td><td>220.87</td><td>196.85</td><td>249.76</td><td>507.41</td><td>617.28</td><td>396.61</td><td>482.39</td></tr><tr><td>DHTR</td><td>220.35</td><td>302.47</td><td>75.19</td><td>98.68</td><td>62.85</td><td>83.43</td><td>80.04</td><td>168.25</td><td>351.20</td><td>415.16</td><td>194.37</td><td>232.59</td></tr><tr><td>Transformer</td><td>130.82</td><td>147.62</td><td>55.23</td><td>62.85</td><td>45.85</td><td>51.96</td><td>94.68</td><td>113.77</td><td>136.58</td><td>163.29</td><td>104.36</td><td>126.96</td></tr><tr><td>DeepMove</td><td>51.16</td><td>62.29</td><td>29.32</td><td>39.02</td><td>27.31</td><td>35.67</td><td>86.38</td><td>107.78</td><td>126.93</td><td>168.07</td><td>136.66</td><td>174.96</td></tr><tr><td>TrajBERT</td><td>58.13</td><td>70.14</td><td>26.48</td><td>33.83</td><td>19.45</td><td>25.13</td><td>34.53</td><td>43.24</td><td>112.68</td><td>136.24</td><td>78.77</td><td>99.23</td></tr><tr><td>TrajFM</td><td>47.64</td><td>58.92</td><td>19.10</td><td>25.09</td><td>18.86</td><td>24.13</td><td>59.34</td><td>64.24</td><td>107.64</td><td>130.69</td><td>71.15</td><td>92.96</td></tr><tr><td>UniTraj (zero-shot)</td><td>10.22</td><td>13.56</td><td>11.98</td><td>20.94</td><td>8.93</td><td>13.83</td><td>37.21</td><td>63.89</td><td>114.07</td><td>167.01</td><td>78.28</td><td>100.14</td></tr><tr><td>Improvement(%)</td><td>↑78.55</td><td>↑76.99</td><td>↑37.28</td><td>↑16.54</td><td>↑52.65</td><td>↑42.69</td><td>↓7.76</td><td>↓47.46</td><td>↓5.97</td><td>↓27.79</td><td>↓10.02</td><td>↓7.72</td></tr><tr><td>UniTraj (fine-tune)</td><td>6.94</td><td>9.67</td><td>6.92</td><td>10.41</td><td>6.50</td><td>9.93</td><td>23.23</td><td>34.70</td><td>48.95</td><td>69.23</td><td>60.18</td><td>79.76</td></tr><tr><td>Improvement(%)</td><td>↑85.43</td><td>↑83.59</td><td>↑63.77</td><td>↑58.51</td><td>↑65.54</td><td>↑58.85</td><td>↑32.73</td><td>↑19.75</td><td>↑54.52</td><td>↑47.03</td><td>↑15.42</td><td>↑14.20</td></tr></table>
+
+tokens, which are initialized as learnable vectors representing missing positions. The full sequence is created by merging the encoded visible embeddings with the mask tokens, preserving the original structure of the trajectory:
+
+$$
+\mathbf {z} _ {\mathrm {d e c}} = \operatorname {R e o r d e r} \left(\left\{ \begin{array}{l l} z _ {i} = z _ {\text {e n c}, j} & \text {i f} i = \operatorname {I n d e x} (j), i \notin \mathbf {I} \\ [ \text {M A S K} ] & \text {i f} i \in \mathbf {I} \end{array} \right\}\right), \tag {5}
+$$
+
+where  $z_{\mathrm{enc},j}$  corresponds to the  $j$ -th encoder output. The decoder then processes the reordered sequence to predict the missing trajectory points:  $\hat{\tau} = \mathrm{Linear}(\mathbf{D}_{\phi}(\mathbf{z}_{\mathrm{dec}}))$ . The model is trained to minimize the reconstruction loss between the predicted and original points at the masked positions:
+
+$$
+\mathcal {L} = \frac {1}{| \mathbf {I} |} \sum_ {i \in \mathbf {I}} \| f _ {\theta , \phi} (\tilde {\boldsymbol {\tau}}) _ {i} - \boldsymbol {\tau} _ {i} \| ^ {2}, \tag {6}
+$$
+
+where  $f_{\theta ,\phi}(\tilde{\tau})$  represents the encoder-decoder network, and  $i$  refers to the masked positions.
+
+# 5 Experiments
+
+# 5.1 Experimental Setups
+
+Datasets. We evaluate UniTraj on six diverse real-world trajectory datasets representing different collection scenarios, quality levels, motion patterns, and geographic regions. These include WorldTrace, Chengdu, Xi'an, GeoLife, Grab-Posisi, and Porto. Detailed summary are provided in Appendix D.1.
+
+# 5.2 Task Applicability Analysis
+
+We explore the applicability and generalizability of UniTraj to various data and downstream tasks, e.g., trajectory recovery, prediction, classification, and generation tasks. Due to space constraints, we provide the detailed setup and the results of generation task in Appendix D.2. It is important to clarify that our work aims to develop a general-purpose trajectory foundation model that generalizes across diverse geographic regions without region-specific dependencies, validating its effectiveness as a backbone supporting real-world trajectory applications across geographical contexts. Existing trajectory representation learning methods inherently rely on region-bound information (POIs, road networks, etc.) [16, 22, 26, 48], which contradicts our initial goal of region-independent modeling. UniTraj extracts meaningful representations solely from trajectory points without requiring auxiliary geographic context. Therefore, we deliberately excluded these methods from our baseline comparison as their architectural dependency on regional knowledge fundamentally diverges from our objective of developing a globally deployable model.
+
+Trajectory Recovery. Table 2 presents a comprehensive comparison of UniTraj against established baselines across six datasets, revealing patterns that illuminate fundamental capabilities in trajectory reconstruction. The performance disparity between UniTraj and previous methods is particularly pronounced in geographically diverse and quality-variable datasets, where it demonstrates substantial resilience to regional variations. In the zero-shot setting, UniTraj achieves remarkable results, confirming it effectively captures transferable spatio-temporal patterns without requiring additional fine-tuning. The performance difference becomes particularly instructive when analyzing low-quality datasets like GeoLife and Grab-Posisi, with their highly irregular sampling intervals and multiple
+
+travel modes. It demonstrate the effectiveness of our adaptive resampling strategy in handling temporal heterogeneity. The Chengdu and Xi'an datasets reveal another critical aspect of UniTraj's capabilities, models trained on high-quality data exhibit reliable transferability and achieve optimal results even in zero-shot scenarios. When fine-tuned, UniTraj achieves the lowest error scores across all datasets, demonstrating UniTraj's superior generalizability across diverse geographic regions. For instance, on GeoLife, UniTraj's fine-tuned performance (MAE 23.23) reduces error by  $32.73\%$  compared to TrajBERT, showcasing its effectiveness with complex travel patterns and lower-quality data. These results validate WorldTrace's potential as a foundation dataset and UniTraj's consistent superiority in trajectory recovery tasks, with substantial improvements through fine-tuning reinforcing its adaptability and robustness.
+
+Trajectory Prediction. Table 3 shows UniTraj's exceptional performance in trajectory prediction, a different task requiring forward inference rather than reconstruction. The zero-shot results merit particular attention, as they represent the most challenging scenario for trajectory models. On WorldTrace, UniTraj's zero-shot MAE significantly outperforms all baselines, underscoring the model's versatility in capturing universal motion patterns. When fine-tuned, the performance further improves, consistently achieving the best results across all
+
+Table 3: Performance comparison of UniTraj with trajectory prediction tasks.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">WorldTrace</td><td colspan="2">Chengdu</td><td colspan="2">GeoLife</td></tr><tr><td>MAE</td><td>RMSE</td><td>MAE</td><td>RMSE</td><td>MAE</td><td>RMSE</td></tr><tr><td>Linear</td><td>153.12</td><td>159.65</td><td>156.85</td><td>164.58</td><td>189.02</td><td>201.34</td></tr><tr><td>DHTR</td><td>146.48</td><td>151.63</td><td>123.47</td><td>129.73</td><td>180.32</td><td>187.59</td></tr><tr><td>Transformer</td><td>114.25</td><td>117.07</td><td>67.38</td><td>70.86</td><td>165.02</td><td>170.84</td></tr><tr><td>DeepMove</td><td>55.69</td><td>58.67</td><td>36.31</td><td>39.10</td><td>116.46</td><td>123.20</td></tr><tr><td>TrajBERT</td><td>80.57</td><td>86.36</td><td>64.73</td><td>68.92</td><td>113.68</td><td>121.18</td></tr><tr><td>TrajFM</td><td>75.45</td><td>81.32</td><td>77.82</td><td>80.48</td><td>121.94</td><td>128.16</td></tr><tr><td>UniTraj (zero-shot)</td><td>49.85</td><td>55.02</td><td>42.75</td><td>45.93</td><td>108.35</td><td>133.60</td></tr><tr><td>Improvement(%)</td><td>↑10.49</td><td>↑6.22</td><td>↓17.74</td><td>↓17.46</td><td>↑4.69</td><td>↓10.25</td></tr><tr><td>UniTraj (fine-tune)</td><td>30.10</td><td>34.46</td><td>28.78</td><td>32.44</td><td>90.97</td><td>102.88</td></tr><tr><td>Improvement(%)</td><td>↑45.95</td><td>↑41.27</td><td>↑20.74</td><td>↑17.03</td><td>↑19.98</td><td>↑15.10</td></tr></table>
+
+evaluated datasets. This generalization capability stems from our Last-N masking strategy, which explicitly shapes the embedding space to support predictive inference. These results further confirm that UniTraj not only generalizes remarkably well across diverse datasets but also benefits considerably from fine-tuning, making it highly adaptable for real-world applications requiring accurate trajectory predictions.
+
+Trajectory Classification. Figure 2 presents classification accuracy results that reveal UniTraj's capacity to learn discriminative representations of movement modalities. Notably, even without fine-tuning, UniTraj achieves  $71.3\%$  accuracy on GeoLife, outperforming several supervised baselines. This zero-shot performance demonstrates that the pre-trained representations inherently capture transportation mode signatures, where movement modality emerges as a natural organizing principle. On the Grab-Posisi
+
+![](images/198ee5a090a4aed303318259839f7bf3556fd92a860e4aa6c860e2879039d8d6.jpg)  
+Figure 2: Performance comparison of classification task with GeoLife and Grab-posisi dataset.
+
+dataset, which presents additional challenges due to similar motion patterns for mixed travel modes (car and motorcycle). UniTraj achieves  $79.3\%$  accuracy after fine-tuning with a substantial improvement over the best baseline. This improvement emphasizes UniTraj's ability to capture subtle kinematic signatures that differentiate travel modes with complex or similar patterns.
+
+# 5.3 Dataset Study
+
+This section analyzes the impact of dataset scale, quality, and diversity on model performance of UniTraj, particularly its generalization capability across different data sources. We focus on two main experiments: (1) examining the effect of dataset scale and quality within WorldTrace, with varying data volumes ( $\{0.01, 0.5, 1\}$  millions) and a high-quality (obtained by further removing loops, staying dense trajectories) subset, and (2) assessing UniTraj's adaptability and effectiveness by training it on these datasets beyond WorldTrace, thus showing its potential as a foundation model.
+
+Effect of Dataset Scale and Quality. Figure 3(a) illustrates the relationship between training data volume and model performance, revealing a phenomenon that goes beyond simple scaling laws. With increasing trajectory count from WorldTrace (from 0.5M to 2.45M), the MAE on the in-domain test set decreases dramatically, showing substantial improvement up to approximately 1M trajectories before beginning to exhibit diminishing returns. The above result indicates that larger datasets enable
+
+Table 4: Ablation study on different resampling and masking strategies on six datasets.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">WorldTrace</td><td colspan="2">Chengdu</td><td colspan="2">Xi&#x27;an</td><td colspan="2">GeoLife</td><td colspan="2">Grab-posisi</td><td colspan="2">Porto</td></tr><tr><td>MAE</td><td>RMSE</td><td>MAE</td><td>RMSE</td><td>MAE</td><td>RMSE</td><td>MAE</td><td>RMSE</td><td>MAE</td><td>RMSE</td><td>MAE</td><td>RMSE</td></tr><tr><td>w/o Dynamic Multi-scale resampl.</td><td>426.80</td><td>482.37</td><td>192.54</td><td>272.42</td><td>157.85</td><td>223.96</td><td>499.95</td><td>671.69</td><td>1933.28</td><td>2504.16</td><td>93.14</td><td>119.93</td></tr><tr><td>w/o Interval Consistent resampl.</td><td>21.30</td><td>24.76</td><td>12.98</td><td>20.61</td><td>9.34</td><td>13.90</td><td>69.41</td><td>115.33</td><td>102.45</td><td>149.60</td><td>1724.12</td><td>2016.61</td></tr><tr><td>w/o Key points masking</td><td>25.49</td><td>28.91</td><td>14.46</td><td>21.98</td><td>11.10</td><td>15.17</td><td>45.94</td><td>72.84</td><td>113.65</td><td>162.57</td><td>76.51</td><td>101.18</td></tr><tr><td>w/o Block masking</td><td>7.79</td><td>10.47</td><td>9.22</td><td>15.36</td><td>7.16</td><td>11.18</td><td>48.59</td><td>77.73</td><td>89.34</td><td>128.72</td><td>198.41</td><td>238.88</td></tr><tr><td>UniTraj</td><td>10.22</td><td>13.56</td><td>11.98</td><td>20.94</td><td>8.93</td><td>13.83</td><td>37.21</td><td>63.89</td><td>114.07</td><td>167.01</td><td>78.28</td><td>100.14</td></tr></table>
+
+the model to capture a wider range of spatio-temporal patterns. However, while increasing the dataset size from 1 million to 2.45 million trajectories results in better coverage, the model's MAE slightly increases due to the introduction of more noise in the full dataset. In contrast, training on a high-quality subset of 1 million trajectories, which includes curated, noise-free data, yields more reliable and consistent learning. This highlights the importance of both dataset scale and quality, with quality being especially crucial when data volume is limited.
+
+![](images/6622c1febeef20eefef516bf41b6b528d355cd7d4315afa4c8aec66f3a06c136.jpg)  
+(a) Training data volume.
+
+![](images/16efda25cf3148fc7c45483e077178ce3b74b51c88292d8edb3a8896db939231.jpg)  
+(b) Training dataset.  
+Figure 3: The effect of amount of data volume, diversity dataset, and different parameter settings.
+
+![](images/82f76225e3e062dcbf49c5658aa3893faebf098f4c4feaa587fe0c6e530bf32e.jpg)  
+(c) Number of Encoders
+
+![](images/f0a8cb34d258b0750e92bf3f59dd352bb6dc4b6dd2890bf9642b7cc18e4a1d19.jpg)  
+(d) Mask ratio  $(\%)$ .
+
+Effect of Dataset Diversity. In Figure 3(b), we compare UniTraj's zero-shot performance when trained on WorldTrace and Chengdu (the highest quality dataset available), evaluated across multiple real-world datasets. Models trained on WorldTrace exhibit superior generalization across diverse datasets (e.g., GeoLife and Porto), reflecting the broad geographic and contextual coverage of WorldTrace. Conversely, models trained on Chengdu perform best on datasets with similar density and travel modes, such as Xi'an. However, proprietary datasets like Chengdu, while offering high quality, are not publicly available, limiting their applicability for universal tasks. These results demonstrate UniTraj's robustness and adaptability, validating WorldTrace as an ideal training resource for building a universal trajectory foundation model. At the same time, the findings confirm that UniTraj can effectively leverage other datasets when necessary, further enhancing its versatility.
+
+# 5.4 Model Study
+
+We investigate architectural components, parameter settings, and pre-training strategies to assess sensitivity to parameter choices and the contributions of their core components.
+
+Effect of Parameter Settings. Figure 3(c) Figure 3(d) and presents the results of our parameter sensitivity analysis, examining how the number of encoder blocks and the masking ratio influence model performance. As shown in Figure 3(c), increasing the number of encoder blocks from 2 to 8 significantly reduces MAE, with performance plateauing beyond 8 blocks. This plateau suggests that while deeper architectures can improve model capacity, the benefits diminish without corresponding adjustments in data or hyperparameters [17]. Figure 3(d) demonstrates that a masking ratio of  $50\%$  yields the best performance. Low masking ratios (e.g.,  $5\% -10\%$ ) result in higher MAE due to insufficient training signal, while higher ratios (e.g.,  $75\%$ ) lead to increased MAE from excessive information loss. A  $50\%$  masking ratio strikes a balance, providing the model with a strong training signal without sacrificing the context needed for effective trajectory reconstruction.
+
+Ablation Study. Table 4 presents an ablation study, showing how different pre-training strategies affect UniTraj's performance across datasets. The performance varies across datasets, indicating the effectiveness and limitations of them depending on the specific data and task scenarios. Dynamic Multi-scale Resampling significantly improves performance across most datasets, especially GeoLife and Grab-Posisi, which have inconsistent sampling intervals and lower data quality. This suggests that dynamic resampling helps the model to adapt to heterogeneous dataset scenarios and to be adaptive for information preservation (see Appendix B.1.1 for more details). The Interval Consistent Resampling has a notable positive effect on datasets with consistent sampling rates, such as Porto
+
+and WorldTrace. It indicates that the integration of this strategy strategy can effectively separate the temporal sampling pattern from the region, it enhances the generalization of the model to data sets with different sampling rates (analysis presented in Appendix B.1.2). Key Points Masking leads to substantial performance drops on high-quality datasets like Chengdu and Xi'an but appears to offer minimal benefits, or even slight disadvantages, for certain datasets. This finding suggests that adjusting adaptive masking strategies based on trajectory complexity, potentially applying it selectively to trajectories with significant directional changes, while using alternative strategies for smoother paths. Block Masking shows significant effects on GeoLife and Porto, where it helps the model handle low sampling frequencies. However, its impact on other datasets is more inconsistent, suggesting that it introduces an artificial challenge that may increase complexity in high-frequency datasets. (we provide a robustness analysis in Appendix B.2) Overall, the varying impact of UniTraj's pre-training strategies across datasets highlights its adaptability to different tasks and scenarios. While not all of them universally enhance performance, their combined use provides a balanced training strategy, allowing for flexible configuration depending on specific dataset requirements. Fine-tuning further optimizes performance, ensuring stability and robustness across diverse tasks.
+
+# 6 Conclusion
+
+In this work, we presented UniTraj, a universal trajectory foundation model designed to overcome the task specificity, regional dependency, and data quality limitations of current approaches. UniTraj acts as a robust backbone that generalizes effectively across diverse tasks and regions. To support its development, we introduced WorldTrace, a high-quality global dataset with 2.45 million trajectories from 70 countries, offering broad geographic coverage, varied sampling rates, and open accessibility. Together, UniTraj and WorldTrace provide a versatile, high-performing foundation for trajectory analysis, paving the new solution for more adaptable and efficient models in trajectory-based research. Future work will focus on expanding the geographic and modal diversity of the WorldTrace dataset to better cover underrepresented regions and non-motorized travel. We also aim to enhance the UniTraj model by integrating contextual information, such as road networks and points of interest, to improve its predictive accuracy and real-world applicability. Further optimizations to the model architecture and pre-training strategies will also be explored to boost performance and efficiency.
+
+# 7 Acknowledgement
+
+This work was mainly supported by the National Natural Science Foundation of China under Grant No. 62506097, No. 62402414 and No.62502404. This work is also supported by the Guangdong Basic and Applied Basic Research Foundation (No. 2025A1515011994), Tencent (CCF-Tencent Open Fund, Tencent Rhino-Bird Focused Research Program), Didi (CCF-DiDi GAIA Collaborative Research Funds), Guangzhou Municipal Science and Technology Project (No. 2023A03J0011) and Guangzhou-HKUST(GZ) Joint Funding Program (No. 2024A03J0620), Hong Kong Research Grants Council's Research Impact Fund (No.R1015-23), Research Grants Council's Collaborative Research Fund (No.C1043-24GF), Research Grants Council's General Research Fund (No.11218325), Institute of Digital Medicine of City University of Hong Kong (No.9229503), Huawei (Huawei Innovation Research Program), Tencent (CCF-Tencent Open Fund, Tencent Rhino-Bird Focused Research Program), Alibaba (CCF-Alimama Tech Kangaroo Fund No. 2024002), Ant Group (CCF-Ant Research Fund), Didi (CCF-Didi Gaia Scholars Research Fund), Kuaishou, and Bytedance.
+
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+
+# SUPPLEMENTARY MATERIAL
+
+UNITRAJ: LEARNING A UNIVERSAL TRAJECTORY FOUNDATION MODEL FROM BILLION-SCALE WORLDWIDE TRACES
+
+TABLE OF CONTENTS
+
+# A Details of WorldTrace Dataset 15
+
+A.1 Data Collection 15  
+A.2 Data Processing 16  
+A.3 Data Statistics and Analysis 16  
+A.4 Data Privacy and Copyright 17
+
+# B Pre-training Strategies 18
+
+B.1 Adaptive Trajectory Resampling 18  
+B.2 Self-supervised Trajectory Masking 21
+
+# C Details of UniTraj 24
+
+C.1 Overall Architecture 24  
+C.2 Input Representation and Embedding 24  
+C.3 Adaptive Representation Learning 25  
+C.4 Task-Specific Adaptation 26  
+C.5 Implementation Details 27
+
+# D Experiments Details 27
+
+D.1 Datasets 27  
+D.2 Tasks Applicability Study Settings 28  
+D.3 Dataset Study Settings 30  
+D.4 Model Study Settings 30
+
+# E More Discussion 30
+
+E.1 Limitation 30  
+E.2 Broader Impact 31  
+E.3 Ethics Issues 31
+
+# A Details of WorldTrace Dataset
+
+In this section, we detail the collection of the dataset, the processing, and provide a detailed analysis of the resulting dataset.
+
+![](images/f3321d21fd8a59dd766f5c2e21613128fcaeeefb9ac80f4ee4554c64e4882d90.jpg)  
+Figure 4: The process pipeline of WorldTrace dataset construction.
+
+# A.1 Data Collection
+
+Data Source. As shown in Figure 4, the raw data for WorldTrace is sourced from the shared trajectory data platform on OpenStreetMap (OSM) [30]5. This platform, a public sharing project, hosts over 11 million GPS trajectories uploaded by contributors worldwide from 2004 to the present. To ensure data quality and reliability, we specifically targeted contributions tagged for motorized movement to ensure data currency and relevance to modern transportation networks. This approach helps minimize device heterogeneity and avoids outdated data that might not reflect current infrastructure. The raw data is stored in the standardized GPX (GPS Exchange Format), an XML schema designed for exchanging GPS data between applications and web services https://www.topografia.com/GPX/1/1/. Each GPX file contains sequences of trackpoints with the following attributes:
+
+- Latitude (decimal degrees)  
+- Longitude (decimal degrees)  
+- Altitude (decimal numbers)  
+- Timestamp (ISO 8601 format)  
+- Optional metadata (version, tags, etc.)
+
+In addition, while crawling the original trajectory, we also crawled the basic information about the trajectory descriptions, such as the starting point, markers, time, creator, etc., which was saved as a JSON file.
+
+Collection Process. Prior to integration, our collection pipeline involved the following steps: Our collection pipeline involved the following steps:
+
+1. API-based Retrieval: We use the OSM API to systematically query and download GPX traces based on selected filters to ensure global coverage. In order not to increase the burden on server providers, we did not use concurrent crawling, and the whole collection process lasted about 6 months, yielding about 4.5 million raw traces.  
+2. Initial Filtering: During acquisition, we implemented preliminary filtering to exclude trajectories with obvious anomalies such as: Coordinates outside valid ranges (-90° to 90° for latitude, -180° to 180° for longitude); Duplicate or long duration consecutive points; Empty or near-empty traces (fewer than 60 seconds).  
+3. Format Standardization: All collected data was parsed from the original GPX format and converted to a unified internal format for subsequent processing.
+
+# A.2 Data Processing
+
+Our preprocessing pipeline was designed to balance preserving authentic movement patterns with removing noise and inconsistencies. The process consists of three main stages:
+
+Normalization. The raw data exhibited highly variable sampling frequencies, ranging from subsecond intervals up to several seconds between consecutive points. This heterogeneity creates challenges for modeling and increases storage requirements unnecessarily. We therefore applied the following normalization procedures:
+
+- Temporal Resampling: We resampled all trajectories to a uniform rate of one point per second (1 Hz). For segments with sampling rates higher than 1 Hz, we select the first occurrence of a trajectory point within each one-second window. For segments with lower sampling rates, we used linear interpolation between available points to estimate positions at one-second intervals.  
+- Coordinate Standardization: All coordinates were converted to the WGS84 datum for consistency, and we ensured uniform precision across the dataset (6 decimal places for both latitude and longitude, providing 0.1m precision at the equator).
+
+Filtering. After normalization, we implemented a multi-stage filtering process to meticulously remove trajectories that were deemed unsuitable for our analysis. This comprehensive filtering approach involved several key steps:
+
+- Length-based Filtering: We discarded trajectories with fewer than 32 points (equivalent to 32 seconds after resampling) or covering distances below 100 meters, as these typically represent stationary periods or very short movements with limited analytical value.  
+- Speed-based Filtering: We calculated point-to-point speeds and removed trajectories containing implausible values (e.g., exceeding  $120\mathrm{km/h}$  or lower  $0.5\mathrm{km/h}$  in urban environments), typically caused by GPS errors or anomalies.  
+- Distance-based Outlier Detection: We calculated the distance between the original trajectory and the map-matched trajectory. Trajectories that were too far away (indicating large deviations in motion) were flagged for further inspection or removal.  
+- Loop Detection: We identify and remove trajectories that form perfect or near-perfect loops with no apparent destination by their geometry, which usually indicates the presence of clearly anomalous patterns.
+
+Calibration. GPS signals can suffer from various errors due to atmospheric conditions, satellite geometry, and physical obstructions. To improve data quality, we applied map-matching techniques to align raw GPS points with underlying road networks, using a Hidden Markov Model-based approach (or using online API) with a custom emission probability function that accounts for both point-to-road distance and heading consistency. Besides, each trajectory point was enriched with derived attributes.
+
+# A.3 Data Statistics and Analysis
+
+Overall Statistics. The final WorldTrace dataset contains:
+
+- Approximately 2.45 million trajectories.  
+8.8 billion raw GPS points (before normalization).  
+- Coverage across 70 countries on all inhabited continents.  
+- Temporal span from August 2021 to December 2023.  
+- Average trajectory duration of approximately 6 minutes.  
+- Average trajectory distance of 5.73 kilometers.  
+Average travel speed of  $48.0\mathrm{km / h}$  
+- Points per trajectory ranging from 32 to over 600, with an average of 358 points.
+
+Geographic Distribution. WorldTrace offers extensive geographic coverage, as illustrated in Figure 5, encompassing trajectory data from 70 countries and spanning diverse environments and
+
+![](images/15e7325965e3456e8f76bef2410df23ee9c8a3653468ef0fe88ac1ffe700b65a.jpg)  
+(a) Geographic distribution across the world.
+
+![](images/7684c2800f1fd39dc0218890ded23ae9bcf45b762579c5708440aaa8810e1615.jpg)  
+(b) Trajectory counts of top 10 countries.
+
+![](images/af5144eda61e0531ccfa5011ac131fe2e353fa966ef26ce69be1dfa6535c6120.jpg)  
+(c) Distribution within contiguous USA.  
+Figure 5: The distribution details of WorldTrace dataset.
+
+infrastructure types. This global distribution is visualized in Figure 5(a), highlighting dense concentrations in North America, East Asia, and parts of Europe, with trajectory counts exceeding in the most represented regions. Figure 5(b) further details the top 10 countries by trajectory counts, with the United States, China, and Canada leading in data volume. Notably, it exhibits substantial geographic diversity, with varying densities across urban, suburban, and rural environments. The top 10 countries by trajectory count, namely, the USA, China, Canada, Germany, UK, Japan, Brazil, Australia, South Korea, and Hungary, represent a wide range of urban forms, road networks, and mobility cultures. Additionally, Figure 5(c) provides a closer look at the data density within the contiguous United States, demonstrating high-resolution coverage along major road networks and urban centers. This detailed distribution underscores the dataset's ability to capture nuanced variations in trajectory data across different regions. Collectively, these figures emphasize the potential of WorldTrace to serve as a robust foundation for developing region-independent and universal trajectory models. Its extensive geographic coverage and diverse environmental representation make it well-suited for applications that require broad and adaptable trajectory data.
+
+# A.4 Data Privacy and Copyright
+
+To protect privacy and comply with international data protection regulations, all data collection adhered strictly to privacy regulations and ethical guidelines. Trajectories were anonymized, and any personally identifiable information was excluded to protect user privacy. In addition, all raw data follows the Open Data Commons Open Database License (ODbL) license from OSM: http://opendatacommons.org/licenses/odbl/1.0/. We will share derived datasets under the same license terms to respect the data use policies of the community.
+
+# B Pre-training Strategies
+
+In this section, we provide specific details on the adaptive trajectory resampling strategy and the self-supervised trajectory masking strategy, and we will provide the design motivation and theoretical analysis for these two strategies.
+
+# B.1 Adaptive Trajectory Resampling
+
+Trajectory data heterogeneous is one of the main challenges in cross-regional and cross-device trajectory modeling. The Adaptive Trajectory Resampling strategy solves this problem through two complementary components: Dynamic Multi-Scale Resampling and Interval Consistent Resampling. We designed these two strategies with the motivation of fitting different regions and dataset qualities through diversified trajectory sampling frequencies and motion patterns. Dynamic Multi-Scale Resampling ensures an optimal balance between information preservation and computational efficiency across different trajectory lengths, prioritizing the retention of key motion patterns. Interval Consistent Resampling enhances the model's generalization ability across datasets with different sampling rates by normalizing the time dimension.
+
+# B.1.1 Dynamic Multi-Scale Resampling
+
+As discussed in Section 4.2, we adopted a logarithmic resampling ratio that adjusts the sampling rate according to the trajectory length. The resampling ratio function  $R(n)$  is designed to decrease logarithmically as the trajectory length  $n$  increases:
+
+$$
+R (n) = \left\{ \begin{array}{l l} R _ {\min }, & n \geq n _ {\max } \\ 1 - \left(1 - R _ {\min }\right) \phi (n), & n _ {\min } <   n <   n _ {\max } \\ 1, & n \leq n _ {\min } \end{array} \right. \tag {7}
+$$
+
+where  $R_{\mathrm{min}}$  is the minimum sampling ratio, and  $n_{\mathrm{min}}$  and  $n_{\mathrm{max}}$  denotes the shortest and longest length thresholds, respectively. The normalization factor  $\phi(n)$  is computed as follows:
+
+$$
+\phi (n) = \frac {\ln \left(n - n _ {\min } + 1\right)}{\ln \left(n _ {\max } - n _ {\min } + 1\right)}. \tag {8}
+$$
+
+Formal Definition. Here, we provide a formal definition and theoretical analysis of the above empirical results through information theory and computational efficiency perspectives. For any trajectory  $\pmb{\tau} = \{p_1, p_2, \dots, p_n\}$  consists of  $n$  spatio-temporal points. The number of points for resampled trajectory  $\pmb{\tau}' = \{p_1, p_2, \dots, p_m\}$  is:
+
+$$
+m = R (n) \cdot n, \tag {9}
+$$
+
+where function  $R(n)$  that determines what proportion of points to retain. The logarithmic sampling strategy guarantees bounded sample sizes for arbitrarily long trajectories while preserving critical minimum information content. Specifically:
+
+- For  $n \leq n_{\min}$ :  $R(n) = 1$ , so  $m = n$ ;  
+- For  $n \geq n_{\max}$ :  $R(n) = R_{\min}$ , we set  $m = m_{\max}$  as a constant. Clearly, the number of sampled points is bounded above by  $m_{\max}$ .
+
+To ensure boundedness, we analyze  $m(n)$  in the intermediate domain  $n \in (n_{\min}, n_{\max})$ .
+
+$$
+m = \left[ 1 - \left(1 - R _ {\min }\right) \cdot \phi (n) \right] \cdot n. \tag {10}
+$$
+
+Taking derivative:
+
+$$
+\frac {d (R (n) \cdot n)}{d n} = 0 \tag {11}
+$$
+
+Solving this equation yields a value  $n^* < n_{\max}$ , ensuring that  $m_{\max}$  is bounded. Since  $R(n)$  becomes constant for  $n \geq n_{\max}$ , and  $m$  increases linearly in that region, the global maximum occurs at either  $n^*$  or  $n_{\max}$ . However, due to the logarithmic decay of  $R(n)$ , the growth of  $m$  slows, and the maximum value is achieved at a finite  $n^* < n_{\max}$ . Hence,  $m$  is bounded for all  $n$ .
+
+# Corollary 1: Information Preservation and Computing Efficiency Optimization
+
+**Standpoint:** The logarithmic sampling function provides an optimal balance between information preservation and computational efficiency across varying trajectory lengths.
+
+Proof: Let  $I(\tau)$  represent the information content of trajectory  $\tau$ . Empirical studies in spatiotemporal data analysis suggest that information content typically scales sub-linearly with trajectory length, following approximately:
+
+$$
+I (\boldsymbol {\tau}) \propto n ^ {\alpha}, \tag {12}
+$$
+
+where  $0 < \alpha < 1$ . For example,  $\alpha \approx 0.7$  indicates that only  $70\%$  of the trajectory points contain valid feature information, and the remaining  $30\%$  are redundant. For a resampled trajectory  $\tau'$  with  $m = R(n) \cdot n$  points, the information preservation ratio  $\eta$  can be approximated as:
+
+$$
+\eta = \frac {I \left(\boldsymbol {\tau} ^ {\prime}\right)}{I (\boldsymbol {\tau})} \approx \left(\frac {m}{n}\right) ^ {\alpha} = R (n) ^ {\alpha}. \tag {13}
+$$
+
+The computational cost  $C$  of processing trajectory typically scales linearly with length:
+
+$$
+C (\tau) \propto n ^ {\beta}, \tag {14}
+$$
+
+where  $\beta \geq 1$ , typically  $\beta \approx 2$  for Transformer-based models. After resampling, the computational efficiency gain  $\gamma$  is:
+
+$$
+\gamma = \frac {C (\boldsymbol {\tau})}{C \left(\boldsymbol {\tau} ^ {\prime}\right)} \approx \left(\frac {n}{m}\right) ^ {\beta} = \frac {1}{R (n) ^ {\beta}}. \tag {15}
+$$
+
+The optimal sampling function maximizes the product of information preservation and computational efficiency:
+
+$$
+\max  _ {R (n)} \eta \cdot \gamma = \max  _ {R (n)} R (n) ^ {\alpha} \cdot \frac {1}{R (n) ^ {\beta}} = \max  _ {R (n)} R (n) ^ {\alpha - \beta}. \tag {16}
+$$
+
+Since  $\alpha < \beta$  for typical trajectory data, this is a decreasing function  $R(n)$ . However, we must maintain a minimum level of information, hence the constraint  $R(n) \geq R_{\min}$ .
+
+When we examine the information density:
+
+$$
+D (n) = \frac {I (\tau)}{n} \propto n ^ {\alpha - 1}, \tag {17}
+$$
+
+we observe that it decreases as  $n$  increases, indicating diminishing information return per point in longer trajectories. An optimal sampling ratio should proportionally track this information density:
+
+$$
+R _ {\text {o p t}} (n) \propto D (n) \propto n ^ {\alpha - 1}. \tag {18}
+$$
+
+Our logarithmic resampling function's derivative in the intermediate domain  $(n_{\mathrm{min}} < n < n_{\mathrm{max}})$  is:
+
+$$
+\frac {d R (n)}{d n} = - \frac {1 - R _ {\operatorname* {m i n}}}{\ln \left(n _ {\operatorname* {m a x}} - n _ {\operatorname* {m i n}} + 1\right)} \cdot \frac {1}{n - n _ {\operatorname* {m i n}} + 1} \propto \frac {1}{n} \tag {19}
+$$
+
+As  $n$  increases, the growth rate of the logarithmic function slows down, causing the rate at which the sampling rate  $R(n)$  decreases to also slow down. This is closely proportional to the derivative of the theoretical optimal sampling rate:
+
+$$
+\frac {d R _ {\text {o p t}} (n)}{d n} \propto (\alpha - 1) n ^ {\alpha - 2} \propto \frac {1}{n ^ {2 - \alpha}}. \tag {20}
+$$
+
+For example, when  $\alpha \approx 0.7$ , we have  $\frac{dR_{\mathrm{opt}}(n)}{dn} \propto \frac{1}{n^{1.3}}$ . This property of logarithmic functions (their rate of change is inversely proportional to the input value), making them naturally suited to this task. Therefore, Logarithmic resampling provides a theoretically reasonable compromise: it preserves almost all of the information from short trajectories (where every point may be significant) while reducing redundancy in long trajectories (where redundancy is highest). Compared to linear functions, logarithmic functions can more naturally adapt to the information density curve across the entire trajectory length range.
+
+![](images/e3d8118ffe6d1cd56afdbec600f14b9d8b000f637baf088f960c2b9aa49a4141.jpg)  
+Figure 6: Illustration of the difference between dynamic resampling strategies with linear method.
+
+Visualization. As shown in Figure 6, we compare in detail the proposed dynamic resampling strategy with a linear resampling strategy (where the sampling ratio  $R(n)$  decreases linearly with the length of the trajectory) regarding the sampling ratio and the sampled points. Specifically, this figure illustrates the dynamic resampling strategy compared to a linear resampling approach. The top plot displays how the sampling ratio  $R(n)$  decreases with trajectory length  $n$ . The dynamic strategy (orange curve) follows a logarithmic decrease, ensuring a smoother transition from retaining all points for short trajectories ( $n \leq n_{\mathrm{min}}$ ) to reducing redundancy for long trajectories ( $n \geq n_{\mathrm{max}}$ ), with a minimum sampling ratio  $R_{\mathrm{min}}$ . In contrast, the linear resampling strategy (blue curve) decreases the sampling ratio at a constant rate. The bottom plot shows the relationship between the number of sampled points and trajectory length for both strategies. The dynamic approach adjusts sampling more gradually, preserving detail for intermediate trajectories while minimizing redundancy in longer trajectories. However, linear sampling methods instead suffer from redundancy of sampling points due to the smoothly decreasing sampling rate. This dynamic resampling strategy ensures a balance between data volume reduction and the retention of critical movement details. The visual comparison highlights the adaptive nature of the dynamic strategy.
+
+# B.1.2 Interval Consistent Resampling
+
+Consider different cities may exhibit drastically different sampling intervals due to: Varying data collection protocols (e.g., 1s in City A vs. 5s in City B) and technical limitations or regional preferences in tracking technologies. This heterogeneity poses a serious challenge for developing universal trajectory models, as models trained on data from one region may fail to generalize to regions with different sampling characteristics. Therefore, we performed consistent interval sampling (at random time intervals) on the original dataset to ensure its generalizability across different datasets. Specifically, ICR standardizes the temporal intervals between trajectory points, transforming a trajectory  $\pmb{\tau} = \{(x_1,y_1,t_1),(x_2,y_2,t_2),\ldots ,(x_n,y_n,t_n)\}$  with irregular time intervals into a trajectory  $\pmb{\tau}' = \{(x_{1},y_{1},\Delta t),(x_{2},y_{2},\Delta t),\ldots ,(x_{m},y_{m},\Delta t)\}$  with uniform time intervals  $\Delta t = t_{i + 1} - t_i$ , for all  $i\in [1,m - 1]$ .
+
+# Corollary 2: Temporal Regularity for Cross-Dataset Generalization
+
+**Standpoint:** Interval consistent resampling regularizes the temporal dimension of trajectory samples, enhancing the model's ability to generalize across datasets with heterogeneous sampling rates.
+
+Proof: Let  $\mathcal{D}_1$  and  $\mathcal{D}_2$  be two datasets of region with average sampling intervals  $\mu_{\Delta t}^{(1)}$  and  $\mu_{\Delta t}^{(2)}$ . Assume the temporal pattern recognition task can be formalized as learning a function  $f_{\theta}: \tau \to Y$  where the learned parameters  $\theta$  should ideally be robust to sampling rate variations. For trajectories with irregular sampling, the model must learn the relationship:
+
+$$
+y = f _ {\theta} \left(\left(x _ {1}, y _ {1}, t _ {1}\right), \left(x _ {2}, y _ {2}, t _ {2}\right), \dots , \left(x _ {n}, y _ {n}, t _ {n}\right)\right). \tag {21}
+$$
+
+This requires implicitly learning the distribution of time intervals  $P(\Delta T)$ , which varies across datasets. With ICR, the learning problem becomes:
+
+$$
+y = f _ {\theta} \left(\left(x _ {1} ^ {\prime}, y _ {1} ^ {\prime}, t _ {1} ^ {\prime}\right), \left(x _ {2} ^ {\prime}, y _ {2} ^ {\prime}, t _ {2} ^ {\prime}\right), \dots , \left(x _ {m}, y _ {m}, t _ {m}\right)\right), \quad \text {w i t h} t _ {i + 1} ^ {\prime} - t _ {i} ^ {\prime} = \Delta t _ {\text {f i x e d}} \tag {22}
+$$
+
+where temporal intervals are now consistently fixed, eliminating the need to learn dataset-specific temporal distributions.
+
+Information Entropy Analysis. From the entropy perspective, consider trajectories from different regions  $r$  with characteristic sampling intervals  $\Delta t^r$ , where the distribution of intervals can be modeled as:
+
+$$
+P (\Delta t \mid r) \sim \mathcal {N} \left(\mu_ {r}, \sigma_ {r} ^ {2}\right), \tag {23}
+$$
+
+where  $\mathcal{N}$  is a dataset distribution with region-specific mean  $\mu_r$  and variance  $\sigma_r^2$ . The entropy of the joint distribution of regions (or dataset) and sampling intervals is:
+
+$$
+H (\mathcal {D}, \Delta T) = H (\mathcal {D}) + H (\Delta T \mid \mathcal {D}). \tag {24}
+$$
+
+This high conditional entropy  $H(\Delta T \mid \mathcal{D})$  creates a strong statistical correlation between regions and temporal patterns, forcing region-specific model adaptations. Interval Consistent Resampling transforms the original trajectory  $\pmb{\tau}$  into  $\pmb{\tau}'$  where  $t_{i+1}' - t_i' = \Delta t_{\mathrm{fixed}} \forall i \in [1,m-1]$ . This transformation minimizes the conditional entropy:
+
+$$
+H \left(\Delta T ^ {\prime} \mid \mathcal {D}\right) \approx 0, \tag {25}
+$$
+
+which effectively decoupling the temporal sampling pattern from the region. This transformation reduces dataset-specific temporal variability, thereby bringing the conditional distributions of trajectories across datasets closer in distributional space:
+
+$$
+P \left(\boldsymbol {\tau} ^ {\prime} \mid \mathcal {D} _ {1}\right) \approx P \left(\boldsymbol {\tau} ^ {\prime} \mid \mathcal {D} _ {2}\right). \tag {26}
+$$
+
+The reduction means the model sees more consistent input distributions, thus reducing the domain gap in learning.
+
+For trajectory modeling tasks that focus on spatial patterns rather than absolute temporal dynamics, information loss is minimal when resampling preserves relative temporal order and approximate speed relationships. For a trajectory with velocity profile  $v(t) = (p_{i+1} - p_i) / (t_{i+1} - t_i)$ , the constraint:
+
+$$
+\frac {\left\| p _ {i + 1} ^ {\prime} - p _ {i} ^ {\prime} \right\|}{\left\| p _ {i + 1} - p _ {i} \right\|} \approx \frac {\Delta t _ {\text {f i x e d}}}{\Delta t _ {i}} \tag {27}
+$$
+
+ensures that relative speed information is preserved even as absolute time intervals are normalized.
+
+# B.2 Self-supervised Trajectory Masking
+
+Self-supervised Trajectory Masking (STM) forms a critical component of UniTraj's pre-training strategy, enabling the model to learn robust representations from incomplete trajectory data. While we introduced the concept in the main paper, this appendix provides a more detailed examination of the theoretical foundations, implementation details, and empirical justifications for our masking approach. Our Self-supervised Trajectory Masking framework implements four complementary masking strategies (as illustrated in Figure 7), each designed to simulate different types of real-world data incompleteness and encourage specific learning objectives:
+
+# B.2.1 Random Masking
+
+Random Masking applies a uniform probability distribution to select trajectory points for masking, where each point has an equal chance of being masked regardless of its position or significance. Formally, we select a subset of indices:
+
+$$
+\mathbf {I} _ {\text {r a n d}} \sim \operatorname {U n i f o r m} (\{1, 2, \dots , n \}). \tag {28}
+$$
+
+![](images/87b7e6409bdcbec1be356142861c54417b245ca5446467d8be894172710f345d.jpg)  
+Figure 7: Illustration of the difference masking strategies.
+
+to mask. This strategy forces the model to develop both local and global dependencies, as it must learn to infer missing points from surrounding context without relying on predictable patterns. Random masking is a general masking strategy used to simulate sensor failures or temporary GPS signal loss that often occur in random trajectories.
+
+# B.2.2 Block Masking
+
+Block Masking conceals consecutive segments of the trajectory by selecting a starting point  $k$  and masking  $b$  consecutive points:
+
+$$
+\mathbf {I} _ {\text {b l o c k}} = \{k, k + 1, \dots , k + b - 1 \}, \text {f o r s o m e} k. \tag {29}
+$$
+
+This approach simulates extended sensor failures, tunnels, or urban canyons where trajectory data may be unavailable for continuous periods. The strategy challenges the model to reconstruct substantial missing segments by understanding the broader movement context, encouraging the development of long-range dependencies and trajectory continuity reasoning.
+
+# B.2.3 Key Points Masking
+
+Algorithm 1 Ramer-Douglas-Peucker (RDP) Algorithm  
+1: RDP(τ, s, e, ε)  
+2: Initialize max distance  $d_{\mathrm{max}} \gets 0$   
+3: Initialize index  $k \gets -1$   
+4: for  $i = s + 1$  to  $e - 1$  do  
+5: Calculate the distance from  $p_i$  to  $\overline{pspe}$ :  $d_i$   
+6: if  $d_i > d_{\mathrm{max}}$  then  
+7: Update max distance  $d_{\mathrm{max}} \gets d_i$   
+8: Update index  $k \gets i$   
+9: end if  
+10: end for  
+11: if  $d_{\mathrm{max}} > \epsilon$  then  
+12:  $\tau_{\mathrm{left}} \gets \mathrm{RDP}(\tau, s, k, \epsilon)$   
+13:  $\tau_{\mathrm{right}} \gets \mathrm{RDP}(\tau, k, e, \epsilon)$   
+14: return  $\{p_k\} \cup \tau_{\mathrm{left}} \cup \tau_{\mathrm{right}}$   
+15: else  
+16: return  $\{p_s, p_e\}$   
+17: end if
+
+The key points masking adopt the Ramer-Douglas-Peucker (RDP) algorithm [10], which simplifies a trajectory by retaining points that are farthest from the line  $\overline{p_1p_n}$  connecting the first and last points. The indices are determined by
+
+$$
+\mathbf {I} _ {\text {k e y}} = \left\{p _ {k} \mid d _ {\max } \left(p _ {k}, \overline {{p _ {1} p _ {n}}}\right) > \epsilon \right\}, \tag {30}
+$$
+
+where  $\epsilon$  is a predefined threshold, and  $d_{\mathrm{max}} = \max \left\{d(p_k,\overline{p_1p_n})\mid 2\leq k\leq n - 1\right\}$  is the maximum distance measures deviation from this line. As summarized in Algorithm 1, the RDP algorithm iteratively identifies the point  $p_k$  that maximizes  $d_{\mathrm{max}} = d(p_k,\overline{p_1p_n})$ . If  $d_{\mathrm{max}} > \epsilon$ , the corresponding point  $p_k$  is treated as a key point and included in the mask set Ikey. This process is recursively applied to the trajectory segments  $\pmb{\tau}_{\mathrm{left}} = \{p_1,\dots ,p_k\}$  and  $\pmb{\tau}_{\mathrm{right}} = \{p_k,\dots ,p_n\}$ , isolating critical points
+
+for masking. By focusing on these pivotal points, the model is challenged to reconstruct essential trajectory segments, reinforcing its understanding of key structural patterns within trajectories.
+
+# B.2.4 Last N Masking
+
+Last N Masking systematically removes the final N points of each trajectory:
+
+$$
+\mathbf {I} _ {\text {l a s t}} = \{n - N + 1, n - N + 2, \dots , n \}. \tag {31}
+$$
+
+This strategy explicitly simulates trajectory prediction scenarios where future positions must be forecasted based on historical observations. By incorporating this masking approach during pretraining, the model develops capabilities directly applicable to trajectory prediction tasks, creating a natural bridge between self-supervised pre-training and downstream forecasting applications.
+
+# Corollary 3: Robustness through Comprehensive Masking
+
+**Standpoint:** Self-supervised Trajectory Masking improves the robustness and generalization ability of the model to incomplete and heterogeneous trajectory data through a comprehensive masking strategy, enabling the model to learn more effective trajectory representations.
+
+Proof: Let the trajectory data space be  $\mathcal{D}$ , with a true data distribution denoted as  $P(\tau)$ . In real-world applications, due to device limitations, communication failures, and environmental factors, the observed trajectories are often incomplete or irregular, and their distribution is denoted as  $P(\tilde{\tau})$ . The incompleteness of trajectory data can be formalized as a conditional distribution  $P(\tilde{\tau} \mid \tau)$ , representing the probability of observing an incomplete  $\tilde{\tau}$  given a complete trajectory  $\tau$ .
+
+STM can be formalized as a set of masking functions  $\{\mathcal{M}_1,\mathcal{M}_2,\ldots ,\mathcal{M}_k\}$ , each corresponding to a different masking strategy. For a resampled trajectory  $\pmb {\tau}' = \{p_1,p_2,\dots ,p_n\}$ , the masking function  $\mathcal{M}_i$  transforms it as:
+
+$$
+\tilde {\boldsymbol {\tau}} _ {i} = \mathcal {M} _ {i} \left(\boldsymbol {\tau} ^ {\prime}, r _ {i}\right) = \left\{p _ {1}, \dots , \left[ \operatorname {M A S K} \right] _ {j \in \mathbf {I} _ {i}}, \dots , p _ {n} \right\} \tag {32}
+$$
+
+where  $\mathbf{I}_i\subseteq \{1,2,\ldots ,n\}$  is the index set of masked positions and  $r_i = |\mathbf{I}_i| / n$  is the masking ratio.
+
+Information-Theoretic Analysis. From an information-theoretic perspective, STM introduces an artificial information bottleneck that forces the model to learn efficient representations. We define the model objective as minimizing the reconstruction loss:
+
+$$
+\mathcal {L} (\theta) = \mathbb {E} _ {\boldsymbol {\tau} \sim P (\boldsymbol {\tau}), i \sim \mathcal {U} (1, k)} \left[ d \left(f _ {\theta} \left(\mathcal {M} _ {i} (\boldsymbol {\tau}, r _ {i})\right), \boldsymbol {\tau}\right) \right], \tag {33}
+$$
+
+where  $d$  is a chosen distance metric.
+
+During training, the model needs to learn the joint distribution  $P(\pmb{\tau}, \tilde{\pmb{\tau}}_i)$  and estimate the conditional distribution  $P(\pmb{\tau} \mid \tilde{\pmb{\tau}}_i)$ . By Bayes' theorem:
+
+$$
+P (\boldsymbol {\tau} \mid \tilde {\boldsymbol {\tau}} _ {i}) = \frac {P \left(\tilde {\boldsymbol {\tau}} _ {i} \mid \boldsymbol {\tau}\right) P (\boldsymbol {\tau})}{P \left(\tilde {\boldsymbol {\tau}} _ {i}\right)}. \tag {34}
+$$
+
+By using diverse masking strategies, the model learns to estimate  $P(\pmb{\tau} \mid \tilde{\pmb{\tau}}_i)$  across different types of masked trajectories, which is equivalent to learning the true trajectory distribution  $P(\pmb{\tau})$  and the various degradation mechanisms  $P(\tilde{\pmb{\tau}}_i \mid \pmb{\tau})$ .
+
+Optimality Theory of Diversity Complementary Masking Strategies. A key innovation in STM is the use of multiple complementary masking strategies. We define the coverage region of the union of masking strategies as:
+
+$$
+\mathcal {C} \left(\left\{\mathcal {M} _ {1}, \dots , \mathcal {M} _ {k} \right\}\right) = \int_ {\tilde {\boldsymbol {\tau}} \in \mathcal {D}} \max  _ {i \in \{1, \dots , k \}} P _ {\mathcal {M} _ {i}} (\tilde {\boldsymbol {\tau}}) d \tilde {\boldsymbol {\tau}}, \tag {35}
+$$
+
+where  $P_{\mathcal{M}_i}(\tilde{\tau})$  denotes the distribution of incomplete trajectories generated by masking strategy  $\mathcal{M}_i$ .
+
+We assert that for a suitable masking ratio and a diverse set of masking strategies  $\{\mathcal{M}_1,\dots ,\mathcal{M}_k\}$ , the combined coverage region satisfies:
+
+$$
+\mathcal {C} \left(\left\{\mathcal {M} _ {1}, \dots , \mathcal {M} _ {k} \right\}\right) > \max  _ {i \in \{1, \dots , k \}} \mathcal {C} \left(\left\{\mathcal {M} _ {i} \right\}\right). \tag {36}
+$$
+
+This inequality indicates that the joint use of diverse masking functions provides strictly better coverage over possible incomplete trajectories than any individual strategy.
+
+The advantage of combining multiple masking strategies in STM over using a single masking strategy can also be theoretically justified by comparing the expected reconstruction error. Assume the real-world conditional distribution of incomplete trajectories is  $P_{\mathrm{real}}(\tilde{\tau} \mid \tau)$ . For a single masking strategy  $\mathcal{M}_i$ , let the generated distribution be  $P_{\mathcal{M}_i}(\tilde{\tau} \mid \tau)$ . Then the expected reconstruction error under this distribution is:
+
+$$
+\mathbb {E} _ {\boldsymbol {\tau} \sim P (\boldsymbol {\tau}), \tilde {\boldsymbol {\tau}} \sim P _ {\text {r e a l}} (\tilde {\boldsymbol {\tau}} | \boldsymbol {\tau})} [ d (f _ {\theta} (\tilde {\boldsymbol {\tau}}), \boldsymbol {\tau}) ] \tag {37}
+$$
+
+It can be shown that training with a mixture of multiple masking strategies leads to a lower bound on this error compared to using any single strategy. This is because the mixture of diverse masking strategies better approximates the true real-world distribution of incomplete trajectories:
+
+$$
+K L \left(P _ {\text {r e a l}} (\tilde {\boldsymbol {\tau}} \mid \boldsymbol {\tau}) \left\| \frac {1}{k} \sum_ {i = 1} ^ {k} P _ {\mathcal {M} _ {i}} (\tilde {\boldsymbol {\tau}} \mid \boldsymbol {\tau}) \right.\right) <   \min  _ {i \in \{1, \dots , k \}} K L \left(P _ {\text {r e a l}} (\tilde {\boldsymbol {\tau}} \mid \boldsymbol {\tau}) \| P _ {\mathcal {M} _ {i}} (\tilde {\boldsymbol {\tau}} \mid \boldsymbol {\tau})\right) \tag {38}
+$$
+
+Here,  $KL(\cdot \| \cdot)$  denotes the Kullback-Leibler divergence.
+
+# C Details of UniTraj
+
+In this section, we provide a detailed implementation of UniTraj, including the architecture and parameter settings.
+
+# C.1 Overall Architecture
+
+The UniTraj model adopts an encoder-decoder architecture based on transformer blocks, designed to process trajectory data with minimal regional dependency and maximum task adaptability. Figure 8 illustrates the overall framework of UniTraj, which consists of several key components: spatiotemporal tokenization, encoder, decoder, and rotary embedding layers.
+
+Our model takes trajectory points that have already undergone adaptive resampling (ATR) and masking (STM) as described in Appendix B. The input trajectories are represented as sequences of latitude-longitude coordinates and timestamps:  $\tau = \{(\mathrm{lng}_i, \mathrm{lat}_i, t_i) | i = 1, 2, \dots, n\}$ , where  $n$  is the total number of points after resampling. Unlike previous approaches that rely on region-specific features or road network information, UniTraj operates solely on these basic coordinates, enhancing its universal applicability across diverse geographic contexts.
+
+![](images/c742590ad9054b985cf38f60673425632328f7a07a3ebe46fdab8823d19efac9.jpg)  
+Figure 8: The main architecture and components of UniTraj.
+
+# C.2 Input Representation and Embedding
+
+Spatio-Temporal Tokenization. To enhance numerical stability and generalization, all input coordinates are normalized relative to the first point in the trajectory  $(x_{i},y_{i}) = (\mathrm{lng}_{i} - \mathrm{lng}_{1},\mathrm{lat}_{i} - \mathrm{lat}_{1})$
+
+For the spatial component, we project the normalized coordinates into a  $d$ -dimensional space using a 1D convolutional neural network, yielding a spatial embedding
+
+$$
+\boldsymbol {h} _ {i} ^ {s} = \operatorname {C o n v 1 D} \left(\left[ x _ {i}, y _ {i} \right]; \theta_ {s}\right), \tag {39}
+$$
+
+where  $\theta_{s}$  represents the learnable parameters of the convolutional layer. We use a kernel size of 1 with no stride to capture local spatial dependencies. Similarly, the temporal component, based on the time intervals  $\Delta t_i$ , is embedded into the same  $d$ -dimensional space via a linear layer, resulting in a temporal embedding:
+
+$$
+\boldsymbol {h} _ {i} ^ {t} = W _ {t} \cdot \Delta t _ {i} + b _ {t}, \tag {40}
+$$
+
+where  $W_{t} \in \mathbb{R}^{d \times 1}$  and  $b_{t} \in \mathbb{R}^{d}$  are learnable parameters. The final embedding for each trajectory point is obtained by element-wise addition of the spatial and temporal components:
+
+$$
+\boldsymbol {h} _ {i} = \boldsymbol {h} _ {i} ^ {s} + \boldsymbol {h} _ {i} ^ {t} \tag {41}
+$$
+
+This dual-tokenization captures both spatial and temporal dynamics, enabling the model to learn relative movement and temporal dependencies effectively.
+
+Rotary Positional Encoding (RoPE). In addition to encoding the spatial and temporal details of each trajectory point, it is essential to capture the relative positional relationships between points. These relationships enable the model to comprehend the movement sequence and the timing between points, both crucial for accurate trajectory modeling. To achieve this, we employ Rotary Position Encoding (RoPE) [32], which maintains the relative positional information between points by rotating the trajectory embedding vectors. Given the combined spatial-temporal embeddings  $h_i$  for point  $i$  in the trajectory, RoPE applies a rotational transformation:
+
+$$
+\operatorname {R o P E} \left(\boldsymbol {h} _ {i}\right) = \left( \begin{array}{c c} \cos \theta_ {i} & - \sin \theta_ {i} \\ \sin \theta_ {i} & \cos \theta_ {i} \end{array} \right) \left( \begin{array}{l} \boldsymbol {h} _ {i} ^ {(1)} \\ \boldsymbol {h} _ {i} ^ {(2)} \end{array} \right), \tag {42}
+$$
+
+where  $\pmb{h}_i^{(1)}$  and  $\pmb{h}_i^{(2)}$  are the first and second halves of the embedding  $\pmb{h}_i$ , and  $\theta_i$  is a rotation angle that varies proportionally with the position index  $i$ . Specifically,  $\theta_i$  is calculated as  $\theta_i = \frac{i}{10000^{2k / d}}$ , where  $k$  is the index of the embedding dimension, and  $d$  is the total dimension of the embedding.
+
+The main advantage of RoPE is its ability to preserve relative positional information through rotational symmetry. This ensures that the relative distance and directional relationships between points are maintained, enabling the model to capture both local patterns (e.g., short-term movements) and global patterns (e.g., long-range directionality) within a trajectory. By encoding these relative positions, RoPE strengthens the model's capacity to understand movement dynamics across varying scales.
+
+# C.3 Adaptive Representation Learning
+
+The UniTraj employs an encoder-decoder architecture [13] tailored for trajectory data. The encoder and decoder use Transformer blocks [33] with RoPE-powered self-attention mechanisms to capture dependencies within trajectory embeddings.
+
+Encoder. Given a masked trajectory  $\tilde{\pmb{\tau}} = \{p_1,\dots ,[\mathrm{MASK}]_{i\in \mathbf{I}},\dots ,p_n\}$ , we first extract the embedding representations of the unmasked points  $\mathbf{H} = \{h_1,h_2,\dots ,h_m\}$  (where  $m\leq n$  and  $i\notin \mathbf{I}$ ) through the tokenizer and positional encoding steps. The encoder  $\mathbf{E}_{\theta}$  processes the visible (unmasked) points in a trajectory to generate contextualized representations. It consists of  $L_{e}$  transformer blocks, each incorporating:
+
+1. Multi-head Self-attention with RoPE: As described previously, we apply RoPE to the self-attention mechanism:
+
+$$
+\operatorname {A t t e n t i o n} (Q, K, V) = \operatorname {s o f t m a x} \left(\frac {Q _ {\mathrm {R o P E}} \cdot K _ {\mathrm {R o P E}} ^ {T}}{\sqrt {d _ {k}}}\right) \cdot V \tag {43}
+$$
+
+where  $Q_{\mathrm{RoPE}}$  and  $K_{\mathrm{RoPE}}$  are the query and key matrices with RoPE applied.
+
+2. Feed-forward Network (FFN): A two-layer FFN with GELU activation:
+
+$$
+\operatorname {F F N} (x) = W _ {2} \cdot \operatorname {G E L U} \left(W _ {1} \cdot x + b _ {1}\right) + b _ {2} \tag {44}
+$$
+
+3. Layer Normalization and Residual Connections: Each sub-block is wrapped with layer normalization (Pre-LN) and residual connections:
+
+$$
+\mathbf {H} ^ {\prime} = \operatorname {L a y e r N o r m} (\mathbf {H} + \operatorname {A t t e n t i o n} (\mathbf {H})) \tag {45}
+$$
+
+$$
+\mathbf {H} ^ {\prime} = \operatorname {L a y e r N o r m} \left(\mathbf {H} ^ {\prime} + \operatorname {F F N} \left(\mathbf {H} ^ {\prime}\right)\right) \tag {46}
+$$
+
+The encoder's output is a set of hidden representations  $\mathbf{H}^e = \{\pmb {h}_i^e |i = 1,2,\dots ,m\}$  for the  $m$  visible points.
+
+Decoder. The decoder reconstructs the masked points based on the contextualized representations from the encoder. It operates by combining the encoder's embeddings with mask tokens and processing them through  $L_{d}$  transformer layers:
+
+1. Input Combination: The decoder input consists of both the encoder outputs for visible points and the mask token embeddings for masked positions:
+
+$$
+\mathbf {H} _ {0} ^ {d} = \operatorname {R e o r d e r} \left(\left\{ \begin{array}{l l} h _ {i} = \mathbf {h} _ {j} ^ {e} & \text {i f} i = \operatorname {I n d e x} (j), i \notin \mathbf {I} \\ h ^ {\text {m a s k}} & \text {i f} i \in \mathbf {I} \end{array} \right\}\right), \tag {47}
+$$
+
+where  $h^{\mathrm{mask}}$  represents the mask token embeddings for all masked positions.
+
+2. Decoder Transformer Blocks: The combined input is processed through  $L_{d}$  transformer blocks, each with the same structure as the encoder blocks (self-attention with RoPE, FFN, layer normalization, and residual connections). The self-attention mechanism allows information to flow between visible and masked positions:
+
+$$
+\mathbf {H} _ {l} ^ {d} = \operatorname {T r a n s f o r m e r B l o c k} \left(\mathbf {H} _ {l - 1} ^ {d}\right) \tag {48}
+$$
+
+for  $l\in \{1,2,\ldots ,L_d\}$
+
+3. Output Projection: The final layer projects the decoder's representations for the masked positions back to coordinate space:
+
+$$
+\left(\hat {x} _ {j}, \hat {y} _ {j}\right) = W _ {o} \cdot \boldsymbol {h} _ {L _ {d}, j} ^ {d} + b _ {o} \tag {49}
+$$
+
+where  $j$  indexes the masked positions, and  $W_{o}\in \mathbb{R}^{2\times d}$  and  $b_{o}\in \mathbb{R}^{2}$  are learnable parameters. These projected coordinates are then transformed back to the original coordinate system  $(\hat{\mathrm{lng}}_j,\hat{\mathrm{lat}}_j) = (\hat{x}_j + \mathrm{lng}_1,\hat{y}_j + \mathrm{lat}_1)$ .
+
+UniTraj is trained using a self-supervised learning approach with a reconstruction loss function. For each trajectory, we apply our masking strategies (random, block, key points, or last N), and the model is trained to reconstruct these masked points:
+
+$$
+\mathcal {L} = \frac {1}{| \mathbf {I} |} \sum_ {j \in \mathbf {I}} \| (\hat {x} _ {j}, \hat {y} _ {j}) - (x _ {j}, y _ {j}) \| _ {2} ^ {2}, \tag {50}
+$$
+
+where  $\mathbf{I}$  is the set of masked positions, and  $\| \cdot \|$  denotes the L2 norm.
+
+# C.4 Task-Specific Adaptation
+
+For downstream applications, UniTraj can be used in two primary ways:
+
+1. Zero-shot Transfer: The pre-trained model's encoder can be directly applied to extract trajectory representations for various tasks without further training. We use the pre-trained UniTraj as a backbone and attach task-specific Multi-Layer Perceptron (MLP) adapters to the output:
+
+$$
+\mathbf {H} ^ {\text {f i n a l}} = \operatorname {M L P} \left(\operatorname {U n i T r a j} _ {\text {e n c o d e r}} (\tau)\right) \tag {51}
+$$
+
+The MLP adapter typically consists of 2-3 layers with non-linear activations:
+
+$$
+\mathbf {H} ^ {\mathrm {a d a}} = W _ {2} \cdot \operatorname {R e L U} \left(W _ {1} \cdot \mathbf {H} ^ {e} + b _ {1}\right) + b _ {2} \tag {52}
+$$
+
+where  $\mathbf{H}^e$  is the output from the UniTraj encoder.
+
+2. Fine-tuning: Update all parameters of the backbone and adapters with specific dataset.
+
+For different downstream tasks, we design specific adapter architectures:
+
+- For Trajectory Recovery/Prediction: We can directly use UniTraj's decoder as an adapter without any additional modifications.  
+- For Trajectory Classification: The adapter includes pooling operations followed by fully connected layers to produce class logits.  
+- For Trajectory Generation: The adapter interfaces with generative models by providing conditioned trajectory embeddings.
+
+# C.5 Implementation Details
+
+Additionally, we summarize the list of key hyperparameters and implementation-specific settings that may be used in the implementation of UniTraj in Table 5. Specifically, our model contains 8 encoders and 4 decoders, each using 4 heads in the attention layer. The model has approximately 2.38 million parameters, allowing it to balance complexity and computational efficiency. We set the embedding dimension to 128 and employ RoPE to capture spatial and temporal relationships effectively. Our model can handle an arbitrary length of the number of trajectory points and pad it to a length of 200. Naturally, due to the use of rotational positional embedding, our model holds extension capability and supports a maximum length of 512. In addition, when performing the dynamic resampling strategy, we set the minimum number of sampling points to 36 and the maximum to 600, and its minimum sampling rate is 0.35. Finally, we provide the probability of using various masking strategies during training, which can be further adapted to the specific task as we discussed in Section 5.4 and Table 4.
+
+Table 5: General parameters setting for UniTraj.  
+
+<table><tr><td>Parameter</td><td>Setting value</td><td>Refer range</td></tr><tr><td>Encoder Blocks</td><td>8</td><td>≥ 2</td></tr><tr><td>Decoder Blocks</td><td>4</td><td>≥ 2</td></tr><tr><td>Attention Heads</td><td>4</td><td>≥ 1</td></tr><tr><td>Encode Dim</td><td>128</td><td>64 ~ 256</td></tr><tr><td>Parameters of Model (Millions)</td><td>2.38</td><td>-</td></tr><tr><td>Mask ratio</td><td>0.5</td><td>0.25 ~ 0.75</td></tr><tr><td>Trajectory Length Padding</td><td>200</td><td>36 ~ 256</td></tr><tr><td>Maximum Length Padding</td><td>512</td><td>-</td></tr><tr><td>Minimum Trajectory Points</td><td>36</td><td>-</td></tr><tr><td>Maximum Trajectory Points</td><td>600</td><td>-</td></tr><tr><td>Minimum Sampling ratio</td><td>0.35</td><td>-</td></tr><tr><td>Random Masking</td><td>0.7</td><td>-</td></tr><tr><td>Key Points Masking</td><td>0.15</td><td>-</td></tr><tr><td>Block Masking</td><td>0.05</td><td>-</td></tr><tr><td>Last N Masking</td><td>0.1</td><td>-</td></tr></table>
+
+# D Experiments Details
+
+We use the Adam optimizer and mean square error loss with an initial learning rate of  $1 \times 10^{-3}$  with a learning rate scheduler. The model is trained for 200 epochs with a batch size of 1024, and early stopping is applied based on validation performance. All experiments were conducted using PyTorch, where the foundation model is trained on NVIDIA A100/L40s 40GB GPUs and the baseline experiments are performed on RTX 2080 Ti.
+
+# D.1 Datasets
+
+We evaluate the performance of the proposed model using six diverse real-world trajectory datasets. Each dataset represents different data collection scenarios, quality levels, motion patterns, and geographic regions, providing a comprehensive test of the capabilities of UniTraj.
+
+- **WorldTrace:** WorldTrace is our proposed large-scale, globally distributed dataset, which we describe in detail in Section 4.1. We curated a high-quality subset of 1.1 million trajectories from
+
+the original dataset, which have been filtered to remove long stops and loops. Of this subset, 1 million trajectories are designated for model training combined with resampling or masking strategies, with the remaining 100,000 reserved for testing without any operation. To ensure consistency and enable independent zero-shot evaluations, the testing dataset is normalized to a sampling interval of 3 seconds per point.
+
+- Chengdu [8]: The Chengdu dataset comprises over one million urban mobility trajectories collected from taxis operating in Chengdu, China, reflecting daily commuting and transportation patterns in a densely urbanized area. It features dense, high-frequency (3-second for most trajectories) sampling points that provide detailed insights into active urban environments.  
+- Xi'an [8]: Similar to Chengdu, the Xi'an dataset includes millions of taxi trajectories gathered in Xi'an, China, focusing on movement patterns within another densely populated Chinese city. The data, collected during November 2016, captures the traffic dynamics and urban mobility behaviors specific to this region.  
+- GeoLife [47]: The GeoLife dataset is a widely used trajectory dataset collected over three years by 182 users, primarily in Beijing, China. It is mainly distinguished by a wide variety of travel modes, including walking, cycling and driving. With this data, we can study the trajectory movement patterns and behavioral habits of different travel modes. Besides, this dataset suffers from irregular and often long sampling intervals, which limit its granularity and quality for trajectory analysis.  
+- Grab-Posisi [15]: Sourced from Southeast Asia, this dataset contains 84,000 ride-hailing trajectories, predominantly from the Grab service in cities such as Jakarta and Singapore. The variable sampling intervals across these trajectories provide insights into urban mobility patterns unique to Southeast Asian metropolises.  
+- Porto [28]: The Porto dataset consists of taxi trajectories collected in Porto, Portugal, capturing trips between different areas of the city. Although it provides valuable insight into taxi mobility within the city, the dataset has a relatively low sampling frequency, with long intervals (15 seconds) between data points.
+
+# D.2 Tasks Applicability Study Settings
+
+# D.2.1 Trajectory Recovery
+
+In this experiment, we randomly mask  $50\%$  of trajectory points and test the recovery performance. Specifically, we evaluate UniTraj in both zero-shot (trained solely on WorldTrace) and fine-tuned settings (trained on WorldTrace and then fine-tuned on each respective dataset), aiming to understand its adaptability with and without task-specific training. Additionally, we compare UniTraj against a diverse range of baselines, including traditional deep learning models (Linear, DHTR [35], Transformer [33], and DeepMove [11]) and pre-trained models (TrajBERT [31] and TrajFM [23]). Performance metrics include Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE) with meters, computed based on geographic distance:
+
+$$
+\mathrm {M A E} = \frac {1}{n} \sum_ {i} ^ {n} \left| y _ {i} - \hat {y} _ {i} \right|, \tag {53}
+$$
+
+$$
+\mathrm {R M S E} = \sqrt {\frac {1}{n} \sum_ {i} ^ {n} \left(y _ {i} - \hat {y} _ {i}\right) ^ {2}} \tag {54}
+$$
+
+where  $y_{i}$  and  $\hat{y}_i$  are the real and recovered coordinates, respectively.
+
+# D.2.2 Trajectory Prediction
+
+In this task, we focus on predicting future trajectories based on historical trajectory points. Following the setup [23] in previous work, we predicted the locations of five future points. The baseline settings and evaluation metrics are consistent with those used for the trajectory recovery task, and experiments were conducted on WorldTrace, Chengdu, and GeoLife datasets.
+
+# D.2.3 Trajectory Classification
+
+The Trajectory Classification task is conducted on two datasets, GeoLife and Grab-Posisi. In this task, we will only use the encoder module of the UniTraj as a backbone and then add a classification header.
+
+We compare UniTraj in two settings: without fine-tuning (wo/ft), where only the classifier head is trained, and with fine-tuning (ft), where the entire model is updated. For baselines, we following prior literature [21] use representative classification models including GRU, LSTM, STGN [45], and TrajFormer [21]. Performance is reported by classification accuracy:
+
+$$
+\operatorname {A c c} = \frac {1}{n} \sum_ {i} ^ {n} \mathbf {I} \left(y _ {i}, \hat {y} _ {i}\right), \tag {55}
+$$
+
+where  $y_{i}$  and  $\hat{y}_i$  are the predicted and true labels, respectively, and  $\mathbf{I}(\cdot)$  is a indicator function. Following the general settings of previous work, we selected four travel modes from the Geolife dataset, namely walking, bus, bike, and driving. For the Grab-Posisi dataset, there are two travel modes: car and motorcycle.
+
+# D.2.4 Trajectory Generation
+
+In this task, we follow the approach in prior work [50], assessing trajectory generation using sequences of road segments that represent trajectories without explicit temporal attributes. Specifically, we use ControlTraj as a downstream task for trajectory generation, where we replace the road segment extraction component (RoadMAE) of the ControlTraj with UniTraj's encoder, testing the effectiveness of the embedded representation. The evaluation includes density error metrics [50]:
+
+$$
+\operatorname {D e n s i t y} \operatorname {E r r o r} = \operatorname {J S D} (G \| O) = \frac {1}{2} \mathbb {D} (\| \frac {(G + O)}{2}) + \frac {1}{2} \mathbb {D} (G \| \frac {(G + O)}{2}), \tag {56}
+$$
+
+where  $G$  is the distribution of the generated trajectories in the city (which divides each city into grids of  $16 \times 16$  size and calculates the count of trajectory points associated with each grid), and  $O$  is the distribution of the original trajectories. JSD(·) is the Jenson-Shannon divergence for two distributions.
+
+![](images/759c8322f52614a766164f734da2065d1746da46e59af78d2e53b28291587875.jpg)  
+(a) Original.
+
+![](images/685de70df41311a9efbbbe462c36836a7ea56e0c6bb2d3cb69021b5516205d1d.jpg)  
+(b) ControlTraj.
+
+![](images/b33fd407bc090da2bca4c855df843dc576a6d0cafbe46fc9d4c2a8a9f3a3cba6.jpg)  
+(c) ControlTraj + UniTraj.
+
+![](images/2208f2292a44ef7cf313a3a5af7a19483b7689a3b03f2a171f4cb66b2cd67160.jpg)  
+(d) Original.
+
+![](images/c07355c90896e42bc68a9f4a9b837815f8c7baf926f5a8437b1b1aceb98089c3.jpg)  
+(e) ControlTraj.
+
+![](images/f80b35e90e14b6995ae6a721f2cbc45e9c900b0ea591db15c81e6d33fb7cc3b0.jpg)  
+(f) ControlTraj + UniTraj.  
+Figure 9: Performance comparison of trajectory generation task with Chengdu dataset (first row), and transfer to Xi'an dataset (second row).
+
+For the this task, UniTraj demonstrates its versatility through integration with existing generative frameworks. By replacing ControlTraj's road segment extraction module with UniTraj, we achieved
+
+a  $5.1\%$  reduction in density error (from 0.0039 to 0.0037) when trained and generated on the Chengdu dataset. This improvement, though modest in magnitude, represents a significant advance in trajectory fidelity. More impressively, when transferring the generation capability to Xi'an without retraining—a challenging cross-region scenario—the UniTraj-enhanced generator maintains a density error of 0.0152. In contrast, the baseline ControlTraj experiences a 0.0171 density error when transferred across regions. This cross-region resilience further validates UniTraj's ability to capture universal trajectory patterns that transcend specific geographic contexts. We also show the heatmap visualizations to measure the accuracy and realism of generated trajectories in Figure 9, where brighter regions indicate denser trajectories and darker regions indicate sparser ones. Detailed analysis of the generated trajectories reveals that UniTraj-enhanced generation produces more realistic speed variations, particularly in complex road segments such as intersections, sparse or dense areas. In summary, the above results underscore UniTraj's potential for robust and transferable trajectory generation, proving its effectiveness in both familiar and novel geographic settings.
+
+# D.3 Dataset Study Settings
+
+Effect of Dataset Scale and Quality. This task focuses on the impact of dataset size and quality on UniTraj performance. We analyze WorldTrace for the effects of different amounts and qualities of training data. Specifically, we further process the complete WorldTrace dataset by removing cyclic trajectories, removing trajectories with too many stopping points and sparse trajectories. In total, we partitioned a subset of high-quality trajectory data numbering 1 million items, and further partitioned a subset of 10,000, 500,000 trajectory data for UniTraj training.
+
+Effect of Dataset Diversity. The task assessed the impact of using different data coverage (i.e., geographic diversity) on the model. We evaluate the zero-shot performance of UniTraj trained on the WorldTrace and Chengdu datasets, respectively, and tested on multiple real-world trajectory datasets. We chose the Chengdu dataset for comparison because it has very high data quality and has the identical me collection standards as the Xi'an dataset.
+
+# D.4 Model Study Settings
+
+For setting the number of encoders decoders for the model, we adopt the following scheme {encoders: 2,4,6,8,12}, {decoders:2,2,4,4,6}, {attention heads:2,2,2,4,8}. We believe that an asymmetric encoder-decoder architecture can significantly reduce the number of parameters while maximizing the performance of the model. And the scaling law between the number of model parameters and the size of the data will be one of the considerations in our future research and model architecture design.
+
+# E More Discussion
+
+# E.1 Limitation
+
+While UniTraj represents a significant advancement in universal trajectory modeling, several limitations remain that warrant acknowledgment and future investigation. Despite WorldTrace's unprecedented geographic coverage spanning 70 countries, data distribution remains uneven, with certain regions (particularly in Africa and parts of Asia) underrepresented, potentially limiting model performance in these areas. Additionally, our focus on motorized movement may restrict generalization to non-motorized mobility patterns, such as pedestrian trajectories with distinctly different motion properties. The computational resources required for training and deploying UniTraj at scale present practical challenges for resource-constrained environments, necessitating more efficient architectures or distillation approaches. From a technical perspective, UniTraj relies solely on coordinate and temporal information, lacking integration of contextual features like road networks, traffic conditions, and points of interest that could further enhance predictive accuracy. Addressing these limitations represents promising directions for future research, potentially through expanded geographic coverage, multimodal trajectory integration, architecture optimization, context-aware modeling, and continual learning techniques. Nonetheless, we believe that the proposed UniTraj and WorldTrace datasets will contribute to the development of the entire community towards a more generalized, global view of trajectory analysis.
+
+# E.2 Broader Impact
+
+This work presents both promising opportunities and notable concerns for society. Positively, this universal trajectory model could popularize mobility intelligence across diverse regions, enabling improved transportation systems in underserved areas without extensive local data collection. The model could drive more efficient urban planning, reduce traffic congestion and emissions, and enhance logistics optimization globally. However, this technology could also enable more pervasive monitoring capabilities, raising surveillance concerns if misused. Additionally, there exists potential for widening technological disparities between resource-rich and resource-constrained organizations. Balancing these implications requires commitment to privacy-preserving techniques and equitable access policies to ensure this technology advances social welfare while minimizing potential harms.
+
+# E.3 Ethics Issues
+
+The development and application of large-scale trajectory foundation models raise important ethical considerations, which have been carefully addressed throughout this research. A primary concern is data privacy and the risk of re-identification. To mitigate this, we implemented a robust anonymization protocol: during data collection from OpenStreetMap, we deliberately excluded user identifiers, and the data was subsequently processed through filtering, resampling, and map-matching. This multi-step process significantly abstracts the trajectories from their original form, ensuring the publicly shared dataset is thoroughly anonymized.
+
+Another critical issue is the potential for misuse, particularly for large-scale surveillance. The design of UniTraj inherently addresses this risk at the model level by learning from aggregated, generalized movement patterns rather than individual user paths, making it incapable of reconstructing original trajectories or tracing personal identities. To further prevent the learning of localized patterns, we will not release any model versions fine-tuned on private datasets and will advocate for clear guidelines for responsible use.
+
+Finally, we acknowledge the potential for biases arising from the uneven geographical distribution of training data. To address this, we are employing data augmentation techniques to mitigate the effects of data scarcity in certain regions. We analyze these inherent biases and their potential societal consequences, and we are committed to ongoing efforts to enhance data diversity and model fairness in future work to ensure our research aligns with best practices in responsible AI.
+
+# NeurIPS Paper Checklist
+
+The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit.
+
+Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:
+
+- You should answer [Yes], [No], or [NA].  
+- [NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available.  
+- Please provide a short (1–2 sentence) justification right after your answer (even for NA).
+
+The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper.
+
+The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes]" is generally preferable to "[No]", it is perfectly acceptable to answer "[No]" provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No]" or "[NA]" is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found.
+
+IMPORTANT, please:
+
+- Delete this instruction block, but keep the section heading "NeurIPS Paper Checklist",  
+- Keep the checklist subsection headings, questions/answers and guidelines below.  
+- Do not modify the questions and only use the provided macros for your answers.
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: All claims have been confirmed in the methods and experiments section.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: We have discussed this in detail in Appendix E of the supplementary materials.
+
+# Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: We provide analysis and proof in Appendix of the supplementary materials.
+
+Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: We provide code and sample data in the supplementary materials. In addition, we provide detailed parameter settings in the Appendix C.5.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.
+
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.  
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.  
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.  
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example  
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: We provide code and data at https://github.com/Yasoz/UniTraj.
+
+# Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).  
+- The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.  
+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.  
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
+
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: We provide detailed data and hyperparameters settings in the Appendix C.5.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [NA]
+
+Justification: We report results by taking the average of multiple rounds of experiments.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).  
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We declare the computing resources used in the experiments in Appendix D.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
+
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more computer than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: We ensure that no personally identifiable information is included in the data.
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: We discussed this topic in Discussion at Appendix.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: This paper is not related to this risk.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.
+
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: We declare the license of the used data in Appendix A.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [Yes]
+
+Justification: licensing, acquisition, and processing of the dataset introduced in this paper, and ensure that there are no privacy risks.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: This paper does not involve crowdsourcing nor research with human subjects.
+
+# Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: This paper does not involve crowdsourcing nor research with human subjects.
+
+# Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [Yes]
+
+Justification: We use LLM to improve language expression and visualization results.
+
+# Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
\ No newline at end of file
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+# UniTransfer: Video Concept Transfer via Progressive Spatial and Timestep Decomposition
+
+Guojun Lei $^{1*}$ , Rong Zhang $^{2*}$ , Tianhang Liu $^{1}$ , Hong Li $^{4}$ , Zhiyuan Ma $^{3\dagger}$ , Chi Wang $^{5}$ , Weiwei Xu $^{1\dagger}$
+
+$^{1}$  State Key Lab of CAD&CG, Zhejiang University,  $^{2}$  Zhejiang Gongshang University,  $^{3}$  Huazhong University of Science and Technology,  $^{4}$  Beihang University,
+
+<sup>5</sup> Ningbo Global Innovation Center, Zhejiang University  
+guojunlei@zju.edu.cn, zhangrong@zjgsu.edu.cn, link0502@buaa.edu.cn, tianhang@zju.edu.cn, mzyth@hust.edu.cn, wangchi1995@zju.edu.cn, xww@cad.zju.edu.cn
+
+[Web Page]
+
+![](images/3c08302cac3a9ed546184d51e5c481406d2f0bceb14e0fcc1d926a8a750bf02d.jpg)
+
+![](images/6672b18abf6bca75915dbfb0f32fb6460e1df5960f98776f103bdb2e4a1161b3.jpg)
+
+![](images/417652e5526cc510813911d955192f98aee27d7f4f381b2216bc6464eb9761be.jpg)  
+Figure 1: Results of our UniTransfer. These qualitative results exhibit the superior performance of our approach in transferring various reference components, including characters, garments, backgrounds, and motions, to synthesize the new target videos.
+
+![](images/be441be8280de8fc038521d219ef128645be60d6c7b2fa1e48a1c713ea897934.jpg)
+
+# Abstract
+
+Recent advancements in video generation models have enabled the creation of diverse and realistic videos, with promising applications in advertising and film production. However, as one of the essential tasks of video generation models, video concept transfer remains significantly challenging. Existing methods generally model video as an entirety, leading to limited flexibility and precision when solely editing specific regions or concepts. To mitigate this dilemma, we propose a novel architecture UniTransfer, which introduces both spatial and diffusion timestep decomposition in a progressive paradigm, achieving precise and controllable video concept transfer. Specifically, in terms of spatial decomposition, we decouple videos into three key components: the
+
+foreground subject, the background, and the motion flow. Building upon this decomposed formulation, we further introduce a dual-to-single-stream DiT-based architecture for supporting fine-grained control over different components in the videos. We also introduce a self-supervised pretraining strategy based on random masking to enhance the decomposed representation learning from large-scale unlabeled video data. Inspired by the Chain-of-Thought reasoning paradigm, we further revisit the denoising diffusion process and propose a Chain-of-Prompt (CoP) mechanism to achieve the timestep decomposition. We decompose the denoising process into three stages of different granularity and leverage large language models (LLMs) for stage-specific instructions to guide the generation progressively. We also curate an animal-centric video dataset called OpenAnimal to facilitate the advancement and benchmarking of research in video concept transfer. Extensive experiments demonstrate that our method achieves high-quality and controllable video concept transfer across diverse reference images and scenes, surpassing existing baselines in both visual fidelity and editability. Web Page: https://yu-shaonian.github.io/UniTransfer-Web/
+
+# 1 Introduction
+
+Recent years, the rapid advancement of generative AI technologies [13, 36, 33] has greatly reformed the community of video editing, opening up new potentials for diverse, fine-grained, and user-controllable content manipulation tasks. Video Concept Transfer (VCT) is one of the most important downstream tasks in video editing, aiming to substitute various user-specified target concepts in a video, such as objects, characters, backgrounds, or subject motions, to enable personalized content manipulation. Its applications span across diverse areas such as film production, game development, virtual reality, etc., drawing increasing attention from both academia and industry.
+
+Despite its potential, achieving high-quality video concept transfer is still challenging as it requires seamless integration of different components, preservation of the identity of the target object, and visual fidelity of the generated videos. Some recent approaches rely on text-based guidance to control the transfer [10, 4, 41]. However, the inherent ambiguity of natural language descriptions often results in imprecise manipulation of object attributes and behaviors, restricting their application scenarios. In contrast, image-guided methods offer a more accurate way to encode the appearance and identity of target objects. For example, methods likeAnimateAnyone2[15] and MIMO [29] have demonstrated the ability to replace human subjects in videos with specific reference images. However, these existing approaches mainly focus on human transfer and struggle to generalize to broader editing tasks [25] involving arbitrary objects, backgrounds, garments or motion patterns. VideoSwap [11] and AnyV2V [20] rely on personalized modeling or image editing techniques to address this dilemma. But the ability of their foundation models limits their scalability and flexibility in complex videos.
+
+This paper focuses on image-based video concept transfer, including animals, characters, backgrounds, and motions. This task inherently involves manipulations and integrations of different components within a video, which often exhibit substantial variation in terms of visual appearance, motion pattern, or semantic attributes. However, most existing approaches [23, 41, 11, 20, 28] generally model the video as a unified whole without considering the heterogeneous nature of its constituents, which may introduce undesired artifacts. MIMO [29] introduced spatial decomposed modeling to VCT by decomposing a video into three predefined components: human, scene, and occlusion. It helps improve the quality of video character replacement. However, MIMO only decomposes videos in the spatial dimension, which is not sufficient for high-quality video generation, as it takes all the timesteps in the denoising process equally. Motivated by ProSpect [53], diffusion models actually generate images in the progressive order of "layout  $\rightarrow$  content  $\rightarrow$  texture", and this work also exhibits that different stages in the diffusion models require guidance at different granularities.
+
+In this work, we propose UniTransfer, a Diffusion Transformer (DiT) based video concept transfer framework via progressive decomposition of both the spatial dimension and the denoising process. In the spatial dimension, we decompose videos into three core components: foreground, background, and motion dynamics, enabling our model to adapt to general concept transfer flexibly. To achieve this, we allow the model to learn the decomposition from coarse foreground masks to detailed ones. Specifically, we first introduce a random masking-based self-supervised pretraining strategy to
+
+strengthen the decomposed representation learning, which enables the model to capture disentangled features without requiring fine-grained annotations. Then we design a dual-to-single-stream Dit architecture to realize further decomposition with delicate semantic annotations. In this stage, individual branches are responsible for encoding different video components, and their features are later integrated into a unified representation through a single-stream network. This design enhances the model's capacity to manipulate different objects and maintain the temporal consistency.
+
+In the denoising process, we decompose the timestep into coarse-grained, mid-grained, and fine-grained stages instead of modeling it equally. Inspired by the Chain-of-Thought (CoT) strategy, we develop a Chain-of-Prompt (CoP) mechanism and leverage Large Language Models (LLMs) to produce hierarchical prompts at different granularities and utilize them to guide the generation, enabling progressive refinement from noise to detailed textures. The main contributions are summarized as follows:
+
+- We propose a DiT-based image-guided video concept transfer framework UniTransfer, which incorporates progressive spatial and timestep decomposition.  
+- We introduce a self-supervised pretraining strategy based on randomized masking to enhance the disentangled representation learning and design a dual-to-single-stream architecture to achieve spatial decomposition.  
+- We further introduce an LLMs-guided chain-of-prompt mechanism to achieve the timestep decomposition. This progressive prompting strategy guides the generation process with stage-specific instructions, improving the VCT generation quality.  
+- We collect an animal-centric video dataset called OpenAnimal to facilitate the training and benchmarking of research in video concept transfer. Extensive experiments demonstrate that our method outperforms state-of-the-art methods in various video concept transfer scenarios.
+
+# 2 Related Work
+
+Text-driven Video Editing. Video editing has witnessed remarkable advances in conditional content synthesis and manipulation through diffusion-based architectures [27]. Recently, video editing typically relies on textual prompts to control object attributes or behaviors [10, 4, 1]. Video-P2P [23, 32] achieves preservation of motion dynamics through attention modulation. RF-Edit [41] preserves structural integrity and temporal consistency by rectified flow ODE solving with reduced error. MotionDirector [55] requires separate training for each motion type. When adapting to new scenarios, it only supports text-guided scene transitions, which significantly limits its flexibility. VMC [18] also needs separate training for different scenes. However, due to the inherent ambiguity and underspecification of natural language, such methods often struggle to achieve fine-grained and accurate control over video content.
+
+Image-guided Video Editing. To overcome these limitations, researchers introduce additional reference images to guide the editing process of the video subject. MIMO [29] and MovieCharacter [34] decompose the video into elements such as foregrounds and preprocessed backgrounds, and perform character transfer through video composition techniques.AnimateAnyone2 [15] proposes a framework to animate characters while considering environmental affordances. Despite their effectiveness, they are designed for character video editing, which limits the application scenarios.
+
+In contrast, VideoSwap [11] pioneers a more general approach to image-guided concept transfer through semantic correspondence learning, enabling more versatile and accurate video edits beyond character animation. Yet critically, its reliance on multi-image concept anchoring introduces semantic abstraction, achieving category-level consistency (e.g., kitten, airplane) but failing to preserve instance-specific attributes. AnyV2V [20] leverages arbitrary image editing tools [3, 42, 7] to modify the first frame of a video and then propagates the modifications to subsequent frames, enabling instance-based object-driven editing and identity manipulation. However, the two-stage pipeline heavily depends on the results of off-the-shelf image editing techniques, and the temporal consistency can not be guaranteed. To address these limitations, we propose a novel framework that does not rely on external image editing tools. Instead, our method decomposes the video into three disentangled components: foreground, background, and motion. Integrating with a carefully-tailored network architecture in conjunction with a large language model (LLM)-based chain-of-thought reasoning mechanism, our method can achieve precise and flexible video concept transfer.
+
+![](images/ea4f934ee795c8870fc38f922e5c7acb6558768a593f50a0992ff1c5434fc613.jpg)  
+Figure 2: Our progressive spatial and timestep decomposition modeling.
+
+Chain-of-Thought Prompting [19] aims to substantially improve the reasoning capabilities of large language models (LLMs) [6]. It provides a framework for complex multi-step inference through explicit intermediate reasoning steps. The CoT concept evolves into CoX [44], where X denotes swappable nodes (e.g. intermediates [19, 56, 21, 40], augmentation [12, 49, 9, 52], feedback [46, 24, 2, 8], and models [47, 5, 45]) for task-specific adaptation. We further propose Chain-of-Prompt (CoP), which shifts the CoX paradigm to the iterative denoising steps of diffusion models. This systematic approach decomposes the video generation into sequential coarse-to-fine steps, where hierarchical prompt guidance progressively refines temporal features through successive diffusion iterations.
+
+# 3 Method
+
+In this section, we present UniTransfer, a DiT-based image-guided video concept transfer framework with progressive decomposed video modeling. Our goal is to generate high-quality videos with user-specified concepts in the referenced images, including objects, characters, animals, backgrounds, or motion dynamics. To achieve this, we decompose the video both in the spatial and the timestep dimension. The overview of the proposed framework is illustrated in Figure 2.
+
+# 3.1 Spatial Decomposition
+
+Existing video generation methods typically treat the video as a holistic entity and attempt to model it under the guidance of text prompts or reference images. Although such strategies are effective for general video synthesis, they fall short in the context of video concept transfer, which requires compositional control over different parts of the video, including foreground objects, background scenes, and motion dynamics. Encoding the entire video into a single latent space makes it difficult to independently manipulate these components during generation. As a result, existing image-based video object transfer methods are often limited to transferring only the main subject, instead of manipulating different components, including background appearance or motion dynamics.
+
+To achieve more flexible and controllable video concept transfer, we propose to spatially disentangle the video generation process. Specifically, a video can be decomposed into three distinct components: the foreground  $M$ , the background  $B$ , and the corresponding motion flow  $F$ . In general, the foreground component may consist of different objects (e.g., characters, animals, or objects). Under this setting, we redefine the video denoising process as follows:
+
+$$
+\mathcal {L} (\theta) = \mathbb {E} _ {x _ {0}, \epsilon , \mathcal {U}, t} \left[ \| \epsilon - \hat {\epsilon} _ {\theta} (x _ {t}, \tau , \mathcal {U}, t) \| _ {2} ^ {2} \right], \tag {1}
+$$
+
+![](images/a2448abffcb7cd62965d6ff982a0c091d1d69901b3ddea6b8610ab6b544d98f3.jpg)  
+Figure 3: The architecture of our UniTransfer.
+
+where  $\tau$  is the condition of text prompts,  $\mathcal{U} = (M,B,F)$ . This decomposition enables us to treat each component independently and recombine them flexibly in the generation pipeline, laying the foundation for video concept transfer.
+
+Building upon this decomposed formulation, we propose a progressive learning scheme that enables the model to effectively capture and utilize the decomposed components for video generation. To achieve this, we design two modules: (1) a random masking-based self-supervised learning mechanism to model the coarse relationships between the foregrounds and the backgrounds without delicate semantic annotations. (2) a carefully designed dual-to-single stream architecture, UniTransfer, tailored to learn detailed interactions of three different components. In the following, we will introduce them respectively.
+
+# 3.1.1 The Architecture of UniTransfer
+
+In this stage, we propose a DiT-based architecture named UniTransfer to decompose the video into three components: the foreground appearance, the background scene, and the motion dynamics to enable flexible and fine-grained control over VCT. Unlike previous approaches that treat the video as a monolithic entity, UniTransfer is designed to disentangle and recompose these components in a controllable diffusion-based generation process.
+
+In the training process, given the input video  $V$ , we first randomly sample a single frame to serve as a reference image  $I$ . A pretrained semantic segmentation model is then utilized to predict a foreground mask  $M \in \{0,1\}^{H \times W}$ , which partitions the reference frame into a foreground region  $F = I \odot M$  and a background region  $B = I \odot (1 - BBox(M))$ , where  $\odot$  represents element-wise multiplication and  $BBox(M)$  means all the pixels in the bounding box are setting to 1. With the unaligned masks, we force the model to learn shape-agnostic interactions between the foreground and the background. Meanwhile, motion dynamics are extracted from the entire video using a pretrained optical flow model RAFT [39], which captures the temporal dynamics independent of appearance content. UniTransfer takes  $F, B, O$  along with the video description  $\tau$  as guidance and iteratively denoises from a randomly sampled Gaussian noise map to generate a new video  $\hat{V}$ . Figure 3 illustrates the framework of the network, which is in a dual-to-single-stream paradigm consisting of three collaborative branches: a foreground branch, a background branch, and a fusion branch.
+
+The foreground and background branches are built based on the CogVideoX architecture[48]. Each of them is composed of an image VAE encoder to project  $F$  and  $B$  into latent codes  $z_{f}$  and  $z_{b}$ , a text encoder to provide shared high-level text embedding  $z_{\tau}$ , and a stack of  $N$  DiT blocks for temporal feature modeling. In the background branch, a random Gaussian noise vector  $z_{t}$  of the timestep  $t$  is concatenated with  $z_{b}$  and  $z_{\tau}$ , and the combined representation is passed through the DiT blocks  $h_{b}^{i}$ , for  $i = 1, \dots, N$ . In contrast, the foreground branch is treated as a conditional stream with no noise input. It adopts a symmetric structure and produces intermediate feature maps for each DiT block  $h_{f}^{i}$ , for  $i = 1, \dots, N$ . Afterwards, we design a feature injection module that introduces foreground features into the background stream inspired by ControlNet [51]. Specifically, the foreground feature map is processed by a zero-initialized convolutional projection layer and then element-wise added to the corresponding layer in the background branch at each DiT block. It allows the model to inject spatially aligned foreground appearance into the generative path.
+
+Moreover, the enhanced background features are combined with the motion dynamics in the fusion branch, which consists of a motion encoder to project the optical flow to a latent code  $z_{o}$ , and  $N$
+
+DiT blocks for further feature fusion. The flow encoder adopts four symmetrically arranged stages, aligning with the structure in 3D VAE encoder [48]. Inspired by Tora[54], we also design an adaptive norm layer before adding it to the dit block. To enable the model to handle the misalignments between the motion and the input images and improve the generalization ability, we inject random noise into the input optical flow.  $z_{o}$  is fused with the background stream through element-wise addition in each fusion DiT block  $h^{i}$ , for  $i = 1,\dots ,N$ . The model outputs the predicted noise at a given denoising timestep  $t$  as follows:
+
+$$
+\hat {\epsilon} _ {\theta} = h \left[ Z C o n v \left(h _ {f} \left(z _ {f} \odot z _ {\tau}\right)\right) \oplus h _ {b} \left(z _ {t} \odot z _ {b} \odot z _ {\tau}\right) \oplus z _ {o} \right] \tag {2}
+$$
+
+where  $ZConv$  represents the zero-initialized convolution.  $\odot$  and  $\oplus$  represent the concatenation operation and element-wise addition, respectively.
+
+# 3.1.2 Self-Supervised Pre-Training via Random Masking
+
+Learning robust video representations for concept-aware generation typically requires large-scale datasets with annotated semantic masks or motions. However, existing video datasets rarely provide sufficient annotations. Relying on small-scale annotated datasets is not enough to directly learn spatially decoupled representations for video concept transfer. To address this limitation, we follow LMD [26] to introduce a random masking strategy before training with fine-grained annotations. It leverages large-scale unlabeled video datasets for learning initial disentangled representations. In this stage, we only learn the foreground and background decomposition for coarse initialization. In this stage, the binary foreground mask  $M \in \{0,1\}^{H \times W}$  is generated through random masking to arbitrarily partition the reference image  $I$  into two regions  $F$  and  $B$ . The mask is initialized as an all-zero matrix and is iteratively updated by randomly drawing rectangles of random size and position on it. The pixels inside each rectangle are set to 1, and the process continues until the foreground coverage exceeds  $50\%$  of the image area. This masking strategy ensures spatial diversity and balance between foreground and background regions. Then the reference frame is partitioned into a foreground region  $F = I \odot M$  and a background region  $B = I \odot (1 - \text{Dilate}(M))$ , where  $\text{Dilate}(M)$  represents the morphological dilation operation to enable the model to learn the boundary interactions. Then  $F$  and  $B$  are fed into the foreground branch and a background branch, which inject the features into the denoising network independently. This allows the model to learn how to reconstruct coherent video content from partial and spatially isolated cues, without requiring ground-truth foreground-background labels. Through extensive self-supervised training on large-scale data, our model acquires strong prior knowledge, enabling efficient adaptation to downstream tasks via precise supervised fine-tuning.
+
+# 3.2 Timestep Decomposition
+
+Revisiting the preliminary of diffusion models reveals another limitation in current video generation pipelines: the same conditioning prompt is applied across all timesteps during the denoising process. It is difficult to generate grained texture corresponding to the grained prompts at the initial stage. Inspired from ProSpect [53], the role of each timestep varies significantly in the generation process. In the early stages of denoising, the model focuses primarily on recovering the global structure and semantic layout of the video. In contrast, the later stages are responsible for learning fine-grained details such as textures, colors, and subtle appearance attributes. Applying a static, single-level prompt across all timesteps ignores this progression in modeling focus. For instance, using overly complex or fine-grained textual prompts during early timesteps may misguide the model, leading to the omission of some semantic attributes or misalignments between the generated content and the input prompt.
+
+To address this, we introduce a timestep decomposition mechanism named Chain-of-Prompt (CoP) inspired by the Chain-of-Thought reasoning paradigm in large language models. Specifically, we decompose the denoising timesteps into three granularity levels—coarse, medium, and fine—and leverage a large language model (LLM) to automatically generate three levels of prompts that progressively reflect the abstraction and detail required at different denoising stages. These prompts are then injected into the diffusion model in a stage-aware manner, ensuring that early timesteps receive high-level, structural guidance, while later steps benefit from detailed, appearance-level
+
+![](images/94ff65bb3bd4b84bb24668b29c24e296a062a60e951ff3090b855adf3c1beb3d.jpg)  
+Figure 4: Comparison of video character transfer. Note that our backgrounds are better preserved. control. Then the denoising loss can be defined as follows:
+
+$$
+\mathcal {L} (\theta) = \left\{ \begin{array}{l l} \| \epsilon - \hat {\epsilon} _ {\theta} \left(z _ {t}, \tau_ {c r s}, \mathcal {U}, t\right) \| _ {2} ^ {2}, & t \in [ t _ {c}, T - 1 ] \\ \| \epsilon - \hat {\epsilon} _ {\theta} \left(z _ {t}, \tau_ {m i d}, \mathcal {U}, t\right) \| _ {2} ^ {2}, & t \in [ t _ {f}, t _ {c}) \\ \| \epsilon - \hat {\epsilon} _ {\theta} \left(z _ {t}, \tau_ {f i n e}, \mathcal {U}, t\right) \| _ {2} ^ {2}, & t \in [ 0, t _ {f}) \end{array} \right. \tag {3}
+$$
+
+where  $\tau_{crs},\tau_{mid},\tau_{fine}$  are coarse-grained, mid-grained and fine-grained text prompts, respectively.  $t_c$  and  $t_f$  are the end points of the corresponding stages.  $T$  is the total timestep of the diffusion process. In our paper, as the dataset provides detailed video descriptions, we utilize them as the fine-grained prompts. Then we employ an LLM Qwen-QWQ-32B [38] to summarize  $\tau_{fine}$  to  $\tau_{crs},\tau_{mid}$ . This decomposition strategy helps to align the complexity of text guidance with the focus of the generation phase, improving both semantic consistency and visual fidelity in the final video output.
+
+# 4 Experiments
+
+To better demonstrate the strengths of our approach, we conduct experiments to validate foreground transfer, background transfer, and optical flow transfer across videos. The implementation details can be referred to the appendix.
+
+# 4.1 Animal-centric Dataset
+
+To promote the broader applicability of video editing models, we introduced a new dataset OpenAnimal, which focused on single-animal video sequences across a wide range of species and diverse motion patterns. While following a similar structure to human-centric datasets (e.g., TikTok, UBC), OpenAnimal is specifically tailored for animal subjects with 10000 video clips.
+
+# 4.2 Comparison and Analysis
+
+As current state-of-the-art methods are primarily trained on human datasets, they often struggle to generalize to non-human scenarios, such as animal motion transfer or object-level editing in nature-based scenes. For fair comparison, we conducted qualitative and quantitative experiments on UBC[50] and TikTok[17] to evaluate the video character transfer performance. Besides, to further evaluate our model's ability to generalize to other concepts, we also demonstrate qualitative results of animal transfer, cloth transfer, background transfer, and motion transfer.
+
+# 4.2.1 Video Character Transfer
+
+To evaluate the effectiveness of our method in video character transfer, we compare it with state-of-the-art character animation and video editing baselines, includingAnimate Anyone [14],Champ [57],
+
+Table 1: Quantitative comparison of video quality for video character transfer.  
+
+<table><tr><td rowspan="2"></td><td colspan="5">TikTok</td><td colspan="5">UBC</td></tr><tr><td>LPIPS↓</td><td>PSNR↑</td><td>SSIM↑</td><td>FID↓</td><td>FVD↓</td><td>LPIPS↓</td><td>PSNR↑</td><td>SSIM↑</td><td>FID↓</td><td>FVD↓</td></tr><tr><td>MRAA</td><td>0.513</td><td>25.31</td><td>0.425</td><td>134.23</td><td>634</td><td>0.589</td><td>23.32</td><td>0.537</td><td>89.13</td><td>438</td></tr><tr><td>DisCo</td><td>0.674</td><td>21.56</td><td>0.532</td><td>112.24</td><td>571</td><td>0.467</td><td>23.45</td><td>0.412</td><td>94.15</td><td>483</td></tr><tr><td>MagicAnimate</td><td>0.259</td><td>24.56</td><td>0.657</td><td>89.23</td><td>428</td><td>0.143</td><td>27.08</td><td>0.742</td><td>48.19</td><td>391</td></tr><tr><td>Animate Anyone</td><td>0.198</td><td>25.89</td><td>0.713</td><td>79.18</td><td>398</td><td>0.132</td><td>26.54</td><td>0.798</td><td>44.18</td><td>378</td></tr><tr><td>Champ</td><td>0.241</td><td>26.57</td><td>0.787</td><td>67.14</td><td>401</td><td>0.159</td><td>27.03</td><td>0.811</td><td>41.02</td><td>324</td></tr><tr><td>UniAnimate</td><td>0.191</td><td>24.34</td><td>0.724</td><td>58.19</td><td>378</td><td>0.127</td><td>27.09</td><td>0.798</td><td>43.24</td><td>334</td></tr><tr><td>Ours</td><td>0.152</td><td>26.78</td><td>0.803</td><td>46.74</td><td>345</td><td>0.125</td><td>27.12</td><td>0.814</td><td>39.73</td><td>312</td></tr></table>
+
+Table 2: Video consistency quality comparison. SubC: Subject Consistency; BkgC: Background Consistency; MoS: Motion Smoothness; AesQ: Aesthetic Quality; DyaD: Dynamic Degree.  
+
+<table><tr><td rowspan="2"></td><td colspan="5">TikTok</td><td colspan="5">UBC</td></tr><tr><td>SubC↑</td><td>BkgC↑</td><td>MoS↑</td><td>AesQ↑</td><td>DyaD↑</td><td>SubC↑</td><td>BkgC↑</td><td>MoS↑</td><td>AesQ↑</td><td>DyaD↑</td></tr><tr><td>Video-P2P</td><td>0.842</td><td>0.853</td><td>0.782</td><td>0.443</td><td>0.710</td><td>0.813</td><td>0.867</td><td>0.734</td><td>0.497</td><td>0.371</td></tr><tr><td>ControlVideo</td><td>0.712</td><td>0.419</td><td>0.747</td><td>0.519</td><td>0.823</td><td>0.814</td><td>0.773</td><td>0.895</td><td>0.624</td><td>0.241</td></tr><tr><td>RF-Editor</td><td>0.911</td><td>0.873</td><td>0.581</td><td>0.423</td><td>0.765</td><td>0.825</td><td>0.648</td><td>0.789</td><td>0.434</td><td>0.627</td></tr><tr><td>MotionClone</td><td>0.784</td><td>0.865</td><td>0.891</td><td>0.651</td><td>0.830</td><td>0.924</td><td>0.943</td><td>0.871</td><td>0.613</td><td>0.679</td></tr><tr><td>CogVideoX</td><td>0.927</td><td>0.904</td><td>0.926</td><td>0.557</td><td>0.842</td><td>0.911</td><td>0.917</td><td>0.911</td><td>0.663</td><td>0.902</td></tr><tr><td>Mofa-Video</td><td>0.923</td><td>0.917</td><td>0.962</td><td>0.593</td><td>0.837</td><td>0.923</td><td>0.879</td><td>0.957</td><td>0.612</td><td>0.829</td></tr><tr><td>Ours</td><td>0.945</td><td>0.931</td><td>0.962</td><td>0.651</td><td>0.903</td><td>0.939</td><td>0.942</td><td>0.971</td><td>0.664</td><td>0.911</td></tr></table>
+
+UniAnimate [43], AnyV2V [20], Video-P2P [23], etc.. Qualitative results are presented in Figure 4. Given an input video, these existing approaches typically modify or replace objects based on textual or image-based prompts. As illustrated in Figure 4, the text-guided transfer methods fail to transfer the reference image to the video, while the image-guided baselines struggle to preserve the structural features or maintain the appearance of the backgrounds. In contrast, our approach leverages reference image guidance and incorporates a progressive spatial and timestep decomposition, which aligns more naturally with real-world editing workflows. Our approach can achieve better preservation of the reference appearance, higher visual quality of the video, and greater inter-frame consistency than all baselines. These improvements are attributed to our novel decomposition strategy, which enables fine-grained control and more structured video generation.
+
+Besides, we perform further quantitative evaluation from two perspectives: video quality and temporal consistency. As shown in Table 1 and Table 2, our method achieves superior performance across multiple metrics, including FID, LPIPS, and subject consistency, aesthetic quality, etc. from vbench[16], significantly outperforming the compared baselines. This demonstrates the robustness and generalization ability of our approach in complex video character transfer tasks.
+
+# 4.2.2 Adaptation to Various Video Concept Transfer Tasks.
+
+Our proposed decomposition-based modeling and random masking pre-training mechanism enable our framework to effectively isolate and manipulate individual visual factors within a video, which can generalize beyond conventional foreground transfer. In this section, we provide more diverse qualitative results of a wide range of video concept transfer tasks, including motion transfer, background transfer, animal transfer, and regional foreground transfer, such as clothing replacement. These results demonstrate the flexibility and adaptability of our approach in handling diverse transformations, which are typically challenging for conventional video editing methods.
+
+Motion transfer. Our framework supports motion transfer by applying the motion dynamics extracted from a driving video onto a reference image, similar to pioneer works[37]. To evaluate the effectiveness of our method on this task, we conduct comparisons with state-of-the-art motion-guided video generation approaches, including AnyV2V [20], MotionClone [22], MotionI2V [37], AnyV2V [20] and Mofa-Video [31]. The results are shown in Figure 5. As illustrated, MotionI2V struggles with maintaining the video consistency as time progresses, often introducing noticeable artifacts or drift in later frames. AnyV2V fails to preserve the appearance of the reference image. MotionClone primarily relies on text guidance for controlling video appearance, which limits its ability to precisely align the generated content with the reference image. Mofa-Video may generate blurry results. In contrast, our method produces videos that not only maintain high visual fidelity but also exhibit coherent and realistic motion consistent with the driving video. This advantage stems from our explicit spatial decomposition and motion-conditioned generation design, which enables better disentanglement and control over appearance and temporal dynamics.
+
+![](images/870cd7dd582e9212ca9d451b5f0b6e7aa5b08b8721c1da42a0bd4393af4c6961.jpg)  
+Figure 5: Comparison of video motion transfer.
+
+Table 3: Ablation study.  
+
+<table><tr><td rowspan="2"></td><td colspan="5">Visual Quality</td><td colspan="2">Video Smoothness</td></tr><tr><td>LPIPS ↓</td><td>PSNR ↑</td><td>SSIM ↑</td><td>FID ↓</td><td>FVD ↓</td><td>SubC ↑</td><td>BkgC ↑</td></tr><tr><td>Early Motion Injection</td><td>0.234</td><td>25.16</td><td>0.743</td><td>56.79</td><td>374</td><td>0.871</td><td>0.736</td></tr><tr><td>w/o Flow Noise Injection</td><td>0.293</td><td>24.31</td><td>0.659</td><td>69.31</td><td>391</td><td>0.915</td><td>0.893</td></tr><tr><td>w/o Self-Supervised Pre-training</td><td>0.498</td><td>19.87</td><td>0.546</td><td>101.34</td><td>487</td><td>0.735</td><td>0.892</td></tr><tr><td>w/o dual-stream decomposition</td><td>0.341</td><td>23.12</td><td>0.694</td><td>78.13</td><td>423</td><td>0.894</td><td>0.639</td></tr><tr><td>w/o CoP Timestep Decomposition</td><td>0.192</td><td>25.87</td><td>0.712</td><td>51.32</td><td>357</td><td>0.881</td><td>0.924</td></tr><tr><td>Full Model</td><td>0.152</td><td>26.78</td><td>0.803</td><td>46.74</td><td>345</td><td>0.945</td><td>0.931</td></tr></table>
+
+Foreground transfer. Our framework enables flexible foreground transfer, including part-level object replacement, such as garment transfer. It poses significant challenges due to the need to selectively modify localized visual regions while preserving the subject's identity and overall video harmonization. This is particularly important in fashion, virtual try-on, and personalization applications. As shown in Figure 7, our model successfully replaces the clothing items in the target videos, with the referenced garments, without disrupting the consistency of facial features, body motion, or background. These results highlight the strength of our decomposition framework in enabling precise and high-quality object-level modifications within complex video generation tasks.
+
+Background transfer. In most video editing approaches, modifications are typically limited to the main subject or local elements. In contrast, our method supports background replacement by leveraging the spatial disentanglement. As illustrated in Figure 7, our framework enables users to seamlessly replace the entire background of a video. Our approach ensures that background changes—such as scene transitions from indoor to outdoor—are consistent across all frames, while the foreground subject remains temporally coherent and unaffected in terms of identity and motion.
+
+# 4.3 Ablation Study.
+
+To verify the effectiveness of the components in our video generation pipeline, we designed several sets of ablation experiments on TikTok[17]. Specifically, we evaluate the following configurations: (1) Early Motion Injection: Instead of injecting motion in the intermediate network layers, we input motion information alongside the foreground and background features in the early layers, and apply attention-based fusion within the denoising branch. (2) Feature Fusion (w/o dual-stream decomposition): We directly concatenate foreground and background features and feed them into the denoising module. (3) Without Self-Supervised Pre-training. (4) Without CoP Timestep Decomposition. (5) Without Flow Noise Injection. The ablation study results are shown in Table 3. From the first two rows, we can see the superiority of the design of our motion branch, where the added noise improves the generalization ability and the injection strategy provides more accurate guidance. The third row emphasizes that the self-supervised training is critical for enhancing the
+
+decomposed representation learning. The last two rows illustrate that our modeling of the spatial and timestep decomposition both are essential for flexible and high-quality video concept manipulation.
+
+# 5 Conclusion
+
+In this work, we present a novel framework for controllable video concept transfer by introducing a progressive spatial and timestep decomposition modeling strategy. Unlike existing methods that treat the video as a holistic entity, our approach explicitly disentangles the video into foreground, background, and motion components, and further decomposes the denoising process into hierarchical stages via chain-of-prompt guidance. This design allows for more fine-grained control over video synthesis, enabling diverse video concept transfer tasks, such as character transfer, motion transfer, background transfer, and garment-level editing. We also incorporate a self-supervised pretraining scheme based on random masking to support robust learning under no additional supervision, which effectively bootstraps spatial disentanglement using large-scale unlabeled video data. Besides, we curate a new dataset featuring animal subjects to advance research in video generation. Extensive experiments on different transfer tasks demonstrate that our method significantly outperforms state-of-the-art baselines in terms of visual quality, consistency, and controllability.
+
+# Acknowledgments and Disclosure of Funding
+
+Weiwei Xu is partially supported by NSFC (No. 92570206, No.62421003). Rong Zhang is partially supported by NSFC (No. 62302449) and Zhejiang Provincial Natural Science Foundation of China (No. LQ23F020009). Zhiyuan Ma is supported by the National Natural Science Foundation of China (No. 62406161), China Postdoctoral Science Foundation (No. 2023M741950), and the Postdoctoral Fellowship Program of CPSF (No. GZB20230347). This paper is partially supported by the Yongjiang Innovation Project No. 2025Z062.
+
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+
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+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We have reported the information on the computer resources in the Section 4.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: We have read the NeurIPS Code of Ethics and confirm that the research in this article fully complies with its content.
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: The paper has discussed both potential positive social impacts and negative societal impacts of the work performed in the supplemental material.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.
+
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: The paper poses no such risks.
+
+# Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: We have cited the original paper that provided the code and dataset used in the paper.
+
+# Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
+
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: The paper does not release new assets.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
+
+We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components.
+
+Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Video Diffusion Model Preliminary
+
+Video diffusion models generalize the concept of image diffusion probabilistic models [13] to the temporal domain. Formally, let  $z_0 \in \mathbb{V}^{f \times h \times w \times c}$  represent a video latent variable, where  $f$  denotes the number of frames, with  $h \times w$  dimension,  $c$  channels. The forward diffusion process is defined as a Markov chain that gradually corrupts the original video into Gaussian noise:
+
+$$
+z _ {t} = \sqrt {\bar {\alpha} _ {t}} z _ {t - 1} + \sqrt {1 - \bar {\alpha} _ {t}} \epsilon , \quad \epsilon \sim \mathcal {N} (0, \mathbf {I}), \tag {4}
+$$
+
+where  $t \in \{1, \dots, T\}$  indexes the diffusion timestep,  $\bar{\alpha}_t$  controls the noise intensity, and  $\epsilon$  represents standard Gaussian noise. In the reverse process, a denoising network is used to estimate the noise from  $z_{t-1}$  to  $z_t$ , typically called a Diffusion Model  $\theta$ . The training objective minimizes the following loss function:
+
+$$
+\mathcal {L} (\theta) = \mathbb {E} _ {z _ {0}, \epsilon , \mathcal {C}, t} \left[ \| \epsilon - \hat {\epsilon} _ {\theta} (z _ {t}, \mathcal {C}, t) \| _ {2} ^ {2} \right], \tag {5}
+$$
+
+where  $\mathcal{C}$  represents conditioning information such as text prompts or reference images.
+
+# B Implementation Details
+
+In this paper, we focus on image-guided video concept transfer, where the foregrounds, backgrounds, or motion dynamics can be manipulated. We perform experiments including character, animal, object, background and motion transfer to evaluate our approach. The weights of the model are initialized from a pretrained CogVideoX-5B [48]. The experiments are conducted on a server equipped with  $8 \times$  NVIDIA Tesla H100 80G GPUs. In our paper, we generate 49-frame videos at the resolution of  $720 \times 480$ .
+
+Our training pipeline consists of two stages: a large-scale self-supervised pretraining phase and a supervised fine-tuning phase. In the pre-training phase, we trained our model on the OpenVid dataset[30], a large-scale collection of approximately 12 million videos covering a wide range of categories including humans, animals, vehicles, and diverse backgrounds. The training used a learning rate of  $1 \times 10^{-4}$  and ran for 200,000 iterations. The OpenVid dataset comprises approximately 12 million samples.
+
+In the supervised training phase, we adapt the pretrained model to specific video concept transfer scenarios. For the human-centric editing task, we fine-tune the model on a combination of the TikTok [17] and UBC Fashion [50] datasets, which include rich annotations for character-level transfer tasks. The model is finetuned for 10,000 iterations with a learning rate of  $5 \times 10^{-5}$ . For the animal-centric task, we performed an additional 10,000 fine-tuning iterations on OpenAnimal using the same learning rate.
+
+# C More Details about Self-Supervised Pretraining
+
+In practice, obtaining large amounts of high-quality annotated data remains challenging. Although efficient segmentation tools like SAM2[35] are available, they still require extensive manual interactions, such as point prompts, bounding boxes, or object-specific filtering. To address this limitation and reduce the reliance on manual annotations, we introduce a self-supervised pretraining approach, the detailed algorithmic pipeline of which is outlined below Algorithm 1 and Figure 6.
+
+# D More Details about CoP Guidance
+
+The detailed algorithmic pipeline of our Chain-of-Prompt (CoP) guidance is demonstrated in Algorithm 2.
+
+To assess the effectiveness of CoP, we conducted ablation experiments comparing it with static prompts, including: (i) All Coarse Prompt: using a coarse prompt across all timesteps, and (ii) All Detailed Prompt: using a detailed prompt across all timesteps. The results demonstrate that the allocoarse setting tends to produce results lacking fine-grained appearance fidelity and exhibiting texture degradation, while the all-detailed setting often introduces misguidance during early denoising steps, leading to distorted structures. In contrast, our CoP strategy, which progressively schedules coarse-to-fine prompts, strikes a balance between structural stability and appearance precision, achieving better performance. The quantitative evaluation can be referred to in Table 4.
+
+![](images/c381fbb8aae59c318b46198f911ceccd358f996e7ec4d118ec1e1299dc83347f.jpg)  
+Figure 6: Self-supervised Pretraining.
+
+# E More Results
+
+Our framework enables flexible foreground and background transfer, including part-level object replacement, such as garment transfer. The results of different transfer tasks are shown in Figure 10, Figure 11, Figure 12, Figure 7. Our curated animal-centric dataset OpenAnimal is demonstrated in Figure 13.
+
+![](images/35d93655e468110916dbde259db5f3480766a21f013e70f57be8753da8447053.jpg)  
+Figure 7: More UniTransfer results of different transfer tasks.
+
+# F Limitation
+
+Although our model can achieve subject transfer and background replacement in videos, there are still cases where the subject and background appear with artifacts. This issue might be due to current segmentation models cannot fully separate foreground and background elements, leading to imperfect composite results in the final output. In the future, we plan to address this by leveraging large-scale models for enhanced video scene understanding, further improving the quality of generated videos.
+
+For scenes involving multiple moving entities, we can fuse their individual optical flows into a composite representation and sequentially condition the network on each object through multiple inference passes. For interacting subjects, our method remains effective when the occlusion or overlap between them is relatively small, as each subject's region can still be identified and modeled with reasonable independence. However, cases with large overlapping areas or complex physical interactions pose challenges to our models due to ambiguity in motion attribution and visual entanglement. Joint modeling of interacting subjects is an interesting direction for future work.
+
+Algorithm 1: The pipeline of our self-supervised pretraining.  
+Input: image: input image coverage: target coverage ratio (default: 0.5) min_block_size: minimum block size (default: 320) max_block_size: maximum block size (default: 640) Output: foreground: processed image with coverage background: inverse masked image   
+Function random_white_blocks(image, coverage, min_block_size, max_block_size) if image is None then raise ValueError("Input image is empty"); if image is grayscale then result  $\leftarrow$  convert image to BGR color space; else result  $\leftarrow$  copy of image;  $(h,w)\gets$  height and width of result; total_area  $\leftarrow h\times w$  covered_area  $\leftarrow 0$  target_area  $\leftarrow$  total_area  $\times$  coverage; mask  $\leftarrow$  zero matrix of size  $(h,w)$  . result_2  $\leftarrow$  gray matrix (127.5) with same size as result; max_iter  $\leftarrow$  500; while covered_area  $<  <$  target_area and max_iter  $>0$  do max_iter  $\leftarrow$  max_iter-1; block_size  $\leftarrow$  random integer between min_block_size and max_block_size;  $x\gets$  random integer between 0 and  $w -$  block_size;  $y\gets$  random integer between 0 and  $h -$  block_size; if mask[y:y+block_size,x:x+block_size] contains any 1 then continue; >Special overlapping effect result_2[y:y+block_size-5,x :  $x +$  block_size-5]  $\leftarrow$  result[y:y+block_size-5,x:x+block_size-5]; result[y:y+block_size+5,x:x+block_size+5]  $\leftarrow$  [127.5,127.5,127.5]; mask[y:y+block_size,x:x+block_size]  $\leftarrow$  1; covered_area  $\leftarrow$  covered_area + block_size  $\times$  block_size; if covered_area  $>1.1\times$  target_area then break; return foreground, background;
+
+Table 4: CoP mechanism with different level prompts.  
+
+<table><tr><td rowspan="2"></td><td colspan="5">Visual Quality</td><td colspan="2">Video Smoothness</td></tr><tr><td>LPIPS ↓</td><td>PSNR ↑</td><td>SSIM ↑</td><td>FID ↓</td><td>FVD ↓</td><td>SubC ↑</td><td>BkgC ↑</td></tr><tr><td>All Coarse Prompt</td><td>0.174</td><td>23.15</td><td>0.701</td><td>72.46</td><td>433</td><td>0.714</td><td>0.865</td></tr><tr><td>All Detailed Prompt</td><td>0.165</td><td>24.62</td><td>0.734</td><td>49.26</td><td>367</td><td>0.832</td><td>0.901</td></tr><tr><td>Full Model</td><td>0.152</td><td>26.78</td><td>0.803</td><td>46.74</td><td>345</td><td>0.945</td><td>0.931</td></tr></table>
+
+# G Social Impact
+
+Our research decouples video generation into foreground, background, and their corresponding motion. This technology will enhance the efficiency of video production, drive innovation in industries such as film and gaming, and enable more immersive entertainment and educational experiences. However, as this technology becomes more widespread, society may face ethical and legal challenges, including concerns over the authenticity of video content, characters, and backgrounds. Therefore, establishing appropriate regulatory frameworks to ensure responsible use of this technology is a critical task that demands our attention.
+
+Algorithm 2: Chain-of-Prompt-Guided Video Denoising  
+Input: initial_noisy视频：Initial noisy video  
+base_prompt: Original text description  
+total_steps: Total denoising steps (default: 50)  
+T1, T2: Stage transition steps  
+Output: generated视频：Final generated video  
+Function hierarchical_denoising(initial_noisy视频，base_prompt，total_steps = 50)  
+// Phase partitioning (all steps)  $T_{1} \leftarrow 35$  // First phase  
+ $T_{2} \leftarrow 15$  // Second phase  
+// Generate hierarchical prompts using LLM  
+prompts  $\leftarrow$  QwQ32B_GenerateHierarchicalPrompts(base_prompt);  
+// Returns: {'stage1': coarse, 'stage2': detailed, 'stage3': fine}  
+current Video  $\leftarrow$  initial_noisy Video;  
+for step  $\leftarrow$  total_steps to 1 do  
+if step  $\geq T_{1}$  then  
+    guidance_prompt  $\leftarrow$  prompts['stage1']; // Coarse prompt  
+    guidance_weight  $\leftarrow 1.5$  // Strong guidance  
+else if step  $\geq T_{2}$  then  
+    guidance_prompt  $\leftarrow$  prompts['stage2']; // Detailed prompt  
+    guidance_weight  $\leftarrow 2.0$   
+else  
+    guidance_prompt  $\leftarrow$  prompts['stage3']; // Fine prompt  
+    guidance_weight  $\leftarrow 1.0$   
+// Execute denoising step current Video  $\leftarrow$   
+DenoiseStep(current Video, guidance_prompt, guidance_weight, step);  
+return current Video;
+
+![](images/e7219758d52cccfb32850423cbd14b7758571405e2ef65ec2d49d125e2a4e299.jpg)  
+Figure 8: More results compared with other methods.
+
+![](images/580a61e3cd806b6891643912a3371836b379e107cbb060f267510cf78d903732.jpg)  
+Figure 9: More Visual Results (Left is reference video, middle is reference image, right is output).
+
+![](images/974aacfccffbc2ec842676859042a3db7bc76e1663c34fd877501906507b31b2.jpg)  
+Figure 10: More Visual Results.(Left is reference video, middle is reference image, right is output)
+
+![](images/096e4478110851bc18ad429237c77365c98dcc875cc077b3c20849a82dbfe46e.jpg)  
+Figure 11: More Visual Results.(Left is reference video, middle is reference image, right is output)
+
+![](images/9d93a947d659a1ee3b85481f5d9a208ee1a8115039a76dc1c24c01dd359b281b.jpg)  
+Figure 12: Animal Motion Transfer.(Left is reference video, right is output)
+
+![](images/267018cf69e84d845e7bba8fee97409ceef3af9d379c7aa0c9f04fe262dc77c1.jpg)  
+Figure 13: OpenAnimals Datasets.
\ No newline at end of file
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+# UniViT: Unifying Image and Video Understanding in One Vision Encoder
+
+Feilong Tang $^{1,3}$ , Xiang An $^{2*}$ , Haolin Yang $^{3*}$ , Yin Xie $^{2}$ , Kaicheng Yang $^{2}$ , Ming Hu $^{1,3}$ , Zheng Cheng $^{2}$ , Xingyu Zhou $^{2}$ , Zimin Ran $^{4}$ , Imran Razzak $^{3}$ , Ziyong Feng $^{2}$ , Behzad Bozorgtabar $^{5}$ , Jiankang Deng $^{6}$ , Zongyuan Ge $^{1}$
+
+<sup>1</sup>Monash University, <sup>2</sup>DeepGlint, <sup>3</sup>MBZUAI, <sup>4</sup>UTS, <sup>5</sup>EPFL, <sup>6</sup>Imperial College London Feilong.Tang@monash.edu, xiangan@deepglint.com
+
+# Abstract
+
+Despite the impressive progress of recent pretraining methods on multimodal tasks, existing methods are inherently biased towards either spatial modeling (e.g., CLIP) or temporal modeling (e.g., V-JEPA), limiting their joint capture of spatial details and temporal dynamics. To this end, we propose UniViT, a cluster-driven unified self-supervised learning framework that effectively captures the structured semantics of both image spatial content and video temporal dynamics through event-level and object-level clustering and discrimination. Specifically, we leverage offline clustering to generate semantic clusters across both modalities. For videos, multi-granularity event-level clustering progressively expands from single-event to structured multi-event segments, capturing coarse-to-fine temporal semantics; for images, object-level clustering captures fine-grained spatial semantics. However, while global clustering provides semantically consistent clusters, it lacks modeling of structured semantic relations (e.g., temporal event structures). To address this, we introduce a contrastive objective that leverages these semantic clusters as pseudo-label supervision to explicitly enforce structural constraints, including temporal event relations and spatial object co-occurrences, capturing structured semantics beyond categories. Meanwhile, UniViT jointly embeds structured object-level and event-level semantics into a unified representation space. Furthermore, UniViT introduces two key components: (i) Unified Rotary Position Embedding integrates relative positional embedding with frequency-aware dimension allocation to support position-invariant semantic learning and enhance the stability of structured semantics in the discrimination stage; and (ii) Variable Spatiotemporal Streams adapt to inputs of varying frame lengths, addressing the rigidity of conventional fixed-input approaches. Extensive experiments across varying model scales demonstrate that UniViT achieves state-of-the-art performance on linear probing, attentive probing, question answering, and spatial understanding tasks.
+
+# 1 Introduction
+
+Visual representations are fundamental to the success of various downstream tasks. Contrastive [48, 71] and self-supervised [6, 10, 44] frameworks have strong spatial semantic modeling and cross-modal alignment. However, existing methods exhibit inherent biases toward specific modalities, limiting their joint capture of spatial details and temporal dynamics: Image-centric models (e.g., CLIP [48]) capture static semantics but inadequately model temporal dynamics, while video-centric models (e.g., V-JEPA [10]) incorporate temporal cues but exhibit deficiencies in fine-grained spatial modeling.
+
+![](images/baef14c5a1719305c511345702b6f7629510ca74c7961a843a0143c97e66eafa.jpg)  
+(a) Image Discrimination
+
+![](images/3c3c12553e67f7a5b7aee390df8c4dbb6eb7f732f43d34ad2074c4aadaba827f.jpg)  
+(b) Video Discrimination  
+Figure 1: Comparisons of cluster discrimination in image, video, and unified representation. (a) Image multi-label cluster discrimination improves semantic cohesion by assigning multiple samples to cluster centers, capturing various granularities of visual signals at the object level, but is limited to image representations. (b) Video cluster discrimination assigns discrete event-level labels to video segments, modeling dynamic semantics but lacking fine-grained spatial structures. (c) The proposed UniViT adopts unified multi-label discrimination at event and object levels with shared clusters and encoders, bridging spatial semantics and temporal structures within a unified representation.
+
+![](images/1ee9dfde80869eb6d7cae54d90a13858cda583a897a8cb8b0b84454af8913619.jpg)  
+(c) UniViT Discrimination
+
+The key to unified visual pretraining lies in jointly modeling static and dynamic semantics through structured, semantically coherent representations. Recent approaches such as UNICOM [2], MLCD [4] and RICE [64] enhance perception of structured semantics in images by introducing clustering mechanisms. MLCD further advances this approach by employing multi-label clustering to capture multiple semantic components in images, as depicted in Fig. 1 (a). Furthermore, ChatUnivi [34] introduces video event clustering for semantic modeling; however, it remains primarily focused on static objects and isolated events, lacking semantic modeling at the structured event level, thus failing to capture structured temporal dynamics. For instance, although actions such as "grabbing a cup" and "grabbing a phone" differ in their visual manifestation, a model with event-level abstraction can generalize them into a unified "grabbing" event category and capture the semantic relationship between the action and the target object. Moreover, such a model can infer the structural role of an action within an event sequence, such as recognizing that "grabbing a bowl" often precedes the event of "serving food." This capability signifies a transition from isolated event recognition to structured event understanding. Therefore, we argue that transitioning from instance semantic recognition to cross-modal structured modeling is essential for constructing a unified visual pretraining framework.
+
+In this work, we propose UniViT, a cluster-driven unified self-supervised learning framework that effectively captures the structured semantics of both image spatial content and video temporal dynamics through event-level and object-level clustering and discrimination, as depicted in Fig. 1(c). Specifically, we design a two-stage cluster-discrimination training paradigm. In the clustering stage, we employ offline clustering to generate semantic for both modalities. For videos, we perform multi-granularity event-level clustering by densely sampling frames at multiple temporal scales, progressively organizing individual events into structured segments of multiple events, thus capturing coarse-to-fine temporal semantics. For static images and individual video frames, we employ object-level clustering extracts fine-grained spatial semantics. Subsequently, in the discrimination stage, we introduce a contrastive objective that utilizes these semantics as pseudo-label supervision to explicitly enforce structural constraints, including temporal relations among adjacent events and spatial co-occurrences among objects. This approach captures structured semantics beyond isolated categories, aligning dynamic and static semantic content within a unified representation space.
+
+The core of this method is to jointly model structured semantics across image and video modalities within a unified representation space, effectively bridging static spatial details and dynamic temporal relations. Therefore, UniViT introduces two critical components: (i) Unified Rotary Position Embedding (U-RoPE), decomposing positional embeddings into distinct spatial and temporal components through frequency-aware allocation, thereby facilitating position-invariant representation learning and enhancing stability of structured semantic representations during the discrimination stage; and (ii) Variable Spatiotemporal Streams  $(\mathrm{VS}^2)$ , adapting to varying frame lengths, enabling the model to flexibly capture fine-grained spatial details and diverse temporal scales simultaneously. Notably, UniViT retains the original vision encoder without incurring extra inference costs.
+
+![](images/18dd4619c0034749523d493869bf1887f2a867eda3fd703f3f64439c92981848.jpg)  
+Figure 2: Overview of the UniViT framework for unified representation learning. Image and video inputs are processed by 2D/3D convolutions, with video features segmented into events. The resulting tokens are fed into a shared UniViT backbone with U-RoPE, followed by Attentive Pooling to produce a class-level representation for contrastive learning against memory bank class centers.
+
+Extensive experiments across varying model scales demonstrate that UniViT achieves state-of-the-art performance on multiple downstream tasks, including linear probing, attentive probing, image and video question answering, and spatial understanding. The contributions are summarized as follows: (i): We propose UniViT, a cluster-driven unified self-supervised learning framework that effectively captures the structured semantics of both image spatial content and video temporal dynamics through event-level and object-level clustering and discrimination. (ii): We introduce U-RoPE and VS² strategies to explicitly disentangle spatial-temporal positional embeddings and adaptively handle varying frame lengths, facilitating position-invariant and structured spatiotemporal representations. (iii): Extensive experiments across diverse model scales and tasks demonstrate that UniViT achieves state-of-the-art performance on multiple downstream tasks.
+
+# 2 Methodology
+
+Our goal is to achieve unified representation learning across image and video modalities (Fig. 2).
+
+Variable Spatiotemporal Streams  $(\mathbf{VS}^2)$ . Given visual samples  $X = \{x_{i}\}_{i = 1}^{N}$ , each sample  $x_{i}$  is uniformly divided into non-overlapping patches, accommodating both images and videos in a unified manner. Specifically, each image sample  $x_{i}\in \mathbb{R}^{H\times W\times 3}$  is partitioned into spatial patches of size  $P\times P$ , while each video sample  $x_{i}\in \mathbb{R}^{T\times H\times W\times 3}$  is partitioned into variable-length frame sequences  $\{\mathcal{F}_{i,s}\}_{s = 1}^{S_i}$ , with each sequence  $\mathcal{F}_{i,s}\in \mathbb{R}^{T_{i,s}\times H\times W\times 3}$  covering consecutive RGB frames (e.g., 1, 2, 4, 8, or 16 frames) over the same spatial regions, where  $S_{i}$  denotes the number of sequences within the  $i$ -th video. Subsequently, each sequence  $\mathcal{F}_{i,s}$  is independently encoded via a shared Transformer-based encoder  $\phi (\cdot)$  into spatiotemporal embedding tokens  $Z_{i}$ :
+
+$$
+Z _ {i} = \left[ \phi \left(\mathcal {F} _ {i, 1}\right); \phi \left(\mathcal {F} _ {i, 2}\right); \dots ; \phi \left(\mathcal {F} _ {i, S _ {i}}\right) \right] \in \mathbb {R} ^ {M \times C}, \quad \text {w i t h} \quad M = \sum_ {s = 1} ^ {S _ {i}} M _ {s}. \tag {1}
+$$
+
+where  $M_{s}$  and  $C$  denote the number of embedding tokens and the embedding dimension for the  $s$ -th sequence, respectively. The resulting tokens are projected into  $D$ -dimensional vectors, forming token embeddings  $E_{i} = \{e_{i,j}\}_{j=1}^{M} \in \mathbb{R}^{M \times D}$  that encode local visual features [21]. Subsequently, positional encodings are dynamically assigned using the VS $^2$  strategy. Therefore, this design enhances flexibility and spatiotemporal representational capacity.
+
+Event-level and Object-level Clustering. Iterative clustering-discrimination approaches commonly suffer from substantial computational overhead [14]. To address this issue, we adopt a single-step offline clustering to efficiently capture both object-level semantics from images and event-level semantics from videos. Specifically, image embeddings are obtained by pooling the features extracted from local object patches,  $e_i^{obj} = e_i \in \mathbb{R}^D$ , while video embeddings which are derived from a fixed-length 16-frame input are obtained by concatenating frame-level features within each segment, yielding  $e_{i,s}^{\mathrm{evt}} = [e_{i,1};\ldots ;e_{i,s}] \in \mathbb{R}^{s\times D}$  from variable-length sub-clips of  $s \in \{1;2;\dots ;S_i\}$ . We define a set of shared semantic centroids  $\mathcal{C} = \{c_k^{\mathrm{obj}}\}_{k = 1}^{K_{\mathrm{obj}}} \cup \{c_k^{\mathrm{evt}}\}_{k = 1}^{K_{\mathrm{evt}}} \subseteq \mathbb{R}^D$ , where  $K = K_{\mathrm{obj}} + K_{\mathrm{evt}}$  represents the total number of clusters across both modalities. The clustering objective is then
+
+formulated separately for object and event embeddings:
+
+$$
+\mathcal {C} _ {\mathrm {u n i}} = \arg \min  _ {\left\{c _ {k} ^ {\mathrm {o b j}}, c _ {k} ^ {\mathrm {e v t}} \right\}} \sum_ {i = 1} ^ {N} \left(\min  _ {k \in [ 1, K _ {\mathrm {o b j}} ]} \left\| e _ {i} ^ {\mathrm {o b j}} - c _ {k} ^ {\mathrm {o b j}} \right\| _ {2} ^ {2} + \sum_ {s = 1} ^ {S _ {i}} \min  _ {k \in [ 1, K _ {\mathrm {e v t}} ]} \left\| e _ {i, s} ^ {\mathrm {e v t}} - c _ {k} ^ {\mathrm {e v t}} \right\| _ {2} ^ {2}\right), \tag {2}
+$$
+
+where  $N$  is the number of samples.  $\mathcal{C}_{\mathrm{uni}}$  integrates object-level and event-level semantics for consistent representation learning.
+
+Unified Rotary Position Embedding (U-RoPE). Unlike traditional absolute position encoding defined as  $p = (t,x,y)$ , U-RoPE adopts a relative scheme [53]  $\Delta p = (t_1 - t_2, x_1 - x_2, y_1 - y_2)$  that supports position-invariant semantic learning, enabling better modeling of multi-event structures in videos. Specifically, the rotary position embedding is applied directly to the query-key dot-product attention matrix, i.e.,  $\mathbf{A}_{i,j} = (\mathbf{q}_i R_i)(\mathbf{k}_j R_j)^\top$ . For image inputs, the temporal position  $t$  is fixed across the spatial grid  $(x,y)$ . For video inputs, temporal positions vary across frames while spatial positions are computed the same way as for images. Existing methods, such as M-RoPE [60], typically allocate temporal position encodings with high-frequency components, determined by the rotary frequency  $\theta_n = \beta^{-\frac{2n}{C}}$ . This allocation causes periodic oscillations, leading to unstable frame representations that conflict with dense label discrimination. Therefore, we propose a unified frequency allocation strategy that assigns global event-related temporal structures to smoother low-frequency components, while retaining high-frequency components for local spatial details:
+
+$$
+\Phi_ {S} = \left\{\beta^ {- \frac {2 (2 j + k)}{C}} \mid j \in [ 0, \frac {3}{4} L), k \in \{0, 1 \} \right\}, \quad \Phi_ {T} = \left\{\beta^ {- \frac {2 j}{C}} \mid j \in [ \frac {3}{4} L, L) \right\}, \tag {3}
+$$
+
+where  $\Phi_S$  and  $\Phi_T$  respectively denote the rotation frequencies used for spatial and temporal rotary applying to the  $2L$ -dimensional embedding space, with  $L = \frac{C}{2}$ , and  $\beta$  represents the base frequency.
+
+Joint Training Objective. Visual samples commonly exhibit multiple semantic components, including object-level semantics from images and event-level semantics from videos, rendering single-label assignments inadequate for unified multimodal representation learning. To capture both object-level and event-level semantic structures, we introduce a contrastive objective that leverages these semantic clusters as pseudo-label supervision to explicitly enforce structural constraints. Specifically, for each visual embedding  $e_i \in \mathbb{R}^D$ , we identify multiple positive semantic labels from the unified semantic centroid set  $\mathcal{C}_{uni} \in \mathbb{R}^{(|\mathcal{C}_{obj}| + |\mathcal{C}_{evt}|) \times D}$ , consisting of both object-level  $\mathcal{C}_{obj}$  and event-level  $\mathcal{C}_{evt}$  centroids. The remaining centroids in this unified set are treated as negative labels. Subsequently, the joint multi-label semantic discrimination objective is formulated as:
+
+$$
+\mathcal {L} _ {\text {J o i n t}} = \sum_ {m \in \{o b j, e v t \}} \left[ \log \left(1 + \sum_ {j \in \Omega_ {n} ^ {m}} \exp \left(\sigma_ {j} ^ {m}\right)\right) + \log \left(1 + \sum_ {i \in \Omega_ {p} ^ {m}} \exp \left(- \sigma_ {i} ^ {m}\right)\right) \right], \tag {4}
+$$
+
+where  $m \in \{obj,evt\}$  denotes the semantic granularity level, corresponding respectively to object-level (images or single frames) and event-level (video segments).  $\Omega_p^m$  and  $\Omega_n^m$  represent the sets of positive and negative semantic labels for granularity level  $m$ , while  $\sigma_i^m$  and  $\sigma_j^m$  indicate embedding similarity scores to positive and negative semantic centroids, respectively. The embedding similarity score  $\sigma_{u,k}^m$  is computed as  $\sigma_{u,k}^m = e_u^\top c_k^m$ , where  $u$  and  $k$  index visual embeddings and semantic centroids within the corresponding positive or negative sets, respectively. This unified formulation leverages semantic clustering to capture spatiotemporal structures for discriminative embeddings.
+
+Unified Image and Video Understanding. As shown in Fig.3a, the frame-to-frame similarity under the multi-event setting is significantly lower than that of the single-event counterpart, indicating finer-grained temporal discrimination and better action segmentation. The lower training loss in Fig. 3b indicates that fine-grained temporal modeling in the multi-event setting benefits video understanding. In Fig. 3c, U-RoPE demonstrates faster convergence and improved stability over absolute position encoding and M-RoPE. By decoupling spatial and temporal dimensions and avoiding high-frequency temporal encoding, U-RoPE enables robust position-invariant learning across both images and videos. Fig. 3d illustrates the distribution of cosine similarity between image and video feature embeddings, indicating that video and image representations are related yet significantly different.
+
+# 3 Experiments
+
+# 3.1 Implementation Details.
+
+Pretraining Setup. Our models are pre-trained on the LAION400M[50], COYO700M[13], and InternVid[61]. We use 80 H800 GPUs for the training process. During training, we maintained a 1:1
+
+![](images/d1d76a4be003c257d90a1c1e0b2c8f00cd854643ddc7e991d6e0f9a13f3e1012.jpg)  
+(a) Mean frame similarity.
+
+![](images/3290489635510111d01d7e32a4517bdb4090210d8cd25f1e0ec5d387d913d9ba.jpg)  
+(b) Loss cruve.
+
+![](images/cd2eb2bc9e10fb3b61a539f55eb4ebfee3ab7483f7141fb0e6b1bb35440c700d.jpg)  
+(c) U-RoPE  
+Figure 3: Analysis of multi-event approach. (a) Multi-event frames demonstrate significantly lower frame similarity (0.65 vs 0.75, indicating greater inter-frame distinctiveness that improves feature discrimination. (b) Multi-event processing substantially accelerates convergence speed, with higher frame counts (8-16) showing faster loss reduction compared to single or fewer frames. (c) U-RoPE enables better local semantic learning, resulting in lower loss compared to other methods. (d) Distribution of cosine similarity between image and video feature embeddings.
+
+![](images/ffefb9196320c358ba9e2690738f0e13896c1eaf0f22807c2d43d77eae4bd9a0.jpg)  
+(d) Image Video Similarity
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Arch.</td><td rowspan="2">Resolution</td><td rowspan="2">Params</td><td rowspan="2">Data Images/Video</td><td rowspan="2">Language pre.</td><td colspan="9">Image Tasks</td><td colspan="7">Video Tasks</td></tr><tr><td>Avg.</td><td>Food [12]</td><td>CIFAR [36]</td><td>EuroSAT [32]</td><td>Resc45 [18]</td><td>Caltech101 [62]</td><td>Imagenet [49]</td><td>Sum397 [63]</td><td>Avg.</td><td>K400 [35]</td><td>K600 [16]</td><td>K700 [51]</td><td>HMD851 [37]</td><td>UCF101 [52]</td><td>RareAct [42]</td><td>SSV2 [27]</td></tr><tr><td colspan="21">Methods pretrained on Images</td><td></td></tr><tr><td rowspan="3">Sigclipv2 [58]</td><td>ViT-B/16</td><td>256</td><td>86M</td><td>10B</td><td>✓</td><td>90.5</td><td>95.2</td><td>94.7</td><td>93.1</td><td>90.9</td><td>95.4</td><td>83.2</td><td>81.2</td><td>58.2</td><td>71.3</td><td>70.9</td><td>59.0</td><td>66.3</td><td>92.2</td><td>31.3</td><td>16.3</td></tr><tr><td>ViT-L/16</td><td>256</td><td>303M</td><td>10B</td><td>✓</td><td>92.6</td><td>96.5</td><td>97.2</td><td>94.5</td><td>94.3</td><td>96.0</td><td>84.1</td><td>85.3</td><td>63.9</td><td>77.3</td><td>77.5</td><td>66.5</td><td>71.7</td><td>95.0</td><td>39.8</td><td>19.4</td></tr><tr><td>ViT-SO400M</td><td>224</td><td>428M</td><td>10B</td><td>✓</td><td>93.4</td><td>97.1</td><td>97.8</td><td>95.0</td><td>95.8</td><td>96.1</td><td>85.5</td><td>86.4</td><td>66.4</td><td>78.7</td><td>79.1</td><td>68.8</td><td>76.3</td><td>95.0</td><td>46.1</td><td>20.9</td></tr><tr><td rowspan="3">DINOv2 [44]</td><td>ViT-S/14</td><td>224</td><td>22M</td><td>145M</td><td>✗</td><td>85.9</td><td>89.3</td><td>90.2</td><td>92.4</td><td>85.6</td><td>91.0</td><td>82.3</td><td>70.3</td><td>49.2</td><td>60.3</td><td>60.2</td><td>49.8</td><td>56.5</td><td>87.2</td><td>16.4</td><td>14.3</td></tr><tr><td>ViT-B/14</td><td>224</td><td>87M</td><td>145M</td><td>✗</td><td>89.3</td><td>95.1</td><td>94.7</td><td>92.1</td><td>90.4</td><td>93.5</td><td>83.1</td><td>76.4</td><td>56.7</td><td>66.9</td><td>66.9</td><td>54.3</td><td>63.8</td><td>91.6</td><td>38.8</td><td>14.5</td></tr><tr><td>ViT-L/14</td><td>224</td><td>311M</td><td>145M</td><td>✗</td><td>91.4</td><td>95.2</td><td>96.4</td><td>94.3</td><td>91.2</td><td>95.4</td><td>84.7</td><td>82.4</td><td>56.5</td><td>67.5</td><td>67.5</td><td>54.2</td><td>64.3</td><td>91.8</td><td>35.6</td><td>14.6</td></tr><tr><td rowspan="2">CLIP [48]</td><td>ViT-B/16</td><td>224</td><td>86M</td><td>400M</td><td>✓</td><td>89.0</td><td>93.0</td><td>92.4</td><td>92.4</td><td>90.8</td><td>95.6</td><td>81.4</td><td>77.1</td><td>56.5</td><td>67.4</td><td>67.1</td><td>54.2</td><td>66.9</td><td>88.1</td><td>37.5</td><td>14.0</td></tr><tr><td>ViT-L/14</td><td>224</td><td>304M</td><td>400M</td><td>✓</td><td>91.6</td><td>95.7</td><td>94.2</td><td>94.5</td><td>93.4</td><td>95.7</td><td>83.2</td><td>84.7</td><td>63.7</td><td>74.8</td><td>74.2</td><td>63.2</td><td>71.3</td><td>92.6</td><td>53.1</td><td>16.8</td></tr><tr><td rowspan="2">I-JEPA [7]</td><td>ViT-H/14</td><td>224</td><td>631M</td><td>22K</td><td>✗</td><td>83.0</td><td>76.3</td><td>94.7</td><td>89.7</td><td>79.5</td><td>92.3</td><td>80.1</td><td>68.4</td><td>40.6</td><td>48.6</td><td>46.6</td><td>34.1</td><td>56.3</td><td>83.9</td><td>2.0</td><td>12.4</td></tr><tr><td>ViT-g/16</td><td>224</td><td>1011M</td><td>22K</td><td>✗</td><td>85.0</td><td>78.9</td><td>95.1</td><td>90.4</td><td>83.7</td><td>93.7</td><td>83.3</td><td>70.2</td><td>41.5</td><td>48.9</td><td>47.2</td><td>35.3</td><td>58.2</td><td>83.1</td><td>3.9</td><td>13.8</td></tr><tr><td>MLCD [4]</td><td>ViT-L/14</td><td>336</td><td>304M</td><td>1.1B</td><td>✗</td><td>92.9</td><td>96.8</td><td>97.8</td><td>95.0</td><td>95.2</td><td>95.7</td><td>84.3</td><td>85.4</td><td>66.8</td><td>78.8</td><td>78.9</td><td>68.2</td><td>80.4</td><td>97.1</td><td>46.9</td><td>17.4</td></tr><tr><td colspan="21">Methods pretrained on Videos</td><td></td></tr><tr><td rowspan="2">V-JEPA [10]</td><td>ViT-L/16</td><td>224</td><td>312M</td><td>2M</td><td>✗</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>57.0</td><td>62.3</td><td>63.2</td><td>49.2</td><td>77.7</td><td>95.4</td><td>6.3</td><td>44.9</td></tr><tr><td>ViT-H/16</td><td>224</td><td>649M</td><td>2M</td><td>✗</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>56.1</td><td>59.3</td><td>60.5</td><td>46.9</td><td>83.0</td><td>94.4</td><td>4.0</td><td>44.8</td></tr><tr><td>LanguageBind [73]</td><td>ViT-L/14</td><td>224</td><td>407M</td><td>3M</td><td>✓</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>60.0</td><td>72.1</td><td>72.3</td><td>60.3</td><td>71.9</td><td>93.3</td><td>33.8</td><td>16.4</td></tr><tr><td>VideoMAEV2 [59]</td><td>ViT-g/14</td><td>224</td><td>1012M</td><td>1.35M</td><td>✗</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>35.3</td><td>39.8</td><td>42.6</td><td>29.4</td><td>30.6</td><td>75.0</td><td>1.6</td><td>27.9</td></tr><tr><td colspan="21">Methods pretrained on Image and Video</td><td></td></tr><tr><td rowspan="3">PE-Core [11]</td><td>ViT-B/16</td><td>224</td><td>93M</td><td>5.4B/22M</td><td>✓</td><td>89.9</td><td>94.0</td><td>92.7</td><td>92.9</td><td>91.2</td><td>94.4</td><td>84.2</td><td>80.1</td><td>54.0</td><td>66.6</td><td>66.2</td><td>53.6</td><td>65.9</td><td>90.3</td><td>19.4</td><td>15.9</td></tr><tr><td>ViT-L/14</td><td>336</td><td>317M</td><td>5.4B/22M</td><td>✓</td><td>92.1</td><td>96.7</td><td>94.5</td><td>94.8</td><td>93.6</td><td>95.3</td><td>85.3</td><td>84.6</td><td>67.6</td><td>79.6</td><td>79.8</td><td>69.4</td><td>77.7</td><td>96.8</td><td>47.5</td><td>22.5</td></tr><tr><td>ViT-G/14</td><td>448</td><td>1882M</td><td>5.4B/22M</td><td>✓</td><td>93.1</td><td>96.9</td><td>97.7</td><td>95.6</td><td>93.9</td><td>95.7</td><td>86.2</td><td>85.4</td><td>68.4</td><td>81.5</td><td>81.7</td><td>72.1</td><td>80.5</td><td>97.3</td><td>42.2</td><td>23.7</td></tr><tr><td rowspan="4">UniViT</td><td>ViT-S/16</td><td>224</td><td>26M</td><td>1.1B/60M</td><td>✗</td><td>85.5</td><td>88.9</td><td>85.5</td><td>92.5</td><td>87.4</td><td>91.4</td><td>81.3</td><td>71.5</td><td>52.0</td><td>65.6</td><td>64.9</td><td>50.9</td><td>59.4</td><td>89.6</td><td>18.1</td><td>15.6</td></tr><tr><td>ViT-B/16</td><td>224</td><td>99M</td><td>1.1B/60M</td><td>✗</td><td>89.0</td><td>91.4</td><td>92.6</td><td>93.1</td><td>90.2</td><td>94.2</td><td>83.1</td><td>78.2</td><td>61.9</td><td>74.7</td><td>74.8</td><td>61.5</td><td>70.2</td><td>94.8</td><td>32.0</td><td>25.1</td></tr><tr><td>ViT-L/14</td><td>224</td><td>334M</td><td>1.1B/60M</td><td>✗</td><td>91.8</td><td>95.2</td><td>95.5</td><td>94.7</td><td>93.1</td><td>95.4</td><td>84.2</td><td>84.5</td><td>68.3</td><td>82.9</td><td>83.0</td><td>72.4</td><td>81.4</td><td>97.3</td><td>33.8</td><td>27.3</td></tr><tr><td>ViT-L/14</td><td>336</td><td>334M</td><td>1.1B/60M</td><td>✗</td><td>92.8</td><td>95.1</td><td>97.7</td><td>95.1</td><td>93.7</td><td>95.7</td><td>85.4</td><td>86.5</td><td>73.1</td><td>84.1</td><td>84.2</td><td>73.3</td><td>83.5</td><td>98.2</td><td>56.3</td><td>32.1</td></tr></table>
+
+Table 1: Attentive Probe Evaluation under few-shot settings for label efficiency analysis. Bold indicates the best performance.
+
+ratio between images and video frames, with an image batch size of 16K and a video batch size of 2K (each video containing 16 frames). In total, our model is exposed to approximately 20B image frames throughout the training. For our standard model, we use 224 resolution images. For the 336 resolution variant, we first train the model at 224 resolution, then increase it to 336 and continue training for an additional 1B frames. We utilize the AdamW optimizer with a learning rate of 0.001 and weight decay of 0.2. The number of classes  $(k)$  is one million, the ratio of sampled negative class centers  $(r)$  is 0.1, and the number of positive labels  $(l)$  assigned to each image and video is 8.
+
+Multimodal Setup. For our multimodal large language model evaluations, we adopt the LLaVA-NeXT [41] framework while maintaining experimental consistency. All training methodologies precisely follow the original LLaVA-NeXT-Video implementation, utilizing identical pretraining datasets and instruction-tuning data. We employ Qwen2.5-7B [68] as our language model backbone, which effectively mitigates potential hyperparameter biases that might favor OpenAI-CLIP in the original LLaVA-NeXT-Video configuration. This controlled experimental design ensures fair comparison when evaluating the performance of our vision encoders within multimodal systems.
+
+# 3.2 Comparisons with Existing Vision Encoders
+
+Attentive Probing Results. We evaluate the comprehensive ability of UniViT across 14 standard benchmarks, covering a wide range of semantic and vision-centric tasks, using a 50-shot attentive
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Arch.</td><td rowspan="2">Resolution</td><td rowspan="2">Params</td><td rowspan="2">Delta Images/Videos</td><td rowspan="2">Language pre.</td><td colspan="9">Image Tasks</td><td colspan="7">Video Tasks</td></tr><tr><td>Avg.</td><td>Food [12]</td><td>CIFAR [36]</td><td>EuroSAT [32]</td><td>Resc45 [18]</td><td>Caltech101 [62]</td><td>Imagenet [49]</td><td>Sun397 [63]</td><td>Avg.</td><td>K400 [35]</td><td>K600 [16]</td><td>K700 [51]</td><td>HMDB51 [37]</td><td>UCF101 [52]</td><td>RareAct [42]</td><td>SSV2 [27]</td></tr><tr><td colspan="21">Methods pretrained on Images</td><td></td></tr><tr><td rowspan="3">Sigclipv2 [58]</td><td>ViT-B/16</td><td>256</td><td>86M</td><td>10B</td><td>✓</td><td>93.7</td><td>94.6</td><td>96.7</td><td>98.3</td><td>93.1</td><td>98.5</td><td>83.5</td><td>91.3</td><td>56.3</td><td>72.4</td><td>73.6</td><td>60.2</td><td>63.7</td><td>90.6</td><td>12.5</td><td>21.1</td></tr><tr><td>ViT-L/16</td><td>256</td><td>303M</td><td>10B</td><td>✓</td><td>94.3</td><td>96.2</td><td>97.8</td><td>98.2</td><td>93.1</td><td>98.6</td><td>85.2</td><td>91.3</td><td>59.4</td><td>76.5</td><td>78.2</td><td>66.2</td><td>66.8</td><td>93.1</td><td>12.5</td><td>22.7</td></tr><tr><td>ViT-SO400M</td><td>224</td><td>428M</td><td>10B</td><td>✓</td><td>94.7</td><td>96.4</td><td>98.2</td><td>97.8</td><td>94.0</td><td>98.6</td><td>85.7</td><td>92.2</td><td>66.5</td><td>78.6</td><td>79.7</td><td>68.8</td><td>68.0</td><td>94.4</td><td>53.1</td><td>23.2</td></tr><tr><td rowspan="3">DINOv2 [44]</td><td>ViT-S/14</td><td>224</td><td>22M</td><td>145M</td><td>✗</td><td>90.6</td><td>89.1</td><td>97.7</td><td>98.1</td><td>84.4</td><td>97.0</td><td>81.1</td><td>86.9</td><td>52.2</td><td>62.6</td><td>62.9</td><td>49.0</td><td>53.5</td><td>82.9</td><td>37.5</td><td>17.2</td></tr><tr><td>ViT-B/14</td><td>224</td><td>87M</td><td>145M</td><td>✗</td><td>92.1</td><td>92.8</td><td>98.7</td><td>98.1</td><td>86.1</td><td>96.1</td><td>84.5</td><td>88.1</td><td>59.0</td><td>68.9</td><td>70.6</td><td>57.3</td><td>61.7</td><td>89.9</td><td>46.9</td><td>18.0</td></tr><tr><td>ViT-L/14</td><td>224</td><td>311M</td><td>145M</td><td>✗</td><td>93.6</td><td>94.3</td><td>99.4</td><td>98.5</td><td>90.1</td><td>97.5</td><td>86.3</td><td>89.0</td><td>61.0</td><td>73.7</td><td>74.1</td><td>63.1</td><td>64.4</td><td>91.8</td><td>40.6</td><td>19.6</td></tr><tr><td rowspan="2">CLIP [48]</td><td>ViT-B/16</td><td>224</td><td>86M</td><td>400M</td><td>✓</td><td>91.8</td><td>92.2</td><td>95.9</td><td>97.8</td><td>90.9</td><td>96.3</td><td>80.2</td><td>89.2</td><td>58.3</td><td>70.3</td><td>71.7</td><td>57.7</td><td>64.4</td><td>88.5</td><td>37.5</td><td>18.0</td></tr><tr><td>ViT-L/14</td><td>224</td><td>304M</td><td>400M</td><td>✓</td><td>93.8</td><td>95.0</td><td>98.1</td><td>98.6</td><td>93.3</td><td>97.4</td><td>83.9</td><td>90.5</td><td>64.4</td><td>76.4</td><td>77.8</td><td>65.8</td><td>64.8</td><td>92.8</td><td>53.1</td><td>19.8</td></tr><tr><td rowspan="2">I-JEPA [7]</td><td>ViT-H/14</td><td>224</td><td>631M</td><td>22K</td><td>✗</td><td>86.1</td><td>74.1</td><td>98.3</td><td>98.6</td><td>78.6</td><td>95.6</td><td>79.3</td><td>78.0</td><td>41.8</td><td>49.4</td><td>50.2</td><td>37.5</td><td>46.9</td><td>74.1</td><td>18.8</td><td>15.5</td></tr><tr><td>ViT-g/16</td><td>224</td><td>1011M</td><td>22K</td><td>✗</td><td>87.9</td><td>77.7</td><td>98.2</td><td>98.7</td><td>82.5</td><td>95.8</td><td>82.1</td><td>80.6</td><td>42.0</td><td>49.6</td><td>50.4</td><td>37.7</td><td>47.7</td><td>74.7</td><td>18.8</td><td>15.1</td></tr><tr><td rowspan="4">MLCD [4]</td><td>ViT-S/16</td><td>224</td><td>22M</td><td>1.1B</td><td>✗</td><td>88.7</td><td>84.0</td><td>94.2</td><td>98.5</td><td>87.9</td><td>92.9</td><td>79.1</td><td>84.6</td><td>38.9</td><td>48.1</td><td>49.6</td><td>38.3</td><td>36.7</td><td>61.4</td><td>25.0</td><td>13.0</td></tr><tr><td>ViT-B/16</td><td>224</td><td>86M</td><td>1.1B</td><td>✗</td><td>91.7</td><td>89.8</td><td>97.6</td><td>98.7</td><td>90.8</td><td>95.8</td><td>82.3</td><td>86.9</td><td>48.3</td><td>58.9</td><td>60.7</td><td>48.4</td><td>46.5</td><td>79.9</td><td>28.1</td><td>15.6</td></tr><tr><td>ViT-L/14</td><td>224</td><td>304M</td><td>1.1B</td><td>✗</td><td>91.4</td><td>87.2</td><td>97.2</td><td>98.8</td><td>89.3</td><td>95.6</td><td>85.4</td><td>86.0</td><td>58.1</td><td>71.4</td><td>72.8</td><td>60.0</td><td>60.2</td><td>88.3</td><td>37.5</td><td>16.7</td></tr><tr><td>ViT-L/14</td><td>336</td><td>304M</td><td>1.1B</td><td>✗</td><td>94.9</td><td>96.2</td><td>99.4</td><td>99.1</td><td>94.5</td><td>97.9</td><td>86.3</td><td>91.0</td><td>62.2</td><td>76.0</td><td>76.2</td><td>64.1</td><td>62.5</td><td>92.2</td><td>46.9</td><td>17.5</td></tr><tr><td colspan="21">Methods pretrained on Videos</td><td></td></tr><tr><td>LanguageBind [73]</td><td>ViT-L/14</td><td>224</td><td>407M</td><td>3M</td><td>✓</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>63.6</td><td>74.4</td><td>75.1</td><td>62.5</td><td>71.1</td><td>92.9</td><td>46.9</td><td>22.1</td></tr><tr><td colspan="21">Methods pretrained on Image and Videos</td><td></td></tr><tr><td rowspan="3">PE-Core [11]</td><td>ViT-B/16</td><td>224</td><td>93M</td><td>5.4B/22M</td><td>✓</td><td>93.4</td><td>93.2</td><td>98.1</td><td>98.8</td><td>92.9</td><td>97.2</td><td>83.4</td><td>90.5</td><td>58.1</td><td>69.5</td><td>71.2</td><td>56.8</td><td>66.0</td><td>88.7</td><td>37.5</td><td>16.9</td></tr><tr><td>ViT-L/14</td><td>336</td><td>317M</td><td>5.4B/22M</td><td>✓</td><td>95.0</td><td>96.2</td><td>99.4</td><td>98.8</td><td>93.6</td><td>98.0</td><td>86.7</td><td>92.0</td><td>63.7</td><td>74.5</td><td>75.4</td><td>62.2</td><td>69.1</td><td>92.4</td><td>53.1</td><td>19.0</td></tr><tr><td>ViT-G/14</td><td>448</td><td>1882M</td><td>5.4B/22M</td><td>✓</td><td>95.2</td><td>96.3</td><td>99.3</td><td>98.1</td><td>93.5</td><td>97.9</td><td>89.5</td><td>92.1</td><td>68.6</td><td>80.8</td><td>81.1</td><td>70.4</td><td>73.4</td><td>94.7</td><td>56.3</td><td>23.2</td></tr><tr><td rowspan="4">UniViT</td><td>ViT-S/16</td><td>224</td><td>26M</td><td>1.1B/60M</td><td>✗</td><td>91.0</td><td>87.7</td><td>95.8</td><td>98.8</td><td>90.5</td><td>96.4</td><td>80.4</td><td>87.3</td><td>56.0</td><td>67.6</td><td>68.8</td><td>54.0</td><td>56.6</td><td>87.6</td><td>40.6</td><td>16.8</td></tr><tr><td>ViT-B/16</td><td>224</td><td>99M</td><td>1.1B/60M</td><td>✗</td><td>92.2</td><td>90.1</td><td>97.1</td><td>98.9</td><td>92.2</td><td>96.3</td><td>83.1</td><td>87.5</td><td>60.8</td><td>75.7</td><td>77.3</td><td>63.2</td><td>66.0</td><td>93.1</td><td>28.1</td><td>22.3</td></tr><tr><td>ViT-L/14</td><td>224</td><td>334M</td><td>1.1B/60M</td><td>✗</td><td>94.3</td><td>94.8</td><td>98.8</td><td>99.0</td><td>93.8</td><td>98.1</td><td>85.6</td><td>90.2</td><td>67.1</td><td>84.3</td><td>85.3</td><td>74.5</td><td>64.1</td><td>92.2</td><td>43.8</td><td>25.5</td></tr><tr><td>ViT-L/14</td><td>336</td><td>334M</td><td>1.1B/60M</td><td>✗</td><td>94.9</td><td>96.6</td><td>98.9</td><td>99.0</td><td>94.9</td><td>98.4</td><td>86.5</td><td>90.2</td><td>72.0</td><td>85.3</td><td>85.4</td><td>74.9</td><td>78.1</td><td>96.6</td><td>56.3</td><td>27.4</td></tr></table>
+
+Table 2: Linear Probe Evaluation. Bold indicates the best performance.
+
+![](images/4ead013ea478fec156416f0a30fc51c55501f21855c8ef7cc4209bed27e8d8b7.jpg)  
+Figure 4: Comparisons of UniViT and existing vision encoders across diverse multimodal image and video benchmarks, highlighting UniViT's superior unified representation capabilities.
+
+probing setup, which allows us to measure feature quality without extensive task-specific finetuning. As shown in Table 1, our model achieves state-of-the-art performance on multiple standard benchmarks, including object classification, scene recognition, and fine-grained categorization.
+
+Linear Probing Results. As shown in Table 2, we report performance on the same set of benchmarks as in attentive probe, this time using a standard linear probing setup instead. The trend aligns closely with the attentive probe results; our UniViT achieves state-of-the-art performance on both semantic and non-semantic tasks, further validating the quality and generality of the learned representations.
+
+All experimental settings for the compared models strictly follow their original implementations to ensure fair comparison. Under this consistent protocol, despite being pretrained without pixel-level or feature-level supervision (e.g., MAE, JEP), our UniViT still achieves strong visual understanding across a wide range of vision-centric tasks. Notably, previous video encoders cannot handle static images, while image encoders struggle with temporally dependent tasks (e.g., SSV2). In contrast, UniViT leverages a unified representation space that captures both spatial and temporal patterns, enabling strong performance across image and video domains. This demonstrates the generalization capability of our architecture, which is designed for unified representation learning across modalities.
+
+UniViT as a Vision Encoder for MLLMs. In this section, we evaluate our unified vision encoder, UniViT, which is designed to seamlessly handle both image and video modalities. We conduct comprehensive experiments on a diverse set of benchmarks to assess the model's ability to learn shared semantic representations across static and temporal inputs. The evaluation is performed within the LLaVA-NeXT-Video framework under consistent and controlled settings to ensure fair comparison. We benchmark UniViT on 18 datasets spanning four major domains: General VQA and Knowledge VQA. These datasets cover both image-based and video-based VQA tasks, providing
+
+![](images/7463b6141d3dff518d3fd04b2c94e5fada937eae1463be5170d4215131c17ee6.jpg)  
+Figure 6: PCA visualization of patch features extracted by our UniViT. Patches displaying similar colors indicate semantic similarities, reflecting that they embody analogous elements or attributes.
+
+a holistic view of the model's multimodal reasoning capability. As summarized in Fig. 4, UniViT achieves strong and consistent performance across all categories, notably outperforming conventional single-modality baselines such as CLIP and SigLIP. At a resolution of  $336\mathrm{px}$ , UniViT surpasses CLIP on 17 out of 18 benchmarks. Importantly, UniViT demonstrates robust performance across both Image-VQA and Video-VQA, highlighting its versatility and effectiveness in learning generalizable multimodal semantics. Notably, these results are achieved without any language supervision, further demonstrating the strength of our unified framework in capturing cross-modal visual understanding. This positions UniViT as a practical and scalable solution for real-world multimodal applications.
+
+Scaling Behavior. To investigate the scalability of our unified vision encoder, we conduct a systematic analysis of its performance across varying configurations using an attentive probe protocol on both image and video tasks. Rather than comparing against other paradigms, we focus on the internal scaling behavior of our framework along three progressive dimensions: (1) increasing training data volume, (2) expanding model capacity, and (3) increasing input resolution.
+
+As illustrated in Fig. 5, we analyze the scaling behavior of UniViT on video tasks by averaging performance scores from seven video datasets. This provides a stable estimate of performance trends specific to video understanding. This controlled design allows us to isolate and examine the contribution of each scaling factor. We observe consistent and meaningful improvements at all stages. Increasing data volume leads to noticeable gains for both modalities, suggesting that enhanced data diversity improves unified semantic modeling. Expanding model capacity, from small to large variants, further boosts representation quality, with more pronounced benefits on fine-grained tasks. Higher input resolution also contributes positively to overall performance, especially in video tasks where capturing spatial continuity across frames is crucial for robust representation learning. Compared to existing models, our approach exhibits more efficient scaling behavior, with larger variants
+
+![](images/8652d09a8513f96f5fa7deecdf6687b9fdb3a9184be11287eb05b81d2a2d16d3.jpg)  
+Figure 5: Scaling Behavior: Performance scalability of UniViT across model capacities, training data volumes, and input resolutions. Average accuracy across multiple video tasks demonstrates consistent improvements with increased scale.
+
+delivering consistently stronger improvements across video tasks. These results demonstrate that our model scales effectively with standard training resources while maintaining strong generalization across diverse video benchmarks.
+
+# 4 Ablation Study
+
+Pretraining Strategy. As shown in Table 3a and 3b, we conduct ablation studies to analyze the impact of robust pretraining strategies and position embedding designs. In Table 3a, we progressively apply modifications including attentive pooling, and stronger data augmentation. Each contributes positively to K400 and SSV2, highlighting the importance of stable pretraining for video understanding. In Table 3b, we compare different designs for position embedding. We find that moving from absolute
+
+<table><tr><td>Data</td><td>K400</td><td>SSV2</td></tr><tr><td>Baseline</td><td>64.1</td><td>15.0</td></tr><tr><td>+Atten Pool</td><td>65.2</td><td>15.5</td></tr><tr><td>+Data Aug</td><td>65.6</td><td>15.6</td></tr></table>
+
+(a) Pretraining Strategy. The effects attentive pooling, and data augmentation.
+
+<table><tr><td>Method</td><td>K400</td><td>SSV2</td></tr><tr><td>Abs + ViT</td><td>64.2</td><td>13.9</td></tr><tr><td>Abs + 2DRoPE</td><td>64.7</td><td>14.3</td></tr><tr><td>M-RoPE</td><td>65.1</td><td>15.2</td></tr><tr><td>U-RoPE</td><td>65.6</td><td>15.6</td></tr></table>
+
+(b) Position Embedding. Comparison of absolute position, 2D, 3D, and unified RoPE strategies.
+
+<table><tr><td>Frames</td><td>K400</td><td>SSV2</td></tr><tr><td>Baseline</td><td>62.7</td><td>13.2</td></tr><tr><td>+8</td><td>64.4</td><td>14.2</td></tr><tr><td>+4,8</td><td>65.2</td><td>15.3</td></tr><tr><td>+1,2,4,8</td><td>65.6</td><td>15.6</td></tr></table>
+
+(c) Varing Frames with Multi-Event. Pretraining at varying frames with multi-event.
+
+<table><tr><td>Method</td><td>K400</td><td>SSV2</td></tr><tr><td>only Obj.</td><td>62.3</td><td>12.5</td></tr><tr><td>only Evt.</td><td>61.9</td><td>11.3</td></tr><tr><td>Obj.+Evt.</td><td>62.7</td><td>13.2</td></tr><tr><td>Obj.+Multi-Evt.</td><td>65.6</td><td>15.6</td></tr></table>
+
+(d) Clustering Strategy. Perform clustering at the Event-level and Object-level.
+
+<table><tr><td>Num Classes</td><td>K400</td><td>SSV2</td></tr><tr><td>500k</td><td>65.2</td><td>15.5</td></tr><tr><td>1M</td><td>65.6</td><td>15.6</td></tr><tr><td>2M</td><td>65.8</td><td>15.5</td></tr><tr><td>5M</td><td>65.2</td><td>15.4</td></tr></table>
+
+(e) Classes Number. The number of classes during event-level and object-level clustering.  
+Table 3: Ablation experiments on K400 and SSV2 under few-shot settings. (a) Pretraining strategies. (b) Position embeddings. (c) Multi-event frame sampling. (d) Semantic clustering strategies. (e) Number of clustering classes. The entries marked in gray are the same, which specify the default settings.
+
+position embedding to 2D-RoPE with 1d-absolutioe position and 3D-RoPE leads to substantial performance gains, with 3D-RoPE achieving better results. This suggests that temporal embedding better captures motion dynamics. Our U-RoPE decouples spatial and temporal dimensions during embedding, enabling more flexible handling of both image and video modalities, and achieves the best overall performance.
+
+Effect of Variable Spatiotemporal Streams. We begin with a baseline where the model is pretrained using only 16-frame video clips. To enhance temporal modeling, we introduce variable-length frame inputs (e.g., 1, 2, 4, 8, 16 frames) with dense labels, as shown in Table 3c. On K400, performance remains stable across frame combinations. In contrast, the more temporally complex SSV2 benefits notably from dense supervision with diverse frame counts, suggesting it helps capture short-range temporal dynamics. To qualitatively assess the short-range temporal robustness of our model, we visualize patch-level features using PCA, as shown in Fig. 6. The model produces consistent local semantic representations across a variety of image and video inputs, demonstrating strong spatial grounding and frame-invariant semantic encoding. Together, these findings validate the effectiveness of our variable-frame pretraining strategy and highlight the model's generalization capability.
+
+Qualitative Analysis of Frame-Agnostic Semantics. We visualize the object-level feature distribution using T-SNE projection on K400, comparing both image and video samples. As shown in Fig. 7, our model produces well-formed and compact clusters, where image and video instances from the same semantic class are consistently grouped together. This indicates a strong alignment of visual representations across modalities. The tight intra-class clusters and clear inter-class boundaries demonstrate the model's ability to abstract high-level semantics that are shared between static and temporal visual data. Compared to the State of the Art model, such as SigLIP2 and DINOv2, our method achieves superior intra-class compactness and inter-class separation, highlighting its strength in learning modality-invariant and semantically consistent features.
+
+The Effect of Clustering. To investigate the impact of different clustering strategies on video understanding, we perform ablation studies at both the event and object levels. As shown in Table 3d, using only object-level clustering achieves  $62.3\%$  on K400 and  $12.5\%$  on SSV2. Clustering only at the event level yields similar performance. However, combining object-level and event-level clustering leads to consistent improvements on both benchmarks (62.7% and  $13.2\%$ ), indicating that the two levels provide complementary information. Furthermore, when we extend to multi-granularity event clustering (Obj.+Multi-Evt.), the performance improves further, achieving the best results on both datasets (65.6% and  $15.6\%$ ). These results demonstrate that clustering at varying event granularities, in combination with object-level semantics, significantly enhances video understanding.
+
+![](images/ffe97fe4cfa76c182d432d94bd06c421ca242b61dea6619fc2aa571874398fba.jpg)  
+UniViT
+
+![](images/61f5c9d2ff1cca2c6a1523cf593e34a5d9a3f418e398b60bc457e1c5436bc99e.jpg)  
+MCLD
+
+![](images/c657acc5097cd6d89610c34d0696e8e38681a637d960e1252068b11cd672b562.jpg)  
+V-JEPA  
+Figure 7: Visualization of object feature distributions using T-SNE projection onto a spherical space on K400 and K700 dataset. Class tokens serve as query vectors for QKV computation.
+
+![](images/d2f32ef9040b296d6a6935c1f27bed1e1cbdc838eed275320130521a1bc1dfc6.jpg)  
+I-JEPA
+
+![](images/e534d0ba2def34ab8921001d073c2aad46c457052d5ea8839717c4859680e003.jpg)  
+DINOv2
+
+![](images/dc35d64d4651035a0c485aeb3beef92b97adbbfd2c46a1615f2bab5c759ca1fe.jpg)  
+SigLip2
+
+Impact of Number of Class Choices. As shown in Table 3e, we jointly vary the number of training classes. We observe that performance improves as both values increase, peaking at 1M classes. Interestingly, despite the significant increase in video data, the optimal number of classes aligns with that used in large-scale image pretraining, suggesting that current video datasets exhibit relatively low semantic diversity. At the same time, assigning multiple positives per sample proves beneficial for learning richer and more robust representations, especially in complex temporal tasks such as SSV2.
+
+# 5 Related Work
+
+Advances in Visual Representation Learning. The adoption of Vision Transformers [21, 39] has become a prevailing paradigm in the field of visual representation learning. Concurrently, equivariant self-supervised learning approaches [20, 45, 26, 29, 19] have emerged to predict structured data transformations consistent with group-theoretic formulations. Masked image modeling methods [30, 9, 22, 65] acquire visual representations by reconstructing masked regions of the input image in the pixel space. Moreover, JEPAs [6, 8] predict masked regions within a learned latent space rather than in the raw pixel domain. Contrastive Language-Image Pretraining (CLIP) [11, 55, 40, 24, 67, 50] aligns images and texts within a shared embedding space through instance-level contrastive supervision. However, existing approaches predominantly focus on either static image understanding or spatiotemporal modeling in isolation. In this work, UniViT structuring a shared semantic space for images and videos by modeling intra- and inter-instance, effective transfer to downstream tasks.
+
+Cluster Discrimination. Instance discrimination methods [17, 31, 47], exemplified by CLIP [47, 69, 28], leverage instance-level contrastive supervision but neglect semantic similarities across instances, whereas cluster-based approaches [14, 5, 15] assign single pseudo-labels per sample, failing to adequately represent images containing multiple visual elements. To better capture semantic structures, cluster discrimination methods [14, 5, 72, 15] typically iteratively assign pseudo-labels through clustering and train classifiers based on these labels, grouping visually similar instances to encourage semantic coherence. However, conventional approaches assign only a single pseudo-label per image, limiting their capacity to represent multiple semantic concepts within one instance, an issue recently addressed by multi-label clustering methods such as Unicom [2] and MLCD [4]. In this work, we adopt multi-label clustering to unify the representation learning of images and videos, effectively enhancing the semantic coherence.
+
+Efficient Training. Recent literature has explored various strategies for efficient CLIP training, such as large-batch optimization (up to 160K) [46, 50] and specialized optimizers like LAMB[56, 70]. RoPE originally designed for language models [54], has also been adapted to vision transformers via two-dimensional extensions [33, 1]. Additionally, significant efforts have focused on effective data curation and filtering at scale [25, 50, 24, 67], as well as image recaption using MLLMs [23, 38, 43, 66, 57, 3]. Motivated by these advances, we extend these methodologies to video data, constructing a unified data engine that facilitates robust representation learning across both images and videos.
+
+# 6 Conclusion & Limitation
+
+In this paper, we introduced UniViT, a cluster-driven unified self-supervised learning framework designed to jointly capture structured semantics across both spatial and temporal modalities through clustering and discrimination. Leveraging multi-granularity event-level clustering for videos and object-level clustering for images, UniViT first constructs structured semantic clusters across modalities during the clustering stage. Subsequently, in the discrimination stage, UniViT explicitly incorporates these clusters as pseudo-labels into a contrastive objective, effectively integrating structured semantic representations into a unified embedding space. To address limitations inherent to traditional
+
+position encoding and fixed-input approaches, we further introduced U-RoPE and VS $^2$ , enhancing semantic stability and flexibility across modalities. Extensive evaluations across multiple benchmarks demonstrate UniViT's superior scalability and its ability to achieve state-of-the-art performance on various downstream tasks, highlighting its effectiveness in unified visual representation learning.
+
+**Limitation:** We clarify the limitations of our proposed UniViT:  $(i)$ : UniViT relies on offline clustering with pretrained embeddings, potentially introducing biases from the initial feature extraction model and limiting its ability to adaptively update cluster assignments during training.  $(ii)$ : Although our VS $^2$  strategy supports flexible temporal lengths and arbitrary input resolutions, long sequences or extremely high-resolution inputs may still incur substantial computational costs, potentially constraining practical scalability in resource-limited scenarios.
+
+# 7 Acknowledgments
+
+This work was supported by the Center of Excellence for Antimicrobial Therapeutics Discovery and Innovation (CEATDI), Grant No. 8002003.
+
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+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: We claim our contributions and scope in Introduction.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
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+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
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+
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+
+Guidelines:
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+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
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+
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+
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+
+# 4. Experimental result reproducibility
+
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+
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+
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+
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+
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+
+# 7. Experiment statistical significance
+
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+
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+
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+
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+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
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+
+# 9. Code of ethics
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+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [NA]
+
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+
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+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
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+# 12. Licenses for existing assets
+
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+
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+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
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+
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+
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+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
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+
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+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
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+
+Guidelines:
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+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+# Answer: [Yes]
+
+Justification: We describe the details of how our model is used as the vision encoder in the LLM-based MLLM, along with the corresponding experimental settings, in UniViT as an Encoder of MLLM.
+
+# Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
\ No newline at end of file
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+# UniZyme: A Unified Protein Cleavage Site Predictor Enhanced with Enzyme Active-Site Knowledge
+
+Chenao Li, Shuo Yan, Enyan Dai*
+
+Hong Kong University of Science and Technology (Guangzhou)
+
+# Abstract
+
+Enzyme-catalyzed protein cleavage is essential for many biological functions. Accurate prediction of cleavage sites can facilitate various applications such as drug development, enzyme design, and a deeper understanding of biological mechanisms. However, most existing models are restricted to an individual enzyme, which neglects shared knowledge of enzymes and fails to generalize to novel enzymes. Thus, we introduce a unified protein cleavage site predictor named UniZyme, which can generalize across diverse enzymes. To enhance the enzyme encoding for the protein cleavage site prediction, UniZyme employs a novel biochemically-informed model architecture along with active-site knowledge of proteolytic enzymes. Extensive experiments demonstrate that UniZyme achieves high accuracy in predicting cleavage sites across a range of proteolytic enzymes, including unseen enzymes. The code is available in https://github.com/Ao-LiChen/UniZyme.
+
+# 1 Introduction
+
+During enzyme-catalyzed protein hydrolysis, proteolytic enzymes cleave proteins at specific cleavage sites. This process is illustrated in Fig. 1, and it is crucial for a variety of physiological processes, including cell proliferation, immune response, and cell death [1, 2]. Accurate prediction of enzymecatalyzed cleavage sites in the substrate proteins facilitates the identification of therapeutic targets and guides drug design [3]. For instance, abnormal protein hydrolysis is closely associated with cancer, viral infections, and neurodegenerative diseases, and predicting the cleavage sites of abnormal proteins under pathological conditions can reveal biomarkers or intervention targets [4, 5]. Additionally, in the design of enzyme inhibitors or prodrugs, identifying key cleavage peptides, such as those cleaved under the catalysis of HIV enzyme, helps enhance drug specificity and minimize off-target effects [6, 7].
+
+Mapping cleavage sites experimentally via peptide assays or high-throughput mass spectrometry is arduous and costly [8]. Therefore, recent studies have employed machine learning methods to advance the prediction of protein cleavage sites. For example, CAT3 [9] predicts caspase-3 cleavage sites based on position-specific scoring matrices (PSSM), and ProsperousPlus [10] integrates multiple methods to comprehensively evaluate cleavage site predictions. However, these methods generally focus on an individual failing to generalize to proteases without labele
+
+![](images/dfdda77515ea184c95f825470a4593676e1a3d73b329a37d93b69a805ceb67fa.jpg)  
+Figure 1: Enzyme-catalyzed protein hydrolysis. Enzyme system, overlooking shared patterns and data. This limitation impedes tasks like off-target
+
+assessment of therapeutic proteins in human body [11, 12]. Therefore, it is crucial to develop a unified protein cleavage site predictor that can generalize across a diverse range of proteolytic enzymes.
+
+To develop a unified protein cleavage site predictor for diverse proteolytic enzymes, the information of enzyme should be extracted and incorporated for the prediction. However, due to substantial cost of biological experiments, existing cleavage site databases only cover a small number of proteolytic enzymes (Tab. 1), which significantly challenges the learning of enzyme information encoder. Despite the limited coverage of enzymes in existing cleavage site datasets, many proteolytic enzymes are annotated with their active sites, which is the core functional region for catalyzing the protein hydrolysis. Specifically, the unique physicochemical environment of these active sites enables recognition of target substrates and lowers the activation energy required for cleaving specific peptide bonds. Therefore, we propose to leverage redundant knowledge of enzyme active sites to enhance the modeling of enzymes in enzyme-catalyzed protein cleavage sites.
+
+However, it is non-trivial to achieve a unified cleavage site predictor enhanced with enzyme active-site knowledge. Two major challenges remain to be resolved. First, the cleavage process is influenced by various factors of enzymes such as 3D structures and environments of active sites. Hence, how to design the architecture of enzyme encoder to effectively capture useful information for enzyme-catalyzed cleavage site prediction? Second, the active-site regions of enzymes determine the specificity and efficiency of enzymatic hydrolysis. How can we leverage this rich information of enzyme active sites to improve cleavage site prediction? In an attempt to address the challenges, we propose a novel framework named UniZyme. More specifically, a biochemically-informed enzyme encoder is deployed along with the active site-aware pooling to produce high-quality enzyme representations. We further augment the enzyme encoder by pretraining on a supplemented enzyme set for active-site prediction. Furthermore, a joint training of enzyme active-site prediction and substrate cleavage site prediction is applied in UniZyme. In summary, our main contributions are as follows:
+
+- We investigate a novel and crucial problem of building a unified protein cleavage site predictor that generalizes across diverse proteolytic enzymes;  
+- We propose a novel framework UniZyme that effectively integrates the enzyme active-site knowledge to enhance the cleavage site prediction in enzyme-protein interaction;  
+- Extensive experiments demonstrate the effectiveness of our UniZyme in predicting cleavage sites of substrate proteins for both seen and unseen proteolytic enzymes.
+
+# 2 Problem Formulation
+
+In this section, we first introduce the preliminaries of enzyme-catalyzed protein hydrolysis. Then, we present the problem definition of protein cleavage site prediction with enzyme active-site knowledge.
+
+# 2.1 Preliminaries of Enzyme-Catalyzed Protein Hydrolysis
+
+Cleavage Sites in Enzyme-Catalyzed Protein Hydrolysis. Protein hydrolysis is a biochemical process where proteins are broken down into smaller fragments such as amino acids and peptides under the catalysis of proteolytic enzymes. As illustrated in Fig. 1, during the protein hydrolysis, proteolytic enzymes will firstly recognize specific amino acid sequences or structural motifs within substrate proteins. Then, the enzymes catalyze the cleavage of peptide bonds at the cleavage site, leading to the formation of smaller peptide fragments or individual amino acids. The positions of cleavage sites are governed by various factors including substrate's amino acid composition, spatial conformation, and unique properties of the enzyme [13-15].
+
+Active Sites of Enzymes. The active sites in enzyme provide an environment that lowers the activation energy required for peptide bond cleavage. As shown in Fig. 1, with the active sites, enzymes can recognize and bind to target substrates, enabling the cleavage of specific peptide bonds within substrate proteins [16]. Hence, active site information can benefit the modeling of enzyme-catalyzed protein hydrolysis.
+
+Current Framework of Cleavage Site Prediction. Recent studies have employed machine learning models to predict cleavage sites [17-21]. However, these methods generally train an independent model for each enzyme, which only predicts the cleavage sites of substrate proteins under the catalysis
+
+of one specific enzyme. Specifically, let  $\mathcal{P}^s$  denote the substrate protein, this enzyme-specific cleavage site predictor aims to learn the  $f:\mathcal{P}^s\to \mathbf{c}^{e,s}$ , where  $\mathbf{c}^{e,s}\in \{0,1\}^{|\mathcal{P}^s|}$  denotes the labels of cleavage site with the enzyme  $\mathcal{P}^e$ . However, the training of enzyme-specific model excludes the valuable interaction knowledge from other enzyme-protein systems. In addition, the enzyme-specific model cannot generalize to unseen enzymes, which limits its application on unseen enzymes and other enzymes with limited annotations. Therefore, it is crucial to develop a unified cleavage site predictor capable of identifying cleavage sites in substrate proteins across various enzymes.
+
+Limited Enzyme Coverage in Cleavage Site Database. Tab. 1 presents statistics from the MEROPS, which is the most comprehensive cleavage site
+
+<table><tr><td colspan="3">Table 1: Statistics of Cleavage Sites Data in MEROPS.</td></tr><tr><td colspan="3"># Proteolytic Enzyme # Substrate Enzyme-Substrate Ratio</td></tr><tr><td>866</td><td>10,146</td><td>1:11.7</td></tr></table>
+
+database. Due to the high cost of experimental assays, MEROPS [22] only includes 866 commonly used proteolytic enzymes. This results in a striking enzyme–substrate ratio of 1:11.7. The limited enzyme coverage poses a significant challenge for developing a unified cleavage site predictor that generalizes across diverse enzyme–substrate systems. Despite the limited enzyme coverage in cleavage site databases, the lower cost of annotating enzyme active sites has enabled UniProt [23] to provide 10,749 high-quality proteolytic enzymes with labeled active sites across multiple organisms. The rich information of active sites can be helpful in enzyme modeling to facilitate the cleavage site prediction in protein hydrolysis.
+
+# 2.2 Problem Definition
+
+In protein hydrolysis, both enzyme  $\mathcal{P}^e$  and substrate  $\mathcal{P}^s$  are proteins composed of amino acid residues that fold into 3D structures. We denote a protein of length  $N$  by  $\mathcal{P} = (\mathbf{X},\mathbf{R})$ , where  $\mathbf{X}\in \mathbb{R}^{N\times d}$  is the feature matrix of  $N$  residues,  $\mathbf{R}\in \mathbb{R}^{N\times 3}$  denotes the 3D positions of residues. We denote  $\mathbf{c}^{e,s}\in \{0,1\}^{|\mathcal{P}_s|}$  as the cleavage site label for the substrate protein  $\mathcal{P}^s$  under the catalysis of enzyme  $\mathcal{P}^e$ . The training data for cleavage site prediction can be represented as  $\mathcal{D}_c = \{(\mathcal{P}_i^e,\mathcal{P}_i^s,\mathbf{c}_i^{e,s})\}_{i = 1}^{|\mathcal{D}_c|}$ . In this work, we propose to enhance the cleavage site prediction with the active site information of enzymes. Hence, we will also utilize a set of enzymes labeled with active sites  $\mathbf{a}\in \{0,1\}^{|\mathcal{P}_e|}$ , which is denoted as  $\mathcal{D}_a = \{(\mathcal{P}_i^e,\mathbf{a}_i)\}_{i = 1}^{|\mathcal{D}_a|}$ .
+
+During the test phase, we will predict the cleavage site for each pair of test enzymes and substrates  $(\mathcal{P}_t^e,\mathcal{P}_t^s)$ . The active sites of test enzymes will not be available for inference. And the test enzyme  $\mathcal{P}_t^e$  can be either seen, i.e.,  $\mathcal{P}_t^e\in \mathcal{D}_c$  or unseen, i.e.,  $\mathcal{P}_t^e\notin \mathcal{D}_c$ , which correspond to supervised setting and zero-shot setting, respectively. For each test substrate protein  $\mathcal{P}_t^s$ , its cleavage site with the test enzyme  $\mathcal{P}_t^e$  is not included in the training set  $\mathcal{D}_c$ . With the above notations and descriptions, the formal definition of building a unified cleavage site predictor can be given by:
+
+Problem 1. Given the dataset  $\mathcal{D}_c$  annotated for cleavage site prediction and the supplemented dataset  $\mathcal{D}_a$  containing enzymes active sites, we aim to obtain a unified cleavage site predictor:
+
+$$
+f: \left(\mathcal {P} ^ {e}, \mathcal {P} ^ {s}\right)\rightarrow \mathbf {c} ^ {e, s}, \tag {1}
+$$
+
+which can accurately predict cleavage sites of test proteins  $\mathcal{P}_t^e$  under the catalysis of test proteolytic enzymes  $\mathcal{P}_t^s$ . Note that test enzymes can be either seen or unseen during the training phase.
+
+# 3 Methodology
+
+In this section, we give the details of the proposed UniZyme. As the Fig. 2 shows, apart from the substrate encoder, UniZyme deploys an enzyme encoder to enable the generalization of cleavage site prediction across various enzymes. In addition, active site information in protein hydrolysis is incorporated into enzyme encoder training to enhance cleavage site prediction. Two main challenges remain to be addressed: (i) how to design the enzyme encoder to preserve critical information for cleavage site prediction? (ii) how to leverage the rich information of enzyme active sites to improve the cleavage site prediction? To tackle the above challenges, our UniZyme deploys a biochemically-informed enzyme encoder which augments the graph transformer with enzyme energy frustration. Furthermore, UniZyme employs active site-aware pooling to preserve the enzyme's information crucial for protein hydrolysis. To facilitate enzyme representation learning, UniZyme first pretrains the enzyme encoder using active site prediction with the supplemented enzyme set. Then, a joint
+
+![](images/2377069b562b673a90b97161a91657469491e8b0f6a5930867d8638ebf9834ea.jpg)  
+Figure 2: Architecture of biochemically-informed enzyme encoder and framework of UniZyme.
+
+![](images/2de6de6054762352466e246f9280f79b8cef64bbbdbbd5e72c5a4ad2c91aad96.jpg)
+
+loss of active site prediction and cleavage site prediction is employed to optimize the UniZyme for accurate cleavage site prediction. Next, we introduce each component in detail.
+
+# 3.1 Biochemically-Informed Enzyme Encoder
+
+Enzymes' 3D structures, especially the local environments around their active sites, are crucial for catalyzing protein hydrolysis. Although direct active-site annotations are often unavailable for test data, recent studies indicate that local energetic frustration can identify functionally important regions [24]. Building on these insights, we propose a biochemically-informed enzyme encoder that integrates both the spatial positions of residues and their energetic frustration scores [25].
+
+Encoding Energetic Frustration. Previous studies indicate that local energetic frustration, referring to regions in a protein not optimized for minimal energy, is commonly observed around enzyme active sites and can significantly influence catalysis [24]. To quantify this phenomenon, a frustration score  $\mathbf{F}(i,j)$  is computed for each residue pair  $(i,j)$  within an enzyme  $\mathcal{P}^e$  following [24]:
+
+$$
+\mathbf {F} (i, j) = \frac {\mathbf {E} (i , j) - \mu_ {\text {r a n d}} (i , j)}{\sigma_ {\text {r a n d}} (i , j)}, \tag {2}
+$$
+
+where  $\mathbf{E}(i,j)$  is the actual interaction energy between residues  $(i,j)$  in the enzyme  $\mathcal{P}^e$ .  $\mu_{\mathrm{rand}}(i,j)$  and  $\sigma_{\mathrm{rand}}(i,j)$  represent the mean and standard deviation of interaction energies that derived from randomized configurations (see Appendix C for details). A higher  $\mathbf{F}(i,j)$  implies stronger local energetic frustration, suggesting that the residue pair is more likely to belong to a functionally important region. Therefore, we incorporate this frustration score to provide useful biochemical information to the enzyme encoder.
+
+Integrating Energy and 3D Position in Transformer. Following prior works [26], we encode the 3D positions by computing pair-wise distance between residues:  $\mathbf{D}(i,j) = \| \mathbf{r}_i - \mathbf{r}_j\|_2$ , where  $\mathbf{r}_i \in \mathbb{R}^3$  is the  $\mathrm{C}\alpha$ -atom coordinate of residue  $i$ . Both energetic frustration score and distance matrix capture pairwise relationships akin to the spatial encoding in graph transformers. Therefore, we locate those pair-wise signals to provide complementary information for the self-attention score computation. Concretely, for an enzyme  $\mathcal{P}^e = (\mathbf{X},\mathbf{R})$ , we process both  $\mathbf{F}(i,j)$  and distance matrix  $\mathbf{D}(i,j)$  with a Gaussian Basis Kernel function followed by a MLP:
+
+$$
+\Phi_ {i, j} ^ {\text {e n e r g y}} = \operatorname {M L P} \left(\phi_ {\text {e n e r g y}} (\mathbf {F} (i, j))\right), \quad \Phi_ {i, j} ^ {\text {d i s t}} = \operatorname {M L P} \left(\phi_ {\text {d i s t}} (\mathbf {D} (i, j))\right), \tag {3}
+$$
+
+where  $\phi_{\mathrm{energy}}$  and  $\phi_{\mathrm{dist}}$  denote the learnable Gaussian Basis Kernel function that can map energy frustration score and distance score to a  $d$ -dimensional vector (See Appendix C for more details). MLP is further deployed to transform these vectors to the space of attention scores. We then add the resulting  $\Phi_{(i,j)}^{\mathrm{dist}}$  and  $\Phi_{(i,j)}^{\mathrm{energy}}$  as bias terms to the self-attention mechanism. Denote  $\mathbf{A}_{i,j}^{k}$  as the  $(i,j)$ -element of the Query-Key product matrix in  $k$ -th attention layer, we have:
+
+$$
+\mathbf {A} _ {i, j} ^ {k} = \frac {\left(\mathbf {h} _ {i} ^ {k - 1} \mathbf {W} _ {Q}\right) \left(\mathbf {h} _ {j} ^ {k - 1} \mathbf {W} _ {K}\right) ^ {T}}{d} + \Phi_ {i, j} ^ {\text {e n e r g y}} + \Phi_ {i, j} ^ {\text {d i s t}} \tag {4}
+$$
+
+$$
+\mathbf {H} ^ {k} = \operatorname {s o f t m a x} \left(\mathbf {A} ^ {k}\right) \mathbf {H} ^ {k - 1} \mathbf {W} _ {V},
+$$
+
+where  $\mathbf{H}^k\in \mathbb{R}^{N\times d}$  denotes the updated representation matrix. And  $\mathbf{W}_Q,\mathbf{W}_K$  , and  $\mathbf{W}_V$  are projection matrices for the query, key, and value transformations.
+
+# 3.2 Enhancing Enzyme Representation Learning with Active Site Knowledge
+
+To incorporate crucial active-site knowledge into enzyme representation learning, we use three strategies: (i) an auxiliary active-site prediction task to strengthen the enzyme encoder, (ii) large-scale pretraining for active-site prediction to capture general catalytic patterns, and (iii) an active site-aware pooling mechanism that emphasizes catalysis-related residues. Next, we give more details.
+
+Active Site Prediction. Active sites play a key role in catalyzing the protein cleavage. Hence, active-site information can provide essential understandings of enzyme functions. As a result, we deploy the active site prediction as the auxiliary task to benefit the enzyme encoder training by:
+
+$$
+\hat {a} _ {i} = \operatorname {s i g m o i d} \left(\mathbf {h} _ {i} \cdot \mathbf {w} _ {a}\right), \tag {5}
+$$
+
+where  $\hat{a}_i\in [0,1]$  denotes the probability of the  $i$  -th residue being the active site,  $\mathbf{h}_i\in \mathbb{R}^d$  is the representation of  $i$  -th residues in the enzyme, and  $\mathbf{w}_a\in \mathbb{R}^d$  denotes the learnable parameters for active site prediction.
+
+Pretraining with the Supplemented Enzyme Set  $\mathcal{D}_a$ . The number of enzymes annotated in cleavage site database is limited, which poses a significant challenge for the effective training of enzyme encoders [27]. Despite the limited enzyme coverage in cleavage site database, abundant enzymes are annotated with active sites. Therefore, we select enzymes highly homologous to the target proteolytic enzymes in biological function to expand the pretraining dataset. Formally, the objective function of pretraining the enzyme encoder on the supplemented enzyme set  $\mathcal{D}_a$  can be written as:
+
+$$
+\min  _ {\theta_ {e}, \mathbf {w} _ {a}} \mathcal {L} _ {a} (\mathcal {D} _ {a}) = \frac {1}{| \mathcal {D} _ {a} |} \sum_ {(\mathcal {P} ^ {e}, \mathbf {a}) \in \mathcal {D} _ {a}} l _ {\mathrm {B C E}} (\hat {\mathbf {a}}, \mathbf {a}), \tag {6}
+$$
+
+where  $\theta_{e}$  denotes the parameters of enzyme encoder.  $\hat{\mathbf{a}} = [\hat{a}_1,\dots \hat{a}_N]$  denotes the probability vector of active site on the enzyme  $\mathcal{P}^e$ .  $l_{\mathrm{BCE}}$  denotes the element-wise binary cross entropy loss. By pretraining on a large corpus of enzyme sequences, we allow the model to capture broader structural and functional patterns common across enzymes.
+
+Active Site-Aware Pooling. To obtain enzyme representation from a sequence of residue representations, a pooling operation such as mean pooling is required. However, residues that are active sites are more critical for enzymatic activity. Intuitively, these active sites should contribute more in the aggregated enzyme representation [28]. Therefore, we design an active site-aware pooling mechanism, whose pooling weights are based on the predicted active site probabilities. Let  $\hat{a}_i \in \mathbb{R}^N$  be the predicted probability that  $i$ -th residue is an active site in an  $N$ -residue enzyme. The active site-aware pooling can be written as:
+
+$$
+\mathbf {h} ^ {e} = \operatorname {s o f t m a x} \left(\left[ w _ {1}, \dots , w _ {N} \right]\right) \mathbf {H}, \quad w _ {i} = f \left(\hat {a} _ {i}\right), \tag {7}
+$$
+
+where  $\mathbf{H} \in \mathbb{R}^{N \times d}$  is residue representation matrix from by the enzyme encoder.  $f(\cdot)$  is a learnable function that will map each  $\hat{a}_i$  to the pooling weight  $w_i$ , see in appendix C. With the active site-aware pooling, we would be able to encourages the model to focus on catalytically relevant segments of the enzyme.
+
+# 3.3 Cleavage Site Prediction
+
+Substrate Protein Encoding. As annotations tying substrate residues to intrinsic energetic states are unavailable, we omit the energetic frustration of the substrate protein. And we only input residue feature matrix  $\mathbf{X}^s$  and distance matrix  $\mathbf{D}^s$  of the substrate  $\mathcal{P}^s$  are integrated in the transformer:  $\mathbf{H}^s = \mathrm{Transformer}(\mathbf{X}^s,\mathbf{D}^s)$ , where  $\mathbf{H}^s\in \mathbb{R}^{|\mathcal{P}^s| \times d}$  is the substrate representation. Further details are provided in Appendix C.
+
+Cleavage Site Prediction. During protein hydrolysis, enzymes generally recognize local residue sequences about 15-30 residues in length. To reflect this biological behavior, we predict whether a subsequence of length  $l$  in substrate  $\mathcal{P}^e$  will be cleaved by enzyme  $\mathcal{P}^e$ . Formally, this process can be written by:
+
+$$
+\hat {c} _ {t} ^ {e, s} = \operatorname {M L P} \left(\operatorname {C O N C A T} \left(\mathbf {H} _ {t: t + l} ^ {s}, \mathbf {h} ^ {e}\right)\right) \tag {8}
+$$
+
+where  $\mathbf{H}_{t:t + l}^s$  is a contiguous slice taken directly from the substrate representation matrix  $\mathbf{H}^s$  and  $\mathbf{h}^e$  is the enzyme representation obtained by Eq.(7). The length of the subsequence is set as 31 (15
+
+residues on each side.). The optimization function of cleavage site prediction can be written as:
+
+$$
+\mathcal {L} _ {c} \left(\mathcal {D} _ {c}\right) = \frac {1}{\left| \mathcal {D} _ {c} \right|} \sum_ {\left(\mathcal {P} ^ {e}, \mathcal {P} ^ {s}, \mathbf {c} ^ {e, s}\right) \in \mathcal {D} _ {c}} l _ {\mathrm {B C E}} \left(\mathbf {c} ^ {e, s}, \hat {\mathbf {c}} ^ {e, s}\right) \tag {9}
+$$
+
+where  $\hat{\mathbf{c}}^{e,s} = [\hat{c}_1^{e,s},\dots,\hat{c}_{|\mathcal{P}^s|}^{e,s}]$  denotes the probability vector of cleavage site within the substrate  $\mathcal{P}^s$  given the enzyme  $\mathcal{P}^e$ .  $l_{\mathrm{BCE}}$  is the element-wise binary cross entropy loss.
+
+# 3.4 Final Objective Function
+
+For each enzyme  $\mathcal{P}^e\in \mathcal{D}_c$  in the cleavage site database, their active sites are also included in the  $\mathcal{D}_a$ . Consequently, we combine the cleavage site prediction loss and the active site prediction to jointly train the whole framework by:
+
+$$
+\min  _ {\theta} \mathcal {L} _ {c} \left(\mathcal {D} _ {c}\right) + \lambda \mathcal {L} _ {a} \left(\mathcal {D} _ {a} ^ {c}\right), \tag {10}
+$$
+
+where  $\theta$  denotes all parameters in UniZyme including the enzyme encoder, substrate encoder, active prediction module and cleavage site prediction module.  $\mathcal{D}_a^c\subset \mathcal{D}_a$  provides the active-site annotations for the enzymes in  $\mathcal{D}_c$ .
+
+# 4 Experiments
+
+In this section, we conduct experiments to answer the following research questions:
+
+- RQ1: How does the UniZyme perform in supervised cleavage site prediction?  
+- RQ2: How well does UniZyme generalize to cleavage site prediction for zero-shot enzymes?  
+- RQ3: How do the design of biochemically-informed enzyme encoder and utilization of active-site knowledge contribute to the performance of UniZyme?
+
+# 4.1 Experimental Setup
+
+Dataset. The cleavage site dataset  $\mathcal{D}_c$  is sourced from the MEROPS database, which provides annotations for roughly 10k substrate proteins across 876 enzymes. With a standard dataset expansion procedure commonly used in cleavage site prediction[10, 18, 20, 21], which propagates substrate-site annotations across enzymes within the same family, we obtain 220k valid enzyme-substrate pairs. The supplemented dataset  $\mathcal{D}_a$  is constructed by combining enzyme active-site annotations in MEROPS [22] and UniProt [23]. Specifically, MEROPS provides active sites for the enzymes already included in  $\mathcal{D}_c$ . UniProt provides hydrolase enzymes with the EC number of  $3.4.*$ * that are highly homologous to the proteolytic enzymes in  $\mathcal{D}_c$ , resulting in 11,530 enzymes with active sites.
+
+Evaluation. To demonstrate the generalization ability of UniZyme, we evaluate its performance on protein cleavage site prediction for both seen enzymes (supervised) and unseen enzymes (zero-shot). The
+
+Table 2: Splits of MEROPS for evaluation.  
+
+<table><tr><td></td><td>Training</td><td>Supervised Test</td><td>Zero-Shot Test</td></tr><tr><td>Enzyme Families</td><td>677</td><td>69</td><td>23</td></tr><tr><td>Substrate-Enzyme Pairs</td><td>197,613</td><td>20,360</td><td>5,345</td></tr></table>
+
+dataset split of the MEROPS for supervised and zero-shot setting are in Tab. 2. The dataset construction details can be found in Appendix A.
+
+- Supervised Setting: In this scenario, target enzymes are paired with novel substrates that were not present in the training data. We restrict our evaluation to enzyme families with at least five unique substrates, yielding 69 enzyme families and a total of 21k enzyme-substrate pairs. For each family, we randomly split the data 70/10/20 into training, validation, and test sets.  
+- Zero-shot Setting: In the zero-shot setting, target enzymes are entirely held out during both training and pretraining. We consider only families with at least five distinct enzymes, reserving  $20\%$  of each family's enzymes as a test set. In total, this yields 23 enzyme families (5.3k enzyme-substrate pairs), with all test enzymes sharing under  $60\%$  sequence identity with any enzyme seen during training or pretraining.
+
+Baseline Methods. To evaluate our model, we compare with the three categories of baselines. (i) Specialized models: CAT3 [9] and ScreenCap3 [18] are two models specifically designed for
+
+C14.003 enzyme family. (ii) Deep models for an individual enzyme: ProsperousPlus [10], DeepCleave [20], and DeepDigest [29] are state-of-the-art deep learning methods focusing an individual enzyme system. To compare with UniZyme across multiple enzyme families, multiple predictors are trained for each baseline method, where each predictor is trained with data of an individual enzyme. Thus, these baselines are limited to supervised setting where an enzyme is provided with substantial experimental data. (iii) Models revising UniZyme with baseline enzyme encoders: Research on cleavage site prediction for zero-shot enzymes remains limited. The closest works are enzyme-substrate reaction predictors, namely ClipZyme [30] and ReactZyme [31], which encode both enzymes and substrates for reaction prediction. However, these models operate at the substrate level and cannot directly predict specific cleavage sites. Therefore, to demonstrate the superiority of the proposed enzyme encoder and enhancement strategies in Unizyme, we substitute the Unizyme's enzyme encoder with enzyme encoders from ClipZyme and ReactZyme. This yields two baselines for comparison in both supervised and zero-shot settings. For ClipZyme, we utilize their EGNN enzyme encoder pretrained for enzyme-substrate reaction prediction. See Appendix B for more details of baselines.
+
+Implementation Details. We utilized the esm2-t12-35M-UR50D model to generate 480-dimensional residue features for both enzymes and substrates. The hyperparameter  $\lambda$  is selected based on the validation set under supervised setting. Hyperparameter analysis are in Sec 4.5. Each experiment is conducted with 5 runs with different random seeds. To ensure a fair evaluation, hyperparameters of trainable baselines were selected by validation set. More details are in Appendix C.
+
+# 4.2 Supervised Cleavage Site Prediction
+
+To answer RQ1, we compare our UniZyme with various existing methods in supervised cleavage site prediction, ensuring that all models are trained and evaluated on the same data. As mentioned in Sec. 4.1, baselines designed for individual enzymes require training a separate predictor for each enzyme family. ClipZyme and ReactZyme are baselines revising our UniZyme with their enzyme encoders, avoiding repeated training. The overall comparisons on all 69 enzymes in supervised setting are given in Tab. 3 and Fig. 3. Results on 8 represent
+
+tative enzymes are given in Tab. 4. Specifically, we can observe that: (i) Methods focusing on a single enzyme generally show poor performance; whereas those trained on multiple enzymes such as Our UniZyme and ReactZyme achieve significantly better results. This highlights the advantage of developing a unified cleavage site predictor across diverse proteolytic enzymes. (ii) Compared with ClipZyme which adopts an enzyme encoder pretrained with the enzyme-substrate reaction task, the proposed UniZyme achieves much better performance. It implies the effectiveness of active-site information in enhancing the enzyme encoder. (iii) Our UniZyme also consistently outperforms the ReactZyme by a large margin. This is because of the deployment of biochemically-informed enzyme encoder and the active-site knowledge.
+
+Table 3: Performance comparisons on overall 69 enzyme families under supervised setting.  
+
+<table><tr><td>Model</td><td>Average PR-AUC (%)</td><td>Rate of Rank 1 (%)</td><td>Average Rank</td></tr><tr><td>UniZyme</td><td>79.3</td><td>75.0</td><td>1.26</td></tr><tr><td>ReactZyme</td><td>70.0</td><td>7.5</td><td>2.51</td></tr><tr><td>ClipZyme</td><td>74.7</td><td>17.5</td><td>1.91</td></tr><tr><td>ProsperousPlus</td><td>7.2</td><td>0.0</td><td>4.65</td></tr><tr><td>DeepCleave</td><td>6.5</td><td>0.0</td><td>4.61</td></tr><tr><td>DeepDigest</td><td>4.2</td><td>0.0</td><td>5.52</td></tr></table>
+
+Table 4: PR-AUC (%) on 8 out of 69 enzyme families under supervised setting. Note that ScreenCap3 and CAT3 are specialized models for the C14.003 enzyme family.  
+
+<table><tr><td>Model</td><td>C01.034</td><td>C14.003</td><td>C14.005</td><td>M13.001</td><td>M24.026</td><td>S01.001</td><td>S01.224</td><td>S08.070</td></tr><tr><td>UniZyme</td><td>78.1±1.3</td><td>45.9±1.2</td><td>52.2±0.9</td><td>60.1±1.1</td><td>87.2±0.8</td><td>82.1±1.2</td><td>62.4±1.1</td><td>85.0±1.3</td></tr><tr><td>ReactZyme</td><td>64.4±0.7</td><td>43.8±0.8</td><td>47.6±0.9</td><td>48.0±0.7</td><td>70.3±1.4</td><td>69.6±1.2</td><td>27.0±0.9</td><td>66.0±0.8</td></tr><tr><td>ClipZyme</td><td>71.1±0.7</td><td>35.3±0.9</td><td>43.2±1.0</td><td>37.1±1.3</td><td>73.1±0.8</td><td>73.8±0.9</td><td>41.7±0.6</td><td>73.8±0.9</td></tr><tr><td>DeepDigest</td><td>3.2±0.5</td><td>0.5±1.1</td><td>2.4±1.1</td><td>16.7±1.3</td><td>1.4±0.4</td><td>21.1±0.9</td><td>1.2±1.0</td><td>2.2±1.2</td></tr><tr><td>DeepCleave</td><td>4.8±1.1</td><td>1.0±1.1</td><td>4.2±0.9</td><td>19.0±1.0</td><td>7.1±1.2</td><td>26.9±1.4</td><td>5.0±0.5</td><td>18.3±0.6</td></tr><tr><td>ProsperousPlus</td><td>4.6±1.3</td><td>26.6±2.1</td><td>15.9±0.5</td><td>21.1±0.8</td><td>2.4±0.8</td><td>16.3±1.0</td><td>4.3±0.9</td><td>43.6±1.0</td></tr><tr><td>CAT3</td><td>-</td><td>18.5±6.2</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>ScreenCap3</td><td>-</td><td>29.2±16.0</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></table>
+
+![](images/054263a55d445789d95e867b50909e09b41b90b5a9342ad76cf3ef979620bba3.jpg)  
+Figure 3: Per-family PR-AUC (\%) across 69 supervised enzymes (Part 1: Enzymes 1-35). Results for the remaining 34 enzymes are provided in Appendix M (Fig. 9).
+
+# 4.3 Cleavage Site Prediction for Zero-Shot Enzymes
+
+To answer RQ2, we evaluate the performance of UniZyme on the zero-shot benchmarks, where enzymes are unseen during the training/pretraining phase. We adopt Needleman-Wunsch algorithm to ensure all zero-shot enzymes have under  $60\%$  sequence similarity with any enzyme used for training and pretraining. Since enzyme-specific models cannot handle novel enzymes, we only compare UniZyme to ReactZyme and ClipZyme, both of which are modified to predict cleavage sites for novel enzymes. The summarized results are given in Tab. 5, 6 and Fig. 5. From the results, we can observe that our UniZyme consistently outperform the baseline methods. In particular, UniZyme exceeds ReactZyme and ClipZyme by more than  $7\%$  in PR-AUC across most enzyme families. This improvement stems from the utilization of active-site knowledge in the enzyme modeling and the biochemical-informed encoder, promoting the generalization ability of UniZyme to unseen enzymes.
+
+To further demonstrate the generalization ability of our model on unseen enzymes, we applied it to identify potential HIV-1 enzyme substrates and predict their cleavage sites as shown in Fig. 4. We present the model's prediction on an unseen HIV-1 enzyme acting on a 147-residue substrate (P62157) with four experimentally validated HIV-1 cleavage sites. As shown in Fig. 4, UniZyme can accurately predict the four annotated cleavage sites, achieving  $100\%$  accuracy. Additionally, UniZyme successfully predicts the cleavage sites of a test case (Uniprot: P00698). This demonstrates the ability to analyze cleavage sites for any potential protein of HIV-1 enzymes. This result provides valuable insights for therapeutic intervention and the development of inhibitors targeting HIV-1 enzymes.
+
+Table 5: PR-AUC (%) on 8 out of 23 zero-shot enzyme families.  
+
+<table><tr><td>Model</td><td>A01.009</td><td>C01.060</td><td>C02.002</td><td>M10.001</td><td>M10.004</td><td>M12.217</td><td>S01.010</td><td>S01.217</td></tr><tr><td>UniZyme</td><td>37.5±0.6</td><td>84.3±1.3</td><td>66.4±0.6</td><td>81.1±1.0</td><td>82.8±1.8</td><td>93.2±1.3</td><td>61.0±0.8</td><td>65.0±1.2</td></tr><tr><td>ReactZyme</td><td>18.0±0.3</td><td>62.7±0.8</td><td>49.7±1.2</td><td>76.0±1.2</td><td>71.0±2.8</td><td>75.0±1.1</td><td>23.0±1.1</td><td>42.9±0.8</td></tr><tr><td>ClipZyme</td><td>25.2±0.6</td><td>59.2±0.9</td><td>48.6±1.2</td><td>76.8±0.6</td><td>56.5±3.5</td><td>84.4±0.9</td><td>18.9±0.9</td><td>41.3±0.6</td></tr></table>
+
+Table 6: Performance comparisons on overall 23 enzyme families under zero-shot setting.  
+
+<table><tr><td>Model</td><td>Average PR-AUC (%)</td><td>Rate of Rank 1 (%)</td><td>Average Rank</td></tr><tr><td>UniZyme</td><td>71.1</td><td>78.3</td><td>1.30</td></tr><tr><td>ReactZyme</td><td>64.7</td><td>21.7</td><td>2.04</td></tr><tr><td>ClipZyme</td><td>61.7</td><td>0.0</td><td>2.65</td></tr></table>
+
+![](images/c324ecd03d6f393898738243198d7b2c3b1743f7b976e380d76b5e6c3224d23e.jpg)  
+(a) Substrate P62157 (Accuracy  $= 1.0$
+
+![](images/56a2cbb429d6e50001a2a266879b80b6ab7dc4d97ca88430d32cb3195b124f9e.jpg)  
+(b)Substrate P00698 (No Ground Truth)  
+Figure 4: Predicted HIV-1 enzyme cleavage sites highlighted in red (threshold = 0.5).
+
+![](images/c8362bd7aaa813d7595dfe398212edbdde9373d55fa063b9e056d9b67ed68b2b.jpg)  
+Figure 5: PR-AUC  $(\%)$  on the zero-shot benchmark.
+
+# 4.4 Ablation Studies
+
+To answer RQ3, we conduct a series of ablation studies to understand the contributions of biochemically-informed enzyme encoder and the active-site knowledge. To demonstrate the effectiveness of the biochemically-informed enzyme encoder, we remove energy frustration and 3D structure, resulting in a variant named UniZyme\SE. To show the benefits brought by pretraining on general enzymes with active site prediction, we trained a variant, UniZyme\P, which excludes the enzyme pretraining phase. To further demonstrate the enhancement of active-site knowledge to the model, we remove the active site prediction in the pretraining/training phase. Additionally, the active site-aware pooling is replaced with average pooling, resulting in a variant named UniZyme\A. Fig. 6 shows the PR-AUC scores across different enzyme families in both zero-shot and supervised settings. More details can be found in the Tab. 17. From these results, we observe:
+
+- UniZyme consistently achieves better results than UniZyme\SE. This indicates that the incorporation of structural-energy features in the biochemically-informed enzyme encoder can enable stronger generalization and performance in cleavage site prediction.  
+- UniZyme\P significantly performs worse than UniZyme on both supervised and zero-shot setting. This verifies that pretraining on a supplemented enzyme set with active site prediction can produce a more transferable enzyme encoder for cleavage site prediction.  
+- UniZyme outperforms UniZyme\A by a large margin. This demonstrates that the active-site knowledge can enhance the enzyme-catalyzed cleavage site prediction.
+
+# 4.5 Hyperparameter Analysis
+
+In this subsection, we investigate how the hyperparameter  $\lambda$  affects the UniZyme.  $\lambda$  controls to contribution of active site prediction loss to the training of UniZyme. To explore the hyperparameter analysis, we vary  $\lambda$  as  $\{100, 10, 1, 0.1, 0.01\}$  in the training phase of UniZyme. Due to the expensive computational cost in training on the full dataset  $\mathcal{D}_c$ , we conduct the hyperparameter analysis with  $3\%$  of training data in various enzyme families. Performance on these enzymes are given in Fig. 7. We can find that while  $\lambda = 100$  produces competitive ROC-AUC results, it leads to suboptimal PR-AUC. Small values like 0.1 and 0.01 cause a noticeable drop in ROC AUC (e.g. C14.005). Among the tested values,  $\lambda = 10$  demonstrates the most consistent performance, achieving strong PR-AUC (e.g., M10.004) while maintaining competitive ROC AUC across datasets such as C14.003 and A01.009. Thus, we selected  $\lambda = 10$  as the optimal choice for final training.
+
+![](images/b6ee2df39ecbf5165852c8bd0f16d595dbc88d3c1e42e13640b3904164e389f7.jpg)  
+(a) Supervised
+
+![](images/3836f6a5ce18953203c7321750adc57cf9d4bd3f7e8c84ef139c38fe3335704b.jpg)  
+(b) Zero-shot
+
+![](images/16497ba72f05ad3a8b90490b18c01cbdcad2fae131c64617220f11462fd7d80f.jpg)  
+Figure 6: Ablation studies of UniZyme.  
+(a) ROC-AUC  
+Figure 7: Hyperparameter sensitivity analysis.
+
+![](images/344309db6f25df0ff77d0fafd60ca21538c797f2805f256bfca2d7ccc01a55e8.jpg)  
+(b) PR-AUC
+
+# 5 Related Works
+
+Protein Representation Learning. Protein representation learning aims to effectively capture and represent the structural and functional features of proteins for downstream tasks. Inspired by large language models, recent years have seen the emergence of sequence-based pre-trained models such as ESM [32] and ProtTrans [33]. In terms of methods that utilize structural information, geometric graph neural networks [34-38] and transformers with structural constraints [26, 39, 40] have become widely used architectures. These approaches show that structural pretraining can significantly benefit downstream performance.
+
+Beyond individual model designs, systematic benchmarks have begun to evaluate how different pretraining paradigms transfer to real-world applications. In particular, Protap [41] establishes a comprehensive benchmark that jointly compares backbone architectures, pretraining objectives, and domain-specific models across both general and specialized protein tasks. It covers five representative applications—including enzyme-catalyzed cleavage site prediction and PROTAC-mediated degradation—and provides unified data splits, evaluation protocols, and analysis of how structural inductive biases interact with pretraining scale. Within this framework, UniZyme is integrated as the representative enzyme-substrate model, demonstrating how incorporating biochemical priors such as energy frustration and active-site cues can complement large-scale foundation encoders.
+
+Cleavage Site Prediction. Early prediction of enzyme-catalyzed cleavage sites relied on substrate sequence motifs, such as CAT3 [9] and Screen-Cap3 [18]. Subsequent works including Procleave [21] and ProsperousPlus [10] began to integrate substrate structural features to better capture enzymatic preferences. Deep learning approaches such as DeepCleave [20], DeepDigest [29], and DeepNeuropred [17] further advanced the field by leveraging CNNs and protein language models. However, these are enzyme-specific models, applicable only to individual targets and often ignoring active-site information. Building on these developments and contextualized by the Protap benchmark, we propose a unified cleavage-site predictor enhanced with explicit active-site knowledge.
+
+# 6 Conclusion and Future work
+
+In this paper, we study a novel problem of developing a unified protein cleavage site predictor for diverse proteolytic enzymes. Specifically, we design a biochemically-informed enzyme encoder and incorporate redundant enzyme active-site information. Our experimental results demonstrate that UniZyme outperforms baselines by a large margin across various enzyme-substrate families, particularly excelling in zero-shot scenarios. Ablation studies further demonstrate the effectiveness of each proposed module in UniZyme. There are two directions that need further investigation. First, while this study focuses on proteolytic enzymes, we will extend to other categories of enzymes and substrates, and investigate whether enzyme-catalyzed reactions follow scaling law. Second, if more hydrolysis process data becomes available, incorporating dynamic structural information may improve prediction accuracy.
+
+# 7 Acknowledgment
+
+This material is based upon work supported by, or in part by, the National Natural Science Foundation of China (NSFC) under grant number 62506316. The findings in this paper do not necessarily reflect the view of the funding agencies.
+
+# References
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+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: The abstract and introduction clearly enumerate the three main contributions of UniZyme and these match the experimental results demonstrating its superior performance and generalization.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: We conduct a failure case analysis in Appendix K to analyze the limitation of our work. It implies the future directions to improve the proposed UniZyme.
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [NA]
+
+Justification: No formal theorems or proofs are presented in this empirical work.
+
+Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: Data curation procedures, training/test splits, model architectures, and hyperparameters are fully specified in the main text and Appendix, and code is made available.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.  
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.  
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.  
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example  
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: The UniZyme code repository URL is provided in the abstract, and all source data (MEROPS, UniProt) are publicly accessible.
+
+Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).  
+- The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.  
+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.  
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: Dataset statistics, data splits, optimizer settings, batch size, pretraining protocol, and evaluation metrics are described in Sections 4 and Appendix A, B and C.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: All reported metrics (ROC-AUC, PR-AUC) include mean  $\pm$  standard deviation over five runs.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
+
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).  
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We detail the exact GPU (NVIDIA A6000), the number of GPUs used  $(8\times \mathrm{A}6000$  for training,  $1\times \mathrm{A}6000$  for inference), per-task runtimes (hours for training, seconds per inference), and aggregate compute cost in Appendix I.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: Our work involves no human subjects or other sensitive domains, and we have followed all NeurIPS ethical guidelines, preserving anonymity and data integrity throughout.
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: The Broader Impacts section L describes concrete benefits—faster drug discovery, enzyme re-engineering, and pandemic preparedness—and notes the potential
+
+misuse of the model for creating harmful proteases, thereby covering both positive and negative societal impacts.
+
+# Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: UniZyme does not release high-risk models or restricted data.
+
+# Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: We explicitly state in Appendix A the license terms, and cite all related sources.
+
+# Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.
+
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [Yes]
+
+Justification: The supplemented enzyme dataset is documented in Appendix A with detailed curation steps, and the code/data release is described in abstract.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: No human-subject or crowdsourced data is involved.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: No human-subject or crowdsourced data is involved.
+
+# Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+# Answer: [NA]
+
+Justification: No large-language models were used in the core methodology or manuscript preparation.
+
+# Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+Algorithm 1 Training algorithm of UniZyme  
+Require: Supplemented enzyme set  $\mathcal{D}_a$ , Cleavage site prediction dataset  $\mathcal{D}_c$ , hyperparameters  $\lambda$   
+Ensure: A unified cleavage site predictor  $f_{\theta}$   
+1: Initialize features of enzyme and substrate protein by ESM-2  
+2: Pretrain enzyme encoder on  $\mathcal{D}_a$  with  $\mathcal{L}_a$  by Eq.(6)  
+3: for epoch = 1 to  $N$  do  
+4: for each batch  $(\mathcal{P}^e, \mathcal{P}^s)$  in  $\mathcal{D}_c$  do  
+5: Compute the distance matrix  $\mathbf{D}^e$ , and energetic frustration matrix  $\mathbf{F}^e$  from enzyme structure by Eq.(2)  
+6: Encode the enzyme and substrate protein by Eq.(4)  
+7: Obtain the enzyme representation with active site-aware pooling by Eq.(7)  
+8: Predict active sites of enzymes by Eq.(5)  
+9: Predict cleavage sites of substrate proteins by Eq.(8)  
+10: Update  $\theta$  via  $\nabla (\mathcal{L}_c + \lambda \mathcal{L}_a)$   
+11: end for  
+12: if validation loss increases for 3 epochs then  
+13: break  
+14: end if  
+15: end for
+
+# A Details of Data Curation and Benchmark Construction
+
+# A.1 Data Curation and Preprocessing
+
+For the cleavage dataset, we downloaded enzyme-substrate pairs from the MEROPS [22] database, collected substrate sequences from the UniProt database, and retrieved enzyme sequences recorded in MEROPS. Additionally, we compared the enzyme sequences between MEROPS and UniProt, excluding those with discrepancies, as such inconsistencies often result from asynchronous updates. To maintain controllable sequence lengths, we filtered out all enzyme and substrate sequences exceeding 1,500 residues.
+
+Regarding the supplemented enzyme set with active sets, we first searched in the UniProt [23] database for enzymes with EC numbers starting with 3.4.*.* and filtered for reviewed data. Then, we selected entries with annotated active sites as our pretraining dataset. In addition, proteolytic enzymes in MEROPS are all annotated with active sites, and are combined as the supplemented enzyme set.
+
+Protein structures were collected from the PDB [42] and AlphaFoldDB [43]. For proteins without available structures in these databases, we generated their structures using OmegaFold [44].
+
+Additionally, all datasets are used in accordance with their respective licenses: MEROPS is distributed under the GNU Lesser General Public License (LGPL), UniProt under Creative Commons Attribution 4.0 (CC BY 4.0), the Protein Data Bank under CC0, AlphaFold DB under CC BY 4.0, and OmegaFold under the MIT License.
+
+# A.2 Data Expansion
+
+The MEROPS database classifies enzymes into categories based on their substrate cleavage sites. Enzymes belonging to the same MEROPS category typically share highly similar cleavage-site characteristics[22]. Drawing on previous work, we assume that minor sequence differences among enzymes of the same category can be disregarded. Consequently, the hydrolysis information from a substrate-enzyme pair is extended to all enzymes in that category.
+
+Therefore, we expanded our dataset by matching each substrate not only with the originally mapped enzyme but also with other enzymes in the same MEROPS category. Through this procedure, we obtained approximately 220K valid enzyme-substrate pairings, involving 677 unique enzymes. Detailed distributions of enzyme and substrate pairs are provided in Tab. 7.
+
+Table 7: Dataset statistics of training datasets.  
+
+<table><tr><td>Utilization</td><td>Datasets</td><td>#Substrate-Enzyme Pairs</td><td>Enzymes</td><td>Substrates</td></tr><tr><td>Active site dataset</td><td>UniProt</td><td>NA</td><td>11,530</td><td>NA</td></tr><tr><td>Cleavage site dataset</td><td>MEROPS</td><td>197,613</td><td>677</td><td>7,475</td></tr></table>
+
+# A.3 Construction of Supervised and Zero-shot Benchmarks
+
+Supervised Setting. We selected MEROPS enzyme families containing at least five distinct substrates, yielding 69 families and approximately  $21\mathrm{K}$  enzyme-substrate pairs. All pairs in each family were randomly split 70/10/20 into training, validation and test sets. To ensure the test substrates were sufficiently distinct from those in training, we collected all substrates per family and computed pairwise sequence similarity using the Needleman-Wunsch algorithm (BLOSUM62, gap opening penalty=10, gap extension penalty=0.5). Substrates exhibiting less than  $50\%$  similarity to any other were deemed independent, and  $20\%$  of independent substrates were sampled to form the final test set. This procedure evaluates model generalization on more divergent substrates within each MEROPS family.
+
+Zero-shot Setting. Following the same selection criteria, we identified MEROPS families with at least five enzymes, yielding 23 families and roughly  $5.3\mathrm{K}$  enzyme-substrate pairs. In each family,  $20\%$  of the enzymes were randomly set aside as a zero-shot test set. To guarantee that these test enzymes were truly unseen, we computed pairwise sequence identities against all training and pretraining enzymes using the Needleman-Wunsch algorithm (BLOSUM62, gap opening penalty  $= 10$ , gap extension penalty  $= 0.5$ ) and retained only those enzymes whose identity fell below a  $60\%$  threshold. This procedure prevents any overlap between zero-shot test enzymes and the training/pretraining pool, rigorously evaluating the model's ability to generalize to novel enzymes.
+
+# B Details of Baselines
+
+Below, we provide additional details on how we adapt, retrain, or utilize each baseline for comparison. Unless otherwise specified, all default hyperparameters are used as in the original implementations of these methods. For any required data, we convert our data format accordingly.
+
+ProsperousPlus [10] and DeepDigest [29] all provide publicly available code, enabling us to retrain their models within our supervised setting. We use the same training and test sets as those used for our method, specifically for the supervised benchmark. We adopt the default training code from each repository while ensuring that all other settings remain consistent.
+
+ScreenCap3 [18] and CAT3 [9], specialized for the C14.003 enzyme, do not provide publicly available datasets or source code for retraining. Instead, they each offer a prediction platform: a web server for ScreenCap3 and standalone software for CAT3. We use these platforms to generate predictions on our test set. Since their training data are not publicly accessible, we can only report their performance as is, with the caveat that neither model can be applied to other enzymes.
+
+We also compare with two recent enzyme-substrate interaction models, ClipZyme [30] and Re-actZyme [31], which were originally proposed for reaction rather than cleavage prediction. Re-actZyme encodes enzymes with an ESM-2 plus MLP pipeline, but since its trained weights are unavailable, we retrain it from scratch on our dataset. ClipZyme employs an E(n) Equivariant Graph Neural Network (EGNN) to incorporate structural information into its enzyme encoder. Both models use average-pooling to aggregate the extracted enzyme features and are trained without activation-site loss. To highlight the effect of leveraging active-site knowledge, we keep the original pretrained EGNN for ClipZyme as is and integrate it into our cleavage-site prediction framework, adding only a linear projection layer to interface with the cleavage-site prediction module.
+
+# C Implementation Details
+
+Framework and Hardware. We implemented our models in PyTorch and trained using the Adam optimizer with a learning rate of  $1 \times 10^{-4}$  and a batch size of 48. All experiments were conducted
+
+on eight NVIDIA A6000 GPUs(48G). We adopted an early stopping strategy with a patience of 3 epochs, monitoring the validation loss to prevent overfitting.
+
+Substrate Representations. Similar to the enzyme pipeline, but without energetic frustration, each residue is embedded by ESM-2 padded to 1500 length. We compute pairwise  $\mathrm{C}\alpha$ -distances  $\mathbf{D}^s (i,j) = \| \mathbf{r}_i - \mathbf{r}_j\| _2$ , then applying a reciprocal transform. Each distance entry is processed by a Gaussian basis kernel and MLP, yielding a bias term  $\Phi_{i,j}^{\mathrm{dist}}$  added to the attention score:
+
+$$
+\mathbf {A} _ {i, j} ^ {k} = \frac {\left(\mathbf {h} _ {i} ^ {k - 1} \mathbf {W} _ {Q}\right) \left(\mathbf {h} _ {j} ^ {k - 1} \mathbf {W} _ {K}\right) ^ {T}}{d} + \Phi_ {i, j} ^ {\text {d i s t}}, \tag {11}
+$$
+
+thus incorporating structural information. The substrate representation  $\mathbf{H}^s\in \mathbb{R}^{|\mathcal{P}^s |\times d}$  is obtained via
+
+$$
+\mathbf {H} ^ {s} = \operatorname {T r a n s f o r m e r} \left(\mathbf {X} ^ {s}, \mathbf {D} ^ {s}\right), \tag {12}
+$$
+
+with the same architecture as the enzyme encoder but omitting energy-related parameters.
+
+Pooling Weight Function. In Active Site-Aware Pooling module,  $f(\cdot)$  is a learnable mapping that transforms each predicted active-site probability into its final pooling weight, in direct analogy to how we use Gaussian kernels to map energy and distance into attention biases.
+
+Concretely, we first pass the scalar probability  $\hat{a}_i$  through a Gaussian basis expansion:
+
+$$
+\phi_ {\mathrm {a c t}} (\hat {a} _ {i}) = \left[ \phi_ {\mathrm {a c t}, 1} (\hat {a} _ {i}), \dots , \phi_ {\mathrm {a c t}, K} (\hat {a} _ {i}) \right] \in \mathbb {R} ^ {K}, \tag {13}
+$$
+
+which produces a richer, multi-dimensional embedding. We then apply an MLP to collapse this embedding to a single weight:
+
+$$
+w _ {i} = f \left(\hat {a} _ {i}\right) = \operatorname {M L P} \left(\phi_ {\text {a c t}} \left(\hat {a} _ {i}\right)\right). \tag {14}
+$$
+
+Energy Frustration Calculation. We computed residue-pair frustration using the Frustratometer tool [45] with AWSEM (Associative Water-mediated Structure and Energy Model) potentials [46], disabling electrostatic interactions ( $k_{\mathrm{electrostatics}} = 0$ ) and enforcing a minimum sequence separation of 12 residues between residue pairs. Specifically, for each pair of residues  $(i,j)$  in enzyme  $\mathcal{P}^e$ , the actual interaction energy  $\mathbf{E}(i,j)$  was extracted from the AWSEM potential. To capture local energetic fluctuations, we generated an ensemble of randomized configurations (where the sequence or side-chain identities are shuffled while preserving the protein backbone), thereby obtaining a distribution of interaction energies for each pair.
+
+Let  $\mu_{\mathrm{rand}}(i,j)$  and  $\sigma_{\mathrm{rand}}(i,j)$  be the mean and standard deviation of these interaction energies over the randomized ensemble. The frustration score  $\mathbf{F}(i,j)$  is then computed as:
+
+$$
+\mathbf {F} (i, j) = \frac {\mathbf {E} (i , j) - \mu_ {\text {r a n d}} (i , j)}{\sigma_ {\text {r a n d}} (i , j)}. \tag {15}
+$$
+
+A higher  $\mathbf{F}(i,j)$  indicates that the local region around residues  $(i,j)$  is more frustrated (i.e., further from minimal AWSEM-derived energy). Such regions often correspond to sites of functional importance in enzymes.
+
+To estimate how  $\mathbf{E}(i,j)$  deviates from an energetically minimal arrangement, we generated an ensemble of randomized "decoy" configurations for the same residue pair. These decoys preserve global geometry (e.g. backbone coordinates) but shuffle aspects such as side-chain packing or local environment, depending on the chosen protocol within the Frustratometer. Each decoy thus provides a distinct pairwise interaction energy. By sampling multiple decoys, we obtain an approximate distribution of energies  $\tilde{E}_k(i,j)$ , from which we compute:
+
+$$
+\mu_ {\text {r a n d}} (i, j) = \frac {1}{K} \sum_ {k = 1} ^ {K} \tilde {E} _ {k} (i, j), \tag {16}
+$$
+
+$$
+\sigma_ {\text {r a n d}} (i, j) = \sqrt {\frac {1}{K - 1} \sum_ {k = 1} ^ {K} \left(\tilde {E} _ {k} (i , j) - \mu_ {\text {r a n d}} (i , j)\right) ^ {2}}, \tag {17}
+$$
+
+where  $K$  is the number of randomized decoys (typically on the order of a few hundred in the Frustratometer).
+
+Gaussian Basis Kernel Function. Following Transformer-M [26], we employ a set of learnable Gaussian basis kernels to transform a scalar input (e.g., the distance  $\mathbf{D}(i,j)$  or the frustration score  $\mathbf{F}(i,j)$ ) into a fixed-dimensional embedding. Concretely, suppose we have  $K$  Gaussian kernels parameterized by  $\{\mu^k,\sigma^k\}_{k = 1}^K$ . For an input scalar  $x$ , the Gaussian basis kernel function  $\phi (x)$  is defined as:
+
+$$
+\phi (x) = \left[ \exp \left(- \frac {1}{2} \left(\frac {x - \mu^ {1}}{\sigma^ {1}}\right) ^ {2}\right), \exp \left(- \frac {1}{2} \left(\frac {x - \mu^ {2}}{\sigma^ {2}}\right) ^ {2}\right), \dots , \exp \left(- \frac {1}{2} \left(\frac {x - \mu^ {K}}{\sigma^ {K}}\right) ^ {2}\right) \right] ^ {\top}. \tag {18}
+$$
+
+Each kernel center  $\mu^k$  and width  $\sigma^k$  is learnable, allowing the model to adaptively capture different regions of the input space. We apply this basis expansion to both  $\mathbf{D}(i,j)$  and  $\mathbf{F}(i,j)$ , producing a  $K$ -dimensional vector for each pair  $(i,j)$ . An MLP then projects this kernel output into the space of attention biases. We set the number of Gaussian basis functions to  $K = 10$ , each parameterized by learnable centers  $\mu^k$  and widths  $\sigma^k$ . Notably, we maintain separate sets of Gaussian parameters for the energy and structure channels, ensuring that the model can adaptively learn distinct representations for each.
+
+Training Algorithm. Each sample's ESM-2 embeddings (packed to length 1500), along with distance and energy frustration matrices, are fed into our model to predict both active-site and cleavage-site residues. We use a weighted binary cross-entropy loss and optimize with Adam for up to 15 epochs, applying early stopping (patience  $= 3$ ) based on validation loss.
+
+# D Comparison of Active-Site Prediction Module
+
+To comprehensively evaluate the active-site prediction module of UniZyme, we compared it with two representative structure-based models that utilize enzyme structural and sequence information: GraphEC [47] and NodeCoder [48]. Both models were implemented using their official repositories and retrained on the same large-scale active-site prediction dataset employed by UniZyme. The dataset statistics are summarized in Table 8. The performance of different models is reported in Table 9. UniZyme achieves consistently superior results compared to the baselines.
+
+Table 8: Data statistics for the active-site prediction task.  
+
+<table><tr><td>Dataset</td><td>Number of Enzymes</td><td>Number of Active Sites</td></tr><tr><td>Training</td><td>9220</td><td>24891</td></tr><tr><td>Test</td><td>2349</td><td>6459</td></tr></table>
+
+Table 9: Comparison of active-site prediction performance.  
+
+<table><tr><td>Model</td><td>AUROC (%)</td><td>AUPR (%)</td><td>Precision (%)</td><td>Recall (%)</td><td>F1 (%)</td></tr><tr><td>UniZyme</td><td>89.5</td><td>35.1</td><td>65.3</td><td>45.6</td><td>53.7</td></tr><tr><td>GraphEC</td><td>80.3</td><td>28.0</td><td>52.1</td><td>38.7</td><td>44.4</td></tr><tr><td>NodeCoder</td><td>67.4</td><td>17.8</td><td>32.6</td><td>22.3</td><td>26.5</td></tr><tr><td>Random Guess</td><td>50.0</td><td>3.2</td><td>3.2</td><td>50.0</td><td>6.0</td></tr></table>
+
+# E Structural Source Sensitivity Analysis
+
+To evaluate the robustness of UniZyme with respect to the source of structural data, we partitioned the test set into four quadrants based on whether the enzyme and substrate structures were obtained from experimental (natural) or predicted sources. Table 10 reports the PR-AUC results under both supervised and zero-shot settings.
+
+Table 10: PR-AUC (%) of UniZyme under different structural-source combinations.  
+
+<table><tr><td>Structure Source</td><td>Zero-shot PR-AUC (%)</td><td>Supervised PR-AUC (%)</td></tr><tr><td>Both Natural Structures (3%)</td><td>72.2</td><td>80.3</td></tr><tr><td>Natural Enzyme + Generated Substrate (7%)</td><td>71.9</td><td>77.9</td></tr><tr><td>Natural Substrate + Generated Enzyme (8%)</td><td>70.5</td><td>81.9</td></tr><tr><td>Both Generated Structures (82%)</td><td>69.4</td><td>78.3</td></tr></table>
+
+# F Statistical Significance Testing
+
+To confirm that the performance improvements of UniZyme over baseline models are statistically significant, we conducted two-sample t-tests on PR-AUC scores across enzyme families under both supervised and zero-shot settings. Table 11 summarizes the results.
+
+Table 11: Two-sample t-test results comparing UniZyme with baseline models.  
+
+<table><tr><td>Setting</td><td>Comparison</td><td>t-value</td><td>p-value</td></tr><tr><td>Supervised</td><td>UniZyme vs ReactZyme</td><td>7.18</td><td>7.8e-10</td></tr><tr><td>Supervised</td><td>UniZyme vs ClipZyme</td><td>5.09</td><td>3.0e-06</td></tr><tr><td>Zero-shot</td><td>UniZyme vs ReactZyme</td><td>2.61</td><td>1.6e-02</td></tr><tr><td>Zero-shot</td><td>UniZyme vs ClipZyme</td><td>4.27</td><td>3.1e-04</td></tr></table>
+
+# G Cross-Task Transferability: EC Number Classification
+
+To assess whether the enzyme encoder of UniZyme captures generalizable biochemical signals, we evaluated it on the EC number classification task for proteases (EC 3.4.*.*). The same dataset split used in the cleavage-site prediction task was reused here. Table 12 presents the results in terms of AUROC.
+
+Table 12: Performance of enzyme encoders on EC number classification.  
+
+<table><tr><td>Model</td><td>AUROC (%)</td></tr><tr><td>UniZyme</td><td>94.1</td></tr><tr><td>ClipZyme</td><td>90.2</td></tr><tr><td>ReactZyme</td><td>82.3</td></tr></table>
+
+# H Interpretability and Mechanistic Consistency
+
+We conducted interpretability analyses to understand how UniZyme utilizes active-site information for cleavage-site prediction. Higher predicted active-site confidence consistently corresponds to better PR-AUC, indicating that UniZyme effectively leverages active-site cues (Table 13). Gradient-based attribution analysis further shows that active-site residues contribute more strongly to cleavage prediction than background residues (Table 14), confirming that the model's attention is aligned with catalytic regions. Moreover, perturbation experiments demonstrate that masking top predicted active-site residues causes a substantial PR-AUC drop, whereas perturbing random residues has minimal effect (Table 15), verifying that UniZyme's predictions are causally linked to the identified catalytic sites.
+
+Mechanistic Basis of Enzyme Cleavage. During protein hydrolysis, enzyme active sites provide a specific geometric and electrochemical environment that enables cleavage only at substrate residues exhibiting optimal complementarity. Thus, active-site geometry and chemistry directly influence cleavage-site specificity. Structural biology and mutational evidence strongly support this relationship: a single residue change in the S1 pocket of canonical serine proteases (e.g., trypsin vs. chymotrypsin)
+
+can dramatically alter specificity by reshaping charge preference and side-chain accommodation [49, 50]. Loop variations near the active-site pocket can also reshape neighboring subsites and modulate substrate scope [51]. Furthermore, structural studies of Alzheimer's  $\gamma$ -secretase show that the architecture of the binding cleft and distal exosomes critically determine substrate recognition, where mutations at the interface shift cleavage patterns [52]. Energetic and mutational scanning analyses of viral proteases (e.g., HCV NS3/4A) also reveal that substrates optimally filling the active-site groove undergo efficient catalysis, while suboptimal packing results in weak or absent cleavage [53].
+
+These mechanistic observations provide biological grounding for UniZyme's interpretability analyses: the model's high sensitivity to predicted active-site residues mirrors the physicochemical principles underlying real enzymatic catalysis, reinforcing that UniZyme captures not only statistical correlations but also mechanistic causality.
+
+Table 13: PR-AUC (%) across bins of predicted active-site confidence.  
+
+<table><tr><td>Active-Site Probability Range</td><td>Zero-shot PR-AUC (%)</td><td>Supervised PR-AUC (%)</td></tr><tr><td>[0.8, 1.0]</td><td>78.6</td><td>85.4</td></tr><tr><td>[0.6, 0.8)</td><td>73.8</td><td>80.3</td></tr><tr><td>[0.4, 0.6)</td><td>63.1</td><td>73.9</td></tr><tr><td>[0.2, 0.4)</td><td>57.7</td><td>61.6</td></tr><tr><td>[0, 0.2)</td><td>52.1</td><td>56.2</td></tr></table>
+
+Table 14: Average attribution magnitude for active-site and background residues.  
+
+<table><tr><td>Residue Type</td><td>Embed. Sens. (Sup.)</td><td>Upstream Attr. (Sup.)</td><td>Embed. Sens. (Zero-shot)</td><td>Upstream Attr. (Zero-shot)</td></tr><tr><td>Active-site residues</td><td>0.68</td><td>0.74</td><td>0.55</td><td>0.60</td></tr><tr><td>Background residues</td><td>0.23</td><td>0.10</td><td>0.19</td><td>0.08</td></tr></table>
+
+Table 15: Effect of perturbing pooling weights on PR-AUC  $(\%)$  
+
+<table><tr><td>Perturbation Target</td><td>Sup. Orig.</td><td>Sup. Post</td><td>Δ (Sup.)</td><td>Zero Orig.</td><td>Zero Post</td><td>Δ (Zero)</td></tr><tr><td>Predicted active-site residues (Top-5)</td><td>79.3</td><td>66.2</td><td>13.1</td><td>71.1</td><td>57.4</td><td>13.7</td></tr><tr><td>Random non-active-site residues</td><td>79.3</td><td>78.8</td><td>1.5</td><td>71.1</td><td>69.4</td><td>1.7</td></tr></table>
+
+# I Computational Effectiveness
+
+Tab. 16 reports wall-clock times measured on NVIDIA A6000 GPUs. Training was conducted on 8xA6000 GPUs; inference was profiled on a single A6000 GPU. As shown, UniZyme's end-to-end training cost is comparable to baselines, and its average per-pair inference latency remains within practical bounds (around 1s overhead).
+
+Table 16: Training and inference times.  
+
+<table><tr><td>Model</td><td>Total Training Time (h)</td><td>Inference per Pair (s)</td></tr><tr><td>Pretraining of UniZyme</td><td>12.5</td><td>—</td></tr><tr><td>UniZyme</td><td>30.9</td><td>5.2</td></tr><tr><td>ClipZyme</td><td>31.4</td><td>3.5</td></tr><tr><td>ReactZyme</td><td>30.3</td><td>4.4</td></tr></table>
+
+# J Ablation Studies
+
+The Tab. 17 reports the average PR-AUC across 69 supervised families and 23 zero-shot families. Full UniZyme achieves the best results in both settings, and performance progressively declines when the SE, A, or P modules are ablated.
+
+Table 17: Ablation Studies on 69 supervised and 23 zero-shot enzyme families.  
+
+<table><tr><td>Model</td><td>Supervised</td><td>Zero-shot</td></tr><tr><td>UniZyme</td><td>79.3±1.2</td><td>71.1±2.3</td></tr><tr><td>UniZyme\SE</td><td>75.3±1.9</td><td>65.3±1.4</td></tr><tr><td>UniZyme\A</td><td>72.5±1.5</td><td>60.6±1.8</td></tr><tr><td>UniZyme\P</td><td>73.0±2.2</td><td>62.1±2.7</td></tr></table>
+
+# K Failure Case Analysis (Limitation)
+
+In our supervised experiments, the M10.003 family consistently lagged behind C14.003 and C14.005, despite having comparable dataset sizes. To understand this, we computed the Shannon entropy of the amino acid distribution at positions P1-P6 around the cleavage site (P3-P4) in Tab. 18. Lower entropy indicates more conserved residues, which simplifies pattern recognition.
+
+Table 18: Shannon entropy at positions around the cleavage site (P3-P4).  
+
+<table><tr><td>Clan</td><td>P1</td><td>P2</td><td>P3 (cleavage site)</td><td>P4 (cleavage site)</td><td>P5</td><td>P6</td></tr><tr><td>C14.005</td><td>4.10</td><td>4.09</td><td>3.20</td><td>3.99</td><td>4.11</td><td>4.11</td></tr><tr><td>C14.003</td><td>4.11</td><td>4.01</td><td>3.40</td><td>3.96</td><td>4.05</td><td>4.11</td></tr><tr><td>M10.003</td><td>3.97</td><td>4.00</td><td>3.95</td><td>3.66</td><td>4.03</td><td>3.90</td></tr></table>
+
+Both C14.003 and C14.005 exhibit markedly lower entropy at the cleavage-site residues (P3, P4), indicating conserved amino acids that aid substrate recognition. By contrast, M10.003 shows uniformly higher entropy, reflecting greater sequence diversity and fewer distinctive cleavage motifs. Moreover, as a metalloprotease, M10.003's activity depends on complex metal-ion coordination in its active-site, further complicating its cleavage specificity.
+
+These observations suggest that M10.003's higher substrate diversity and more intricate catalytic mechanism underlie its lower prediction accuracy. In future work, we plan to integrate metal-ion binding data and more detailed active-site pocket features to better capture the unique determinants of metalloprotease specificity.
+
+# L Broader Impacts
+
+Accurately predicting protease-substrate cleavage sites across a wide enzymatic landscape is pivotal for therapeutic molecule design, industrial biocatalysis, and systematic studies of disease-associated proteolysis. By performing large-scale virtual mapping of these interactions, UniZyme enlarges the set of candidate targets that experimentalists can pursue, thereby accelerating lead prioritization in early-stage drug discovery, guiding the engineering of enzymes with enhanced specificity and stability, and enabling rapid evaluation of emerging viral proteases to bolster pandemic preparedness. Like any computational framework that substantially improves the efficiency of protease screening and functional prediction, UniZyme could be misapplied—for instance, to create proteases that undermine existing biologics or aid in the synthesis of harmful compounds.
+
+# M Additional Experiments and Visualizations
+
+As shown in Fig. 8, we conducted zero-shot testing on all enzymes not included in the training data to evaluate the model's capability in predicting enzyme active sites.
+
+![](images/e067b71fdbba07281125b56755eede0295dca25e41112fc380f3b69a282de22a.jpg)  
+(a) ROC Curve
+
+![](images/c9104701edecbc10cd1cee4a424a21b575fb18022002431ab40fdd9bda399b04.jpg)  
+(b) PR Curve  
+Figure 8: Model performance of active-site prediction
+
+![](images/2d63650b236681725872155f9065d70f3abd3b565fae1971679771c986bcd471.jpg)  
+Figure 9: Per-family PR-AUC (\%) across 69 supervised enzymes(Part 2: Enzymes 36-69) corresponding to Fig. 3.
\ No newline at end of file
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+# Unified 2D-3D Discrete Priors for Noise-Robust and Calibration-Free Multiview 3D Human Pose Estimation
+
+Geng Chen* Pengfei Ren* Xufeng Jian Haifeng Sun† Menghao Zhang
+
+Qi Qi Zirui Zhuang Jing Wang Jianxin Liao Jingyu Wang
+
+State Key Laboratory of Networking and Switching Technology,
+
+Beijing University of Posts and Telecommunications
+
+{chengeng, rpf, jianxf, hfsun, zhangmenghao, qiqi8266, zhuangzirui,
+
+wangjing, liaojx, wangjingyu}@bupt.edu.cn
+
+# Abstract
+
+Multi-view 3D human pose estimation (HPE) leverages complementary information across views to improve accuracy and robustness. Traditional methods rely on camera calibration to establish geometric correspondences, which is sensitive to calibration accuracy and lacks flexibility in dynamic settings. Calibration-free approaches address these limitations by learning adaptive view interactions, typically leveraging expressive and flexible continuous representations. However, as the multiview interaction relationship is learned entirely from data without constraint, they are vulnerable to noisy input, which can propagate, amplify and accumulate errors across all views, severely corrupting the final estimated pose. To mitigate this, we propose a novel framework that integrates a noise-resilient discrete prior into the continuous representation-based model. Specifically, we introduce the UniCodebook, a unified, compact, robust, and discrete representation complementary to continuous features, allowing the model to benefit from robustness to noise while preserving regression capability. Furthermore, we propose an attribute-preserving and complementarity-enhancing Discrete-Continuous Spatial Attention (DCSA) mechanism to facilitate interaction between discrete priors and continuous pose features. Extensive experiments on three representative datasets demonstrate that our approach outperforms both calibration-required and calibration-free methods, achieving state-of-the-art performance.
+
+# 1 Introduction
+
+3D human pose estimation aims to estimate the 3D locations of keypoints from images or videos. It is important because 3D human skeletons are widely used in scenarios such as action recognition [21, 25], human mesh recovery [8, 12], and robotics manipulation [7, 34]. Although monocular pose estimation offers greater convenience in usage, its performance is limited by depth ambiguity and occlusion issues, which hinder its broader application. Multiview systems [36, 13, 22, 31] offer a promising solution by leveraging complementary information from different views, enhancing accuracy and robustness in pose estimation. Existing multiview methods utilize the geometric constraints provided by camera extrinsics to establish correspondences between views, facilitating the fusion of complementary information to enhance 3D pose prediction. There are several drawbacks to using camera calibration-based methods. First, due to the complexity and computational intensity of camera
+
+![](images/60575749b015084c04df5503ec93b6d948bc6a5da64987b1be1c8791f7da966d.jpg)  
+Figure 1: (a) Continuous transformer-based lifting method, which directly processes 2D pose inputs to estimate 3D poses. (b) Proposed method, which integrates discrete features as a robust prior within a continuous transformer-based framework, enhancing robustness to noisy 2D inputs and improving pose estimation accuracy.
+
+calibration, these methods incur higher costs and exhibit limited flexibility. Second, their performance is sensitive to the precision of camera extrinsics, which can be unsatisfactory in complex or dynamic environments. Therefore, uncalibrated multi-view pose estimation has emerged as a significant trend, as it avoids the impact of unreliable camera extrinsics and is more flexible in deployment, leading to better generalization capability and broader applicability in real-world scenarios.
+
+Many uncalibrated methods leverage adaptive multiview interaction patterns, relying on semantic intra-view and inter-view relationships to eliminate reliance on rigid geometric constraints. In recent years, purely transformer-based methods [32, 1, 2] have attracted increasing attention in calibration-free pose estimation, as they provide a unified and scalable framework for modeling spatial, view, and temporal relationships using attention mechanisms. Notably, FusionFormer [1] and PoseIRM [2] have exhibited performance that surpasses even that of calibrated methods. This observation suggests that uncalibrated methods, rather than being constrained by predefined geometric relationships, can instead exploit data-driven adaptive semantic constraints. Such flexibility enables them to more effectively capture context-aware dependencies across views, thereby learning richer and more robust feature representations. For example, calibrated methods depend heavily on geometric information from external calibration, which occlusions can easily disrupt. In contrast, without calibration, uncalibrated methods are forced to leverage multi-view interactions to dynamically correct errors.
+
+The superiority of transformer-based calibration-free framework can be attributed to their expressive and flexible continuous representations, which enable fine-grained modeling of pose dependencies when equipped with attention mechanism. Their continuity aligns with the continuous nature of both input and output spaces in 2D-to-3D pose lifting, enabling smooth, consistent, and precise mappings that are free from quantization errors and easy to optimize. However, this flexibility comes at the cost of increased susceptibility to overfitting when trained on limited data [5], making them sensitive to noisy 2D poses caused by occlusions and other common real-world challenges. Such errors can propagate, amplify and accumulate across views, affecting the accuracy of predictions from other views. Human pose is fundamentally constrained by inherent anatomical priors. These biological constraints, such as skeletal structure, joint motion limitations, and other biomechanical properties, define a natural boundary for valid poses. By encoding such anatomical knowledge as biomechanical priors in pose estimation models, we can intrinsically restrict the solution space to exclude implausible poses (e.g., hyperextended elbows or inverted knees). Explicitly inject priors for guidance has been proven highly effective in other domains, such as image generation [44] and 3D reconstruction [40, 41]. This prior-based regularization complements data-driven approaches by injecting domain-specific knowledge, significantly improving robustness against noisy inputs, and is particularly valuable in uncalibrated scenarios where semantic constraints are entirely learned from the input data.
+
+Despite the promise of integrating biomechanical prior-based regularization, effectively incorporating them into pose estimation models remains a non-trivial challenge. First, a core difficulty lies in how to represent such priors in a way that is structurally robust in representation. Discrete representations offer a promising solution by clustering large amount of body configurations into a limited set of prototypical poses, which encourages the learning of high-quality dominant pose prototypes. This compression enforces structural regularity by restricting the model to select from a predefined pose vocabulary, thereby reducing overfitting to spurious or implausible patterns. Moreover, such many-to-one mappings, i.e., multiple inputs can be mapped to the same prototype, naturally introduces tolerance to
+
+noisy variations, as structurally similar but perturbed poses are treated equivalently. Although it is data-driven, such quantized space can serve as compact, interpretable, and noise-resilient priors-especially beneficial in ambiguous or weakly constrained estimation scenarios. Second, integrating discrete priors into continuous regression models introduces a representational mismatch. While discrete pose spaces are optimized for compactness and structural regularity, pose regressors demand fine-grained flexibility. Directly constraining outputs to lie in the discrete space may hinder expressiveness.
+
+As illustrated in Figure 1, to address the challenges of constructing and leveraging human pose priors, we propose a unified, compact, and noise-resilient discrete prior-enhanced pipeline for 2D-to-3D human pose estimation. Central to our framework is the UniCodebook, which encodes both 2D and 3D poses as compositions of discrete sub-structure tokens within a shared discrete space to server as an informative prior that captures essential pose patterns and promotes resilience to noise. While discrete priors provide strong structural regularization and robustness to noise, their quantized nature inherently mismatches the continuous solution space required for accurate pose regression. Rather than directly regularizing the regressed output, we propose a Discrete-Continuous Spatial Attention (DCSA) module to enable continuous pose features to selectively attend to relevant discrete prototypes, which preserves the continuity of the original regression pipeline while softly integrating structural guidance from the discrete prior. By avoiding hard projection into the discrete space, DCSA mitigates representational mismatch and allows the model to inherit noise robustness from the discrete prior without sacrificing fine-grained precision, thereby effectively reconciling the gap between discrete structure and continuous prediction. We demonstrate the effectiveness by achieving state-of-the-art results on three benchmark datasets covering multiple complex scenarios.
+
+The main contributions of the article are as follows:
+
+- VQ-VAE as a Robust Prior for Pose Estimation: We propose the use of VQ-VAE to serve as a robust prior for pose estimation. By unifying 2D and 3D pose representations within a shared codebook, VQ-VAE further enhances its robustness to noisy pose, ensuring more accurate and consistent predictions, especially in challenging scenarios such as occlusion or 2D detection errors.  
+- Discrete-Continuous Representation Interaction: We introduce an attribute-preserving and complementarity-enhancing mechanism for the interaction between discrete (codebook-based) and continuous (transformer-based) representations. This allows the discrete tokens to guide the continuous features, infusing noise-resilient properties into the backbone while preserving the regression capability of the continuous representation in continuous tasks (i.e., pose estimation).  
+- State-of-the-Art Performance: The proposed approach is capable of calibration-free human pose estimation even under significant noise, achieving SOTA performance on three large datasets.
+
+# 2 Related Work
+
+# 2.1 Multiview Calibration-Free 3D HPE
+
+Methods with unknown camera extrinsics typically rely on semantic information rather than geometric information to enable better multiview feature interaction. FLEX [11] exploits the view-invariance characteristics of skeleton representations, such as bone lengths and rotation angles, to reconstruct human poses. MTF-TransFormer [32] and FusionFormer [1] leverage transformers to elegantly and efficiently model multiview, temporal, and spatial information, with the latter notably surpassing the performance of calibrated methods for the first time. PoseIRM [2] treats the uncalibrated task as a domain generalization problem and employs the invariant risk minimization paradigm to enhance performance in unseen camera settings.  $A^3$ -Net [3] leverages pre-defined alignment proxies, such as meshes and joints, to automatically discover inter-view interaction patterns.
+
+# 2.2 VQ-VAE in Pose-Related Tasks
+
+Recent advancements in pose-related tasks have leveraged VQ-VAE [37] to enhance representation learning and improve task performance. In PCT [10], they introduced a two-step framework for 2D pose estimation, where VQ-VAE is first used to learn a compact, discrete pose representation space, followed by a translator that maps image features to this space. This approach effectively prevents unrealistic poses and provides prior knowledge for occluded pose prediction. In TokenHMR [6], they applied a similar pipeline, focusing on learning pose parameters of the SMPL [23] body model using VQ-VAE. In MEGA [9], they utilized VQ-VAE to discretize 3D human meshes, achieving enhanced
+
+![](images/f632d62c55c77ce6731c5deef54ab6beac4ad5e1f1de964c022237da6b5e7111.jpg)
+
+![](images/f3ae7963c72f46a8dbf4d5b6e0b27a81a9ff862e7ad027a5002694e81fe6e906.jpg)
+
+![](images/cd82df236ae8d9f7b521b34cfc98b21c9285bed43525984cd653208acd6aee53.jpg)  
+Figure 2: Two stages of the proposed calibration-free multiview 3D human pose lifting pipeline (a, b) and the detailed structure of the Spatial Multi-Head Self-Attention (MHSA) with Discrete-Continuous Spatial Attention (DCSA) (c). In Stage I, we construct the UniCodebook, a unified discrete representation space, through a multi-strategy training scheme (2Dto2D, 2Dto3D, 3Dto2D, and 3Dto3D). In this space, both 2D and 3D poses are encoded as sets of discrete tokens in this shared space to bridge the representation gap between 2D and 3D data. In Stage II, a transformer-based continuous model is employed for pose lifting, where codebook tokens generated from the UniCodebook are injected into the hybrid spatial attention block. Here, the proposed DCSA mechanism is integrated with conventional MHSA to facilitate effective fusion between the noise-resilient discrete priors and expressive continuous pose features, which enhances the robustness to noisy 2D input.
+
+![](images/5740362cae4d1a656c31162c1a7223d56bcb864fc824f1ad53e114a759423671.jpg)
+
+mesh recovery from single RGB images through both deterministic and stochastic generation modes, significantly improving uncertainty estimation and multi-output predictions. In DVQVAE [48], they proposed a decomposed VQ-VAE approach for generating realistic human grasps, where hand components are separately encoded to enhance interaction with objects.  $\mathrm{Di}^{2}\mathrm{Pose}$  [39] is a discrete diffusion model for occluded 3D pose estimation, which combines pose quantization with a discrete diffusion process to model 3D poses in latent space, leading to superior performance on occlusion handling and physical plausibility. These methods highlight the versatility of VQVAE, demonstrating its potential in improving robustness and generalization across a wide range of pose related problems. However, these methods either rely on hard mappings between continuous features and discrete representations [9], or constrain the entire prediction or generation process to discrete latent spaces [48, 39, 10, 6], where quantization can obscure fine-grained details. In contrast, continuous representations excel at capturing subtle spatial variations, which are critical for precise and smooth pose reconstruction. To fully exploit the complementary strengths of discrete structure and continuous precision, our method softly integrates discrete structural priors into the continuous regression stream, enabling noise-robust yet precise pose prediction.
+
+# 3 Methodology
+
+Figure 2 depicts the overall architecture of the proposed methods. In Section 3.1, we describe how to learn an UniCodebook which represents 2D and 3D data simultaneously. Section 3.2 explains how the pretrained the UniCodebook is used in the human pose lifting task.
+
+# 3.1 Stage I: Pretrain of the UniCodebook
+
+In this section, we first introduce the key components used in Stage I. We begin with the Pose Encoder, which processes the original pose into several tokens, followed by the proposed UniCodebook, which retrieves features from this unified compact feature space. Finally, we introduce the Pose Decoder and the loss function used to train the entire pipeline.
+
+Existing codebook-based pose representation methods suffer from modality isolation, i.e., they encode either 2D poses [10] or 3D poses [6] in separate models. This isolation creates two critical issues: (1) underutilization of heterogeneous pose data that inherently contains complementary 2D-3D correlations, and (2) failure to align the representation gap between 2D observation and 3D geometry. Since the lifting task involves both 2D and 3D features, this gap forces the backbone model to allocate additional capacity to transform unimodal features into a more suitable representation for lifting, rather than focusing entirely on learning robust 2D-to-3D mappings.
+
+To more effectively utilize 2D and 3D data and bridge the gap between them, we propose a unified cross-modal representation learning framework that establishes shared latent embeddings between 2D and 3D human poses. Specifically, while maintaining dedicated modality-specific encoders and decoders for 2D/3D data processing, we use a shared codebook, i.e., the UniCodebook, that jointly represents both 2D and 3D pose. We optimize the whole framework through four complementary objectives: 2Dto2D self-reconstruction, 3Dto3D self-reconstruction, cross-modal 2Dto3D projection, and inverse 3Dto2D back-projection. The first two auto-encoding tasks preserve modality-specific reconstruction fidelity and the latter two projection objectives enforce a smaller representation gap through bidirectional translation between modalities.
+
+Pose Encoder. Given a human pose  $\mathbf{P} \in \mathbb{R}^{J \times D}$  where  $D = 2$  for 2D pose  $\mathbf{P}_{2d}$  and  $D = 3$  for 3D pose  $\mathbf{P}_{3d}$ , we adopt the compositional encoding strategy to capture human priors and structural relationships following [10]. Unlike joint-wise encoding that processes each joint independently, we decompose the pose into  $N$  overlapping sub-structures  $\mathbf{f}_i$  (e.g., limbs, torso combinations), where each sub-structure corresponds to a group of biomechanically correlated joints. It is noted that the compositional representation contains lots of redundancy due to the fact that different tokens may share overlapping joints. The redundancy, in turn, enhances the robustness of the representation against occlusions of individual parts [10].
+
+We utilize two separate encoders, both identical in structure, to encode 2D and 3D poses, respectively. Following the approach in [10], we adopt the MLP-Mixer architecture [35] as our encoder  $f_{e}$ , which is particularly adept at capturing both local joint interactions and global pose semantics. Given an input pose  $\mathbf{P}$ , we first embed it into  $\mathbf{P}_{\mathrm{emb}} \in \mathbb{R}^{J \times D_{\mathrm{feat}}}$  using a linear layer  $f_{\mathrm{emb}}$ . Subsequently, the embedded features are passed through four MLP-Mixer blocks to effectively model the relationships between different joints. Finally, we employ another linear layer  $f_{\mathrm{trans}}$  to map the features to  $N$  tokens. The overall process is described by  $\mathbf{F}_{\mathrm{en}} = [\mathbf{f}_1, \mathbf{f}_2, \dots, \mathbf{f}_N] = f_{\mathrm{trans}}(f_e(f_{\mathrm{emb}}(\mathbf{P}))) \in \mathbb{R}^{N \times d}$  where each  $\mathbf{f}_i \in \mathbb{R}^{D_{feat}}$ . The number of tokens is set to  $N = 63$ , which is larger than the 51 input dimensions (17 joints × 3) of a Human3.6M pose, to introduce representational redundancy.
+
+Quantization Process With the UniCodebook. We define a shared UniCodebook space  $\mathbf{C} = [\mathbf{c}_1, \dots, \mathbf{c}_U]^\top \in \mathbb{R}^{U \times D_{\mathrm{feat}}}$  that jointly represents both 2D and 3D poses. Specifically, each codebook entry  $\mathbf{c}_u$  encodes pose patterns applicable to both modalities. This shared representation of discrete tokens helps to minimize the disparity between heterogeneous pose representations.
+
+For each sub-structure token  $\mathbf{f}_i$ , we perform quantization via nearest neighbor lookup in the codebook:
+
+$$
+q \left(\mathbf {f} _ {i} = u \mid \mathbf {P}\right) = \left\{ \begin{array}{l l} 1 & \text {i f} u = \arg \min  _ {j} \| \mathbf {f} _ {i} - \mathbf {c} _ {j} \| _ {2} \\ 0 & \text {o t h e r w i s e} \end{array} \right. \tag {1}
+$$
+
+We abuse  $q(\mathbf{f}_i)$  to denote the index of the closest codebook entry for token  $\mathbf{f}_i$ . The quantized pose representation is given by  $\mathbf{F}_{\mathrm{quant}} = [\mathbf{c}_{q(\mathbf{f}_1)},\mathbf{c}_{q(\mathbf{f}_2)},\dots,\mathbf{c}_{q(\mathbf{f}_N)}]\in \mathbb{R}^{N\times D_{\mathrm{feat}}}$ .
+
+Pose Decoder. The pose decoder is designed to recover pose  $\hat{\mathbf{P}}$  from quantized feature  $\mathbf{F}_{\mathrm{quant}}$ . It mirrors the architecture of the pose encoder but in a reverse manner.
+
+Loss. For each strategy, we optimize the entire framework using the following loss function:
+
+$$
+\mathcal {L} _ {\text {p r e t r a i n}} = \operatorname {s m o o t h} _ {\mathcal {L} _ {1}} (\hat {\mathbf {P}}, \mathbf {P}) + \beta \sum_ {i = 1} ^ {M} \| \mathbf {f} _ {i} - \mathrm {s g} [ \mathbf {c} _ {q \left(\mathbf {f} _ {i}\right)} ] \| _ {2} ^ {2} \tag {2}
+$$
+
+where  $\mathrm{sg}$  denotes stopping gradient, and  $\beta$  is a hyperparameter which we set to 10.
+
+# 3.2 Stage II: Human Pose Lifting
+
+Our framework consists of three main components at Stage II: (1) 2D pose discretization and initial continuous feature embedding, (2) dual-stream spatial-temporal feature interaction, and (3) cross-view and temporal feature refinement, and 3D pose regression.
+
+Given a 2D input skeleton sequence  $\mathbf{P}_{2\mathrm{D}} \in \mathbb{R}^{V \times T \times J \times D_{2\mathrm{D}}}$ , we first feed it into the 2D Pose Encoder of the VQ-VAE trained in Stage I and quantize it into discrete codebook features  $\mathbf{F}_{\mathrm{quant}} \in \mathbb{R}^{V \times T \times N \times D_{\mathrm{feat}}}$ . Subsequently, within the transformer backbone, we initially project the 2D pose  $\mathbf{P}_{2\mathrm{D}}$  into a higher-dimensional feature space  $\mathbf{F}^0 \in \mathbb{R}^{V \times T \times J \times D_{\mathrm{feat}}}$  via a fully-connected (FC) layer. We then employ a dual-stream transformer equipped with spatial and temporal Multi-Head Self-Attention (MHSA) mechanisms to derive the feature  $\mathbf{F}_{\mathrm{pose}} \in \mathbb{R}^{V \times T \times J \times D_{\mathrm{feat}}}$ . Within the spatial MHSA, we implement Discrete-Continuous Spatial Attention (DCSA) to facilitate interaction between continuous features and the discrete features  $\mathbf{F}_{\mathrm{quant}}$  along the spatial dimension.
+
+Next, we apply spatial pooling to obtain  $\mathbf{F}_{\mathrm{pool}} = \mathrm{AveragePooling}(\mathbf{F}_{\mathrm{pose}}) \in \mathbb{R}^{V \times T \times D_{\mathrm{feat}}}$ . This is followed by the application of View MHSA and Temporal MHSA to yield the final continuous pose features  $\mathbf{F}_{\mathrm{pose}}' \in \mathbb{R}^{V \times T \times D_{\mathrm{feat}}}$ . Finally, we regress the predicted 3D pose  $\hat{\mathbf{P}}_{\mathrm{3D}} \in \mathbb{R}^{V \times T \times J \times D_{\mathrm{3D}}}$  through an FC layer. We only apply L1 loss between ground truth(GT) 3D pose  $\mathbf{P}_{\mathrm{3D}}$  and  $\hat{\mathbf{P}}_{\mathrm{3D}}$ . For simplicity, in the following descriptions, we omit layer notations unless cross-layer feature interaction is involved.
+
+# 3.2.1 Hybrid Spatial Attention Block
+
+The spatial block contains a Spatial Multihead Self-Attention (MHSA) and a Discrete-Continuous Spatial Attention (DCSA) module. For simplicity, we describe the computation for a single attention head. The multi-head version follows standard practice by computing heads in parallel and projecting the concatenated results.
+
+Spatial MHSA models relationships between joints within the same view and frame. Given a spatial feature  $\mathbf{F}_S\in \mathbb{R}^{J\times D_{\mathrm{feat}}}$  from  $\mathbf{F}_{\mathrm{pose}}$  , we first compute its query, key, and value vectors:
+
+$$
+\mathbf {Q} _ {S} = \mathbf {F} _ {S} \mathbf {W} _ {S} ^ {Q}, \quad \mathbf {K} _ {S} = \mathbf {F} _ {S} \mathbf {W} _ {S} ^ {K}, \quad \mathbf {V} _ {S} = \mathbf {F} _ {S} \mathbf {W} _ {S} ^ {V}, \tag {3}
+$$
+
+where  $\mathbf{W}_S^Q, \mathbf{W}_S^K, \mathbf{W}_S^V \in \mathbb{R}^{D_{\mathrm{feat}} \times D_K}$  are learnable projection matrices. The self-attention output is then computed as:
+
+$$
+\operatorname {S} - \operatorname {M H S A} \left(\mathbf {F} _ {S}\right) = \operatorname {s o f t m a x} \left(\frac {\mathbf {Q} _ {S} \mathbf {K} _ {S} ^ {\top}}{\sqrt {D _ {K}}}\right) \mathbf {V} _ {S}, \tag {4}
+$$
+
+where  $D_K$  is the dimension of the key vectors. The temporal MHSA follows the same computation but applies to temporal tokens.
+
+Discrete-Continuous Spatial Attention (DCSA) augments the continuous joint interactions with semantic guidance from discrete codebook tokens. We implement this module using a cross-attention mechanism. As detailed in our ablation study in Table 4b and Table 7 in the Supplementary Materials, this approach proved to be the most effective and straightforward compared to other alternatives. The mechanism derives queries from the continuous features  $\mathbf{F}_S$  and keys/values from the discrete codebook tokens  $\mathbf{F}_{\mathrm{quant}} \in \mathbb{R}^{N \times D_{\mathrm{feat}}}$ :
+
+$$
+\mathbf {Q} _ {D} = \mathbf {F} _ {S} \mathbf {W} _ {D} ^ {Q}, \quad \mathbf {K} _ {D} = \mathbf {F} _ {\text {q u a n t}} \mathbf {W} _ {D} ^ {K}, \quad \mathbf {V} _ {D} = \mathbf {F} _ {\text {q u a n t}} \mathbf {W} _ {D} ^ {V}. \tag {5}
+$$
+
+The discrete-to-continuous attention is then computed as:
+
+$$
+\operatorname {D C S A} \left(\mathbf {F} _ {S}, \mathbf {F} _ {\text {q u a n t}}\right) = \operatorname {s o f t m a x} \left(\frac {\mathbf {Q} _ {D} \mathbf {K} _ {D} ^ {\top}}{\sqrt {D _ {K}}}\right) \mathbf {V} _ {D}. \tag {6}
+$$
+
+The final spatial representation combines features from both modules through addition:
+
+$$
+\mathbf {F} _ {S} ^ {\text {o u t}} = \underbrace {\mathbf {S} - \operatorname {M H S A} \left(\mathbf {F} _ {S}\right)} _ {\text {C o n t i n u o u s - t o - C o n t i n u o u s}} + \underbrace {\operatorname {D C S A} \left(\mathbf {F} _ {S} , \mathbf {F} _ {\text {q u a n t}}\right)} _ {\text {D i s c r e t e - t o - C o n t i n u o u s}}. \tag {7}
+$$
+
+The output features undergo residual connections and layer normalization (LayerNorm), followed by an multilayer perceptron (MLP) block with another residual connection and LayerNorm. We denote the entire spatial block by  $S$ .
+
+# 3.2.2 Dual-Stream Spatial-Temporal Transformer
+
+Following MotionBert [49], we construct a dual-stream spatial-temporal transformer by alternately stacking temporal and hybrid spatial blocks in different order, thereby creating two parallel branches.
+
+To enable the model to dynamically adjust the importance of different branches, we compute an adaptive weight for each branch. Adaptive fusion weights  $\alpha_{ST}^{i},\alpha_{TS}^{i}\in \mathbb{R}^{L\times V\times T\times J}$  are predicted by linear projection, as  $\alpha_{ST}^i,\alpha_{TS}^i = \mathrm{softmax}(\mathcal{W}([\mathcal{T}_1^i (\mathcal{S}_1^i (\mathbf{F}_{\mathrm{pose}}^{i - 1})),\mathcal{S}_2^i (\mathcal{T}_2^i (\mathbf{F}_{\mathrm{pose}}^{i - 1})))]$ ), where  $\mathcal{W}$  is a learnable linear transformation,  $[\cdot ,\cdot ]$  denotes concatenation,  $i\in \{1,\dots ,L\}$  is the current layer number, and  $L$  is the number of layers of the dual-stream module.
+
+The features from the two branches are fused additively via predicted weights as  $\mathbf{F}_{\mathrm{pose}}^i = \alpha_{ST}^i\circ$ $\mathcal{T}_1^i (\mathcal{S}_1^i (\mathbf{F}_{\mathrm{pose}}^{i - 1})) + \alpha_{TS}^i\circ \mathcal{S}_2^i (\mathcal{T}_2^i (\mathbf{F}_{\mathrm{pose}}^{i - 1}))$ , where  $\circ$  denotes element-wise multiplication.
+
+# 4 Experiments
+
+# 4.1 Datasets and Metrics
+
+Human3.6M [14] is a widely used 3D human pose dataset with 3.6 million frames captured by 4 synchronized cameras, covering 15 actions performed by 11 subjects. And the poses are represented by 17 human joints. Following common protocol, we use subjects S1, S5, S6, S7, and S8 for training, and S9 and S11 for testing. We evaluate under two settings: using ground-truth (GT) 2D poses and CPN-detected 2D poses. The CPN detector introduces significant 2D pose noise: the average 2D MPJPE between CPN-detected and ground-truth keypoints is 3.9 pixels on the training set and 6.5 pixels on the test set, with maximum errors reaching 55 and 124 pixels, respectively.
+
+MPI-INF-3DHP [27] is a challenging multiview dataset for 3D human pose estimation, consisting of 1.3 million frames performed by 8 subjects in diverse indoor and outdoor scenarios, which is widely used to evaluate the generalization performance in complex real-world scenarios. Following previous studies [20, 16], we train on subjects S1-S7 and test on subject S8.
+
+FreeMan [38] is a large-scale multiview 3D human pose dataset collected under real-world conditions, containing 11 million frames from 8,000 sequences captured by 8 synchronized smartphones. It covers 40 subjects across 10 diverse indoor and outdoor scenarios with varying lighting. To address the limited distribution gap between the training and testing sets in the original protocol, we re-split the dataset that introduces clearer separation in both scene and camera configurations: scenes 1-8 and cameras (1,3,5,7) are used for training, while scenes 10-12 and cameras (1,3,5,7) (2,4,6,8) are reserved for testing. The resulting training set consists of 566,484 image pairs, and the test set includes 646,608 pairs. This revised split enables a more realistic and challenging evaluation of model generalization.
+
+AMASS [26] is a large-scale motion capture dataset with over 40 hours of motion data from  $300+$  subjects and  $11,000+$  motions. Since it does not provide direct 2D-3D annotations, we render SMPL meshes and apply official Human3.6M and COCO regressors to extract 3D joints, which are then projected onto 2D planes from 8 virtual views to form 2D-3D pairs. We use the official validation and test splits for training and evaluating the VQ-VAE in Stage I only.
+
+Evaluation Metrics. We report Mean Per Joint Position Error (MPJPE), measured in millimeters, Percentage of Correct Keypoints (PCK), and Area Under Curve (AUC) with a 150mm threshold.
+
+# 4.2 Implementation Details
+
+All individual experiments were completed within 1 day using one server equipped with an Nvidia RTX 3090 GPU, 64GB RAM, and a 28-core CPU. We employed the AdamW [24] optimizer with a learning rate of 2e-4 for both the Stage I and Stage II. We used a cosine annealing learning scheduler with 100 warmup steps. The model was trained for 50 and 100 epochs with a batch size of 256 at Stage I and Stage II, respectively. No data augmentation techniques were applied during training and test. For VQ-VAE, we use  $N = 63$  tokens to represent a pose, and the codebook is in shape  $U \times D_{\mathrm{feat}} = 2048 \times 256$ . The original VQ-VAE suffers from codebook collapse, where a significant portion of codebook entries remain underutilized. Following [6], we employ exponential moving average (EMA) and code reset tricks to enhance codebook activation rates.
+
+Table 1: Results on Human3.6M are reported using MPJPE as the evaluation metric. CPN, HRNet and ResNet152 are different 2D pose detectors. GT means using ground truth 2D pose. * means this is an image-to-3d method. † indicates our reimplementation. T represents the number of frames.  
+
+<table><tr><td>Method</td><td>Venue</td><td>Input Setting</td><td>Dir.</td><td>Disc.</td><td>Eat.</td><td>Greet</td><td>Phone</td><td>Photo</td><td>Pose</td><td>Purch.</td><td>Sit.</td><td>SitD.</td><td>Smoke</td><td>Wait</td><td>WalkD.</td><td>Walk</td><td>WalkT.</td><td>Avg.</td></tr><tr><td colspan="19">Monocular Methods</td></tr><tr><td>MHFormer [19]</td><td>CVPR&#x27;2022</td><td>(CPN, T = 351)</td><td>39.2</td><td>43.1</td><td>40.1</td><td>40.9</td><td>44.9</td><td>51.2</td><td>40.6</td><td>41.3</td><td>53.5</td><td>60.3</td><td>43.7</td><td>41.1</td><td>43.8</td><td>29.8</td><td>30.6</td><td>43.0</td></tr><tr><td>MixSTE [43]</td><td>CVPR&#x27;2022</td><td>(CPN, T = 243)</td><td>37.6</td><td>40.9</td><td>37.3</td><td>39.7</td><td>42.3</td><td>49.9</td><td>40.1</td><td>39.8</td><td>51.7</td><td>55.0</td><td>42.1</td><td>39.8</td><td>41.0</td><td>27.9</td><td>27.9</td><td>40.9</td></tr><tr><td>KTPFormer [29]</td><td>CVPR&#x27;2024</td><td>(CPN, T = 243)</td><td>30.1</td><td>32.1</td><td>29.1</td><td>30.6</td><td>35.4</td><td>39.3</td><td>32.8</td><td>30.9</td><td>43.1</td><td>45.5</td><td>34.7</td><td>33.2</td><td>32.7</td><td>22.1</td><td>23.0</td><td>33.0</td></tr><tr><td colspan="19">Multi-View Methods With Camera Parameter</td></tr><tr><td>Epiolar Transform [13]</td><td>CVPR&#x27;2020</td><td>(*, T = 1)</td><td>25.7</td><td>27.7</td><td>23.7</td><td>24.8</td><td>26.9</td><td>31.4</td><td>24.9</td><td>26.5</td><td>28.8</td><td>31.7</td><td>28.2</td><td>26.4</td><td>23.6</td><td>28.3</td><td>23.5</td><td>26.9</td></tr><tr><td>Crossview Fusion [30]</td><td>ICCV&#x27;2019</td><td>(*, T = 1)</td><td>24.0</td><td>26.7</td><td>23.2</td><td>24.3</td><td>24.8</td><td>22.8</td><td>24.1</td><td>28.6</td><td>32.1</td><td>26.9</td><td>31.0</td><td>25.6</td><td>25.0</td><td>28.0</td><td>24.4</td><td>26.2</td></tr><tr><td>Geometry-Biased Transformer [28]</td><td>FG&#x27;2024</td><td>(HRNet, T=27)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>26.0</td></tr><tr><td>MTF-Transformer+ [32]</td><td>TPAMI&#x27;2022</td><td>(CPN, T = 27)</td><td>23.4</td><td>25.2</td><td>23.1</td><td>24.4</td><td>27.4</td><td>28.5</td><td>22.8</td><td>25.2</td><td>28.7</td><td>36.2</td><td>25.9</td><td>23.6</td><td>26.6</td><td>22.6</td><td>22.7</td><td>25.8</td></tr><tr><td>LearnTriangulation [15]</td><td>ICCV&#x27;2019</td><td>(*, T = 1)</td><td>19.9</td><td>20.0</td><td>18.9</td><td>18.5</td><td>20.5</td><td>19.4</td><td>18.4</td><td>22.1</td><td>22.5</td><td>28.7</td><td>21.2</td><td>20.8</td><td>19.7</td><td>22.1</td><td>20.2</td><td>20.8</td></tr><tr><td>AdaFuse [47]</td><td>JCV&#x27;2021</td><td>(*, T = 1)</td><td>17.8</td><td>19.5</td><td>17.6</td><td>20.7</td><td>19.3</td><td>16.8</td><td>18.9</td><td>20.2</td><td>25.7</td><td>20.1</td><td>19.2</td><td>20.5</td><td>17.2</td><td>20.5</td><td>17.3</td><td>19.5</td></tr><tr><td>MvP [42]</td><td>NeuIPS&#x27;2021</td><td>(*, T = 1)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>18.6</td></tr><tr><td colspan="19">Multi-View Methods Without Camera Parameter</td></tr><tr><td>ProTriangulation [17]</td><td>ICCV&#x27;2023</td><td>(*, T = 1)</td><td>24.0</td><td>25.4</td><td>26.6</td><td>30.4</td><td>32.1</td><td>20.1</td><td>20.5</td><td>36.5</td><td>40.1</td><td>29.5</td><td>27.4</td><td>27.6</td><td>20.8</td><td>24.1</td><td>22.0</td><td>27.8</td></tr><tr><td>FLEX [11]</td><td>ECCV&#x27;2022</td><td>(ResNet152, T = 27)</td><td>23.1</td><td>28.8</td><td>26.8</td><td>28.1</td><td>31.6</td><td>37.1</td><td>25.7</td><td>31.4</td><td>36.5</td><td>39.6</td><td>35.0</td><td>29.5</td><td>35.6</td><td>26.8</td><td>26.4</td><td>30.9</td></tr><tr><td>FLEX [11]</td><td>ECCV&#x27;2022</td><td>(CPN, T = 27)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>31.7</td></tr><tr><td>SGraFormer [45]</td><td>AAAI&#x27;2024</td><td>(CPN, T = 27)</td><td>26.5</td><td>28.3</td><td>23.0</td><td>25.9</td><td>27.2</td><td>31.0</td><td>25.4</td><td>27.2</td><td>28.6</td><td>33.8</td><td>28.6</td><td>25.6</td><td>30.1</td><td>27.1</td><td>26.5</td><td>27.6</td></tr><tr><td>FusionFormer [1] †</td><td>AAAI&#x27;2024</td><td>(CPN, T = 27)</td><td>23.7</td><td>26.3</td><td>24.8</td><td>25.3</td><td>27.7</td><td>27.1</td><td>23.9</td><td>26.6</td><td>32.3</td><td>32.7</td><td>25.9</td><td>25.2</td><td>27.9</td><td>23.5</td><td>23.8</td><td>26.6</td></tr><tr><td>Ours</td><td></td><td>(CPN, T = 27)</td><td>23.0</td><td>25.2</td><td>24.8</td><td>24.4</td><td>27.5</td><td>26.4</td><td>23.4</td><td>25.6</td><td>31.4</td><td>32.3</td><td>25.7</td><td>24.0</td><td>27.3</td><td>22.9</td><td>22.9</td><td>26.0</td></tr><tr><td>MTF-Transformer [32]</td><td>TPAMI&#x27;2022</td><td>(GT, T = 27)</td><td>15.5</td><td>17.1</td><td>13.7</td><td>15.5</td><td>14.0</td><td>16.2</td><td>15.8</td><td>16.5</td><td>15.8</td><td>16.1</td><td>14.5</td><td>14.5</td><td>16.9</td><td>14.3</td><td>13.7</td><td>15.3</td></tr><tr><td>SGraFormer [45]</td><td>AAAI&#x27;2024</td><td>(GT, T = 27)</td><td>11.7</td><td>13.0</td><td>10.1</td><td>12.1</td><td>10.7</td><td>13.0</td><td>12.1</td><td>10.7</td><td>10.8</td><td>11.9</td><td>11.0</td><td>11.6</td><td>12.8</td><td>11.1</td><td>12.0</td><td>11.7</td></tr><tr><td>SVTFormer [46]</td><td>AAAI&#x27;2025</td><td>(GT, T = 27)</td><td>11.6</td><td>12.3</td><td>11.4</td><td>12.2</td><td>10.6</td><td>12.1</td><td>12.5</td><td>11.7</td><td>10.3</td><td>10.7</td><td>10.7</td><td>12.1</td><td>10.8</td><td>10.9</td><td>11.4</td><td>11.4</td></tr><tr><td>FusionFormer [1]</td><td>AAAI&#x27;2024</td><td>(GT, T = 27)</td><td>7.84</td><td>8.04</td><td>7.39</td><td>8.33</td><td>7.13</td><td>9.02</td><td>8.00</td><td>8.19</td><td>7.57</td><td></td><td>7.37</td><td>7.83</td><td></td><td>7.26</td><td></td><td>7.90</td></tr><tr><td>Ours</td><td></td><td>(GT, T = 27)</td><td>7.77</td><td>7.82</td><td>7.70</td><td>7.71</td><td>7.80</td><td>7.76</td><td>7.87</td><td>7.60</td><td>7.81</td><td>7.66</td><td>7.76</td><td>7.92</td><td>7.63</td><td>7.66</td><td>7.74</td><td>7.74</td></tr></table>
+
+Table 2: Comparison results on the MPI-INF-3DHP.  
+
+<table><tr><td>Method</td><td>Extra Data for Finetune</td><td>MPJPE↓</td><td>PCK↑</td><td>AUC↑</td></tr><tr><td>ShapeAware [20]</td><td>syn data</td><td>62</td><td>95</td><td>63</td></tr><tr><td>PaFF [16]</td><td>×</td><td>48.4</td><td>98.6</td><td>67.3</td></tr><tr><td>SGraFormer [45]</td><td>×</td><td>16.9</td><td>98.7</td><td>90.2</td></tr><tr><td>MTF-Transformer [32]</td><td>h36m</td><td>14.6</td><td>-</td><td>-</td></tr><tr><td>SGraFormer [45]</td><td>h36m</td><td>10.6</td><td>99.9</td><td>91.7</td></tr><tr><td>FusionFormer [1]</td><td>h36m</td><td>5.4</td><td>-</td><td>-</td></tr><tr><td>Ours</td><td>×</td><td>3.37</td><td>99.9</td><td>95.94</td></tr></table>
+
+Table 3: Comparison results on the FreeMan dataset.  $\dagger$  indicates our reimplementation.  
+
+<table><tr><td></td><td>seen camera MPJPE ↓</td><td>unseen camera MPJPE ↓</td><td>mean MPJPE ↓</td></tr><tr><td>SGraFormer [45] †</td><td>22.20</td><td>28.69</td><td>25.44</td></tr><tr><td>MTF [32] †</td><td>10.85</td><td>13.51</td><td>12.18</td></tr><tr><td>Fusionformer [1] †</td><td>9.89</td><td>12.72</td><td>11.30</td></tr><tr><td>Ours</td><td>9.59</td><td>12.71</td><td>11.15</td></tr></table>
+
+# 4.3 Comparisons With SOTAs
+
+Human 3.6M. Table 1 compares our method with several state-of-the-art (SOTA) monocular and multi-view approaches on the Human 3.6M dataset for 3D human pose estimation (HPE). Our method outperforms all the calibrated methods due to that our method are not limited by predefined geometric relationships between views but instead utilizes data-driven adaptive semantic constraints, allowing for more effective multi-view feature fusion. When compared to calibration-free 3D HPE methods using CPN [4] or GT 2D poses, our approach achieves SOTA performance, with MPJPE values of  $26.0\mathrm{mm}$  and  $7.74\mathrm{mm}$ , respectively, demonstrating its effectiveness. Notably, our method does not rely on any additional tricks but simply adopts L1 Loss.
+
+MPI-INF-3DHP. Table 2 presents a detailed comparison with other SOTA calibration-free methods on the MPI-INF-3DHP dataset. We use GT 2D poses as input and keep the same settings as [20]. As shown in the table, our method consistently outperforms all other methods across various metrics. Specifically, we achieve a significant reduction in the MPJPE, outperforming the second-best method by at least  $2.03\mathrm{mm}$ $(37.5\%)$ . Additionally, our method shows a notable increase in the AUC, improving it by at least  $4.24\%$ . These indicate that our model not only achieves higher accuracy but also maintains robust performance across different thresholds. It also demonstrates that the UniCodebook not only enhances robustness but also improves accuracy without noise.
+
+FreeMan. Table 3 reports the evaluation results on the FreeMan dataset under seen and unseen camera settings using HRNet-w48 [33] 2D poses. Our method achieves the best overall performance with a mean MPJPE of  $11.15\mathrm{mm}$ , outperforming recent approaches such as Fusionformer [1] (11.30mm) and MTF [32] (12.18mm). In particular, our method attains the lowest error under unseen camera settings (9.59mm), demonstrating strong generalization to novel viewpoints. These results confirm the robustness and adaptability of our approach in real-world scenarios with cross-camera configurations.
+
+# 4.4 Ablation Study
+
+Impact of Codebook Training Strategy. Table 4a shows how different codebook training strategies in Stage I affect the lifting performance at Stage II. The best performance is achieved when all strategies are used. Without the codebook, performance drops significantly, highlighting its importance as a robust prior for the continuous model. The MPJPE are around 26.8, 26.7, 26.4 and 26.3 when
+
+Table 4: Ablation studies on key components of the UniCodebook, including training strategies, feature interaction methods, and data sources. MPJPE is calculated at Stage II. "CPN" and "GT" denote 2D poses from CPN [4] and ground truth, respectively.
+
+(a) Ablation study on different training strategy combinations for the UniCodebook where  $\mathrm{T} = 1$ .
+
+(b) Ablation experiments on different interaction methods between discrete codebook features and continuous features.  $T = 27$ .
+
+<table><tr><td rowspan="2">Number of Strategy</td><td colspan="4">Strategy</td><td rowspan="2">MPJPE↓</td></tr><tr><td>2dto2d</td><td>2dto3d</td><td>3dto2d</td><td>3dto3d</td></tr><tr><td>0</td><td></td><td></td><td></td><td></td><td>27.01</td></tr><tr><td rowspan="2">1</td><td>✓</td><td></td><td></td><td></td><td>26.74</td></tr><tr><td></td><td>✓</td><td></td><td></td><td>26.86</td></tr><tr><td rowspan="5">2</td><td>✓</td><td>✓</td><td></td><td></td><td>26.70</td></tr><tr><td>✓</td><td></td><td>✓</td><td></td><td>26.78</td></tr><tr><td>✓</td><td></td><td></td><td>✓</td><td>26.74</td></tr><tr><td></td><td>✓</td><td>✓</td><td></td><td>26.73</td></tr><tr><td></td><td>✓</td><td></td><td>✓</td><td>26.75</td></tr><tr><td rowspan="4">3</td><td>✓</td><td>✓</td><td>✓</td><td></td><td>26.40</td></tr><tr><td>✓</td><td>✓</td><td></td><td>✓</td><td>26.47</td></tr><tr><td>✓</td><td></td><td>✓</td><td>✓</td><td>26.53</td></tr><tr><td></td><td>✓</td><td>✓</td><td>✓</td><td>26.45</td></tr><tr><td>4</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>26.34</td></tr></table>
+
+<table><tr><td>Method</td><td>MPJPE↓</td></tr><tr><td>Baseline</td><td>26.7</td></tr><tr><td>Conv + Addition</td><td>26.6</td></tr><tr><td>Q-Former [18]</td><td>34.1</td></tr><tr><td>Concat + MHSA</td><td>26.4</td></tr><tr><td>DCSA</td><td>26.0</td></tr></table>
+
+(c) Ablation on training data of VQ-VAE.  $T = 27$ .
+
+<table><tr><td>Data Source</td><td>MPJPE↓</td></tr><tr><td>H36M CPN</td><td>26.43</td></tr><tr><td>H36M GT</td><td>26.35</td></tr><tr><td>AMASS</td><td>26.02</td></tr></table>
+
+![](images/c3f2dbb28ba86b1decb653b1253f6e66c6016de72a960dcea54ff5570d87887c.jpg)  
+Figure 3: Comparison of MPJPE error across four models (i.e., baseline trained on H36M CPN, baseline with codebook trained on H36M CPN, baseline trained on H36M GT, and baseline with codebook trained on H36M GT) under varying noise intensities without retraining. For each instance (consisting of multi-view 2D poses of the same person at the same timestamp), we randomly select 1 to 4 views and add Gaussian noise with zero mean and a standard deviation of "Noise Intensity" pixels to each 2D joint. For models trained on H36M CPN, we evaluate them using H36M CPN test data with extra noise. Similarly, models trained on H36M GT are evaluated with H36M GT test data with extra noise. The results show that models with the codebook exhibit robustness across all noise levels, with greater robustness observed at higher noise intensities.
+
+![](images/789e99ef128ae75f15f9773dc1800cc2194459adc204a2117e5b52c5db66488d.jpg)
+
+![](images/bfcdbf9351a38fe821dbe492fb35bbb1762d6904e72fa64672202bbbc155dd37.jpg)
+
+![](images/a21e4052a4e0e0c909a5789a16da63ccc198a9cb2a9d8d1aa5db70ce5f77eb5d.jpg)
+
+applying one, two, three and four strategies, respectively. This decreasing trend demonstrates that as more strategies are used, the UniCodebook can better capture the relationships between 2D and 3D poses and data utilization rate, making it a robuster prior for lifting tasks.
+
+Impact of the Interaction Method of Discrete Codebook Features and Continuous Features. In Table 4b, we aim to investigate the impact of different interaction methods between discrete and continuous features on model performance. "Baseline" means no discrete features are used. "Conv + Addition" involves aligning the token count of the codebook features using a convolutional layer, followed by adding them to the continuous features. "Q-Former [18]" employs learned queries with the same shape as continuous features and maps discrete feature information onto the queries using a transformer layer. The queries and continuous features then interact via cross-attention. "Concat + MHSA" first concatenates discrete and continuous features along the token dimension and then applies self-attention. All methods are applied in multiple spatial blocks. The results demonstrate that our DCSA method achieves the best performance, improving MPJPE by  $0.7\mathrm{mm}$ .
+
+Impact of Noise. To evaluate the robustness of our model to noisy inputs, we present in Figure 3 the MPJPE performance of four models under varying noise intensities. The results demonstrate that
+
+Table 5: Ablation on the number of pose tokens and codebook size in UniCodebook where the frame number is 1. The active rate represents the mean activation rate in Stage I, while MPJPE is evaluated in Stage II on Human3.6M [14].  
+
+<table><tr><td>pose tokens</td><td>codebook size</td><td>Active Rate(%)↑</td><td>MPJPE↓</td></tr><tr><td>21</td><td></td><td>61.8</td><td>26.82</td></tr><tr><td>63</td><td rowspan="2">2048</td><td>83.5</td><td>26.34</td></tr><tr><td>105</td><td>84.3</td><td>26.35</td></tr><tr><td rowspan="3">63</td><td>1024</td><td>84.5</td><td>26.70</td></tr><tr><td>2048</td><td>83.5</td><td>26.34</td></tr><tr><td>4096</td><td>75.0</td><td>26.47</td></tr></table>
+
+our method—whether trained on H36M CPN or H36M GT—consistently outperforms the baseline in the presence of noise. Moreover, the performance gap widens as the noise intensity increases, indicating that the codebook-based model exhibits stronger resilience to noisy 2D observations than the continuous transformer-based baseline. This robustness stems from the discrete nature of the codebook, which provides a form of structural regularization by anchoring predictions to a set of stable pose prototypes. Such discrete priors may help correct noisy or ambiguous joint inputs by guiding the model toward plausible configurations. These findings highlight the effectiveness of our approach in real-world scenarios, where noisy 2D detections are common, and underscore the practical advantages of incorporating discrete structural knowledge into continuous pose estimation models.
+
+Impact of Codebook Training Data Quality. To evaluate the impact of data quality on model performance, we conducted experiments in Table 4c using three data sources: H36M CPN, H36M GT, and AMASS. The results show a consistent performance increase among them. As data quality improves, the model performance also increases, indicating that higher-quality data leads to better performance. It is worth noting that even when using only H36M CPN data, i.e., without any external data, our model still achieves near-SOTA performance on the Human3.6M benchmark.
+
+Impact of Codebook Configurations. We analyze two key hyperparameters: the codebook size, which defines the capacity of the discrete latent space, and the number of pose tokens, which reflects the granularity of the pose representation. (1) First, we fixed the codebook size to 2048 and varied the number of pose tokens. Increasing the tokens from 21 to 63 significantly boosted the active rate (from  $61.8\%$  to  $83.5\%$ ) and improved MPJPE (from 26.82 to 26.34). A further increase to 105 tokens yielded no improvement, suggesting that 63 tokens offer a favorable balance between expressiveness and efficiency. (2) Next, we fixed the number of pose tokens at 63 and varied the codebook size. A codebook of 2048 achieved the best MPJPE (26.34). A smaller size (1024) suffered from higher error (MPJPE: 26.70) due to limited capacity, while a larger one (4096) slightly degraded performance (MPJPE: 26.47). Overall, our selected configuration of 63 pose tokens and a codebook size of 2048 achieves the best trade-off between token utilization and pose estimation accuracy.
+
+# 5 Conclusion
+
+In this paper, we propose a novel approach for uncalibrated multi-view 3D human pose estimation by integrating a noise-resilient prior, i.e., the UniCodebook, into a transformer-based framework. By employing a multi-strategy training scheme, the UniCodebook effectively bridges the gap between 2D and 3D pose representations. Furthermore, we introduce Discrete-Continuous Spatial Attention (DCSA) to facilitate interaction between discrete codebook tokens and continuous pose features, ensuring the benefits of noise-resilient priors without disrupting the continuity of the backbone model. Extensive experiments on three benchmark datasets demonstrate that our approach outperforms both calibration-required and calibration-free methods, achieving state-of-the-art performance. These results highlight the effectiveness of integrating discrete priors into multi-view pose estimation, demonstrating their potential to enhance the robustness of continuous models.
+
+# Acknowledgement
+
+This work was supported in part by the National Natural Science Foundation of China under Grants (62406039, 62321001, 62471055, U23B2001, 62171057, 62201072, 62071067, 62406039, 62101064), the High-Quality Development Project of the MIIT (2440STCZB2584), the Ministry of Education and China Mobile Joint Fund (MCM20200202, MCM20180101), the Project funded by China Postdoctoral Science Foundation (2023TQ0039, 2024M750257, GZC20230320), the Fundamental Research Funds for the Central Universities (2024PTB-004), and the 2025 Education and Teaching Reform Project Funding at Beijing University of Posts and Telecommunications (2025YZ005).
+
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+
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+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
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+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
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+
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+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
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+
+Answer: [NA]
+
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\ No newline at end of file
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+# Unified Multimodal Chain-of-Thought Reward Model through Reinforcement Fine-Tuning
+
+Yibin Wang $^{1,2,4\dagger}$ , Zhimin Li $^{4*}$ , Yuhang Zang $^{3*}$ , Chunyu Wang $^{4}$ , Qinglin Lu $^{4\dagger}$ , Cheng Jin $^{1,2\dagger}$ , Jiaqi Wang $^{2,3\dagger}$
+
+<sup>1</sup>College of Computer Science and Artificial Intelligence, Fudan University,
+
+$^{2}$ Shanghai Innovation Institute  $^{3}$ Shanghai AI Lab,  $^{4}$ Hunyuan, Tencent
+
+https://codegoat24.github.io/UnifiedReward/think
+
+![](images/fe938932eac07193e6ca2a42f81b266f64b29db5ecf841a6c91c3c3df2e311c1.jpg)  
+(a) Qualitative Case Study (Image Generation)  
+Figure 1: Overview of Comparison Results. (a) Our method enables multi-dimensional long CoT reasoning to improve reward signal accuracy. (b) Extensive quantitative results demonstrate our superiority in both vision understanding and generation reward tasks.
+
+# Abstract
+
+Recent advances in multimodal Reward Models (RMs) have shown significant promise in delivering reward signals to align vision models with human preferences. However, current RMs are generally restricted to providing direct responses or engaging in shallow reasoning processes with limited depth, often leading to inaccurate reward signals. We posit that incorporating explicit long chains of thought (CoT) into the reward reasoning process can significantly strengthen their reliability and robustness. Furthermore, we believe that once RMs internalize CoT reasoning, their direct response accuracy can also be improved through implicit reasoning capabilities. To this end, this paper proposes UNIFIED REWARD-THINK, the first unified multimodal CoT-based reward model, capable of multi-dimensional, step-by-step long-chain reasoning for both visual understanding and generation reward tasks. Specifically, we adopt an exploration-driven reinforcement finetuning approach to elicit and incentivize the model's latent complex reasoning
+
+ability: (1) We first use a small amount of image generation preference data to distill the reasoning process of GPT-4o, which is then used for the model's cold start to learn the format and structure of CoT reasoning. (2) Subsequently, by leveraging the model's prior knowledge and generalization capabilities, we prepare large-scale unified multimodal preference data to elicit the model's reasoning process across various vision tasks. During this phase, correct reasoning outputs are retained for rejection sampling to refine the model (3) while incorrect predicted samples are finally used for Group Relative Policy Optimization (GRPO) based reinforcement fine-tuning, enabling the model to explore diverse reasoning paths and optimize for correct and robust solutions. Extensive experiments confirm that incorporating long CoT reasoning significantly enhances the accuracy of reward signals. Notably, after mastering CoT reasoning, the model exhibits implicit reasoning capabilities, allowing it to surpass existing baselines even without explicit reasoning traces.
+
+# 1 Introduction
+
+In recent years, multimodal reward models (RMs) [Wang et al., 2024, 2025d, Zang et al., 2025, Xiong et al., 2024, He et al., 2024, Xu et al., 2024, Liu et al., 2025a, Li et al., 2025, Wang et al., 2025c] have excelled at aligning vision model outputs with human preferences, providing crucial reward signals to guide model training [Wang et al., 2024, 2025d, Liu et al., 2025a, Ouyang et al., 2022, Rafailov et al., 2023, Schulman et al., 2017] and inference [Gulcehre et al., 2023, Snell et al., 2024]. Traditional reward models are typically trained on large-scale human-annotated preference data through supervised fine-tuning (SFT). At test time, most methods [He et al., 2024, Liu et al., 2025a, Xu et al., 2024] directly assign scores or provide pairwise ranking for vision model outputs, relying on the knowledge and intuitions acquired from the training data. While effective, these methods tend to lack interpretability, which makes it difficult for users to understand the underlying reasoning process behind the assigned scores or rankings. To this end, recent studies [Wang et al., 2025d, Xiong et al., 2024, Wang et al., 2024, Li et al., 2025] leverage the generative capabilities of Visual-Language Models (VLMs), enabling RMs to provide concise justifications alongside the assigned reward signals. Despite their success, their reasoning processes often lack rigorous logical structure and the capacity for multi-dimensional, deep analysis, which may result in inaccurate reward signals in complex scenarios or misguided conclusions arising from flawed reasoning processes.
+
+In light of these issues, we posit two key hypotheses: (1) Incorporating explicit long Chains-of-Thought (CoT) into the reward reasoning process is essential for significantly enhancing RM's reliability and robustness; (2) Once the model internalizes this ability, the accuracy of its directly provided reward signals, even without CoT reasoning traces, can also be improved by leveraging its implicit logical thinking capabilities. However, equipping RMs with CoT reasoning using traditional training schemes like SFT poses a highly challenge due to the scarcity of large-scale CoT reward data, as manual annotation requires substantial human resources and time.
+
+In this work, we argue that this challenge can be effectively addressed, as VLMs inherently possess prior knowledge of complex reasoning; what is needed is an effective strategy to elicit and incentivize this capability [Guo et al., 2025]. Therefore, this paper proposes UNIFIED REWARD-THINK, the first unified multimodal CoT-based reward model, capable of performing multi-dimensional, step-by-step long-chain reasoning across both visual understanding and generation reward tasks. The core idea is to activate the model's latent long-chain reasoning capabilities through limited CoT reward data and to progressively reinforce and refine this capability through exploration-driven reinforcement fine-tuning that optimizes for accurate and robust reasoning patterns. Specifically, our training pipeline, as shown in Fig. 2, consists of three stages: (1) Cold Start. We first use a small amount of labeled image generation preference data to distill the reasoning process from GPT-4o [Hurst et al., 2024], filtering for reasoning traces whose final answers align with the ground-truth (GT) labels. These samples serve to cold-start the model training, enabling it to learn the structure and format of long CoT reasoning. (2) Rejection Sampling. Next, we prepare large-scale unified multimodal labeled preference data to incentivize the model's CoT reasoning outputs across various visual reward tasks. Reasoning trajectories whose final answers match the GT label are retained for rejection sampling, reinforcing the distribution of correct reasoning patterns. (3) Group Relative Policy Optimization (GRPO). Finally, incorrectly reasoned samples are leveraged for GRPO-based reinforcement fine-tuning, enabling the model to explore diverse reasoning paths and optimize toward desirable outcomes defined by our
+
+verified rewards (i.e., format and accuracy rewards). Unlike SFT, which merely imitates predefined answers, GRPO promotes trial-and-error learning by evaluating and refining the model's reasoning outputs based on verified rewards, thus encouraging deeper reasoning discovery rather than passive memorization. Notably, we also propose a multidimensional CoT reward scoring strategy to address a widely observed issue, i.e., inconsistency between reasoning and final decisions, where reasoning may appear implausible despite a correct prediction. Specifically, as shown in Fig. 1, by scoring each candidate across various task-relevant dimensions and aggregating the scores from each dimension into a final decision, our approach enforces alignment between reasoning and outcomes, enhancing both the accuracy and reliability of the reward signal. This design allows us to rely solely on the final answer when filtering data during cold-start and rejection sampling, and applying reward supervision in GRPO, while implicitly ensuring the correctness of the reasoning process.
+
+Extensive experiments demonstrate that incorporating long CoT reasoning significantly improves the accuracy and reliability of reward signals. Remarkably, experimental results also prove that once the model internalizes the CoT reasoning ability, it also exhibits strong implicit reasoning capabilities: even when providing direct reward outputs without explicit reasoning traces, it consistently outperforms existing baselines across all vision reward tasks.
+
+Contributions: (1) We propose the first unified multimodal CoT reward model, UNIFIEDREWARD-THINK, capable of multi-dimensional, step-by-step long-chain reasoning across both visual understanding and generation tasks; (2) We demonstrate that explicit long CoT reasoning substantially enhances reward model reliability, and once mastered, also strengthens implicit reasoning, enabling more accurate reward signals even without explicit reasoning traces; (3) Extensive experiments validate the superiority of our method compared with existing baselines across all vision reward tasks. We hope our work can unlock reward models' reasoning potential to enhance interpretability, generalization, and alignment, enabling more trustworthy and human-aligned reward signals for multimodal generation and understanding.
+
+# 2 Related work
+
+# 2.1 Multimodal reward models
+
+Multimodal reward models have become increasingly important for aligning vision understanding and generation models with human preferences. A dominant approach is to fine-tune visual-language models (VLMs) [Li et al., 2024a, Bai et al., 2025b], exploiting their powerful multimodal alignment capabilities to learn human judgment-based reward functions. Earlier studies have explored reward models on visual generation [Wang et al., 2024, Liu et al., 2025a, He et al., 2024] and understanding [Zang et al., 2025, Xiong et al., 2024] tasks. For instance, [Wang et al., 2024] collects human feedback and constructs human-rated video datasets to train the reward model, LiFT-Critic, which measures how well the generated videos align with human expectations. [Zang et al., 2025] develops an effective pipeline for constructing multimodal preference datasets and leverages existing high-quality data to train the reward model, IXC-2.5-Reward, enabling accurate evaluation of visual understanding outputs. However, these reward models are task-specific, limiting their adaptability across diverse visual understanding and generation tasks. Therefore, [Wang et al., 2025d] introduces UnifiedReward, a unified reward model capable of assessing both image and video generation and understanding tasks, demonstrating that joint learning across diverse visual tasks can yield substantial mutual benefits.
+
+Despite their effectiveness, these reward models are largely limited to providing direct responses [Liu et al., 2025a, Xu et al., 2024, He et al., 2024, Xu et al., 2023] or engaging in shallow reasoning with limited depth [Wang et al., 2024, 2025d, Xiong et al., 2024], often resulting in inaccurate or unreliable reward signals in complex scenarios or misguided conclusions arising from flawed reasoning processes. To this end, we propose UNIFIED REWARD-THINK, the first unified multimodal long CoT-based reward model, enabling multi-dimensional long-chain reasoning for both visual understanding and generation tasks. The core idea is to use reinforcement learning to activate and enhance VLMs' latent reasoning capabilities, which will be discussed in the following section.
+
+# 2.2 Reinforcement learning
+
+Recently, reinforcement learning (RL) techniques have been extensively used to enhance the reasoning capabilities of Large-Language Models (LLMs), enabling them to effectively solve complex problems
+
+![](images/96f8792b80010500d276a34b491daf710d5ed9b255f92d9c824b4751e262ba65.jpg)  
+Stage 1. Cold Start: Learning CoT Reward Format
+
+![](images/42451526cfa40eb4aa279b2d951fe75822d4f66e5162bc3db4648796b9155c8e.jpg)  
+Stage 2. Rejection Sampling: Unified CoT Reward Generalization Fine-tuning
+
+![](images/dae1c6ec4142147e5afe9183b895980abe4336d5742665db286fadeafe09e9b2.jpg)  
+Stage 3. GRPO: Unified CoT Reward Reinforcement Fine-tuning  
+Figure 2: Method Overview. (1) Cold Start: We first distill GPT-4o's reasoning process on a small amount of labeled image generation preference data to initialize the model's CoT reasoning format; (2) Rejection Sampling: Then, we leverage the model's generalization capabilities on large-scale unified multimodal preference data to elicit its CoT reasoning process across various vision tasks. Correctly predicted samples whose final answers match the ground-truth are retained for rejection sampling to further refine the model. (3) GRPO: Finally, incorrectly predicted samples are utilized for GRPO-based reinforcement fine-tuning to further enhance step-by-step reasoning capabilities.
+
+[Jaech et al., 2024, Luong et al., 2024, Shao et al., 2024, Yang et al., 2024, Ying et al., 2024, Hui et al., 2024, Jiao et al., 2024, Zhang et al., 2024a,c, Wang et al., 2025a,b]. A significant advancement in this field is Deepseek-R1-Zero [Guo et al., 2025], which introduced a novel approach for developing robust reasoning capabilities using Group Relative Policy Optimization (GRPO). By leveraging rule-based rewards, it enhances reasoning without the need for supervised fine-tuning (SFT). For VLMs, RL has mainly been applied to tasks such as mitigating hallucinations [Sun et al., 2023, Yu et al., 2024a], aligning models with human preferences [Yu et al., 2024b, Zhou et al., 2024], improving visual perception [Liu et al., 2025b], and quality assessment [Li et al., 2025].
+
+However, the application of RL in multimodal reward models is still under exploration. To address this gap, our work introduces a reinforcement fine-tuning strategy that leverages verifiable rewards with GRPO-based RL to integrate long CoT reasoning, enhancing the accuracy of reward signals by enabling multi-dimensional and step-by-step reasoning processes.
+
+# 3 Method
+
+# 3.1 Overview
+
+This work aims to incorporate long Chain-of-Thought (CoT) reasoning into the reward model's decision-making process to enhance the reliability and robustness of reward signals. However, achieving this with traditional training methods like Supervised Fine-Tuning (SFT) remains highly challenging due to the scarcity of large-scale CoT-based reward data. This work posits that Visual-
+
+![](images/3c729576ae8dfa20e2324484d88096405abea21a6c78398a898a921ca672ff1b.jpg)  
+Figure 3: Qualitative Results of Video Generation CoT Reward Reasoning. Given a pair of videos and the corresponding caption, our model performs quality assessment across semantic consistency, temporal coherence, and authenticity through CoT reasoning.
+
+![](images/13c21e6f145ed8815fa84fc295b329fc0020bbc27ea05854006cc55d1175b42b.jpg)
+
+Language Models (VLMs) inherently possess prior knowledge of complex reasoning; the key challenge lies in devising an effective strategy to elicit and incentivize this capability. Therefore, we take the first step to propose the unified multimodal CoT-based reward model, UNIFIEDREWARD-THINK, adopting exploration-driven reinforcement fine-tuning to activate and refine the VLM's multi-dimensional and step-by-step long chain reasoning across various vision reward tasks.
+
+Specifically, as shown in Fig. 2, our pipeline includes three key stages: (1) Cold Start: use a small amount of distilled CoT reward data to initialize the reward model with the format and structure of multi-step reasoning (Sec. 3.2); (2) Rejection Sampling: utilize large-scale unified preference data to elicit the model's generalizable CoT reasoning across diverse vision tasks; correctly reasoned samples are retained for rejection sampling to reinforce accurate reasoning patterns (Sec. 3.3); (3) Group Relative Policy Optimization (GRPO): leverage incorrectly reasoned samples for GRPO-based reinforcement fine-tuning to further improve the model's CoT reasoning capabilities (Sec. 3.4). To further ensure consistency between the reasoning process and final decisions, we design an effective multidimensional CoT reward score strategy (Sec. 3.5).
+
+# 3.2 Cold start: learning CoT reward format
+
+In this work, we hypothesize that VLMs inherently possess the potential for complex, long-chain reasoning. However, due to the absence of exposure to reward modeling during pre-training, they often lack a suitable format to articulate such reasoning in this context, which leads to inconsistent or shallow outputs. To address this, we initiate the training with a cold start phase, where a small amount of high-quality CoT reward data distilled from GPT-4o [Hurst et al., 2024] is used to explicitly teach the model the desired reasoning format and structure. Specifically, we begin by preparing a small set of image generation preference data, each consisting of an image pair, a prompt (i.e., instruction and image caption), and the ground-truth (GT) label. These samples are then fed into GPT-4o to generate the detailed long-chain reasoning process and the corresponding final answer. Among the generated samples, we retain only those reasoning trajectories whose final answers align with the GT label, denoted as  $(x,y)$  where  $x$  is the original input, and  $y = \{y_1,y_2,\dots ,y_T\}$  is the distilled output, which are subsequently used to cold-start the training. The objective function is defined as:
+
+$$
+\mathcal {L} _ {\text {c o l d \_ s t a r t}} (\theta) = - \sum_ {i = 1} ^ {T} \log p \left(\boldsymbol {y} _ {i} \mid \boldsymbol {x}, \boldsymbol {y} _ {<   i}; \theta\right), \tag {1}
+$$
+
+where  $\theta$  represents the parameters of the reward model. This stage serves to initialize the model's ability to follow a structured CoT reasoning format.
+
+# 3.3 Rejection sampling: unified CoT reward generalization fine-tuning
+
+After learning the CoT reasoning format through cold-start training, we further elicit and expand the model's reasoning capabilities to vision understanding and generation tasks. This is inspired by a
+
+prior study [Wang et al., 2025d], which highlights the benefits of multi-task joint training: reward reasoning abilities learned in one task can generalize well and effectively enhance performance in other vision tasks. Therefore, we apply rejection sampling to reinforce the learning of correct reasoning patterns to improve both reliability and accuracy in vision understanding and generation tasks. Specifically, we first prompt the model to perform generalization CoT reasoning on large-scale unified preference datasets, leveraging its prior knowledge across various tasks (e.g., evaluating video temporal consistency and vision-question-answer accuracy). In vision generation tasks, each input sample consists of a textual prompt (instruction and caption) and a corresponding image/video pair, whereas in vision understanding tasks, it comprises a textual prompt (instruction and query) along with an image/video. Then, we retain only the reasoning samples whose trajectories lead to a final answer consistent with the ground-truth label, while discarding incorrect reasoning. The training objective for this stage is similar to Eqn. 1, but it utilizes the filtered reasoning data of diverse vision tasks  $(x', y')$  obtained through rejection sampling.
+
+This process concentrates the training distribution around accurate reasoning patterns and enhances the model's generalization ability across diverse visual domains.
+
+# 3.4 GRPO: unified CoT reward reinforcement fine-tuning
+
+In the rejection sampling stage, a subset of challenging data featuring intricate reasoning patterns is filtered, which the model has yet to fully comprehend and master. To ensure the model fully learns the underlying knowledge in the training dataset and further enhances its reasoning ability, we introduce GRPO-based reinforcement fine-tuning [Guo et al., 2025]. Specifically, in GRPO, the policy model  $\pi_{\theta}$  generates multiple candidate responses for a given input, which are evaluated using predefined verifiable reward functions, providing corresponding reward signals. These signals guide policy updates, encouraging alignment with high-quality reasoning while constraining deviations from the reference model  $\pi_{\mathrm{ref}}$ . This approach enables the model to explore diverse reasoning processes, guiding it toward the correct reasoning trajectory and improving its ability to handle complex scenarios. In the following, we will first introduce the design of verifiable rewards in our work, followed by a detailed description of the GRPO training process.
+
+# 3.4.1 Verifiable reward
+
+In GRPO, verifiable rewards are essential for guiding the model's learning by offering rule-based feedback. In this work, we employ two types of verifiable rewards i.e., format reward and accuracy reward, to ensure both the quality and accuracy of the model's responses:
+
+Format reward ensures that the generated response follows a specific reasoning structure, which is critical for maintaining clarity and consistency in the reasoning process. In this case, the response is expected to contain two key tags:  $\langle$ think  $\rangle$  and  $\langle$ answer  $\rangle$ . These tags are used to delineate the model's reasoning process and the final answer, respectively. If both tags are present and properly formatted in the response, the reward  $R_{fmt}$  is set to 1. Otherwise, the reward  $R_{fmt}$  is 0. This mechanism helps reinforce the importance of structuring the response. Although most VLMs already exhibit strong instruction-following capabilities, the format reward can still serve as a safety check to filter out occasional formatting errors, which, though minor, may negatively affect model performance.
+
+Accuracy reward evaluates whether the final answer output  $o$  (i.e., Image/Video/Answer  $X$  is better), enclosed within the <answer> tag, exactly matches the ground truth. This reward serves as a reliable signal to ensure that the model produces correct answers. If  $o$  matches the ground truth precisely, the reward is set to 1; otherwise, it is 0. Formally, the accuracy reward is defined as:
+
+$$
+R _ {a c c} = \left\{ \begin{array}{l l} 1 & \text {i f o = g r o u n d t r u t h} \\ 0 & \text {o t h e r w i s e .} \end{array} \right.
+$$
+
+This is crucial for reinforcing accurate reasoning and encouraging the model to generate correct responses. Finally, the overall verifiable reward  $R$  is formulated as:
+
+$$
+R = R _ {f m t} + R _ {a c c}.
+$$
+
+By incorporating both the format and accuracy rewards into the GRPO training, we provide the model with explicit feedback that encourages it to generate responses that are both well-structured and factually accurate.
+
+![](images/330194cec3676960b957e360a524c0994f3900758f6b62e48c0ec99e3aa9ad94.jpg)
+
+Can you elaborate on the details of the image?
+
+![](images/849d7d1c4347e914cd088f17107ba1ff0f93a4af88010d40fcfa36b0345518bb.jpg)
+
+![](images/ba46e2d3909b31701ee9dc890d9679d822f8a0598ab11dfe278d95af57392d52.jpg)
+
+# Image Understanding
+
+Answer 1: The image shows a male tennis player in mid-action, executing a serve. He is dressed in a teal athletic shirt with short sleeves, suitable for a sports activity. The player has dark curly hair and is looking upwards, concentrating on the tennis ball he is tossing into the air with his left hand. His right arm is drawn back, holding the racket in preparation to swing at the ball. The background features a green barrier with the word \`MASTERS\` written across in white lettering, suggesting this is a professional level tennis tournament.
+
+Answer 2: The image captures a young man playing tennis on a court. He is wearing a blue shirt and black shorts, and is in the process of serving a tennis ball. The ball is in mid-air, and the man is fully engaged in the game. There are several tennis rackets scattered around the court, indicating that the game is in progress. Additionally, there are several chairs placed around the court, suggesting that spectators are watching the game.
+
+![](images/fd7a97c6ce70725b24901389575fbe3ba95eb49a1a6ddd9768f416a20fc2df26.jpg)
+
+<think>
+
+# 1. Semantic accuracy:
+
+- Answer 1 (9/10): Accurately describes the player's attire, action, and background details such as the "MASTERS" sign. The description of the player's focus and posture is precise.
+
+- Answer 2 (6/10): Incorrectly mentions several tennis jackets and chairs around the court, which are not visible in the image. Hirt color is described inaccurately as blue instead of teal.
+
+# 2. Correctness:
+
+- Answer 1 (9/10): Logically consistent with the image content, correctly identifying the player's action and attire.
+
+- Answer 2 (5/10): Contains factual inaccuracies regarding the presence of multiple rackets and chairs, which are not depicted in the image.
+
+# 3. Clarity:
+
+- Answer 1 (8/10): Clearly and fluently expressed, providing a vivid depiction of the scene.
+
+- Answer 2 (7/10): Generally clear but includes incorrect details that could confuse the reader.
+
+Total score: Answer 1:  $9 + 9 + 8 = 26$  \\nAnswer 2:  $6 + 5 + 7 = 18$  </think>
+
+<answer>Answer 1 is better</answer>
+
+![](images/177812fc352b399bf6b12cc4282744be246f615732f82ca371c379e69cd5ef07.jpg)  
+Figure 4: Qualitative Cases of Image and Video Understanding CoT Reward Reasoning. Given an image or video, a query, and a pair of candidate answers, our model performs quality assessment across semantic accuracy, factual correctness, and clarity through CoT reasoning.
+
+What emotions are conveyed to the video, and how do visual details support your concept?
+
+![](images/760575a5c5115d932ae7c96f295db6b416d50b54872054c14d6296c800169cca.jpg)
+
+![](images/77a8ad8898b055fe7be052877bebc967db98391315e69ea363253b8480c3d254.jpg)
+
+# Video Understanding
+
+Answer 1: The video conveys a sense of happiness and freedom. The boy's big smile and relaxed posture show he is enjoying himself. The bright sunlight and the clean, quiet neighborhood make the scene feel peaceful and safe, highlighting the carefree nature of childhood.
+
+Answer 2: The video captures a strong theme of youthful freedom and innocence. The boy's genuine smile, casual clothing, and confident grip on the handlebars suggest both comfort and excitement. The warm sunlight enhances the feeling of joy and security, while the neatly maintained suburban setting—with white fences and wide sidewalks—emphasizes a safe environment that nurtures exploration and independence. These visual elements collectively celebrate the simple, unburdened pleasures of childhood.
+
+![](images/925447ad9a5b01f3e2f663594aa704a2f39425577ddc2235cba72f1cc0311dee.jpg)
+
+<think>
+
+# 1. Semantic accuracy:
+
+- Answer 1 (8/10) - The answer captures the emotions of happiness and freedom well, supported by the boy's smile and the peaceful suburban setting. However, it could delve deeper into the visual details.
+
+- Answer 2 (9/10) - This answer effectively conveys the themes of youthful freedom and innocence, supported by the boy's expression and the suburban environment. It provides a more comprehensive interpretation.
+
+# 2. Correctness:
+
+- Answer 1 (7/10) - The interpretation is correct but somewhat simplistic. It focuses mainly on the boy's emotions without much depth.
+- Answer 2 (9/10) - The answer is factually correct and provides a nuanced understanding of the scene, including the setting and the boy's demeanor.
+
+# 3. Clarity:
+
+- Answer 1 (8/10) - The answer is clear and easy to understand, with a straightforward explanation of the emotions and setting.
+
+- Answer 2 (9/10) - The answer is very clear and well-articulated, with detailed observations that enhance the interpretation.
+
+Total score: Answer 1:  $8 + 7 + 8 = 23$  nAnswer 2:  $9 + 9 + 9 = 27$  <think>
+
+<answer>Answer 2 is better</answer>
+
+# 3.4.2 Reinforcement fine-tuning
+
+Given an input  $x$ , GRPO first samples  $N$  distinct responses  $\{o^{(1)}, o^{(2)}, \ldots, o^{(N)}\}$  from the previous policy model  $\pi_{\theta_{\mathrm{old}}}$ . Each response is evaluated using our verifiable rewards, i.e., format and accuracy rewards, resulting in corresponding reward scores  $\{R^{(1)}, R^{(2)}, \ldots, R^{(N)}\}$ . Then, GRPO normalizes these scores and quantifies the relative quality of each response by computing the advantage of each response using the standardized reward:
+
+$$
+\hat {A} ^ {(i)} = \frac {R ^ {(i)} - \operatorname {m e a n} (\{R ^ {(1)} , \ldots , R ^ {(N)} \})}{\operatorname {s t d} (\{R ^ {(1)} , \ldots , R ^ {(N)} \})},
+$$
+
+where  $\hat{A}^{(i)}$  quantifies the relative quality of the  $i$ -th response in comparison to other candidates within the same sampled group.
+
+Then, GRPO estimates magnitude of the policy update by computing the likelihood ratio of each response under the new policy  $\pi_{\theta_{\mathrm{new}}}$  relative to the old policy  $\pi_{\theta_{\mathrm{old}}}$ , defined as:
+
+$$
+r ^ {(i)} = \frac {\pi_ {\theta_ {\text {n e w}}} \left(o ^ {(i)} \mid \boldsymbol {x}\right)}{\pi_ {\theta_ {\text {o l d}}} \left(o ^ {(i)} \mid \boldsymbol {x}\right)}.
+$$
+
+To stabilize training and avoid overly aggressive updates, the ratio is clipped to a bounded interval  $[1 - \delta, 1 + \delta]$ . Moreover, to ensure that the updated policy does not diverge significantly from a fixed reference model  $\pi_{\mathrm{ref}}$ , a KL divergence penalty term is introduced with a weighting factor  $\beta$ . The final optimization objective is defined as:
+
+$$
+L _ {g r p o} (\theta) = \mathbb {E} _ {\mathbf {x} \sim \mathcal {X}, o ^ {(i)} \sim \pi_ {\theta_ {\text {o l d}}}} \left[ \min  \left(r ^ {(i)} \hat {A} ^ {(i)}, \operatorname {c l i p} (r ^ {(i)}, 1 - \delta , 1 + \delta) \hat {A} ^ {(i)}\right) - \beta \cdot D _ {\mathrm {K L}} \left(\pi_ {\theta_ {\text {n e w}}} \| \pi_ {\text {r e f}}\right) \right],
+$$
+
+where  $\mathcal{X}$  denotes the set of training sample input and  $D_{\mathrm{KL}}(\cdot \| \cdot)$  denotes the KL divergence.
+
+By integrating normalized reward advantages, clipped importance sampling, and reference-model regularization, GRPO enables stable and effective policy optimization, driving our model toward generating higher-quality, verifiably correct CoT reasoning paths.
+
+# 3.5 Multidimensional CoT reward scoring strategy
+
+In the cold-start and rejection sampling stages, we filter data by retaining only those samples whose final answers align with the ground truth, and in the GRPO stage, supervision is likewise applied solely to the final answers. However, both GPT-4o and existing reward models often suffer from inconsistencies between reasoning traces and final decisions, where reasoning may appear implausible even if the prediction is correct, leading to unreliable supervision. To address this issue, we propose a multidimensional CoT reward scoring strategy that tightly couples the reasoning process with the final score. Specifically, the model evaluates each candidate (image, video, or textual response) along multiple task-relevant dimensions, aggregates the dimension-wise scores into a final decision, and thereby enforces consistency between reasoning and outcomes (Figs. 1, 3, 4). By grounding the final prediction in multidimensional and interpretable reasoning processes, our design implicitly ensures reasoning correctness even when data filtering or supervision relies solely on the final answer, as in all training stages, thereby enhancing both the accuracy and reliability of our reward model.
+
+# 4 Experiments
+
+# 4.1 Experimental setup
+
+Datasets. For Image Generation, we utilize HPD (25.6K) [Christodoulou and Kuhlmann-Jorgensen, 2024], OIP  $(7.4\mathrm{K})^{1}$ , EvalMuse (3K) [Han et al., 2024], all preprocessed by [Wang et al., 2025d], as well as OpenAI-4o_t2i_human Preference (6.7K) collected by Rapidata². For Video Generation, we employ VideoDPO (10K) [Liu et al., 2024] and Text2Video-Human Preferences (5.7K), also collected by Rapidata. For Image Understanding, we sample 30K data from LLaVA-Critic-113K [Xiong et al., 2024]. For Video Understanding, we adopt ShareGPTVideo-DPO (17K) [Zhang et al., 2024b]. In the cold-start stage, we distill image generation CoT reward reasoning samples from GPT-4o, constructing our ImageGen-CoT-Reward-5K. The input data are randomly sampled from the image generation datasets, with the remaining data reserved for the subsequent training stages.
+
+Reward model. We adopt UnifiedReward [Wang et al., 2025d] as our base model, which is capable of assessing both image/video generation and understanding. We leverage its strong performance and extensive prior knowledge in visual perception and generation, and further activate its latent capacity for long-chain CoT reasoning.
+
+Training details. For both the cold-start and rejection sampling stages, training is performed with a batch size of 1, 16 gradient accumulation steps, a learning rate of  $2.5 \times 10^{-6}$ , and a warm-up ratio of 0.3, using 8 NVIDIA H100 (80GB) GPUs. For GRPO, training is conducted with a batch size of 1, a single gradient accumulation step, a learning rate of  $1 \times 10^{-6}$ , and a KL penalty coefficient of  $\beta = 0.04$ . The number of generated responses  $N$  is set to 8, using 64 NVIDIA H20 (97GB) GPUs.
+
+**Evaluations.** We evaluate image and video understanding reward assessment on VLRewardBench [Li et al., 2024b] and ShareGPTVideo [Zhang et al., 2024b], using 5K test samples, respectively. For generation evaluation, we adopt GenAI-Bench [Jiang et al., 2024], which covers both image and video reward benchmarks. Additionally, we utilize VideoGen-RewardBench [Liu et al., 2025a] to further assess video generation.
+
+Table 1: Image Understanding Assessment Comparison. We evaluate baselines across different understanding aspects on VLRewardBench.  
+
+<table><tr><td>Models</td><td>General</td><td>Hallu.</td><td>Reason.</td><td>Overall Accuracy</td><td>Macro Accuracy</td></tr><tr><td>Gemini-1.5-Pro</td><td>50.8</td><td>72.5</td><td>64.2</td><td>67.2</td><td>62.5</td></tr><tr><td>GPT-4o</td><td>49.1</td><td>67.6</td><td>70.5</td><td>65.8</td><td>62.4</td></tr><tr><td>LLaVA-Critic</td><td>47.4</td><td>38.5</td><td>53.8</td><td>46.9</td><td>46.6</td></tr><tr><td>UnifiedReward</td><td>76.5</td><td>58.1</td><td>65.1</td><td>67.5</td><td>66.6</td></tr><tr><td>Ours (w/o CoT)</td><td>77.9</td><td>70.5</td><td>65.4</td><td>73.1</td><td>71.3</td></tr><tr><td>Ours</td><td>78.1</td><td>72.7</td><td>66.0</td><td>73.8</td><td>72.3</td></tr></table>
+
+Table 2: Image and Video Generation Assessment Comparison. "tau" indicates that accuracy is calculated with ties, and "diff" excludes tied pairs when calculating accuracy.  
+
+<table><tr><td rowspan="3">Method</td><td colspan="2">Image Generation</td><td rowspan="3">Method</td><td colspan="4">Video Generation</td></tr><tr><td colspan="2">GenAI-Bench</td><td colspan="2">GenAI-Bench</td><td colspan="2">VideoGen-Reward</td></tr><tr><td>tau</td><td>diff</td><td>tau</td><td>diff</td><td>tau</td><td>diff</td></tr><tr><td>PickScore</td><td>53.2</td><td>67.2</td><td>VideoScore</td><td>46.2</td><td>70.6</td><td>42.1</td><td>49.9</td></tr><tr><td>HPSv2</td><td>51.6</td><td>68.4</td><td>LiFT</td><td>41.2</td><td>60.1</td><td>40.6</td><td>58.3</td></tr><tr><td>ImageReward</td><td>47.8</td><td>65.0</td><td>VisionReward</td><td>52.1</td><td>73.1</td><td>57.4</td><td>68.2</td></tr><tr><td>VisionReward</td><td>46.8</td><td>66.4</td><td>VideoReward</td><td>50.2</td><td>73.3</td><td>60.1</td><td>73.9</td></tr><tr><td>UnifiedReward</td><td>54.8</td><td>70.9</td><td>UnifiedReward</td><td>60.7</td><td>77.2</td><td>66.6</td><td>79.3</td></tr><tr><td>Ours (w/o CoT)</td><td>54.1</td><td>71.9</td><td>Ours (w/o CoT)</td><td>57.8</td><td>81.6</td><td>64.9</td><td>79.9</td></tr><tr><td>Ours</td><td>-</td><td>72.5</td><td>Ours</td><td>-</td><td>82.3</td><td>-</td><td>80.5</td></tr></table>
+
+# 4.2 Comparison results
+
+The quantitative results, presented in Tabs. 1, 2, and Fig. 1, demonstrate the superiority of our model across visual generation and understanding tasks. In image and video generation tasks, our model does not account for tie scenarios during evaluation, since such cases were not included in the training data. Nevertheless, it consistently outperforms existing methods across all other evaluations. Notably, compared to the base model UnifiedReward, incorporating multi-dimensional and multi-step reasoning yields substantial performance gains across all tasks. In particular, our model achieves significant improvements in image understanding reward tasks. This is intuitive, as CoT reasoning in vision tasks fundamentally centers on a deeper understanding of the visual content. Additionally, we explore whether our model, after learning long CoT reasoning, can improve the accuracy of direct reward signals, i.e., without explicit CoT reasoning. Experimental results show that the model still outperforms existing methods by leveraging implicit logical reasoning, with only a slight drop in performance compared to explicit CoT reasoning. These results strongly validate the effectiveness and superiority of our approach. Our qualitative results are provided in Figs. 1, 3, and 4.
+
+# 4.3 Ablation studies
+
+Ablation of each training stage. We conduct ablation studies to assess the effectiveness of each training stage. As shown in Tabs. 3 and 4, after the cold start phase, though the model learns the CoT reasoning format, it still struggles with accurate reward prediction. Notably, introducing rejection sampling leads to clear improvements by retaining correctly reasoned samples for supervised fine-tuning, thereby reinforcing desirable reasoning patterns. Further, the GRPO stage yields the most substantial gains, as it focuses on previously mispredicted cases, allowing the model to explore multiple reasoning paths and converge on more accurate solutions. These results highlight the complementary roles of each stage and demonstrate how our staged training strategy progressively enhances CoT-based reward modeling.
+
+Ablation of GRPO without CoT reasoning. To further validate the necessity of learning the CoT reasoning process, we evaluate a GRPO variant that removes CoT and directly optimizes reward predictions based on final answers. As shown in Tabs. 3 and 4, although this approach yields slight improvements over the baseline, the gains are significantly limited. This suggests that learning
+
+Table 3: Ablation Results of Image Understanding Assessment. We conduct ablation experiments under different settings and evaluate them across multiple aspects on VLRewardBench.  
+
+<table><tr><td>Models</td><td>General</td><td>Hallu.</td><td>Reason.</td><td>Overall Accuracy</td><td>Macro Accuracy</td></tr><tr><td>UnifiedReward</td><td>76.5</td><td>58.1</td><td>65.1</td><td>67.5</td><td>66.6</td></tr><tr><td>+GRPO (w/o CoT)</td><td>77.8</td><td>59.1</td><td>65.5</td><td>69.0</td><td>67.4</td></tr><tr><td>UnifiedReward</td><td>76.5</td><td>58.1</td><td>65.1</td><td>67.5</td><td>66.6</td></tr><tr><td>+cold start</td><td>76.0</td><td>56.7</td><td>65.2</td><td>66.9</td><td>66.0</td></tr><tr><td>+rejection sampling</td><td>77.6</td><td>64.9</td><td>65.4</td><td>72.1</td><td>69.3</td></tr><tr><td>+GRPO(Ours)</td><td>78.1</td><td>72.7</td><td>66.0</td><td>73.8</td><td>72.3</td></tr></table>
+
+Table 4: Ablation Results of Image and Video Generation Assessment. "tau" indicates that accuracy is calculated with ties, and "diff" excludes tied pairs when calculating accuracy.  
+
+<table><tr><td rowspan="3">Method</td><td colspan="2">Image Generation</td><td rowspan="3">Method</td><td colspan="4">Video Generation</td></tr><tr><td colspan="2">GenAI-Bench</td><td colspan="2">GenAI-Bench</td><td colspan="2">VideoGen-Reward</td></tr><tr><td>tau</td><td>diff</td><td>tau</td><td>diff</td><td>tau</td><td>diff</td></tr><tr><td>UnifiedReward</td><td>54.8</td><td>70.9</td><td>UnifiedReward</td><td>60.7</td><td>77.2</td><td>66.6</td><td>79.3</td></tr><tr><td>+GRPO (w/o CoT)</td><td>54.1</td><td>71.3</td><td>+GRPO (w/o CoT)</td><td>56.7</td><td>78.4</td><td>64.8</td><td>79.5</td></tr><tr><td>UnifiedReward</td><td>54.8</td><td>70.9</td><td>UnifiedReward</td><td>60.7</td><td>77.2</td><td>66.6</td><td>79.3</td></tr><tr><td>+cold start</td><td>-</td><td>68.6</td><td>+cold start</td><td>-</td><td>75.3</td><td>-</td><td>76.8</td></tr><tr><td>+rejection sampling</td><td>-</td><td>72.0</td><td>+rejection sampling</td><td>-</td><td>78.9</td><td>-</td><td>79.7</td></tr><tr><td>+GRPO(Ours)</td><td>-</td><td>72.5</td><td>+GRPO(Ours)</td><td>-</td><td>82.3</td><td>-</td><td>80.5</td></tr></table>
+
+from final answers alone fails to teach the model the underlying reasoning process. In contrast, our CoT-based GRPO guides the model to explore multiple reasoning trajectories and gradually converge toward the correct path, leading to deeper understanding and more robust generalization. These results show that the effectiveness of our GRPO-based reinforcement fine-tuning stems from explicitly modeling the reasoning process, rather than simply reinforcing the final answer.
+
+# 5 Limitations and future works
+
+While our method introduces long-form CoT reasoning to improve reward modeling, this inevitably increases inference time during reasoning. However, we show that once the model has mastered CoT reasoning, it can leverage implicit reasoning to enhance answer accuracy even without explicitly generating CoT traces. This suggests strong internalization of the reasoning process. Besides, although our reinforcement fine-tuning strategy successfully activates the model's latent long CoT reasoning ability using only a small amount of high-quality data, the prior study [Yue et al., 2025] has shown that reinforcement learning cannot fundamentally extend a model's capability: it can only amplify the potential already acquired during supervised fine-tuning (SFT). Therefore, to further push the boundaries of CoT-based reward reasoning, scaling up high-quality CoT supervision still remains a promising direction.
+
+# 6 Conclusion
+
+We propose UNIFIED REWARD-THINK, the first unified multimodal CoT reward model capable of multi-dimensional, step-by-step reliable reward reasoning for both visual understanding and generation tasks. Specifically, we adopt an exploration-driven reinforcement fine-tuning approach to elicit and incentivize the model's latent complex reasoning ability, including three key stages: cold start, rejection sampling, and GRPO-based reinforcement fine-tuning. Our experiments demonstrate that CoT reasoning not only improves the accuracy and robustness of reward signals but also equips the model with strong implicit reasoning capabilities, enabling superior performance even without explicit CoT outputs. We hope our work can unlock reward models' reasoning potential to enhance interpretability and generalization, enabling more trustworthy and human-aligned reward signals for multimodal understanding and generation.
+
+# 7 Acknowledgments
+
+This work was supported by the National Natural Science Foundation of China (Grant No. 62176064), AI for Science Foundation of Fudan University (FudanX24AI028), and Shanghai Innovation Institute.
+
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+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
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+
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+
+# 2. Limitations
+
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+
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+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [NA]
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+# Guidelines:
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+
+# 4. Experimental result reproducibility
+
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+
+# Answer: [Yes]
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+
+# Guidelines:
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+
+# 5. Open access to data and code
+
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+
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+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.  
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+# Answer: [Yes]
+
+Justification: This paper specifies all the training and test details in Experiments section.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+# Answer: [No]
+
+Justification: Error bars are not reported because it would be too computationally expensive.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
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+- The assumptions made should be given (e.g., Normally distributed errors).
+
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: This paper provides sufficient information on the computer resources in Experiments section.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: The research conducted in the paper conforms, in every respect, with the NeurIPS Code of Ethics.
+
+Guidelines:
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+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: This paper has discussed both potential positive societal impacts and negative societal impacts of the work performed in the Supplementary Materials.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
+
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: This paper poses no such risks.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
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+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: The creators or original owners of assets are properly credited and the license and terms of use are explicitly mentioned and properly respected.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
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+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
+
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: This paper does not release new assets.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: This paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: This paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
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+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+# Answer: [NA]
+
+Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components.
+
+# Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+Table 5: Ablation of Rejection Sampling Training Stage. We conduct an ablation study to assess the effect of the rejection sampling stage on different benchmarks.  
+
+<table><tr><td>Method</td><td>VLReward</td><td>GenAI-Image</td><td>GenAI-Video</td><td>VideoGen</td></tr><tr><td>Baseline</td><td>67.5</td><td>70.9</td><td>77.2</td><td>79.3</td></tr><tr><td>UnifiedReward-Think</td><td>73.8</td><td>72.5</td><td>82.3</td><td>80.5</td></tr><tr><td>w/o Rejection Sampling</td><td>70.4</td><td>71.6</td><td>79.2</td><td>79.8</td></tr></table>
+
+# A Further methodological insights
+
+Cold start only with image generation preference data: why it works. Our experiments demonstrate that using a small amount of high-quality image generation CoT reward reasoning data, instead of distilling data for every task, is sufficient for the model to learn the CoT reasoning format and structure across all visual reward tasks. The underlying reason lies in the fact that video tasks can be seen as multi-image understanding problems. Video frames, like images, also require recognizing objects, spatial relationships, and context. Once the model masters CoT reasoning for images, it can naturally extend this reasoning to videos by leveraging its inherent prior knowledge of temporal dynamics and sequential visual understanding. Therefore, by learning CoT reasoning from images, the model can seamlessly generalize to both static and dynamic visual tasks, eliminating the need for separate distillation for each task.
+
+Rejection sampling for unified reward generalization fine-tuning: why we need it. After the cold start, the model has already internalized the format and structure of CoT reasoning. With prior knowledge across tasks, it can generate accurate CoT-based reward analyses for many simple scenarios. However, directly applying GRPO to the entire training set would be inefficient and computationally costly. Besides, GRPO offers limited gains on samples that the model has already mastered. Therefore, we first apply rejection sampling to filter out cases that the model already performs well on. This not only reduces training cost but also reinforces the distribution of correct reasoning patterns by prioritizing high-confidence outputs. More challenging or ambiguous samples are passed to the GRPO stage, where the model explores diverse reasoning trajectories and gradually learns to prefer more accurate solutions. Ablation results of this training stage are provided in Tab. 5.
+
+Why we trust the CoT reasoning when the final answer is correct during GRPO. In MLLMs, a common failure mode during CoT reasoning is the disconnect between the reasoning steps and the final answer: models may produce plausible reasoning chains but ultimately rely on shortcuts or intuition to generate the conclusion. To mitigate this, we explicitly structure the CoT process by having the model score each image across multiple dimensions and then aggregate these scores to derive the final decision. This enforces a step-by-step alignment between intermediate reasoning and outcome, ensuring that a correct final answer emerges from a coherent and interpretable reasoning process. As a result, in GRPO, verifying the correctness of the final answer implicitly validates the reasoning trajectory, offering a principled yet efficient way to supervise complex CoT generation.
+
+Why learning can be generalized to different tasks. In this work, our unified multimodal CoT reward model is designed to support both vision generation and understanding tasks across both images and videos. Although the downstream tasks may differ, the input-output structure to the reward model is unified: in all cases, the model receives a combination of visual content (image or video) and caption or textual response, and outputs a reward judgment signal indicating the relative quality or preference between candidate outputs. Because of this consistent structure, the reward model learns to evaluate visual-text pairs in a task-agnostic manner, allowing it to generalize across different modalities and task types. This design enables us to train the reward model jointly on diverse datasets and apply it effectively across vision reward tasks.
+
+# B Robustness on different baselines
+
+To further demonstrate the robustness of our method across different base models, we additionally train UNIFIEDREWARD-THINK on Qwen2.5-VL-7b [Bai et al., 2025a]. As shown in Tab. 6, leveraging the stronger capability of the base model, the Qwen2.5-VL-based model achieves consistent improvements.
+
+Table 6: Performance Comparison on Different Backbones. We compare the performance of UNIFIEDREWARD-THINK trained on LLaVA-OneVision and Qwen2.5-VL.  
+
+<table><tr><td rowspan="2">UnifiedReward-Think</td><td colspan="2">GenAI-Bench</td><td colspan="5">VLRewardBench</td></tr><tr><td>Image</td><td>Video</td><td>General</td><td>Hallu.</td><td>Reason.</td><td>Overall Accuracy</td><td>Macro Accuracy</td></tr><tr><td>LLaVA-OneVision-7B</td><td>72.5</td><td>82.3</td><td>78.1</td><td>72.7</td><td>66.0</td><td>73.8</td><td>72.3</td></tr><tr><td>Qwen2.5-VL-7B</td><td>76.6</td><td>84.0</td><td>78.3</td><td>75.4</td><td>74.2</td><td>75.9</td><td>74.6</td></tr></table>
+
+Table 7: Exploring CoT Distillation from Different Models. We compare the impact of distilling CoT samples from GPT-4o versus Qwen2.5-VL.  
+
+<table><tr><td>Method</td><td>VLReward</td><td>GenAI-Image</td><td>GenAI-Video</td><td>VideoGen</td></tr><tr><td>Baseline</td><td>67.5</td><td>70.9</td><td>77.2</td><td>79.3</td></tr><tr><td>Ours (distilling GPT-4o)</td><td>73.8</td><td>72.5</td><td>82.3</td><td>80.5</td></tr><tr><td>Ours (distilling Qwen2.5-VL-72b)</td><td>73.2</td><td>73.6</td><td>81.7</td><td>80.2</td></tr></table>
+
+# C Role of distilled CoT samples on cold-start stage
+
+In our pipeline, the cold-start CoT data serves primarily to teach the model the structure and format of CoT-style reasoning, rather than to enhance reward decision quality. The generalization and strengthening of the model's reward discrimination capability are achieved through the subsequent stages of rejection sampling and GRPO, which operate on large-scale unified multimodal vision preference data and iteratively reinforce correct reasoning trajectories. Our experimental results in Tab. 7 also show that even when using distilled CoT data from weaker models (Qwen2.5VL-72b) instead of GPT-4o for cold-start, the model can still achieve comparable performance after subsequent reinforcement through rejection sampling and GRPO.
+
+# D Details of our cold-start dataset: ImageGen-CoT-Reward-5K
+
+In this work, we distill a small set of CoT reward reasoning samples for image generation from GPT-4o [Hurst et al., 2024], which are used to cold-start our model and teach it the format and structure of CoT reasoning. Specifically, we randomly select 10K samples from HPD, EvalMuse, and OIP datasets and input them into GPT-4o for distillation. To ensure consistency between the long-chain reasoning process and final decision, we carefully designed the CoT reasoning format: the model is prompted to independently score each image along multiple evaluation dimensions, and the final judgment is made based on the aggregated scores across all dimensions. This structured approach mitigates inconsistencies between intermediate reasoning and final conclusions. An example prompt template is shown in Fig. 5. We retain only those CoT outputs whose final conclusion matches the ground truth preference label. After filtering, we obtain a total of 5K high-quality CoT reward reasoning samples for image generation, constructing our cold-start dataset: ImageGen-CoT-Reward-5K.
+
+# E More experimental details
+
+# E.1 Base model.
+
+This work adopts UnifiedReward [Wang et al., 2025d] as our base architecture, which unifies image/video generation and understanding reward tasks, demonstrating the mutual benefits of multi-task learning and achieving strong performance across various vision reward benchmarks. However, its reasoning is limited to direct responses or shallow thinking, lacking the capability for long and structured CoT reasoning, which may lead to lower accuracy and weaker interpretability in complex scenarios. Inspired by this work, we build upon UnifiedReward and further activate its latent long CoT reasoning capabilities across different vision reward tasks, aiming to enhance the accuracy and robustness of the reward signals.
+
+# E.2 Reward model baselines.
+
+We compare our method against a series of strong reward model baselines across both image and video domains, covering generation and understanding tasks.
+
+PickScore [Kirstain et al., 2023] is a text-to-image preference model that integrates CLIP-based vision-language features with a reward modeling strategy inspired by InstructGPT. It is trained on the Pick-a-Pic dataset to align image generation outputs with human preferences.
+
+HPSv2 [Christodoulou and Kuhlmann-Jorgensen, 2024] builds upon CLIP and is fine-tuned using the HPD_v2 [Christodoulou and Kuhlmann-Jorgensen, 2024] to predict human preferences over generated images. It demonstrates strong performance in pairwise ranking tasks and serves as a representative image generation reward baseline.
+
+ImageReward [Xu et al., 2023] is trained on a large-scale preference dataset containing 137k human expert comparisons through both rating and ranking. It is specifically designed to capture subtle aspects of human preferences in text-to-image generation quality.
+
+LLaVA-Critic [Xiong et al., 2024] extends large language models for evaluating image understanding through a critic-style framework. It is trained on high-quality instruction-following data covering diverse criteria such as accuracy, relevance, and hallucination, supporting both pointwise and pairwise evaluation.
+
+VisionReward [Xu et al., 2024] introduces a fine-grained, multi-dimensional evaluation framework for both image and video domains. It trains separate reward models tailored to human preferences collected through carefully curated datasets, offering strong baselines for visual content assessment.
+
+VideoScore [He et al., 2024] focuses on assessing video generation quality. It is trained on the VideoFeedback dataset comprising human-annotated scores over 37.6K videos, each evaluated across multiple aspects including fidelity, consistency, and alignment.
+
+LiFT-Critic [Wang et al., 2024] is a reward model developed under the LiFT framework, which aligns text-to-video models using human feedback. Trained on LiFT-HRA—a dataset containing over 10K human-labeled samples with both scores and rationales—it captures detailed human evaluation signals across multiple dimensions.
+
+VideoReward [Liu et al., 2025a] offers a multi-dimensional assessment for video generation tasks. It is trained on a large-scale 182K dataset of human-labeled comparisons collected from outputs of 12 video generation models, providing strong performance on complex video benchmarks.
+
+UnifiedReward Wang et al. [2025d] serves as our base architecture. It leverages multi-task learning across diverse image and video generation and understanding datasets. By unifying multimodal reward tasks into a single framework, UnifiedReward demonstrates mutual enhancement effects and establishes a solid baseline for holistic visual reward modeling.
+
+Our Unified Reward-Think extends UnifiedReward by integrating explicit long CoT reasoning across both visual understanding and generation tasks. Through a three-stage training pipeline—including cold start to learning CoT reward format, rejection sampling for unified CoT reward generalization fine-tuning, and GRPO for unified CoT reward reinforcement fine-tuning, the model achieves stronger accuracy and interpretability in reward assessment. It also generalizes well without explicit reasoning, leveraging implicit CoT capabilities for robust performance.
+
+# E.3 Evaluation benchmarks
+
+VLRewardBench [Li et al., 2024b] serves as a diverse benchmark for evaluating image understanding capabilities, featuring 1,250 carefully curated samples across general vision-language queries, hallucination detection, and complex reasoning. To ensure robust evaluation, response orders are randomly shuffled during testing.
+
+ShareGPTVideo [Zhang et al., 2024b] provides large-scale video-caption pairs and human preference data, covering various aspects of video understanding such as temporal reasoning, spatial relations, and factual grounding. We 3K for evaluation in our reward modeling experiments.
+
+GenAI-Bench [Jiang et al., 2024] is a multimodal generation benchmark designed to assess how well models align with human preferences across image and video generation tasks. We adopt its image and video generation subsets for evaluating generative reward performance.
+
+VideoGen-RewardBench [Liu et al., 2025a] offers a large-scale benchmark tailored for evaluating video reward models, consisting of  $26.5\mathrm{k}$  video pairs labeled by humans. Each pair is ranked according to multiple criteria, and we use the Overall Quality scores to benchmark the performance of the model.
+
+# F Regarding performance on "Tau" metric
+
+In both GenAI-Bench and VideoGen benchmarks shown in Tab. 2, the "Tau" metric is specifically designed to evaluate model behavior in tie cases, where two images or videos are equally preferred. In this work, our approach is explicitly focused on enhancing the model's discriminative ability, that is, its capacity to distinguish between better and worse samples. While existing reward models are generally effective at identifying outputs with large quality differences, they often struggle to make fine-grained distinctions between two high-quality candidates. To address this gap, we introduce a multidimensional CoT reasoning process, aimed at improving the model's sensitivity to subtle differences in quality. As a result, we intentionally exclude tie cases from our training data, and our method is not optimized for these scenarios. This design choice is deliberate and aligns with practical objectives in vision-based reinforcement learning, where reward models are expected to make clear and decisive judgments between competing outputs in order to effectively guide policy optimization. In contrast, baseline models were specially trained on datasets that include such tie scenarios, enabling them to perform well in such situations. Although baseline models may achieve higher "Tau" scores, our method, both with and without CoT reasoning, consistently demonstrates superior performance in "Diff." metrics where a quality difference exists between the two candidates. We believe these results convincingly demonstrate the effectiveness of our proposed training approach.
+
+# G Prompting templates and more qualitative cases
+
+We provide more qualitative cases across diverse vision tasks with prompting templates in Figs. 5, 6, 7 and 8.
+
+# H Societal impacts
+
+Our work introduces a unified multimodal CoT reward model capable of high-quality, interpretable assessment across diverse multimodal tasks. This advancement can significantly enhance the alignment of generative models with human preferences in real-world applications such as AI-assisted content creation, and education. By improving both the accuracy and interpretability of reward signals, our method contributes more transparent and controllable AI behaviors, potentially increasing public trust in generative technologies. However, as reward models become more capable and general, they may also be misused to reinforce harmful biases in generation models, especially if the training data or preference annotations reflect subjective or skewed human values. We encourage future work to further examine the ethical implications of large-scale reward modeling and to include fairness-aware training strategies.
+
+# I Ethical statement
+
+In this work, we affirm our commitment to ethical research practices and responsible innovation. To the best of our knowledge, this study does not involve any data, methodologies, or applications that raise ethical concerns. All experiments and analyses were conducted in compliance with established ethical guidelines, ensuring the integrity and transparency of our research process.
+
+# Image Generation
+
+a photo of a boy eating ramen in a restaurant
+
+![](images/100758316228bae96ea4d769d1f434924fdaf4ae1a82686c6d7f9bbf077afd74.jpg)
+
+![](images/1e7d328cd148bc9676c6cf1aaf22c77c076d2793b3c3b3eef5e3d222f4211622.jpg)  
+Figure 5: More Qualitative Results of Image Generation CoT Reward Reasoning. Given a pair of images and the corresponding caption, our model performs quality assessment across semantic consistency, aesthetics, and authenticity through CoT reasoning.
+
+![](images/10a0322bfbac6a35766b5244f91d2ce7e8081ebc58353f83ffa18d97b260b049.jpg)
+
+![](images/2611aa9eb0ccef1f003836fa1422e17fe9c13169aaf1d30b2cb7b042fd9cdd41.jpg)
+
+Given a caption and two images generated based on this caption, please analyze in detail the two provided images. Evaluate them on various dimensions such as semantic consistency (how closely the image content aligns with the caption), aesthetics (composition, color usage, artistic expression), authenticity (realism and attention to detail), and any other factors you deem relevant. For each evaluation dimension, provide a score between 1-10 for both images (e.g., Image 1: 8/10, Image 2: 6/10) and provide a concise rationale for the score. Calculate the total score for each image by summing all dimension scores. Use a chain-of-thought process to detail your reasoning steps, and enclose all your detailed reasoning within <think> and </think> tags. Then, in the <answer> tag, output exactly one of the following strings:  $\backslash$  Image 1 is better' or  $\backslash$  Image 2 is better' based on the total scores. No additional text is allowed in the <answer> section.
+
+Example output format:
+
+<Think> 1. Semantic consistency: Image 1 (9/10) - ...; Image 2 (7/10) - ...
+
+2. Aesthetics: Image 2 (8/10) - ...; Image 1 (8/10) - ...
+
+3. Authenticity: Image 1 (8/10) - ...; Image 2 (5/10) - ...  
+[Additional dimensions if any]: Image 2 (8/10) - ...; Image 1 (6/10) - ...
+
+Total score:
+
+Image 1:  $9 + 8 + 8 + 6 = 31$
+
+Image 2:  $7 + 8 + 5 + 8 = 28$
+
+</think>
+
+<answer>Image 1 is better</answer>
+
+Note: In the example above, scores and the final answer are placeholders meant only to demonstrate the format. Your actual evaluation should be based on the quality of two given images.
+
+Your task is provided as follows:
+
+Text Caption: {{prompt}}
+
+# 1. Semantic consistency:
+
+- Image 1 (9/10): The image shows a boy eating ramen in a restaurant setting, which aligns well with the caption. The presence of a bowl of ramen and the restaurant environment are clear.  
+- Image 2 (8/10): This image also depicts a boy eating ramen in a restaurant. However, the focus on the boy's face is more intense, slightly reducing the emphasis on the restaurant setting.
+
+# 2. Aesthetics:
+
+- Image 1 (8/10): The composition is balanced, with warm lighting that enhances the dining atmosphere. The focus on the boy and the ramen is clear and visually appealing.  
+- Image 2 (7/10): The image has a good composition, but the focus on the boy's face might distract from the overall aesthetic. The lighting is slightly harsher, affecting the ambiance.
+
+# 3. Authenticity:
+
+- Image 1 (9/10): The details in the image, such as the texture of the ramen and the restaurant background, are realistic and convincing.  
+- Image 2 (7/10): While the image is realistic, the focus on the boy's face and the lighting make it slightly less authentic in terms of capturing a typical restaurant dining experience.
+
+# Total score:
+
+Image 1:  $9 + 8 + 9 = 26$
+
+Image 2:  $8 + 7 + 7 = 22$
+
+</think>
+
+<answer>Image 1 is better</answer>
+
+# Video Generation
+
+The camera remains still, a woman with long blonde hair and wearing a green blouse turns on the faucet with her hands, and water flows out from the faucet, the background is a modern kitchen, bright natural lighting.
+
+![](images/6af2488a85e3100db4e2f22a36483f3a03f01f1fed211c972f1a4795e5a831a5.jpg)  
+Figure 6: More Qualitative Results of Video Generation CoT Reward Reasoning. Given a pair of videos and the corresponding caption, our model performs quality assessment across semantic consistency, temporal coherence, and authenticity through CoT reasoning.
+
+![](images/7f824ef20c8bce250863b3a8bbee34c51f941e1885f0233706a5d58588fd3ff2.jpg)
+
+![](images/c922f1f171f7487977a5d90dd56790e3505de79c2529afed6ed3940e8cc7f7bb.jpg)
+
+![](images/34a8874ae675a06d9bdbb17541fed7ed911cf73739ca3485118bac1298ecc55f.jpg)
+
+Given a caption and two videos generated based on this caption, please analyze in detail the two provided videos. Evaluate them on various dimensions such as semantic consistency (how closely the video content aligns with the caption), temporal coherence (smoothness and logical flow of motion across frames), authenticity (realism and attention to detail), and any other factors you deem relevant. For each evaluation dimension, provide a score between 1-10 for both videos (e.g., Video 1: 8/10, Video 2: 6/10) and provide a concise rationale for the score. Calculate the total score for each video by summing all dimension scores. Use a chain-of-thought process to detail your reasoning steps, and enclose all your detailed reasoning within <think> and </think> tags. Then, in the <answer> tag, output exactly one of the following strings: 'Video 1 is better' or 'Video 2 is better' based on the total scores. No additional text is allowed in the <answer> section.
+
+Example output format:
+
+1. Semantic consistency: Video 1 (9/10) - ...; Video 2 (7/10) - ...
+
+2. Temporal coherence: Video 1 (8/10) - ...; Video 2 (6/10) - ...
+
+3. Authenticity: Video 1 (7/10) - ...; Video 2 (5/10) -
+
+[Additional dimensions if any]: Video 2 (8/10) - ...; Video 1 (6/10) - ...
+
+Total score:
+
+Video 1:  $9 + 8 + 7 + 6 = 30$
+
+Video 2:  $7 + 6 + 5 + 8 = 26$
+
+<answer>Video 1 is better</answer>
+
+Note: In the example above, scores and the final answer are placeholders meant only to demonstrate the format. Your actual evaluation should be based on the quality of two given videos.
+
+Your task is provided as follows:
+
+Text Caption: {{prompt}}
+
+# 1. Semantic consistency:
+
+Video 1 (5/10): The video shows a woman in a green blouse, but her hair is dark, not blonde. The faucet is turned on, but the water flow is not visible. The setting is a modern kitchen, aligning with the caption.
+
+- Video 2 (9/10): The video features a woman with long blonde hair and a green blouse, matching the caption. The faucet is turned on, and water flows out, consistent with the description.
+
+# 2. Temporal coherence:
+
+- Video 1 (4/10): The woman's actions are not smooth, and the faucet turning on is not clearly depicted. The scene lacks a coherent flow.
+
+- Video 2 (9/10): The video shows a clear, smooth action of turning on the faucet with water flowing, maintaining a logical sequence.
+
+# 3. Authenticity:
+
+- Video 1 (6/10): The setting and attire are realistic, but the lack of visible water flow reduces authenticity.  
+- Video 2 (9/10): The video is realistic, with natural lighting and a believable scene of water flowing from the faucet.
+
+# Total score:
+
+Video 1:  $5 + 4 + 6 = 15$
+
+Video 2:  $9 + 9 + 9 = 27$
+
+</think>
+
+<answer>Video 2 is better</answer>
+
+![](images/e1b0bdef63a66343c41b021b66b9ab9298504eb9bbf700b36f5b35c36b448aa4.jpg)  
+Describe this image in detail.
+
+![](images/0523066fa2b5c688bde83ceb8fbd290db8e2069b53ec253e155d236e4b294f39.jpg)  
+Figure 7: More Qualitative Cases of Image Understanding CoT Reward Reasoning. Given an image, a query, and a pair of candidate answers, our model performs quality assessment across semantic accuracy, factual correctness, and clarity through CoT reasoning.
+
+![](images/fe6a3002275614c40551a4d0ca302e19a1305c573fa45ba2c650b52df6a3bb61.jpg)
+
+# Image Understanding
+
+Answer 1: The image features a table filled with several cakes, including a birthday cake and numerous cupcakes. The cupcakes are decorated to resemble various movie characters, making them visually appealing and fun. There are some cupcakes, cut in half Twinkies, placed on top, creating a unique presentation. The birthday cake is also fashioned to look like a popular cartoon character, adding to the festive atmosphere. In The dining table is in the center of the room, with a refrigerator visible in the background. Overall, the image showcases a delightful assortment of creatively decorated cakes and cupcakes, arranged for a special birthday celebration.
+
+Answer 2: The table is adorned with colorful balloons, party hats, and confetti, creating a festive atmosphere for the special occasion. Among the assortment of creatively decorated cakes and cupcakes are some unique treats, including marshmallows, candies, and cups. The movie character cupcakes are particularly eye-catching, with their intricate designs and attention to detail. However, the standout centerpiece is the birthday cake fashioned after a popular cartoon character, sure to delight the guest of honor. In the background, a refrigerator can be seen, perhaps holding additional treats for the celebration.
+
+![](images/bbad9a7b136c8814390f144258530060158e6a40816a2201d2592067f90abd6b.jpg)
+
+Given a question and a reference image, please analyze in detail the two provided answers (Answer 1 and Answer 2).
+
+Evaluate them based on the following three core dimensions:
+
+1. Semantic accuracy: How well the answer reflects the visual content of the image
+
+2. Correctness: Whether the answer is logically and factually correct
+
+3. Clarity: Whether the answer is clearly and fluently expressed
+
+You may also consider additional dimensions if you find them relevant (e.g., reasoning ability, attention to detail, multimodal grounding, etc.).
+
+For each dimension, provide a score from 1 to 10 for both answers, and briefly explain your reasoning. Then, compute the total score for each answer by explicitly adding the scores for all dimensions and showing the full calculation. Enclose your full reasoning within <think> and </think> tags. Then, in the <answer> tag, output exactly one of the following: 'Answer 1 is better' or 'Answer 2 is better'. No other text is allowed in the <answer> section.
+
+Example format
+
+# <think>
+
+1. Semantic accuracy: Answer 1 (9/10) - ...; Answer 2 (7/10) - ...
+
+2. Correctness: Answer 1 (8/10) - ...; Answer 2 (7/10) - ...
+
+3. Clarity: Answer 1 (9/10) - ...; Answer 2 (8/10) -
+
+[Additional dimensions if any]: Answer 1 (6/10) - ...; Answer 2 (7/10) - ...
+
+Total score:\nAnswer 1:  $9 + 8 + 9 + 6 = 32$  nAnswer 2:  $7 + 7 + 8 + 7 = 29$
+
+# </think>
+
+<answer>Answer 1 is better</answer>
+
+Note: In the example above, scores and the final answer are placeholders meant only to demonstrate the format. Your actual evaluation should be based on the quality of two given answers.
+
+Your task is provided as follows:
+
+Question: {{Query}}\nAnswer 1: {{R1}}\nAnswer 2: {{R2}"}
+
+![](images/a2b452d7aa07d7ea5ae5a49fc3b984a429bcc159a41be0c745d1344857992639.jpg)
+
+# <think>
+
+# 1. Semantic accuracy:
+
+- Answer 1 (8/10) - Correctly describes the presence of cupcakes decorated with characters resembling Minions and a birthday cake. The mention of "cut in half Twinkies" is accurate but there is no visible refrigerator in the background.
+
+- Answer 2 (6/10) - Describes a festive scene with balloons, party hats, and confetti, none of which are visible in the image. Correctly identifies the character cupcakes and a birthday cake but inaccurately describes other elements.
+
+# 2. Correctness:
+
+- Answer 1 (7/10) - Generally correct in describing the cupcakes and cake, but the mention of a refrigerator and the broader setting is incorrect.
+
+- Answer 2 (5/10) - Incorrectly adds elements like balloons and party hats that are not present, affecting overall correctness.
+
+# 3. Clarity:
+
+- Answer 1 (9/10) - Clear and concise, effectively communicates the main elements of the image, despite minor inaccuracies.  
+- Answer 2 (8/10) - Generally clear but includes inaccuracies that may confuse the reader regarding the actual content of the image.
+
+# Total score:
+
+Answer 1:  $8 + 7 + 9 = 24$
+
+Answer 2:  $6 + 5 + 8 = 19$
+
+</think>
+
+<answer>Answer 1 is better</answer>
+
+![](images/34085c0cbf192ac683bf6b4a76dde57fe7bbe8d69ff332ff84a450eb06507821.jpg)  
+Figure 8: More Qualitative Cases of Video Understanding CoT Reward Reasoning. Given a video, a query, and a pair of candidate answers, our model performs quality assessment across semantic accuracy, factual correctness, and clarity through CoT reasoning.
+
+![](images/2f6d9993be9c5a07661c4d51ace5fcff94a05fc339395cfdd8f8be8bdfc21498.jpg)
+
+# Video Understanding
+
+Answer 1: Fireworks add a spectacular visual element to the lion dance performance. They enhance the festive atmosphere, making the event more exciting and eye - catching for the large crowd gathered. The bright bursts of fireworks also symbolize good luck and prosperity, which aligns with the cultural significance of lion dances in many celebrations.
+
+Answer 2: Fireworks play a multi - faceted role in the lion dance performance. Visually, they create a dazzling backdrop that heightens the drama and grandeur of the lion's movements, captivating the audience's attention. Culturally, fireworks are deeply rooted in traditions of warding off evil spirits and welcoming good fortune, just as the lion dance itself is believed to bring luck and drive away bad omens. Their combination intensifies the symbolic power of the performance, making it a more profound and memorable cultural experience for all present. This answer is more in - depth, covering both the visual and cultural aspects in greater detail and showing a better understanding of the cultural significance and the synergy between the two elements.
+
+![](images/ced9429fca5f3d4d8c34a3123dbe585bc5ce3cf0b69b9d71a92db6890f4f8c67.jpg)
+
+Given a question and a reference video, please evaluate the two provided answers (Answer 1 and Answer 2). Judge them based on the following key dimensions:
+
+1. Semantic accuracy: Does the answer align with the visual and temporal content in the video?  
+2. Correctness: Is the answer factually and logically correct?  
+3. Clarity: Is the answer expressed fluently, clearly, and coherently?  
+You are encouraged to consider any additional dimensions if relevant (e.g., temporal reasoning, causal understanding, visual detail, emotional perception, etc.).
+
+For each dimension, assign a score from 1 to 10 for both answers and explain briefly. Then, compute and explicitly show the total score as an addition of all dimension scores. Wrap your full reasoning in <think> tags. In the <answer> tag, output exactly one of the following: 'Answer 1 is better' or 'Answer 2 is better'. No additional commentary is allowed in the <answer> section.
+
+Example format:
+
+<
+
+1. Semantic accuracy: Answer 1 (8/10) - ...; Answer 2 (9/10) - ...
+
+2. Correctness: Answer 1 (7/10) - ...; Answer 2 (6/10) - ...
+
+3. Clarity: Answer 1 (9/10) - ...; Answer 2 (8/10) - ...
+
+[Additional dimensions if any]: Answer 1 (7/10) - ...; Answer 2 (6/10) - ...
+
+Total score: \Answer 1:  $8 + 7 + 9 + 7 = 31$  \Answer 2:  $9 + 6 + 8 + 6 = 29$
+
+</think>
+
+<answer>Answer 1 is better</answer>
+
+Your task is provided as follows:
+
+Question:  $\{\text{Query}\} \backslash \text{nAnswer 1: \{R1\} \backslash nAnswer 2: \{R2\} \backslash n}$
+
+![](images/a728dafed09f732345c44d499763761aa127724bf5f4149a30d85b1e8cf10058.jpg)
+
+# <think:
+
+# 1. Semantic accuracy:
+
+- Answer 1 (8/10): The answer accurately describes the visual and cultural significance of fireworks in the lion dance performance. It mentions the enhancement of the festive atmosphere and the symbolic meaning of good luck and prosperity.
+
+- Answer 2 (9/10): This answer provides a more detailed explanation, covering both the visual impact and cultural significance. It elaborates on the symbolic power of fireworks in relation to warding off evil spirits and welcoming good fortune.
+
+# 2. Correctness:
+
+- Answer 1 (9/10): The explanation is factually correct, aligning well with the cultural context of lion dances and the role of fireworks.
+
+- Answer 2 (10/10): This answer is also factually correct and provides a comprehensive understanding of the cultural and symbolic aspects of fireworks in the context of the lion dance.
+
+# 3. Clarity:
+
+- Answer 1 (8/10): The answer is clear and well-structured, effectively conveying the importance of fireworks in the performance.  
+- Answer 2 (9/10): The answer is very clear and detailed, offering a thorough explanation that enhances understanding.
+
+# Total score:
+
+Answer 1:  $8 + 9 + 8 = 25$
+
+Answer 2:  $9 + 10 + 9 = 28$
+
+</think
+
+<answer>Answer 2 is better</answer>
\ No newline at end of file
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+# Unified Reinforcement and Imitation Learning for Vision-Language Models
+
+Byung-Kwan Lee†  
+NVIDIA, KAIST
+
+byungkwanl@nvidia.com
+
+Ryo Hachiuma  
+NVIDIA
+
+rhachiuma@nvidia.com
+
+Yong Man Ro KAIST
+
+ymro@kaist.ac.kr
+
+Yu-Chiang Frank Wang
+
+NVIDIA, National Taiwan University
+
+frankwang@nvidia.com
+
+Yueh-Hua Wu
+
+NVIDIA
+
+krisw@nvidia.com
+
+# Abstract
+
+Vision-Language Models (VLMs) have achieved remarkable progress, yet their large scale often renders them impractical for resource-constrained environments. This paper introduces Unified Reinforcement and Imitation Learning (RIL), a novel and efficient training algorithm designed to create powerful, lightweight VLMs. RIL distinctively combines the strengths of reinforcement learning with adversarial imitation learning. This enables smaller student VLMs not only to mimic the sophisticated text generation of large teacher models but also to systematically improve their generative capabilities through reinforcement signals. Key to our imitation framework is an LLM-based discriminator that adeptly distinguishes between student and teacher outputs, complemented by guidance from multiple large teacher VLMs to ensure diverse learning. This unified learning strategy, leveraging both reinforcement and imitation, empowers student models to achieve significant performance gains, making them competitive with leading closed-source VLMs. Extensive experiments on diverse vision-language benchmarks demonstrate that RIL significantly narrows the performance gap with state-of-the-art open- and closed-source VLMs and, in several instances, surpasses them.
+
+![](images/d35fa69a876e598fb61eedeeba83c06ba11eba07eb013a35fe3ed3073860bfdc.jpg)  
+Performance Improvements  $(\%)$  across Evaluation Benchmarks  
+Figure 1: Showing the performance improvements  $(\%)$  of Qwen2.5-VL-7B [1] across vision-language evaluation benchmarks for AI2D [2], MathVista [3], MM-Vet [4], MMMU [5], BLINK [6], and the average scores for 14 evaluation benchmarks used in Table 1. Note that, we conduct RL on GRPO [7] and advanced GRPO, Dr.GRPO [8], with only answer rewards from LLM-as-a-Judge [9] (see Algorithm 1), and we present RIL based on similarity rewards from single or multi large teacher VLMs and simultaneously answer rewards (see Algorithm 2).
+
+![](images/7297799c593f6450fc1d8055a89fb19845f0cad44a4761bc301442d6ac0933d3.jpg)  
+Figure 2: Comparing RIL-applied VLMs based on multi large VLMs with diverse open- and closed-source VLMs, under average performance of numerous vision-language evaluation benchmarks: AI2D [2], ChartQA [19], MathVista [3], MMB [20], MM-Vet [4], MMMU [5], MMMU-Pro [21], MMStar [22], BLINK [6], SEED [23], SEED2+ [24], and RealWorldQA (RWQA).
+
+# 1 Introduction
+
+The pursuit of artificial general intelligence (AGI) has gained momentum, with much of the initial progress driven by instruction-tuned large language models (LLMs) [10-15]. Vision-Language Models (VLMs) [16, 1, 17, 18] represent a significant step forward, extending the LLM framework to integrate visual data and thereby achieve multimodal understanding. Their ability to generate insightful, visually-grounded textual responses that increasingly emulate human-like comprehension has made VLMs a focal point of considerable attention. Recognizing their transformative potential, numerous organizations are heavily investing to push the boundaries of what VLMs can achieve.
+
+Historically, the enhancements in vision-language performance have primarily relied on traditional strategies such as scaling up model sizes [15, 25, 26] and expanding visual instruction tuning datasets [27-30, 18]. More recent advancements have explored architectural modifications—for instance, directly altering model structures [31, 32] or integrating auxiliary vision [33, 27, 34] and reasoning modules [35]. Alongside these efforts, the 'think-answer' paradigm [36-42] has gained prominence following DeepSeek-R1's [7] application of reinforcement learning (RL) via GRPO [43]. However, such architectural changes and the verbose 'think' responses preceding answers can significantly increase inference latency and computational memory requirements. These challenges render many powerful VLMs impractical for resource-limited settings like smartphones and augmented reality (AR) devices. As a result, there is a growing imperative within the research community to develop VLM designs that achieve a compelling balance between strong vision-language capabilities, low inference latency, and a lightweight model footprint.
+
+To overcome these challenges, we introduce an approach for developing high-performing, efficient VLMs that avoids architectural modifications and the need for lengthy 'think' responses. Our method, termed Unified Reinforcement and Imitation Learning (RIL), is an efficient training algorithm integrating principles from both GRPO [43] and the GAIL framework [44]. The central objective of RIL is to enable smaller VLMs (e.g., 7B models) to effectively mimic the text generation style of significantly larger VLMs (e.g., 72B models). Prior works [45, 46] have argued that natural language response-based distillation is more effective than high-dimensional feature distillation, emphasizing the importance of utilizing the language head—referred to as the verbalization effect. Motivated by this insight, we began by leveraging the similarity in natural language responses between teacher and student models. Drawing inspiration from how models articulate responses differently, RIL employs an LLM-based discriminator trained to distinguish between the text responses generated by the student (small VLMs) and the teacher (large VLMs). This discriminator's output provides a similarity reward signal, guiding the student VLMs, which act as a generator, to produce responses increasingly akin to those of the teacher. To maintain training stability and prevent the generator or discriminator from overpowering the other, a common issue known as the balance problem [47, 48], the discriminator's architecture initially mirrors that of the generator.
+
+However, the practical implementation of RIL for VLMs entails several specific challenges. Firstly, relying on continuous discriminator scores (ranging from zero to one) to assess similarity to large VLM outputs can introduce ambiguity into the learning signal. To ensure a clearer and more decisive reward, and drawing inspiration from prior works that binarize answer rewards [7, 43], we convert the discriminator's similarity score into a binary value. Secondly, the discriminator, by design, focuses on stylistic similarity and does not inherently verify factual correctness against ground truth
+
+SYSTEM: You are a discriminator assistant.
+
+You are provided with Response in <response> </response> tag.
+
+Your task is to determine whether the Response was generated by teacher or student model.
+
+Return 0 if the Response was generated by the teacher model.
+
+Return 1 if the Response was generated by the student model.
+
+USER: <response> {response} </response>
+
+Determine the source of the Response.
+
+Provide your answer as one of the following: 0 (Teacher model) 1 (Student model)
+
+Figure 3: Input prompt for Discriminator
+
+answers, which could otherwise lead to performance degradation. We mitigate this by incorporating an LLM-as-a-Judge [9] to provide a separate binary reward signal based on the accuracy of the generated response. A third challenge arises during the GRPO [43] phase: if the student VLMs, due to their more limited knowledge, fail to generate any correct responses for a given task, they struggle to identify effective learning pathways. To provide clearer guidance, especially when the student VLM lacks sufficient domain-specific knowledge compared to larger models, we incorporate text responses from both student and teacher VLMs during this GRPO training step. This strategy not only stabilizes RIL training but also creates opportunities for the student VLM to potentially surpass the performance of larger teachers.
+
+Several key aspects of RIL contribute to its effectiveness. Notably, we observe that sourcing text responses from multiple large teacher VLMs significantly boosts student performance beyond what a single teacher can achieve, owing to the richer diversity of responses and the consequently more robust discriminator. This enriched learning environment is further enhanced during GRPO [43] step: by incorporating text responses from both student and teacher VLMs, we provide the student with clearer exemplars for reaching correct answers. RIL also demonstrates particular synergy with distillation-based VLMs; these models show marked performance improvements because their inherent feature alignment through distillation complements RIL's objectives. Beyond these learning dynamics, a crucial practical advantage of RIL is its inference efficiency: trained models do not require an explicit 'think' phase before generating answers, thereby maintaining the same inference speed as the original student VLM.
+
+Furthermore, RIL's reliance on LLM-as-a-Judge [9] for accuracy evaluation offers substantial advantages in applicability. Unlike methods such as DeepSeek-R1 [7] and its variants [39, 49, 41, 37], which often use conventional answer parsing for reward computation and are thus limited to domains with predefined metrics (e.g., math or image grounding), RIL transcends these limitations. For example, conventional parsing struggles with open-ended general visual questions, such as "How to cook this dish?" based on a food image, or "Summarize this movie" from its poster. In addition, the parsing poses a challenge to flexibly compare different types of answers, if a response is "The answer is twenty percent" but its ground truth is "20". The versatile evaluation power of LLM-as-a-Judge [9] enables RIL's successful application to a broad spectrum of visual question answering tasks, fostering strong performance in areas like recognition, OCR, common-sense reasoning, mathematical problem-solving, and chart/document understanding. Finally, because RIL operates purely on generated text responses, it is agnostic to the specific image embedding strategies or language tokenizers used by the student or teacher VLMs, ensuring broad compatibility.
+
+To validate our approach, we conduct extensive experiments across diverse vision-language benchmarks. The results compellingly demonstrate that RIL not only significantly narrows the performance gap to state-of-the-art open- and closed-source VLMs but, in several instances, surpasses them. The primary contributions of this work are:
+
+- Unified Reinforcement and Imitation Learning (RIL): We introduce RIL, an efficient and novel training framework for VLMs. It empowers smaller models to emulate the text generation patterns of significantly larger VLMs, leading to substantial enhancements in their overall performance.  
+- Broad Applicability and Flexibility: RIL exhibits wide applicability, functioning effectively with diverse VLMs irrespective of their underlying image embedding strategies or language tokenizers. Furthermore, it preserves the original inference speed by avoiding lengthy intermediate reasoning steps and capably addresses general visual question answering tasks through its integration with LLM-as-a-Judge.
+
+![](images/a6e7764a0d2c529bb6bd709a43a0827438ef42a9eaf175054c3809e6948de49d.jpg)  
+Figure 4: Input prompt for LLM-as-a-Judge [9]
+
+- Extensive Validation and Significant Performance Gains: Our comprehensive experimental results consistently show that RIL delivers substantial performance improvements across a variety of vision-language benchmarks. These gains position RIL-trained models as highly competitive with, and sometimes superior to, existing state-of-the-art VLMs.
+
+# 2 Related Works
+
+Evolution of Vision-Language Models (VLMs). Conventional remedies for advancing VLMs have largely centered on scaling up model sizes and expanding datasets to push performance boundaries. Consequently, promising VLMs have emerged, for example, closed-source VLMs: GPT-4o [50], Gemini [51], and Claude-3.5 Sonnet [52], and open-source VLMs: Molmo-72B [53], LLaVA-OneVision-72B [28], NVLM-72B [54], Qwen2.5-VL-72B [1], and InternVL3-78B [18] have followed this paradigm, incorporating large-scale language models such as Qwen2/Qwen2.5-72B [13]. However, the sheer size and computational demands of these models present significant barriers to deployment in resource-constrained environments such as mobile and embedded devices. To address these challenges, CoLLaVO [33] and MoAI [34] utilize computer vision models directly for visual capability, and Mini-Gemini [27], MoVA [55], and Eagle [56] employ multiple vision encoders such as CLIP [57], ConvNeXt [58], DINO-v2 [59], and SAM [60]. In parallel, Meteor [35] explores the efficient way of learning complex reasoning abilities, and TroL [31] and Phantom [32] investigate propagation modification for how we can embed vision-language knowledge as much as possible despite using the same architectures. More recently, since DeepSeek-R1 [7] introduces think-answer process and shows its dramatic performance improvements by using reinforcement learning (RL) such as GRPO [43], many variant models for VLMs has presented LMM-R1 [61], Vision-R1 [37], R1-V [42], OpenVLThinker [62], R1-OneVision [63], R1-Zero [64], and MM-Eureka [65]. They employ think-answer process to VLMs in specific areas such as math problems and object counting tasks. In addition, NoisyRollout [40] injects noise into clean images for distortion and train VLMs to strongly improve visual robustness, thereby understanding visual properties more than before.
+
+Imitation Learning (IL). It originates from the concept of learning expert behavior in robotics. Generative adversarial imitation learning (GAIL) [44] is a foundational framework in this domain. It employs a generator and a discriminator where the generator (e.g., small student VLMs) attempts to replicate expert behavior (e.g., large teacher VLMs) by producing outputs such as actions or trajectories (e.g., text responses) that are indistinguishable from those generated by experts. Meanwhile, the discriminator aims to distinguish between the generator and the expert. By framing imitation learning (IL) as a minimax game, GAIL [44] leverages adversarial training to align the behavior of the generator with that of the expert without explicitly defining a reward function. Instead, GAIL [44] only relies on discriminator scores to evaluate how the generator similarly produces outputs of an expert. This approach has demonstrated strong performance in complex, high-dimensional tasks by directly learning from expert data, making it a widely adopted strategy in IL [66]. Based on its effectiveness, we integrate reinforcement and imitation learning (RIL) for VLMs with four key modifications: (1) combining GRPO [43] and GAIL [44] framework with explicitly reward design, (2) making the output scores of the discriminator binary to stabilize training, (3) incorporating LLM-as-a-Judge [9] to evaluate answer rewards that assess whether the generated text responses from
+
+Table 1: Comparing the performances by using answer rewards from LLM-as-a-Judge [9] with purely RL on GRPO [7] and advanced GRPO [8], and by using similarity and answer rewards from RIL on GAIL [44], under numerous evaluation benchmarks: AI2D [2], ChartQA [19], MathVista [3], MMB/MMBCN [20], MM-Vet [4], MM-Vet-v2 [67], MMMU [5], MMMU-Pro [21], MMStar [22], BLINK [6], SEED [23], SEED2+ [24], and RealWorldQA (RWQA). Note that, for RIL, we set large VLMs as the largest version of same small VLMs, such as Qwen2.5-VL-7B  $\leftarrow$  Qwen2.5-VL-72B and InternVL3-8B  $\leftarrow$  InternVL3-78B.
+
+<table><tr><td>VLMs</td><td colspan="2">AI2D ChartQA</td><td colspan="2">MathVista</td><td colspan="2">MMB</td><td colspan="2">\(MMBC^{CN}\)</td><td colspan="2">MM-Vet MM-Vet-v2</td><td colspan="2">MMMU</td><td colspan="2">MMMU-Pro</td><td colspan="2">MMStar BLINK</td><td colspan="2">SEED SEED2+</td><td>RWQA</td></tr><tr><td>Qwen2.5-VL-7B</td><td>83.9</td><td>87.3</td><td>67.8</td><td>83.5</td><td>83.4</td><td>71.8</td><td>63.7</td><td>55.0</td><td>38.3</td><td>63.9</td><td>56.4</td><td>77.0</td><td>70.4</td><td>68.5</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>w. RL (GRPO)</td><td>84.1</td><td>88.8</td><td>69.0</td><td>83.8</td><td>83.9</td><td>72.8</td><td>63.9</td><td>56.2</td><td>40.3</td><td>65.2</td><td>58.9</td><td>77.4</td><td>70.6</td><td>69.4</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>w. RL (Dr.GRPO)</td><td>84.5</td><td>90.0</td><td>69.5</td><td>84.3</td><td>84.4</td><td>73.5</td><td>64.2</td><td>57.2</td><td>41.8</td><td>66.3</td><td>60.7</td><td>78.0</td><td>70.9</td><td>70.3</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>w. RIL (Dr.GRPO + GAIL)</td><td>86.7</td><td>95.4</td><td>74.5</td><td>86.8</td><td>87.2</td><td>77.3</td><td>66.1</td><td>61.8</td><td>48.2</td><td>71.1</td><td>68.5</td><td>80.7</td><td>73.0</td><td>74.2</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Qwen2.5-VL-3B</td><td>81.6</td><td>84.0</td><td>61.2</td><td>79.1</td><td>78.1</td><td>61.8</td><td>57.6</td><td>51.2</td><td>31.6</td><td>55.9</td><td>47.6</td><td>74.0</td><td>67.6</td><td>65.4</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>w. RL (GRPO)</td><td>81.9</td><td>87.8</td><td>61.8</td><td>80.3</td><td>79.7</td><td>62.8</td><td>55.4</td><td>52.0</td><td>33.7</td><td>57.7</td><td>52.6</td><td>75.7</td><td>68.6</td><td>66.4</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>w. RL (Dr.GRPO)</td><td>82.4</td><td>89.7</td><td>62.5</td><td>81.8</td><td>81.2</td><td>63.8</td><td>55.9</td><td>52.5</td><td>34.3</td><td>59.1</td><td>53.7</td><td>76.5</td><td>69.2</td><td>67.0</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>w. RIL (Dr.GRPO + GAIL)</td><td>83.0</td><td>95.4</td><td>65.2</td><td>84.8</td><td>84.2</td><td>67.4</td><td>57.6</td><td>53.7</td><td>36.8</td><td>61.2</td><td>55.5</td><td>78.7</td><td>70.3</td><td>67.7</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>InternVL3-8B</td><td>85.2</td><td>86.6</td><td>71.6</td><td>83.4</td><td>82.2</td><td>78.5</td><td>66.3</td><td>62.7</td><td>41.3</td><td>68.2</td><td>55.5</td><td>77.1</td><td>69.7</td><td>70.8</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>w. RL (GRPO)</td><td>85.9</td><td>89.6</td><td>72.3</td><td>84.6</td><td>84.2</td><td>78.7</td><td>66.4</td><td>63.8</td><td>42.5</td><td>70.0</td><td>57.1</td><td>77.9</td><td>70.6</td><td>71.3</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>w. RL (Dr.GRPO)</td><td>86.3</td><td>91.2</td><td>72.9</td><td>85.2</td><td>85.3</td><td>79.0</td><td>66.5</td><td>64.9</td><td>42.8</td><td>70.7</td><td>57.5</td><td>78.1</td><td>71.1</td><td>72.0</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>w. RIL (Dr.GRPO + GAIL)</td><td>87.4</td><td>95.5</td><td>74.1</td><td>88.7</td><td>89.3</td><td>78.4</td><td>66.5</td><td>66.8</td><td>44.8</td><td>75.4</td><td>60.1</td><td>80.6</td><td>73.0</td><td>73.3</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>InternVL3-2B</td><td>78.7</td><td>80.2</td><td>57.0</td><td>81.1</td><td>78.4</td><td>62.2</td><td>53.9</td><td>48.6</td><td>24.9</td><td>60.7</td><td>47.0</td><td>75.0</td><td>64.6</td><td>64.3</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>w. RL (GRPO)</td><td>79.1</td><td>86.9</td><td>57.8</td><td>82.4</td><td>80.0</td><td>61.8</td><td>53.6</td><td>49.6</td><td>25.7</td><td>61.5</td><td>47.3</td><td>75.8</td><td>64.9</td><td>64.6</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>w. RL (Dr.GRPO)</td><td>79.8</td><td>90.3</td><td>58.5</td><td>83.2</td><td>81.3</td><td>62.1</td><td>53.8</td><td>50.5</td><td>26.4</td><td>62.4</td><td>47.9</td><td>76.5</td><td>65.4</td><td>65.2</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>w. RIL (Dr.GRPO + GAIL)</td><td>81.1</td><td>93.6</td><td>63.0</td><td>85.6</td><td>83.2</td><td>61.5</td><td>54.2</td><td>52.9</td><td>27.5</td><td>63.2</td><td>48.3</td><td>78.2</td><td>66.7</td><td>66.8</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>InternVL3-1B</td><td>69.4</td><td>75.3</td><td>45.8</td><td>72.6</td><td>67.9</td><td>58.7</td><td>47.5</td><td>43.4</td><td>17.5</td><td>51.5</td><td>42.9</td><td>71.2</td><td>58.2</td><td>58.2</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>w. RL (GRPO)</td><td>69.6</td><td>83.0</td><td>46.7</td><td>73.8</td><td>69.4</td><td>57.7</td><td>47.0</td><td>42.8</td><td>17.2</td><td>52.5</td><td>42.7</td><td>71.5</td><td>58.5</td><td>58.5</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>w. RL (Dr.GRPO)</td><td>69.9</td><td>86.7</td><td>47.6</td><td>74.9</td><td>70.5</td><td>57.5</td><td>47.0</td><td>42.8</td><td>17.3</td><td>53.4</td><td>42.9</td><td>71.9</td><td>59.4</td><td>59.4</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>w. RIL (Dr.GRPO + GAIL)</td><td>72.0</td><td>93.6</td><td>51.6</td><td>79.6</td><td>75.7</td><td>56.9</td><td>47.4</td><td>43.1</td><td>18.3</td><td>57.4</td><td>44.1</td><td>74.2</td><td>61.4</td><td>61.4</td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+student VLMs are correct, and (4) utilizing the generated text responses from teacher VLMs when GRPO [43] updates student VLMs, serving as a direct guidance to reach the correct text responses. These approach not only stabilizes RIL training but also provides a mechanism for student VLMs to potentially exceed the performance of teacher VLMs. Moreover, RIL do not require an explicit think-answer process, thereby maintaining the same inference, and it is independent of any particular image embedding methods or language tokenizers used in the student or the teacher VLMs.
+
+# 3 Unified Reinforcement and Imitation Learning for VLMs
+
+# 3.1 Core Model Components for RIL
+
+Our RIL framework utilizes several key model components. For the student models—which are the VLMs we aim to enhance—we employ recently released architectures such as Qwen2.5-VL (3B and 7B variants) [1] and InternVL3 (1B, 2B, and 8B variants) [18]. Starting from these pre-trained checkpoints, RIL directly updates their parameters with the goal of developing efficient, high-performing VLMs capable of rivaling leading closed-source systems like GPT-4o [50], Gemini [51], and Claude-3.5 Sonnet [52]. For the teacher models, which provide the target behavior, we select powerful large teacher VLMs recognized for their strong performance on benchmarks like VLM Leaderboard [68], specifically Qwen2.5-VL-72B [1] and InternVL3-78B [18]. The student VLMs are trained to mimic the text generation patterns and response styles of these teacher models. Central to the imitation learning aspect is a discriminator, designed to assess the similarity between student-generated responses and those from the teacher VLMs. To promote training stability and mitigate the balance problem common in adversarial setups [47, 48]—where one component might overpower the other—the discriminator's architecture and initial parameters deliberately mirror those of the student VLMs. Lastly, to evaluate the factual correctness of the generated text, we utilize an LLM-as-a-Judge [9], specifically Qwen2.5-32B [13], following the methodology of Zheng et al. [9]. This model is not trained but is used solely to determine whether the predicted response accurately reflects the meaning of the ground truth.
+
+The subsequent sections will first elaborate on the discriminator's architecture and pre-training (Section 3.2), followed by a detailed exposition of the RIL algorithm and its reward design (Section 3.3).
+
+# 3.2 Discriminator Architecture and Pre-training
+
+With the student (small) VLMs serving as the generator in our framework, the effective pre-training of the discriminator is crucial. This initial training phase equips the discriminator  $(D_{\phi})$  with the ability to reliably differentiate between text responses originating from student VLMs versus teacher
+
+Table 2: Showing the effectiveness of employing multiple large VLMs more than a single VLM, under the numerous evaluation benchmarks equally used in Table 1.  
+
+<table><tr><td>VLMs</td><td colspan="2">AI2D ChartQA</td><td colspan="2">MathVista</td><td colspan="2">MMB</td><td colspan="2">MMBCN</td><td colspan="2">MM-Vet</td><td colspan="2">MM-Met-v2</td><td colspan="2">MMU</td><td colspan="2">MMU-Pro</td><td colspan="2">MMStar</td><td colspan="2">BLINK</td><td colspan="2">SEED</td><td colspan="2">SEED2+</td><td>RWQA</td></tr><tr><td>Qwen2.5-VL-7B</td><td>83.9</td><td>87.3</td><td>67.8</td><td>83.5</td><td>83.4</td><td>71.8</td><td>63.7</td><td>55.0</td><td>38.3</td><td>63.9</td><td>56.4</td><td>77.0</td><td>70.4</td><td>68.5</td><td>70.4</td><td>70.4</td><td>70.4</td><td>70.4</td><td>70.4</td><td>70.4</td><td>68.5</td><td>68.5</td><td>68.5</td><td>68.5</td><td>68.5</td></tr><tr><td>w. RIL (Qwen2.5-VL-72B)</td><td>86.7</td><td>95.4</td><td>74.5</td><td>86.8</td><td>87.2</td><td>77.3</td><td>66.1</td><td>61.8</td><td>48.2</td><td>71.1</td><td>68.5</td><td>80.8</td><td>74.2</td><td>70.4</td><td>70.4</td><td>70.4</td><td>70.4</td><td>70.4</td><td>70.4</td><td>70.4</td><td>68.5</td><td>68.5</td><td>68.5</td><td>68.5</td><td>68.5</td></tr><tr><td>w. RIL (InternVL3-78B)</td><td>86.8</td><td>95.5</td><td>74.6</td><td>86.7</td><td>87.1</td><td>75.8</td><td>66.0</td><td>60.9</td><td>47.1</td><td>71.1</td><td>68.1</td><td>80.8</td><td>72.7</td><td>70.4</td><td>70.4</td><td>70.4</td><td>70.4</td><td>70.4</td><td>70.4</td><td>70.4</td><td>68.5</td><td>68.5</td><td>68.5</td><td>68.5</td><td>68.5</td></tr><tr><td>w. RIL (Both)</td><td>86.1</td><td>95.6</td><td>79.7</td><td>86.3</td><td>86.5</td><td>80.4</td><td>71.1</td><td>65.7</td><td>48.5</td><td>71.1</td><td>70.0</td><td>80.5</td><td>72.8</td><td>70.4</td><td>70.4</td><td>70.4</td><td>70.4</td><td>70.4</td><td>70.4</td><td>70.4</td><td>68.5</td><td>68.5</td><td>68.5</td><td>68.5</td><td>68.5</td></tr><tr><td>Qwen2.5-VL-3B</td><td>81.6</td><td>84.0</td><td>61.2</td><td>79.1</td><td>78.1</td><td>61.8</td><td>57.6</td><td>51.2</td><td>31.6</td><td>55.9</td><td>47.6</td><td>74.0</td><td>67.6</td><td>66.5</td><td>61.2</td><td>55.5</td><td>78.7</td><td>70.3</td><td>67.6</td><td>65.4</td><td>61.2</td><td>68.5</td><td>68.5</td><td>68.5</td><td>65.4</td></tr><tr><td>w. RIL (Qwen2.5-VL-72B)</td><td>83.0</td><td>95.4</td><td>65.2</td><td>84.8</td><td>84.2</td><td>67.4</td><td>57.6</td><td>53.7</td><td>36.8</td><td>61.2</td><td>55.5</td><td>78.7</td><td>70.3</td><td>67.6</td><td>65.2</td><td>55.5</td><td>78.7</td><td>70.3</td><td>67.6</td><td>65.4</td><td>61.2</td><td>68.5</td><td>68.5</td><td>65.4</td><td></td></tr><tr><td>w. RIL (InternVL3-78B)</td><td>83.3</td><td>95.4</td><td>65.1</td><td>85.0</td><td>84.7</td><td>66.6</td><td>56.3</td><td>54.9</td><td>37.7</td><td>61.1</td><td>55.3</td><td>78.8</td><td>70.2</td><td>68.2</td><td>61.2</td><td>55.3</td><td>78.8</td><td>70.2</td><td>68.2</td><td>62.7</td><td>60.8</td><td>78.7</td><td>71.4</td><td>73.2</td><td></td></tr><tr><td>w. RIL (Both)</td><td>83.9</td><td>95.7</td><td>71.7</td><td>85.2</td><td>84.5</td><td>74.7</td><td>62.8</td><td>60.7</td><td>43.7</td><td>65.2</td><td>60.8</td><td>78.7</td><td>71.4</td><td>68.5</td><td>60.8</td><td>78.7</td><td>71.4</td><td>68.5</td><td>73.2</td><td>73.2</td><td>68.5</td><td>68.5</td><td>68.5</td><td>73.2</td><td></td></tr><tr><td>InternVL3-8B</td><td>85.2</td><td>86.6</td><td>71.6</td><td>83.4</td><td>82.2</td><td>78.5</td><td>66.3</td><td>62.7</td><td>41.3</td><td>68.2</td><td>55.5</td><td>77.1</td><td>69.7</td><td>70.4</td><td>68.5</td><td>71.4</td><td>69.7</td><td>70.8</td><td>70.4</td><td>68.5</td><td>68.5</td><td>68.5</td><td>68.5</td><td>70.8</td><td></td></tr><tr><td>w. RIL (Qwen2.5-VL-72B)</td><td>87.5</td><td>95.2</td><td>74.3</td><td>88.8</td><td>89.8</td><td>77.6</td><td>65.9</td><td>67.3</td><td>45.7</td><td>75.2</td><td>59.1</td><td>80.4</td><td>72.9</td><td>70.4</td><td>68.5</td><td>71.4</td><td>69.7</td><td>70.8</td><td>70.4</td><td>68.5</td><td>68.5</td><td>68.5</td><td>68.5</td><td>72.5</td><td></td></tr><tr><td>w. RIL (InternVL3-78B)</td><td>87.4</td><td>95.5</td><td>74.1</td><td>88.7</td><td>89.3</td><td>78.4</td><td>66.5</td><td>66.8</td><td>44.8</td><td>75.4</td><td>60.1</td><td>80.6</td><td>73.0</td><td>70.4</td><td>68.5</td><td>60.1</td><td>80.6</td><td>73.0</td><td>73.3</td><td>68.5</td><td>68.5</td><td>68.5</td><td>68.5</td><td>73.3</td><td></td></tr><tr><td>w. RIL (Both)</td><td>87.3</td><td>95.3</td><td>77.8</td><td>88.1</td><td>89.6</td><td>80.1</td><td>67.6</td><td>68.6</td><td>47.8</td><td>74.8</td><td>62.7</td><td>80.5</td><td>73.8</td><td>70.4</td><td>68.5</td><td>60.1</td><td>80.6</td><td>73.8</td><td>73.7</td><td>68.5</td><td>68.5</td><td>68.5</td><td>68.5</td><td>73.7</td><td></td></tr><tr><td>InternVL3-2B</td><td>78.7</td><td>80.2</td><td>57.0</td><td>81.1</td><td>78.4</td><td>62.2</td><td>53.9</td><td>48.6</td><td>24.9</td><td>60.7</td><td>47.0</td><td>75.0</td><td>64.6</td><td>64.3</td><td>60.7</td><td>47.0</td><td>75.0</td><td>64.6</td><td>64.3</td><td>64.3</td><td>64.3</td><td>64.3</td><td>64.3</td><td>64.3</td><td></td></tr><tr><td>w. RIL (Qwen2.5-VL-72B)</td><td>81.1</td><td>93.6</td><td>61.9</td><td>85.3</td><td>83.0</td><td>62.6</td><td>52.8</td><td>52.6</td><td>26.6</td><td>63.6</td><td>48.8</td><td>78.0</td><td>67.1</td><td>66.3</td><td>61.2</td><td>48.3</td><td>78.0</td><td>67.1</td><td>66.3</td><td>61.2</td><td>66.3</td><td>61.2</td><td>66.3</td><td>66.3</td><td></td></tr><tr><td>w. RIL (InternVL3-78B)</td><td>81.1</td><td>93.6</td><td>63.0</td><td>85.6</td><td>83.2</td><td>61.5</td><td>54.2</td><td>52.9</td><td>27.5</td><td>63.2</td><td>48.3</td><td>78.2</td><td>66.7</td><td>66.8</td><td>61.2</td><td>48.3</td><td>78.2</td><td>66.7</td><td>66.8</td><td>66.8</td><td>66.8</td><td>66.8</td><td>66.8</td><td>66.8</td><td></td></tr><tr><td>w. RIL (Both)</td><td>80.7</td><td>94.0</td><td>67.4</td><td>84.7</td><td>82.2</td><td>70.7</td><td>58.4</td><td>56.6</td><td>32.3</td><td>63.9</td><td>51.5</td><td>77.8</td><td>66.8</td><td>69.3</td><td>61.2</td><td>57.8</td><td>76.8</td><td>66.8</td><td>69.3</td><td>69.3</td><td>69.3</td><td>69.3</td><td>69.3</td><td>69.3</td><td></td></tr><tr><td>InternVL3-1B</td><td>69.4</td><td>75.3</td><td>45.8</td><td>72.6</td><td>67.9</td><td>58.7</td><td>47.5</td><td>43.4</td><td>17.5</td><td>51.5</td><td>42.9</td><td>71.2</td><td>58.2</td><td>58.2</td><td>52.9</td><td>42.9</td><td>71.2</td><td>58.2</td><td>58.2</td><td>58.2</td><td>58.2</td><td>58.2</td><td>58.2</td><td>58.2</td><td></td></tr><tr><td>w. RIL (Qwen2.5-VL-72B)</td><td>72.2</td><td>93.6</td><td>50.9</td><td>79.9</td><td>75.8</td><td>56.1</td><td>47.3</td><td>43.6</td><td>18.0</td><td>58.0</td><td>43.6</td><td>74.6</td><td>60.7</td><td>61.0</td><td>50.9</td><td>44.1</td><td>74.2</td><td>61.4</td><td>61.4</td><td>61.4</td><td>61.4</td><td>61.4</td><td>61.4</td><td>61.4</td><td></td></tr><tr><td>w. RIL (InternVL3-78B)</td><td>72.0</td><td>93.6</td><td>51.6</td><td>79.6</td><td>75.7</td><td>56.9</td><td>47.4</td><td>43.1</td><td>18.3</td><td>57.4</td><td>44.1</td><td>74.2</td><td>61.4</td><td>61.4</td><td>50.9</td><td>44.1</td><td>74.2</td><td>61.4</td><td>61.4</td><td>61.4</td><td>61.4</td><td>61.4</td><td>61.4</td><td>61.4</td><td></td></tr><tr><td>w. RIL (Both)</td><td>73.0</td><td>94.1</td><td>55.5</td><td>79.1</td><td>75.9</td><td>62.9</td><td>50.7</td><td>49.7</td><td>19.0</td><td>60.5</td><td>46.9</td><td>74.7</td><td>61.9</td><td>63.7</td><td>60.5</td><td>46.9</td><td>74.7</td><td>61.9</td><td>61.9</td><td>61.9</td><td>61.9</td><td>61.9</td><td>61.9</td><td>61.9</td><td></td></tr></table>
+
+(large) VLMs. Without adequate pre-training,  $D_{\phi}$  would yield unreliable scores, undermining its utility and potentially degrading overall model performance. To prepare for pre-training, we first collect a dataset of responses. For each given question  $q$ ,  $N$  text responses are generated from both the student VLMs, denoted as  $\{\mathbf{o}_i^{(\mathbf{s})}\}_{i=1}^N$ , and the teacher VLMs, denoted as  $\{\mathbf{o}_i^{(\mathbf{t})}\}_{i=1}^N$ . Using this collection, the discriminator  $D_{\phi}$ , parameterized by  $\phi$ , is trained to maximize the following objective:
+
+$$
+\max  _ {\phi} \mathcal {L} (\phi) = \frac {1}{N} \sum_ {i = 1} ^ {N} \left\{\log D _ {\phi} \left(q, o _ {i} ^ {(s)}\right) + \log \left(1 - D _ {\phi} \left(q, o _ {i} ^ {(t)}\right)\right) \right\}. \tag {1}
+$$
+
+According to this objective,  $D_{\phi}$  learns to output a score approaching 'one value' for responses from student VLMs and 'zero value' for those from teacher VLMs. For practical implementation, given that the discriminator's backbone is language-based (mirroring the student VLMs), the generated responses  $o$  are formatted using a specific prompt (detailed in Fig. 3) to ensure proper textual understanding. To elicit the scalar discrimination score, the standard language head of the VLM (typically mapping to vocabulary logits,  $(\mathbb{R}^{d\times v})$  is replaced with a linear discriminator head  $(\mathbb{R}^{d\times 1})$ . The input to this head is the representation of the final sequence token from the last layer of the backbone model, from which the score is directly computed with the sigmoid function.
+
+# 3.3 Mimicking Large Teacher VLMs with Similarity and Answer Rewards
+
+The RIL process commences after an initial supervised finetuning (SFT) phase for the student (small) VLMs  $\pi_{\theta}$  using a comprehensive visual instruction tuning dataset (see Appendix C). This SFT stage acts as a crucial warm-up, acclimating  $\pi_{\theta}$  to the target data distribution. Once the student VLMs and the pre-trained discriminator  $D_{\phi}$  (from Section 3.2) are prepared, the RIL loop begins. For each question  $q$  in a training batch, we generate  $G$  text responses from the current student VLMs,  $\{o_i^{(s)}\}_{i = 1}^G$ , and retrieve  $G$  responses corresponding the question  $q$  from the teacher (large) VLMs,  $\{o_i^{(t)}\}_{i = 1}^G$ . We note that pre-generating and caching the responses from teacher VLMs (e.g., during the discriminator's pre-training) can significantly reduce RIL training time. In the end, this yields a combined set of  $2G$  responses,  $\{o_i\}_{i = 1}^{2G}$ , for each question. Within each RIL iteration, we first update the discriminator  $D_{\phi}$  using Eq. (1) on these  $2G$  responses. This step ensures that  $D_{\phi}$  maintains its ability to distinguish between student and teacher outputs as the student VLMs evolve. Next, the student VLMs  $\pi_{\phi}$  are updated using an objective derived from Dr.GRPO [8], an advanced variant of GRPO that provides unbiased advantage estimates  $\{\hat{A}_i\}_{i = 1}^{2G}$ . The objective function is:
+
+$$
+\begin{array}{l} \max _ {\theta} \mathcal {L} (\theta) = \frac {1}{2 G} \sum_ {i = 1} ^ {2 G} \left\{\min \left[ r _ {i} (\theta) \hat {A} _ {i}, \operatorname {c l i p} (r _ {i} (\theta), 1 - \epsilon , 1 + \epsilon) \hat {A} _ {i} \right] - \beta \mathcal {D} _ {\mathrm {K L}} (\pi_ {\theta} | \pi_ {\mathrm {r e f}}) \right\}, \\ \text {w h e r e} \quad r _ {i} (\theta) = \frac {\pi_ {\theta} (o _ {i} \mid q)}{\pi_ {\theta_ {\mathrm {o l d}}} (o _ {i} \mid q)}, \quad \hat {A} _ {i} = R (q, o _ {i}) - \frac {1}{2 G} \sum_ {j = 1} ^ {2 G} R (q, o _ {j}). \\ \end{array}
+$$
+
+Here,  $r_i(\theta)$  is the probability ratio between the current policy  $\pi_{\theta}$  and the policy before the update  $\pi_{\mathrm{old}}$ ,  $\pi_{\mathrm{ref}}$  is typically the initial SFT model,  $\epsilon$  is a clipping hyperparameter, and  $\beta$  controls the KL-
+
+Table 3: Comparing RIL-applied VLMs with standard or smaller model size open-source VLMs.  
+
+<table><tr><td>VLMs</td><td>AI2D</td><td>ChartQA</td><td>MathVista</td><td>MMB</td><td>\( {\mathbf{{MMB}}}^{\mathbf{{CN}}} \)</td><td>MM-Vet</td><td>MMMU</td><td>MMMU-Pro</td><td>MMStar</td><td>BLINK</td><td>SEED</td><td>SEED2+</td><td>RWQA</td></tr><tr><td>LLaVA-OneVision-7B [28]</td><td>81.4</td><td>80.0</td><td>63.2</td><td>80.8</td><td>-</td><td>57.5</td><td>48.8</td><td>24.1</td><td>61.9</td><td>53.0</td><td>76.7</td><td>65.4</td><td>69.9</td></tr><tr><td>InternVL2-8B [69]</td><td>83.8</td><td>83.3</td><td>58.3</td><td>81.7</td><td>81.2</td><td>54.2</td><td>49.3</td><td>29.0</td><td>61.5</td><td>50.9</td><td>75.4</td><td>67.3</td><td>64.2</td></tr><tr><td>MiniCPM-V2.5-8B [70]</td><td>78.4</td><td>-</td><td>54.3</td><td>77.2</td><td>74.2</td><td>52.8</td><td>45.8</td><td>-</td><td>51.8</td><td>-</td><td>72.3</td><td>61.4</td><td>63.5</td></tr><tr><td>MiniCPM-V2.6-8B [70]</td><td>82.1</td><td>-</td><td>60.6</td><td>-</td><td>-</td><td>60.0</td><td>49.8</td><td>27.2</td><td>57.5</td><td>55.2</td><td>74.0</td><td>65.7</td><td>65.0</td></tr><tr><td>MiniCPM-o2.6-8B [70]</td><td>86.1</td><td>86.9</td><td>73.3</td><td>-</td><td>-</td><td>67.2</td><td>50.9</td><td>-</td><td>63.3</td><td>53.9</td><td>75.5</td><td>67.9</td><td>68.0</td></tr><tr><td>Ovis2-8B [71]</td><td>86.6</td><td>-</td><td>71.8</td><td>-</td><td>-</td><td>65.1</td><td>57.4</td><td>-</td><td>64.6</td><td>54.3</td><td>77.2</td><td>70.4</td><td>72.5</td></tr><tr><td>Ovis2-16B [71]</td><td>86.3</td><td>-</td><td>73.7</td><td>-</td><td>-</td><td>68.4</td><td>60.7</td><td>-</td><td>67.2</td><td>59.0</td><td>77.7</td><td>72.1</td><td>74.1</td></tr><tr><td>Qwen2-VL-7B [16]</td><td>77.5</td><td>83.0</td><td>58.2</td><td>83.0</td><td>80.5</td><td>62.0</td><td>54.1</td><td>30.5</td><td>60.7</td><td>53.8</td><td>76.0</td><td>68.6</td><td>68.5</td></tr><tr><td>InternVL2.5-8B [17]</td><td>84.8</td><td>84.8</td><td>64.4</td><td>84.6</td><td>82.6</td><td>62.8</td><td>56.0</td><td>34.3</td><td>63.2</td><td>54.8</td><td>76.8</td><td>69.7</td><td>70.1</td></tr><tr><td>Qwen2.5-VL-7B [1]</td><td>83.9</td><td>87.3</td><td>67.8</td><td>83.5</td><td>83.4</td><td>71.8</td><td>55.0</td><td>38.3</td><td>63.9</td><td>56.4</td><td>77.0</td><td>70.4</td><td>68.5</td></tr><tr><td>InternVL3-8B [18]</td><td>85.2</td><td>86.6</td><td>71.6</td><td>83.4</td><td>82.2</td><td>78.5</td><td>62.7</td><td>41.3</td><td>68.2</td><td>55.5</td><td>77.1</td><td>69.7</td><td>70.8</td></tr><tr><td>Qwen2.5-VL-RL-7B (Both)</td><td>86.1</td><td>95.6</td><td>79.7</td><td>86.3</td><td>86.5</td><td>80.4</td><td>65.7</td><td>48.5</td><td>71.1</td><td>70.0</td><td>80.5</td><td>72.8</td><td>72.8</td></tr><tr><td>InternVL3-RL-8B (Both)</td><td>87.3</td><td>95.3</td><td>77.8</td><td>88.1</td><td>89.6</td><td>80.1</td><td>68.6</td><td>47.8</td><td>74.8</td><td>62.7</td><td>80.5</td><td>73.8</td><td>73.7</td></tr><tr><td>Phi-3.5-Vision-4B [72]</td><td>77.8</td><td>81.8</td><td>43.9</td><td>76.0</td><td>66.1</td><td>43.2</td><td>43.0</td><td>19.7</td><td>47.5</td><td>58.3</td><td>69.7</td><td>62.2</td><td>53.6</td></tr><tr><td>Phi-4-Multimodal-5.6B [73]</td><td>83.0</td><td>81.4</td><td>65.8</td><td>86.7</td><td>-</td><td>51.9</td><td>56.0</td><td>38.5</td><td>58.9</td><td>42.1</td><td>73.2</td><td>68.5</td><td>64.1</td></tr><tr><td>Ovis2-4B [71]</td><td>85.7</td><td>-</td><td>69.6</td><td>-</td><td>-</td><td>65.5</td><td>49.0</td><td>-</td><td>61.9</td><td>53.0</td><td>76.2</td><td>69.3</td><td>71.1</td></tr><tr><td>InternVL2-4B [69]</td><td>78.9</td><td>81.5</td><td>58.6</td><td>78.6</td><td>73.9</td><td>51.0</td><td>34.3</td><td>32.7</td><td>53.9</td><td>46.1</td><td>73.2</td><td>63.9</td><td>60.7</td></tr><tr><td>InternVL2.5-4B [69]</td><td>81.4</td><td>84.0</td><td>60.5</td><td>81.1</td><td>79.3</td><td>60.6</td><td>52.3</td><td>32.7</td><td>58.7</td><td>50.8</td><td>74.8</td><td>66.9</td><td>64.3</td></tr><tr><td>Qwen2.5-VL-3B [1]</td><td>81.6</td><td>84.0</td><td>61.2</td><td>79.1</td><td>78.1</td><td>61.8</td><td>51.2</td><td>31.6</td><td>55.9</td><td>47.6</td><td>74.0</td><td>67.6</td><td>65.4</td></tr><tr><td>Qwen2.5-VL-RL-3B (Both)</td><td>83.9</td><td>95.7</td><td>71.7</td><td>85.2</td><td>84.5</td><td>74.7</td><td>60.7</td><td>43.7</td><td>65.2</td><td>60.8</td><td>78.7</td><td>71.4</td><td>73.2</td></tr><tr><td>InternVL2-2B [69]</td><td>74.1</td><td>76.2</td><td>46.3</td><td>73.2</td><td>70.9</td><td>39.5</td><td>34.3</td><td>18.2</td><td>49.8</td><td>43.8</td><td>70.9</td><td>60.0</td><td>57.3</td></tr><tr><td>Qwen2-VL-2B [16]</td><td>60.2</td><td>73.5</td><td>43.0</td><td>74.9</td><td>73.5</td><td>49.5</td><td>41.1</td><td>21.2</td><td>47.5</td><td>45.2</td><td>72.4</td><td>61.2</td><td>62.6</td></tr><tr><td>Aquila-VL-2B [17]</td><td>75.0</td><td>76.5</td><td>59.0</td><td>-</td><td>-</td><td>43.8</td><td>47.4</td><td>-</td><td>54.9</td><td>34.1</td><td>73.9</td><td>63.0</td><td>65.0</td></tr><tr><td>Ovis2-2B [71]</td><td>82.7</td><td>-</td><td>64.1</td><td>-</td><td>-</td><td>58.3</td><td>45.6</td><td>-</td><td>56.7</td><td>47.9</td><td>74.4</td><td>67.4</td><td>66.0</td></tr><tr><td>InternVL2.5-2B [17]</td><td>74.9</td><td>79.2</td><td>51.3</td><td>74.7</td><td>71.9</td><td>60.8</td><td>43.6</td><td>23.7</td><td>54.3</td><td>44.0</td><td>73.2</td><td>60.0</td><td>60.1</td></tr><tr><td>InternVL3-2B [18]</td><td>78.7</td><td>80.2</td><td>57.0</td><td>81.1</td><td>78.4</td><td>62.2</td><td>48.6</td><td>24.9</td><td>60.7</td><td>47.0</td><td>75.0</td><td>64.6</td><td>64.3</td></tr><tr><td>InternVL3-RL-2B (Both)</td><td>80.7</td><td>94.0</td><td>67.4</td><td>84.7</td><td>82.2</td><td>70.7</td><td>56.6</td><td>32.3</td><td>63.9</td><td>51.5</td><td>77.8</td><td>66.8</td><td>69.3</td></tr><tr><td>LLaVA-OneVision-0.5B [28]</td><td>57.1</td><td>61.4</td><td>34.8</td><td>61.6</td><td>55.5</td><td>32.2</td><td>31.4</td><td>-</td><td>37.7</td><td>52.1</td><td>66.6</td><td>45.7</td><td>55.6</td></tr><tr><td>Ovis2-1B [71]</td><td>76.4</td><td>-</td><td>59.4</td><td>-</td><td>-</td><td>50.0</td><td>36.1</td><td>-</td><td>52.1</td><td>44.0</td><td>71.7</td><td>61.4</td><td>63.9</td></tr><tr><td>InternVL2-1B [69]</td><td>64.1</td><td>72.9</td><td>37.7</td><td>65.4</td><td>60.7</td><td>32.7</td><td>36.7</td><td>14.8</td><td>45.6</td><td>38.6</td><td>65.2</td><td>54.3</td><td>50.3</td></tr><tr><td>InternVL2.5-1B [17]</td><td>69.3</td><td>75.9</td><td>43.2</td><td>70.7</td><td>66.3</td><td>48.8</td><td>40.9</td><td>18.8</td><td>51.3</td><td>42.0</td><td>70.4</td><td>59.0</td><td>57.5</td></tr><tr><td>InternVL3-1B [18]</td><td>69.4</td><td>75.3</td><td>45.8</td><td>72.6</td><td>67.9</td><td>58.7</td><td>43.4</td><td>17.5</td><td>51.5</td><td>42.9</td><td>71.2</td><td>58.2</td><td>58.2</td></tr><tr><td>InternVL3-RL-1B (Both)</td><td>73.0</td><td>94.1</td><td>55.5</td><td>79.1</td><td>75.9</td><td>62.9</td><td>49.7</td><td>19.0</td><td>60.5</td><td>46.9</td><td>74.7</td><td>61.9</td><td>63.7</td></tr></table>
+
+divergence penalty. The total reward  $R(q, o_i)$  for each response  $o_i$  is a crucial composite signal, comprising two binary components:
+
+1. Similarity Reward: This indicates if the response  $o_i$  resembles those from teacher VLMs, denoted by  $\mathbb{1}(D_{\phi}(q,o_i) < 0.5)$ . A discriminator score below 0.5 suggests higher similarity to teacher responses, consistent with  $D_{\phi}$  being trained to output 'zero value' for teacher samples (Eq. 1).  
+2. Answer Reward: It assesses the factual correctness of  $o_i$  against a ground truth answer  $a_i$ , determined by LLM-as-a-Judge  $(q, a, o_i)$ . This evaluation uses the prompt detailed in Fig. 4.
+
+These two rewards are summed to form the overall reward  $R(q, o_i)$ . The inclusion of the answer reward is essential, as the discriminator alone primarily captures stylistic similarity and does not guarantee factual accuracy, the absence of which could lead to significant performance issues. This dual-reward strategy ensures that mimicking large teacher VLMs involves not just replicating surface-level text patterns but also encompasses the deeper 'verbalization effect' [45], which characterizes how proficient VLMs articulate correct and contextually appropriate answers. The comprehensive RIL procedure is detailed in Algorithm 2.
+
+# 4 Experiments
+
+# 4.1 Dissecting RIL
+
+To understand the distinct contributions of its components, we first analyze RIL by comparing it against reinforcement learning (RL) baselines that utilize only answer rewards. Specifically, we evaluate RL agents trained solely with GRPO [43] or its advanced variant, Dr.GRPO [8], without the discriminator or the associated similarity rewards. As shown in Table 1, RIL, which integrates reinforcement and imitation learning elements based on Dr.GRPO and GAIL [44], demonstrably outperforms these RL-only approaches, confirming the efficacy of our unified training algorithm.
+
+A key finding is the substantial benefit of employing multiple large teacher VLMs. As detailed in Table 2, using responses from a combination of large models (specifically Qwen2.5-VL-72B [1] and InternVL3-78B [18]) yields significantly better vision-language performance than relying on any single teacher. This advantage arises from the richer diversity of textual responses, which strengthens the discriminator's training. Furthermore, incorporating this diverse set of high-quality responses from teacher VLMs directly into the GRPO optimization step provides student VLMs with clearer and
+
+Table 4: Comparing RIL-applied VLMs with large size open-source and closed-source VLMs.  
+
+<table><tr><td>VLMs</td><td colspan="2">AI2D ChartQA</td><td colspan="2">MathVista</td><td>MMB</td><td>MM-Vet</td><td>MM-Vet-v2</td><td>MMMU</td><td>MMMU-Pro</td><td>MMStar</td><td>BLINK</td><td>SEED</td><td>SEED2+</td><td>RWQA</td></tr><tr><td>NVLM-72B [54]</td><td>85.2</td><td>86.0</td><td>66.6</td><td>-</td><td>58.9</td><td>-</td><td>59.7</td><td>-</td><td>63.7</td><td>48.0</td><td>75.5</td><td>68.4</td><td>69.9</td><td></td></tr><tr><td>LLaVA-OV-72B [28]</td><td>85.6</td><td>83.7</td><td>67.5</td><td>85.8</td><td>60.6</td><td>-</td><td>56.8</td><td>31.0</td><td>65.8</td><td>55.4</td><td>77.5</td><td>-</td><td>71.9</td><td></td></tr><tr><td>Molmo-72B [53]</td><td>83.4</td><td>87.3</td><td>58.6</td><td>-</td><td>61.1</td><td>-</td><td>54.1</td><td>-</td><td>63.3</td><td>49.7</td><td>74.6</td><td>67.6</td><td>73.7</td><td></td></tr><tr><td>Qwen2-VL-72B [16]</td><td>88.1</td><td>88.3</td><td>70.5</td><td>86.5</td><td>74.0</td><td>68.7</td><td>64.5</td><td>46.2</td><td>68.3</td><td>60.5</td><td>77.9</td><td>72.3</td><td>77.8</td><td></td></tr><tr><td>Qwen2.5-VL-72B [1]</td><td>88.7</td><td>89.5</td><td>74.2</td><td>88.6</td><td>76.9</td><td>76.2</td><td>68.2</td><td>46.2</td><td>70.8</td><td>64.4</td><td>79.5</td><td>73.0</td><td>75.7</td><td></td></tr><tr><td>InternVL2-76B [69]</td><td>87.6</td><td>88.4</td><td>65.5</td><td>86.5</td><td>65.7</td><td>68.4</td><td>62.7</td><td>40.0</td><td>67.4</td><td>56.8</td><td>77.6</td><td>69.7</td><td>72.2</td><td></td></tr><tr><td>InternVL2.5-78B [28]</td><td>89.1</td><td>88.3</td><td>72.3</td><td>88.3</td><td>72.3</td><td>65.5</td><td>70.1</td><td>48.6</td><td>69.5</td><td>63.8</td><td>77.0</td><td>71.3</td><td>78.7</td><td></td></tr><tr><td>InternVL3-78B [18]</td><td>89.7</td><td>89.7</td><td>79.0</td><td>89.0</td><td>81.3</td><td>70.0</td><td>72.2</td><td>47.3</td><td>72.5</td><td>66.3</td><td>78.7</td><td>71.9</td><td>78.0</td><td></td></tr><tr><td>Claude-3.5-Sonnet [52]</td><td>81.2</td><td>90.8</td><td>67.7</td><td>82.6</td><td>70.1</td><td>71.8</td><td>68.3</td><td>51.5</td><td>65.1</td><td>60.1</td><td>61.7</td><td>71.7</td><td>65.8</td><td></td></tr><tr><td>Claude-3.7-Sonnet [52]</td><td>82.5</td><td>-</td><td>66.8</td><td>-</td><td>70.0</td><td>-</td><td>71.0</td><td>-</td><td>65.1</td><td>56.6</td><td>74.3</td><td>67.6</td><td>55.4</td><td></td></tr><tr><td>Gemini-1.5-Pro [51]</td><td>79.1</td><td>87.2</td><td>63.9</td><td>73.9</td><td>64.0</td><td>66.9</td><td>62.2</td><td>46.9</td><td>59.1</td><td>59.1</td><td>76.0</td><td>70.8</td><td>67.5</td><td></td></tr><tr><td>Gemini-2.0-Flash [51]</td><td>83.1</td><td>-</td><td>70.4</td><td>90.0</td><td>73.6</td><td>-</td><td>69.9</td><td>54.4</td><td>69.4</td><td>64.0</td><td>-</td><td>-</td><td>72.3</td><td></td></tr><tr><td>GPT-4o (24.05.13) [74]</td><td>84.6</td><td>85.7</td><td>63.8</td><td>83.4</td><td>69.1</td><td>71.0</td><td>69.1</td><td>51.9</td><td>64.7</td><td>68.0</td><td>77.1</td><td>72.0</td><td>75.4</td><td></td></tr><tr><td>GPT-4.1 (25.04.14) [74]</td><td>85.9</td><td>-</td><td>70.4</td><td>-</td><td>78.8</td><td>-</td><td>74.0</td><td>-</td><td>69.8</td><td>68.5</td><td>78.0</td><td>73.1</td><td>78.7</td><td></td></tr><tr><td>Qwen2.5-VL-RIL-7B (Both)</td><td>86.1</td><td>95.6</td><td>79.7</td><td>86.3</td><td>80.4</td><td>71.1</td><td>65.7</td><td>48.5</td><td>71.1</td><td>70.0</td><td>80.5</td><td>72.8</td><td>72.8</td><td></td></tr><tr><td>InternVL3-RIL-8B (Both)</td><td>87.3</td><td>95.3</td><td>77.8</td><td>88.1</td><td>80.1</td><td>67.6</td><td>68.6</td><td>47.8</td><td>74.8</td><td>62.7</td><td>80.5</td><td>73.8</td><td>73.7</td><td></td></tr></table>
+
+![](images/b5a4d06e174faf9a3dad41b2c07e3ea9a88e161cdfb0dee7f2759bb755cdd232.jpg)  
+Figure 5: Illustrating training dynamics of small VLMs during RIL, where (Left) showing the evolution of similarity rewards over training iterations and (Mid) accuracy rewards obtained using LLM-as-a-Judge [9], ensuring that generated responses are both contextually appropriate and factually correct. (Right) displaying the overall average performance of evaluation benchmarks used in Table 1.
+
+![](images/1b73832260f89ffb723dc1d45f797fdf30045cf4c2a84834a2a79a9b7d521cd2.jpg)
+
+![](images/092eb7935e1f9ad00d8a578d5e8d3389ff2de785144bc8c959be18594c917f3a.jpg)
+
+more varied exemplars of correct answer generation. To be simplified in the manner of an algorithm, given a response 'T1' from one teacher VLM and 'T2' from another teacher VLM, and a response 'S' from the student VLM, then the discriminator is trained to output zero when 'S' is provided and one when either 'T1' or 'T2' is provided. Additionally, the rewards and their advantages for 'S', 'T1', and 'T2' are computed based on similarity and answer quality in order to train the student VLM by using RIL's objective loss.
+
+Beyond these component analyses, our RIL-trained student VLMs exhibit highly competitive performance against a range of recent open-source and closed-source models. Comparisons across various model sizes (detailed in Table 3 for smaller models and Table 4 for larger ones) and diverse vision-language benchmarks (illustrated in Figures 1 and 2) show that RIL not only substantially narrows the performance gap to state-of-the-art VLMs but, in several instances, surpasses them.
+
+Finally, the training dynamics of RIL, depicted in Figure 5, offer further insight. Over the course of training, RIL consistently increases both the similarity rewards (from the discriminator) and the answer rewards (from LLM-as-a-Judge [9]). It indicates that the student VLMs are progressively generating responses that are both more stylistically aligned with teacher models and factually correct, with these improving reward signals directly translating into significant overall performance gains.
+
+For a comprehensive account of the specific training procedures, hyperparameter settings, evaluation setups, and dataset details that ensure the reproducibility of these, please refer to Appendix C. Using 256 NVIDIA A100 GPUs, pre-training the discriminator on 1.2M samples (40K [number of samples]  $\times$  16 [generated responses]  $\times$  2 [for both teacher and student]) takes approximately 1 to 3 days. The SFT step on the 4M-sample SFT dataset takes around 3 to 5 days. Conducting the RIL loop for the sampled 40K data requires an additional 3 to 5 days using 8 NVIDIA A100 GPUs.
+
+# 4.2 Impacts of Hyperparameters, Techniques, and Distillation
+
+We systematically investigate the sources of RIL's effectiveness by analyzing key training hyperparameters, component design choices, and its interplay with other learning techniques. Note that, for the optimization objective of student VLMs in Eq. (2), the clipping hyperparameter  $\epsilon$  is consistently set to 0.2. Our first analyses focus on core optimization settings. We examine the number of update iterations,  $\mu$ , for the discriminator and student VLMs within each RIL cycle (related to Algorithm 2).
+
+Table 5: Analyzing the impacts of training hyperparameters, techniques for stabilization, and the effect of distillation. Note that, these tables (a)-(c) set a backbone model as Qwen2.5-VL-7B [1] and apply RIL to this model in order to validate it by controlling multiple factors.
+
+(a) Discriminator & small VLMs Iteration  $(\mu)$  
+
+<table><tr><td>μ</td><td>AI2D</td><td>MathVista</td><td>MMB</td><td>MM-Vet</td><td>MMMU</td><td>BLINK</td></tr><tr><td>1</td><td>86.1</td><td>79.7</td><td>86.3</td><td>80.4</td><td>65.7</td><td>70.0</td></tr><tr><td>2</td><td>86.1</td><td>79.0</td><td>86.3</td><td>80.4</td><td>65.3</td><td>69.2</td></tr><tr><td>4</td><td>85.5</td><td>78.8</td><td>85.5</td><td>79.5</td><td>65.0</td><td>68.4</td></tr><tr><td>6</td><td>85.1</td><td>77.5</td><td>85.2</td><td>78.0</td><td>63.8</td><td>68.0</td></tr><tr><td>8</td><td>84.5</td><td>76.2</td><td>84.7</td><td>77.4</td><td>62.5</td><td>66.7</td></tr><tr><td>10</td><td>84.3</td><td>75.0</td><td>84.5</td><td>77.2</td><td>61.9</td><td>65.5</td></tr><tr><td>12</td><td>84.0</td><td>73.8</td><td>84.4</td><td>76.9</td><td>61.0</td><td>64.2</td></tr></table>
+
+(b) KL-divergence Penalty Coefficient  $(\beta)$  
+
+<table><tr><td>\( \beta \)</td><td>AI2D</td><td>MathVista</td><td>MMB</td><td>MM-Vet</td><td>MMMU</td><td>BLINK</td></tr><tr><td>0.00</td><td>77.1</td><td>59.9</td><td>73.2</td><td>60.2</td><td>42.9</td><td>45.0</td></tr><tr><td>0.01</td><td>86.1</td><td>79.7</td><td>86.3</td><td>80.4</td><td>65.7</td><td>70.0</td></tr><tr><td>0.05</td><td>85.7</td><td>77.3</td><td>85.7</td><td>78.5</td><td>63.5</td><td>68.3</td></tr><tr><td>0.10</td><td>85.2</td><td>74.9</td><td>85.2</td><td>76.6</td><td>63.0</td><td>65.8</td></tr><tr><td>0.20</td><td>84.8</td><td>74.4</td><td>84.5</td><td>75.8</td><td>62.5</td><td>64.2</td></tr><tr><td>0.50</td><td>84.3</td><td>74.0</td><td>84.8</td><td>75.9</td><td>61.3</td><td>62.0</td></tr><tr><td>1.00</td><td>84.2</td><td>73.9</td><td>84.3</td><td>74.8</td><td>60.4</td><td>61.7</td></tr></table>
+
+(c) Importance of VLM Modules  
+
+<table><tr><td>Module</td><td>MathVista</td><td>MMB</td><td>MM-Vet</td><td>MMMU</td></tr><tr><td>Full Training</td><td>79.7</td><td>86.3</td><td>80.4</td><td>65.7</td></tr><tr><td>w.o. Word-Embed</td><td>75.4</td><td>85.5</td><td>75.1</td><td>62.2</td></tr><tr><td>w.o. Self-Attn</td><td>73.7</td><td>84.3</td><td>75.2</td><td>61.3</td></tr><tr><td>w.o. FFN-Gate</td><td>76.5</td><td>86.0</td><td>78.8</td><td>63.5</td></tr><tr><td>w.o. FFN-Up</td><td>77.3</td><td>86.0</td><td>79.0</td><td>64.1</td></tr><tr><td>w.o. FFN-Down</td><td>78.4</td><td>86.1</td><td>79.6</td><td>64.8</td></tr><tr><td>w.o. Layer-Norm</td><td>79.7</td><td>86.3</td><td>80.4</td><td>65.7</td></tr><tr><td>w.o. Lang-Head</td><td>74.9</td><td>85.2</td><td>73.9</td><td>62.0</td></tr></table>
+
+(d) Importance of discriminator  $D_{\phi}$  
+
+<table><tr><td>VLMs</td><td>Dφ</td><td>MathVista</td><td>MMB</td><td>MM-Vet</td><td>MMMU</td></tr><tr><td rowspan="3">Qwen2.5-VL-7B</td><td>X</td><td>69.5</td><td>84.3</td><td>73.5</td><td>57.2</td></tr><tr><td>△</td><td>75.3</td><td>85.5</td><td>75.1</td><td>61.2</td></tr><tr><td>✓</td><td>79.7</td><td>86.3</td><td>80.4</td><td>65.7</td></tr><tr><td rowspan="3">InternVL3-8B</td><td>X</td><td>72.9</td><td>85.2</td><td>79.0</td><td>64.9</td></tr><tr><td>△</td><td>75.3</td><td>86.6</td><td>79.5</td><td>66.5</td></tr><tr><td>✓</td><td>77.8</td><td>88.1</td><td>80.1</td><td>68.6</td></tr></table>
+
+(e) Effect of Answer Rewards (AR) and SFT  
+
+<table><tr><td>VLMs</td><td>AR</td><td>MathVista</td><td>MMB</td><td>MM-Vet</td><td>MMMU</td></tr><tr><td rowspan="4">Qwen2.5-VL-7B</td><td>✘</td><td>65.2</td><td>82.1</td><td>69.9</td><td>53.6</td></tr><tr><td>\( \bigtriangleup \)</td><td>65.3</td><td>82.5</td><td>70.3</td><td>53.8</td></tr><tr><td>✓</td><td>79.7</td><td>86.3</td><td>80.4</td><td>65.7</td></tr><tr><td>✓w.o. SFT</td><td>73.5</td><td>84.1</td><td>76.4</td><td>60.7</td></tr><tr><td rowspan="4">InternVL3-8B</td><td>✘</td><td>69.3</td><td>81.4</td><td>75.4</td><td>59.3</td></tr><tr><td>\( \bigtriangleup \)</td><td>70.0</td><td>82.1</td><td>76.2</td><td>60.2</td></tr><tr><td>✓</td><td>77.8</td><td>88.1</td><td>80.1</td><td>68.6</td></tr><tr><td>✓w.o. SFT</td><td>75.2</td><td>85.3</td><td>79.4</td><td>64.9</td></tr></table>
+
+(f) Effect of Distillation  
+
+<table><tr><td>VLMs</td><td>RIL</td><td>MathVista</td><td>MMB</td><td>MM-Vet</td><td>MMMU</td></tr><tr><td rowspan="2">MiniLLM-7B</td><td>X</td><td>61.3</td><td>84.4</td><td>65.1</td><td>57.4</td></tr><tr><td>✓</td><td>70.8</td><td>85.3</td><td>73.3</td><td>66.1</td></tr><tr><td rowspan="2">DistilLLM-7B</td><td>X</td><td>61.5</td><td>84.8</td><td>65.3</td><td>57.9</td></tr><tr><td>✓</td><td>71.0</td><td>85.4</td><td>73.9</td><td>67.2</td></tr><tr><td rowspan="2">LLaVA-KD-7B</td><td>X</td><td>61.8</td><td>85.0</td><td>65.9</td><td>58.2</td></tr><tr><td>✓</td><td>71.9</td><td>85.8</td><td>74.0</td><td>68.7</td></tr><tr><td rowspan="2">VLs1-7B</td><td>X</td><td>74.7</td><td>85.8</td><td>75.2</td><td>69.3</td></tr><tr><td>✓</td><td>76.8</td><td>86.2</td><td>81.2</td><td>73.4</td></tr></table>
+
+As shown in Table 5(a), a single update iteration  $(\mu = 1)$  for both is sufficient for strong performance; more iterations, especially for student VLM updates, can risk overfitting. Subsequently, we explore the impact of the KL-divergence penalty coefficient  $\beta$  (Eq. 2) in Table 5(b). These results confirm that while constraining policy updates is crucial for stability, an overly restrictive penalty (large  $\beta$ ) can hinder performance.
+
+Next, we assess decisions related to model component updates. Table 5(c) details the importance of updating different parameter groups within the student VLMs, revealing that training the self-attention layers, word embeddings, and the language head yields more significant gains than updating feed-forward networks (FFN) or layer normalization parameters. We also investigate the necessity of continuously updating the discriminator during RIL training (Table 5(d)). Interestingly, Table 5(d) shows that using a fixed (not updated:  $\triangle$ ), pre-trained discriminator can outperform RL-only baselines (denoted by  $X$ ), but overall RIL efficacy still critically depends on leveraging a continuously trained discriminator (denoted by  $\checkmark$ ) during training RIL.
+
+In addition, we evaluate RIL's interaction with evaluation methodologies of the predicted answer and its synergy with other training paradigms. Table 5(e) underscores the importance of LLM-as-a-Judge [9] for answer rewards; replacing it with conventional answer parsing (denoted  $\triangle$ ) significantly degrades performance, particularly on general visual questions (e.g., "How to cook this dish?" from an image) where fixed parsing rules are inadequate. This highlights the need for flexible, LLM-based evaluation. The same table also affirms that while initial SFT is beneficial for capturing the training data distribution, RIL subsequently delivers substantially larger improvements.
+
+On Table 5(f), we examine RIL's effectiveness on student VLMs that are already trained on distillation methods like MiniLLM [75], DistillLLM [76], LLaVA-KD [77], and VLsI [45] (on same backbones like Qwen2-VL [16] and training datasets C). As shown in Table 5(f), RIL proves particularly potent for these models compared to non-distillation counterparts, likely because prior feature alignment through distillation primes them for RIL's alignment mechanisms.
+
+Lastly, we demonstrate that the binary similarity reward introduced in Sec. 3.3 enables more efficient and stable training of RIL. As shown in Fig. 6, the binary reward consistently outperforms continuous and multi-level discretizations, suggesting that binary feedback provides clearer and more reliable learning signals. This improvement arises because models often struggle to interpret subtle differences in continuous rewards (e.g., why a score of 0.21 should be preferred over 0.20).
+
+# 4.3 Discussion and Limitation
+
+RIL leverages the diversity of responses generated by large VLMs, which tend to produce more varied and higher-quality answers beyond the ground truth answers used in SFT training datasets. As illustrated in Fig. 7, increasing the number of teacher-generated responses consistently improves the student model's performance, demonstrating that richer supervision from diverse teacher outputs enhances the student VLM's overall generalization ability. In summary, RIL exploits these diverse, high-quality teacher responses to guide smaller VLMs toward learning more flexible responses beyond fixed ground truth answers.
+
+Furthermore, RIL offers distinct advantages over traditional knowledge distillation techniques [78]. Unlike conventional distillation, which typically requires student VLMs to replicate the high-
+
+![](images/267815d77bf0beee47d3fe54a396f238182a1c3e89698adc861caeeff0d42ee3.jpg)  
+Figure 6: Comparison of RIL performance using different discretization levels of the similarity reward on four benchmarks: MathVista [3], MMB [20], MM-Vet [4], and MMMU [5]) with Qwen2.5- VL-7B [1] and InternVL3-8B [18].
+
+![](images/73a965d0db0faffed0a14d34200b5ce7df0a92fc0d73f7ebec7f8b0fb07e542f.jpg)  
+Figure 7: Performance of RIL with varying numbers of generated responses from a teacher VLM across four benchmarks and their average. They are evaluated on Qwen2.5-VL-7B [1].
+
+![](images/0e2dd88f74a6c0d34c2e5e1b1ff4540a84d3ae5a9d236c87fae0de1ffb75458e.jpg)
+
+![](images/951112592ad5d8819efda166d743f46bd4620a8b5de6a75efb07ec55e5c4b909.jpg)
+
+![](images/3ddebd27a96b1830b644d0d35d3f0aab62fc92ead8eda543b7c706b664f1d12d.jpg)
+
+![](images/69c6f016318f9dc3dafcc94ea0f9b204723ddc19b1dce3df62307c30b76bb3da.jpg)
+
+dimensional logit features of teacher models, RIL operates exclusively on generated text responses in natural language. This fundamental difference liberates RIL from restrictive feature-level constraints, such as the need for identical image embedding strategies or perfectly aligned language tokenizers (including vocabulary, index order, and sequence length) between student and teacher VLMs. Common distillation methods [75-77, 45] often rely on KL divergence—a static, non-trainable metric devoid of contextual understanding—to measure feature similarity, but RIL employs a trainable discriminator which assesses natural language responses directly, learning to discern nuanced similarities and stylistic differences between student and teacher outputs. The adaptability of the trainable discriminator in capturing contextual relationships makes RIL a particularly flexible and powerful approach for lightweight VLM and enhancement.
+
+While RIL demonstrates these significant strengths, its current implementation has been focused primarily on the post-instruction tuning alignment phase. A promising avenue for future research, therefore, involves extending the utility of our discriminator to the initial visual instruction tuning stage itself. We hypothesize that the discriminator's ability to learn from natural language signals and overcome conventional architectural constraints could also offer significant benefits in this earlier phase of VLM training, potentially leading to more efficient instruction following from the outset.
+
+# 5 Conclusion
+
+In this work, we have introduced Unified Reinforcement and Imitation Learning (RIL), a novel framework enabling smaller VLMs to emulate the sophisticated text-generation capabilities of substantially larger counterparts, often matching or even exceeding their performance. RIL directly addresses the pressing need for high-performing, lightweight VLMs deployable in resource-constrained settings. By leveraging a dual reward system—combining similarity scores from a discriminator with answer accuracy assessments from an LLM-as-a-Judge—RIL effectively guides student VLMs towards superior alignment and overall efficacy. Key findings from our research include the significant performance boost achieved by using multiple large teacher VLMs, which provides a richer diversity of training signals, and the notable success of RIL in enhancing distillation-based VLMs. Comprehensive experiments across a wide array of vision-language benchmarks confirm that RIL substantially narrows, and in several cases surpasses, the performance gap to state-of-the-art open-and closed-source VLMs.
+
+# References
+
+[1] S. Bai, K. Chen, X. Liu, J. Wang, W. Ge, S. Song, K. Dang, P. Wang, S. Wang, J. Tang, et al., "Qwen2. 5-vl technical report," arXiv preprint arXiv:2502.13923, 2025.  
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+
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+
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+
+# A Algorithm of RL only with Answer Rewards from LLM-as-a-Judge
+
+Algorithm 1 RL purely with GRPO or Dr.GRPO based on accuracy rewards from LLM-as-a-Judge
+
+Require: Pre-trained VLMs  $\pi_{\theta_{\mathrm{init}}}$    
+1: Set reference model  $\pi_{\mathrm{ref}}\gets \pi_{\theta_{\mathrm{init}}}$    
+2: Set the training model  $\pi_{\theta}\leftarrow \pi_{\theta_{\mathrm{init}}}$    
+3: for sample a batch  $\mathcal{B}$  in Dataset do   
+4: Copy and freeze model  $\pi_{\theta_{\mathrm{old}}}\gets \pi_{\theta}$    
+5: Sample  $G$  outputs  $\{o_i\}_{i = 1}^G\sim \pi_{\theta_{\mathrm{old}}}(\cdot |q)$  for each question  $q\in \mathcal{B}$    
+6: Compute Rewards and Advantages for all  $G$  outputs by LLM-as-a-Judge   
+7: for VLMs Updating iteration  $= 1,2,\dots ,\mu$  do   
+8: Update  $\pi_{\theta}$  by using the objective of GRPO [43] or Dr.GRPO [8]   
+9: end for   
+10: end for
+
+# B Algorithm of RIL for VLMs
+
+Algorithm 2 RIL for VLMs  
+Require: Pre-trained discriminator  $D_{\phi}$  and Pre-trained student VLMs  $\pi_{\theta_{\mathrm{init}}}$    
+Require: Saved collection for the generated text responses  $\{o^{(1)}\}$  from teacher VLMs   
+1: Set reference model  $\pi_{\mathrm{ref}}\gets \pi_{\theta_{\mathrm{init}}}$    
+2: Set the training model  $\pi_{\theta}\gets \pi_{\theta_{\mathrm{init}}}$    
+3: for sample a batch  $\mathcal{B}$  in Dataset do   
+4: Copy and freeze model  $\pi_{\theta_{\mathrm{old}}}\gets \pi_{\theta}$    
+5: Sample  $G$  outputs  $\{o_i^{(s)}\}_{i = 1}^G\sim \pi_{\theta_{\mathrm{old}}}(\cdot |q)$  for each question  $q\in \mathcal{B}$    
+6: Extract  $G$  outputs  $\{o_i^{(t)}\}_{i = 1}^G$  in the saved collection for each question  $q\in \mathcal{B}$    
+7: for Discriminator Updating iteration  $= 1,2,\dots ,\mu$  do   
+8: Update  $D_{\phi}$  by using Equation 1   
+9: end for   
+10: Compute Rewards and Advantages for all  $2G$  outputs by Discriminator and LLM-as-a-Judge   
+11: for Student VLMs Updating iteration  $= 1,2,\dots ,\mu$  do   
+12: Update  $\pi_{\theta}$  by using Equation 2   
+13: end for   
+14: end for
+
+# C Implementation Detail
+
+Details of Training and Evaluation. We train and evaluate RIL, mainly on NVIDIA A100 80GB GPUs. Because it is practically critical point to get fast text generation during training, we utilize vLLM [79] built on PagedAttention. For the pre-training step of discriminator, we assign vLLM [79] to 8 GPUs for fast text generation, and we generate  $N = 16$  text responses for each question at student and teacher VLMs, respectively. Once it is finished, we use DeepSpeed engine with ZeRO-3 [80] for 8 GPUs, and we use AdamW optimizer [81] and apply a linearly decayed learning rate from 1e-5 to 1e-6 to pre-training discriminator and SFT of student VLMs. In subsequent step, when training both discriminator and student VLMs, one of 8 GPUs is assigned to conduct online generation of student VLMs by vLLM [79]. In addition, another GPU is assigned to conduct LLM-as-a-Judge [9] by vLLM [79]. The other 6 GPUs are assigned to use DeepSpeed engine with ZeRO-3 [80]. Mimicking step requires static learning rate 1e-6 and  $\mu = 1$  iteration to train both student VLMs and the discriminator. Note that, when we generate text responses, we generate  $G = 4$  responses for each question, by setting temperature to 1.0, top-p to 0.95, top-k to 50, and repetition penalty to 1.05, in order to get diverse text responses. For stable training, we handle large batch sizes by using gradient accumulation with 6 steps. At every step, we use 4 batches per one GPU, leading to total 144 batches. For evaluation, we use distributed data parallel to load student VLMs for all 8 GPUs and
+
+score the correctness or generation quality of their text responses by using their default generation hyperparameter.
+
+Computational Cost. Compared to GRPO [43], RIL requires increased computational costs during training due to teacher, student, discriminator, and LLM-as-a-Judge [9]. However, we would like to emphasize that discriminator and student costs can be at least mitigated. While pretraining the discriminator is a necessary step and cannot be avoided from a computational standpoint, we can reduce training costs of discriminator and student during the RIL loop (Algorithm 2) by managing model weights efficiently between CPU and GPU. Specifically, we offload the weights of the discriminator and student to the CPU when they are not in use. When needed, we load the appropriate weights from the CPU to the GPU using DeepSpeed API [80]. This allows us to load or unload model weights without uploading both architecture to GPU, thereby minimizing computational overhead. This approach is feasible because we use the same model architecture for both the discriminator and student, enabling seamless weight replacement and resource-efficient training.
+
+Visual Instruction Tuning Dataset. We assemble a 4M-sample visual instruction tuning dataset that encompasses a diverse array of vision-language tasks, including general visual question answering, dense image captioning, chart, diagram, and document understanding, common-sense knowledge, scientific and mathematical problem-solving, and multidimensional reasoning. For SFT, we utilize the entire 4M-sample dataset, and then we curate a 40K-sample dataset for RIL of VLMs based on log-probability sampling [82] and overlong filtering [36]. Our dataset integrates both real-world and synthetic sources: COCO-ReM [83], iNaturalist2018 [84], VQA-v2 [85], Super-CLEVR [86], MAVIS [87], Geometry3K [88], SQA [89], AI2D [2], SA-1B [60], LLaVAR [90], VSR [91], TallyQA [92], TabMWP [93], KonIQ [94], InternVL [95]-filtered synthetic knowledge dataset covering politics, math, physics, chemistry, RLAI-F [96], CLEVR-Math [97], SROIE [98], ChartQA [19], DocVQA [99], FigureQA [100], GQA [101], InfoVQA [102], M3CoT [103], MapQA [104], OK-VQA [105], TextVQA [106], WildVision [107], DVQA [108], GeoQA+ [109], GeOS [110], IconQA [111], UniGEO [112], GeomVerse [113], Geo170K [114], MathV360K [115], multimodal wikipedia knowledge [116], InfoSeek [117], and RAM++ [118]-filtered synthetic data of Infinity-MM [30] covering coarse and fine-grained perception, relation, attribute, and logic reasoning.
+
+# D Broader Impacts
+
+Our reinforcement and imitation learning (RIL) approach democratises access to advanced vision-language capabilities by enabling lightweight, efficient models to run on edge devices and in low-resource settings—thereby lowering both deployment costs and environmental impact. RIL fosters seamless interoperability across diverse model architectures and tokenization schemes. The use of multiple large teacher VLMs to generate varied training samples further accelerates performance gains, underscoring the value of collaborative model development. We believe RIL will inspire future research into efficient multimodal learning techniques and support the widespread, sustainable adoption of lightweight [119] and robust [120-122] AI solutions across a broad range of applications even with different tokenizers [123]. We hope that by bridging efficiency and performance, RIL become a landscape of building sophisticated multimodal AI technologies and accelerating innovation across industries and academia.
\ No newline at end of file
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+# Unified Scaling Laws for Compressed Representations
+
+<table><tr><td>Andrei Panferov*</td><td>Alexandra Volkova*</td><td>Ionut-Vlad Modoranu</td><td>Vage Egiazarian</td></tr><tr><td>ISTA</td><td>ISTA</td><td>ISTA</td><td>ISTA</td></tr><tr><td colspan="2">Mher Safaryan</td><td colspan="2">Dan Alistarh†</td></tr><tr><td colspan="2">ISTA</td><td colspan="2">ISTA &amp; Red Hat AI</td></tr></table>
+
+# Abstract
+
+Scaling laws have shaped recent advances in machine learning by enabling predictable scaling of model performance based on model size, computation, and data volume. Concurrently, the rise in computational cost for AI has motivated model compression techniques, notably quantization and sparsification, which have emerged to mitigate the steep computational demands associated with large-scale training and inference. This paper investigates the interplay between scaling laws and compression formats, exploring whether a unified scaling framework can accurately predict model performance when training occurs over various compressed representations, such as sparse, scalar-quantized, sparse-quantized or even vector-quantized formats. Our key contributions include validating a general scaling law formulation and showing that it is applicable both individually but also compositably across compression types. Based on this, our main finding is demonstrating both theoretically and empirically that there exists a simple "capacity" metric—based on the representation's ability to fit random Gaussian data—which can robustly predict parameter efficiency across multiple compressed representations. On the practical side, we extend our formulation to directly compare the accuracy potential of different compressed formats, and to derive better algorithms for training over sparse-quantized formats. Our source code is available at: IST-DASLab/unified-sc-laws
+
+# 1 Introduction
+
+A key recent advance in machine learning has been the idea of predictable scaling of learning performance with respect to model, computation and data sizes. This approach is encompassed by the scaling laws Kaplan et al. [17], which allow researchers to predict the values of these three parameters required to reach a certain model performance. This powerful idea has been expanded upon in several directions, e.g. [15; 5; 22], and is a key ingredient behind the massive expansion of computational power for AI [12].
+
+A parallel research direction, motivated by this massive increase in computational cost, has been model compression, which proposes a series of techniques to reduce the computational and memory footprint of model inference and training, via techniques such as sparsification [14] and quantization [11]. In this paper, we focus on the interplay between scaling laws and the degree of compression of the representation over which learning occurs. While there is significant emerging work in this direction, e.g. [9; 19; 33; 31], current scaling laws are specialized to single representations (e.g., quantization or sparsity) and/or formats (e.g., integer quantization), and cannot yet address the question of predicting model scaling behavior when training over general compressed representations.
+
+Contributions. This paper is structured two main questions, and their practical ramifications:
+
+Q1: Is there a unified compression scaling law? First, we wish to find a single general law that not only applies to sparse [9] or quantized [19] representations in isolation, but that also provides a good fit for hybrid formats, such as sparse-and-quantized weights, or compound compression, i.e. sparse weights and activations. Through extensive experimentation, we identify this law to be of the form
+
+$$
+L o s s (N, D) \sim A \cdot (N \cdot \rho (R)) ^ {- \alpha} + B \cdot D ^ {- \beta} + E, \tag {1}
+$$
+
+where  $N$  is the number of model parameters,  $D$  is the dataset size,  $E$  is the irreducible error,  $A$ ,  $B$ ,  $\alpha$  and  $\beta$  are constants, and  $\rho$  is a parametric function of the representation  $R$ . Crucially, we find that, even for very complex representations—e.g. 3-bit quantization with group size 32 and  $1\%$  outliers in full-precision—the parametric function  $\rho$  can still predict the scaling of model performance w.r.t. the parameter count  $N$ . We call  $\rho(R)$  the representation capacity of  $R$ . Consequently, there is always a "dense equivalent" parameter count  $N' = N \cdot \rho(R)$  which would yield the same loss during training. The capacity  $\rho(R)$  lies naturally in the interval  $(0, 1]$ , and the key goal of compression is to maximize the trade-off between model accuracy and the size and computational cost of the representation.
+
+Q2: Is capacity an "intrinsic" property of the representation? While related forms of the above law have been proposed in prior work [10; 19], we are the first show that capacity is an intrinsic property of the representation, independent of the model and task for which the scaling law is obtained, but relatable to standard information-theoretic measures. Moreover, we establish the applicability of the law across hybrid (e.g. sparse-quantized weights) or composite (e.g. quantized weights-and-activations) representations.
+
+More precisely, our main finding is that capacity is tightly-correlated with the representation's ability to fit random Gaussian data, measured in terms of minimal mean-squared error (MSE). Concretely,  $\rho(R)$  is a simple parametric function of the MSE of the representation  $R$  when fitting random Gaussian data, i.e.  $\rho(R) = \tilde{\rho}(MSE(R))$ , where instances of the same representation  $R$ , e.g. 3 and 4-bit integer quantization, share the same parametric form  $\tilde{\rho}$ . This finding, which we validate across quantized, sparse, quantized-sparse, and even vector-quantized representations, provides a simple metric to "rank" different formats implementing the same representation. In addition, this also allows us to determine the "optimal" capacity at a certain bit-width, which is given by theoretical bounds on Gaussian fitting for a given support, which can be easily estimated via Monte Carlo algorithms. In addition, we also provide a non-trivial theoretical justification for this relationship in Theorem 1, for Adam-optimized compressed models: we relate the convergence of Adam over compressed representations with the product between the number of parameters  $N$  and the average root mean-squared error of compression across optimization, which connects to our notion of capacity.
+
+Our second finding is that, except for pathological cases, capacity factorizes across composite representations: concretely, the capacity of a 4-bit and 2:4 sparse model is the product between the capacity of the 4-bit dense model, and that of a 2:4-sparse but unquantized model. Factorization allows us to evaluate the capacity of complex representations based on simple ones, and also holds when compressing different model representations, e.g. both weights and activations.
+
+Practical Implications. The analytical metrics suggested by representation capacity also have non-trivial practical applications. First, the fact that we are able to relate the predictive parameter  $\rho$  to intrinsic properties of the underlying representation gives us the ability to analytically predict the representational power of different compressed numerical formats. This way, we can accurately compare and predict the efficacy of various formats such as Floating-Point, Integer (INT with and without grouping), or sparse-quantized formats  $(2:4 + \mathrm{INT})$  at different compression budgets. Second, this framework inspires an improved approach for sparse training, which we show provides significant improvements (above  $20\%$  in some sparsity regimes) in capacity at the same number of parameters.
+
+Overall, our results provide a new lens to view the scaling properties of compressed models, with respect to intrinsic properties of the representation over which training is performed. Thus, we believe that capacity-aware scaling has the potential to become a practical design principle for the next generation of efficient foundation models.
+
+# 2 Preliminaries
+
+Scaling Laws. We start from the "Chinchilla" scaling law formulation [15] that proposed to model loss scaling as a function of the number of parameters in the model  $N$  and the number of data points
+
+Table 1: Representation scaling laws (rows) versus the quantities of interest (columns). For all laws,  $N$  represents the number of parameters,  $D$  is the data, and  $E$  is the irreducible error. For the sparsity scaling law of Frantar et al. [8],  $S$  is the sparsity and the lowercase parameters are learnable constants. For the precision scaling law of Kumar et al. [19],  $P_w$  is the weight precision, and  $\gamma_P$  is a learnable weight sensitivity parameter. For the law of Frantar et al. [10],  $\mathrm{eff}_C$  is the "effective parameter multiplier," that is explicitly fitted for every instance of compression  $C$ . By contrast, our formulation postulates that the parameter efficiency is a simple parametric function of the representation's capacity to fit random Gaussian data  $(GMSE(R))$ .
+
+<table><tr><td>Parametrization</td><td>Formulation for Loss(N,D)</td><td>Sparsity fit
+(Error)</td><td>Quantization fit
+(Error)</td></tr><tr><td>Sparsity S
+Frantar et al. [9]</td><td>aS(1-S)bS+cS/NbN + (aD/D)bD+E</td><td>5.7·10-4</td><td>N/A</td></tr><tr><td>Quantization to Pw
+Kumar et al. [19]</td><td>A[N(1-e-Pw/γw)]-α+BD-β+E</td><td>N/A</td><td>4.5·10-3</td></tr><tr><td>Compression C
+Frantar et al. [10]</td><td>A/(N·effC)α+B/Dβ+E</td><td>4.2·10-4</td><td>1.9·10-3</td></tr><tr><td>Representation R
+(OURS)</td><td>A/(N·ρ(GMSE(R)))α+B/Dβ+E</td><td>4.7·10-4</td><td>2.1·10-3</td></tr></table>
+
+$D$  the model was trained on, in the form the parametric function:
+
+$$
+\operatorname {L o s s} (N, D) = A N ^ {- \alpha} + B D ^ {- \beta} + E, \tag {2}
+$$
+
+where  $A, B, E, \alpha$ , and  $\beta$  are the scaling law parameters that can be fit empirically. It is important to note that such scaling laws assume an ideal, well-tuned training setup, and that the parameter may vary slightly depending on architecture, optimizer, and hyper-parameters.
+
+Compressed Representations. For sparsity, we assume that a specific fraction, within each parameter group of a certain size  $G$ , is set to zero. Sparsity is unstructured if the group is the whole tensor, whereas it is semi-structured (N:M) if  $N$  parameters out of every  $M$  are set to zero. For quantization, unless otherwise stated, we assume that parameters are mapped onto a scalar, symmetric grid corresponding to the number of bits available for quantization, as is standard [11]. (We will also consider vector quantization in Section 4.1.) For sparse-quantized representations, we follow [13] by first applying sparsification, and then quantization, to map continuous parameters onto this format.
+
+Prior Scaling Laws. The relationship between the learning representation and the scaling law formulation was considered by Frantar et al. [9] for sparsity, and by Kumar et al. [19] for quantization. The scaling laws they propose are described in Table 1, together with their parametrization, for the special case of weight-only compression. While both these laws can predict loss with respect to training over their target representations, their formulation is not designed to generalize to other representations, or to hybrid ones (e.g. sparse-quantized).
+
+The unified law we consider extends preliminary work by Frantar et al. [10], who, assuming that training happens over weights compressed in representation  $C$ , proposed a simple parametric law similar to Equation 1, but which is fitted independently for each instance of compressed training, yielding a value of the corresponding parameter efficiency factor, called  $\mathrm{eff}_C$ . Frantar et al. [10] focuses on quantization; they fit sparsity in limited experiments, and do not consider hybrid formats.
+
+Our Approach. By contrast, our focus is on relating parameter efficiency to intrinsic properties of the representation  $R$ : in their parlance, we show that, across all instances of a given compressed representation  $R$ , e.g. uniform integer (INT) quantization, the parameter efficiency has the same parametric form  $\rho(R)$ , and, in fact, this parametric form is simply a function of the MSE for the representation  $R$  w.r.t. random Gaussian data, i.e.  $\rho(R) = \tilde{\rho}(GMSE(R))$ . Importantly,  $GMSE(R)$  is an intrinsic property of  $R$ , and only depends on its own parametrization: for INT, this would be the number of bits we employ per parameter.
+
+The fact that this parametric form is shared across instances of the same representation (Section 4.1), is powerful since it allows us to compare and transfer parameters between instances of the same representation  $R$ . Clearly, if  $GMSE \simeq 0$ , then  $\rho(P) \simeq 1$ , and we recover the original "dense"
+
+scaling law [15]. Interestingly, Table 1 shows that our unified law can provide a better fit than the representation-specific formulations of Frantar et al. [9] and Kumar et al. [19], and (almost) matches the formulation of [10], which is fitted for each compression instantiation  $C$ .
+
+Setting for Experimental Validation. For our scaling law investigations, we pretrained decoder-only Transformers following the Llama architecture [34] for 30M, 50M, 100M and 200M non-embedding parameters. The models were trained on the C4 dataset [28], using the Llama-2 tokenizer [34]. To ensure we operate in a data-rich regime, we use 50, 100, and 200 training tokens per model parameter for each training configuration, and train on fixed-length context windows of 512 tokens. We used AdamW [18; 23] with a 0.1 ratio of warm-up epochs with cosine scheduler. Our experimental setup is very similar to that of [9; 19; 10]. More details are provided in Appendix A.
+
+We follow standard quantization-aware training (QAT) methods, combined with various levels of unstructured weight sparsity. For quantization we employ the gradient estimator of [27], a per-layer uniform quantizer with static scaling factors and gradient masking. Quantization levels range from 1-bit to 8-bit precision. We consider configurations with quantized weights only, activations only, and both simultaneously. For sparsity, we apply unstructured magnitude pruning via top-k thresholding on a per-layer basis. The sparsity mask is recomputed dynamically at each optimization step. For Vector Quantization (VQ), we follow QuEST scalar quantization and apply it to 2- and 4-dimensional HIGGS grids [24]. To restrain outliers we use the trust estimation method [27] that zeros out gradients for any point lying outside a hypersphere of a certain radius.
+
+# 3 Theoretical Analysis
+
+One key focus of our work is whether, given a compressed representation  $R$  over which learning is performed, we can identify a predictive metric that correlates with the representation's efficiency  $\rho(R)$ . To identify this metric, we first model the standard weight-compressed LLM optimization process, which combines the Adam algorithm [18] with the straight-through estimator (STE) [2]. We have:
+
+$$
+\text {S T E G r a d i e n t :} \quad g _ {t} = \widetilde {\nabla} f (\widehat {\theta} _ {t}), \qquad \text {w h e r e} \widehat {\theta} _ {t} = \mathcal {C} (\theta_ {t}),
+$$
+
+$$
+\text {O p t i m i z e r s t a t e s :} \quad m _ {t} = \beta_ {1} m _ {t - 1} + (1 - \beta_ {1}) g _ {t}
+$$
+
+$$
+v _ {t} = \beta_ {2} v _ {t - 1} + (1 - \beta_ {2}) g _ {t} ^ {2}, \quad \widetilde {v} _ {t} = \max  (v _ {t}, \widetilde {v} _ {t - 1})
+$$
+
+$$
+\text {M o d e l} \theta_ {t + 1} = \theta_ {t} - \eta \frac {m _ {t}}{\sqrt {\tilde {v} _ {t} + \epsilon}},
+$$
+
+where  $\widetilde{\nabla}$  represents stochastic mini-batch gradient operator,  $\mathcal{C} \colon \mathbb{R}^N \to \mathbb{R}^N$  is the compression scheme,  $\beta_1, \beta_2 \in (0, 1)$  are momentum parameters,  $\epsilon > 0$  is a small constant for numerical stability and  $\eta > 0$  is the learning rate or the step-size. All vector operations are element-wise, including the max operation. Our analysis relies on the following assumptions, which are standard in adaptive optimization [36; 29; 4; 20; 6; 25; 30]:
+
+Assumption 1 (Lower bound and smoothness). The loss function  $f\colon \mathbb{R}^N\to \mathbb{R}$  is lower bounded by some  $f^{*}\in \mathbb{R}$  and is  $L$ -smooth, namely,  $\| \nabla f(\theta) - \nabla f(\theta^{\prime})\|_{2}\leq L\| \theta -\theta^{\prime}\|_{2}$ , for any  $\theta ,\theta^{\prime}\in \mathbb{R}^{N}$ .
+
+Assumption 2 (Unbiased and bounded stochastic gradient). For all iterates  $t \geq 1$ , the stochastic gradient  $g_t$  at  $\widehat{\theta}_t$  is unbiased,  $\mathbb{E}[g_t] = \nabla f(\widehat{\theta}_t)$ , and uniformly bounded,  $\| g_t \|_{\infty} \leq G_{\infty}$ , by some constant  $G_{\infty} \geq 0$ .
+
+Assumption 3 (Bounded variance). For all iterates  $t \geq 1$ , the variance of the stochastic gradient  $g_{t}$  at  $\widehat{\theta}_{t}$  is uniformly bounded by some constant  $\sigma^2 \geq 0$ , namely  $\mathbb{E}[\| g_t - \nabla f(\widehat{\theta}_t)\| _2^2 ] \leq \sigma^2$ .
+
+In this context, our main claim is the following:
+
+Theorem 1 (Non-convex convergence analysis). Let Assumptions 1, 2 and 3 hold. Then, choosing step-size  $\eta = \min (\eta_0,\frac{1}{\sqrt{T}})$  with  $\eta_0 = \frac{\epsilon(1 - \beta_1)}{2LC\sqrt{N}}$  and  $C = 2\sqrt{G_{\infty}^2 + \epsilon / N}$ , a randomly chosen compressed iterate  $\widehat{\theta}$  from  $\{\widehat{\theta}_1,\dots ,\widehat{\theta}_T\}$  satisfies
+
+$$
+\mathbb {E} [ \| \nabla f (\widehat {\theta}) \| ^ {2} ] \leq \frac {C L G _ {\infty}}{\sqrt {\epsilon}} \cdot \mathbb {E} \left[ \frac {1}{T} \sum_ {t = 1} ^ {T} \| \widehat {\theta} _ {t} - \theta_ {t} \| _ {2} \right] \cdot N + \frac {C \sqrt {N}}{\sqrt {T}} \left(f (\theta_ {1}) - f ^ {*} + \frac {L \sigma^ {2}}{\epsilon}\right) + \mathcal {O} \left(\frac {N ^ {3 / 2}}{T}\right).
+$$
+
+Discussion. Similar convergence analysis for Adam under compressed iterates  $\widehat{\theta}_t$  was performed in the setup of convex online learning with bounded domain condition [16], and in nonconvex
+
+![](images/3e986721ab7b6cc17c724b8a4075387711daa80a6ba6922f90a613ef46dfa107.jpg)  
+Figure 1: Comparison of  $\rho$  fits for scaling law forms from Table 1: (a, left) shows quantizations scaling laws, (b, middle) and (c, right) demonstrate the match between noise injection and QuEST quantization for weight-only and weights+activations quantization.
+
+![](images/c9a93615f30ca05ba6d7ec1bae95d9fd037a964c8f8b1dbfdc2a6d3febbceefe.jpg)
+
+![](images/214de42f64532d779adecd6c1129d5250f9c648827a6441e171ddbbdc9353b10.jpg)
+
+optimization with restricted hyperparameter choices and slower rate in terms of constants and extra log-terms [3]. We now interpret this bound, whose complete proof can be found in the Supplementary. Specifically, the bound shows the ergodic convergence of the gradients taken at compressed iterates, which is the strongest property shown even in the uncompressed case [36]. In turn, this term is bounded by 3 terms on the RHS. The second and third bounding terms are standard in the analysis of uncompressed Adam [36], showing that our analysis is fairly tight. We focus our attention on the first term, whose key part is highlighted in blue: this term consists of absolute constants, multiplying the critical term  $N \cdot \mathbb{E} \left[ \frac{1}{T} \sum_{t=1}^{T} \| \widehat{\theta}_t - \theta_t \|_2 \right]$ . Specifically, this term consists of the number of parameters  $N$  times the average  $\ell_2$  compression error over the model parameters throughout the execution. Thus, this analysis suggests that the parameter efficiency of such models may depend multiplicatively on both the number of parameters, and the compression root-mean-square-error (RMSE) throughout the execution. Importantly, this bound is independent of the compression type.
+
+# 4 Findings
+
+# 4.1 Finding 1: Gaussian RMSE Predicts Representation Capacity
+
+Table 1 presents a number of scaling laws that model the same functions via different parametrizations. One can notice, that both the Sparsity form of Frantar et al. [9] and the Quantization form of Kumar et al. [19] can be reduced to the Decoupled form of Frantar et al. [10] in the third row, by imposing additional constraints (e.g.  $\mathrm{eff}_P = 1 - e^{-P_w / \gamma_w}$  for quantization). Naturally, the Decoupled form can achieve lower fit error, but it does not provide any information about the interpretation of the capacity term, which we call  $\rho(R)$ , across different representations  $R$ . The Sparsity form and the Quantization form, on the other hand, feature intertwining and interpretable parameters. For simplicity, we first focus on the Quantization form for now.
+
+The Functional Form. Kumar et al. [19] choose the functional form  $\rho(P_w) = 1 - e^{-P_w / \gamma_w}$  to model quantization efficiency. By contrast, we propose a different form to model  $\rho(R)$ :
+
+$$
+\widetilde {\rho} (G M S E (R)) = L \cdot \tanh  (F \cdot \log_ {1 / 4} (G M S E (R))) ^ {C}, \tag {3}
+$$
+
+which depends only on the representation  $R$ 's Gaussian-MSE fit, denoted by  $GMSE(R)$ , and on the scalars  $L$ ,  $F$ , and  $C$ , detailed below. The  $GMSE(R)$  is easily computable for any representation, and allows us to bypass the dependency on representation-specific parametrization, such as bit-width or sparsity. Specifically, we fit the scalar parameters for each compression type, e.g. scalar quantization, and then re-use these parameters while varying  $GMSE(R)$  w.r.t. compression parameters, e.g. bit-width. The scalar parameters  $L$ ,  $F$ , and  $C$  allow us to accurately model observed effects such as:
+
+- Imperfect convergence in high-precision: While modern QAT algorithms such as QuEST reach efficiency  $\rho = 1$  for low quantization error, older algorithms such as LSQ (Figure 1 (a)), have an efficiency limit strictly below 1, since for instance its gradient estimator introduces consistent bias. The factor  $L$ , defaulting to 1 for saturating representations, allows us to model this imperfection.  
+- Various low-precision curvature: As seen in Figures 1 (b) and (c), different representations behave differently around  $GMSE = 1$ , with some have noticeably higher curvature ("breakdown").
+
+From Figure 1 (a), one can see how that region disproportionally affects the law of Kumar et al. [19], leading to a very poor fit at higher bitwidths. The factor  $C$ , closer to 1 for representations "more linear" around  $GMSE = 1$ , allows us to more accurately model  $\rho(R)$ .
+
+Quality of Fit. Table 1 shows that our approach leads similar or better quality-of-fit relative to prior laws, covering both scalar quantization and sparsity, while Figure 1 shows  $\rho(R)$  alignment between scaling law forms, compared to Kumar et al. [19], for the QuEST and LSQ quantizers. Again, our approach provides significantly better fit. In Figure 2(a), we show that our method can also provide a good fit for models trained with vector-quantized (VQ) weights, using the projection method of [24], for lattice dimensions 2 and 4. This shows both the versatility of our approach, and the necessity of the  $L$  term, since higher-dimensional VQ appears to have clear sub-unit saturation due to higher bias. We provide further examples in Section 4.3.
+
+# 4.2 Finding 2: Noise Injection as a Scaling Law Predictor
+
+Next, we turn our scaling law on its head, and ask: what if we plug the optimal achievable GMSE for a certain bit-width into the scaling law? In that case, the scaling law should allow us to compute a lower bound on the achievable parameter efficiency given a certain type of representation. In turn, we can find out how close existing quantization- or sparsity-aware training techniques, or numerical formats, are to the information-theoretic lower bound for that specific representation.
+
+Figure 1 (b) illustrates the "optimality gap" for the QuEST algorithm for scalar weight-only quantization across bit-widths, suggesting that this approach is fairly close to optimal. In Figure 1 (c), we compare the fit between actual runs of this QAT algorithm across bit-widths, and the predicted values via noise injection [1] (plugging in the equivalent  $GMSE$ ) into the scaling law, showing a near-perfect fit.
+
+# 4.3 Finding 3: Representation Capacity Is Multiplicative Across Compression Types
+
+In prior work, [19] have claimed that, for their formulation of the law, the representation capacity factorizes independently for quantization of weights and activations. Our experimental findings extend this result, showing that representation capacity,  $\rho(R)$ , also factorizes naturally across a wide range of compression approaches, whether for the same tensor (sparse-and-quantized weights) or for different state tensors (sparse weights and sparse activations). We follow the experimental setup from Section 2, training models with sparse weights and activations, or sparse-quantized weights, or both. We fit a scaling law in the 100 toks/param regime. We show that representation capacity factorizes for the following scenarios:
+
+1. Sparse weights and activations: For sparsity, independently applied to weight and activations,
+
+$$
+\rho \left(R _ {s _ {w}, s _ {a}}\right) = \rho \left(R _ {s _ {w}}\right) \cdot \rho \left(R _ {s _ {a}}\right). \tag {4}
+$$
+
+We summarize the fitted values of  $\rho(R)$  levels in a matrix  $M$  (Figure 2(b)), where each entry corresponds to the fitted efficiency for a model trained with a specific sparsity configuration. Remarkably, the matrix can be accurately approximated by a rank-1 outer product of the first column  $M_{0,:}$  (weight-only) and the top row  $M:,0$  (activations-only) elements, i.e.  $M\approx M_{0,:}\otimes M_{0,:}$ . The resulting parameter efficiencies closely match the product of efficiencies obtained for runs with weight-only and activations-only configurations.
+
+2. Sparse and quantized weights: Given a weight sparsity level  $s$  combined with  $q$ -bit quantization, we claim that the representation capacity can be represented as the product:  $\rho(R_{q,s}) = \rho(R_q) \cdot \rho(R_s)$ . We report the results for different sparsity levels and bit width in Figure 9. Similarly, the matrix  $\rho(R)$  factorizes into the outer product of marginal vectors for quantization-only and sparsity-only representation. Apart from extreme quantization to 2-bit precision, the approximation maintains the error of order of  $10^{-2}$ .  
+3. Sparse and quantized weights, and quantized activations: Finally, we observe that factorization extends to quantization of activations as well. In supplementary experiments, we apply quantization to activation tensors alongside with weight sparsity and quantization. Our results indicate that the representation capacity with weight sparsity  $s_w$  and quantization bitwidth  $q_w$ , and activation sparsity  $q_a$  follows  $\rho(R_{q_w, s_w, q_a}) = \rho(R_{q_w}) \cdot \rho(R_{s_w}) \cdot \rho(R_{q_a})$ .
+
+This result allows for low-cost comparison across compression comparison. Moreover, it facilitates compression hyperparameter tuning and thus predictable model training in a compressed regime.
+
+![](images/ded99225a4fb9074dab3b22ae6ca2cac6455d80b2e1b2a1b219c5227d0b02305.jpg)  
+(a)
+
+![](images/3f7e63b21eba86fe0830b1356d5f4bc1011b605a2660b3a19dcc6004051df4a9.jpg)  
+Figure 2: (a) Scaling law for 2- and 4-dimensional vector quantization. (b) Representation capacity across weight and activation sparsity levels: baseline, factorized prediction, and relative errors. Note the low errors for the factorized predictions, with slight increases at the larger sparsity levels.
+
+![](images/2ae2f1397440737740961b67219500ce4cbad4104e4baef79e58cec6ae3cfbe7.jpg)  
+(b)
+
+![](images/8fe8d0a010ceb403472f8e8e357a876e1233685d79a924f6fba7cb7a5c9e0684.jpg)
+
+![](images/5f38895a2df75a6702208adcaf348e85bd54d88ac6cba06a7b9fe2ef6c9b1e8a.jpg)  
+Figure 3: Comparison of floating point and integer data-types in terms of GMSE, and C4 Validation Loss when trained using the corresponding formats via QuEST, and the resulting capacity  $\rho(R)$ . Observe the high correlation between ranking in terms of GMSE (top), and Val. Loss (bottom).
+
+![](images/2d12e84fd82829ce345616a72d413ad1e51ec12a953ea2cf59922eecb758e1e4.jpg)
+
+# 5 Applications
+
+# 5.1 Application 1: Comparing Compressed Numerical Formats
+
+Practical Formats. The scaling law for compressed representations enables systematic comparison of numerical formats used in quantization, such as INT8, INT4, FP4, or custom low-precision representations, based just on their GMSE, which can be determined via fast Monte Carlo algorithms. Thus, it provides a clear guidance on which low-precision format delivers the best performance for given resource constraints. Figure 3 illustrates this for a number of floating-point and integer
+
+![](images/d290bf0376e9b5b649741710b0b064db338038a1de67e8cc7e68334ac8cbb3eb.jpg)  
+Figure 4: Representation capacity  $\rho(R)$  versus MSE for (a) group-wise quantization, with markers indicate group counts (color encodes quantization bitwidth), and (b) outlier-aware quantization.
+
+![](images/4a5b3fde886adb9dfc35e39b11c52ec34f041a32e88d4bed813706c7df2d49c5.jpg)
+
+![](images/99d517052ee05e4ed438677febab632444b2bd347d61875a63ba43c457b11cd8.jpg)  
+(a)
+
+![](images/5bb3229d8311febcb2774246033caec9b452b917541ca56b1b964b43e6ffdc72.jpg)  
+(b)  
+Figure 5: Backward heuristics: (a) forward mask  $M_{FW}$  determined by the  $T_{k}$  threshold, (b) backward mask determined by threshold  $T_{p}$ , (c) banded backward mask determined by both  $T_{k}$  and  $T_{p}$ , (d) capacity plot compared to the Magnitude Pruning baseline with  $M_{BW} = M_{FW}$ .
+
+![](images/d934186037973d5f17f8d12e136386d46b850623bdc68c2de3a85d8a282c1dcc.jpg)  
+(c)
+
+![](images/5225687fe4268f280455be0d9d8149afd9430aeac755375e9e613f1ba3320467.jpg)  
+(d)
+
+data-types. Specifically, we observe a direct correlation between the ranking of GMSE values (top) and the actual C4 validation loss obtained in actual experiments (bottom). This suggests that our GMSE metric is an accurate predictor of compressed pre-training performance. For instance, it suggests that switching to FP4 (E2M1) will not bring gains relative to INT4 training, and that both formats are close to the theoretical lower bound at 4 bits.
+
+The Impact of Parameter Grouping and Outlier Preservation. A related question regarding formats is whether more complex approaches, such as group-wise quantization, or outlier preservation in higher precision, can disrupt the scaling law. We examine this in Figure 4, which it shows that preserving no outliers (0%) lies on the Pareto-optimal boundary: higher outlier ratios achieve a worse trade-off between the MSE and the representation capacity  $\rho(R)$ . This suggests that, for pre-training it is more effective to allocate bits to encoding the values distribution rather than outlier preservation or careful grouping. This further demonstrates that the proposed RMSE dependency is a general property and remains valid even under diverse structured compression techniques.
+
+Compositionality. An immediate practical application of the multiplicative behavior of the law (Section 4.3) is the ability to estimate the model's performance in advance for arbitrary compression configuration. Given the individual efficiencies of different compression methods, such as quantization or sparsity, applied to weights or activations, one can predict the combined effect without spending additional compute for training.
+
+# 5.2 Application 2: Increasing the Capacity of Sparse Training
+
+The Sparse Training Problem. For our second application, we investigate implications of the RMSE law to maximize the capacity of a sparse representation during training. Specifically, standard sparse training methods such as gradual magnitude pruning (GMP) [37; 9] compute a forward sparsity mask, which we denote by  $M_{FW}$  during the forward pass based on the absolute-magnitude Top-K operation applied to the model parameters  $\theta$  with respect to a target sparsity. Then, a gradient  $\nabla L(TopK(\theta))$  is taken w.r.t. the sparsified weights. Standard baselines, such as the ones we use for sparse training, re-use the forward sparsity mask for the backward, preventing the pruned weights from receiving any gradient. We are interested in heuristics to improve the parameter efficiency of this standard approach, increasing capacity at the same sparsity level.
+
+RMSE-Banded Gradient Masking. For this, we follow the RMSE law and align the parameters  $\theta \in \mathbb{R}^N$  with the standard normal distribution by dividing  $\theta$  by its root mean square  $RMS(\theta) = \sqrt{\frac{1}{N}\sum_{i=1}^{N}\theta_i^2}$ , which results in  $||\theta / RMS(\theta)||_2^2 = N$ . We allow the user to provide a median deviation parameter  $p \in (0, 0.5)$ , which determines the threshold for the backward mask  $T_p = RMS(\theta) \cdot ppf(0.5 + p)$ , where  $ppf$  is the inverse cumulative distribution function of the standard Normal distribution. The multiplication by  $RMS(x)$  "converts" the threshold for the standard Normal distribution to the threshold for the vector  $\theta$ . As a result,  $M_{BW} = |\theta| > T_p$ .
+
+Effectively, our approach, which we call RMSE-Banded Backward Masking (RBBM), sets a backward mask  $M_{BW}$  that may be different than the TopK mask for the forward, whose sparsity is controllable via the parameter  $p$ . To address the fact that it may not allow gradient flow for small parameter values, we allow gradients to flow for the smallest and largest parameters, and create a band between  $T_{p}$  and the TopK threshold  $T_{k}$ , where we do not allow gradient flow. Let  $m = \min(T_{p}, T_{k})$  and  $M = \max(T_{p}, T_{k})$  and define  $M_{BW} = (|\theta| < m) \vee (|\theta| > M)$ . Since we do not control the
+
+relationship between  $T_{p}$  and  $T_{k}$ , we need to ensure that the band is defined correctly. Concretely, the values  $\theta_{i} < m$  and values  $\theta_{i} > M$  will receive gradients, while the values  $\theta_{i} \in [m, M]$  will not.
+
+To illustrate, in Figure 5a we show the structure of forward mask  $M_{FW}$ , were the red region corresponds to values  $|\theta_i| < T_k$  that will not receive any gradient, while the green region corresponds to the values  $|\theta_i| \geq T_k$  which will receive gradient. The top-k threshold  $T_k$  is fixed. In Figure 5b, we have a similar behavior for the backward mask  $M_{BW}$  determined by the threshold  $T_p$ , which is now user-controlled via the median deviation parameter  $p$ . In Figure 5c we show an example for  $T_p < T_k$  where we obtain a banded-mask: the values  $|\theta_i| \in [T_p, T_k]$  will not receive any gradient (red region), while the other values will receive gradient (green region). The band width can be controlled via the parameter  $p \in (0, 0.5)$ . When  $p$  is close to zero, the  $T_p$  value will decrease, having the effect of increasing the width of the red band, where the corresponding weights do not get gradient. When  $p = 0$ , the value of  $T_p$  will be equal to the median and this will be equivalent to the baseline (e.g.  $M_{BW} = M_{FW}$ ) illustrated in Figure 5a.
+
+Results. We apply RBBM for sparse training in our pretraining scenario, for the 30M parameter Llama model, using our training setup from Sec. 2, and for unstructured sparsities between  $10\%$  and  $90\%$ . We compute the capacity of the sparse representation. The results for our RMSE-based heuristic and the standard sparse training baseline (Magnitude Pruning) are provided in Figure 5d. The results show that our RMSE-based approach enables consistently higher capacity than the baseline.
+
+# 6 Related Work
+
+We focus on studies that extended classical scaling laws [17; 15] to model compression. Frantar et al. [9] presented the first scaling law for weight-sparse Transformers, across vision and language and unstructured and semi-structured sparsity. Earlier work by Clark et al. [5] studied mixture-of-experts sparsity, deriving scaling laws in terms of total parameters and compute per token, reinforcing the idea that only effective parameters govern scaling.
+
+A recent breakthrough by Kumar et al. [19] introduced scaling laws that incorporate numerical quantization during training and inference, showing that, as for sparsity, a model trained in low precision behaves like a smaller high-precision model. They also apply their approach to post-training quantization (PTQ), showing that PTQ quality worsens as training data increases. For training-time quantization, their laws suggest that using lower precision allows training larger models on the same compute budget. Relative to their pioneering work, we bring the following improvements. First, we investigate a different and arguably simpler scaling law, showing that it yields a considerably better fit for quantization itself (see Table 1). Second, our key focus is different, we provide a first interpretation of the notion of representation "capacity", together with a theoretical justification, and ample experimental validation. Finally, we validate the factorization property posited by Kumar et al. [19], as well as extensions to hybrid formats. Follow-up work by Sun et al. [33] examines scaling laws for floating-point (FP) formats, finding that the law of Kumar et al. [19] does not provide a good fit in this case, and investigates an extension of the law via additional parametrization.
+
+Preliminary work by Frantar et al. [10] proposed the single-parameter scaling law on which we build, and showed that it can be applied to instances of weight quantization and sparsity, by directly fitting the efficiency parameter. By contrast, we identify a general law, in the sense that the same parametric form can transfer between compression types, to hybrid sparse-quantized formats, as well as to instances where both weights and activations are compressed. More interestingly, we equate the representation capacity factor in the law with a natural notion of representation capacity, show that the law factorizes across representations. In concurrent work, ParetoQ [22] aimed to unify the fragmented landscape of LLM quantization by systematically evaluating the interplay between training strategy, quantization function design, and bit selection. Our results complement their findings: for instance, we obtain that, for the architectures we consider, 2-bit weight-only quantization is Pareto optimal.
+
+# 7 Discussion and Limitations
+
+Our study introduces representation capacity—roughly defined as a simple monotone transform of the Gaussian MSE—as a unified metric when training compressed models across various representations. Capacity enables format comparisons without retraining or exhaustive grid searches, so that future hardware designers can expose any format whose capacity  $\rho$  dominates the Pareto frontier, confident
+
+that software will exploit it optimally. Moreover, our law factorizes, further simplifying the search for the "optimal" training format.
+
+A few caveats remain. First, in line with prior work in this area, our experiments are limited to decoder-only Llama-style architectures trained on C4 in the data-rich regime (100 toks/param); we plan to extend this at larger scale. Second, the law may need specific fits for ultra-low precision (e.g. 2-bit or ternary formats) and for vector-quantization codebooks below 8 entries, suggesting second-order effects may need to be taken into account. Third, while our theoretical evidence uses standard assumptions, it could be extended to more complex representation types.
+
+# Acknowledgments
+
+We would like to thank Lambda Cloud for their generous computational grant. We thank the NVIDIA and Google corporation for their grants, which supported part of this research. Alexandra Volkova and Andrei Panferov were supported in part by the BILAI Cluster of Excellence program.
+
+# References
+
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+
+# Appendix Roadmap
+
+This appendix provides supporting material organized as follows:
+
+- Experimental Setup (Appendix A): Model architectures, hyperparameters, and training configurations.  
+- Factorization of Representation Capacity (Appendix B): Detailed analysis showing how representation capacity matrices can be factorized for various compression techniques including quantization, sparsity, and their combinations.  
+- Ablation Studies on Law Formulation (Appendix C): Investigation of different noise distributions (Gaussian, Logistic, Student's t, Laplace) and functional forms (tanh, logistic) for the scaling law formulation.  
+- Scaling Laws for Vector Quantization (Appendix D): Implementation details and algorithms for vector quantization approaches, including forward and backward pass descriptions for HIGGS-based training.  
+- Theoretical Support (Appendix E): Convergence analysis for Adam optimizer with Straight Through Estimation (STE), including complete proofs and supporting lemmas.  
+- Improved Sparse Training via RBBM (Appendix F): Comparison of our backward mask heuristics against RigL and Gradual Magnitude Pruning, with detailed descriptions of different masking strategies.
+
+# A Experimental setup
+
+Hyperparameters. Table 2 summarizes the architectural and optimization hyperparameters used in this study.
+
+<table><tr><td>Model size</td><td># Layers</td><td># Heads</td><td># Embeddings</td><td>Learning rate</td></tr><tr><td>30 M</td><td>6</td><td>5</td><td>640</td><td>1.2 · 10-3</td></tr><tr><td>50 M</td><td>7</td><td>6</td><td>768</td><td>1.2 · 10-3</td></tr><tr><td>100 M</td><td>8</td><td>8</td><td>1024</td><td>6 · 10-4</td></tr><tr><td>200 M</td><td>10</td><td>10</td><td>1280</td><td>3 · 10-4</td></tr></table>
+
+Table 2: Key architectural and training hyperparameters for Llama family models.
+
+Generalization Across Model Families and Datasets. To validate that our findings are independent of the model family and dataset, we conducted comparable experiments on the OLMo2 [26] family models trained on the ClimbMix dataset [7]. OLMo2 models use no biases, employ rotary positional embeddings [32], RMSNorm [35], and reordered pre-normalization [21; 26]. We also used  $\mathrm{ReLU}^2$  activation function for linear layers.
+
+We matched the model size range, data-model size ratios, and optimizer hyperparameters to those used in our original setup. Scaling laws of the same functional form as in Table 1, where  $\rho (\mathrm{GMSE})$  takes the form of Eq. 3, were refitted from scratch. We report the corresponding estimates for parameters  $(\alpha ,\beta ,E,L,F,C)$  in Table 3. Confidence intervals of one standard deviation were obtained by the bootstrapping procedure. The results match closely with those from the Llama / C4 setup, with overlapping confidence intervals.
+
+We use 8x80GB H100 machines for efficient training, and training one model takes on average 1 hour. To produce the full set of results we ran in total approximately 250 such training runs for various compression configurations.
+
+# B Factorization of Representation Capacity
+
+Figures 6-9 show factorization of the representation capacity matrix for various in-training compression techniques:
+
+1. Quantized weights and activations (Fig. 6).
+
+<table><tr><td></td><td>Llama family / C4</td><td>OLMo2 family / ClimbMix</td></tr><tr><td>α</td><td>0.13 ± 0.05</td><td>0.18 ± 0.03</td></tr><tr><td>β</td><td>0.33 ± 0.06</td><td>0.26 ± 0.02</td></tr><tr><td>E</td><td>1.3 ± 0.5</td><td>1.4 ± 0.3</td></tr><tr><td>L</td><td>1.0 ± 0.7</td><td>0.84 ± 0.10</td></tr><tr><td>F</td><td>0.41 ± 0.02</td><td>0.37 ± 0.04</td></tr><tr><td>C</td><td>1.39 ± 0.08</td><td>1.24 ± 0.13</td></tr></table>
+
+Table 3: Scaling Law hyperparameters for Llama and OMLo2 model families.
+
+2. Sparsity + QuEST quantizer (Fig. 7).  
+3. Joint sparse & quantized weights + activations (Fig. 8), for all combinations  $(s_a, q_a, q_b)$  for sparsity  $s_a \in [0.25, 0.5, 0.75]$  and bit widths  $q_a, q_b \in [2, 4, 6]$ .  
+4. Sparsity + uniform quantizer with maximum absolute value as a scale (Fig. 9).
+
+From the factorized representation-capacity matrices we observe the following:
+
+1. The element-wise error of the fitted coefficients  $\rho$  (from our scaling law) is of order  $10^{-3}10^{-2}$ .  
+2. A rank-1 row-column outer product accurately approximates the matrix, confirming the multiplicative property of representation capacity  $\rho$  in various scenarios.  
+3. Approximation error remains of the order  $10^{-2}$ , except for the cases of extreme 2-bit quantization, where  $\rho \lesssim 0.1$ . We explain this gap due to the poorer performance of the optimizer in these extreme compression regimes, which is not taken into account currently by our model (as it uses the same coefficients for both 16 and 2 bits).
+
+![](images/e428c7aca6028af348400fd9901b0e1c678ddc8201cd03a9c64fb2f11c647a0f.jpg)  
+Figure 6: Representation capacity coefficients for independent quantization of weights and activations. Element-wise  $\rho$  fitting error is not greater than  $5 \cdot 10^{-3}$ .
+
+![](images/64cbcb963d27ac0f792ae3f3b3125e94c0dfa744161d8685edd6d3725110ab42.jpg)
+
+![](images/c3eea744e082739dd4e195f842ce6d1677842a783dae4c5679b2abf71ba04642.jpg)
+
+# C Ablation studies on Law Formulation
+
+# C.1 Evaluating RMSE across Different Distributions
+
+We investigate how the choice of noise distribution used in our law formulation from Sec. 4.1 affects the predicted representation capacity. In Figure 10a we plot the mapping  $\rho (MSE)$  for different bit widths using Logistic, Student's t, and Laplace noise distributions. Each distribution is rescaled to have zero mean and unit variance.
+
+We observe that, no matter which noise distribution we choose, the mapping  $\rho (MSE)$  always remains strictly monotonically decreasing. In principle, one could use heavilytailed distributions (for example, Student-t or Laplace) to give more weight to extreme outlier errors. However, this leads to a smaller range of MSE values. By contrast, assuming Gaussian noise—which we propose—produces the widest spread of MSE, which in turn allows for a better fit for the scaling law. In short, although
+
+![](images/d473c6ef6342b1f1ae137e6c2eb3e1ab8a0cbb05775fff6b11412c71f81222e5.jpg)  
+Figure 7: Representation capacity coefficients with fit errors in case of sparsity combined with the QuEST quantization.
+
+![](images/62fe00bf37c3eb9861b7d900989d2f22ecfe832d8a6bce17da39abdf3b2bfaf8.jpg)
+
+![](images/d902333752ab783928663f0640c28af8c875fa605d2483a03e2f00dbc415b1cf.jpg)
+
+![](images/67e163bc1d541f29cf98ffc4ec1a47fe8c987f7f2e70469fdbce5edfcd646e37.jpg)  
+Figure 8: Representation capacity fit errors for sparse+quantized weights and quantized activations. Error bars denote  $\pm 1$  standard deviation from the mean.
+
+![](images/2c4186f792dadcb2b0e1e43daaea0c9c2b5926a85f4f9192f32cc48585e0111f.jpg)  
+Figure 9: Representation capacity coefficients matrix for sparsity applied with uniform quantization. Element-wise  $\rho$  fitting error is not greater than  $2 \cdot 10^{-3}$ .
+
+![](images/ef31678d2d2809f8a462fcebb1a23ab4b3c51452cb1ea7acbd56dea02eca6ca4.jpg)
+
+![](images/4e9803c42811673ac8c7db2bda62608c0ce3bef988b2350a1ee544b8582e0c24.jpg)
+
+monotonicity is preserved under various distributions, the Gaussian MSE delivers the best overall representation capacity prediction, so we adopt it as our default formulation.
+
+Throughout this work, unless specified otherwise, MSE is computed over standard Gaussian input.
+
+# C.2 Functional form of the Law
+
+The behavior of  $\rho (GMSE)$  observed in our experiments can be captured by fitting multiple smooth, monotonically decreasing functions, with no more than 3 additional parameters. In principle, a wide range of such functions can be used to model this relationship, depending on the desired fit properties.
+
+For lower overall fitting error, we found it beneficial to constrain the function to satisfy boundary conditions  $f(0) = 1$  and  $f(1) = 0$ . This way the correct behavior in the high-error region  $MSE \lesssim 1$  is enforced, which is critical for stable predictions in the extreme compression cases. The corresponding fits are summarized in Table 4, the fitting error is calculated for the combined scaling law  $L(MSE) = L(\rho (MSE))$ .
+
+![](images/ba1b9e8159e35352f508e7e3e67c04116fd8079bba81aacd304ff9392c6840de.jpg)  
+(a) Effect of input noise distribution on the mapping  $\rho (MSE)$
+
+![](images/418ff15f672ff6d5f53184e4ba7604b79a47f52d80341bc47dd46cb3960c6a1b.jpg)  
+(b) Different functions used to fit  $\rho (MSE)$
+
+<table><tr><td></td><td>Functional form</td><td>Fitting error (RMSE)</td></tr><tr><td>Decoupled</td><td>Independently fitted ρ</td><td>4.2 · 10-4</td></tr><tr><td>Tanh</td><td>ρ = tanh(F · log1/4 MSE)C</td><td>4.7 · 10-4</td></tr><tr><td>Logistic</td><td>ρ = (1 + B · MSEA)-1</td><td>4.9 · 10-4</td></tr><tr><td>Logistic (1, 0)</td><td>ρ = 1 - MSEA/1 + B · MSEA</td><td>1.1 · 10-3</td></tr></table>
+
+Table 4: Functional form choices and associated fitting error.
+
+Throughout this work, we adopt the functional form of hyperbolic tangent as it provides the smallest fitting error.
+
+# D Scaling Laws for Vector Quantization
+
+In this section, we provide detailed information about the Vector Quantization approach used to produce the results in Figure 2(a). Algorithms 1 and 2 describe the forward and backward passes over a linear layer actively quantized with HIGGS for row-major weights. As was described earlier, our method is combines ideas from Panferov et al. [27] for the gradient estimator, and Malinovskii et al. [24] for the lattice representation. We use the trust estimation method that zeros out gradients for any point lying outside a hypersphere of radius  $R$ :  $\| x\| _2^2 >R^2$ . Our experiments were conducted on 30M and 50M models using the same set of hyperparameters as in Sec. 2.
+
+# Algorithm 1 VQ Training Forward
+
+1: Input: Input activations  $\mathbf{x}$ , row-major weight  $\mathbf{w}$  
+2:  $\mathbf{w}_h = \mathrm{HT}(\mathbf{w})$  
+3:  $\hat{\mathbf{w}}_h = \mathrm{proj}_{grid}\mathbf{w}_h$  
+4:  $\mathbf{y} = \mathbf{x}\hat{\mathbf{w}}_h^T$  
+5: Return:  $\mathbf{y}, \mathbf{x}, \hat{\mathbf{w}}_h, M_{grid}(\mathbf{w}_h; \hat{\mathbf{w}}_h)$
+
+# E Theoretical Support
+
+Here we provide the full proof of Theorem 1 giving a convergence analysis of the Adam optimizer when used with STE. For completeness, the description of the algorithm is presented in the Algorithm 3.
+
+Algorithm 2 VQ Training Backward
+
+1: Input:  $\frac{\partial L}{\partial \mathbf{y}}$ ,  $\mathbf{x}$ ,  $\hat{\mathbf{w}}_h$ ,  $M_{grid}(\mathbf{w}_h; \hat{\mathbf{w}}_h)$  
+2:  $\frac{\partial L}{\partial \mathbf{x}} = \frac{\partial L}{\partial \mathbf{y}} \hat{\mathbf{w}}_h$  
+3:  $\frac{\partial L}{\partial \hat{\mathbf{w}}_h} = \mathbf{x}^T \frac{\partial L}{\partial \mathbf{y}}$  
+4:  $\frac{\partial L}{\partial \mathbf{w}} = \mathrm{IHT}\left(M_{grid}(\mathbf{w}_h; \hat{\mathbf{w}}_h) \odot \frac{\partial L}{\partial \hat{\mathbf{w}}_h}\right)$  
+5: Return:  $\frac{\partial L}{\partial \mathbf{x}}, \frac{\partial L}{\partial \mathbf{w}}$
+
+Algorithm 3 Adam with Straight Through Estimation (STE) and AMSGrad normalization
+
+1: Input: parameters  $\beta_{1}$ ,  $\beta_{2} \in (0, 1)$ ,  $\epsilon > 0$ , step-size  $\eta > 0$ ,  $\theta_{1} \in \mathbb{R}^{d}$ ,  $m_{0} = v_{0} = \widetilde{v}_{0} = 0_{N}$  
+2: for  $t = \{1, 2, \dots, T\}$  do  
+3:  $\widehat{\theta}_t = \mathcal{C}(\theta_t)$  Compress the model via quantization and/or sparsification  
+4:  $g_{t} = \widetilde{\nabla}_{\theta}f(\widehat{\theta}_{t})$  Compute STE for compressed model  
+5:  $m_{t} = \beta_{1}m_{t - 1} + (1 - \beta_{1})g_{t}$  Update first-order gradient momentum  
+6:  $v_{t} = \beta_{2}v_{t - 1} + (1 - \beta_{2})g_{t}^{2}$  Update second-order gradient momentum  
+7:  $\tilde{v}_t = \max (v_t,\tilde{v}_{t - 1})$  Apply AMSGrad normalization  
+8:  $\theta_{t + 1} = \theta_t - \eta \frac{m_t}{\sqrt{\tilde{\nu}_t + \epsilon}}$  Update the uncompressed model parameters  
+9: end for
+
+Proof. Let  $G$  be the gradient bound with respect to  $\ell_2$  norm, that is,  $\| g_t\| _2\leq G$ . Using the relationship between  $\ell_2$  and  $\ell_{\infty}$  norms, we conclude  $G\leq \sqrt{d} G_{\infty}$ . Let  $\Gamma_t = \mathrm{Diag}^{-1 / 2}(\tilde{v}_t + \epsilon)$  be the preconditioning (diagonal) matrix and rewrite the main update rule as
+
+$$
+\theta_ {t + 1} = \theta_ {t} - \eta \Gamma_ {t} m _ {t}.
+$$
+
+Letting  $\theta_0 = \theta_1$ , define virtual iterates  $x_{t}$  as follows:
+
+$$
+x _ {t} = \frac {1}{1 - \beta_ {1}} \theta_ {t} - \frac {\beta_ {1}}{1 - \beta_ {1}} \theta_ {t - 1}.
+$$
+
+In particular,  $x_{1} = \theta_{1}$ . Then, the update rule for the virtual iterates becomes
+
+$$
+\begin{array}{l} x _ {t + 1} - x _ {t} = \frac {1}{1 - \beta_ {1}} (\theta_ {t + 1} - \theta_ {t}) - \frac {\beta_ {1}}{1 - \beta_ {1}} (\theta_ {t} - \theta_ {t - 1}) \\ { = } { - \frac { \eta } { 1 - \beta _ { 1 } } \Gamma _ { t } m _ { t } + \frac { \eta \beta _ { 1 } } { 1 - \beta _ { 1 } } \Gamma _ { t - 1 } m _ { t - 1 } } \\ = - \frac {\eta}{1 - \beta_ {1}} \Gamma_ {t} m _ {t} + \frac {\eta \beta_ {1}}{1 - \beta_ {1}} \Gamma_ {t - 1} m _ {t - 1} \pm \frac {\eta \beta_ {1}}{1 - \beta_ {1}} \Gamma_ {t} m _ {t - 1} \\ = - \frac {\eta}{1 - \beta_ {1}} \Gamma_ {t} (m _ {t} - \beta m _ {t - 1}) + \frac {\eta \beta_ {1}}{1 - \beta_ {1}} \underbrace {(\Gamma_ {t - 1} - \Gamma_ {t})} _ {\stackrel {\mathrm {d e f}} {=}} m _ {t - 1} \\ = - \eta \Gamma_ {t} g _ {t} + \frac {\eta \beta_ {1}}{1 - \beta_ {1}} \Delta \Gamma_ {t} m _ {t - 1}. \\ \end{array}
+$$
+
+Next we apply smoothness (Assumption 1) of the loss function  $f$  over the iterates  $x_{t}$ :
+
+$$
+f (x _ {t + 1}) \leq f (x _ {t}) + \langle \nabla f (x _ {t}), x _ {t + 1} - x _ {t} \rangle + \frac {L}{2} \| x _ {t + 1} - x _ {t} \| ^ {2}.
+$$
+
+Taking expectation and splitting the inner product into two parts, we obtain
+
+$$
+\begin{array}{l} \mathbb {E} [ f (x _ {t + 1}) ] - \mathbb {E} [ f (x _ {t}) ] \\ \leq - \eta \mathbb {E} \left[ \left\langle \nabla f (x _ {t}), \Gamma_ {t} g _ {t} \right\rangle \right] + \eta \mathbb {E} \left[ \left\langle \nabla f (x _ {t}), \frac {\beta_ {1}}{1 - \beta_ {1}} \Delta \Gamma_ {t} m _ {t - 1} \right\rangle \right] \\ + \frac {\eta^ {2} L}{2} \mathbb {E} \left[ \left\| \Gamma_ {t} g _ {t} - \frac {\beta_ {1}}{1 - \beta_ {1}} \Delta \Gamma_ {t} m _ {t - 1} \right\| ^ {2} \right] \\ = \underbrace {- \eta \mathbb {E} \left[ \left\langle \nabla f \left(\theta_ {t}\right) , \Gamma_ {t} g _ {t} \right\rangle \right]} _ {I} + \underbrace {\eta \mathbb {E} \left[ \left\langle \nabla f (x _ {t}) , \frac {\beta_ {1}}{1 - \beta_ {1}} \Delta \Gamma_ {t} m _ {t - 1} \right\rangle \right]} _ {I I} \\ + \underbrace {\frac {\eta^ {2} L}{2} \mathbb {E} \left[ \left\| \Gamma_ {t} g _ {t} - \frac {\beta_ {1}}{1 - \beta_ {1}} \Delta \Gamma_ {t} m _ {t - 1} \right\| ^ {2} \right]} _ {I I I} + \underbrace {\eta \mathbb {E} \left[ \langle \nabla f (\theta_ {t}) - \nabla f (x _ {t}) , \Gamma_ {t} g _ {t} \rangle \right]} _ {I V}. \tag {5} \\ \end{array}
+$$
+
+In the following, we bound all the four terms mentioned above.
+
+Bounding term I. Let  $\|\Delta \Gamma_t\|$  be the operator norm (with respect to  $\ell_2$  norm) of the matrix  $\Delta \Gamma_t$ . Since  $\Delta \Gamma_t$  is diagonal, the spectral norm coincides with the largest diagonal value in magnitude. Using unbiasedness of the stochastic gradients, we have
+
+$$
+\begin{array}{l} I = - \eta \mathbb {E} \left[ \left\langle \nabla f \left(\theta_ {t}\right), \Gamma_ {t - 1} g _ {t} \right\rangle \right] - \eta \mathbb {E} \left[ \left\langle \nabla f \left(\theta_ {t}\right), \Delta \Gamma_ {t} g _ {t} \right\rangle \right] \\ \leq - \eta \mathbb {E} \left[ \left\langle \nabla f (\theta_ {t}), \Gamma_ {t - 1} \nabla f (\widehat {\theta} _ {t}) \right\rangle \right] + \eta G ^ {2} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ]. \\ = - \eta \mathbb {E} \left[ \left\langle \nabla f (\widehat {\theta} _ {t}), \Gamma_ {t - 1} \nabla f (\widehat {\theta} _ {t}) \right\rangle \right] + \eta \mathbb {E} \left[ \left\langle \nabla f (\widehat {\theta} _ {t}) - \nabla f (\theta_ {t}), \Gamma_ {t - 1} \nabla f (\widehat {\theta} _ {t}) \right\rangle \right] + \eta G ^ {2} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ]. \\ \leq - \eta \lambda_ {\min } (\Gamma_ {t - 1}) \mathbb {E} [ \| \nabla f (\widehat {\theta_ {t}}) \| ^ {2} ] + \eta L G \mathbb {E} [ \| \Gamma_ {t - 1} \| \| \widehat {\theta_ {t}} - \theta_ {t} \| ] + \eta G ^ {2} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ] \\ \leq - \frac {\eta}{C _ {0}} \mathbb {E} [ \| \nabla f (\widehat {\theta_ {t}}) \| ^ {2} ] + \eta L G \mathbb {E} [ \| \widehat {\theta_ {t}} - \theta_ {t} \| \cdot \| \Gamma_ {t - 1} \| ] + \eta G ^ {2} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ], \tag {6} \\ \end{array}
+$$
+
+where we used Assumption 2 and Lemma 3 to bound
+
+$$
+\lambda_ {\min } \left(\Gamma_ {t - 1}\right) \geq \left(\| \tilde {v} _ {t - 1} \| _ {\max } + \epsilon\right) ^ {- 1 / 2} \geq \left(G ^ {2} + \epsilon\right) ^ {- 1 / 2} \stackrel {\text {d e f}} {=} \frac {1}{C _ {0}}.
+$$
+
+Bounding term II. Splitting the inner product again and bounded each term, we get
+
+$$
+\begin{array}{l} I I = \eta \mathbb {E} \left[ \left\langle \nabla f (\theta_ {t}), \frac {\beta_ {1}}{1 - \beta_ {1}} \Delta \Gamma_ {t} m _ {t - 1} \right\rangle \right] + \eta \mathbb {E} \left[ \left\langle \nabla f (x _ {t}) - \nabla f (\theta_ {t}), \frac {\beta_ {1}}{1 - \beta_ {1}} \Delta \Gamma_ {t} m _ {t - 1} \right\rangle \right] \\ \leq \frac {\eta \beta_ {1}}{1 - \beta_ {1}} \mathbb {E} \left[ \| \nabla f (\theta_ {t}) \| \| \Delta \Gamma_ {t} m _ {t - 1} \| \right] + \frac {\eta^ {2} L \beta_ {1} ^ {2}}{(1 - \beta_ {1}) ^ {2}} \mathbb {E} \left[ \| \Gamma_ {t - 1} m _ {t - 1} \| \cdot \| \Delta \Gamma_ {t} m _ {t - 1} \| \right] \\ \leq \frac {\eta \beta_ {1}}{1 - \beta_ {1}} G ^ {2} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ] + \frac {\eta^ {2} \beta_ {1} ^ {2} L G ^ {2}}{(1 - \beta_ {1}) ^ {2} \sqrt {\epsilon}} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ], \tag {7} \\ \end{array}
+$$
+
+where we used the fact that the largest eigenvalue  $\lambda_{\mathrm{max}}(\Gamma_t) = \| \Gamma_t\| = (\| \tilde{v}_t\|_{\min} + \epsilon)^{-1 / 2}\leq \epsilon^{-1 / 2}$ . The second inequality is due to the smoothness of  $f$ , and the last inequality is due to Lemma 1, Assumption 2 and the property of norms.
+
+Bounding term III. This term can be bounded as follows:
+
+$$
+\begin{array}{l} I I I \leq \eta^ {2} L \mathbb {E} \left[ \| \Gamma_ {t} g _ {t} \| ^ {2} \right] + \frac {\eta^ {2} L \beta_ {1} ^ {2}}{(1 - \beta_ {1}) ^ {2}} \mathbb {E} \left[ \| \Delta \Gamma_ {t} m _ {t - 1} \| ^ {2} \right] \\ \leq \frac {\eta^ {2} L}{\epsilon} \mathbb {E} \left[ \| g _ {t} - \nabla f (\widehat {\theta} _ {t}) + \nabla f (\widehat {\theta} _ {t}) \| ^ {2} \right] + \frac {\eta^ {2} L \beta_ {1} ^ {2}}{(1 - \beta_ {1}) ^ {2}} \mathbb {E} \left[ \| \Delta \Gamma_ {t} m _ {t - 1} \| ^ {2} \right] \\ \leq \frac {\eta^ {2} L}{\epsilon} \left(\mathbb {E} [ \| \nabla f (\widehat {\theta} _ {t}) \| ^ {2} ] + \sigma^ {2}\right) + \frac {\eta^ {2} L \beta_ {1} ^ {2} G ^ {2}}{(1 - \beta_ {1}) ^ {2}} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ^ {2} ] \\ \leq \frac {\eta^ {2} L}{\epsilon} \mathbb {E} [ \| \nabla f (\widehat {\theta_ {t}}) \| ^ {2} ] + \frac {\eta^ {2} L \sigma^ {2}}{\epsilon} + \frac {\eta^ {2} L \beta_ {1} ^ {2} G ^ {2}}{(1 - \beta_ {1}) ^ {2}} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ^ {2} ], \tag {8} \\ \end{array}
+$$
+
+where we used Assumption 3 that  $g_{t}$  is unbiased with bounded variance  $\sigma^2$ .
+
+Bounding term IV. Finally, for the fourth term, we have
+
+$$
+\begin{array}{l} I V = \eta \mathbb {E} \left[ \langle \nabla f (\theta_ {t}) - \nabla f (x _ {t}), \Gamma_ {t - 1} g _ {t} \rangle \right] + \eta \mathbb {E} \left[ \langle \nabla f (\theta_ {t}) - \nabla f (x _ {t}), \Delta \Gamma_ {t} g _ {t} \rangle \right] \\ \leq \eta \mathbb {E} \left[ \left\langle \nabla f (\theta_ {t}) - \nabla f (x _ {t}), \Gamma_ {t - 1} \nabla f (\widehat {\theta} _ {t}) \right\rangle \right] + \frac {\eta^ {2} L \beta_ {1}}{1 - \beta_ {1}} \mathbb {E} \left[ \| \Gamma_ {t} m _ {t - 1} \| \| \Delta \Gamma_ {t} g _ {t} \| \right] \\ \stackrel {(a)} {\leq} \frac {\eta \rho}{2 \epsilon} \mathbb {E} [ \| \nabla f (\widehat {\theta} _ {t}) \| ^ {2} ] + \frac {\eta}{2 \rho} \mathbb {E} [ \| \nabla f (\theta_ {t}) - \nabla f (x _ {t}) \| ^ {2} ] + \frac {\eta^ {2} \beta_ {1} L G ^ {2}}{(1 - \beta_ {1}) \sqrt {\epsilon}} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ] \\ \stackrel {(b)} {\leq} \frac {\eta \rho}{2 \epsilon} \mathbb {E} [ \| \nabla f (\widehat {\theta_ {t}}) \| ^ {2} ] + \frac {\eta^ {3} \beta_ {1} ^ {2} L ^ {2}}{2 (1 - \beta_ {1}) ^ {2} \rho} \mathbb {E} \left[ \| \Gamma_ {t} m _ {t - 1} \| ^ {2} \right] + \frac {\eta^ {2} \beta_ {1} L G ^ {2}}{(1 - \beta_ {1}) \sqrt {\epsilon}} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ] \\ \leq \frac {\eta \rho}{2 \epsilon} \mathbb {E} [ \| \nabla f (\widehat {\theta} _ {t}) \| ^ {2} ] + \frac {\eta^ {3} \beta_ {1} ^ {2} L ^ {2}}{2 (1 - \beta_ {1}) ^ {2} \rho \epsilon} \mathbb {E} \left[ \| m _ {t - 1} \| ^ {2} \right] + \frac {\eta^ {2} L \beta_ {1} G ^ {2}}{(1 - \beta_ {1}) \sqrt {\epsilon}} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ], \tag {9} \\ \end{array}
+$$
+
+where (a) is due to Young's inequality and (b) is based on Assumption 1. Now integrating (6), (7), (8), (9) into (5),
+
+$$
+{ I } { \leq } { - \frac { \eta } { C _ { 0 } } \mathbb { E } [ \| \nabla f ( \widehat { \theta } _ { t } ) \| ^ { 2 } ] + \eta L G \mathbb { E } [ \| \widehat { \theta } _ { t } - \theta _ { t } \| \cdot \| \Gamma _ { t - 1 } \| ] + \eta G ^ { 2 } \mathbb { E } [ \| \Delta \Gamma _ { t } \| ] }
+$$
+
+$$
+{ I I } { \leq } { \frac { \eta \beta _ { 1 } } { 1 - \beta _ { 1 } } G ^ { 2 } \mathbb { E } [ \| \Delta \Gamma _ { t } \| ] + \frac { \eta ^ { 2 } \beta _ { 1 } ^ { 2 } L G ^ { 2 } } { ( 1 - \beta _ { 1 } ) ^ { 2 } \sqrt { \epsilon } } \mathbb { E } [ \| \Delta \Gamma _ { t } \| ] }
+$$
+
+$$
+I I I \leq \frac {\eta^ {2} L}{\epsilon} \mathbb {E} [ \| \nabla f (\widehat {\theta} _ {t}) \| ^ {2} ] + \frac {\eta^ {2} L \sigma^ {2}}{\epsilon} + \frac {\eta^ {2} \beta_ {1} ^ {2} L G ^ {2}}{(1 - \beta_ {1}) ^ {2}} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ^ {2} ]
+$$
+
+$$
+{ I V } { \leq } { \frac { \eta \rho } { 2 \epsilon } \mathbb { E } [ \| \nabla f ( \widehat { \theta _ { t } } ) \| ^ { 2 } ] + \frac { \eta ^ { 3 } \beta _ { 1 } ^ { 2 } L ^ { 2 } } { 2 ( 1 - \beta _ { 1 } ) ^ { 2 } \rho \epsilon } \mathbb { E } \left[ \| m _ { t - 1 } \| ^ { 2 } \right] + \frac { \eta ^ { 2 } L \beta _ { 1 } G ^ { 2 } } { ( 1 - \beta _ { 1 } ) \sqrt { \epsilon } } \mathbb { E } [ \| \Delta \Gamma _ { t } \| ] , }
+$$
+
+and taking the telescoping summation over  $t = 1,\ldots ,T$  , we obtain
+
+$$
+\begin{array}{l} \mathbb {E} [ f (x _ {T + 1}) ] - \mathbb {E} [ f (x _ {1}) ] \\ \leq \left(- \frac {\eta}{C _ {0}} + \frac {\eta^ {2} L}{\epsilon} + \frac {\eta \rho}{2 \epsilon}\right) \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \nabla f (\widehat {\theta} _ {t}) \| ^ {2} ] + \frac {T \eta^ {2} L \sigma^ {2}}{\epsilon} + \frac {\eta^ {3} \beta_ {1} ^ {2} L ^ {2}}{2 (1 - \beta_ {1}) ^ {2} \rho \epsilon} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \| m _ {t - 1} \| ^ {2} \right] \\ + \left(\frac {\eta G ^ {2}}{1 - \beta_ {1}} + \frac {\eta^ {2} \beta_ {1} L G ^ {2}}{(1 - \beta_ {1}) ^ {2} \sqrt {\epsilon}}\right) \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ] + \frac {\eta^ {2} \beta_ {1} ^ {2} L G ^ {2}}{(1 - \beta_ {1}) ^ {2}} \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ^ {2} ] + \frac {\eta L G}{T} \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \widehat {\theta} _ {t} - \theta_ {t} \| \cdot \| \Gamma_ {t - 1} \| ] \\ \leq \left(- \frac {\eta}{C _ {0}} + \frac {\eta^ {2} L}{\epsilon} + \frac {\eta \rho}{2 \epsilon} + \frac {\eta^ {3} \beta_ {2} ^ {2} L ^ {2}}{2 (1 - \beta_ {1}) ^ {2} \rho \epsilon}\right) \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \nabla f (\widehat {\theta_ {t}}) \| ^ {2} ] + \frac {T \eta^ {2} L \sigma^ {2}}{\epsilon} + \frac {T \eta^ {3} L ^ {2} \beta_ {1} ^ {2} \sigma^ {2}}{2 (1 - \beta_ {1}) ^ {2} \rho \epsilon} \\ + \left(\frac {\eta G ^ {2}}{1 - \beta_ {1}} + \frac {\eta^ {2} \beta_ {1} L G ^ {2}}{(1 - \beta_ {1}) ^ {2} \sqrt {\epsilon}}\right) \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ] + \frac {\eta^ {2} \beta_ {1} ^ {2} L G ^ {2}}{(1 - \beta_ {1}) ^ {2}} \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \Delta \Gamma_ {t} \| ^ {2} ] + \frac {\eta L G}{T} \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \widehat {\theta_ {t}} - \theta_ {t} \| \cdot \| \Gamma_ {t - 1} \| ], \\ \end{array}
+$$
+
+where we used Lemma 1. Choosing  $\rho = \frac{\epsilon}{2C_0}$  and  $\eta \leq \eta_0 \stackrel{\mathrm{def}}{=} \frac{\epsilon(1 - \beta_1)}{4LC_0}$  and using Lemma 2, we get
+
+$$
+\begin{array}{l} \mathbb {E} [ f (x _ {T + 1}) - f (x _ {1}) ] \leq - \frac {\eta}{2 C _ {0}} \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \nabla f (\widehat {\theta_ {t}}) \| ^ {2} ] + \frac {T \eta^ {2} L \sigma^ {2}}{\epsilon} + \frac {T \eta^ {3} L ^ {2} C _ {0} \beta_ {1} ^ {2} \sigma^ {2}}{(1 - \beta_ {1}) ^ {2} \epsilon^ {2}} \\ + \frac {2 \eta G ^ {2}}{(1 - \beta_ {1}) \sqrt {\epsilon}} + \frac {4 \eta^ {2} \beta_ {1} L G ^ {2}}{(1 - \beta_ {1}) ^ {2} \epsilon} + \frac {\eta L G}{T} \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \widehat {\theta} _ {t} - \theta_ {t} \| \cdot \| \Gamma_ {t - 1} \| ]. \\ \end{array}
+$$
+
+Re-arranging terms, we get
+
+$$
+\begin{array}{l} \frac {1}{T} \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \nabla f (\widehat {\theta} _ {t}) \| ^ {2} ] \leq 2 C _ {0} \left(\frac {f (\theta_ {1}) - f ^ {*}}{T \eta} + \frac {\eta L \sigma^ {2}}{\epsilon} + \frac {\eta^ {2} L ^ {2} C _ {0} \beta_ {1} ^ {2} \sigma^ {2}}{(1 - \beta_ {1}) ^ {2} \epsilon^ {2}}\right) \\ + 4 C _ {0} \left(\frac {G ^ {2}}{T (1 - \beta_ {1}) \sqrt {\epsilon}} + \frac {\eta \beta_ {1} L G ^ {2}}{T (1 - \beta_ {1}) ^ {2} \epsilon}\right) + \frac {2 C _ {0} L G}{T} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \frac {\| \widehat {\theta} _ {t} - \theta_ {t} \| _ {2}}{\sqrt {\epsilon + \| \widetilde {v} _ {t - 1} \| _ {\min}}} \right], \\ \end{array}
+$$
+
+where in the last inequality we used  $x_{1} = \theta_{1}$  and the lower bound  $f^{*}\leq f(\theta)$  for all  $\theta \in \mathbb{R}^d$ . Finally, choosing  $\eta = \min (\eta_0,\frac{1}{\sqrt{T}})$  and considering the two cases, we continue
+
+$$
+\begin{array}{l} \frac {1}{T} \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \nabla f (\widehat {\theta} _ {t}) \| ^ {2} ] \leq 2 C _ {0} \left(\max  \left(1, \frac {1}{\eta_ {0} \sqrt {T}}\right) \frac {f (\theta_ {1}) - f ^ {*}}{\sqrt {T}} + \frac {L \sigma^ {2}}{\epsilon \sqrt {T}} + \frac {L ^ {2} C _ {0} \beta_ {1} ^ {2} \sigma^ {2}}{(1 - \beta_ {1}) ^ {2} \epsilon^ {2} T}\right) \\ + 4 C _ {0} \left(\frac {G ^ {2}}{T (1 - \beta_ {1}) \sqrt {\epsilon}} + \frac {\beta_ {1} L G ^ {2}}{T ^ {3 / 2} (1 - \beta_ {1}) ^ {2} \epsilon}\right) + \frac {2 C _ {0} L G}{T} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \frac {\| \widehat {\theta} _ {t} - \theta_ {t} \| _ {2}}{\sqrt {\epsilon + \| \tilde {v} _ {t - 1} \| _ {\min}}} \right] \\ \leq 2 C _ {0} \left(\frac {f \left(\theta_ {1}\right) - f ^ {*}}{\sqrt {T}} + \frac {L \sigma^ {2}}{\epsilon \sqrt {T}} + \frac {L ^ {2} C _ {0} \beta_ {1} ^ {2} \sigma^ {2}}{\left(1 - \beta_ {1}\right) ^ {2} \epsilon^ {2} T}\right) \\ + 4 C _ {0} \left(\frac {f \left(\theta_ {1}\right) - f ^ {*}}{2 \eta_ {0} T} + \frac {G ^ {2}}{T \left(1 - \beta_ {1}\right) \sqrt {\epsilon}} + \frac {\beta_ {1} L G ^ {2}}{T ^ {3 / 2} \left(1 - \beta_ {1}\right) ^ {2} \epsilon}\right) \\ + \frac {2 C _ {0} L G}{\sqrt {\epsilon}} \mathbb {E} \left[ \frac {1}{T} \sum_ {t = 1} ^ {T} \| \widehat {\theta_ {t}} - \theta_ {t} \| _ {2} \right], \\ \end{array}
+$$
+
+Using the bounds  $G \leq \sqrt{N} G_{\infty}$ ,  $C_0 \leq \frac{\sqrt{N}}{2} C$  and surpassing higher order terms, we simplify the bound to
+
+$$
+\frac {1}{T} \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \nabla f (\widehat {\theta} _ {t}) \| ^ {2} ] \leq \frac {C L G _ {\infty}}{\sqrt {\epsilon}} \mathbb {E} \left[ \frac {1}{T} \sum_ {t = 1} ^ {T} \| \widehat {\theta} _ {t} - \theta_ {t} \| _ {2} \right] \cdot N + \frac {C \sqrt {N}}{\sqrt {T}} \left(f (\theta_ {1}) - f ^ {*} + \frac {L \sigma^ {2}}{\epsilon}\right) + \mathcal {O} \left(\frac {N ^ {3 / 2}}{T}\right),
+$$
+
+which completes the proof of the theorem.
+
+![](images/d6527139f08518c9c0f252a52b549a35bd32c8f3718abb4acf4684e981cc59d2.jpg)
+
+Lemma 1. For any  $t \geq 1$  the following bounds hold:
+
+$$
+\left\| m _ {t} \right\| \leq G, \quad \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \left\| m _ {t} \right\| ^ {2} \right] \leq T \sigma^ {2} + \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \left\| \nabla f (\widehat {\theta} _ {t}) \right\| ^ {2} \right] \tag {10}
+$$
+
+Proof. Let us start with the proof of the first bound on  $m_t$ .
+
+$$
+\begin{array}{l} \left\| m _ {t + 1} \right\| ^ {2} = \left\| \beta_ {1} m _ {t} + (1 - \beta_ {1}) g _ {t + 1} \right\| ^ {2} \\ \leq \beta_ {1} \| m _ {t} \| ^ {2} + (1 - \beta_ {1}) \| g _ {t + 1} \| ^ {2} \\ \leq \beta_ {1} ^ {t} \| m _ {1} \| + (1 - \beta_ {1}) \sum_ {\tau = 2} ^ {t + 1} \beta_ {1} ^ {t + 1 - \tau} \| g _ {\tau} \| ^ {2} = (1 - \beta_ {1}) \sum_ {\tau = 1} ^ {t + 1} \beta_ {1} ^ {t + 1 - \tau} \| g _ {\tau} \| ^ {2}. \\ \end{array}
+$$
+
+Using the bounded gradient assumption, we get
+
+$$
+\left\| m _ {t} \right\| ^ {2} \leq (1 - \beta_ {1}) G ^ {2} \sum_ {\tau = 1} ^ {t} \beta_ {1} ^ {t - \tau} \leq G ^ {2}.
+$$
+
+To derive the bound with expectation, we apply Cauchy-Schwartz inequality and the bounded variance assumption:
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \| m _ {t} \| ^ {2} \right] \leq (1 - \beta_ {1}) \sum_ {t = 1} ^ {T} \sum_ {\tau = 1} ^ {t} \beta_ {1} ^ {t - \tau} \mathbb {E} \left[ \| g _ {\tau} \| ^ {2} \right] \\ \leq \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \| g _ {t} \| ^ {2} \right] = \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \| g _ {t} - \nabla f (\widehat {\theta} _ {t}) + \nabla f (\widehat {\theta} _ {t}) \| ^ {2} \right] \\ \leq \sum_ {t = 1} ^ {T} \left(\sigma^ {2} + \mathbb {E} \left[ \| \nabla f (\widehat {\theta} _ {t}) \| ^ {2} \right]\right) = T \sigma^ {2} + \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \| \nabla f (\widehat {\theta} _ {t}) \| ^ {2} \right]. \\ \end{array}
+$$
+
+![](images/a260dd6644c47a41ba0d0379a1f4caf1f4fe79216b22ea6404bb6050f76417ef.jpg)
+
+Lemma 2. For  $\Delta \Gamma_t = \Gamma_{t-1} - \Gamma_t$  we have
+
+$$
+\sum_ {t = 1} ^ {T} \| \Delta \Gamma_ {t} \| \leq \frac {1}{\sqrt {\epsilon}}, \quad \sum_ {t = 1} ^ {T} \| \Delta \Gamma_ {t} \| ^ {2} \leq \frac {1}{\epsilon}.
+$$
+
+Proof. From the definitions of  $\Gamma_t = \mathrm{Diag}^{-1/2}(\tilde{v}_t + \epsilon)$  and  $\tilde{v}_t = \max(v_t, \tilde{v}_{t-1})$  imply that  $\Delta \Gamma_t = \Gamma_{t-1} - \Gamma_t$  is positive semidefinite. Hence,  $\|\Delta \Gamma_t\| = \lambda_{\max}(\Delta \Gamma_t) \geq 0$ . Using the convexity of  $\lambda_{\max}$  over symmetric matrices, we get
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \| \Delta \Gamma_ {t} \| = \max  _ {i} \sum_ {t = 1} ^ {T} \Delta \Gamma_ {t, i} \\ = \max  _ {i} \sum_ {t = 1} ^ {T} \left(\frac {1}{\sqrt {\tilde {v} _ {t - 1 , i} + \epsilon}} - \frac {1}{\sqrt {\tilde {v} _ {t , i} + \epsilon}}\right) = \max  _ {i} \left(\frac {1}{\sqrt {\tilde {v} _ {0 , i} + \epsilon}} - \frac {1}{\sqrt {\tilde {v} _ {T , i} + \epsilon}}\right) \leq \frac {1}{\sqrt {\epsilon}} \\ \end{array}
+$$
+
+For the second sum of squared norms, notice that for scalars  $a \geq b \geq 0$ , it holds that
+
+$$
+(a - b) ^ {2} \leq (a - b) (a + b) = a ^ {2} - b ^ {2}.
+$$
+
+Therefore, the above derivation can be repeated without the square roots as follows:
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \| \Delta \Gamma_ {t} \| ^ {2} = \max _ {i} \sum_ {t = 1} ^ {T} \Delta \Gamma_ {t, i} ^ {2} \\ = \max  _ {i} \sum_ {t = 1} ^ {T} \left(\frac {1}{\sqrt {\tilde {v} _ {t - 1 , i} + \epsilon}} - \frac {1}{\sqrt {\tilde {v} _ {t , i} + \epsilon}}\right) ^ {2} \\ = \max  _ {i} \sum_ {t = 1} ^ {T} \left(\frac {1}{\tilde {v} _ {t - 1 , i} + \epsilon} - \frac {1}{\tilde {v} _ {t , i} + \epsilon}\right) = \max  _ {i} \left(\frac {1}{\tilde {v} _ {0 , i} + \epsilon} - \frac {1}{\tilde {v} _ {T , i} + \epsilon}\right) \leq \frac {1}{\epsilon}, \\ \end{array}
+$$
+
+which completes the proof.
+
+Lemma 3. For all iterates  $t \geq 1$  the following bound holds
+
+$$
+\| \tilde {v} _ {t} \| _ {\infty} \leq G ^ {2}.
+$$
+
+Proof. From the update rules we get the bound for  $v_{t}$  using the initialization  $v_{0} = 0$ :
+
+$$
+\begin{array}{l} \left\| v _ {t} \right\| _ {\infty} \leq \left\| v _ {t} \right\| _ {1} \leq \beta_ {2} \left\| v _ {t - 1} \right\| _ {1} + \left(1 - \beta_ {2}\right) \left\| g _ {t} \right\| ^ {2} \\ \leq \beta_ {2} \| v _ {t - 1} \| _ {1} + (1 - \beta_ {2}) G ^ {2} \\ \leq \beta_ {2} ^ {t} \| v _ {0} \| _ {1} + (1 - \beta_ {2}) G ^ {2} \sum_ {\tau = 0} ^ {t - 1} \beta_ {2} ^ {\tau} \leq G ^ {2}. \\ \end{array}
+$$
+
+Hence, using the update rule of  $\tilde{v}_t$  and initialization  $\tilde{v}_0 = 0$ , we conclude
+
+$$
+\left\| \tilde {v} _ {t} \right\| _ {\infty} \leq \max  (\left\| v _ {t} \right\| _ {\infty}, \left\| \tilde {v} _ {t - 1} \right\| _ {\infty}) \leq G ^ {2}.
+$$
+
+Next, we simplify the optimization setup by considering SGD optimizer over (still generally nonconvex) quadratics. In this special case, we provide improved and generally optimal asymptotic convergence rate. Moreover, we do not use the bounded gradient condition (i.e.,  $\| g_t\|_{\infty}\leq G_{\infty}$ ) of Assumption 2 in this analysis.
+
+More formally, consider iterates  $\theta_{t + 1} = \theta_t - \eta \widetilde{\nabla}_\theta f(\widehat{\theta}_t)$ , where  $\widehat{\theta}_t = \mathcal{C}(\theta_t)$  is the compressed model. Suppose that the loss function is quadratic with Hessian matrix  $\mathbf{H}\in \mathbb{R}^{N\times N}$  and our compression scheme  $\mathcal{C}\colon \mathbb{R}^N\to \mathbb{R}^N$  is unbiased, namely  $\mathbb{E}_{\mathcal{C}}[\widehat{\theta}_t] = \theta_t$ . Since the loss is quadratic, we have
+
+$$
+\nabla f (\widehat {\theta} _ {t}) = \nabla f \left(\theta_ {t} + \left(\widehat {\theta} _ {t} - \theta_ {t}\right)\right) = \nabla f \left(\theta_ {t}\right) + \mathbf {H} \left(\widehat {\theta} _ {t} - \theta_ {t}\right).
+$$
+
+Denote by  $\mathbb{E}_t = \mathbb{E}[\cdot |\theta_t]$  the conditional expectation conditioned on iterate  $\theta_{t}$ , and apply unbiasedness of the compression to get
+
+$$
+\mathbb {E} _ {t} \| \nabla f (\widehat {\theta} _ {t}) \| ^ {2} = \| \nabla f (\theta_ {t}) \| ^ {2} + \mathbb {E} _ {t} \| \mathbf {H} (\widehat {\theta} _ {t} - \theta_ {t}) \| ^ {2} \tag {11}
+$$
+
+Therefore,
+
+$$
+\begin{array}{l} \mathbb {E} _ {t} \left[ f \left(\theta_ {t + 1}\right) - f ^ {*} \right] \\ \leq \left(f (\theta_ {t}) - f ^ {*}\right) - \eta \mathbb {E} _ {t} [ \langle \nabla f (\theta_ {t}), \widetilde {\nabla} f (\widehat {\theta} _ {t}) \rangle ] + \frac {L \eta^ {2}}{2} \mathbb {E} _ {t} [ \| \widetilde {\nabla} f (\widehat {\theta} _ {t}) \| ^ {2} ] \\ \leq \left(f (\theta_ {t}) - f ^ {*}\right) - \eta \mathbb {E} _ {t} [ \langle \nabla f (\theta_ {t}), \nabla f (\widehat {\theta} _ {t}) \rangle ] + \frac {L \eta^ {2}}{2} \mathbb {E} _ {t} [ \| \nabla f (\widehat {\theta} _ {t}) \| ^ {2} ] + \frac {L \eta^ {2}}{2} \sigma^ {2} \\ = \left(f \left(\theta_ {t}\right) - f ^ {*}\right) - \eta \mathbb {E} _ {t} \left[ \| \nabla f \left(\theta_ {t}\right) \| ^ {2} \right] + \frac {L \eta^ {2}}{2} \mathbb {E} _ {t} \left[ \| \nabla f \left(\theta_ {t}\right) \| ^ {2} \right] + \frac {L \eta^ {2}}{2} \mathbb {E} _ {t} \left[ \| \mathbf {H} \left(\widehat {\theta} _ {t} - \theta_ {t}\right) \| ^ {2} \right] + \frac {L \eta^ {2}}{2} \sigma^ {2} \\ = \left(f \left(\theta_ {t}\right) - f ^ {*}\right) - \eta (1 - L \eta / 2) \mathbb {E} _ {t} \left[ \| \nabla f \left(\theta_ {t}\right) \| ^ {2} \right] + \frac {L \eta^ {2}}{2} \left(\mathbb {E} _ {t} \left[ \| \widehat {\theta} _ {t} - \theta_ {t} \| _ {\mathbf {H} ^ {2}} ^ {2} \right] + \sigma^ {2}\right) \\ \leq (f (\theta_ {t}) - f ^ {*}) - \frac {\eta}{2} \mathbb {E} _ {t} [ \| \nabla f (\theta_ {t}) \| ^ {2} ] + \frac {L \eta^ {2}}{2} (\mathbb {E} _ {t} [ \| \widehat {\theta} _ {t} - \theta_ {t} \| _ {\mathbf {H} ^ {2}} ^ {2} ] + \sigma^ {2}), \\ \end{array}
+$$
+
+where we used  $\mathbb{E}_t[\nabla f(\widehat{\theta}_t)] = \nabla f(\theta_t)$  due to the unbiasedness of compression and enforced the bound  $\eta \leq \frac{1}{L}$  in the last step. Hence,
+
+$$
+\frac {1}{T} \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \nabla f (\theta_ {t}) \| ^ {2} ] \leq \frac {2 (f (x _ {1}) - f ^ {*})}{\eta T} + \eta L \left(\mathbb {E} \left[ \frac {1}{T} \sum_ {t = 1} ^ {T} \| \widehat {\theta} _ {t} - \theta_ {t} \| _ {\mathbf {H} ^ {2}} ^ {2} \right] + \sigma^ {2}\right).
+$$
+
+Choosing the step size  $\eta = \min \left(\frac{1}{L},\frac{1}{\sqrt{T}}\right)$  and applying  $L$ -smoothness, we get  $\mathcal{O}(1 / \sqrt{T})$  convergence rate for the uncompressed iterates  $\theta_t$ :
+
+$$
+\frac {1}{T} \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \nabla f (\theta_ {t}) \| ^ {2} ] \leq \frac {1}{\sqrt {T}} \left(2 (f (x _ {1}) - f ^ {*}) + L \sigma^ {2} + L ^ {3} \mathbb {E} \left[ \frac {1}{T} \sum_ {t = 1} ^ {T} \| \widehat {\theta_ {t}} - \theta_ {t} \| _ {2} ^ {2} \right]\right) \max  \left(1, \frac {L}{\sqrt {T}}\right).
+$$
+
+For the convergence bound with respect to the compressed iterates  $\widehat{\theta}_t$ , we apply (11) to quantify the exact difference in average gradient norms with the following identity:
+
+$$
+\frac {1}{T} \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \nabla f (\widehat {\theta} _ {t}) \| _ {2} ^ {2} ] = \frac {1}{T} \sum_ {t = 1} ^ {T} \mathbb {E} [ \| \nabla f (\theta_ {t}) \| _ {2} ^ {2} ] + \mathbb {E} \left[ \frac {1}{T} \sum_ {t = 1} ^ {T} \| \mathbf {H} (\widehat {\theta} _ {t} - \theta_ {t}) \| _ {2} ^ {2} \right].
+$$
+
+Thus, a randomly chosen compressed iterate  $\widehat{\theta}$  from  $\{\widehat{\theta}_1,\dots ,\widehat{\theta}_T\}$  satisfies
+
+$$
+\mathbb {E} [ \| \nabla f (\widehat {\theta}) \| ^ {2} ] \leq L ^ {2} \cdot \mathbb {E} \left[ \frac {1}{T} \sum_ {t = 1} ^ {T} \| \widehat {\theta} _ {t} - \theta_ {t} \| _ {2} ^ {2} \right] + \mathcal {O} \left(\frac {1}{\sqrt {T}}\right).
+$$
+
+# F Improved Sparse Training via RBBM
+
+# F.1 Comparison against RigL
+
+In this subsection we compare our backward mask heuristic in Figure5d with the RigL method of (Evci et al., 2020). We run two instances of RigL: 1) the default one that updates the mask once at
+
+100 steps (i.e.  $\Delta = 100$ ) and updates the mask for the last time at  $75\%$  of training and 2) a version of RigL that is closer to our RBBM setup, which changes the mask at each step (i.e.  $\Delta = 1$ ) during the entire training. In Figure 11a we observe that both versions of RigL induce lower capacity than our naive baseline for a fixed sparsity.
+
+![](images/72de9613677a164319ad0102b18adaacf93ee3f6e2b75da8982080d8fe1d440a.jpg)  
+(a) Pre-training Llama-30M with different sparsities using our MP baseline, RBBM heuristic and RigL variations.  
+Figure 11: Comparison of sparse training methods for Llama-30M.
+
+![](images/6bdf3447b19bc3b57270b0ea156230ece20c55632ace6d4c7e5a9277818e6c84.jpg)  
+(b) Pre-training Llama-30M with different sparsities using our constant MP (CMP) baseline, GMP and RBBM heuristic with GMP schedule.
+
+# F.2 Comparison against Gradual Magnitude Pruning (GMP)
+
+In this section we show our results for applying the GMP sparsity schedule [37] for our setup in Figure 11b. Our first baseline is the constant Magnitude Pruning (CMP), where the backward mask is identical to the forward mask (determined by Top-K) and the sparsity is kept fixed during training. The second baseline is the original GMP where sparsity increases gradually and we compare against the gradual sparsity schedule applied to our b-rms heuristic.
+
+We observe our RBBM heuristic with GMP schedule has lower capacity than both CMP and GMP baselines when sparsity is  $< 40\%$ . However, for sparsities  $\geq 40\%$  there is no significant difference in capacity between CMP and GMP schedules.
+
+# F.3 Backward Mask Heuristics
+
+In this section we provide more details about our backward heuristics.
+
+Our purpose is to perform sparse training for both forward and backward passes. All models are trained with the same learning rates as in the Quest project.
+
+Notations. Let  $\theta$  be the model parameters,  $M_{FW}$  be the mask for the forward pass, and  $M_{BW}$  be the mask for the backward pass.
+
+Forward Pass. The mask for the forward pass is computed using Top-K operator, where  $K$  is chosen based on the target sparsity. Supposing Top-K returns the indices of largest entries by magnitude, the  $i^{th}$  entry in the forward mask for a tensor  $x$  is computed using the indicator function  $\mathbb{I}$  as follows:
+
+$$
+M _ {F W} ^ {i} (x) = \mathbb {I} [ i \in \operatorname {T o p} K (| x |) ] \tag {12}
+$$
+
+Backward Pass. The mask for the backward pass is computed using a few heuristics described below.
+
+1. fw: the backward mask is simply set to the forward mask:  $M_{BW} = M_{FW}$ . This heuristic allows gradients to flow only through the largest parameters by magnitude selected by Top-K, while the low-magnitude parameters will have zero gradient  
+2. rms: we align the tensor  $x$  with the standard normal distribution by dividing  $x$  by its root mean square  $RMS(x) = \sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}$ , which results in  $||x / RMS(x)||_2^2 = n$ . For this heuristic, the
+
+user sets a median deviation parameter  $p \in (0, 0.5)$ , which is used to determine the threshold for the backward mask  $T_{RMS}(x, p) = RMS(x) \cdot ppf(0.5 + p)$ , where  $ppf$  is the inverse cumulative distribution function of the standard normal distribution (see the scipy.stats(norm.ppf function). The multiplication with  $RMS(x)$  has the purpose of converting the threshold for the standard normal distribution to the threshold for the vector  $x$ . As a result,  $M_{BW} = |x| > T_{RMS}(x, p)$ .
+
+3. banded-rms (b-rms): the rms heuristic has the property that the absolute values of  $x$  that are larger than the threshold  $T_{RMS}(x,p)$  will have value 1 in the mask, while the smaller ones will have value 0. This banded heuristic determines the backward mask using the threshold  $T_{RMS}(x,p)$  computed for the rms heuristic in conjunction with the Top-K threshold (which we denote by  $T_{k}$ ). We want to allow gradients to flow for the small parameters and create a band between  $T_{RMS}(x,p)$  and  $T_{k}$  where we do not allow gradients. Concretely, the backward mask is set as follows:  $M_{BW} = (|x| < \min(T_{RMS}(x,p), T_{k})) \vee (\max(T_{RMS}(x,p), T_{k}) < |x|)$ . Since the median deviation  $p$  is a hyper-parameter, we do not have any control over the relationship between  $T_{RMS}(x,p)$  and  $T_{k}$  and we are using the min and max functions to make sure the band is valid, e.g. the parameters do not receive gradient if they lie in the interval  $[\min(T_{k}, T_{RMS}(x,p)), \max(T_{k}, T_{RMS}(x,p))]$ .
+
+4. area-banded-rms (a-b-rms): in the b-rms heuristic we do not have any control over the relationship between the Top-K threshold  $T_{k}$  and  $T_{RMS}(x,p)$ . Let us discuss the two possible cases:
+
+(a)  $T_{k} < T_{RMS}(x,p): M_{BW} = (|x| < T_{k}) \lor (T_{RMS}(x,p) < |x|)$ , which means that all values from  $x$  with a lower magnitude than  $T_{k}$  and larger magnitude than  $T_{RMS}(x,p)$  will get gradient, while the values in the range  $[T_{k}, T_{RMS}(x,p)]$  will not receive gradient, even though they were selected among the Top-K during the forward pass.  
+(b)  $T_{RMS}(x,p) < T_k : M_{BW} = (|x| < T_{RMS}(x,p)) \vee (T_k < |x|)$ , which is the desired case we developed the b-rms heuristic for: the largest entries from  $x$  according to the Top-K rule will receive gradient, as well as the entries smaller than  $T_{RMS}(x,p)$ . The entries lying in the interval  $[T_{RMS}(x,p), T_k]$  will not receive gradient.
+
+We want to make sure that case (a) above does not happen in practice and force the heuristic to behave as in the case (b). For this, we need to change the way we compute the threshold  $T_{RMS}(x,p)$ .
+
+The area-b-rms heuristic uses the area hyper-parameter  $a \in [0,1]$  (instead of median deviation  $p$ ) and expresses the width of the band starting from the Top-K parameter  $T_{k}$  towards zero to compute the threshold  $T_{a}$  to make sure the condition  $T_{a} < T_{k}$  always holds. As a result,  $M_{BW} = (|x| < T_{a}) \vee (T_{k} < |x|)$ . For example,  $a = 0$  yields  $T_{k} = T_{a}$  and this heuristic turns into  $\mathbf{f}\mathbf{w}$ , while  $a = 1$  yields  $T_{a} = 0$  and is equivalent to  $M_{BW} = \mathbf{1}_{d}$  (all entries set to 1, meaning all parameters get gradients). When  $a \in (0,1)$ , the parameters smaller than  $T_{a}$  or larger than  $T_{k}$  get gradients, while the parameters lying in the interval  $[T_{a}, T_{k}]$  do not get gradients.
+
+How to compute the threshold  $T_{a}$ ? Compared to the threshold computation for the previous heuristic, the definition for  $T_{a}$  is slightly more complicated and it was computed graphically. Let  $f$  be the cdf function and  $f^{-1}$  be the ppf function (inverse cdf) for the standard normal distribution.
+
+$$
+T _ {a} = R M S (x) \cdot f ^ {- 1} \left(0. 5 + (1 - a) \cdot \left(f \left(\frac {T _ {k}}{R M S (x)}\right) - 0. 5\right)\right) \tag {13}
+$$
+
+Explanations for the formula above. Suppose the Top-K threshold  $T_{k}$  has a corresponding cdf of 0.8 and we set  $a = 0.5$  (which means  $50\%$ ). We need to set the threshold  $T_{a}$  such that  $(f(T_{k}) - f(T_{a})) / (f(T_{k}) - 0.5) = a$ , where 0.5 is the cdf of the mean (which is identical to the median for a standard normal distribution). This ratio expresses the length of the band  $[0.5, f(T_{k})]$  in the cdf space starting from  $f(T_{k})$  towards the median. As a consequence, the threshold  $T_{a} = ppf(0.65)$  because the quantile 0.65 is the center of the interval  $[0.5, cdf(T_{k})] = [0.5, 0.8]$ . The explanations of each term follow:
+
+![](images/38aadc8776179ecd5dc181256fdd7c46aec58663e272cd75f36141f86d91a9bb.jpg)
+
+- A:  $f = cdf$  computes the corresponding quantile of the Top-K threshold  $T_{k}$  normalized by  $RMS(x)$  
+- B: subtract 0.5 from term  $A$  to compute the length of the interval  $[0.5, f(T_k^{RMS})]$  
+- C: multiply by  $1 - a$  because we take into consideration the band length that starts at  $f(T_{k}^{RMS})$  towards 0  
+- D: compute the cdf of  $T_{a}$  by offsetting again by 0.5 (the quantile of the median)  
+- E: use  $p p f = f^{-1}$  to obtain the value that corresponds to  $c d f(T_{a})$  for the standard normal distribution  
+- F: multiply by  $RMS(x)$  to obtain  $T_{a}$  in the same space as  $x$
+
+Technical note. One could determine the threshold  $T_{a}$  naively by employing the formula  $T_{a}^{naive} = (1 - a)T_{k}$ . Despite simpler, this naive approach leads to a narrower band because the cdf space is non-linear.
+
+Conclusion. The mask computed using the a-b-rms heuristic is more straightforward to understand because the parameter  $a$  describes the area of the red band (where parameters do not receive gradients) as a percentage of the area between 0 and the Top-K threshold  $T_{k}$ . This heuristic can be used as a replacement for b-rms and the parameter  $a$  should be tuned, similarly to parameter  $p$  for b-rms, with the distinction that  $a \in [0,1]$  (for a-b-rms) and  $p \in (0,0.5)$  (for b-rms).
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: See final section.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: See Section 3.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: Code for reproducing the results is provided.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: Code is provided.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: This is standard and specified. Code is provided.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: We provide quality of fit for the scaling laws.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We provide this in the supplementary material.
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [NA]
+
+Justification: The paper is technical in nature, so there isn't anything significant to discuss on this point.
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [Yes]
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [Yes]
\ No newline at end of file
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+# Unified Transferability Metrics for Time Series Foundation Models
+
+Weiyang Zhang $^{1}$  Xinyang Chen $^{1}$  Xiucheng Li $^{1}$  Kehai Chen $^{1}$  Weili Guan $^{2}$  Liqiang Nie $^{1}$ \
+ $^{1}$ School of Computer Science and Technology, Harbin Institute of Technology (Shenzhen)\
+ $^{2}$ School of Information Science and Technology, Harbin Institute of Technology (Shenzhen)\
+{zhangweiyang12138, chenxinyang95, nieliqiang}@gmail.com\
+{lxiucheng, chenkehai, guanweili}@hit.edu.cn
+
+# Abstract
+
+With the increasing number of time series pre-trained models, designing transferability evaluation metrics for time series has become an urgent problem to address. While transferability evaluation has been extensively studied in computer vision, we aim to address a critical gap by developing tailored metrics for time series analysis. In this paper, we introduce TEMPLATE, a transferability estimation framework specifically tailored for versatile time series analysis, comprising three complementary metrics: (1) Dependency Learning Score quantifies a model's capacity to capture temporal dependencies. (2) Pattern Learning Score evaluates the representation quality in extracting discriminative temporal patterns. (3) Task Adaptation Score assesses cross-task generalization capability, enabling versatile time series analysis. TEMPLATE presents a versatile framework compatible with both classification and regression paradigms. Through comprehensive benchmarking across 5 distinct downstream tasks, our method demonstrates superior capability in identifying optimal pre-trained models from heterogeneous model pools for transfer learning. Compared to the state-of-the-art method ETran, our approach improves the weighted Kendall's  $\tau_w$  across 5 downstream tasks by  $35\%$ . The code is available at https://github.com/TEMPLATE.
+
+# 1 Introduction
+
+Recently, pre-trained models have drawn increasing attention in the time series domain due to their exceptional performance in computer vision and natural language processing [1]. These models have achieved significant success across various time series downstream tasks [2, 3] and are readily available on platforms like HuggingFace [4] and TensorFlow Hub [5]. However, no single model consistently outperforms others across all datasets. Therefore, selecting the most suitable pre-trained time series model for a given target task has become a pressing challenge. A time-consuming solution is to fine-tune all pre-trained models on the target dataset and then select the best-performing fine-tuned model. But compared to pre-trained models in computer vision such as ResNet [6] and MobileNet [7], time-series pre-trained models have much larger parameter scales [8, 9], making direct brute-force fine-tuning incur enormous time costs and high computational resource requirements [10], as shown in the left part of Figure 1.
+
+Recent studies propose fast transferability evaluation methods to efficiently rank models and select the optimal one. Existing methods can generally be categorized into static and dynamic approaches [11]. Static methods calculate scores directly based on the statistical information of the model, such as LEEP [12], NLEEP [13], H-score [14] and TMI [15]. In contrast, dynamic methods transform this statistical information using certain learning frameworks or representation space mapping algorithms before calculating scores, such as SFDA [16], LogME [17], and ETran [18]. These approaches are empirically validated as effective metrics for selecting computer vision models.
+
+![](images/4c441d475447ff9db47362994cec277267cecf9fb200e33ef1eeeca40bb6edd7.jpg)  
+Figure 1: Pre-trained models' fine-tune overhead (left) and transferability metric framework (right).
+
+![](images/9d3d35887870d43b395c487e4e0a5d4779f485e9388a667f2af8aada482a7dce.jpg)
+
+Nevertheless, existing research primarily focuses on image-based tasks. When applied to time series, several challenges emerge, as shown in the right part of Figure 1. First, mainstream time series tasks exhibit significant diversity, encompassing forecasting, classification, imputation, and anomaly detection, among others. While existing methods predominantly concentrate on classification-based tasks, achieving cross-task generalization capability remains the primary challenge. Second, downstream time series datasets frequently exhibit large-scale characteristics, resulting in high-dimensional computational infeasibility. This consequently imposes stringent requirements on evaluation metrics, demanding both computational efficiency and measurement accuracy. Last but not least, computer vision models consistently employ standardized data preprocessing protocols, consequently maintaining minimal variance in feature dimensionality. But in time series analysis, the architectural heterogeneity of models and their divergent approaches to processing inter-channel dependencies lead to substantial discrepancies in feature dimensionality. This consequently induces feature space mismatch, rendering evaluation metrics that rely on numerical magnitudes such as intra-class variance [15, 19] or singular value [17] ineffective. Table 1 summarizes the challenges in developing transferability metrics for time series. Prior to this paper, there is no ideal solution for pre-trained model selection specially designed to tackle these challenges.
+
+To address these fundamental challenges while accounting for the intrinsic characteristics of time series, we propose TEMPLATE (TEMPoraL representAtion Transferability Estimation), which quantitatively assesses the transferability of time series pre-trained models through three complementary evaluation metrics: Firstly, an effective time series pre-trained model must demonstrate robust capability in capturing and representing temporal dependencies. Moreover, we empirically demonstrate that the principal components constitute the most transferable elements within learned representations. Consequently, we propose the Dependency Learning Score as a quantitative metric to assess dependency learning capability through comparative analysis of principal components between original representations and their corresponding representations in trend. Secondly, to assess the core ability of time series pre-trained models in capturing key temporal patterns, we introduce the Pattern Learning Score as a quantitative evaluation metric. This score precisely measures the model's focus on key temporal patterns through the ratio of the principal component to the overall pattern complexity. Finally, to ensure adaptability across diverse downstream tasks and enable versatile time series analysis, we introduce the Task Adaptation Score as a task-specific evaluation metric for assessing representation quality. For high-level tasks, hierarchical representations are preferred. Thus we calculate the similarity between the features output by the first and last layers of the encoder. Through the integration of these three complementary evaluation metrics, we establish a comprehensive framework for assessing time series pre-trained models. Experimental results demonstrate strong correlation between our proposed metric and actual model transferability, consistently outperforming existing evaluation methods in benchmark tests across 5 downstream tasks.
+
+Our contributions are summarized as follows:
+
+- We evaluated the challenges in designing transferability metrics for time series and, from the model's dependency learning ability, pattern learning ability, and task adaptation ability, proposed the TEMPLATE pre-trained model transferability evaluation framework.
+
+Table 1: Time series design challenges and methods comparison.  
+
+<table><tr><td>Challenges</td><td>NCE</td><td>LEEP</td><td>LogME</td><td>SFDA</td><td>NLEEP</td><td>PACTran</td><td>TMI</td><td>ETran</td><td>NCTI</td><td>Ours</td></tr><tr><td>Cross-task generalization bottleneck</td><td>X</td><td>X</td><td>✓</td><td>X</td><td>X</td><td>X</td><td>X</td><td>✓</td><td>X</td><td>✓</td></tr><tr><td>High-dimensional infeasibility</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Feature space mismatch</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>✓</td></tr></table>
+
+- Our proposed framework is highly flexible and generalizable, supporting both classification and regression tasks. Moreover, the metrics can be applied to 5 mainstream time series analysis tasks, demonstrating our broad applicability and task-specific adaptability.  
+- Experiments on 32 datasets across 5 mainstream time-series tasks demonstrate that our method achieves state-of-the-art (SOTA) performance in evaluating the transferability of time-series pretrained models, with an average improvement of  $35\%$  compared to previous SOTA methods.
+
+# 2 Related Work
+
+# 2.1 Time Series Pre-trained Model
+
+In the image and vision domains, pre-training on large-scale sequence datasets has significantly advanced modality understanding [20, 21]. Building on this, pre-trained models have been widely developed in the time series domain [1]. LLM4TS [22] enhances forecasting accuracy by fine-tuning large language models, while Tempo [23] improves cross-modal alignment using multimodal prompts. Lag-Llama [24] conducts pre-training on multi-domain time series data and uses lag features as covariates for probabilistic univariate forecasting. Moment [25] conducts masked reconstruction pre-training on various time series datasets. Similarly, Timer [9] undergoes autoregressive pre-training on large-scale collected datasets. UniTS [26] integrates downstream tasks into a single framework through task tokenization. Time-MOE [27], which combines the mixture-of-experts method, is the first to scale time series pre-trained models to 2.4 billion parameters. Therefore, as the scale of time-series pre-trained models continues to grow, how to evaluate the transferability of these models has become an urgent issue that needs to be addressed.
+
+# 2.2 Model Transferability Estimation
+
+Model Transferability Estimation (MTE) aims to provide fast and efficient methods for quantifying a model's performance on downstream tasks without fine-tuning [11]. Recent MTE advancements focus on vision models. Early works, such as NCE [28] and LEEP [12], focused on probabilistic methods based on the expected empirical distribution of target labels. LogME [17] estimates the maximum marginalized likelihood of labels, while NLEEP [13] extends LEEP by replacing the output layer with a Gaussian Mixture Model. SFDA [16] projects features into Fisher space and enhances class separability using physics-inspired models. TMI [15] uses intra-class feature variance as a performance indicator, assuming that lower variance reflects tighter clustering of class features. NCTI [19], inspired by the neural collapse phenomenon, develops metrics to measure the distance between the current state of the pre-trained model and its hypothetical fine-tuned state. ETran [18] introduces energy scores to quantify whether the target dataset is in-distribution or out-of-distribution for a candidate model, assuming models with higher in-distribution levels for the target dataset exhibit greater transferability. Existing studies primarily focus on vision and have not fully considered the characteristics of time-series data and pre-trained models. Therefore, we propose TEMPLATE, a transferability evaluation framework specifically designed for time-series pre-trained models.
+
+# 3 Approach
+
+# 3.1 Problem Formulation
+
+Given  $M$  pretrained models  $\{\phi_m\}_{m=1}^M$  and a target dataset  $\mathcal{D}$  with  $N$  samples, where each model consists of a backbone  $f$  that outputs encoded features, and the target dataset is associated with an evaluation metric (accuracy, MSE, F-score etc.), we initialize a predictor head on the backbone  $f$  and fine-tune the entire model on the target dataset. The feature extracted by the  $l$ -th layer of the pretrained model  $\phi_m(\cdot)$  is denoted as  $\mathbf{H}^l$ , where  $\mathbf{H}^l = \phi_m(\mathbf{X}) \in \mathbb{R}^{N \times d}$  and  $d$  is the feature dimension. In the
+
+following, we omit  $l$  using  $\mathbf{H}$  to denote the feature output of the last layer of the model's encoder. By performing brute-force fine-tuning on all models, we obtain the true performance  $\{P_m\}_{m=1}^M$  for the model hub. A practical model selection method should generate a score for each pretrained model. Ideally, the score should correlate strongly with the actual fine-tuning performance  $\{P_m\}_{m=1}^M$ , allowing the best-performing pretrained model to be selected based on the evaluation score alone.
+
+# 3.2 Transferability Assessment through Temporal Representation Transferability Estimation
+
+To effectively evaluate the transferability of time series pre-trained models, it is necessary to address the dimensional gap in feature matrices caused by the model's different handling of inter-channel relationships, as well as the varying requirements of various downstream tasks. Therefore, we avoid directly using statistical quantities of the feature matrix to measure the transferability of pretrained models. Specifically, we propose TEMPLATE, a hybrid transferability metric that includes Dependency Learning Score, Pattern Learning Score, and Task Adaptation Score, aiming to assess transferability of the learned representation in pre-trained models from the perspective of temporal dependency learning, temporal pattern learning and hierarchical features.
+
+To better understand how fine-tuning affects the representation learned by pre-trained models, we explore the changes in the components of feature matrices before and after fine-tuning. Specifically, we use Singular Value Decomposition (SVD) to analyze this process as it efficiently decomposes the feature matrix and analyzes its dominant patterns. We fine-tune MOMENT-small [25], Timer-small [9], and UniTS-small [26] on EthanolConcentration [29], Handwriting [29], and UWaveGestureLibrary [29] dataset, respectively, and compare the cosine similarity of the feature vectors corresponding to singular values before and after fine-tuning. The results are shown in Figure 2 and it is observed that the cosine similarity of feature vectors decreases as their corresponding singular values decrease. This indicates that larger singular values encapsulate more knowledge and are more transferable to downstream tasks.
+
+![](images/c09d740a9e3e177ade529d3d10db0c02b066615bbad271bf67935bfb528a87c4.jpg)  
+Figure 2: The cosine similarity of feature vectors, computed based on feature vectors before and after fine-tuning, on three datasets (sorted in descending order by singular values): EthanolConcentration (Left), Handwriting (Middle), and UWaveGestureLibrary (Right).
+
+![](images/eeb8d05796879c2a5cdc099d43ed0856f08983e4cb291791f88ccbb5d36bdf0d.jpg)
+
+![](images/03b0831cfbb9fb37b52b7e04c85c9233c95f81fa5968ea8245097d13ffdd2a96.jpg)
+
+Dependency Learning Score. The core of time series modeling is to capture temporal dependency. Time series pre-trained models should be able to capture long-term dependencies, which represent stable and persistent patterns crucial for effective modeling [30]. To assess this ability, we employ series decomposition [31] to extract the trend component, which inherently contains long-term dependencies. This component is fed into the model, yielding a feature matrix  $\mathbf{T}$  that is focused on capturing long-term dependencies, as follows:
+
+$$
+\operatorname {t r e n d} (\mathbf {X}) = \operatorname {A v g P o o l} (\text {p a d d i n g} (\mathbf {X})) , \mathbf {T} = \phi_ {m} (\operatorname {t r e n d} (\mathbf {X})) ， \tag {1}
+$$
+
+where we use padding to maintain the original series length, and then apply the AvgPool layer for moving average calculations. A model proficient in learning long-term dependencies should exhibit a high degree of consistency between the feature matrix  $\mathbf{H}$  and the feature matrix  $\mathbf{T}$  of the trend component. To this end, we propose the Dependency Learning Score  $S_{dl}$ . Specifically, considering that the largest singular value and its corresponding feature vector demonstrate superior transferability, we prioritize the use of the feature vectors corresponding to the largest singular values of  $\mathbf{H}$  and  $\mathbf{T}$ . We perform SVD to decompose the features:
+
+$$
+\mathbf {H} = \mathbf {U} _ {h} \Sigma_ {h} \mathbf {V} _ {h} ^ {T}, \mathbf {T} = \mathbf {U} _ {t} \Sigma_ {t} \mathbf {V} _ {t} ^ {T}. \tag {2}
+$$
+
+We denote the largest singular values of  $\mathbf{H}$  and  $\mathbf{T}$  as  $\sigma_h$  and  $\sigma_t$ , and their corresponding feature vectors as  $u_h$  and  $u_t$ , respectively. With the feature vectors,  $S_{dl}$  is formulated to quantify the pre-trained model's capability in capturing long-term temporal dependencies through computation of the Pearson correlation coefficient [32] between the  $u_h$  and  $u_t$ , as follows:
+
+$$
+S _ {d l} = \rho \left(u _ {h}, u _ {t}\right) = \frac {\operatorname {C o v} \left(u _ {h} , u _ {t}\right)}{\lambda_ {h} \lambda_ {t}}, \tag {3}
+$$
+
+where  $\mathrm{Cov}(\cdot)$  represents the covariance between feature vectors, and  $\lambda_h, \lambda_t$  represents their respective variances. Since  $S_{dl}$  represents the pre-trained model's ability to learn long-term dependencies, a pre-trained model that effectively learns long-term dependencies should achieve a larger  $S_{dl}$ .
+
+Pattern Learning Score. Effective temporal pattern learning is critical for time series pre-trained models to handle complex, multi-scale behaviors, which allows the model to learn the temporal dynamics in time series data, thereby achieving superior performance on downstream tasks.
+
+To this end, we propose the Pattern Learning Score  $S_{pl}$  to measure the pre-trained model's ability to learn the primary temporal patterns. Within the framework of matrix decomposition, singular values quantitatively characterize the relative significance of distinct patterns embedded in the feature matrix. The largest singular value typically corresponds to the primary temporal pattern in time series, reflecting the most significant dynamic behavior captured by the model.
+
+While the dominant singular value provides an intuitive measure of a model's capacity to capture primary temporal patterns, the inherent feature space mismatch resulting from heterogeneous inter-channel relationship processing across pre-trained models may significantly compromise the reliability of transferability assessment. Therefore, we introduce the nuclear norm as a measure of the overall importance of the patterns, defined as the sum of all singular values of the matrix. Specifically, we use the ratio of the largest singular value to the nuclear norm to measure the pre-trained model's ability to learn primary temporal patterns. A higher ratio signifies enhanced capability of the pretrained model in capturing dominant temporal patterns, whereas a lower ratio potentially indicates insufficient representation of these critical temporal patterns. Moreover, compared to the raw data, the temporal patterns of the trend component are simpler, with the primary patterns being more prominent. Therefore, the ratio of the largest singular value to the nuclear norm using the trend component provides a more accurate evaluation of the pre-trained model's ability to learn primary temporal patterns. The formulation of  $S_{pl}$  is as follows:
+
+$$
+S _ {p l} = \frac {\sigma_ {t}}{\| \mathbf {T} \| _ {*}}, \tag {4}
+$$
+
+where  $\sigma_{t}$  denotes the largest singular value of the trend component feature matrix, and  $\| \mathbf{T}\| _*$  represents its nuclear norm. Since  $S_{pl}$  represents the pre-trained model's ability to learn primary temporal patterns, a pre-trained model that effectively learns these patterns should achieve a larger  $S_{pl}$ .
+
+Task Adaptation Score. Time series downstream tasks exhibit substantial heterogeneity, with distinct tasks imposing markedly divergent requirements on feature representations [33]. Classification tasks require high-level semantics, and imputation tasks require the model to learn from missing data. Consequently, hierarchical representations become essential. In contrast, forecasting tasks and anomaly detection tasks place more emphasis on the precise extraction of low-level features from the raw data itself, focusing on local patterns and temporal dependencies [34, 33]. Consequently, an effectively designed pre-trained model should demonstrate the capability to learn task-adaptive representations that dynamically adjust to varying downstream task requirements. [35]. Moreover, due to the diverse designs of time-series pre-trained models and the significant differences in network architectures, it is necessary to consider the overall evolution of features. We measure the task adaptation ability of the pretrained model by calculating the similarity between the output features  $\mathbf{H}^1$  from the first layer of the encoder and the output features  $\mathbf{H}$  from the last layer. A simple approach is to directly compute the dot product-based similarity [36] between them:
+
+$$
+\langle \operatorname {v e c} \left(H ^ {1} \left(H ^ {1}\right) ^ {T}\right), \operatorname {v e c} \left(H H ^ {T}\right) \rangle = \operatorname {t r} \left(H ^ {1} \left(H ^ {1}\right) ^ {T} H H ^ {T}\right) = \| H ^ {T} H ^ {1} \| _ {F} ^ {2}, \tag {5}
+$$
+
+Due to the differences in how time series pre-trained models handle inter-channel relationships, there is a significant gap in feature matrix dimensions across different pre-trained models. Directly calculating squared dot product can lead to inaccurate estimates of model's transferability. Therefore, we use centered kernel alignment [37, 38, 36] to measure similarity, which further optimizes dot product-based similarity. Specifically, Equation 5 means that, for the centered  $\mathbf{H}^1$  and  $\mathbf{H}$ :
+
+$$
+\frac {1}{(n - 1) ^ {2}} \operatorname {t r} \left(H ^ {1} \left(H ^ {1}\right) ^ {T} H H ^ {T}\right) = \| \operatorname {c o v} \left(\left(H ^ {1}\right) ^ {T}, H ^ {T}\right) \| _ {F} ^ {2}, \tag {6}
+$$
+
+The Hilbert-Schmidt Independence Criterion [39] generalizes Equations 5 and 6 to inner products from reproducing kernel Hilbert spaces, where the squared Frobenius norm of the cross-covariance matrix
+
+becomes the squared Hilbert-Schmidt norm of the cross-covariance operator. Let  $K_{ij} = k(\mathbf{x}_i,\mathbf{x}_j)$  and  $L_{ij} = l(\mathbf{y}_i,\mathbf{y}_j)$  where  $k$  and  $l$  are two kernels. The empirical estimator of HSIC is:
+
+$$
+\operatorname {H S I C} (K, L) = \frac {1}{(n - 1) ^ {2}} \operatorname {t r} (K \mathbf {Q} L \mathbf {Q}), \tag {7}
+$$
+
+where  $\mathbf{Q}$  is the centering matrix  $\mathbf{Q}_n = I_n - \frac{1}{n}\mathbf{1}\mathbf{1}^{\mathrm{T}}$ . Finally, we normalize the HSIC to obtain our score function,  $S_{ra}$ :
+
+$$
+S _ {t a} = \frac {\operatorname {H S I C} (K , L)}{\sqrt {\operatorname {H S I C} (K , K) \operatorname {H S I C} (L , L)}}, \tag {8}
+$$
+
+where we use the RBF kernel  $k(\mathbf{x}_i, \mathbf{x}_j) = \exp(-||\mathbf{x}_i - \mathbf{x}_j||_2^2 / (2\gamma^2))$  for mapping, with several possible strategies for selecting the bandwidth. This strategy controls the extent to which similarity at small distances is emphasized over large distances. We set  $\gamma$  to the median distance between samples. For classification and imputation task, it is essential for the model to learn hierarchical representations, making a lower  $S_{ta}$  more desirable. In contrast, for forecasting and anomaly detection task, the model needs to focus on capturing fundamental features, making a higher  $S_{ta}$  more appropriate.
+
+Overall Score Function. In this paper, we expect pre-trained models to learn valuable temporal dependencies and patterns from the time series data and output task adaptive ability tailored to different downstream tasks, enabling versatile time series analysis. To this end, we design three score functions:  $S_{dl}$ ,  $S_{pl}$ , and  $S_{ta}$ .  $S_{dl}$  and  $S_{pl}$  measure the dependency learning ability of the model and primary temporal pattern learning, respectively.  $S_{ta}$  evaluates the task-adaptive ability of the pre-trained model to various downstream tasks. Since time series often involves large-scale datasets, directly applying SVD can result in high computational costs. To accelerate the computation, we use the power iteration method [40, 41] to approximate the largest singular value of the model and its corresponding eigenvector. For detailed information about the power iteration method, please refer to Appendix C.4. Secondly, we quickly calculate the nuclear norm using equation 9, as shown below:
+
+$$
+\left\| \mathbf {T} \right\| _ {*} = \operatorname {t r} \left(\sqrt {(\mathbf {T}) ^ {T} \mathbf {T}}\right). \tag {9}
+$$
+
+By employing these two methods, we avoid the high computational cost associated with SVD, enabling a fast evaluation of the transferability of pre-trained models. In addition, since the score functions have different scales, directly summing these scores may negatively impact the performance of transferability evaluation. Therefore, we normalize each score to a unit range (0 to 1) instead of manually defining hyperparameters to balance the contributions of each score, as follows:
+
+$$
+S _ {d l} \leftarrow \frac {S _ {d l} - \min \left(S _ {d l}\right)}{\max \left(S _ {d l}\right) - \min \left(S _ {d l}\right)}. \tag {10}
+$$
+
+Similarly, we can obtain the normalized scores  $S_{pl}(\mathbf{T})$  and  $S_{ta}(\mathbf{H}^1,\mathbf{H})$ . By combining these three scores, we derive the final transferability evaluation metric:
+
+$$
+S _ {t o t a l} = S _ {d l} (\mathbf {H}, \mathbf {T}) + S _ {p l} (\mathbf {T}) + \lambda S _ {t a} \left(\mathbf {H} ^ {1}, \mathbf {H}\right), \tag {11}
+$$
+
+where  $\lambda$  is determined by the downstream task type. When evaluating on classification task and imputation task,  $\lambda$  is set to -1. For forecasting task and anomaly detection task,  $\lambda$  is set to 1. Note that the three scores are equally weighted, and  $\lambda$  acts as a sign function determined by the task type, without requiring fine-tuning for specific downstream dataset. The pre-trained model with a higher overall score  $S_{total}$  indicates better transferability within the model pool for the target dataset  $\mathcal{D}$ .
+
+# 4 Experiments
+
+In this section, we examine the effectiveness of TEMPLATE in assessing the transfer learning performance of pretrained models across 5 mainstream downstream time series tasks, including classification, short- and long-term forecasting, imputation, and anomaly detection.
+
+Implementation Details. To quantify the correlation between estimated assessment scores and actual fine-tuning results, we employ the weighted Kendall's  $\tau_w$  (see Appendix C.3), which measures ranking agreement with higher weights for higher ranks. More details about the dataset and experiment implementation can be found in Appendix C.
+
+Table 2: Comparison of SOTA Rate and Relative Improvement for Different Methods.  
+
+<table><tr><td>Method</td><td>GBC [42]</td><td>CC-FV [43]</td><td>NCTI [19]</td><td>ETran [18]</td><td>Ours</td></tr><tr><td>SOTA Rate in Original Paper</td><td>37%</td><td>36%</td><td>40%</td><td>45%</td><td>46%</td></tr><tr><td>Improvement (vs Previous SOTA)</td><td>20%</td><td>9%</td><td>21%</td><td>18%</td><td>35%</td></tr></table>
+
+Pre-trained Models. We select 10 pre-trained models from 4 model families to form the pre-trained model pool, including MOMENT-small [25], MOMENT-base [25], MOMENT-large [25], Timersmall [9], Timer-base [9], TimerXL-small [44], TimerXL-base [44], UniTS-small [26], UniTS-base [26], and UniTS-large [26]. We fine-tune all source models on the target dataset to obtain the actual ranking. More details about the pre-trained model pool can be found in the AppendixC.2.
+
+# 4.1 Main Results.
+
+As a transferability evaluation metric designed for pretrained models in time series, TEMPLATE achieves consistently state-of-the-art accuracy across 5 mainstream analytical tasks, as shown in Figure 3. Notably, no method has achieved SOTA on all dataset, and as illustrated in Table 2, our method attains a remarkably high SOTA rate. We show more experimental results in Appendix D to further demonstrate the effectiveness of the method.
+
+![](images/4f506fdab4e75541a99c3751d7fccc8cc240c2437ae6da38dafe8c3ab40e6761.jpg)  
+Figure 3: Performance comparison.
+
+# 4.2 Classification
+
+Setup. Time series classification can be applied to tasks such as recognition and medical diagnosis [45]. Specifically, We verify all methods on 9 multivariate datasets from the UEA classification archive [29], covering gesture, audio recognition, etc., across multiple task categories. We preprocess the datasets according to [46], where different subsets have varying sequence lengths.
+
+Table 3: Classification Benchmark: Performance (Weighted Kendall's  $\tau_w$ ) of different methods. The best results are highlighted in bold while the second best are underlined.  
+
+<table><tr><td>Datasets</td><td>LogME (2021)</td><td>NLEEP (2022)</td><td>SFDA (2022)</td><td>PACTran (2022)</td><td>TMI (2023)</td><td>NCTI (2023)</td><td>ETran (2023)</td><td>RetMMD (2024)</td><td>TEMPLATE (Ours)</td></tr><tr><td>EthanolConcentration</td><td>0.567</td><td>0.432</td><td>0.120</td><td>0.488</td><td>-0.430</td><td>-0.32</td><td>0.686</td><td>0.512</td><td>0.724</td></tr><tr><td>FaceDetection</td><td>-0.203</td><td>0.092</td><td>0.598</td><td>0.306</td><td>-0.600</td><td>0.109</td><td>-0.359</td><td>0.310</td><td>0.597</td></tr><tr><td>Handwriting</td><td>-0.445</td><td>-0.104</td><td>0.314</td><td>0.596</td><td>0.768</td><td>0.700</td><td>0.478</td><td>0.365</td><td>0.822</td></tr><tr><td>JapaneseVowels</td><td>0.231</td><td>0.213</td><td>0.021</td><td>0.306</td><td>0.302</td><td>0.340</td><td>-0.196</td><td>0.654</td><td>0.447</td></tr><tr><td>PEMS-SF</td><td>-0.472</td><td>-0.612</td><td>-0.312</td><td>0.306</td><td>-0.300</td><td>0.053</td><td>0.076</td><td>0.520</td><td>0.470</td></tr><tr><td>SelfRegulationSCP1</td><td>0.356</td><td>0.529</td><td>0.459</td><td>0.619</td><td>0.300</td><td>0.310</td><td>0.651</td><td>0.450</td><td>0.484</td></tr><tr><td>SelfRegulationSCP2</td><td>0.268</td><td>0.241</td><td>-0.198</td><td>0.457</td><td>0.455</td><td>0.450</td><td>0.667</td><td>0.397</td><td>0.551</td></tr><tr><td>SpokenArabicDigits</td><td>0.342</td><td>0.321</td><td>-0.367</td><td>0.744</td><td>0.647</td><td>-0.210</td><td>0.479</td><td>0.201</td><td>0.637</td></tr><tr><td>UWaveGestureLibrary</td><td>0.584</td><td>0.127</td><td>0.440</td><td>0.592</td><td>0.576</td><td>0.245</td><td>0.624</td><td>0.362</td><td>0.719</td></tr><tr><td>Average</td><td>0.136</td><td>0.138</td><td>0.119</td><td>0.490</td><td>0.191</td><td>0.186</td><td>0.345</td><td>0.419</td><td>0.608</td></tr></table>
+
+Performance Comparison. We compare our method with existing transferability evaluation metrics, including LogME [17], NLEEP [13], SFDA [16], PACTran [47], TMI [15], NCTI [19], ETran [18] and RetMMD [48]. Table 3 shows the results of our method compared to previous work on classification benchmarks. Our method outperforms all previous methods, achieving a state-of-the-art (SOTA)  $\tau_w$  of 0.608, which is  $24\%$  ahead of the second-best method. Due to the different ways models handle inter-channel relationships, feature-statistics-based methods, such as LogME [17], TMI [15], and NCTI [19], generally perform poorly on datasets with a large number of channels, like FaceDetection and PEMS-SF. In contrast, our method achieves outstanding results on these two datasets, with  $\tau_w$  of 0.597 and 0.470, respectively. These results highlight the effectiveness of our approach in evaluating pre-trained time series models for transfer learning.
+
+# 4.3 Short- and Long-term Forecasting
+
+Setup. Time series forecasting plays a crucial role in areas such as weather forecasting, traffic management, and energy consumption. To thoroughly evaluate the generalizability of our approach in forecasting, we employ two types of benchmarks: long-term forecasting and short-term forecasting. For long-term forecasting, we use seven widely recognized long-term time series forecasting datasets [31], covering practical applications across various domains. For short-term forecasting, we adopt the M4 dataset [49], which includes univariate marketing data collected annually, quarterly, and monthly.
+
+Table 4: Long-term Forecasting Benchmark: All the results are averaged from 4 different prediction lengths. - means out of memory. See Table 12 in Appendix for the full results.  
+
+<table><tr><td>Methods</td><td>ETTh1</td><td>ETTh2</td><td>ETTm1</td><td>ETTm2</td><td>Weather</td><td>Electricity</td><td>Traffic</td><td>Average</td></tr><tr><td>LogME</td><td>0.215</td><td>0.167</td><td>0.400</td><td>0.565</td><td>0.114</td><td>-0.130</td><td>-</td><td>0.190</td></tr><tr><td>ETran</td><td>0.138</td><td>0.212</td><td>0.189</td><td>0.351</td><td>0.474</td><td>0.302</td><td>0.192</td><td>0.265</td></tr><tr><td>Ours</td><td>0.240</td><td>-0.003</td><td>0.518</td><td>0.576</td><td>0.412</td><td>0.361</td><td>0.432</td><td>0.362</td></tr></table>
+
+Performance Comparison. Tables 4 and 11 present the experimental results of our method in long-term and short-term forecasting scenarios. We compare our method with existing transferability evaluation metrics for regression tasks, including LogME [17] and ETran [18]. The core of long-term forecasting task lies in the model's ability to effectively capture complex temporal dependencies and long-term trend. By contrast, short-term forecasting task focuses more on the precise capture of fine-grained temporal patterns. Overall, our method demonstrates outstanding performance across both scenarios. Notably, LogME performs poorly on multi-channel datasets like Electricity and cannot evaluate the performance of pre-trained models on Traffic due to out of memory issues. In contrast, our method achieves excellent results on both datasets. These results further validate the effectiveness of our method in assessing time series pre-trained models for transfer learning.
+
+# 4.4 Imputation
+
+Setup. Real-time systems are continuously operating and monitored by automated observation devices. However, due to failures, the collected time series may be partially missing, makes downstream analysis challenging. Therefore, imputation has become widely used in practical applications. In this paper, we select datasets from electricity, weather, and traffic scenarios as benchmarks.
+
+Table 5: Imputation Benchmark: All results are averaged from 4 mask ratios. \* means out of memory. See Table 13 in Appendix for the full results.  
+
+<table><tr><td>Methods</td><td>ETTh1</td><td>ETTh2</td><td>ETTm1</td><td>ETTm2</td><td>Weather</td><td>Electricity</td><td>Traffic</td><td>Average</td></tr><tr><td>LogME</td><td>0.322</td><td>0.238</td><td>0.607</td><td>0.388</td><td>0.312</td><td>-0.061</td><td>-</td><td>0.258</td></tr><tr><td>ETran</td><td>0.324</td><td>0.419</td><td>0.254</td><td>0.310</td><td>0.441</td><td>0.456</td><td>0.398</td><td>0.372</td></tr><tr><td>Ours</td><td>0.406</td><td>0.317</td><td>0.664</td><td>0.650</td><td>0.645</td><td>0.412</td><td>0.529</td><td>0.518</td></tr></table>
+
+Performance Comparison. Effective time series imputation places high demands on the model's ability to capture temporal dependencies and understand contextual information. Specifically, when imputing missing values, the model needs to utilize the implicit temporal patterns within data, including modeling dynamic relationships between time points and accurately inferring both the overall trends and local features of time series. Table 5 presents the experimental results of our method in the time series imputation scenario. In the experiments, we evaluate the model's performance under different missing ratios. The results show that, whether in scenarios with a low or high proportion of missing data, our method demonstrates a high level of model transferability evaluation accuracy.
+
+# 4.5 Anomaly Detection
+
+Setup. Anomaly detection in sensor data is crucial for industrial maintenance, as anomalies are often hidden within large-scale datasets, making data labeling challenging. We compare 5 widely used anomaly detection benchmarks: SMD [50], MSL [51], SMAP [51], SWaT [52], and PSM [53], covering applications in service monitoring, spatial data, earth exploration, and water treatment.
+
+Table 6: Anomaly Detection Benchmark: Performance (Weighted Kendall's  $\tau_w$ ) of different methods. The best results are highlighted in **bold** and ' -' means out of memory.  
+
+<table><tr><td>Methods</td><td>PSM</td><td>MSL</td><td>SMAP</td><td>SMD</td><td>SWaT</td><td>Average</td></tr><tr><td>LogME</td><td>0.230</td><td>0.468</td><td>0.340</td><td>-</td><td>-</td><td>0.207</td></tr><tr><td>ETran</td><td>0.433</td><td>-0.024</td><td>0.121</td><td>0.126</td><td>0.277</td><td>0.187</td></tr><tr><td>Ours</td><td>0.142</td><td>0.515</td><td>0.431</td><td>-0.08</td><td>0.173</td><td>0.236</td></tr></table>
+
+Performance Comparison. Effective time series anomaly detection requires the model to identify normal and abnormal patterns in the data and accurately distinguish between the two. Table 6 presents the results of our method in the time series anomaly detection scenario. The experimental results show that our method effectively evaluates the model's transferability in anomaly detection scenarios.
+
+![](images/30b5ada7a5cb5912c4de90652b5fac2e0e28da16a9a5e613c4086bf30f34b74e.jpg)  
+Figure 4: Effectiveness of component in TEM-PLATE on Classification and Imputation task.
+
+![](images/78a8707c8f2f8268e689a12530f2eac777baa3fe2af3ca76f767c555538c1aad.jpg)
+
+![](images/bf5358008b7c8d27db8ebe66ecc6187a2c9c858b5ffbd06bf8077e7228694fa5.jpg)  
+Figure 5: Time complexity analysis.
+
+# 4.6 Analysis
+
+Ablation Study. Our TEMPLATE consists of three sub-items designed to evaluate the performance of pre-trained models on downstream tasks. To explore the individual contribution of each sub-item to the final metric and analyze its impact on overall performance, we conduct an ablation study. Figure 4 presents the results of the ablation study on the classification and imputation task, with the complete results available in Appendix D.2. It can be observed that all three metrics achieve positive ranking correlations, and their combination yields the highest average ranking correlation, indicating that the three metrics complement each other and can comprehensively measure the performance of pre-trained models.
+
+Time Complexity. In the Figure 5, we present a comparison of the average runtime and  $\tau_w$  on the classification task across 9 datasets using different model selection metrics. While NCTI [19] and TMI [15] are fast, they perform poorly in evaluating the transferability of time series pre-trained models due to their reliance on statistical metrics of feature matrices. LogME [17] is efficient, but it cannot be applied to large datasets like Traffic and SMD because it relies on the magnitude of singular values. In contrast, our method uses the power iteration method to avoid high computational costs and enabling its application to large datasets. Although ETran [18] efficiently obtains the  $S_{en}$  score, the LDA-based score calculation incurs significant time costs, resulting in slower overall speed.
+
+# 5 Conclusion
+
+In this paper, we propose TEMPLATE, a pre-trained model evaluation framework specifically designed for time series. We are the first to design a transferability evaluation metric tailored for time series. TEMPLATE consists of three scores, aimed at measuring the transferability of pre-trained models from two perspectives: dependency and pattern learning, and task adaptation. Our approach has been successfully applied to 5 mainstream time series downstream tasks. Extensive experiments demonstrate that our method achieves state-of-the-art (SOTA) performance in evaluating time series pre-trained models for transfer learning.
+
+# Acknowledgements
+
+This work was supported by the National Natural Science Foundation of China (62306085, 62206074, 62476071, U23B2055, U24A20328), Shenzhen College Stability Support Plan (GXWD20231130151329002, GXWD20220811173233001), CCF-ALIMAMA TECH Kangaroo Fund (CCF-ALIMAMA OF 2025001), Guangdong Basic and Applied Basic Research Foundation (2025A1515012932, 2025A1515011732), Shenzhen Science and Technology Program (KQTD20240729102154066, ZDSYS20230626091203008).
+
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+
+# A Limitations
+
+Our method may perform poorly on some datasets with severe distribution shifts or structural breaks, as the results on the training set cannot accurately reflect the model's performance on the test set. This remains a challenging problem, and no existing method has effectively addressed it yet.
+
+# B Broader impacts
+
+The work in this paper contributes to facilitating the evaluation of transfer capability for time series pre-trained models, thereby further saving time and training resources and enabling the rapid selection of suitable time series pre-trained models for downstream tasks.
+
+# C Experiment Details
+
+# C.1 Implement Details
+
+We provide detailed descriptions of the datasets in Tables 8. For all 5 downstream tasks, we follow the experimental setup of [34]. To compute the transfer performance values, we carefully fine-tuned the pre-trained models through hyperparameter grid search. As [54] highlighted, learning rate and weight decay are the two most critical parameters. Therefore, we performed grid search over learning rates and weight decay values (6 learning rates ranging from  $10^{-3}$  to  $10^{-5}$ , and 3 weight decay values from  $10^{-3}$  to  $10^{-5}$ ) to select the optimal hyperparameters. The fine-tuning experiments of the pre-trained models were conducted on an NVIDIA H20 GPU with 96GB of memory. It is important to note that we only fine-tuned the pre-trained models to validate the effectiveness of the proposed method; the method itself does not require fine-tuning of the pre-trained models. All the results of pre-trained model transferability evaluation metrics were obtained on an AMD EPYC 7513 32-Core CPU.
+
+# C.2 Model Pool Details
+
+We selected 10 pre-trained models from four model families to form the model pool, including MOMENT-small [25], MOMENT-base [25], MOMENT-large [25], Timer-small [9], Timer-base [9], TimerXL-small [44], TimerXL-base [44], UniTS-small [26], UniTS-base [26], and UniTS-large [26]. As shown in Table 7, the selected pre-trained models differ in structure, model parameters, and the scale of pre-training. The MOMENT models were reconstructed and trained on the large hybrid dataset Time-series Pile; Timer-small and TimerXL-small were pre-trained on the ERA5 dataset, while Timer-base and TimerXL-base were pre-trained on the large dataset UTSD; the UniTS series models were pre-trained on a multi-domain hybrid dataset for time series. This highlights the differences between the models and further demonstrates that the proposed method is sufficient to support general transferability.
+
+Table 7: The Architecture, Model size, and Scale of the pre-trained data of the model pool.  
+
+<table><tr><td>Method</td><td>Timer</td><td>Timer-XL</td><td>MOMENT</td><td>UniTS</td></tr><tr><td>Architecture</td><td>Decoder</td><td>Decoder</td><td>Encoder</td><td>Encoder</td></tr><tr><td>Model size</td><td>11M~67M</td><td>14M~69M</td><td>40M~380M</td><td>3M~8.4M</td></tr><tr><td>Pre-training Scale</td><td>0.5B, 28B</td><td>0.5B, 28B</td><td>1.13B</td><td>0.5B</td></tr></table>
+
+# C.3 Evaluation Metric
+
+To evaluate the effectiveness of the proposed method, we refer to previous work [17, 18] and use Kendall's  $\tau$  [55] to measure the correlation between the estimated evaluation scores and the ground-truth ranking scores. This metric assesses ranking consistency, where a larger  $\tau$  value indicates greater effectiveness in ranking models. It is defined as the number of concordant pairs minus the
+
+number of discordant pairs, divided by the overall number of pairs  $\binom{M}{2}$ , as follows:
+
+$$
+\tau = \frac {2}{M (M - 1)} \sum_ {1 \leq i <   j \leq M} \operatorname {s g n} \left(G _ {i} - G _ {j}\right). \operatorname {s g n} \left(T _ {i} - T _ {j}\right). \tag {12}
+$$
+
+Furthermore, we adopt a weighted version of Kendall's  $\tau$  [56], denoted as  $\tau_w$ , to assign higher weights to models ranked higher, to quantify this correlation.
+
+# C.4 Power Iteration
+
+Due to the limited space in the main text, we show the detailed procedure of the power iteration method here, as shown in Algorithm 1:
+
+Algorithm 1 Power Iteration  
+Input: Matrix  $\mathbf{H}$  , initial vector  $v_{0}$  , number of iterations  $k$  , here we set 10.   
+Output: Dominant eigenvalue  $\lambda_{\mathrm{max}}$  , corresponding eigenvector  $v_{k}$    
+Initialize  $v_{0}$  randomly   
+for  $i = 1$  to  $k$  do   
+ $v_{i + 1} = \mathbf{H}v_{i}$    
+Normalize  $v_{i + 1}$  to unit length   
+end for   
+Estimate the dominant eigenvalue  $\lambda_{\mathrm{max}} = \frac{v_k^T\mathbf{H}v_k}{v_k^T v_k}$
+
+# D More Experimental Results
+
+# D.1 Statistical Significance Testing
+
+To validate the effectiveness of TEMPLATE over ETran, we conducted a statistical significance test. The results show that the mean of TEMPLATE (0.483) is significantly higher than that of ETran (0.359), with a mean difference of 0.124, which is statistically significant (T_statistic = 2.679, P_value = 0.012). Therefore, TEMPLATE demonstrates significantly better performance across all tasks compared to ETran, confirming its effectiveness improvement.
+
+# D.2 Ablation Study
+
+![](images/5336960a091384c7342d83a9e523de8ef6d654621c24fcf458f7546e78aac5f6.jpg)  
+(a) Classification.
+
+![](images/17788737730123cba4ae1197c7e2023de496632e9aa4de0440227e66237d5823.jpg)  
+(b) Imputation.
+
+![](images/f89b765e2d9acf1999ffdce9fc27f3a747c9ed4352d2c3ed6065df931f20e737.jpg)  
+(c) Forecasting.  
+Figure 6: Effectiveness of each individual component in TEMPLATE. We use the three terms individually or remove them one at a time from the full system. The results are averaged across all datasets.
+
+![](images/16676890c300ac1f373f16206ebaeb692b0fd8085689015330ef768f79cc676e.jpg)  
+(d) anomaly detection.
+
+Due to space limitations, we have included the complete ablation experiment results in Figure 6. The experimental results show that, compared to the three individual metrics, TEMPLATE achieves the best performance, demonstrating the effectiveness of TEMPLATE. Additionally, we have also validated the effectiveness of combining these metrics in pairs. The combination of  $S_{dl}$  and  $S_{pl}$  outperforms each metric individually, proving their complementary nature. This demonstrates that evaluating the pre-trained model's ability for time series semantic learning from the perspectives of
+
+Table 8: Dataset descriptions. The dataset size is organized in (Train, Validation, Test).  
+
+<table><tr><td>Tasks</td><td>Dataset</td><td>Dim</td><td>Series Length</td><td>Dataset Size</td><td>Information (Frequency)</td></tr><tr><td rowspan="5">Forecasting (Long-term)</td><td>ETTm1, ETTm2</td><td>7</td><td>{96, 192, 336, 720}</td><td>(34465, 11521, 11521)</td><td>Electricity (15 mins)</td></tr><tr><td>ETTh1, ETTh2</td><td>7</td><td>{96, 192, 336, 720}</td><td>(8545, 2881, 2881)</td><td>Electricity (15 mins)</td></tr><tr><td>Electricity</td><td>321</td><td>{96, 192, 336, 720}</td><td>(18317, 2633, 5261)</td><td>Electricity (Hourly)</td></tr><tr><td>Traffic</td><td>862</td><td>{96, 192, 336, 720}</td><td>(12185, 1757, 3509)</td><td>Transportation (Hourly)</td></tr><tr><td>Weather</td><td>21</td><td>{96, 192, 336, 720}</td><td>(36792, 5271, 10540)</td><td>Weather (10 mins)</td></tr><tr><td rowspan="6">Forecasting (short-term)</td><td>M4-Yearly</td><td>1</td><td>6</td><td>(23000, 0, 23000)</td><td>Demographic</td></tr><tr><td>M4-Quarterly</td><td>1</td><td>8</td><td>(24000, 0, 24000)</td><td>Finance</td></tr><tr><td>M4-Monthly</td><td>1</td><td>18</td><td>(48000, 0, 48000)</td><td>Industry</td></tr><tr><td>M4-Weakly</td><td>1</td><td>13</td><td>(359, 0, 359)</td><td>Macro</td></tr><tr><td>M4-Daily</td><td>1</td><td>14</td><td>(4227, 0, 4227)</td><td>Micro</td></tr><tr><td>M4-Hourly</td><td>1</td><td>48</td><td>(414, 0, 414)</td><td>Other</td></tr><tr><td rowspan="5">Imputation</td><td>ETTm1, ETTm2</td><td>7</td><td>96</td><td>(34465, 11521, 11521)</td><td>Electricity (15 mins)</td></tr><tr><td>ETTh1, ETTh2</td><td>7</td><td>96</td><td>(8545, 2881, 2881)</td><td>Electricity (15 mins)</td></tr><tr><td>Electricity</td><td>321</td><td>96</td><td>(18317, 2633, 5261)</td><td>Electricity (15 mins)</td></tr><tr><td>Traffic</td><td>862</td><td>96</td><td>(12185, 1757, 3509)</td><td>Transportation (Hourly)</td></tr><tr><td>Weather</td><td>21</td><td>96</td><td>(36792, 5271, 10540)</td><td>Weather (10 mins)</td></tr><tr><td rowspan="9">Classification (UEA)</td><td>EthanolConcentration</td><td>3</td><td>1751</td><td>(261, 0, 263)</td><td>Alcohol Industry</td></tr><tr><td>FaceDetection</td><td>144</td><td>62</td><td>(5890, 0, 3524)</td><td>Face (250Hz)</td></tr><tr><td>Handwriting</td><td>3</td><td>152</td><td>(150, 0, 850)</td><td>Handwriting</td></tr><tr><td>JapaneseVowels</td><td>12</td><td>29</td><td>(270, 0, 370)</td><td>Voice</td></tr><tr><td>PEMS-SF</td><td>963</td><td>144</td><td>(267, 0, 173)</td><td>Transportation (Daily)</td></tr><tr><td>SelfRegulationSCP1</td><td>6</td><td>896</td><td>(268, 0, 293)</td><td>Health (256Hz)</td></tr><tr><td>SelfRegulationSCP2</td><td>7</td><td>1152</td><td>(200, 0, 180)</td><td>Health (256Hz)</td></tr><tr><td>SpokenArabicDigits</td><td>13</td><td>93</td><td>(6599, 0, 2199)</td><td>Voice (11025Hz)</td></tr><tr><td>UWaveGestureLibrary</td><td>3</td><td>315</td><td>(120, 0, 320)</td><td>Gesture</td></tr><tr><td rowspan="5">Anomaly Detection</td><td>SMD</td><td>38</td><td>100</td><td>(566724, 141681, 708420)</td><td>Server Machine</td></tr><tr><td>MSL</td><td>55</td><td>100</td><td>(44653, 11664, 73729)</td><td>Spacecraft</td></tr><tr><td>SMAP</td><td>25</td><td>100</td><td>(108146, 27037, 427617)</td><td>Spacecraft</td></tr><tr><td>SWaT</td><td>51</td><td>100</td><td>(396000, 99000, 449919)</td><td>Infrastructure</td></tr><tr><td>PSM</td><td>25</td><td>100</td><td>(105984, 26497, 87841)</td><td>Server Machine</td></tr></table>
+
+dependency learning and pattern learning is an effective and reasonable approach. Although  $S_{ta}$  did not achieve the best correlation in the anomaly detection task, compared to W/O  $S_{ta}$ , TEMPLATE showed an  $8\%$  average improvement in  $t_w$  across 5 downstream tasks, effectively proving the validity of  $S_{ta}$ .
+
+Additionally, it is important to note that TEMPLATE is a very robust method. We conducted additional ablation experiments to explore the specific implementation methods for the scores. For  $S_{dl}$ , the main idea is to measure the alignment of the eigenvector corresponding to the largest singular value between the trend component and the original sequence. In the table, we also implemented  $S_{dl}$  based on Euclidean distance, and it can be seen that  $S_{dl}$  based on Euclidean distance achieves a positive transferability evaluation. For  $S_{ta}$ , the main idea is to measure the similarity between features
+
+Table 9: Ablation study on the specific implementation methods of  ${S}_{dl}$  and  ${S}_{ta}$  .  
+
+<table><tr><td>Method</td><td>EthanolConcentration</td><td>Handwriting</td><td>PEMS - SF</td><td>FaceDetection</td></tr><tr><td>Sdl(euclidean_distance)</td><td>0.332</td><td>0.597</td><td>0.370</td><td>0.361</td></tr><tr><td>Sdl(pearsonr)</td><td>0.349</td><td>0.608</td><td>0.381</td><td>0.366</td></tr><tr><td>Sta.dot product )</td><td>0.421</td><td>0.654</td><td>0.276</td><td>0.450</td></tr><tr><td>Sta(HSIC)</td><td>0.436</td><td>0.698</td><td>0.350</td><td>0.453</td></tr></table>
+
+to meet the requirements of different task types for features. In the following table, we implemented  $S_{t}$  based on squared dot product, and it can be observed that  $S_{ta}$  based on squared dot product also achieves positive transferability evaluation. HSIC maps the features to a higher-dimensional space and then computes the similarity. The method of measuring feature similarity has good properties [36]. Therefore, we use HSIC to measure similarity. The experimental results in the table further demonstrate the effectiveness and robustness of TEMPLATE.
+
+Table 10: Ablation Experiment on the Sensitivity of Hyperparameter  $\lambda$  .  
+
+<table><tr><td>λ</td><td>EthanolConcentration</td><td>Handwriting</td><td>PEMS - SF</td><td>SelfRegulationSCP2</td></tr><tr><td>-0.6</td><td>0.716</td><td>0.822</td><td>0.461</td><td>0.551</td></tr><tr><td>-0.8</td><td>0.724</td><td>0.822</td><td>0.461</td><td>0.551</td></tr><tr><td>-1</td><td>0.724</td><td>0.822</td><td>0.470</td><td>0.551</td></tr></table>
+
+Finally, we note that the three scores in the final score are equally important, and  $\lambda$  is only related to the task type rather than the specific downstream task dataset. To demonstrate the robustness of our method, we add a parameter sensitivity experiment for  $\lambda$  in Table 10. The experimental results show that on specific downstream datasets, there is no need to fine-tune  $\lambda$ .
+
+Table 11: Short-Term Benchmark: Performance (Weighted Kendall's  $\tau_w$ ) of different methods. The best results are highlighted in bold.  
+
+<table><tr><td>Datasets</td><td>Yearly</td><td>Quarterly</td><td>Monthly</td><td>Others</td><td>Average</td></tr><tr><td>LogME</td><td>0.492</td><td>0.332</td><td>0.562</td><td>0.468</td><td>0.464</td></tr><tr><td>ETran</td><td>0.235</td><td>0.763</td><td>0.624</td><td>0.318</td><td>0.485</td></tr><tr><td>Ours</td><td>0.534</td><td>0.667</td><td>0.712</td><td>0.417</td><td>0.583</td></tr></table>
+
+# D.3 Full Result
+
+Due to space limitations in the main text, we have included the results of the short-term forecasting, long-term forecasting and imputation in Tables 11, Tables 12 and 13. The experimental results demonstrate that compared to LogME and ETran, TEMPLATE achieves the best average performance.
+
+Table 12: The full result for forecasting task.  
+
+<table><tr><td>Datasets</td><td colspan="3">ETTh1</td><td colspan="3">ETTh2</td><td colspan="3">ETTm1</td><td colspan="3">ETTm2</td><td colspan="3">weather</td><td colspan="3">ECL</td><td colspan="3">Traffic</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Length</td><td>96</td><td>192</td><td>336</td><td>720</td><td>96</td><td>192</td><td>336</td><td>720</td><td>96</td><td>192</td><td>336</td><td>720</td><td>96</td><td>192</td><td>336</td><td>720</td><td>96</td><td>192</td><td>336</td><td>720</td><td>96</td><td>192</td><td>336</td><td>720</td><td></td><td></td><td></td><td></td></tr><tr><td>LogME</td><td>0.169</td><td>0.375</td><td>0.198</td><td>0.121</td><td>0.103</td><td>0.234</td><td>0.132</td><td>0.199</td><td>0.596</td><td>0.232</td><td>0.364</td><td>0.411</td><td>0.522</td><td>0.516</td><td>0.700</td><td>0.522</td><td>0.310</td><td>0.021</td><td>-0.110</td><td>0.233</td><td>0.031</td><td>-0.332</td><td>0.060</td><td>-0.280</td><td>-</td><td>-</td><td>-</td><td></td></tr><tr><td>ETran</td><td>-0.007</td><td>0.276</td><td>0.282</td><td>0.001</td><td>0.321</td><td>0.003</td><td>0.180</td><td>0.345</td><td>0.473</td><td>0.126</td><td>0.314</td><td>-0.154</td><td>0.210</td><td>0.393</td><td>0.312</td><td>0.489</td><td>0.531</td><td>0.642</td><td>0.384</td><td>0.341</td><td>0.461</td><td>0.184</td><td>0.296</td><td>0.268</td><td>0.102</td><td>0.168</td><td>0.314</td><td>0.184</td></tr><tr><td>Ours</td><td>0.234</td><td>0.309</td><td>0.243</td><td>0.175</td><td>-0.012</td><td>0.076</td><td>0.131</td><td>-0.199</td><td>0.746</td><td>0.317</td><td>0.670</td><td>0.340</td><td>0.438</td><td>0.598</td><td>0.680</td><td>0.589</td><td>0.496</td><td>0.523</td><td>0.342</td><td>0.291</td><td>0.180</td><td>0.346</td><td>0.458</td><td>0.458</td><td>0.642</td><td>0.334</td><td>0.395</td><td>0.361</td></tr></table>
+
+Table 13: The full result for imputation task.  
+
+<table><tr><td>Datasets</td><td colspan="3">ETTh1</td><td colspan="3">ETTh2</td><td colspan="3">ETTm1</td><td colspan="3">ETTm2</td><td colspan="3">weather</td><td colspan="3">ECL</td><td colspan="3">Traffic</td><td></td><td></td><td></td><td></td></tr><tr><td>Mask Ratio</td><td>12.5%</td><td>25%</td><td>37.5%</td><td>50%</td><td>12.5%</td><td>25%</td><td>37.5%</td><td>50%</td><td>12.5%</td><td>25%</td><td>37.5%</td><td>50%</td><td>12.5%</td><td>25%</td><td>37.5%</td><td>50%</td><td>12.5%</td><td>25%</td><td>37.5%</td><td>50%</td><td>12.5%</td><td>25%</td><td>37.5%</td><td>50%
+- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -</td><td></td></tr><tr><td>LogME</td><td>0.193</td><td>0.441</td><td>0.251</td><td>0.405</td><td>0.214</td><td>0.242</td><td>0.196</td><td>0.301</td><td>0.342</td><td>0.613</td><td>0.658</td><td>0.815</td><td>0.417</td><td>0.365</td><td>0.423</td><td>0.348</td><td>0.402</td><td>0.195</td><td>0.297</td><td>0.351</td><td>0.0390</td><td>-0.150</td><td>-0.091</td><td>-0.051</td><td>- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -</td></tr></table>
\ No newline at end of file
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+# Unified all-atom molecule generation with neural fields
+
+Matthieu Kirchmeyer $^{1,\dagger}$  Pedro O. Pinheiro $^{1,\dagger}$  Emma Willett $^{1}$  Karolis Martinkus $^{1,\ddagger}$  Joseph Kleinhenz $^{1}$  Emily K. Makowski $^{2}$  Andrew M. Watkins $^{1}$  Vladimir Gligorijevic $^{1}$  Richard Bonneau $^{1}$  Saeed Saremi $^{1}$
+
+$^{1}$ Prescient Design, Genentech  $^{2}$ Antibody Engineering, Genentech
+
+# Abstract
+
+Generative models for structure-based drug design are often limited to a specific modality, restricting their broader applicability. To address this challenge, we introduce FuncBind, a framework based on computer vision to generate target-conditioned, all-atom molecules across atomic systems. FuncBind uses neural fields to represent molecules as continuous atomic densities and employs score-based generative models with modern architectures adapted from the computer vision literature. This modality-agnostic representation allows a single unified model to be trained on diverse atomic systems, from small to large molecules, and handle variable atom/residue counts, including non-canonical amino acids. FuncBind achieves competitive in silico performance in generating small molecules, macrocyclic peptides, and antibody complementarity-determining region loops, conditioned on target structures. FuncBind also generated in vitro novel antibody binders via de novo redesign of the complementarity-determining region H3 loop of two chosen co-crystal structures. As a final contribution, we introduce a new dataset and benchmark for structure-conditioned macrocyclic peptide generation*.
+
+# 1 Introduction
+
+A central challenge in drug discovery is designing molecules that bind specifically to a target protein [1]. This task involves navigating a diverse landscape of molecular modalities, from small organic compounds to large biomolecules, each with unique chemical properties. Structure-based approaches are frequently employed to meet this challenge, utilizing the 3D structure of a target site (often an accessible protein region) to generate novel molecules with high affinity. Generative models are emerging as a powerful data-driven alternative to established traditional techniques such as virtual screening and physics-based simulations. These newer models can explore vast chemical spaces more effectively to identify molecules with desired binding properties [2].
+
+Most structure-based generative models specialize on a single molecule modality to better account for specific physicochemical properties. Focusing on a single molecular modality also simplifies data gathering and augmentation, training, representation choice, and metrics used for validation. Generative models of small molecules typically represent molecules as point cloud of atoms [3, 4, 5] or discretized atomic densities [6, 7]. Most protein generative models leverage the fact that proteins are sequences of amino acids to represent them with point clouds of residues, where each residue contains multiple atoms [8], recovering their sequences with, e.g., co-generation [9] or inverse folding [10]. Many protein-centric models also rely on large sequence databases and self-supervised generative models for sequence that can help in scoring and generating/proposing mutations.
+
+Domain-specific representations limit generalization, as models are not transferable from one modality to another. This narrow focus also limits utility, as most key applications involve interfaces and catalysis across multiple modalities. We argue that modality-agnostic representations are better suited to a wide range of tasks and can learn physical properties across diverse atomic systems, thus leveraging more training data and more challenging metrics. These representations are more expressive [11] as evidenced by the successes of cross-modality models on structure prediction [12, 13], inverse folding [14] or molecular interaction prediction [15].
+
+Here, we introduce FuncBind, a unified and scalable framework for generating all-atom molecular systems conditioned on target structures. Following Kirchmeyer et al. [16], we represent molecules with neural fields: functions that map 3D coordinates to atomic densities. Neural fields are a compact and scalable alternative to voxels, while sharing many common advantages over point cloud-based representations: they (i) are compatible with expressive neural network architectures (such as CNNs and transformers), (ii) account for variable number of atoms and residues implicitly, and (iii) can represent molecules across modalities with an all-atom formulation. Using this new representation, we build a latent conditional generative model compatible with any score-based approaches. We tested denoising diffusion [17] and walk-jump sampling (WJS) [18], a sampling approach that enjoys fast-mixing and training simplicity when compared to diffusion models, largely because it only relies on one or few noise levels.
+
+We train FuncBind on structures from three drug modalities: small molecules, macrocyclic peptides (MCPs), and antibody complementarity-determining region (CDR) loops in complex with a target protein. These modalities encompass a range of chemical matter with challenging constraints, such as cyclic backbones and non-canonical amino acids. FuncBind achieves competitive results on in silico benchmarks, matching or outperforming modality-specific baselines. On in vitro experiments, we show that FuncBind can produce novel antibody binders by redesigning the CDR H3 loop of two chosen co-crystal structures. We also create a new dataset†, containing  $\sim 190,000$  synthetic MCP/protein complexes derived from 641 RCSB PDB structures [19], particularly relevant for this work, as cyclic peptides exhibit chemistry and function that span small and large molecule modalities.
+
+# 2 Related work
+
+Pocket-conditioned small molecule generation. Several approaches have framed structure-based drug design as a generative modeling problem [20]. These methods commonly represent molecules—including both ligands and targets—either as point clouds of atoms or as voxel grids. Point-cloud approaches represent atoms as points in 3D space, along with their atomic types, and typically use graph neural network architectures. Point-cloud approaches have been used to generate molecules using autoregressive models [21, 22, 3], iterative sampling approaches [23, 24, 25], normalizing flows [26], diffusion models [27, 4, 28], and Bayesian flow networks [5]. Voxel-based approaches map atomic densities to 3D discrete voxel grids and apply computer vision techniques for generation [29, 6, 30, 31]. VoxBind [7] demonstrates that voxel-based representations achieve state-of-the-art results using expressive vision-based networks and score-based generative models. However, raw voxel-based models do not currently scale to larger molecules due to high memory requirements. Neural fields serve as the continuous analogue to discrete voxels, a technique widely adopted in 3D computer vision [32]. When applied to molecular generation, these fields match existing performance levels while demonstrating superior memory and computational efficiency [16].
+
+Antigen-conditioned CDR loop generation. Antibody design is an active research topic and antibody-based treatments represented  $26\%$  of 2024 FDA approvals [33] (with related biologics approvals an even higher fraction). A key line of antibody engineering work re-designs the complementarity-determining regions (CDRs), a subset of the heavy and light chains that totals 48 to 82 residues [34] and represents most of the protein's affinity determining variability. A recent approach is to co-design the CDRs sequences and structures using residue cloud representations and equivariant graph neural networks [35, 36, 37, 38], combining these representations with diffusion models for generation [39, 40]. Other works leverage protein language models [41, 42], or revisit the problem by proposing new representations that incorporate domain knowledge and physics-based constraints [43]. As is the case for most models with demonstrated redesign capabilities, we tackle the task where the pose (docking) of the framework is provided. This assumption is relaxed in [44],
+
+![](images/0a7fc9a49647dac8b1bac23cd70dd941ba037808c599edd001c15c52f0493d60.jpg)  
+Figure 1: Neural field architectures. (a) Architecture used in [16] where a global embedding is used as input to the neural field decoder. (b) Our proposed neural field architecture, where embeddings are spatially arranged into a feature map grid. The latter allows us to better capture local signal information from input space and is compatible with expressive architectures for denoising.
+
+where the authors finetuned RFDiffusion [45] to generate the positions of backbone CDR atoms, followed by an inverse folding step [10]. Unlike other methods, FuncBind is the first approach based on neural fields that is also applicable to different data modalities simultaneously.
+
+Pocket-conditioned macrocyclic peptide generation. Occupying a unique chemical space between small and large molecules, cyclic peptides are a rapidly growing class of therapeutics that can access biological targets often challenging for both small molecules and antibodies [46]. De novo generation of target-specific cyclic peptides is therefore highly desirable, in part due to the power and utility of high throughput screens for cyclic and linear peptides and peptoid binders, yet only few works have investigated this. Rettie et al. [47] propose an approach based on RFDiffusion [45] that successfully designed cyclic peptides conditioned on a target protein structure. Yet this prior method is unable to handle non-canonical amino acids, an essential component for both compatibility with industry standard high-throughput screens and for enhancing the therapeutic properties of these peptides. The work by Tang et al. [48] is the only work we are aware of that can handle non-canonical amino acids; this model operates on tokenized smiles and performs target conditioning via classifier guidance with ML-based property predictors that are known to generalize poorly out-of-distribution [49].
+
+# 3 Method
+
+FuncBind is a latent score-based generative model that consists of two training steps: we first learn a latent representation for each molecule that modulates the parameters of a neural field decoder (Section 3.1), then we train a conditional denoiser on these latents (Section 3.2.1). The denoiser is used to sample molecules, conditioned on a given target (Section 3.2.2).
+
+# 3.1 Neural field-based latent representation
+
+We consider a dataset of  $N$  molecular complex tuples  $\mathcal{D} = \{(v,v^{\mathrm{tar}},c)_i\}_{i = 1}^N$ , where  $v$  and  $v^{\mathrm{tar}}$  are the binder and target, respectively, and  $c$  is the modality of the binder. In this work, we focus on three modalities: small molecules, macrocyclic peptides, and CDR loops, though the framework accommodates any atomic system. Atoms are represented as continuous Gaussian-like densities in 3D space and molecules as functions mapping coordinates  $x$  to  $n$ -dimensional atomic occupancy values,  $v:\mathbb{R}^3\to [0,1]^n$  (where  $n$  is the number of atom types) [50, 51, 52]. See Section B for details.
+
+Similar to [16], an encoder embeds a molecule into a latent  $z$ , which is used to decode back an atomic density field. Decoding consists in modulating the parameters of a shared neural field decoder based on the latent representation. However, instead of representing the latent  $z$  with a global embedding (Figure 1a) used in [16], we consider a spatially arranged feature map (Figure 1b). This approach has been successfully applied in other domains [53, 54, 55] and provides two key advantages: (i) each spatial feature captures local information helping to scale the model to larger molecules, (ii) it is compatible with expressive architectures (e.g. U-Nets [56]) for denoising.
+
+The encoder  $E_{\psi} : \mathbb{R}^{n \times L^3} \to \mathbb{R}^d$ ,  $d = C \times L^3$ , is a 3D CNN parameterized by  $\psi$  that maps a voxel grid  $G_v$ , generated by discretizing  $v$  at a fixed low resolution (for computational efficiency) set by the integer  $L$ , into a latent space with  $C$  channels. For decoding, we use nearest neighbor interpolation as in [54]: from the feature map  $z$ , we extract position-dependent vectors  $z_x \in \mathbb{R}^C$ . The embedding  $z_x$  is constant over a 3D patch in coordinate space. The decoder  $D_{\phi} : \mathbb{R}^C \times \mathbb{R}^3 \to \mathbb{R}^n$ , parameterized
+
+![](images/bd4b439d2a9e56db06c936d86924e4ebd7365e55aecc70c34de81013036cf133.jpg)  
+Figure 2: Conditional denoiser training overview. We voxelize separately the binder  $v$  and the target  $v_{\mathrm{tar}}$  of a given complex and encode them into  $z, z^{\mathrm{tar}}$  using encoders  $E_{\psi}, E_{\psi'}$ , respectively. We train a denoiser  $\hat{z}_{\theta}(y \mid z^{\mathrm{tar}}, \sigma, c)$  to remove the noise from  $y$  conditioned on  $z_{\mathrm{tar}}$ , the noise level  $\sigma$  and the one-hot modality class  $c$  (e.g. a cyclic peptide). The denoised latent representation is fed into a neural field decoder  $D_{\phi}$ ; this gives a reconstructed field  $\hat{v}$ .  $\hat{v}$  undergoes some additional postprocessing to recover the bonds and residue identities (if applicable); see Section 3.3.
+
+by  $\phi$ , then computes the molecular density field at coordinates  $x \in \mathbb{R}^3$ , given a local modulation embedding  $z_x \in \mathbb{R}^C$ . We use a conditional neural field based on multiplicative filter networks [57, 58] with Gabor filters, a natural choice for modeling the sparse atomic density fields [16].
+
+The neural field is trained across modalities with the objective proposed in [16]. A KL-regularization term [59] is added, following common practice in latent generation [60]:
+
+$$
+\mathcal {L} _ {\mathrm {A E}} (\psi , \phi) = \sum_ {v \in \mathcal {D}} \mathbb {E} _ {z \sim q _ {\psi} (z | v)} \left[ \int \| D _ {\phi} (x, z) - v (x) \| _ {2} ^ {2} \mathrm {d} x \right] + \beta \operatorname {K L} \left(q _ {\psi} (z \mid v) \mid \mathcal {N} (z; 0, I _ {d})\right), \tag {1}
+$$
+
+where  $q_{\psi}(z \mid v) = \mathcal{N}(z; \mu(v), \mathrm{diag}(\sigma(v))^2 I_d)$ ,  $\mu(v)$  and  $\sigma(v)$  are parameterized by  $E_{\psi}$  and  $\beta$  is a regularization weight. As [16], when optimizing for Equation (1), we upsample coordinates  $x$  close to the center of each atom to focus training on non-empty spaces. Since our model does not have equivariance built in the architecture, we apply data augmentation (translation and rotation).
+
+# 3.2 Conditional latent score-based generation
+
+We train a conditional denoiser on the neural-field based representations (Section 3.2.1). The denoiser is used to sample molecules from the aggregate posterior of the VAE encoder [61], conditioned on a given target, with conditional diffusion and walk-jump sampling (Section 3.2.2).
+
+# 3.2.1 Conditioned denoiser
+
+Our denoiser takes as input a noisy latent representation and a set of conditioning information, and outputs the "clean" version of the latent representation. In this work, we condition the denoiser on three signals: (i) the target structure  $v^{\mathrm{tar}}$ , (ii) the molecule modality  $c$  and (iii) the noise level  $\sigma$ .
+
+More formally, let  $(v,v^{\mathrm{tar}},c)$  be a (binder, target, modality) tuple from the dataset,  $(z,z^{\mathrm{tar}}):= (E_{\psi}(G_v),E_{\psi '}(G_v^{\mathrm{tar}}))$  their latent representations and  $y = z + \sigma \varepsilon$ ,  $\varepsilon \sim \mathcal{N}(0,I_d)$ , a noisy version of  $z$ . The target encoder  $E_{\psi '}$  is a 3D CNN with similar architecture as  $E_{\psi}$  but different parameters  $\psi'$ . Following the preconditioning pre-processing proposed by [62], our denoiser  $\hat{z}_{\theta}$  is defined as:
+
+$$
+\hat {z} _ {\theta} (y \mid z ^ {\mathrm {t a r}}, \sigma , c) = \frac {1}{\sigma^ {2} + 1} y + \frac {\sigma}{\sqrt {\sigma^ {2} + 1}} U _ {\theta} \left(\frac {1}{\sqrt {\sigma^ {2} + 1}} y, z ^ {\mathrm {t a r}}, \frac {1}{4} \log (\sigma), c\right),
+$$
+
+where  $U_{\theta}$  is a neural network parameterized by  $\theta$  and the embeddings  $z$  and  $z^{\mathrm{tar}}$  are normalized to unit variance and zero mean per channel, similar to [60]. Figure 2 shows an overview of the model architecture. The spatial structure of the latent space allows us to model  $U_{\theta}$  with 3D U-Nets, a standard architecture for generative models in computer vision. In particular, we adapt the network of Karras et al. [62]—designed to generate 2D images—to our 3D generation setting. Crucially, similar to recent works [52, 7, 63], we do not use any type of SE(3) equivariance constraints. Instead, we replace these constraints with data augmentation (rotations and translations) during training.
+
+The conditional denoiser is trained by minimizing the following loss at a given noise level  $\sigma$ :
+
+$$
+\mathcal {L} _ {\sigma} (\theta , \psi^ {\prime}) = \mathbb {E} _ {(v, v ^ {\operatorname {t a r}}, c) \sim \mathcal {D}, z \sim q _ {\psi} (z | v), \varepsilon \sim \mathfrak {N} (0, I _ {d})} \left[ \| \hat {z} _ {\theta} (z + \sigma \varepsilon \mid z ^ {\operatorname {t a r}}, \sigma , c) - z \| _ {2} ^ {2} \right], \tag {2}
+$$
+
+where  $z^{\mathrm{tar}} = E_{\psi'}(G_v^{\mathrm{tar}})$  is the encoding of the low-resolution voxel of the target. We apply the reweighting scheme in [62] across noise levels, i.e.:
+
+$$
+\mathcal {L} _ {\mathrm {d e n o i s e r}} (\theta , \psi^ {\prime}) = \mathbb {E} _ {\sigma \sim p (\sigma)} \left[ \frac {\sigma^ {2} + 1}{\sigma^ {2}} \frac {1}{e ^ {u (\sigma)}} \mathcal {L} _ {\sigma} (\theta , \psi^ {\prime}) + u (\sigma) \right],
+$$
+
+where  $\sigma$  is sampled along some pre-determined distribution  $p(\sigma)$  (see Section 3.2.2) and  $u(\sigma)$  is a one-layer MLP trained with the denoiser. This effectively reweights the loss based on the noise level.
+
+# 3.2.2 Sampling strategies
+
+We experimented with various score-based sampling strategies in the conditional setting, including the SDE formulation of denoising diffusion [64, 65], widely recognized for its state-of-the-art performance in image generation and walk-jump sampling (WJS), based on a probabilistic formulation of least-squares denoising [18]. Diffusion models operate over a continuous range of noise levels in contrast to WJS which considers only one noise level.
+
+These models rely on the Tweedie-Miyasawa formula (TMF) [66, 67], which relates the least-squares denoiser at a noise level  $\sigma$  with the score function at the noise level. Given  $y = z + \sigma \varepsilon$ ,  $\varepsilon \sim \mathcal{N}(0, I_d)$ , the conditional extension of TMF was derived in [7]. In our notation, it takes the form:
+
+$$
+\nabla_ {y} \log p (y \mid z ^ {\operatorname {t a r}}, \sigma , c) \approx s _ {\theta} (y \mid z ^ {\operatorname {t a r}}, \sigma , c) := \left(\hat {z} _ {\theta} (y \mid z ^ {\operatorname {t a r}}, \sigma , c) - y\right) / \sigma^ {2}, \tag {3}
+$$
+
+where  $\hat{z}_{\theta}$  is the minimizer of Equation (2);  $s_{\theta}(y \mid z^{\mathrm{tar}}, \sigma, c)$  is the learned conditional score function.
+
+For diffusion, we follow [65] and generate samples by numerically integrating the reverse-time SDE from noise level  $\sigma_{\mathrm{max}}$  to  $\sigma_{\mathrm{min}}$ , approximating the score function with the learned denoiser  $\hat{z}_{\theta}(y \mid z^{\mathrm{tar}}, \sigma, c)$  and TMF. We adopt the variance exploding formulation and the stochastic SDE sampler from EDM [62]. For WJS, we proceed as in Section B.4.2 and report the results on CDR H3 redesign in Section C.1.2.
+
+# 3.3 Recovering molecules from generated atomic-density fields
+
+To recover the underlying molecular structures from sampled latent codes  $z$ , we employ a post-processing pipeline inspired by [16]. The initial phase determines atom coordinates by identifying local optima in the neural field. This is achieved by first rendering the latent code  $z$  into a 0.25Å resolution voxel, then performing peak detection with MaxPooling filters, and finally refining the coordinates through gradient ascent, which takes advantage of the neural field's differentiability. The second phase involves inferring bonds and, when applicable, amino acid identities from the generated point cloud (coordinates and atom types) using OpenBabel software [68]. This yields .sdf files for molecules and peptides and .pdb files for proteins. A specific approach for identifying non-canonical amino acids, which are not recognized by OpenBabel, is described in Section D.
+
+# 4 Experiments
+
+We test our model on the following in silico settings, covering the three modalities discussed above: (i) small molecule generation conditioned on a protein pocket (Section 4.1); (ii) antibody CDR loops redesign conditioned on an epitope (Section 4.2); and (iii) macrocyclic peptides generation conditioned on a protein pocket (Section 4.3). We also performed in vitro validation of antibody CDR loops redesign conditioned on an epitope.
+
+For these tasks, the neural field is jointly trained on all three modalities. We train a 5B parameter model across modalities. Samples are generated via conditioning on the target structure. Note that our network is significantly larger than alternative models; we have observed improved performance in the unified setting for larger networks. See Section B for additional model details and Figure 3 for qualitative samples.
+
+We compared our unified model against specialized models trained independently for each modality. Overall, performance parity was observed across most metrics, with the key exception being uniqueness, which was significantly higher in the unified model. See Section C.1.3 for a comparison on CDR H3 inpainting. Further research exploring transfer learning across a broader set of modalities represents an exciting avenue for future work.
+
+![](images/2e2b5047872ee62532bcf04246da8b37e5e45248e83a290e6fdd480001f94ca1.jpg)  
+Figure 3: Examples of generated molecules given a target structure for different modalities: (top) small molecules against  $2\mathsf{r}\mathsf{ma}$ , (middle) macrocyclic peptides against 5ooC and (bottom) CDR H3 loop against  $5\mathsf{t}1\mathsf{k}$ . The seed binders are shown on the right.
+
+# 4.1 Small molecule generation
+
+Data. We consider the standard CrossDocked2020 [69] benchmark, with the pre-processing and splitting strategy of [70]. Pockets are clustered at a sequence identity of  $< 30\%$  using MMseqs2 and are split into 99,900 train ligand pockets pairs, 100 validation pairs and 100 test pairs.
+
+Baselines. We compare FuncBind to various pocket-conditioned ligand generative models: these include point cloud approaches based on autoregressive models (AR [21] and Pocket2Mol [3]), diffusion (DiffSBDD [27], TargetDiff [4], DecompDiff [28]), Bayesian Flow networks (MolCraft [5]) and a voxel-based approach based on walk-jump sampling (VoxBind [7]). FuncBind can be seen as a more scalable generalization of VoxBind and closely matches its performance. All methods but DecompDiff and MolCraft rely OpenBabel [68] to assign bonds from generated atom coordinates.
+
+Metrics. We evaluate performance using similar metrics as previous work [4]. For each method, we sample 100 ligands per pocket. We measure affinity with three metrics using AutoDock Vina [71]: VinaScore is the docking score of the generated molecule, VinaMin is the docking score after a local energy minimization, VinaDock fully re-docks the generated molecule, including both search and scoring steps. We also compute the drug-likeness, QED [72], and synthesizeability, SA [73], score of the generated molecules with RDKit [74]. Diversity is the average Tanimoto distance (in RDKit fingerprints) per pocket across pairs of generated ligands [75]. # atoms is the average number of (heavy) atoms per molecule. We also compute the PoseCheck metrics [76]: Steric clash computes the number of clashes between the generated ligands and their pockets, Strain energy (SE) measures the difference between the internal energy of the generated molecule's pose (without pocket) and a relaxed pose (computed using Universal Force Field [77] within RDKit); we report the median value.
+
+Results. The results are reported in Table 1. FuncBind is competitive with the current state of the art, slightly underperforming on docking-related metrics and strain energy compared to VoxBind and Molcraft and on number of clashes compared to VoxBind. This experiment demonstrates FuncBind's ability to generate highly-variable small molecules. Next, we demonstrate that it can also handle the more regular structures of amino acid-based molecules.
+
+# 4.2 Antibody CDR redesign
+
+Data. We consider the SabDab dataset [78], which comprises antibody-protein co-crystal structures and the data splits from DiffAb [39]. This non-i.i.d. split ensures that antibodies similar to those of the test set (i.e. more than  $50\%$  CDR H3 identity) are removed from the training set. The test split includes 19 targets, for which we redesign each CDR loop individually. As our baselines, we consider the Chothia numbering scheme [79] for the CDR definition.
+
+Baselines. We compare FuncBind to representative baselines: RAbD [80], a Rosetta-based method and two ML-based models, DiffAb [39] and AbDiffuser [43]. We consider the variation of AbDiffuser
+
+Table 1: Results on CrossDocked2020 test set.  $\uparrow / \downarrow$  denote that higher/ lower average (Avg.) or median (Med.) is better. For # atoms, numbers close to Reference are better. Baseline results are from [7, 28]. FuncBind's results are shown with mean/standard deviation obtained over 1,000 bootstraps.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">VinaScore ↓</td><td colspan="2">VinaMin ↓</td><td colspan="2">VinaDock ↓</td><td>QED ↑</td><td>SA ↑</td><td>Div. ↑</td><td>S.E. ↓</td><td>Clash ↓</td><td>#atoms</td></tr><tr><td>Avg.</td><td>Med.</td><td>Avg.</td><td>Med.</td><td>Avg.</td><td>Med.</td><td>Avg.</td><td>Avg.</td><td>Avg.</td><td>Med.</td><td>Avg.</td><td>Avg.</td></tr><tr><td>Reference</td><td>-6.36</td><td>-6.46</td><td>-6.71</td><td>-6.49</td><td>-7.45</td><td>-7.26</td><td>.48</td><td>.73</td><td>-</td><td>103</td><td>4.7</td><td>22.8</td></tr><tr><td>AR</td><td>-5.75</td><td>-5.64</td><td>-6.18</td><td>-5.88</td><td>-6.75</td><td>-6.62</td><td>.51</td><td>.63</td><td>.70</td><td>595</td><td>4.2</td><td>17.6</td></tr><tr><td>Pocket2mol</td><td>-5.14</td><td>-4.70</td><td>-6.42</td><td>-5.82</td><td>-7.15</td><td>-6.79</td><td>.56</td><td>.74</td><td>.69</td><td>206</td><td>5.8</td><td>17.7</td></tr><tr><td>DiffSBDD</td><td>-1.94</td><td>-4.24</td><td>-5.85</td><td>-5.94</td><td>-7.00</td><td>-6.90</td><td>.48</td><td>.58</td><td>.73</td><td>1193</td><td>15.4</td><td>24.0</td></tr><tr><td>TargetDiff</td><td>-5.47</td><td>-6.30</td><td>-6.64</td><td>-6.83</td><td>-7.80</td><td>-7.91</td><td>.48</td><td>.58</td><td>.72</td><td>1243</td><td>10.8</td><td>24.2</td></tr><tr><td>DecompDiff</td><td>-5.67</td><td>-6.04</td><td>-7.04</td><td>-7.09</td><td>-8.39</td><td>-8.43</td><td>.45</td><td>.61</td><td>.68</td><td>N/A</td><td>7.1</td><td>20.9</td></tr><tr><td>MolCraft</td><td>-6.59</td><td>-7.04</td><td>-7.27</td><td>-7.26</td><td>-7.92</td><td>-8.01</td><td>.50</td><td>.69</td><td>.72</td><td>195</td><td>7.1</td><td>22.7</td></tr><tr><td>VoxBind</td><td>-6.94</td><td>-7.11</td><td>-7.54</td><td>-7.55</td><td>-8.30</td><td>-8.41</td><td>.57</td><td>.70</td><td>.73</td><td>162</td><td>5.1</td><td>23.4</td></tr><tr><td>FuncBind</td><td>-5.71 (±.03)</td><td>-5.64 (±.03)</td><td>-6.34 (±.03)</td><td>-6.18 (±.03)</td><td>-7.26 (±.03)</td><td>-7.28 (±.03)</td><td>.50 (±.002)</td><td>.65 (±.001)</td><td>.70 (±.0)</td><td>217 (±.12)</td><td>7.4 (±.06)</td><td>19.0 (±.09)</td></tr></table>
+
+with side chain generation to better match FuncBind's all-atom setting; AbDiffuser in contrast to other baselines, generates all 6 loops jointly. We also compare to AbX [81] and the reproduction of dyMEAN [38] from [81] for H3 design, where the DiffAb splits were considered.
+
+Metrics. We compute the following metrics, measuring the similarity of the generated designs to the seed: amino acid recovery (AAR), the sequence identity between the seed and the generated CDRs; RMSD, the  $C_{\alpha}$  root-mean-square deviation between the seed and generated structure and IMP, the percentage of designs with lower binding energy  $(\Delta G)$  than the seed, as calculated by InterfaceAnalyzer in Rosetta [80]. Baselines apply Rosetta-based relaxations prior to computing IMP to improve the energy scores: DiffAb refines the generated structure with OpenMM [82] and AbX uses fast-relax [80]. We report metrics for 100 generated samples per target. Note that our model, unlike most baselines, generates samples with diverse sequence lengths; to compute these metrics we consider samples with the same length as the original seed. We found that uniqueness impacts AAR and RMSD, particularly for non-H3 loop designs which exhibited low uniqueness. This presents a challenge for fair model comparison, as baselines do not report uniqueness. For completeness, we also report the metrics for the unique samples on Table 4 (Section C.1).
+
+Results. The results in Table 2 indicate that our model is state of the art both on amino acid recovery (AAR) and  $C_{\alpha}$  RMSD values, outperforming other baselines by 1.5 to 3 times across all loops. This performance can be attributed to our all-atom formulation and neural-field representation, which enable the model to better capture the molecular conformation and conditioning context. AbDiffuser also leverages side-chain information but underperforms in RMSD, highlighting the distinct advantages of our approach. Finally, FuncBind's interface energy improvement (IMP) without backbone minimization is competitive to the IMP of approaches that apply minimization. This showcases the quality and fidelity of the generated structures, as energy is very sensitive to wrong atom placement. As the baselines [81, 39], when applying Rosetta's fast-relax backbone minimization on the generated loops, IMP greatly improves as expected, outperforming even the Rosetta RAbD protocol that directly optimizes the energy function. This refinement procedure slightly increases RMSD, as it changes the loop to minimize strain, while FuncBind is trained to mimic patterns in ground-truth crystal structures.
+
+Length distributions generated. The above evaluation restricted designs to the seed's length, a common prior in many generative models for this task. However, in many settings, we do not know what is a reasonable length. FuncBind is designed to sample designs across various lengths, a useful capability for de novo CDR generation. To demonstrate this flexibility, we analyzed histograms of sequence lengths and atom counts for CDR H3s designed for a de novo target, comparing them against the original seed's values (see Figure 4). For this specific target, while the generated designs exhibited a range of lengths, their distributions were centered on the seed's reference values. Further validation of designs with other lengths is left for future research.
+
+Table 2: CDR inpainting on SAbDab [78] with DiffAb splits [70]. RMSD is in Å and AAR, IMP are in %. † indicates additional relaxation / optimization with Rosetta.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">H1</td><td colspan="3">L1</td></tr><tr><td>AAR↑</td><td>RMSD↓</td><td>IMP↑</td><td>AAR↑</td><td>RMSD↓</td><td>IMP↑</td></tr><tr><td>\( RAbD^{\dagger} \)</td><td>22.9</td><td>2.26</td><td>43.9</td><td>34.3</td><td>1.20</td><td>46.8</td></tr><tr><td>\( DiffAb^{\dagger} \)</td><td>65.8</td><td>1.19</td><td>53.6</td><td>55.7</td><td>1.39</td><td>45.6</td></tr><tr><td>AbDiffuser</td><td>76.3</td><td>1.58</td><td>-</td><td>81.4</td><td>1.46</td><td>-</td></tr><tr><td>FuncBind</td><td>86.9</td><td>\( 0.41 / 0.44^{\dagger} \)</td><td>\( 35.0 / 77.2^{\dagger} \)</td><td>86.4</td><td>\( 0.68 / 0.73^{\dagger} \)</td><td>\( 45.0 / 80.4^{\dagger} \)</td></tr><tr><td></td><td colspan="3">H2</td><td colspan="3">L2</td></tr><tr><td>\( RAbD^{\dagger} \)</td><td>25.5</td><td>1.64</td><td>53.5</td><td>26.3</td><td>1.77</td><td>56.9</td></tr><tr><td>\( DiffAb^{\dagger} \)</td><td>49.3</td><td>1.08</td><td>29.8</td><td>59.3</td><td>1.37</td><td>50.0</td></tr><tr><td>AbDiffuser</td><td>65.7</td><td>1.45</td><td>-</td><td>83.2</td><td>1.40</td><td>-</td></tr><tr><td>FuncBind</td><td>78.2</td><td>\( 0.52 / 0.54^{\dagger} \)</td><td>\( 31.7 / 61.4^{\dagger} \)</td><td>86.2</td><td>\( 0.83 / 0.84^{\dagger} \)</td><td>\( 39.5 / 66.0^{\dagger} \)</td></tr><tr><td></td><td colspan="3">H3</td><td colspan="3">L3</td></tr><tr><td>\( RAbD^{\dagger} \)</td><td>22.1</td><td>2.90</td><td>23.3</td><td>20.7</td><td>1.62</td><td>55.6</td></tr><tr><td>\( DiffAb^{\dagger} \)</td><td>26.8</td><td>3.60</td><td>23.6</td><td>46.5</td><td>1.63</td><td>47.3</td></tr><tr><td>\( dyMEAN^{\dagger} \)</td><td>29.3</td><td>4.80</td><td>5.26</td><td>-</td><td>-</td><td>-</td></tr><tr><td>\( AbX^{\dagger} \)</td><td>30.3</td><td>3.41</td><td>42.9</td><td>-</td><td>-</td><td>-</td></tr><tr><td>AbDiffuser</td><td>34.1</td><td>3.35</td><td>-</td><td>73.2</td><td>1.59</td><td>-</td></tr><tr><td>FuncBind</td><td>47.5</td><td>\( 2.04 / 2.10^{\dagger} \)</td><td>\( 19.4 / 49.9^{\dagger} \)</td><td>80.8</td><td>\( 0.68 / 0.73^{\dagger} \)</td><td>\( 32.7 / 67.5^{\dagger} \)</td></tr></table>
+
+![](images/4e385a36ada18b2891fe9c0cccfe6f3e8abdea502735f743687ceda7b60040b1.jpg)  
+Figure 4: CDR H3 length (left) and atom count (right) histogram on the de-novo 4cni target. Red is the seed H3's reference numbers.
+
+![](images/9d4170e8231daa18a37915be6014e0536494697cec3ce53c4d9c19f0dfe22d53.jpg)
+
+In vitro evaluation. We performed wet-lab validation of H3 loop redesigns based on the co-crystal structure of an antibody bound to a rigid and a flexible epitope $^{\ddagger}$ . We selected the H3 loop for its important contribution to the antibody's functional properties. We consider a de novo setting, where interfaces similar to the two complexes, identified using Ab-Ligity [83], were excluded from training.
+
+From an initial pool of 10,000 unique generated H3 designs (all matching the original seed's length), 190 were selected for experimental testing. This selection involved two steps: 1) The top 500 designs were shortlisted based on model confidence, as indicated by their repeated generation counts. Our in silico validation showed that repeats is an useful proxy for high amino acid recovery of the seed. 2) These 500 designs were then clustered into 190 groups using weighted K-means based on sequence edit distance, where the weights were defined by the repeat generation count. The design with the highest repeat count (highest confidence) from each of these 190 clusters was chosen for synthesis and characterization. The selected antibody designs were expressed and purified in the wet lab. Binding affinity was then determined using surface plasmon resonance (SPR) measurements. Section C.2 presents some detailed analysis. FuncBind achieves a binding rate of  $45\%$  on the rigid epitope and  $2\%$  on the flexible epitope, which increases to  $4\%$  with a relaxed binding threshold.
+
+Table 3: Results on our MCP benchmark. RMSD is in Å, Residues-TS≥0.5, Vina Dock are in %.  
+
+<table><tr><td></td><td>TS ↑</td><td>Residues-TS≥0.5 ↑</td><td>L-RMSD ↓</td><td>I-RMSD ↓</td><td>TM-Score ↑</td><td>Vina dock ↑</td></tr><tr><td>RFPeptide</td><td>0.31</td><td>29</td><td>12</td><td>3.3</td><td>0.33</td><td>8.8</td></tr><tr><td>AfCycDesign</td><td>0.34</td><td>29</td><td>7.6</td><td>3.7</td><td>0.33</td><td>29</td></tr><tr><td>FuncBind</td><td>0.33</td><td>25</td><td>2.6</td><td>1.8</td><td>0.36</td><td>41</td></tr></table>
+
+# 4.3 Macrocyclic peptide generation
+
+Data. Given the scarcity of established baselines, benchmarks, and available data for macrocyclic peptide (MCP)-protein complexes, we introduce a novel benchmark to facilitate the evaluation of generative models for MCPs. To address the data limitation, we have curated a dataset of 186,685 MCP-protein complexes using a "mutate then relax" strategy detailed in Section E. Taking as input an original set of 641 protein-MCP complexes sourced from RCSB PDB [19], this strategy consists in (i) randomly mutating the MCPs at 1 to 8 different sites, using a list of 213 distinct amino acids, (ii) relaxing them using fast-relax, which involves iterative cycles of side-chain packing and all-atom minimization [84] and (iii) selecting the lowest interface scores. The source dataset comprises lengths ranging from 4 to 25 amino acids with an average of 10 (Section E Figure 13a).  $78\%$  of the MCPs contain one or more non-canonical amino acids, i.e. any amino acid that is neither L-canonical nor D-canonical. We split the dataset into train, test and validation subsets using a clustering approach detailed in Section E that aims at creating a non-i.i.d. test set consisting of 85 protein pockets.
+
+Baselines. MCPs pose significant challenges for generative models due to their non-canonical amino acids, cyclization, and scarce training data. To our knowledge, no other target-conditioned, structure-based MCP generative models handles non-canonical amino acids, precluding direct comparisons. For reference, we compare nonetheless FuncBind with AfCycDesign [85] and RFPeptide [47], two models generating MCPs exclusively with canonical amino acids and N-to-C cyclization.
+
+Metrics. As part of this new benchmark, we define and compute relevant metrics. Tanimoto similarity (TS) assesses the resemblance between the ground-truth seed and the sampled MCP structure. Ligand RMSD (L-RMSD) is the RMSD between sample MCP to seed MCP, and template modeling (TM) score, a length independent similarity metric, based on the Kabsch alignment of the backbone atoms  $(N,C_{\alpha},C,O)$ . TM score was calculated by maximizing the scaling factor. The same backbone logic is applied to compute interface RMSD (I-RMSD), the RMSD in the pocket (which are for the most part slightly lower since the pockets are identical). The sample and the seed were not aligned for I-RMSD since this is based on where the MCPs are in the pocket. Binding affinity was calculated through Autodock Vina [71].
+
+Results. We observe a correlation between the generated designs with the MCP seeds. Qualitatively, Section E.1 Figure 10 illustrates the close alignment of the backbone between the sampled structures and the seed. Most of the sampled molecules display consistent repeating peptide bonds, linking the  $C_1$  carbon of one  $\alpha$ -amino acid to the  $N_2$  nitrogen of the next. Closure bonds (such as disulfide in Figure 10a,b and N to C cyclization in Figure 10c) are also often maintained in the sampled sets.
+
+Metrics are reported in Table 3. The mid-range TS and TM scores reflects a strong similarity to the peptide backbone, with variability occurring at the functional groups of the residues. An example of per-residue TS for molecules sampled with the seed mutant (Section E.1 Table 10a) and the crystal MCP (Section E.1 Table 10b) shows that the highest TS occurs at the disulfide closure bond. This elevated per-residue similarity results from the preservation of closure bond residues throughout the curated dataset. Furthermore, the per-residue TS is higher across the crystal MCP residues than at the mutated residues of the seed mutant (PRO3A20 and GLU4B60). Because all mutants originate from the crystal MCP, the dataset is closely tied to the crystal sequence, and the sampled structures similarly reflect this connection. Low RMSD results show generally good alignment with the seed MCP, with many samples exhibiting RMSDs below  $1\AA$ , particularly for I-RMSD. These results are consistent with the TM scores, where  $\sim 20\%$  of the samples exhibit TM scores greater than 0.5. Finally, Autodock Vina binding affinity reveals that nearly half of the generated samples, both before and after minimization, have better binding affinity in the pocket compared to the seed.
+
+![](images/881b9e6e9d262bcaee3277524da4f059013bbd885b5f7701d4f6fa37220c3386.jpg)  
+Figure 5: Per-residue energy scores at the same position were calculated using Rosetta's residue energy breakdown for a seed and two samples. We analyzed: (a) the seed's serine, (b) 3-hydroxycyclopentylalanine (C1O) from sample 11 (Section E.1 Figure 12), (c) tyrosine from sample 35.
+
+![](images/1f1bb3103549d21f48ce1b947fff508de7b656f557e6cffe570edc1cb5890f5f.jpg)
+
+![](images/3a4586493923c8689e1b99c486ebf2f4cf3ecb67232906f3cc285fd60c3dda55.jpg)
+
+Compared to the baselines, FuncBind achieves superior or similar metrics, notably lower L/I-RMSDs. FuncBind also yields the highest proportion of designs with better docking scores. The only underperformance was in Tanimoto similarity (Residues-TS  $\geq 0.5$  in particular), expected as FuncBind accesses a larger set of non-canonical amino acids and sequence lengths as opposed to these baselines. We encourage future comparisons on this new benchmark, especially for models handling non-canonical amino acids.
+
+Analysis of generated non-canonicals. In Figure 11 (Section E.1), amino acids are categorized into known canonical and non-canonical amino acids (seen in the training set), and unknown non-canonical amino acids, which represent newly generated amino acids not previously seen. Fewer than  $1\%$  of all categorized amino acids were labeled as "unreasonable", a designation applied when a bond was shorter than  $0.8\AA$  or when invalid oxygen-oxygen or nitrogen-nitrogen bonds were present. Some reasonable and novel generated amino acids are presented in Figure 12 (Section E.1). Generating novel, chemically plausible amino acids without restrictions from a predefined library or initial cyclic backbone allows broader exploration of the binding pocket. This is demonstrated in Figure 5b, where an amino acid absent from our library interacts with pocket residues that neither the seed (Figure 5a) nor a chemically similar amino acid at the same position (Figure 5c) engage. The absence of constraints in MCP generation promotes greater sequence diversity and deeper investigation of the binding pocket.
+
+# 5 Conclusion
+
+We presented FuncBind, a new framework for all-atom, structure-conditioned de novo molecular design. FuncBind is based on a new modality-agnostic representation, that enables a single model to be trained across diverse drug modalities; we focused on small molecules, macrocyclic peptides and antibody CDRs. FuncBind handles variable atom and residue counts and is based on recent advancements in computer vision, replacing equivariance constraints with data augmentation. FuncBind demonstrates competitive in silico performance, matching or outperforming specialized baselines. In vitro, we demonstrate that FuncBind generates binders against de novo targets. It generates novel and chemically plausible molecules, including new non canonical amino acids. Future directions include extending FuncBind to larger biomolecular systems and to more data modalities. Furthermore, the scaling behaviour of this model remains an interesting direction for future study, particularly given the absence of overfitting as the denoiser increased in size (we tested up to 5B parameters).
+
+It is important to note that, like other structure-based methods, FuncBind relies on the availability of an accurate model of the molecular interface to be designed. This can be a limitation, as these models are costly to obtain and their availability is often restricted in the drug discovery process, particularly for large molecules. Finally, real-world application of generative models in drug design requires addressing a range of properties beyond binding, e.g. synthesizeability for small molecules and developability for antibodies, considerations not handled in this work.
+
+Acknowledgements We thank President Design and the following colleagues: Jan Ludwiczak for processing the SabDab dataset. Tamica D'Souza for performing the antibody wetlab validation.
+
+Max Shen, Namuk Park, Nathan Frey, Sidney Lisanza, Rob Alberstein, Ewa Nowara, Natasa Tagasovska, Chen Cheng, Pan Kessel, Sarah Robinson, Joshua Yao-Yu Lin for insightful discussions and Genentech's legal team.
+
+# References
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+
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+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+# Answer: [Yes]
+
+Justification: Sections B and 4
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+# Answer: [Yes]
+
+Justification: Error bars reported where it is common to do so (Section 4.1)
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
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+
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: Section B
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
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+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes].
+
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+
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+
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+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes].
+
+Justification: Section A
+
+Guidelines:
+
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+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: The paper poses no high risk for misuse.
+
+Guidelines:
+
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+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
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+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
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+
+Guidelines:
+
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+
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [Yes]
+
+Justification: See Sections B and E
+
+Guidelines:
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+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
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+
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+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: The paper does not involve research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
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+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+# Answer: [NA]
+
+Justification: The core method development does not involve LLMs.
+
+# Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# Appendices
+
+This supplementary material is organized as follows:
+
+1. Section A includes a broader impact statement.  
+2. Section B includes model and implementation details.  
+3. Section C provides some additional results for antibody CDR redesign.
+
+Section C.1 presents some additional in silico results  
+- Section C.2 presents our in vitro results.
+
+4. Section D explain how we inferred non canonical amino acids without OpenBabel.
+
+5. Section E presents some additional results for macro-cyclic peptide generation
+
+- Section E.1 presents some additional results.  
+- Section E.2 presents how we curated the dataset for training FuncBind.  
+- Section E.3 presents the train / val / test splitting logic.
+
+# A Broader impacts
+
+This work introduces FuncBind, a novel framework for all-atom, structure-conditioned molecular generation, whose primary positive impact lies in its potential to accelerate and enhance the discovery of new therapeutics across diverse modalities, from small molecules to complex biologics like peptides and antibodies. While the underlying principles could find applications in other scientific fields like materials science, its deployment in drug discovery requires addressing significant challenges, including the validation gap between in silico predictions and experimental testing (in vitro, in vivo, and clinical trials).
+
+# B Model details
+
+# B.1 Representation
+
+FuncBind is based on a neural field representation that models an atomic density field, a smooth function taking values between 0 (far away from all atoms) and 1 (at the center of atoms). This field takes the following form [50, 51]:
+
+$$
+\forall x \in \mathbb {R} ^ {3}, v _ {a} (x) = 1 - \prod_ {i = 1} ^ {n _ {a}} \left(1 - \exp \left(- \left(\frac {\| x - x _ {a _ {i}} \|}{. 9 3 r}\right) ^ {2}\right)\right), \tag {4}
+$$
+
+where  $a_{i}$  is the  $i^{\mathrm{th}}$  atom of type  $a$  (among  $n$  choices), for a total of  $n_a$  atoms and  $r$  is the atoms' radius set to  $r = 1.0\AA$  for all atom types.
+
+We consider  $n = 8$  element types  $C, O, N, S, F, Cl, P, Br$  that cover all major atom types across small molecule, macrocyclic peptides and proteins. Note that protein-specific atom types (e.g.  $C_{\alpha}, C_{\beta}$  etc.) are merged into a single element type (e.g.  $C$ ). This helps transfer learning across modalities.
+
+Finally, the field is defined over a volume of  $(32\mathring{\mathrm{A}})^{3}$ . It is the continuous version of a voxel grid of spatial dimension 128 and resolution of  $0.25\mathring{\mathrm{A}}$ , in  $\mathbb{R}^{8\times 128^3}$ .
+
+# B.2 Neural Field
+
+The encoder  $E_{\psi}$  is a 3D CNN containing 4 residual blocks (number of hidden units 256, 512, 1024, 2048 for each block), where each block contains 3 convolutional layers followed by BatchNorm, ReLU and pooling layers (we use max pooling on the first three blocks). The encoder has 59M parameters. The input to the encoder is a low-resolution grid of spatial grid dimension  $L = 16$  corresponding to a resolution of  $2\mathring{\mathrm{A}}$ . Before voxelizing the molecules, we first center the atoms around the tightest bounding box encapsulating the molecule, apply a random rotation to the atoms
+
+(each Euler angle rotated randomly between  $[0,2\pi)$ ) and random translation between  $[-1,1]\mathring{\mathrm{A}}$  then normalize their coordinates to the range of  $[-1,1]$ .
+
+The decoder  $D_{\phi}$  is a conditional Multiplicative Filter Network (MFN) [57, 58] with Gabor filters and 6 FiLM-modulated layers, where each fully-connected layer has 2048 hidden units. The decoder has 59M parameters.
+
+The auto-encoder is trained with Adam Optimizer [86] with learning rate  $10^{-2}$ ,  $\beta_{1} = 0.9$ ,  $\beta_{2} = 0.999$ . We apply a KL regularization weight of  $\lambda = 10^{-5}$ . Batch size is 32 over 1 B200 GPU; we sample 15000 coordinates per batch.
+
+# B.3 Denoiser
+
+The hyperparameters of the Karras et al. [62] UNet architecture are as follows: 5B model with 8 blocks with 512 channels, channel multipliers [1,2,3,4], attention resolutions [4, 2]. This model follows the XXL setting of EDM2 [62] and increased the number of channels from 448 to 512. For reference, Proteina [63], the largest protein backbone generative model, comprises 400M parameters and RFDiffusion [8] roughly 100M parameters. As [65], we apply preconditioning to learn the denoiser across noise levels. Moreover, we sample the noise levels from a log-normal distribution with mean 1.2 and standard deviation 0.8.
+
+The target encoder  $E_{\psi'}$  is a 3D CNN which takes as input a voxel grid of dimension  $\mathbb{R}^{4 \times 32^3}$  of the target protein (resolution  $1.0\AA$ ), considering atom elements  $C, O, N, S$ , and consists of a magnitude preserving [62] CNN layer with kernel  $3 \times 3 \times 3$  and output channels 64, then a downsampling layer to a spatial grid in  $\mathbb{R}^{64 \times 16^3}$  then a magnitude preserving U-Net block [62] with output channels  $C = 128$  leading to a voxel of size  $\mathbb{R}^{C \times 16^3}$ .
+
+The parameters are optimized with Adam optimizer [86] with learning rate  $\alpha_{\mathrm{ref}} = 10^{-2}$ ,  $\beta_{1} = 0.9$ ,  $\beta_{2} = 0.95$  using an aggregated batch size of 768 over 8 B200 GPUs. We perform early stopping on the validation loss. We use the power function exponential moving average from EDM2 [62] with an EMA length of  $5\%$ . Moreover, we adopt the inverse square root decay schedule of [86], also used in [62] which sets  $\alpha(t) = \frac{\alpha_{\mathrm{ref}}}{\sqrt{\max(t / t_{\mathrm{ref}}, 1)}}$ , where we set  $t_{\mathrm{ref}} = 20040$ . Finally, the networks are trained by randomly dropping the conditioning information  $10\%$  of the time.
+
+# B.4 Sampling
+
+To improve uniqueness, we apply different rotations to the pocket on each MCMC chain, in a similar fashion to how rotation-based data augmentation is performed at training time.
+
+# B.4.1 Denoising diffusion
+
+We set as follows the sampling parameters of EDM2 [62]:
+
+-  $N = 128$  steps  
+-  $\sigma_{\mathrm{min}} = 0.01$ ,  $\sigma_{\mathrm{max}} = 10$  
+$S_{\mathrm{min}} = 5.0,S_{\mathrm{max}} = 7.0$  
+$S_{\mathrm{churn}} = 30.0$  
+$S_{\mathrm{noise}} = 1.003$  
+$\bullet \rho = 7$
+
+Moreover, we apply a temperature scaling with  $\tau = 0.5$  on Crossdocked and  $\tau = 0.33$  on MCP-protein complexes.
+
+# B.4.2 Walk-Jump sampling
+
+We also implemented a conditional form of the Walk-Jump Sampling (WJS), a score-based generative model that is based on a probabilistic formulation of least-squares denoising [18]. The framework is based on the Tweedie-Miyasawa formula (TMF) [66, 67], which relates the least-squares denoiser
+
+at a noise level  $\sigma$  with the score function at the noise level. Given  $y = z + \sigma \varepsilon, \varepsilon \sim \mathcal{N}(0, I_d)$ , the conditional extension of TMF was derived in [7]. In our notation, it takes the form:
+
+$$
+\nabla_ {y} \log p (y \mid z ^ {\operatorname {t a r}}, \sigma , c) \approx s _ {\theta} (y \mid z ^ {\operatorname {t a r}}, \sigma , c) := (\hat {z} _ {\theta} (y \mid z ^ {\operatorname {t a r}}, \sigma , c) - y) / \sigma^ {2}, \tag {5}
+$$
+
+where  $\hat{z}_{\theta}$  is the minimizer of Equation (2);  $s_{\theta}(y \mid z^{\mathrm{tar}}, \sigma, c)$  is the learned conditional score function.
+
+Walk-jump sampling. WJS [18] is composed of two stages: (i) (walk) samples the noisy latent variables conditioned on  $z^{\mathrm{tar}}$ ,  $c$  using Langevin Markov chain Monte Carlo (MCMC) via the learned score function (Equation (5)), (ii) (jump) estimates "clean"  $z$  by single-step denoising. There is a fundamental trade-off in this sampling strategy: for larger  $\sigma$ , sampling from the smoother density becomes easier, but the denoised samples move farther away from the distribution of interest [87].
+
+Multimeasurement walk-jump sampling. The sampling trade-off in WJS is addressed in multi-measurement denoising models [88, 87], in which the problem is framed as sampling from the distribution  $p_{\sigma}(y_{1:m})$  associated with  $y_{1:m} \coloneqq (y_1,\dots ,y_m)$ , where  $y_{k} = z + \sigma \varepsilon_{k}$ ,  $k \in \{1,\ldots ,m\}$ , and  $\varepsilon_{k} \sim \mathcal{N}(0,I_{d})$  all independent of  $z$ . Saremi et al. [87] studied a sequential scheme for sampling from  $p_{\sigma}(y_{1:m})$  and showed that the noise level effectively decreases (as far as the denoiser is concerned) at the rate  $\sigma /\sqrt{m}$ . Furthermore, it was shown that sampling becomes easier upon accumulation of measurements. The general sampling problem is therefore mapped to a sequence of sampling noisy data at a fixed noise scale, while the effective noise decreases via accumulation of measurements. We refer to this scheme as WJS-  $m$ , which involves two hyperparameters: the noise level  $\sigma$ , and the number of measurements  $m$ . In this construction, we only need to keep track of the empirical mean of noisy samples. In particular, we have (see [87, Eq. 4.9]):
+
+$$
+\nabla_ {y _ {m}} \log p _ {\theta} (y _ {m} | y _ {1: m - 1}, z ^ {\mathrm {t a r}}, \sigma , c) = \frac {1}{m} s _ {\theta} (\overline {{y}} _ {1: m} | z ^ {\mathrm {t a r}}, \frac {\sigma}{\sqrt {m}}, c) + \frac {1}{\sigma^ {2}} (\overline {{y}} _ {1: m} - y _ {m}),
+$$
+
+where  $\overline{y}_{1:m}$  is the empirical mean of the measurements  $(y_{1},\ldots ,y_{m})$ . The score function above is used in sampling  $(y_{1},\dots ,y_{m})$  iteratively using Langevin MCMC [87, Algorithm 1]. Finally, the denoising "jump" in WJS-  $m$  is achieved via (single-measurement) TMF using the sufficient statistics  $\overline{y}_{1:m}$  at the noise scale  $\sigma /\sqrt{m}$ . It is clear that the vanilla WJS discussed above reduces to WJS-1. Although there is a flavor of diffusion in this scheme due to its sequential strategy, WJS-  $m$  is arguably more "surgical" in that, by construction, we do not need to learn score functions over a continuum of noise levels, but only a finite one identified by  $m$ . This is especially appealing for applications where WJS-1 already shows reasonable performance and  $m$  is therefore taken to be small. This work contains the first experimental validation of WJS-  $m$  in generative modeling applications.
+
+We report the results for WJS for CDR H3 inpainting in Section C.1.2. The parameters are set to  $\sigma = 7.0$  and  $m = 16$ . We use underdamped Langevin MCMC from Sachs et al. [89] in the BAOAB scheme with  $K = 50$  steps, friction  $\gamma = 1.0$ , discretization step  $\delta = \sigma /2$  ([87, Algorithm 1]).
+
+# C CDR redesign
+
+# C.1 In silico evaluation
+
+# C.1.1 Uniqueness
+
+Table 4 reports our CDR sampling results over unique sequences.
+
+We observe that higher uniqueness usually leads to lower Amino Acid Recovery (AAR). In other words, repeated sequences tend to correlate more with the seed. We use this simple heuristic to select H3 designs for in vitro evaluation (see Section C.2).
+
+# C.1.2 Ablation with Walk Jump Sampling
+
+We report the performance of multimeasurement WJS and diffusion. Overall the models are comparable with slightly higher uniqueness for diffusion.
+
+# C.1.3 Comparison to specialized model
+
+We observe that the unified model has higher uniqueness than the specialized model, with slightly better CDR loop inpainting performance on unique samples. We observe this trend on the other data modalities as well.
+
+Table 4: Impact of uniqueness on CDR inpainting performance. RMSD is in Å and AAR and Uniqueness are in %. * indicates designs with unique sequences.  
+
+<table><tr><td>Method</td><td>AAR↑</td><td>RMSD↓</td><td>Unique↑</td><td>AAR↑</td><td>RMSD↓</td><td>Unique↑</td></tr><tr><td></td><td colspan="3">H1</td><td colspan="3">L1</td></tr><tr><td>AbDiffuser</td><td>76.3</td><td>1.58</td><td>-</td><td>81.4</td><td>1.46</td><td>-</td></tr><tr><td>FuncBind</td><td>86.9</td><td>0.41</td><td>23.5</td><td>86.4</td><td>0.68</td><td>38.9</td></tr><tr><td>FuncBind*</td><td>75.9</td><td>0.45</td><td>100</td><td>79.3</td><td>0.85</td><td>100</td></tr><tr><td></td><td colspan="3">H2</td><td colspan="3">L2</td></tr><tr><td>AbDiffuser</td><td>65.7</td><td>1.45</td><td>-</td><td>83.2</td><td>1.40</td><td>-</td></tr><tr><td>FuncBind</td><td>78.2</td><td>0.52</td><td>20.6</td><td>86.2</td><td>0.83</td><td>19.4</td></tr><tr><td>FuncBind*</td><td>59.5</td><td>0.57</td><td>100</td><td>54.4</td><td>2.39</td><td>100</td></tr><tr><td></td><td colspan="3">H3</td><td colspan="3">L3</td></tr><tr><td>AbDiffuser</td><td>34.1</td><td>3.35</td><td>-</td><td>73.2</td><td>1.59</td><td>-</td></tr><tr><td>FuncBind</td><td>47.5</td><td>2.04</td><td>85.5</td><td>80.8</td><td>0.68</td><td>44.1</td></tr><tr><td>FuncBind*</td><td>44.1</td><td>2.16</td><td>100</td><td>68.9</td><td>0.94</td><td>100</td></tr></table>
+
+Table 5: Diffusion vs WJS on H3 loop inpainting. \* indicates designs with unique sequences.  
+
+<table><tr><td>Method</td><td>AAR ↑</td><td>RMSD ↓</td><td>Unique ↑</td></tr><tr><td>FuncBinddiff</td><td>47.5</td><td>2.04</td><td>85.5</td></tr><tr><td>FuncBind*diff</td><td>44.1</td><td>2.16</td><td>100</td></tr><tr><td>FuncBindWJS-16</td><td>51.0</td><td>1.89</td><td>73.8</td></tr><tr><td>FuncBind*WJS-16</td><td>41.4</td><td>2.18</td><td>100</td></tr></table>
+
+Table 6: Loop uniqueness comparison between Unified and Specialized models.  
+
+<table><tr><td>Loop</td><td>Uniqueness Unified</td><td>Uniqueness Specialized</td></tr><tr><td>H1</td><td>23.5</td><td>9.6</td></tr><tr><td>H2</td><td>20.6</td><td>12.6</td></tr><tr><td>H3</td><td>85.5</td><td>69.2</td></tr><tr><td>L1</td><td>38.9</td><td>10.6</td></tr><tr><td>L2</td><td>19.4</td><td>14.0</td></tr><tr><td>L3</td><td>44.1</td><td>22.3</td></tr></table>
+
+Table 7: H3 loop performance on unique samples.  
+
+<table><tr><td>H3 loop</td><td>AAR</td><td>RMSD</td></tr><tr><td>Unified</td><td>0.441</td><td>2.16</td></tr><tr><td>Specialized</td><td>0.406</td><td>2.07</td></tr></table>
+
+# C.2 In vitro validation
+
+To experimentally validate FuncBind's capabilities, we performed wet-lab validation of H3 loop redesigns based on the co-crystal structure of an antibody bound to a rigid and a flexible epitope. We selected H3 loop redesign due to H3's important contribution to the antibody's functional properties. This study was conducted in a de novo setting: interfaces similar to the two complexes, identified using Ab-Ligity [83], were excluded from training.
+
+From an initial pool of 10,000 unique generated H3 designs (all matching the original seed's length), 190 (i.e. 2 SPR plates) were selected for experimental testing. This selection involved two steps:
+
+1. The top 500 designs were shortlisted based on model confidence, as indicated by their repeated generation counts. Our in-silico validation showed that repeats is an useful proxy for high amino acid recovery of the seed.  
+2. These 500 designs were then clustered into 190 groups using weighted K-means based on sequence edit distance, where the weights were defined by the repeat generation count. The design with the highest repeat count (highest confidence) from each of these 190 clusters was chosen for synthesis and characterization.
+
+The selected antibody designs were expressed and purified in the wet lab. Binding affinity was then determined using surface plasmon resonance (SPR) measurements.
+
+# C.2.1 Rigid epitope
+
+An analysis of Amino Acid Recovery (AAR) and Root Mean Square Deviation (RMSD) for the selected designs are provided in Figure 6:
+
+![](images/f80c4b11f275c28644cdb370c16a7017cad83d0930b6a2beafa01dd6b14a65fa.jpg)  
+Figure 6: All generated CDR H3 designs on rigid epitope: AAR (left) and RMSD (right) histogram.
+
+![](images/25e0638d75430f1185e912af423f8bb88ca5e9f0d4d0c79e6a941f9b718516f5.jpg)
+
+FuncBind successfully generated novel antibody binders in this de novo redesign problem; in fact  $54\%$  were binders (42% with pKD values). 94% of all 190 submitted designs were successfully expressed and purified. Experimental results confirmed that  $42\%$  of all submitted designs were binders, with pKD values in the range of [7.55, 11.29] ( $K_{D} \in [5.08 \times 10^{-12}, 2.84 \times 10^{-8}]M$ ) and an average pKD of 9.56 ( $K_{D} = 2.00 \times 10^{-9}M$ ). For comparison, the pKD of the parent antibody is 10.20 ( $K_{D} = 2.63 \times 10^{-11}M$ ).  $12\%$  were binders with no pKD ("bad" binders). We identified a 5X binder in that set.
+
+Table 8: Average RMSD and AAR for binders and non binders on rigid epitope  
+
+<table><tr><td></td><td>Binders</td><td>Unassigned</td><td>Non-Binders</td><td>Global</td></tr><tr><td>RMSD</td><td>0.50</td><td>0.53</td><td>1.31</td><td>0.88</td></tr><tr><td>AAR</td><td>55.1</td><td>50.8</td><td>37.9</td><td>46.7</td></tr></table>
+
+Table 9: Average RMSD and AAR for binders and non binders on flexible epitope  
+
+<table><tr><td></td><td>Binders</td><td>Unassigned</td><td>Non-Binders</td><td>Global</td></tr><tr><td>RMSD</td><td>1.93</td><td>2.08</td><td>1.94</td><td>1.95</td></tr><tr><td>AAR</td><td>40.4</td><td>35.6</td><td>34.2</td><td>34.5</td></tr></table>
+
+![](images/945ffe9089e0a98b413aa57b304487a3c0ac28edfc35b20de0329466b277731e.jpg)  
+Figure 7: In vitro validation of FuncBind's designs against a rigid epitope; expression (left), binding affinity (center), binding rate (right). Expression rate is  $93.68\%$ .
+
+![](images/290b30c692b79c0eb926954b4e35ed34bf0c51d3bf70fbc440c65c1ba1a845ea.jpg)
+
+# C.3 Flexible epitope
+
+An analysis of Amino Acid Recovery (AAR) and Root Mean Square Deviation (RMSD) for the selected designs are provided in Figure 8:
+
+![](images/713903ce70b9e3d24afa077835e61781964a1600abb067cc41b856f9d2ecade4.jpg)  
+Figure 8: All generated CDR H3 designs against a flexible epitope; target: AAR (top left) and RMSD (top right) histogram. Bottom: Logo of generated designs.
+
+![](images/5fc925dcc0963501b03bc161b72195eb6b4396ef56b0d3bdf313637f56d0e56e.jpg)
+
+Experimental results confirmed that  $100\%$  of the designs expressed and  $2\%$  of all submitted designs were binders, with pKDs [6.95, 7.44, 8.08, 8.47] ( $K_{D} \in [3.38 \times 10^{-9}, 1.13 \times 10^{-7}]M$ ) and an average pKD of 7.74 ( $K_{D} = 4.03 \times 10^{-8}M$ ). For comparison, the pKD of the parent antibody is 10.58 ( $K_{D} = 6.32 \times 10^{-11}M$ ).  $10\%$  were binders with no pKD ("bad" binders); around 3 Unassigned binders had a very reasonable SPR curves.
+
+Looking at Table 9, no specific correlation between binders and non binders were found based on RMSD on the limited set of binders we had. Though higher AAR seemed to be better (based on 4 binder samples only).
+
+Table 10: Per-residue TS between the (a) seed MCP and (b) crystal MCP of the 20 sampled molecules in the pocket relaxed to 1vwe mutant 90.  
+
+<table><tr><td>(a)</td><td>CYS</td><td>HIS</td><td>A20</td><td>B67</td><td>PHE</td><td>CYS</td></tr><tr><td>1vwe_mut90</td><td>0.63 ± 0.24</td><td>0.24 ± 0.17</td><td>0.26 ± 0.07</td><td>0.14 ± 0.12</td><td>0.29 ± 0.14</td><td>0.53 ± 0.21</td></tr><tr><td>(b)</td><td>CYS</td><td>HIS</td><td>PRO</td><td>GLU</td><td>PHE</td><td>CYS</td></tr><tr><td>1vwe-CP</td><td>0.61 ± 0.23</td><td>0.24 ± 0.12</td><td>0.31 ± 0.14</td><td>0.28 ± 0.10</td><td>0.33 ± 0.08</td><td>0.57 ± 0.23</td></tr></table>
+
+![](images/4b05b3f1d8f80a804670af09284496594e676402215f1122bb1251333434fd2d.jpg)  
+Figure 9: In vitro validation of FuncBind's designs against a flexible epitope; expression (left), binding affinity (center), binding rate (right). Expression rate is  $100\%$ .
+
+![](images/02672709b885edceeb0ac66d2f3296dbb3bd20ed1b8e147062a96204fe577ea2.jpg)
+
+# D Identifying non canonical amino acids
+
+Unidentified amino acids were determined by recognizing repeated patterns of peptide backbone atoms around a chiral carbon. All atoms stemming from a  $C_{\alpha}$ , including those in the side chain, were identified and labeled per canonical atom naming conventions. SMILES strings of unidentified amino acids were compared with a non-canonical amino acid library, assigning residue names when a match was found. If no match was found, the amino acid was labeled as unknown and added to the non-canonical library. In the output PDB file, a canonical residue name was assigned based on the closest alignment of atom naming patterns (e.g.,  $C_{\gamma}$ ,  $C_{\delta_1}$ ,  $N_{\epsilon_1}$ ) to a known canonical residue.
+
+# E Macro cyclic peptides
+
+# E.1 Additional results
+
+We perform the following studies:
+
+Figure 10: qualitative comparison between some MCP samples and the seed.  
+- Table 10: per residue tanimoto similarity (TS) for molecules samples with the seed mutant (a) and the crystal MCP (b).  
+- Figure 11 shows the categorization of the generated amino acids.  
+- Figure 12 shows some reasonable and novel generated non canonical amino acids.  
+- Figure 5 shows how a newly generated amino acid interacts with pocket residues that neither the seed nor a chemically similar amino acid at the same position engage.
+
+![](images/8cac46191104af35ccc6de245d7ee002351acc895a3f2b9d8ed15e45da19403d.jpg)  
+a.  
+4gw5 mutant 104  
+b.  
+1vwe mutant 90  
+c.  
+1wb0 mutant 389  
+Figure 10: Results of sampling in pocket relaxed around MCP seed (grey) are in (a) blue in 4gw5 mutant 104 pocket, (b) purple for 1vwe mutant 90 pocket, and (c) 1wb0 mutant 389 pocket
+
+![](images/1121d3805d6b92e2b06b5b85ad839168ad2e81f9528c293f8158a8a75b9e3ffc.jpg)  
+Figure 11: Proportion of amino acid types classified as L-canonical, D-canonical, N-methylated, other known non-canonical amino acids (as annotated in our library), and unknown non-canonical amino acids in the sampled set.
+
+![](images/a914458ba746b1f53e2486af74a510ada603734b7c2d273fd255c925ff6060cd.jpg)  
+Figure 12: Examples of unknown or novel NCAAs that appeared in the sampled set but were not present in the initial test set library. Novel NCAAs are labeled with a formal name and an assigned 3-letter AA code.
+
+![](images/09c4abd2ab52e7956314eaa6a6b89f93699c0002f48b05024cdab978b9c70e7a.jpg)  
+Figure 13: (a) Count of MCPs with each amino acid length in the source MCP-protein dataset. (b) Percentage of the number of mutations in the curated MCP-protein bound dataset.
+
+![](images/f11e554683a499b11e37a7ffd6aedcc6183e097c7fe10e6a6d8afdd81276fb81.jpg)
+
+# E.2 Data curation
+
+The MCP-protein pair dataset was curated by randomly mutating the MCPs from the source dataset at 1-8 different sites, using a list of 213 distinct amino acids (Figure 13b). The closure bonds in the source dataset consist of  $55\%$  N-to-C (head-to-tail),  $28\%$  disulfide (cysteine-cysteine), and  $23\%$  S-acetyl-cysteine. The remaining  $4\%$  contain other closure bonds, such as linkers, and were mainly avoided or modified in the curated MCP dataset so that mutations can be easily implemented in Rosetta. Mutations were avoided in amino acids involved in disulfide bonds and S-acetyl-cysteine linkage. The amino acid list used for random mutation of the MCPs included L-canonical, D-canonical, N-methylated, and other non-canonical types—such as alpha-modified, beta-modified, and peptoid amino acids. Many of these non-canonical residues were pre-parameterized and available in the Rosetta non-canonical rotamer libraries [90]. Following mutation, the MCPs were relaxed using the fast-relax protocol, which involves iterative cycles of side-chain packing and all-atom minimization [84]. The Rosetta interface energy scores—representing the binding energy of the protein-peptide complex at each position—were calculated using the ref_2015-cart energy function. From each source MCP-protein structure, over 2,000 mutated and relaxed complexes were generated, and approximately 500 with the lowest Rosetta interface scores were selected for the curated dataset. Therefore, we were able to expand the source dataset to 186,685 total MCP-protein structures in the curated dataset.
+
+# E.3 Clustering and Splitting
+
+The pairwise similarity of protein sequences from the 641 protein targets in the curated dataset was evaluated using the Longest Common Subsequence (LCS) method [91]. Similarity between each
+
+pair of sequences was calculated as the ratio of the length of their LCS to the length of the longer sequence. The target proteins were clustered together under a representative if their similarity score was greater than 0.5. If none of the similarity scores met the 0.5 threshold, a new cluster was created with that protein as the representative. 208 distinct protein clusters were used for training, validation, and testing. Clusters containing more than 100 MCP-protein pair structures in total were included in the training set. From the remaining clusters, 100 were randomly assigned to the test set, while the rest were added to the validation set.
\ No newline at end of file
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+# Uniform Wrappers: Bridging Concave to Quadratizable Functions in Online Optimization
+
+Mohammad Pedramfar  
+Mila - Quebec AI Institute/McGill University  
+mohammad.pedramfar@mila.quebec
+
+Christopher J. Quinn  
+Iowa State University  
+cjquinn@iastate.edu
+
+Vaneet Aggarwal Purdue University vaneet@purdue.edu
+
+# Abstract
+
+This paper presents novel contributions to the field of online optimization, particularly focusing on the adaptation of algorithms from concave optimization to more challenging classes of functions. Key contributions include the introduction of uniform wrappers, a class of meta-algorithms that could be used for algorithmic conversions such as converting algorithms for convex optimization into those for quadratizable optimization. Moreover, we propose a guideline that, given a base algorithm  $\mathcal{A}$  for concave optimization and a uniform wrapper  $\mathcal{W}$ , describes how to convert a proof of the regret bound of  $\mathcal{A}$  in the concave setting into a proof of the regret bound of  $\mathcal{W}(\bar{\mathcal{A}})$  for quadratizable setting. Through this framework, the paper demonstrates improved regret guarantees for various classes of DR-submodular functions under zeroth-order feedback. Furthermore, the paper extends zeroth-order online algorithms to bandit feedback and offline counterparts, achieving notable improvements in regret/sample complexity compared to existing approaches.
+
+# 1 Introduction
+
+The optimization of continuous DR-submodular functions has become increasingly prominent in recent years. This form of optimization represents an important subset of non-convex optimization problems at the forefront of machine learning and statistics. These challenges have numerous real-world applications like revenue maximization, mean-field inference, and recommendation systems, among others [Bian et al., 2019, Hassani et al., 2017, Mitra et al., 2021, Djolonga and Krause, 2014, Ito and Fujimaki, 2016, Gu et al., 2023, Li et al., 2023].
+
+A natural starting point for DR-submodular maximization is to begin with a convex optimization algorithm and adapt it to the setting of DR-submodular functions. Online Convex Optimization (OCO) is extensively utilized across various fields due to its numerous practical applications and robust theoretical underpinnings. The tools from the area of online convex optimization have been applied to many online non-concave optimization algorithms, e.g., to converge to stationary points in online non-concave optimization [Yang et al., 2018], or algorithms with approximation guarantees for DR-submodular optimization [Chen et al., 2018, Niazadeh et al., 2020, Zhang et al., 2022, Pedramfar et al., 2023].
+
+In this paper, we focus on a large class of functions, namely the class of quadratizable functions, first introduced in [Pedramfar and Aggarwal, 2024a]. Quadratizable functions include special subclasses of non-convex/non-concave functions where the offline constrained optimization problem is NP-hard to solve but we can find an  $\alpha$ -approximation of the optimal value in polynomial time. Indeed, it is shown that this class of online upper quadratizable optimization includes up-concave
+
+optimization (a generalization of DR-submodular and concave optimization) in the following cases: (i) monotone  $\gamma$ -weakly  $\mu$ -strongly DR-submodular functions with curvature  $c$  over general convex sets, (ii) monotone  $\gamma$ -weakly DR-submodular functions over convex sets containing the origin, and (iii) non-monotone DR-submodular optimization over general convex sets.
+
+Even though the tools from OCO have proven effective in more challenging classes, much of the past work along these lines take inspiration from OCO and manually designing new algorithms and analyzing them specific to each problem setting. This raises the following question
+
+When and how can we adapt algorithms from the (simpler) setup of online convex optimization into algorithms for online optimization over more general classes of functions?
+
+In this paper, we try to provide partial solutions to this question for adapting OCO algorithms to algorithms for online quadratizable optimization. The notion of quadratizability is built upon a generalization of the defining condition  $f(\mathbf{x}) - f(\mathbf{y}) \geq \langle \nabla f(\mathbf{y}), \mathbf{x} - \mathbf{y} \rangle$  of convex functions. This similarity with convex functions is a starting point which allows us to define a class of meta-algorithms called "uniform wrappers". Uniform wrappers provide a straightforward way to convert OCO algorithms into algorithms that can handle quadratizable functions. We also develop a guideline to convert the existing proofs for regret bounds of the base algorithms in the convex setting into regret bounds of the new algorithms over quadratizable functions.
+
+We note that for a specific class of algorithms, this question was partly addressed in [Pedramfar and Aggarwal, 2024a]. Specifically, as we will discuss in Appendix E, their result can be formulated as a special case of ours, where they assume that the starting algorithm is a deterministic first order online algorithm with semi-bandit feedback. This condition is too restrictive to facilitate adapting many of the ideas in the OCO literature. For example, their result could not be applied to Follow-The-Perturbed-Leader type algorithms, or more generally any algorithm that is not deterministic. Another important setting where their result could not be applied is the setting with zeroth-order feedback. Without using a zeroth-order algorithm as the base algorithm, we can obtain zeroth-order results by converting first-order algorithms to zeroth-order ones. However, such conversion comes at a cost. For example, in the convex case, taking an optimal first-order algorithm such as Online Gradient Descent and converting it into a zeroth-order algorithm will result in sub-optimal  $O(T^{3/4})$  regret [Flaxman et al., 2005]. In contrast, using our framework, starting from a zeroth-order base algorithm, we obtain  $O(T^{2/3})$  regret for zeroth-order feedback. In particular, our approach recovers the result of [Wan et al., 2023] while their approach could not. (See Table 1.)
+
+In this paper, we take a step further and can handle broader classes of algorithms, including the more challenging setting of zeroth order feedback. Note that, without limiting ourselves to classes of algorithms such as the one discussed in [Pedramfar and Aggarwal, 2024a], there is not much hope of obtaining a single theorem that captures conversions of any OCO algorithm (and proof) into one for online quadratizable optimization. Instead, we provide a guideline that captures the process of converting algorithms and proofs for convex optimization into ones for quadratizable optimization.
+
+As an application of our framework, we propose a variant of a bandit convex optimization algorithm that was introduced in [Saha and Tewari, 2011] as the base algorithm, namely Zeroth Order Follow the Regularized Leader (ZO-FTRL) and demonstrate how it can be converted using uniform wrappers (denoted by  $\mathcal{W}$ ) to obtain 3 algorithms for function classes (i)-(iii) mentioned above. See Tables 1 and 2 for details. Note that ZO-FTRL and  $\mathcal{W}(\mathrm{ZO - FTRL})$  are zeroth order, but they are not bandit feedback algorithms. We also extend the results to those with bandit feedback, as well as derive sample complexity guarantees for the offline algorithm.
+
+We consider 3 classes of weakly DR-submodular functions, specifically (i) monotone  $\gamma$ -weakly  $\mu$ -strongly DR-submodular functions with curvature  $c$  over general convex sets, (ii) monotone  $\gamma$ -weakly DR-submodular functions over convex sets containing the origin, and (iii) non-monotone DR-submodular optimization over general convex sets. The main contributions in this work include:
+
+1. We develop a general framework for converting algorithms and their regret guarantees from online convex concave optimization to online quadratizable optimization. Conversion of the algorithm could be applied to any online optimization algorithm, and the conversion of the proof is described using a general guideline.
+
+Table 1: Online up-concave maximization  
+
+<table><tr><td>F</td><td>Set</td><td colspan="3">Feedback</td><td>Reference</td><td>Appx.</td><td># of queries</td><td>logT(α-regret)</td></tr><tr><td rowspan="24">Monotone</td><td rowspan="14">0 ∈ K</td><td rowspan="5">∇F</td><td rowspan="3">Full Information</td><td rowspan="3">stoch.</td><td>[Zhang et al., 2022] (*)</td><td>1 - e-γ</td><td>1</td><td>1/2</td></tr><tr><td>[Pedramfar et al., 2024a]</td><td>1 - e-1</td><td>Tθ(θ ∈ [0,1/2])</td><td>2/3 - θ/3</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>1 - e-γ</td><td>1</td><td>1/2</td></tr><tr><td rowspan="2">Semi-bandit</td><td rowspan="2">stoch.</td><td>[Pedramfar et al., 2024a]</td><td>1 - e-1</td><td>-</td><td>3/4</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>1 - e-γ</td><td>-</td><td>2/3</td></tr><tr><td rowspan="9">F</td><td rowspan="4">Full Information</td><td>det.</td><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>1 - e-1</td><td>2</td><td>1/2</td></tr><tr><td>stoch. †</td><td>Theorem 5</td><td>1 - e-1</td><td>1</td><td>2/3</td></tr><tr><td rowspan="2">stoch.</td><td>[Pedramfar et al., 2024a]</td><td>1 - e-1</td><td>Tθ(θ ∈ [0,1/4])</td><td>4/5 - θ/5</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>1 - e-γ</td><td>1</td><td>3/4</td></tr><tr><td rowspan="5">Bandit</td><td rowspan="2">det.</td><td>[Wan et al., 2023] (*)</td><td>1 - e-1</td><td>-</td><td>3/4</td></tr><tr><td>[Zhang et al., 2024] (*)</td><td>1 - e-γ</td><td>-</td><td>4/5</td></tr><tr><td>stoch. †</td><td>Theorem 6</td><td>1 - e-1</td><td>-</td><td>3/4</td></tr><tr><td rowspan="2">stoch.</td><td>[Pedramfar et al., 2024a]</td><td>1 - e-1</td><td>-</td><td>5/6</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>1 - e-γ</td><td>-</td><td>4/5</td></tr><tr><td rowspan="10">general</td><td rowspan="4">∇F</td><td>Full Information</td><td>stoch.</td><td>[Pedramfar et al., 2024a]</td><td>1/2</td><td>Tθ(θ ∈ [0,1/2])</td><td>2/3 - θ/3</td></tr><tr><td rowspan="3">Semi-bandit</td><td rowspan="3">stoch.</td><td>[Chen et al., 2018] (*)</td><td>γ2/(1 + γ2)</td><td>-</td><td>1/2</td></tr><tr><td>[Pedramfar et al., 2024a]</td><td>1/2</td><td>-</td><td>3/4</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>γ2/(1 + cγ2)</td><td>-</td><td>1/2</td></tr><tr><td rowspan="6">F</td><td rowspan="3">Full Information</td><td>det.</td><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>γ2/(1 + cγ2)</td><td>2</td><td>1/2</td></tr><tr><td rowspan="2">stoch.</td><td>[Pedramfar et al., 2024a]</td><td>1/2</td><td>Tθ(θ ∈ [0,1/4])</td><td>4/5 - θ/5</td></tr><tr><td>Theorem 5</td><td>γ2/(1 + cγ2)</td><td>1</td><td>2/3</td></tr><tr><td rowspan="3">Bandit</td><td rowspan="3">stoch.</td><td>[Pedramfar et al., 2024a]</td><td>1/2</td><td>-</td><td>5/6</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>γ2/(1 + cγ2)</td><td>-</td><td>3/4</td></tr><tr><td>Theorem 6</td><td>γ2/(1 + cγ2)</td><td>-</td><td>3/4</td></tr><tr><td rowspan="13">Non-Monotone</td><td rowspan="13">general</td><td rowspan="5">∇F</td><td rowspan="3">Full Information</td><td rowspan="3">stoch.</td><td>[Pedramfar et al., 2024a]</td><td>(1 - h)/4</td><td>Tθ(θ ∈ [0,1/2])</td><td>2/3 - θ/3</td></tr><tr><td>[Zhang et al., 2024] (*)</td><td>(1 - h)/4</td><td>1</td><td>1/2</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>(1 - h)/4</td><td>1</td><td>1/2</td></tr><tr><td rowspan="2">Semi-bandit</td><td rowspan="2">stoch.</td><td>[Pedramfar et al., 2024a]</td><td>(1 - h)/4</td><td>-</td><td>3/4</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>(1 - h)/4</td><td>-</td><td>2/3</td></tr><tr><td rowspan="4">F</td><td rowspan="4">Full Information</td><td>det.</td><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>(1 - h)/4</td><td>2</td><td>1/2</td></tr><tr><td>stoch. †</td><td>Theorem 5</td><td>(1 - h)/4</td><td>1</td><td>2/3</td></tr><tr><td rowspan="2">stoch.</td><td>[Pedramfar et al., 2024a]</td><td>(1 - h)/4</td><td>Tθ(θ ∈ [0,1/4])</td><td>4/5 - θ/5</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>(1 - h)/4</td><td>1</td><td>3/4</td></tr><tr><td rowspan="4">Bandit</td><td rowspan="4">det.</td><td>[Zhang et al., 2024] (*)</td><td>(1 - h)/4</td><td>-</td><td>-</td><td>4/5</td></tr><tr><td>stoch. †</td><td>Theorem 6</td><td>(1 - h)/4</td><td>-</td><td>3/4</td></tr><tr><td rowspan="2">stoch.</td><td>[Pedramfar et al., 2024a]</td><td>(1 - h)/4</td><td>-</td><td>5/6</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>(1 - h)/4</td><td>-</td><td>4/5</td></tr></table>
+
+This table compares different static regret results for the online up-concave maximization. The logarithmic terms in regret are ignored. Here  $h \coloneqq \min_{\mathbf{z} \in \mathcal{K}} \| \mathbf{z} \|_{\infty}$ . Rows marked with (*) are results in the literature that are special cases of the results stated here and therefore fit within the framework described in this paper. The rows describing results with stochastic feedback that are marked with † assume that the random query oracle is contained with a cone, as detailed in Theorem 5. Note that such oracles cover deterministic oracles as a special case.
+
+2. To the best our knowledge, in all cases considered, our framework recovers all algorithms in the literature for continuous DR-submodular maximization that are not Frank-Wolfe type. (See Appendix B for more details.) This includes all results marked with  $(^{*})$  in Tables 1 and 2.  
+3. In all online optimization settings considered, our framework either beats the SOTA or successfully recovers it (see Table 1). Note that our framework also recovers all known results for non-stationary DR-submodular maximization. (See Remark 11 and Table 3 in [Pedramfar and Aggarwal, 2024a].)  
+4. In all offline optimization settings considered—except for deterministic first-order feedback and the special case of  $\gamma$ -weakly non-monotone functions with  $\gamma < 1$ —our framework either beats the SOTA or successfully recovers it. (See Table 2.)  
+5. When considering either deterministic or fully stochastic query oracles, we obtain superior regret guarantees for 3 settings. Specifically, for online or offline zeroth order feedback, we improve the SOTA for (ii) and for bandit feedback, we improve the SOTA for (iii). (See Tables 1 and 2)  
+6. We define a new notion of query oracles that are stochastic, but contained within a cone. (See Section 8) When considering these types of oracles (together with deterministic and fully stochastic query oracles), we improve SOTA for 6 more settings. Specifically, for online zeroth order, offline zeroth order and bandit feedback settings for (i) and (iii). (See Tables 1 and 2)
+
+To simplify the notation and statements, we define regret for maximization problems and focus on concave maximization and DR-submodular maximization.
+
+# 2 Background and Notation
+
+For a set  $\mathcal{D} \subseteq \mathbb{R}^d$ , we define its affine hull  $\operatorname{aff}(\mathcal{D})$  to be the set of  $\alpha \mathbf{x} + (1 - \alpha) \mathbf{y}$  for all  $\mathbf{x}, \mathbf{y}$  in  $\mathcal{D}$  and  $\alpha \in \mathbb{R}$ . The relative interior of  $\mathcal{D}$  is defined as  $\operatorname{relint}(\mathcal{D}) := \{\mathbf{x} \in \mathcal{D} \mid \exists r > 0, \mathbb{B}_r(\mathbf{x}) \cap \operatorname{aff}(\mathcal{D}) \subseteq \mathcal{D}\}$ . All convex functions are continuous on any point in the relative interior of their domains. In this work, we will only focus on (everywhere) continuous functions. If  $\mathbf{x} \in \operatorname{relint}(\mathcal{K})$  and  $f$  is convex
+
+and is differentiable at  $\mathbf{x}$ , then we have  $f(\mathbf{y}) - f(\mathbf{x}) \geq \langle \nabla f(\mathbf{x}), \mathbf{y} - \mathbf{x} \rangle$ , for all  $\mathbf{y} \in \mathcal{K}$ . More generally, given  $\mu \geq 0$ , we say a vector  $\mathbf{o} \in \mathbb{R}^d$  is a  $\mu$ -subgradient of  $f$  at  $\mathbf{x}$  if  $f(\mathbf{y}) - f(\mathbf{x}) \geq \langle \mathbf{o}, \mathbf{y} - \mathbf{x} \rangle + \frac{\mu}{2} \| \mathbf{y} - \mathbf{x} \|^2$ . For all  $\mathbf{y} \in \mathcal{K}$ . Given a convex set  $\mathcal{K}$ , a function  $f: \mathcal{K} \to \mathbb{R}$  is  $\mu$ -strongly convex if and only if it has a  $\mu$ -subgradient at all points  $\mathbf{x} \in \mathcal{K}$ . A function  $F: \mathcal{D} \to \mathbb{R}^+$  is  $G$ -Lipschitz continuous if for all  $\mathbf{x}, \mathbf{y} \in \mathcal{D}$ ,  $\| F(\mathbf{x}) - F(\mathbf{y}) \| \leq G \| \mathbf{x} - \mathbf{y} \|$ . A differentiable function  $F: \mathcal{D} \to \mathbb{R}^+$  is  $L$ -smooth if for all  $\mathbf{x}, \mathbf{y} \in \mathcal{D}$ ,  $\| \nabla F(\mathbf{x}) - \nabla F(\mathbf{y}) \| \leq L \| \mathbf{x} - \mathbf{y} \|$ . Given a continuous monotone function  $f: \mathcal{K} \to \mathbb{R}$ , its curvature is defined as the smallest number  $c \in [0,1]$  such that  $f(\mathbf{y} + \mathbf{z}) - f(\mathbf{y}) \geq (1 - c)(f(\mathbf{x} + \mathbf{z}) - f(\mathbf{x}))$ , for all  $\mathbf{x}, \mathbf{y} \in \mathcal{K}$  and  $\mathbf{z} \geq 0$  such that  $\mathbf{x} + \mathbf{z}, \mathbf{y} + \mathbf{z} \in \mathcal{K}$ . We define the curvature of a function class  $\mathbf{F}$  as the supremum of the curvature of functions in  $\mathbf{F}$ .
+
+We say  $\tilde{\nabla} f: \mathcal{K} \to \mathbb{R}^d$  is a  $\mu$ -strongly  $\gamma$ -weakly up-super-gradient of  $f$  if for all  $\mathbf{x} \leq \mathbf{y}$  in  $\mathcal{K}$ , we have  $\gamma (\langle \tilde{\nabla} f(\mathbf{y}), \mathbf{y} - \mathbf{x} \rangle + \frac{\mu}{2} \| \mathbf{y} - \mathbf{x} \|^2) \leq f(\mathbf{y}) - f(\mathbf{x}) \leq \frac{1}{\gamma} (\langle \tilde{\nabla} f(\mathbf{x}), \mathbf{y} - \mathbf{x} \rangle - \frac{\mu}{2} \| \mathbf{y} - \mathbf{x} \|^2)$ . Then we say  $f$  is  $\mu$ -strongly  $\gamma$ -weakly up-concave if it is continuous and it has a  $\mu$ -strongly  $\gamma$ -weakly up-super-gradient. When  $\gamma = 1$  and the above inequality holds for all  $\mathbf{x}, \mathbf{y} \in \mathcal{K}$ , we say  $f$  is  $\mu$ -strongly concave. A differentiable function  $f: \mathcal{K} \to \mathbb{R}$  is called continuous  $DR$ -submodular if for all  $\mathbf{x} \leq \mathbf{y}$ , we have  $\nabla f(\mathbf{x}) \geq \nabla f(\mathbf{y})$ . More generally, we say  $f$  is  $\gamma$ -weakly continuous  $DR$ -submodular if for all  $\mathbf{x} \leq \mathbf{y}$ , we have  $\nabla f(\mathbf{x}) \geq \gamma \nabla f(\mathbf{y})$ . It follows that any  $\gamma$ -weakly continuous  $DR$ -submodular functions are  $\gamma$ -weakly up-concave.
+
+# 3 Problem setup
+
+Online optimization problems can be formalized as a repeated game between an agent and an adversary. The game lasts for  $T$  rounds on a convex domain  $\mathcal{K}$  where  $T$  and  $\mathcal{K}$  are known to both players. In the  $t$ -th round, the agent chooses an action  $\mathbf{x}_t$  from an action set  $\mathcal{K} \subseteq \mathbb{R}^d$ , then the adversary chooses a loss function  $f_t \in \mathbf{F}$  and a query oracle for the function  $f_t$ . Then, for some  $k_t \geq 0$ , and  $1 \leq i \leq k_t$ , the agent chooses a points  $\mathbf{y}_{t,i}$  and receives the outputs of the query oracle.
+
+To be more precise, an agent consists of a tuple  $(\Omega^{\mathcal{A}},\mathcal{A}^{\mathrm{action}},\mathcal{A}^{\mathrm{query}})$ , where  $\Omega^{\mathcal{A}}$  is a probability space that captures all the randomness of  $\mathcal{A}$ . We assume that, before the first action, the agent samples  $\omega \in \Omega$ . The next element in the tuple,  $\mathcal{A}^{\mathrm{action}} = (\mathcal{A}_1^{\mathrm{action}},\dots,\mathcal{A}_T^{\mathrm{action}})$  is a sequence of functions such that  $\mathcal{A}_t$  that maps the history  $\Omega^{\mathcal{A}}\times \mathcal{K}^{t - 1}\times \prod_{s = 1}^{t - 1}(\mathcal{K}\times \mathcal{O})^{k_s}$  to  $\mathbf{x}_t\in \mathcal{K}$  where we use  $\mathcal{O}$  to denote range of the query oracle. The last element in the tuple,  $\mathcal{A}^{\mathrm{query}}$ , is the query policy. For each  $1\leq t\leq T$  and  $1\leq i\leq k_t$ ,  $\mathcal{A}_{t,i}^{\mathrm{query}}:\Omega^{\mathcal{A}}\times \mathcal{K}^{t}\times \prod_{s = 1}^{t - 1}(\mathcal{K}\times \mathcal{O})^{k_{s}}\times (\mathcal{K}\times \mathcal{O})^{i - 1}$  is a function that, given previous actions and observations, either selects a point  $\mathbf{y}_t^i\in \mathcal{K}$ , i.e., query, or signals that the query policy at this time-step is terminated. We may drop  $\omega$  as one of the inputs of the above functions when there is no ambiguity. We say the agent query function is trivial if  $k_{t} = 1$  and  $\mathbf{y}_{t,1} = \mathbf{x}_t$  for all  $1\leq t\leq T$ . In this case, we simplify the notation and use the notation  $\mathcal{A} = \mathcal{A}^{\mathrm{action}} = (\mathcal{A}_1,\dots ,\mathcal{A}_T)$  to denote the agent action functions and assume that the domain of  $\mathcal{A}_t$  is  $\Omega^{\mathcal{A}}\times (\mathcal{K}\times \mathcal{O})^{t - 1}$ .
+
+A query oracle is a function that provides the observation to the agent. Formally, a query oracle for a function  $f$  is a map  $\mathcal{Q}$  defined on  $\mathcal{K}$  such that for each  $\mathbf{x} \in \mathcal{K}$ , the  $\mathcal{Q}(\mathbf{x})$  is a random variable taking value in the observation space  $\mathcal{O}$ . The query oracle is called a stochastic value oracle or stochastic zeroth order oracle if  $\mathcal{O} = \mathbb{R}$  and  $f(\mathbf{x}) = \mathbb{E}[\mathcal{Q}(\mathbf{x})]$ . Similarly, it is called a stochastic up-super-gradient oracle or stochastic first order oracle if  $\mathcal{O} = \mathbb{R}^d$  and  $\mathbb{E}[\mathcal{Q}(\mathbf{x})]$  is a up-super-gradient of  $f$  at  $\mathbf{x}$ . In all cases, if the random variable takes a single value with probability one, we refer to it as a deterministic oracle. Note that, given a function, there is at most a single deterministic gradient oracle, but there may be many deterministic up-super-gradient oracles. We will use  $\nabla$  to denote the deterministic gradient oracle. We say an oracle is bounded by  $B$  if its output is always within the Euclidean ball of radius  $B$  centered at the origin. We say the agent takes semi-bandit feedback if the oracle is first-order and the agent query function is trivial. Similarly, it takes bandit feedback if the oracle is zeroth-order and the agent query function is trivial. If the agent query function is non-trivial, then we say the agent requires full-information feedback.
+
+An adversary Adv is a set such that each element  $\mathcal{B} \in \mathrm{Adv}$ , referred to as a realized adversary, is a sequence  $(\mathcal{B}_1, \dots, \mathcal{B}_T)$  of functions where each  $\mathcal{B}_t$  maps a tuple  $(\mathbf{x}_1, \dots, \mathbf{x}_t) \in \mathcal{K}^t$  to a tuple  $(f_t, \mathcal{Q}_t)$  where  $f_t \in \mathbf{F}$  and  $\mathcal{Q}_t$  is a query oracle for  $f_t$ . We say an adversary Adv is oblivious if for any realization  $\mathcal{B} = (\mathcal{B}_1, \dots, \mathcal{B}_T)$ , all functions  $\mathcal{B}_t$  are constant, i.e., they are independent of  $(\mathbf{x}_1, \dots, \mathbf{x}_t)$ . In this case, a realized adversary may be simply represented by a sequence of func
+
+tions  $(f_{1},\dots ,f_{T})\in \mathbf{F}^{T}$  and a sequence of query oracles  $(Q_{1},\dots ,Q_{T})$  for these functions. In this work we also consider adversaries that are fully adaptive, i.e., adversaries with no restriction. Given a function class  $\mathbf{F}$  and  $i\in \{0,1\}$ , we use  $\mathrm{Adv}_i^{\mathrm{f}}(\mathbf{F})$  to denote the set of all possible realized adversaries with deterministic  $i$ -th order oracles. If the oracle is instead stochastic and bounded by  $B$ , we use  $\mathrm{Adv}_i^{\mathrm{f}}(\mathbf{F},B)$  to denote such an adversary. Finally, we use  $\mathrm{Adv}_i^{\mathrm{o}}(\mathbf{F})$  and  $\mathrm{Adv}_i^{\mathrm{o}}(\mathbf{F},B)$  to denote all oblivious realized adversaries with  $i$ -th order deterministic and stochastic oracles, respectively.
+
+In order to handle different notions of regret with the same approach, for an agent  $\mathcal{A}$ , adversary Adv, compact set  $\mathcal{U} \subseteq \mathcal{K}^T$ , approximation coefficient  $0 < \alpha \leq 1$  and  $1 \leq a \leq b \leq T$ , we define regret
+
+$$
+\mathcal{R}^{\mathcal{A}}_{\alpha ,\mathrm{Adv}}(\mathcal{U})[a,b]:= \sup_{\mathcal{B}\in \mathrm{Adv}}\mathbb{E}\left[\alpha \max_{\mathbf{u} = (\mathbf{u}_{1},\dots ,\mathbf{u}_{T})\in \mathcal{U}}\sum_{t = a}^{b}f_{t}(\mathbf{u}_{t}) - \sum_{t = a}^{b}f_{t}(\mathbf{x}_{t})\right],
+$$
+
+where the expectation in the definition of the regret is over the randomness of the algorithm and the query oracle. We use the notation  $\mathcal{R}_{\alpha, \mathcal{B}}^{\mathcal{A}}(\mathcal{U})[a, b] := \mathcal{R}_{\alpha, \mathrm{Adv}}^{\mathcal{A}}(\mathcal{U})[a, b]$  when  $\mathrm{Adv} = \{\mathcal{B}\}$  is a singleton. We may drop  $\alpha$  when it is equal to 1. When  $\alpha < 1$ , we often assume that the functions are non-negative. Static adversarial regret or simply adversarial regret corresponds to  $a = 1, b = T$  and  $\mathcal{U} = \mathcal{K}_\star^T := \{(\mathbf{x}, \dots, \mathbf{x}) \mid \mathbf{x} \in \mathcal{K}\}$ . When  $a = 1$ ,  $b = T$  and  $\mathcal{U}$  contains only a single element then it is referred to as the dynamic regret [Zinkevich, 2003, Zhang et al., 2018]. Adaptive regret, is defined as  $\max_{1 \leq a \leq b \leq T} \mathcal{R}_{\alpha, \mathrm{Adv}}^{\mathcal{A}}(\mathcal{K}_\star^T)[a, b]$  [Hazan and Seshadhri, 2009]. We drop  $a$ ,  $b$  and  $\mathcal{U}$  when the statement is independent of their value or their value is clear from the context.
+
+# 4 Uniform wrappers
+
+We next introduce a class of meta-algorithms that will be a central element of our proposed framework for adapting algorithms. At a high level, the meta-algorithms we consider wrap around the base algorithm and translate each action and feedback signal between the base algorithm and the adversary. The qualifier "uniform" highlights that the translations are one-to-one and independent of time.
+
+Definition 1. Given a function class  $\mathbf{F}$  and a family of query oracles  $\mathcal{Q}$  over  $\mathbf{F}$ , we say a uniform wrapper  $\mathcal{W} = (\mathcal{W}^{\text{action}}, \mathcal{W}^{\text{function}}, \mathcal{W}^{\text{query}})$  is a tuple of maps where  $\mathcal{W}^{\text{action}}: \mathcal{K} \to \mathcal{K}$ ,  $\mathcal{W}^{\text{function}}: \mathbf{F} \to \mathbf{H}$  for a function class  $\mathbf{H}$  and for any  $f \in \mathbf{F}$  and any query oracle  $\mathcal{Q}_f \in \mathcal{Q}$ ,  $\mathcal{W}^{\text{query}}(\mathcal{Q}_f)$  is a query oracle for  $\mathcal{W}^{\text{function}}(f) \in \mathbf{H}$ .<sup>1</sup> Given an oblivious adversary Adv choosing functions in  $\mathbf{F}$  and query oracles in  $\mathcal{Q}$ , we define  $\mathcal{W}(\text{Adv})$  to be the adversary over  $\mathbf{H}$  where the selected function and query by the adversary are transformed according to  $\mathcal{W}^{\text{function}}$  and  $\mathcal{W}^{\text{query}}$ .<sup>2</sup> We say  $\mathcal{W} = \text{Id}$  if
+
+all the maps in  $\mathcal{W}$  are identity.
+
+In Appendix C we will discuss several examples of uniform wrappers for up-concave optimization. We drop the superscripts and use  $\mathcal{W}(\mathbf{x})$ ,  $\mathcal{W}(f)$  and  $\mathcal{W}(\mathcal{Q}_f)$  to denote  $\mathcal{W}^{\mathrm{action}}(\mathbf{x})$ ,  $\mathcal{W}^{\mathrm{function}}(f)$  and  $\mathcal{W}^{\mathrm{query}}(\mathcal{Q}_f)$ , respectively, when there is no ambiguity.
+
+Meta-algorithm 1 details the pseudo-code for  $\mathcal{W}(\mathcal{A})$  for a uniform wrapper  $\mathcal{W}$  and an online optimization algorithm  $\mathcal{A}$ . Note that, when  $\mathcal{W} = \mathrm{Id}$ , the meta-algorithm also reduces to the identity meta-algorithm and we see that  $\mathcal{W}(\mathcal{A}) = \mathcal{A}$ . Note that in the special case where  $\mathcal{A}$  is an online algorithm with semi-bandit feedback, Meta-algorithm 1 reduces to Algorithm 1
+
+# Meta-algorithm 1: Application of a uniform wrapper to the base algorithm -  $\mathcal{W}(\mathcal{A})$
+
+Input : horizon  $T$  , algorithm  $\mathcal{A}$  uniform wrapper W
+
+for  $t = 1,2,\dots ,T$  do
+
+Play  $\mathcal{W}^{\mathrm{action}}(\mathbf{x}_t)$  where  $\mathbf{x}_t$  is the action chosen by  $\mathcal{A}^{\mathrm{action}}$
+
+The adversary selects  $f_{t}$  and a query oracle  $\mathcal{Q}_t$  for  $f_{t}$
+
+for  $i$  starting from 1, while  $\mathcal{A}^{\text{query}}$  is not terminated for this time-step do
+
+Let  $\mathbf{y}_{t,i}$  be the query chosen by  $\mathcal{A}^{\mathrm{query}}$
+
+Return  $\mathbf{o}_{t,i} = \mathcal{W}^{\mathrm{query}}(\mathcal{Q}_t)(\mathbf{y}_{t,i})$  as the output of the query oracle to  $\mathcal{A}$  query
+
+end
+
+end
+
+in [Pedramfar and Aggarwal, 2024a]. See Appendix A for more details.
+
+In this paper, we will design uniform wrappers that could convert algorithms for concave optimization into algorithms for more general class of functions that contains many DR-submodular functions. Specifically, we consider upper-quadratic/linearizable functions which we will discuss in the following section.
+
+# 5 Linearizable and quadratizable functions classes
+
+We next define an important function class significantly generalizes concavity but preserves enough structure that will enable us to obtain improved regret bounds for various problems.
+
+Definition 2 ([Pedramfar and Aggarwal, 2024a]). Let  $\mathcal{K} \subseteq \mathbb{R}^d$  be a convex set,  $\mathbf{F}$  be a function class over  $\mathcal{K}$ . We say the function class  $\mathbf{F}$  is upper quadratizable if there are maps  $\mathfrak{g}: \mathbf{F} \times \mathcal{K} \to \mathbb{R}^d$  and  $h: \mathcal{K} \to \mathcal{K}$  and constants  $\mu \geq 0$ ,  $0 < \alpha \leq 1$  and  $\beta > 0$  such that
+
+$$
+\alpha f (\mathbf {y}) - f (h (\mathbf {x})) \leq \beta \left(\langle \mathfrak {g} (f, \mathbf {x}), \mathbf {y} - \mathbf {x} \rangle - \frac {\mu}{2} \| \mathbf {y} - \mathbf {x} \| ^ {2}\right). \tag {1}
+$$
+
+As a special case, when  $\mu = 0$ , we say  $\mathbf{F}$  is upper linearizable. By setting  $\mathfrak{g}(f,\mathbf{x}) = \nabla f(\mathbf{x})$ ,  $h = \mathrm{Id}_{\mathcal{K}}$  and  $\alpha = \beta = 1$ , we see that the notion of upper linearizability generalizes concavity and upper quadratizability generalizes strong concavity. It was shown in [Pedramfar and Aggarwal, 2024a] that several classes of DR-submodular (and up-concave) functions are upper quadratizable. (see Lemmas 1, 2 and 3) A similar notion of lower-quadratizable/linearizable may be similarly defined for minimization problems such as convex minimization<sup>3</sup>.
+
+Definition 3. We say  $\mathbf{F}$  is upper quadratizable with a uniform wrapper  $\mathcal{W}$  if  $\mathcal{W}(\mathbf{F})$  is defined and differentiable over  $\mathcal{K}$  and, for all  $f\in \mathbf{F}$ , we have
+
+$$
+\alpha f (\mathbf {y}) - f (\mathcal {W} (\mathbf {x})) \leq \beta \left(\langle \nabla \mathcal {W} (f) (\mathbf {x}), \mathbf {y} - \mathbf {x} \rangle - \frac {\mu}{2} \| \mathbf {y} - \mathbf {x} \| ^ {2}\right). \tag {2}
+$$
+
+Note that a uniform wrapper is not uniquely determined by  $h$  and  $\mathfrak{g}$  in the definition of upper quadraticizable functions as it also needs to describe transformations of query oracles. The special case with  $\alpha = \beta = 1$ ,  $\mathcal{W} = \mathrm{Id}$  reduces to the definition of (strong) concavity. In Appendix C, we will construct uniform wrappers for several classes of upper quadratizable functions, specifically, monotone  $\mu$ -strongly  $\gamma$ -weakly up-concave functions with bounded curvature  $(\mathbf{F}^{\mathrm{M}})$ , monotone  $\gamma$ -weakly up-concave functions over convex sets containing the origin  $(\mathbf{F}^{\mathrm{M0}})$ , and non-monotone up-concave functions over general convex sets  $(\mathbf{F}^{\mathrm{NM}})$ .
+
+# 6 When Is Concave Optimization Enough?
+
+As can be seen in Meta-algorithm 1, we may apply a uniform wrapper  $\mathcal{W}$  to any online optimization algorithm  $\mathcal{A}$ . However, even if the original algorithm has a sublinear regret over concave functions and  $\mathbf{F}$  is a function class that is upper quadratizable with  $\mathcal{W}$ , this does not guarantee that the resulting algorithm  $\mathcal{W}(\mathcal{A})$  has a sublinear regret over  $\mathbf{F}$ . In this section we discuss how we might convert the proofs of the regret bound for  $\mathcal{A}$  over concave functions into a proof of a similar regret bound for  $\mathcal{W}(\mathcal{A})$  over  $\mathbf{F}$ . That starting point is to notice that a game played between  $\mathcal{W}(\mathcal{A})$  and Adv is mathematically equivalent to one played between  $\mathcal{A}$  and  $\mathcal{W}(\mathrm{Adv})$ . Therefore we need to convert the regret guarantees of  $\mathcal{A}$  against a concave adversary into guarantees for  $\mathcal{A}$  against  $\mathcal{W}(\mathrm{Adv})$ . See Appendix A for more details.
+
+The guideline for converting proof for concave optimization into proofs for upper-quadratizable optimization can be informally summarized in a few steps:
+
+(0) Sometimes, if the algorithm  $\mathcal{A}$  is the result of application of a meta-algorithm to another algorithm  $\mathcal{B}$ , e.g.  $\mathcal{A} = \mathrm{SFTT}(\mathcal{B})$  (the meta-algorithm SFTT converts algorithms that require full-information feedback to ones that work with (semi)-bandit feedback; see Appendix M), we may
+
+need to consider the base algorithm instead. For example, in the example of SFTT, we might want to consider SFTT  $(\mathcal{W}(\mathcal{B}))$  instead of  $\mathcal{W}(\mathrm{SFTT}(\mathcal{B})) = \mathcal{W}(\mathcal{A})$ .
+
+(1) Rewrite the parts of proof (after possibly adapting the algorithm) of the original regret bound without assuming that the function class in concave, in order to isolate the use on concavity in the proof. In this step, we hope to obtain a result that would only require a single use of an inequality of the type  $f(\mathbf{y}) - f(\mathbf{x}) \leq \langle \nabla f(\mathbf{x}), \mathbf{y} - \mathbf{x} \rangle - \frac{\mu}{2} \| \mathbf{y} - \mathbf{x} \|^2$  to complete the proof for the concave case. See Theorems 1, 3 (as an example of a family of zeroth order results), and 11 (as an example of a family of first order results) for examples of this step.  
+(2) Verify that the results of the previous step could be adapted to upper-quadratizable setting. See the proof of Theorems 2, 4, and 12 for examples of this step.
+
+In the following section, we discuss a version of Follow The Regularized Leader (FTRL) algorithm for concave optimization and adapt it to fit the guidelines discussed above. As another application of the guideline, we refer to Appendix E for a discussion of applying this guideline to recover some previous results in the literature, including all the results in Tables 1 and 2 that are marked with  $(^{*})$ .
+
+# 7 Examples
+
+In this section, we provide examples of applying uniform wrappers and the guideline described earlier. The first example is illustrative and recovers some existing results, while the second example provides new results, as we will discuss in Section 8.
+
+# 7.1 Online Gradient Ascent
+
+Here we clarify the guideline in Section 6 by applying it to a simple base algorithm, namely Online Gradient Ascent (OGA).
+
+Note that this is only an illustrative example and a much more general result is proven in Appendix E.
+
+In Algorithm 2, we use  $Q_{t}$  to denote the stochastic first order query oracle for  $f_{t}$  which returns unbiased estimates of  $\nabla f_{t}$ . The effect of Meta-algorithm 1 is that it transforms the action of the base algorithm using  $\mathcal{W}^{\mathrm{action}}$  and query oracle of the base algorithm using  $\mathcal{W}^{\mathrm{query}}$ . If we apply the first-order to first-order uniform wrapper  $\mathcal{W}_{1}^{\mathrm{NM}}$  (for non-monotone up-concave functions over general convex sets; see Appendix C.3 for additional details), we get a first-order algorithm for online optimization of non-monotone DR-submodular function over general convex sets. The resulting algorithm is
+
+By plugging in the definitions of  $(\mathcal{W}_1^{\mathrm{NM}})^{\mathrm{action}}$  and  $(\mathcal{W}_1^{\mathrm{NM}})^{\mathrm{query}}$  as per Appendix C.3, we get the Algorithm 4 which is Algorithm 1 in [Zhang et al., 2024] for the case of non-monotone functions.
+
+The conversion of the regret bound is the focus of Section 6, in the form of a three-step guideline. Since that is a guideline that depends on the structure and the details of the original proof, any explanation of the application of the guideline requires an understanding of the original proof. For OGA specifically, we may simplify the proofs and the guideline in the following manner:
+
+Algorithm 2: Online Gradient Ascent (OGA)  
+Input: horizon  $T$ , convex set  $\mathcal{K}$ ,  $\mathbf{x}_1 \in \mathcal{K}$ , step-sizes  $\{\eta_t\}_{t=1}^T$   
+for  $t = 1, 2, \dots, T$  do  
+    Play  $\mathbf{x}_t$   
+    The adversary selects  $f_t$  and a query oracle  $\mathcal{Q}_t$  for  $f_t$   
+    Observe  $\mathbf{o}_t$ , a sample of  $\mathcal{Q}_t(\mathbf{x}_t)$ , an unbiased estimate of  $\nabla f_t(\mathbf{x}_t)$ $\mathbf{x}_{t+1} \gets P_{\mathcal{K}}(\mathbf{x}_t + \eta_t \mathbf{o}_t)$   
+end
+
+Algorithm 3: Application of  $\mathcal{W}_1^{\mathrm{NM}}$  to Online Gradient Ascent  $(\mathcal{W}_1^{\mathrm{NM}}(\mathrm{OGA}))$    
+Input : horizon  $T$  , convex set  $\kappa$ $\mathbf{x}_1,\underline{\mathbf{x}}\in \mathcal{K}$  step-sizes  $\{\eta_t\}_{t = 1}^T$    
+for  $t = 1,2,\ldots ,T$  do Play  $(\mathcal{W}_1^{\mathrm{NM}})^{\mathrm{action}}(\mathbf{x}_t)$  The adversary selects  $f_{t}$  and a query oracle  $\mathcal{Q}_t$  for  $f_{t}$  Observe  $\mathbf{o}_t$  , a sample of  $(\mathcal{W}_1^{\mathrm{NM}})^{\mathrm{query}}(\mathcal{Q}_t)(\mathbf{x}_t)$ $\mathbf{x}_{t + 1}\gets P_K(\mathbf{x}_t + \eta_t\mathbf{o}_t)$    
+end
+
+As per Step 1 of the guideline, we first rewrite a part of the proof of the OGA, in a way that does not require concavity, but the proof of the regret bound for the concave case can be recovered using the inequality  $f(\mathbf{y}) - f(\mathbf{x}) \leq \langle \nabla f_t(\mathbf{x}), \mathbf{y} - \mathbf{x} \rangle$ .
+
+Theorem 1. For any differentiable function class  $\mathbf{F}$ , with stochastic first-order query oracles bounded by  $G$  and step-sizes  $\eta_t = \frac{D}{G\sqrt{t}}$ , the algorithm OGA guarantees that  $\max_{\mathbf{x} \in \mathcal{K}} \sum_{t=1}^{T} \mathbb{E}[\langle \nabla f_t(\mathbf{x}_t), \mathbf{x} - \mathbf{x}_t \rangle] \leq \frac{3}{2}GD\sqrt{T}$ .
+
+The proof of Theorem 1 is a simple rewriting of the classical results. We include a proof in
+
+Appendix D for completeness. Note that the boundedness of the stochastic first-order query oracle by  $G$  implies that the gradients of functions in  $\mathbf{F}$  are bounded by  $G$ , i.e., functions in  $\mathbf{F}$  are  $G$ -Lipschitz continuous.
+
+# Algorithm 4: Application of  $\mathcal{W}_1^{\mathrm{NM}}$  to Online Gradient Ascent (W<sub>1</sub>NM(OGA))
+
+Input: horizon  $T$ , convex set  $\mathcal{K}$ ,  $\mathbf{x}_1, \underline{\mathbf{x}} \in \mathcal{K}$ , step-sizes  $\{\eta_t\}_{t=1}^T$   
+for  $t = 1, 2, \dots, T$  do
+
+$$
+\text {P l a y} (\mathbf {x} _ {t} + \underline {{\mathbf {x}}}) / 2
+$$
+
+$$
+\text {S a m p l e} z \text {a c c o r d i n g} \mathcal {Z} ^ {\mathrm {N M}} (\text {a s d e f i n e d i n}
+$$
+
+$$
+\begin{array}{l} \text {T h e a d v a r s a r y s e l e c t s f _ {t} a n d a q u e y o r a l c e \mathcal {Q} _ {t}} \\ \text {f o r f _ {t}} \end{array}
+$$
+
+$$
+\begin{array}{l} \text {O b s e r v e} \mathbf {o} _ {t}, \text {a s p a r a m e l o f} \mathcal {Q} _ {t} \big (\frac {z}{2} \big (\mathbf {x} _ {t} - \underline {{\mathbf {x}}} \big) + \underline {{\mathbf {x}}} \big), \\ \text {a n u n b i a s e d e s t i m a t e o f} \end{array}
+$$
+
+$$
+\nabla f _ {t} \left(\frac {z}{2} (\mathbf {x} _ {t} - \underline {{\mathbf {x}}}) + \underline {{\mathbf {x}}}\right)
+$$
+
+$$
+\mathbf {x} _ {t + 1} \leftarrow P _ {\mathcal {K}} \left(\mathbf {x} _ {t} + \eta_ {t} \mathbf {o} _ {t}\right)
+$$
+
+# end
+
+In Step 2 of the guideline, we adapt the result of Theorem 1 to the upper-linearizable setting.
+
+Theorem 2. If  $\mathbf{F}$  is a function class over  $\mathcal{K}$  that is upper-linearizable with  $0 < \alpha \leq 1$  and  $\beta \geq 0$  and a first-order uniform wrapper  $\mathcal{W}$ , then with stochastic first-order query oracles bounded by  $G$  and step-sizes  $\eta_t = \frac{D}{G\sqrt{t}}$ , the algorithm  $\mathcal{W}(\mathrm{OGA})$  guarantees that  $\max_{\mathbf{x} \in \mathcal{K}} \sum_{t=1}^{T} \mathbb{E}[f_t(\mathbf{x}) - f_t(\mathcal{W}^{\mathrm{action}}(\mathbf{x}_t))] \leq \frac{3}{2}GD\sqrt{T}$ .
+
+We refer to Appendix D for a proof.
+
+# 7.2 Zeroth Order Follow The Regularized Leader
+
+Follow The Regularized Leader is a popular online optimization algorithm. When applied to a sequence of vectors  $\{\mathbf{g}_t\}_{t=1}^T$  in  $\mathcal{K}$ , FTRL outputs a sequence of points  $\{\mathbf{x}_t\}_{t=1}^T$ , where
+
+$$
+\mathbf {x} _ {1} = \underset {\mathbf {x} \in \mathcal {K}} {\operatorname {a r g m i n}} \Phi (\mathbf {x}), \quad \mathbf {x} _ {t + 1} = \underset {\mathbf {x} \in \mathcal {K}} {\operatorname {a r g m i n}} \eta \sum_ {s = 1} ^ {t} \left\langle - \mathbf {g} _ {s}, \mathbf {x} \right\rangle + \Phi (\mathbf {x}). \tag {3}
+$$
+
+Here  $\Phi (\mathbf{x})$  is an arbitrary regularizer and  $\eta$  is a parameter. In this paper, we use a self-concordant barrier of  $\kappa$  as the regularizer of FTRL. Self-concordant barriers were first proposed in the convex optimization literature, with [Abernethy et al., 2008] the first use in bandit feedback setting. We refer to Appendix H for an overview of the main ideas present in FTRL, including the definition of self-concordant barrier  $\Phi$ , the Minkowski set  $\kappa_{\gamma ,\mathbf{x}_1}$ , and  $\Sigma$ -smoothing of function  $f$  to obtain  $f^{\Sigma}$ .
+
+Here we propose a FTRL variant for zeroth-order feedback, based on [Saha and Tewari, 2011], which will be a key base algorithm for our framework. See Algorithm 5 for pseudocode.
+
+# Algorithm 5: Zeroth Order Follow The Regularized Leader - ZO-FTRL
+
+Input: Horizon  $T$ , smoothing radius  $\delta$ , learning rate  $\eta$ ,  $\nu$ -self-concordant barrier  $\Phi$ $\mathbf{x}_1 \gets \mathrm{argmin}_{\mathbf{x} \in \mathcal{K}} \Phi(\mathbf{x})$   
+for  $t = 1, 2, \dots, T$  do
+
+Play  $\mathbf{x}_t$  The adversary selects  $f_{t}$  and reveals a zeroth-order query oracle  $\mathcal{Q}_t$  for  $f_{t}$ $\Sigma_t\gets \left(\nabla^2\Phi (\mathbf{x}_t)\right)^{-1 / 2}$
+
+Draw  $\mathbf{v}_t$  uniformly from  $\mathbb{S}^{d - 1}$
+
+$$
+\begin{array}{l} y _ {t} \leftarrow \text {a s p a m e o f} Q _ {t} \text {a t} \mathbf {x} _ {t} + \delta \Sigma_ {t} \mathbf {v} _ {t} \\ \mathbf {O} _ {t} \leftarrow \frac {d}{\delta} y _ {t} \Sigma_ {t} ^ {- 1} \mathbf {V} _ {t} \\ \mathbf{x}_{t + 1}\leftarrow \operatorname *{argmin}_{\mathbf{x}\in \mathcal{K}}\sum_{s = 1}^{t} - \eta \langle \mathbf{o}_{t},\mathbf{x}\rangle +\Phi (\mathbf{x}) \\ \end{array}
+$$
+
+# end
+
+The following theorems demonstrate how to apply the guideline described in the beginning of Section 6 to the results of [Saha and Tewari, 2011]. The first step is to analyze the proof and modify the base algorithm so that we could obtain a result that is valid for non-convex functions and would only require a single use of an inequality similar to  $f(\mathbf{y}) - f(\mathbf{x}) \leq \langle \nabla f(\mathbf{x}), \mathbf{y} - \mathbf{x} \rangle$  to obtain a regret bound for concave case. By a small modification in the original algorithm, we get ZO-FTRL which
+
+differs from the original in that it is no longer a bandit algorithm. While the agent plays  $\mathbf{x}_t$  it queries the oracle at  $\mathbf{x}_t + \delta \Sigma_t\mathbf{v}_t \neq \mathbf{x}_t$ . This modification allows us to obtain the following result.
+
+Theorem 3. Let  $\mathbf{F}$  be an  $M_1$ -Lipschitz  $M_2$ -smooth function class that is bounded by  $M_0$  and let  $B_0 \geq M_0$ . Also let  $\mathcal{B} \in \mathrm{Adv}_0^o(\mathbf{F}, B_0)$  be a realized adversary that returns  $f_1, \dots, f_T$ , let  $\mathbf{u}_* \in \operatorname{argmax}_{\mathbf{u} \in \mathcal{K}} \sum_{t=1}^{T} f_t(\mathbf{u})$  and  $\hat{\mathbf{u}}_* \in \operatorname{argmin}_{\mathbf{x} \in \mathcal{K}_{\gamma, \mathbf{x}_1}} \| \mathbf{u}_* - \mathbf{x} \|$  where  $\gamma = T^{-1}$ . Then, when running Algorithm 5 against  $\mathcal{B}$ , we have
+
+$$
+\sum_ {t = 1} ^ {T} \mathbb {E} \left[ f _ {t} (\mathbf {u} _ {*}) - f _ {t} (\mathbf {x} _ {t}) \right] - O (\delta^ {2} T) \leq \sum_ {t = 1} ^ {T} \mathbb {E} \left[ f _ {t} ^ {\delta \Sigma_ {t}} (\hat {\mathbf {u}} _ {*}) - f _ {t} ^ {\delta \Sigma_ {t}} (\mathbf {x} _ {t}) \right],
+$$
+
+$$
+a n d \quad \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \langle \nabla f _ {t} ^ {\delta \Sigma_ {t}} (\mathbf {x} _ {t}), \hat {\mathbf {u}} _ {*} - \mathbf {x} _ {t} \rangle \right] \leq O \left(\eta \delta^ {- 2} T + \eta^ {- 1} \log T\right).
+$$
+
+See Appendix J for the proof. Note that if  $f$  is concave, then we use Lemma 5 to see that the right hand side of the first inequality is bounded by the left hand side of the second inequality and obtain the regret bound for the concave case. See Appendix K for the proof.
+
+Theorem 4. Let  $\mathbf{F}$  be an  $M_1$ -Lipschitz  $M_2$ -smooth function class over  $\mathcal{K}$  that is upper-linearizable with  $0 < \alpha \leq 1$ ,  $\beta \geq 0$  and a zeroth-order uniform wrapper  $\mathcal{W}$ . Also assume that  $\mathcal{W}^{\text{action}}$  is  $M_1'$ -Lipschitz and  $M_2'$ -smooth. If  $\text{Adv}$  is a zeroth order oblivious adversary over  $\mathbf{F}$  such that for any  $f \in \mathbf{F}$  and any query oracle  $\mathcal{Q}_f$  returned by  $\text{Adv}$ ,  $\mathcal{W}(\mathcal{Q}_f)$  is a stochastic zeroth order query oracle for  $\mathcal{W}(f)$  that is bounded by  $B_0$ , then
+
+$$
+\mathcal {R} _ {\alpha , \mathrm {A d v}} ^ {\mathcal {W} (\mathrm {Z O - F T R L})} = O \left(\eta \delta^ {- 2} T + \eta^ {- 1} \log T + \delta^ {2} T\right),
+$$
+
+In particular, by setting  $\eta = T^{-2/3}$  and  $\delta = T^{-1/6}$ , we see that  $\mathcal{R}_{\alpha, \mathrm{Adv}}^{\mathcal{W}(\mathrm{ZO - FTRL})} = \tilde{O}(T^{2/3})$ .
+
+We emphasize that Theorems 3 and 4 correspond to steps 1 and 2 in the guideline discussed in Section 6 respectively. Specifically, for step 1, we note that Theorem 3 is NOT for concave functions. Moreover, if we assume that the functions are concave, then we may use Theorem 3 and the inequality
+
+$$
+f _ {t} ^ {\delta \Sigma_ {t}} (\hat {\mathbf {u}} _ {*}) - f _ {t} ^ {\delta \Sigma_ {t}} (\mathbf {x} _ {t}) \leq \left\langle \nabla f _ {t} ^ {\delta \Sigma_ {t}} (\mathbf {x} _ {t}), \hat {\mathbf {u}} _ {*} - \mathbf {x} _ {t} \right\rangle
+$$
+
+to immediately obtain a regret bound. For step 2, we note that Theorem 3 adapts the result of previous step to uniform wrappers to obtain a regret bound for linearizable functions.
+
+# 8 Applications
+
+We next discuss some specific online and offline non-convex/non-concave optimization problems for which we can use our new framework to derive improved regret and sample complexity bounds respectively by applying uniform wrappers proposed in Section C to the zeroth order feedback OCO base algorithm ZO-FTRL (Algorithm 5). We note that we can also apply our framework to other base algorithms to recover many existing results in the literature. (See Appendix E for more details).
+
+We start with a definition. For  $\mathbf{x} \in \mathcal{K}$  and  $C > 0$ , we say a zeroth order query oracle  $\mathcal{Q}_f$  is contained in a  $(\mathbf{x}, C)$  cone if we have  $|Q_f(\mathbf{z}) - f(\mathbf{x})| \leq C\|\mathbf{z} - \mathbf{x}\|$  for all  $\mathbf{z} \in \mathcal{K}$ . In other words, the randomness of the query oracle approaches to zero at least linearly as we approach the point  $\mathbf{x}$ . We use the notation  $\mathrm{Adv}_0^{\mathrm{o}}(\mathbf{F}, \mathrm{Cone}(\mathbf{x}, C))$  to denote the oblivious adversary over  $\mathbf{F}$  with query oracles that are contained within this cone. Note that  $\mathcal{Q}_f \in \mathrm{Adv}_0^{\mathrm{o}}(\mathbf{F}, \mathrm{Cone}(\mathbf{x}, C))$  is equivalent to  $\mathcal{W}_0^{\mathrm{NM}}(\mathcal{Q}_f)$  being bounded. See condition (iii) of Definition 5 for details. If  $\mathcal{Q}_f$  does not belong to a cone as described above, we can see that the term  $\left(\frac{z}{2}\right)^{-1}$  causes  $\mathcal{W}_0^{\mathrm{NM}}(\mathcal{Q}_f)$  to blow up. Similarly, in the special case when  $\underline{x} = \mathbf{0}$  and  $f(\mathbf{0}) = 0$ , it is also equivalent to  $\mathcal{W}_0^{\mathrm{M0}}(\mathcal{Q}_f)$  being bounded.
+
+We begin by showing  $\tilde{O}(T^{2/3})$ $\alpha$ -regret bounds for online optimization problems for the three function classes discussed in Appendix C under zeroth order feedback. See Appendix L for the proof.
+
+Theorem 5. Let  $\mathbf{F}_0^{\mathrm{M}},\mathbf{F}_0^{\mathrm{M0}}$  and  $\mathbf{F}_0^{\mathrm{NM}}$  denote the function classes described in Lemmas 1, 2 and 3 respectively and let  $\alpha^{\mathrm{M}},\alpha^{\mathrm{M0}}$  and  $\alpha^{\mathrm{NM}}$  be the values of  $\alpha$ . If the function classes are  $M_{1}$ -Lipschitz and  $M_{2}$ -smooth, then for any  $C > 0$  and  $B_0\geq M_0 = \max_{\mathbf{x}\in \mathcal{K}}f(\mathbf{x})$ , the following are  $\tilde{O} (T^{2 / 3})$ :
+
+$$
+\mathcal {R} _ {\alpha^ {\mathrm {M}}, \operatorname {A d v} _ {0} ^ {o} (\mathbf {F} _ {0} ^ {\mathrm {M}}, B _ {0})} ^ {\mathcal {W} _ {0} ^ {\mathrm {M}} (\mathrm {Z O - F T R L})}, \quad \mathcal {R} _ {\alpha^ {\mathrm {M} 0}, \operatorname {A d v} _ {0} ^ {o} (\mathbf {F} _ {0} ^ {\mathrm {M} 0}, \operatorname {C o n e} (\mathbf {0}, C))} ^ {\mathcal {W} _ {0} ^ {\mathrm {M} 0} (\mathrm {Z O - F T R L})}, \quad \mathcal {R} _ {\alpha^ {\mathrm {N M}}, \operatorname {A d v} _ {0} ^ {o} (\mathbf {F} _ {0} ^ {\mathrm {N M}}, \operatorname {C o n e} (\underline {{\mathbf {x}}}, C))} ^ {\mathcal {W} _ {0} ^ {\mathrm {N M}} (\mathrm {Z O - F T R L})}.
+$$
+
+Remark 1. For each function class, the SOTA for noisy zeroth order feedback achieved  $\tilde{O}(T^{3/4})$ $\alpha$ -regret bounds while we achieve  $\tilde{O}(T^{2/3})$ . For the special case of exact zeroth order feedback, the SOTA is  $\tilde{O}(\sqrt{T})$ . All the SOTA algorithms mentioned are special cases of our framework.
+
+We next show  $\tilde{O}(T^{3/4})$ $\alpha$ -regret bounds for online optimization problems for the three function classes discussed in Appendix C under bandit feedback. For full information zeroth order algorithms, the query location may differ from the action taken. Here we convert them into bandit algorithms using the meta-algorithm Stochastic Full-information To Trivial query (SFTT) from [Pedramfar and Aggarwal, 2024a] (see Appendix M for details). The proof is in Appendix N.
+
+Theorem 6. Under the assumptions of Theorem 5, the following are  $\tilde{O}(T^{3/4})$ :
+
+$$
+\mathcal {R} _ {\alpha^ {\mathrm {M}}, \operatorname {A d v} _ {0} ^ {o} (\mathbf {F} _ {0} ^ {\mathrm {M}}, B _ {0})} ^ {\text {S F T T} (\mathcal {W} _ {0} ^ {\mathrm {M}} (\text {Z O - F T R L}))}, \quad \mathcal {R} _ {\alpha^ {\mathrm {M 0}}, \operatorname {A d v} _ {0} ^ {o} (\mathbf {F} _ {0} ^ {\mathrm {M 0}}, \operatorname {C o n e} (\mathbf {0}, C))} ^ {\text {S F T T} (\mathcal {W} _ {0} ^ {\mathrm {M 0}} (\text {Z O - F T R L}))}, \quad \mathcal {R} _ {\alpha^ {\mathrm {N M}}, \operatorname {A d v} _ {0} ^ {o} (\mathbf {F} _ {0} ^ {\mathrm {N M}}, \operatorname {C o n e} (\underline {{\mathbf {x}}}, C))} ^ {\text {S F T T} (\mathcal {W} _ {0} ^ {\mathrm {N M}} (\text {Z O - F T R L}))},
+$$
+
+where SFTT is Algorithm 4 in [Pedramfar and Aggarwal, 2024a] with  $L = T^{1/4}$ .
+
+Remark 2. Note that Algorithm 3 in [Wan et al., 2023] is in fact  $\mathrm{SFTT}(\mathcal{W}_0^{\mathrm{M0}}(\mathrm{ZO - FTRL}))$ . However, our analysis simplifies the proof and generalizes the result to allow for stochastic feedback.
+
+Remark 3. For the class  $\mathbf{F}^{\mathrm{NM}}$  of non-monotone up-concave functions over general convex sets, our  $\tilde{O}(T^{3/4})$  bound beats the SOTA  $\tilde{O}(T^{4/5})$  bounds for exact and for noisy bandit feedback. For the class  $\mathbf{F}^{\mathrm{M0}}$  of monotone  $\gamma$ -weakly up-concave functions over convex sets containing the origin, our  $\tilde{O}(T^{3/4})$  bound beats the SOTA  $\tilde{O}(T^{4/5})$  bound for noisy bandit feedback and matches the bound for exact bandit feedback. For the third class  $\mathbf{F}^{\mathrm{M}}$  of monotone  $\mu$ -strongly  $\gamma$ -weakly up-concave functions with bounded curvature, our results match the SOTA. All of the SOTA algorithms mentioned here are special cases of our framework.
+
+Conversions of online algorithms to offline are referred to online-to-batch techniques and are well-known in the literature (See [Shalev-Shwartz, 2012]). A simple approach is to simply run the online algorithm and if the actions chosen by the algorithm are  $\mathbf{x}_1,\dots ,\mathbf{x}_T$ , return  $\mathbf{x}_t$  for  $1\leq t\leq T$  with probability  $1 / T$ . We use OTB to denote the meta-algorithm that uses this approach to convert online algorithms to offline algorithms.
+
+We next show that using OTB conversion (on top of  $\mathcal{W}(\mathrm{ZO - FTRL})$ ), we obtain  $\tilde{O}(1 / \epsilon^3)$  sample complexity for finding an  $\alpha$ -approximate solution in each function class under a noisy value oracle model, beating the SOTA  $\tilde{O}(1 / \epsilon^4)$  sample complexity. The proof is in Appendix O
+
+Theorem 7. Under the assumptions of Theorem 5, the following is true.
+
+(i) If the stochastic query oracle is bounded by  $B_0$ , then the sample complexity of the offline algorithm  $\mathrm{OTB}(\mathcal{W}_0^{\mathrm{M}}(\mathrm{ZO - FTRL}))$  over  $\mathbf{F}_0^{\mathrm{M}}$  is  $\tilde{O} (\epsilon^{-3})$ .  
+(ii) If the stochastic query oracle is contained in the cone  $\mathrm{Cone}(\mathbf{0}, C)$ , then the sample complexity of the offline algorithm  $\mathrm{OTB}(\mathcal{W}_0^{\mathrm{M0}}(\mathrm{ZO - FTRL}))$  over  $\mathbf{F}_0^{\mathrm{M0}}$  is  $\tilde{O}(\epsilon^{-3})$ .  
+(iii) If the stochastic query oracle is contained in the cone  $\mathrm{Cone}(\underline{\mathbf{x}}, C)$ , then the sample complexity of the offline algorithm  $\mathrm{OTB}(\mathcal{W}_0^{\mathrm{NM}}(\mathrm{ZO - FTRL}))$  over  $\mathbf{F}_0^{\mathrm{NM}}$  is  $\tilde{O}(\epsilon^{-3})$ .
+
+# 9 Conclusion
+
+We proposed a class of meta-algorithms that can be used to convert convex optimization algorithms (with various types of feedback) into algorithms for quadratizable optimization. Using this framework, we improved regret guarantees for various classes of DR-submodular functions under zeroth-order feedback. Furthermore, we extended zeroth-order online algorithms to bandit feedback and offline counterparts, achieving notable improvements in regret/sample complexity compared to existing approaches.
+
+# Acknowledgments
+
+This work was supported in part by the National Science Foundation under grants CCF-2149588 and CCF-2149617.
+
+# References
+
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+An Bian, Kfir Levy, Andreas Krause, and Joachim M Buhmann. Continuous DR-submodular maximization: Structure and algorithms. In Advances in Neural Information Processing Systems, 2017a.  
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+
+# Table of Contents
+
+1 Introduction 1  
+2 Background and Notation 3  
+3 Problem setup 4  
+4 Uniform wrappers 5  
+5 Linearizable and quadratizable functions classes 6  
+6 When Is Concave Optimization Enough? 6
+
+7 Examples 7
+
+7.1 Online Gradient Ascent 7  
+7.2 Zeroth Order Follow The Regularized Leader 8
+
+8 Applications 9  
+9 Conclusion 10
+
+Appendices 13
+
+A How uniform wrappers work 15  
+B Additional Related Works 15  
+C Uniform wrappers for up-concave optimization 17  
+D Proof of Theorems 1 and 2 19  
+E Recovering previous results in the literature 20  
+F Proof of Theorem 11 21  
+G Proof of Theorem 12 23  
+H Follow The Regularized Leader 23
+
+I Technical Lemmas 24  
+J Proof of Theorem 3 26  
+K Proof of Theorem 4 27  
+L Proof of Theorem 5 29  
+M Stochastic Full-information To Trivial query - SFTT 29  
+N Proof of Theorem 6 30  
+O Proof of Theorem 7 30
+
+![](images/ab2c8aa8eda4bbd5a900903fe8acdf5ba36ce5c6305191079258143f53d199ca.jpg)  
+Figure 1: Effect of a uniform wrapper on the interaction between the agent and the adversary.
+
+# A How uniform wrappers work
+
+Figure A demonstrates the effect of a uniform wrapper on the interaction of the agent and the adversary. The vertical line depicts the uniform wrapper, acting as a translation layer, where any information passing through is changed accordingly. The action of the agent is changed from  $\mathbf{x}_t$  to  $\mathcal{W}^{\mathrm{action}}(\mathbf{x}_t)$  and the query oracle selected by the adversary is changed from  $\mathcal{Q}_t$  to  $\mathcal{W}^{\mathrm{query}}(\mathcal{Q}_t)$ .
+
+Remark 4. Note that neither Figure A, nor Meta-Algorithm 1 require  $\mathcal{W}^{\text {function }}$ . Thus, we define an abstract uniform wrapper as a tuple of maps  $\mathcal{W} = (\mathcal{W}^{\text {action }}, \mathcal{W}^{\text {query }})$  where  $\mathcal{W}^{\text {action }}$  is defined as before. On the other hand,  $\mathcal{W}^{\text {query }}$  is defined as a function that, for any query oracle  $\mathcal{Q}_f \in \mathcal{Q}$ , the value of  $\mathcal{W}(\mathcal{Q}_f)$  behaves similar to a query oracle, but is not necessarily one. Specifically, if  $i, j \geq 0$  and  $\mathcal{W}$  is an  $i$ -order to  $j$ -order abstract uniform wrapper, then for any  $i$ -th order query oracle  $\mathcal{Q}_f$ ,  $\mathcal{W}(\mathcal{Q}_f)$  is just function from  $\mathcal{K}$  to  $\mathbb{R}^{d^j}$ . In particular, there may not exist a function  $F: \mathcal{K} \to \mathbb{R}$  such that  $D^j F(\mathbf{x}) = \mathbb{E}[\mathcal{W}(\mathcal{Q}_f)(\mathbf{x})]$ , where  $D^j$  denotes the  $j$ -th order derivative. The generality of abstract uniform wrappers allow for considering quadratizable function classes where  $\mathfrak{g}$ , as described in Definition 2, might not satisfy  $\mathfrak{g}(f, \mathbf{x}) = \nabla F(\mathbf{x})$  for any function  $F: \mathcal{K} \to \mathbb{R}$ . Note that, unlike uniform wrappers, we may no longer be able to define  $\mathcal{W}(\mathrm{Adv})$  for abstract uniform wrappers.
+
+The figure may be interpreted in two equivalent ways. If we think of  $\mathcal{W}$  as a meta-algorithm that wraps around the agent, then we get  $\mathcal{W}(\mathcal{A})$ , as described in Meta-Algorithm 1, playing against the adversary Adv. Alternatively, we may think of  $\mathcal{W}$  as wrapping around the adversary. In this interpretation, the agent  $\mathcal{A}$  is playing against  $\mathcal{W}(\mathrm{Adv})$  Since these two interpretations are equivalent, we may use one to better understand the other. Specifically, we may modify the proof of the regret bounds of the game between  $\mathcal{A}$  and a concave adversary to bound the regret of  $\mathcal{A}$  against  $\mathcal{W}(\mathrm{Adv})$  and therefore use that to obtain regret bounds for  $\mathcal{W}(\mathcal{A})$  against Adv. The guideline in Section 6 describes how this might be achieved.
+
+# B Additional Related Works
+
+DR-submodular maximization Two of the main methods for continuous DR-submodular maximization are Frank-Wolfe type methods and Boosting based methods. This division is based on how the approximation coefficient appears in the proof. To the best of knowledge, the only other algorithms in the literature are those that are designed for specific choices of domain and use techniques such as discretization and rely heavily on the detailed structures on the constraint set.
+
+In Frank-Wolfe type algorithms, the approximation coefficient appears by specific choices of the Frank-Wolfe update rules. (See Lemma 8 in [Pedramfar et al., 2024a]) The specific choices of the update rules for different settings have been proposed in [Bian et al., 2017b,a, Mualem and Feldman, 2023, Pedramfar et al., 2023, Chen et al., 2023]. The momentum technique of [Mokhtari et al., 2020] has been used to convert algorithms designed for deterministic feedback to stochastic feedback setting. [Hassani et al., 2020] proposed a Frank-Wolfe variant with access to a stochastic gradient oracle with known distribution. Frank-Wolfe type algorithms been adapted to the online setting using Meta-Frank-Wolfe [Chen et al., 2018, 2019] or using Blackwell approachability [Niazadeh et al., 2023]. Later [Zhang et al., 2019] used a Meta-Frank-Wolfe with random permutation technique to obtain full-information results that only require a single query per function and also bandit results.
+
+Table 2: Offline up-concave maximization  
+
+<table><tr><td>F</td><td>Set</td><td colspan="2">Feedback</td><td>Reference</td><td>Appx.</td><td>Complexity</td></tr><tr><td rowspan="17">Monotone</td><td rowspan="9">0∈K</td><td rowspan="4">∇F</td><td rowspan="4">stoch.</td><td>[Mokhtari et al., 2020]</td><td>1-e-γ</td><td>O(1/ε3)</td></tr><tr><td>[Hassani et al., 2020] (*)</td><td>1-e-γ</td><td>O(1/ε2)</td></tr><tr><td>[Zhang et al., 2022] (*)</td><td>1-e-γ</td><td>O(1/ε2)</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>1-e-γ</td><td>O(1/ε2)</td></tr><tr><td rowspan="5">F</td><td rowspan="2">det.</td><td>[Pedramfar et al., 2024b]</td><td>1-e-γ</td><td>O(1/ε3)</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>1-e-γ</td><td>O(1/ε2)</td></tr><tr><td>stoch. †</td><td>Theorem 7</td><td>1-e-γ</td><td>O(1/ε3)</td></tr><tr><td rowspan="2">stoch.</td><td>[Pedramfar et al., 2024b]</td><td>1-e-γ</td><td>O(1/ε5)</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>1-e-γ</td><td>O(1/ε4)</td></tr><tr><td rowspan="8">general</td><td rowspan="3">∇F</td><td rowspan="3">stoch.</td><td>[Hassani et al., 2017] (*)</td><td>γ2/(1+γ2)</td><td>O(1/ε2)</td></tr><tr><td>[Pedramfar et al., 2024b]</td><td>γ2/(1+γ2)</td><td>O(1/ε3)</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>γ2/(1+cy2)</td><td>O(1/ε2)</td></tr><tr><td rowspan="5">F</td><td rowspan="2">det.</td><td>[Pedramfar et al., 2023]</td><td>γ2/(1+γ2)</td><td>O(1/ε3)</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>γ2/(1+cy2)</td><td>O(1/ε2)</td></tr><tr><td rowspan="3">stoch.</td><td>[Pedramfar et al., 2023]</td><td>γ2/(1+γ2)</td><td>O(1/ε5)</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>γ2/(1+cy2)</td><td>O(1/ε4)</td></tr><tr><td>Theorem 7</td><td>γ2/(1+cy2)</td><td>O(1/ε3)</td></tr><tr><td rowspan="8">Non-Monotone</td><td rowspan="8">general</td><td rowspan="3">∇F</td><td rowspan="3">stoch.</td><td>[Pedramfar et al., 2024b]</td><td>γ(1-γh)/γ&#x27;-1 (1/2 - 1/2γ&#x27;)</td><td>O(1/ε3)</td></tr><tr><td>[Zhang et al., 2024] (*)</td><td>(1-h)/4</td><td>O(1/ε2)</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>(1-h)/4</td><td>O(1/ε2)</td></tr><tr><td rowspan="5">F</td><td rowspan="2">det.</td><td>[Pedramfar et al., 2024b]</td><td>γ(1-γh)/γ&#x27;-1 (1/2 - 1/2γ&#x27;)</td><td>O(1/ε3)</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>(1-h)/4</td><td>O(1/ε2)</td></tr><tr><td>stoch. †</td><td>Theorem 7</td><td>(1-h)/4</td><td>O(1/ε3)</td></tr><tr><td rowspan="2">stoch.</td><td>[Pedramfar et al., 2024b]</td><td>γ(1-γh)/γ&#x27;-1 (1/2 - 1/2γ&#x27;)</td><td>O(1/ε5)</td></tr><tr><td>[Pedramfar and Aggarwal, 2024a] (*)</td><td>(1-h)/4</td><td>O(1/ε4)</td></tr></table>
+
+This table compares the different results for the number of oracle calls (complexity) within the constraint set for up-concave maximization. We refer to [Pedramfar et al., 2024b] for a more comprehensive table that includes results for deterministic first order feedback. Here  $h \coloneqq \min_{\mathbf{z} \in \mathcal{K}} \| \mathbf{z} \|_{\infty}$  and  $\gamma' \coloneqq \gamma + 1 / \gamma$ . Rows marked with (*) are results in the literature that fit within the framework described in this paper. The rows describing results with stochastic feedback that are marked with  $\dagger$  assume that the random query oracle is contained with a cone, as detailed in Theorem 5.
+
+This was extended to another settings by [Zhang et al., 2023] and generalized to many different settings with improved regret bounds by [Pedramfar et al., 2024a].
+
+Some techniques construct an alternative function such that maximization of this function results in approximate maximization of the original function. Given this definition, we may consider the result of [Hassani et al., 2017, Chen et al., 2018, Fazel and Sadeghi, 2023] as the first boosting based results. However, in the case of monotone DR-submodular functions over general convex sets, the alternative function is identical to the original function. The term boosting in this context was first used in [Zhang et al., 2022], based on ideas presented in [Filmus and Ward, 2012, Mitra et al., 2021], for monotone functions over convex sets containing the origin. This idea has been used later in [Wan et al., 2023, Liao et al., 2023] in bandit and projection-free full-information settings. Finally, in [Zhang et al., 2024] a boosting based method was introduced for non-monotone functions over general convex sets.
+
+Up-concave maximization Not all continuous DR-submodular functions are concave and not all concave functions are continuous DR-submodular. [Mitra et al., 2021] considers functions that are the sum of a concave and a continuous DR-submodular function. It is well-known that continuous DR-submodular functions are concave along positive directions [Calinescu et al., 2011, Bian et al., 2017b]. Based on this idea, [Wilder, 2018] defined an up-concave function as a function that is concave along positive directions. Up-concave maximization has been considered in the offline
+
+setting before, e.g. in [Lee et al., 2023] and [Pedramfar and Aggarwal, 2024a]. In this work, we focus on up-concave maximization which is a generalization of DR-submodular maximization.
+
+# C Uniform wrappers for up-concave optimization
+
+In this section, we study three classes of up-concave functions and show that they are upper-quadratic with appropriate uniform wrappers.
+
+Remark 5. As we will see in the following, all of these uniform wrappers fit into the following pattern. There is a random variable  $\mathcal{Z}$  taking values in  $[0,1]$  and a parametrized family of functions  $h_z:\mathcal{K}\to \mathcal{K}$  for  $z\in [0,1]$ , such that, for some  $\alpha \in (0,1]$  and  $\beta ,\mu \geq 0$ , we have
+
+$$
+\alpha f (\mathbf {y}) - f (\mathbf {x}) \leq \beta \left(\langle \nabla F (\mathbf {x}), \mathbf {y} - \mathbf {x} \rangle - \frac {\mu}{2} \| \mathbf {y} - \mathbf {x} \| ^ {2}\right),
+$$
+
+where  $F(\mathbf{x}) = \mathbb{E}_{z\sim \mathcal{Z}}[f(h_z(\mathbf{x})) - f(h_0(\mathbf{x}))]$ . If such  $h_z$  and  $\mathcal{Z}$  exist, we define  $\mathcal{W}^{\mathrm{action}} = h_1$ , and  $\mathcal{W}^{\mathrm{function}}(f) = F$ . We also need to define  $\mathcal{W}^{\mathrm{query}}(Q_f)$  such that it converts a query oracle for  $f$  into a query oracle for  $F$  using the following equality.
+
+$$
+\nabla F (\mathbf {x}) = \mathbb {E} _ {z \sim \mathcal {Z}} [ \nabla f (h _ {z} (\mathbf {x})) J _ {h _ {z}} (\mathbf {x}) - \nabla f (h _ {0} (\mathbf {x})) J _ {h _ {0}} (\mathbf {x}) ].
+$$
+
+Note that the identity wrapper, where  $F = f$ , can be recovered by choosing  $h_z(x) \coloneqq z * \mathbf{x}$  and  $\mathcal{Z} = \delta_1$ , the Dirac delta distribution with mass at 1.
+
+# C.1 Monotone  $\mu$ -strongly  $\gamma$ -weakly up-concave functions with bounded curvature  $(\mathbf{F}^{\mathrm{M}})$
+
+For differentiable DR-submodular functions, the following lemma is proved for the case  $\gamma = 1$  in [Fazel and Sadeghi, 2023] and for the case  $\mu = 0$  in [Hassani et al., 2017]. The general form we use here is proved in Lemma 1 in [Pedramfar and Aggarwal, 2024a].
+
+Lemma 1. Let  $f:[0,1]^d \to \mathbb{R}$  be a non-negative monotone  $\mu$ -strongly  $\gamma$ -weakly up-concave function with curvature bounded by  $c$ . Then, for all  $\mathbf{x},\mathbf{y}\in [0,1]^d$ , we have
+
+$$
+\frac {\gamma^ {2}}{1 + c \gamma^ {2}} f (\mathbf {y}) - f (\mathbf {x}) \leq \frac {\gamma}{1 + c \gamma^ {2}} \big (\langle \tilde {\nabla} f (\mathbf {x}), \mathbf {y} - \mathbf {x} \rangle - \frac {\mu}{2} \| \mathbf {y} - \mathbf {x} \| ^ {2} \big),
+$$
+
+where  $\tilde{\nabla} f$  is an up-super-gradient for  $f$ .
+
+Lemma 1, together with Definition 1 of uniform wrappers, immediately imply the following.
+
+Theorem 8. Let  $\mathbf{F}^{\mathrm{M}}$  be the class of functions over  $\mathcal{K}$  where every  $f\in \mathbf{F}^{\mathrm{M}}$  may be extended to a nonnegative differentiable monotone  $\mu$ -strongly  $\gamma$ -weakly up-concave function with curvature bounded by  $c$  defined over  $[0,1]^d$ . Then  $\mathbf{F}^{\mathrm{M}}$  is upper-quadratizable with uniform wrapper  $\mathcal{W}^{\mathrm{M}} = \mathrm{Id}$ .
+
+If  $\mathcal{A}$  is one of the algorithms described in Theorem 12, using the above theorem recovers Theorem 2 in [Pedramfar and Aggarwal, 2024a] which itself is a generalization of Theorem 2 in [Chen et al., 2018] and Theorem 3 in [Fazel and Sadeghi, 2023].
+
+# C.2 Monotone  $\gamma$ -weakly up-concave functions over convex sets containing the origin ( $\mathbf{F}^{\mathrm{MO}}$ )
+
+For differentiable monotone DR-submodular functions, the following lemma is proved in [Zhang et al., 2022]. The general form here is proved in Lemma 2 in [Pedramfar and Aggarwal, 2024a].
+
+Lemma 2. Let  $f:[0,1]^d \to \mathbb{R}$  be a non-negative monotone  $\gamma$ -weakly up-concave differentiable function and let  $F:[0,1]^d \to \mathbb{R}$  be the function defined by  $F(\mathbf{x}) := \int_0^1 \frac{\gamma e^{\gamma(z - 1)}}{(1 - e^{-\gamma})z} (f(z*\mathbf{x}) - f(\mathbf{0}))dz$ .
+
+Then  $F$  is differentiable and  $(1 - e^{-\gamma})f(\mathbf{y}) - f(\mathbf{x})\leq \frac{1 - e^{-\gamma}}{\gamma}\langle \nabla F(\mathbf{x}),\mathbf{y} - \mathbf{x}\rangle .$
+
+Let the random variable  $\mathcal{Z}^{\mathrm{M0}}\in [0,1]$  be defined by the law  $\forall z\in [0,1],\quad \mathbb{P}(\mathcal{Z}^{\mathrm{M0}}\leq z) = \int_0^z\frac{\gamma e^{\gamma(u - 1)}}{1 - e^{-\gamma}} du$ . Then we have  $\mathbb{E}_{z\sim \mathcal{Z}^{\mathrm{M0}}}\left[z^{-1}(f(z*\mathbf{x}) - f(\mathbf{0}))\right] = F(\mathbf{x})$ . Moreover, for  $i\geq 1$ , if  $f$  is  $i$  times differentiable then we also have  $\mathbb{E}_{z\sim \mathcal{Z}^{\mathrm{M0}}}\left[z^{i - 1}\nabla^{i}f(z*\mathbf{x})\right] = \nabla^{i}F(\mathbf{x})$ .
+
+Definition 4. Let  $\mathcal{K} \subseteq [0,1]^d$  be a convex set containing the origin and, for any  $i \geq 0$ , let  $\mathbf{F}_i^{\mathrm{M0}}$  be the class of functions over  $\mathcal{K}$  that are  $\max\{i,1\}$  times differentiable and where every  $f \in \mathbf{F}_i^{\mathrm{M0}}$  may be extended to a non-negative monotone  $\gamma$ -weakly up-concave function defined over  $[0,1]^d$ . We also assume that  $f(\mathbf{0}) = 0$  for all  $f \in \mathbf{F}_0^{\mathrm{M0}}$ . We define  $\mathcal{W}_i^{\mathrm{M0}} := ((\mathcal{W}_i^{\mathrm{M0}})^{\mathrm{action}}, (\mathcal{W}_i^{\mathrm{M0}})^{\mathrm{function}}, (\mathcal{W}_i^{\mathrm{M0}})^{\mathrm{query}})$  to be the uniform wrapper with (i)  $(\mathcal{W}_i^{\mathrm{M0}})^{\mathrm{action}} := \operatorname{Id}_{\mathcal{K}}$ ; (ii) for any  $f \in \mathbf{F}_i^{\mathrm{M0}}$ ,
+
+$(\mathcal{W}_i^{\mathrm{M0}})^{\mathrm{function}}(f)\coloneqq \mathbf{x}\mapsto \mathbb{E}_{z\sim \mathcal{Z}^{\mathrm{M0}}}\left[z^{-1}(f(z*\mathbf{x}) - f(\mathbf{0}))\right]:\mathcal{K}\to \mathbb{R};$  and
+
+(iii) for any  $f \in \mathbf{F}_i^{\mathrm{M0}}$  and any  $i$ -th order query oracle  $\mathcal{Q}_f$  for  $f$ , we have  $(\mathcal{W}_i^{\mathrm{M0}})^{\mathrm{query}}(\mathcal{Q}_f)(\mathbf{x}) := z^{i-1} * Q_f(z * \mathbf{x})$ , where  $z$  is sampled according to  $\mathbb{P}(\mathcal{Z}^{\mathrm{M0}} \leq z)$ .
+
+Remark 6. This wrapper fits into the pattern described in Remark 5 by defining  $h_z(\mathbf{x}) \coloneqq z * \mathbf{x}$  and  $\mathcal{Z} = \mathcal{Z}^{\mathrm{M0}}$ .
+
+Theorem 9. For any  $i \geq 0$ , the function class  $\mathbf{F}_i^{\mathrm{M0}}$  defined above is upper-linearizable with the uniform wrapper  $\mathcal{W}_i^{\mathrm{M0}}$ .
+
+Remark 7. The meta-algorithm  $\mathcal{A} \mapsto \mathrm{OMBQ}(\mathcal{A}, \mathrm{BQM0}, \mathrm{Id})$ , described in [Pedramfar and Aggarwal, 2024a], is identical to  $\mathcal{A} \mapsto \mathcal{W}_1^{\mathrm{M0}}(\mathcal{A})$ . In other words, the results of Theorem 3 in [Pedramfar and Aggarwal, 2024a] are about the first order uniform wrapper  $\mathcal{W}_1^{\mathrm{M0}}$ . Here we consider a more general case where we are not necessarily limited to first order.
+
+# C.3 Non-monotone up-concave functions over general convex sets  $(\mathbf{F}^{\mathrm{NM}})$
+
+For differentiable monotone DR-submodular functions, the following lemma is proved in [Zhang et al., 2024]. The general form we use is proven in Lemma 3 in [Pedramfar and Aggarwal, 2024a].
+
+Lemma 3. Let  $f:[0,1]^d \to \mathbb{R}$  be a non-negative continuous up-concave differentiable function and let  $\underline{\mathbf{x}} \in \mathcal{K}$ . Define  $F:[0,1]^d \to \mathbb{R}$  as the function  $F(\mathbf{x}) \coloneqq \int_0^1 \frac{2}{3z(1 - \frac{z}{2})^3} (f\left(\frac{z}{2} * (\mathbf{x} - \underline{\mathbf{x}}) + \underline{\mathbf{x}}\right) - f(\underline{\mathbf{x}})) dz$ , then  $F$  is differentiable and we have
+
+$$
+\frac {1 - \| \underline {{\mathbf {x}}} \| _ {\infty}}{4} f (\mathbf {y}) - f \left(\frac {\mathbf {x} + \underline {{\mathbf {x}}}}{2}\right) \leq \frac {3}{8} \langle \nabla F (\mathbf {x}), \mathbf {y} - \mathbf {x} \rangle .
+$$
+
+Let the random variable  $\mathcal{Z}^{\mathrm{NM}}\in [0,1]$  be defined by the law  $\forall z\in [0,1]$ ,  $\mathbb{P}(\mathcal{Z}^{\mathrm{NM}}\leq z) = \int_0^z\frac{1}{3(1 - \frac{u}{2})^3} du$ . Then we have  $\mathbb{E}_{z\sim \mathcal{Z}^{\mathrm{NM}}}\big[(\frac{z}{2})^{-1}*(f(\frac{z}{2} *(x - \underline{x}) + \underline{x}) - f(\underline{x}))\big] = F(x)$ . Moreover, if  $i\geq 1$  and  $f$  is  $i$  times differentiable, then  $\mathbb{E}_{z\sim \mathcal{Z}^{\mathrm{NM}}}\big[(\frac{z}{2})^{i - 1}*\nabla^i f(\frac{z}{2} *(x - \underline{x}) + \underline{x})\big] = \nabla^i F(x)$ .
+
+Definition 5. Let  $\mathcal{K} \subseteq [0,1]^d$  be a convex set and, for any  $i \geq 0$ , let  $\mathbf{F}_i^{\mathrm{NM}}$  be the class of functions over  $\mathcal{K}$  where every  $f \in \mathbf{F}_i^{\mathrm{NM}}$  may be extended to a non-negative up-concave function defined over  $[0,1]^d$ . We also assume that  $\mathbf{F}_i^{\mathrm{NM}}$  is  $\max\{i,1\}$  times differentiable for all  $i \geq 0$  and, for some known constant  $c \geq 0$  and all  $f \in \mathbf{F}_0^{\mathrm{NM}}$ ,  $f(\underline{\mathbf{x}}) = c$ . For  $i \geq 0$ , we define  $\mathcal{W}_i^{\mathrm{NM}} = ((\mathcal{W}_i^{\mathrm{NM}})^{\mathrm{action}}, (\mathcal{W}_i^{\mathrm{NM}})^{\mathrm{function}}, (\mathcal{W}_i^{\mathrm{NM}})^{\mathrm{query}})$  to be the uniform wrapper with
+
+(i)  $(\mathcal{W}_i^{\mathrm{NM}})^{\mathrm{action}}\coloneqq \mathbf{x}\mapsto \frac{\mathbf{x} + \mathbf{x}}{2}:\mathcal{K}\to \mathcal{K};$  (ii) for any  $f\in \mathbf{F}_i^{\mathrm{NM}}$
+
+$(\mathcal{W}_i^{\mathrm{NM}})^{\mathrm{function}}(f)\coloneqq \mathbf{x}\mapsto \overline{\mathbb{E}}_{z\sim \mathcal{Z}^{\mathrm{NM}}}[(\frac{z}{2})^{-1}*(f(\frac{z}{2} *(\mathbf{x} - \underline{\mathbf{x}}) + \underline{\mathbf{x}}) - f(\underline{\mathbf{x}}))]:\mathcal{K}\to \mathbb{R};$  and (iii) for any  $f\in \mathbf{F}_i^{\mathrm{NM}}$  and any  $i$  -th order query oracle  $\mathcal{Q}_f$  for  $f$
+
+$$
+(\mathcal {W} _ {i} ^ {\mathrm {N M}}) ^ {\mathrm {q u e r y}} (\mathcal {Q} _ {f}) (\mathbf {x}) := \left\{ \begin{array}{l l} \left(\frac {z}{2}\right) ^ {i - 1} * Q _ {f} \left(\frac {z}{2} * (\mathbf {x} - \underline {{\mathbf {x}}}) + \underline {{\mathbf {x}}}\right) & \text {i f} i \geq 1 \\ \left(\frac {z}{2}\right) ^ {- 1} * \left(Q _ {f} \left(\frac {z}{2} * (\mathbf {x} - \underline {{\mathbf {x}}}) + \underline {{\mathbf {x}}}\right) - c\right) & \text {i f} i = 0 \end{array} \right.
+$$
+
+where  $z$  is sampled according to  $\mathbb{P}(\mathcal{Z}^{\mathrm{NM}}\leq z)$
+
+Remark 8. This wrapper fits into the pattern described in Remark 5 by defining  $h_z(\mathbf{x}) \coloneqq \frac{z}{2} * (\mathbf{x} - \underline{\mathbf{x}}) + \underline{\mathbf{x}}$  and  $\mathcal{Z} = \mathcal{Z}^{\mathrm{NM}}$ .
+
+Theorem 10. For any  $i \geq 0$ , the function class  $\mathbf{F}_i^{\mathrm{NM}}$  defined above is upper-linearizable with the uniform wrapper  $\mathcal{W}_i^{\mathrm{NM}}$ .
+
+Remark 9. The meta-algorithm  $\mathcal{A} \mapsto \mathrm{OMBQ}(\mathcal{A}, \mathrm{BQN}, \mathbf{x} \mapsto \frac{\mathbf{x} + \mathbf{x}}{2})$ , described in [Pedramfar and Aggarwal, 2024a], is identical to  $\mathcal{A} \mapsto \mathcal{W}_1^{\mathrm{NM}}(\mathcal{A})$ . In other words, the results of Theorem 4 in [Pedramfar and Aggarwal, 2024a] are about the first order uniform wrapper  $\mathcal{W}_1^{\mathrm{NM}}$ . Here we consider a more general case where we are not necessarily limited to first order.
+
+# D Proof of Theorems 1 and 2
+
+The proof of Theorem 1 is a simple rewriting of the classical results. We start with a lemma which contains the main idea commonly used in the proof of the regret bound of Online Gradient Descent. Lemma 4. Let  $\mathbf{x},\mathbf{x}_1\in \mathcal{K}$ , let  $G > 0$  and let  $(\mathbf{g}_t)_{t = 1}^T$  be a sequence of vectors such that  $\| \mathbf{g}_t\| \leq G$ . Define the sequence  $(\mathbf{x}_t)_{t = 1}^T$  such that  $\mathbf{x}_{t + 1}\gets P_{\mathcal{K}}(\mathbf{x}_t - \eta_t\mathbf{g}_t)$ , where  $\eta_t = \frac{D}{G\sqrt{t}}$ . Then we have
+
+$$
+\sum_ {t = 1} ^ {T} \langle \mathbf {g} _ {t}, \mathbf {x} _ {t} - \mathbf {x} \rangle \leq \frac {3}{2} G D \sqrt {T}.
+$$
+
+The following proof is copied from the proof of Theorem 3.1 in [Hazan et al., 2016].
+
+Proof. Using Pythagorean theorem, we have
+
+$$
+\left\| \mathbf {x} _ {t + 1} - \mathbf {x} \right\| ^ {2} = \left\| P _ {\mathcal {K}} (\mathbf {x} _ {t} - \eta_ {t} \mathbf {g} _ {t}) - \mathbf {x} \right\| ^ {2} \leq \left\| \mathbf {x} _ {t} - \eta_ {t} \mathbf {g} _ {t} - \mathbf {x} \right\| ^ {2}.
+$$
+
+Hence,
+
+$$
+\left\| \mathbf {x} _ {t + 1} - \mathbf {x} \right\| ^ {2} \leq \left\| \mathbf {x} _ {t} - \mathbf {x} \right\| ^ {2} + \eta_ {t} ^ {2} \left\| \mathbf {g} _ {t} \right\| ^ {2} - 2 \eta_ {t} \langle \mathbf {g} _ {t}, \mathbf {x} _ {t} - \mathbf {x} \rangle .
+$$
+
+$$
+2 \eta_ {t} \langle \mathbf {g} _ {t}, \mathbf {x} _ {t} - \mathbf {x} \rangle \leq \frac {1}{\eta_ {t}} \left(\| \mathbf {x} _ {t} - \mathbf {x} \| ^ {2} - \| \mathbf {x} _ {t + 1} - \mathbf {x} \| ^ {2}\right) + \eta_ {t} G ^ {2}.
+$$
+
+Summing this from  $t = 1$  to  $T$ , and setting  $\eta_t = \frac{D}{G\sqrt{t}}$ , we see that
+
+$$
+\begin{array}{l} 2 \sum_ {t = 1} ^ {T} \eta_ {t} \langle \mathbf {g} _ {t}, \mathbf {x} _ {t} - \mathbf {x} \rangle \leq \sum_ {t = 1} ^ {T} \left[ \frac {1}{\eta_ {t}} \left(\| \mathbf {x} _ {t} - \mathbf {x} \| ^ {2} - \| \mathbf {x} _ {t + 1} - \mathbf {x} \| ^ {2}\right) + \eta_ {t} G ^ {2} \right] \\ = \frac {1}{\eta_ {1}} \| \mathbf {x} _ {1} - \mathbf {x} \| ^ {2} + \sum_ {t = 2} ^ {T} \| \mathbf {x} _ {t} - \mathbf {x} \| ^ {2} \left(\frac {1}{\eta_ {t}} - \frac {1}{\eta_ {t - 1}}\right) + G ^ {2} \sum_ {t = 2} ^ {T} \eta_ {t} \\ \leq D ^ {2} \left[ \frac {1}{\eta_ {1}} + \sum_ {t = 2} ^ {T} \left(\frac {1}{\eta_ {t}} - \frac {1}{\eta_ {t - 1}}\right) \right] + G ^ {2} \sum_ {t = 2} ^ {T} \eta_ {t} \\ = D ^ {2} \frac {1}{\eta_ {T}} + G ^ {2} \sum_ {t = 2} ^ {T} \eta_ {t} \\ \leq 3 D G \sqrt {T}, \\ \end{array}
+$$
+
+where the second inequality follows from the facts that  $\mathrm{diam}(\mathcal{K})\leq D$  and  $\eta_t$  is decreasing and therefore  $\frac{1}{\eta_t} -\frac{1}{\eta_{t - 1}}$  is always positive; and the last inequality follows from the facts that  $\eta_t = \frac{D}{G\sqrt{t}}$  and  $\sum_{t = 1}^{T}\frac{1}{\sqrt{t}}\leq 2\sqrt{T}$
+
+Using this lemma, we can easily prove Theorem 1.
+
+Proof. By applying Lemma 4 to  $\mathbf{o}_t = -\mathbf{g}_t$ , (to account for the fact, unlike OGD, that we update according to  $\mathbf{x}_{t + 1}\gets P_{\mathcal{K}}(\mathbf{x}_t + \eta_t\mathbf{o}_t)$ ) we see that
+
+$$
+\sum_ {t = 1} ^ {T} \left\langle \mathbf {o} _ {t}, \mathbf {x} - \mathbf {x} _ {t} \right\rangle \leq \frac {3}{2} G D \sqrt {T}.
+$$
+
+If we take the expectation, we see that
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \mathbb {E} [ \langle \nabla f _ {t} (\mathbf {x} _ {t}), \mathbf {x} - \mathbf {x} _ {t} \rangle ] = \sum_ {t = 1} ^ {T} \mathbb {E} [ \langle \mathbb {E} [ \mathbf {o} _ {t} | \mathbf {x} _ {t} ], \mathbf {x} - \mathbf {x} _ {t} \rangle ] \\ = \sum_ {t = 1} ^ {T} \mathbb {E} [ \mathbb {E} [ \langle \mathbf {o} _ {t}, \mathbf {x} - \mathbf {x} _ {t} \rangle | \mathbf {x} _ {t} ] ] \\ = \sum_ {t = 1} ^ {T} \mathbb {E} [ \left\langle \mathbf {o} _ {t}, \mathbf {x} - \mathbf {x} _ {t} \right\rangle ] \leq \frac {3}{2} G D \sqrt {T}. \\ \end{array}
+$$
+
+Next we prove Theorem 2.
+
+Proof. At first glance, it may seem that moving from Theorem 1 to Theorem 2 might be non-trivial. However, as elaborated in Appendix A, whether we apply  $\mathcal{W}(\mathrm{OGA})$  to an adversary Adv that selects functions from  $\mathbf{F}$  or apply OGA to the adversary  $\mathcal{W}(\mathrm{Adv})$ , the sequence  $\mathbf{x}_t$  remains the same. Thus, by using Theorem 1, we immediately see that
+
+$$
+\max _ {\mathbf {x} \in \mathcal {K}} \sum_ {t = 1} ^ {T} \mathbb {E} [ \langle \nabla \mathcal {W} (f _ {t}) (\mathbf {x} _ {t}), \mathbf {x} - \mathbf {x} _ {t} \rangle ] \leq \frac {3}{2} G D \sqrt {T}.
+$$
+
+Another way to arrive at this inequality is to follow the proof of Theorem 2 and adapt is here. Specifically, for all  $\mathbf{x} \in \mathcal{K}$ , we have
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \mathbb {E} [ \langle \nabla \mathcal {W} (f _ {t}) (\mathbf {x} _ {t}), \mathbf {x} - \mathbf {x} _ {t} \rangle ] = \sum_ {t = 1} ^ {T} \mathbb {E} [ \langle \mathbb {E} [ \mathbf {o} _ {t} | \mathbf {x} _ {t} ], \mathbf {x} - \mathbf {x} _ {t} \rangle ] \\ = \sum_ {t = 1} ^ {T} \mathbb {E} [ \mathbb {E} [ \langle \mathbf {o} _ {t}, \mathbf {x} - \mathbf {x} _ {t} \rangle | \mathbf {x} _ {t} ] ] \\ = \sum_ {t = 1} ^ {T} \mathbb {E} [ \langle \mathbf {o} _ {t}, \mathbf {x} - \mathbf {x} _ {t} \rangle ] \leq \frac {3}{2} G D \sqrt {T}, \\ \end{array}
+$$
+
+where the last inequality follows from Lemma 4. Now we may use the definition of uniform wrappers to complete the proof of Theorem 2.
+
+# E Recovering previous results in the literature
+
+As mentioned in Remark 2, Algorithm 3 in [Wan et al., 2023] is in fact SFTT  $(\mathcal{W}_0^{\mathrm{M0}}(\mathrm{ZO - FTRL}))$  and therefore their result fits within our framework. The way the remaining results in the tables that are marked with  $(^{*})$  is discussed in the following.
+
+We demonstrate how to apply the guideline described in the beginning of Section 6 to Theorem 2 in [Pedramfar and Aggarwal, 2024b]. This allows us to obtain a generalized version of Theorems 1 in [Pedramfar and Aggarwal, 2024a]. As we will discuss below, this will allow us to recover all the remaining results in Tables 1 and 2 that are marked with (*) and all the results of [Pedramfar and Aggarwal, 2024a]. Note that the results of [Pedramfar and Aggarwal, 2024a] in non-stationary setting are not discussed in this paper, but they are also recovered.
+
+We start with some definitions. Given a function class  $\mathbf{F}$ , we use the notation  $\mathbf{F}_{\mu, \mathbf{g}}$  to denote the class of functions  $q(\mathbf{y}) := \langle \mathfrak{g}(f, \mathbf{x}), \mathbf{y} - \mathbf{x} \rangle - \frac{\mu}{2} \| \mathbf{y} - \mathbf{x} \|^2 : \mathcal{K} \to \mathbb{R}$ , for all  $f \in \mathbf{F}$  and  $\mathbf{x} \in \mathcal{K}$ . This is the class of quadratic (or linear, when  $\mu = 0$ ) functions that form the upper bound in Equation 1. Similarly, for any  $B_1 > 0$ , we use the notation  $\mathbf{Q}_{\mu}[B_1]$  to denote the class of functions  $q(\mathbf{y}) := \langle \mathbf{o}, \mathbf{y} - \mathbf{x} \rangle - \frac{\mu}{2} \| \mathbf{y} - \mathbf{x} \|^2 : \mathcal{K} \to \mathbb{R}$ , for all  $\mathbf{x} \in \mathcal{K}$  and  $\mathbf{o} \in \mathbb{B}_{B_1}(\mathbf{0})$ . In the following theorems, we will obtain results that allow us to reduce the problem of online optimization over  $\mathbf{F}$  to the problem of online optimization over the quadratic (or linear) function class  $\mathbf{F}_{\mu, \mathbf{g}}$ .
+
+Theorem 11. Let  $\mathcal{A}$  be algorithm for online optimization with semi-bandit feedback. Also let  $\mathbf{F}$  be a differentiable function class over  $\mathcal{K}$  and  $\mu \geq 0$ . Then the following are true.
+
+- If query oracles in Adv are deterministic gradient oracles, then we have
+
+$$
+\sup  _ {\mathcal {B} \in \mathrm {A d v}} \mathbb {E} \left[ \max  _ {\mathbf {u} \in \mathcal {U}} \sum_ {t = a} ^ {b} \left(\langle \nabla f _ {t} (\mathbf {x} _ {t}), \mathbf {u} _ {t} - \mathbf {x} _ {t} \rangle - \frac {\mu}{2} \| \mathbf {u} _ {t} - \mathbf {x} _ {t} \|\right) \right] \leq \mathcal {R} _ {1, \mathrm {A d v} _ {1} ^ {f} (\mathbf {F} _ {\mu , \nabla})} ^ {\mathcal {A}}.
+$$
+
+- On the other hand, if  $\mathbf{F}$  is  $M_1$ -Lipschitz and query oracles in Adv are stochastic gradient oracles that are bounded by  $B_1 \geq M_1$ , then we have
+
+$$
+\sup  _ {\mathcal {B} \in \mathrm {A d v}} \max  _ {\mathbf {u} \in \mathcal {U}} \mathbb {E} \left[ \sum_ {t = a} ^ {b} \left(\langle \nabla f _ {t} (\mathbf {x} _ {t}), \mathbf {u} _ {t} - \mathbf {x} _ {t} \rangle - \frac {\mu}{2} \| \mathbf {u} _ {t} - \mathbf {x} _ {t} \|\right) \right] \leq \mathcal {R} _ {1, \mathrm {A d v} _ {1} ^ {f} (\mathbf {Q} _ {\mu} [ B _ {1} ])} ^ {\mathcal {A}}.
+$$
+
+See Appendix F for proof. Note that if  $f_{t}$  are  $\mu$ -strongly concave, then this result reduces to Theorem 2 in [Pedramfar and Aggarwal, 2024b]. Next, we follow step (2) in the guideline to obtain the following result.
+
+Theorem 12. Let  $\mathbf{F}$  be a function class over  $\mathcal{K}$  that is upper-quadratizable with  $\mu \geq 0$ ,  $0 < \alpha \leq 1$  and  $\beta \geq 0$  and a first-order uniform wrapper  $\mathcal{W}$ .
+
+- If  $\mathcal{W}(\nabla) = \nabla$ , i.e., it maps deterministic gradient oracles into deterministic gradient oracles, then we have  $\mathcal{R}_{\alpha, \mathrm{Adv}_1^f(\mathbf{F})}^{\mathcal{W}(\mathcal{A})} \leq \beta \mathcal{R}_{1, \mathrm{Adv}_1^f(\mathbf{F}_{\mu, \nabla})}^{\mathcal{A}}$ .  
+- If, for any  $f \in \mathbf{F}$  and any query oracle  $\mathcal{Q}_f$  bounded by  $B_1$ ,  $\mathcal{W}(\mathcal{Q}_f)$  is a stochastic query oracle for  $\mathcal{W}(f)$  that is bounded by  $B_1'$ , then we have  $\mathcal{R}_{\alpha, \mathrm{Adv}_1^q(\mathbf{F}, B_1)}^{\mathcal{W}(\mathcal{A})} \leq \beta \mathcal{R}_{1, \mathrm{Adv}_1^f(\mathbf{Q}_\mu[B_1'])}^{\mathcal{A}}$ .
+
+See Appendix G for proof. In this theorem, by using the uniform wrappers described in Section C, in the special case of  $i = 1$ , we recover Theorems 2, 3 and 4 in [Pedramfar and Aggarwal, 2024a]. (See Remarks 7 and 9) In other words, we recover all meta-algorithms in [Pedramfar and Aggarwal, 2024a] that are used to convert concave optimization algorithms into up-concave optimization algorithms.
+
+Remark 10. Online Gradient Ascent requires semi-bandit feedback. Thus, Theorems 1 and 2 are special cases of Theorems 11 and 12 respectively.
+
+Remark 11. By applying these uniform wrappers to base algorithms SO-OGA ([Garber and Kretzu, 2022]) or IA ([Zhang et al., 2018]), we recover all the results of [Pedramfar and Aggarwal, 2024a]. In particular, we also recover the results for non-stationary regret described in Table 3 in [Pedramfar and Aggarwal, 2024a].
+
+# F Proof of Theorem 11
+
+Proof.
+
+# Deterministic oracle:
+
+For any realization  $\mathcal{B} = (\mathcal{B}_1,\dots ,\mathcal{B}_T)\in \mathrm{Adv}\subseteq \mathrm{Adv}_1^{\mathrm{f}}(\mathbf{F})$  , we define  $\mathcal{B}_t^\prime (\mathbf{x}_1,\dots ,\mathbf{x}_t)$  to be the tuple  $(q_{t},\nabla)$  where
+
+$$
+\mathcal {B} _ {t} ^ {\prime} \left(\mathbf {x} _ {1}, \dots , \mathbf {x} _ {t}\right) := q _ {t} := \mathbf {y} \mapsto \left\langle \nabla f _ {t} \left(\mathbf {x} _ {t}\right), \mathbf {y} - \mathbf {x} _ {t} \right\rangle - \frac {\mu}{2} \| \mathbf {y} - \mathbf {x} _ {t} \| ^ {2},
+$$
+
+and  $\mathcal{B}' = (\mathcal{B}_1',\dots ,\mathcal{B}_T')$ . Note that each  $\mathcal{B}_t^\prime$  is a deterministic function of  $\mathbf{x}_1,\dots ,\mathbf{x}_t$  and therefore  $\mathcal{B}^{\prime}\in \mathrm{Adv}_{1}^{\mathrm{f}}(\mathbf{F}_{\mu ,\nabla})$ . Since the algorithm uses semi-bandit feedback, the sequence of random vectors  $(\mathbf{x}_1,\dots ,\mathbf{x}_T)$  chosen by  $\mathcal{A}$  is identical between the game with  $\mathcal{B}$  and  $\mathcal{B}^{\prime}$ . Hence
+
+$$
+\begin{array}{l} \sup _ {\mathcal {B} \in \mathrm {A d v}} \mathbb {E} \left[ \max _ {\mathbf {u} \in \mathcal {U}} \sum_ {t = a} ^ {b} \left(\langle \nabla f _ {t} (\mathbf {x} _ {t}), \mathbf {u} _ {t} - \mathbf {x} _ {t} \rangle - \frac {\mu}{2} \| \mathbf {u} _ {t} - \mathbf {x} _ {t} \| ^ {2}\right) \right] \\ = \sup _ {\mathcal {B} \in \mathrm {A d v}} \mathbb {E} \left[ \max _ {\mathbf {u} \in \mathcal {U}} \left(\sum_ {t = a} ^ {b} q _ {t} (\mathbf {u} _ {t}) - \sum_ {t = a} ^ {b} q _ {t} (\mathbf {x} _ {t})\right) \right] \\ \leq \sup_{\mathcal{B}^{\prime}\in \operatorname{Adv}_{1}^{\mathrm{f}}(\mathbf{F}_{\mu ,\nabla})}\mathcal{R}_{1,\mathcal{B}^{\prime}}^{\mathcal{A}} = \mathcal{R}_{1,\operatorname{Adv}_{1}^{\mathrm{f}}(\mathbf{F}_{\mu ,\nabla})}^{\mathcal{A}}. \\ \end{array}
+$$
+
+# Stochastic oracle:
+
+Let  $\Omega^{\mathcal{Q}} = \Omega_1^{\mathcal{Q}} \times \dots \times \Omega_T^{\mathcal{Q}}$  capture all sources of randomness in the query oracles of  $\mathrm{Adv}_1^{\mathrm{o}}(\mathbf{F}, B_1)$ , i.e., for any choice of  $\theta \in \Omega^{\mathcal{Q}}$ , the query oracle is deterministic. Hence for any  $\theta \in \Omega^{\mathcal{Q}}$  and realized adversary  $\mathcal{B} \in \mathrm{Adv} \subseteq \mathrm{Adv}_1^{\mathrm{f}}(\mathbf{F}, B_1)$ , we may consider  $\mathcal{B}_{\theta}$  as an object similar to an adversary with a deterministic oracle. However, note that  $\mathcal{B}_{\theta}$  does not satisfy the unbiasedness condition of the oracle, i.e., the returned value of the oracle is not necessarily the gradient of the function at that point. Recall that  $\mathcal{B}_t$  maps a tuple  $(\mathbf{x}_1, \dots, \mathbf{x}_t)$  to a tuple of  $f_t$  and a stochastic query oracle for  $f_t$ . We will use  $\mathbb{E}_{\Omega^{\mathcal{Q}}}$  to denote the expectation with respect to the randomness of query oracle and
+
+$\mathbb{E}_{\Omega_t^{\mathcal{Q}}}[\cdot ]\coloneqq \mathbb{E}_{\Omega^{\mathcal{Q}}}[\cdot |f_t,\mathbf{x}_t]$  to denote the expectation conditioned on the action of the agent and the adversary. Similarly, let  $\mathbb{E}_{\Omega^A}$  denote the expectation with respect to the randomness of the agent. Let  $\mathbf{o}_t$  be the random variable denoting the output of  $\mathcal{Q}$  at time-step  $t$  and let
+
+$$
+\bar {\mathbf {o}} _ {t} := \mathbb {E} [ \mathbf {o} _ {t} \mid f _ {t}, \mathbf {x} _ {t} ] = \mathbb {E} _ {\Omega_ {t} ^ {\mathcal {Q}}} [ \mathbf {o} _ {t} ] = \nabla f _ {t} (\mathbf {x} _ {t}).
+$$
+
+Similar to the deterministic case, for any realization  $\mathcal{B} = (\mathcal{B}_1,\dots ,\mathcal{B}_T)\in \mathrm{Adv}$  and any  $\theta \in \Omega^{\mathcal{Q}}$  we define  $\mathcal{B}_{\theta ,t}^{\prime}(\mathbf{x}_1,\dots ,\mathbf{x}_t)$  to be the pair  $(q_{t},\nabla)$  where
+
+$$
+q _ {t} := \mathbf {y} \mapsto \left\langle \mathbf {o} _ {t}, \mathbf {y} - \mathbf {x} _ {t} \right\rangle - \frac {\mu}{2} \| \mathbf {y} - \mathbf {x} _ {t} \| ^ {2}.
+$$
+
+We also define  $\mathcal{B}_{\theta}^{\prime} := (\mathcal{B}_{\theta,1}^{\prime}, \dots, \mathcal{B}_{\theta,T}^{\prime})$ . Note that a specific choice of  $\theta$  is necessary to make sure that the function returned by  $\mathcal{B}_{\theta,t}^{\prime}$  is a deterministic function of  $\mathbf{x}_1, \dots, \mathbf{x}_t$  and not a random variable and therefore  $\mathcal{B}_{\theta}^{\prime}$  belongs to  $\mathrm{Adv}_{1}^{\mathrm{f}}(\mathbf{F}_{\mu}[B_1])$ .
+
+Since the algorithm uses (semi-)bandit feedback, given a specific value of  $\theta$ , the sequence of random vectors  $(\mathbf{x}_1,\dots ,\mathbf{x}_T)$  chosen by  $\mathcal{A}$  is identical between the game with  $\mathcal{B}_{\theta}$  and  $\mathcal{B}_{\theta}^{\prime}$ . Therefore, for any  $\mathbf{u}\in \mathcal{U}$ , we have
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \sum_ {t = a} ^ {b} \left(\langle \nabla f _ {t} (\mathbf {x} _ {t}), \mathbf {u} _ {t} - \mathbf {x} _ {t} \rangle - \frac {\mu}{2} \| \mathbf {u} _ {t} - \mathbf {x} _ {t} \| ^ {2}\right) \right] \\ = \mathbb {E} \left[ \sum_ {t = a} ^ {b} \left(\langle \mathbb {E} \left[ \mathbf {o} _ {t} \mid f _ {t}, \mathbf {x} _ {t} \right], \mathbf {u} _ {t} - \mathbf {x} _ {t} \rangle - \frac {\mu}{2} \| \mathbf {u} _ {t} - \mathbf {x} _ {t} \| ^ {2}\right) \right] \\ = \mathbb {E} \left[ \sum_ {t = a} ^ {b} \left(\mathbb {E} \left[ \langle \mathbf {o} _ {t}, \mathbf {u} _ {t} - \mathbf {x} _ {t} \rangle - \frac {\mu}{2} \| \mathbf {u} _ {t} - \mathbf {x} _ {t} \| ^ {2} \mid f _ {t}, \mathbf {x} _ {t} \right]\right) \right] \\ = \mathbb {E} \left[ \sum_ {t = a} ^ {b} \left(\mathbb {E} \left[ q _ {t} \left(\mathbf {u} _ {t}\right) - q _ {t} \left(\mathbf {x} _ {t}\right) \mid f _ {t}, \mathbf {x} _ {t} \right]\right) \right] \\ = \mathbb {E} \left[ \sum_ {t = a} ^ {b} \left(q _ {t} (\mathbf {u} _ {t}) - q _ {t} (\mathbf {x} _ {t})\right) \right]. \\ \end{array}
+$$
+
+Hence we have
+
+$$
+\begin{array}{l} \max _ {\mathbf {u} \in \mathcal {U}} \mathbb {E} \left[ \sum_ {t = a} ^ {b} \left(\langle \nabla f _ {t} (\mathbf {x} _ {t}), \mathbf {u} _ {t} - \mathbf {x} _ {t} \rangle - \frac {\mu}{2} \| \mathbf {u} _ {t} - \mathbf {x} _ {t} \|\right) \right] = \max _ {\mathbf {u} \in \mathcal {U}} \mathbb {E} \left[ \sum_ {t = a} ^ {b} \left(q _ {t} (\mathbf {u} _ {t}) - q _ {t} (\mathbf {x} _ {t})\right) \right] \\ \leq \mathbb {E} \left[ \max  _ {\mathbf {u} = \left(\mathbf {u} _ {1}, \dots , \mathbf {u} _ {T}\right) \in \mathcal {U}} \sum_ {t = a} ^ {b} \left(q _ {t} \left(\mathbf {u} _ {t}\right) - q _ {t} \left(\mathbf {x} _ {t}\right)\right) \right] \\ = \mathcal {R} _ {\mathcal {B} _ {\theta} ^ {\prime}} ^ {\mathcal {A}} (\mathcal {U}) [ a, b ] \\ \end{array}
+$$
+
+where the inequality follows from Jensen's inequality. Therefore
+
+$$
+\begin{array}{l} \sup  _ {\mathcal {B} \in \mathrm {A d v}} \max  _ {\mathbf {u} \in \mathcal {U}} \mathbb {E} \left[ \sum_ {t = a} ^ {b} \left(\left\langle \nabla f _ {t} (\mathbf {x} _ {t}), \mathbf {u} _ {t} - \mathbf {x} _ {t} \right\rangle - \frac {\mu}{2} \| \mathbf {u} _ {t} - \mathbf {x} _ {t} \|\right) \right] \\ \leq \sup  _ {\mathcal {B} \in \operatorname {A d v}, \theta \in \Omega^ {\mathcal {Q}}} \mathcal {R} _ {\mathcal {B} _ {\theta} ^ {\prime}} ^ {\mathcal {A}} \\ \leq \sup_{\mathcal{B}^{\prime}\in \operatorname{Adv}_{1}^{f}(\mathbf{F}_{\mu}[B_{1}])}\mathcal{R}_{\mathcal{B}^{\prime}}^{A}, \\ = \mathcal {R} _ {\operatorname {A d v} _ {1} ^ {\mathrm {f}} (\mathbf {F} _ {\mu} [ B _ {1} ])} ^ {\mathcal {A}} \\ \end{array}
+$$
+
+![](images/9a1d6cbe25b5acb9eb267a56d8cb03bf9b49030275a9e7b7fe69fb4451c5bab9.jpg)
+
+# G Proof of Theorem 12
+
+Proof.
+
+(i):
+
+We have
+
+$$
+\begin{array}{l} \mathcal {R} _ {\alpha , \operatorname {A d v} _ {1} ^ {\mathrm {f}} (\mathbf {F})} ^ {\mathcal {W} (\mathcal {A})} = \sup  _ {\mathcal {B} \in \operatorname {A d v} _ {1} ^ {\mathrm {f}} (\mathbf {F})} \mathbb {E} \left[ \max  _ {\mathbf {u} = (\mathbf {u} _ {1}, \dots , \mathbf {u} _ {T}) \in \mathcal {U}} \sum_ {t = a} ^ {b} \left(\alpha f _ {t} (\mathbf {u} _ {t}) - f _ {t} (\mathcal {W} (\mathbf {x} _ {t}))\right) \right] \\ \leq \sup  _ {\mathcal {B} \in \operatorname {A d v} _ {1} ^ {\mathrm {f}} (\mathbf {F})} \max  _ {\mathbf {u} = (\mathbf {u} _ {1}, \dots , \mathbf {u} _ {T}) \in \mathcal {U}} \mathbb {E} \left[ \sum_ {t = a} ^ {b} \beta \left(\left\langle \nabla \mathcal {W} (f _ {t}) (\mathbf {x} _ {t}), \mathbf {u} _ {t} - \mathbf {x} _ {t} \right\rangle - \frac {\mu}{2} \| \mathbf {u} _ {t} - \mathbf {x} _ {t} \|\right) \right] \\ = \beta \mathcal {R} _ {1, \mathrm {A d v} _ {1} ^ {\mathrm {f}}} ^ {\mathcal {A}} (\mathbf {H} _ {\mu , \nabla}). \\ \end{array}
+$$
+
+(ii):
+
+Since Adv is oblivious, the sequence of functions  $(f_{1},\dots ,f_{T})$  is not random and we have
+
+$$
+\begin{array}{l} \mathcal {R} _ {\alpha , \operatorname {A d v} _ {1} ^ {\circ} (\mathbf {F}, B _ {1})} ^ {\mathcal {W} (\mathcal {A})} = \sup  _ {\mathcal {B} \in \operatorname {A d v} _ {1} ^ {\circ} (\mathbf {F}, B _ {1})} \mathbb {E} \left[ \max  _ {\mathbf {u} = (\mathbf {u} _ {1}, \dots , \mathbf {u} _ {T}) \in \mathcal {U}} \sum_ {t = a} ^ {b} \left(\alpha f _ {t} (\mathbf {u} _ {t}) - f _ {t} (\mathcal {W} (\mathbf {x} _ {t}))\right) \right] \\ = \sup  _ {\mathcal {B} \in \operatorname {A d v} _ {1} ^ {\mathrm {o}} (\mathbf {F}, B _ {1})} \max  _ {\mathbf {u} = (\mathbf {u} _ {1}, \dots , \mathbf {u} _ {T}) \in \mathcal {U}} \mathbb {E} \left[ \sum_ {t = a} ^ {b} \left(\alpha f _ {t} (\mathbf {u} _ {t}) - f _ {t} (\mathcal {W} (\mathbf {x} _ {t}))\right) \right] \\ \leq \sup  _ {\mathcal {B} \in \operatorname {A d v} _ {1} ^ {\mathrm {o}} (\mathbf {F}, B _ {1})} \max  _ {\mathbf {u} = (\mathbf {u} _ {1}, \dots , \mathbf {u} _ {T}) \in \mathcal {U}} \mathbb {E} \left[ \sum_ {t = a} ^ {b} \beta \left(\langle \nabla \mathcal {W} (f _ {t}) (\mathbf {x} _ {t}), \mathbf {u} _ {t} - \mathbf {x} _ {t} \rangle - \frac {\mu}{2} \| \mathbf {u} _ {t} - \mathbf {x} _ {t} \|\right) \right] \\ = \beta \mathcal {R} _ {1, \mathrm {A d v} _ {1} ^ {\mathrm {f}} (\mathbf {Q} _ {\mu} [ B _ {1} ^ {\prime} ])}. \\ \end{array}
+$$
+
+![](images/1609515706c6d44efcd40c4e81c7770e3f4e2035f095e6a5e1f7da7945665bbd.jpg)
+
+# H Follow The Regularized Leader
+
+We start by defining the notion of self-concordant barrier.
+
+Definition 6 ([Hazan et al., 2016]). Let  $\mathcal{K} \in \mathbb{R}^d$  be a convex set with non empty interior  $\operatorname{int}(\mathcal{K})$ . We call a function  $\Phi: \operatorname{int}(\mathcal{K}) \longrightarrow \mathbb{R}$  a  $\nu$ -self-concordant barrier of  $\mathcal{K}$  if:
+
+(i)  $\Phi$  is three-times continuously differentiable, convex, and tends to infinity along any sequence of points approaching the boundary of  $\mathcal{K}$ ;  
+(ii) For every  $\mathbf{h}\in \mathbb{R}^d$  and  $\mathbf{x}\in \mathrm{int}(\mathcal{K})$ , we have:
+
+$$
+| \nabla^ {3} \Phi (\mathbf {x}) [ \mathbf {h}, \mathbf {h}, \mathbf {h} ] | \leq 2 (\nabla^ {2} \Phi (\mathbf {x}) [ \mathbf {h}, \mathbf {h} ]) ^ {3 / 2}, | \nabla \Phi (x) [ \mathbf {h} ] | \leq \nu^ {1 / 2} (\nabla^ {2} \Phi (\mathbf {x}) [ \mathbf {h}, \mathbf {h} ]) ^ {1 / 2}
+$$
+
+where the third-order differential is defined as  $\nabla^3\Phi (\mathbf{x})[\mathbf{h},\mathbf{h},\mathbf{h}]\coloneqq \frac{\partial^3}{\partial t_1\partial t_2\partial t_3}\Phi (x + t_1\mathbf{h} + t_2\mathbf{h} + t_3\mathbf{h})|_{t_1 = t_2 = t_3 = 0}$ .
+
+Next we define the notion of local norm and dual norm with respect to a self-concordant barrier.
+
+Definition 7. For every  $x \in \mathrm{int}(\mathcal{K})$ , the Hessian of the self-concordant barrier induces a local norm, denoted as  $\| \cdot \|_{\Phi ,x}$ , and a dual norm, denoted as  $\| \cdot \|_{\Phi ,\mathbf{x},*}$ , where for any  $\mathbf{v} \in \mathbb{R}^d$
+
+$$
+\| \mathbf {v} \| _ {\Phi , \mathbf {x}} = \sqrt {\mathbf {v} ^ {T} \nabla^ {2} \Phi (\mathbf {x}) \mathbf {v}}, \qquad \| \mathbf {v} \| _ {\Phi , \mathbf {x}, *} = \sqrt {\mathbf {v} ^ {T} (\nabla^ {2} \Phi (\mathbf {x})) ^ {- 1} \mathbf {v}}.
+$$
+
+An important result for FTRL is the following theorem which was proved in [Abernethy et al., 2008]. It shows that if we set the regularizer to be a self-concordant barrier of  $\mathcal{K}$  and the algorithm can access the unbiased estimator of  $\mathbf{g}_t$ , then the regret of the generated solution sequence  $\{\boldsymbol{x}_t\}_{t=1}^T$  can be bounded in terms of the local norm of the estimator.
+
+Theorem 13 ([Abernethy et al., 2008]). Let  $\mathcal{K} \subseteq \mathbb{R}^d$  be a convex set,  $\Phi(x)$  be a self-concordant barrier on  $\mathcal{K}$ ,  $\{\mathbf{g}_t\}_{t=1}^T$  be a sequence of random vectors in  $\mathbb{R}^d$ . Then running FTRL (described in Equation 3) on a vector sequence  $\{\mathbf{g}_t\}_{t=1}^T$  in  $\mathbb{R}^d$  with  $\Phi(x)$  as the regularizer will produce a sequence of points  $\{\mathbf{x}_t\}_{t=1}^T$  in  $\mathcal{K}$  where
+
+$$
+\sum_ {t = 1} ^ {T} \langle \mathbf {g} _ {t}, \mathbf {y} - \mathbf {x} _ {t} \rangle \leq \eta \sum_ {t = 1} ^ {T} \| \mathbf {g} _ {t} \| _ {\Phi , \mathbf {x} _ {t}, *} ^ {2} + \frac {\Phi (\mathbf {y}) - \Phi (\mathbf {x} _ {1})}{\eta},
+$$
+
+for any  $\mathbf{y}\in \mathcal{K}$
+
+The ellipsoid gradient estimator was proposed in [Abernethy et al., 2008], where the authors use it along with Theorem 13 to design an  $\widetilde{O} (\sqrt{T})$  regret algorithm for bandit linear optimization. For a continuous function but possibly non-smooth  $f:\mathbb{R}^d\to \mathbb{R}$  and an invertible matrix  $\Sigma \in \mathbb{R}^{d\times d}$ , we define the  $\Sigma$ -smoothed version of  $f$ .
+
+Definition 8. For function  $f(\mathbf{x}):\mathbb{R}^d\to \mathbb{R}$  and invertible matrix  $\Sigma \in \mathbb{R}^{d\times d}$ , we call  $f^{\Sigma}(\mathbf{x})$  a  $\Sigma$ -smoothed version of  $f(\mathbf{x})$ , where  $f^{\Sigma}(\mathbf{x}) = \mathbb{E}_{\mathbf{v}\sim \mathbb{B}^{d}}\left[f(\mathbf{x} + \Sigma \mathbf{v})\right]$ . Here  $\mathbf{v}\sim \mathbb{B}^d$  means that  $\mathbf{v}$  is sampled from the unit ball  $\mathbb{B}^d$  uniformly at random.
+
+There is a surprising fact that there is an unbiased estimator of  $\nabla f^{\Sigma}(\mathbf{x})$  for any  $\mathbf{x}$ , and the estimator uses only a single query to the value oracle of  $f$ .
+
+Lemma 5 ([Abernethy et al., 2008]). Let  $\Sigma \in \mathcal{R}^{d\times d}$  be an invertible matrix,  $f(\mathbf{x}): \mathbb{R}^d \to \mathbb{R}$  be an arbitrary function. Then  $\nabla f^{\Sigma}(\mathbf{x}) = d\mathbb{E}_{\mathbf{v} \sim \mathbb{S}^{d-1}}\left[f(\mathbf{x} + \Sigma \mathbf{v})\Sigma^{-1}\mathbf{v}\right]$ . Here  $\mathbf{v} \sim \mathbb{S}^{d-1}$  means that  $\mathbf{v}$  is sampled from the  $(d-1)$ -dimensional unit sphere  $\mathbb{S}^{d-1}$  uniformly at random.
+
+If  $f$  is a linear function,  $f^{\Sigma}(\mathbf{x}) = f(\mathbf{x})$ , so Lemma 5 provides a one-sample unbiased estimator of the gradient of the linear function. The ellipsoid gradient estimator is usually used along with FTRL with a self-concordant regularizer  $\Phi$  of  $\mathcal{K}$ . When the invertible matrix  $\Sigma$  is set to be  $(\nabla^{2}\Phi (\mathbf{x}))^{-1 / 2}$  and  $\mathbf{x}\in \mathrm{int}(\mathcal{K})$ , the sampled action  $\mathbf{x} + \Sigma \mathbf{v}$  is located in the surface of a so-called Dikin ellipsoid centered at  $\mathbf{x}$ , i.e.  $\{\mathbf{x}'\mid \| \mathbf{x}' - \mathbf{x}\|_{\Phi ,\mathbf{x}}\leq 1\}$ . The fact that Dikin ellipsoid is entirely contained in  $\mathcal{K}$  allows us to define  $f^{\Sigma}$  at  $\mathbf{x}$ .
+
+We finish this section with quick overview of the concept of the Minkowski function, the Minkowski set and some of their useful properties.
+
+Definition 9. Let  $\mathcal{K}$  be a compact convex set, the Minkowski function  $\pi_{\mathbf{x}}: \mathcal{K} \to \mathbb{R}$  parameterized by a pole  $\mathbf{x} \in \mathrm{int}(\mathcal{K})$  is defined as  $\pi_{\mathbf{x}}(\mathbf{y}) := \inf \{t \geq 0 \mid x + t^{-1}(y - x) \in \mathcal{K}\}$ . Given  $\delta \in \mathbb{R}^{+}$  and  $\mathbf{x}_1 \in \mathrm{int}(\mathcal{K})$ , we define the Minkowski set
+
+$$
+\mathcal {K} _ {\gamma , \mathbf {x} _ {1}} := \left\{\mathbf {x} \in \mathcal {K} \mid \pi_ {\mathbf {x} _ {1}} (\mathbf {x}) \leq (1 + \gamma) ^ {- 1} \right\}.
+$$
+
+Lemma 6 ([Abernethy et al., 2008]). Let  $\mathcal{K}$  be a compact convex set,  $\mathbf{x} \in \operatorname{int}(\mathcal{K})$  with diameter  $D$ ,  $\mathbf{u}_* \in \mathcal{K}$  and  $\hat{\mathbf{u}}_* := \operatorname{argmin}_{\mathbf{z} \in \mathcal{K}_{\gamma, \mathbf{x}}} \| \mathbf{z} - \mathbf{u}_* \|$  be the projection of  $\mathbf{u}_*$  onto the Minkowski set  $\mathcal{K}_{\gamma, \mathbf{x}}$ , then
+
+$$
+\left\| \mathbf {u} _ {*} - \hat {\mathbf {u}} _ {*} \right\| \leq \gamma D.
+$$
+
+The following lemma provides an upper bound of the difference between the function value of a self-concordant barrier at two different points.
+
+Lemma 7 ([Nesterov and Nemirovskii, 1994]). Let  $\Phi$  be a  $\nu$ -self-concordant barrier over a compact convex set  $\mathcal{K}$ , then for all  $\mathbf{x}, \mathbf{y} \in \operatorname{int}(\mathcal{K})$ :
+
+$$
+\Phi (\mathbf {y}) - \Phi (\mathbf {x}) \leq \nu \log \frac {1}{1 - \pi_ {\mathbf {x}} (\mathbf {y})}.
+$$
+
+# I Technical Lemmas
+
+This section provides some technical lemmas that will be used in the proofs later.
+
+Lemma 8. Let  $\mathcal{K}$  be a compact set and let  $f:\mathcal{K}\to \mathbb{R}^d$  be an  $M_2$ -smooth function. Then  $f$  may be extended to an  $M_2$ -smooth function  $\tilde{f}:\mathbb{R}^d\rightarrow \mathbb{R}$ .
+
+Proof. The function  $\nabla F$  is an  $M_2$ -Lipschitz function defined on  $\mathcal{K}$ . Therefore, according to Kirschbraun theorem [Kirschbraun, 1934] it may be extended to a function  $g: \mathbb{R}^d \to \mathbb{R}^d$  that is  $M_2$ -Lipschitz. Now the result follows directly from Whitney's extension theorem [Whitney, 1934].
+
+Parts (i)-(iii) of the following lemma are well-known in the literature. (See Lemma A.5 in [Wan et al., 2023] for a proof). Here we provide a proof for part (iv).
+
+Lemma 9. Following properties hold for  $\Sigma$ -smoothed version of a function  $f(\mathbf{x})$  for an invertible matrix  $\Sigma$ .
+
+(i) If  $f(\mathbf{x})$  is a monotone function, then so is  $f^{\Sigma}(\mathbf{x})$ .  
+(ii) If  $f(\mathbf{x})$  is  $M_1$ -Lipschitz, then so is  $f^{\Sigma}(\mathbf{x})$ .  
+(iii) If  $f(\mathbf{x})$  is  $M_2$ -smooth, then so is  $f^{\Sigma}(\mathbf{x})$ .  
+(iv) If  $f$  is upper-quadratic with a uniform wrapper  $\mathcal{W}$  and  $\alpha, \beta$  and  $\mu$ , then we have
+
+$$
+\alpha f ^ {\Sigma} (\mathbf {y}) - (f \circ \mathcal {W}) ^ {\Sigma} (\mathbf {x}) \leq \beta \left(\left\langle \nabla (\mathcal {W} (f)) ^ {\Sigma} (\mathbf {x}), \mathbf {y} - \mathbf {x} \right\rangle - \frac {\mu}{2} \| \mathbf {y} - \mathbf {x} \| ^ {2}\right).
+$$
+
+Proof. We have
+
+$$
+\begin{array}{l} \alpha f ^ {\Sigma} (\mathbf {y}) - (f \circ \mathcal {W}) ^ {\Sigma} (\mathbf {x}) = \mathbb {E} _ {\mathbf {v} \sim \mathbb {B} ^ {d}} \left[ \alpha f (\mathbf {y} + \Sigma \mathbf {v}) - f (\mathcal {W} (\mathbf {x} + \Sigma \mathbf {v})) \right] \\ \leq \mathbb {E} _ {\mathbf {v} \sim \mathbb {B} ^ {d}} \left[ \beta \left(\langle \nabla \mathcal {W} (f) (\mathbf {x} + \Sigma \mathbf {v}), \mathbf {y} - \mathbf {x} \rangle - \frac {\mu}{2} \| \mathbf {y} - \mathbf {x} \| ^ {2}\right) \right] \\ = \beta \left(\left\langle \mathbb {E} _ {\mathbf {v} \sim \mathbb {B} ^ {d}} [ \nabla \mathcal {W} (f) (\mathbf {x} + \Sigma \mathbf {v}) ], \mathbf {y} - \mathbf {x} \right\rangle - \frac {\mu}{2} \| \mathbf {y} - \mathbf {x} \| ^ {2}\right) \\ = \beta \left(\left\langle \nabla \mathbb {E} _ {\mathbf {v} \sim \mathbb {B} ^ {d}} \left[ \mathcal {W} (f) (\mathbf {x} + \Sigma \mathbf {v}) \right], \mathbf {y} - \mathbf {x} \right\rangle - \frac {\mu}{2} \| \mathbf {y} - \mathbf {x} \| ^ {2}\right) \\ = \beta \left(\left\langle \nabla (\mathcal {W} (f)) ^ {\Sigma} (\mathbf {x}), \mathbf {y} - \mathbf {x} \right\rangle - \frac {\mu}{2} \| \mathbf {y} - \mathbf {x} \| ^ {2}\right). \\ \end{array}
+$$
+
+Lemma 10. If  $f: \mathcal{K} \to \mathbb{R}$  is  $M_1$ -Lipschitz and  $M_2$ -smooth and  $g: \mathcal{K} \to \mathcal{K}$  is  $M_1'$ -Lipschitz and  $M_2'$ -smooth, then  $f \circ g$  is  $M_1''$ -Lipschitz and  $M_2''$ -smooth where  $M_1'' := M_1M_1'$  and  $M_2'' := M_1M_2' + M_2M_1'^2$ .
+
+Proof. We have
+
+$$
+\left\| D (f \circ g) (\mathbf {x}) \right\| = \left\| D f (g (\mathbf {x})) \cdot D g (\mathbf {x}) \right\| \leq M _ {1} M _ {1} ^ {\prime},
+$$
+
+and therefore for all  $\mathbf{x},\mathbf{y}\in \mathcal{K}$  , we have
+
+$$
+\begin{array}{l} \| D (f \circ g) (\mathbf {x}) - D (f \circ g) (\mathbf {y}) \| = \| D f (g (\mathbf {x})) \cdot D g (\mathbf {x}) - D f (g (\mathbf {y})) \cdot D g (\mathbf {y}) \| \\ \leq \left\| D f (g (\mathbf {x})) \cdot D g (\mathbf {x}) - D f (g (\mathbf {x})) \cdot D g (\mathbf {y}) \right\| \\ + \left\| D f (g (\mathbf {x})) \cdot D g (\mathbf {y}) - D f (g (\mathbf {y})) \cdot D g (\mathbf {y}) \right\| \\ = \| D f (g (\mathbf {x})) \| \| D g (\mathbf {x}) - D g (\mathbf {y}) \| \\ + \| D f (g (\mathbf {x})) - D f (g (\mathbf {y})) \| \| D g (\mathbf {y}) \| \\ \leq M _ {1} M _ {2} ^ {\prime} \| \mathbf {x} - \mathbf {y} \| + M _ {2} M _ {1} ^ {\prime} \| g (\mathbf {x}) - g (\mathbf {y}) \| \\ \leq \left(M _ {1} M _ {2} ^ {\prime} + M _ {2} M _ {1} ^ {\prime 2}\right) \| \mathbf {x} - \mathbf {y} \|. \\ \end{array}
+$$
+
+# J Proof of Theorem 3
+
+Proof. We have
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ f _ {t} (\mathbf {u} _ {*}) - f _ {t} (\mathbf {x} _ {t}) \right] \\ = \sum_ {t = 1} ^ {T} \mathbb {E} \left[ f _ {t} ^ {\delta \Sigma_ {t}} \left(\hat {\mathbf {u}} _ {*}\right) - f _ {t} ^ {\delta \Sigma_ {t}} \left(\mathbf {x} _ {t}\right) \right] + \underbrace {\sum_ {t = 1} ^ {T} \mathbb {E} \left[ f _ {t} ^ {\delta \Sigma_ {t}} \left(\mathbf {u} _ {*}\right) - f _ {t} ^ {\delta \Sigma_ {t}} \left(\hat {\mathbf {u}} _ {*}\right) \right]} _ {(A)} \tag {4} \\ + \underbrace {\sum_ {t = 1} ^ {T} \mathbb {E} \left[ f _ {t} (\mathbf {u} _ {*}) - f _ {t} ^ {\delta \Sigma_ {t}} (\mathbf {u} _ {*}) \right]} _ {(B)} + \underbrace {\sum_ {t = 1} ^ {T} \mathbb {E} \left[ f _ {t} ^ {\delta \Sigma_ {t}} (\mathbf {x} _ {t}) - f _ {t} (\mathbf {x} _ {t}) \right]} _ {(C)} \\ \end{array}
+$$
+
+Note that, for the terms above to be well-defined, we need to be able to define  $f_{t}^{\delta \Sigma_{t}}$  over  $\mathcal{K}$  which requires computing  $f_{t}$  over a set that is slightly larger than  $\mathcal{K}$ . Using Lemma 8, we assume that all functions  $f_{t}$  are well-defined and  $M_2$ -smooth over  $\mathbb{R}^d$ .
+
+Bounding (A): Since  $f_{t}(\mathbf{x})$  is  $M_1$ -Lipschitz continuous,  $f_{t}^{\delta \Sigma_{t}}$  is also  $M_1$ -Lipschitz continuous by Lemma 9. Since  $\| \hat{\mathbf{u}}_* - \mathbf{u}_* \| \leq \gamma D$  by Lemma 6,
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ f _ {t} ^ {\delta \Sigma_ {t}} \left(\mathbf {u} _ {*}\right) - f _ {t} ^ {\delta \Sigma_ {t}} \left(\hat {\mathbf {u}} _ {*}\right) \right] \leq \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \left| f _ {t} ^ {\delta \Sigma_ {t}} \left(\hat {\mathbf {u}} _ {*}\right) - f _ {t} ^ {\delta \Sigma_ {t}} \left(\mathbf {u} _ {*}\right) \right| \right] \tag {5} \\ \leq \sum_ {t = 1} ^ {T} M _ {1} \gamma D = M _ {1} \gamma D T. \\ \end{array}
+$$
+
+Bounding (B): Since  $f_{t}(\mathbf{x})$  is  $M_2$ -smooth, by Lemma 9,  $f_{t}^{\delta \Sigma_{t}}$  is  $M_2$ -smooth. Thus,
+
+$$
+\begin{array}{l} f _ {t} \left(\mathbf {u} _ {*}\right) - f _ {t} ^ {\delta \Sigma_ {t}} \left(\mathbf {u} _ {*}\right) = \mathbb {E} _ {\mathbf {v} \sim \mathbb {B} ^ {d}} \left[ f _ {t} \left(\mathbf {u} _ {*}\right) - f _ {t} \left(\mathbf {u} _ {*} + \delta \Sigma_ {t} \mathbf {v}\right) \right] \\ \leq \mathbb {E} _ {\mathbf {v} \sim \mathbb {B} ^ {d}} \left[ - \langle \nabla f _ {t} (\mathbf {u} _ {*}), \delta \Sigma_ {t} \mathbf {v} \rangle + \frac {M _ {2}}{2} \| \delta \Sigma_ {t} \mathbf {v} \| ^ {2} \right] \\ = \mathbb {E} _ {\mathbf {v} \sim \mathbb {B} ^ {d}} \left[ - \langle \nabla f _ {t} (\mathbf {u} _ {*}), \delta \Sigma_ {t} \mathbf {v} \rangle \right] + \mathbb {E} _ {\mathbf {v} \sim \mathbb {B} ^ {d}} \left[ \frac {M _ {2}}{2} \| \delta \Sigma_ {t} \mathbf {v} \| ^ {2} \right] \\ = \mathbb {E} _ {\mathbf {v} \sim \mathbb {B} ^ {d}} \left[ \frac {M _ {2}}{2} \| \delta \Sigma_ {t} \mathbf {v} \| ^ {2} \right] \\ \leq \frac {M _ {2} \delta^ {2} D ^ {2}}{2}. \\ \end{array}
+$$
+
+Note that in the last inequality, we used the fact that the Dikin ellipsoid centered at  $\mathbf{x}_t$  is contained in  $\kappa$  which implies that  $\mathbf{x}_t + \Sigma_t\mathbf{v} \in \kappa$  and therefore  $\| \Sigma_t\mathbf{v}\| \leq D$ . It follows that,
+
+$$
+\sum_ {t = 1} ^ {T} \mathbb {E} \left[ f _ {t} \left(\hat {\mathbf {u}} _ {*}\right) - f _ {t} ^ {\delta \Sigma_ {t}} \left(\hat {\mathbf {u}} _ {*}\right) \right] \leq \frac {M _ {2} \delta^ {2} D ^ {2} T}{2}. \tag {6}
+$$
+
+Bounding  $(C)$  : Similarly,
+
+$$
+\begin{array}{l} f _ {t} ^ {\delta \Sigma_ {t}} (\mathbf {x} _ {t}) - f _ {t} (\mathbf {x} _ {t}) = \mathbb {E} _ {\mathbf {v} \sim \mathbb {B} ^ {d}} \left[ f _ {t} (\mathbf {x} _ {t} + \delta \Sigma_ {t} \mathbf {v}) - f _ {t} (\mathbf {x} _ {t}) \right] \\ \leq \mathbb {E} _ {\mathbf {v} \sim \mathbb {B} ^ {d}} \left[ - \left\langle \nabla f _ {t} (\mathbf {x} _ {t}), \delta \Sigma_ {t} \mathbf {v} \right\rangle + \frac {M _ {2}}{2} \| \delta \Sigma_ {t} \mathbf {v} \| ^ {2} \right] \leq \frac {M _ {2} \delta^ {2} D ^ {2}}{2}. \\ \end{array}
+$$
+
+Therefore,
+
+$$
+\sum_ {t = 1} ^ {T} \mathbb {E} \left[ f _ {t} ^ {\delta \Sigma_ {t}} \left(\mathbf {x} _ {t}\right) - f _ {t} \left(\mathbf {x} _ {t}\right) \right] \leq \frac {M _ {2} \delta^ {2} D ^ {2} T}{2} \tag {7}
+$$
+
+Putting 5,6,7 in 4, we see that
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ f _ {t} \left(\mathbf {u} _ {*}\right) - f _ {t} \left(\mathbf {x} _ {t}\right) \right] \leq \sum_ {t = 1} ^ {T} \mathbb {E} \left[ f _ {t} ^ {\delta \Sigma_ {t}} \left(\hat {\mathbf {u}} _ {*}\right) - f _ {t} ^ {\delta \Sigma_ {t}} \left(\mathbf {x} _ {t}\right) \right] \\ + \alpha M _ {1} \gamma D T + \frac {M _ {2} \delta^ {2} D ^ {2} T}{2} + \frac {M _ {2} \delta^ {2} D ^ {2} T}{2}, \\ \end{array}
+$$
+
+which completes the proof of the first claim.
+
+To prove the second claim, we first use Lemma 5, with  $\Sigma = \delta \Sigma_{t}$ , to see that  $\mathbb{E}\left[\mathbf{o}_t\mid \mathbf{x}_t\right] = \nabla f_t^{\delta \Sigma_t}(\mathbf{x}_t)$ . On the other hand, since  $\mathcal{Q}_t$  is bounded by  $B_0$ , we have
+
+$$
+\| \mathbf {o} _ {t} \| _ {\mathbf {x} _ {t}, *} ^ {2} = \left\| \frac {d}{\delta} y _ {t} \Sigma_ {t} ^ {- 1} \mathbf {v} _ {t} \right\| _ {\mathbf {x} _ {t}, *} ^ {2} = \frac {d ^ {2}}{\delta^ {2}} | y _ {t} | ^ {2} \mathbf {v} _ {t} ^ {T} \Sigma_ {t} ^ {- 1} \left(\nabla^ {2} \Phi (\mathbf {x} _ {t})\right) ^ {- 1} \Sigma_ {t} ^ {- 1} \mathbf {v} _ {t} \leq \frac {d ^ {2}}{\delta^ {2}} B _ {0} ^ {2} \| \mathbf {v} _ {t} \| ^ {2} \leq \frac {d ^ {2} B _ {0} ^ {2}}{\delta^ {2}}
+$$
+
+Hence, using Theorem 13 with  $\mathbf{g}_t = \mathbf{o}_t$  and  $\mathbf{y} = \hat{\mathbf{u}}_*$ , we see that
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \left\langle \nabla f _ {t} ^ {\delta \Sigma_ {q}} \left(\mathbf {x} _ {t}\right), \hat {\mathbf {u}} _ {*} - \mathbf {x} _ {t} \right\rangle \right] = \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \left\langle \mathbb {E} \left[ \mathbf {o} _ {t} \mid \mathbf {x} _ {t} \right], \hat {\mathbf {u}} _ {*} - \mathbf {x} _ {t} \right\rangle \right] \\ = \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \mathbb {E} \left[ \langle \mathbf {o} _ {t}, \hat {\mathbf {u}} _ {*} - \mathbf {x} _ {t} \rangle \mid \mathbf {x} _ {t} \right] \right] \\ = \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \langle \mathbf {o} _ {t}, \hat {\mathbf {u}} _ {*} - \mathbf {x} _ {t} \rangle \right] \\ \leq \mathbb {E} \left[ \eta \sum_ {t = 1} ^ {T} \| \mathbf {o} _ {t} \| _ {\Phi , \mathbf {x} _ {t}, *} ^ {2} + \frac {\Phi (\hat {\mathbf {u}} _ {*}) - \Phi (\mathbf {x} _ {1})}{\eta} \right] \\ \leq \eta \sum_ {t = 1} ^ {T} \frac {d ^ {2} B _ {0} ^ {2}}{\delta^ {2}} + \frac {\Phi (\hat {\mathbf {u}} _ {*}) - \Phi (\mathbf {x} _ {1})}{\eta} \\ \leq \frac {\eta d ^ {2} B _ {0} ^ {2} T}{\delta^ {2}} + \frac {\nu \log (\frac {1}{1 - (1 + \gamma) ^ {- 1}})}{\eta}, \\ \end{array}
+$$
+
+where we used Lemma 7 in the last inequality.
+
+![](images/6dc188f8ee6bc7bc955bda8041ec7a0baf697e2eeb51a547809875e2b386bb87.jpg)
+
+# K Proof of Theorem 4
+
+Proof. Let  $\mathcal{B} \in \mathrm{Adv}$  be a realized adversary and let  $f_1, \dots, f_T$  be the sequence of functions selected by  $\mathcal{B}$ . Also let  $\mathbf{u}_* \in \operatorname{argmax}_{\mathbf{u} \in \mathcal{K}} \sum_{t=1}^{T} f_t(\mathbf{u})$  and  $\hat{\mathbf{u}}_* \in \operatorname{argmin}_{\mathbf{x} \in \mathcal{K}_{\gamma, \mathbf{x}_1}} \| \mathbf{u}_* - \mathbf{x} \|$  where  $\gamma = T^{-1}$ . We have
+
+$$
+\begin{array}{l} \mathcal {R} _ {\alpha , \mathcal {B}} ^ {\mathcal {W} (\mathrm {Z O - F T R L})} = \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \alpha f _ {t} \left(\mathbf {u} _ {*}\right) - f _ {t} \left(\mathcal {W} \left(\mathbf {x} _ {t}\right)\right) \right] \\ = \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \alpha f _ {t} ^ {\delta \Sigma_ {t}} (\hat {\mathbf {u}} _ {*}) - (f _ {t} \circ \mathcal {W}) ^ {\delta \Sigma_ {t}} (\mathbf {x} _ {t}) \right] + \alpha \underbrace {\sum_ {t = 1} ^ {T} \mathbb {E} \left[ f _ {t} ^ {\delta \Sigma_ {t}} (\mathbf {u} _ {*}) - f _ {t} ^ {\delta \Sigma_ {t}} (\hat {\mathbf {u}} _ {*}) \right]} _ {(A)} \\ + \alpha \underbrace {\sum_ {t = 1} ^ {T} \mathbb {E} \left[ f _ {t} (\mathbf {u} _ {*}) - f _ {t} ^ {\delta \Sigma_ {t}} (\mathbf {u} _ {*}) \right]} _ {(B)} + \underbrace {\sum_ {t = 1} ^ {T} \mathbb {E} \left[ (f _ {t} \circ \mathcal {W}) ^ {\delta \Sigma_ {t}} (\mathbf {x} _ {t}) - f _ {t} (\mathcal {W} (\mathbf {x} _ {t})) \right]} _ {(C)} \\ \end{array}
+$$
+
+As in the proof of Theorem 3, we use Lemma 8 to extend all functions  $f_{t}$  to  $M_2$ -smooth functions over  $\mathbb{R}^d$  and we bound the terms (A) and (B) by  $M_1\gamma DT$  and  $\frac{M_2\delta^2D^2T}{2}$ , respectively. To bound (C),
+
+we first use Lemma 10 to see that  $f_{t} \circ \mathcal{W}$  is  $M_2^{\prime \prime}$ -smooth, where  $M_2^{\prime \prime} = M_1M_2^{\prime} + M_2M_1^{\prime 2}$ . Hence, we see that
+
+$$
+\begin{array}{l} \left(f _ {t} \circ \mathcal {W}\right) ^ {\delta \Sigma_ {t}} \left(\mathbf {x} _ {t}\right) - f _ {t} \left(\mathcal {W} \left(\mathbf {x} _ {t}\right)\right) = \mathbb {E} _ {\mathbf {v} \sim \mathbb {B} ^ {d}} \left[ f _ {t} \left(\mathcal {W} \left(\mathbf {x} _ {t} + \delta \Sigma_ {t} \mathbf {v}\right)\right) - f _ {t} \left(\mathcal {W} \left(\mathbf {x} _ {t}\right)\right) \right] \\ \leq \mathbb {E} _ {\mathbf {v} \sim \mathbb {B} ^ {d}} \left[ - \left\langle \nabla f _ {t} (\mathcal {W} (\mathbf {x} _ {t})), \delta \Sigma_ {t} \mathbf {v} \right\rangle + \frac {M _ {2} ^ {\prime \prime}}{2} \| \delta \Sigma_ {t} \mathbf {v} \| ^ {2} \right] \leq \frac {M _ {2} ^ {\prime \prime} \delta^ {2} D ^ {2}}{2}. \\ \end{array}
+$$
+
+Therefore,
+
+$$
+\sum_ {t = 1} ^ {T} \mathbb {E} \left[ \left(f _ {t} \circ \mathcal {W}\right) ^ {\delta \Sigma_ {t}} \left(\mathbf {x} _ {t}\right) - f _ {t} \left(\mathcal {W} \left(\mathbf {x} _ {t}\right)\right) \right] \leq \frac {M _ {2} ^ {\prime \prime} \delta^ {2} D ^ {2} T}{2}
+$$
+
+Putting the bounds for (A), (B) and (C) together, we see that
+
+$$
+\begin{array}{l} \mathcal {R} _ {\alpha , \mathcal {B}} ^ {\mathcal {W} (\mathrm {Z O - F T R L})} = \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \alpha f _ {t} \left(\mathbf {u} _ {*}\right) - f _ {t} \left(\mathcal {W} \left(\mathbf {x} _ {t}\right)\right) \right] \\ \leq \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \alpha f _ {t} ^ {\delta \Sigma_ {t}} (\hat {\mathbf {u}} _ {*}) - \left(f _ {t} \circ \mathcal {W}\right) ^ {\delta \Sigma_ {t}} \left(\mathbf {x} _ {t}\right) \right] + \alpha M _ {1} \gamma D T + \frac {\left(\alpha M _ {2} + M _ {2} ^ {\prime \prime}\right) \delta^ {2} D ^ {2} T}{2} \\ \leq \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \beta \left\langle \nabla \left(\mathcal {W} \left(f _ {t}\right)\right) ^ {\delta \Sigma_ {t}} \left(\mathbf {x} _ {t}\right), \hat {\mathbf {u}} _ {*} - \mathbf {x} _ {t} \right\rangle \right] + \alpha M _ {1} \gamma D T + \frac {\left(\alpha M _ {2} + M _ {2} ^ {\prime \prime}\right) \delta^ {2} D ^ {2} T}{2}, \tag {8} \\ \end{array}
+$$
+
+where the second inequality follows from Lemma 9.
+
+To bound the remaining term, we use an argument similar to the one used in the proof of Theorem 3 again. Using Lemma 5 with  $\Sigma = \delta \Sigma_{t}$  and the fact that  $y_{t}$  is an unbiased sample of  $\mathcal{W}(f_t)$  at  $\mathbf{x}_t + \delta \Sigma_t\mathbf{v}_t$ , we see that  $\mathbb{E}\left[\mathbf{o}_t\mid \mathbf{x}_t\right] = \nabla (\mathcal{W}(f_t))^{\delta \Sigma_t}(\mathbf{x}_t)$ . On the other hand, since  $\mathcal{W}(\mathcal{Q}_t)$  is bounded by  $B_0$ , we have  $|y_{t}|\leq B_{0}$ , which implies that
+
+$$
+\left\| \mathbf {o} _ {t} \right\| _ {\mathbf {x} _ {t}, *} ^ {2} = \left\| \frac {d}{\delta} y _ {t} \Sigma_ {t} ^ {- 1} \mathbf {v} _ {t} \right\| _ {\mathbf {x} _ {t}, *} ^ {2} = \frac {d ^ {2}}{\delta^ {2}} \left| y _ {t} \right| ^ {2} \mathbf {v} _ {t} ^ {T} \Sigma_ {t} ^ {- 1} \left(\nabla^ {2} \Phi (\mathbf {x} _ {t})\right) ^ {- 1} \Sigma_ {t} ^ {- 1} \mathbf {v} _ {t} \leq \frac {d ^ {2}}{\delta^ {2}} B _ {0} ^ {2} \| \mathbf {v} _ {t} \| ^ {2} \leq \frac {d ^ {2} B _ {0} ^ {2}}{\delta^ {2}}.
+$$
+
+Hence, using Theorem 13 with  $\mathbf{g}_t = \mathbf{o}_t$  and  $\mathbf{y} = \hat{\mathbf{u}}_*$ , we see that
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \beta \langle \nabla (\mathcal {W} (f _ {t})) ^ {\delta \Sigma_ {t}} (\mathbf {x} _ {t}), \hat {\mathbf {u}} _ {*} - \mathbf {x} _ {t} \rangle \right] = \beta \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \langle \mathbb {E} [ \mathbf {o} _ {t} | \mathbf {x} _ {t} ], \hat {\mathbf {u}} _ {*} - \mathbf {x} _ {t} \rangle \right] \\ = \beta \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \mathbb {E} \left[ \left\langle \mathbf {o} _ {t}, \hat {\mathbf {u}} _ {*} - \mathbf {x} _ {t} \right\rangle \mid \mathbf {x} _ {t} \right] \right] \\ = \beta \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \langle \mathbf {o} _ {t}, \hat {\mathbf {u}} _ {*} - \mathbf {x} _ {t} \rangle \right] \\ \leq \beta \mathbb {E} \left[ \eta \sum_ {t = 1} ^ {T} \| \mathbf {o} _ {t} \| _ {\Phi , \mathbf {x} _ {t}, *} ^ {2} + \frac {\Phi (\hat {\mathbf {u}} _ {*}) - \Phi (\mathbf {x} _ {1})}{\eta} \right] \\ \leq \beta \eta \sum_ {t = 1} ^ {T} \frac {d ^ {2} B _ {0} ^ {2}}{\delta^ {2}} + \beta \frac {\Phi (\hat {\mathbf {u}} _ {*}) - \Phi (\mathbf {x} _ {1})}{\eta} \\ \leq \frac {\beta \eta d ^ {2} B _ {0} ^ {2} T}{\delta^ {2}} + \frac {\beta \nu \log \left(\frac {1}{1 - (1 + \gamma) ^ {- 1}}\right)}{\eta}, \\ \end{array}
+$$
+
+where we used Lemma 7 in the last inequality. Plugging this into Equation 8 and using  $M_2'' = M_1M_2' + M_2M_1'^2$  and  $\gamma = T^{-1}$ , we see that
+
+$$
+\begin{array}{l} \mathcal {R} _ {\alpha , \mathcal {B}} ^ {\mathcal {W} (\mathrm {Z O - F T R L})} \leq \frac {\beta \eta d ^ {2} B _ {0} ^ {2} T}{\delta^ {2}} + \frac {\beta \nu \log \left(\frac {1}{1 - (1 + \gamma) ^ {- 1}}\right)}{\eta} \\ + \alpha M _ {1} \gamma D T + \frac {\left(\alpha M _ {2} + M _ {1} M _ {2} ^ {\prime} + M _ {2} M _ {1} ^ {\prime 2}\right) \delta^ {2} D ^ {2} T}{2} \\ = O \left(\eta \delta^ {- 2} T + \eta^ {- 1} \log T + \delta^ {2} T\right). \\ \end{array}
+$$
+
+# L Proof of Theorem 5
+
+Proof. Note that in all three cases,  $\mathcal{W}^{\mathrm{action}}$  is 1-Lipschitz and 0-smooth. Now the result for the first case follows immediately from the fact that  $\mathcal{W}^{\mathrm{M}} = \mathrm{Id}$ . Also note that for any zeroth order query oracle  $Q_{f}$  for a function  $f\in \mathbf{F}^{\mathrm{M}0}$  and any  $\mathbf{y}\in \mathcal{K}$
+
+$$
+\left| \mathcal {W} ^ {\mathrm {M} 0} \left(\mathcal {Q} _ {f}\right) (\mathbf {y}) \right| = \left| z ^ {- 1} \mathcal {Q} _ {f} (z * \mathbf {y}) \right| \leq z ^ {- 1} \cdot C \| z * \mathbf {y} \| = \| \mathbf {y} \| \leq D.
+$$
+
+Thus the query oracle  $\mathcal{W}(\mathcal{Q}_f)$  is bounded by  $D$  and the assumptions of Theorem 4 are satisfied. The proof of boundedness of  $\mathcal{W}^{\mathrm{NM}}(\mathcal{Q}_f)$  for any  $f\in \mathbf{F}^{\mathrm{NM}}$  is similar.
+
+# M Stochastic Full-information To Trivial query - SFTT
+
+In this section, we discuss the SFTT meta-algorithm (Algorithm 4 in [Pedramfar and Aggarwal, 2024a]) which converts algorithms that require full-information feedback into algorithms that have a trivial query oracle. In particular, it converts algorithms that require zeroth-order full-information feedback into bandit algorithms.
+
+We say a function class  $\mathbf{F}$  is closed under convex combination if for any  $f_{1},\dots ,f_{k}\in \mathbf{F}$  and any  $\delta_1,\dots ,\delta_k\geq 0$  with  $\sum_{i}\delta_{i} = 1$ , we have  $\sum_{i}\delta_{i}f_{i}\in \mathbf{F}$ .
+
+Theorem 14 (Theorem 7 and Remark 1 and Corollary 6 in [Pedramfar and Aggarwal, 2024a]). Let  $\mathcal{A}$  be an online optimization algorithm with full-information feedback and with  $K$  queries at each time-step where  $\mathcal{A}^{query}$  does not depend on the observations in the current round and  $\mathcal{A}' = \mathrm{SFTT}(\mathcal{A})$ . Then, for any  $M_1$ -Lipschitz function class  $\mathbf{F}$  that is closed under convex combination and any  $B_1 \geq M_1$ ,  $0 < \alpha \leq 1$  and  $1 \leq a \leq b \leq T$ , let  $a' = \lfloor (a - 1) / L \rfloor + 1$ ,  $b' = \lceil b / L \rceil$ ,  $D = \mathrm{diam}(\mathcal{K})$  and let  $\{T\}$  and  $\{T / L\}$  denote the horizon of the adversary. If we also have  $\mathcal{R}_{\alpha, \mathrm{Adv}_i^o (\mathbf{F}, B)}^{\mathcal{A}'}(\mathcal{K}_\star^T)[a, b] = O(BT^\eta)$ ,  $K = O(1)$  and  $L = O(T^{\frac{1 - \eta}{2 - \eta}})$ , then
+
+$$
+\mathcal {R} _ {\alpha , \mathrm {A d v} _ {i} ^ {o} (\mathbf {F}, B)} ^ {\mathcal {A} ^ {\prime}} (\mathcal {K} _ {\star} ^ {T}) [ a, b ] = O \left(B T ^ {\frac {1}{2 - \eta}}\right).
+$$
+
+More generally, the above result holds even if the query oracles are not bounded. Specifically, what we require is that the set of query oracles to be closed under convex combinations.
+
+Algorithm 6: Stochastic Full-information To Trivial query - SFTT  $(\mathcal{A})$  
+Input: base algorithm  $\mathcal{A}$  , horizon  $T$  , block size  $L > K$    
+for  $q = 1,2,\dots ,T / L$  do Let  $\hat{\mathbf{x}}_q$  be the action chosen by  $\mathcal{A}^{\mathrm{action}}$  Let  $(\hat{\mathbf{y}}_q^i)_{i = 1}^K$  be the queries selected by  $\mathcal{A}^{\mathrm{query}}$  Let  $(t_{q,1},\ldots ,t_{q,L})$  be a random permutation of  $\{(q - 1)L + 1,\dots ,qL\}$  for  $t = (q - 1)L + 1,\ldots ,qL$  do if  $t = t_{q,i}$  for some  $1\le i\le K$  then Play the action  $\mathbf{x}_t = \hat{\mathbf{y}}_q^i$  Return the observation to the query oracle as the response to the  $i$  -th query else Play the action  $\mathbf{x}_t = \hat{\mathbf{x}}_q$  end end
+
+# N Proof of Theorem 6
+
+Proof. All three class of functions considered are closed under convex combination. Therefore we may directly apply Theorems 5 and 14 to obtain this result for the first case.
+
+For any sequence of functions  $f_{1}, \dots, f_{k}$  and query oracles  $\mathcal{Q}_{1}, \dots, \mathcal{Q}_{k}$  for these functions that are contained within a cone  $\operatorname{Cone}(\mathbf{0}, C)$  and non-negative numbers  $\delta_{1}, \dots, \delta_{k}$  such that  $\sum_{i} \delta_{i} = 0$ , the query oracle  $\overline{Q}$  that uses  $\mathcal{Q}_{i}$  with probability  $\delta_{i}$  is trivially a query oracle for  $\sum_{i} \delta_{i} f_{i}$  that is also contained within this cone. Therefore, we may apply Theorem 14 to obtain this result for the second case as well. The proof of the last case is similar.
+
+# O Proof of Theorem 7
+
+First we state the following simple result about OTB.
+
+Theorem 15 (Theorem 8 in [Pedramfar and Aggarwal, 2024a]). If  $\mathcal{A}$  is an online algorithm that queries no more than  $K = T^{\theta}$  times per time-step and obtains an  $\alpha$ -regret bound of  $O(T^{\delta})$ , then the sample complexity of  $\mathrm{OTB}(\mathcal{A})$  is  $\Omega(\epsilon^{-\frac{1 + \theta}{1 - \delta}})$ .
+
+Proof of Theorem 7. This is an immediate corollary of Theorem 5 and the guarantees for the OTB meta-algorithm stated in Theorem 15.  $\square$
+
+# NeurIPS Paper Checklist
+
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+
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+
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+
+Guidelines:
+
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+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
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+
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+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
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+
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+
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+
+# 4. Experimental result reproducibility
+
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+
+Answer: [NA]
+
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+
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+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
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+
+# 5. Open access to data and code
+
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+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
\ No newline at end of file
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+# Unifying Appearance Codes and Bilateral Grids for Driving Scene Gaussian Splatting
+
+Nan Wang $^{1}$ , Yuantao Chen $^{1}$ , Lixing Xiao $^{1}$ , Weiqing Xiao $^{1}$ , Bohan Li $^{3,4}$ , Zhaoxi Chen $^{1}$ , Chongjie Ye $^{1}$ , Shaocong Xu $^{1}$ , Saining Zhang $^{1}$ , Ziyang Yan $^{1}$ , Pierre Merriaux $^{6}$ , Lei Lei $^{6}$ , Tianfan Xue $^{5}$ , Hao Zhao $^{1,2‡}$
+
+$^{1}$ BAAI;  $^{2}$ AIR, THU;  $^{3}$ SJTU;  $^{4}$ EIT(Ningbo);  $^{5}$ CUHK;  $^{6}$ LeddarTech
+
+![](images/0a62bbec9ab6c5c3c24592033831e582759261031eb76742b8e0f418624c4be6.jpg)  
+Figure 1: (a) Appearance codes rely on global affine transformations but have limited modeling capability. (b) Bilateral grids perform pixel-wise transformations, improving photometric consistency but are challenging to optimize. (c) The proposed multi-scale bilateral grid unifies appearance codes and bilateral grids, enabling patch-wise transformations.
+
+# Abstract
+
+Neural rendering techniques, including NeRF and Gaussian Splatting (GS), rely on photometric consistency to produce high-quality reconstructions. However, in real-world driving scenarios, it is challenging to guarantee perfect photometric consistency in acquired images. Appearance codes have been widely used to address this issue, but their modeling capability is limited, as a single code is applied to the entire image. Recently, the bilateral grid was introduced to perform pixel-wise color mapping, but it is difficult to optimize and constrain effectively. In this paper, we propose a novel multi-scale bilateral grid that unifies appearance codes and bilateral grids. We demonstrate that this approach significantly improves geometric accuracy in dynamic, decoupled autonomous driving scene reconstruction, outperforming both appearance codes and bilateral grids. This is crucial for autonomous driving, where accurate geometry is important for obstacle avoidance and control. Our method shows strong results across four datasets: Waymo, NuScenes, Argoverse, and PandaSet. We further demonstrate that the improvement in geometry is driven by the multi-scale bilateral grid, which effectively reduces floaters caused by photometric inconsistency. Our code is open-sourced at https://bigcileng.github.io/bilateral-driving.
+
+# 1 Introduction
+
+Neural rendering techniques, such as NeRFs [24, 37, 54, 20, 55] and Gaussian Splitting (GS) [14, 2, 6, 57, 29, 53, 28], have demonstrated significant potential in producing high-quality 3D reconstructions by leveraging photometric consistency. However, ensuring photometric consistency across images in real-world scenarios remains a challenge, as lighting conditions, viewpoints, and camera settings can vary considerably [23, 56, 15, 36, 8]. In autonomous driving applications [46, 41, 35, 5, 17, 13, 34], these challenges are amplified due to the presence of multiple cameras capturing the same scene from different angles over time. As illustrated in the left half of Teaser. 1, we observe that these variations—whether temporal (across different time steps) or inter-camera (across different viewpoints)—can introduce significant discrepancies in image appearance. These inconsistencies pose difficulties in accurately reconstructing dynamic driving environments.
+
+To address this issue, appearance codes [35, 23, 5] have been introduced to encode per-image information and assist in correcting the photometric discrepancies. These codes, while effective, have limitations in modeling capability since they apply a single transformation across the entire image, often neglecting local variations that may occur in complex scenes. For example, in dynamic scenes, such as those encountered in autonomous driving environments, lighting and viewpoints can change across different camera angles and time frames. As shown in the right top of Teaser. 1, a global affine transformation based on a single appearance code may not be sufficient to handle such variations (In Teaser. 1 NuScenes (a), the fence is blurred).
+
+Recent advancements [38, 10] have introduced bilateral grids to perform pixel-wise transformations, enabling improved photometric consistency by allowing more localized adjustments. However, these grids face significant challenges in optimization, as they require complex constraints to avoid instability during training (As shown in Teaser. 1 (b), bilateral grids fail to effectively optimize the reconstruction of large, complex scenes.). In this work, we propose a novel solution—a multiscale bilateral grid—integrating the strengths of appearance codes and bilateral grids. As shown in Teaser. 1, this new approach allows for patch-wise transformations. Interestingly, in extreme cases, the multi-scale bilateral grid naturally converges to either the bilateral grid or appearance code, depending on the scale. At the finest scale, the multi-scale grid behaves like a traditional bilateral grid, performing pixel-wise transformations to adjust for local variations. In contrast, at the coarsest scale, it effectively reverts to a more global transformation, resembling the behavior of appearance codes. This flexible, scale-dependent approach offers the best of both worlds, providing localized fine-tuning where necessary, while maintaining global consistency when appropriate.
+
+We demonstrate that the multi-scale bilateral grid significantly enhances the geometric accuracy of dynamic, decoupled driving scenes, which is essential for autonomous driving applications where precise geometry plays a crucial role in tasks like obstacle avoidance and path planning. This is evidenced by our evaluation on multiple widely used datasets, including Waymo [30], NuScenes [1], Argoverse [40], and PandaSet [42]. The corresponding Chamfer Distance (CD) values, displayed in the Teaser. 1, highlight a marked reduction in error and improvement in reconstruction quality. Additionally, our method reduces photometric inconsistency, mitigating the appearance of floaters and enhancing the overall realism of the scene.
+
+The contributions of this paper are as follows: First, we introduce a novel multi-scale bilateral grid that unifies appearance codes and bilateral grids, transitioning to either of the two paradigms in extreme cases, thus enhancing both modeling capability and optimization efficiency. Second, we show that by addressing photometric inconsistencies, it improves the geometric accuracy of dynamic, decoupled driving scene reconstructions. Third, we provide extensive benchmarking across four widely used datasets—Waymo, NuScenes, Argoverse, and PandaSet—where our method outperforms previous approaches, showcasing notable improvements in both qualitative and quantitative results.
+
+# 2 Related Works
+
+# 2.1 Bilateral Grids and Appearance Codes
+
+Recent advancements in bilateral grids and appearance codes have significantly influenced techniques for addressing photometric inconsistencies, which are particularly critical in dynamic environments such as autonomous driving. The bilateral grids, first introduced for real-time edge-aware image processing [4], enable efficient manipulation of spatial and intensity variations, forming the foundation
+
+![](images/76712b84aa6c117663d60c33f592d045860ca0146fda6ceb2ba954820d27321b.jpg)  
+Figure 2: Overview of our method. We unify appearance codes with multi-scale bilateral grids. Initially, a coarse rendering is obtained from a Gaussian scene graph. This rendered image is then processed by our multi-scale bilateral grids to perform detailed per-pixel color modeling, guided by a luminance-based map through slice and fusion operations.
+
+for many modern techniques [10, 22, 3, 44, 58]. Extensions like deep bilateral learning [10] further enhance this framework by predicting local affine transformations in bilateral space, achieving real-time image enhancement on resource-constrained devices. Similarly, bilateral-guided upsampling [3] and cost volume refinement [44] have proven effective in tasks ranging from high-dynamic-range imaging to stereo matching, demonstrating the versatility of bilateral representations. In the context of neural rendering, methods such as NeRF [31, 12, 23] and Gaussian Splatting (GS) [56, 15] have achieved groundbreaking results in novel view synthesis but photometric inconsistency caused by varying illumination or transient occluders [23, 56] remains a challenge. Techniques like bilateral guided radiance field processing [38] address these issues by disentangling camera-specific enhancements and reapplying them consistently in 3D space. WildGaussians [15] integrates DINO-based appearance codes into 3D GS to robustly handle occlusions and dynamic lighting in uncontrolled scenes. Cross-Ray NeRF [52] employs cross-ray feature covariance and grid-based transient masking to harmonize appearance variations and suppress occlusions in unconstrained image collections. Despite these strides, existing approaches often struggle to balance global consistency with localized adaptability. For instance, while appearance codes provide global adjustments, they lack the granularity to model fine-grained variations [7]. Conversely, bilateral grids excel at pixel-wise transformations but are challenging to optimize effectively [27]. To address this issue, we introduce a multi-scale bilateral grid which unifies bilateral grids and appearance codes, facilitating high-quality dynamic autonomous driving scene reconstruction with enhanced modeling capability and optimization efficiency.
+
+# 2.2 Autonomous Driving Simulation
+
+Autonomous driving simulation has emerged as a critical tool for developing perception, planning, and control systems by generating diverse, realistic driving scenarios [41, 26, 45, 19, 33, 17, 47, 9]. Recent advancements focus on photorealistic rendering, dynamic scene modeling, and multi-modal sensor simulation [41, 16, 60, 35, 45, 51, 50, 43]. For instance, MARS [41] employs a modular NeRF-based framework to independently control static and dynamic scene elements, while Street Gaussian [45] achieves real-time urban scene rendering using explicit 3D Gaussian representations. DrivingGaussian [60] enhances dynamic scene reconstruction via composite Gaussian Splatting, ensuring occlusion accuracy and multi-camera consistency. NeuRAD [35] integrates sensor-specific effects (e.g., rolling shutter, LiDAR beam divergence) to improve novel view synthesis. Holistic scene understanding and editing have also seen progress. HUGS [59] combines static and dynamic 3D Gaussians for real-time semantic parsing. ChatSim [39] enables language-driven scene editing with external asset integration. LiDAR data integration has advanced through methods like LiDAR-NeRF [32], which uses structural regularization for low-texture regions. HO-Gaussian [18] merges grid-based volumes with Gaussian Splatting to eliminate Structure-from-Motion dependencies.
+
+Despite progress, gaps persist between simulated and real-world data [19, 33, 49, 48]. AlignMiF [33] addresses LiDAR-camera misalignments via geometry-aligned implicit fields, while RodUS [25] decomposes urban scenes into static and dynamic components using 4D semantics to reduce artifacts. Innovations in large-scale generation include InfiniCube [21], which leverages sparse-voxel representations for unbounded dynamic scenes, and Omnire [5], which reconstructs diverse dynamic objects (e.g., pedestrians) for human-vehicle interaction simulations. Furthermore, recent works like GaussianPro [6] and SplatAD [11] explore to refine 3D GS for real-time rendering of dynamic scenes. GaussianPro [6] introduces progressive propagation for texture-less surfaces, and SplatAD [11] models sensor-specific phenomena (e.g., rolling shutter, LiDAR intensity). In this work, we aim to reconstruct high-quality driving scenes by unifying appearance codes and bilateral grids to enhance photometric consistency and modeling capability.
+
+# 3 Methodology
+
+# 3.1 Problem Formulation
+
+We aim to reconstruct a 3D scene representation  $\mathbb{G}$  from multi-view images and LiDAR depth maps, commonly available in autonomous driving datasets. Given a set of images  $\{I_{c,t}\}$  and corresponding depths  $\{D_{c,t}\}$  captured by  $N$  cameras over  $T$  time steps, we formulate our objective as minimizing the discrepancy between rendered and observed images and depths:
+
+$$
+\min  _ {\mathbb {G}} \sum_ {c, t} \left(\left\| I _ {c, t} ^ {r} - I _ {c, t} \right\| _ {2} ^ {2} + \lambda_ {D} \left\| D _ {c, t} ^ {r} - D _ {c, t} \right\| _ {2} ^ {2}\right), \tag {1}
+$$
+
+where  $I_{c,t}^{r}$  and  $D_{c,t}^{r}$  are the rendered appearance and geometry of the scene  $\mathbb{G}$  from camera  $c$  at time  $t$ , and  $\lambda_{D}$  is a weight balancing the depth term.
+
+However, dynamic driving scenes and varying camera properties cause inconsistent appearances, leading to geometric and texture artifacts. To address inconsistency, we decompose each image  $I_{c,t}$  into consistent  $\mathcal{C}_{c,t}$  (e.g., intrinsic scene colors and constant sensor adjustments like normalization exposure), and non-consistent  $\mathcal{N}_{c,t}$  (e.g., components-varying lighting conditions, different camera settings and diverse image ISP effects):
+
+$$
+I _ {c, t} = \mathcal {C} _ {c, t} + \mathcal {N} _ {c, t}, \tag {2}
+$$
+
+We model  $\mathcal{N}_{c,t}$  as a non-linear transformation  $\mathcal{F}(\mathcal{C}_{c,t})$ . Substituting into Eq. (2):
+
+$$
+I _ {c, t} = \mathcal {C} _ {c, t} + \mathcal {F} \left(\mathcal {C} _ {c, t}\right) ， \tag {3}
+$$
+
+Let  $\mathcal{E}(\mathcal{C}_{c,t}) = \mathcal{C}_{c,t} + \mathcal{F}(\mathcal{C}_{c,t})$ , we can reformulate the optimization objective in Eq. (1) as:
+
+$$
+\min  _ {\mathbb {G}} \sum_ {c, t} \left(\left\| \mathcal {E} \left(\mathcal {C} _ {c, t} ^ {r}\right) - I _ {c, t} \right\| _ {2} ^ {2} + \lambda_ {D} \left\| D _ {c, t} ^ {r} - D _ {c, t} \right\| _ {2} ^ {2}\right), \tag {4}
+$$
+
+This reformulation links photometric and geometric consistency optimization within a joint objective, guiding  $\mathcal{C}_{c,t}^{r}$  adjustments by geometric constraints  $(D_{c,t}^{r})$ . This mitigates texture-geometry ambiguities (e.g., shadows on road surfaces misinterpretable as geometry changes).  $\mathcal{F}(\mathcal{C}_{c,t})$  models transient appearance, while  $\mathcal{C}_{c,t}^{r}$  enforces consistent scene properties. Experiments (Sec. 4, Tab. 1) show this joint optimization reduces both photometric error and geometric drift in comparison to baselines.
+
+# 3.2 Gaussian Splitting for Autonomous Driving Environments
+
+Following prior methodologies [5, 35], we represent autonomous driving environments using a hybrid scene graph, decomposing the scene into sky, background, and dynamic object models. Each dynamic object is represented by a 3D model in canonical space and transformed to the scene with an associated sequence of SE(3) transformations. We derive the transformation matrixes from existing object detection pipelines or ground-truth annotations.
+
+For the sky, we use an environment map to model sky color based on viewing direction, while static background is represented as a set of semi-transparent 3D Gaussians. Each Gaussian is characterized by a learnable opacity parameter  $o \in (0,1)$ , a mean position  $\mu \in \mathbb{R}^3$ , and an anisotropic covariance matrix  $\Sigma \in \mathbb{R}^{3\times 3}$  which is parameterized by a scale vector  $S \in \mathbb{R}^3$  and a rotation quaternion  $q \in \mathbb{R}^4$ . Additionally, Spherical harmonics coefficients  $c \in \mathbb{R}^F$  are used to model appearance.
+
+For the moving object models, we further distinguish between non-deformable (e.g., vehicles) and deformable objects (e.g., pedestrians). Non-deformable objects  $\{\mathbb{G}_i^N | i \in \{1, \dots, n_n\}\}$  are optimized in their local coordinate space and transformed into the global world space via their pose  $T$ :
+
+$$
+\mathbb {G} _ {i} ^ {N} = T \otimes \hat {\mathbb {G}} _ {i} ^ {N}, \tag {5}
+$$
+
+For deformable objects  $\{\mathbb{G}_i^D | i \in \{1, \dots, n_d\}\}$ , we employ a deformation network  $\mathcal{F}_{\Phi}$  (parameterized by  $\Phi$ ) to adapt the Gaussian representation based on latent variable  $e$  and time  $t$ :
+
+$$
+\mathbb {G} _ {i} ^ {D} = T \otimes \left(\hat {\mathbb {G}} _ {i} ^ {D} \oplus \mathcal {F} _ {\Phi} \left(\hat {\mathbb {G}} _ {i} ^ {D}, e, t\right)\right). \tag {6}
+$$
+
+# 3.3 Multi-Scale Bilateral Grid for Appearance Enhancement
+
+Solving the photometric correction problem formulated in Eq. (4) fundamentally depends on the ability to effectively model the complex and spatially-varying photometric transformations present in real-world driving scenes. The function  $\mathcal{E}(\cdot)$  in our formulation represents this crucial photometric enhancement process. To address this, we propose the Multi-Scale Bilateral Grid architecture, detailed in this section, as a novel grounded solution for approximating  $\mathcal{E}(\cdot)$  and achieving high-quality, consistent appearance enhancement.
+
+# 3.3.1 Multi-Scale Bilateral Grid Architecture
+
+To address limitations of existing methods in handling diverse photometric variations in driving scenes, we propose a multi-scale bilateral grid framework. Our framework achieves this unification by employing a hierarchical pyramid of bilateral grids, organized across multiple scales. This multi-scale design is directly inspired by the nature of photometric inconsistencies in real-world environments, which range from global scene-level changes to highly localized variations (e.g., from overall lighting shifts to fine texture-level shadows). By utilizing this grid hierarchy, our framework aims to capture and effectively correct photometric variations at their corresponding scales, thus enabling a more comprehensive and spatially adaptable photometric correction.
+
+The multi-scale bilateral grid transformation can be formulated as  $I^{e} = \bar{A}\odot I^{r}$ , where  $\bar{A}$  represents a composite, scale-dependent photometric transformation. Crucially, this composite transformation is constructed hierarchically, combining transformations learned by individual bilateral grids at different scales, progressing from coarse to fine. This staged composition is key to achieving scale-dependent adaptation to photometric inconsistencies and allows us to move beyond the inherent limitations of single-scale methods. Further details are provided in the subsequent sections.
+
+Our framework achieves this hierarchical representation using a three-level bilateral grid pyramid (Fig. 2): (1) Coarse Level  $(2 \times 2 \times 1 \times 12$  grid) captures global scene structure and approximate global appearance codes; (2) Intermediate Level  $(4 \times 4 \times 2 \times 12$  grid) represents regional features and capture mid-range photometric variations; (3) Fine Level  $(8 \times 8 \times 4 \times 12$  grid) encodes local details and approximate pixel-wise bilateral grids within the multi-scale framework.
+
+At each level  $l$ , the grid tensor is defined as  $\mathcal{A}^{(l)} \in \mathbb{R}^{H^{(l)} \times W^{(l)} \times D^{(l)} \times C^{(l)}}$ , where Spatial Dimension  $(H^{(l)}, W^{(l)})$  represents spatial resolution (height, width) at level  $l$ , Guidance Dimension  $D^{(l)}$  specifies the guidance intensity levels at level  $l$  and Coefficient Channel  $C = 12$  represents a flattened  $3 \times 4$  affine color transformation matrix.
+
+# 3.3.2 Guidance Map, Slice Operation, and Multi-Scale Fusion
+
+To achieve adaptive and spatially-varying photometric correction, our multi-scale bilateral grid framework employs a guidance map and a slicing operation [4] to retrieve localized transformations, followed by a hierarchical fusion strategy to combine transformations across scales.
+
+We first derive a luminance-based guidance map  $I^{g}(u,v) = \mathrm{GrayScale}(I^{r}(u,v))$  from the rendered image  $I^{r}(u,v)$ , following the approach of [4, 38, 10]. This map encodes spatial brightness variations like shadows and highlights, serving as a spatially-varying query to our multi-scale grid. For each level  $l$  and pixel  $(u,v)$ , the luminance  $d$  from  $I^{g}$  is used to perform a slicing operation, querying the grid's intensity dimension  $D^{(l)}$  to locate an intensity bin. Affine transformation coefficients  $\mathcal{A}_{i,j,k}^{(l)}$  around this bin are then combined via trilinear interpolation for a level-specific transformation
+
+$\bar{A}^{(l)}(u,v)$ . The slicing operation retrieves per-pixel transformations from each grid level, written as:
+
+$$
+\bar {A} ^ {(l)} (u, v) = \sum_ {i, j, k} w _ {i, j, k} (u, v, d) \mathcal {A} ^ {(l)} (i, j, k), \tag {7}
+$$
+
+To efficiently fuse transformations across scales, we utilize hierarchical function composition. A naive approach of full-resolution slicing at each scale is computationally expensive and redundant due to local photometric coherence. To address this, we employ downsampled guidance maps  $I^{g(l)}$  for each scale  $l$ , performing slicing to obtain low-resolution coefficient fields, which are then upsampled. This approach explicitly links scale-aware guidance to patch operations-the reduced-resolution maps enable efficient spatial aggregation of photometric transformations while enhances efficiency (See ablation study in Tab. 5). The level-specific transformations  $\mathcal{T}^{(l)}$ , decomposed into linear matrices  $\bar{M}^{(l)}$  and translations  $\bar{T}^{(l)}$ , are then sequentially composed from coarse to fine to produce the final composite transformation  $\bar{A}$ . The refined image  $I^{e}(u,v)$  is obtained by applying  $\bar{A}(u,v)$  to  $I^{r}(u,v)$ , with the fusion process expressed as:
+
+$$
+I ^ {e} = \mathcal {T} ^ {(L - 1)} \circ \mathcal {T} ^ {(L - 2)} \circ \dots \circ \mathcal {T} ^ {(0)} (I ^ {r}), \tag {8}
+$$
+
+This hierarchical fusion decomposes the photometric transformation into scale-dependent residual refinements. Coarse scales capture global transformations, intermediate scales refine regional variations, and fine scales address local details (Further elaboration in Sec. 4.3). Further details on the interpolation kernel and computational efficiency considerations are provided in Appendix A1.1.
+
+Table 1: Ablation studies and comparisons on four large-scale driving datasets. Our proposed method consistently outperforms OmniRe across all metrics and datasets. w/AC denotes appearance codes; w/BG denotes single bilateral grids (size of  $16 \times 16 \times 8$ ).  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Method</td><td colspan="3">Reconstruction</td><td colspan="3">Novel View Synthesis</td><td colspan="3">Geometry</td></tr><tr><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>CD ↓</td><td>RMSE ↓</td><td>Depth ↓</td></tr><tr><td rowspan="4">Waymo</td><td>OmniRe</td><td>28.92</td><td>0.833</td><td>0.295</td><td>26.40</td><td>0.761</td><td>0.311</td><td>1.482</td><td>2.785</td><td>0.540</td></tr><tr><td>OmniRe w/AC</td><td>28.95</td><td>0.835</td><td>0.293</td><td>26.44</td><td>0.759</td><td>0.315</td><td>1.378</td><td>2.760</td><td>0.519</td></tr><tr><td>OmniRe w/BG</td><td>28.19</td><td>0.831</td><td>0.292</td><td>21.51</td><td>0.743</td><td>0.333</td><td>1.523</td><td>2.798</td><td>0.492</td></tr><tr><td>Ours</td><td>29.23</td><td>0.836</td><td>0.289</td><td>26.55</td><td>0.762</td><td>0.310</td><td>0.989</td><td>2.744</td><td>0.477</td></tr><tr><td rowspan="4">NuScenes</td><td>OmniRe</td><td>26.37</td><td>0.837</td><td>0.209</td><td>23.74</td><td>0.733</td><td>0.232</td><td>1.458</td><td>3.420</td><td>0.110</td></tr><tr><td>OmniRe w/AC</td><td>26.38</td><td>0.840</td><td>0.204</td><td>23.74</td><td>0.732</td><td>0.229</td><td>1.437</td><td>3.415</td><td>0.106</td></tr><tr><td>OmniRe w/BG</td><td>25.98</td><td>0.837</td><td>0.209</td><td>23.60</td><td>0.705</td><td>0.262</td><td>1.380</td><td>3.390</td><td>0.097</td></tr><tr><td>Ours</td><td>27.69</td><td>0.847</td><td>0.193</td><td>24.64</td><td>0.739</td><td>0.216</td><td>1.161</td><td>3.340</td><td>0.059</td></tr><tr><td rowspan="4">Argoverse</td><td>OmniRe</td><td>24.59</td><td>0.848</td><td>0.202</td><td>22.53</td><td>0.755</td><td>0.220</td><td>0.954</td><td>4.208</td><td>0.050</td></tr><tr><td>OmniRe w/AC</td><td>24.58</td><td>0.848</td><td>0.201</td><td>22.51</td><td>0.756</td><td>0.219</td><td>0.959</td><td>4.215</td><td>0.051</td></tr><tr><td>OmniRe w/BG</td><td>23.31</td><td>0.842</td><td>0.216</td><td>21.70</td><td>0.725</td><td>0.254</td><td>0.901</td><td>4.218</td><td>0.049</td></tr><tr><td>Ours</td><td>24.68</td><td>0.849</td><td>0.200</td><td>22.58</td><td>0.756</td><td>0.217</td><td>0.807</td><td>4.199</td><td>0.040</td></tr><tr><td rowspan="4">Pandaset</td><td>OmniRe</td><td>30.20</td><td>0.903</td><td>0.219</td><td>27.49</td><td>0.835</td><td>0.240</td><td>0.503</td><td>2.874</td><td>0.018</td></tr><tr><td>OmniRe w/AC</td><td>30.20</td><td>0.903</td><td>0.220</td><td>27.51</td><td>0.841</td><td>0.242</td><td>0.496</td><td>2.871</td><td>0.020</td></tr><tr><td>OmniRe w/BG</td><td>29.73</td><td>0.904</td><td>0.220</td><td>27.38</td><td>0.830</td><td>0.246</td><td>0.484</td><td>2.867</td><td>0.013</td></tr><tr><td>Ours</td><td>30.75</td><td>0.906</td><td>0.213</td><td>27.89</td><td>0.847</td><td>0.235</td><td>0.453</td><td>2.852</td><td>0.011</td></tr></table>
+
+# 4 Experiments
+
+# 4.1 Experimental Setup
+
+Datasets. We evaluate on four autonomous driving datasets: Waymo [30], NuScenes [1], Argov- verse [40], and PandaSet [42], selected for their diversity in sensor configurations (LiDAR, camera specifications), environmental conditions (lighting, seasons), and geographic locations. Identical model hyperparameters are used across all datasets to test generalization capability. Technical specifications including camera counts and ego-vehicle view cropping details are provided in Appendix A2.
+
+Baseline. Our focus is on modeling appearance variance across viewpoints, we build upon the opensource works DriveStudio [5], ChatSim [39], StreetGS [45] and test three approaches: appearance codes, standalone bilateral grids, and our multi-scale bilateral grids. This allows for direct comparison of how each method handles photometric inconsistencies across viewpoints.
+
+Implementation details. We evaluate our method on many autonomous driving datasets to demonstrate its robustness. For each dataset, We train all the methods on a single NVIDIA L40 GPU and
+
+Table 2: Comparison with different baseline approaches for scene reconstruction. Performance is reported as the average over the nuScenes [1] scenarios.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Reconstruction</td><td colspan="3">Novel View Synthesis</td><td colspan="3">Geometry</td></tr><tr><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>CD ↓</td><td>RMSE ↓</td><td>Depth ↓</td></tr><tr><td>ChatSim [39]</td><td>25.10</td><td>0.805</td><td>0.252</td><td>23.27</td><td>0.725</td><td>0.270</td><td>1.557</td><td>3.509</td><td>0.106</td></tr><tr><td>ChatSim (Ours)</td><td>27.04</td><td>0.819</td><td>0.231</td><td>24.67</td><td>0.735</td><td>0.249</td><td>1.236</td><td>3.412</td><td>0.053</td></tr><tr><td>StreetGS [45]</td><td>25.74</td><td>0.822</td><td>0.240</td><td>23.64</td><td>0.736</td><td>0.259</td><td>1.604</td><td>3.544</td><td>0.107</td></tr><tr><td>StreetGS (Ours)</td><td>27.90</td><td>0.836</td><td>0.219</td><td>25.10</td><td>0.747</td><td>0.238</td><td>1.272</td><td>3.458</td><td>0.055</td></tr></table>
+
+compare the 3D reconstruction and novel view synthesis (NVS) results between camera and LiDAR data. Additionally, we ablate key components of our method and quantify their impact on both appearance and geometry quality.
+
+# Training and Dynamic Rendering.
+
+1) Training. To optimize our multi-scale Gaussian scene representation, we employ a joint training strategy that minimizes a composite reconstruction loss:
+
+$$
+\mathcal {L} _ {\text {r e c o n}} = \lambda_ {r} \mathcal {L} _ {1} + \lambda_ {s} \mathcal {L} _ {\mathrm {S S I M}} + \lambda_ {d} \mathcal {L} _ {d} + \lambda_ {o} \mathcal {L} _ {o}, \tag {9}
+$$
+
+Furthermore, we introduce two regularization terms to enhance image fidelity: Adaptive Total Variation Regularization. This term encourages smoothness and reduces noise while preserving image details, and Circle Regularization Loss. This loss applies inverse transformation to the ground-truth images, preventing discrepancies and image quality degradation. As visually demonstrated in Fig. 5, these terms effectively improve image fidelity. Detailed are provided in Appendix A1.
+
+2) Rendering. For dynamic rendering using our multi-scale bilateral grid framework, we employ a specific interpolation strategy when encountering novel test images, especially those simulating dynamic ISP conditions. We first conducts a temporal proximity search to identify the most relevant grids. Following this, we perform scale-specific interpolation using the two nearest grids found.
+
+Table 3: Evaluation on Challenging Scene Subsets. Results on systematically selected extreme cases. Our method shows substantially improvements when photometric inconsistencies are severe.  
+
+<table><tr><td rowspan="2">Scene Type</td><td rowspan="2">Method</td><td colspan="3">Reconstruction</td><td colspan="3">Novel View Synthesis</td><td colspan="3">Geometry</td></tr><tr><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>CD ↓</td><td>RMSE ↓</td><td>Depth ↓</td></tr><tr><td rowspan="2">Challenging Scenes (18 scenes)</td><td>OmniRe</td><td>26.84</td><td>0.847</td><td>0.239</td><td>24.46</td><td>0.758</td><td>0.263</td><td>0.650</td><td>3.176</td><td>0.063</td></tr><tr><td>Ours</td><td>28.13</td><td>0.856</td><td>0.223</td><td>25.17</td><td>0.763</td><td>0.250</td><td>0.523</td><td>3.120</td><td>0.023</td></tr><tr><td rowspan="2">Night Scenes (4 scenes)</td><td>OmniRe</td><td>28.34</td><td>0.805</td><td>0.338</td><td>26.54</td><td>0.753</td><td>0.360</td><td>0.586</td><td>2.767</td><td>0.041</td></tr><tr><td>Ours</td><td>29.13</td><td>0.812</td><td>0.330</td><td>27.21</td><td>0.762</td><td>0.349</td><td>0.500</td><td>2.731</td><td>0.016</td></tr><tr><td rowspan="2">Sun Flare Scenes (5 scenes)</td><td>OmniRe</td><td>26.68</td><td>0.866</td><td>0.195</td><td>24.98</td><td>0.796</td><td>0.212</td><td>0.781</td><td>3.073</td><td>0.079</td></tr><tr><td>Ours</td><td>28.65</td><td>0.879</td><td>0.173</td><td>25.84</td><td>0.803</td><td>0.192</td><td>0.593</td><td>2.994</td><td>0.022</td></tr></table>
+
+# 4.2 Quantitative Evaluation
+
+Reconstruction and Geometric Quality. We present a comprehensive quantitative analysis in Tab. 1, comparing our full model against the OmniRe and its variants with appearance codes (w/AC) and a single bilateral grid (w/BG). Our method consistently sets a new state-of-the-art across all four datasets in both appearance and, most critically, geometric metrics.
+
+In terms of appearance fidelity, our model achieves the highest PSNR and SSIM scores. For example, on NuScenes, our method reaches a PSNR of 27.69, a significant jump from 26.37 (OmniRe) and 26.38 (OmniRe w/AC). This demonstrates the superior ability of the multi-scale grid to model complex photometric variations.
+
+However, the most striking result is the substantial improvement in geometric accuracy. Our method drastically reduces the Chamfer Distance, a key indicator of geometric fidelity. On the Waymo dataset, our model hits a CD of 0.989, outperforming the best baseline (OmniRe w/AC, 1.378) by a remarkable  $28.2\%$ . This trend holds across all datasets, with significant reductions in Chamfer Distance (CD), Root Mean Squared Error (RMSE), and Median Squared Depth Error (Depth). This result provides strong evidence for our central hypothesis: by accurately resolving photometric ambiguities, our multi-scale approach effectively eliminates geometric artifacts like "floaters" and
+
+![](images/c11051e2c15525cb4d28ef5dae6006ec4ce7dace953faab0791dd5e213f1b18f.jpg)  
+(a) Specular highlights
+
+![](images/a9ae0a2455645557411ae0097e1c0057e2818c06f529c2f0597b75b77b5e498c.jpg)  
+(b) Motion Blur  
+Ground Truth  
+Baseline + Ours
+
+![](images/f6d2c71469d43be4f5ca7106c28c2ffbba404ddcbc87772c10b1643daca5879d.jpg)  
+(c) Incomplete Geometry  
+(d) Duplication Artifacts
+
+![](images/3090bcb033a5afbd2dced199736a8d35caf58c6f0f424edf0e3f2a533951645d.jpg)
+
+![](images/7ab706385d81b67f181e366a7f36b422e6529673510adf5db00f840a26cbb1cf.jpg)  
+(e) Low-Light  
+Ground Truth  
+Baseline + Ours  
+Baseline
+
+![](images/1c6677be190f909cc09092bf55ee25f4da998fa87755e7bbccee455d79f797b5.jpg)  
+Figure 3: Our Method vs. Baseline Methods on Waymo, NuScenes, Argoverse, and PandaSet.  
+Figure 4: Our proposed framework (c) outperforms appearance codes (a) and single bilateral grids (b) by simultaneously addressing optimization challenges and enhancing geometric modeling. This delivers significant geometric accuracy improvements (lower Chamfer Distance (CD) via multi-scale bilateral grid, reducing error and improving reconstruction) and reduced photometric inconsistency (fewer floaters) in dynamic, decoupled driving scenes. Yellow indicates a high lidar error, while Purple indicates a low lidar error.
+
+produces a much cleaner, more accurate 3D reconstruction. The single bilateral grid (w/BG), despite its high expressiveness, often fails to converge and even degrades performance, highlighting the optimization challenges that our multi-scale, coarse-to-fine framework successfully overcomes.
+
+NVS. As shown in Tab. 1, our model's superior reconstruction quality translates directly to improved NVS. Our method consistently outperforms all baselines in NVS metrics across the four datasets. For instance, on PandaSet, we achieve a PSNR of 27.89, surpassing the next best (OmniRe w/AC at 27.51). This indicates that the cleaner geometry and more consistent appearance learned by our model generalize better to unseen viewpoints, producing higher-fidelity renderings.
+
+Generalization to Other Methods. To demonstrate the broad applicability of our approach, we integrated our multi-scale bilateral grid into two other state-of-the-art methods, ChatSim [39] and StreetGS [45]. The results on NuScenes, presented in Tab. 2, are compelling. Our module brings substantial gains to both frameworks. For StreetGS, incorporating our method boosts the reconstruction PSNR from 25.74 to 27.90 and slashes the Chamfer Distance from 1.604 to 1.272. These significant improvements underscore that our multi-scale grid is not just a bespoke enhancement for one architecture but a generalizable and powerful module for improving photometric and geometric consistency in Gaussian Splitting-based scene reconstruction.
+
+Performance on Challenging Scenarios. To validate robustness, we curated a subset of 18 challenging scenarios with extreme photometric inconsistencies. As shown in Table 3, our method's advantages are substantially more pronounced here than on the standard datasets. Across this entire
+
+subset, our model achieves a  $+1.29$  dB PSNR gain. This improvement is even more significant in specific difficult sub-categories; for instance, in "Sun Flare Scenes," the gain reaches  $+1.97$  dB. This demonstrates our model's strong effectiveness in resolving severe, real-world edge cases.
+
+# 4.3 Qualitative Evaluation
+
+**Rendering Results.** Fig. 3 and Fig. 4 present visualization results of our method on scenes from Waymo [30], NuScenes [1], Argoverse [40], and PandaSet [42], demonstrating its capability to handle diverse conditions that might otherwise lead to inconsistencies.
+
+![](images/1ca10d8b5b2a63d92e666af4be96e36440afbab0bc5e7c457149af69432f34aa.jpg)  
+w/ Circle Regularization Loss
+
+![](images/931774a9e9cefe0cfbe147d7bdc300e10b57446d55f19dbe5a00c5d9db76c53f.jpg)  
+w/o Circle Regularization Loss
+
+![](images/8746bbaad485db1a0a35f439d59e1c49b2e7f9c907fc2b091d91b27346fcc0d2.jpg)  
+w/ Adaptive TV Loss
+
+![](images/353fc7ebcc6307334d8402016b337f87d489a09b64a6e366d229c8e26efe821d.jpg)  
+w/o Adaptive TV Loss
+
+![](images/59ad8f55eb206094fdd69322cb35c0584107986fdfa900776187a0223a600dfa.jpg)  
+Figure 5: Visualizations of ablations on Circle Regularization loss and Adaptive TV loss.
+
+![](images/54c1bcff1cf990e77b43338eec65b0bab89bd46d2a8982fa48662dc6b59abb48.jpg)  
+Figure 6: Visualization of Affine Transformations and Distributions. The first three rows display the affine transformation and corresponding distributions when applying the first level, the first two levels, and all three levels of our multi-scale approach. The final row compares these to results from a single bilateral grid, demonstrating that the multi-scale bilateral grid achieves a coarse-to-fine, smooth image enhancement with more diverse processing capabilities.
+
+![](images/7fad4b11df2edf5db9d381038e6a5f68e9636a17766c70a9c5c04b7b2fa31780.jpg)
+
+![](images/bb12bdfdcdff93c8e4f1eb90781308be81bca6ea49a653b332b618918c5f8808.jpg)
+
+![](images/27bd43f07de977df9a5986b6447b7f135c68fbbb8ad5806aff85e47b04992b11.jpg)
+
+![](images/a30aeb09f1641e5d2b7f5ad6e40f4cec576d4a04dcfab083dde7fb71ad027b7a.jpg)
+
+![](images/7ce94f7b5adca4d1462a44b5ba3540258913cd28764fa62b5509fc0a7f421751.jpg)
+
+![](images/c47411229b3a4084f6f25df0c39f04ca5171851b6b95455e34a04e48253785bf.jpg)
+
+# Analysis of Learned Affine
+
+Transformations. To gain deeper insight into how our model functions, we analyze the distribution of the learned affine transformations in Fig. 6. The histograms for the original single-scale bilateral grid (BG) exhibit a distinct bi-modal, peaky
+
+<table><tr><td>Configuration</td><td>Grid Params (M)</td><td>Training Time (h)</td><td>Testing time (FPS)</td></tr><tr><td>OmniRe w/AC</td><td>0.072</td><td>1.93</td><td>53</td></tr><tr><td>OmniRe w/BG</td><td>27.843</td><td>2.85</td><td>54 / 38†</td></tr><tr><td>Ours</td><td>3.969</td><td>2.10</td><td>54 / 42†</td></tr></table>
+
+Table 4: Computational Efficiency Analysis. † denotes rendering with bilateral grid processing active.
+
+distribution for each viewpoint. This suggests that the single grid learns a limited set of dominant photometric corrections for each view, struggling to capture the full spectrum of variations. In stark contrast, the aggregated histogram from our multi-scale grid (across all levels and viewpoints) is significantly flatter and more dispersed. This is a direct visualization of our framework's core strength. The coarse-level grid first establishes a view-dependent baseline appearance, capturing the dominant global photometric shift (interestingly, its own histogram is also peaky, but varies across views). Then, the medium and fine grids learn residual transformations in a coarse-to-fine manner,
+
+correcting the errors from the preceding levels. This hierarchical, residual refinement process allows the model to represent a much broader and more diverse range of photometric transformations. The flatter histogram is empirical evidence of this enhanced representational power, which is crucial for achieving consistent, high-quality renderings across the diverse and challenging views encountered in autonomous driving.
+
+# 4.4 Ablation Studies
+
+As shown in Tab. 4, Tab. 5 and Fig. 5, we systematically evaluate component contributions through:
+
+Guidance Map Resolution. We test different downsampling factors for the guidance map at each grid level. The results in Tab. 5(a) show that a moderate downsampling of (2,2,1) from coarse to fine yields the best balance, achieving the highest NVS PSNR (24.64). Using the full resolution (1,1,1) slightly degrades performance, likely due to overfitting to noise, validating our patch-based aggregation strategy.
+
+Grid Size Combinations. The combination of grid sizes is crucial. As shown in Tab. 5(b), removing the coarsest grid level (e.g., using only  $(4,4,2) + (8,8,4)$ ) leads to a dramatic increase in CD, highlighting the importance of the coarse grid for establishing global geometric consistency. Crucially, a baseline with only a single fine-resolution grid  $(8,8,4)$  performs poorly, confirming that finer grids fail to converge without the stable, coarse-level initialization our method provides.
+
+<table><tr><td>Settings</td><td>Recon.↑</td><td>NVS.↑</td><td>CD ↓</td></tr><tr><td>Full model</td><td>27.69</td><td>24.64</td><td>1.161</td></tr><tr><td colspan="4">(a) Guidance Map Resolution</td></tr><tr><td>(1,1,1)</td><td>27.62</td><td>24.34</td><td>1.271</td></tr><tr><td>(2,2,1)</td><td>27.68</td><td>24.64</td><td>1.170</td></tr><tr><td>(8,8,4)</td><td>27.52</td><td>24.53</td><td>1.187</td></tr><tr><td>(16,16,8)</td><td>27.50</td><td>24.54</td><td>1.202</td></tr><tr><td colspan="4">(b) Grid Size Combinations</td></tr><tr><td>Single Grid (8,8,4)</td><td>26.56</td><td>23.90</td><td>1.376</td></tr><tr><td>(4,4,2)+(8,8,4)</td><td>27.49</td><td>24.56</td><td>1.963</td></tr><tr><td>(2,2,1)+(8,8,4)</td><td>27.57</td><td>24.57</td><td>1.940</td></tr><tr><td>(2,2,1)+(4,4,2)</td><td>27.33</td><td>24.42</td><td>1.213</td></tr><tr><td colspan="4">(c) Loss Functions</td></tr><tr><td>w/o circle regularization loss</td><td>27.47</td><td>24.52</td><td>1.164</td></tr><tr><td>w/o adaptive total variance loss</td><td>27.64</td><td>24.61</td><td>1.169</td></tr></table>
+
+Table 5: Ablations. Numbers in Guidance Map Resolution indicate downsample factors from coarse to fine level, and numbers in Grid Size Combinations indicate grid size.
+
+Loss Functions. Ablating our proposed regularization losses in Tab. 5(c) shows their contribution. Removing the circle regularization loss slightly degrades both reconstruction and NVS performance. The effect of the adaptive total variance loss is more subtle in metrics but visibly improves rendering quality by reducing noise, as shown in Fig. 5.
+
+Efficiency. As detailed in Tab. 4, our method strikes an effective balance between performance and computational cost. The single large bilateral grid (w/BG) leads to a massive parameter count (27.8M) and the longest training time (2.85h). In contrast, our multi-scale approach requires only 3.9M parameters and a modest  $9\%$  increase in training time (2.10h vs. 1.93h) compared to the appearance code baseline, while delivering substantially better results. During inference, our method maintains high frame rates, demonstrating its practicality.
+
+# 5 Conclusion
+
+We introduced a novel multi-scale bilateral grid that unifies global appearance codes and pixel-wise bilateral grids into a single, hierarchical framework. Our extensive experiments demonstrate that this approach not only effectively models and corrects complex photometric inconsistencies in autonomous driving scenes but, more importantly, leads to a significant and crucial improvement in the geometric accuracy of the final reconstruction. By establishing this strong link between advanced appearance modeling and geometric fidelity, our work paves the way for more robust and reliable neural rendering systems in safety-critical, real-world applications.
+
+Limitations. While our method shows significant improvements, some limitations remain. (1) The computational overhead, though much lower than a single large bilateral grid, is still higher than simple appearance codes, which could be a consideration for resource-constrained scenarios. (2) Modeling extremely fast-moving or highly non-rigid objects where LiDAR and camera data misalign remains an open challenge.
+
+# References
+
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+
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+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+# Answer: [Yes]
+
+Justification: All results are accompanied by statistical significance tests.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).
+
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: They are described in the implementation details part in Section 4.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: [NA]
+
+# Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: We have discussed both positive and negative impacts.
+
+# Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
+
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: We identify our paper as having no such risks.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: We have given the creators or original owners of assets used in the paper proper credits.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
+
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: We have not released new assets at the submission time. We will carefully document our data and model when we release the code and data.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: The paper does not involve LLM.
+
+Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# Appendix: Unifying Appearance Codes and Bilateral Grids for Driving Scene Gaussian Splatting
+
+# A Additional Implementation Details
+
+# A.1 Loss Functions and Optimization
+
+We jointly optimize our multi-scale Gaussian scene representation by minimizing a composite loss function  $\mathcal{L}_{\mathrm{total}}$ :
+
+$$
+\mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {\text {r e c o n}} + \lambda_ {\mathrm {T V}} \mathcal {L} _ {\mathrm {T V}} + \lambda_ {\text {c i r c l e}} \mathcal {L} _ {\text {c i r c l e}} \tag {A1}
+$$
+
+where  $\mathcal{L}_{\mathrm{recon}}$  is the primary reconstruction loss, and  $\mathcal{L}_{\mathrm{TV}}$  and  $\mathcal{L}_{\mathrm{circle}}$  are regularization terms.
+
+# A.1.1 Reconstruction Loss
+
+The core reconstruction loss,  $\mathcal{L}_{\mathrm{recon}}$ , drives the accurate reproduction of both RGB and depth information:
+
+$$
+\mathcal {L} _ {\text {r e c o n}} = \lambda_ {r} \mathcal {L} _ {1} + \left(1 - \lambda_ {r}\right) \mathcal {L} _ {\text {S S I M}} + \lambda_ {d} \mathcal {L} _ {d} + \lambda_ {o} \mathcal {L} _ {o} \tag {A2}
+$$
+
+Here,  $\mathcal{L}_1$  and  $\mathcal{L}_{\mathrm{SSIM}}$  measure the image-space difference.  $\mathcal{L}_d$  represents the loss between the rendered depth and the ground truth LiDAR depth.  $\mathcal{L}_o$  is an opacity regularization term that encourages alignment with a non-sky mask.
+
+# A.1.2 Adaptive Total Variation (TV) Regularization
+
+To encourage smoothness and reduce noise in the optimized grid representations, we incorporate a TV regularization term,  $\mathcal{L}_{\mathrm{TV}}$ , at each level  $l$  of the multi-scale representation:
+
+$$
+\mathcal {L} _ {\mathrm {T V}} = \sum_ {l} k ^ {(l)} \cdot \frac {1}{| \mathcal {A} ^ {(l)} |} \sum_ {i, j, k} \sum_ {\mathbb {D} \in \{x, y, z \}} \left\| \Delta_ {\mathbb {D}} \mathcal {A} ^ {(l)} (i, j, k) \right\| _ {2} ^ {2} \tag {A3}
+$$
+
+The adaptive weight  $k^{(l)}$  for each level  $l$  is proportional to the grid size, applying stronger smoothing to finer, higher-resolution grids and lighter regularization to coarser grids.
+
+$$
+k ^ {(l)} = a \sqrt {H ^ {(l)} \cdot W ^ {(l)} \cdot D ^ {(l)}} + b \tag {A4}
+$$
+
+# A.1.3 Circle Regularization for Photometric Consistency
+
+To constrain the noise introduced by photometric corrections and prevent overly aggressive alterations, we introduce a circle regularization loss,  $\mathcal{L}_{\mathrm{circle}}$ :
+
+$$
+\mathcal {L} _ {\text {c i r c l e}} = \sum_ {(u, v) \in \mathcal {S}} \left\| I ^ {r} (u, v) - \bar {A} ^ {- 1} \left(I ^ {g t} (u, v)\right) \right\| _ {2} ^ {2} \tag {A5}
+$$
+
+where  $S$  denotes the set of pixel coordinates. This loss encourages the rendered image  $I^{r}$  to be reconstructible from the ground truth image  $I^{gt}$  via an inverse appearance transformation  $A^{-1}$ .
+
+# A.1.4 Coarse-to-Fine Optimization Strategy
+
+We employ a coarse-to-fine optimization strategy by utilizing level-dependent learning rates (e.g.,  $1 \times 10^{-5}, 3 \times 10^{-5}, 1 \times 10^{-4}$  from coarse to fine). Coarser grids, assigned higher learning rates, rapidly learn the global scene illumination, while finer grids, with lower learning rates, hierarchically refine high-frequency photometric details. This staged optimization enhances stability.
+
+# A.2 Dynamic Rendering for Real-World ISP Adaptation
+
+To bridge the domain gap between the fixed ISP used during training and dynamic real-world ISP pipelines encountered during inference, we employ an interpolation strategy. For a novel test image with timestamp  $t_{\mathrm{novel}}$ , we proceed as follows:
+
+1. Temporal Proximity Search: We identify the two training timestamps from the same camera,  $t_1$  and  $t_2$ , that are temporally closest to  $t_{\mathrm{novel}}$  (such that  $t_1 \leq t_{\mathrm{novel}} \leq t_2$ ).
+
+2. Scale-Specific Grid Interpolation: We perform linear interpolation specifically on the coarse and medium-scale bilateral grids  $(\mathcal{A}^{(0)},\mathcal{A}^{(1)})$  derived from these two timestamps.
+
+The interpolated grid  $\hat{\mathcal{A}}^{(l)}$  for  $l\in \{0,1\}$  is formulated as:
+
+$$
+\hat {\mathcal {A}} ^ {(l)} = \omega \mathcal {A} _ {t _ {1}} ^ {(l)} + (1 - \omega) \mathcal {A} _ {t _ {2}} ^ {(l)} \tag {A6}
+$$
+
+The temporal interpolation weight,  $\omega$ , is determined by proximity:
+
+$$
+\omega = \frac {t _ {2} - t _ {\text {n o v e l}}}{t _ {2} - t _ {1}} \tag {A7}
+$$
+
+The fine-scale grid  $(l = 2)$  is not used during interpolation, as it is primarily intended to capture scene-intrinsic details rather than global ISP variations.
+
+# B Dataset Details
+
+This section provides dataset-specific details regarding our evaluation protocol, including sequence IDs for reproducibility.
+
+# B.1 Waymo Open Dataset [2]
+
+We use all five cameras and all LiDAR sensors. We conduct our experiments on the following 6 sequences, selected according to [3, 4]:
+
+- segment-10017090168044687777  
+- segment-10061305430875486848  
+- segment-10584247114982259878  
+- segment-15090871771939393635  
+- segment-4458730539804900192  
+- segment-5835049423600303130
+
+# B.2 NuScenes [1]
+
+We utilize all six available cameras and all LiDAR sensors. We select the following 8 sequences: 152, 164, 171, 200, 209, 359, 529, 916, which is an extension of the dataset used by [3]. To address ego-vehicle visibility, we crop the bottom 80 pixels from the back camera images.
+
+# B.3 PandaSet [6]
+
+We utilize six cameras and one LiDAR unit. We specifically evaluate on the following 10 challenging nighttime sequences: 063, 066, 070, 073, 074, 077, 078, 079, 088, 149. To mitigate ego-vehicle artifacts, we apply a bottom crop of 260 pixels to the back camera images.
+
+# B.4 Argoverse2 [5]
+
+We leverage the seven ring cameras and both LiDAR sensors available in the dataset. We conduct our experiments on the following 9 sequences, in line with [3]:
+
+- 05fa5048-f355-3274-b565-c0ddc547b315  
+- 0b86f508-5df9-4a46-bc59-5b9536dbde9f  
+185d3943-dd15-397a-8b2e-69cd86628fb7  
+- 25e5c600-36fe-3245-9cc0-40ef91620c22  
+- 27be7d34-ecb4-377b-8477-ccfd7cf4d0bc  
+280269f9-6111-311d-b351-ce9f63f88c81  
+- 2f2321d2-7912-3567-a789-25e46a145bda  
+44adf4c4-6064-362f-94d3-323ed42cfda9  
+- 5589de60-1727-3e3f-9423-33437fc5da4b
+
+To minimize ego-vehicle interference, we apply a bottom crop of 250 pixels to the front center, rear left, and rear right camera views.
+
+# C Additional Quantitative Results
+
+This section presents a comprehensive breakdown of our model's performance, including per-scene metrics, integration results with SOTA methods, and robustness checks on challenging subsets.
+
+# C.1 Per-Scene Geometry and Appearance Evaluation
+
+We provide detailed metrics for geometry (Tab. A1 - A4) and appearance (Tab. A5 - A8).
+
+![](images/b7edb6e2a2d8a67a6ff14d9f8fc5debe58279f29e12e244354d630352efd6197.jpg)  
+Ground Truth
+
+![](images/0a3de11e56d7408d3f81ea5992b3912a26069b0a7d0bb0a4d28fe08098e6d763.jpg)  
+Baseline+AP
+
+![](images/4b70bcc65b91a129b464187be568d0a03a1a06363fca297f674a1eac30b8f3ea.jpg)
+
+![](images/11244e8b747db8cb95da74a49a6dbb399107fe0a2636bc428c6262f3cce877e5.jpg)  
+Baseline+BG
+
+![](images/1201b12b25a03bac6cd179530303abe4f0ccfcd0444681a759b3d1b4a63700a3.jpg)
+
+![](images/0b22af04c2efa7bc82977676fd7d9a6c898f01766949ddbab7f25dba55754a31.jpg)  
+Baseline+Ours  
+Figure A1: Qualitative Comparison of Photometric Correction with Baseline Methods. These figures visualize the output of our method ('Baseline+Ours') against the ground truth, a baseline with appearance codes ('Baseline+AP'), and a baseline with a single bilateral grid ('Baseline+BG'). The accompanying error maps (blue indicating lower error, red higher) and highlighted red boxes demonstrate our method's superior ability to handle complex illumination and reduce artifacts compared to traditional approaches.
+
+![](images/72b90911c1ab36a5329cec5bfcb11723f4db7e019312fbeea2651d52ff2c602b.jpg)
+
+# C.2 Integration Evaluation with SOTA Methods
+
+Tab. A9 and A10 provide comparisons with SOTA methods like ChatSim [4] and StreetGS [7] on the NuScenes dataset.
+
+![](images/ac9dca440cb7bbfba8d2b81e085c7386743d8cdd1141388a3570069cfe9758fd.jpg)  
+Ground Truth
+
+![](images/4df31c1cd31cc8f6c342f126fa27c5864abc9b0c22d7ddc777d0661a42e72f33.jpg)
+
+![](images/832a94a19041c2df2c29a55e660e43bff9fdcdeaa01f6327670fbf86ce353755.jpg)  
+Baseline+AP
+
+![](images/b447db24dccb2b25f3b60c234822f98747c6c5c12fdb5166bc60cd0d27701377.jpg)
+
+![](images/aa67547e2142bf465e13112139a3a6d925e3002d0b851ab07e99733d1d54bb67.jpg)
+
+![](images/21efbeb1143a2b6a0aa661718a794f6f9f16d672d3c03710ac67495755abb5c8.jpg)  
+Baseline+BG
+
+![](images/fde6f98f0ea116f9ab8bf5ce9f56a62d4f015dcf3b32f9d9c1478516e9e672a2.jpg)
+
+![](images/58916539a5fcba3681401e3f55cc80bf97ff705a827689884a134aa9d33ada58.jpg)  
+Baseline+Ours  
+Figure A2: Qualitative Comparison of Photometric Correction with Baseline Methods.
+
+![](images/83ea4e2eab1a766e4be26f77b3453d270ff922191f2ef2e596e6c091b5807c65.jpg)
+
+<table><tr><td rowspan="2">Scene</td><td colspan="3">Geometry Evaluation</td></tr><tr><td>CD↓</td><td>RMSE↓</td><td>Depth↓</td></tr><tr><td>152</td><td>1.227</td><td>3.463</td><td>0.042</td></tr><tr><td>164</td><td>0.889</td><td>2.469</td><td>0.103</td></tr><tr><td>171</td><td>1.042</td><td>2.813</td><td>0.075</td></tr><tr><td>200</td><td>1.300</td><td>3.617</td><td>0.020</td></tr><tr><td>209</td><td>1.294</td><td>3.524</td><td>0.065</td></tr><tr><td>359</td><td>1.238</td><td>3.542</td><td>0.036</td></tr><tr><td>529</td><td>0.678</td><td>2.675</td><td>0.021</td></tr><tr><td>916</td><td>1.623</td><td>4.617</td><td>0.109</td></tr><tr><td>Average</td><td>1.161</td><td>3.340</td><td>0.059</td></tr></table>
+
+Table A1: Detailed Scene-by-Scene Geometry Evaluation on the NuScenes [1]. This table presents key geometry metrics—Chamfer Distance (CD), Root Mean Square Error (RMSE), and Depth error—for individual scenes and averaged across the evaluated sequences from the NuScenes dataset.
+
+![](images/ba2fef644e54ab049ed40831ede11ad42b7b1aba9071c7b12b9a414f4f5f0a67.jpg)  
+Figure A3: Qualitative Comparison with the OmniRe. This figure showcases a side-by-side visual comparison of renderings and depth maps produced by the original OmniRe framework versus OmniRe integrated with our proposed method ('OmniRe+Ours'). The results highlight the enhancements in image fidelity and depth map coherence achieved by our approach.
+
+<table><tr><td rowspan="2">Scene</td><td colspan="3">Geometry Evaluation</td></tr><tr><td>CD↓</td><td>RMSE↓</td><td>Depth↓</td></tr><tr><td>0</td><td>1.638</td><td>3.217</td><td>1.733</td></tr><tr><td>3</td><td>1.427</td><td>2.905</td><td>0.350</td></tr><tr><td>31</td><td>0.278</td><td>1.636</td><td>0.026</td></tr><tr><td>233</td><td>1.149</td><td>3.465</td><td>0.594</td></tr><tr><td>551</td><td>1.339</td><td>3.362</td><td>0.095</td></tr><tr><td>621</td><td>0.103</td><td>1.879</td><td>0.007</td></tr><tr><td>Average</td><td>0.989</td><td>2.744</td><td>0.467</td></tr></table>
+
+Table A2: Detailed Scene-by-Scene Geometry Evaluation on the Waymo Open Dataset [2]. This table showcases the Chamfer Distance (CD), Root Mean Square Error (RMSE), and Depth error for specific scenes and their average, evaluating the geometric reconstruction accuracy on the Waymo Open Dataset.
+
+![](images/d5c7b1381ad36ff9375a2754685321c87b4b9035962bb8f1b6ad14256a60ab91.jpg)  
+Figure A4: Qualitative Comparison with the ChatSim. Visual results comparing the ChatSim framework with ChatSim augmented by our method ('ChatSim+Ours'). The rendered images and corresponding depth maps illustrate the improvements in visual quality and geometric detail provided by our multi-scale bilateral grid framework.
+
+![](images/f486a8e0f362b7e36ed68f24e888fd104c1f405830924496ab965c89670998da.jpg)  
+Figure A5: Qualitative Comparison with the StreetGS. This figure presents a visual comparison between the StreetGS framework and StreetGS enhanced with our method ('StreetGS+Ours'). The displayed images and depth maps demonstrate the capability of our approach to improve rendering realism and depth accuracy in challenging driving scenarios.
+
+<table><tr><td rowspan="2">Scene</td><td colspan="3">Geometry Evaluation</td></tr><tr><td>CD↓</td><td>RMSE↓</td><td>Depth↓</td></tr><tr><td>63</td><td>0.760</td><td>3.812</td><td>0.015</td></tr><tr><td>66</td><td>0.517</td><td>2.657</td><td>0.008</td></tr><tr><td>70</td><td>0.310</td><td>2.244</td><td>0.027</td></tr><tr><td>73</td><td>0.280</td><td>2.012</td><td>0.007</td></tr><tr><td>74</td><td>0.680</td><td>3.240</td><td>0.009</td></tr><tr><td>77</td><td>0.190</td><td>2.926</td><td>0.004</td></tr><tr><td>78</td><td>0.403</td><td>2.819</td><td>0.008</td></tr><tr><td>79</td><td>0.198</td><td>2.081</td><td>0.004</td></tr><tr><td>88</td><td>0.968</td><td>4.674</td><td>0.039</td></tr><tr><td>149</td><td>0.228</td><td>2.052</td><td>0.007</td></tr><tr><td>Average</td><td>0.453</td><td>2.852</td><td>0.013</td></tr></table>
+
+Table A3: Detailed Scene-by-Scene Geometry Evaluation on the PandaSet [6]. This table provides a per-scene breakdown and average of Chamfer Distance (CD), Root Mean Square Error (RMSE), and Depth error, assessing geometric reconstruction performance on challenging nighttime scenarios from PandaSet.  
+
+<table><tr><td rowspan="2">Scene</td><td colspan="3">Geometry Evaluation</td></tr><tr><td>CD↓</td><td>RMSE↓</td><td>Depth↓</td></tr><tr><td>0</td><td>0.522</td><td>3.001</td><td>0.023</td></tr><tr><td>1</td><td>2.071</td><td>5.845</td><td>0.085</td></tr><tr><td>2</td><td>0.461</td><td>3.226</td><td>0.011</td></tr><tr><td>3</td><td>0.348</td><td>3.677</td><td>0.014</td></tr><tr><td>4</td><td>1.198</td><td>4.485</td><td>0.059</td></tr><tr><td>5</td><td>0.694</td><td>5.069</td><td>0.040</td></tr><tr><td>6</td><td>0.967</td><td>6.193</td><td>0.075</td></tr><tr><td>8</td><td>0.743</td><td>4.436</td><td>0.045</td></tr><tr><td>9</td><td>0.257</td><td>1.952</td><td>0.005</td></tr><tr><td>Average</td><td>0.807</td><td>4.209</td><td>0.040</td></tr></table>
+
+Table A4: Detailed Scene-by-Scene Geometry Evaluation on the Argoverse2 Dataset [5]. This table displays Chamfer Distance (CD), Root Mean Square Error (RMSE), and Depth error for individual sequences and their average, evaluating geometric accuracy on the Argoverse2 dataset.  
+
+<table><tr><td rowspan="3">Scene</td><td colspan="6">Scene Reconstruction</td><td colspan="7">Novel View Synthesis</td></tr><tr><td colspan="3">Full Image</td><td colspan="2">human</td><td>vehicle</td><td colspan="3">Full Image</td><td colspan="2">human</td><td colspan="2">vehicle</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>152</td><td>27.39</td><td>0.839</td><td>0.204</td><td>29.33</td><td>0.873</td><td>27.76</td><td>0.862</td><td>23.57</td><td>0.695</td><td>0.234</td><td>25.04</td><td>0.655</td><td>22.07</td></tr><tr><td>164</td><td>26.52</td><td>0.829</td><td>0.232</td><td>24.75</td><td>0.789</td><td>25.60</td><td>0.793</td><td>23.16</td><td>0.711</td><td>0.262</td><td>19.86</td><td>0.464</td><td>21.74</td></tr><tr><td>171</td><td>26.97</td><td>0.833</td><td>0.249</td><td>24.05</td><td>0.652</td><td>26.22</td><td>0.812</td><td>24.39</td><td>0.748</td><td>0.273</td><td>22.28</td><td>0.568</td><td>22.51</td></tr><tr><td>200</td><td>27.80</td><td>0.835</td><td>0.193</td><td>N/A</td><td>N/A</td><td>28.08</td><td>0.832</td><td>25.28</td><td>0.745</td><td>0.211</td><td>N/A</td><td>N/A</td><td>24.11</td></tr><tr><td>209</td><td>29.07</td><td>0.866</td><td>0.185</td><td>28.20</td><td>0.802</td><td>28.84</td><td>0.836</td><td>26.18</td><td>0.785</td><td>0.204</td><td>25.42</td><td>0.694</td><td>25.07</td></tr><tr><td>359</td><td>27.95</td><td>0.859</td><td>0.168</td><td>26.98</td><td>0.795</td><td>27.25</td><td>0.839</td><td>24.43</td><td>0.723</td><td>0.190</td><td>24.06</td><td>0.629</td><td>22.72</td></tr><tr><td>529</td><td>29.90</td><td>0.893</td><td>0.133</td><td>24.25</td><td>0.736</td><td>27.80</td><td>0.856</td><td>27.40</td><td>0.826</td><td>0.146</td><td>24.18</td><td>0.712</td><td>24.84</td></tr><tr><td>916</td><td>25.88</td><td>0.819</td><td>0.181</td><td>28.16</td><td>0.792</td><td>26.95</td><td>0.839</td><td>22.73</td><td>0.678</td><td>0.208</td><td>25.44</td><td>0.637</td><td>22.75</td></tr><tr><td>Average</td><td>27.69</td><td>0.847</td><td>0.193</td><td>26.53</td><td>0.777</td><td>27.31</td><td>0.834</td><td>24.64</td><td>0.739</td><td>0.216</td><td>23.75</td><td>0.623</td><td>23.23</td></tr></table>
+
+Table A5: Detailed Appearance Evaluation for Scene Reconstruction and Novel View Synthesis on the NuScenes Dataset [1]. This table presents PSNR, SSIM, and LPIPS metrics for the full image, as well as for 'human' and 'vehicle' classes, for both scene reconstruction and novel view synthesis tasks on various NuScenes sequences.  
+
+<table><tr><td rowspan="3">Scene</td><td colspan="7">Scene Reconstruction</td><td colspan="7">Novel View Synthesis</td></tr><tr><td colspan="3">Full Image</td><td colspan="2">human</td><td colspan="2">vehicle</td><td colspan="3">Full Image</td><td colspan="2">human</td><td colspan="2">vehicle</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>0</td><td>29.26</td><td>0.840</td><td>0.311</td><td>N/A</td><td>N/A</td><td>23.33</td><td>0.640</td><td>25.94</td><td>0.770</td><td>0.341</td><td>N/A</td><td>N/A</td><td>22.29</td><td>0.531</td></tr><tr><td>3</td><td>27.94</td><td>0.841</td><td>0.241</td><td>21.82</td><td>0.632</td><td>N/A</td><td>N/A</td><td>24.30</td><td>0.737</td><td>0.269</td><td>20.45</td><td>0.505</td><td>N/A</td><td>N/A</td></tr><tr><td>31</td><td>27.65</td><td>0.849</td><td>0.224</td><td>32.93</td><td>0.784</td><td>22.54</td><td>0.683</td><td>23.98</td><td>0.723</td><td>0.250</td><td>30.48</td><td>0.657</td><td>19.05</td><td>0.421</td></tr><tr><td>233</td><td>33.71</td><td>0.801</td><td>0.482</td><td>N/A</td><td>N/A</td><td>23.55</td><td>0.686</td><td>32.81</td><td>0.777</td><td>0.488</td><td>N/A</td><td>N/A</td><td>22.37</td><td>0.645</td></tr><tr><td>551</td><td>24.89</td><td>0.748</td><td>0.409</td><td>20.77</td><td>0.470</td><td>21.66</td><td>0.683</td><td>22.54</td><td>0.670</td><td>0.437</td><td>19.99</td><td>0.407</td><td>18.76</td><td>0.471</td></tr><tr><td>621</td><td>31.91</td><td>0.939</td><td>0.067</td><td>26.68</td><td>0.846</td><td>26.40</td><td>0.845</td><td>29.73</td><td>0.895</td><td>0.078</td><td>24.94</td><td>0.789</td><td>24.35</td><td>0.765</td></tr><tr><td>Average</td><td>29.23</td><td>0.836</td><td>0.289</td><td>25.55</td><td>0.683</td><td>23.50</td><td>0.707</td><td>26.55</td><td>0.762</td><td>0.310</td><td>23.97</td><td>0.590</td><td>21.36</td><td>0.567</td></tr></table>
+
+Table A6: Detailed Appearance Evaluation for Scene Reconstruction and Novel View Synthesis on the Waymo Open Dataset [2]. This table details PSNR, SSIM, and LPIPS metrics for full images and specific object classes ('human', 'vehicle') during scene reconstruction and novel view synthesis on selected Waymo sequences.
+
+<table><tr><td rowspan="3">Scene</td><td colspan="7">Scene Reconstruction</td><td colspan="7">Novel View Synthesis</td></tr><tr><td colspan="3">Full Image</td><td colspan="2">human</td><td colspan="2">vehicle</td><td colspan="3">Full Image</td><td colspan="2">human</td><td colspan="2">vehicle</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>63</td><td>29.95</td><td>0.905</td><td>0.226</td><td>29.44</td><td>0.783</td><td>22.32</td><td>0.726</td><td>27.32</td><td>0.854</td><td>0.258</td><td>26.93</td><td>0.613</td><td>18.49</td><td>0.447</td></tr><tr><td>66</td><td>31.73</td><td>0.931</td><td>0.198</td><td>29.94</td><td>0.845</td><td>20.78</td><td>0.681</td><td>28.92</td><td>0.890</td><td>0.215</td><td>26.84</td><td>0.738</td><td>19.55</td><td>0.591</td></tr><tr><td>70</td><td>31.09</td><td>0.909</td><td>0.260</td><td>N/A</td><td>N/A</td><td>21.35</td><td>0.740</td><td>28.35</td><td>0.862</td><td>0.287</td><td>N/A</td><td>N/A</td><td>21.20</td><td>0.674</td></tr><tr><td>73</td><td>32.13</td><td>0.931</td><td>0.198</td><td>31.26</td><td>0.868</td><td>23.33</td><td>0.806</td><td>29.16</td><td>0.889</td><td>0.214</td><td>N/A</td><td>N/A</td><td>21.84</td><td>0.706</td></tr><tr><td>74</td><td>29.85</td><td>0.914</td><td>0.207</td><td>30.99</td><td>0.874</td><td>24.44</td><td>0.849</td><td>27.13</td><td>0.862</td><td>0.228</td><td>27.82</td><td>0.740</td><td>22.23</td><td>0.740</td></tr><tr><td>77</td><td>32.60</td><td>0.937</td><td>0.165</td><td>32.28</td><td>0.852</td><td>25.42</td><td>0.855</td><td>29.77</td><td>0.895</td><td>0.178</td><td>29.02</td><td>0.730</td><td>22.12</td><td>0.721</td></tr><tr><td>78</td><td>31.30</td><td>0.922</td><td>0.181</td><td>33.54</td><td>0.896</td><td>25.27</td><td>0.812</td><td>28.39</td><td>0.869</td><td>0.197</td><td>30.74</td><td>0.809</td><td>22.86</td><td>0.683</td></tr><tr><td>79</td><td>31.42</td><td>0.902</td><td>0.228</td><td>31.91</td><td>0.850</td><td>26.06</td><td>0.843</td><td>28.14</td><td>0.832</td><td>0.251</td><td>28.98</td><td>0.726</td><td>23.08</td><td>0.702</td></tr><tr><td>88</td><td>25.09</td><td>0.787</td><td>0.269</td><td>22.10</td><td>0.670</td><td>21.90</td><td>0.738</td><td>21.95</td><td>0.623</td><td>0.305</td><td>20.19</td><td>0.362</td><td>18.46</td><td>0.444</td></tr><tr><td>149</td><td>32.31</td><td>0.926</td><td>0.202</td><td>N/A</td><td>N/A</td><td>24.04</td><td>0.791</td><td>29.73</td><td>0.891</td><td>0.215</td><td>N/A</td><td>N/A</td><td>23.37</td><td>0.728</td></tr><tr><td>Average</td><td>30.75</td><td>0.906</td><td>0.213</td><td>30.18</td><td>0.830</td><td>23.49</td><td>0.784</td><td>27.89</td><td>0.847</td><td>0.235</td><td>27.22</td><td>0.674</td><td>21.32</td><td>0.644</td></tr></table>
+
+Table A7: Detailed Appearance Evaluation for Scene Reconstruction and Novel View Synthesis on PandaSet [6]. This table outlines PSNR, SSIM, and LPIPS metrics for full image and object-specific ('human', 'vehicle') evaluations in both scene reconstruction and novel view synthesis tasks using PandaSet nighttime sequences.  
+
+<table><tr><td rowspan="3">Scene</td><td colspan="7">Scene Reconstruction</td><td colspan="7">Novel View Synthesis</td></tr><tr><td colspan="3">Full Image</td><td colspan="2">human</td><td colspan="2">vehicle</td><td colspan="3">Full Image</td><td colspan="2">human</td><td colspan="2">vehicle</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>PSNR↑</td><td>SSIM↑</td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>0</td><td>25.72</td><td>0.871</td><td>0.181</td><td>25.54</td><td>0.822</td><td>26.58</td><td>0.825</td><td>22.83</td><td>0.765</td><td>0.199</td><td>23.73</td><td>0.734</td><td>23.31</td><td>0.682</td></tr><tr><td>1</td><td>22.18</td><td>0.769</td><td>0.302</td><td>21.76</td><td>0.681</td><td>22.87</td><td>0.798</td><td>20.20</td><td>0.655</td><td>0.323</td><td>18.22</td><td>0.472</td><td>18.48</td><td>0.478</td></tr><tr><td>2</td><td>27.36</td><td>0.888</td><td>0.166</td><td>25.33</td><td>0.785</td><td>28.93</td><td>0.884</td><td>24.83</td><td>0.803</td><td>0.183</td><td>22.32</td><td>0.620</td><td>23.63</td><td>0.669</td></tr><tr><td>3</td><td>26.55</td><td>0.893</td><td>0.123</td><td>23.24</td><td>0.811</td><td>25.58</td><td>0.875</td><td>24.69</td><td>0.826</td><td>0.141</td><td>21.99</td><td>0.756</td><td>23.10</td><td>0.769</td></tr><tr><td>4</td><td>22.65</td><td>0.804</td><td>0.272</td><td>22.73</td><td>0.720</td><td>23.76</td><td>0.812</td><td>20.55</td><td>0.685</td><td>0.290</td><td>19.95</td><td>0.518</td><td>19.58</td><td>0.493</td></tr><tr><td>5</td><td>24.19</td><td>0.848</td><td>0.207</td><td>23.69</td><td>0.760</td><td>23.75</td><td>0.790</td><td>22.07</td><td>0.738</td><td>0.226</td><td>21.12</td><td>0.598</td><td>20.32</td><td>0.580</td></tr><tr><td>6</td><td>23.56</td><td>0.795</td><td>0.275</td><td>19.80</td><td>0.588</td><td>21.05</td><td>0.757</td><td>21.86</td><td>0.703</td><td>0.292</td><td>18.15</td><td>0.446</td><td>17.76</td><td>0.498</td></tr><tr><td>8</td><td>23.47</td><td>0.850</td><td>0.183</td><td>25.63</td><td>0.787</td><td>23.66</td><td>0.851</td><td>21.25</td><td>0.756</td><td>0.199</td><td>20.57</td><td>0.531</td><td>19.58</td><td>0.628</td></tr><tr><td>9</td><td>26.48</td><td>0.925</td><td>0.091</td><td>22.44</td><td>0.700</td><td>24.42</td><td>0.853</td><td>24.91</td><td>0.875</td><td>0.100</td><td>21.04</td><td>0.614</td><td>21.96</td><td>0.732</td></tr><tr><td>Average</td><td>24.68</td><td>0.849</td><td>0.200</td><td>23.35</td><td>0.739</td><td>24.51</td><td>0.827</td><td>22.58</td><td>0.756</td><td>0.217</td><td>20.79</td><td>0.588</td><td>20.86</td><td>0.615</td></tr></table>
+
+Table A8: Detailed Appearance Evaluation for Scene Reconstruction and Novel View Synthesis on the Argoverse2 Dataset [5]. This table shows PSNR, SSIM, and LPIPS for full image and object-focused ('human', 'vehicle') assessments across scene reconstruction and novel view synthesis on the Argoverse2 dataset.  
+
+<table><tr><td rowspan="2">Scene</td><td rowspan="2">Method</td><td colspan="3">Reconstruction</td><td colspan="3">Novel View Synthesis</td><td colspan="3">Geometry</td></tr><tr><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>CD ↓</td><td>RMSE ↓</td><td>Depth ↓</td></tr><tr><td rowspan="2">152</td><td>ChatSim</td><td>25.37</td><td>0.811</td><td>0.251</td><td>1.592</td><td>3.637</td><td>0.085</td><td>22.72</td><td>0.694</td><td>0.276</td></tr><tr><td>Ours</td><td>27.09</td><td>0.825</td><td>0.231</td><td>1.309</td><td>3.540</td><td>0.040</td><td>23.80</td><td>0.703</td><td>0.256</td></tr><tr><td rowspan="2">164</td><td>ChatSim</td><td>24.08</td><td>0.788</td><td>0.299</td><td>1.198</td><td>2.615</td><td>0.186</td><td>22.07</td><td>0.708</td><td>0.321</td></tr><tr><td>Ours</td><td>26.16</td><td>0.802</td><td>0.273</td><td>0.972</td><td>2.523</td><td>0.101</td><td>23.79</td><td>0.718</td><td>0.293</td></tr><tr><td rowspan="2">171</td><td>ChatSim</td><td>25.00</td><td>0.801</td><td>0.310</td><td>1.420</td><td>2.981</td><td>0.105</td><td>23.31</td><td>0.736</td><td>0.326</td></tr><tr><td>Ours</td><td>26.95</td><td>0.813</td><td>0.289</td><td>1.114</td><td>2.895</td><td>0.057</td><td>24.77</td><td>0.744</td><td>0.307</td></tr><tr><td rowspan="2">200</td><td>ChatSim</td><td>24.85</td><td>0.799</td><td>0.230</td><td>1.690</td><td>3.740</td><td>0.035</td><td>23.47</td><td>0.732</td><td>0.246</td></tr><tr><td>Ours</td><td>27.54</td><td>0.816</td><td>0.220</td><td>1.289</td><td>3.619</td><td>0.017</td><td>25.43</td><td>0.745</td><td>0.235</td></tr><tr><td rowspan="2">209</td><td>ChatSim</td><td>25.77</td><td>0.818</td><td>0.268</td><td>1.841</td><td>3.805</td><td>0.118</td><td>24.30</td><td>0.763</td><td>0.281</td></tr><tr><td>Ours</td><td>27.93</td><td>0.833</td><td>0.243</td><td>1.445</td><td>3.691</td><td>0.061</td><td>25.93</td><td>0.776</td><td>0.255</td></tr><tr><td rowspan="2">359</td><td>ChatSim</td><td>25.76</td><td>0.823</td><td>0.220</td><td>1.498</td><td>3.701</td><td>0.073</td><td>23.46</td><td>0.714</td><td>0.238</td></tr><tr><td>Ours</td><td>27.17</td><td>0.834</td><td>0.208</td><td>1.317</td><td>3.647</td><td>0.036</td><td>24.31</td><td>0.719</td><td>0.227</td></tr><tr><td rowspan="2">529</td><td>ChatSim</td><td>27.19</td><td>0.864</td><td>0.191</td><td>1.071</td><td>2.853</td><td>0.077</td><td>25.70</td><td>0.812</td><td>0.201</td></tr><tr><td>Ours</td><td>29.60</td><td>0.878</td><td>0.159</td><td>0.692</td><td>2.749</td><td>0.018</td><td>27.35</td><td>0.822</td><td>0.169</td></tr><tr><td rowspan="2">916</td><td>ChatSim</td><td>22.81</td><td>0.742</td><td>0.249</td><td>2.145</td><td>4.745</td><td>0.173</td><td>21.19</td><td>0.641</td><td>0.271</td></tr><tr><td>Ours</td><td>23.95</td><td>0.759</td><td>0.232</td><td>1.750</td><td>4.635</td><td>0.102</td><td>21.96</td><td>0.655</td><td>0.253</td></tr></table>
+
+Table A9: Comparative Performance Analysis against ChatSim on the NuScenes Dataset. This table provides a scene-by-scene comparison of our method ('Ours') with the ChatSim baseline across scene reconstruction (PSNR, SSIM, LPIPS), novel view synthesis (PSNR, SSIM, LPIPS), and geometry (CD, RMSE, Depth) metrics.
+
+<table><tr><td rowspan="2">Scene</td><td rowspan="2">Method</td><td colspan="3">Reconstruction</td><td colspan="3">Novel View Synthesis</td><td colspan="3">Geometry</td></tr><tr><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>CD ↓</td><td>RMSE ↓</td><td>Depth ↓</td></tr><tr><td rowspan="2">152</td><td>StreetGS</td><td>25.49</td><td>0.813</td><td>0.254</td><td>1.611</td><td>3.648</td><td>0.089</td><td>22.73</td><td>0.695</td><td>0.280</td></tr><tr><td>Ours</td><td>27.34</td><td>0.828</td><td>0.231</td><td>1.343</td><td>3.560</td><td>0.038</td><td>23.90</td><td>0.706</td><td>0.256</td></tr><tr><td rowspan="2">164</td><td>StreetGS</td><td>25.04</td><td>0.813</td><td>0.273</td><td>1.187</td><td>2.605</td><td>0.179</td><td>22.55</td><td>0.724</td><td>0.296</td></tr><tr><td>Ours</td><td>27.41</td><td>0.827</td><td>0.248</td><td>0.956</td><td>2.523</td><td>0.096</td><td>24.29</td><td>0.734</td><td>0.269</td></tr><tr><td rowspan="2">171</td><td>StreetGS</td><td>25.62</td><td>0.811</td><td>0.305</td><td>1.355</td><td>2.958</td><td>0.115</td><td>23.63</td><td>0.743</td><td>0.323</td></tr><tr><td>Ours</td><td>27.79</td><td>0.822</td><td>0.281</td><td>1.184</td><td>2.910</td><td>0.067</td><td>25.21</td><td>0.751</td><td>0.300</td></tr><tr><td rowspan="2">200</td><td>StreetGS</td><td>25.32</td><td>0.811</td><td>0.230</td><td>1.878</td><td>3.925</td><td>0.041</td><td>23.75</td><td>0.739</td><td>0.247</td></tr><tr><td>Ours</td><td>28.12</td><td>0.829</td><td>0.219</td><td>1.418</td><td>3.817</td><td>0.020</td><td>25.67</td><td>0.752</td><td>0.235</td></tr><tr><td rowspan="2">209</td><td>StreetGS</td><td>26.53</td><td>0.840</td><td>0.243</td><td>1.932</td><td>3.813</td><td>0.125</td><td>24.77</td><td>0.779</td><td>0.258</td></tr><tr><td>Ours</td><td>29.05</td><td>0.854</td><td>0.217</td><td>1.481</td><td>3.697</td><td>0.061</td><td>26.53</td><td>0.791</td><td>0.231</td></tr><tr><td rowspan="2">359</td><td>StreetGS</td><td>26.08</td><td>0.831</td><td>0.211</td><td>1.539</td><td>3.710</td><td>0.075</td><td>23.53</td><td>0.716</td><td>0.230</td></tr><tr><td>Ours</td><td>27.63</td><td>0.843</td><td>0.199</td><td>1.299</td><td>3.659</td><td>0.039</td><td>24.45</td><td>0.723</td><td>0.218</td></tr><tr><td rowspan="2">529</td><td>StreetGS</td><td>27.50</td><td>0.871</td><td>0.186</td><td>1.065</td><td>2.883</td><td>0.063</td><td>25.96</td><td>0.819</td><td>0.197</td></tr><tr><td>Ours</td><td>30.09</td><td>0.884</td><td>0.158</td><td>0.718</td><td>2.795</td><td>0.020</td><td>27.69</td><td>0.827</td><td>0.169</td></tr><tr><td rowspan="2">916</td><td>StreetGS</td><td>24.34</td><td>0.787</td><td>0.223</td><td>2.268</td><td>4.813</td><td>0.175</td><td>22.21</td><td>0.680</td><td>0.245</td></tr><tr><td>Ours</td><td>25.80</td><td>0.804</td><td>0.204</td><td>1.781</td><td>4.709</td><td>0.106</td><td>23.11</td><td>0.692</td><td>0.226</td></tr></table>
+
+Table A10: Comparative Performance Analysis against StreetGS on the NuScenes Dataset. This table presents a detailed per-scene comparison of our approach ('Ours') with the StreetGS baseline, evaluating scene reconstruction (PSNR, SSIM, LPIPS), novel view synthesis (PSNR, SSIM, LPIPS), and geometry (CD, RMSE, Depth) metrics.
+
+Table A11: Detailed comparison on Challenging Argoverse Scenarios (6 scenes).  
+
+<table><tr><td rowspan="2">Method</td><td colspan="5">Reconstruction</td><td colspan="2">Novel View Synthesis</td></tr><tr><td>PSNR ↑</td><td>SSIM ↑</td><td>CD ↓</td><td>RMSE ↓</td><td>Depth ↓</td><td>PSNR ↑</td><td>SSIM ↑</td></tr><tr><td>OmniRe (Baseline)</td><td>23.95</td><td>0.850</td><td>0.964</td><td>3.647</td><td>0.098</td><td>22.10</td><td>0.761</td></tr><tr><td>OmniRe w/ AC</td><td>23.97</td><td>0.848</td><td>0.923</td><td>3.642</td><td>0.097</td><td>22.09</td><td>0.759</td></tr><tr><td>OmniRe w/ BG</td><td>24.29</td><td>0.855</td><td>0.807</td><td>3.586</td><td>0.047</td><td>22.32</td><td>0.768</td></tr><tr><td>Ours</td><td>25.29</td><td>0.863</td><td>0.726</td><td>3.564</td><td>0.027</td><td>22.96</td><td>0.773</td></tr></table>
+
+Table A12: Detailed comparison on Challenging Waymo Scenarios (6 scenes).  
+
+<table><tr><td rowspan="2">Method</td><td colspan="5">Reconstruction</td><td colspan="2">Novel View Synthesis</td></tr><tr><td>PSNR ↑</td><td>SSIM ↑</td><td>CD ↓</td><td>RMSE ↓</td><td>Depth ↓</td><td>PSNR ↑</td><td>SSIM ↑</td></tr><tr><td>OmniRe (Baseline)</td><td>29.28</td><td>0.833</td><td>0.352</td><td>1.917</td><td>0.061</td><td>27.37</td><td>0.791</td></tr><tr><td>OmniRe w/ AC</td><td>29.31</td><td>0.833</td><td>0.343</td><td>1.915</td><td>0.053</td><td>27.30</td><td>0.789</td></tr><tr><td>OmniRe w/ BG</td><td>29.61</td><td>0.836</td><td>0.333</td><td>1.893</td><td>0.042</td><td>27.56</td><td>0.792</td></tr><tr><td>Ours</td><td>31.23</td><td>0.841</td><td>0.272</td><td>1.846</td><td>0.021</td><td>28.34</td><td>0.793</td></tr></table>
+
+Table A13: Detailed comparison on Challenging Pandaset Scenarios (6 scenes).  
+
+<table><tr><td rowspan="2">Method</td><td colspan="5">Reconstruction</td><td colspan="2">Novel View Synthesis</td></tr><tr><td>PSNR ↑</td><td>SSIM ↑</td><td>CD ↓</td><td>RMSE ↓</td><td>Depth ↓</td><td>PSNR ↑</td><td>SSIM ↑</td></tr><tr><td>OmniRe (Baseline)</td><td>27.28</td><td>0.859</td><td>0.634</td><td>3.963</td><td>0.029</td><td>23.91</td><td>0.721</td></tr><tr><td>Ours</td><td>27.88</td><td>0.863</td><td>0.572</td><td>3.951</td><td>0.022</td><td>24.22</td><td>0.724</td></tr></table>
+
+Table A14: Comparison results of vehicle-specific metrics, averaged across 20 scenes from all four datasets.  
+
+<table><tr><td>Method</td><td>PSNR ↑</td><td>SSIM ↑</td><td>CD ↓</td></tr><tr><td>OmniRe (Baseline)</td><td>24.24</td><td>0.784</td><td>8.675</td></tr><tr><td>Ours</td><td>24.80</td><td>0.791</td><td>7.400</td></tr></table>
+
+# C.3 Performance on Challenging Scenarios
+
+To validate robustness, we curated subsets of challenging scenarios with extreme photometric inconsistencies (e.g., night scenes, sun glare, rain, and complex reflections). As shown in Tables A11, A12, and A13, our method's performance gap widens significantly compared to all baselines (OmniRe, w/ AC, w/ BG) in these difficult conditions, demonstrating the superior robustness of our multi-scale design.
+
+- Challenging Argoverse (6 scenes): 49e970c4 (Extreme Lighting, Overexposure, Lens Flare), 8184872e (Complex Shadows, Direct Sunlight), 91923e20 (Extreme Lighting, Camera Artifacts), a8a2fbc2 (Low-Angle Sun), b403f8a3 (Overpass Structure, Extreme Lighting Transitions), eaaf5ad3 (Multiple Artificial Lights, Wet Surfaces, Rainy Conditions).  
+- Challenging Waymo (6 scenes): segment-10 (Night Scene, Artificial Lighting, Lens Flare), segment-30 (Night Scene, Rainy Conditions, Specular Reflections), segment-539 (High Dynamic Range, Low-Angle Sun, Sun Glare), segment-550 (High Dynamic Range, Storefront Reflections, Hard Shadows), segment-561 (Low-Angle Sun, Lens Flare, Over-exposure), segment-570 (Rainy Conditions, Low Light, Motion Blur).  
+- Challenging Pandaset (6 scenes): 19, 21, 29, 48, 52 (all Low-Angle Sun with Hard/Deep Shadows or High Dynamic Range), 63 (Night Scene, Multiple Light Sources, Glare).
+
+# C.4 Vehicle-Specific Metrics
+
+To specifically evaluate performance on challenging, highly reflective surfaces such as car windows, we used semantic masks to isolate vehicles. Table A14 shows that our method improves not only appearance (PSNR) but also, crucially, geometry (CD) for these specific objects. This supports our claim that our method mitigates the negative geometric impact of such severe view-dependent effects.
+
+# D Quantitative Analysis of Photometric Inconsistency
+
+To rigorously validate the learned affine transformations discussed in the main paper, we compute the Kullback-Leibler (KL) Divergence between our model's learned distribution and a ground-truth (GT) distribution of real-world photometric inconsistency.
+
+Experimental Setup: We construct the GT distribution  $P(\Delta L)$  by sampling corresponding pixels representing the same 3D world points across different camera views in the nuScenes dataset, using ground-truth LiDAR points and camera parameters. The luminance difference  $\Delta L$  for all corresponding pairs forms the GT histogram. We then measure the KL Divergence between this  $P(\Delta L)$  and the distributions learned by our model and the baseline.
+
+Results: As shown in Table A15, the distribution learned by our multi-scale method achieves a significantly lower KL Divergence (0.79) compared to the single-scale baseline (1.25). This quantitatively confirms that our model learns a distribution that more accurately reflects real-world photometric inconsistencies.
+
+Table A15: Comparison of KL Divergence against the ground-truth (GT) photometric inconsistency distribution from the nuScenes dataset. Lower is better, indicating a more realistic learned distribution.  
+
+<table><tr><td>Method</td><td>KL Divergence (vs. Dataset GT) ↓</td></tr><tr><td>Single Bilateral Grids&#x27; Affine Transformation</td><td>1.25</td></tr><tr><td>Multi-scale Bilateral Grids&#x27; Affine Transformation (Ours)</td><td>0.79</td></tr></table>
+
+# References
+
+[1] Holger Caesar, Varun Bankiti, Alex H Lang, Sourabh Vora, Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Giancarlo Baldan, and Oscar Beijbom. nuscenes: A multimodal dataset for autonomous driving. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 11621-11631, 2020.
+
+[2] Pei Sun, Henrik Kretzschmar, Xerxes Dotiwalla, Aurelien Chouard, Vijaysai Patnaik, Paul Tsui, James Guo, Yin Zhou, Yuning Chai, Benjamin Caine, et al. Scalability in perception for autonomous driving: Waymo open dataset. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 2446-2454, 2020.  
+[3] Adam Tondersski, Carl Lindström, Georg Hess, William Ljungbergh, Lennart Svensson, and Christoffer Petersson. Neurad: Neural rendering for autonomous driving. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 14895-14904, 2024.  
+[4] Yuxi Wei, Zi Wang, Yifan Lu, Chenxin Xu, Changxing Liu, Hao Zhao, Siheng Chen, and Yanfeng Wang. Editable scene simulation for autonomous driving via collaborative lvm-agents. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 15077-15087, 2024.  
+[5] Benjamin Wilson, William Qi, Tanmay Agarwal, John Lambert, Jagjeet Singh, Siddhesh Khandelwal, Bowen Pan, Ratnesh Kumar, Andrew Hartnett, Jhony Kaesemodel Pontes, et al. Argoverse 2: Next generation datasets for self-driving perception and forecasting. arXiv preprint arXiv:2301.00493, 2023.  
+[6] Pengchuan Xiao, Zhenlei Shao, Steven Hao, Zishuo Zhang, Xiaolin Chai, Judy Jiao, Zesong Li, Jian Wu, Kai Sun, Kun Jiang, et al. Pandaset: Advanced sensor suite dataset for autonomous driving. In 2021 IEEE international intelligent transportation systems conference (ITSC), pages 3095-3101. IEEE, 2021.  
+[7] Yunzhi Yan, Haotong Lin, Chenxu Zhou, Weijie Wang, Haiyang Sun, Kun Zhan, Xianpeng Lang, Xiaowei Zhou, and Sida Peng. Street gaussians: Modeling dynamic urban scenes with gaussian splatting. In European Conference on Computer Vision, pages 156-173. Springer, 2024.
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+# Unifying Attention Heads and Task Vectors via Hidden State Geometry in In-Context Learning
+
+Haolin Yang<sup>1</sup> Hakaze Cho<sup>2</sup> Yiqiao Zhong<sup>3</sup> Naoya Inoue<sup>2,4</sup>  
+<sup>1</sup>University of Chicago <sup>2</sup>JAIST <sup>3</sup>University of Wisconsin - Madison <sup>4</sup>RIKEN haolinyang2001@uchicago.edu, yfzhao@jaist.ac.jp yiqiao.zhong@wisc.edu, naoya-i@jaist.ac.jp
+
+# Abstract
+
+The unusual properties of in-context learning (ICL) have prompted investigations into the internal mechanisms of large language models. Prior work typically focuses on either special attention heads or task vectors at specific layers, but lacks a unified framework linking these components to the evolution of hidden states across layers that ultimately produce the model's output. In this paper, we propose such a framework for ICL primarily in classification tasks by analyzing two geometric factors that govern performance: the separability and alignment of query hidden states. A fine-grained analysis of layer-wise dynamics reveals a striking two-stage mechanism—separability emerges in early layers, while alignment develops in later layers. Ablation studies further show that Previous Token Heads drive separability, while Induction Heads and task vectors enhance alignment. Our findings thus bridge the gap between attention heads and task vectors, offering a unified account of ICL's underlying mechanisms. $^{1}$
+
+# 1 Introduction
+
+One of the most remarkable features of Large Language Models (LLMs) is their ability to respond to novel queries in user-desired mannerisms on-the-fly solely by learning from demonstrations provided in the input (as shown in Figure 1 (A))—without any additional training. This capability is known as In-context Learning (ICL) [Brown et al., 2020, Radford et al., 2019]. ICL has revolutionized natural language processing by reducing dependence on burdensome data collection and costly finetuning, enabling swift and seamless adaptation of LLMs to a variety of downstream tasks [Dong et al., 2024].
+
+Due to its stark departure from traditional gradient-based learning paradigms, ICL has attracted significant academic interest. One line of research aims to develop a mechanistic understanding of ICL by analyzing the internal behavior of LLMs in ICL settings [Reddy, 2024]. Some studies highlight the importance of key structural components in LLMs, such as Induction Heads (IH) [Elhage et al., 2021, Olsson et al., 2022, Cho et al., 2025a], which retrieve information from demonstrations and perform copy-like operations within Transformer-based architectures. Other studies adopt a hidden-state-centric view, interpreting ICL as a process in which LLMs construct vector representations of the task from demonstrations (so-called task vectors [Hendel et al., 2023, Liu et al., 2024]) using the hidden states at an intermediate layer, and then use the vectors to steer hidden states and produce better next-token predictions. Both lines of work support their claims primarily through ablation and intervention experiments that assess the impact on model outputs.
+
+However, treating model components and hidden states as separate entities in ICL analysis is both unnecessary and misleading. The two are inherently intertwined—components like attention heads transform hidden states layer by layer, and these intermediate updates ultimately determine the
+
+![](images/eefadb2478d9cdfcad4f4cb0b38ade54fa3a365bf136cdf73911b0314f714655.jpg)  
+(A) I like this. Sentiment: positive \n It is bad. Sentiment: negative \n Fantastic! Sentiment:  
+Figure 1: (A) An example for ICL input. (B) In early layers, LLMs promote separability among the last tokens' hidden state  $(\pmb{h})$  clusters w.r.t. the ground-truth labels of the queries through Previous Token Heads (PTHs). (C) In early layers or zero-shot scenarios, the direction where the hidden are maximally separated is insufficiently aligned with the output direction (i.e., the difference vector of the label-token unembedding vectors), increasing cluster overlap after mapping and inducing higher classification errors, and (D) in later layers, Induction Heads (IHs) align these clusters towards the output direction, with the same underlying mechanism of task vectors.
+
+model's output. As such, analyzing only the influence of attention heads or task vectors on the final prediction reduces ICL to a black-box process. A true mechanistic understanding of ICL requires tracing how these components progressively shape hidden states throughout the model, not just their end effects. A unified framework that synthesizes both perspectives and assesses their importance in the layer-wise evolution of hidden states is thus highly essential.
+
+Therefore, in this work, we propose a framework, shown in Figure 1, based on a key geometric insight into LLM classification: the success of LLMs in classification depends on whether the hidden states of queries with different ground-truth labels are (1) sufficiently separable and (2) well aligned with the unembedding vectors of labels. In other words, hidden states should be separated along a direction that aligns with the difference between the unembedding vectors of different labels. To investigate this, we collect hidden states in both ICL and zero-shot settings and analyze them using a range of geometric measures to assess how ICL improves hidden states in terms of separability and alignment. Our approach operates on three levels: (1) We examine the layer-wise trends in these measures and compare zero-shot with ICL (and across different ICL settings) to quantify the effect of ICL on the hidden state geometry. (2) We observe how changes in measure values reflect the evolution of semantic information represented in the hidden states that ultimately inform model outputs in ICL. (3) We probe specific layers to examine the role of different types of attention heads and how they influence both the values and trajectories of these geometric measures.
+
+Our experiments find that ICL improves classification performance primarily by enhancing alignment between query hidden states and unembedding vectors, with less impact on separability. Results further suggest that ICL proceeds as a two-stage process: early layers mainly increase hidden-state separability, while in middle-to-late layers, separability plateaus and alignment improves. The refinement of alignment arises through layer updates that filter out task-irrelevant semantics and preserve label-relevant content, thereby semantically refining query hidden states. Such updates qualify ICL query hidden states as task vectors. We also test this two-stage characterization on ICL for a generation task and confirm its explanatory power across settings.
+
+To build the bridge between the task vector and model components, we analyze the effect of attention heads on these geometric measures and find that (as demonstrated by Figure 1): (1) The early increase in separability is driven by a proliferation of Previous Token Heads (PTHs)—attention heads that attend to the previous token at each position—in early layers. (2) In middle-to-late layers, Induction Heads (IHs) enhance alignment by amplifying hidden state components along the unembedding directions of the labels. Thus, our findings not only explain why attention heads like PTHs and IHs are essential for ICL performance, but also clarify why induction head outputs empirically serve as effective task vectors [Todd et al., 2024, Hendel et al., 2023].
+
+# 2 Related works
+
+IH and PTH Previous works on the mechanistic cause of ICL identify structural circuits in Transformers comprising a Previous Token Head (PTH) and an Induction Head (IH), where the PTH gathers information at demonstration label tokens, and the IH copies it to query positions for prediction [Olsson et al., 2022, Singh et al., 2024, Song et al., 2025]. Their importance in classification has been confirmed in large-scale models as well [Cho et al., 2025a]. However, existing analyses focus on end-to-end effects, leaving open how these heads shape query hidden states.
+
+Task vectors An alternative theory proposes that LLMs summarize demonstrations into task vectors, which can be used to steer zero-shot hidden states to increase prediction accuracy. These vectors may be ICL query hidden states extracted at an intermediate layer [Hendel et al., 2023, Liu et al., 2024], attention head outputs [Todd et al., 2024, Li et al., 2024], or MLP outputs [Merullo et al., 2024]. Decoding task vectors through the unembedding matrix reveals that they encode task-related tokens and support semantic manipulations akin to "word2vec"-style algebra [Mikolov et al., 2013]. Despite intriguing empirical findings, explanations for their mechanism remain preliminary. Kahardipraja et al. [2025] show that heads expressing contextual information or model's parametric knowledge produce task vectors with distinct function. Jiang et al. [2025b] and Han et al. [2025] study how the geometry of model's task representation impacts the effects of injecting task vectors. Jiang et al. [2025a] reveal the correlation between model's catastrophic forgetting and the change in task vectors. However, none of these studies answer the question of how task vectors influence the concrete internal computation process of models which directly and ultimately determine the outputs.
+
+Geometry of hidden states LLMs encode inputs in geometrically structured ways [Park et al., 2024]. For instance, they encode cities and countries by geographic relations [Gurnee and Tegmark, 2024] and numbers in rings under modular arithmetic [Liu et al., 2023]. For classification tasks, they can separate hidden states into clusters by attributes such as truth value [Marks and Tegmark, 2024] or toxicity [Lee et al., 2024]. However, the alignment of such clusters with the unembedding vectors of their labels—an equally crucial geometric factor—has received limited attention, with only a few studies noting suboptimal alignment at the output layer [Cho et al., 2025b, Kirsanov et al., 2025].
+
+# 3 Theoretical framework
+
+We begin our framework with an analysis of classification tasks, where an LLM is prompted with a query  $x$  and its ground-truth label  $y$ , e.g.,  $x =$  "I like this movie. Sentiment:" and  $y =$  "positive". The LLM with  $L$  layers encodes  $x$  into a vector  $\pmb{h}_x \in \mathbb{R}^d$ , taken from the hidden state at the final token (":"). Given a vocabulary  $\mathbb{V} = \{v_1, \dots, v_{|\mathbb{V}|}\}$  and an output embedding matrix  $\pmb{E} \in \mathbb{R}^{|\mathbb{V}| \times d}$ , the model predicts  $p(v_j|x) = \exp(\pmb{E}_{v_j} \pmb{h}_x) / \sum_{j'=1}^{|\mathbb{V}|} \exp(\pmb{E}_{v_j'} \pmb{h}_x)$ , where  $\pmb{E}_{v_j}$  is the unembedding vector of  $v_j$ . The model outputs the token  $v_j$  with the highest probability as the predicted label for  $x$ .
+
+In the ICL scenario, as shown in Figure 1 (A),  $k$  demonstration-label pairs  $(x_{1},y_{1}),\ldots ,(x_{k},y_{k})$  are pretended to form the prompt  $[x_{1},y_{1},\dots,x_{k},y_{k},x]$ , which modifies the hidden state of  $x$  and results in a different distribution  $p(\nu_j|x_1,y_1,\dots,x_k,y_k,x)$ , potentially improving prediction accuracy.
+
+Rather than a single query, we more often evaluate classification accuracy over a dataset of  $n$  queries  $\{x_{1},\ldots ,x_{n}\}$ , with hidden states  $\{\pmb {h}_1,\dots ,\pmb {h}_n\}$  and ground-truth labels  $\{y_1,\dots ,y_n\}$ . We will show that the classification accuracy depends on two geometric properties of the hidden state collection  $H = [h_i]_{i = 1}^n\in \mathbb{R}^{n\times d}$ : (1) its separability, and (2) its alignment with the unembedding vectors.
+
+As mentioned before, an LLM performs classification using the unembedding matrix  $\mathbf{E}$ , which is a multiclass classifier on the hidden states. We define its accuracy on the dataset as:
+
+$$
+\operatorname {A c c} = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {1} \left(y _ {i} = \underset {\nu_ {j}} {\arg \max } \frac {\exp \left(\boldsymbol {E} _ {\nu_ {j}} \boldsymbol {h} _ {i}\right)}{\sum_ {j ^ {\prime} = 1} ^ {| \mathrm {V} |} \exp \left(\boldsymbol {E} _ {\nu_ {j} ^ {\prime}} \boldsymbol {h} _ {i}\right)}\right) = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {1} \left(y _ {i} = \underset {\nu_ {j}} {\arg \max } \frac {\boldsymbol {E} _ {\nu_ {j}} \boldsymbol {h} _ {i}}{\sum_ {j ^ {\prime} = 1} ^ {| \mathrm {V} |} \boldsymbol {E} _ {\nu_ {j} ^ {\prime}} \boldsymbol {h} _ {i}}\right), \tag {1}
+$$
+
+where the first equality states that classification accuracy equals the proportion of the labels being predicted as the most probable token, and the second equality uses the monotonicity of softmax.
+
+We first consider the simple case where  $\{x_1, \dots, x_n\}$  only has 2 different labels,  $y_A$  and  $y_B$ , i.e.  $y_i \in \{y_A, y_B\}, \forall i$ , and leave the generalization to the case of more labels to Appendix B. For a unit vector  $\mathbf{u} \in \mathbb{R}^d$ , we define the separability of  $\mathbf{H}$  along direction  $\mathbf{u}$  with labels  $y_A, y_B$  to be:
+
+$$
+S (\boldsymbol {u}) = \frac {1}{n} \left(\sum_ {i \in \mathbb {N} _ {A}} \mathbb {1} \left(\boldsymbol {u} ^ {\top} \boldsymbol {h} _ {i} \geq 0\right) + \sum_ {i \in \mathbb {N} _ {B}} \mathbb {1} \left(\boldsymbol {u} ^ {\top} \boldsymbol {h} _ {i} <   0\right)\right), \tag {2}
+$$
+
+where  $\mathbb{N}_A = \{i:y_i = y_A\}$  and  $\mathbb{N}_B = \{i:y_i = y_B\}$ .  $S(\pmb{u})$  measures the fraction of hidden states that can be correctly classified by a linear decision boundary orthogonal to  $\pmb{u}$ . By considering all possible directions, we obtain the maximum separability of  $\pmb{H}$ , i.e.  $S^{*} = \sup_{\pmb{u}\in \mathbb{R}^{d},\| \pmb{u}\|_{2} = 1}S(\pmb{u})$ . Though  $S(\pmb{u})$
+
+is not a continuous function,  $S^*$  as a supremum can be attained on  $\mathbb{S}^{d-1} = \{\pmb{u} : \pmb{u} \in \mathbb{R}^d, \| \pmb{u} \|_2 = 1\}$  (See Appendix A). Denote  $\pmb{u}^*$  as the  $\pmb{u}$  where  $S$  attains the supremum, then:
+
+Theorem 1 Acc  $\leq S^{*}$ . The equality is achieved when  $\max_{v\in \mathbb{V}, v\notin \{y_A, y_B\}} \pmb{E}_v \pmb{h}_i < \max (\pmb{E}_{y_A} \pmb{h}_i, \pmb{E}_{y_B} \pmb{h}_i), \forall i$  and  $\frac{\pmb{E}_{y_A} - \pmb{E}_{y_B}}{\|\pmb{E}_{y_A} - \pmb{E}_{y_B}\|_2} = c\pmb{u}^*$  for some positive constant  $c$ .
+
+This theorem shows that accuracy is upper-bounded by the maximum separability of  $\pmb{H}$  and that the bound is achieved when: (1) each  $\pmb{h}_i$  is maximally aligned with either  $\pmb{E}_{y_A}$  or  $\pmb{E}_{y_B}$  — the unembedding vectors of  $y_A$  and  $y_B$  — while excluding interference from other tokens, a condition we call output alignment; and (2) the direction of maximum separation  $\pmb{u}^*$  aligns with  $\pmb{E}_{y_A} - \pmb{E}_{y_B}$ , which we refer to as directional alignment. Thus, separability and alignment of hidden states critically determine accuracy. Since query hidden states in ICL differ from those in the zero-shot setting due to the presence of demonstrations, we conclude that ICL can improve classification accuracy by enhancing both their separability and alignment.
+
+Attention heads influence query hidden states as well by their outputs to the residual stream. At layer  $l$ , the  $h^{\mathrm{th}}$  attention head adds  $\pmb{a}_{l,l}^{h}$  to the hidden state  $\pmb{h}_{i,l}$  of the  $i^{\mathrm{th}}$  query, affecting the transition to  $\pmb{h}_{i,l+1}$ . Ablating a head (i.e., setting  $\pmb{a}_{i,l}^{h} = 0$ ) alters all downstream hidden states and can degrade accuracy if the final-layer hidden states  $\pmb{H}_L$  lose separability or alignment.
+
+Task-vector-based experiments operate similarly by steering hidden states at intermediate layers. In those experiments, a task vector  $\pmb{t}_i$  is added to  $^2\pmb{h}_{i,l}$  and the influence likewise propagates through subsequent layers. This intervention is effective if the final  $H_{L}$  has better separability and is better aligned with the labels of the task encoded in  $\pmb{t}_i$ .
+
+# 4 Method: Measuring separability and alignment of hidden states
+
+Given the aforementioned influence of ICL, attention heads, and task vectors on the separability and alignment of hidden states across layers, we introduce several measures<sup>3</sup> to quantify their influence. The formal definitions and mathematical details of all measures are provided in Appendix C.
+
+# A. Separability measure: Separability score
+
+Theorem 1 shows that the classification is affected by the maximum separability of hidden states. However, finding maximum separability requires evaluating separability along infinitely many directions in  $\mathbb{S}^{d - 1}$  and is intractable. As a practical proxy, we train a logistic classifier on a subset of  $H$  and evaluate its accuracy on held-out data. We call the accuracy separability score as it reflects the empirical maximum separability of hidden states achieved by the classifier.
+
+# B. Alignment measures
+
+In Theorem 1 we show that both output alignment and directional alignment affect accuracy. Below, we introduce measures for both.
+
+# B.1 Output alignment
+
+Output alignment is defined as the classification accuracy obtained by applying the unembedding matrix  $\pmb{E}$  directly to hidden states  $\pmb{H}$ , skipping subsequent layers. It measures how well  $\pmb{H}$  aligns with the label unembedding vectors  $\pmb{E}_{y_A}$  and  $\pmb{E}_{y_B}$ , and is also known as logit lens accuracy [nostalgebraist, 2020]. When  $\pmb{H}$  comes from the final layer, it equals standard classification accuracy.
+
+# B.2-B.5 Directional alignment
+
+Directional alignment is the alignment between  $\pmb{u}^*$  and the label unembedding difference direction  $\frac{\pmb{E}_{y_A} - \pmb{E}_{y_B}}{\|\pmb{E}_{y_A} - \pmb{E}_{y_B}\|_2}$ . Since finding  $\pmb{u}^*$  is intractable, we introduce the following proxies based on approximating  $\pmb{u}^*$  or measuring separability along the label-difference direction.
+
+- Singular alignment (B.2). To approximate  $\pmb{u}^{*}$ , we apply Singular Value Decomposition (SVD) to the mean-centered  $\pmb{H}$  and extract the top- $r$  right singular vectors. These are the directions where  $\pmb{H}$  has the greatest spread and can approximate  $\pmb{u}^{*}$ . We compute the maximum absolute cosine similarities between each of these vectors and  $\frac{\pmb{E}_{y_A} - \pmb{E}_{y_B}}{\|\pmb{E}_{y_A} - \pmb{E}_{y_B}\|_2}$  as a proxy for directional alignment.  
+- Variance-based alignment (B.3). To measure separability along  $\frac{\pmb{E}_{y_A} - \pmb{E}_{y_B}}{\|\pmb{E}_{y_A} - \pmb{E}_{y_B}\|_2}$ , we first measure the proportion of total variance in  $\pmb{H}$  that lies along this direction. A higher value indicates that this direction is a principal axis of variation—suggesting stronger directional alignment.  
+- Mean-based alignment (B.4). We project the difference in means of label clusters—hidden states with labels  $y_{A}$  and  $y_{B}$ —onto  $\frac{\pmb{E}_{y_{A}} - \pmb{E}_{y_{B}}}{\|\pmb{E}_{y_{A}} - \pmb{E}_{y_{B}}\|_{2}}$  and divide it by the pooled projected variance. A higher value implies better separation margin along  $\frac{\pmb{E}_{y_{A}} - \pmb{E}_{y_{B}}}{\|\pmb{E}_{y_{A}} - \pmb{E}_{y_{B}}\|_{2}}$  and better directional alignment.  
+- Composite alignment (B.5). To unify the mean- and variance-based alignment, we define composite alignment as their product. It captures both the magnitude and the relative significance of separation along the label-difference direction, serving as a robust proxy for directional alignment.
+
+# C. Indirect measure: Effective dimension
+
+As an indirect indicator of separability and alignment, we compute the effective dimension of  $\pmb{H}$  — the number of directions along which the hidden states exhibit significant variance. A low effective dimension suggests that hidden states concentrate along a small number of directions, potentially those associated with the label unembedding vectors, and are thus easier to separate.
+
+# 5 Experiments and Results
+
+Models We experiment on the following 7 models: Llama2-7B, Llama2-13B, Llama2-70B [Touvron et al., 2023], Llama3-8B, Llama3-70B [Grattafori et al., 2024], Gemma-2B, and Gemma-7B [Gemini Team et al., 2024]. Unless otherwise stated, we report the results on Llama2-70B.
+
+Datasets We conduct experiments on two major types of classification datasets: text classification (SUBJ [Wang and Manning, 2012], SST-2 [Socher et al., 2013], TREC [Li and Roth, 2002]) and natural language inference (SNLI [MacCartney and Manning, 2008], RTE [Dagan et al., 2005], and CB [De Marneffte et al., 2019]). We further include a generation dataset (detailed in Subsection 5.1) to test the generalizability of our results in settings other than classification. We also use a generated dataset to explore the scenario where ground-truth labels of queries are not presented in the demonstrations (discussed in Appendix I.5).
+
+ICL setting We set the number of in-context demonstrations to be 8 unless otherwise stated. The demonstration samples are chosen randomly, except for one experiment that investigates the effect of using demonstrations procured by kNN retrieval [Liu et al., 2022]. For a detailed exposition of the implementation of models, datasets, prompt templates, etc., refer to Appendix D.
+
+# 5.1 Layer-wise trends of separability and alignment
+
+ICL as a two-stage process Figure 2 shows the values of six geometric measures at all layers of Llama2-70B in both zero-shot and ICL settings averaged across datasets (output alignment is deferred to Subsection 5.2 due to its strong link to the semantic information of hidden states; results for other models are in Appendix F). A key observation is that Figure 2 reveals a striking phase transition in the evolution along the layers of ICL hidden states. Phase 1 (Figure 1 (B)): In the early layers, directional alignment measures take low values, whereas separability increases rapidly. Phase 2 (Figure 1 (D)): In the subsequent layers, this trend reverses: separability plateaus, while all directional alignment measures spike simultaneously, which suggests that the label unembedding difference direction starts to capture more variance and separate label clusters more effectively. It causes the semantic information of query labels to be predominant in  $H$ , which ultimately leads to final output alignment. These findings are consistent with prior work showing that task information emerges only after the initial layers [Sia et al., 2024]. The alternation between two phases is also captured by the effective dimension of  $H$ , which first rises and then declines as alignment starts to improve, reflecting a gradual concentration of hidden states along label unembedding directions. In contrast, this transition is much weaker in the zero-shot setting, where effective dimension rises
+
+![](images/dc5b408d50adb8f02fd7050cb1013102a2555da03062a0b4bdcb8bd107d6c6c1.jpg)
+
+![](images/932ff5bd8ca8640e273c4e2e640a7045bec800c96acdae16ce40e6b01f43bf84.jpg)
+
+![](images/d6d3866eaad2922525251cba7146204e2052fbd3a82aab84fbdcffe6d58d09e7.jpg)
+
+![](images/dc1d04d3840de22f7025afc306e80e5f24e46aa94c40075b67750d959f81f6f0.jpg)  
+Figure 2: Comparison of trends in separability and alignment measures: ICL vs. zero-shot. Under ICL, a clear phase transition emerges: separability increases first and then alignment surges. The effective dimension first rises and then declines. This pattern is missing in the zero-shot setting. Accuracy gains from ICL over zero-shot are reflected in alignment, not separability measures.
+
+![](images/82495cb6081b4887eaa06b0b0c1d367651d477528616a676a38a8f2a244b2903.jpg)
+
+![](images/03f9264bfc640437367af6398f136fb1963c5c49260e27c4f676126aff86c31b.jpg)
+
+monotonously and the rise in alignment measures is not nearly as pronounced. Hence, this phase transition pattern characterizes ICL as a two-stage process: an initial phase enhancing separability, followed by a refinement phase that improves alignment.
+
+ICL improves accuracy primarily through alignment. A key observation is that ICL and zero-shot settings show only a small gap in separability scores—despite an  $80\%$  (Figure 2, upper middle) difference in classification accuracy. This implies that the semantic separation of query hidden states is intrinsic to LLM inference rather than a main effect of ICL (see Appendix E.1 for the practical implication of this), making alignment the primary bottleneck in zero-shot performance. As shown in Figure 1 (C), the high-separability direction in zero-shot hidden states is misaligned with the label-difference vector, as reflected in low singular and composite alignment scores. Along this direction, label clusters remain poorly separated (low mean-based alignment) and explain little variance (low variance-based alignment). These findings align with prior work indicating that unembedding vectors often fail to capture the dominant separation axes in hidden state space [Cho et al., 2025b].
+
+Given the volatility of ICL accuracy across settings with different demonstration numbers and input formats, we investigate whether the phase transition persists across settings, and whether accuracy differences can similarly be attributed to alignment properties of hidden states. We consider three settings: (1) Varying the number of demonstrations from 0 to 24 in increments of 4; (2) Changing the demonstration selection strategy to kNN retrieval where demonstrations with the closest zero-shot embeddings to each query are chosen; (3) Replacing original demonstration labels with semantically uninformative symbols like "@" and "#". Figure 3 reports the results averaged across datasets.
+
+Phase transition is a robust hallmark of ICL Figure 3 shows that the phase transition pattern persists across all three settings. In Figure 3 (A), it is evident at all demonstration counts except 0. Figure 3 (B) confirms that principled demonstration selection (via kNN) preserves instead of interfering with the transition. Even in Figure 3 (C), where labels are replaced by symbols, the two-stage pattern in effective dimension remains—distinguishing it from zero-shot, where effective dimension rises monotonically—though the post-transition alignment surge is dampened.
+
+Phase transition holds in open-ended settings To test whether the phase transition pattern occurs in tasks beyond classification, i.e., open-ended generative tasks, where ICL is also frequently applied, we consider the following task: the queries assume the format "Praise/Critique {subject}", where {subject} is the name of a food (e.g., tapas). The labels are reviews for the subject satisfying the sentiment requirement, generated using GPT-4o [OpenAI et al., 2024], for instance, "Praise tapas  $\rightarrow$  The tapas assortment at this restaurant is absolutely delightful!". We track the layer-wise dynamics of the separability score and composite alignment $^{4}$  as representatives of the metrics evaluated on the 8-shot and 0-shot hidden states of this task. The results in Figure 4 mirror those in Figure 2, where
+
+![](images/aab129ff24dc9448d64082fc33a1aae590df5fe09113cf2a676cffe1761a74ab.jpg)  
+(a) Varying number of demonstrations
+
+![](images/f0d15df0fdb219d5ea24004bfa05c4de144f61b1fe786c11e84a7bf0a117c2e6.jpg)  
+Llama2-70B: Random Selection Acc.: 80.57%; KNN Selection Acc.: 88.01%  
+(b) KNN demonstration selection  
+(c) Symbolized demonstration labels  
+Figure 3: Layer-wise trends of separability and alignment measures in different ICL settings. (A) Phase transition is evident under demonstration numbers from 4 to 24. Accuracy improvements of increasing demonstrations are reflected by consistently improving alignment measures. (B) Changing the demonstration selection method to kNN retrieval preserves phase transition and improves accuracy through enhancing alignment. (C) Using uninformative demonstration labels hurts accuracy due to decreased alignment of hidden states, yet a similar phase transition is evident.
+
+![](images/c9e4e386cca605607424cec2138ce5cdab4237bba6acaa0898cea45f53f15db2.jpg)
+
+the few-shot and zero-shot hidden states exhibit similar separability properties across layers, but the surge in composite alignment beginning from mid-layers is specific to few-shot hidden states.
+
+![](images/f719ca2148d8c2e3d0600cb46997bb9aeb864f21e0bc776e4a55e0b233ccd8c9.jpg)  
+(a) Layer-wise separability score: 0-shot vs. 8-shot  
+Figure 4: (A) In the generation setting, the separability score exhibits almost identical trends in the 0-shot and 8-shot cases. (B) In the 8-shot case, the phase-transition of composite alignment is pronounced.
+
+![](images/fa8cb97558debb18fe3992592f2d6cf28e561fbf6e6fdcd8d95a45bc1c86ae93.jpg)  
+(b) Layer-wise composite alignment: 0-shot vs. 8-shot
+
+Alignment explains accuracy differences across ICL settings We find that: accuracy differences between ICL settings are consistently reflected in alignment measures. Specifically, in Figure 3 (A), increasing the number of demonstrations from 4 to 24 produces minimal changes
+
+in separability, while alignment measures improve in a consistent manner as the trend curves are shifted versions of one another. Effective dimension also decreases steadily, suggesting a higher concentration of hidden states along label-unembedding directions. Figure 3 (B) similarly shows that the accuracy gains from kNN-selected demonstrations correspond to stronger alignment and lower effective dimension. In Figure 3 (C), the accuracy drop from symbolic label replacement aligns with a flattened post-transition rise in alignment measures, resembling the case of zero-shot.
+
+# 5.2 Semantic interpretation of alignment dynamics
+
+To understand how the surge in directional alignment after the phase transition leads to high output alignment and successful decoding of task-related labels at the final layer, we conduct a case study on the SST-2 dataset to interpret hidden-state alignment dynamics.
+
+Surge in alignment concurs with the encoding of label-relevant semantics First, we compute output alignment (logit lens accuracy) across all layers and compare its trajectory to directional alignment measures. As shown in Figure 5, their strong correlation indicates that output alignment rises in sync with directional alignment. This suggests that as alignment improves, label-relevant
+
+semantics are injected into hidden states via consecutive layer updates, enabling correct label decoding even at intermediate layers. To support this, we decode the top right singular vectors of hidden states near the transition point. Figure 5 (B) reveals a sharp semantic shift: while layer 20 produces irrelevant tokens, the SST-2 label token negative emerges as the second most probable token in layer 21—immediately post-transition. This confirms that post-transition layers elevate label-relevant directions in both the semantic and variance structure of the hidden states.
+
+![](images/d572204f9a6d23636eb017624b0b3a33ab757291289ae6f82460fe1a2c7f2577.jpg)
+
+![](images/c026706f532aa1d7181b834995cb4fd817ca3a32f1fd6299fd280d9567195d27.jpg)
+
+![](images/b76e4f7ad30fd3a553b4726a697f6b5948ea938d7d7a457447d58ee9060a26c4.jpg)  
+Figure 5: Dynamics of alignment measures and semantics of post-transition hidden states. (A) Strong correlation between output and directional alignment; (B) Surge in alignment measures concur with encoding of label-related semantics; (C) Post-transition layers, except for the last ones, filter out unrelated semantics and retain the related ones. Refer to Appendix G for more semantics decoding cases.
+
+tions are filtered, leading to a progressive amplification of label-relevant information in the hidden states and a suppression of the irrelevant semantics. However, in the final layers, this pattern reverses: label-related directions are filtered, and label-irrelevant semantics are reintroduced. This shift corresponds to the slight drop in output alignment in the final layers in Figure 5 (A), and aligns with prior findings that late layers in LLMs tend to encode high-frequency, semantically uninformative tokens [Sharma et al., 2023], which may interfere with the expression of task-relevant content. The same phenomenon also occurs in other datasets as presented in Appendix G.4.
+
+Layer-wise pattern of semantic retention and filtering To better understand how exactly layer updates after the phase transition encode relevant semantics into hidden states, we analyze the semantic information retained or removed by each layer. Given the SVDs of the centered hidden states at two consecutive layers  $\bar{H}_{l - 1} = U_{l - 1}\Sigma_{l - 1}V_{l - 1}^{\top}$  and  $\bar{H}_l = U_l\Sigma_lV_l^\top$  , we compute the projection of the previous layer's singular directions onto the next layer's as  $V_{l - 1}^{\top}V_{l}V_{l}^{\top}\in \mathbb{R}^{n\times d}$  . The rows of this matrix with larger norms are the principal directions of  $H_{l - 1}$  that are retained in  $H_{l}$  , while those with smaller norms indicate filtered directions. By decoding these directions, we identify the semantic information retained or removed by each layer. The results are reported in Figure 5 (C). In most layers after the phase transition, label-related semantic directions are consistently retained, and unrelated direc
+
+![](images/53cb3bbfbb78fe6a6452b2de0160fdf77f0818bad4f2190405e73b0c11a11b36.jpg)  
+Figure 6: Accuracy gains of rank-10 denoising. See Appendix G.5 for results in other settings.
+
+Low-rank denoising enhances label-relevant semantics To better understand the retention-and-filtering process, we test whether filtering can be accelerated—and label-relevant semantics enhanced—by denoising the hidden states. Specifically, we apply a rank-10 SVD approximation to the centered hidden states  $\bar{H}$ , retaining only the top 10 directions of variation and adding the mean back. This denoising should
+
+theoretically remove less informative components and amplify label-relevant semantics. We then feed the denoised hidden states into the unembedding layer and measure output accuracy. As shown in Figure 6, low-rank denoising yields accuracy gains exceeding  $10\%$ , especially in early post-transition layers where irrelevant semantics are just beginning to be filtered. These results validate low-rank denoising and support semantic retention and filtering as a core mechanism of post-transition layers. We explore the practical implication of this in Appendix E.2.
+
+Task vector properties of post-transition hidden states As visualized by Figure 1 (D), the simultaneous enhancement of both alignment properties and label-relevant semantics after the phase transition qualifies ICL query hidden states in middle-to-late layers as task vectors [Hendel et al., 2023]. Due to their improved alignment properties, steering or replacing zero-shot hidden states
+
+Normal Acc.:  $80.57\%$  | w/o PTH Acc.:  $33.11\%$  | w/o IH Acc.:  $2.50\%$  | Random Heads Ablated Acc.:  $79.13\%$
+
+![](images/a14ddbbf19dbc60679267516814327e58e17a07efd6bb6d34fc9e1ef14509226.jpg)  
+Effective dimension
+
+![](images/780c3f74a9dbe96cf93c6507be938a990f27b3fd05626dd975b8fb39f3dc69f0.jpg)  
+Separability score
+
+![](images/faef4c044321d8134aeae939bbb2e2f8998c32eef6da64201dd3b06b3f036920.jpg)  
+Singular alignment
+
+Composite alignment  
+![](images/8cebe125da8fc024c25f3dc2c2cecabb3775f0cb1af936337de68f702c3f7aef.jpg)  
+Normal w/o PTH w/o IH Random Heads Ablated
+
+![](images/fe66a802fb1cf47f4e62f14ac99a6537cb6bf709ee18e82620dd3dfb19b1ad4e.jpg)  
+Mean-based alignment  
+Figure 7: Effects of attention heads ablation on the layer-wise separability and alignment measures. Ablating PTH significantly reduces separability, and ablating IH significantly reduces alignment.
+
+![](images/da7e96ea3e5976993144e906ba3de2b3e0fcf4658c496adc2fc6eaf9be74a567.jpg)  
+Var.-based alignment
+
+with ICL hidden states can align zero-shot hidden states better with the label unembedding vectors. Similarly, modifying zero-shot hidden states with ICL hidden states, which are label-informative, makes them more suited for the prediction of the correct labels.
+
+# 5.3 Significance of critical attention heads
+
+PTHs and IHs are attention heads with distinctive attention patterns. Given a token sequence like  $\mathrm{[A][B_1][A][B_2]...[A][B_n][A]}$ , a PTH at the final [A] attends to the immediately preceding  $\left[\mathbf{B}_n\right]$ , while an IH attends to tokens such as  $\left[\mathbf{B}_1\right]-\left[\mathbf{B}_n\right]$  following earlier [A] positions. For example, in the ICL prompt "I don't like it. Answer: negative. I like it. Answer:" at the final " $\vdots$ ", a PTH attends to the preceding "Answer", while an IH attends to "negative" after the earlier " $\vdots$ ", effectively copying semantics of the demonstration label into the query's hidden state. By analyzing their effects on separability and alignment across layers, we gain a more fine-grained understanding of their role in ICL—beyond simply measuring their impact on final predictions.
+
+For each dataset and model, we rank all attention heads by their PTH and IH scores, which quantify how strongly they exhibit corresponding attention patterns on test prompts. We then ablate (zero out) the outputs of the top  $10\%$  identified IHs and PTHs and measure the resulting effects on hidden state separability and alignment across layers as well as the output accuracy. As a control, we repeat the procedure with an equal number of randomly selected heads (excluding the top PTH/ih heads). Details on identifying PTHs and IHs are provided in Appendix H.
+
+PTHs induce separability, IHs induce alignment The results in Figure 7 confirm that PTHs and IHs are crucial not only for verbalizing ICL outputs [Cho et al., 2025a] but also for structuring hidden states both separability-wise and alignment-wise. Ablating random attention heads has a negligible effect (as shown by overlapping black and gray curves). In contrast, ablating PTHs or IHs significantly alters the measures, but with distinct effects. Ablating PTHs substantially reduces separability and also influences mean-based alignment (and thus composite alignment) which concerns the separation margin of the label clusters, but has a minimal impact on other alignment measures. Conversely, ablating IHs leaves separability largely intact but severely disrupts all alignment measures, and has a far greater impact on output alignment as measured by the accuracy.
+
+These results not only confirm a two-stage ICL process—first driven by separability, then by alignment—but also clarify its mechanism. PTHs, concentrated in early layers [Cho et al., 2025a], enhance separability by attending to query tokens and encoding distinct semantics into hidden states. IHs, emerging after a critical depth (30–40% of total layers [Halawi et al., 2024]), copy demonstration label embeddings into query hidden states via the residual stream, improving alignment with label unembedding directions.
+
+Table 1: Dataset-mean layer positions of top IH and PTH heads by percentage levels with statistically significant differences.
+
+<table><tr><td>%</td><td>IH mean</td><td>PTH mean</td><td>p-value</td></tr><tr><td>1%</td><td>40.203</td><td>33.379</td><td>1.01 × 10-7</td></tr><tr><td>2%</td><td>40.884</td><td>36.459</td><td>1.30 × 10-6</td></tr><tr><td>5%</td><td>42.238</td><td>36.422</td><td>2.06 × 10-17</td></tr><tr><td>10%</td><td>41.535</td><td>35.850</td><td>9.72 × 10-29</td></tr></table>
+
+Furthermore, in Appendix I.5, we show that
+
+PTHs and IHs contribute to ICL through the same mechanism even when the query's ground-truth label—previously considered essential for IHs [Cho et al., 2025a—is absent from demonstrations.
+
+PTHs in early layers, IHs in late layers To test whether PTHs and IHs align with our two-stage ICL characterization—inducing separability and alignment respectively—we examine their layer distributions. Specifically, we locate the top  $1\%$ ,  $2\%$ ,  $5\%$ , and  $10\%$  identified PTHs and IHs and perform Mann-Whitney U tests to assess differences across layers. As shown in Table 1, PTHs mainly occur in earlier layers, consistent with their role in inducing separability, while IHs appear later, with the differences being highly significant under the Mann-Whitney test.
+
+![](images/d8ba71deead826e472bd3aac7598d8b0103a6515909791c7c7dcfbc3e4d4f48c.jpg)  
+(a) Layer-wise separability score: PTH task vector  
+Figure 8: (A) PTH task vector increases separability after injection; (B) Injecting IH task vector recreates the phase-transition in alignment.
+
+![](images/c57b8c55d09ff0f091e0bf0070c025b05031774cbfb14602d64f4054aa436852.jpg)  
+(b) Layer-wise composite alignment: IH task vector
+
+Geometric effects of IH and PTH outputs as task vectors The ablation in Figure 7, though revealing how removing PTHs and IHs affects separability and alignment, does not show how they actively drive the geometric evolution of hidden states during inference.
+
+To address this, we follow Todd et al. [2024] and inject their outputs as task vectors into zero-shot hidden states at selected layers to observe induced metric changes. We construct task vectors using outputs of the top  $10\%$  IHs and PTHs as in Appendix J. We inject the PTH-derived vector at layer  $l = 2$  to track separability, and the IH-derived vector at  $\frac{3}{8} L$  to track composite alignment. These layers correspond to where separability and alignment begin to shift markedly in Figure 2. Figure 8a shows that injecting PTH outputs as task vectors boosts separability (the green line surpassing the magenta line before convergence; see Appendix I.2 for similar trends across models), while injecting IH task vectors restores the ICL-specific spike in alignment metrics in zero-shot hidden states.
+
+![](images/37b607c6d79b544d9ce606575895056d56bdceb972733a17c235c3fd7b8cfe6f.jpg)  
+Figure 9: Average effect across datasets of steering zero-shot hidden states using IH outputs as task vectors.
+
+IH outputs as task vectors Since we have shown that IH outputs with ICL demonstrations can compensate for alignment deficits in hidden states, we have clarified how IH outputs as task vectors influence the forward computation of models: they steer hidden states to align with label-token unembeddings so these tokens can be decoded. This also explains why IH outputs themselves decode label-related tokens [Todd et al., 2024], as they must align with label unembedding directions to perform such steering. To validate this conclusion on our classification datasets (not considered in the original study), we extract task vectors from identified IH outputs and add them to layer-30 zero-shot hidden states of Llama2-70B. The substantial accuracy gains in Figure 9 support this conclusion.
+
+# 6 Conclusions and Limitations
+
+Conclusions This work unifies two major perspectives on in-context learning (ICL)—the roles of special attention heads and task vectors—within a geometric framework centered on separability and alignment of hidden states. We theoretically show that these two properties fully determine ICL classification accuracy and empirically demonstrate that ICL across classification and generation tasks improves accuracy over zero-shot mainly by enhancing alignment, explaining why ICL query hidden states serve effectively as task vectors. Our analysis reveals a phase transition: early layers boost separability, while later layers refine alignment with label unembedding vectors by filtering out label-irrelevant semantics and preserving task-relevant directions, coinciding with the emergence of label semantics in hidden states. Zooming into layers, we identify complementary roles of PTHs and IHs: PTHs in early layers enhance separability, while IHs emerging post-transition promote alignment—clarifying both the phase transition and the effectiveness of IH outputs as task vectors.
+
+**Limitations** Our analysis of the ICL hidden states of generation tasks is preliminary. In addition, our analysis focuses on inference-time behavior in pretrained models. Future work should investigate how the phase transition pattern of ICL hidden states emerges during training, which could shed light on the emergence of ICL capability.
+
+# Acknowledgments
+
+This work was supported by JST FOREST Program (Grant Number JPMJFR232K, Japan) and the Nakajima Foundation. We used ABCI 3.0 provided by AIST and AIST Solutions with support from "ABCI 3.0 Development Acceleration Use".
+
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+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
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+
+Answer: [Yes]
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+
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+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: We discuss the limitations of our work in Section 6.
+
+Guidelines:
+
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+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: For our theoretical claims put forward in Section 3, we provide the complete proofs in Appendix A. In Appendix B we consider a generalization of the theory we proposed, the proof of which is also in Appendix B.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+# Answer: [Yes]
+
+Justification: We present the details of the implementations we used for the models, datasets, as well our procedures of conducting in-context learning in Appendix D.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
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+
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
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+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
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+Justification: We have submitted the data and code as the supplementary material.
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+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: The experiment settings and presented in Section 5 Appendix D.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [No]
+
+Justification: Most of the experiment results presented in this work are obtained using fully deterministic procedures with no randomness involved. The only randomness is in the random selection of demonstrations in ICL, but its impact on the results is minimal, so we do not construct error bars over random seeds.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
+
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).  
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We provide such information in Appendix D.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: We conduct the research according to the NeurIPS Code of Ethics in every respect.
+
+Guidelines:
+
+The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: We discuss the potential societal impacts of our findings in Appendix K.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.
+
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
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+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: The paper poses no such risks in any circumstances.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: We provide full reference to the assets we used, primarily in Section 5
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URI.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
+
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [Yes]
+
+Justification: We provide extensive details to the dataset we generated in Appendix I and Appendix D.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: The paper does not involve human subjects and no IRB approvals are required.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
+
+We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+# Answer: [Yes]
+
+Justification: LLM is used to generate one of the datasets used in the experiments. We introduce this dataset in detail in Appendix D and Appendix I. Apart from that, LLM is not related to any aspects of the core methodology in this paper.
+
+# Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# Appendices
+
+# A Proofs related to Theorem 1
+
+We first show that  $S^*$  is attained at some point  $v^*$  in the hypersphere  $\mathbb{S}^{d-1} = \{ \boldsymbol{u} : \boldsymbol{u} \in \mathbb{R}^d, \| \boldsymbol{u} \|_2 = 1 \}$  though  $\mathbb{S}^{d-1}$  is an infinite set and  $S(\boldsymbol{u})$  is not continuous. Observe that each  $\boldsymbol{h}_i$  in  $H$  partitions  $\mathbb{S}^{d-1}$  into two hyperhemispheres. One is  $\{ \boldsymbol{u} : \boldsymbol{u} \in \mathbb{R}^d, \| \boldsymbol{u} \|_2 = 1, \boldsymbol{u}^T \boldsymbol{h}_i \geq 0 \}$  and the other is  $\{ \boldsymbol{u} : \boldsymbol{u} \in \mathbb{R}^d, \| \boldsymbol{u} \|_2 = 1, \boldsymbol{u}^T \boldsymbol{h}_i < 0 \}$ . Hence, as long as the number of queries in the dataset  $n$  is finite,  $\boldsymbol{h}_1, \dots, \boldsymbol{h}_n$  will partition  $\mathbb{S}^{d-1}$  into a finite number of "cells". Each cell corresponds to the elements in  $\mathbb{S}^{d-1}$  that satisfy a certain sign sequence  $[\operatorname{sgn}(\boldsymbol{u}^T \boldsymbol{h}_1), \dots, \operatorname{sgn}(\boldsymbol{u}^T \boldsymbol{h}_n)] \in \{1, 0, -1\}^n$ . Hence,  $S(\boldsymbol{u})$  is constant on each of the cells, and the supremum (or the maximum) value  $S^*$  is attained on one of the cells.
+
+We then show the proof of Theorem 1:
+
+Proof First, consider the case where  $\mathbb{V}$  only consists of  $y_{A}$  and  $y_{B}$ . Then  $\pmb{E} \in \mathbb{R}^{2 \times d}$  and the two rows are  $\pmb{E}_{y_A}$  and  $\pmb{E}_{y_B}$ . Then we have
+
+$$
+\begin{array}{l} \operatorname {A c c} = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {1} \left(y _ {i} = \arg \max  _ {v _ {j}} \frac {\boldsymbol {E} _ {v _ {j}} \boldsymbol {h} _ {i}}{\sum_ {v _ {j} ^ {\prime} \in \left\{y _ {A} , y _ {B} \right\}}} \boldsymbol {E} _ {v _ {j} ^ {\prime}} \boldsymbol {h} _ {i}\right) (3) \\ = \frac {1}{n} \left(\sum_ {i \in \mathbb {N} _ {1}} \mathbb {1} \left(\boldsymbol {E} _ {y _ {A}} \boldsymbol {h} _ {i} \geq \boldsymbol {E} _ {y _ {B}} \boldsymbol {h} _ {i}\right) + \sum_ {i \in \mathbb {N} _ {2}} \mathbb {1} \left(\boldsymbol {E} _ {y _ {A}} \boldsymbol {h} _ {i} <   \boldsymbol {E} _ {y _ {B}} \boldsymbol {h} _ {i}\right)\right) \quad \text {d e f i n i t i o n o f a r g m a x} (4) \\ = \frac {1}{n} \left(\sum_ {i \in \mathbb {N} _ {1}} \mathbb {1} \left[ \left(\boldsymbol {E} _ {y _ {A}} - \boldsymbol {E} _ {y _ {B}}\right) \boldsymbol {h} _ {i} \geq 0 \right] + \sum_ {i \in \mathbb {N} _ {2}} \mathbb {1} \left[ \left(\boldsymbol {E} _ {y _ {A}} - \boldsymbol {E} _ {y _ {B}}\right) \boldsymbol {h} _ {i} <   0 \right] \quad \text {r e o r g a n i z i n g t e r m s} \right. (5) \\ = \frac {1}{n} \left(\sum_ {i \in \mathbb {N} _ {1}} \mathbb {1} \left[ \left(\frac {\boldsymbol {E} _ {y _ {A}} - \boldsymbol {E} _ {y _ {B}}}{\| \boldsymbol {E} _ {y _ {A}} - \boldsymbol {E} _ {y _ {B}} \| _ {2}}\right) \boldsymbol {h} _ {i} \geq 0 \right] + \sum_ {i \in \mathbb {N} _ {2}} \mathbb {1} \left[ \left(\frac {\boldsymbol {E} _ {y _ {A}} - \boldsymbol {E} _ {y _ {B}}}{\| \boldsymbol {E} _ {y _ {A}} - \boldsymbol {E} _ {y _ {B}} \| _ {2}}\right) \boldsymbol {h} _ {i} <   0 \right] \quad \text {l i n e a r i t y} \right. (6) \\ \leq \sup  _ {\boldsymbol {u} \in R ^ {d}, \left\| \boldsymbol {u} \right\| _ {2} = 1} \frac {1}{n} \left(\sum_ {i \in \mathbb {N} _ {1}} \mathbb {1} \left(\boldsymbol {u} ^ {T} \boldsymbol {h} _ {i} \geq 0\right) + \sum_ {i \in \mathbb {N} _ {2}} \mathbb {1} \left(\boldsymbol {u} ^ {T} \boldsymbol {h} _ {i} <   0\right)\right) \quad \text {s u p r e m u m} (7) \\ = S ^ {*}. (8) \\ \end{array}
+$$
+
+Here, when  $\mathbf{E}_{y_A}\mathbf{h}_i = \mathbf{E}_{y_B}\mathbf{h}_i$ , without loss of generality, we break the tie by letting the predicted label to be  $y_A$  for  $x_i$ .
+
+When there are more tokens in  $\mathbb{V}$ , it holds true that  $\mathbb{1}(y_i = \arg \max_{\nu_j} \frac{\pmb{E}_{\nu_j} \pmb{h}_i}{\sum_{\nu_j'} \{\pmb{y}_A, \pmb{y}_B\} \pmb{E}_{\nu_j'} \pmb{h}_i}) \leq \mathbb{1}(y_i = \arg \max_{\nu_j} \frac{\pmb{E}_{\nu_j} \pmb{h}_i}{\sum_{\nu_j' \in \{Y_A, Y_B\}} \pmb{E}_{\nu_j'} \pmb{h}_i})$ . Hence, we have  $Acc = \frac{1}{n} \sum_{i=1}^{n} \mathbb{1}(y_i = \arg \max_{\nu_j} \frac{\pmb{E}_{\nu_j} \pmb{h}_i}{\sum_{\nu_j' \in \{Y_A, Y_B\}} \pmb{E}_{\nu_j'} \pmb{h}_i}) \leq S^*$ .
+
+By the definition of  $\pmb{u}^{*}$ , we have
+
+$$
+\frac {1}{n} \left(\sum_ {i \in \mathbb {N} _ {1}} \mathbb {1} \left(\boldsymbol {u} ^ {*}, T \boldsymbol {h} _ {i} \geq 0\right) + \sum_ {i \in \mathbb {N} _ {2}} \mathbb {1} \left(\boldsymbol {u} ^ {*}, T \boldsymbol {h} _ {i} <   0\right)\right) = S ^ {*}. \tag {9}
+$$
+
+If  $\frac{\pmb{E}y_A - \pmb{E}y_B}{|\pmb{E}y_A - \pmb{E}y_B|_2} = c\pmb{u}^*$  for some positive constant  $c$ , then
+
+$$
+\begin{array}{l} \frac {1}{n} \left(\sum_ {i \in \mathbb {N} _ {1}} \mathbb {1} \left[ \left(\boldsymbol {E} _ {y _ {A}} - \boldsymbol {E} _ {y _ {B}}\right) ^ {T} \boldsymbol {h} _ {i} \geq 0 \right] + \sum_ {i \in \mathbb {N} _ {2}} \mathbb {1} \left[ \left(\boldsymbol {E} _ {y _ {A}} - \boldsymbol {E} _ {y _ {B}}\right) ^ {T} \boldsymbol {h} _ {i} <   0) \right]\right) (10) \\ = \frac {1}{n} \left(\sum_ {i \in \mathbb {N} _ {1}} \mathbb {1} \left(\boldsymbol {u} ^ {*}, T \boldsymbol {h} _ {i} \geq 0\right) + \sum_ {i \in \mathbb {N} _ {2}} \mathbb {1} \left(\boldsymbol {u} ^ {*}, T \boldsymbol {h} _ {i} <   0\right)\right). (11) \\ \end{array}
+$$
+
+We already showed that
+
+$$
+\begin{array}{l} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {1} \left(y _ {i} = \underset {v _ {j}} {\arg \max } \frac {\boldsymbol {E} _ {v _ {j}} \boldsymbol {h} _ {i}}{\sum_ {v _ {j} ^ {\prime} \in \left\{y _ {A} , y _ {B} \right\}} \boldsymbol {E} _ {v _ {j} ^ {\prime}} \boldsymbol {h} _ {i}}\right) (12) \\ = \frac {1}{n} \left(\sum_ {i \in \mathbb {N} _ {1}} \mathbb {1} \left[ \left(\boldsymbol {E} _ {y _ {A}} - \boldsymbol {E} _ {y _ {B}}\right) ^ {T} \boldsymbol {h} _ {i} \geq 0 \right] + \sum_ {i \in \mathbb {N} _ {2}} \mathbb {1} \left[ \left(\boldsymbol {E} _ {y _ {A}} - \boldsymbol {E} _ {y _ {B}}\right) ^ {T} \boldsymbol {h} _ {i} <   0) \right]. \right. (13) \\ \end{array}
+$$
+
+When  $\max_{v\in \mathbb{V},v\notin y_{A},y_{B}}\pmb{E}_{v}\pmb{h}_{i} <   \max (\pmb{E}_{y_{A}}\pmb{h}_{i},\pmb{E}_{y_{B}}\pmb{h}_{i}),\forall i,$  we have
+
+$$
+\begin{array}{l} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {1} \left(y _ {i} = \arg \max  _ {v _ {j}} \frac {\boldsymbol {E} _ {v _ {j}} \boldsymbol {h} _ {i}}{\sum_ {v _ {j} ^ {\prime} \in \{y _ {A} , y _ {B} \}} \boldsymbol {E} _ {v _ {j} ^ {\prime}} \boldsymbol {h} _ {i}}\right) (14) \\ = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {1} \left(y _ {i} = \arg \max  _ {v _ {j}} \frac {\boldsymbol {E} _ {v _ {j}} \boldsymbol {h} _ {i}}{\sum_ {v _ {j} \in 1 , \dots , | \mathrm {V} |} \boldsymbol {E} _ {v _ {j} ^ {\prime}} \boldsymbol {h} _ {i}}\right) (15) \\ = \text {A c c .} (16) \\ \end{array}
+$$
+
+Hence, we have  $\mathrm{Acc} = S^{*}$  when  $\max_{v\in \mathbb{V},v\notin y_A,y_B}\pmb {E}_v\pmb {h}_i <   \max (\pmb {E}_{y_A}\pmb {h}_i,\pmb {E}_{y_B}\pmb {h}_i),\forall i$  and  $\frac{\pmb{E}_{y_A} - \pmb{E}_{y_B}}{\|\pmb{E}_{y_A} - \pmb{E}_{y_B}\|_2} = c\pmb{u}^*$ .
+
+# B Generalization of Theorem 1
+
+In this section, we consider a generalization of Theorem 1 where  $x_{1},\ldots ,x_{n}$  are allowed to have  $m < n$  labels, i.e.  $y_{i}\in \{y_{1},\dots,y_{m}\} ,\forall i.$  Same as before, let  $\mathbb{N}_j = \{i:y_i = y_j\} ,j = 1,\dots,m.$ $\mathbb{N}_j,j = 1,\dots,m$  are pairwise disjoint and  $\bigcup_{j = 1}^{m}|\mathbb{N}_{j}| = \{1,\dots,n\}$  . Let  $S_{j,k}(\pmb {u}) = \frac{1}{|\mathbb{N}_j| + |\mathbb{N}_k|} (\sum_{i\in \mathbb{N}_j}\mathbb{1}(\pmb {u}^T\pmb {h}_i\geq 0) + \sum_{i\in \mathbb{N}_k}\mathbb{1}(\pmb {u}^T\pmb {h}_i < 0))$  and let  $\pmb{u}_{j,k}^{*} = \arg \max_{\pmb {u}\in \mathbb{S}^{d - 1}}S_{j,k}(\pmb {u})$  for  $j,k\in 1,\dots,m$  , and  $S_{j,k}(\pmb{u}_{j,k}^{*}) = S_{j,k}^{*}$  For each  $j$  and  $k$ $u_{j,k}^{*}$  is guaranteed to be in  $\mathbb{S}^{d - 1}$  because only  $|\mathbb{N}_j| + |\mathbb{N}_k| <   n$  hyperplanes are involved in separating separating  $\mathbb{S}^{d - 1}$  , resulting in a smaller number of cells (guaranteed to be finite) compared to the case of  $x_{1},\ldots ,x_{n}$  having only  $y_{1},y_{2}$  as labels. Then, we have the following result.
+
+Theorem 2 Acc  $\leq \frac{1}{n(m - 1)} (\sum_{k = 1}^{m}\sum_{j = k + 1}^{m}(|\mathbb{N}_k| + |\mathbb{N}_j|)S_{j,k}^*)$ . The equality is achieved when for all unordered pairs  $(j,k), j,k \in 1, \ldots, m$ ,  $\max_{v \in \mathbb{V}, v \notin \{y_j, y_k\}} \pmb{E}_v \pmb{h}_i < \max(\pmb{E}_{y_j} \pmb{h}_i, \pmb{E}_{y_k} \pmb{h}_i), \forall i \in N_j \cup N_k$  and  $\pmb{E}_{y_j} - \pmb{E}_{y_k} = c_{j,k} \pmb{u}_{j,k}^*$  for some positive constant  $c_{j,k}$ .
+
+The theorem states that in the case of the queries having more than two labels, the dataset classification accuracy is controlled by the pairwise separability and alignment properties of the hidden states. After calculating the separability of the hidden states of queries with labels  $y_{j}$  and  $y_{k}$  for each  $(j,k)$ , the dataset classification accuracy is then upper-bounded by these separability values weighted by the number of queries with different labels.
+
+Similarly, the accuracy can match this upper bound if each pairwise hidden states collection is maximally separated along a direction parallel to the difference of the unembedding vectors of the corresponding labels. This theorem shows that the otherwise difficult multiclass classification problem is geometrically equivalent to the combination of a series of binary classification subproblems. And the geometric properties of the multiclass classification problem can be understood by studying the much more tractable geometry of the constituent binary subproblems.
+
+Proof First, note the following relations
+
+$$
+\begin{array}{l} \operatorname {A c c} = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {1} \left(y _ {i} = \arg \max  _ {v _ {j}} \frac {\boldsymbol {E} _ {v _ {j}} \boldsymbol {h} _ {i}}{\sum_ {j ^ {\prime} \in 1 , \dots , | \mathbb {V} |} \boldsymbol {E} _ {v ^ {\prime} j} \boldsymbol {h} _ {i}}\right) \\ \leq \frac{1}{n}\sum_{i = 1}^{n}\mathbb{1}(y_{i} = \arg \max_{v_{j}}\frac{\boldsymbol{E}_{v_{j}}\boldsymbol{h}_{i}}{\sum_{v_{j}^{\prime}\in\{y_{1},\ldots,y_{k}\}}\boldsymbol{E}_{v_{j}^{\prime}}\boldsymbol{h}_{i}}) \\ = \frac {1}{n} \left(\sum_ {k = 1} ^ {m} \sum_ {x _ {i} \in \mathbb {N} _ {k}} \mathbb {1} \left(\boldsymbol {E} _ {y _ {k}} \boldsymbol {h} _ {i} \geq \boldsymbol {E} _ {y _ {j}} \boldsymbol {h} _ {i}, \forall j \neq k, j \in 1, \dots , m\right)\right) \\ = \frac {1}{n} \left(\sum_ {k = 1} ^ {m} \sum_ {x _ {i} \in \mathbb {N} _ {k}} \min  _ {j \neq k, j \in 1, \dots , m} \mathbb {1} \left(\boldsymbol {E} _ {y _ {k}} \boldsymbol {h} _ {i} \geq \boldsymbol {E} _ {y _ {j}} \boldsymbol {h} _ {i}\right)\right) \quad \text {i n d i c a t o r f u n c t i o n i s e i t h e r 1 o r 0} \\ \leq \frac {1}{n} \left(\sum_ {k = 1} ^ {m} \sum_ {x _ {i} \in \mathbb {N} _ {k}} \frac {1}{m - 1} \sum_ {j, j \neq k, j \in 1, \dots , m} \mathbb {1} \left(\boldsymbol {E} _ {y _ {k}} \boldsymbol {h} _ {i} \geq \boldsymbol {E} _ {y _ {j}} \boldsymbol {h} _ {i}\right)\right) \\ = \frac {1}{n (m - 1)} \left(\sum_ {k = 1} ^ {m} \sum_ {j, j \neq k, j \in 1, \dots , m x _ {i} \in \mathbb {N} _ {k}} \mathbb {1} \left(\boldsymbol {E} _ {y _ {k}} \boldsymbol {h} _ {i} \geq \boldsymbol {E} _ {y _ {j}} \boldsymbol {h} _ {i}\right)\right) \\ = \frac {1}{n (m - 1)} \left(\sum_ {k = 1} ^ {m} \sum_ {j, j \neq k, j \in 1, \dots , m x _ {i} \in \mathbb {N} _ {k}} \sum_ {i} \mathbb {1} \left(\frac {\boldsymbol {E} _ {y _ {k}} - \boldsymbol {E} _ {y _ {j}}}{\| \boldsymbol {E} _ {y _ {k}} - \boldsymbol {E} _ {y _ {j}} \| _ {2}} \boldsymbol {h} _ {i} \geq 0\right)\right). \\ \end{array}
+$$
+
+Note that the sum is over all ordered pairs  $(j,k)$  s.t.  $j,k\in 1,\dots,m,j\neq k$ . Consider the case of  $(k = 1,j = 2)$  and  $(k = 2,j = 1)$ , then both  $\mathbb{1}(\pmb{E}_{y_k}\pmb{h}_i\geq \pmb{E}_{y_j}\pmb{h}_i)$  and  $\mathbb{1}(\pmb{E}_{y_j}\pmb{h}_i\geq \pmb{E}_{y_k}\pmb{h}_i)$  appear in the sum. Hence, the sum can be taken over all unordered pairs  $(j,k)$  and rewritten as:
+
+$$
+\begin{array}{l} \frac {1}{n (m - 1)} \left(\sum_ {k = 1} ^ {m - 1} \sum_ {j = k + 1} ^ {m} \sum_ {x _ {i} \in \mathbb {N} _ {k} \cup \mathbb {N} _ {j}} \left(\mathbb {1} \left(\frac {\boldsymbol {E} _ {y _ {k}} - \boldsymbol {E} _ {y _ {j}}}{\| \boldsymbol {E} _ {y _ {k}} - \boldsymbol {E} _ {y _ {j}} \| _ {2}} \boldsymbol {h} _ {i} \geq 0\right) + \mathbb {1} \left(\frac {\boldsymbol {E} _ {y _ {k}} - \boldsymbol {E} _ {y _ {j}}}{\| \boldsymbol {E} _ {y _ {k}} - \boldsymbol {E} _ {y _ {j}} \| _ {2}} \boldsymbol {h} _ {i} <   0\right)\right)\right) (17) \\ \leq \frac {1}{n (m - 1)} \left(\sum_ {k = 1} ^ {m - 1} \sum_ {j = k + 1} ^ {m} \left(| \mathbb {N} _ {k} | + | \mathbb {N} _ {j} |\right) S _ {j, k} ^ {*}\right). (18) \\ \end{array}
+$$
+
+Hence, we have
+
+$$
+\operatorname {A c c} \leq \frac {1}{n (m - 1)} \left(\sum_ {k = 1} ^ {m - 1} \sum_ {j = k + 1} ^ {m} \left(| \mathbb {N} _ {k} | + | \mathbb {N} _ {j} |\right) S _ {j, k} ^ {*}\right). \tag {19}
+$$
+
+As a proof of concept, plug-in  $m = 2$ , then the bound becomes  $\frac{1}{n} (|N_1| + |N_2|)S^* = S^*$  since  $|N_1| + |N_2| = n$ , which matches the result in Theorem 1.
+
+To establish the equality conditions, as in the proof of Theorem 1, if  $\pmb{E}_{y_j} - \pmb{E}_{y_k} = c_{j,k}\pmb{u}_{j,k}^*$  for some positive constant  $c_{j,k}$ , then
+
+$$
+\begin{array}{l} \sum_ {i \in \mathbb {N} _ {k}} \mathbb {1} \left[ \left(\boldsymbol {E} _ {y _ {k}} - \boldsymbol {E} _ {y _ {j}}\right) ^ {T} \boldsymbol {h} _ {i} \geq 0 \right] + \sum_ {i \in \mathbb {N} _ {j}} \mathbb {1} \left[ \left(\boldsymbol {E} _ {y _ {k}} - \boldsymbol {E} _ {y _ {j}}\right) ^ {T} \boldsymbol {h} _ {i} <   0) \right] (20) \\ = \sum_ {i \in \mathbb {N} _ {k}} \mathbb {1} \left(\boldsymbol {u} _ {j, k} ^ {*}, T \boldsymbol {h} _ {i} \geq 0\right) + \sum_ {i \in \mathbb {N} _ {j}} \mathbb {1} \left(\boldsymbol {u} _ {j, k} ^ {*}, T \boldsymbol {h} _ {i} <   0\right) = \left(| N _ {j} | + | N _ {k} |\right) S _ {j, k} ^ {*}. (21) \\ \end{array}
+$$
+
+We also know that if  $\max_{\nu \in \mathbb{V}, \nu \notin y_j, y_k} \pmb{E}_{\nu} \pmb{h}_i < \max(\pmb{E}_{y_j} \pmb{h}_i, \pmb{E}_{y_k} \pmb{h}_i), \forall i \in N_j \cup N_k$ , then
+
+$$
+\begin{array}{l} \sum_ {i \in \mathbb {N} _ {k}} \mathbb {1} \left[ \left(\boldsymbol {E} _ {y _ {k}} - \boldsymbol {E} _ {y _ {j}}\right) ^ {T} \boldsymbol {h} _ {i} \geq 0 \right] + \sum_ {i \in \mathbb {N} _ {j}} \mathbb {1} \left[ \left(\boldsymbol {E} _ {y _ {k}} - \boldsymbol {E} _ {y _ {j}}\right) ^ {T} \boldsymbol {h} _ {i} <   0) \right] (22) \\ = \sum_ {i \in \mathbb {N} _ {k} \cup \mathbb {N} _ {j}} \mathbb {1} \left(y _ {i} = \arg \max  _ {v _ {j}} \frac {\boldsymbol {E} _ {v _ {j}} \boldsymbol {h} _ {i}}{\sum v _ {j} ^ {\prime} \in \left\{y _ {j} , y _ {k} \right\} \boldsymbol {E} _ {v _ {j} ^ {\prime}} \boldsymbol {h} _ {i}}\right) (23) \\ = \sum_ {i \in \mathbb {N} _ {k} \cup \mathbb {N} _ {j}} \mathbb {1} \left(y _ {i} = \arg \max  _ {v _ {j}} \frac {\boldsymbol {E} _ {v _ {j}} \boldsymbol {h} _ {i}}{\sum_ {v _ {j} \in 1 , \dots , | \mathbb {V} |} \boldsymbol {E} _ {v _ {j} ^ {\prime}} \boldsymbol {h} _ {i}}\right). (24) \\ \end{array}
+$$
+
+Finally, note that
+
+$$
+\begin{array}{l} \operatorname {A c c} = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {1} \left(y _ {i} = \underset {v _ {j}} {\arg \max } \frac {\boldsymbol {E} _ {v _ {j}} \boldsymbol {h} _ {i}}{\sum_ {v _ {j} ^ {\prime} \in \{y _ {1} , y _ {2} \}} \boldsymbol {E} _ {v _ {j} ^ {\prime}} \boldsymbol {h} _ {i}}\right) (25) \\ = \frac {1}{n} \frac {1}{m - 1} \sum_ {k = 1} ^ {m - 1} \sum_ {j = l + 1} ^ {m} \sum_ {i \in \mathbb {N} _ {j} \cup \mathbb {N} _ {k}} \mathbb {1} \left(y _ {i} = \arg \max  _ {v _ {j}} \frac {\boldsymbol {E} _ {v _ {j}} \cdot \boldsymbol {h} _ {i}}{\sum_ {v _ {j} \in 1 , \dots , | \mathbb {V} |} \boldsymbol {E} _ {v _ {j} ^ {\prime}} \cdot \boldsymbol {h} _ {i}}\right). (26) \\ \end{array}
+$$
+
+because for each  $\mathbb{N}_j$ , the term  $\mathbb{1}(y_i = \arg \max_{v_j} \frac{\pmb{E}_{v_j} \pmb{h}_i}{\sum_{v_j \in 1, \dots, |\mathbb{V}|} \pmb{E}_{v_j'} \pmb{h}_i})$  appears  $(m - 1)$  times in the summation for each  $i \in \mathbb{N}_j$ . Hence, combining everything, we have
+
+$$
+\begin{array}{l} \operatorname {A c c} = \frac {1}{n} \frac {1}{m - 1} \sum_ {k = 1} ^ {m - 1} \sum_ {j = k + 1} ^ {m} \sum_ {i \in \mathbb {N} _ {j} \cup \mathbb {N} _ {k}} \mathbb {1} \left(y _ {i} = \arg \max  _ {v _ {j}} \frac {\boldsymbol {E} _ {v _ {j}} \boldsymbol {h} _ {i}}{\sum_ {v _ {j} \in 1 , \dots , | \mathbb {V} |} \boldsymbol {E} _ {v _ {j} ^ {\prime}} \boldsymbol {h} _ {i}}\right) (27) \\ = \frac {1}{n (m - 1)} \sum_ {k = 1} ^ {m - 1} \sum_ {j = k + 1} ^ {m} \left(| N _ {j} | + | N _ {k} |\right) S _ {j, k} ^ {*}. (28) \\ \end{array}
+$$
+
+The proof is thus complete.
+
+![](images/55422ba6c3d34660d9ac463572dba673fe4a64461d656c6ecb6cbd7251884fb0.jpg)
+
+# C Calculation of geometric measures
+
+1. Separability score We train the logistic classifier on half of the points in  $\pmb{H}$  and take its prediction accuracy on the other half as the estimated maximum separability of  $\pmb{H}$ . We use the scikit-learn [Pedregosa et al., 2011] default implementation of the logistic classifier, and set the number of iterations to be 100.  
+2. Output alignment We feed  $\pmb{H}$  into the output unembedding layer of the model to get the model's predictions of the query labels based on  $\pmb{H}$  and calculate the percentage of correct predictions.  
+3. Singular alignment First obtain the centered version of  $\pmb{H}$ , denoted as  $\bar{H}$ . Then calculate its singular value decomposition, namely  $\bar{\pmb{H}} = \pmb{U}\pmb{\Sigma}\pmb{V}^T$ , and compute the absolute values of cosine similarities between the top-r singular vectors and the difference of label unembedding vectors  $\pmb{E}_{y_A} - \pmb{E}_{y_B}$ . The absolute value is taken since the sign of the singular vectors is arbitrary. In practice, we calculate the values using the top-2 singular vectors and report the maximum of the two.  
+4. Variance-based alignment First normalize  $\pmb{E}_{y_A} - \pmb{E}_{y_B}$  as  $\frac{\pmb{E}_{y_A} - \pmb{E}_{y_B}}{\|\pmb{E}_{y_A} - \pmb{E}_{y_B}\|_2}$ . Then the ratio of variance (w.r.t. its mean) of  $\pmb{H}$  explained along  $\frac{\pmb{E}_{y_A} - \pmb{E}_{y_B}}{\|\pmb{E}_{y_A} - \pmb{E}_{y_B}\|_2}$  is  $\frac{\pmb{E}_{y_A} - \pmb{E}_{y_B}}{\|\pmb{E}_{y_A} - \pmb{E}_{y_B}\|_2} \frac{T}{n} \frac{\bar{\pmb{H}}^T \bar{\pmb{H}}}{\|\pmb{E}_{y_A} - \pmb{E}_{y_B}\|_2}$ , which we take to be the variance-based alignment of  $\pmb{H}$ .  
+5. Mean-based alignment First compute the means of label clusters (subsets of  $\pmb{H}$  with each of the two different labels), i.e.  $\frac{1}{|\mathbb{N}_A|}\sum_{i\in \mathbb{N}_A}\pmb {h}_i$  and  $\frac{1}{|\mathbb{N}_B|}\sum_{i\in \mathbb{N}_B}\pmb {h}_i$ . Then take the difference and compute the value of its projection onto  $\frac{\pmb{E}_{y_A} - \pmb{E}_{y_B}}{\|\pmb{E}_{y_A} - \pmb{E}_{y_B}\|_2}$ , i.e.  $(\frac{\pmb{E}_{y_A} - \pmb{E}_{y_B}}{\|\pmb{E}_{y_A} - \pmb{E}_{y_B}\|_2})^T (\frac{1}{\mathbb{N}_A}\sum_{i\in |\mathbb{N}_A|}\pmb {h}_i - \frac{1}{|\mathbb{N}_B|}\sum_{i\in \mathbb{N}_B}\pmb {h}_i)$ . Then, calculate the weighted average of the variances of the two label clusters along  $\frac{\pmb{E}_{y_A} - \pmb{E}_{y_B}}{\|\pmb{E}_{y_A} - \pmb{E}_{y_B}\|_2}$ , i.e.  $\frac{|\mathbb{N}_A|\operatorname{Var}(\{\pmb{h}_i^T\frac{\pmb{E}_{y_A} - \pmb{E}_{y_B}}{\|\pmb{E}_{y_A} - \pmb{E}_{y_B}\|_2}:i\in\mathbb{N}_A\}) + |\mathbb{N}_B|\operatorname{Var}(\{\pmb{h}_i^T\frac{\pmb{E}_{y_A} - \pmb{E}_{y_B}}{\|\pmb{E}_{y_A} - \pmb{E}_{y_B}\|_2}:i\in\mathbb{N}_B\})}{|\mathbb{N}_A| + |\mathbb{N}_B|}$ . Finally, divide the projected difference between means by the square root of the weighted average of the projected variance.  
+6. Composite alignment Multiply the variance-based alignment by the mean-based alignment.  
+7. Effective dimension Given the SVD  $\bar{H} = U\Sigma V^T$  and the singular values  $\sigma_{1},\ldots,\sigma_{n}$ , the effective dimension of  $\pmb{H}$  is  $\frac{\left(\sum_{i=1}^{n}\sigma_i^2\right)^2}{\sum_{i=1}^{n}\sigma_i^4}$ .
+
+Table 2: Prompt templates and labels for different datasets.  
+
+<table><tr><td>Dataset</td><td colspan="3">Template</td><td>Label</td></tr><tr><td>SST-2</td><td colspan="3">{Sentence} Sentiment: {Label}</td><td>positive / negative</td></tr><tr><td>SUBJ</td><td colspan="3">{Sentence} Type: {Label}</td><td>subjective / objective</td></tr><tr><td>TREC</td><td colspan="3">Question: {Sentence} Type: {Label}</td><td>abbreviation / entity / description / human / location / number</td></tr><tr><td>SNLI</td><td colspan="3">The question is: {Premise}? The answer is: {Hypothesis} {Label}</td><td>true / maybe / false</td></tr><tr><td>RTE</td><td colspan="3">The question is: {Premise}? The answer is: {Hypothesis} {Label}</td><td>true / false</td></tr><tr><td>CB</td><td colspan="3">The question is: {Premise}? The answer is: {Hypothesis} {Label}</td><td>true / maybe / false</td></tr><tr><td>Famous People</td><td colspan="3">{Person Name} is a: {Label}</td><td>actor / politician / singer / scientist / writer / athlete</td></tr></table>
+
+Table 3: Mappings used to replace ground-truth labels of different datasets to symbols  
+
+<table><tr><td>Dataset</td><td>Label Mapping</td></tr><tr><td>SST-2</td><td>negative/positive → @/#</td></tr><tr><td>SUBJ</td><td>objective/subjective → @/#</td></tr><tr><td>TREC</td><td>abbreviation / entity / description / person / number / location / → @/#!/$/&amp;//*</td></tr><tr><td>SNLI</td><td>true/maybe/false → @/#!</td></tr><tr><td>RTE</td><td>true/false → @/#</td></tr><tr><td>CB</td><td>true/maybe/false → @/#!</td></tr></table>
+
+# D Implementation details
+
+Models We use the official huggingface implementation of all models. All models with more than 10B parameters are quantized to 4bit.
+
+Datasets We use the official huggingface implementation of all datasets except for the Famous People dataset crafted by ourselves. The Famous People dataset is generated using ChatGPT-4o [OpenAI et al., 2024]. The dataset has 180 datapoints, each is the name of a famous person with a profession label from one of six categories: actor, politician, singer, scientist, writer, athlete. The dataset is balanced as there are 30 data points for each label.
+
+ICL setting For each dataset except for the Famous People dataset, we select demonstrations from the train set and queries from the test set, or the validation set if the ground-truth labels for the test set are not provided. We keep only the first 10000 datapoints in the train set for demonstration selection if the train set has more than 10000 entries, and the first 1000 data points in the test set/Validation set for accuracy evaluation. For the famous people dataset, we use the first 90 data points for demonstration selection and the remaining 90 data points for testing. For experiments involving kNN demonstration selection, we first use the respective LLMs to encode the demonstrations and queries, and select the demonstrations with embeddings closer to that of the query in  $l_{2}$  distance for each query. We use the prompt formats detailed in Table 2.
+
+Devices All experiments with Llama2-7B, Llama2-13B, Gemma-2B, and Gemma-7B are conducted with an A800 GPU. All experiments with Llama3-8B, Llama2-70B, and Llama3-70B are conducted with an H200 GPU.
+
+For the label mapping experiment in Subsection 5.1. We use the mappings listed in Table 3 to map the ground-truth labels of each dataset to symbols.
+
+# E Practical implications of the mechanistic findings
+
+We discuss in this section how our geometric characterization of the model's inference process in the zero-shot or ICL setting can be converted to actionable and efficient methods to improve model performance.
+
+# E.1 Improving classification by capitalizing on hidden states separability
+
+In Figure 2 we see the high separability of hidden states of prompts from classification tasks which surpasses the accuracy of the actual label prediction by far, irrespective of whether demonstrations are
+
+prepended to the labels. This implies that rather than direct decoding we can instead use the model as a feature extractor with which to train classifiers to perform the classification task, which is exactly how we evaluate the separability score. To demonstrate this, we train a logistic classifier on  $50\%$  of the last layer hidden states and evaluate the classification accuracy on the rest half. The results in Table 22 demonstrate that such a practice improves the classification accuracy with zero-shot prompts by over  $70\%$  across models, while only incurring the minimal cost of training an extra logistic classifier. Moreover, in Table 23 we see that training the classifier over 8-shot ICL hidden states also brings a solid  $\sim 10\%$  increase over the original ICL accuracy, which shows the generalizability of this lightweight method that leverages hidden states separability.
+
+# E.2 Unsupervised enhancement of classification performance through low-rank denoising
+
+In Figure 6 we see that low-rank denoising of the ICL hidden states matrix reveals the task-related directions as its dominant directions and enhances its alignment properties. We can thus apply this technique to the final layer hidden states instead of the intermediate layer ones (which is the focus of Figure 6) to directly improve classification performance. In Table 24 we apply rank-5 denoising to final layer hidden states before feeding them into the unembedding layer. This method is different from the classifier-based method in that it assumes no knowledge of the ground-truth label of each hidden state. The dataset-average results demonstrate that the rank-5 denoising can indeed improve the classification performance of all models with the only exception of Gemma-7B.
+
+# F Supplementary materials for Subsection 5.1
+
+# F.1 Replication of Figure 2 for other models
+
+In Figure 10-15, we provide the visualizations of the experiments presented in Figure 2 for other models, from which similar conclusions to those in Subsection 5.1 can be drawn. Alignment measures explain the accuracy difference between ICL and zero-shot across models, and the layer-wise trends of the measures explain the phase transition pattern for all models.
+
+# F.2 Replication of Figure 3 for other models
+
+In Figure 16-21 we provide the visualizations of the experiments presented in Figure 3 for other models to establish the generality of our conclusions. It is clear that the phase transition pattern is evident in different ICL settings across models, and alignment measures capture accuracy differences across ICL settings for all models.
+
+# F.3 Replication of Figure 4 for other models
+
+In Figure 53-58 we repeat the visualizations in Figure 4 for other models to examine where the generation task fits well into our framework. The results demonstrate that for other models, the phase-transition pattern in the composite alignment metric is specific to the 8-shot ICL hidden states, while ICL and zero-shot hidden states do not differ much in terms of the dynamics of the separability score.
+
+# G Supplementary materials for Subsection 5.2
+
+# G.1 Replication of subplot (A) of Figure 5 for other datasets and models
+
+In Figure 22-28 we provide the visualizations of the experiments presented in subplot (A) of Figure 5, to establish the generalizability of our conclusion that the directional alignment and the encoding of label-related semantics into the hidden states happen almost simultaneously after the phase transition. The layer-wise trends of the directional alignment measures and output alignment exhibit high correlation across all datasets and models.
+
+# G.2 Replication of subplot (B) of Figure 5 for other datasets
+
+In Table 4-8, we present the tokens decoded from top singular directions near the phase transition breakpoint for other datasets. The results demonstrate that the sudden semantic shift toward label-related tokens in the hidden states (as demonstrated by subplot (B) of Figure 5) is present in other datasets as well. Note that the phase transition happens at different layers for different datasets.
+
+# G.3 Replication of subplot (B) of Figure 5 with replaced demonstration labels
+
+To further examine the relationship between alignment and the encoding of task-related semantics, we design a scenario in which the ground-truth demonstration labels are consistently mapped to symbols, and the expected predictions for the query label becomes the mapped symbols as well. This forces a separation between semantic encoding and alignment, as the task-related semantics remain unchanged, but alignment must now occur with respect to the unembedding vectors of the symbols. Specifically, we conduct experiments on SST-2 and SNLI where we map the SST-2 labels (negative/positive) to @ and #, and the SNLI labels (true/maybe/false) to @, #, and ! within the demonstrations. We then inspect the tokens decoded from the top singular directions of the hidden states at each layer.
+
+As shown in Table 9, the results reveal an interesting pattern. The first major shift in the layer-wise semantic content occurs at layer 24, where the ground-truth label tokens suddenly become dominant. This closely mirrors the pattern observed with normal demonstration labels in Table 6, and aligns with the point where output alignment begins to rise in Figure 22 (C). However, another drastic shift is observed at layer 45, where the decoded top tokens begin to include the symbol tokens. These symbols suddenly appear and start to compete with semantically relevant task tokens, marking the onset of alignment between the hidden states and the unembedding vectors of the symbols. This delayed alignment corresponds with the pattern seen in subplot (C) of Figure 3, where the phase transition under demonstrations with symbolized labels is postponed to later layers.
+
+The competition between the true labels and the injected symbol labels persists through to the final layers. This likely contributes to the reduced accuracy in this setting, as the semantic influence of the substituted labels fails to fully override that of the ground-truth labels. Similar results for SST-2 are presented in Table 10, showing similar trends.
+
+# G.4 Replication of subplot (C) of Figure 5 for other datasets
+
+Subplot (C) of Figure 5 reveals a two-stage pattern in the layer-wise retainment and filtering of semantic information, where most post-transition layers retain label-related semantics and filter out irrelevant ones, and the final layers do the opposite. We show in Table 11-15 that it exists in other datasets besides SST-2 as well.
+
+# G.5 Replication of Figure 6 for other datasets and models
+
+We first provide the results of rank-10 approximations on all 6 datasets for Llama2-70B in Figure 42. The conclusion is that low-rank denoising significantly increase the output alignment significantly across all datasets, particularly in the initial layers after the phase transition. This confirms the generality of our findings in Subsection 5.2. Additionally, in the final layers, the output alignment also increases on on SST-2, SUBJ, TREC, and SNLI, though not on RTE and CB.
+
+In Figure 43-48, we present results for other models. These are consistent with the findings from Llama2-70B. Specifically, the pattern that low-rank denoising improves output alignment more in text classification datasets than in natural language inference datasets holds across models. This suggests a higher level of linguistic complexity in natural language inference tasks—likely because such tasks require reasoning over both a premise and a hypothesis, whose combined semantics are more difficult to compress and more prone to information loss during low-rank projection.
+
+# G.6 Ablation experiments for the rank hyperparameter in low-rank approximations
+
+In this section, we explore the effect of varying the number of retained ranks in rank-  $r$  SVD approximations of Llama2-70B hidden states across all datasets. Specifically, we investigate how increasingly
+
+aggressive rank reductions—setting  $r = 1, 2$ , and 5—impact performance. The results, shown in Figure 49, Figure 50, and Figure 51, yield several notable findings.
+
+For binary classification tasks such as SST-2, SUBJ, and RTE, more aggressive rank reduction sometimes leads to better output alignment, particularly in layers after the phase transition where alignment with label unembedding vectors begins to emerge. In some cases, even a rank-1 approximation preserves or enhances alignment-related performance. However, for three-way classification tasks like SNLI and RTE, aggressive reductions (especially at  $r = 1$  or 2) can cause accuracy drops in later layers. For TREC, which involves six label classes, rank-2 or rank-1 approximations degrade accuracy across all layers.
+
+These results align with the intuition that datasets with more labels require higher-dimensional hidden state representations to preserve the necessary separability and alignment structures among label clusters.
+
+# H Experiment details concerning the identification of PTHs and IHs
+
+For each dataset, we use the first 50 queries to identify the set of IHs and PTHs. Denote the queries as  $x_{1}, \ldots, x_{50}$  each with token length  $s(x_{1}), \ldots, s(x_{50})$ .
+
+Identification of PTHs For each  $x_{i}$ , the LLM will generate an attention tensor  $\mathbf{Att}\mathbf{n}_{i} \in \mathbb{R}^{L \times N_{h} \times s(x_{i}) \times s(x_{i})}$ , where  $L$  is the total number of layers,  $N_{h}$  is the number of attention heads per layer. Hence, for the  $n_{h}^{th}$  attention head at layer  $l$ , there is a corresponding attention matrix  $\mathbf{Att}\mathbf{n}(l, n_{h})_{i} \in \mathbb{R}^{s(x_{i}) \times s(x_{i})}$ , where  $\mathbf{Att}\mathbf{n}(l, n_{h})_{i,j,k}$  is the attention of this attention head at the  $k^{th}$  token to the  $j^{th}$  token in  $x_{i}$ . The PTH score of this attention head on  $x_{i}$  is thus defined as  $\sum_{k=2}^{s(x_{i})} \mathbf{Att}\mathbf{n}(l, n_{h})_{i,k-1,k}$  which measures the total attention the attention head assigns at each token position to the previous token. Consequently, the PTH score of the  $n_{h}^{th}$  attention head at layer  $l$  on all the queries is  $\sum_{i=1}^{50} \sum_{k=2}^{s(x_{i})} \mathbf{Att}\mathbf{n}(l, n_{h})_{i,k-1,k}$ . We calculate the PTH scores for all  $(l, n_{h})$  pairs and choose the top 10% attention heads as the identified PTHs.
+
+Identification of IHs For each  $x_{i}$ , randomly select 8 demonstrations and prepare them to  $x_{i}$ . The resultant ICL prompt (denoted as  $X_{i}$  with length  $s(X_{i})$ ), with a slight abuse of notation, takes the format of  $\langle x_{i,1}\rangle : \langle y_{i,1}\rangle, \dots, \langle x_{i,8}\rangle : \langle y_{i,8}\rangle, \langle x_{i}\rangle$ , where  $\langle x_{i,k}\rangle$  represents the sentence part of demonstration  $k$  ("I like this movie. Sentiment") and  $\langle y_{i,k}\rangle$  the label ("positive") separated from the sentence part by a colon (:), and  $\langle x_{i}\rangle$  is the sentence part for the query. At the position of the final "::", an IH will attend to tokens after the previous "::"s, i.e. the label tokens  $\langle y_{i,1}\rangle, \dots, \langle y_{i,8}\rangle$ . Let  $\mathbb{I}_i$  denote the indices of the label tokens in  $X_{i}$ . The IH score of the  $n_h^{th}$  attention head at layer  $l$  over the 50 queries is thus  $\sum_{i=1}^{50} \sum_{k\in\mathbb{I}_i} \in\text{Attr}(l,n_h)_{i,k,s(X_i)}$ , i.e. we are measuring the total attention a head assigns at the final "::" position to the positions of all the label tokens summed over all 50 queries. We calculate the IH scores for all  $(l,n_h)$  pairs and choose the top 10% attention heads as the identified IHs.
+
+# I Supplementary materials for Subsection 5.3
+
+# I.1 Replication of Figure 7 for other models
+
+In Figure 29-34, we present the results of attention heads ablations across models. The results support our conclusions regarding the significance of PTHs in enhancing the separability of the ICL hidden states and the significance of IHs in aligning them with the label unembedding vectors.
+
+# I.2 Replication of Figure 8 for other models
+
+In Figure 59-64, we present the results of analyzing the geometric contribution of PTHs and IHs in other models to the evolution of hidden states by injecting their outputs as task vectors. The results clearly demonstrate their respective significance in fostering the separability and alignment of hidden states. In particular, the PTH task vectors of other models exhibit greater potential to increase hidden-state separability than the result shown in Figure 8.
+
+# I.3 Replication of Table 1 for other models
+
+In Table 16-21, we compare the layer distributions of the top  $1\%$ ,  $2\%$ ,  $5\%$ , and  $10\%$  identified IHs and PTHs in other models and conduct Mann-Whitney U tests to assess the significance of the differences. The results demonstrate a consistent difference in the layer positions of PTHs and IHs across models, with PTHs significantly preceding IHs on average—the difference being most prominent in the two Gemma-family models, Gemma-2B and Gemma-7B.
+
+# I.4 Replication of Figure 9 for other models
+
+In Figure 65-70, we show the results of injecting task vectors derived from IH outputs for other models. The task vectors are similarly injected at layer  $\frac{3}{8} L$ , where  $L$  is the total number of layers of the respective models. The results demonstrate that IH outputs qualify as effective task vectors across models, with the exception of Gemma-2B, as they considerably improve prediction accuracy in the zero-shot setting after being injected into the zero-shot hidden states.
+
+# I.5 Results of attention heads ablation in the ICL setting where query label is not in context
+
+The results for other models, reported in Figure 29-34, confirm the generality of our conclusions across model sizes and architectures. To test whether these findings hold under different ICL settings as well, we replicate the attention head ablation experiments in a special ICL scenario ("In-weight Learning") studied by Reddy [2024] and Chan et al. [2022], where none of the in-context demonstrations share the query's label. This setting is useful for analyzing attention heads—especially IHs, which are known for copy-like behavior—when direct label copying is not possible.
+
+To this end, we construct a Famous People dataset following the procedures in Appendix D. Each data point is a sentence in the format " $\{\text{Famous Person Name}\}$  is a: ", and the label is the profession of the famous person. Once combined, they form an input sequence like, e.g., "Taylor Swift is a: Singer." that can be used as either a query or a demonstration. For each query, we sample  $k = 4$  demonstrations, ensuring that none share the query's profession label. We then ablate PTHs and IHs to examine their influence on model performance. As a control, we also evaluate the measures in the setting where at least one demonstration does share the query's label, allowing us to assess how the availability of directly copyable labels affects the impact of head ablation.
+
+The results in Figure 35 resemble those in standard ICL settings in that ablating PTHs significantly impairs separability, while ablating IHs affects alignment. However, there are notable differences. Most importantly, the accuracy drop from ablating PTHs is greater than that from ablating IHs—opposite to the standard case, where IH ablation has a larger impact. This highlights the critical role of having the query label present in the context for IHs to function effectively. The importance of label presence is further confirmed by the substantial accuracy gain observed when the query label is forced into the context.
+
+We provide results analogous to Figure 35 for other models in Figure 36-41, which demonstrate similar results.
+
+# I.6 Results of attention heads ablation in the zero-shot setting
+
+In this section, we investigate the effect of attention head ablation on the separability and alignment measures of zero-shot hidden states. The results averaged across datasets, as provided in Figure 52, reveal that the zero-shot setting is fundamentally different from ICL. In the zero-shot setting, ablating IHs has practically no effect on either separability or alignment measures, similar to ablating random heads, because there are no demonstrations from which to copy. On the contrary, ablating PTHs substantially compromises all separability and alignment measures. This illustrates that the contribution of PTHs to the separability of hidden states is by no means restricted to the ICL setting and highlights the significance of PTHs as key structural components of LLMs.
+
+# J Experiment details of using IH outputs to construct task vectors
+
+For each dataset, we first construct 8-shot ICL prompts using the first 50 queries. The demonstrations used are exactly the same as the ones used to evaluate the 8-shot ICL accuracy for each dataset. Then,
+
+following the procedures described in Todd et al. [2024], we obtain the average output (across the 50 prompts) of each identified top  $10\%$  IHs and PTHs at the final token positions, and sum the average outputs of all IHs to obtain the task vector.
+
+In terms of the steering experiment, we add the task vector to the hidden state of the final token of each query at Layer-30 of Llama2-70B, and let the modified hidden state flow through subsequent layers. While calculating the accuracy, we exclude the first 50 queries and only evaluate accuracy on the remaining queries in the test set to ensure an independence between the task vector and the queries involved.
+
+# K Broader impacts
+
+Understanding the internal mechanisms of large language models is increasingly critical as these models are deployed in high-stakes applications ranging from education and healthcare to legal and governmental decision-making. This work contributes to that understanding by providing a geometric framework that explains in-context learning through the lens of hidden state dynamics, specifically separability and alignment. By identifying the role of specific architectural components—such as Previous Token Heads and Induction Heads—in shaping model behavior, our study brings interpretability to a domain often criticized for opacity.
+
+The primary positive societal impact of this research lies in its potential to improve the transparency, controllability, and safety of LLMs. Better mechanistic understanding can inform the development of more robust models that are less reliant on spurious correlations and more capable of structured generalization. For example, insights into how semantic information is injected and refined layer by layer may support diagnostic tools that detect when a model is failing to align its internal representations with task-relevant signals. Additionally, the SVD-based techniques we propose for identifying and amplifying task-relevant components could be used to improve the efficiency or reliability of model outputs, particularly in low-resource or privacy-sensitive settings where retraining is infeasible.
+
+However, a deeper understanding of model internals also introduces risks. The ability to precisely manipulate hidden states or attention mechanisms may enable adversarial behaviors, such as constructing prompts that selectively suppress or amplify certain outputs for political or financial gain. Moreover, techniques for filtering and steering semantic content could be misused to hide bias or simulate alignment without genuine safety improvements. As such, we encourage future work to pair interpretability research with rigorous safety and ethics evaluations, and to involve interdisciplinary expertise when applying these findings to real-world systems.
+
+Overall, we view our contributions as a step toward more interpretable and accountable AI, but we emphasize that interpretability alone is not a guarantee of ethical deployment. Responsible application of these insights requires careful consideration of both technical and societal factors.
+
+![](images/8c5ee7f62bc44e53da38c9fdf0a03ce11c5b34b4c0554c177d82c31e63fbb3a4.jpg)  
+Effective dimension
+
+![](images/62e8a68cd3682c2c2101d0be251627bc61aa44e758e2b64bf9aad68c320ba037.jpg)  
+Separability score
+
+![](images/df43ecc0574ee1b7419312ffb3832e305b590d77154910754173d002f5e17c6d.jpg)  
+Singular alignment
+
+![](images/9177d06e1a0fc49da75dee23ba7b7f6ba3eb75a866c70ebd482925b6d9065f0e.jpg)  
+Composite alignment  
+Figure 10: Comparison of trends in separability and alignment measures of Llama3-8B hidden states between ICL and zero-shot
+
+![](images/46fb68b9f88f2b7264625bd0ae3f3b85c4421f810239c152dc6f99eca8a8b485.jpg)  
+Mean-based alignment
+
+![](images/19e1d535febb278d65fabad0b957b61d7d522fe09a65a3c775d7e88f41c5af2b.jpg)  
+Var.-based alignment
+
+Llama3-70B: 8-shot Acc.:  $81.09\%$ ; 0-shot Acc.:  $0.00\%$
+
+![](images/92a43f933d9c7eb85be6be82c48929ff4152c65afac7effa3ce02bbcf90f8942.jpg)  
+Effective dimension
+
+![](images/d372c6fb871e1a4724048a09bbb8bb1664d67abe1dfb9b0d9883a25841e026b5.jpg)  
+Separability score
+
+![](images/a1132d999cebcb7c98a1cb5df8a3ca14f7a11de534946d70372e1311ff2fe581.jpg)  
+Singular alignment
+
+![](images/9dd0c6e52821ea708fd5c3eca5e256611d45c433b2a567c3de7c78b59dcd5e10.jpg)  
+Composite alignment  
+Figure 11: Comparison of trends in separability and alignment measures of Llama3-70B hidden states between ICL and zero-shot
+
+![](images/ca5a1f1a934bbe8415893b90be015d05cff22cf1e6459aab611a1a7a8f565f8d.jpg)  
+Mean-based alignment
+
+![](images/af3a0b0c7805d36a641834ba570a97c5b00fa0d6af3273a0a4c5ff4d76b8fdf2.jpg)  
+Var.-based alignment
+
+Llama2-7B: 8-shot Acc.:  $75.13\%$ ; 0-shot Acc.:  $0.31\%$
+
+![](images/d2bf179a640d109b507cfc32a7e664c29526d32d32831207729d859117f9e05c.jpg)  
+Effective dimension
+
+![](images/de217ff92d3659a674cac3ac7495ea96b2dd8b44b1a5c0c2bd1526dc8ca1137c.jpg)  
+Separability score
+
+![](images/e58d634cacc4d92f8562da2ff6e3e40a11395fa1db3bc2e083a6d05575563ad1.jpg)  
+Singular alignment
+
+![](images/057647f6a61f99fda7dba68db3d48f792dac424abca3bf1935d38f55ab4aa873.jpg)  
+Composite alignment  
+Figure 12: Comparison of trends in separability and alignment measures of Llama2-7B hidden states between ICL and zero-shot
+
+![](images/64893dc8feda3333207ed4f2204cd4cf0f20fbc24f9a0f4a38348bed3b2ebbc6.jpg)  
+Mean-based alignment
+
+![](images/5d825f6c8df431a03a22f5e4c23502d8c0b5e0f8ce6d9d0958254002f8dae764.jpg)  
+Var.-based alignment
+
+![](images/741349db13a65cbaaecd9b53c715cfefbccdf38d1485a09b6dc791b0b60adf3e.jpg)  
+Effective dimension
+
+![](images/4ed2f638ce4743ef201b0c7b7e4eb22eda3b23eca308b0108dfb06ae1fbb433f.jpg)  
+Separability score
+
+![](images/2fd36602a39418ea5c1895f6fba797a3b5fe30b386ebb05d5d8d24dc01e9c70e.jpg)  
+Singular alignment
+
+![](images/a73a663f9e0b99dcdd3168516d9672682e4a492d4a521aa392deadf64152e1d7.jpg)  
+Composite alignment  
+Figure 13: Comparison of trends in separability and alignment measures of Llama2-13B hidden states between ICL and zero-shot
+
+![](images/127e855910f990c747460ded626c5dbf7746a96967ee3cc93714fbab7cb03864.jpg)  
+Mean-based alignment
+
+![](images/985ab1e16d435d9b3ce963e9b2183bc25907afd256e03dedd58d8fac01e826ba.jpg)  
+Var.-based alignment
+
+Gemma-2B: 8-shot Acc.:  $66.16\%$ ; 0-shot Acc.:  $0.00\%$
+
+![](images/8a9965118e651a6391816351809ffad1125c0bfcf97455c44240593221bf1578.jpg)  
+Effective dimension
+
+![](images/9f50879d6d4ae944702644d4adf6d9b6bc10a523925f4694acb8f1c204fe0e53.jpg)  
+Separability score
+
+![](images/5f7e8b964057efa00a6f9c3a729d3b965256d13a7b773a4be2b23c36af37526b.jpg)  
+Singular alignment
+
+![](images/ecf788e6ab973253cc85335698e3c9815f4673deba8e3653a31bf11749b76d64.jpg)  
+Composite alignment  
+Figure 14: Comparison of trends in separability and alignment measures of Gemma-2B hidden states between ICL and zero-shot
+
+![](images/dba8c01f3053d56b9d1e8d2193a4a063fbea56145b4d66d3735126165d603d73.jpg)  
+Mean-based alignment
+
+![](images/029240c570f48e7c73a4063e8b24d7c6b44b9a6f5a6327541d51068aeb2b8921.jpg)  
+Var.-based alignment
+
+Gemma-7B: 8-shot Acc.:  $79.06\%$ ; 0-shot Acc.:  $0.02\%$
+
+![](images/e0dd065f0cc51832b2f2633290566564745666c3696b09f001dfdf753534920f.jpg)  
+Effective dimension
+
+![](images/d609f356854175dc2fd8211b97b1651d66be9bfc15abf39e25ddfc99ca183d62.jpg)  
+Separability score
+
+![](images/2265e6568907c2491f3061bc0bce4f7015d10e9bd41b6d70f8718cebbcb9297a.jpg)  
+Singular alignment
+
+![](images/5c5be13a3bbc27824159256427567d066a3f9d4cd5fc83a70ae49419623eca9e.jpg)  
+Composite alignment  
+Figure 15: Comparison of trends in separability and alignment measures of Gemma-7B hidden states between ICL and zero-shot
+
+![](images/76485114e13df31275d147f40ca1c65c558cda503c8f112c21dc14a61bab8357.jpg)  
+Mean-based alignment
+
+![](images/8902727cab227a14d3931ae6178c760c51669447e06fdd9a68f43d435aab2b52.jpg)  
+Var.-based alignment
+
+![](images/8cca11aa22157361d91dcf467f66a62ed69e9bf4fddf27a889e0999d714dd6e4.jpg)  
+(a) Varying number of demonstrations
+
+![](images/534398df9ff6cc2c2c2fbf0780704143f8628ee9b59234876cbca7203d93bbf8.jpg)  
+Llama3-8B: Random Selection Acc.: 76.20%; KNN Selection Acc.: 84.03%
+
+![](images/d054b92a606101d5da3cd532368980e21b4edb813654c218fcfb827600c03014.jpg)
+
+![](images/19dad17113e7b543019908872c293cebbbb1171ad608faeeaf3e5a931cd36809.jpg)
+
+![](images/f0a40c81936e3c224ce7427ac156835fe6b58cd34c19c502b51118d06d7ab9eb.jpg)  
+(b) kNN Demonstration selection  
+Figure 16: Layer-wise trends of separability and alignment measures of Llama3-8B hidden states in different ICL settings
+
+![](images/d9354815ed63b0d83d2a4886ee61da3cc3aea65ca8337c9b3fd876f2a99eaf0e.jpg)
+
+![](images/2df7ac2cf85c19dd6399c5e7ff228ffc269c01e067ec0b580d5ae4cee997b568.jpg)
+
+![](images/b62c1fadb9f36eb6b8eda56f2dc49bf2032710c3d2b9b7cb96bf97bd9e605e01.jpg)  
+Llama3-8B: Normal Label Acc.: 76.20%; Symbol Label Acc.: 47.05%
+
+![](images/6f41e4caabd4c9d08f5c267d738b4f457100843ffd433bb411819a60fed68df5.jpg)  
+(c) Random symbols as demonstration labels
+
+![](images/c2c994aaf2c7bfdbf29352d3bc5bc61971e3271df1f709f54cf891a93e8ac0ee.jpg)  
+(a) Varying number of demonstrations
+
+![](images/a9fba76b08b75c3c8108ab9a8286b113b9621eb163e7499367c0dcc5fd993582.jpg)  
+Llama3-70B: Random Selection Acc.: 81.09%; KNN Selection Acc.: 89.44%
+
+![](images/0a931810dc9991d94078844def272353254952dedc7b2e2849b66cb3688bcae0.jpg)
+
+![](images/25fc024222245ce48594e583c84ceaf55bf48d9dcdb67496f7ec96bec55cf31d.jpg)
+
+![](images/9088b1e5d8dbfb087fed2c9dbe9b63a74bf0b076170e2f8f3e248b3ca52e26f4.jpg)  
+(b) KNN Demonstration selection  
+Figure 17: Layer-wise trends of separability and alignment measures of Llama3-70B hidden states in different ICL settings
+
+![](images/b3ef056fd8b3024f4329f1dc0f1fe050d0ae410a127370ffcc85df7aa20285c0.jpg)
+
+![](images/2dbd6e42761bee63844fd525da3c9ff5c52c94bc1846a599977db94885fa27ce.jpg)  
+Llama3-70B: Normal Label Acc.: 81.09%; Symbol Label Acc.: 64.67%  
+(c) Random symbols as demonstration labels
+
+![](images/e29c08d8062a724688b106c34e31e2381fd95a04c2e58246dbc27f53ee3940c3.jpg)  
+(a) Varying number of demonstrations
+
+![](images/eacb08fee2115dcc43d335190c84b91e2de26d91a682a9642661fe3a8422ed6d.jpg)  
+Llama2-7B: Random Selection Acc.: 75.13%; KNN Selection Acc.: 84.07%
+
+![](images/98197934337926a377ffc37ed989f4acfa96a28ec3d8428dd6abf30cdcacb1b7.jpg)
+
+![](images/089a8c707fb13700774d859e22507c9dd134645aabf3cf40778b13927b4a4801.jpg)
+
+![](images/810b0223d2b1bda2cb83a2aef3eb216b1177fa20aa6ea8947b1e22100c4a8ab2.jpg)  
+(b) KNN Demonstration selection  
+Figure 18: Layer-wise trends of separability and alignment measures of Llama2-7B hidden states in different ICL settings
+
+![](images/19fb398c90376a091de520e9ad0de3615ca2090c128b3d90ede5df810f812a4f.jpg)
+
+![](images/fface778757493ac382b7331443818fe592bd76219a58d0b29b6bd9316d1b586.jpg)
+
+![](images/8a5927e665e3e219bf4f2f098ebab5723ff52545e7dde89ef0c41da8505e26b3.jpg)  
+Llama2-7B: Normal Label Acc.: 75.13%; Symbol Label Acc.: 44.73%  
+(c) Random symbols as demonstration labels
+
+![](images/6b3762ba1f02b8c2f5281e68cf5b1d557cc50b20704d98ad360d44eb72fee142.jpg)
+
+![](images/08353a653bdad51612bec6e3c94c174f0f368be4020fe75585c6d4a27c92a1de.jpg)  
+(a) Varying number of demonstrations
+
+![](images/714dc69f0a5aefb829d653f3d51c423ca98f410edf484c1847fbd4338b8c2441.jpg)
+
+![](images/edde4f48e9231dd772c9885a1379ab9b139c1f3982e47b1c4d7944fbad828306.jpg)  
+Llama2-13B: Random Selection Acc.: 74.79%; KNN Selection Acc.: 86.18%
+
+![](images/ac05deb03c5a83ed40b9d871d9c7ccce7a31ea41901b1b79107feab85501f53e.jpg)
+
+![](images/b54cf7a4472bf38cd6137e394dad37f19e5a2e0b0fcb557150d89caed4e77fd3.jpg)
+
+![](images/a937cfef44cc9a0336540cbf87e92afb1d25a14d9594445c3e83fdab93ada6b8.jpg)  
+(b) KNN Demonstration selection
+
+![](images/77723b8ddd0faea921d64d1b341e5d8fcab9abbeb5e7143478e8da2286d8bd9c.jpg)
+
+![](images/b836dc8cdd693eaa203b1b8eb8c202a2972331004f9f972e3f5a247979d18dd1.jpg)  
+Llama2-13B: Normal Label Acc.: 74.79%; Symbol Label Acc.: 51.78%  
+(c) Random symbols as demonstration labels  
+Figure 19: Layer-wise trends of separability and alignment measures of Llama2-13B hidden states in different ICL settings
+
+![](images/80d284e475f954cc6eeba32211ef697ff7f496f1d20c562aa069f1a3261f63cf.jpg)  
+(a) Varying number of demonstrations
+
+![](images/5bdd54d6acfaf537eb2a7064025e3ee585dd29dff80dc9febf87c7dbfd970d9e.jpg)  
+Gemma-2B: Random Selection Acc.: 66.16%; KNN Selection Acc.: 77.99%
+
+![](images/3c8ab5c2cf4512db96fbd461ad66404f205ad3a128562785f1100f78c8782678.jpg)
+
+![](images/725a2bdd96d86f8f38895aad5c3ae0ae2aaf6668c770fd97b59a095dd518a390.jpg)  
+Lemma-2B: Normal Label Acc.: 66.16%; Symbol Label Acc.: 40.06%
+
+![](images/b2a16aa43549eda2c029d007e280861047ad719b1bfb5eee230f3795fd72f98d.jpg)
+
+![](images/ce2d81bab9b5fce6d75de3b5f80f518b5f6e5204ae354db472db2ba831121d39.jpg)  
+(b) KNN Demonstration selection
+
+![](images/b7164f39e66c911240210d59444ce80978ba7ba88485657df65cece1362811f7.jpg)  
+(c) Random symbols as demonstration labels  
+Figure 20: Layer-wise trends of separability and alignment measures of Gemma-2B hidden states in different ICL settings
+
+![](images/6cf1083a2962abc58d398353ccf7629ddd207f13e941d836dd28bbbb749efc99.jpg)  
+(a) Varying number of demonstrations
+
+![](images/988a55adbe5e26100128410aaeb61d3007fcb46ca4efdd0f4a4d867ee114e330.jpg)  
+Gemma-7B: Random Selection Acc.: 79.06%; KNN Selection Acc.: 86.23%
+
+![](images/22d5ee7bb66fbe340eb6fe6d1850491b4c3fbd0de50617834dfb1aca75d2f1c3.jpg)
+
+![](images/30c95978753539e08f4973c81d063b55600f2105110150f734416cd458bd567f.jpg)
+
+![](images/f0b9bb94bd695ef8821dea148d4d861c6da397446f05fe47a5f6a253ab993874.jpg)  
+(b) KNN Demonstration selection  
+Figure 21: Layer-wise trends of separability and alignment measures of Gemma-7B hidden states in different ICL settings
+
+![](images/c5c0909e9dd4d4f8293ecee8fa277a29416b2ffa2c1b5bbf3e886727ef5d5d68.jpg)  
+Gemma-7B: Normal Label Acc.: 79.06%; Symbol Label Acc.: 42.99%  
+(c) Random symbols as demonstration labels
+
+![](images/2eea7458b8a1395b233b549d799c1e8f58dcae4942eb6da96ab599dc395a72df.jpg)  
+(a)SUBJ
+
+![](images/3ab01efb5f1cf84cdc3e98a21aea091a0c35e3b5c82b5f5e76ef592e5dd9ca1e.jpg)
+
+![](images/70e715c28e045a028d40f72c2b6fd92088b70803cce8162da5f01f1deea131fb.jpg)  
+(b) TREC
+
+![](images/d0e53c69f17cfa635f86897eaaaa6def9b8da277d577616524050db6edd8db45.jpg)
+
+![](images/279fb01c5b8d9b33dd708827e8d2884e5629793188a1c391be3edb79d7d2feae.jpg)  
+(c) SNLI
+
+![](images/3906be370287d9776f5152729e55560361b96aad88b9f78bca42756dbc5809e1.jpg)
+
+![](images/435c22d24b6f53346f481ea422b50d0f2f04545b27511f4af62a8debfe4ce8ed.jpg)  
+(d) RTE
+
+![](images/c5f90281dd8c06a0347b3889dc9f362f2708ec182cff9295e65b01ecc6b83f10.jpg)
+
+![](images/ae135ba3861beeb9c107e0e9ec0ff6c593d854756a668cb3da65ba5145890e01.jpg)  
+(e) CB
+
+![](images/862cb44391481b5865b7f558cb664e7228a9f120d3f716aa612b175cff0946c9.jpg)  
+Figure 22: Output and directional alignment of Llama2-70B ICL hidden states with label unembedding vectors on various datasets.
+
+Table 4: Tokens decoded from top 10 singular directions from layers near the phase transition on SUBJ  
+
+<table><tr><td>Layer</td><td>Tokens</td></tr><tr><td>Layer 51</td><td>opinion, humor, genre, adm, facts, plot, description, explanation, pont, humor</td></tr><tr><td>Layer 52</td><td>objective, humor,., adm, dram, plot, enthus, categor, pont, subject</td></tr><tr><td>Layer 53</td><td>objective, humor,., plot, fact, humor, enthus, explanation, character, humor</td></tr></table>
+
+Table 5: Tokens decoded from top 10 singular directions from layers near the phase transition on TREC  
+
+<table><tr><td>Layer</td><td>Tokens</td></tr><tr><td>Layer 18</td><td>Ras, Float, Physics, Heimat, Zob, rela, ijd, lbl, alphabet, uther</td></tr><tr><td>Layer 19</td><td>Geography, name, Physics, map, &gt;, Ele, location, Vall, alphabet, Tests</td></tr><tr><td>Layer 20</td><td>definitions, histor, maps, Geography, amos, inn, map, babel, etr, letters</td></tr></table>
+
+Table 6: Tokens decoded from top 10 singular directions of layers near the phase transition on SNLI  
+
+<table><tr><td>Layer</td><td>Tokens</td></tr><tr><td>Layer 24</td><td>imagination, inen, exceptions, hner, igin, amerikanischer, Fich, float, opposite, orf</td></tr><tr><td>Layer 25</td><td>dup, unknown, similarity, Dy, Sci, ete, nost, Fich, endorf, Correct</td></tr><tr><td>Layer 26</td><td>erde, unknown, similarity, context, apper, architecture, unity, ambigu, alg, py</td></tr></table>
+
+Table 7: Tokens decoded from top 10 singular directions from layers near the phase transition on RTE  
+
+<table><tr><td>Layer</td><td>Tokens</td></tr><tr><td>Layer 21</td><td>Eva, pul, ikel, orp, Mol, enk, Dal, adrat, hor, üng</td></tr><tr><td>Layer 22</td><td>correct, nor, urch, Boh, VP, aqua, Dal, subscribe, currency, sign</td></tr><tr><td>Layer 23</td><td>correction, zero, tempo, Boh, VP, lak, FA, iva, currency, trag</td></tr></table>
+
+![](images/a8f7dffa13216c909a498ff6d6b8edf8a3cf5b332f7b28a306128c4f5d0bc24f.jpg)  
+(a)SUBJ
+
+![](images/0188675c03dc8d594e3f1d44b23f5073a5bac0a004abd0e4620e609f2789eafb.jpg)
+
+![](images/173634d1f13de0a75fb13eb58f91b9ef948c8b57d003c62c99335be5659be208.jpg)  
+(b) TREC
+
+![](images/ed352896f0064b5437d39fdf422b19b2cb7ca48b12de9867d5892066c2a5a0cf.jpg)
+
+![](images/ed00f978418d3d7cdf1a31ef12b0eae8a562e5fe485bc5db585b40819a717904.jpg)  
+(c) SNLI
+
+![](images/30c1b13be7667d89a9ea5253a849b574085e07502f2ca3dcc2109dff44714785.jpg)
+
+![](images/e2c7eaad299760bc1ac23d9c05da6cbfde49625a758dbda62c49d7870e3c10e4.jpg)  
+(d) RTE
+
+![](images/e0a2e35ed50c8280e2b6fc0e0db3d05279b023b36198ae29d81ecfbc8e17cfdc.jpg)
+
+![](images/05e8f0a5247331c63cd42f01a040a6286a9bd4a9cffb6df721ec5e9fccb38949.jpg)  
+(e) CB
+
+![](images/d445dad533bd22e25295d4c62875f130fcf37bd8b4e9564e5aba47fc5cc942b3.jpg)  
+Figure 23: Output and directional alignment of Llama3-8B ICL hidden states with label unembedding vectors on various datasets.
+
+![](images/21679b2895fc2a0489ae187f1d661fb4b5cd5a1baf048ddef6cf7672898ee76f.jpg)  
+(f) SST-2
+
+![](images/8c8cd1c65eea6d9a454d294614058d6339083e03c488b2d82d65d5687fbb638e.jpg)
+
+Table 8: Tokens decoded from top 10 singular directions from layers near the phase transition on CB  
+
+<table><tr><td>Layer</td><td>Tokens</td></tr><tr><td>Layer 22</td><td>TR, eign, &gt;, qual, Joan, Icon, ebol, pHT, āž, Kontrola</td></tr><tr><td>Layer 23</td><td>deg, Cet, Bou, SV, ando, Simon, True, lav, ibōl, opposition</td></tr><tr><td>Layer 24</td><td>negative, schap, ftrag, court, ando, iada, rib, Martf, counter, answer</td></tr></table>
+
+Table 9: Tokens decoded from top 10 singular directions with labels replaced on SNLI  
+
+<table><tr><td>Layer</td><td>Tokens</td></tr><tr><td>Layer 23</td><td>Tier, aro, cum, Vector, FB, Eur, Fon, pending, Dre, Fant</td></tr><tr><td>Layer 24</td><td>False, aten, beck, ardo, Ú/3, Sci, attan, Kaiser, tant, Fant</td></tr><tr><td>Layer 25</td><td>False, Cop,/cgi, Dou, Ú/3, Ra, hur, TF, Dre, nehm</td></tr><tr><td>Layer 44</td><td>false, combination, inois, multiply, scenario, group, action, group, prec, Solo</td></tr><tr><td>Layer 45</td><td>False, #, xspace, @, #, neutral, @, large, sentiment, False</td></tr><tr><td>Layer 46</td><td>False, @, contr, #, @, neutral, cheer, group, ppo, monot</td></tr><tr><td>Layer 78</td><td>false, False, $., @@, @, false, $., false, False, #</td></tr><tr><td>Layer 79</td><td>false,!, .#, True, &lt;s&gt;,!, Hash, Street,群</td></tr><tr><td>Layer 80</td><td>twe, , X, !:, #: ##, action,!, #, Street</td></tr></table>
+
+![](images/fb83fdb75fd3e8e34f402c48e4eddff6e26e47a380615fe2f7a9765b116a28ab.jpg)  
+(a)SUBJ
+
+![](images/fe99eca16a22088ec3f557298fdfb769d2de5571f1817d487e26ff3509e472d7.jpg)
+
+![](images/54b7a27024a75bd5158464380351d9345642faedd425608dfaa4fa6b08214b09.jpg)  
+(b) TREC
+
+![](images/d1b8b0f578d9613b7961b4e95a3852a38773fc02c28c3d736dbf67e1dc063fb4.jpg)
+
+![](images/c726bc8c02014079f5e390e889143a7bce56ffed3d15a4700b420473a0cd68cd.jpg)  
+(c) SNLI
+
+![](images/df4198cada6382afa9ecdd1947b28c163e01525ecf86bd29098133fe7b927da8.jpg)
+
+![](images/fdc34c154e5ca43a778b3937606be2648463670fb89a51bd92db2d5f6ce03b2d.jpg)  
+(d) RTE
+
+![](images/ad066a4b9953d6ec51ecd3336ae3394f67e07a7bf70d55d99997e8a8c140d452.jpg)
+
+![](images/8d6cc7c2556d5bec4c9be54607466dcb88c3eed66899daa4a74c50aa9c3c9c7b.jpg)  
+(e) CB
+
+![](images/28f95b52ef868ff61a61ed5b3c7f2701c0df8117fe9d9bc2ee15d151c77de74d.jpg)  
+Figure 24: Output and directional alignment of Llama3-70B ICL hidden states with label unembedding vectors on various datasets.
+
+![](images/95375f608d2ce09942546558fa2433bd48bb6db183d1151b2e7cb6b54ce54b3f.jpg)  
+(f) SST-2
+
+![](images/262cf541020416a49afe247c03a4de551e0289f673a331b982fee12c781d8373.jpg)
+
+Table 10: Tokens decoded from top 10 singular directions with labels replaced on SST-2  
+
+<table><tr><td>Layer</td><td>Tokens</td></tr><tr><td>Layer 24</td><td>poste, Cant, Jug, Lars, &gt;, zak, zig, geh, eg, alus</td></tr><tr><td>Layer 25</td><td>rott, igi, endi, noreferrer, positive, zak, Aur, Mans, ù, humor</td></tr><tr><td>Layer 26</td><td>rott, Armen, Lau, noreferrer, negative, mee, cot, tu, f, citing</td></tr><tr><td>Layer 40</td><td>ipage, pra, Kle, estaven, Maj, CURLOPT, extreme, humor, ⊙, anel</td></tr><tr><td>Layer 41</td><td>ViewByld, pra, meno, Einzelnach,ziel, @, disappoint, Meyer, L, humor</td></tr><tr><td>Layer 42</td><td>negative, pra, nar, Einzelnach, #, humor, extreme, humor, embargo, #</td></tr><tr><td>Layer 78</td><td>negative, πούημι, elt, neutral, hyper, $., &lt;s&gt;, positive, inea, introsros</td></tr><tr><td>Layer 79</td><td>positive, { , *, neutral, Tru, ???, &lt;s&gt;, ?, humor, Sent</td></tr><tr><td>Layer 80</td><td>{, positive, , !, disappoint, positive, about, positive, wod, introsros</td></tr></table>
+
+Table 11: Tokens decoded from top retained and filtered directions in layer updates on SUBJ  
+
+<table><tr><td colspan="3">Retained Tokens</td></tr><tr><td>Stage</td><td>Layer</td><td>Tokens</td></tr><tr><td rowspan="3">Stage 1</td><td>Layer 54</td><td>humor, description, explanation, objective, humor, shock, subject, character, comparison, humor</td></tr><tr><td>Layer 59</td><td>humor, plot, pra, facts, objective, hypoth, subject, met, explanation, subject</td></tr><tr><td>Layer 66</td><td>., objective, Plot, explain, prom, subject, character, fact, objective, sci</td></tr><tr><td>Stage 2</td><td>Layer 80</td><td>plot, observation, IM, Thom, verb, hyper, quar, Em, ps, Guillaume</td></tr><tr><td colspan="3">Filtered Tokens</td></tr><tr><td>Stage</td><td>Layer</td><td>Tokens</td></tr><tr><td rowspan="3">Stage 1</td><td>Layer 54</td><td>oro, Kant, Harr, Werner, Bast, desert, Kent, Uns, ola, Fall</td></tr><tr><td>Layer 59</td><td>iter, Arg,halten, enser, Lok, Kontrola, tring, atif, ias, Sof</td></tr><tr><td>Layer 66</td><td>conj, dens, shal, amo, Lisa, lish, Sob, narrow, UMN, co</td></tr><tr><td>Stage 2</td><td>Layer 80</td><td>subject, Met, chron, intros, mix, emot, humor, Type, jective, subject</td></tr></table>
+
+![](images/e90bbc53828bbc2a034e7d8371d055ba841f327de698580b2761bf685cc28c67.jpg)  
+(a)SUBJ
+
+![](images/e98e85b5e0e3b4f20d1d594807f9a0523f9598c3e896093fb56d87be3d2f1d40.jpg)
+
+![](images/8282b63aa2db054f73fe00696d04e4c271f4343604c236a97f48629737b270c6.jpg)  
+(b) TREC
+
+![](images/8464cd74920c797ad48184e04525fab206f5e1311672a8af42a12c9106c5d81a.jpg)
+
+![](images/614e0d7debb511f1e2bb68be1d7fcc699279f5effa413481aa34c2772a072526.jpg)  
+(c) SNLI
+
+![](images/9997c902a4984c0791845b3b5b35b8c76d989c5ef690c739c0c35420b3b94d25.jpg)
+
+![](images/07896b83fc0b9a813241f15bc5467baf9960309b97261c525f8a7cc7aaf026ab.jpg)  
+(d) RTE
+
+![](images/2377b11a3d7655b3dfa10e7504510b34632aa921cc2e50a8302e0c159e4f6844.jpg)
+
+![](images/f1ef48167e51b8c2236d84d894257fdf19db39fcc95f5a2da16d0e4147fcf20f.jpg)  
+(e) CB
+
+![](images/1530e9729394a2a0052ae56ea3146f03ec6cfdd9ae3ab12a5fed9e2a0682aa4f.jpg)  
+Figure 25: Output and directional alignment of Llama2-7B ICL hidden states with label unembedding vectors on various datasets.
+
+![](images/cfdb6b213d39e4f80636bb361fef96b185f21e3e0f45db51a9ecbd4d70897a3d.jpg)  
+(f) SST-2
+
+![](images/52dcc5c7d5687fc3892756a1f7239791c9580d2065974eebc0aec00bf52459ef.jpg)
+
+Table 12: Tokens decoded from top retained and filtered directions in layer updates on TREC  
+
+<table><tr><td colspan="3">Retained Tokens</td></tr><tr><td>Stage</td><td>Layer</td><td>Tokens</td></tr><tr><td rowspan="3">Stage 1</td><td>Layer 30</td><td>pione, taste, definition, temporal, precision, science, amen, lich, irectory, enta</td></tr><tr><td>Layer 50</td><td>description, subst, person, numbers, interpret, location, wars, location, answer, date</td></tr><tr><td>Layer 70</td><td>hay, date, number, location, abbre, currency, location, number, description, Creative</td></tr><tr><td>Stage 2</td><td>Layer 80</td><td>hol, Heidel, Desp, eth, olis, description, origin, Type, description, arto</td></tr><tr><td colspan="3">Filtered Tokens</td></tr><tr><td>Stage</td><td>Layer</td><td>Tokens</td></tr><tr><td rowspan="3">Stage 1</td><td>Layer 30</td><td>ako, dn, amen, ainer, overlay, aar, CHAP, storm, synchron, sober</td></tr><tr><td>Layer 50</td><td>Gordon, Chinese, acci, Grace, penas, zak, gas, Kü, ps, prototype</td></tr><tr><td>Layer 70</td><td>mas, fit, ioni, Salt, ru, oci, âtre, quet,thro, Sever</td></tr><tr><td>Stage 2</td><td>Layer 80</td><td>, number, list, discipline, organ, entertain, weather, bum, poll, coff</td></tr></table>
+
+Table 13: Tokens decoded from top retained and filtered directions in layer updates on SNLI  
+
+<table><tr><td colspan="3">Retained Tokens</td></tr><tr><td>Stage</td><td>Layer</td><td>Tokens</td></tr><tr><td rowspan="3">Stage 1</td><td>Layer 25</td><td>pac, fácio, charm, beck, hist, rap, iella, fact, contradiction, engol</td></tr><tr><td>Layer 50</td><td>., probable, false, uncertain, solo, neutral, tender, unknown, gender, unlikely</td></tr><tr><td>Layer 70</td><td>., reverse, false, scenario, maybe, reverse, identity, gender, conce, Gram</td></tr><tr><td>Stage 2</td><td>Layer 80</td><td>rola, Dic, Kra, Column, Common, motor, ..., ge, Salt, fern</td></tr><tr><td colspan="3">Filtered Tokens</td></tr><tr><td>Stage</td><td>Layer</td><td>Tokens</td></tr><tr><td rowspan="3">Stage 1</td><td>Layer 25</td><td>brief, uclude, Ath, bul, refêrences, EC, asta, Rot, nia, Tamil</td></tr><tr><td>Layer 50</td><td>aho, oba, zu,umble, prepared, Mey, jer, Bras, irre, Maj</td></tr><tr><td>Layer 70</td><td>Yu, Mare, iner, artificial, overlap, ilder, aret, elev, constru, Storm</td></tr><tr><td>Stage 2</td><td>Layer 80</td><td>im, &lt;s&gt;, fl, maybe , similarity, sem, habit, Ta, gender, action</td></tr></table>
+
+![](images/c3644302173d5d0d40c1570bf9971ee8533649d702a1116ae4ae0d4d6e748496.jpg)  
+(a)SUBJ
+
+![](images/7867b3e98d955e8688d51d79ef79ffeebea7e83b5a3f86b572995f1b143bf73a.jpg)
+
+![](images/84ee2d722bcee06a68d79345f2604230d7e3d37dc4bc6e293ab1935c3a13c713.jpg)  
+(b) TREC
+
+![](images/8fae65cd89aef4011345fe810fd70deb4defa6a2efec1a9aa91a4929c77f4d93.jpg)
+
+![](images/ca858f125ef3240c782f46c413e532522706872ec43b51dd20b390a8ad4c852f.jpg)  
+(c) SNLI
+
+![](images/60ab9b51d7c0c4f9b2bd2f9356ed623bb2aec6a8b01b5e690ad02e521df915b2.jpg)
+
+![](images/b706470dee13e05023a285ff30262a76d4f162d58f6f248b9037bf3190da00b9.jpg)  
+(d) RTE
+
+![](images/95f8b5b9783cd3ec379b146b11a8d36aa1f0b89d4c2fe399fd2da7f2bd11dad0.jpg)
+
+![](images/73ab5af417a05065f26032f92d9e03d03c3204c38fe3cdf4dbce1f3a2dbbda6c.jpg)  
+(e) CB
+
+![](images/75298f4a0ea765669fa8b33f5738304151b8daed4bbf08bedbdcccca8ab2bebd.jpg)  
+Figure 26: Output and directional alignment of Llama2-13B ICL hidden states with label unembedding vectors on various datasets.
+
+![](images/21e47ddd67f53aab036cab6e0c2e225afe971588051e32306e6487efa3cc0c37.jpg)  
+(f) SST-2
+
+![](images/efa7a850a3be0784e7bc69467e453b314904931a3cd8170b16c14f7ba065326e.jpg)
+
+Table 14: Tokens decoded from top retained and filtered directions in layer updates on RTE  
+
+<table><tr><td colspan="3">Retained Tokens</td></tr><tr><td>Stage</td><td>Layer</td><td>Tokens</td></tr><tr><td rowspan="3">Stage 1</td><td>Layer 33</td><td>Wikipedia, disput, yes, Jakob, alias, Ven, essel, indirect, False, isu</td></tr><tr><td>Layer 50</td><td>false, alarm, irrelevant, opposite, noct, naming, duplicates, misunder, Geography, location</td></tr><tr><td>Layer 68</td><td>:,(Geography, lament, sto, impossible, opinion, ken, numer, negative, true</td></tr><tr><td>Stage 2</td><td>Layer 80</td><td>Ale, Ad, Hub, Ke, lub, Terra, Haz, Ta, part, Andre</td></tr><tr><td colspan="3">Filtered Tokens</td></tr><tr><td>Stage</td><td>Layer</td><td>Tokens</td></tr><tr><td rowspan="3">Stage 1</td><td>Layer 33</td><td>cust,▼, Mün, Lucas, itel, Barb, inha, Hed, FI, nica</td></tr><tr><td>Layer 50</td><td>Лови, CA, Um, wob, prot, jär, 望, ang, 孝, Roman</td></tr><tr><td>Layer 68</td><td>fert, CA, bach, Word, RL, gon, Charlie, MS, nen, Feld</td></tr><tr><td>Stage 2</td><td>Layer 80</td><td>SR, techni, gr, unclear, trag, opposite, Susan, estimate, rare, murder</td></tr></table>
+
+Table 15: Tokens decoded from top retained and filtered directions in layer updates on CB  
+
+<table><tr><td colspan="3">Retained Tokens</td></tr><tr><td>Stage</td><td>Layer</td><td>Tokens</td></tr><tr><td rowspan="3">Stage 1</td><td>Layer 25</td><td>Bou, True, &lt;s&gt;, Cet, and, elin, JIOM, Pear, stabil, lav</td></tr><tr><td>Layer 49</td><td>true, unlikely, uncertain, absence, False, predictions, avoided, myth, predictions, asym</td></tr><tr><td>Layer 70</td><td>true, false, agreement, maybe, predictions, silent, negative, disapp, av, predictions</td></tr><tr><td>Stage 2</td><td>Layer 80</td><td>Giov, future, often, Frank, tend, imet, Sid, fran, Intent, maybe</td></tr><tr><td colspan="3">Filtered Tokens</td></tr><tr><td>Stage</td><td>Layer</td><td>Tokens</td></tr><tr><td rowspan="3">Stage 1</td><td>Layer 25</td><td>slant, acci, Laurent, cht, brie, chamber, olare, gart, una, vie</td></tr><tr><td>Layer 49</td><td>Dra, cito, nick, Verm, moz, neglect, abel, WR, anger, uchar</td></tr><tr><td>Layer 70</td><td>ottom, nation, Cooper, exhaust, reign, hem, Um, diag, Ri, wa</td></tr><tr><td>Stage 2</td><td>Layer 80</td><td>false, esi, , False, Yes, often, aer, „, Mod, George</td></tr></table>
+
+![](images/a2a09b7c552b4b13ae256badb1cba350cb78c602d5905decc45981696bdc8544.jpg)  
+(a)SUBJ
+
+![](images/6f13c672f67485f6ec2ded112369fcaff3b4234f6020732f201b078f59516cfb.jpg)
+
+![](images/37a5fdef34d2d30c35abc876bb5fb3c614f9981784b0e31d8e0f36e242a6f5d2.jpg)  
+(b) TREC
+
+![](images/54ad3e52c1c784d6f04bfb88374519ce50de3e222b2e8db9bf70f4c17b578f86.jpg)
+
+![](images/80efbc7e3a39ef95b22d7f32044abdfa31d4c4f24f4c7e180f44105e0bb57117.jpg)  
+(c) SNLI
+
+![](images/3a812c06b2300e7615a8122cce637840bc4a3e9ccf1e8abb8611aad0a00e1d98.jpg)
+
+![](images/68bd79fb2edb0674b15f77d1a9fe67ec7a76e68fff936fd732489b4d6d16e3ee.jpg)  
+(d) RTE
+
+![](images/30f546bd335685e18d1a27dc378c69bde641b5ad732d11caaa64d1e88f3d86ce.jpg)
+
+![](images/06e619b4a81263ba78e804652e6b9cacbf6ecc508ef5458489c08edd94b4c9b3.jpg)  
+(e) CB
+
+![](images/0c77b5c873d550e2f39c69540ab1a98b791866165e627f8f1cfd64811e17dff9.jpg)  
+Figure 27: Output and directional alignment of Gemma-2B ICL hidden states with label unembedding vectors on various datasets.
+
+![](images/fadd42707158bd8c4e442b32c45bc7ffed553b9528ce819f2c1f2972d74b19b6.jpg)  
+(f) SST-2
+
+![](images/7decca0f8250cbdf7251c43c56dfc0061a0d12b6963b50aca3d1082b336e7c17.jpg)
+
+![](images/5e160bc16f02f58398a58f9e21e74040a795cd572815b77f7b513390130a58a9.jpg)  
+(a)SUBJ
+
+![](images/4fa4b66038b9086b10ca757094f817837daf12bfe3c2a7d1a7e1598f419a0ef1.jpg)
+
+![](images/fed3253217ff6c2bb3313603a7b832429118b5916f37d7392096dc475449553b.jpg)  
+(b) TREC
+
+![](images/7847624313134020f373843bd6d2ac91a6c05ac9d52d28539205eaaddcfce1e8.jpg)
+
+![](images/1b684c728bb2f3e9b6a9240d84288b732f1859b53ce440e39f87f05c06cad984.jpg)  
+(c) SNLI
+
+![](images/024a59b68fc7faa6c498c4b867422f65ac911d318e7135659fdf2bb6888ef75e.jpg)
+
+![](images/44d37029f8acd55ebd2450e28efbb78c013dd80345961c67fd10cd3ac5cc5576.jpg)  
+(d) RTE
+
+![](images/998bf694ea007756da1b721a4258f175f0e0e57b1189880c34943f3834a48091.jpg)
+
+![](images/22b459ecd77b0b4c3d01af3384fa3e9565317c50810f66ec695d4a9551668f96.jpg)  
+(e) CB
+
+![](images/aa37c728ea163618ec8eac8fbce7b286cee18fc33879d37da05ffd84d38fbfec.jpg)  
+Figure 28: Output and directional alignment of Gemma-7B ICL hidden states with label unembedding vectors on various datasets.
+
+![](images/3a1cc93849e31d1d8949b860b9b2105736fe120a870eab5755c32fb3ec8ec655.jpg)  
+(f) SST-2
+
+![](images/81168e09fac00a52949af33a38be4792bfc1e2168c6d302b7ab6ab614b0a4f9f.jpg)
+
+Llama3-8B, average across datasets
+
+Normal Acc.: 76.20% | w/o PTH Acc.: 31.49% | w/o IH Acc.: 0.00% | Random Heads Ablated Acc.: 47.86%
+
+![](images/dafa0929bd1de0d107b8480022de236f1f71951cd9932858467180e81696c8f6.jpg)  
+Effective dimension
+
+![](images/9cc6831a472a71ffd29230b18c6335845b856d0e89f0ee88f221cf4091c5d052.jpg)  
+Separability score
+
+![](images/17782336bc3b82fdd303658d87d9322e640b18efa537083372e67c1e1bc152e0.jpg)  
+Singular alignment
+
+![](images/d245ecbf1e0cff767a7b35c40ad67aba8c29e9dadf44c811b16d0f1a767ef327.jpg)  
+Composite alignment
+
+![](images/34708522f64a89bb4dd176a58b9e8315dcaceb3c96ac73b69fd6f5f4bbfdf6a0.jpg)  
+Mean-based alignment
+
+![](images/174779d0454a199be11becfc7169a284405fcd276617645c3ac1d931aede332a.jpg)  
+Var.-based alignment
+
+![](images/b030f8c5f37a7f077a0c530fc7536bb8e1a8aea1489d8543aa23e038df03a982.jpg)  
+Figure 29: Effects of attention heads ablation on the layer-wise separability and alignment measures of Llama3-8B hidden states averaged across all datasets
+
+Llama3-70B, average across datasets
+
+Normal Acc.: 81.09% | w/o PTH Acc.: 26.89% | w/o IH Acc.: 0.92% | Random Heads Ablated Acc.: 77.89%
+
+![](images/803c8170c80d9cc75e9437c0e0b7427e051f929a7746c3d25fb5c97c77cf2887.jpg)  
+Effective dimension
+
+![](images/8b144ae69009737199ea30eb0fd109bd0aaf557fbd3e4dbdc0f1b9d352156a23.jpg)  
+Separability score
+
+![](images/ab92bca6ed4312951216bac440d2daa2915e344008a6edf552fc189cc7443020.jpg)  
+Singular alignment
+
+Composite alignment  
+![](images/a96abb5fd1cfda7ef8209f54aec1bce35fe0c1438635242ddf0352831a669c91.jpg)  
+Normal w/o PTH w/o IH Random Heads Ablated
+
+![](images/fc3fc5fea30c1636576eefde2065fe32feb2d93dd9a628091e4ebe654bd99521.jpg)  
+Mean-based alignment  
+Figure 30: Effects of attention heads ablation on the layer-wise separability and alignment measures of Llama3-70B hidden states averaged across all datasets
+
+![](images/791d2f8127424bbb60d5e30aa08fe3a67f0f5f4bb9444c924237d44807c0ec10.jpg)  
+Var.-based alignment
+
+Llama2-7B, average across datasets
+
+Normal Acc.: 75.13% | w/o PTH Acc.: 10.59% | w/o IH Acc.: 0.70% | Random Heads Ablated Acc.: 48.15%
+
+![](images/e99101aa87ec14c810dd265fb18d1ade3d42b206fa28bd56840e72f2f5d5576b.jpg)  
+Effective dimension
+
+![](images/3d6f814e9ac70dd5055da424541948364ce950ccd027158f1097951c1bbcb203.jpg)  
+Separability score
+
+![](images/70fbd9cd2bbe5cc036914752cc887918f5ed0344af00c1eb2bd671e5ca50e0bd.jpg)  
+Singular alignment
+
+Composite alignment  
+Figure 31: Effects of attention heads ablation on the layer-wise separability and alignment measures of Llama2-7B hidden states averaged across all datasets  
+![](images/004fd98e8324c6f31d1bf466a2aededf224d3e519039e085fdead8aee290f529.jpg)  
+Normal w/o PTH w/o IH Random Heads Ablated
+
+![](images/50e37c7d92f7dac4dfd259d5ec55af22d2135b65c97a792e344d468ecbc496f5.jpg)  
+Mean-based alignment
+
+![](images/9a0f4e997c43fd86399d30f5e9f686efb68237ed19f05ca41aa14a2d42fb259f.jpg)  
+Var.-based alignment
+
+Llama2-13B, average across datasets
+
+Normal Acc.: 74.79% | w/o PTH Acc.: 23.19% | w/o IH Acc.: 1.16% | Random Heads Ablated Acc.: 73.49%
+
+![](images/4887e793dd67aa8118f6fca4770c311f33d7a0cb27eeb0283129dc38f030e640.jpg)  
+Effective dimension
+
+![](images/20e3f6a8d352123b18aa29b5447550073a5e39115fafccad0438e3cc7ad2fc9b.jpg)  
+Separability score
+
+![](images/1689918ea693a18d5809e144c4ade90d608c6fa6cb5092204480846a1c5716b1.jpg)  
+Singular alignment
+
+Composite alignment  
+Figure 32: Effects of attention heads ablation on the layer-wise separability and alignment measures of Llama2-13B hidden states averaged across all datasets  
+![](images/64e20aaaa77a0afc6070c645d80317b1eb1693b69a2abc232295afc18a84b273.jpg)  
+Normal w/o PTH w/o IH Random Heads Ablated
+
+![](images/b72b0bd3ef7dd005d9a510cc84ddadb8f10e4921c14e148756435367b9ccb69c.jpg)  
+Mean-based alignment
+
+![](images/ef3810908fde2b46d4ea53d1e1921ad6c5b049bb12855036ebe27fa3572cf716.jpg)  
+Var.-based alignment
+
+Gemma-2B, average across datasets
+
+Normal Acc.: 66.16% | w/o PTH Acc.: 16.24% | w/o IH Acc.: 1.20% | Random Heads Ablated Acc.: 36.48%
+
+![](images/f81e441930fe63fe1267df2bc805326c1e75daddf3ad80d4b264376ff9717fd1.jpg)  
+Effective dimension
+
+![](images/4c1c87cdf3c98b82ab05bd680526641cc226b17c4a227bc8c7410cb482dac6ba.jpg)  
+Separability score
+
+![](images/4a774ffb7d0da3a0dde6ab9f2606278ac0e1e93d45621250365c217310cd9c2b.jpg)  
+Singular alignment
+
+Composite alignment  
+Figure 33: Effects of attention heads ablation on the layer-wise separability and alignment measures of Gemma-2B hidden states averaged across all datasets  
+![](images/e4b18c177c75b3beaed66337fb7519a121d08a030bd59ebc232ba006867fe2b1.jpg)  
+Normal w/o PTH w/o IH Random Heads Ablated
+
+![](images/9469aa5fa35ce007aeab0d8095daf93bb6a92b8a245a1bb6f4243c705fd493e8.jpg)  
+Mean-based alignment
+
+![](images/2b42f726576c8251e15be8e4326d850af9f8d54221ffce3e2c6c4e22c3cb63fb.jpg)  
+Var.-based alignment
+
+Gemma-7B, average across datasets
+
+Normal Acc.: 79.06% | w/o PTH Acc.: 25.60% | w/o IH Acc.: 5.66% | Random Heads Ablated Acc.: 40.82%
+
+![](images/a510ceeb6d8a4ac7378cfc2570777c3689965546f76daf630c1bb9a923729c0c.jpg)  
+Effective dimension
+
+![](images/83411ffec1d4fee9f978e889134e58fedfb57ad76e9bbb63a4b8aefa616d4a21.jpg)  
+Separability score
+
+![](images/df69b4d1eeebc6ab95c9d362707f764898377946f91c9cb202a54209d02b9bfe.jpg)  
+Singular alignment
+
+Composite alignment  
+Figure 34: Effects of attention heads ablation on the layer-wise separability and alignment measures of Gemma-7B hidden states averaged across all datasets  
+![](images/a9753b248aeaee2f09b51feb272a802f749f853c08fbb49bb6343f8dc975d9d9.jpg)  
+Normal w/o PTH w/o IH Random Heads Ablated
+
+![](images/395c7d5be139a4578fcbbd077762166b9fcb3e5f9695f79a3c804a2e385a2afd.jpg)  
+Mean-based alignment
+
+![](images/d12a7519bfd4b16640aea20a3f1c983eeecb6b9daa207de5044a84e663a3ec11.jpg)  
+Var.-based alignment
+
+# Llama2-70B Famous
+
+Unseen label Acc.: 57.78% | w/o PTH Acc.: 1.11% | w/o IH Acc.: 37.78% | Seen label Acc.: 78.89%
+
+![](images/4c7d8650f2821c33e18803c14ed466efc4165cfc2a2391d6fc2526aa593911e4.jpg)  
+Effective dimension
+
+![](images/ae0be881f77f99fc1d7c7458e2f7135d0022345dfbaa0506ca8c506fbd58e1dd.jpg)  
+Separability score
+
+![](images/42f3882967f1e33acc92ff9fa8ed4cdcfe2bbaaab65ee5ee1fef6273a4ed6e5c.jpg)  
+Singular alignment
+
+Composite alignment  
+Figure 35: Effects of attention heads ablation on the layer-wise separability and alignment measures of Llama2-70B hidden states on Famous People dataset.  
+![](images/4e157c5afdc0b083048dd51250d087dce202fbc256388faca2344bce92639978.jpg)  
+Unseen label w/o PTH w/o IH Seen label
+
+![](images/2e03028a2e436f39ab7caf978136a140fb251d8d349def8208a55bb8913e0a98.jpg)  
+Mean-based alignment
+
+![](images/fa59eba04884e8454549a5f62d2f6ac1f6aeb25b886af751b211d5ce96985263.jpg)  
+Var.-based alignment
+
+Llama3-8B Famous
+
+Unseen label Acc.: 68.89% | w/o PTH Acc.: 15.56% | w/o IH Acc.: 6.67% | Seen label Acc.: 83.33%
+
+![](images/2f0a9430e96d277c2a4e468301322bd7d71eb97db21f59f3de52d69fa222f962.jpg)  
+Effective dimension
+
+![](images/fde317b72c792d81e46a6660ff09d6d0d45b5fc4172535ffaf2b9e68a33d5c22.jpg)  
+Separability score
+
+![](images/09d3a0ce32fefc496afb04af0bdfb3f1e5ce9a16bcd1c86dc7b1498ca7e260d5.jpg)  
+Singular alignment
+
+Composite alignment  
+Figure 36: Effects of attention heads ablation on the layer-wise separability and alignment measures of Llama3-8B hidden states on Famous People dataset.  
+![](images/76885a7679ef9afce8718da2995b3fec0d7fb42771aa044aa20cf63510326faf.jpg)  
+Unseen label w/o PTH w/o IH Seen label
+
+![](images/04bfc416cf7521b9120739b09776859b59a24b38829851047685797c5b9acc26.jpg)  
+Mean-based alignment
+
+![](images/a8a4076c3b3dc7b7bc72c7ae4cd01ab5d0837ea0d746be03d298a0166ddfd623.jpg)  
+Var.-based alignment
+
+Llama3-70B Famous
+
+Unseen label Acc.: 61.11% | w/o PTH Acc.: 16.67% | w/o IH Acc.: 26.67% | Seen label Acc.: 83.33%
+
+![](images/207487b9538503de68e081aca37564c33d910c9e8e92f32650c656931e277b4e.jpg)  
+Effective dimension
+
+![](images/8b359cd9a54a7de05cda953ca9a6b4267918ca7ac7ac6e2041c82e7267c98e3c.jpg)  
+Separability score
+
+![](images/dfb133a736d0bcd9a0b255b4d8fb43f20c39b47457ebfa49d98d276b70e5be7e.jpg)  
+Singular alignment
+
+Composite alignment  
+Figure 37: Effects of attention heads ablation on the layer-wise separability and alignment measures of Llama3-70B hidden states on Famous People dataset.  
+![](images/ea9085483928f18ce949db54f16f82c71d264072293bee8ce72af7eaaa2a3005.jpg)  
+Unseen label w/o PTH w/o IH Seen label
+
+![](images/458ab0904a57b0674ca05fcd0a866eea480f46396fa354f8036f31cfa51891e1.jpg)  
+Mean-based alignment
+
+![](images/f2211350d6c77560cab7bf5beab5500d9dae7e04c03a1a99149f4703a9aff70c.jpg)  
+Var.-based alignment
+
+Llama2-7B Famous
+
+Unseen label Acc.: 54.44% | w/o PTH Acc.: 1.11% | w/o IH Acc.: 3.33% | Seen label Acc.: 72.22%
+
+![](images/e34ba0fc6e3ad009d80688a4783c24231935d3dd404845ee4e46ab5633638665.jpg)  
+Effective dimension
+
+![](images/65b16149f0dff167bde1d575036017014dd896859ab91f034a86451d18475f5d.jpg)  
+Separability score
+
+![](images/d899c5ee1e82bf7b782ce88f38a8c9ec4bac97570166bb9203ea97c83dad5639.jpg)  
+Singular alignment
+
+Composite alignment  
+Figure 38: Effects of attention heads ablation on the layer-wise separability and alignment measures of Llama2-7B hidden states on Famous People dataset.  
+![](images/659f1280b38bbed4700b37bc048e2aaeb040291c901fe04e919104001e9db7e5.jpg)  
+Unseen label w/o PTH w/o IH Seen label
+
+![](images/4cb5fb6f214e4c93e4bf80ee9deb12c099b6bc5aacb704fa7cb0659a1be01d6b.jpg)  
+Mean-based alignment
+
+![](images/2ffb0eeb6be0c454a1dd60361b3ab8a4b2f77db6b9bdb36e3b43c78e517a7ec5.jpg)  
+Var.-based alignment
+
+Llama2-13B Famous
+
+Unseen label Acc.: 52.22% | w/o PTH Acc.: 0.00% | w/o IH Acc.: 0.00% | Seen label Acc.: 67.78%
+
+![](images/5dd4b9a3032c528c15b3eb8415845d78c5e5957de8febf0e6b516ff01c91a7ba.jpg)  
+Effective dimension
+
+![](images/2d4950e9826d1e17106c95f94dcfda099c68a1e67e46ceced5b60496d19cd1b3.jpg)  
+Separability score
+
+![](images/6f6fcbb2e246e04f3f13e811104b2416441ec1f829b2377c8462aa321b53cb50.jpg)  
+Singular alignment
+
+Composite alignment  
+Figure 39: Effects of attention heads ablation on the layer-wise separability and alignment measures of Llama2-13B hidden states on Famous People dataset.  
+![](images/3e00c7673db0c9b2e614d1c3826e45aa8b3ee8c582b0102a5e76b6f4767449be.jpg)  
+Unseen label w/o PTH w/o IH Seen label
+
+![](images/4f10519a0138c84186d56fb98ba53f4e30a9a15f27dae09fcc9b543f91462a1b.jpg)  
+Mean-based alignment
+
+![](images/62a45cfa9668cdc601c4555a08770f5be687af906373a1d9c5abb11f5d286b7d.jpg)  
+Var.-based alignment
+
+Gemma-2B Famous Unseen label Acc.:  $60.00\%$  | w/o PTH Acc.:  $0.00\%$  | w/o IH Acc.:  $0.00\%$  | Seen label Acc.:  $73.33\%$
+
+![](images/98b399fd5056771fd9bb11a79770e9fa9bbfaabbc8630a4616607f00b3cb61f8.jpg)
+
+![](images/2b9dfc761c84f7d385f0bf8e862210426d7c22997015c71834457711f85c8081.jpg)
+
+![](images/46afda1d810d61d6f7ad58b1e7c49bb9ca16b07512a1c559bff68712459054f5.jpg)
+
+![](images/eea9c90b327f041c45465ea5a6e3c0fcbcb60d639e5154225e4b4e140ceab83a.jpg)
+
+![](images/0dfc680ff3015058ea8b43146b559ff7bc2e55b3e2f8b551bad63c7f38c7700c.jpg)
+
+![](images/b4e5dedf1f80fa4d9b5133a209793c44948f379184ce23f9ec1e9a31df408843.jpg)
+
+![](images/b1dc4ca197f0b6c1f682c1d7c9a61d143cd5fb16747a5f5354f8955c7ec4a384.jpg)  
+Figure 40: Effects of attention heads ablation on the layer-wise separability and alignment measures of Gemma-2B hidden states on Famous People dataset.
+
+# Gemma-7B Famous
+
+Unseen label Acc.: 50.00% | w/o PTH Acc.: 26.67% | w/o IH Acc.: 25.56% | Seen label Acc.: 75.56%
+
+![](images/6058dbe15e6f69a1ba802b55548b7c34ef774f5364b7775ee8e8437645979d0a.jpg)
+
+![](images/e3e5edc1c841787414b855746d73970d6e01cf7c41c579b42be4e15965acceb7.jpg)
+
+![](images/34b8a3dd1093fc7f2c7b8b5eeeb10da197eaf92f95a46ef9624b709604244395.jpg)
+
+![](images/f1bdbf551d9bb8fcb02f625b95bba34d139e8f5d3546769f0aa249ebff8b312e.jpg)
+
+![](images/1f81de87f1e8aebda06f363deb50a0e7c6d29a87ebd1baee54dab2182ff178f2.jpg)
+
+![](images/fc2c2f6d5dc7b111679477c8d8b11864249688113a5375f3a14142eec8281152.jpg)
+
+![](images/d983e4efb9ab3e36c1a851c856e60d1dad8bd9192474bba10579dbf2cb725c25.jpg)  
+Figure 41: Effects of attention heads ablation on the layer-wise separability and alignment measures of Gemma-7B hidden states on Famous People dataset.
+
+![](images/c206586a53cd43b8d38f0674249bb63431e393a58b79e75158a1aff58c607d6a.jpg)  
+(a) SST-2
+
+![](images/0af4694ae3f5a34fad346ccead498d0774b80e7b132b028f420f65ae3405ae33.jpg)  
+(b)SUBJ
+
+![](images/c80fbe250665fd2a52e8adc172f9dc615a97cd97b48d406c031c3d90ada731db.jpg)  
+(c) TREC
+
+![](images/e6b2cd5d21d3cfd71446eeda65aa4d80b513708d77baad528ab65d4f67d82fc2.jpg)  
+(d) SNLI
+
+![](images/8ba3e0cb4b28d8a008803d437395091b24ed6db22e238fc1fbc5a1b54350b805.jpg)  
+(e) RTE
+
+![](images/45c0c4d24e06465b5871048c2e78720dfb018695748720c462707e1318414412.jpg)  
+(f) CB  
+Figure 42: Accuracy gains of rank-10 denoising over layers on all datasets with Llama2-70B
+
+![](images/1b9eefa663ed06117ec6de14c6aa1fbb0647c0d009536cc8e41ae8fcc22b9867.jpg)  
+(a) SST-2
+
+![](images/9389fafa8af40dda5dcf99ed905dc8b16f0f247f5f54aeda46c27540cf623477.jpg)  
+(b)SUBJ
+
+![](images/b446fd71e0442d9d594428c1efcfed4cf9eeffc9d723aee2abf776582f5c1d26.jpg)  
+(c) TREC
+
+![](images/64a4092b93cdbe9e1016f2ba8fa2428d7548ba9335fc0da0ff812ea2cd1b6975.jpg)  
+(d) SNLI
+
+![](images/3175001eb774480b07cb2f746e279baa064c18164ca5565341cedec1f263a1b6.jpg)  
+(e) RTE
+
+![](images/d1ebe295551afa3f4db4f281c0bd895abdd87186e8b844fbf067bbc97c8cc6be.jpg)  
+(f) CB
+
+![](images/d047ca6a1e1b939765b5c436e8b5bdedaa75de60ce09e7ac971ff677c3a8cfd9.jpg)  
+Figure 43: Accuracy gains of rank-10 denoising over layers on all datasets with Llama3-8B
+
+![](images/9135b252ce9c0c09dbf9a8ccfc72191cf78eeea4a895e84921e7c0692a47fda4.jpg)  
+(b)SUBJ
+
+![](images/89a070234f6c2283129ad4f9f434fd27bea7157172fe3129214d206a4133fd1b.jpg)  
+(c) TREC
+
+![](images/b48821d4df9d532d4d3d2adcd723d79a0dbcad1fe7c7ecb057239514d1e517b5.jpg)  
+(a) SST-2  
+(d) SNLI
+
+![](images/78192d12d54e8b7dbb1dbbe5cd7825bf132b3f369e16950cdf65137486136368.jpg)  
+(e) RTE
+
+![](images/beb7b100932493068ae32316b73fe9e3ddf840961c04db1c583381697907e1ff.jpg)  
+(f) CB
+
+![](images/780f0f67c05a803a5da053cc1d8e92a92ecb3b82744f52511e14fb5dc48beee2.jpg)  
+Figure 44: Accuracy gains of rank-10 denoising over layers on all datasets with Llama3-70B  
+(a) SST-2
+
+![](images/bd88b420516ae10e42fb78365fb92c2a6a765a4d5d743d681b46ef66c4f79319.jpg)  
+(b)SUBJ
+
+![](images/59f3798dd0dbe1c3ccbb5b813b6534d3a1b22ae14f6d4444ac449e36eddbba8d.jpg)  
+(c) TREC
+
+![](images/b31cccadb1326299cb536d09d90446848b03fd19e8866eacf4c2bebf66c5b96f.jpg)  
+(d) SNLI
+
+![](images/f6f4d39acda1b5fa4943e96e61e1fd3695608a541e07a03f55efef4acb430db5.jpg)  
+(e) RTE
+
+![](images/0f45fe374c3a401f2a825af673705f2900280dbf968eb458bf265a3c2b58d2a3.jpg)  
+(f) CB
+
+![](images/98f0031d3577e14b222f36e998f908fc3c3c495d1d7a6392fdfa723688d93e92.jpg)  
+Figure 45: Accuracy gains of rank-10 denoising over layers on all datasets with Llama2-7B  
+(a) SST-2
+
+![](images/e3503d17fdc38b59dd448c55dac586a3849c7a71a6a29c148847e1e2a77c3626.jpg)  
+(b)SUBJ
+
+![](images/873340fdff53b68d1529b36e458515f95c48d23ac3d36ce5b3f5e2678d8087f3.jpg)  
+(c) TREC
+
+![](images/0579887af81486e654c3865a012c280809063b2dfe8c4d195a6d57eedb54254f.jpg)  
+(d) SNLI  
+Figure 46: Accuracy gains of rank-10 denoising over layers on all datasets with Llama2-13B
+
+![](images/b3bd45602da5616e7398271cae82446d27fc24e0399755ff92759de5877cca1b.jpg)  
+(e) RTE
+
+![](images/1da02fb1768d87b55d4f100cefde2ba3cf1a618b026d9ee37c8fc1aedbbd1d78.jpg)  
+(f) CB
+
+![](images/d74d6ff2102154976a481157291a6a3cd57838533222f1f48a5d1c513fd5d264.jpg)  
+(a) SST-2
+
+![](images/3b85663606d11a718cc39848246369799b26b5d7334bbaf6fdf07193f0229030.jpg)  
+(b)SUBJ
+
+![](images/9ac8a4c35e999b150df6335a812d6460b53b0c93405560730b38ca30467c525b.jpg)  
+(c) TREC
+
+![](images/46557a4af374077bf35a854ddf5dbbe82c35ac31bb7967f65546fa5677636097.jpg)  
+(d) SNLI
+
+![](images/dc912b2ef336f45ae2a47de7fde8d4b12bacbd40261b7337e6f2a84225b4f463.jpg)  
+(e) RTE
+
+![](images/fe09bae2afe82e4c97024af825a1ff0bccd75098bb3e67c783caa2ad36046604.jpg)  
+(f) CB
+
+![](images/db7a371197dd00e534f7d6737b67c334e0157ee7c8c3fcaff6b5bf42bbf25236.jpg)  
+Figure 47: Accuracy gains of rank-10 denoising over layers on all datasets with Gemma-2B
+
+![](images/cd2f9128bc28c9199d495d349d543027f3da8739b2345b6275cd8da984f626d4.jpg)  
+(b)SUBJ
+
+![](images/662076dbd6a2a47f8105ac417f4d1bf7c12b14f8ce0c96bd4c2b54887535a331.jpg)  
+(c) TREC
+
+![](images/7c7758b3c5ead4af064a6191461a69d5b2e74d9f9e3641e8289979f35b42d1de.jpg)  
+(a) SST-2  
+(d) SNLI
+
+![](images/ea3539a727bf5aa463ac60b028efe9745959945577a49d052ce3a24e9413de64.jpg)  
+(e) RTE
+
+![](images/d55be268c0c26501727710db219894bf8184cc5c77af6710fdfac3f2f054e36f.jpg)  
+(f) CB
+
+![](images/0fa52d46cace7d95cffa604bc11329c1b5216d6d3e49e8ca2de0cc9acb8277ba.jpg)  
+Figure 48: Accuracy gains of rank-10 denoising over layers on all datasets with Gemma-7B  
+(a) SST-2
+
+![](images/4a47cbab4d9f41f40c8d6ec5e549bcafff7c9c78c759c6f8fa19c18901fd0ff2.jpg)  
+(b)SUBJ
+
+![](images/879f4aa2e4ecb3bc7df51396e7b3933d12fd532766a81425675ded86d421998f.jpg)  
+(c) TREC
+
+![](images/4fa0f27d807ab508051ce7cc147a7570ef1fb2e98e3832897efb17a811ac3dec.jpg)  
+(d) SNLI
+
+![](images/6f86560a8a0eb946a66dfe25fb9e7d26b72f766872530a2f9735dcf9042469dd.jpg)  
+(e) RTE
+
+![](images/d97cd73e001c2f3bcc7cde4abb24697d218817813618b2dbbf3768f5eba5333a.jpg)  
+(f) CB
+
+![](images/96501f52fc178c9307845a1ba0013d2593a0cf65acba0f9d67e51fdc782505cb.jpg)  
+Figure 49: Accuracy gains of rank-5 denoising over layers on all datasets with Llama2-70B  
+(a) SST-2
+
+![](images/7992cd8ff2fabf376e23559005ee97145c247be41324fb237e389e6a8f7bf677.jpg)  
+(b)SUBJ
+
+![](images/37514c60b03d80450742c4a4c37806f910d496d6f858308478d4c573f3aec461.jpg)  
+(c) TREC
+
+![](images/f3b1e1a918af5ec6ee5cd15123062414283583e869360acf06edeb32f5d59c19.jpg)  
+(d) SNLI  
+Figure 50: Accuracy gains of rank-2 denoising over layers on all datasets with Llama2-70B
+
+![](images/bb522dfb4c1bc011a0b3694d30b68e5b2f8bd6600ef8915d5186581729b99ca8.jpg)  
+(e) RTE
+
+![](images/6ef4a737c9e0b16e8ccba11d18595744fdc1de0da5a2c5ccf27d7711c91f81e7.jpg)  
+(f) CB
+
+![](images/2140920fb5b69ed1f50e5b67c1f6ba895136390d4e8f23e79dbbc5d62e6b12fb.jpg)  
+(a) SST-2
+
+![](images/988e32558d48fb97dffad52bf7d2f2b22a97b7f44ff19ef8762f3a542ff94fcc.jpg)  
+(b)SUBJ
+
+![](images/4e6d8b9a55c8d0789a136506168fbe2659f03a07817f173225cd27bdc7dca9b6.jpg)  
+(c) TREC
+
+![](images/1153315dbee5ccd316babdc686cc0e48941e98c052ee3b154ca56bac680c6309.jpg)  
+(d) SNLI
+
+![](images/d393f7b1b70106164996d5bbef2cc2147725b69dd931d0201ec342ced70ff632.jpg)  
+(e) RTE  
+Figure 51: Accuracy gains of rank-1 denoising over layers on all datasets with Llama2-70B
+
+![](images/dff28a4b83132697e7b422d390851c79e0824e650e36fea45c327269c631834c.jpg)  
+(f) CB
+
+![](images/e4d145a2a8eb4bf1e21e078d7ac12575c4e7951c54368d51faae494d277928ea.jpg)  
+Llama2-70B, average across datasets  
+Normal Acc.:  $0.24\%$  | w/o PTH Acc.:  $0.00\%$  | w/o IH Acc.:  $0.05\%$  | Random Heads Ablated Acc.:  $0.08\%$  
+Effective dimension
+
+![](images/069d5c6ff307d2359fca813c83fcfe81089088fd1337552d7a993bdb37a89b15.jpg)  
+Separability score
+
+![](images/ed8da17c1494203b4b26d6bded4de238493bb7115cbcc028b1d84c3dcb372b88.jpg)  
+Singular alignment
+
+Composite alignment  
+Figure 52: Average effects across datasets of attention heads ablation on the layer-wise separability and alignment measures of Llama2-70B hidden states in the zero-shot setting.  
+![](images/f5c4f53c5703c8753d3f28a8db45878180cb6cba0b4e0390c1bb6c5f89c3320d.jpg)  
+Normal w/o PTH w/o IH Random Heads Ablated
+
+![](images/a0f6540860b97a1c0b26d5c51aa64001be3318593e89b9aa4f8fe2ef38f1125f.jpg)  
+Mean-based alignment
+
+![](images/5671719a3116a878d49cece9691ee8419c9df35a0056e8988102c55604cf0e97.jpg)  
+Var.-based alignment
+
+Table 16: Mean layer values of IH and PTH along with p-values of Mann-Whitney U test on Llama3-8B.  
+
+<table><tr><td>% Level</td><td>IH mean layer</td><td>PTH mean layer</td><td>p-value of Mann-Whitney U test</td></tr><tr><td>1%</td><td>15.883</td><td>9.5667</td><td>5.06 × 10-7</td></tr><tr><td>2%</td><td>16.392</td><td>12.658</td><td>3.99 × 10-5</td></tr><tr><td>5%</td><td>16.493</td><td>13.833</td><td>2.29 × 10-5</td></tr><tr><td>10%</td><td>16.435</td><td>14.364</td><td>4.46 × 10-6</td></tr></table>
+
+Table 17: Mean layer values of IH and PTH along with p-values of Mann-Whitney U test on Llama3-70B.  
+
+<table><tr><td>% Level</td><td>IH mean layer</td><td>PTH mean layer</td><td>p-value of Mann-Whitney U test</td></tr><tr><td>1%</td><td>47.624</td><td>45.755</td><td>0.5573</td></tr><tr><td>2%</td><td>46.273</td><td>46.977</td><td>0.1347</td></tr><tr><td>5%</td><td>44.354</td><td>47.277</td><td>1.18 × 10-7</td></tr><tr><td>10%</td><td>43.278</td><td>44.257</td><td>8.38 × 10-4</td></tr></table>
+
+![](images/eb1fd7b9771f6d6ec556fb5fbab21f41a9e4ee63048a336001bf148fc64885c6.jpg)  
+(a) Layer-wise separability score: 0-shot vs. 8-shot
+
+![](images/53bfcc1c2aa0297bd6e94a9b79ddc3386e98333baf44d98295ac272e7b3994ad.jpg)  
+(b) Layer-wise composite alignment: 0-shot vs. 8-shot  
+Figure 53: Layer-wise trends of separability score and composite alignment of 0-shot and 8-shot hidden states of Llama3-8B.
+
+![](images/4708c75cf002b1d28e5c0d83e5b45f1d72a6fe0256e1080bbf24cce9be600032.jpg)  
+(a) Layer-wise separability score: 0-shot vs. 8-shot
+
+![](images/d67a66dcf9343859752142cd928975706c74cc976dfba4f74e7c0c670b6f64ee.jpg)  
+(b) Layer-wise composite alignment: 0-shot vs. 8-shot  
+Figure 54: Layer-wise trends of separability score and composite alignment of 0-shot and 8-shot hidden states of Llama3-70B.
+
+![](images/4860d1c898b866a0fee65624b738113da01f58e1412983bca1c9da1c8194d6dc.jpg)  
+(a) Layer-wise separability score: 0-shot vs. 8-shot  
+Figure 55: Layer-wise trends of separability score and composite alignment of 0-shot and 8-shot hidden states of Llama2-7B.
+
+![](images/62cb30e65069ace983235aca55f8f0a918a3fdccd9609749a9d2cc9d59093134.jpg)  
+(b) Layer-wise composite alignment: 0-shot vs. 8-shot
+
+![](images/0c7444739dfca8e74219ed45e120eaa5bd9957b04980f03a565115ef0b00f79b.jpg)  
+(a) Layer-wise separability score: 0-shot vs. 8-shot  
+Figure 56: Layer-wise trends of separability score and composite alignment of 0-shot and 8-shot hidden states of Llama2-13B.
+
+![](images/2a272750a0562b6ed24b5bc0c1ea50019fb6ceadac13dbeea36ee591cab07b5f.jpg)  
+(b) Layer-wise composite alignment: 0-shot vs. 8-shot
+
+![](images/081e52ebd3779620c78fe75c3d7b4c7c60b174ba3c0a14fc63b77f82b1b38cef.jpg)  
+(a) Layer-wise separability score: 0-shot vs. 8-shot
+
+![](images/a650b3916f9e531f9a318ffb28e6ce3756ff88dab62baf3cbee6292424db5c7f.jpg)  
+(b) Layer-wise composite alignment: 0-shot vs. 8-shot  
+Figure 57: Layer-wise trends of separability score and composite alignment of 0-shot and 8-shot hidden states of Gemma-2B.
+
+![](images/30028af47790fc7cfa66fc4c7bee9b27b6ff4b0192a5cea6f61fa46533a24ee1.jpg)  
+(a) Layer-wise separability score: 0-shot vs. 8-shot
+
+![](images/8752dd5b89e5a561ba604ec1edb0f8e812a71c785364295af088f2aa1ed0de73.jpg)  
+(b) Layer-wise composite alignment: 0-shot vs. 8-shot  
+Figure 58: Layer-wise trends of separability score and composite alignment of 0-shot and 8-shot hidden states of Gemma-7B.
+
+![](images/eb347c45e682dd7702d9223350fef6d66274df368259cf51bb0ef5c2d02f05bf.jpg)  
+(a) Layer-wise separability score: PTH task vector  
+Figure 59: Effects of injecting PTH and IH outputs as task vectors on the geometric properties of Llama3-8B zero-shot hidden states.
+
+![](images/6c639b77ca1c748c6b41ffb36f8fa3489b1bb2109090b183cd629fe7b38dcdba.jpg)  
+(b) Layer-wise composite alignment: IH task vector
+
+![](images/475c17581ffee4b97bc3ae36eca659b7c658d4937ea3fb00bf5b0f2dbe0d8807.jpg)  
+(a) Layer-wise separability score: PTH task vector  
+Figure 60: Effects of injecting PTH and IH outputs as task vectors on the geometric properties of Llama3-70B zero-shot hidden states.
+
+![](images/8ed3afc9088a2e0cfa68f01fc616c5b3c4fa192e70b13467615f08a5cb713c19.jpg)  
+(b) Layer-wise composite alignment: IH task vector
+
+![](images/f872b90528ecaab93eb4db048c2ce2a847aed5ba4bbea896e7bb2bf948b59260.jpg)  
+(a) Layer-wise separability score: PTH task vector
+
+![](images/9289869b22865e74877cb40b37668431e7ef190006e218467083bebe9ac338fb.jpg)  
+(b) Layer-wise composite alignment: IH task vector  
+Figure 61: Effects of injecting PTH and IH outputs as task vectors on the geometric properties of Llama2-7B zero-shot hidden states.
+
+![](images/a1be3b336a80978b95ea63bcef601da4bb4e613379a46694209ad4403ef7cb1a.jpg)  
+(a) Layer-wise separability score: PTH task vector
+
+![](images/6572782123a74767384d90dcdb5ca41d9422780adf2aeae93fe21194c4a2f4bf.jpg)  
+(b) Layer-wise composite alignment: IH task vector  
+Figure 62: Effects of injecting PTH and IH outputs as task vectors on the geometric properties of Llama2-13B zero-shot hidden states.
+
+![](images/ddd2856b42515050ba1c3f92e9ed1a70ca585a09fdb4426dd142529d39780ec0.jpg)  
+(a) Layer-wise separability score: PTH task vector  
+Figure 63: Effects of injecting PTH and IH outputs as task vectors on the geometric properties of Gemma-2B zero-shot hidden states.
+
+![](images/d8a5e21daa25a95d28952c4afd073264bb02cd2eb1b744343a2644f6b99c33e3.jpg)  
+(b) Layer-wise composite alignment: IH task vector
+
+![](images/0a87eb92bd42f107dbd06f2e4c599ca17d7fac28c20a12f3da64f2e99f704efb.jpg)  
+(a) Layer-wise separability score: PTH task vector  
+Figure 64: Effects of injecting PTH and IH outputs as task vectors on the geometric properties of Gemma-7B zero-shot hidden states.
+
+![](images/9913c298fa17824f9775d35f273d2e40e474e77cf92bb9dee54afb62c40d4cdb.jpg)  
+(b) Layer-wise composite alignment: IH task vector
+
+Table 18: Mean layer values of IH and PTH along with p-values of Mann-Whitney U test on Llama2-7B.  
+
+<table><tr><td>% Level</td><td>IH mean layer</td><td>PTH mean layer</td><td>p-value of Mann-Whitney U test</td></tr><tr><td>1%</td><td>15.883</td><td>8.5833</td><td>5.62 × 10-7</td></tr><tr><td>2%</td><td>16.133</td><td>10.108</td><td>9.78 × 10-9</td></tr><tr><td>5%</td><td>16.301</td><td>12.092</td><td>4.12 × 10-8</td></tr><tr><td>10%</td><td>16.405</td><td>12.559</td><td>7.27 × 10-13</td></tr></table>
+
+Table 19: Mean layer values of IH and PTH along with p-values of Mann-Whitney U test on Llama2-13B.  
+
+<table><tr><td>% Level</td><td>IH mean layer</td><td>PTH mean layer</td><td>p-value of Mann-Whitney U test</td></tr><tr><td>1%</td><td>21.312</td><td>17.094</td><td>1.75 × 10-2</td></tr><tr><td>2%</td><td>20.135</td><td>18.026</td><td>9.44 × 10-2</td></tr><tr><td>5%</td><td>19.960</td><td>16.927</td><td>6.08 × 10-6</td></tr><tr><td>10%</td><td>19.888</td><td>16.343</td><td>3.08 × 10-13</td></tr></table>
+
+Table 20: Mean layer values of IH and PTH along with p-values of Mann-Whitney U test on Gemma-2B.  
+
+<table><tr><td>% Level</td><td>IH mean layer</td><td>PTH mean layer</td><td>p-value of Mann-Whitney U test</td></tr><tr><td>1%</td><td>9.0000</td><td>1.6667</td><td>8.58 × 10-3</td></tr><tr><td>2%</td><td>10.917</td><td>5.0000</td><td>7.91 × 10-4</td></tr><tr><td>5%</td><td>11.571</td><td>4.4286</td><td>2.22 × 10-10</td></tr><tr><td>10%</td><td>12.214</td><td>6.1667</td><td>2.48 × 10-12</td></tr></table>
+
+Table 21: Mean layer values of IH and PTH along with p-values of Mann-Whitney U test on Gemma7B.  
+
+<table><tr><td>% Level</td><td>IH mean layer</td><td>PTH mean layer</td><td>p-value of Mann-Whitney U test</td></tr><tr><td>1%</td><td>18.708</td><td>9.0833</td><td>2.01 × 10-5</td></tr><tr><td>2%</td><td>19.062</td><td>9.1250</td><td>1.96 × 10-6</td></tr><tr><td>5%</td><td>18.742</td><td>7.6212</td><td>6.31 × 10-21</td></tr><tr><td>10%</td><td>18.523</td><td>9.0530</td><td>4.44 × 10-32</td></tr></table>
+
+![](images/6d42750c176a8545546b6a8bfc71de5407c2c00beca211bd7bdd62a33054eee5.jpg)  
+Figure 65: Average effect across datasets of steering Llama3-8B zero-shot hidden states using task vectors created from IH outputs.
+
+![](images/2c1feb3dbf0c20731674696f9a85d56f23e632f9e823d77db23ec499518e148b.jpg)  
+Figure 66: Average effect across datasets of steering Llama3-70B zero-shot hidden states using task vectors created from IH outputs.
+
+![](images/87aca166d92e6d5ef2847248f699b9dac357543c0073d6cb579f20ae85bb967e.jpg)  
+Figure 67: Average effect across datasets of steering Llama2-7B zero-shot hidden states using task vectors created from IH outputs.
+
+![](images/e1701160bd0bd8c6641efda7c984078f6383930ce36424e5d2fe6476c3a1f34b.jpg)  
+Figure 68: Average effect across datasets of steering Llama2-13B zero-shot hidden states using task vectors created from IH outputs.
+
+![](images/bc67ada341605886cf1136ea1f68828c571b8f343aa23b4b038ed79a6abd4db5.jpg)  
+Figure 69: Average effect across datasets of steering Gemma-2B zero-shot hidden states using task vectors created from IH outputs.
+
+![](images/3b860f3fc69dccb74758bed00b7089878c819b5414cf5b207a9a923a17d593eb.jpg)  
+Figure 70: Average effect across datasets of steering Gemma-7B zero-shot hidden states using task vectors created from IH outputs.
+
+Table 22: Comparison of classification accuracy: training classifier v.s. direct decoding with zero-shot prompts.  
+
+<table><tr><td>Model</td><td>0-shot (%)</td><td>0-shot+classifier (%)</td><td>Improvement</td></tr><tr><td>Llama3-8B</td><td>0.30%</td><td>73.71%</td><td>+73.41%</td></tr><tr><td>Llama3-70B</td><td>0.00%</td><td>78.97%</td><td>+78.97%</td></tr><tr><td>Llama2-7B</td><td>0.31%</td><td>75.63%</td><td>+75.32%</td></tr><tr><td>Llama2-13B</td><td>0.36%</td><td>79.71%</td><td>+79.35%</td></tr><tr><td>Llama2-70B</td><td>0.24%</td><td>79.02%</td><td>+78.78%</td></tr><tr><td>Gemma-2B</td><td>0.00%</td><td>75.56%</td><td>+75.56%</td></tr><tr><td>Gemma-7B</td><td>0.02%</td><td>75.43%</td><td>+75.41%</td></tr></table>
+
+Table 23: Comparison of classification accuracy: training classifier v.s. direct decoding with zero-shot prompts.  
+
+<table><tr><td>Model</td><td>8-shot (%)</td><td>8-shot+classifier (%)</td><td>Improvement</td></tr><tr><td>Llama3-8B</td><td>76.20%</td><td>85.47%</td><td>+9.27%</td></tr><tr><td>Llama3-70B</td><td>81.01%</td><td>84.46%</td><td>+3.45%</td></tr><tr><td>Llama2-7B</td><td>74.83%</td><td>85.59%</td><td>+10.76%</td></tr><tr><td>Llama2-13B</td><td>74.82%</td><td>85.44%</td><td>+10.62%</td></tr><tr><td>Llama2-70B</td><td>80.52%</td><td>87.76%</td><td>+7.24%</td></tr><tr><td>Gemma-2B</td><td>67.03%</td><td>77.87%</td><td>+10.84%</td></tr><tr><td>Gemma-7B</td><td>79.19%</td><td>84.94%</td><td>+5.75%</td></tr></table>
+
+Table 24: Comparison of classification accuracy: regular ICL v.s. low-rank denoising of hidden states  
+
+<table><tr><td>Model</td><td>8-shot (%)</td><td>8-shot+denoising (%)</td><td>Improvement</td></tr><tr><td>Llama3-8B</td><td>76.20%</td><td>78.58%</td><td>+2.38%</td></tr><tr><td>Llama3-70B</td><td>81.01%</td><td>82.40%</td><td>+1.39%</td></tr><tr><td>Llama2-7B</td><td>74.83%</td><td>77.68%</td><td>+2.85%</td></tr><tr><td>Llama2-13B</td><td>74.82%</td><td>79.52%</td><td>+4.70%</td></tr><tr><td>Llama2-70B</td><td>80.52%</td><td>80.72%</td><td>+0.20%</td></tr><tr><td>Gemma-2B</td><td>67.03%</td><td>67.25%</td><td>+0.22%</td></tr><tr><td>Gemma-7B</td><td>79.19%</td><td>66.11%</td><td>-13.08%</td></tr></table>
\ No newline at end of file
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@@ -0,0 +1,1274 @@
+# Unifying Proportional Fairness in Centroid and Non-Centroid Clustering
+
+Benjamin Cookson
+
+Department of Computer Science
+
+University of Toronto
+
+bcookson@cs.toronto.edu
+
+Nisarg Shah
+
+Department of Computer Science
+
+University of Toronto
+
+nisarg@cs.toronto.edu
+
+Ziqi Yu
+
+Department of Computer Science
+
+University of Toronto
+
+ziqiyu@cs.toronto.edu
+
+# Abstract
+
+Proportional fairness criteria inspired by democratic ideals of proportional representation have received growing attention in the clustering literature. Prior work has investigated them in two separate paradigms. Chen et al. [1] study centroid clustering, in which each data point's loss is determined by its distance to a representative point (centroid) chosen in its cluster. Caragiannis et al. [2] study non-centroid clustering, in which each data point's loss is determined by its maximum distance to any other data point in its cluster.
+
+We generalize both paradigms to introduce semi-centroid clustering, in which each data point's loss is a combination of its centroid and non-centroid losses, and study two proportional fairness criteria—the core and, its relaxation, fully justified representation (FJR). Our main result is a novel algorithm which achieves a constant approximation to the core, in polynomial time, even when the distance metrics used for centroid and non-centroid loss measurements are different. We also derive improved results for more restricted loss functions and the weaker FJR criterion, and establish lower bounds in each case.
+
+# 1 Introduction
+
+Clustering is a central task in AI, in which the goal is to either partition a set of  $n$  data points into  $k$  clusters  $C_1, \ldots, C_k$ , or return  $k$  centroids  $x_1, \ldots, x_k$ , or perhaps return both the clusters and their assigned centroids. In many cases, partitioning the data points and choosing centroids are viewed as interchangeable. For example, in the traditional clustering setting, the popular  $k$ -means loss function (to be minimized) can be written in two equivalent forms:
+
+$$
+\min  _ {C _ {1}, \dots , C _ {k}} \sum_ {t = 1} ^ {k} \frac {1}{| C _ {t} |} \sum_ {i, j \in C _ {t}} d (i, j) ^ {2} \equiv \min  _ {C _ {1}, \dots , C _ {k}, x _ {1}, \dots , x _ {k}} \sum_ {t = 1} ^ {k} \sum_ {i \in C _ {t}} d (i, x _ {t}) ^ {2}.
+$$
+
+This is because for a given partition  $(C_1, \ldots, C_k)$ , the optimal choice of the centroid  $x_t$  for each cluster  $C_t$  is the average of its points (this can be generalized to non-Euclidean metrics too) and plugging in these optimal choices yields the formulation on the left.
+
+An emerging line of research at the intersection of economics and AI focuses on applications in which every data point represents an agent, and studies facets such as dealing with strategic manipulations
+
+by selfish agents [3-7] or ensuring fairness to the agents [8-10]. Here, it is not sufficient to have only a global loss function to be minimized; one needs to know the loss functions (i.e., preferences) of the individual agents and different formulations of loss functions at the agent level, even if equivalent in the aggregate, lead to different results.
+
+Our focus is on fairness—specifically, on proportional fairness. Informally, it provides a fairness guarantee to every possible group of agents—not just those determined by a limited set of sensitive attributes such as race and gender, or their intersections—with the strength of the guarantee scaling proportionally to the group's size and cohesiveness of its members' preferences. Proportional fairness criteria have been applied to a growing number of problems in AI [1, 9-13].
+
+For clustering, Chen et al. [1] initiate the study of proportional fairness for centroid clustering, in which the loss of an agent is its distance to the centroid of its cluster. They motivate it through facility location, in which the goal is to build  $k$  facilities; each agent naturally prefers to be close to its assigned facility. Caragiannis et al. [2] study proportional fairness for non-centroid clustering, in which the loss of an agent is an aggregate (e.g., average or maximum) of its distances to the other agents in its cluster. Their motivation stems from clustered federated learning, in which data sources (agents) are partitioned into clusters and agents in each cluster collaboratively learn a model; each agent then wishes to be close (in terms of their data distributions) to the other agents in its cluster, so that the model they collaboratively learn is accurate for the agent personally.
+
+But in both these applications, and in many others, it may be natural for agents to care about both their distance to the centroids of their clusters and their distances to the other agents in their clusters. For example, in facility location, an individual may want their assigned facility being close to their house as well as other individuals using the same facility to also be from their neighborhood, which makes it more likely to bump into friends and acquaintances. In clustered federated learning, the centroids may represent the models collaboratively learned by the different clusters of agents, and an agent may care about both the model being effective for their own data distribution and the other agents in their cluster having data distributions similar to their own as such cohesion can be important to inducing trust and cooperation within the cluster. It may even be the case that agents' preferences are formed out of a combination of centroid and non-centroid losses that are induced by completely different metric spaces. When a teacher groups students together for final class projects, students preferences for which group they are placed in would depend on the project topic that group is assigned (centroid) and which other students are in that group (non-centroid). Two students may be very close friends but have completely different opinions on project topics.
+
+This motivates us to initiate the study of proportional fairness for a natural generalization of centroid and non-centroid clustering that we term semi-centroid clustering, in which the loss of each agent  $i$  is a function of both the cluster of agents  $C_t$  it is a part of (i.e.,  $i \in C_t$ ) and the centroid  $x_t$  assigned to this cluster.
+
+By leveraging approaches common in the works of Chen et al. [1] and Caragiannis et al. [2], along with novel and intricate techniques we develop, we are able to attain desirable proportional fairness guarantees for semi-centroid clustering.
+
+# 1.1 Our Results
+
+<table><tr><td></td><td colspan="2">Dual Metric Loss</td><td colspan="2">Weighted Single Metric Loss</td></tr><tr><td></td><td>Core</td><td>FJR</td><td>Core</td><td>FJR</td></tr><tr><td>Existential Upper Bound</td><td>3</td><td>1</td><td>min{2/λ, 3}</td><td>1</td></tr><tr><td>Efficient Upper Bound</td><td>3 + 2√3</td><td>4</td><td>min{2/λ, fλ}</td><td>min{4, 2/λ, fλ}</td></tr><tr><td>Lower Bound</td><td>2</td><td>1</td><td>max{gλ, (2(1-λ)/2λ+1}</td><td>1</td></tr></table>
+
+Table 1: Core and FJR approximations for semi-centroid clustering under dual metric loss and weighted single metric loss. Here,  $f_{\lambda} = \frac{\sqrt{2\lambda - 11\lambda^2 + 13} + 3 - \lambda}{2 - 2\lambda}$  and  $g_{\lambda} = \frac{\sqrt{\lambda^2 - 2\lambda + 5} - \lambda + 1}{2}$ .
+
+A semi-centroid clustering algorithm takes a set of  $n$  data points (agents) as input and returns both a partition  $(C_1, \ldots, C_k)$  of the data points and the corresponding centroids  $(x_1, \ldots, x_k)$ . The loss of each agent  $i \in C_t$  is measured by a loss function  $\ell_i(C_t, x_t)$ , which depends both on the cluster  $C_t$  it belongs to and its centroid  $x_t$ . While some of our results apply to arbitrary loss functions, most focus on two structured families. Dual metric loss measures the sum of agent  $i$ 's distance to its centroid  $x_t$  according to a metric  $d^c$  and its maximum distance to any agent  $j \in C_t$  in its cluster according to a different metric  $d^m$ . Weighted single metric loss is the special case in which both metrics are scaled versions of a common metric, i.e.,  $d^c = (1 - \lambda) \cdot d$  and  $d^m = \lambda \cdot d$  for some metric  $d$  and  $\lambda \in [0, 1]$ .
+
+We investigate (multiplicative approximations of) two proportional fairness criteria studied in prior work on centroid and non-centroid clustering: the core and, its relaxation, fully justified representation (FJR). Intuitively, when forming  $k$  clusters given  $n$  data points, the core demands that there is no set  $S$  of at least  $n / k$  agents, who proportionally deserve to be able to form a cluster, and a feasible centroid  $y$  such that every member of  $S$  prefers  $y$  to its currently assigned centroid. FJR puts a greater demand on successful violations: even the maximum loss of any agent  $i \in S$  for  $(S, y)$  should be lower than the minimum loss of any agent  $i \in S$  under the algorithm's clustering.
+
+First, we show that none of the algorithms developed in prior work for centroid or non-centroid clustering work in our more general paradigm of semi-centroid clustering, even for the restricted family of weighted single metric loss. Then, for both loss families, we obtain core and FJR approximation guarantees, both existential and polynomial-time attainable, by designing novel algorithms, and prove (existential) lower bounds. Our main results are summarized in Table 1.
+
+Finally, we also evaluate the performance of our algorithms on real-world datasets in Appendix E. We observe that our algorithms consistently achieve near-perfect approximations of the fairness notions we consider, beating both the theoretical worst-case bounds we establish, and the performance of classical clustering algorithms such as k-means++. We also show that they achieve these guarantees while sacrificing classical clustering objectives only slightly. This generalizes similar observations for centroid and non-centroid clustering in prior work [1, 2].
+
+# 1.2 Related Work
+
+As mentioned above, there are several existing papers studying proportional fairness in clustering. Chen et al. [1] are the first to introduce the notion of the core for centroid clustering, and use the Greedy Capture algorithm to achieve a  $(1 + \sqrt{2})$ -approximation to the core. Micha and Shah [14] improve the approximation factor to 2 for the Euclidean  $L^2$  metric in the "unconstrained setting" where the centroids can be placed anywhere in  $\mathbb{R}^t$ . Aziz et al. [15] observe that in this unconstrained setting, instead of the infinitely many possible centroid locations, one can focus on just the  $n$  agent locations and achieve a constant approximation to the core in polynomial time. Li et al. [16] study a stronger criterion than the core. Caragiannis et al. [2] study proportional fairness in non-centroid clustering, and use a variation of the Greedy Capture algorithm to achieve 2-core with respect to the maximum-distance loss. They also study FJR as a weakening of the core.
+
+Li et al. [17] study a similar graph-based problem of partitioning friends into groups, which is equivalent to non-centroid clustering, albeit when each person has a utility (rather than a loss) of 1 for every friend of theirs placed in their cluster. Arkin et al. [18] look at the multi-dimensional stable roommates problem, which is equivalent to the non-centroid clustering model of [2], but focus on special cases such as  $k = n/3$ .
+
+Finally, other fairness notions have also been investigated for clustering (see the survey by Chhabra et al. [19]). The most prominent one assumes that every data point belongs to one of many classes, and certain protected classes must have equal representation across the clusters [20, 21].
+
+# 2 Preliminaries
+
+# 2.1 Clustering Model
+
+A semi-centroid clustering instance is a tuple  $(\mathcal{N},\mathcal{M},\{\ell_i\}_{i\in \mathcal{N}},k)$ , where  $\mathcal{N}$  is a set of  $n$  agents;  $\mathcal{M}$  is a set of possible cluster centers;  $\ell_i:\{C\subseteq \mathcal{N}:i\in C\} \times \mathcal{M}\to \mathbb{R}_{\geqslant 0}$  is the loss function of agent  $i\in \mathcal{N}$ , with  $\ell_i(C,x)$  being the loss incurred when it is part of cluster  $C$  and assigned cluster center  $x$ ; and  $k\in \mathbb{N}$  is the number of clusters to be returned by the algorithm.
+
+A clustering algorithm takes such an instance and returns a clustering  $\mathcal{X} = \{(C_1, x_1), \ldots, (C_k, x_k)\}$ , where  $(C_1, \ldots, C_k)$  is a partition of the set of agents  $\mathcal{N}$  (so,  $\cup_{t \in [k]} C_t = \mathcal{N}$  and  $C_t \cap C_{t'} = \emptyset$  for all distinct  $t, t' \in [k]$ ). We refer to each tuple  $(C_t, x_t) \in \mathcal{X}$  as a cluster, with  $C_t$  as the member set (and the agents in  $C_t$  as the members) of the cluster and  $x_t$  as the center assigned to the cluster.
+
+Given a clustering  $\mathcal{X}$  and an agent  $i\in \mathcal{N}$ , let  $\mathcal{X}(i)$  denote the (unique) index of the cluster of which  $i$  is a member (i.e.,  $i\in C_{\mathcal{X}(i)}$ ). The loss incurred by each agent  $i\in \mathcal{N}$  under clustering  $\mathcal{X}$  is  $\ell_i(C_{\mathcal{X}(i)},x_{\mathcal{X}(i)})$ , which, with slight abuse of notation, we also write as  $\ell_i(\mathcal{X})$  for brevity.
+
+# 2.2 Loss Functions
+
+Some of our results hold for arbitrary loss functions as defined above. However, the more interesting results are obtained for two structured families of loss functions that naturally combine previously-studied loss functions for centroid and non-centroid clustering.
+
+Dual metric loss: We are given two distance metrics, a centroid metric  $d^{c}:\mathcal{N}\times \mathcal{M}\to \mathbb{R}_{\geqslant 0}$  and a non-centroid metric  $d^m:\mathcal{N}\times \mathcal{N}\rightarrow \mathbb{R}_{\geqslant 0}$ . Given a clustering  $\mathcal{X}$  and an agent  $i\in \mathcal{N}$ , the centroid metric induces its centroid loss  $\ell_i^c (\mathcal{X})\triangleq d^c (i,x_{\mathcal{X}(i)})$ , which measures the distance of agent  $i$  to its assigned center, and the non-centroid metric induces its non-centroid loss  $\ell_i^m (\mathcal{X}) = \max_{j\in C_{\mathcal{X}(i)}}d^m (i,j)$ , which measures the maximum distance of agent  $i$  to any other agent in its cluster. The former is the canonical loss function used for centroid clustering [1, 14, 15, 22] while the latter is a loss function for which appealing fairness guarantees have been derived for non-centroid clustering [2].
+
+The overall dual metric loss of agent  $i$  is the sum of the centroid and non-centroid parts:
+
+$$
+\ell_ {i} (\mathcal {X}) = \ell_ {i} ^ {m} (\mathcal {X}) + \ell_ {i} ^ {c} (\mathcal {X}) = \max  _ {j \in C _ {\mathcal {X} (i)}} d ^ {m} (i, j) + d ^ {c} (i, x _ {\mathcal {X} (i)}).
+$$
+
+It is worth remarking that adding scaling factors so that  $\ell_i(\mathcal{X}) = \alpha \cdot \ell_i^m (\mathcal{X}) + \beta \cdot \ell_i^c (\mathcal{X})$  for some  $\alpha ,\beta \geqslant 0$  would not make the family any more general; one can instead just use  $d^{m} = \alpha \cdot d^{m}$  and  $\widehat{d}^c = \beta \cdot d^c$  as the new non-centroid and centroid metrics, respectively.
+
+Weighted single metric loss: A special case of dual metric loss is where both centroid and noncentroid losses use (scaled versions) of the same metric  $d:(\mathcal{N}\cup \mathcal{M})\times (\mathcal{N}\cup \mathcal{M})\to \mathbb{R}_{\geqslant 0}$ . That is, for some  $\lambda \in [0,1]$ , we have  $d^{m}(i,j) = \lambda \cdot d(i,j)$  for all  $i,j\in \mathcal{N}$  and  $d^{c}(i,x) = (1 - \lambda)\cdot d(i,x)$  for all  $i\in \mathcal{N}$  and  $x\in \mathcal{M}$ , so that the overall loss is given by
+
+$$
+\ell_ {i} (\mathcal {X}) = \lambda \cdot \max  _ {j \in C _ {\mathcal {X} (i)}} d (i, j) + (1 - \lambda) \cdot d (i, x _ {\mathcal {X} (i)}).
+$$
+
+Once again, note that using arbitrary non-negative scaling factors  $\alpha$  and  $\beta$  instead of  $\lambda$  and  $1 - \lambda$  does not offer any further generalization as one can equivalently use  $\lambda = \frac{\alpha}{\alpha + \beta}$ .
+
+Non-centroid and centroid clustering are easily seen as special cases of the weighted single metric loss (which is, in turn, a special case of the dual metric loss) with  $\lambda = 1$  and  $\lambda = 0$ , respectively. Hence, the semi-centroid clustering algorithms we design for dual metric loss and for weighted single metric loss can be used for both centroid and non-centroid clustering.
+
+# 2.3 Proportional Fairness Guarantees
+
+We focus on two proportional fairness criteria prominently studied by prior work.
+
+Definition 1 (The Core). For  $\alpha \geqslant 1$ , a clustering  $\mathcal{X} = \{(C_1, x_1), \ldots, (C_k, x_k)\}$  is in the  $\alpha$ -core if there are no group of agents  $S \subseteq \mathcal{N}$  with  $|S| \geqslant n / k$  and feasible center  $y \in \mathcal{M}$  such that  $\alpha \cdot \ell_i(S, y) < \ell_i(\mathcal{X})$  for all  $i \in S$  (if this happens, we say that coalition  $S$  deviates with center  $y$ ).
+
+In words, no group  $S$  of at least  $n / k$  agents, which proportionately deserves a cluster center representing it, can unilaterally improve by being a cluster of its own with a feasible center  $y$ .
+
+The second criterion is a relaxation that imposes a higher bar on deviating coalitions.
+
+Definition 2 (Fully Justified Representation (FJR)). For  $\alpha \geqslant 1$ , a clustering  $\mathcal{X} = \{(C_1, x_1), \ldots, (C_k, x_k)\}$  is  $\alpha$ -FJR if there are no groups of agents  $S \subseteq \mathcal{N}$  with  $|S| \geqslant n / k$  and feasible center  $y \in \mathcal{M}$  such that  $\alpha \cdot \ell_i(S, y) < \min_{j \in S} \ell_j(\mathcal{X})$  for all  $i \in S$ .
+
+That is, the new loss for each member of the deviating coalition, scaled by  $\alpha$ , should not only be less than its own loss prior to deviation, but less than the loss of any deviating member prior to deviation. It is then clear that  $\alpha$ -FJR is a relaxation of  $\alpha$ -core for all  $\alpha \geqslant 1$ .
+
+Proposition 1. For every  $\alpha \geqslant 1$ , every clustering in the  $\alpha$ -core is also  $\alpha$ -FJR.
+
+Due to space constraints, proofs missing from each section of the main body are presented in the correspondingly titled section of the appendix.
+
+# 3 Greedy Capture Algorithms
+
+In prior work on proportionally fair clustering, the Greedy Capture algorithm has been the go-to way to achieve a constant approximation of proportional fairness criteria in both the centroid and non-centroid worlds. Chen et al. [1] introduce it for centroid clustering and Caragiannis et al. [2] adapt it to non-centroid clustering. The centroid and non-centroid variants, presented formally as Algorithms 2 and 3 respectively in Appendix A, share many similarities. In both variants, every agent starts off as "uncaptured" and a number of balls start growing at the same rate. As soon as a ball covers at least  $n / k$  uncaptured agents, it captures them and becomes "open" (and in the centroid variant, the center of the open ball is added as one of at most  $k$  cluster centers to be selected by the algorithm). But the two variants differ in three key dimensions.
+
+1. Where the balls grow: In the centroid version, a ball grows around every feasible cluster center. In the non-centroid version, which has no centers, one grows around each agent.  
+2. Whether open balls keep growing: In the centroid variant, a ball continues to grow after it opens, capturing any uncaptured agents as soon as it covers them. In the non-centroid variant, a ball stops growing once it opens. This is intuitive because an open ball that grows and captures an agent far away would not worsen previously-captured agents in the centroid paradigm, but would do so in the non-centroid paradigm.  
+3. Whether agents switch clusters in the end: In the centroid variant, the algorithm simply returns a set of (up to)  $k$  cluster centers and each agent's loss is their distance to the nearest cluster center. Note that this may not be the center of the ball that captured it as the algorithm may have added a closer center afterwards. Hence, the set of agents captured by a ball when it opens may not form a single cluster. In the non-centroid variant, there is no "switching" in the end and the set of agents captured by a ball together do form a cluster. This is again because allowing an agent to switch to a different cluster she prefers could significantly worsen existing agents in that cluster.
+
+Chen et al. [1] prove that centroid greedy capture yields a clustering in the  $(1 + \sqrt{2})$ -core in the centroid paradigm, and Caragiannis et al. [2] prove that non-centroid greedy capture yields a clustering in the 2-core in the non-centroid paradigm. This naturally raises the question: Can some variant of greedy capture get a constant approximation to the core in the semi-centroid clustering paradigm?
+
+Unfortunately, we can show that solving the semi-centroid paradigm is not that simple. To see why, it is sufficient to consider the restricted case of the weighted single metric loss, in which the loss function is characterized by a single metric  $d$  and a parameter  $\lambda \in [0,1]$ . Note that the greedy capture algorithms only take in  $d$  as input, and not  $\lambda$ . Thus, for (some variant of) greedy capture to achieve  $\alpha$ -core with respect to the weighted single metric loss (for any  $\lambda \in [0,1]$ ), it must achieve  $\alpha$ -core with respect to the non-centroid loss  $(\lambda = 1)$  and the centroid loss  $(\lambda = 0)$  simultaneously. In our first result, we show that no algorithm can achieve such a simultaneous guarantee.
+
+Theorem 1. For every  $\alpha \geqslant 1$  and  $\beta \geqslant 1$ , there exists a semi-centroid clustering instance in which no clustering is simultaneously in the  $\alpha$ -core with respect to the centroid loss and in the  $\beta$ -core with respect to the non-centroid loss.
+
+# 4 The Core
+
+Given Theorem 1, we switch our attention to rules that do not simply take a single metric  $d$  as input; that is, for the weighted single metric loss, they also take  $\lambda$  as input, and for the more general dual metric loss, they take both the centroid and non-centroid metrics,  $d^{c}$  and  $d^{m}$ , as inputs.
+
+In Section 4.1, we present results for the dual metric loss. Later, in Section 4.2, we present improvements for the restricted case of the weighted single metric loss.
+
+# 4.1 Dual Metric Loss
+
+Algorithm 1: Dual Metric Algorithm  
+Input: Set of agents  $\mathcal{N}$  , set of feasible cluster centers  $\mathcal{M}$  , non-centroid metric  $d^{m}$  , centroid metric  $d^c$  , number of centers  $k$  , growth factor  $c$  , MCC approximation factor  $\alpha$  Output: Clustering  $\mathcal{X} = \{(C_1,x_1),\ldots ,(C_k,x_k)\}$    
+1 Initialize:  $\mathcal{N}'\gets \mathcal{N},C\gets \emptyset ,\widehat{\mathcal{X}}\gets \emptyset ,t\gets 1;$    
+// Phase 1: Build a tentative clustering X   
+2 while  $\mathcal{N}'\neq \emptyset$  do // Uncaptured agents remain   
+3 // Find an  $\alpha$  -MCC cluster  $(\widehat{C}_t,x_t)$  and save its maximum loss as  $r_t$  if  $\mathcal{N}'\geqslant n / k$  then   
+4  $(\widehat{C}_t,x_t)\gets \mathrm{An}$ $\alpha$  -most cohesive cluster from  $\mathcal{N}'\times \mathcal{M}$  .   
+5 else   
+6 // Edge-case to capture the last cluster  $(\widehat{C}_t,x_t)\gets (\mathcal{N}',\arg \min_{x\in M}\max_{j\in \mathcal{N}'}d^c (j,x))$    
+7  $r_t\gets \max_{i\in \widehat{C}_t}\ell_i(\widehat{C}_t,x_t);$    
+8 // Add the cluster to  $\widehat{\mathcal{X}}$  and mark its agents as captured   
+9  $\mathcal{N}'\gets \mathcal{N}'\setminus \widehat{C}_t$    
+10 t t+1;   
+// Phase 2: Selectively allow agents to switch to a better cluster   
+11  $\forall t\in [k],C_t\gets \widehat{C}_t$    
+for i e N do   
+13 t  $\leftarrow \widehat{\chi} (i)$  .. // Clusters agent i can switch to without harming its members too much   
+14  $V_{i}\gets \{t^{\prime}\in [k]:\forall j\in \widehat{C}_{t^{\prime}},d^{c}(j,x_{t^{\prime}}) + d^{m}(j,i)\leqslant c\cdot r_{t^{\prime}}\}$  .. // Among them, agent i picks one minimizing an upper bound on its loss   
+15 t*  $\leftarrow$  arg min  $t^{\prime}\in V_{i}\left(d^{c}(i,x_{t^{\prime}}) + c\cdot r_{t^{\prime}} + \min_{j\in \widehat{C}_{t^{\prime}}}(d^{m}(i,j) - d^{c}(j,x_{t^{\prime}}))\right);$    
+16 // And switches to it if that improves the upper bound on its loss if  $d^{c}(i,x_{t^{*}}) + c\cdot r_{t^{*}} + \min_{j\in \widehat{C}_{t^{*}}}(d^{m}(i,j) - d^{c}(j,x_{t^{*}})) <   c\cdot r_{t}$  then   
+17 C t C t* C t* U {i};   
+18
+
+The following two are the most significant and the most intricate results of our work.
+
+Theorem 2. For the dual metric loss, there always exists a clustering in the 3-core.
+
+Theorem 3. For the dual metric loss, a clustering in the  $(3 + 2\sqrt{3})$ -core can be computed in polynomial time.
+
+To understand Algorithm 1, first we need to introduce the Most Cohesive Cluster problem, defined by Caragiannis et al. [2] for non-centroid clustering. Adapted to our semi-centroid clustering, it simply requires finding the cluster  $(C,x)$  with  $|C|\geqslant n / k$  that minimizes the maximum loss incurred by any agent in  $C$ . This is defined and approximated for arbitrary loss functions as follows.
+
+Definition 3 ( $\alpha$ -MCC). For  $\alpha \geqslant 1$ , a cluster  $(C,x)$  with  $|C| \geqslant n / k$  is an  $\alpha$ -approximate most cohesive cluster (in short, an  $\alpha$ -MCC cluster) if  $\max_{i \in C} \ell_i(C,x) \leqslant \alpha \cdot \max_{i \in S} \ell_i(S,y)$  for every cluster  $(S,y)$  with  $|S| \geqslant n / k$ . When  $\alpha = 1$ , it is simply called a most cohesive cluster.
+
+Caragiannis et al. [2] use this in place of the greedy capture subroutine, which finds the cluster  $(C,x)$  with the smallest radius  $\max_{i\in C}d(i,x)$ , in order to obtain improved fairness guarantees. Theorems 2 and 3 are both obtained using Algorithm 1, which calls a subroutine that finds an  $\alpha$ -MCC cluster; Theorem 2 is obtained by an inefficient subroutine that finds an MCC cluster, while Theorem 3 is obtained by plugging in an efficient subroutine we design for computing a 4-MCC cluster.
+
+Algorithm description: Algorithm 1 is split into two phases. Phase 1 works by iteratively finding an  $\alpha$ -MCC cluster with respect to the dual metric loss and adding it to a tentative solution. We denote the clustering found in Phase 1 by  $\widehat{\mathcal{X}} = \{(\widehat{C}_1, x_1), \ldots, (\widehat{C}_k, x_k)\}$ . For each cluster  $(\widehat{C}_t, x_t) \in \widehat{\mathcal{X}}$ , the algorithm stores the maximum loss of any agent in that cluster as  $r_t = \max_{i \in \widehat{C}_t} \ell_i(\widehat{C}_t, x_t)$ .
+
+Phase 2 takes the clustering  $\widehat{\mathcal{X}}$  as a starting point and allows each agent  $i\in \mathcal{N}$  to switch from its  $t$ -th cluster to any  $t^\prime$ -th cluster subject to two careful constraints:
+
+- Bound the loss of the agents in the new cluster: We know the original loss of every  $j \in \widehat{C}_{t'}$  was at most  $r_{t'}$ . If agent  $i$  switches to this cluster, it can increase  $j$ 's loss to  $d^{c}(j, x_{t'}) + d^{m}(j, i)$ . We need this to be at most  $c \cdot r_{t'}$  for some constant  $c$  given as input to the algorithm.  
+- Bound own loss: We show that the expression in Line 15— $d^{c}(i, x_{t'}) + c \cdot r_{t'} + \min_{j \in \widehat{C}_{t'}} (d^{m}(i, j) - d^{c}(j, x_{t'}))$ —serves as an upper bound on the loss of agent  $i$  when switching to the  $t'$ -th cluster, regardless of which other agents switch to the  $t'$ -th cluster during Phase 2. This ensures that agent  $i$ 's decision to switch (or not) is independent of the decisions made by the other agents in Phase 2, and the loss of agent  $i$ , who was originally in  $\widehat{C}_{t}$ , remains bounded by  $c \cdot r_{t}$  throughout Phase 2, regardless of Phase 2 decisions. Agent  $i$  makes a switch only if the new loss upper bound is lower than  $c \cdot r_{t}$ .
+
+The key to establishing an approximate core guarantee lies in this selective switching administered in Phase 2. Intuitively, this can be seen as striking a middle ground between the centroid greedy capture, which allows an agent to switch to any other cluster because it cannot increase the centroid loss of the agents in that cluster, and the non-centroid greedy capture, which forbids such switches completely lest it might harm the agents in the cluster being switched to. The growth factor  $c$  carefully controls the balance, and optimally setting this factor allows us to derive the following bound. Its proof, our most technically deep result, is presented in Appendix C.1.
+
+Lemma 1. Given a subroutine for computing an  $\alpha$ -approximate most cohesive cluster, Algorithm 1, with the growth factor set to  $c = \frac{3\alpha + \sqrt{\alpha(\alpha + 8)}}{4\alpha}$ , finds a clustering in the  $\frac{1}{2} (\alpha + \sqrt{\alpha(\alpha + 8)} + 2)$ -core with respect to the dual metric loss.
+
+From Lemma 1, the proof of Theorem 2 follows easily.
+
+While Theorem 2 ensures the existence of a clustering in the 3-core, its computation relies on iteratively finding the most cohesive cluster, which is already NP-hard in the non-centroid paradigm [2]. Luckily, Lemma 1 yields a fairness guarantee even if we can find only an approximate most cohesive cluster during Phase 1 of Algorithm 1.
+
+Next, we show how to compute a 4-approximate most cohesive cluster in polynomial time. The algorithm, presented as Algorithm 4 in Appendix C.1, works as follows. For each feasible cluster center  $y \in \mathcal{M}$ , it runs the non-centroid greedy capture with a crafted distance metric  $d_y$ , given by  $d_y(i,j) = d^m(i,j) + d^c(i,y) + d^c(j,y)$  for all  $i,j \in \mathcal{N}$ , until it produces the first cluster  $C_y$ . Then, it returns  $(C_{y^*}, y^*)$  with the lowest maximum loss, i.e., with  $y^* \in \arg \min_{y \in \mathcal{M}} \max_{i \in C_y} \ell_i(C_y,y)$ .
+
+Lemma 2. For the dual metric loss, Algorithm 4 computes a 4-approximate most cohesive cluster in polynomial time.
+
+Plugging this into Lemma 1 gives us Theorem 3, a constant approximation to the core in polynomial time.
+
+Lower bounds: One may wonder how good these approximations are. As stated previously, semicentroid clustering is more challenging than both centroid and non-centroid clustering. Specifically,
+
+one can trivially import a lower bound from either paradigm by setting  $d^{m} = 0$  or  $d^{c} = 0$  (where a distance metric being zero means all pairwise distances are zero), which makes the dual metric loss coincide with the centroid or non-centroid loss, respectively. For the non-centroid loss, Caragiannis et al. [2] do not offer any lower bounds; that is, the existence of an exact core clustering with respect to the (maximum distance) non-centroid loss remains an open question. However, Chen et al. [1] establish a lower bound of 2-core with respect to the centroid loss, which we can import. Somewhat surprisingly, we are unable to derive a lower bound better than 2 despite reasonable effort.
+
+# 4.2 Weighted Single Metric Loss
+
+In this section, we focus on improving the core approximation guarantees for the restricted case of the weighted single metric loss. Recall that this is parametrized by a single metric  $d$  and a parameter  $\lambda \in [0,1]$  such that, for any  $i\in \mathcal{N}$  and cluster  $(C,x)$ , we have  $\ell_i(C,x) = \lambda \cdot \max_{j\in C}d(i,j) + (1 - \lambda)\cdot d(i,x)$ . We devise two algorithms that offer improvements.
+
+The first algorithm takes non-centroid greedy capture (Algorithm 3), which grows balls around agents, and adapts it to the semi-centroid world by assigning each cluster to the cluster center nearest to the agent whose ball captured the cluster. The resulting efficient algorithm is presented as Algorithm 5 in Appendix C.2, and achieves the following guarantee.
+
+Lemma 3. For the weighted single metric loss with parameter  $\lambda \in (0,1]$ , Algorithm 5 finds a clustering in the  $\frac{2}{\lambda}$ -core in polynomial time.
+
+The second algorithm, presented formally as Algorithm 6 in Appendix C.2, is similar to Algorithm 1: in Phase 1, agents are partitioned into an initial clustering, and in Phase 2, agents are allowed to switch to a different cluster that reduces their loss and does not significantly increase the loss of the agents in that cluster. However, there are two key differences. First, Algorithm 6 uses the centroid greedy capture subroutine instead of the (approximate) most cohesive cluster subroutine used by Algorithm 1 to iteratively find clusters in Phase 1. Second, due to having a single metric, the switching condition is much simpler and leads to better bounds for some values of  $\lambda$ . A careful analysis yields the following guarantee.
+
+Lemma 4. For the weighted single metric loss with parameter  $\lambda \in [0,1)$ , Algorithm 6 finds a clustering in the  $\frac{\sqrt{2\lambda - 11\lambda^2 + 13} + 3 - \lambda}{2 - 2\lambda}$ -core in polynomial time.
+
+Combining the bounds from Lemmas 3 and 4, together with the existential 3-core (Theorem 2) and the polynomial-time  $(3 + 2\sqrt{3})$ -core (Theorem 3) guarantees that we can import from dual metric loss, yields the following existential and efficiently attainable bounds.
+
+Theorem 4. For the weighted single metric loss with parameter  $\lambda \in [0,1]$ , a clustering in the  $\min\{\frac{2}{\lambda},3\}$ -core always exists, and a clustering in the  $\min\{\frac{2}{\lambda},\frac{\sqrt{2\lambda - 11\lambda^2 + 13 + 3 - \lambda}}{2 - 2\lambda}\}$ -core can be found in polynomial time.
+
+Lower bounds: Similarly, we also derive two lower bounds on the core approximation under the weighted single metric loss. The first bound interpolates between  $(\sqrt{5} + 1)/2$  at  $\lambda = 0$  (centroid) and  $1$  at  $\lambda = 1$  (non-centroid). Since Chen et al. [1] prove a lower bound of 2 for  $\lambda = 0$  (centroid), which is better than  $(\sqrt{5} + 1)/2$ , adapting their counterexample to semi-centroid clustering yields a better lower bound when  $\lambda$  is close to 0.
+
+Theorem 5. For the weighted single metric loss with parameter  $\lambda \in [0,1]$ , no clustering algorithm can achieve a core approximation better than  $\max \left\{\frac{\sqrt{\lambda^2 - 2\lambda + 5} - \lambda + 1}{2}, \frac{2(1 - \lambda)}{2\lambda + 1}\right\}$ .
+
+# 5 Fully Justified Representation
+
+Let us turn our attention to the weaker criterion of fully justified representation (FJR), a relaxation of the core originally introduced by Peters et al. [23] in the context of approval-based committee selection. While we can already achieve a (small) constant approximation to the core even under the dual metric loss (Theorems 2 and 3), we show that this relaxation offers several advantages.
+
+Caragiannis et al. [2] study FJR for non-centroid clustering, and show that iteratively finding an  $\alpha$ -approximate most cohesive cluster produces an  $\alpha$ -FJR clustering, even under arbitrary loss functions.
+
+The same result, with an almost identical proof, holds in our more general semi-centroid clustering setup. For completeness, in Appendix D, we formally present the algorithm as Algorithm 7 and provide a complete proof. This gives us the existence of a 1-FJR clustering under arbitrary loss functions, and polynomial-time computation of a 4-FJR clustering under the dual metric loss when plugging in our polynomial-time algorithm for computing a 4-MCC cluster (Lemma 2).
+
+Theorem 6. For arbitrary losses and  $\alpha \geqslant 1$ , Algorithm 7, which iteratively finds an  $\alpha$ -approximate most cohesive cluster, returns an  $\alpha$ -FJR clustering. Hence, a 1-FJR clustering exists for arbitrary losses and, due to Lemma 2, a 4-FJR clustering can be computed in polynomial time for the dual metric loss.
+
+In Section 3, we noticed that for a clustering, being in the  $\alpha$ -core with respect to both centroid and non-centroid losses simultaneously is (trivially) a necessary condition for it to be in the  $\alpha$ -core with respect to the weighted single metric loss for all  $\lambda \in [0,1]$ , but it is not a sufficient condition (i.e., it is strictly weaker). Nonetheless, it is already impossible to achieve (Theorem 1).
+
+For FJR, it turns out to also be a sufficient condition (and hence, an equivalent condition), even for a broader family of (weighted) dual metric losses.
+
+Lemma 5. For any  $\alpha \geqslant 1$ , a clustering is  $\alpha$ -FJR with respect to the centroid loss with metric  $d^c$  and with respect to the non-centroid loss with metric  $d^m$  simultaneously if and only if it is  $\alpha$ -FJR with respect to all dual metric loss functions with centroid metric  $\lambda \cdot d^c$  and non-centroid metric  $(1 - \lambda) \cdot d^m$ , for all  $\lambda \in [0,1]$ .
+
+Unfortunately, a simultaneous FJR approximation with respect to centroid and non-centroid losses induced by two different metrics is still infeasible.
+
+Theorem 7. For any  $\alpha \geqslant 1$  and  $\beta \geqslant 1$ , there exists an instance in which no clustering is both  $\alpha$ -FJR with respect to the centroid loss and  $\beta$ -FJR with respect to the non-centroid loss, when the two loss functions are allowed to use different metrics.
+
+When both losses are induced by a common metric  $d$ , a surprisingly simple algorithm turns out to achieve a constant approximation to FJR with respect to both centroid and non-centroid losses (and thus, with respect to the weighted single metric loss for every  $\lambda \in [0,1]$ ). This is in contrast to the core, for which no simultaneous finite approximation is feasible, even for a single metric (Theorem 1).
+
+The algorithm is Algorithm 5, which runs non-centroid greedy capture with metric  $d$  followed by a greedy centroid selection step. We presented it in Section 4.2 and proved that it yields  $2 / \lambda$ -core for all  $\lambda \in [0,1]$ . Note that this achieves 2-core (and hence, 2-FJR) with respect to the non-centroid loss  $(\lambda = 1)$ . However, its approximation to the core becomes unbounded as  $\lambda$  approaches 0 (the centroid loss), which is inevitable given Theorem 1. However, we show that its approximation to the weaker criterion of FJR still remains bounded by 5.
+
+Theorem 8. Algorithm 5, given metric  $d$ , finds, in polynomial time, a clustering that is 5-FJR with respect to the centroid loss and 2-FJR with respect to the non-centroid loss, both defined using  $d$ .
+
+Applying Lemma 5 tells us that Algorithm 5 achieves 5-FJR with respect to the weighted single metric loss for all  $\lambda \in [0,1]$  simultaneously; if one wishes to achieve approximate FJR with respect to a given loss function, Theorem 6 yields 4-FJR with respect to all dual metric losses and Theorem 4 yields a better approximation to the core (and, hence, to FJR) for weighted single metric losses.
+
+# 6 Discussion
+
+Our work leaves open a number of immediate technical questions, which can be gleaned from the gaps between upper and lower bounds in Table 1. Future work can also investigate generalizations such as non-additive combinations of centroid and non-centroid losses and several exciting applications.
+
+One particularly important special case of this model that we leave largely unexplored is when  $\mathcal{M} = \mathcal{N}$ . This case generalizes many problems where one wishes to group a set of  $n$  datapoints in  $k$  clusters, and return a single point to serve as a representative point for that cluster. This can have numerous applications such as coreset selection [24], scalable LLM fine-tuning [25], feature selection [26], and dimensionality reduction [27]. However, in these cases, our semi-centroid algorithms offer no guarantees that the centroid it selects for a cluster will be a point that is in that cluster. Exploring algorithms with this additional constraint would be an interesting future direction.
+
+# Acknowledgments and Disclosure of Funding
+
+This work was supported by an NSERC Discovery Grant and an NSERC-CSE Research Communities Grant. Researchers funded through the NSERC-CSE Research Communities Grants do not represent the Communications Security Establishment Canada or the Government of Canada. Any research, opinions or positions they produce as part of this initiative do not represent the official views of the Government of Canada.
+
+# References
+
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+
+# Appendix
+
+# A Greedy Capture Algorithms
+
+Algorithm 2: Centroid Greedy Capture [1]  
+Input:  $\mathcal{N},\mathcal{M},d,k$    
+Output:  $\mathcal{X} = \{(C_1,x_1),\ldots ,(C_k,x_k)\}$  . Initialize:  $\mathcal{N}'\gets \mathcal{N},\mathcal{X}\gets \emptyset ,\delta \gets 0$    
+// Uncaptured agents remain while  $\mathcal{N}'\neq \emptyset$  do Increase  $\delta$  smoothly // Ball opens at a center if  $\exists x\in \mathcal{M},C\subseteq \mathcal{N}'$  s.t.  $|C|\geqslant n / k$  and  $d(i,x)\leqslant \delta$  for all  $i\in C$  then  $\begin{array}{r}\mathcal{X}\leftarrow \mathcal{X}\cup \{(C,x)\} \\ \mathcal{N}'\leftarrow \mathcal{N}'\setminus C \end{array}$  // Open balls keep growing if  $\exists (C,x)\in \mathcal{X},i\in \mathcal{N}'$  s.t.  $d(i,x)\leqslant \delta$  then  $\begin{array}{r}(C,x)\gets (C\cup \{i\} ,x)\\ \mathcal{N}'\gets \mathcal{N}'\setminus \{i\} \end{array}$    
+// Agents switch clusters   
+ $P\gets \{x\in \mathcal{M}:(C,x)\in \mathcal{X}$  for some  $C\}$ $\forall i\in \mathcal{N},x_i\in \arg \min_{x\in P}d(i,x)$ $\forall x\in P,\widehat{C}_x\gets \{i\in \mathcal{N}:x_i = x\}$ $\mathcal{X}\gets \{(\widehat{C}_x,x):x\in P\}$    
+return X
+
+Algorithm 3: Non-Centroid Greedy Capture [2]  
+Input:  $\mathcal{N},d,k$    
+Output:  $\mathcal{X} = \{C_1,\dots ,C_k\}$    
+Initialize:  $\mathcal{N}'\gets \mathcal{N},\mathcal{X}\gets \emptyset ,\delta \gets 0$    
+// Uncaptured agents remain   
+while  $\mathcal{N}'\neq \emptyset$  do   
+Increase  $\delta$  smoothly if  $|\mathcal{N}^{\prime}| <   n / k$  then // Edge-case to capture the last cluster  $\mathcal{X}\leftarrow \mathcal{X}\cup \{C\leftarrow \mathcal{N}^{\prime}\}$ $\mathcal{N}^{\prime}\gets \emptyset$    
+// Ball opens at an agent if  $\exists i\in \mathcal{N}',C\subseteq \mathcal{N}'$  s.t.  $|C|\geqslant n / k$  and  $d(i,i^{\prime})\leqslant \delta$  for all  $i^\prime \in C$  then  $\begin{array}{r}\mathcal{X}\leftarrow \mathcal{X}\cup \{C\} \\ \mathcal{N}^{\prime}\leftarrow \mathcal{N}^{\prime}\setminus C\\ \end{array}$    
+// Open balls don't grow   
+// Agents don't switch clusters   
+return X
+
+In the three dimensions discussed in Section 3, which induce eight variants of greedy capture, centroid greedy capture (Algorithm 2) makes the following choices:
+
+(1) The balls grow around the centroids.  
+(2) After a ball "captures" a group of  $\frac{n}{k}$  points and opens, it continues to grow and capture any uncaptured points as soon as it contains them.  
+(3) After all agents have been captured, each agent gets an opportunity to switch to the cluster containing the centroid that is closest to it.
+
+We can show that the  $(1 + \sqrt{2})$ -core guarantee of this algorithm with respect to the centroid loss is retained even when disabling one of the features (2) and (3).
+
+Theorem 9. Consider a variant of greedy capture in which balls grow around the agents, and open balls keep growing and/or agents are allowed to switch clusters at the end of the algorithm (at least one of the two holds). Then, it always returns a clustering in the  $(1 + \sqrt{2})$ -core with respect to the centroid loss.
+
+Proof. For contradiction assume this is false, let  $\mathcal{X} = \{(C_1,x_1),\ldots ,(C_k,x_k)\}$  be the clustering found by such an algorithm, and suppose that there is a deviating cluster  $(S,y)$  where  $|S|\geqslant \lceil \frac{n}{k}\rceil$  and  $(1 + \sqrt{2})\cdot d(i,y) <   d(i,x_{\mathcal{X}(i)})$  for every  $i\in S$
+
+Let  $i = \arg \min_{i' \in S} \mathcal{X}(i')$  be the first agent in  $S$  that was captured by the algorithm, and let  $t = \mathcal{X}(i)$ . We know that  $x_t \neq y$ , or  $i$  would clearly not be a part of the deviating cluster  $S$ . Let  $j = \arg \max_{j' \in S} d(j', y)$  be the agent in  $S$  that is farthest from  $y$ , and let  $\mathcal{X}(j) = t'$ .
+
+First, note that we must have  $d(j,y) \geqslant d(i,x_t)$ . If this were false, and  $d(j,y) < d(i,x_t)$ , then by the way that greedy capture functions, the cluster  $(S,y)$  would be captured prior to  $i$  being captured by  $x_{t}$ , which would give a contradiction.
+
+With the above inequality, we can show that  $t \neq t'$ . If this were not the case, and  $t = t'$ , then we could say the following:
+
+$$
+\begin{array}{l} (1 + \sqrt {2}) \cdot d (j, y) <   d (j, x _ {t ^ {\prime}}) = d (j, x _ {t}) \\ \leqslant d (i, x _ {t}) + d (i, y) + d (j, y) / / \text {T r i a n g l e} \\ <   d (i, x _ {t}) + \frac {1}{1 + \sqrt {2}} \cdot d (i, x _ {t}) + d (j, y) \\ \leqslant \left(\frac {1}{1 + \sqrt {2}} + 2\right) \cdot d (j, y). \\ \end{array}
+$$
+
+Hence,  $1 + \sqrt{2} < \frac{1}{1 + \sqrt{2}} + 2 = 1 + \sqrt{2}$ , which is a contradiction.
+
+We can also show that  $d(j, x_{t}) \geqslant d(j, x_{t'})$ . We can do this by considering two cases.
+
+Case 1: Open balls keep growing. If open balls keep growing in the greedy capture variant under consideration, and if  $d(j, x_t) < d(j, x_{t'})$ , then the fact that  $x_t$  captures  $i$  prior to  $x_{t'}$  capturing  $j$  implies that at the point in the algorithm when  $x_{t'}$  captures  $j$ ,  $x_t$  will have already captured  $i$ , and  $j$  will have already been in the the radius of  $x_t$ 's ball at a previous value of  $\delta$ . This would mean that  $x_t$  must have initially captured  $j$ . But this would contradict  $\mathcal{X}(j) = t'$ , since regardless of if agents are allowed to switch clusters in the last step of the algorithm,  $j$  could not end up in the  $t'$ -th cluster at the end of the algorithm.
+
+Case 2: Agents can switch clusters. If agents can switch clusters at the end of the greedy capture variant under consideration, and if  $d(j, x_{t}) < d(j, x_{t'})$ , then regardless of whether open balls continue to grow,  $j$  would always want to switch to  $x_{t}$  over  $x_{t'}$ , contradicting the fact that  $\mathcal{X}(j) = t'$ .
+
+With this inequality, we can say the following:
+
+$$
+\begin{array}{l} (1 + \sqrt {2}) \cdot d (j, y) <   d (j, x _ {t ^ {\prime}}) \\ \leqslant d (j, x _ {t}) \\ \leqslant d (i, x _ {t}) + d (i, y) + d (y, j) \\ <   d (i, x _ {t}) + \frac {1}{1 + \sqrt {2}} \cdot d (i, x _ {t}) + d (j, y) \\ \leqslant \left(\frac {1}{1 + \sqrt {2}} + 2\right) \cdot d (j, y), \\ \end{array}
+$$
+
+Again yielding  $1 + \sqrt{2} < \frac{1}{1 + \sqrt{2}} + 2 = 1 + \sqrt{2}$ , which is a contradiction.
+
+![](images/8cf4dba24685c2f7e582e5c125a5e5b1ac647a2a058434b11511db7311f8fcc7.jpg)
+
+# B Missing Proofs from Section 3
+
+$$
+\begin{array}{c}a\\ 1\Big| \hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hgap \\ b\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\end{array} c\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\begin{array}{c}a\\ 1\Big| \hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\begin{array}{c}p\\ 1\\ p\\ f \end{array} \end{array} d\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt\hskip -14.226378pt
+$$
+
+Figure 1: A semi-centroid clustering instance in which no clustering yields a finite approximation to the core with respect to both the centroid and non-centroid losses simultaneously.
+
+Theorem 1. For every  $\alpha \geqslant 1$  and  $\beta \geqslant 1$ , there exists a semi-centroid clustering instance in which no clustering is simultaneously in the  $\alpha$ -core with respect to the centroid loss and in the  $\beta$ -core with respect to the non-centroid loss.
+
+Proof. Fix any  $\alpha \geqslant 1$ ,  $\beta \geqslant 1$ , and  $p > \max \{\alpha, \beta\}$ . Consider the instance in Figure 1. Here  $\mathcal{N} = \{a, b, c, d, e, f\}$ ,  $\mathcal{M} = \mathcal{N}$ ,  $k = 3$ , and let  $d^m = d^c = d$ , where  $d$  is defined by the distances shown in the diagram, with each distance that is not shown assumed to be the maximum possible
+
+allowed under the triangle inequality. Since  $n / k = 2$ , a deviating coalition must contain 2 or more agents.
+
+For contradiction, assume that there exists a clustering in this instance that is simultaneously in the  $\alpha$ -core in the centroid paradigm and in the  $\beta$ -core in the non-centroid paradigm.
+
+Non-centroid  $\beta$ -core: We claim that  $\{a, b\}$  must be one of the clusters. If not, then either  $a$  and  $b$  are each in a cluster with a different agent, or at least one of them is a singleton cluster. In the former case,  $\{a, b\}$  can deviate and improve the non-centroid loss of each member by a factor of  $p > \beta$ , which is a contradiction. In the latter case, since  $k = 3$ , by the pigeonhole principle, there must exist a cluster  $C$  with  $|C| \geqslant 3$  (which means  $C \cap \{d, e, f\} \neq \emptyset$ ). If  $C \cap \{a, b, c\} \neq \emptyset$ , then the larger of  $C \cap \{a, b, c\}$  and  $C \cap \{d, e, f\}$ , which must contain at least two agents, can deviate and each of its members would improve by an infinite factor, again a contradiction. The only remaining possibility is that  $C = \{d, e, f\}$ , in which case  $\{e, f\}$  can deviate and improve each member by a factor of  $p > \beta$ , a contradiction.
+
+Hence,  $\{a,b\}$  must be one of the clusters. By symmetry,  $\{e,f\}$  must also be a cluster, so the remaining cluster must be  $\{c,d\}$ . Thus, the only clustering that can be in the  $\beta$ -core in the non-centroid paradigm is  $\{(\{a,b\},x_1),(\{c,d\},x_2),(\{e,f\},x_3)\}$  for some  $x_{1},x_{2},x_{3}\in \mathcal{N}$ .
+
+Centroid  $\alpha$ -core: If  $x_{1} \notin \{a, b\}$ , then  $\{a, b\}$  can deviate with either of  $a$  or  $b$  as the cluster center and improve the centroid loss of each by a factor at least  $p > \alpha$ , a contradiction. Without loss of generality, assume  $x_{1} = a$ . Similarly, assume without loss of generality that  $x_{3} = e$ . Now, either  $x_{2} \in \{a, b, c\}$  or  $x_{2} \in \{d, e, f\}$ . In the former case,  $\{d, f\}$  can deviate with a center at  $f$ , and in the latter case,  $\{b, c\}$  can deviate with a center at  $b$ , in each case improving each member by a factor at least  $p > \alpha$ , which is a contradiction.
+
+This concludes the proof.
+
+![](images/8e54752932ae7204506ec0e591bf18bf3825d11cd5c19c41263de932061044d4.jpg)
+
+# C Missing Proofs from Section 4
+
+# C.1 Dual Metric Loss
+
+Here, we present the missing proof of our central result, Lemma 1, which establishes an approximate core guarantee for Algorithm 1 with access to an oracle for an  $\alpha$ -approximate maximum cohesive cluster. In the main body, we use it to prove our existential and polynomial-time attainable approximate core guarantees for the dual metric loss (Theorems 2 and 3). Let us recall the statement.
+
+Lemma 1. Given a subroutine for computing an  $\alpha$ -approximate most cohesive cluster, Algorithm 1, with the growth factor set to  $c = \frac{3\alpha + \sqrt{\alpha(\alpha + 8)}}{4\alpha}$ , finds a clustering in the  $\frac{1}{2} (\alpha + \sqrt{\alpha(\alpha + 8)} + 2)$ -core with respect to the dual metric loss.
+
+We begin by establishing several useful properties of Algorithm 1. These properties bound the loss incurred by the agents when they (or other agents) switch (or choose not to switch) clusters during Phase 2 of Algorithm 1.
+
+Recall that Phase 1 produces a tentative clustering  $\widehat{\mathcal{X}}$ , with  $r_t = \max_{i \in \widehat{C}_t} \ell_i(\widehat{C}_t, x_t)$  being the maximum loss in the  $t$ -th tentative cluster,  $(\widehat{C}_t, x_t)$ . Then, in Phase 2, each agent  $i \in \widehat{C}_t$  is given the opportunity to switch to the  $t'$ -th cluster for every  $t' \in V_i$ . For each such cluster, the agent evaluates the expression
+
+$$
+\Phi (i, t ^ {\prime}) \triangleq d ^ {c} (i, x _ {t ^ {\prime}}) + c \cdot r _ {t ^ {\prime}} + \min  _ {j \in \widehat {C} _ {t ^ {\prime}}} \left(d ^ {m} (i, j) - d ^ {c} (j, x _ {t ^ {\prime}})\right),
+$$
+
+picks its minimizer  $t^* \in \arg \min_{t' \in V_i} \Phi(i, t')$ , and switches to it if  $\Phi(i, t') < c \cdot r_t$ . The following claims shed light into the reasoning behind this specific step.
+
+The first lemma establishes that if an agent is part of the  $t$ -th cluster at the end of both Phase 1 and Phase 2 (i.e., it does not switch clusters during Phase 2), then its loss, which is at most  $r_t$  at the end of Phase 1, remains bounded by  $c \cdot r_t$  at the end of Phase 2. The increase may come from other agents possibly switching into the  $t$ -th cluster during Phase 2.
+
+Lemma 6. For each  $i\in \mathcal{N}$  , if  $\widehat{\mathcal{X}} (i) = \mathcal{X}(i) = t$  then  $\ell_i(\mathcal{X}) = \ell_i(C_t,x_t)\leqslant c\cdot r_t$
+
+Proof. Our goal is to show that  $d^{c}(i, x_{t}) + d^{m}(i, j) \leqslant c \cdot r_{t}$  for all  $j \in C_{t}$ . Consider any  $j \in C_{t}$ .
+
+Case 1:  $j \in C_t \cap \widehat{C}_t$ . By the definition of  $r_t$ , we must have  $d^c(i, x_t) + d^m(i, j) \leqslant r_t \leqslant c \cdot r_t$ , where the last inequality holds because  $c \geqslant 1$ . In words, such an agent does not increase the loss of agent  $i$  even slightly.
+
+Case 2:  $j \in C_t \setminus \widehat{C}_t$ . Because agent  $j$  was allowed to switch to the  $t$ -th cluster, we must have  $t \in V_j$ . Then, Line 13 in Algorithm 1 ensures that  $d^c(i, x_t) + d^m(i, j) \leqslant c \cdot r_t$ ; this is the condition that forbids an agent from joining a cluster if it would significantly hurt a "core member" of that cluster.
+
+This concludes the proof.
+
+![](images/ec1f0626df4677a9556e9abe69484158e7a6da45bf41f442a30e545459ba643b.jpg)
+
+The next lemma shows that if agent  $i$  switches to the  $t^*$ -th cluster during Phase 2, its final loss  $\ell_i(\mathcal{X})$  would be bounded by  $\Phi(i, t^*)$ .
+
+Lemma 7. For each  $i\in \mathcal{N}$  , if  $\mathcal{X}(i) = t^{*}\neq \widehat{\mathcal{X}} (i)$  , then
+
+$$
+\ell_ {i} (\mathcal {X}) = \ell_ {i} (C _ {t ^ {*}}, x _ {t ^ {*}}) \leqslant \Phi (i, t ^ {*}) = d ^ {c} (i, x _ {t ^ {*}}) + c \cdot r _ {t ^ {*}} + \min  _ {j \in \widehat {C} _ {t ^ {*}}} (d ^ {m} (i, j) - d ^ {c} (j, x _ {t ^ {*}})).
+$$
+
+Proof. Let  $i' = \arg \max_{j \in C_{t^*}} d^m(i, j)$  be the agent in  $C_{t^*}$  that is the farthest from  $i$ . Then, we have  $\ell_i(C_{t^*}, x_{t^*}) = d^c(i, x_{t^*}) + d^m(i, i')$ . We want to show that this is at most  $\Phi(i, t^*)$ . Let  $j^* = \arg \min_{j \in \widehat{C}_{t^*}} (d^m(i, j) - d^c(j, x_{t^*}))$ , so  $\Phi(i, t^*) = d^c(i, x_{t^*}) + c \cdot r_{t^*} + d^m(i, j^*) - d^c(j^*, x_{t^*})$ . Let us simplify our goal:
+
+$$
+\begin{array}{l} d ^ {c} \left(i, x _ {t ^ {*}}\right) + d ^ {m} \left(i, i ^ {\prime}\right) \leqslant d ^ {c} \left(i, x _ {t ^ {*}}\right) + c \cdot r _ {t ^ {*}} + d ^ {m} \left(i, j ^ {*}\right) - d ^ {c} \left(j ^ {*}, x _ {t ^ {*}}\right) \\ \Leftrightarrow d ^ {m} \left(i, i ^ {\prime}\right) \leqslant c \cdot r _ {t ^ {*}} + d ^ {m} \left(i, j ^ {*}\right) - d ^ {c} \left(j ^ {*}, x _ {t ^ {*}}\right). \tag {1} \\ \end{array}
+$$
+
+Case 1:  $i' \in \widehat{C}_{t^*}$ . Then, by the fact that  $j^* \in \widehat{C}_{t^*}$  is also true, we must have that  $d^{c}(j^{*},x_{t^{*}}) + d^{m}(j^{*},i^{\prime}) \leqslant \ell_{j^{*}}(\widehat{C}_{t^{*}},x_{t^{*}}) \leqslant r_{t^{*}}$ , which yields  $d^{m}(j^{*},i^{\prime}) \leqslant r_{t^{*}} - d^{c}(j^{*},x_{t^{*}})$ .
+
+Case 2:  $i' \notin \widehat{C}_{t^*}$ . Then, since agent  $i'$  switched to the  $t^*$ -th cluster during Phase 2, we must have  $t^* \in V_{i'}$ . Line 13 of Algorithm 1 and the fact that  $j^* \in \widehat{C}_{t^*}$  ensure that  $d^{c}(j^{*},x_{t^{*}}) + d^{m}(j^{*},i^{\prime}) \leqslant c\cdot r_{t^{*}}$ , i.e.,  $d^{m}(j^{*},i^{\prime}) \leqslant c\cdot r_{t^{*}} - d^{c}(j^{*},x_{t^{*}})$ .
+
+Since  $c \geqslant 1$ , we get  $d^{m}(j^{*},i^{\prime}) \leqslant c \cdot r_{t^{*}} - d^{c}(j^{*},x_{t^{*}})$  in both cases. Using the triangle inequality, we get
+
+$$
+d ^ {m} (i, i ^ {\prime}) \leqslant d ^ {m} (i, j ^ {*}) + d ^ {m} (j ^ {*}, i ^ {\prime}) \leqslant d ^ {m} (i, j ^ {*}) + c \cdot r _ {t ^ {*}} - d ^ {c} (j ^ {*}, x _ {t ^ {*}}),
+$$
+
+which is the desired Equation (1), concluding the proof.
+
+![](images/a601396deb5e7a9eb5ce0bdf6648063890ca192710a83f68767141895bbfdf06.jpg)
+
+Here is what we know now. During Phase 2, if agent  $i$  remains in its original cluster  $\widehat{\mathcal{X}}(i) = t$ , its loss remains upper bounded by  $c \cdot r_t$  (Lemma 6), and if it switches to a cluster  $t' \in V_i$ , its loss remains upper bounded by  $\Phi(i, t')$ . This explains why agent  $i$  switches to the cluster  $t^* \in \arg \min_{t' \in V_i} \Phi(i, t')$ , only if  $\Phi(i, t^*) < c \cdot r_t$ : the agent is simply picking the cluster that offers the smallest upper bound on its eventual loss. This is captured by the next lemma.
+
+Lemma 8. For each  $i \in \mathcal{N}$  and  $t' \in V_i$ , we have  $\ell_i(\mathcal{X}) \leqslant \Phi(i, t') = d^c(i, x_{t'}) + c \cdot r_{t'} + \min_{j \in \widehat{C}_{t'}} (d^m(i, j) - d^c(j, x_{t'}))$ .
+
+Proof. Let  $t = \widehat{\mathcal{X}}(i)$  and  $t^* \in \arg \min_{t' \in V_i} \Phi(i, t')$ . If agent  $i$  switches to the  $t^*$ -th cluster during Phase 2 of Algorithm 1, then the claim follows from the fact that  $\Phi(i, t^*) \leqslant \Phi(i, t')$  for all  $t' \in V_i$ . If agent  $i$  does not make a switch during Phase 2 and remains in the  $t$ -th cluster, this must be because  $c \cdot r_t \leqslant \Phi(i, t^*)$  due to Line 15 of Algorithm 1. The claim again follows from  $\Phi(i, t^*) \leqslant \Phi(i, t')$  for all  $t' \in V_i$ .
+
+Similarly to the proof of Lemma 8, we can see that if agent  $i$  was in the  $t$ -th cluster at the end of Phase 1, then  $c \cdot r_t$  also remains an upper bound on its eventual loss, regardless of whether the agent stays in the  $t$ -th cluster or switches to a  $t^*$ -th cluster during Phase 2. This is because the agent only switches to the  $t^*$ -th cluster if it provides a smaller upper bound on the loss.
+
+Lemma 9. For each  $i\in \mathcal{N}$  , if  $\hat{\mathcal{X}} (i) = t$  then  $\ell_i(\mathcal{X})\leqslant c\cdot r_t$
+
+Proof. If  $\mathcal{X}(i) = t$  (i.e., the agent stays in the  $t$ -th cluster during Phase 2), then the claim reduces to Lemma 6. If  $\mathcal{X}(i) = t^{*} \neq t$ , then by Lemma 7, we have  $\ell_{i}(\mathcal{X}) \leqslant \Phi(i, t^{*})$ . However, due to Line 16, agent  $i$  only switches to the  $t^{*}$ -th cluster if  $\Phi(i, t^{*}) < c \cdot r_{t}$ , which implies  $\ell_{i}(\mathcal{X}) \leqslant c \cdot r_{t}$ , as needed.
+
+Together, Lemmas 8 and 9 give us two different ways to upper bound the loss of agent  $i$ , in terms of the  $t$ -th cluster that agent  $i$  was part of at the end of Phase 1, and in terms of every  $t'$ -th cluster in  $V_{i}$  that agent  $i$  had the opportunity to switch to during Phase 2. With these bounds, we are ready to prove the central lemma.
+
+Lemma 1. Given a subroutine for computing an  $\alpha$ -approximate most cohesive cluster, Algorithm 1, with the growth factor set to  $c = \frac{3\alpha + \sqrt{\alpha(\alpha + 8)}}{4\alpha}$ , finds a clustering in the  $\frac{1}{2} (\alpha + \sqrt{\alpha(\alpha + 8)} + 2)$ -core with respect to the dual metric loss.
+
+Proof. Let  $\beta = \frac{1}{2} (\alpha +\sqrt{\alpha(\alpha + 8)} +2)$
+
+For contradiction, assume that in some clustering  $\mathcal{X} = \{(C_1,x_1),\ldots ,(C_k,x_k)\}$  returned by the algorithm, there is some deviating cluster  $(S,y)$  such that for all  $i\in S,\ell_i(S,y) < \frac{1}{\beta}\ell_i(\mathcal{X})$  . Let  $i = \arg \min_{i\in S}\widehat{\mathcal{X}} (i)$  be the agent from this deviating cluster that was captured first during the phase 1 of Algorithm 1. Let  $\widehat{\mathcal{X}} (i) = t$  . Let  $j = \arg \max_{j\in S}\ell_j(S,y)$  be the agent from  $S$  that has the highest loss among all agents in the deviating cluster.
+
+First note that it must be the case that  $\ell_j(S, y) \geqslant \frac{1}{\alpha} r_t$ . If this were not the case, it would contradict the fact that  $(\widehat{C}_t, x_t)$  was a  $\alpha$ -approximation of the most cohesive cluster at that step of the algorithm.
+
+Next, we can show that  $\widehat{\mathcal{X}}(j) \neq t$ , and that  $j$  did not have the opportunity to switch into the  $t$ -th cluster during the second phase of the algorithm.
+
+Showing that  $\widehat{\mathcal{X}}(j) \neq t$  is easy. If  $\widehat{\mathcal{X}}(j) = t$  were true, then by Lemma 9, we must have that  $\ell_j(\mathcal{X}) \leqslant c \cdot r_t$ . Since  $j$  wants to deviate to  $(S, y)$ , we also must have that  $\ell_j(S, y) < \frac{1}{\beta} \ell_j(\mathcal{X}) \leqslant \frac{c}{\beta} r_t$ . By plugging in the correct values of  $\beta$  and  $c$ , it can be seen that  $\frac{c}{\beta} < \frac{1}{\alpha}$  whenever  $\alpha \geqslant 1$ , so this would contradict that  $\ell_j(S, y) \geqslant \frac{1}{\alpha} r_t$ .
+
+With a bit more work, we can also show that  $j$  did not have the opportunity to switch to the  $t$ -th cluster during the second phase of the algorithm. If  $j$  did have this opportunity, then by Lemma 8 we must have that  $\ell_j(\mathcal{X}) \leqslant d^c(j, x_t) + c \cdot r_t + \min_{i' \in \widehat{C}_t} \{d^m(i', j) - d^c(i', x_t)\}$ . Since  $i \in \widehat{C}_t$ , this also gives us  $\ell_j(\mathcal{X}) \leqslant d^c(j, x_t) + c \cdot r_t + d^m(i, j) - d^c(i, x_t)$ .
+
+By the triangle inequality, we have that  $d^{c}(j,x_{t}) \leqslant d^{c}(j,y) + d^{c}(i,y) + d^{c}(i,x_{t})$ . Plugging this into our bound on  $j$ 's loss gives us:
+
+$$
+\ell_ {j} (\mathcal {X}) \leqslant c \cdot r _ {t} + d ^ {m} (i, j) + d ^ {c} (j, y) + d ^ {c} (i, y)
+$$
+
+Notice that since  $i$  and  $j$  are both in  $S$ ,  $d^{m}(i,j) + d^{c}(j,y) \leqslant \ell_{j}(S,y)$ . Also, we must have that  $d^{c}(i,y) \leqslant \ell_{i}(S,y) < \frac{1}{\beta} \ell_{i}(\mathcal{X}) \leqslant \frac{c}{\beta} r_{t}$ , where the last inequality follows from Lemma 9. Plugging these in to our inequality gives us:
+
+$$
+\ell_ {j} (\mathcal {X}) <   \ell_ {j} (S, y) + \left(c + \frac {c}{\beta}\right) \cdot r _ {t}.
+$$
+
+This implies that when  $j$  deviates to  $(S, y)$ , it can only improve its loss by a factor of at most:
+
+$$
+\frac {\ell_ {j} (S , y) + \left(c + \frac {c}{\beta}\right) \cdot r _ {t}}{\ell_ {j} (S , y)}.
+$$
+
+The fact that  $\ell_j(S, y) \geqslant \frac{1}{\alpha} r_t$  allows us to upper-bound this as
+
+$$
+\frac {\frac {1}{\alpha} \cdot r _ {t} + \left(c + \frac {c}{\beta}\right) \cdot r _ {t}}{\frac {1}{\alpha} \cdot r _ {t}} = 1 + c \cdot \alpha \cdot \left(1 + \frac {1}{\beta}\right).
+$$
+
+By substituting in the values of  $c$  and  $\beta$  in terms of  $\alpha$ , this comes out to be  $\frac{1}{2} (\alpha + \sqrt{\alpha(\alpha + 8)} + 2)$  which gives a contradiction.
+
+Thus, it must be the case that  $j$  does not have the opportunity to switch to the  $t$ -th cluster during the second phase of the algorithm. Therefore, there must be some  $i' \in \widehat{C}_t$  such that  $d^c(i', x_t) + d^m(i', j) > c \cdot r_t$ , which can be rearranged to get  $d^m(i', j) > c \cdot r_t - d^c(i', x_t)$ . As the final part of the proof, we can show that this also leads to a contradiction.
+
+By the triangle inequality, we have that  $d^{m}(i', j) \leqslant d^{m}(i', i) + d^{m}(i, j)$ . This can be rearranged to get  $d^{m}(i, j) \geqslant d^{m}(i', j) - d^{m}(i', i)$ . Plugging this and  $d^{m}(i', j) > c \cdot r_{t} - d^{c}(i', x_{t})$  into this inequality gives us  $d^{m}(i, j) > c \cdot r_{t} - (d^{c}(i', x_{t}) + d^{m}(i', i))$ . Since  $i$  and  $i'$  are both in  $\widehat{C}_{t}$ , we have that  $d^{c}(i', x_{t}) + d^{m}(i', i) \leqslant \ell_{i'}(\widehat{C}_{t}, x_{t}) \leqslant r_{t}$ , which gives us  $d^{m}(i, j) > (c - 1)r_{t}$ . Since  $i$  and  $j$  are both in  $S$ , this tells us that  $\ell_{i}(S, y) \geqslant d^{m}(i, j) > (c - 1)r_{t}$ .
+
+Finally, since  $i \in \widehat{C}_t$ , by Lemma 9 we know that  $\ell_i(\mathcal{X}) \leqslant c \cdot r_t$ . This means that when  $i$  deviates to  $(S,y)$ , they are only improving their loss by a factor less than  $\frac{c \cdot r_t}{(c - 1)r_t} = \frac{1}{2} (\alpha + \sqrt{\alpha(\alpha + 8)} + 2)$ , giving another contradiction and completing the proof.
+
+From Lemma 1, the proof of Theorem 2 follows easily.
+
+Theorem 2. For the dual metric loss, there always exists a clustering in the 3-core.
+
+Proof. Set  $\alpha = 1$  (i.e., let Algorithm 1 iteratively compute a most cohesive cluster in Phase 1) in Lemma 1, which sets the growth factor to  $c = 3/2$  and yields a 3-approximation to the core.
+
+Next, we present Algorithm 4, which efficiently computes a 4-MCC cluster for the dual metric loss by running the non-centroid greedy capture algorithm on a carefully crafted distance metric, once for each feasible center.
+
+Algorithm 4: 4-Approximate Most Cohesive Clustering Algorithm  
+Input: Set of agents  $\mathcal{N}$  , set of feasible centers  $\mathcal{M}$  , non-centroid metric  $d^{m}$  , centroid metric  $d^c$  number of clusters  $k$    
+Output: Cluster  $(C,x)$    
+1 for  $y\in \mathcal{M}$  do   
+2  $\begin{array}{rlr}{\forall i,j\in \mathcal{N}:d_y(i,j)\leftarrow d^m (i,j) + d^c (i,y) + d^c (j,y);} & {} & {\mathrm{~ / ~}C\mathrm{~f~a~t~}3\mathrm{~t~h~e~m~e~r~c~}}\\ {C_y\leftarrow \mathrm{First~clust~er~produced~by~Algorithm~3~with~input}(\mathcal{N},d_y);\quad / / \mathrm{Run~greedy~capture}}\\ {r_y\leftarrow \max_{i\in C_y}\ell_i(C_y,y);} & {} & {\mathrm{/ / }\ell_i = \mathrm{dual~metric~loss}} \end{array}$    
+3   
+4   
+5  $y^{*}\gets \arg \min_{y\in \mathcal{M}}r_{y};$  // Best center   
+6 return  $(C_{y^*},y^*)$
+
+The proof of its approximation guarantee follows.
+
+Lemma 2. For the dual metric loss, Algorithm 4 computes a 4-approximate most cohesive cluster in polynomial time.
+
+Proof. For any instance, fix some  $y \in \mathcal{M}$  and let  $d_y$  be a metric distance function over  $\mathcal{N}$ , where for any distinct  $i, j \in \mathcal{N}$ ,  $d_y(i, j) = d^m(i, j) + d^c(i, y) + d^c(j, y)$  (if  $i = j$  then we assign  $d_y(i, j) = 0$ ). We can first show that  $d_y$  indeed induces a metric space over  $\mathcal{N}$ . To show this, we just need to show that the triangle inequality holds, as clearly  $d_y$  will meet to other two metric axioms.
+
+For any  $i, i', j \in \mathcal{N}$ , we should have that  $d_y(i, j) \leqslant d_y(i, i') + d_y(i', j)$ . By the definition of  $d_y$ , this is equivalent to saying  $d^m(i, j) + d^c(i, y) + d^c(j, y) \leqslant d^m(i, i') + d^c(i, y) + d^c(i', y) + d^m(i', j) + d^c(i', y) + d^c(j, y)$ . By the fact that  $d^m$  is a metric, we have that  $d^m(i, j) \leqslant d^m(i, i') + d^m(i', j)$ .
+
+Therefore, we just need to show that  $d^{c}(i,y) + d^{c}(j,y) \leqslant d^{c}(i,y) + d^{c}(i',y) + d^{c}(i',y) + d^{c}(j,y)$ . Clearly, this holds as well, since  $d^{c}(i,y) + d^{c}(j,y)$  appears on the right-hand side of the inequality.
+
+Algorithm 4 runs the non-centroid greedy capture algorithm on the agents, using the distance metric  $d_{y}$ , for every  $y \in \mathcal{M}$ , then considers that cluster in the original dual metric setting with  $y$  as its center, and selects the most cohesive cluster found this way.
+
+For contradiction, assume that Algorithm 4 does not find does not find a 4 approximation of the most cohesive cluster. Let  $(S,y)$  be the most cohesive cluster for an instance inputted to Algorithm 4. Define  $r = \max_{i\in S}\ell_i(S,y)$  as the maximum loss felt by any agent in  $S$ . Assume that Algorithm 4 returns the cluster  $(C^{*},x^{*})$ , where  $r^* = \max_{i\in C^*}\ell_i(C^*,x^*) > 4r$
+
+Take the distance metric  $d_y$ , and let  $C_y$  be the non-centroid cluster that Algorithm 4 finds while using  $d_y$  as its distance function. First, note that  $\max_{i \in C_y} \ell_i(C_y, y) \geqslant r^*$ . Otherwise, Algorithm 4 would have returned  $(C_y, y)$  rather than  $(C^*, x)$ .
+
+Next, consider the maximum non-centroid loss with respect to  $d_{y}$  for any agent in the member set  $S$ . We will have  $\forall i \in S, \max_{j \in S} d_{y}(i,j) = \max_{j \in S} \{d^{m}(i,j) + d^{c}(i,y) + d^{c}(j,y)\} \leqslant \max_{j \in S} d^{m}(i,j) + d^{c}(i,y) + \max_{j \in S} d^{c}(j,y) \leqslant r + \max_{j \in S} d^{c}(j,y)$ . For all  $j \in S$ , we must have that  $d^{c}(j,y) \leqslant r$ , which gives us  $\max_{j \in S} d_{y}(i,j) \leqslant 2r$ .
+
+Caragiannis et al. [2] prove that in any non-centroid clustering instance, Algorithm 3 achieves a 2-approximation of the most cohesive cluster with respect to non-centroid loss for that instance. Since  $C_y$  was found by running Algorithm 3 on  $\mathcal{N}$  with distance function  $d_y$ ,  $C_y$  must be a 2-approximation to the most cohesive non-centroid cluster in that instance. Particularly, this implies that the maximum non-centroid loss with respect to  $d_y$  loss felt by any agent in  $C_y$  must be no more than two times more than the maximum such loss felt by an agent in any member set  $C' \subseteq \mathcal{N}$  with  $|C'| \geqslant n / k$ . With respect to  $S$ , this implies that  $\max_{i,j \in C_y} d_y(i,j) \leqslant 2\max_{i,j \in S} d_y(i,j) \leqslant 2(2r) = 4r$ .
+
+Finally, note that for any member set of agents  $C$ , the maximum non-centroid loss for that member set felt by any agent  $i \in C$  with respect to the metric  $d_{y}$  will be an upper bound on the dual-metric loss agent  $i$  will feel under the cluster  $(C, y)$ . To see this, note that  $i$ 's non-centroid loss under  $C$  with respect to  $d_{y}$  will be equal to  $\max_{j \in C} d_{y}(i, j) = \max_{j \in C} (d^{m}(i, j) + d^{c}(i, y) + d^{c}(j, y)) \geqslant \max_{j \in C} d^{m}(i, j) + d^{c}(i, y) = \ell_{i}(C, y)$ . In the case of  $C_{y}$ , this tells us that  $r^{*} \leqslant \max_{i \in C_{y}} \ell_{i}(C_{y}, y) \leqslant \max_{i \in C_{y}} \max_{j \in C_{y}} d_{y}(i, j) \leqslant 4r$ , which gives a contradiction, concluding the proof.
+
+Plugging this into Lemma 1 gives us a constant approximation to the core in polynomial time.
+
+Theorem 3. For the dual metric loss, a clustering in the  $(3 + 2\sqrt{3})$ -core can be computed in polynomial time.
+
+Proof. From Lemma 2, Algorithm 4 computes a 4-MCC cluster. Let Algorithm 1 iteratively use it in Phase 1 and set  $\alpha = 4$  in Lemma 1. Then, the growth factor is set to  $c = 3 + \sqrt{3}$  and we obtain a core approximation of  $\frac{1}{2} (4 + \sqrt{48} + 2) = 3 + 2\sqrt{3}$ . Each call to Algorithm 4 runs in polynomial time, and it is clear that the rest of Algorithm 1 also runs in polynomial time.
+
+# C.2 Weighted Single Metric Loss
+
+Lemma 3. For the weighted single metric loss with parameter  $\lambda \in (0,1]$ , Algorithm 5 finds a clustering in the  $\frac{2}{\lambda}$ -core in polynomial time.
+
+Proof. For contradiction, assume this is false, and there exists some instance such that the greedy clustering solution  $\mathcal{X} = \{(C_1,x_1),\ldots ,(C_k,x_k)\}$  is not in the  $\frac{2}{\lambda}$ -core. This means there exists some deviating cluster  $(S,y)$  such that for all  $i\in S$ ,  $\frac{2}{\lambda}\cdot \ell_i(S,y) < \ell_i(\mathcal{X})$ .
+
+Let  $i = \arg \min_{i' \in S} \mathcal{X}(i')$  be the first agent from  $S$  captured by Algorithm 5. Let  $\mathcal{X}(i) = t$ . Let  $j \in C_t$  be the agent that "captured" the cluster  $C_t$  (meaning that at the time when  $C_t$  was captured by the algorithm,  $d(i', j) \leqslant \delta$  for all  $i' \in C_t$  was true), and let  $r_t = \max_{i \in C_t} d(i, j)$  be the value of  $\delta$  at the point when  $C_t$  was captured.
+
+Due to the triangle inequality, we have that for any  $i' \in C_t$ ,  $d(i, i') \leqslant d(i, j) + d(j, i') \leqslant 2r_t$ . With this, we have that  $\ell_i(\mathcal{X}) \leqslant (1 - \lambda)d(i, x_t) + \lambda 2r_t$ .
+
+Algorithm 5: Non-Centroid Greedy Capture With Greedy Centroid Selection  
+Input: Set of agents \(\mathcal{N}\) , set of feasible centers \(\mathcal{M}\) , metric \(d\) , number of clusters \(k\) Output: Clustering \(\mathcal{X} = \{(C_1,x_1),\ldots ,(C_k,x_k)\}\)   
+1 Initialize: \(\mathcal{N}'\gets \mathcal{N},\mathcal{X}\gets \emptyset ,\delta \gets 0;\) // Uncaptured agents remain   
+2 while \(\mathcal{N}'\neq \emptyset\) do   
+3 Increase \(\delta\) smoothly; // Edge-case to capture last cluster.   
+4 if \(|\mathcal{N}^{\prime}| <   n / k\) then i \(\leftarrow\) arg min \(i\in \mathcal{N}^{\prime}\) max \(j\in \mathcal{N}^{\prime}\) d(i,j); (C,x) \(\leftarrow\) (N',arg min \(x\in M\) d(i,x)); X \(\leftarrow\) XU {C,x}; N' \(\leftarrow\) N' \\( C; // Non-centroid ball found   
+9 if \(\exists i\in \mathcal{N}',C\subseteq \mathcal{N}'\) s.t. \(|C|\geqslant n / k\) and \(d(i,i')\leqslant \delta\) for all \(i'\in C\) then x \(\leftarrow\) arg min \(x\in \mathcal{M}\) d(i,x); // Centroid closest to the ball's center   
+11 X \(\leftarrow\) XU {C,x}; N' \(\leftarrow\) N' \\( C;   
+13 return X
+
+Note that there must be some  $i' \in S$  with  $d(i, i') \geqslant r_t$ . If this were not the case, then Algorithm 5 would have captured  $S$  prior to capturing  $C_t$ , with  $i$  being the "capturing" agent of  $S$ . Thus, we also have that  $\ell_i(S, y) \geqslant (1 - \lambda)d(i, y) + \lambda r_t$ .
+
+Due to the way that Algorithm 5 greedily selects a center for each cluster, it must be the case that  $d(j,y) \geqslant d(j,x_t)$ , since otherwise,  $y$  would have been selected as the center for  $C_t$  rather than  $x_t$ . From the triangle inequality, we have that  $d(i,x_t) \leqslant d(i,j) + d(j,x_t) \leqslant r_t + d(j,y) \leqslant r_t + d(i,j) + d(i,y) \leqslant 2r_t + d(i,y)$ . Allowing us to conclude that  $d(i,x_t) \leqslant 2r_t + d(i,y)$
+
+The above inequality allows us to bound  $\ell_i(\mathcal{X})\leqslant (1 - \lambda)d(i,x_t) + \lambda 2r_t\leqslant (1 - \lambda)(2r_t + d(i,y)) + \lambda 2r_t = 2r_t + (1 - \lambda)d(i,y)$ . This means that  $i$  improves their loss by deviating by at most a factor of:
+
+$$
+\frac {2 r _ {t} + (1 - \lambda) d (i , y)}{(1 - \lambda) d (i , y) + \lambda r _ {t}} = \frac {(2 - \lambda) r _ {t}}{(1 - \lambda) d (i , y) + \lambda r _ {t}} + 1 \leqslant \frac {(2 - \lambda) r _ {t}}{\lambda r _ {t}} + 1 = \frac {2}{\lambda}
+$$
+
+Which gives the desired contradiction.
+
+![](images/b3d744c3c647e92d35cbff11c5614a0ff92243f2e3125affb7e897b8f8f859e7.jpg)
+
+Lemma 4. For the weighted single metric loss with parameter  $\lambda \in [0,1)$ , Algorithm 6 finds a clustering in the  $\frac{\sqrt{2\lambda - 11\lambda^2 + 13} + 3 - \lambda}{2 - 2\lambda}$ -core in polynomial time.
+
+Proof. When running Algorithm 6, we set the growth factor  $c = \frac{q}{\lambda}$ , where  $q = \frac{\sqrt{2\lambda - 11\lambda^2 + 13} + 5\lambda - 1}{6}$ .
+
+Let  $\mathcal{X} = \{(C_1, x_1), \ldots, (C_k, x_k)\}$  be the clustering returned by Algorithm 6. Further, let  $\widehat{\mathcal{X}} = \{(\widehat{C}_1, x_1), \ldots, (\widehat{C}_k, x_k)\}$  be the clustering found by Algorithm 6 after it has completed phase 1.
+
+First, we will show that for any agent  $i \in N$ , if  $\widehat{\mathcal{X}}(i) = t$ , then  $\ell_i(\mathcal{X})$  will be no more than  $(1 - \lambda)d(i, x_t) + \lambda(d(i, x_t) + \frac{q}{\lambda}r_t) = d(i, x_t) + q \cdot r_t$ . This can be seen by analyzing how the second phase of the algorithm works.
+
+In phase 2 of Algorithm 6, each agent  $i \in \mathcal{N}$  can choose to switch from the cluster they were originally assigned to by the algorithm, to a new cluster. However, they can only switch to a cluster  $(C_{t'}, x_{t'})$  if  $d(i, x_{t'}) \leqslant cr_{t'}$ . In the algorithm,  $c$  is called the "growth factor", as the second phase can be thought of as the the radius of each of the cluster balls increasing by a factor of  $c$ , then each agent choosing to change to a new cluster that they are within the radius of. Also note that each agent
+
+Algorithm 6: Semi-Ball-Growing Algorithm  
+Input: Set of agents  $\mathcal{N}$  , set of feasible centers  $\mathcal{M}$  , distance metric  $d$  , parameter  $\lambda \in [0,1]$  number of clusters  $k$  growth factor c Output: Clustering  $\mathcal{X} = \{(C_1,x_1),\dots ,(C_k,x_k)\}$  1 Initialize:  $\mathcal{N}'\gets \mathcal{N},\widehat{\mathcal{X}}\gets \emptyset \delta \gets 0,t\gets 0;$  // Phase 1: Build a tentative clustering via centroid greedy capture   
+2 while  $\mathcal{N}'\neq \emptyset$  do // Uncaptured agents remain   
+3 Increase  $\delta$  smoothly;   
+4 if  $|\mathcal{N}^{\prime}| <   n / k$  then // Edge-case to capture the last cluster   
+5  $(\widehat{C}_t,x_t)\gets (\mathcal{N}',\arg \min_{x\in M}\max_{j\in \mathcal{N}'}(d(j,x)))$    
+6  $\widehat{\mathcal{X}}\gets \widehat{\mathcal{X}}\cup \{(\widehat{C}_t,x_t)\}$    
+7  $r_t\gets \max_{i\in \widehat{C}_t}d(i,x_i);$    
+8  $\mathcal{N}'\gets \mathcal{N}'\setminus \widehat{C}_t;$    
+9 t← t+1   
+10 if  $\exists x_{t}\in \mathcal{M},\widehat{C}_{t}\subseteq \mathcal{N}^{\prime}$  s.t.  $|\widehat{C}_t|\geqslant n / k$  and  $d(i,x_{t})\leqslant \delta$  for all  $i\in \widehat{C}_t$  then   
+11  $\begin{array}{rl} & {\widehat{\mathcal{X}}\gets \widehat{\mathcal{X}}\cup \{(\widehat{C}_t,x_t)\} ;}\\ & {r_t\gets \delta ;}\\ & {\mathcal{N}'\gets \mathcal{N}'\setminus \widehat{C}_t;}\\ & {t\gets t + 1} \end{array}$    
+12   
+13   
+14 // Phase 2: Selectively allow agents to switch clusters   
+15 Vt E [k], C t - C t;   
+for i E N do   
+17 // Allowed cluster with the best upper bound on loss  $t^\prime \gets \arg \min_{t\in [k]:d(i,x_t)\leqslant c\cdot r_t}\{(1 - \lambda)\cdot d(i,x_t) + c\cdot \lambda \cdot 2r_t\} ;$    
+18 t* e X(i);   
+19 // Switch if it improves the upper bound on loss if (1-λ) d(i,xt)+c·λ·2rt< d(i,xt)+c·λ·rt* then   
+20 Ct* C t* C t' U {i};   
+21 C t' C t' U {i};   
+return X={(C1,x1),..,(Ck,xk)}
+
+judges whether some new cluster  $(C_{t'}, x_{t'})$  will provide it less loss than its current cluster based on the worst-case loss  $i$  could possibility feel after switching to  $(C_{t'}, x_{t'})$ . This is reflected in the term  $c(\lambda) 2r_t$  on line 13 of Algorithm 6. Due to the fact that the non-centroid part of each agent's loss function will change based on which clusters other agents choose to switch into during the second phase of the algorithm, each agent will only switch to a new cluster if, regardless of how other agents switch, their loss under that new cluster is still guaranteed to be less than their current loss.
+
+For any  $i \in \mathcal{N}$  with  $\widehat{\mathcal{X}}(i) = t$ , if  $i$  were to not switch to a new cluster in the second phase of the algorithm, then we can show that its final loss will be less than  $d(i, x_t) + q \cdot r_t$ . To see this, assume  $i$  does not switch clusters in phase 2, and thus  $\mathcal{X}(i) = t$ . Because we know that  $i$  did not switch, it must be the case that the distance from  $i$  to the centroid of its final cluster is  $d(i, x_t) \leqslant r_t$ . Further, from the way that agents were allowed to switch clusters in the second phase, we know that all for any  $j \in C_t$ , we must have  $d(j, x_t) \leqslant c \cdot r_t$ . By the triangle inequality, this means that for  $j = \max_{j \in C_t} d(j, i)$ , the agent from  $C_t$  that is farthest away from  $i$ , we must have  $d(i, j) \leqslant d(i, x_t) + d(j, x_t) \leqslant d(i, x_t) + c \cdot r_t$ .
+
+Putting these together and plugging in that  $c = \frac{q}{\lambda}$  we have that:
+
+$$
+\ell_ {i} (\mathcal {X}) = (1 - \lambda) \cdot d (i, x _ {t}) + \lambda \cdot d (i, j) \leqslant (1 - \lambda) \cdot d (i, x _ {t}) + \lambda \cdot \left(d (i, x _ {t}) + c \cdot r _ {t}\right) = d (i, x _ {t}) + q \cdot r _ {t}
+$$
+
+Similarly, we can bound  $i$ 's loss if they do choose to switch clusters in phase 2. Assume now that  $i$  does switch such that after the second phase of the algorithm  $\mathcal{X}(i) = t'$ .
+
+By similar logic as above, we can upper bound  $i$ 's loss as:
+
+$$
+\ell_ {i} (\mathcal {X}) = (1 - \lambda) d \left(i, x _ {t ^ {\prime}}\right) + \lambda \ell_ {i} ^ {m} (\mathcal {X}) \leqslant (1 - \lambda) d \left(i, x _ {t ^ {\prime}}\right) + \lambda 2 c r _ {t ^ {\prime}} = (1 - \lambda) d \left(i, x _ {t ^ {\prime}}\right) + 2 q r _ {t ^ {\prime}}
+$$
+
+Here, the second transition follows from the fact that every agent in  $C_{t'}$  is at most a distance of  $cr_{t'}$  away from  $x_{t'}$ . This along with a similar triangle inequality argument as above allows us to bound  $\ell_i^m (C_{t'},x_{t'})\leqslant 2cr_{t'}$ . Notice that this is worse than the bound we were able to establish on  $i$ 's non-centroid loss in the case when it did not switch. This is because when  $i$  did not switch, we knew that  $d(i,x_t)\leqslant r_t$ , where in the case where  $i$  did switch, all we know is the weaker bound of  $d(i,x_{t'})\leqslant c\cdot r_t$ .
+
+From line 13 of Algorithm 6, we can see that an agent  $i$  will only switch from their first phase cluster to a new cluster if the upper-bounded loss from this new cluster is less than the upper-bounded loss of their original cluster. Thus, no matter what  $i$  does during the second phase, we can guarantee that their loss will not be more than  $d(i, x_t) + q \cdot r_t$ .
+
+With this bound on each agent's loss established, we can prove the required core guarantee. Assume for contradiction that this is false, and that Algorithm 6 produces a clustering  $\mathcal{X}$  such that there is some deviating cluster  $(S,y)$  with  $S\subseteq \mathcal{N},|S|\geqslant \left\lceil \frac{n}{k}\right\rceil$  , and  $y\in \mathcal{M}$  with  $\frac{\sqrt{2\lambda - 11\lambda^2 + 13} + 3 - \lambda}{2 - 2\lambda}\ell_i(S,y) < \ell_i(\mathcal{X})$  for all  $i\in S$
+
+Let  $i = \min_{i\in S}\widehat{\mathcal{X}} (i)$  be the agent from  $S$  that was captured first by phase 1 of the algorithm, and let  $\widehat{\mathcal{X}} (i) = t$ . We know that  $d(i,x_{t})\leqslant r_{t}$ . Let  $j = \arg \max_{j\in S}d(j,y)$  be the agent in  $S$  that is farthest away from  $y$ . We know that  $d(j,y)\geqslant r_t$  must be true, otherwise, the first phase of the algorithm would have captured  $(S,y)$  prior to capturing  $(\widehat{C}_t,x_t)$ .
+
+First, we will consider the case where  $d(j,x_{t}) \leqslant c\cdot r_{t}$ . This means that in the phase 2 of the algorithm,  $j$  will have the opportunity to switch to the  $t$ -th cluster. The maximum loss that  $j$  could feel under the cluster centered at  $x_{t}$  is  $(1 - \lambda)d(j,x_{t}) + 2qr_{t}$ , and since  $j$  will switch to the cluster with the lowest upper-bounded loss, this means that  $j$  deviating to  $(S,y)$  can improve its loss by at most a factor of  $\frac{(1 - \lambda)d(j,x_{t}) + 2qr_{t}}{(1 - \lambda)d(j,y)}$ . Note that this can be upper bounded as
+
+$$
+\begin{array}{l} \frac {(1 - \lambda) d (j , x _ {t}) + 2 q r _ {t}}{(1 - \lambda) d (j , y)} \leqslant \frac {(1 - \lambda) (d (j , y) + d (i , y) + d (i , x _ {t})) + 2 q r _ {t}}{(1 - \lambda) d (j , y)} \\ \leqslant \frac {(1 - \lambda) d (j , y) + (1 - \lambda) (d (i , y) + r _ {t}) + 2 q r _ {t}}{(1 - \lambda) d (j , y)} \\ \leqslant \frac {(1 - \lambda) r _ {t} + (1 - \lambda) (d (i , y) + r _ {t}) + 2 q r _ {t}}{(1 - \lambda) r _ {t}} \\ = \frac {(2 (1 - \lambda) + 2 q) r _ {t} + (1 - \lambda) d (i , y)}{(1 - \lambda) r _ {t}} \\ \leqslant \frac {2 (1 - \lambda) + 2 q}{(1 - \lambda)} + \frac {d (i , y)}{r _ {t}}. \tag {2} \\ \end{array}
+$$
+
+For  $i$ , we must have that  $\ell_i^m (S,y)\geqslant (1 - \lambda)d(i,y)$ , and  $\ell_i^m (\mathcal{X})$  cannot be more than  $i$ 's upper-bounded loss under  $(\widehat{C}_t,x_t)$ , meaning that  $i$  deviating cannot improve its loss by more than a factor of
+
+$$
+\frac {d \left(i , x _ {t}\right) + q \cdot r _ {t}}{(1 - \lambda) d (i , y)} \leqslant \frac {(1 + q) r _ {t}}{(1 - \lambda) d (i , y)}. \tag {3}
+$$
+
+Combining Equations (2) and (3), we get the following bound:
+
+$$
+\begin{array}{l} \min \left\{\frac {1 + q}{1 - \lambda} \cdot \frac {r _ {t}}{d (i , y)}, \frac {2 (1 - \lambda) + 2 q}{(1 - \lambda)} + \frac {d (i , y)}{r _ {t}} \right\} \leqslant \max _ {z \geqslant 0} \min \left\{\frac {1 + q}{1 - \lambda} z, \frac {2 (1 - \lambda) + 2 q}{(1 - \lambda)} + \frac {1}{z} \right\} \\ = \frac {\sqrt {\lambda^ {2} - 3 \lambda (q + 1) + q ^ {2} + 3 q + 2} - \lambda + q + 1}{1 - \lambda}, \\ \end{array}
+$$
+
+where the last transition can be derived by equating the two expressions in  $z$  and solving the resulting quadratic equation in  $z$ . Substituting the chosen value of  $q$  and simplifying gives us the following bound:
+
+$$
+\frac {\sqrt {2 \lambda - 1 1 \lambda^ {2} + 1 3} + 3 - \lambda}{2 - 2 \lambda}.
+$$
+
+This yields a contradiction. This shows that it must be true that  $d(j, x_{t}) > c \cdot r_{t}$ , meaning that  $j$  cannot switch to the  $t$ -th cluster during the second phase of Algorithm 6.
+
+Using this along with the triangle inequality, we can say that  $\frac{q}{\lambda} r_t < d(j,x_t) \leqslant d(j,i) + d(i,x_t) \leqslant d(j,i) + r_t$ . Rearranging we get that  $d(i,j) > (\frac{q}{\lambda} - 1)r_t = (\frac{q - \lambda}{\lambda})r_t$ .
+
+With this, we can say that  $\ell_i(S,y)\geqslant \lambda d(i,j) > \lambda \frac{q - \lambda}{\lambda} r_t = (q - \lambda)r_t$
+
+Combining this with the fact that  $\ell_i(\mathcal{X})\leqslant d(i,x_t) + q\cdot r_t\leqslant (q + 1)r_t$ , we get that in this case,  $i$  deviating to  $(S,y)$  cannot improve its loss by more than a factor of  $\frac{q + 1}{q - \lambda}$ .
+
+Substituting in our chosen value of  $q$  and simplifying once again gives us:
+
+$$
+\frac {\sqrt {2 \lambda - 1 1 \lambda^ {2} + 1 3} + 3 - \lambda}{2 - 2 \lambda}
+$$
+
+which is a contradiction, and concludes the proof.
+
+![](images/3e68c63cd81325e325e700b0b82eaf7e206eee53a9e519ba4c5a0ff9244e0a50.jpg)
+
+Lemma 10. For the weighted single metric loss with parameter  $\lambda \in (0,1)$ , no clustering algorithm can achieve a core approximation better than  $\frac{\sqrt{\lambda^2 - 2\lambda + 5} - \lambda + 1}{2}$ .
+
+Proof. We will analyze the instance in Figure 1, with  $p = \frac{\sqrt{\lambda^2 - 2\lambda + 5} + \lambda - 1}{2\lambda}$ . As in the proof of Theorem 1, we can, without loss of generality, limit our analysis to the section of the instance containing  $\{a,b,c\}$ , where we will assume that one cluster center is placed.
+
+If the cluster  $(\{a, b, c\}, b)$  is initially selected, then  $a$  will have a loss of  $(1 - \lambda)d(a, b) + \lambda d(a, c) = (1 - \lambda) + \lambda p$ , and  $b$  will have a loss of  $(1 - \lambda)0 + \lambda p = \lambda p$ . If  $a$  and  $b$  deviate to the cluster  $(\{a, b\}, b)$ , then they will improve their loss by a factor of  $(1 - \lambda) + \lambda p$  and  $p$  respectively. This deviation will lead to a violation of core with a factor of  $(1 - \lambda) + \lambda p = \frac{\sqrt{\lambda^2 + 2\lambda + 5} - \lambda + 1}{2}$ . The same argument can show that  $(\{a, b, c\}, a)$  would also not work as a cluster, and clearly,  $(\{a, b, c\}, c)$  would also not work, as  $a$  and  $b$ 's losses would only get worse than before.
+
+If the cluster  $(\{a, b\}, b)$  is initially selected, then note that  $a$  will have a loss of  $(1 - \lambda)d(a, b) + \lambda d(a, b) = 1$ . In contrast, under the clustering  $(\{a, c\}, a)$ ,  $a$  will have a loss of  $(1 - \lambda)0 + \lambda d(a, c) = \lambda p$ . This means that by deviating,  $a$  will decrease its loss by a factor of  $\frac{1}{\lambda p} = \frac{2}{\sqrt{\lambda^2 - 2\lambda + 5} + \lambda - 1} = \frac{\sqrt{\lambda^2 + 2\lambda + 5} - \lambda + 1}{2}$ . Since we assume that only one point from  $\{a, b, c\}$  is selected as a center, this deviation must also improve  $c$ 's loss by an infinite factor. Note that the same analysis holds if  $a$  was selected as the centroid in the initial cluster rather than  $b$ . Again, clearly the center  $c$  would not work with this cluster either, as it would just make both agents no better off.
+
+Next, note that the cluster  $(\{a,c\} ,a)$  cannot be selected.  $a$  would have a loss of  $(1 - \lambda)0 + \lambda p = \lambda p$  But under the clustering  $(\{a,b\} ,a)$ $a$  's loss would be  $(1 - \lambda)0 + \lambda = \lambda$  , so deviating would improve  $a$  's loss by a factor of  $p\geqslant \frac{\sqrt{\lambda^2 + 2\lambda + 5} - \lambda + 1}{2}$ $b$  would also improve their loss by an infinite factor. Clearly trying this cluster with any other center would not help, as it would only make  $a$  's initial loss worse. This argument can also be used to show that the cluster with member set  $\{b,c\}$  will not work.
+
+This eliminates the possibility of any cluster of the 3 agents of size greater than 1. Finally, we cannot have more than 2 of these agents involved in a singleton cluster, as since  $k = 3$ , all the agents on the other side would have to all belong to the same cluster, and we could apply a similar argument as the second paragraph of this proof to find a deviating set there.
+
+Lemma 11. For the weighted single metric loss with parameter  $\lambda \in [0,1]$ , no clustering algorithm can achieve a core approximation better than  $\frac{2(1 - \lambda)}{2\lambda + 1}$ .
+
+Proof. Take the instance from Claim 1 of Chen et al. [1].  $\mathcal{N} = \{a_1, \ldots, a_6\}, \mathcal{M} = \{y_1, \ldots, y_6\}, k = 3$ . The distances between agents are given as  $d(a, b) = d(a, c) = d(b, c) = 3$  and  $d(d, e) = d(d, e) = d(e, f) = 3$ , with all other pairs having distance  $\infty$ . The distances between agents and centers are given by the in the below table (which can also be seen in claim 1 of Chen et al. [1]).
+
+<table><tr><td></td><td>y1</td><td>y2</td><td>y3</td><td>y4</td><td>y5</td><td>y6</td></tr><tr><td>a1</td><td>4</td><td>1</td><td>2</td><td>∞</td><td>∞</td><td>∞</td></tr><tr><td>a2</td><td>2</td><td>4</td><td>1</td><td>∞</td><td>∞</td><td>∞</td></tr><tr><td>a3</td><td>1</td><td>2</td><td>4</td><td>∞</td><td>∞</td><td>∞</td></tr><tr><td>a4</td><td>∞</td><td>∞</td><td>∞</td><td>4</td><td>1</td><td>2</td></tr><tr><td>a5</td><td>∞</td><td>∞</td><td>∞</td><td>2</td><td>4</td><td>1</td></tr><tr><td>a6</td><td>∞</td><td>∞</td><td>∞</td><td>1</td><td>2</td><td>4</td></tr></table>
+
+Chen et al. [1] show that in this instance, it is impossible to achieve a centroid core approximation of more than 2.
+
+Fix some clustering  $\mathcal{X} = \{(C_1, x_1), (C_2, x_2), (C_3, x_3)\}$ . From Chen et al. [1], there must be some pair of agents that would want to deviate in the centroid world for a core improvement of at least 2. More specifically, since there are 3 clusters, and the instance consists of two "groups" of agents/centers, which are separated by a distance of  $\infty$ , one of the groups must have at most 1 center from that group selected as one of the centers in  $\mathcal{X}$ .
+
+Without loss of generality, assume that the group  $\{a_1, a_2, a_3\}$  only has one center from their group selected by  $\mathcal{X}$ , and assume  $y_1$  is that center (because of the symmetry of the instance, we could pick either group, and any center of that group, and an identical argument would hold). In this scenario, the other two centers from  $\mathcal{X}$  are distance  $\infty$  away from  $a_1$  and  $a_2$ .  $a_1$  and  $a_2$  are either in the cluster with  $y_1$ , or in a cluster with a center farther away from them than  $y_1$ . Therefore, we must have that  $\ell_{a_1}(\mathcal{X}) \geqslant (1 - \lambda)d(a_1, y_1) = 4(1 - \lambda)$  and  $\ell_{a_2}(\mathcal{X}) \geqslant (1 - \lambda)d(a_2, y_1) = 2(1 - \lambda)$ .
+
+Now consider the deviating cluster  $(S = \{a_{1}, a_{2}\}, y_{3})$ .  $a_{1}$  and  $a_{2}$  will have the following losses under that cluster:
+
+$$
+\ell_ {a _ {1}} (S, y _ {3}) = \lambda d (a _ {1}, a _ {2}) + (1 - \lambda) d (a _ {1}, y _ {3}) = 3 \lambda + 2 (1 - \lambda)
+$$
+
+$$
+\ell_ {a _ {2}} (S, y _ {3}) = \lambda d (a _ {1}, a _ {2}) + (1 - \lambda) d (a _ {2}, y _ {3}) = 3 \lambda + 1 (1 - \lambda)
+$$
+
+These, along with our lower bounds on the agents' loss under  $\mathcal{X}$  allow us to conclude that  $a_1$  would be able to make an improvement of at least  $\frac{4(1 - \lambda)}{3\lambda + 2(1 - \lambda)}$  by deviating, and  $a_2$  would see an improvement of at least  $\frac{2(1 - \lambda)}{3\lambda + 1(1 - \lambda)}$ .
+
+It can easily be verify that  $\frac{4(1 - \lambda)}{3\lambda + 2(1 - \lambda)} \geqslant \frac{2(1 - \lambda)}{3\lambda + 1(1 - \lambda)}$  for all  $\lambda \in [0,1]$ , meaning that this would lead to a core violation of  $\frac{2(1 - \lambda)}{3\lambda + 1(1 - \lambda)} = \frac{2(1 - \lambda)}{2\lambda + 1}$ , completing the proof.
+
+Theorem 5. For the weighted single metric loss with parameter  $\lambda \in [0,1]$ , no clustering algorithm can achieve a core approximation better than  $\max \left\{\frac{\sqrt{\lambda^2 - 2\lambda + 5} - \lambda + 1}{2}, \frac{2(1 - \lambda)}{2\lambda + 1}\right\}$ .
+
+Proof. The result for  $\lambda \in (0,1)$  follows directly from Lemmas 10 and 11. The edge cases follow from known lower bounds from the centroid (lower bound of 2 [1]) and non-centroid (lower bound of 1 [2]) worlds.
+
+# D Missing Proofs from Section 5
+
+First, we provide the missing proof of Theorem 6, which shows that Algorithm 7 yields an  $\alpha$ -FJR clustering, guaranteeing the existence of an (exact) FJR clustering for arbitrary loss functions and the polynomial-time computability of a 4-FJR clustering for the dual metric loss.
+
+![](images/94421805c63dbef664f0226a012a448e8ff23a46f42fa5cb85c22e2381166e17.jpg)  
+Figure 2: Single Metric Core Bounds Visualized as  $\lambda$  varies from 0 to 1. For exact bounds, see Table 1
+
+Algorithm 7: Iterative  $\alpha$ -MCC Clustering  
+Input: Set of agents  $\mathcal{N}$  , set of feasible centers  $\mathcal{M}$  , agents' loss functions  $(\ell_i)_{i\in \mathcal{N}}$  , number of centers  $k$  , approximation factor  $\alpha \geqslant 1$    
+Output: Clustering  $\mathcal{X} = \{(C_1,x_1),\ldots ,(C_k,x_k)\}$    
+1 Initialize:  $\mathcal{N}'\gets \mathcal{N},\mathcal{X}\gets \emptyset$  .. // Uncaptured agents remain   
+2 while  $\mathcal{N}'\neq \emptyset$  do   
+3  $(C,x)\gets \mathrm{An}\alpha$  -MCC cluster from the agents in  $\mathcal{N}^{\prime}$  ; // Find an  $\alpha$  -MCC cluster   
+4  $\mathcal{X}\gets \mathcal{X}\cup \{(C,x)\}$  // Add it to the clustering   
+5  $\mathcal{N}'\gets \mathcal{N}'\backslash C;$  // Mark its agents as captured   
+6 return X
+
+Theorem 6. For arbitrary losses and  $\alpha \geqslant 1$ , Algorithm 7, which iteratively finds an  $\alpha$ -approximate most cohesive cluster, returns an  $\alpha$ -FJR clustering. Hence, a 1-FJR clustering exists for arbitrary losses and, due to Lemma 2, a 4-FJR clustering can be computed in polynomial time for the dual metric loss.
+
+Proof. For contradiction, assume that Algorithm 7 does not always yield an  $\alpha$ -FJR clustering. Then, there exists an instance in which the clustering  $\mathcal{X} = \{(C_1, x_1), \ldots, (C_k, x_k)\}$  returned by Algorithm 7 admits a deviating cluster  $(S, y)$  such that for all  $i \in S$ ,
+
+$$
+\alpha \cdot \ell_ {i} (S, y) <   \min  _ {j \in S} \ell_ {j} (\mathcal {X}). \tag {4}
+$$
+
+Let  $i^{*} \in \arg \min_{i \in S} \mathcal{X}(i)$  be the first agent in  $S$  that was "captured" (i.e., removed from  $\mathcal{N}'$ ) by the algorithm, ties broken arbitrarily. Let  $t = \mathcal{X}(i^{*})$ . Then, from Equation (4), we get that for all  $i \in S$ ,
+
+$$
+\alpha \cdot \ell_ {i} (S, y) <   \min  _ {j \in S} \ell_ {j} (\mathcal {X}) \leqslant \ell_ {i ^ {*}} (\mathcal {X}) \leqslant \max  _ {i \in C _ {t}} \ell_ {i} (\mathcal {X}) = \max  _ {i \in C _ {t}} \ell_ {i} (C _ {t}, x _ {t}).
+$$
+
+Hence,  $\alpha \cdot \max_{i\in S}\ell_i(S,y) < \max_{i\in C_t}\ell_i(C_t,x_t)$ . Since all agents in  $S$  were uncaptured prior to  $(C_t,x_t)$  being added to  $\mathcal{X}$ , this contradicts  $(C_t,x_t)$  being an  $\alpha$ -MCC cluster at that time. This concludes the proof that Algorithm 7 always yields an  $\alpha$ -FJR clustering.
+
+Since an MCC cluster exists by definition, plugging in  $\alpha = 1$  yields the existence of an FJR clustering. And for the dual metric loss, since a 4-MCC cluster can be computed efficiently via Algorithm 4 (Lemma 2), we obtain that a 4-FJR clustering can be computed in polynomial time.
+
+Next, we provide the missing proof of Lemma 5, which establishes that for FJR, simultaneous approximation with respect to centroid and non-centroid loss functions is equivalent to approximation with respect to all dual metric loss functions that take a weighted combination of them (in contrast to the core, for which the former is only a necessary condition for the latter).
+
+Lemma 5. For any  $\alpha \geqslant 1$ , a clustering is  $\alpha$ -FJR with respect to the centroid loss with metric  $d^c$  and with respect to the non-centroid loss with metric  $d^m$  simultaneously if and only if it is  $\alpha$ -FJR with respect to all dual metric loss functions with centroid metric  $\lambda \cdot d^c$  and non-centroid metric  $(1 - \lambda) \cdot d^m$ , for all  $\lambda \in [0,1]$ .
+
+Proof. The if direction is trivial; one simply substitutes  $\lambda = 0$  and  $\lambda = 1$ . For the only if direction, take any clustering  $\mathcal{X}$  and assume that it is  $\alpha$ -FJR with respect to the centroid loss  $\ell^c$  induced by metric  $d^c$  and the non-centroid loss  $\ell^m$  induced by metric  $d^m$  simultaneously. Then, we have:
+
+$$
+\alpha \cdot \max  _ {i \in S} \ell_ {i} ^ {c} (S, y) \leqslant \min  _ {i \in S} \ell_ {i} ^ {c} (\mathcal {X}), \tag {5}
+$$
+
+$$
+\alpha \cdot \max  _ {i \in S} \ell_ {i} ^ {m} (S, y) \leqslant \min  _ {i \in S} \ell_ {i} ^ {m} (\mathcal {X}). \tag {6}
+$$
+
+Fix any  $\lambda \in (0,1)$ , and consider the dual metric loss  $\ell = \lambda \cdot \ell^c + (1 - \lambda) \cdot \ell^m$ . Our goal is to establish that it is  $\alpha$ -FJR with respect to  $\ell$ , regardless of the value of  $\lambda$ . We have:
+
+$$
+\begin{array}{l} \alpha \cdot \max  _ {i \in S} \ell_ {i} (S, y) = \alpha \cdot \max  _ {i \in S} \left(\lambda \cdot \ell_ {i} ^ {c} (S, y) + (1 - \lambda) \cdot \ell_ {i} ^ {m} (S, y)\right) \\ \leqslant \alpha \cdot \left(\lambda \cdot \max  _ {i \in S} \ell_ {i} ^ {c} (S, y) + (1 - \lambda) \cdot \max  _ {i \in S} \ell_ {i} ^ {m} (S, y)\right) \\ \leqslant \lambda \cdot \min  _ {i \in S} \ell_ {i} ^ {c} (\mathcal {X}) + (1 - \lambda) \cdot \min  _ {i \in S} \ell_ {i} ^ {m} (\mathcal {X}) \\ \leqslant \min  _ {i \in S} \left(\lambda \cdot \ell_ {i} ^ {c} (\mathcal {X}) + (1 - \lambda) \cdot \ell_ {i} ^ {m} (\mathcal {X})\right) = \min  _ {i \in S} \ell_ {o} i (\mathcal {X}), \\ \end{array}
+$$
+
+where the second inequality follows from Equations (5) and (6). This concludes the proof.
+
+![](images/dc5109ace4e04c4f44aed4db06616076768a1d82cc17faefaaf7a5857ed12776.jpg)
+
+Next, we provide the missing proof of Theorem 7, which shows that such a simultaneous approximation is impossible when using different centroid and non-centroid metrics.
+
+Theorem 7. For any  $\alpha \geqslant 1$  and  $\beta \geqslant 1$ , there exists an instance in which no clustering is both  $\alpha$ -FJR with respect to the centroid loss and  $\beta$ -FJR with respect to the non-centroid loss, when the two loss functions are allowed to use different metrics.
+
+Proof. Consider an instance with a set of 6 agents  $\mathcal{N} = \{a, b, c, d, e, f\}$ , a set of two feasible cluster centers  $\mathcal{M} = \{x_1, x_2\}$ , and  $k = 3$ .
+
+Non-centroid metric: Let  $d^{m}(a,b) = d^{m}(c,d) = d^{m}(e,f) = 0$ , and let the non-centroid distance between every other pair of agents be  $\infty$ .<sup>3</sup>
+
+Centroid metric: Let  $d^{c}(a, x_{1}) = d^{c}(c, x_{1}) = d^{c}(e, x_{1}) = 0$ , let  $d^{c}(b, x_{2}) = d^{c}(d, x_{2}) = d^{c}(f, x_{2}) = 0$ , and let the centroid distance between every other agent-center pair be  $\infty$ .
+
+Finite non-centroid FJR approximation: First, note that the only clustering that leads to a finite FJR approximation with respect to the non-centroid loss is the one that partitions the set of agents as  $\mathcal{X} = \{\{a,b\}, \{c,d\}, \{e,f\}\}$ .
+
+To see this, consider any  $C \in \mathcal{X}$ . If  $C$  is "broken" (i.e., not a cluster by itself), then at least one member of  $C$  must be a singleton cluster; the only other possibility would be that  $C$  is a strict subset of a cluster, in which case  $C$  can deviate, improving both its members from infinite loss to zero loss, yielding infinite FJR approximation.
+
+Hence, every unbroken pair in  $\mathcal{X}$  forms a cluster of size two, and (at least) one member from every broken pair in  $\mathcal{X}$  forms a singleton cluster. Since there are three pairs in  $\mathcal{X}$ , this already uses up  $k = 3$  clusters, leaving the other members from the broken pairs in  $\mathcal{X}$  not included in the partition, which is impossible. Hence, there must not be any broken pairs, which shows that the partition must be  $\mathcal{X} = \{\{a,b\}, \{c,d\}, \{e,f\}\}$ .
+
+Finite centroid FJR approximation: Now, let us try to assign a center to each cluster in  $\mathcal{X}$  in a way that yields a finite approximation with respect to the centroid loss. Since there are only two feasible centers  $x_{1}$  and  $x_{2}$ , at least one of them must be assigned to at most one of the clusters in  $\mathcal{X}$ . Without loss of generality, assume this center is  $x_{1}$ . There are three agents,  $a$ ,  $c$ , and  $e$ , which are at distance 0 away from  $x_{1}$  under the centroid metric. However, they are in different clusters. Hence, at least two of them must be assigned to  $x_{2}$ . Hence, they can deviate together with  $x_{1}$  and improve both their centroid losses from non-zero to zero, resulting in an infinite FJR approximation with respect to the centroid loss.
+
+This shows that there is no clustering that achieves finite FJR approximations with respect to both the centroid and non-centroid losses, concluding the proof.
+
+Next, we provide the missing proof of Theorem 8, which shows that simultaneous approximation is possible if both centroid and non-centroid losses use the same underlying metric. Since our algorithm (Algorithm 5) builds on non-centroid greedy capture, its FJR approximation with respect to the non-centroid loss is borrowed easily; the key difficulty is establishing its FJR approximation with respect to the centroid loss.
+
+Lemma 12. Algorithm 5, given metric  $d$ , yields a clustering that is 5-FJR with respect to the centroid loss induced by  $d$ .
+
+Proof. For contradiction, assume this is false. Then, there exists an instance in which Algorithm 5 returns a clustering  $\mathcal{X} = \{(C_1,x_1),\ldots ,(C_k,x_k)\}$  that admits a deviating cluster  $(S,y)$  such that
+
+$$
+\forall q \in S: 5 \cdot d (q, y) <   \min  _ {p \in S} \ell_ {p} ^ {c} (\mathcal {X}). \tag {7}
+$$
+
+Let  $i \in S$  be the agent from  $S$  that was captured first by Algorithm 5. Denote the cluster containing agent  $i$  as  $(C, x) \triangleq (C_{\mathcal{X}(i)}, x_{\mathcal{X}(i)})$  for brevity. Because Algorithm 5 runs non-centroid greedy capture to create the partition of the set of agents,  $C$  must have been captured by a ball centered at some agent  $j \in C$ . Note that the center  $x$  selected by Algorithm 5 for  $C$  must be the closest feasible center to agent  $j$ . Let  $r = \max_{i' \in C} d(j, i')$  be the radius of the ball centered at  $j$  when it captured  $C$ .
+
+First, we show that  $d(i,x) \leqslant 2r + d(i,y)$ . To see this, note that
+
+$$
+d (i, x) \leqslant d (i, j) + d (j, x) \leqslant d (i, j) + d (j, y) \leqslant d (i, j) + d (i, j) + d (i, y) \leqslant 2 r + d (i, y),
+$$
+
+where the first and third inequalities use the triangle inequality, the second inequality uses the fact that  $x$  is the closest center to agent  $j$  (hence,  $d(j,x) \leqslant d(j,y)$ ), and the last inequality follows from the definition of  $r$ .
+
+Now, from Equation (7), we have that
+
+$$
+\forall q \in S: 5 \cdot d (q, y) <   \min  _ {p \in S} \ell_ {p} ^ {c} (\mathcal {X}) \leqslant \ell_ {i} ^ {c} (\mathcal {X}) = d (i, x) \leqslant 2 r + d (i, y).
+$$
+
+In particular, substituting  $q = i$ , we get that  $5 \cdot d(i,y) < 2r + d(i,y)$ , which yields  $d(i,y) < r/2$ . Then, for every other agent  $q \in S$ , we have that  $5 \cdot d(q,y) < 2r + d(i,y) < 5r/2$ , which yields  $d(q,y) < r/2$ .
+
+By the triangle inequality, we have that, for all  $p, q \in S$ ,  $d(p, q) \leqslant d(p, y) + d(q, y) < r/2 + r/2 = r$ . But this is a contradiction because this would imply that in the execution of Algorithm 5, the ball centered at any agent in  $S$  would have captured  $S$  before the ball centered at agent  $j$  captured  $C$ .
+
+The proof of our simultaneous FJR approximation guarantee (Theorem 8) is now straightforward.
+
+Theorem 8. Algorithm 5, given metric  $d$ , finds, in polynomial time, a clustering that is 5-FJR with respect to the centroid loss and 2-FJR with respect to the non-centroid loss, both defined using  $d$ .
+
+Proof. 5-FJR with respect to the centroid loss  $\ell^c$  is established in Lemma 12.
+
+Caragiannis et al. [2] show that non-centroid greedy capture achieves 2-FJR with respect to the non-centroid loss  $\ell^m$ . It is easy to see that because Algorithm 5 runs non-centroid greedy capture and uses the partition  $(C_1, \ldots, C_k)$  of the set of agents  $\mathcal{N}$  that it produces, it retains this 2-FJR guarantee with respect to the non-centroid loss  $\ell^m$ .
+
+# E Experiments
+
+In this section, our goal is to empirically compare proportionally fair algorithms designed in this work to classical clustering methods, in terms of both proportional fairness metrics and traditional metrics that the classical methods optimize.
+
+Because the classical methods take only a single distance metric  $d$  as input, we limit our empirical analysis to weighted single metric loss, which depends on a single metric  $d$  along with a parameter  $\lambda \in [0,1]$ .
+
+Algorithms. We evaluate four algorithms.
+
+- GC: Algorithm 5, which runs non-centroid greedy capture followed by greedy centroid selection, and attains  $\frac{2}{\lambda}$ -core (Lemma 3). We denote it by 'GC'.  
+- SemiBall: Algorithm 6, which mimicks centroid greedy capture but with limited cluster switching, and attains  $\frac{\sqrt{2\lambda - 11\lambda^2 + 13} + 3 - \lambda}{2 - 2\lambda}$ -core (Lemma 4). We denote it by 'SemiBall'.  
+-  $k$ -means++: This classical algorithm approximately optimizes the  $k$ -means objective,  $\sum_{i\in \mathcal{N}}d(i,\mathcal{X}(i))^2$ .  
+-  $k$ -medoids: This classical algorithm approximately optimizes the  $k$ -medoids objective,  $\sum_{i\in \mathcal{N}}d(i,\mathcal{X}(i))$ .
+
+The  $k$ -means++ and  $k$ -medoids implementations are based on the Scikit-learn package in Python.
+
+It is worth remarking that three of the four algorithms—GC,  $k$ -means++, and  $k$ -medoids—take only the distance metric  $d$  as input and not the parameter  $\lambda$ . Hence, the clustering returned by these methods is independent of  $\lambda$ , although its approximations to proportional fairness metrics would depend on  $\lambda$ .
+
+Datasets. We use three datasets from the UCI Machine Learning Repository [28]: Iris, Pima Indians Diabetes, and Adult. These are the three datasets used by Caragiannis et al. [2] for their experiments with non-centroid clustering.
+
+- The Iris dataset contains 150 data points with measurements of sepal and petal dimensions across three species of iris flowers.  
+- The (Pima Indians) Diabetes dataset includes health-related indicators such as insulin dosage, glucose level, and number of pregnancies.  
+- The Adult dataset consists of 48,842 instances, with both categorical features (such as race and education) and numerical features (such as age and hours worked per week). It is typically used for the classification task of predicting whether an individual's income exceeds a specified threshold, but here it is used for clustering the individuals. We apply a one-hot encoding to all categorical features.<sup>5</sup>
+
+Metrics. We evaluate the four algorithms above on five metrics.
+
+The first two metrics are core and FJR violations (i.e., the smallest  $\alpha$  such that the clustering is in  $\alpha$ -core or  $\alpha$ -FJR); these two are proportional fairness metrics. We compute the exact violations of core and FJR by solving integer linear programs, which, given a clustering, find a deviating cluster  $(C, x)$  recording the largest possible violation of the corresponding metric.
+
+The next three metrics are the  $k$ -means objective (stated above), the  $k$ -medoids objective (stated above), and average within-cluster distance,  $\sum_{t \in [k]} \frac{1}{|C_t|} \sum_{i,j \in C_t} d(i,j)$  [29], which is another popular objective; these three can be considered efficiency metrics. These metrics are easy to compute given a clustering.
+
+Experimental setup. Following Chen et al. [1], Caragiannis et al. [2], we assume  $\mathcal{N} = \mathcal{M}$ , and use the Euclidean  $L_{2}$  distance metric. We vary two parameters: the number of clusters  $k\in \{5,6,\ldots ,25\}$  and the weighted loss parameter  $\lambda \in \{0.1,0.2,\dots ,0.9\}$ . Specifically, we report experimental results when varying  $\lambda \in \{0.1,0.2,\dots ,0.9\}$  while fixing  $k = 15$ , and when varying  $k\in \{5,6,\dots ,25\}$  while fixing  $\lambda = 0.5$ .
+
+While our algorithms (and the traditional baselines) scale well, solving integer linear programs to compute the exact core and FJR violations of the clustering they return is computationally intensive. Hence, for the two larger datasets, namely Adult and Diabetes, we randomly sample 100 data points in each of 40 independent trials. For the Iris dataset, since  $k$ -means++ uses a randomized initialization step, we run it 20 times. Our plots show the five metrics above on average along with  $95\%$  confidence intervals.
+
+Results. The results for Adult, Diabetes, and Iris datasets are presented in Figures 3 to 5, respectively. The qualitative takeaways are similar across the three datasets, so we use the Adult dataset (Figure 3) as an example to illustrate them.
+
+First, our two algorithms, GC and SemiBall, achieve near-exact core and FJR, which is significantly better than their worst-case approximation guarantees from Lemmas 3 and 4. The classical algorithms,  $k$ -means++ and  $k$ -medoids, admit notable core and FJR violations, and their fairness deteriorates as  $k$  increases; this may be because  $k$  is still less than  $n/2$ , so an increase in  $k$  increases the number of small coalitions (of size close to  $n/k$ ) that can deviate. In particular,  $k$ -means++ is highly unfair, especially for larger values of  $k$  and smaller values of  $\lambda$  (i.e., when the centroid loss is dominant). Interestingly,  $k$ -medoids is reasonably fair; Caragiannis et al. [2] observe it to be unfair for the setting of  $\lambda = 1$  (non-centroid loss), but our results show that its fairness improves as  $\lambda$  decreases, and it becomes almost as fair as GC and SemiBall, producing near-exact core and FJR clustering, when  $\lambda \rightarrow 0$  (centroid loss).
+
+The increased fairness of GC and SemiBall comes at the cost of decreased efficiency metrics as compared to  $k$ -means++ and  $k$ -medoids. The efficiency loss is, however, small, especially when either  $k$  is large or  $\lambda$  is small. Among our two algorithms, SemiBall seems to consistently perform better in efficiency metrics (and equal in fairness metrics).
+
+Hence, our main takeaway is that SemiBall and  $k$ -medoids are compelling algorithmic choices for semi-centroid clustering; the choice between the two may rely on the value of  $k$  and  $\lambda$ , as well as the principal's desired fairness-efficiency tradeoff.
+
+# F Impossibility Results for Balanced Clustering
+
+In this section, we provide two impossibility results for balanced clustering in our semi-centroid world. Informally, balancedness dictates that the partition of the set of agents contain clusters of roughly equal size. In many applications such as creating groups of students for class projects or designing oral presentation sessions at a conference, this may be a desirable criterion.
+
+Formally, one may refer to a clustering  $\mathcal{X} = \{(C_1,x_1),\ldots ,(C_k,x_k)\}$  as balanced if  $|C_t|\in$ $\{\lfloor n / k\rfloor ,\lceil n / k\rceil \}$  for all  $t\in [k]$  (equivalently,  $||C_t| - |C_{t'}||\leqslant 1$  for all  $t,t'\in [k])$  . However, both our impossibility results hold even under the following weaker criterion.
+
+Definition 4 (Balanced Clustering). We call a clustering  $\mathcal{X} = \{(C_1, x_1), \ldots, (C_k, x_k)\}$  balanced if, when  $k$  divides  $n$ ,  $|C_t| = n / k$  for all  $t \in [k]$ .
+
+It may be encouraging that non-centroid greedy capture indeed produces a balanced clustering under this weak version. The centroid greedy capture, however, offers no such guarantees. This is because, while it still opens balls as soon as they capture  $\lceil n / k\rceil$  uncaptured agents, (i) open balls continue to grow and capture more agents, and (ii) agents are reassigned to the nearest center in the end. At least the second aspect is retained to some degree in both our algorithm for the dual metric loss (Algorithm 1) and one of our algorithms for the weighted single metric loss (Algorithm 6). This makes the final sizes of the clusters rather unpredictable (and not necessarily roughly equal).
+
+That said, Algorithm 5, which runs non-centroid greedy capture followed by a greedy centroid selection step, does in fact produce a balanced clustering and achieves  $\frac{2}{\lambda}$ -core with respect to the weighted single metric loss. Unfortunately, this becomes unbounded when  $\lambda$  approaches 0 (the centroid world). The next result shows that this is impossible to avoid when requiring a balanced clustering. Hence, our use of Algorithm 6, which computes an imbalanced clustering to achieve a finite core approximation in the small  $\lambda$  regime (thus yielding a finite core approximation across the entire spectrum of  $\lambda \in [0,1]$ ) is necessary.
+
+Theorem 10. For the weighted single metric loss with parameter  $\lambda \in [0,1]$ , there exists an instance in which no balanced clustering is in the  $\alpha$ -core for any  $\alpha < \frac{1}{\sqrt{\lambda}}$ .
+
+![](images/6b4936842ccc7adb36653906184e2a4c8dd5a0fcfc6802e587723e8b648ee2a8.jpg)
+
+![](images/98d7742a4ce96734e0f81a044438426fbec5ccbe5eea2c1b6445f6709a2c7a1b.jpg)
+
+![](images/ff06fa2df5ccc66ef95284a99aadec0ceb9fa50521714375e9875445e0c53a4e.jpg)
+
+![](images/8acb821ee9021d44d26ac171af58aadd9c5269999e41405d949b94036fde89dc.jpg)
+
+![](images/dc59b74afe164f406a0ae14994463851ebde125b972bcab470189094162cf019.jpg)
+
+![](images/e5f5949fe93bbe2b28ebd53ee1067f983785da0933734c0fec56656fa1f6754a.jpg)
+
+![](images/d7d53cb1749c6ae3ab1d49cf7670c21a5671657100c1c6e0cfad23637106428e.jpg)
+
+![](images/b1b8461d39cb9969ec298e056b867d4ec05625944a747f2a14bc39bb50a99b66.jpg)  
+Figure 3: Results for the Adults (Census Income) dataset. The common legend appears at the top. The top five plots compare the algorithms on our five metrics when varying  $\lambda \in \{0.1, 0.2, \dots, 0.9\}$  and fixing  $k = 15$ , and the bottom five plots correspond to fixing  $\lambda = 0.5$  and varying  $k \in \{5, 6, \dots, 25\}$ .
+
+![](images/b07a5fa7616e1b96006f901926334ff691f5a636ae5b33a86fde472f6894fa22.jpg)
+
+![](images/c57f661fa11e5b01b96dd096b410908417cf17b7dce690208301b6dd64b3613d.jpg)
+
+![](images/82c3cba95ea6bd1286b99e2379780d4027f36d531fc238d9b7fa322e1da66074.jpg)  
+GC
+
+![](images/19f6e7d1c0a8ea18d5e8364909ab7c501b7d4248e69583ddbe836c853ef090d4.jpg)  
+SemiBall
+
+![](images/5f5439e5fa9eb7b0d23d1db7399d7a0c8231caf33fd47b86c60374ef446b709e.jpg)  
+k-means
+
+![](images/e23cd3492ba1ed2039a28ef9ea9c61bd57737a896c741b163f0c7bc329b7f1ab.jpg)  
+k-medoids
+
+![](images/99b9d9f5d64936b13aaa6c5e37583dd45a9eb3aa8f4b2f660f3e9fdb3b3b8bb4.jpg)
+
+![](images/ba68057532ba74856dddf8998258e666716aedc7d537553c7e263291b2842378.jpg)
+
+![](images/080138fe4be9ae3131943a631d4d5c6e7eac9d83054ba9db9c38f678ed40c9db.jpg)
+
+![](images/318af44d9355801cf88f9ff6b40b060750cad6cb138dc0d661aed8fefc198569.jpg)
+
+![](images/6196aba6b3f86827c01d2e7e6a31a4dbf5e87462b2cc170c631fef95c68a4bce.jpg)
+
+![](images/c9569568e847e1af8da553e0970b35c75c674ce7213823f86ffa6571135116c4.jpg)
+
+![](images/105b1900a6f03412078b1a852a476f3e16402014bf6cbe3a2720d02e1ec5c8ec.jpg)
+
+![](images/4ee438bc6908d296c8d3917191f7e0f9e236704186bcd3ec329f46b5977e35e5.jpg)  
+Figure 4: Results for the Diabetes dataset. The common legend appears at the top. The top five plots compare the algorithms on our five metrics when varying  $\lambda \in \{0.1, 0.2, \dots, 0.9\}$  and fixing  $k = 15$  and the bottom five plots correspond to fixing  $\lambda = 0.5$  and varying  $k \in \{5, 6, \dots, 25\}$ .
+
+![](images/63fe6413c3987f0b84288b958e4f6f4d8761ca8b5374e3574440bf5116a9635c.jpg)
+
+![](images/b970764a4b1c16a98e547c49356d977fcc44b378bce13aeaeb3b8b678c5f5095.jpg)
+
+![](images/66ea5804485f9d2c40790baa838c50a4d5e72ef89d3450db5850299989184173.jpg)  
+GC
+
+![](images/c341246d3dea62d663e1b6a8d9d56975c480720949921689afb451ba50a20c77.jpg)  
+SemiBall
+
+![](images/c4da4bd20eea8dae23dd699b56b332e850b9b9c00e62593645d2429a4df0c5b6.jpg)  
+k-means++
+
+![](images/411857a4c680808dbf25c12700fe699bee77e802994bd290faa3b2e06d010087.jpg)  
+k-medoids
+
+![](images/5cce1b10f8c909200b6141fd5deeac63e892cc4b119865d6c184e0adde1df680.jpg)
+
+![](images/b120bc221ecb72cceb15afba769fd0ab14fae4da8b62e30d76aefdcd4f53421f.jpg)
+
+![](images/a4d4e64814aed7268429a59e2de24d57891c341a18eddc0c18f29564e5b7b9cc.jpg)
+
+![](images/c5a77541283dde714d94146724389626f865d57fa74f9aeb73c838a95ba5af73.jpg)
+
+![](images/c8c1ab2b6dd6fbb6d3117d1eb4b051fe172269e25d1d2d1c99cced571013fada.jpg)
+
+![](images/5cbcd59b42cf17c823f58c3bcce3791af1d273ffefef51de5edb7120d66372ef.jpg)
+
+![](images/340e2789ce987e7a4a1bd11e43b29b0f0b430838e1a4ed01680bed982c21d751.jpg)
+
+![](images/077d0139eeb37fa035ff4d41acb25b270722d76e72625bcc007e77a908cf4640.jpg)  
+Figure 5: Results for the Iris dataset. The common legend appears at the top. The top five plots compare the algorithms on our five metrics when varying  $\lambda \in \{0.1, 0.2, \dots, 0.9\}$  and fixing  $k = 15$  and the bottom five plots correspond to fixing  $\lambda = 0.5$  and varying  $k \in \{5, 6, \dots, 25\}$ .
+
+![](images/c9a91fdbbfdcbee7a3820c6fc0d72f555b4d0d24e2dafda5b33fe9593499442a.jpg)
+
+![](images/3af3d255852f07cd6a71794e8519cc4359808eb2c1afd7f1dce32297217cb2ff.jpg)
+
+Proof. Consider the instance from Figure 1. Note that  $k = 3$  divides  $n = 6$ . Set  $p = \frac{1}{\sqrt{\lambda}}$ . For contradiction, assume that there exists a balanced clustering  $\mathcal{X} = \{(C_1, x_1), (C_2, x_2), (C_3, x_3)\}$  with  $|C_1| = |C_2| = |C_3| = 2$ , which is in the  $\alpha$ -core for some  $\alpha < \frac{1}{\sqrt{\lambda}}$ .
+
+First, note that  $\{a,b\}$  must be one of the clusters. Suppose for contradiction that this is not the case. Then, at most one of agents  $a$  and  $b$  can be clustered with  $c$  (i.e., at most one of  $\mathcal{X}(a) = \mathcal{X}(c)$  and  $\mathcal{X}(b) = \mathcal{X}(c)$  is true). Without loss of generality, suppose that agent  $a$  is clustered with  $c$  (a symmetric argument holds when agent  $b$  is clustered with  $c$ ). Then,  $\ell_a(\mathcal{X}) \geqslant \lambda \cdot p = \sqrt{\lambda}$ ; this lower bound holds due to the non-centroid loss part alone. Since agent  $b$  is clustered with an agent in  $\{d,e,f\}$  that it is infinitely far from,  $\ell_b(\mathcal{X}) = \infty$ . Then, consider the deviating cluster  $(\{a,b\}, a)$ , which yields  $\ell_a(\{a,b\}, a) = \lambda$  and  $\ell_b(\{a,b\}, a) = 1$ . Compared to  $\mathcal{X}$ , agent  $b$  improves its loss by an infinite factor and agent  $a$  improves its loss by a factor of  $\frac{\sqrt{\lambda}}{\lambda} = \frac{1}{\sqrt{\lambda}}$ , which contradicts the fact that  $\mathcal{X}$  is in the  $\alpha$ -core for some  $\alpha < \frac{1}{\sqrt{\lambda}}$ .
+
+Therefore,  $\{a,b\}$  must be one of the clusters and at least one of agents  $a$  and  $b$  is not the centroid of this cluster. Without loss of generality, assume that the centroid is not at agent  $b$ . Then, consider the deviating cluster  $(\{b,c\}, b)$ .  $\ell_b(\{b,c\}, b) = \lambda \cdot p = \sqrt{\lambda}$  and  $\ell_c(\{b,c\}, b) = p$ . In contrast,  $\ell_b(\mathcal{X}) \geqslant 1$ , so agent  $b$  improves by a factor of at least  $\frac{1}{\sqrt{\lambda}}$ . Further, since  $\{a,b\}$  is a cluster, agent  $c$  must be clustered with an agent from  $\{d,e,f\}$  that it is infinitely far from, yielding  $\ell_c(\mathcal{X}) = \infty$ . This means agent  $c$  improves by an infinite factor. This contradicts the fact that  $\mathcal{X}$  is in the  $\alpha$ -core for some  $\alpha < \frac{1}{\sqrt{\lambda}}$ . This completes the proof.
+
+Since the broader class of dual metric loss contains the weighted single metric loss for all  $\lambda \in [0,1]$ , it follows that one cannot hope to design a balanced clustering algorithm that achieves any finite approximation to the core with respect to the dual metric loss.
+
+Theorem 11. Under the dual metric loss, no algorithm that always produces a balanced clustering can achieve  $\alpha$ -core for any  $\alpha \geqslant 1$ .
+
+Proof. Suppose for contradiction that there exists an algorithm that always produces a balanced clustering in the  $\alpha$ -core with respect to the dual metric loss. Then, consider the instance from Figure 1, and set the non-centroid metric as  $d^{m} = \lambda \cdot d$  and the centroid metric as  $d^{c} = (1 - \lambda) \cdot d$  for any  $\lambda < 1 / \alpha^{2}$ . The induced dual metric loss is  $\ell_{i}(C, x) = \lambda \cdot \max_{j \in C} d(i, j) + (1 - \lambda) \cdot d(i, x)$ , which is precisely the weighted single metric loss induced by distance metric  $d$  and parameter  $\lambda$ . From Theorem 10, the core approximation achieved by any balanced clustering, including the one produced by the algorithm, must be at least  $\frac{1}{\sqrt{\lambda}} > \alpha$ , which is the desired contradiction.
+
+# NeurIPS Paper Checklist
+
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+
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+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
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+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
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+
+# Guidelines:
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+
+# 4. Experimental result reproducibility
+
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+
+# Answer: [Yes]
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+
+# Guidelines:
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+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
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+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
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+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
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+Answer: [Yes]
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+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
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+Justification: Yes, our experiments do not require a large amount of compute and can be executed on a personal computer. In the section where we outline our experiments, we mention the steps that were computational bottlenecks, but even these were not a large undertaking to run on a laptop.
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+Guidelines:
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+# 10. Broader impacts
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+# 11. Safeguards
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+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
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+# 12. Licenses for existing assets
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+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
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+# 13. New assets
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+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
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+Answer: [NA]
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+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: Our paper does not involve crowdsourcing or research with human subjects.
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+Guidelines:
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+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: Our paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
+
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: LLMs were only used for editing and formatting purposes.
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+Guidelines:
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+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
\ No newline at end of file
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+# Unifying Re-Identification, Attribute Inference, and Data Reconstruction Risks in Differential Privacy
+
+Bogdan Kulynych  
+Lausanne University Hospital
+
+Juan Felipe Gomez  
+Harvard University
+
+Georgios Kaissis  
+Google DeepMind
+
+Jamie Hayes Google DeepMind
+
+Borja Balle Google DeepMind
+
+Flavio P. Calmon  
+Harvard University
+
+# n Louis Raisaro
+
+Lausanne University Hospital  
+University of Lausanne
+
+# Abstract
+
+Differentially private (DP) mechanisms are difficult to interpret and calibrate because existing methods for mapping standard privacy parameters to concrete privacy risks—re-identification, attribute inference, and data reconstruction—are both overly pessimistic and inconsistent. In this work, we use the hypothesis-testing interpretation of DP ( $f$ -DP), and determine that bounds on attack success can take the same unified form across re-identification, attribute inference, and data reconstruction risks. Our unified bounds are (1) consistent across a multitude of attack settings, and (2) tunable, enabling practitioners to evaluate risk with respect to arbitrary, including worst-case, levels of baseline risk. Empirically, our results are tighter than prior methods using  $\varepsilon$ -DP, Rényi DP, and concentrated DP. As a result, calibrating noise using our bounds can reduce the required noise by  $20\%$  at the same risk level, which yields, e.g., an accuracy increase from  $52\%$  to  $70\%$  in a text classification task. Overall, this unifying perspective provides a principled framework for interpreting and calibrating the degree of protection in DP against specific levels of re-identification, attribute inference, or data reconstruction risk.
+
+# 1 Introduction
+
+Releases of models and statistics derived from personal data—such as de-identified and synthetic data releases, outcomes of statistical analyses, and machine-learning models—can reveal information about individuals in the data. These releases can be classified into record-level (releasing de-identified or obfuscated records) and aggregate (releasing statistical properties and models) (Gadotti et al., 2024). The disclosure of such information incurs different types of privacy risks, which are legislated differently across jurisdictions and regulatory frameworks.
+
+In record-level releases, information leakage can occur via singling out of an individual (Cohen and Nissim, 2020), e.g., determining that there is only one Colombian woman fluent in Mandarin over the age of 60 in the dataset. When combined with external information, singling out can result in a person's re-identification. Re-identification is harmful when it enables inference of previously unknown personal attributes by an attacker, e.g., disease status or voting record (Sweeney, 2002). Although in aggregate releases there is no one-to-one mapping between the released information and individuals, sensitive information can still be inferred via attribute inference or data reconstruction attacks (Fredrikson et al., 2015; Yeom et al., 2018; Balle et al., 2022), in which the adversary could,
+
+![](images/a5d513a6f69c96d0e5e2b61fed0c96fae7466896c45b00d2d7fa623e69871273.jpg)
+
+![](images/da18e3e97382252791ea662888a849a2ae444d9d4c9774eb0f300b55d2258fce.jpg)
+
+![](images/304049f44022ba9480ca68209645b614b55c53c67bf01b1f443ac0d758f0fc6d.jpg)
+
+![](images/80d7511705000f0664dc6289344e1db01fee70358f4310f389fb30e8082b0214.jpg)  
+Figure 1: Our results offer a unified and more precise way to interpret and calibrate DP mechanisms in terms of re-identification, attribute inference, and data reconstruction risks. Top: The success of all these attacks cannot be higher than the power of the worst-case membership inference attack, which we immediately obtain from the  $f$  function in the decision-theoretic characterization of privacy mechanisms— $f$ -DP. Bottom left: Our results enable to add  $\approx 20\%$  less noise at any given level of risk compared to using prior methods (see Section 4 for details). Bottom middle: Less noise translates into  $\approx 18$  pp improved task accuracy (shown: DP-SGD for sentiment classification with GPT-2). Bottom right: The unifying risk measure is tunable—we can either estimate post-release risk for any given level of baseline risk, or measure the worst-case risk (shown: US 2020 Census state-level data).
+
+![](images/2f581b277620187dd46c18f4e6d0dd3591f355cc7ba8bcb1e3fec8f95cce7e8d.jpg)
+
+![](images/199133ea1614393a04061121a962191a992169cab280c144cc9524931470c34e.jpg)
+
+e.g., inspect the confidence outputs of a predictive model to infer one or more unknown attributes of an individual. These notions of risk are interpretable and appear in data-protection guidelines such as ISO standards (ISO/IEC 27559:2022), or guidelines from the European Medicines Agency (European Medicines Agency, 2018).
+
+Differential privacy (DP) (Dwork et al., 2014) is a standard tool for controlling information leakage in record-level and aggregate data disclosures. The goal of DP mechanisms is to mask the contribution of each individual in the release via noise injection and randomization. One of the core challenges in applying DP in practice is its interpretation (Franzen et al., 2022; Nanayakkara et al., 2023), as DP parameters do not trivially bound concrete risks such as re-identification, attribute inference, and data reconstruction. In the standard practice, information leakage in DP is quantified by two parameters:  $\varepsilon$ , with values closer to zero meaning better privacy, and  $\delta$ , which should be kept cryptographically small (Vadhan, 2017).
+
+There are few established guidelines for selecting DP parameters  $(\varepsilon, \delta)$ , and often contradictory standards are adopted in practice. For example, heuristics such as  $\varepsilon < 10$  are common in deep learning (Ponomareva et al., 2023). When translated to concrete privacy risks, such choices can result in vacuous privacy guarantees: in theory,  $\varepsilon \approx 10$  can yield a more than  $99\%$  worst-case attribute inference accuracy. This poor performance is not due to inherent limitations of DP itself—instead, it is a reflection of the imprecise nature of existing methods that convert between DP guarantees and notions of privacy risk. In fact, empirical evidence shows that even  $\varepsilon > 10$  still prevents practical reconstruction attacks (Ziller et al., 2024; Hayes et al., 2024).
+
+In this paper, we use the decision-theoretic characterization of DP known as  $f$ -DP (Dong et al., 2022) to unify and improve existing conversion bounds between DP and different notions of privacy risk. We work under the strong-adversary threat model, in which the adversary has access to the entire dataset
+
+except for one element, and has a certain prior distribution over the remaining element. Inspired by Cohen and Nissim (2020), we characterize the attack success probability of the adversary after observing the output of a DP mechanism as a function of the attack's baseline probability without the DP output. Surprisingly, we find that under the strong threat model, we can simultaneously bound the success probability of re-identification, attribute inference, and reconstruction attacks at once, using the same expression. Importantly, we find that not only this is a convenient way to analyze multiple kinds of risk at once, but also it provides a significantly tighter bound than prior methods, due to our usage of  $f$ -DP. See Fig. 1 for an illustration of these properties, with more details in Section 4.
+
+As a concrete example, consider the release of all state-level US 2020 Census data, which satisfies DP with  $\varepsilon = 10.6$  at  $\delta = 10^{-10}$ . A standard risk analysis (Salem et al., 2023) indicates that the maximum difference between success and baseline probabilities of an attribute inference attack is  $>99$  percentage points (pp), a more involved numeric analysis (Balle et al., 2022) based on the notion of Rényi DP (Mironov, 2017) yields 73pp, whereas our worst-case bound yields 52pp. Moreover, if we assume an adversary with a specific goal and background knowledge, e.g., inferring a disease status with a 1 in 10,000 prevalence in the population, our bound yields  $< 0.001\mathrm{pp}$ . This corroborates and generalizes prior observations (Ziller et al., 2024; Hayes et al., 2024) that even  $\varepsilon \approx 10$  attains meaningful privacy guarantees in certain regimes, to arbitrary  $f$ -DP mechanisms.
+
+To summarize, our contributions are:
+
+- A unifying framework for analyzing re-identification, attribute inference, and data reconstruction risks in the strong threat model of DP. This includes a new formalization of re-identification risk as a variant of the notion of predicate singling out (Cohen and Nissim, 2020), adapted to the standard threat model of DP.  
+- Demonstration that our bounds offer tighter estimates of risk compared to prior approaches (Cohen and Nissim, 2020; Balle et al., 2022).  
+- Case studies demonstrating that our results provide a more precise and comprehensive interpretation of privacy risk for the US 2020 Census than prior methods, and enable noise reduction in DP mechanisms by approximately  $20\%$ , yielding utility improvements in deep learning with DP-SGD of up to 18pp in test accuracy when calibrating noise to a given level of privacy risk (Kulynych et al., 2024).  
+- Novel bounds on generalization and memorization under  $f$ -DP as corollaries of our results.  
+- We release the code as part of the Python package:
+
+https://github.com/Felipe-Gomez/riskcal
+
+Overall, our unifying perspective provides a principled and easy-to-use framework for interpreting and calibrating the degree of privacy protection in DP mechanisms against specific levels of privacy risks that are relevant in practice.
+
+# 2 Background
+
+This section sets up the notation and provides a concise overview of relevant background. We defer a more extensive overview of relevant background to Appendix A.
+
+Suppose that we have a sensitive dataset, e.g., patient healthcare records,  $S \in 2^{\mathbb{D}}$ , where  $\mathbb{D}$  is the data record space. We want to run a randomized algorithm  $M(S)$ , e.g., a statistical analysis, or train a machine learning model, and release its output. We also call  $M(\cdot)$  a mechanism. We denote the space of its outputs as  $\Theta$ , and a specific output, e.g., a model, as  $\theta \in \Theta$ . We say that two datasets  $S, S'$  are neighbouring if they belong to a neighborhood relation denoted as  $S \simeq S'$ . To capture a meaningful notion of training data privacy, this relation must correspond to adding, removing, or replacing all of the data that contains information of a single privacy unit, e.g., individual or secret. In the rest of the paper, we follow the standard practice and assume that the privacy unit corresponds to a single record. We consider two standard neighbourhood relations: the add-remove relation in which either  $S = S' \cup \{z\}$  or  $S' = S \cup \{z\}$  for some  $z \in \mathbb{D}$ , and the replace-one relation in which both  $S$  and  $S'$  have the same size, but differ by one record.
+
+Differential privacy. An algorithm  $M: 2^{\mathbb{D}} \to \Theta$  satisfies  $(\varepsilon, \delta)$ -DP if for any measurable  $E \subseteq \Theta$  and  $S \simeq S'$ , we have  $\operatorname*{Pr}[M(S) \in E] \leq e^{\varepsilon} \operatorname*{Pr}[M(S') \in E] + \delta$  (Dwork et al., 2014). We also make use of the notion of  $\eta$ -total-variation (TV) privacy, which is equivalent to satisfying  $(0, \eta)$ -DP.
+
+Strong-adversary membership inference. DP can be completely characterised via a constraint on the success rate of strong membership inference attacks (SMIA), which aim to determine whether a given record was part of the input dataset  $S$  based on the output of  $M(S)$ . Formally, given a sensitive record  $z \in \mathbb{D}$ , and a partial dataset  $\bar{S} \in \mathbb{D}^{n-1}$ , let  $P = M(\bar{S})$  and  $Q = M(\bar{S} \cup \{z\})$ . SMIA can be seen as a hypothesis test:
+
+$$
+H _ {0}: \theta \sim P, \text {a n d} H _ {1}: \theta \sim Q. \tag {1}
+$$
+
+For a given decision  $\phi : \Theta \to \{0,1\}$  to reject the null hypothesis, we can quantify the adversary's success by characterizing its error rates (Wasserman and Zhou, 2010; Kairouz et al., 2015; Dong et al., 2022):  $\alpha_{\phi} \triangleq \mathbb{E}_P[\phi], \beta_{\phi} \triangleq 1 - \mathbb{E}_Q[\phi]$ , where  $\alpha_{\phi}$  and  $\beta_{\phi}$  are the false positive rate (FPR) and false negative rate (FNR) respectively. For any desired FPR level  $\alpha \in [0,1]$ , the Neyman-Pearson lemma guarantees that there exists an optimal test  $\phi^*$  which achieves the lowest possible FNR  $\beta$ . We can thus characterize the SMIA by a trade-off curve, a function which shows the lowest FNR achieved by the most powerful test for any level of FPR  $\alpha$ :  $T(P,Q)(\alpha) \triangleq \inf_{\phi: \Theta \to [0,1]} \{\beta_{\phi} \mid \alpha_{\phi} \leq \alpha\}$ .
+
+This trade-off curve forms the basis of a more general version of DP called  $f$ -DP: a mechanism  $M(\cdot)$  satisfies  $f$ -DP if for any  $S \simeq S'$  and  $\alpha \in [0,1]$  we have that  $T(M(S), M(S'))(\alpha) \geq f(\alpha)$ . This formulation is more general than DP: a mechanism  $M(\cdot)$  is  $(\varepsilon, \delta)$ -DP iff it satisfies  $f$ -DP with:
+
+$$
+f (\alpha) = \max  \{0, 1 - \delta - e ^ {\varepsilon} \alpha , e ^ {- \varepsilon} (1 - \delta - \alpha) \}. \tag {2}
+$$
+
+Notably,  $f$ -DP is closed under post-processing: if  $M(\cdot)$  satisfies  $f$ -DP, then so does  $g \circ M$  for any deterministic or randomized mapping  $g(\cdot)$ . We make use of a lesser known representation of  $f$ -DP:
+
+Lemma 2.1. An algorithm  $M: 2^{\mathbb{D}} \to \Theta$  satisfies  $f$ -DP iff for any measurable  $E \subseteq \Theta$  and  $S \simeq S'$ :
+
+$$
+\Pr [ M (S) \in E ] \leq 1 - f (\Pr [ M (S ^ {\prime}) \in E ]). \tag {3}
+$$
+
+This form also appears in Kifer et al. (2022), and we provide a self-contained proof in Appendix C. Another property of  $f$ -DP that is relevant to our work is that it implies  $\eta$ -TV privacy with  $\eta = \max_{\alpha \in [0,1]} (1 - f(\alpha) - \alpha)$  (Kaisis et al., 2024). As a result, even though TV privacy on its own is a weak notion of privacy (Vadhan, 2017), any  $f$ -DP algorithm implies TV privacy for some  $\eta \geq 0$ .
+
+# 3 Bounding Operational Privacy Risks
+
+Next, we introduce formalizations of standard notions of risk in privacy-preserving statistics and learning: singling out, attribute inference, and data reconstruction. Our main results enable the analysis of these risks under a unifying framework given by  $f$ -DP.
+
+# 3.1 Threat Model
+
+We focus on the risk within the strong threat model (see, e.g., Balle et al., 2022) in which the adversary has access to the workings of the privacy-preserving algorithm  $M(\cdot)$ , has access to the partial dataset  $\bar{S}$  except for one target record  $z$ , and has side information about the remaining record  $z$  in the form of a prior distribution  $z \sim P$ . Note that this threat model is slightly different from the standard threat model of SMIA (see Section 2) in that we do not assume that the adversary knows the target record exactly. To quantify the adversary's success rate, we consider expectations over the prior distribution  $P$  as well as over the randomness of the algorithm, as we detail next.
+
+Although this might seem like an overly strong model, providing security in this setting is desirable because it represents the worst-case scenario for attacks against privacy of individuals. In particular, as the adversary knows the partial dataset, it captures even the case when the adversary can poison the data (Leemann et al., 2024). By bounding the success of the strong adversary, we also bound the success of attacks against the considered privacy unit under other threat models that are weaker or more realistic for a specific application.
+
+In addition, we discuss existing risk notions which assume that the adversary does not have access to the partial dataset, but knows the data distribution  $S \sim P^n$ . The success is then evaluated over  $S \sim P^n$ . We call this the average-dataset threat model. We summarize the difference between these threat models in Table 1 in Appendix H.
+
+# 3.2 Notions of Risk
+
+Singling-out risk. We introduce formalizations of singling-out risk based on the notion of predicate singling out (PSO) (Cohen and Nissim, 2020). For the purpose of our work, we use singling-out risk and re-identification risk interchangeably for brevity, although singling out is only a necessary condition for re-identification, and other approaches to defining re-identification (e.g., also capturing inference) exist (Article 29 Data Protection Working Party, 2014).
+
+Definition 3.1 (PSO security). For a given  $n > 1$ , mechanism  $M: \mathbb{D}^n \to \Theta$ , data distribution  $P$  over  $\mathbb{D}$ , weight  $w \in [0,1/n]$ , and adversary  $\mathcal{A}_{n,M,P,w}: \Theta \to \mathbb{Q}_{P,w}$  which outputs a predicate that aims to single out one record in the training dataset from the set of admissible predicates  $\mathbb{Q}_{P,w} \triangleq \{p \mid p: \mathbb{D} \to \{0,1\}, \mathbb{E}_P[p] \leq w\}$ , we define the PSO success rate as follows:
+
+$$
+\operatorname {s u c c} _ {P S O} (n, M, P, w; \mathcal {A}) \triangleq \Pr_ {\substack {S \sim P ^ {n} \\ p \leftarrow \mathcal {A} _ {n, M, P, w} (M (S))}} \left[ \sum_ {z \in S} p (z) = 1 \right], \tag{4}
+$$
+
+and the baseline success as:
+
+$$
+\operatorname {b a s e} _ {P S O} (n, P, w) \triangleq \sup  _ {p \in \mathbb {Q} _ {P, w}} \Pr_ {S \sim P ^ {n}} \left[ \sum_ {z \in S} p (z) = 1 \right] = \sup  _ {p \in \mathbb {Q} _ {P, w}} n \mathbb {E} _ {P} [ p ] \cdot (1 - \mathbb {E} _ {P} [ p ]) ^ {n - 1}. \tag {5}
+$$
+
+Observe that PSO security does not assume that the adversary has access to the dataset beyond its distribution  $S \sim P^n$ , thus assumes the average-dataset threat model. As we are interested in the strong-adversary model, we introduce a new notion of PSO security where we assume that the adversary has knowledge of the entire dataset except for one element:
+
+Definition 3.2 (SPSO security). For a given  $n > 1$ ,  $k \geq 2$ , mechanism  $M: 2^{\mathbb{D}} \to \Theta$ , partial dataset  $\bar{S} \in \mathbb{D}^{n-1}$ , data distribution  $P$  over a candidate set  $\mathbb{W} \subseteq \mathbb{D}$  with  $|W| = k$ , weight  $w \in [0,1]$ , and adversary  $\mathcal{A}_{M,\bar{S},P,w}: \Theta \to \mathbb{Q}_{\bar{S},P,w}$ , which outputs a predicate that aims to single out the unknown record in the training dataset from the set of predicates  $\mathbb{Q}_{\bar{S},P,w} \subseteq \{p \mid p: \mathbb{D} \to \{0,1\}, \sum_{z' \in \bar{S}} p(z') = 0, \mathbb{E}_P[p] \leq w\}$ , we define the strong PSO (SPSO) success rate as follows:
+
+$$
+\operatorname {s u c c} _ {S P S O} (M, \bar {S}, P, w; \mathcal {A}) \triangleq \underset { \begin{array}{c} z \sim P \\ p \leftarrow \mathcal {A} _ {M, \bar {S}, P, w} (M (\bar {S} \cup \{z \})) \end{array} } {\Pr} [ p (z) = 1 ], \tag {6}
+$$
+
+and the baseline success as:
+
+$$
+\operatorname {b a s e} _ {S P S O} (\bar {S}, P, w) \triangleq \sup  _ {p \in \mathbb {Q} _ {\bar {S}, P, w}} \Pr_ {z \sim P} [ p (z) = 1 ] = \sup  _ {p \in \mathbb {Q} _ {\bar {S}, P, w}} \mathbb {E} _ {P} [ p ] \leq w. \tag {7}
+$$
+
+Unlike PSO, in which the adversary aims to single out any record in an unknown dataset, in SPSO, the adversary aims to find a predicate that matches only one record from a prior set of candidates. We provide additional background on PSO in Appendix A.2.
+
+Attribute inference and reconstruction attacks. Following Balle et al. (2022), we formalize attribute inference and data reconstruction attacks using the notion of reconstruction robustness:
+
+Definition 3.3 (SRR security). For a given  $n \geq 1$ , mechanism  $M: 2^{\mathbb{D}} \to \Theta$ , data distribution  $P$  over  $\mathbb{D}$ , partial dataset  $\tilde{S} \in \mathbb{D}^{n-1}$ , loss function  $\ell: \mathbb{D} \times \mathbb{D} \to \mathbb{R}$ , threshold  $\gamma \in \mathbb{R}$ , and reconstruction attack  $\mathcal{A}_{M,\tilde{S},\mathcal{D}}: \Theta \to \mathbb{D}$ , we define the strong reconstruction robustness (SRR) success rate as follows:
+
+$$
+\operatorname {s u c c} _ {S R R} (M, \bar {S}, P; \mathcal {A}, \ell , \gamma) \triangleq \underset {\substack {z \sim P \\ \hat {z} \leftarrow \mathcal {A} _ {M, \bar {S}, P} (M (\bar {S} \cup \{z}))}} {\Pr} [ \ell (z, \hat {z}) \leq \gamma ], \tag{8}
+$$
+
+and the baseline success as  $\mathrm{base}_{SRR}(P;\ell ,\gamma)\triangleq \sup_{\hat{z}\in \mathbb{D}}\operatorname *{Pr}_{z\sim P}[\ell (z,\hat{z})\leq \gamma ]$
+
+Robustness against attribute inference attacks can be seen as a special case of SRR:
+
+Definition 3.4 (SAI security). For a given  $n \geq 1$ , mechanism  $M: 2^{\mathbb{D}} \to \Theta$ , set of attributes  $\mathbb{A} = \{1, \dots, k\}$ , mapping from records to attributes  $a: \mathbb{D} \to \mathbb{A}$ , data distribution  $P$  over  $\mathbb{D}$ , partial dataset  $\bar{S} \in \mathbb{D}^{n-1}$ , and attribute inference attack  $\mathcal{A}_{M,\bar{S},P}: \Theta \to \mathbb{A}$ , the strong attribute inference (SAI) success rate is defined as:
+
+$$
+\operatorname {s u c c} _ {S A I} (M, \bar {S}, P; \mathcal {A}, a) \triangleq \underset {\substack {z \sim P \\ \hat {a} \leftarrow \mathcal {A} _ {M, \bar {S}, P} (M (\bar {S} \cup \{z \}))}} {\Pr} [ a (z) = \hat {a} ], \tag{9}
+$$
+
+and the baseline success as  $\mathrm{base}_{SAI}(P, a) \triangleq \sup_{\hat{a} \in \mathbb{A}} \operatorname{Pr}_{z \sim P}\left[a(z) = \hat{a}\right]$ .
+
+We review the prior methods to bound these notions of risk under DP in Appendix A.3.
+
+# 3.3 Main Results: Unifying Bounds
+
+Our core contribution is a unifying relationship between the hypothesis-testing interpretation of DP via  $f$ -DP and the operational notions of privacy risk in the strong adversary model defined in Section 3.2. We achieve this via a general result that connects the expectation of an attack-specific query function  $q(z, \theta)$  over the randomness of  $\theta \sim M(S \cup \{z\})$  and  $z \sim P$  to the maximum "baseline" expectation of the same function over only  $z \sim P$ :
+
+Lemma 3.1. Suppose that  $M:2^{\mathbb{D}}\to \Theta$  satisfies  $f - DP$  w.r.t. either add-remove or replace-one relation. Then, for any bounded function  $q:\mathbb{D}\times \Theta \rightarrow [0,1]$ , any partial dataset  $\bar{S}\in \mathbb{D}^{n - 1}$  with  $n\geq 1$ , and any probability distribution  $P$  over  $\mathbb{D}$ , we have:
+
+$$
+\mathbb {E} _ {z \sim P} \mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z \})} [ q (z; \theta) ] \leq 1 - f \left(\sup  _ {\theta \in \Theta} \mathbb {E} _ {z \sim P} [ q (z; \theta) ]\right). \tag {10}
+$$
+
+Intuitively, the function  $q(z; \theta)$  can be seen as indicating the degree of success of some attack against the privacy of the record  $z$  given the knowledge of a model  $\theta$ , on a scale from 0 to 1. We provide a proof of this statement, as well as of all the following results, in Appendix C. Technically, the core steps of the proof involve the usage of the representation of  $f$ -DP in Lemma 2.1, and an application of Jensen's inequality to push the expectation over  $z \sim P$  inside of  $f(\alpha)$ .
+
+We recover the notions of risk in Section 3.2 by taking relevant instantiations of  $q(z;\theta)$ , e.g., the indicator of a successful reconstruction event,  $q(z;\theta) = \mathbb{1}[\ell (z,\mathcal{A}(\theta))\leq \gamma ]$ , in the case of reconstruction attacks. As a result, we get a unifying bound on attack success:
+
+Theorem 3.1 (Informal). Suppose that  $M(\cdot)$  satisfies  $f$ -DP w.r.t. either add-remove or replace-one relation. For SPSO, SRR, and SAI, it holds that:
+
+$$
+\operatorname {s u c c} _ {[ S P S O, S R R, S A I ]} \leq 1 - f (\text {b a s e} _ {[ S P S O, S R R, S A I ]}). \tag {11}
+$$
+
+We provide a precise statement including the missing arguments for each risk notion in Appendix C.
+
+Normalized success. Multiple prior works have observed that privacy analyses based on attack success alone can be misleading (Guerra-Balboa et al., 2023; Salem et al., 2023; Cohen et al., 2025). Thus, for all notions of risk, we consider an additional representation of risk in terms of success normalized by the baseline via additive advantage:  $\text{adv} \triangleq \text{succ} - \text{base}$ , where we drop the arguments for brevity. As a consequence of Theorem 3.1, we obtain:
+
+Theorem 3.2 (Informal). Suppose that  $M(\cdot)$  satisfies  $f$ -DP w.r.t. either add-remove or replace-one relation. For SPSO, SRR, and SAI, it holds that:
+
+$$
+\operatorname {a d v} _ {[ S P S O, S R R, S A I ]} \leq 1 - f \left(\operatorname {b a s e} _ {[ S P S O, S R R, S A I ]}\right) - \operatorname {b a s e} _ {[ S P S O, S R R, S A I ]}. \tag {12}
+$$
+
+Theorem 3.2 trivially extends to other ways to normalize attack success, such as  $\frac{\mathrm{succ} - \mathrm{base}}{1 - \mathrm{base}}$  (see, e.g., Guo et al., 2023) or  $\frac{1 - \mathrm{succ}}{1 - \mathrm{base}}$  (see, e.g., Chatzikokolakis et al., 2023). In the rest of the paper, we use additive advantage to express risk.
+
+Baseline-independent guarantees. We can also obtain a bound on advantage that is independent of the baseline by just taking the maximum over all baselines. For this, we use the notion of  $\eta$ -TV privacy, which any  $f$ -DP algorithm satisfies for some  $\eta$  (see Section 2).
+
+Theorem 3.3 (Informal). Suppose that the algorithm  $M(\cdot)$  satisfies  $\eta$ -TV privacy w.r.t. either the add-remove or replace-one relation. Then, the advantage of SPSO, SRR, and SAI is bounded:
+
+$$
+\operatorname {a d v} _ {[ S P S O, S R R, S A I ]} \leq \eta . \tag {13}
+$$
+
+This result is useful as (1) it enables to measure and communicate the worst-case risk across all baselines using a single number, and (2) we can use standard software tools for analyzing privacy in common algorithms such as DP-SGD (Abadi et al., 2016) by evaluating the DP guarantee for  $(\varepsilon = 0, \delta = \eta)$ -DP without the need to instantiate the full  $f$  curve. We detail on these properties next.
+
+# 3.4 Discussion of the Theoretical Results
+
+Computing the bounds in practice. Effective usage of our baseline-specific bounds in Theorems 3.1 and 3.2 requires an analysis of a privacy-preserving algorithm in terms of  $f$ -DP. Although different variants of DP imply  $f$ -DP, e.g., via Eq. (2) for a single pair of  $(\varepsilon, \delta)$  guarantees, such conversions are loose (Kulynych et al., 2024). For simple mechanisms such as Gaussian or Laplace (Dwork et al., 2014), exact trade-off curves are known (Dong et al., 2022). For more complex algorithms such as DP-SGD (Abadi et al., 2016), there are two ways for obtaining the corresponding curve.
+
+First, we can estimate it using Eq. (2) from the privacy profile of the algorithm (Balle et al., 2018), i.e., the set of all attainable  $(\varepsilon, \delta)$ -DP pairs. This method has been used for privacy auditing previously (Nasr et al., 2023). Second, it is possible to analyze DP-SGD or other algorithms that are compositions of (subsampled) Gaussian and Laplace mechanisms using a direct method (Kulynych et al., 2024). Both approaches provide tight analyses when using state-of-the-art accounting tools as a backbone, such as the Connect-the-Dots accountant (Doroshenko et al., 2022). In particular, prior work (Nasr et al., 2023) has shown that the upper bounds on privacy leakage obtained with the modern accounting techniques can be nearly reached by empirical attacks in the SMIA threat model.
+
+Tunability. Our bound on advantage in Theorem 3.2 enables practitioners to tune the baseline risk. In certain data release scenarios, it might be useful to simulate relevant re-identification or attribute inference attacks, estimate their plausible baseline risk, and consider the risk of the release with respect to such a baseline, i.e.,  $\text{adv} \leq 1 - f(\text{base}) - \text{base}$ . As an example, consider releasing outputs of a mechanism trained on tabular medical data which contains a column corresponding to a patient's HIV status. In this case, we might want to evaluate the risk of an adversary inferring the HIV status under standard threat models considered when releasing medical data (El Emam, 2010) such as "marker" or "journalist". For instance, the baseline risk of a "marker" adversary—who does not target any specific individual—could be modeled as guessing based on the prevalence of HIV in the general population. The baseline risk of a "journalist" adversary—who is assumed to be able to obtain side information on the target from public sources—could be modeled as guessing based on a target patient's demographics. We leave the derivation of application-specific procedures and guidelines for modeling baseline risk to future work.
+
+In high-risk scenarios, e.g., public model releases, it might be more desirable to ensure security against the worst-case attacks. In this case, it is more appropriate to use the baseline-independent bound of  $\mathsf{adv} \leq \max_{\mathrm{base} \in [0,1]} (1 - f(\mathrm{base}) - \mathrm{base}) \leq \eta$  from Theorem 3.3.
+
+On tightness of the bounds. Our bounds are never vacuous, and are saturated for perfectly private and blatantly non-private mechanisms. However, they can be significantly tightened under additional assumptions on the adversary's threat model. In particular, we show an example of such strengthening in Appendix D for an important case of binary attribute inference with a non-uniform prior, and empirically show that it provides tighter bounds in this setting than prior work (Guo et al., 2023). Although there exist cases under which the unified bound can be tightened, we conjecture that it is tight in the most general case, i.e., there exist non-trivial settings under which Theorem 3.1 holds with equality. We leave the verification of this conjecture, and the identification of settings under which equality is achieved in case it is true, to future work.
+
+Beyond privacy. Our results extend to statistical learning theory. Concretely, in Appendix E, we show that our derivations can be extended to a new kind of generalization bound. Informally, we show that for the on-average train set error  $\mathrm{err}_{\mathrm{tr}}$  and on-average generalization error  $\mathrm{err}_{\mathrm{test}}$  (Shalev-Shwartz et al., 2010) of a learning algorithm satisfying  $f$ -DP, it holds that  $\mathrm{err}_{\mathrm{test}} \leq 1 - f(\mathrm{err}_{\mathrm{tr}})$ . Moreover, in Appendix F, we show that our derivations imply a bound on memorization (Feldman, 2019; Zhang et al., 2023). We believe these results are of independent interest.
+
+# 4 Experimental Evaluation
+
+# 4.1 Bounds Comparison
+
+Singling-out risk. We compare prior bounds on singling-out risk in the average-dataset threat model from Cohen and Nissim (2020) (detailed in Appendix B) to our bound on SPSO in Theorem 3.2, which operates under a different threat model. To provide an apples-to-apples comparison, we fix
+
+![](images/eee46509b88ecb5e34d3e9e544afed48e34dae46d7db011711768c4319f72985.jpg)  
+Figure 2: Our bound on predicate singling out in the strong threat model (SPSO) is always non-vacuous, and, surprisingly, shows significantly lower risk than bounds in the PSO threat model. The risk is  $\text{adv} = \text{succ} - \text{base}$  with fixed given base for Gaussian mechanism with  $\varepsilon$  calculated for  $\delta = 10^{-5}$ .
+
+![](images/38908b2a0d22960cc5416d72d9bb9e8a4df35f8e6964d09f1762be414ff16faf.jpg)
+
+![](images/f9fc18f6a0c4b6513dc05fea4d04b6c5a299c870e3e9af397ea95f1412015b91.jpg)
+
+![](images/a06287fa3ff1e74b4b5cadffdc671febc5a059f172abf0663cef3775886b1d41.jpg)  
+Figure 3: Our bound on reconstruction robustness shows lower risk than prior bounds. We show risk as  $\text{adv} = \text{succ} - \text{base}$  for three different baseline values for Gaussian mechanism with  $\varepsilon$  calculated for  $\delta = 10^{-5}$ .
+
+![](images/85b722cc59ff8a9cf594a7949b02c371e18c8b41eab437e001e298d4f914afcb.jpg)
+
+![](images/ea13a75bf75e02233ca8a96cfc305ba4effc659cbb6eb23ec4cd8eaa86432a0f.jpg)
+
+![](images/5c9dda9c36d3c180b00495058a4b0d7151b54bef581ae785ac518218fafe1025.jpg)
+
+the weight  $w$  for both threat models. The baseline probability for PSO is base  $= n \cdot w \cdot (1 - w)^{n-1}$ , whereas it is base  $= w$  for SPSO. We use standard Laplace and Gaussian mechanisms (Dwork et al., 2014) with varying noise levels, and simulate different dataset sizes  $n \in \{500, 1000, 5000\}$  with a fixed weight  $w = 1/5000$ . We use  $\delta = 10^{-5}$  to derive  $\varepsilon$  for the Gaussian mechanism, and analyze DP under replace-one relation.
+
+We show the results for the Gaussian mechanism in Fig. 2 (the results for Laplace mechanism are similar, shown in Appendix H). Unlike the PSO bounds, our bounds provide meaningful guarantees even in the  $\varepsilon \in [10,20]$  regime, and saturate only around  $\varepsilon \approx 35$ . Note that our SPSO bound based on  $f$ -DP and the strong adversary threat model shows lower risk than the PSO bound in the average-dataset threat model based on  $(\varepsilon ,\delta)$ -DP. In Appendix F, we derive a novel PSO guarantee based on  $f$ -DP, but we observe no meaningful difference from the  $(\varepsilon ,\delta)$ -DP bound in Fig. 2 (see Appendix H). We hypothesize that both phenomena are due to the looseness in the derivations of the average-dataset PSO results that are not present in the SPSO derivation. We leave the derivation of tighter PSO bounds as future work.
+
+Reconstruction robustness. Next, we compare prior bounds on SRR from Balle et al. (2022) that are based on Rényi DP (RDP) (Mironov, 2017) and zero-concentrated DP (zCDP) (Dwork and Rothblum, 2016; Bun and Steinke, 2016), detailed in Appendix A.3, to our unifying bounds in Theorem 3.2. We use the Gaussian mechanism, analyzed under the add-one relation. We vary the noise parameter  $\sigma$ , and compute  $\varepsilon$  at  $\delta = 10^{-5}$ . For the SRR bounds, we evaluate separately the bound for a single RDP guarantee  $(t,\varepsilon)$  with  $t = 2$ , and the bound for the tighter zCDP guarantee. Fig. 3 demonstrates that our bound always shows lower risk than prior approaches.
+
+# 4.2 Case Studies
+
+We present two case studies which showcase how our results can be used to interpret and calibrate realistic DP mechanisms. Additionally, in Appendix G, we discuss another application of our results to a DP statistical-query answering system (Gaboardi et al., 2020).
+
+Calibrating noise to reconstruction risk in deep learning. It is possible to partially reconstruct training-data records from observing outputs or weights of classification models (Balle et al., 2022) and language models (Carlini et al., 2022). In this case study, we assume the modeler fine-tunes a
+
+![](images/a3005f8936507b27a8efbff76642da72ea211783d57055cbdd9ddadd1c9da7eb.jpg)  
+Figure 4: Our method shows up to  $33\%$  lower worst-case reconstruction risk in the US 2020 Census release than the prior method. x axis shows granularity levels of the release, y axis shows risk of attacks as succ - base for the worst-case baseline.
+
+language model for text sentiment classification on the SST-2 dataset (Socher et al., 2013). In order to limit the privacy risk to a given threshold, e.g., ensure at most 0.15 increase in the probability of successful reconstruction under the worst-case baseline, the modeler runs DP-SGD with attack-aware noise calibration (Kulynych et al., 2024). To simulate this, we fine-tune multiple GPT-2 (small) models (Radford et al., 2019) using a DP version of LoRA (Yu et al., 2021) (we provide technical details in Appendix G). We obtain the  $f$  curve under the add-remove relation for each model using the direct method (Kulynych et al., 2024), and apply Theorem 3.2 to measure risk. We compare this to the RDP-based analysis from Balle et al. (2022), where we optimize RDP over a grid (see Appendix A.3).
+
+Fig. 1 shows that if we calibrate the noise scale to a given level of maximum attack risk, our unifying bound enables one to choose a lower noise scale at the same level of risk, which, in turn, results in better classification accuracy. Fig. 1 (left) shows the risk as a function of noise scale, and Fig. 1 (middle) shows the corresponding accuracy when training using that noise scale. The figure demonstrates that for the worst-case risk target of 0.15, we can increase the accuracy from  $52\%$  to  $70\%$  using our analysis, as opposed to an RDP-based one. Notably, the increase in accuracy is due to a more precise privacy calculation alone, thus does not come with additional costs. In Appendix F, we detail on a variant of this case study with CIFAR-10 (Krizhevsky et al., 2009).
+
+Analyzing risk in US Census release. The US Census bureau has released the 2020 Census data using DP (Abowd et al., 2022). In this case study, we aim to analyze this release in terms of operational privacy risks using our bounds, leveraging a recent analysis of the Census algorithm in terms of  $f$ -DP (Su et al., 2024). The Census data consists of eight different levels of granularity, ranging from the US-wide level to block-level, with different privacy guarantees at each level. The standard analysis uses zCDP, thus we compare Theorem 3.2 to the analysis of data reconstruction risk via zCDP, as in our previous experiments in Section 4.1.
+
+In Fig. 4, we present the results for each granularity level under the worst-case baseline, which show that our unifying bound indicates up to  $33\%$  less risk. The results for other baselines are similar, shown in Fig. 9, Appendix H. Moreover, in Fig. 1 (right) we show the risk as a function of baseline for the state-level release as an illustration for analyzing risk at different baselines.
+
+# 5 Related Work
+
+Prior work has extensively studied bounds on attack risks in DP. Balle et al. (2022) introduced a formalization of robustness to reconstruction attacks and provided bounds using  $(\varepsilon, 0)$ -DP, RDP and zCDP. Compared to these, our bounds are tighter, as we demonstrate in Section 4. Under specialized threat models, these bounds were subsequently improved by Guo et al. (2023), and by Guerra-Balboa et al. (2023) for the case of  $(\varepsilon, 0)$ -DP. We study the bounds by Guo et al. (2023), which specifically assume a discrete prior  $P$ , in Appendix D. Cherubin et al. (2024) provided closed-form bounds on data reconstruction and membership inference advantage for DP-SGD using an approximate analysis. Our results do not rely on approximations and instead use the tight privacy analysis in terms of  $f$ -DP based on the state-of-the-art DP-SGD accountants, as detailed in Section 3.4.
+
+Hayes et al. (2024) used a decision-theoretic view of DP via approximations to provide a bound on reconstruction robustness. There are several core differences compared to our results: (1) they provide a Monte-Carlo sampling-based method rather than a provable bound, (2) their method is
+
+specific to DP-SGD, and (3) it only covers reconstruction attacks. In contrast, our work focuses on the fundamental connection between different standard types of risks and the  $f$ -DP framework, as opposed to providing a method to evaluate a specific type of privacy risk in DP-SGD. Our approach enables us to obtain provable bounds on risks for general DP mechanisms. Even though the theoretical underpinnings of our approaches are different, the outputs of the algorithm from Hayes et al. (2024) will converge in the limit of infinite Monte-Carlo samples to our  $f$ -DP bound in the setting of reconstruction risk in DP-SGD.
+
+In Appendix B, we provide an additional overview of the alternative Bayesian approaches to interpret DP and other approaches to unify privacy risks.
+
+Bibliographic note. This paper subsumes the results from a prior workshop paper (Kaisis et al., 2023a) presented at the 2023 Theory and Practice of Differential Privacy (TPDP) workshop.
+
+# 6 Concluding Remarks
+
+This paper presents a unifying framework for analyzing re-identification, attribute inference, and data reconstruction risks—realistic privacy threats facing the releases of statistics, models, or de-identified data—using  $f$ -differential privacy, a decision-theoretic approach to differential privacy. We demonstrated that our unifying framework is applicable to privacy-preserving machine-learning algorithms such as DP-SGD, and statistical applications such as the US Census data release.
+
+Impact. We have shown that the existing approaches to mapping DP parameters to risk are imprecise. As a result, in settings where ensuring a certain maximum level of operational risk is required—as opposed to ensuring a target value of  $\varepsilon$ —practitioners would need to add more noise than is necessary. Not only does this hurt overall utility, but has negative effects on fairness (Bagdasaryan et al., 2019) and prediction consistency (Kulynych et al., 2023). Our approach provides a more precise way to estimate maximum risk, enabling practitioners to obtain better utility at the same level of target risk.
+
+$f$ -DP for analyzing operational risks. Our empirical and theoretical results demonstrated that the decision-theoretic view of DP via the  $f$ -DP framework is a useful tool for analyzing privacy-preserving algorithms in terms of operational risks. As we have shown, it enables both substantially tighter characterizations of operational risk compared to other standard approaches such as Rényi DP or Concentrated DP, and easy-to-use unified analyses of various types of risk.
+
+Optimality. Although our bounds outperform prior methods to measure risk in DP, as we show in Appendix D, they can be significantly strengthened under additional assumptions on the adversarial model, e.g., by assuming the setting of binary attribute inference. We leave finding other practically relevant settings in which the bounds can be further tightened—either under specific attack configurations and priors like in Appendix D or under relaxed threat models (see, e.g., Kaissis et al., 2023b; Swanberg et al., 2025)—to future work.
+
+Importance of the privacy unit. Our privacy risk bounds have to be interpreted in terms of the underlying privacy unit. For instance, when training language models on open internet data, it is often practically infeasible to define a neighborhood relation that protects the privacy of individuals, as an individual's sensitive data can appear in various forms across multiple documents (Brown et al., 2022). In such settings, a more feasible privacy unit is a single document (Sinha et al., 2025). Thus, our bounds in this context would quantify the risk to a single document rather than to an individual.
+
+The need for transparent processes to prevent baseline gaming. Our methods provide a way to adjust the level of risk to a given adversarial setting by modeling the baseline risk. This is a strength as it enables to measure the risk of relevant threats, but also could facilitate privacy theater (Smart et al., 2022) if the communicated or calibrated risk is specific to a baseline that is too low. In this case, the risk estimate could appear lower than it actually is. Trustworthy usage of our baseline-specific bounds requires ensuring that the baseline is well-justified and representative of realistic threats.
+
+# Acknowledgments
+
+This project is supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Department of Energy Computational Science Graduate Fellowship under Award Number DE-SC0022158, by Swiss National Science Foundation under Award Numbers 10003518 and 237378, and is part of the SYNTHIA project. SYNTHIA (Synthetic Data Generation framework for integrated validation of use cases and AI healthcare applications) is supported by the Innovative Health Initiative Joint Undertaking (IHI JU) under grant agreement No. 101172872. Thus, the project is partially funded by the European Union, the private members, and those contributing partners of the IHI JU. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the aforementioned parties. Neither of the aforementioned parties can be held responsible for them.
+
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+
+# A Extended Background
+
+This section presents a more extensive overview of the background.
+
+# A.1 Differential Privacy and its Variants
+
+Classical differential privacy. Differential privacy (DP) is a formal notion of privacy applicable to releases of statistical information or machine learning models:
+
+Definition A.1 (Dwork et al., 2006). A mechanism  $M: 2^{\mathbb{D}} \to \Theta$  satisfies  $(\varepsilon, \delta)$ -DP if for any  $S \simeq S'$  we have  $\mathsf{E}_{e^{\varepsilon}}(P \parallel Q) \leq \delta$ , where  $P$  and  $Q$  are the respective probability distributions of  $M(S)$  and  $M(S')$ , and the hockey-stick divergence between probability distributions is defined as follows:
+
+$$
+\mathsf {E} _ {\gamma} (P \| Q) \triangleq \sup  _ {E \subseteq \Theta} P (E) - \gamma Q (E). \tag {14}
+$$
+
+We refer to the case when  $\delta = 0$  as pure DP.
+
+Realistic DP mechanisms satisfy a potentially infinite collection of different  $(\varepsilon, \delta)$ -DP guarantees. To analyze this collection, we make use of a notion of the privacy profile:
+
+Definition A.2 (Balle et al., 2018). A mechanism  $M(\cdot)$  has a privacy profile  $\delta(\varepsilon)$  if for every  $\varepsilon \in \mathbb{R}$ , it satisfies  $(\varepsilon, \delta(\varepsilon)) - DP$ .
+
+Strong-adversary membership inference. DP can be completely characterised via a constraint on the success rate of membership inference attacks, which aim to determine whether a given record was part of the input dataset  $S$  based on the output of  $M(S)$ . Formally, given a sensitive record  $z \in \mathbb{D}$ , and a partial dataset  $\bar{S} \in \mathbb{D}^{n-1}$ , consider a probability distribution of the algorithm's outputs when the record is not part of the dataset,  $P(E) \triangleq \operatorname*{Pr}_{\theta \sim M(\bar{S})}[\theta \in E]$ , and the respective distribution when the record is present in the data,  $Q(E) \triangleq \operatorname*{Pr}_{\theta \sim M(\bar{S} \cup \{z\})}[\theta \in E]$ . In a strong-adversary membership inference attack (see, e.g., Nasr et al., 2021), given knowledge of the mechanism  $M(\cdot)$ , its output  $\theta$ , a target record  $z$ , and a partial dataset  $\bar{S}$ , the adversary aims to determine whether the record was part of the training dataset used to produce  $\theta$  or not. In other words, the attack is a hypothesis test which aims to distinguish between two hypotheses:
+
+$$
+H _ {0}: \theta \sim P, \text {a n d} H _ {1}: \theta \sim Q. \tag {15}
+$$
+
+"Strong" refers to the fact that the adversary knows all of the partial dataset  $\bar{S}$ . We discuss this threat model further in Section 3.1.
+
+Differential privacy as hypothesis testing. For a given hypothesis test (equivalent to a membership inference attack)  $\phi : \Theta \to \{0,1\}$  which returns 0 when the guess is  $H_0$ , and 1 when the guess is  $H_1$ , we can quantify the adversary's success by characterizing the error rates of the test (Wasserman and Zhou, 2010; Kairouz et al., 2015; Dong et al., 2022):
+
+$$
+\alpha_ {\phi} \triangleq \mathbb {E} _ {P} [ \phi ], \quad \beta_ {\phi} \triangleq 1 - \mathbb {E} _ {Q} [ \phi ], \tag {16}
+$$
+
+where  $\alpha_{\phi}$  and  $\beta_{\phi}$  are the false positive rate (FPR) and false negative rate (FNR) respectively.
+
+For any desired FPR level  $\alpha \in [0,1]$ , the Neyman-Pearson lemma guarantees that there exists an optimal test  $\phi^{*}$  which achieves the lowest possible FNR  $\beta$ . We can thus characterize the attack setting by a trade-off curve, a function which shows the lowest FNR achieved by the most powerful level  $\alpha$  test for any level of FPR  $\alpha \in [0,1]$ :
+
+$$
+T (P, Q) (\alpha) \triangleq \inf  _ {\phi : \Theta \rightarrow [ 0, 1 ]} \left\{\beta_ {\phi} \mid \alpha_ {\phi} \leq \alpha \right\} \tag {17}
+$$
+
+A common way to summarize the success of the optimal strong-adversary membership inference attack (SMIA) is via its advantage, the difference between its TPR and FPR (Yeom et al., 2018):
+
+$$
+\operatorname{adv}_{\mathrm{SMIA}}\triangleq \sup_{\substack{\phi :  \Theta \to [0,1]}}1 - \beta_{\phi} - \alpha_{\phi}.
+$$
+
+We can now define  $f$ -DP, a version of DP which centers the hypothesis-testing interpretation:
+
+Definition A.3 (Dong et al., 2022). A mechanism  $M(\cdot)$  satisfies  $f$ -Differential Privacy ( $f$ -DP) if, for any pair  $S \simeq S'$ , the trade-off curve of the strong membership inference test is bounded:
+
+$$
+T (M (S), M \left(S ^ {\prime}\right)) (\alpha) \geq f (\alpha), \quad \forall \alpha \in [ 0, 1 ], \tag {18}
+$$
+
+where  $f:[0,1] \to [0,1]$  is a given convex, continuous, non-increasing function such that  $f(\alpha) \leq 1 - \alpha$  for all  $\alpha \in [0,1]$ .
+
+Note that  $(\varepsilon, \delta)$ -DP is a special case of  $f$ -DP. Specifically, a mechanism is  $(\varepsilon, \delta)$ -DP if and only if it satisfies  $f$ -DP with the following  $f(\alpha)$ :
+
+$$
+f (\alpha) = \max  \{0, 1 - \delta - e ^ {\varepsilon} \alpha , e ^ {- \varepsilon} (1 - \delta - \alpha) \}. \tag {19}
+$$
+
+We make use of the following equivalent representation of  $f$ -DP:
+
+Lemma 2.1. An algorithm  $M: 2^{\mathbb{D}} \to \Theta$  satisfies  $f$ -DP iff for any measurable  $E \subseteq \Theta$  and  $S \simeq S'$ :
+
+$$
+\Pr [ M (S) \in E ] \leq 1 - f (\Pr [ M (S ^ {\prime}) \in E ]). \tag {3}
+$$
+
+This form is key to deriving our results in a streamlined way.
+
+A useful property of  $f$ -DP is post-processing: if  $M(\cdot)$  satisfies  $f$ -DP, so does  $g \circ M$  for any deterministic or randomized mapping  $g(\cdot)$ . In other words, post-processing the results of an  $f$ -DP mechanism cannot decrease its privacy or, equivalently, cannot increase the success of the aforementioned decision problem. This is a consequence of the Blackwell-Sherman-Stein theorem (Dong et al., 2022; Kaissis et al., 2024; Su, 2024).
+
+Total-variation privacy. We also make use of the notion of total variation (TV) privacy:
+
+Definition A.4. A mechanism  $M: 2^{\mathbb{D}} \to \Theta$  satisfies  $\eta$ -TV privacy if it satisfies  $(0, \eta)$ -DP. Equivalently, the total variation distance, a special case of  $\mathsf{E}_{\gamma}$  with  $\gamma = 1$ , is bounded for all  $S \simeq S'$ :
+
+$$
+\mathsf {E} _ {1} (P \| Q) = \sup  _ {E \subseteq \Theta} P (E) - Q (E) \leq \eta , \tag {20}
+$$
+
+where  $P$  and  $Q$  are the respective probability distributions of  $M(S)$  and  $M(S')$ .
+
+TV privacy has been extensively studied under equivalent forms and different names in literature (Geng and Viswanath, 2015; Bassily et al., 2016; Kulynych et al., 2022a; Chatzikokolakis et al., 2023; Ghazi and Issa, 2023). Although TV privacy is a weak notion of privacy on its own (Vadhan, 2017), any DP mechanism satisfies TV privacy for some  $\eta$ :
+
+Proposition A.1. The following statements hold:
+
+$(\varepsilon, \delta)$ -DP implies  $\eta$ -TV privacy with  $\eta = \frac{e^{\varepsilon} - 1 + 2\delta}{e^{\varepsilon} + 1}$  (Kairouz et al., 2015).  
+-  $f$ -DP implies  $\eta$ -TV privacy with  $\eta = \max_{\alpha \in [0,1]} (1 - f(\alpha) - \alpha)$  (Kaissis et al., 2024).
+
+Privacy via Rényi divergence. Another notion we use is Rényi DP (RDP) (Mironov, 2017).
+
+Definition A.5 (Mironov, 2017). A mechanism  $M(\cdot)$  satisfies  $(t,\varepsilon)$ -RDP iff for all  $S\simeq S'$ :
+
+$$
+D _ {t} (P \| Q) \leq \varepsilon , \tag {21}
+$$
+
+where  $P$  and  $Q$  are the respective probability distributions of  $M(S)$  and  $M(S')$ , and  $D_t(P \parallel Q)$  is the Rényi divergence of order  $t \geq 1$ :
+
+$$
+D _ {t} (P \parallel Q) \triangleq \frac {1}{t - 1} \log \mathbb {E} _ {o \sim Q} \left(\frac {P (o)}{Q (o)}\right) ^ {t},
+$$
+
+where the case of  $t = 1$  is defined by continuous extension.
+
+The mechanism satisfies  $\rho$ -zCDP (Bun and Steinke, 2016; Dwork and Rothblum, 2016) for  $\rho \geq 0$  if it satisfies  $(t,\rho t)$ -RDP for every  $t \geq 1$ .
+
+# A.2 Predicate Singling Out
+
+In this section, we provide a brief overview of the notion of predicate singling out security.
+
+Predicate singling out (PSO) risk, which is defined in Section 3.2, is a privacy concept introduced by Cohen and Nissim (2020) to rigorously interpret the "singing out" criterion mentioned in the EU GDPR (General Data Protection Regulation). Their goal was to bridge the gap between legal expectations of data anonymity and technical guarantees provided by data anonymization mechanisms. At a high level, in the PSO threat model, the adversary uses the output of a data release mechanism  $\theta = M(S)$  to find a predicate  $p$  that matches exactly one individual (row) in the original dataset  $S$  with significantly greater success than would be expected by chance. In this context, a predicate represents a set of attribute values that characterize a person, e.g., "speaks Dutch  $\wedge$  vegan  $\wedge$  concert pianist  $\wedge$  born March 15." If a predicate matches only one individual in a dataset, that individual is singled out. Moreover, such a predicate is unlikely to identify a person by random chance. Formally, the goal of the adversary is to find a predicate  $p$  such that  $\sum_{z\in S}p(z) = 1$ .
+
+Unlike differential privacy, which operates under a strong threat model to provide formal "upper bounds" to more realistic notions of privacy, PSO takes the alternate approach of intentionally being a necessary but not sufficient condition for privacy, i.e. a "lower bound" to realistic notions of privacy. The goal of the original paper was to make claims such as "if you do not satisfy PSO security, you are not GDPR compliant". Hence, Cohen and Nissim consider the i.i.d. setting: the dataset  $S$  is sampled i.i.d. from a distribution  $P$  known to the adversary. Cohen and Nissim reason about this adversary by considering the baseline (before seeing  $M(S)$ ) and success probability (after seeing  $M(S)$ ) of the adversary assuming the adversary's predicate in both cases has a fixed weight  $w$ , where weight is defined as  $w = \mathbb{E}_P[p]$ , the probability that the predicate  $p$  evaluates to 1 on a random sample from  $P$ .
+
+In this setting, the optimal baseline is the random guessing strategy: before observing the output of the mechanism  $M(S)$ , the adversary constructs a predicate  $p$  that has a small weight  $w$ . Intuitively, predicates such as "woman  $\wedge$  over 60 year old" have high weight and predicates such as the Dutch vegan pianist from the previous paragraph have low weight. The best the adversary can do as a baseline is to pick a predicate with weight  $w$  and hope that they single out. This has a probability of base  $= nw(1 - w)^{n-1}$  for a dataset of size  $n$ .
+
+# A.3 Previous Bounds on Privacy Risk Notions
+
+PSO Security Assuming a fixed weight  $w$  and approximate DP, Cohen and Nissim (2020) showed that the optimal strategy for PSO can be upper bounded as a function of the baseline probability:
+
+Theorem A.1 (Cohen and Nissim, 2020). Suppose that  $M: \mathbb{D}^n \to \Theta$  satisfies  $(\varepsilon, \delta)$ -DP w.r.t. replace-one neighbourhood relation. Then, for any given  $n > 1$ , data distribution  $P$  over  $\mathbb{D}$ , and an adversary  $\mathcal{A}_{n,M,P,w}: \Theta \to \mathbb{Q}_P$  with  $\mathbb{Q}_P \triangleq \{p \mid p: \mathbb{D} \to \{0,1\}, \mathbb{E}_P[p] \leq w\}$  for given  $w \in [0,1/n]$ , we have:
+
+$$
+\begin{array}{l} \operatorname {s u c c} _ {P S O} (n, M, P, w; \mathcal {A}) \leq n \left(e ^ {\varepsilon} w + \delta\right) (22) \\ = e ^ {\varepsilon} \cdot \frac {\operatorname {b a s e} _ {P S O} (n , P , w)}{(1 - w) ^ {n - 1}} + n \delta . (23) \\ \end{array}
+$$
+
+This is the bound we use in Fig. 2.
+
+Reconstruction Robustness Next, we review the state-of-the-art results for bounding reconstruction robustness. We start with the Renyi DP bound due to Balle et al. (2022):
+
+Theorem A.2 (Balle et al., 2022). Suppose that  $M(\cdot)$  satisfies  $(t, \varepsilon)$ -RDP w.r.t. either add-remove or replace-one relation. It holds that:
+
+$$
+\operatorname {s u c c} _ {[ S R R ]} \leq \left(\operatorname {b a s e} _ {[ S R R ]} \cdot e ^ {\varepsilon}\right) ^ {\frac {t - 1}{t}}. \tag {24}
+$$
+
+Applying the above result to a  $\rho$ -zCDP mechanism and optimizing  $t$  to minimize the upper bound yields the following:
+
+Corollary A.1 (Balle et al., 2022). Suppose that  $M(\cdot)$  satisfies  $\rho$ -zCDP w.r.t. either add-remove or replace-one relation. It holds that:
+
+$$
+\operatorname {s u c c} _ {[ S R R ]} \leq \exp \left\{- \left(\sqrt {\log \frac {1}{\operatorname {b a s e} _ {[ S R R ]}}} - \sqrt {\rho}\right) ^ {2} \right\}. \tag {25}
+$$
+
+Corollary A.1 is the bound on SRR used in Figs. 3, 4, 5 and 9. We can also apply Theorem A.2 to a mechanism that satisfies a continuum of RDP guarantees and minimize over the upper bound:
+
+Corollary A.2. Suppose that  $M(\cdot)$  satisfies  $(t, \varepsilon(t))$ -RDP for all  $t > 1$  w.r.t. either add-remove or replace-one relation. It holds that:
+
+$$
+\operatorname {s u c c} _ {[ S R R ]} \leq \min  _ {t > 1} \left\{\left(\operatorname {b a s e} _ {[ S R R ]} \cdot e ^ {\varepsilon (t)}\right) ^ {\frac {t - 1}{t}} \right\}. \tag {26}
+$$
+
+We use Corollary A.2 in Fig. 1 as the prior method that we compare our  $f$ -DP bound to.
+
+# B Additional Discussion on Related Work
+
+Related unification efforts. Recently, Cohen et al. (2025) proposed a unifying notion of risk called narcissus resiliency, which averages over dataset sampling. The notions of risk we consider under our unifying framework are all tailored to the strong threat model, which is dataset-specific. We additionally provide bounds on Narcissus resiliency using  $f$ -DP in Appendix F. Salem et al. (2023); Cummings et al. (2024) provided taxonomies and reductions between risk notions. Our focus is providing analysis using  $f$ -DP, as opposed to a taxonomization.
+
+Bayesian semantics of DP. An alternative way to analyze and interpret risk in DP is a Bayesian posterior-to-prior analysis (see, e.g., Wood et al., 2018; Kifer et al., 2022; Kazan and Reiter, 2024). We focus on security-based literature instead, where risk is analyzed from the point of view of relevant adversarial models. See Kifer et al. (2022); Kaissis et al. (2024) for connections between the Bayesian view and  $f$ -DP.
+
+Attack-aware noise calibration. Kulynych et al. (2024) introduced attack-aware noise calibration, the idea of calibrating DP algorithms to a given level of operational attack risk as opposed to the standard practice of calibrating to a given pair of  $(\varepsilon, \delta)$  parameters. They evaluated this approach for the use case of calibrating noise to a given level of membership inference risk. One of the applications of our result is that it extends the toolbox of attack-aware calibration to further attack risks, thus enabling practitioners to calibrate DP algorithms to notions of risk beyond membership inference, which often appear in data protection guidelines (see Section 1).
+
+# C Omitted Proofs
+
+# C.1 Additional and Auxiliary Results
+
+We start with a proof of a useful equivalent form of  $f$ -DP.
+
+Lemma 2.1. An algorithm  $M: 2^{\mathbb{D}} \to \Theta$  satisfies  $f$ -DP iff for any measurable  $E \subseteq \Theta$  and  $S \simeq S'$ :
+
+$$
+\Pr [ M (S) \in E ] \leq 1 - f (\Pr [ M (S ^ {\prime}) \in E ]). \tag {3}
+$$
+
+Proof. Denote by  $P$  the distribution of  $M(S)$  and by  $Q$  the distribution of  $M(S')$ .
+
+$\Longrightarrow$  Suppose that the algorithm satisfies  $f$ -DP. For any  $\alpha \in [0,1]$ , we can find  $\phi : \mathbb{D} \to [0,1]$  such that  $\mathbb{E}_Q[\phi] = \alpha_\phi \leq \alpha$ . For this  $\alpha$ , by the definition of  $f$ -DP we have:
+
+$$
+f (\alpha_ {\phi}) \leq f (\alpha) \leq \beta_ {\phi} = 1 - \mathbb {E} _ {P} [ \phi ] = 1 - P (E), \tag {27}
+$$
+
+where  $E = \{\theta \in \Theta \mid \phi(\theta) = 1\}$ . Plugging  $\alpha_{\phi} = Q(E)$  into Eq. (27), we get:
+
+$$
+f (Q (E)) \leq 1 - P (E),
+$$
+
+from which we immediately get the sought form.
+
+$\Longleftarrow$  Suppose that Eq. (3) holds for any measurable  $E \subseteq \Theta$  and any  $S \simeq S'$ . Then, for any  $\phi: \mathbb{D} \to [0,1]$  and  $E = \{\theta \in \Theta \mid \phi(\theta) = 1\}$ :
+
+$$
+\Pr [ M (S) \in E ] = \mathbb {E} _ {P} [ \phi ] \leq 1 - f (\Pr [ M (S ^ {\prime}) \in E ]) = 1 - f (\mathbb {E} _ {Q} [ \phi ]).
+$$
+
+As  $\alpha_{\phi} = \mathbb{E}_Q[\phi ]$  and  $\beta_{\phi} = 1 - \mathbb{E}_{P}[\phi ]$  , we have:
+
+$$
+1 - \beta_ {\phi} \leq 1 - f (\alpha_ {\phi}),
+$$
+
+which implies  $f(\alpha_{\phi})\leq \beta_{\phi}$  , thus Eq. (18).
+
+To show the main result on the unified bounds, we make use of the following elementary lemmas. First, due to the properties of the trade-off function  $f$ , we can push the expectation inside of  $f(\cdot)$ :
+
+Lemma C.1. Suppose that  $f$  is a valid trade-off function according to Definition A.3. For any random variable  $W$  over  $[0,1]$ , we have:
+
+$$
+\mathbb {E} [ 1 - f (W) ] \leq 1 - f (\mathbb {E} W). \tag {28}
+$$
+
+Proof. By assumption,  $1 - f$  is concave and increasing. The result immediately follows from applying the linearity of expectation and Jensen's inequality.
+
+Second, we can suprime an expectation inside of  $f(\cdot)$ :
+
+Lemma C.2. Suppose that  $f$  is a valid trade-off function according to Definition A.3. For any random variable  $V$  taking values in a set  $\mathbb{V}$ , and any bounded function  $g: \mathbb{V} \to [0,1]$ , we have:
+
+$$
+1 - f (\mathbb {E} g (V)) \leq 1 - f \left(\sup  _ {v \in \mathbb {V}} g (v)\right). \tag {29}
+$$
+
+Proof. Observe that  $\mathbb{E}g(V)\leq \sup_{v\in \mathbb{V}}g(v)$ . We get the result as  $1 - f$  is increasing.
+
+![](images/ad1d1790cedcfc556d1e826a4f1fc8a0916482a4bfa979251901ff602c729623.jpg)
+
+Next, we can show a result which can be seen as a version of a risk bound for on-average baselines.
+
+Lemma C.3. Suppose that  $M: 2^{\mathbb{D}} \to \Theta$  satisfies  $f - DP$  w.r.t add-remove relation. Then, for any bounded function  $q: \mathbb{D} \times \Theta \to [0,1]$ , any partial dataset  $\bar{S} \in \mathbb{D}^n$ , and any probability distribution  $P$  over  $\mathbb{D}$ , we have:
+
+$$
+\mathbb {E} _ {z \sim P} \mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z \})} [ q (z; \theta) ] \leq 1 - f \left(\mathbb {E} _ {z \sim P} \mathbb {E} _ {\theta \sim M (\bar {S})} [ q (z; \theta) ]\right). \tag {30}
+$$
+
+Moreover, if the  $M(\cdot)$  satisfies  $f$ -DP w.r.t. replace-one relation, we have for any  $z' \in \mathbb{D}$ :
+
+$$
+\mathbb {E} _ {z \sim P} \mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z \})} [ q (z; \theta) ] \leq 1 - f \left(\mathbb {E} _ {z \sim P} \mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z ^ {\prime} \})} [ q (z; \theta) ]\right). \tag {31}
+$$
+
+Proof. Fix any  $z \in \mathbb{D}$ . By the form of  $f$ -DP in Lemma 2.1 and the post-processing property of  $f$ -DP, we have:
+
+$$
+\mathbb {E} _ {\theta \sim M (\bar {S})} q (z; \theta) \leq 1 - f \left(\mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z \})} q (z; \theta)\right) \tag {32}
+$$
+
+$$
+\mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z \})} q (z; \theta) \leq 1 - f \left(\mathbb {E} _ {\theta \sim M (\bar {S})} q (z; \theta)\right).
+$$
+
+By Lemma C.1, we have for any  $g:\mathbb{D}\to [0,1]$
+
+$$
+\mathbb {E} _ {z \sim P} [ 1 - f (g (z)) ] \leq 1 - f (\mathbb {E} _ {z \sim P} g (z)). \tag {33}
+$$
+
+We get the sought statement in Eq. (30) by taking the expectation over  $z \sim P$  of both sides in Eq. (32) and applying Eq. (33). Finally, we get the result in Eq. (31) analogously from:
+
+$$
+\mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z \})} q (z; \theta) \leq 1 - f \left(\mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z ^ {\prime} \})} q (z; \theta)\right).
+$$
+
+![](images/8d50aa383124572fdb23a86bb6542f4e6e1ebb47326554fb6ad5a97e18b6c9c8.jpg)
+
+In Appendix E, we also derive an equivalent result with the semantics of standard generalization bounds. Moreover, in Appendix F, we show that this result immediately implies bounds on reconstruction robustness with an alternative on-average definition of baseline success rate (Guerra-Balboa et al., 2023).
+
+We can now show the general version of our risk bound:
+
+Lemma 3.1. Suppose that  $M:2^{\mathbb{D}}\to \Theta$  satisfies  $f$ -DP w.r.t. either add-remove or replace-one relation. Then, for any bounded function  $q:\mathbb{D}\times \Theta \rightarrow [0,1]$ , any partial dataset  $\bar{S}\in \mathbb{D}^{n - 1}$  with  $n\geq 1$ , and any probability distribution  $P$  over  $\mathbb{D}$ , we have:
+
+$$
+\mathbb {E} _ {z \sim P} \mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z \})} [ q (z; \theta) ] \leq 1 - f \left(\sup  _ {\theta \in \Theta} \mathbb {E} _ {z \sim P} [ q (z; \theta) ]\right). \tag {10}
+$$
+
+Proof. For add-remove relation, we immediately get the result by applying Lemma C.2 to the r.h.s. of Eq. (30), and taking  $g(\theta) = \mathbb{E}_{z\sim P}q(z;\theta)$ :
+
+$$
+\mathbb {E} _ {z \sim P} \mathbb {E} _ {\theta \sim M (\bar {S})} q (z; \theta) = \mathbb {E} _ {\theta \sim M (\bar {S})} \mathbb {E} _ {z \sim P} q (z; \theta) = \mathbb {E} _ {\theta \sim M (\bar {S})} g (\theta).
+$$
+
+Analogously, for replace-one relation, we get the result by applying Lemma C.2 to the r.h.s. of Eq. (31) with:
+
+$$
+\mathbb {E} _ {z \sim P} \mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z ^ {\prime} \})} q (z; \theta) = \mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z ^ {\prime} \})} \mathbb {E} _ {z \sim P} q (z; \theta) = \mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z ^ {\prime} \})} g (\theta).
+$$
+
+![](images/8f28b2b7c7abd94f7b6530b1e97fa894330c2221f088865ad6a822680431f1f3.jpg)
+
+We proceed to showing that the difference between the success and the baseline is upper-bounded by TV privacy:
+
+Lemma C.4. Suppose that the algorithm  $M: 2^{\mathbb{D}} \to \Theta$  satisfies  $\eta$ -TV privacy w.r.t. either the add-remove or replace-one relation. Then, for any bounded function  $q: \mathbb{D} \times \Theta \to [0,1]$ , any partial dataset  $\bar{S} \in \mathbb{D}^{n-1}$  with  $n \geq 1$ , and any probability distribution  $P$  over  $\mathbb{D}$ , we have:
+
+$$
+\mathbb {E} _ {z \sim P} \mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z \})} [ q (z; \theta) ] - \sup  _ {\theta \in \Theta} \mathbb {E} _ {z \sim P} [ q (z; \theta) ] \leq \eta . \tag {34}
+$$
+
+Proof. Let us assume that  $M(\cdot)$  satisfies  $f$ -DP for some valid trade-off curve  $f$ . This is always the case under the assumption of the statement, as  $\eta$ -TV implies  $f(\alpha) = 1 - \alpha - \eta$  by Eq. (2). Denoting by  $\mathrm{succ} = \mathbb{E}_{z \sim P} \mathbb{E}_{\theta \sim M(\bar{S} \cup \{z\})}[q(z; \theta)]$  and base  $= \sup_{\theta \in \Theta} \mathbb{E}_{z \sim P}[q(z; \theta)]$ , recall that by Lemma 3.1:
+
+$$
+\operatorname {s u c c} - \operatorname {b a s e} \leq 1 - f (\operatorname {b a s e}) - \operatorname {b a s e}
+$$
+
+To make the r.h.s. independent of the baseline, let us maximize it over all possible baseline values:
+
+$$
+1 - f (\alpha) - \alpha \leq \max  _ {\alpha \in [ 0, 1 ]} 1 - f (\alpha) - \alpha \leq \eta ,
+$$
+
+where the last inequality is by Proposition A.1. Therefore,  $\operatorname{succ} - \operatorname{base} \leq \eta$ , which is exactly the sought result.
+
+![](images/98edc5117911e8ea3e9867e9df5d63a7a96bb7088da067235ff1da1ae34fad6c.jpg)
+
+# C.2 Formal Versions of the Simplified Statements in the Main Body
+
+We can now formally re-state Theorems 3.1 and 3.3:
+
+Theorem C.1 (Formal version of Theorem 3.1). Suppose that the algorithm  $M:2^{\mathbb{D}}\to \Theta$  satisfies  $f - DP$  w.r.t. either the add-remove or replace-one relation. Then, the following hold:
+
+-  $SPSO$ . For any  $n \geq 1, k \geq 2$ , partial dataset  $\bar{S} \in \mathbb{D}^{n-1}$ , data distribution  $P$  over a candidate set  $\mathbb{W} \subseteq \mathbb{D}$  of size  $|\mathbb{W}| = k$ , weight  $w \in [0,1]$ , and any adversary  $\mathcal{A}_{M,\bar{S},P,w}: \Theta \to \mathbb{Q}_{\bar{S},P}$  for  $\mathbb{Q}_{\bar{S},P} \subseteq \{p \mid p: \mathbb{D} \to \{0,1\}, \sum_{z' \in \bar{S}} p(z') = 0, \mathbb{E}_P[p] \leq w\}$ , we have:
+
+$$
+\operatorname {s u c c} _ {S P S O} (M, \bar {S}, P, w; \mathcal {A} _ {M, \bar {S}, P, w}) \leq 1 - f (\operatorname {b a s e} _ {S P S O} (\bar {S}, P, w)).
+$$
+
+- SRR. For any  $n \geq 1$ , partial dataset  $\bar{S} \in \mathbb{D}^{n - 1}$ , data distribution  $P$  over  $\mathbb{D}$ , loss function  $\ell : \mathbb{D} \times \mathbb{D} \to \mathbb{R}$ , threshold  $\gamma \in \mathbb{R}$ , and any adversary  $A_{M, \bar{S}, P}: \Theta \to \mathbb{D}$ , we have:
+
+$$
+\operatorname {s u c c} _ {S R R} (M, \bar {S}, P; \mathcal {A} _ {M, \bar {S}, P}, \ell , \gamma) \leq 1 - f (\operatorname {b a s e} _ {S R R} (P; \ell , \gamma)).
+$$
+
+- SAI. For any  $n \geq 1$ , set of attributes  $\mathbb{A} = \{1, \dots, k\}$ , mapping from records to attributes  $a : \mathbb{D} \to \mathbb{A}$ , partial dataset  $\bar{S} \in \mathbb{D}^{n-1}$ , data distribution  $P$  over  $\mathbb{D}$ , and any adversary  $A_{M, \bar{S}, P} : \Theta \to \mathbb{A}$ , we have:
+
+$$
+\operatorname {s u c c} _ {S A I} (M, \bar {S}, P; \mathcal {A} _ {M, \bar {S}, P}, a) \leq 1 - f (\operatorname {b a s e} _ {S A I} (P, a)).
+$$
+
+Proof. Consider the mechanism  $\mathcal{A}(M(\cdot))$ , where we omit the subscripts for brevity. It satisfies  $f$ -DP by post-processing. Let us denote the output space of this mechanism as  $\mathbb{Y}$ , i.e.,  $\mathbb{Y} = \mathbb{Q}_{\bar{S},P}$  for SPSO,  $\mathbb{Y} = \mathbb{D}$  for SRR, and  $\mathbb{Y} = \mathbb{A}$  for SAI. The upper bounds follow immediately from Lemma 3.1 by taking an appropriate choice of  $q:\mathbb{D}\times \mathbb{Y}\to [0,1]$ :
+
+$$
+S P S O: \quad q (z; p) = \mathbb {1} [ p (z) = 1 ]
+$$
+
+$$
+S R R: \quad q (z; \hat {z}) = \mathbb {1} [ \ell (z, \hat {z}) \leq \gamma ] \tag {35}
+$$
+
+$$
+S A I: \quad q (z; \hat {a}) = \mathbb {1} [ a (z) = \hat {a} ]
+$$
+
+Theorem C.2 (Formal version of Theorem 3.3). Suppose that the algorithm  $M: 2^{\mathbb{D}} \to \Theta$  satisfies  $\eta$ -TV privacy w.r.t. either the add-remove or replace-one relation. Then, the following hold:
+
+-  $SPSO$ . For any  $n \geq 1, k \geq 2$ , partial dataset  $\bar{S} \in \mathbb{D}^{n-1}$ , data distribution  $P$  over a candidate set  $\mathbb{W} \subseteq \mathbb{D}$  of size  $|\mathbb{W}| = k$ , weight  $w \in [0,1]$ , and any adversary  $\mathcal{A}_{M,\bar{S},P,w}: \Theta \to \mathbb{Q}_{\bar{S},P}$  for  $\mathbb{Q}_{\bar{S},P} \subseteq \{p \mid p: \mathbb{D} \to \{0,1\}, \sum_{z' \in \bar{S}} p(z') = 0, \mathbb{E}_P[p] \leq w\}$ , we have:
+
+$$
+\operatorname {s u c c} _ {S P S O} (M, \bar {S}, P, w; \mathcal {A} _ {M, \bar {S}, P, w}) - \operatorname {b a s e} _ {S P S O} (\bar {S}, P, w) \leq \eta .
+$$
+
+- SRR. For any  $n \geq 1$ , partial dataset  $\bar{S} \in \mathbb{D}^{n - 1}$ , data distribution  $P$  over  $\mathbb{D}$ , loss function  $\ell : \mathbb{D} \times \mathbb{D} \to \mathbb{R}$ , threshold  $\gamma \in \mathbb{R}$ , and any adversary  $\mathcal{A}_{M,\bar{S},P}: \Theta \to \mathbb{D}$ , we have:
+
+$$
+\operatorname {s u c c} _ {S R R} (M, \bar {S}, P; \mathcal {A} _ {M, \bar {S}, P}, \ell , \gamma) - \operatorname {b a s e} _ {S R R} (P; \ell , \gamma) \leq \eta .
+$$
+
+- SAI. For any  $n \geq 1$ , set of attributes  $\mathbb{A} = \{1, \dots, k\}$ , mapping from records to attributes  $a : \mathbb{D} \to \mathbb{A}$ , partial dataset  $\bar{S} \in \mathbb{D}^{n-1}$ , data distribution  $P$  over  $\mathbb{D}$ , and any adversary  $\mathcal{A}_{M,\bar{S},P} : \Theta \to \mathbb{A}$ , we have:
+
+$$
+\operatorname {s u c c} _ {S A l} (M, \bar {S}, P; \mathcal {A} _ {M, \bar {S}, P}, a) - \operatorname {b a s e} _ {S A l} (P, a) \leq \eta .
+$$
+
+Proof. As before, we obtain the results by applying Lemma C.4 to a mechanism  $\mathcal{A}(M(\cdot))$ , which satisfies  $f$ -DP by post-processing, and an appropriately chosen  $q(z;\cdot)$  in Eq. (35).
+
+# D Tighter Bounds under a Bernoulli Prior
+
+In this section, we provide a tighter bound than in Theorem 3.1 assuming a known form of the prior distribution  $P$  over  $\mathbb{D}$ , specifically,  $z = z_{b}$  with  $b\sim \mathrm{Bern}(\pi)$  for some  $\pi \in [0,1]$ . This setting can model two threats: (1) SAI of a single binary attribute with a non-uniform prior (see, e.g., Guo et al., 2023) or, equivalently, SRR with only two points in the support of the prior distribution  $P$ , and (2) SMIA with a non-uniform prior probability of membership (Jayaraman et al., 2021).
+
+We make use of the notion of the Bayes error of the attacker (Chatzikokolakis et al., 2023).
+
+Definition D.1 (Bayes error for  $f$ -DP, Kaissis et al., 2024). Suppose that  $f$  is a valid trade-off function according to Definition A.3. For any  $\pi \in [0,1]$ , we define the Bayes error w.r.t. the prior  $\pi$  as follows:
+
+$$
+R_{f}(\pi)\triangleq \min_{\alpha \in [0,1]}\bigl(\pi \alpha +(1 - \pi)f(\alpha)\bigr).
+$$
+
+Given a numeric representation of a trade-off curve  $f$ , Bayes error can be easily computed numerically by, e.g., grid search over a grid of  $\alpha$  values. Additionally, we show an elegant way to obtain the Bayes error directly from the privacy profile  $\delta(\varepsilon)$ , which was previously unknown in the context of DP:
+
+Proposition D.1 (Based on Sason and Verdu, 2016, Eq. (421)). Suppose that a mechanism  $M(\cdot)$  satisfies  $f$ -DP and has an associated privacy profile  $\delta(\varepsilon)$  w.r.t. any neighbourhood relation. Then, the Bayes error is bounded:
+
+$$
+R _ {f} (\pi) \geq (1 - \pi) \cdot \left[ 1 - \delta \left(\operatorname {l o g i t} (\pi)\right) \right], \quad \text {w h e r e} \log_ {1} (\pi) \triangleq \log \left(\frac {\pi}{1 - \pi}\right). \tag {36}
+$$
+
+Next, we show a bound on success of SAI/SMIA under Bernoulli prior using the Bayes error.
+
+Theorem D.1. Fix any partial dataset  $\bar{S} \in \mathbb{D}^{n-1}$  with  $n \geq 1$ . Suppose that  $M: \mathbb{D}^n \to \Theta$  satisfies  $f$ -DP w.r.t. the replace-one relation. Then, for any prior probability  $\pi \in [0,1]$ , a set of any two candidate records  $\{z_0, z_1\} \subseteq \mathbb{D}$ , and any score function  $\hat{b}: \Theta \to [0,1]$  indicating adversary's confidence of whether  $z_1$  or  $z_0$  was used for training, we have:
+
+$$
+\mathbb {E} _ {b \sim \operatorname {B e r n} (\pi)} \mathbb {E} _ {\theta \sim M \left(\bar {S} \cup \{z _ {b} \}\right))} [ q (z _ {b}; \theta) ] \leq 1 - R _ {f} (\pi), \tag {37}
+$$
+
+where we define the success indicator  $q(z;\theta)$  as follows:
+
+$$
+q (z; \theta) \triangleq \left\{ \begin{array}{l l} \hat {b} (\theta), & i f z = z _ {1} \\ 1 - \hat {b} (\theta), & i f z = z _ {0} \end{array} . \right. \tag {38}
+$$
+
+Proof. We can express the l.h.s. in Eq. (37) as follows:
+
+$$
+\underset {\theta \sim M (\bar {S} \cup \{z _ {b} \})} {\mathbb {E}} [ q (z _ {b}; \theta) ] = (1 - \pi) \cdot \mathbb {E} _ {\theta \leftarrow M (\bar {S} \cup \{z _ {0} \})} [ q (z _ {0}; \theta) ] + \pi \cdot \mathbb {E} _ {\theta \leftarrow M (\bar {S} \cup \{z _ {1} \})} [ q (z _ {1}; \theta) ]. \tag {39}
+$$
+
+Observe that in the setup of this statement, we have  $q(z_{1};\theta) = 1 - q(z_{0};\theta)$ . Denote by  $\alpha = \mathbb{E}_{\theta \sim M(\bar{S}\cup \{z_0\})}[q(z_1;\theta)] = 1 - \mathbb{E}_{\theta \sim M(\bar{S}\cup \{z_0\})}[q(z_0;\theta)]$ . By Lemma 2.1, we have:
+
+$$
+\mathbb {E} _ {\theta \leftarrow M (\bar {S} \cup \{z _ {1} \})} [ q (z _ {1}; \theta) ] \leq 1 - f (\alpha).
+$$
+
+Therefore, we can upper bound Eq. (39) as:
+
+$$
+\begin{array}{l} \leq (1 - \pi) \cdot (1 - \alpha) + \pi \cdot (1 - f (\alpha)) \\ = 1 - R _ {f} (\pi), \\ \end{array}
+$$
+
+where the last equality is by Definition D.1.
+
+![](images/50b7e7f76d9d84ba7c548bea332e975a52a3ab1677a7f538f0bf97d13aacd5d1.jpg)
+
+Experimental evaluation. We compare our bounds in Theorem 3.1, which are applicable to any prior, and the specialized result in Theorem D.1 for Bernoulli priors. We additionally compare these results to the bounds on SRR based on RDP as in Section 4, and the bounds on SAI based on the Fano's inequality (Guo et al., 2023).
+
+For this comparison, we use Gaussian mechanism with noise scales  $\sigma \in \{0.5, 1.0, 2.0\}$ , analyzed under replace-one relation. We use  $\delta = 10^{-5}$  to compute  $\varepsilon$ . We use two versions of the bounds from Guo et al. (2023): (1) based on an analytical bound on mutual information (MI) between Gaussians, and a Monte-Carlo (MC) method using 10,000 samples (we repeat each run 10 times). Note that the latter method only provides an estimate of a bound, unlike all other evaluated approaches. We show the result in Fig. 5 (error bars for the MC approach are not visible). We can see that for all noise parameters, and all baselines, the bound in Theorem D.1 outperforms the other bounds.
+
+# E Generalization Bounds
+
+In this section, we extend the previous results to the setting in which the dataset is assumed to be sampled i.i.d. from some data distribution  $P$ . The results in this section assume the replace-one neighbourhood relation.
+
+Theorem E.1 (On-average generalization from  $f$ -DP). Suppose that  $M: \mathbb{D}^n \to \Theta$  satisfies  $f$ -DP w.r.t. replace-one relation. Then, for any bounded loss function  $q: \mathbb{D} \times \Theta \to [0,1]$ , and any
+
+![](images/77762f3cd5f333d49aefa70e684601dca65498466d426fcfd7f1911f5aea9033.jpg)  
+Figure 5: We can significantly tighten the bounds in the setting of binary attribute inference (SAI, via  $f$ -DP), outperforming the prior bound based on Fano's inequality.
+
+probability distribution  $P$  over  $\mathbb{D}$ , we have:
+
+$$
+\underbrace {\mathbb {E} _ {S \sim P ^ {n}} \frac {1}{n} \sum_ {i = 1} ^ {n} q \left(z _ {i} ; \theta\right)} _ {\text {o n - a v e r a g e t r a i n l o s s}} \leq 1 - f \underbrace {\left(\mathbb {E} _ {S \sim P ^ {n} , z \sim P} q (z ; \theta)\right)} _ {\text {o n - a v e r a g e t e s t l o s s}} \tag {40}
+$$
+
+$$
+\underbrace {\mathbb {E} _ {S \sim P ^ {n} , z \sim P} q (z ; \theta)} _ {\text {o n - a v e r a g e t e s t l o s s}} \leq 1 - f \underbrace {\left(\mathbb {E} _ {S \sim P ^ {n}} \frac {1}{n} \sum_ {i = 1} ^ {n} q \left(z _ {i} ; \theta\right)\right)} _ {\text {o n - a v e r a g e t r a i n l o s s}}, \tag {41}
+$$
+
+where  $S = (z_{1},z_{2},\ldots ,z_{n})$
+
+Proof. Fix any  $z, z' \in \mathbb{D}$ , and  $\bar{S} \in \mathbb{D}^{n-1}$ . By Lemma 2.1 and the post-processing property of  $f$ -DP, we have:
+
+$$
+\mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z \})} q (z; \theta) \leq 1 - f \left(\mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z ^ {\prime} \})} q (z; \theta)\right) \tag {42}
+$$
+
+$$
+\mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z ^ {\prime} \})} q (z; \theta) \leq 1 - f \left(\mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z \})} q (z; \theta)\right).
+$$
+
+By using Lemma C.1 after taking the expectation over  $z, z' \sim P$  and  $\bar{S} \sim P^{n-1}$  of both sides in Eq. (42), we have:
+
+$$
+\mathbb{E}_{\substack{S\sim P^{n}\\ z\sim \operatorname {Unif}[S]\\ \theta \sim M(S)}}q(z;\theta)\leq 1 - f\left(\mathbb{E}_{\substack{S\sim P^{n}\\ z\sim P\\ \theta \sim M(S)}}q(z;\theta)\right)
+$$
+
+$$
+\mathbb{E}_{\substack{S\sim P^{n}\\ z\sim P\\ \theta \sim M(S)}}q(z;\theta)\leq 1 - f\left(\mathbb{E}_{\substack{S\sim P^{n}\\ z\sim \operatorname {Unif}[S]\\ \theta \sim M(S)}}q(z;\theta)\right)
+$$
+
+The forms in Eqs. (40) and (41) are equivalent as  $\mathbb{E}_{z\sim \mathrm{Unif}[S]}q(z;\theta) = \frac{1}{n}\sum_{i = 1}^{n}q(z_i;\theta)$ .
+
+By applying Theorem 3.3, we recover the prior result that  $\sup_{q: \mathbb{D} \times \Theta \to [0,1]} |\mathrm{err}_{\mathrm{tr}}(q) - \mathrm{err}_{\mathrm{test}}(q)| \leq \eta$  for  $\eta$ -TV algorithms (Kulynych et al., 2022b). Next, we show a related result which covers non-linear queries, i.e., those that do not decompose linearly over the dataset. For this, we make use of the group privacy property of  $f$ -DP:
+
+Proposition E.1 (Dong et al., 2022). Suppose that  $M(\cdot)$  satisfies  $f$ -DP w.r.t. replace-one relation. Then, the algorithm satisfies  $f^{(k)}$ -DP w.r.t. replace-k relation  $S \simeq S'$  ( $|S| = |S'|$  but differ by exactly  $k$  points), where  $f^{(k)} = 1 - (1 - f)^{\circ k}$ , with  $f^{\circ k}$  denoting repeated  $k$ -fold composition of the function as  $f \underbrace{0 \cdots 0}_{k} f(x)$ . The function  $f^{(k)}$  is a valid trade-off curve as per Definition A.3.
+
+We can now show the result for non-linear queries:
+
+Theorem E.2. Suppose that  $M: \mathbb{D}^n \to \Theta$  satisfies  $f$ -DP w.r.t. replace-one relation. Then, for any bounded function  $R: \mathbb{D}^n \times \Theta \to [0,1]$ , and any probability distribution  $P$  over  $\mathbb{D}$ , we have:
+
+$$
+\mathbb {E} _ {\substack {S \sim P ^ {n} \\ \theta \sim M (S)}} R (S; \theta) \leq 1 - f ^ {(n)} \left(\mathbb {E} _ {\substack {S, T \sim P ^ {n} \\ \theta \sim M (S)}} R (T; \theta)\right) \tag{43}
+$$
+
+$$
+\underset {\theta \sim M (S)} {E _ {S, T \sim P ^ {n}}} R (T; \theta) \leq 1 - f ^ {(n)} \left(\underset {\theta \sim M (S)} {\mathbb {E}} _ {S \sim P ^ {n}} R (T; \theta)\right). \tag {44}
+$$
+
+Proof. Fix any  $S \in \mathbb{D}^n$  and  $T \in \mathbb{D}^n$ . By Lemma 2.1 and the group privacy property in Proposition E.1, we have:
+
+$$
+\mathbb {E} _ {\theta \sim M (S)} R (S; \theta) \leq 1 - f ^ {(n)} \left(\mathbb {E} _ {\theta \sim M (T)} R (S; \theta)\right) \tag {45}
+$$
+
+$$
+\mathbb {E} _ {\theta \sim M (T)} R (S; \theta) \leq 1 - f ^ {(n)} \left(\mathbb {E} _ {\theta \sim M (S)} R (S; \theta)\right).
+$$
+
+By using Lemma C.1 after taking the expectation over  $S \sim P^n$ , and  $T \sim P^n$  of both sides in Eq. (45), we have:
+
+$$
+\mathbb{E}_{\substack{S\sim P^{n}\\ \theta \sim M(S)}}R(S;\theta)\leq 1 - f^{(n)}\left(\mathbb{E}_{\substack{S,T\sim P^{n}\\ \theta \sim M(T)}}R(S;\theta)\right)
+$$
+
+$$
+\underset { \begin{array}{c} \theta \sim M (T) \end{array} } {\mathbb {E}} _ {S, T \sim P ^ {n}} R (S; \theta) \leq 1 - f ^ {(n)} \left(\underset {\theta \sim M (S)} {\mathbb {E}} _ {S \sim P ^ {n}} R (S; \theta)\right)
+$$
+
+We get the desired result by renaming the variables  $S$  and  $T$  in the  $\mathbb{E}_{S,T}[R(S;M(T))]$  terms.
+
+![](images/5c42cac9da04a5cc3a61502b3cc178bfc50f91616cab6894bae7c234c91de5d5.jpg)
+
+In Appendix F, we use this result to bound the notion of narcissus resiliency (Cohen et al., 2025) using  $f$ -DP. Unfortunately, the privacy guarantees obtained using this approach can quickly become vacuous, as they scale with the dataset size.
+
+# F Bounds for Other Risk Notions
+
+In this section, we show additional bounds based on  $f$ -DP for other risk notions beyond the strong-adversary notions in Section 3.2.
+
+Counterfactual memorization and influence. Using the tools in Appendices C and E, we can bound notions of memorization (Feldman, 2019; Zhang et al., 2023). First, we define cross-influence:
+
+Definition F.1 (Cross-influence). Fix a partial dataset  $\bar{S} \in \mathbb{D}^{n-1}$ , two records  $z, z' \in \mathbb{D}$ , and a bounded loss function  $\ell: \mathbb{D} \times \mathbb{D} \to [0,1]$ . We define the cross-influence of  $z'$  on  $z$  as:
+
+$$
+\operatorname {x i n f} \left(z \Leftarrow z ^ {\prime}\right) \triangleq \mathbb {E} _ {\theta \sim M \left(\bar {S} \cup \{z ^ {\prime} \}\right)} \left[ \ell (z; \theta) \right] - \mathbb {E} _ {z ^ {\prime \prime} \sim P, \theta \sim M \left(\bar {S} \cup \{z ^ {\prime \prime} \}\right)} \left[ \ell (z; \theta) \right]. \tag {46}
+$$
+
+When  $z = z'$ , we obtain the special case of counterfactual self-influence, i.e., memorization.
+
+Definition F.2 (Counterfactual memorization). Fix  $\bar{S} \in \mathbb{D}^{n-1}$  to be a partial dataset,  $z \in \mathbb{D}$  to be a fixed record,  $P$  to be a distribution over  $\mathbb{D}$ , and a bounded loss function  $\ell: \mathbb{D} \times \mathbb{D} \to [0,1]$ . We define the memorization of  $z$  as:
+
+$$
+\operatorname {m e m} (z) \triangleq \mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z \})} [ \ell (z; \theta) ] - \mathbb {E} _ {z ^ {\prime} \sim P, \theta \sim M (\bar {S} \cup \{z ^ {\prime} \})} [ \ell (z; \theta) ]. \tag {47}
+$$
+
+For the special case that  $\ell (z;\theta)$  is a decision rule implementing a membership inference attack which aims to infer the membership of  $z$ , we can write the SMIA advantage in a replace-one neighbourhood model as follows:
+
+$$
+\operatorname {a d v} _ {\mathrm {S M I A} ^ {\prime}} (z) \triangleq \underbrace {\mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z \})} [ \phi (z ; \theta) ]} _ {\text {T P R o f} \phi} - \underbrace {\mathbb {E} _ {z ^ {\prime} \sim P , \theta \sim M (\bar {S} \cup \{z ^ {\prime} \})} [ \phi (z ; \theta) ]} _ {\text {F P R o f} \phi}. \tag {48}
+$$
+
+The next result then follows by recognizing that mem recovers the definition of  $\mathsf{adv}_{\mathsf{SMIA}'}$ , and that  $\mathsf{adv}_{\mathsf{SMIA}'}$  is upper-bounded by the TV privacy parameter:
+
+Proposition F.1 ( $\eta$ -TV privacy bounds memorization/advantage). Suppose that  $M: \mathbb{D}^n \to \Theta$  satisfies  $\eta$ -TV privacy w.r.t. the replace-one neighbourhood relation. Then, for any  $n \geq 1$ , probability distribution  $P$  over  $\mathbb{D}$ , partial dataset  $\bar{S} \in \mathbb{D}^{n-1}$ , any bounded  $\ell: \mathbb{D} \times \Theta \to [0,1]$ , and any target record  $z \in \mathbb{D}$ , it holds that:
+
+$$
+\operatorname {m e m} (z) \leq \eta \text {a n d} \operatorname {a d v} _ {S M I A ^ {\prime}} (z) \leq \eta . \tag {49}
+$$
+
+Proof. Recall that  $M$  satisfies  $f$ -DP for some  $f$ . For any fixed  $z, z' \in \mathbb{D}$ , we have by Lemma 2.1:
+
+$$
+\mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z \})} q (z; \theta) \leq 1 - f \left(\mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z ^ {\prime} \})} q (z; \theta)\right),
+$$
+
+Subsequently, taking the expectation over  $z' \sim P$  and applying Lemma 3.1, we get:
+
+$$
+\mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z \})} q (z; \theta) \leq 1 - f \left(\mathbb {E} _ {z ^ {\prime} \sim P} \mathbb {E} _ {\theta \sim M (\bar {S} \cup \{z ^ {\prime} \})} q (z; \theta)\right),
+$$
+
+Finally, we obtain the bound in terms of  $\eta$  by subtracting  $E_{z^{\prime}\sim P\theta \sim M(\bar{S}\cup \{z^{\prime}\})}q(z;\theta)$  from both sides and recalling that  $\max_{\alpha \in [0,1]}(1 - f(\alpha) - \alpha)\leq \eta$  by Proposition A.1.
+
+Note that a similar connection between memorization and the advantage of membership inference attacks in an average-dataset threat model has been observed previously (Kulynych et al., 2022a).
+
+Unbiased reconstruction robustness. Next, we present a variant of reconstruction robustness with an average baseline.
+
+Definition F.3 (Unbiased RR, adapted from Guerra-Balboa et al., 2023). For a given  $n \geq 1$ , mechanism  $M: 2^{\mathbb{D}} \to \Theta$ , data distribution  $P$  over  $\mathbb{D}$ , partial dataset  $\bar{S} \in \mathbb{D}^{n-1}$ , loss function  $\ell: \mathbb{D} \times \mathbb{D} \to \mathbb{R}$ , threshold  $\gamma \in \mathbb{R}$ , and reconstruction attack  $\mathcal{A}_{M,\bar{S},\mathcal{D}}: \Theta \to \mathbb{D}$ , we define the unbiased reconstruction robustness (URR) success rate as follows:
+
+$$
+\mathsf{succ}_{URR}(M,\bar{S},P;\mathcal{A},\ell ,\gamma)\triangleq \operatorname *{Pr}_{\substack{z\sim_{P}\\ \hat{z}\gets \mathcal{A}_{M,\bar{S},P}(M(\bar{S}\cup \{z\}))}}[\ell (z,\hat{z})\leq \gamma ],
+$$
+
+and the baseline success as:
+
+$$
+\operatorname{base}_{URR}(P;\ell ,\gamma)\triangleq \operatorname *{Pr}_{\substack{z,z^{\prime}\sim P\\ \hat{z}\gets \mathcal{A}_{M,\bar{S},P}(M(\bar{S}\cup \{z^{\prime}\}))}}[\ell (z,\hat{z})\leq \gamma ].
+$$
+
+Theorem F.1. Suppose that the algorithm  $M: \mathbb{D}^n \to \Theta$  satisfies  $f$ -DP w.r.t. replace-one relation. Then, for any  $n \geq 1$ , partial dataset  $\bar{S} \in \mathbb{D}^{n-1}$ , data distribution  $P$  over  $\mathbb{D}$ , loss function  $\ell: \mathbb{D} \times \mathbb{D} \to \mathbb{R}$ , threshold  $\gamma \in \mathbb{R}$ , and any adversary  $A_{M,\bar{S},P}: \Theta \to \mathbb{D}$ , we have:
+
+$$
+\operatorname {s u c c} _ {U R R} (M, \bar {S}, P; \mathcal {A} _ {M, \bar {S}, P}, \ell , \gamma) \leq 1 - f (\operatorname {b a s e} _ {U R R} (P; \ell , \gamma)). \tag {50}
+$$
+
+Moreover,
+
+$$
+\operatorname {s u c c} _ {U R R} (M, \bar {S}, P; \mathcal {A} _ {M, S, P}, \ell , \gamma) - \operatorname {b a s e} _ {U R R} (P; \ell , \gamma) \leq \eta , \tag {51}
+$$
+
+where  $\eta$  is the TV privacy guarantee of the mechanism.
+
+Proof. We obtain Eq. (50) by applying Lemma C.3 to a mechanism  $\mathcal{A}_{M,\bar{S},P}(\theta)$ , which satisfies  $f$ -DP by post-processing, and using  $q(z;\cdot)$  chosen as in Eq. (35). We immediately get Eq. (51) by applying Proposition A.1 to Eq. (50) as in the proof of Lemma C.4.
+
+Narcissus resiliency. We present a recent definition of privacy due to Cohen et al. (2025), adapted to our setting:
+
+Definition F.4 (Narcissus resiliency, adapted from Cohen et al., 2025). For a given  $n \geq 1$ , mechanism  $M: 2^{\mathbb{D}} \to \Theta$ , data distribution  $P$  over  $\mathbb{D}$ , an adversary  $A_{n,M,P}: \Theta \to \mathbb{V}$  which outputs an element from an arbitrary set  $\mathbb{V}$ , and a bounded function  $R: \mathbb{D}^n \times \mathbb{V} \to [0,1]$ , we define the narcissus resiliency (NR) success rate as follows:
+
+$$
+\mathsf{succ}_{NR}(n,M,P;\mathcal{A},R)\triangleq \operatorname *{Pr}_{\substack{S\sim P^{n}\\ p\leftarrow \mathcal{A}_{n,M,P}(M(S))}}[R(S,v)],
+$$
+
+and the baseline success as:
+
+$$
+\operatorname{base}_{NR}(n,P;\mathcal{A},R)\triangleq \operatorname *{Pr}_{\substack{S,T\sim P^{n}\\ v\gets \mathcal{A}_{n,M,P}(M(S))}}[R(T,v)].
+$$
+
+Our generalization result for non-linear functions in Theorem E.2 immediately implies a bound on its success:
+
+Theorem F.2. Suppose that  $M: \mathbb{D}^n \to \Theta$  satisfies  $f$ -DP w.r.t. replace-one neighbourhood relation. Then, for any given  $n \geq 1$ , data distribution  $P$  over  $\mathbb{D}$ , and an adversary  $\mathcal{A}_{n,M,P}: \Theta \to \mathbb{Q}$ , we have:
+
+$$
+\operatorname {s u c c} _ {N R} (n, M, P; \mathcal {A}, R) \leq 1 - f ^ {(n)} (\operatorname {b a s e} _ {N R} (n, P; \mathcal {A}, R)).
+$$
+
+Proof. After observing that  $\mathcal{A}(M(S))$  satisfies  $f$ -DP by post-processing, the result is an immediate implication of applying Theorem E.2 to  $\mathcal{A}$  as a mechanism.
+
+Average-dataset predicate singling out. We can show the following bound on PSO based on  $f$ -DP:
+
+Theorem F.3. Suppose that  $M: \mathbb{D}^n \to \Theta$  satisfies  $f$ -DP w.r.t. replace-one neighbourhood relation. Then, for any given  $n > 1$ , data distribution  $P$  over  $\mathbb{D}$ , and an adversary  $\mathcal{A}_{n,M,P}: \Theta \to \mathbb{Q}_{P,w}$  with  $\mathbb{Q}_{P,w} \triangleq \{p \mid p: \mathbb{D} \to \{0,1\}, \mathbb{E}_P[p] = w\}$  for any given  $w \in [0,1/n]$ , we have:
+
+$$
+\operatorname {s u c c} _ {P S O} (n, M, P, w; \mathcal {A}) \leq n \cdot (1 - f (w)) \tag {52}
+$$
+
+Proof. We follow the steps of the proof in Cohen and Nissim (2020). Regardless of the weight of any given predicate  $p$ , we can modify it to have weight less than  $w$  via:
+
+$$
+p ^ {*} (x) = \left\{ \begin{array}{l l} p (x) & \quad \text {i f} \mathbb {E} _ {P} [ p ] \leq w \\ 0 & \quad \text {i f} \mathbb {E} _ {P} [ p ] > w \end{array} \right.
+$$
+
+Note that  $p^*$  is exactly  $p$  if  $p$ 's weight is  $\leq w$ , and is trivially always 0 else. So, the weight of  $p^*$  is either 0 or  $\mathbb{E}_P[p^*]$ . Moreover, by postprocessing we know that  $p^*$  is  $f$ -DP. We investigate the probability of a successful PSO attack on  $S$ , which occurs when  $p^*(S) \triangleq \frac{1}{n} \sum_{i=1}^{n} p^*(x_i) = \frac{1}{n}$  and  $\mathbb{E}_P[p^*] \leq w$ . The second condition is always satisfied via the construction of  $p^*$ , so:
+
+$$
+\begin{array}{l} \operatorname {s u c c} (n, M, P; \mathcal {A}) = \Pr_ {\substack {S \sim P ^ {n} \\ p \leftarrow M (S)}} [ p (S) = 1 / n \text{and}\mathbb {E} _ {P} [ p ]\leq w ] \\ = \operatorname *{Pr}_{\substack{S\sim P^{n}\\ p\leftarrow M(S)}}[p^{*}(S) = 1 / n] \\ \leq \sum_{i = 1}^{n}\operatorname *{Pr}_{\substack{S\sim P^{n}\\ p\leftarrow M(S)}}\left[p^{*}(z_{i}) = 1\text{and}\forall j\neq i,p^{*}(z_{j}) = 0\right], \\ \end{array}
+$$
+
+where  $S = (z_{1},z_{2},\dots ,z_{n})$
+
+Now, as  $p^*$  satisfies  $f$ -DP, we can apply Lemma 2.1 in conjunction with Lemma C.1:
+
+$$
+\leq \sum_{i = 1}^{n}1 - f\left(\operatorname *{Pr}_{\substack{S\sim P^{n}\\ z\sim P\\ p\gets M(S_{i\to z})}}[p^{*}(z_{i}) = 1\text{and}\forall i\neq j, p^{*}(z_{j}) = 0]\right),
+$$
+
+where  $i\to z$  denotes substitution of  $z_{i}$  by  $z$ . We can reparametrize without changing the expectation:
+
+$$
+= \sum_ {i = 1} ^ {n} 1 - f \left(\Pr_ {\substack {S \sim P ^ {n} \\ z \sim P \\ p \leftarrow M (S)}} \left[ p ^ {*} \left(z _ {i}\right) = 1 \text {and} \forall j \neq i, p ^ {*} \left(x _ {j}\right) = 0 \right]\right). \tag{53}
+$$
+
+Now, we can upper bound the probability inside  $f(\cdot)$  by  $w$  by Lemma C.2:
+
+$$
+\begin{array}{l} \leq \sum_ {i = 1} ^ {n} 1 - f (w) \\ = n \cdot (1 - f (w)). \\ \end{array}
+$$
+
+![](images/e2af29dad3e16172aa39b97bed2eae1684d227a58ed5d1081bf3f8fe818f10d2.jpg)
+
+Note that, to tighten the bound, we slightly modified the definition of the admissible predicate in Theorem F.3 to only consider predicates of a specific weight.
+
+![](images/de6cccb1373650d224da6fe3c464252cd162d101f19d2364b4ddd5416c81e7c4.jpg)  
+Figure 6: Our  $f$ -DP based bound enables to perform more statistical queries at the same level of re-identification or other notions of risk.
+
+# G Additional Experimental Details
+
+# G.1 Additional Case Studies
+
+Tight budget analysis in DP query answering in terms of risk. One of the common use cases of DP is answering statistical queries about a dataset (Gaboardi et al., 2020). In real-world instances of such systems, e.g., in healthcare, re-identification risk is a concern (Raisaro et al., 2018).
+
+To evaluate how our bounds apply to risk measurement in this setting, we simulate the release of multiple queries with added Laplace noise. We compare our bound to the default budget computation procedure of smartnoise software framework (Gaboardi et al., 2020), which is based on optimal composition results for  $(\varepsilon, \delta)$ -DP mechanisms (Kairouz et al., 2015). To apply our bounds, we use the direct method from Kulynych et al. (2024) to obtain the  $f$  curve of an adaptive composition of Laplace mechanisms. In this simulation, we assume a fixed noise parameter  $b = 5.0$  for each query, corresponding to  $\varepsilon = 0.2$  pure DP per query. In Fig. 6, we show how our bounds on risk can enable researchers to conduct significantly more queries while satisfying any given risk requirement, in particular, on re-identification. For example, if the requirement is to ensure at most 0.2 attack advantage under the baseline risk of 0.1 (Fig. 6, middle pane), we can issue 15 queries according to our analysis vs. 5 with the standard one.
+
+Calibrating noise to image reconstruction risk in deep learning. In this case study, as in Section 4, we assume the modeler aims to use DP training for CIFAR-10 image classification (Krizhevsky et al., 2009) to limit such risks to a given threshold. We train multiple convolutional networks using the approach of Tramer and Boneh (2021) (we provide further technical details next). We train with standard DP-SGD (Abadi et al., 2016) and use five different levels of noise. We obtain the  $f$  curve under the add-remove relation for each model using the direct method as before (Kulynych et al., 2024), and apply Theorem 3.2 to measure risk. We compare this to the RDP-based analysis from Balle et al. (2022), as described previously. Fig. 7 shows that if we aim calibrate the noise scale to a given level of maximum attack risk, our unifying bound enables to choose a lower noise scale at the same level of risk (left plot), which, in turn, results in better classification accuracy (right plot). E.g., for the risk target of 0.25, we can increase the accuracy from  $65\%$  to  $68\%$  using our analysis, as opposed to an RDP-based one.
+
+# G.2 Experiment Details
+
+We use an Nvidia GeForce RTX 4070 16 GB GPU machine for the deep learning experiments. The experiments take up to four hours to finish.
+
+Text Sentiment Classification case study details We follow Yu et al. (2021) to finetune a GPT-2 (small) (Radford et al., 2019) using LoRA (Hu et al., 2021) with DP-SGD on the SST-2 sentiment classification task (Socher et al., 2013) from the GLUE benchmark (Wang et al., 2018). We summarize the parameters next:
+
+![](images/a4a0cfed0ab53449cc0669b5db2cf30847d072e448086d883d9b7c24ef896445.jpg)  
+Figure 7: Our  $f$ -DP based bound enables us to achieve higher classification accuracy for any level of risk.
+
+![](images/8546e9ac82f44f15fd59190ab3dcf9ad52c71351a1ee090dd0f6de8a2624c75d.jpg)
+
+<table><tr><td>Parameter</td><td>Values</td></tr><tr><td>Poisson subsampling probability</td><td>≈ 0.004</td></tr><tr><td>Expected batch size</td><td>256</td></tr><tr><td>Gradient noise multiplier (σ)</td><td>{0.5715, 0.6072, 0.6366, 0.6945, 0.7498}</td></tr><tr><td>Privacy budget (ε) at δ = 10-5</td><td>{3.95, 3.2, 2.7, 1.9, 1.45}</td></tr><tr><td>Training epochs</td><td>3</td></tr><tr><td>Gradient clipping norm (Δ2)</td><td>1.0</td></tr><tr><td>LoRA dimension</td><td>4</td></tr><tr><td>LoRA scaling factor</td><td>32</td></tr></table>
+
+Image-classification case study details. We use the method of Tramer and Boneh (2021) to train a convolutional neural network. We use CIFAR-10 (Krizhevsky et al., 2009) image classification dataset with a default split. We summarize the parameters next:
+
+<table><tr><td>Parameter</td><td>Values</td></tr><tr><td>Poisson subsampling probability</td><td>≈ 0.16</td></tr><tr><td>Expected batch size</td><td>8192</td></tr><tr><td>Gradient noise multiplier (σ/Δ2)</td><td>{4,5,6,8,10}</td></tr><tr><td>Training epochs</td><td>≤ 100</td></tr><tr><td>Gradient clipping norm (Δ2)</td><td>0.1</td></tr><tr><td>Learning rate</td><td>4</td></tr><tr><td>Momentum (Nesterov)</td><td>0.9</td></tr></table>
+
+Software We use the following key open-source software:
+
+- PyTorch (Paszke et al., 2019) for implementing neural networks.  
+- opacus (Yousefpour et al., 2021) for training PyTorch neural networks with DP-SGD.  
+- numpy (Harris et al., 2020), pandas (pandas development team, 2020), and jupyter (Kluyver et al., 2016) for numeric analyses.  
+- seaborn (Waskom, 2021) for visualizations.
+
+# H Additional Figures
+
+Table 1: Comparison of average-dataset vs. strong-adversary threat models.  $n$ : dataset size,  $P$ : data distribution,  $M$ : mechanism,  $\bar{S}$ : partial dataset, poisoning cap.: whether an adversary has the capability to insert arbitrary records into the dataset, *: the dataset size is implicitly known from the size of the partial dataset  $\bar{S}$ .  
+
+<table><tr><td rowspan="2">Threat model</td><td rowspan="2">Risk notion</td><td rowspan="2">Reference</td><td colspan="4">Adv. knowledge</td><td rowspan="2">Poisoning cap.</td><td rowspan="2">Bounds from f-DP</td></tr><tr><td>n</td><td>P</td><td>M</td><td>S</td></tr><tr><td rowspan="5">Strong</td><td>SPSO</td><td>Definition 3.2</td><td>*</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>|</td></tr><tr><td>SRR (Balle et al., 2022)</td><td>Definition 3.3</td><td>*</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>Theorem 3.2</td></tr><tr><td>SAI (any k)</td><td>Definition 3.4</td><td>*</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>|</td></tr><tr><td>SAI (k=2)</td><td>Definition 3.4</td><td>*</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>Theorem D.1</td></tr><tr><td>URR (Guerra-Balboa et al., 2023)</td><td>Definition F.3</td><td>*</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>Theorem F.1</td></tr><tr><td rowspan="2">Avg. dataset</td><td>NR (Cohen et al., 2025)</td><td>Definition F.4</td><td>✓</td><td>✓</td><td>✓</td><td></td><td></td><td>Theorem F.2</td></tr><tr><td>PSO (Cohen and Nissim, 2020)</td><td>Definition 3.1</td><td>✓</td><td>✓</td><td>✓</td><td></td><td></td><td>Theorem F.3</td></tr></table>
+
+![](images/dfd80f69464ff9003c253e872abcd08982a73f3bd949f81a14613b7c85d6b18c.jpg)  
+Figure 8: PSO bounds for Laplace mechanism. See Fig. 2 for details. We do not show the results with the PSO bound based on  $f$ -DP, as they are identical to the  $(\varepsilon, \delta)$ -based bound for this mechanism.
+
+![](images/3f68362ba9693efe451f5769d75cefe396a03390cbfd8a78b88bbd47ad29822e.jpg)  
+Figure 9: The results for US Census under multiple risk baselines. See Fig. 4 for details.
+
+![](images/b463b466ed2e8a6aa430365f2482a0d5d64840e14be469328414ad228aeb41f7.jpg)
+
+![](images/5065966765d120d4915db1b429097671fc3fb7508f84e345d937047a2cc13c23.jpg)
+
+![](images/1062155bfaf1313fadbc596bd2353e70970b5f04f2fc6e57355222e3cdbabc62.jpg)  
+Method PSO (Cohen and Nissim, via  $(\varepsilon ,\delta)$  
+PSO (via  $f$ -DP)  
+Unifying (via  $f$ -DP)
+
+![](images/987e735cdf87e07903e597dc6a3bb366af2182109e5f80b586ef6ca92a7fa8f4.jpg)  
+Figure 10: Result of redoing Fig. 2 (top) and Fig. 8 (bottom) with the improved  $f$ -DP bound for PSO derived in Theorem F.3. We observe no significant gain in using this bound.
+
+![](images/5a60b26f441a5a26564eeb554f9eab2069a2d596da1870dd8bb55f2a5ee365e9.jpg)
+
+![](images/7b98395c1c0aa7081a42694bb98a2a140f470752ef0e0c6f03c8bcce9300c9a0.jpg)  
+Method PSO (Cohen and Nissim, via  $(\varepsilon ,\delta)$  PSO (via  $f$  -DP) Unifying (via  $f$  -DP)
+
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\ No newline at end of file
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+# Unifying Reconstruction and Density Estimation via Invertible Contraction Mapping in One-Class Classification
+
+Xiaolei Wang $^{1,2}$  Tianhong Dai $^{4}$  Huihui Bai $^{3}$  Yao Zhao $^{3}$  Jimin Xiao $^{1*}$
+
+$^{1}$ Xi'an Jiaotong-Liverpool University
+
+$^{2}$ University of Liverpool
+
+$^{3}$ Beijing Jiaotong University
+
+Applied Computing Technologies
+
+# Abstract
+
+Due to the difficulty in collecting all unexpected abnormal patterns, One-Class Classification (OCC) has become the most popular approach to anomaly detection (AD). Reconstruction-based AD method relies on the discrepancy between inputs and reconstructed results to identify unobserved anomalies. However, recent methods trained only on normal samples may generalize to certain abnormal inputs, leading to well-reconstructed anomalies and degraded performance. To address this, we constrain reconstructions to remain on the normal manifold using a novel AD framework based on contraction mapping. This mapping guarantees that any input converges to a fixed point through iterations of this mapping. Based on this property, training the contraction mapping using only normal data ensures that its fixed point lies within the normal manifold. As a result, abnormal inputs are iteratively transformed toward the normal manifold, increasing the reconstruction error. In addition, the inherent invertibility of contraction mapping enables flow-based density estimation, where a prior distribution learned from the previous reconstruction is used to estimate the input likelihood for anomaly detection, further improving the performance. Using both mechanisms, we propose a bidirectional structure with forward reconstruction and backward density estimation. Extensive experiments on tabular data, natural image, and industrial image data demonstrate the effectiveness of our method. Code is available at URD.
+
+# 1 Introduction
+
+Anomaly detection (AD) focuses on identifying unexpected patterns that deviate from the known normal ones. Due to difficulty of collecting abnormal samples, One-Class Classification (OCC) paradigm [38, 10, 4, 34, 41, 43, 45, 18, 49] has become the primary solution for unsupervised anomaly detection (UAD) tasks. In the OCC paradigm, only normal data is provided during training, and the model is expected to detect previously unseen anomalies during inference. Therefore, the key challenge of the OCC paradigm involves in effectively modeling the normality of the training data to recognize the abnormal pattern. Recently, reconstruction-based [12, 14, 54, 19, 30, 21, 48] and density-estimation-based [39, 26, 55, 10, 58] methods are the two most widely used OCC AD methods. The former relies on the assumption that a model trained solely on normal data will be unable to accurately reconstruct anomalies. In contrast, the latter focuses on fitting a probability distribution over normal data to estimate the likelihood of previously unseen inputs.
+
+![](images/83038f96efe8a5d7a745649620e49ab475e37a621d9a45dce271806d2890b348.jpg)
+
+![](images/76c9540022aae914ee3abf6a8b791421ec3852dabc0bf2a300f323be408f70e9.jpg)
+
+![](images/e9b11e79ba25721b1642ebc9c3f6da17e3ddb48a57ed3d7a84f4b071489c4047.jpg)
+
+![](images/4df441baffa26dbc8fd07d84fc1afd8f932a837799a4e29ccc28e65fae4376ae.jpg)  
+Figure 1: (a): Any input converges to a fixed point after several CM iterations. (b): The fixed point of CM trained by normal data is located within the normal manifold. (c): Abnormal inputs (red) are reconstructed into the normal manifold (green) by several iterations of CM. (d) T-SNE visualization of changes in the reconstructed feature manifold on the 'Ionosphere' tabular AD dataset [36].
+
+![](images/af851d79bde503a295eda4d383fe2cd8ac6d17873b439259c6423323d9b1dc44.jpg)  
+$\times$  Normal + Abnormal (d) Changes of Reconstructed Feature Manifold under Different CM Iterations
+
+![](images/9d96db5718da134449708110006eee94ad8b3366f5be1faa9331a654af9f7a94.jpg)  
++ + Reconstructed Result
+
+Specifically, reconstruction-based methods rely on the discrepancy between the input and its reconstructed output to identify unseen anomalies. However, models trained exclusively on normal data may generalize to certain abnormal inputs, resulting in well-reconstructed anomalies and reduced reconstruction error, which affects the performance. To address this issue, previous AD methods [54, 53] adopt reconstruction strategies based on partially masked data. However, when using handcrafted or random mask strategies, not all abnormal information can be covered, so they still cannot prevent well-reconstructed anomalies and guarantee the effectiveness of all obtained reconstruction discrepancies. Thus, the key question remains: How can reconstruction discrepancies be made sufficiently discriminative for detecting each abnormal input?
+
+An intuitive way to obtain discriminative reconstruction discrepancy is to constrain reconstructed results of abnormal inputs remain on the normal manifold. To this end, we propose a novel reconstruction network based on contraction mapping (CM). Based on the property of CM [16], any input is theoretically guaranteed to converge to a unique fixed point through iterative applications of a CM (see Fig.1 (a)), and this point depends exclusively on the mapping itself. Inspired by the property of CM, we further observe during training that optimizing our CM-based network with a reconstruction loss on normal data encourages the fixed point to lie within the normal manifold (see Fig.1 (b)). Therefore, given abnormal inputs, we can leverage trained CM iteratively to transform them into the normal manifold (see Fig.1 (c)). T-SNE visualization of real data in Fig.1 (d) also shows that the reconstructed results are constrained to remain on the normal manifold after six iterations of CM, resulting in a significant discrepancy between the abnormal input and its reconstruction.
+
+On the other hand, our proposed CM structure is inherently invertible. According to the theory of normalizing flow [13, 27], an invertible mapping flow can transform a Gaussian prior into a real data distribution, which enables the estimation of the likelihood of the input. Therefore, our invertible CM flow is leveraged to perform density estimation to further improve the detection performance. In particular, after the reconstruction stage, the reconstructed output is used to learn a prior Gaussian distribution. Then, we leverage the inverse of the contraction mapping flow on the prior distribution to estimate the likelihood of the inputs. To this end, our framework consists of forward reconstruction and backward density estimation, unifying these two popular paradigms to detect unseen anomalies. The main contributions of this paper are summarized as follows:
+
+(1) To prevent well-reconstructed anomalies, we adopt a novel contraction mapping structure to constrain abnormal inputs onto the normal manifold. (2) A unified framework is introduced that
+
+combines reconstruction and density estimation using an invertible contraction mapping flow, which bridges two popular paradigms in anomaly detection. (3) Extensive experiments on tabular, natural image, and industrial image data demonstrate the effectiveness of our method.
+
+# 2 Related Work
+
+Anomaly Detection Anomaly detection aims to recognize unexpected abnormal patterns in different types of data, such as tabular [53, 9], graph [57], and image data [47, 8, 44]. Since the form and type of anomaly events are unpredictable and cannot be fully collected, One-Class-Classification-based AD paradigm has become the mainstream solution. The key of OCC approaches is to model the normality of training data to detect unseen anomalies. Classic OCC methods [35, 6, 40, 46, 38, 57] learn a profile from normal data, computing the distance between the abnormal input and the center/boundary of the profile as the anomaly score. KNN [35] and LOF [6] directly compute the distance between a test input and its several normal neighbors. OCSVM [40] learn a linear boundary of normal samples to discriminate anomalies. SVDD-based approaches [46, 38, 57] construct hypersphere boundaries to effectively surround normal representations, recognizing abnormal patterns based on the distance between these representations and the center/boundary of the sphere.
+
+Reconstruction-based Method Reconstruction-based methods rely on the discrepancy of inputs and their reconstructed results to identify unseen anomalies. However, a trained reconstruction network may recover some anomalies well, obtaining a small reconstruction error and failing to detect anomalies. To alleviate this issue, the tabular AD method, MCM[53] adopts a learnable mask to cover partial areas of the original input and learning normal reconstruction from masked data. UniAD [54], a classic visual AD method, defines this issue as 'identical shortcut' and similarly adopts the neighbor mask in attention computation to alleviate well-reconstructed anomalies. The graph-based AD method [22] leverages the denoising autoencoder to learn the reconstruction for the noisy perturbation. In this paper, we consider that as long as the reconstructed output can be constrained within the normal manifold, well-reconstructed anomalies will be avoided. Based on the above idea, we propose to apply contraction mapping to conduct reconstruction.
+
+Density-Estimation-based Method Density-estimation-based methods [50, 58] apply normal data to fit a Gaussian distribution or its variants to estimate the likelihood of each test input as the anomaly score. However, the real normal data distribution is much more complex than a Gaussian distribution. Therefore, normalizing-flow-based AD methods [51, 39, 55, 17] adopt invertible mapping to transform a prior distribution of latent variables to the distribution of real inputs, better capturing the probability density of each input. In this paper, the introduced contraction mapping flow is designed to be invertible and can directly estimate the likelihood of input based on a learnable Gaussian prior distribution of reconstructed results. Our framework consists of forward reconstruction and backward density estimation, unifying these two main paradigms.
+
+# 3 Methodology
+
+Preliminary In the one-class classification (OCC) anomaly detection (AD) task, only normal samples are provided during training, and the goal is to detect previously unseen anomalies in the testing set. Formally,  $\mathcal{I} = \mathcal{I}^{\mathrm{train}}\cup \mathcal{I}^{\mathrm{test}}$  is represented as an AD dataset, where the data can be either tabular or image-based. The training set is denoted as  $\mathcal{I}^{\mathrm{train}} = \{I_i,y_i\}_{i = 1}^{N_1}$ , where each  $I_{i}$  is a normal sample with label  $y_{i} = 0$ , and  $N_{1}$  is the number of samples. The testing set is denoted as  $\mathcal{I}^{\mathrm{test}} = \{I_i^t,y_i^t\}_{i = 1}^{N_2}$ , which contains both normal and abnormal samples,  $y_{i}^{t}\in \{0,1\}$ , and  $N_{2}$  is the number of testing samples.
+
+Overview As shown in Fig.2, our proposed framework consists of two main components: 1) forward reconstruction and 2) backward density estimation. This unified design integrates both reconstruction-based and density-based anomaly detection within a single framework, resulting in enhanced detection performance. During training, the normal input  $I$  is first passed through a standard neural network to achieve the corresponding representations  $\mathbf{x} \in \mathbb{R}^d$ , where  $d$  is the dimension of the vector. Next, to mitigate the issues of 'reconstructing anomalies well', a novel contraction mapping (CM) based network  $\varphi(\cdot)$  is proposed. This network is trained with a reconstruction loss on
+
+![](images/3839489b6219a62fc9853f3bffc17587c0d2c9619d8060cfa702c6db6e7a93c6.jpg)  
+Figure 2: Overview of our proposed method. For a normal input, we first extract its deep representation. Next, we apply contraction mapping flow to reconstruct the obtained representation. Then, reconstructed outputs are used to fit a prior distribution. Finally, the reverse of the mapping flow can transform this prior to the input distribution, enabling likelihood estimation of the original input.
+
+normal samples to constrain reconstructed results to lie within the normal manifold. Specifically, the contraction mapping is used in a flow-based manner to iteratively constrain the reconstruction within the normal manifold, and the final reconstructed result is denoted as  $\mathbf{z} \in \mathbb{R}^d$ . To further exploit the inherently invertibility of CM, we leverage the reverse mapping  $\varphi^{-1}(\cdot)$  to estimate the density  $p(\mathbf{x})$  of the input  $\mathbf{x}$  from a learnable prior distribution over the latent variable  $\mathbf{z}$ . Then, the flow-based reverse mapping  $\varphi^{-1}(\cdot)$  gradually transforms the prior Gaussian distribution  $\pi(\cdot)$  to the density  $p(\cdot)$ , which captures the likelihood of the real input  $\mathbf{x}$ . During inference, a combination of reconstruction error and likelihood is adopted as the ultimate anomaly score.
+
+# 3.1 Reconstruction via Contraction Mapping
+
+Anomaly detection via reconstruction relies on the discrepancy between the input and its reconstruction. Typically, reconstruction-based methods employ the following objective function to train the reconstruction network  $\varphi (\cdot)$ :
+
+$$
+\mathcal {L} _ {\text {r e c}} = \left\| \mathbf {z} - \mathbf {x} \right\| _ {2}, \quad \mathbf {z} = \varphi (\mathbf {x}), \tag {1}
+$$
+
+where  $\mathbf{x}$  is the input feature,  $\mathbf{z}$  is the corresponding reconstruction, and  $\| \cdot \| _2$  is  $\ell_2$ -norm. During inference, the reconstruction error  $\| \varphi (\mathbf{x}^t) - \mathbf{x}^t\| _2$  between the input feature  $\mathbf{x}$  and its reconstruction is adopted as the anomaly score for the test data. However, as discussed in Section 1, recent methods often generalize to anomaly inputs, producing low reconstruction error and thus failing to identify anomalies efficiently. To address this issue, we model  $\varphi (\cdot)$  as a contraction mapping with a flow-based reconstruction process, enforcing that the reconstructed output from any input remains on the normal manifold. This leads to larger reconstruction errors when receiving anomaly inputs.
+
+# 3.1.1 Theoretical Analysis of Contraction Mapping Reconstruction
+
+In this section, we provide formal mathematical proofs to demonstrate how the contraction mapping (CM) constrains the reconstruction to remain on the normal manifold. Specifically, we prove two key propositions: 1) training the proposed CM-based network with a reconstruction loss on normal data ensures that the fixed point of the CM lies within the normal manifold, and 2) the trained CM network can transform the reconstruction of abnormal inputs toward the normal manifold.
+
+First, we introduce the formal definition and the property of the contraction mapping.
+
+Definition 1 (Contraction Mapping [25]): A mapping  $f: \mathbb{R}^d \longrightarrow \mathbb{R}^d$  is called a contraction mapping if there exists a constant  $\eta \in [0,1)$  such that  $\| f(x_1) - f(x_2)\|_2 \leq \eta \| x_1 - x_2\|_2$  for all  $x_1, x_2 \in \mathbb{R}^d$ . The constant  $\eta$  is called the Lipschitz constant.
+
+Lemma 1 (Convergence of Iterations [25]): Let  $f(\cdot)$  be a contraction mapping with  $\eta \in [0,1)$ . For any initial input  $x_0 \in \mathbb{R}^d$ , we define the sequence  $\{x_n\}$  by  $x_{n+1} = f(x_n)$  for  $n = 1, 2, \dots$ . Then  $\lim_{n \to \infty} x_n = x^*$ , where  $x^*$  is unique and is called a fixed point of  $f(\cdot)$ .
+
+![](images/5df22482ddc3dd23e192c6730d8ee5e0ac38a49a161c8b5a66d7592ad572c253.jpg)  
+(a) Cardio
+
+![](images/661da9aeec8e197ba6aade6b52a0fa9ad54f151f7e261bb59f278dc0d1a418b9.jpg)  
+(b) Pima  
+Figure 3: T-SNE visualization of the fixed point on four AD datasets. Training  $f(\cdot)$  with  $\mathcal{L}_{\mathrm{rec}}$  on normal data ensures that the fixed point of  $f(\cdot)$  lies within the normal manifold.
+
+![](images/e3a6cb57d865ad26de15f2973957ebda9cc005ff742de4ed5c3e95f2dc12824d.jpg)  
+(c) Thyroid
+
+![](images/d061b34151dfa2f483833d010a7567f75ee268ea4c45d68483a8d6f64ee4ae71.jpg)  
+(d) Wbc
+
+Lemma 1 shows that any input will converge to a fixed point after iterative application of a contraction mapping (see Fig.1 (a)), where the point is only related to that mapping itself. Next, the contraction mapping is modeled by a learnable network  $f^{\theta}(\cdot)$ , where  $\theta$  denotes the parameters. To simplify the notation,  $\theta$  is omitted. Empirically, we observe that training  $f(\cdot)$  on normal data using reconstruction loss  $\mathcal{L}_{\mathrm{rec}} = \| f(\mathbf{x}) - \mathbf{x}\|_2$  result in a fixed point that lies within the normal manifold. To theoretically support this observation, we propose the following proposition:
+
+Proposition 1 Let  $f(\cdot)$  be a contraction mapping with  $\eta \in [0,1)$ , and let  $\mathbf{x} \in \mathbb{R}^d$  be a normal sample. Under the optimization of the reconstruction loss  $\mathcal{L}_{\mathrm{rec}} = \| f(\mathbf{x}) - \mathbf{x}\|_2$ , the fixed point  $\mathbf{x}^* \in \mathbb{R}^d$  of  $f(\cdot)$  is located within a small neighborhood of the normal input  $\mathbf{x}$ , and  $\| \mathbf{x}^* - \mathbf{x}\|_2 \leq \frac{\mathcal{L}_{\mathrm{rec}}}{1 - \eta}$ .
+
+Proof. Let us assume that  $f(\cdot)$  is a contraction mapping iteration with  $L \in [0,1)$ . According to Definition 1 and Lemma 1, one inequality can be obtained:  $\| \mathbf{x}^* - f(\mathbf{x}) \|_2 = \| f(\mathbf{x}^*) - f(\mathbf{x}) \|_2 \leq \eta \| \mathbf{x}^* - \mathbf{x} \|_2$ . Next, using the triangle inequality, we obtain  $\| \mathbf{x}^* - \mathbf{x} \|_2 = \| \mathbf{x}^* - f(\mathbf{x}) + f(\mathbf{x}) - \mathbf{x} \|_2 \leq \| \mathbf{x}^* - f(\mathbf{x}) \|_2 + \| f(\mathbf{x}) - \mathbf{x} \|_2$ . Combining the above two inequalities, one can be obtained:  $\| \mathbf{x}^* - \mathbf{x} \|_2 \leq \frac{1}{1 - \eta} \| f(\mathbf{x}) - \mathbf{x} \|_2 = \frac{\mathcal{L}_{\mathrm{rec}}}{1 - \eta}$ . When  $\mathcal{L}_{\mathrm{rec}}$  is minimized during training, the fixed point  $\mathbf{x}^*$  must be in the vicinity of the normal input  $\mathbf{x}$ .
+
+Proposition 1 provides a theoretical analysis of the reconstruction for a single sample. In practice, during training, all normal samples are used to optimize the network  $f(\cdot)$ , which drives the fixed point  $\mathbf{x}^*$  to converge to the interior of the normal manifold. The empirical visualizations in Fig. 3 on real datasets strongly support this theoretical claim, showing the fixed point landing within the normal data manifold.
+
+Next, based on Proposition 1 and Lemma 1, any input can be iteratively mapped to a fixed point within the normal manifold by applying  $f(\cdot)$  recursively. Therefore, inspired by normalizing flow [13], we adopt a flow-based module  $\varphi_K(\cdot) = f_K \circ f_{K-1} \circ \dots \circ f_1(\cdot)$  to model the iterative process of the contraction mapping, where each  $f_k(\cdot)$  shares the same structure but has its own parameters  $\theta_k$ . Given a test abnormal input  $\mathbf{x}^t$ , the flow-based module  $\varphi_K(\cdot)$  can transform it toward a fixed point  $\mathbf{x}^*$  within the normal manifold, thus inducing a large reconstruction error  $\| \varphi_K(\mathbf{x}^t) - \mathbf{x}^t \|_2$ . To theoretically support this perspective, we introduce a key proposition along with a related lemma:
+
+Lemma 2 Let  $\varphi_K(\cdot) = f_K\circ f_{K - 1}\circ \dots f_1(\cdot)$ , where each  $f_{k}$  is contraction mapping with  $\eta_k\in [0,1)$ , then  $\varphi_K(\cdot)$  is a contraction mapping with Lipschitz constant  $\eta_{\varphi}\leq \eta_{1}\cdot \eta_{2}\cdot \cdot \cdot \eta_{K}$ .
+
+Proposition 2 Let  $\varphi_K(\cdot) = f_K \circ f_{K-1} \circ \dots \circ f_1(\cdot)$  is a contraction mapping flow trained by reconstruction loss  $\mathcal{L}_{\mathrm{rec}} = \|\varphi_K(\mathbf{x}) - \mathbf{x}\|_2$ , where  $\mathbf{x}$  is normal. The fixed point  $\mathbf{x}^*$  of  $\varphi_K(\cdot)$  is located within a small neighborhood of  $\mathbf{x}$ . Additionally, for any abnormal input  $\mathbf{x}^t$ , as the value of  $K$  increases,  $\varphi_K(\mathbf{x}^t)$  is closer to the fixed point  $\mathbf{x}^*$ , thereby being closer to normal objective  $\mathbf{x}$ .
+
+Proof. According to Lemma 2,  $\varphi_K(\cdot)$  is a contraction mapping with Lipschitz constant  $\eta_{\varphi}$ , so it also exists a unique fixed point  $\mathbf{x}^*$ . According to Proposition 1, under optimization of  $\mathcal{L}_{\mathrm{rec}}$ , one inequality can be obtained:  $\| \mathbf{x}^* - \mathbf{x} \|_2 \leq \frac{\mathcal{L}_{\mathrm{rec}}}{1 - \eta_\varphi}$ , where  $\eta_{\varphi} \leq \eta_1 \cdot \eta_2 \cdots \eta_K < 1$ . Therefore, as minimizing  $\mathcal{L}_{\mathrm{rec}}$  the fixed point  $\mathbf{x}^*$  must be in the vicinity of normal  $\mathbf{x}$ . Next, for an abnormal input  $\mathbf{x}^t$ , the distance between its reconstructed result  $\varphi_K(\mathbf{x}^t)$  and the fixed point  $\mathbf{x}^*$  is:  $\| \varphi_K(\mathbf{x}^t) - \mathbf{x}^* \|_2 \leq \eta_{\varphi} \| \mathbf{x}^t - \mathbf{x}^* \|_2$ . Therefore, as  $K$  increases,  $\eta_{\varphi}$  decreases and  $\varphi_K(\mathbf{x}^t)$  is closer to the fixed point. Moreover, according to triangle inequality and above inequality, one can be obtained:  $\| \varphi_K(\mathbf{x}^t) - \mathbf{x} \|_2 \leq \| \varphi_K(\mathbf{x}^t) - \mathbf{x}^* \|_2 + \| \mathbf{x}^* - \mathbf{x} \|_2 \leq \eta_{\varphi} \| \mathbf{x}^t - \mathbf{x}^* \|_2 + \frac{\mathcal{L}_{\mathrm{rec}}}{1 - \eta_{\varphi}}$ . As  $K$  increases,  $\eta_{\varphi}$  decreases, reducing the first term. The second term also decreases due to  $1 / (1 - \eta_{\varphi})$  where  $\mathcal{L}_{\mathrm{rec}}$  is a small constant through training.
+
+With the support of Proposition 2, any abnormal input can be transformed into its corresponding normal representation by  $\varphi_K(\cdot)$ , resulting in more discriminative reconstruction errors and improved detection performance. Fig.1 (d) shows a T-SNE visualization of the iterative mapping results of  $\varphi_K(\cdot)$  for different values of  $K$ . Proofs of Lemma 1 and 2 are provided in the appendix.
+
+Since the mapping  $\varphi_K(\cdot)$  is trained for reconstruction by minimizing the loss on normal samples, it effectively behaves as an approximate identity mapping for normal inputs. It also retains its contraction property, pulling abnormal inputs toward the fixed point, which is clearly illustrated in Fig.4. Moreover, in practice,  $K$  is kept small to prevent all inputs from converging to the fixed point.
+
+# 3.1.2 Modeling of Contraction Mapping Iteration
+
+In this section, based on the above theoretical analysis, a detailed description of the structure of the flow-based contraction mapping  $\varphi(\cdot)$  is provided. First, inspired by [3, 1], each contraction mapping unit is represented as  $f(\cdot)$ :
+
+$$
+f (x) = \left(\frac {\mathbf {I d} + \mathbf {G}}{2}\right) ^ {- 1} (x) - x, \tag {2}
+$$
+
+where  $\mathbf{G}$  is designed as a Lipschitz-continuous invertible mapping with Lipschitz constant  $\eta < 1$ . For practical implementation, a block-wise approach is used to build  $\varphi(\cdot)$ . Specifically, a contraction mapping flow module  $F$  is constructed as a composition of  $K$  identical mapping units, i.e.,  $F(\cdot) = f_{K} \circ f_{K-1} \circ \dots \circ f_{1}(\cdot)$ . Finally, the contraction mapping flow  $\varphi(\cdot)$  can be written as:
+
+$$
+\varphi (\cdot) = F _ {R} \circ F _ {R - 1} \dots F _ {1} (\cdot), \tag {3}
+$$
+
+where  $R$  is the number of flow modules. As a reconstruction network, the primary objective is to learn a contraction mapping flow  $\varphi(\cdot)$  via Eq.(1), which enables the reconstruction of abnormal inputs toward the normal manifold without forcing full convergence to the final fixed point. Thus,  $K$  and  $R$  are kept relatively small to avoid full convergence and maintain the detection performance.
+
+In addition, to implement  $\varphi(\cdot)$ , i-DenseNet [33] is applied to parametrize the mapping  $\mathbf{G}$ . The basic layer of i-DenseNet can be written as  $\psi(\mathbf{W}x + \mathbf{b})$ , where weight  $\|\mathbf{W}\|_2 < 1$ ,  $\|\cdot\|_2$  denotes the spectral norm ( $\mathbf{W}$  is a matrix), and  $\psi$  is 1-Lipschitz activation function. The above constraints ensure that the Lipschitz constant of  $\mathbf{G}$  is less than 1. More details about the implementation of invertible neural network  $\varphi(\cdot)$  are provided in the appendix.
+
+# 3.2 Density Estimation via Reversed Flow
+
+Based on Eq.(2), the contraction mapping  $f(\cdot)$  is an invertible mapping and  $f^{-1}(x) = ((\mathbf{Id} - \mathbf{G}) / 2)^{-1}(x) - x$  (the proof can be found in [1]). Therefore, the flow module  $\varphi(\cdot)$  is also invertible and  $\varphi^{-1}(\cdot)$  can be written as  $\varphi^{-1}(\cdot) = F_1^{-1} \circ F_2^{-1} \cdots F_R^{-1}(\cdot)$ .
+
+As established in the framework of normalizing flows [13, 27],  $\varphi^{-1}(\cdot)$  is able to transform a Gaussian prior distribution into a real data distribution, allowing for estimating the likelihood of the input. In this section, the previous reconstructed output  $\mathbf{z}$  is used to learn a Gaussian prior distribution  $\pi (\mathbf{z})$ . Then, the reversed flow  $\varphi^{-1}(\cdot)$  is applied on the prior distribution  $\pi (\mathbf{z})$  to achieve the log-likelihood  $\log p(\mathbf{x})$  of the input  $\mathbf{x}$ :
+
+$$
+\log p (\mathbf {x}) = \log \pi (\mathbf {z}) - \sum_ {r = 1} ^ {R} \log | \det  \left(\mathbf {J} _ {F _ {r} ^ {- 1}} \left(\mathbf {z} _ {r - 1}\right) \right|, \tag {4}
+$$
+
+where  $|\operatorname{det}(\cdot)|$  is the magnitude of the determinant and  $\mathbf{J}_{F_r^{-1}}$  is the Jacobian matrix of each  $F_r^{-1}$ , each  $\mathbf{z}_r$  is the output of the module  $F_r^{-1}$ , i.e.,  $\mathbf{z}_{r-1} = F_r^{-1}(\mathbf{z}_r)$ , and  $\mathbf{z}_R = \mathbf{z}$ . Next, we will show more details of prior distribution  $\pi(\cdot)$  and optimization objective.
+
+Prior Distribution  $\pi (\cdot)$  According to Eq.(4), we need to compute  $\log \pi (\mathbf{z})$  for the final  $\log p(\mathbf{x})$ . Given a reconstructed output  $\mathbf{z}$ , the learnable parameters  $\mu$  and  $\Sigma$  of the prior Gaussian distribution  $\pi (\cdot)$  are updated through maximizing  $\log \pi (\mathbf{z})$ , where  $\mathbf{z}\sim \pi (\mathbf{z})$ . A stop-gradient operation  $\mathrm{sg}(\cdot)$  is employed during the optimization of prior distribution  $\pi (\cdot)$ , which allows it to be learned independently without affecting the reconstruction training:
+
+$$
+\mathcal {L} _ {\text {p r i o r}} = \mathbb {E} [ - \log \pi (\operatorname {s g} (\mathbf {z})) ], \tag {5}
+$$
+
+where  $\mathbb{E}[\cdot ]$  represents the mathematical expectation operator. Next, the distribution  $p(\mathbf{x})$  is estimated by transforming the prior distribution  $\pi (\mathbf{z})$  through  $\varphi^{-1}(\cdot)$
+
+**Optimization Objective** Based on normalizing flows [13], we adopt maximizing log likelihood to train flow  $\varphi(\cdot)$ . According to Eq.(4),  $\mathcal{L}_{\mathrm{MLE}}$  implicitly optimizes both the prior distribution  $\pi(\cdot)$  and the parameters of our INN  $\varphi(\cdot)$ , i.e.,
+
+$$
+\mathcal {L} _ {\mathrm {M L E}} = \mathbb {E} [ - \log p (\mathbf {x}) ] = \mathbb {E} [ - \log \pi (\operatorname {s g} (\mathbf {z})) ] + \mathbb {E} \left[ \sum_ {r = 1} ^ {R} \log | \det  \left(\mathbf {J} _ {F _ {r} ^ {- 1}} \left(\mathbf {z} _ {r - 1}\right) \right| \right] \tag {6}
+$$
+
+Combining Eq.(1) and (6), the ultimate loss function is defined as:
+
+$$
+\mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {\text {r e c}} + \lambda \mathcal {L} _ {\text {M L E}}, \tag {7}
+$$
+
+where  $\lambda$  is a hyper-parameter. The entire framework is trained in an end-to-end manner.
+
+# 3.3 Inference and Anomaly Score
+
+During inference, the feature vector  $\mathbf{x}^t$  of a given input  $I^t$  is obtained through the feature extraction network. Then, in the forward reconstruction, the reconstruction  $\mathbf{z}^t$  is achieved by using the contraction mapping flow  $\varphi(\cdot)$  to get the reconstruction error. Next, the log-likelihood  $\log p(\mathbf{x}^t)$  of the input is computed according to Eq.(4) and  $\log \pi(\mathbf{x})$ . Therefore, the final anomaly score is represented as:
+
+$$
+S \left(\mathbf {x} ^ {t}\right) = \| \varphi \left(\mathbf {x} ^ {t}\right) - \mathbf {x} ^ {t} \| _ {2} - \alpha \log p \left(\mathbf {x} ^ {t}\right), \tag {8}
+$$
+
+where  $\alpha$  is a weight coefficient to control the scale of  $\log p(\mathbf{x}^t)$ .
+
+# 4 Experiment
+
+Datasets In this paper, our method is evaluated on anomaly detection (AD) tasks across tabular, graph, and image domains. For tabular AD, we select 12 benchmark datasets from OODS [36] and ADBench [20], which include diverse fields, such as healthcare, finance, and social sciences, etc. For visual AD tasks, we conduct experiments on both natural and industrial datasets: CIFAR-10 and MVTec AD [5], respectively. Following previous OCC works [38, 57], only one class in CIFAR-10 is selected as the normal class, while the remaining classes are considered abnormal.
+
+Evaluation Metrics For tabular data, we adopt the Area Under the Receiver Operating Characteristic Curve (AUROC) and Area Under the Precision-Recall Curve (AP) as evaluation metrics. For image data, image-level AUROC (I-AUROC) is used for anomaly classification and pixel-level AUROC is used for anomaly localization.
+
+Implementation Details The implementation is based on PyTorch. In the tabular AD task, we use a three-layer Multilayer Perceptron (MLP) with LeakyReLU activation function as a feature extractor. The flow-based structure is modeled using  $K = 2$  and  $R = 3$ . In particular, the hyperparameters are set to  $\lambda = 0.5$  and  $\alpha = 0.01$  across 20 tabular datasets. During training, Adam optimizer is used to update the network, where learning rate and weight decay are  $1 \times 10^{-5}$  and  $1 \times 10^{-7}$ , respectively. For natural image datasets, CIFIAR-10, we use the same convolutional neural network (CNN) as DO2HSC [57] to extract the deep representation of each image. Same as the tabular setting, we set  $K = 2$ ,  $R = 3$ ,  $\lambda = 0.5$ , and  $\alpha = 0.001$ . The learning rate and weight decay are set to  $2 \times 10^{-5}$  and  $1 \times 10^{-7}$ , respectively. We train the model for 50 epochs with a batch size of 32. For the industrial dataset MVTec, a frozen WideResNet-50 [23] is used as the backbone for both the encoder and decoder. For fair comparison, each image is resized to  $224 \times 224$  without any data augmentation. We select  $K = 3$ ,  $R = 3$ ,  $\lambda = 0.5$ , and  $\alpha = 0.001$  in the framework. During training, the learning rate and the weight decay are selected as  $5 \times 10^{-4}$  and  $1 \times 10^{-6}$ , respectively. We train the model for 100 epochs with a batch size of 16. All experiments are conducted on a single NVIDIA RTX 3090 24GB GPU. More details can be found in the appendix.
+
+# 4.1 Main Results
+
+Tabular OCC Task Tab.1 demonstrates that our framework outperforms previous state-of-the-art (SOTA) methods [29, 6, 40, 38, 58, 42, 53] and achieves a competitive performance with an AP of 78.64 and an AUROC of 91.19. Compared to density-estimation-based methods [6, 38, 58], our framework adopts the property of the invertible mapping to compute the log-likelihood of each input
+
+Table 1: Comparison of AP/AUROC results with the previous methods on 20 different tabular datasets. Following MCM [53], all datasets are implemented with identical dataset partitioning. 'DSVDD' stands for 'DeepSVDD'. The best result is highlighted in bold.  
+
+<table><tr><td>Dataset</td><td>IForest[29]</td><td>LOF[6]</td><td>OCSVM[40]</td><td>DSVDD[38]</td><td>DGMM[58]</td><td>ICL[42]</td><td>MCM[53]</td><td>Ours</td></tr><tr><td>Arrhythmia</td><td>50.97/77.34</td><td>52.77/76.88</td><td>53.39/76.89</td><td>60.36/79.40</td><td>46.68/72.83</td><td>61.55/81.45</td><td>61.07/81.14</td><td>59.63/77.56</td></tr><tr><td>Breastw</td><td>94.49/97.19</td><td>99.23/99.37</td><td>99.34/99.38</td><td>99.24/99.25</td><td>75.84/76.05</td><td>96.56/97.25</td><td>99.52/99.55</td><td>99.33/99.43</td></tr><tr><td>Cardio</td><td>70.18/92.20</td><td>83.60/95.62</td><td>86.14/96.54</td><td>78.80/93.13</td><td>30.89/69.51</td><td>80.37/95.14</td><td>84.89/96.03</td><td>86.35/96.21</td></tr><tr><td>Census</td><td>13.57/60.15</td><td>23.43/71.10</td><td>22.79/71.58</td><td>15.14/63.27</td><td>10.66/45.06</td><td>19.49/68.31</td><td>24.20/75.81</td><td>37.68/80.64</td></tr><tr><td>Campaign</td><td>46.08/72.69</td><td>44.59/70.61</td><td>47.49/76.26</td><td>25.29/52.54</td><td>24.72/57.85</td><td>45.06/72.41</td><td>60.40/89.02</td><td>55.31/84.49</td></tr><tr><td>Cardiotoco.</td><td>60.36/72.49</td><td>57.32/64.49</td><td>66.19/75.22</td><td>46.02/51.49</td><td>44.40/61.01</td><td>59.55/64.78</td><td>69.93/80.01</td><td>73.34/81.17</td></tr><tr><td>Fraud</td><td>69.39/96.30</td><td>40.46/95.73</td><td>34.90/95.48</td><td>33.20/95.19</td><td>0.99/72.71</td><td>59.60/91.07</td><td>51.41/93.57</td><td>65.48/94.25</td></tr><tr><td>Glass</td><td>9.52/57.71</td><td>9.23/56.20</td><td>8.96/54.80</td><td>9.12/55.66</td><td>10.19/59.82</td><td>25.73/83.50</td><td>19.05/72.25</td><td>50.67/86.32</td></tr><tr><td>Ionosphere</td><td>97.68/96.83</td><td>95.91/94.54</td><td>89.69/87.65</td><td>86.70/85.52</td><td>70.46/70.41</td><td>97.77/97.10</td><td>98.02/97.26</td><td>97.96/97.34</td></tr><tr><td>Mammo.</td><td>33.34/82.20</td><td>40.63/89.22</td><td>41.78/90.03</td><td>41.90/88.79</td><td>11.41/74.12</td><td>18.94/65.48</td><td>47.55/90.53</td><td>54.32/89.87</td></tr><tr><td>NSL-KDD</td><td>75.32/73.88</td><td>74.50/54.96</td><td>75.29/57.07</td><td>79.35/75.51</td><td>75.04/61.36</td><td>51.94/16.84</td><td>90.85/87.80</td><td>87.20/85.18</td></tr><tr><td>Optdigits</td><td>15.70/82.39</td><td>43.63/96.65</td><td>6.92/63.38</td><td>11.59/76.03</td><td>5.36/47.06</td><td>16.96/7.87</td><td>88.85/99.47</td><td>84.32/97.61</td></tr><tr><td>Pima</td><td>66.62/67.37</td><td>69.70/69.13</td><td>70.08/71.33</td><td>71.65/73.48</td><td>59.56/61.08</td><td>69.65/67.27</td><td>73.89/76.39</td><td>70.51/74.36</td></tr><tr><td>Pendigits</td><td>51.33/96.66</td><td>78.55/99.05</td><td>51.78/96.36</td><td>6.16/45.63</td><td>4.41/39.82</td><td>40.39/91.42</td><td>82.58/99.19</td><td>86.43/99.24</td></tr><tr><td>Satellite</td><td>85.83/80.26</td><td>80.88/73.91</td><td>77.78/66.63</td><td>82.17/76.59</td><td>68.66/72.59</td><td>87.99/85.49</td><td>85.32/79.62</td><td>84.95/82.00</td></tr><tr><td>Satimage-2</td><td>88.46/99.38</td><td>96.92/99.61</td><td>91.92/98.17</td><td>94.27/98.81</td><td>11.42/89.94</td><td>81.24/97.92</td><td>98.50/99.92</td><td>97.54/99.83</td></tr><tr><td>Shuttle</td><td>91.72/99.61</td><td>96.01/99.83</td><td>94.88/99.69</td><td>98.18/99.52</td><td>48.75/90.49</td><td>98.11/99.35</td><td>94.79/99.75</td><td>98.63/99.92</td></tr><tr><td>Thyroid</td><td>60.55/92.71</td><td>78.92/98.56</td><td>81.34/98.55</td><td>72.82/98.87</td><td>10.95/71.49</td><td>65.75/95.18</td><td>84.17/98.04</td><td>88.37/99.21</td></tr><tr><td>Wbc</td><td>85.73/97.15</td><td>84.12/96.70</td><td>83.91/96.67</td><td>83.40/96.33</td><td>29.59/79.99</td><td>72.18/90.80</td><td>88.87/98.14</td><td>94.72/99.24</td></tr><tr><td>Wine</td><td>24.58/65.71</td><td>12.53/40.83</td><td>14.24/4 .85</td><td>14.76/50.67</td><td>49.07/88.33</td><td>56.59/91.50</td><td>93.35/95.38</td><td>99.99/100.0</td></tr><tr><td>Average</td><td>59.57/83.01</td><td>63.15/82.15</td><td>59.94/81.01</td><td>55.51/77.78</td><td>34.45/68.08</td><td>60.27/81.55</td><td>74.86/90.44</td><td>78.64/91.19</td></tr></table>
+
+Table 2: Anomaly classification results (AUROC) on CIFAR-10 datasets and anomaly classification (image-level AUROC) & localization (pixel-level AUROC) results on MVTec AD dataset. The best and second results are highlighted in bold/underline.  
+
+<table><tr><td>Normal Class</td><td>Airplane</td><td>Auto Mobile</td><td>Bird</td><td>Cat</td><td>Deer</td><td>Dog</td><td>Frog</td><td>Horse</td><td>Ship</td><td>Truck</td><td>ID Method</td><td>MVTec AD</td></tr><tr><td>DSVDD[38]</td><td>61.7</td><td>65.9</td><td>50.8</td><td>59.1</td><td>60.9</td><td>65.7</td><td>67.7</td><td>67.3</td><td>75.9</td><td>73.1</td><td>PSVDD[52]</td><td>92.1 95.7</td></tr><tr><td>OCGAN[32]</td><td>75.7</td><td>53.1</td><td>64.0</td><td>62.0</td><td>72.3</td><td>62.0</td><td>72.3</td><td>57.5</td><td>82.0</td><td>55.4</td><td>PaDiM[11]</td><td>95.8 97.5</td></tr><tr><td>DROCC[15]</td><td>82.1</td><td>64.8</td><td>69.2</td><td>64.4</td><td>72.8</td><td>66.5</td><td>68.6</td><td>67.5</td><td>79.3</td><td>60.6</td><td>CutPaste[28]</td><td>96.1 96.0</td></tr><tr><td>HRN-L2[24]</td><td>80.6</td><td>48.2</td><td>64.9</td><td>57.4</td><td>73.3</td><td>61.0</td><td>74.1</td><td>55.5</td><td>79.9</td><td>71.6</td><td>DRAEM[56]</td><td>98.0 97.3</td></tr><tr><td>HRN[24]</td><td>77.3</td><td>69.9</td><td>60.6</td><td>64.4</td><td>71.5</td><td>67.4</td><td>77.4</td><td>64.9</td><td>82.5</td><td>77.3</td><td>RD4AD[12]</td><td>98.5 97.8</td></tr><tr><td>PLAD[7]</td><td>82.5</td><td>80.8</td><td>68.8</td><td>65.2</td><td>71.6</td><td>71.2</td><td>76.4</td><td>73.5</td><td>80.6</td><td>80.5</td><td>PatchCore[37]</td><td>99.1 98.1</td></tr><tr><td>DO2HSC[57]</td><td>81.3</td><td>82.7</td><td>71.3</td><td>71.2</td><td>72.9</td><td>72.8</td><td>83.0</td><td>75.5</td><td>84.4</td><td>82.0</td><td>CFLOW[17]</td><td>98.3 98.6</td></tr><tr><td>Ours</td><td>83.1</td><td>78.4</td><td>69.8</td><td>73.4</td><td>75.1</td><td>70.9</td><td>85.2</td><td>76.2</td><td>82.3</td><td>80.2</td><td>Ours</td><td>99.5 98.3</td></tr></table>
+
+from a prior distribution, achieving superior performance. Compared to the reconstruction-based work [53], our method improves AP by  $+3.78$  and AUROC by  $+0.75$ , benefiting from the proposed forward reconstruction that transforms abnormal inputs toward the normal manifold.
+
+Visual OCC Task Tab.2 shows the performance of our model on the natural image dataset, CIFAR-10, and the industrial defect detection dataset, MVtec AD. For CIFAR-10, we follow the standard OCC setting, where only one class is selected as the normal one, and the remaining classes are treated as abnormal. Under this setting, our method achieves the best performance compared to previous SOTA methods [38, 32, 15, 24, 7, 57], particularly when 'airplane', 'cat', 'deer', 'frog', and 'horse' are used as the normal class. For industrial data, our method also achieves competitive performance compared to widely used methods [52, 11, 28, 56, 12, 37, 17]. Specifically, compared to the reconstruction-based method [12], our method improves image-level and pixel-level AUROC by  $+1.0$  and  $+0.5$ , respectively, as the proposed contraction mapping can constrain reconstructed results to the normal manifold to enhance the performance.
+
+# 4.2 Ablation Study
+
+In this section, we investigate the contributions of the main components and impact of key hyperparameters of our approach on both tabular and visual datasets.
+
+Table 3: Ablation study of our method on tabular datasets using AP/AUROC metrics. We show results on 5 selective datasets and mean results on 20 used datasets. 'RE': reconstruction via contraction mapping. 'DE': density estimation via reversed flow. The best result is highlighted in bold.  
+
+<table><tr><td>Model ↓</td><td>RE</td><td>DE</td><td>Cardio</td><td>Glass</td><td>Thyroid</td><td>Wbc</td><td>Mammo.</td><td>All Tabular Datasets</td></tr><tr><td>A</td><td>-</td><td>-</td><td>80.84/93.71</td><td>15.30/71.08</td><td>82.31/98.17</td><td>71.21/93.53</td><td>31.42/87.21</td><td>67.44/86.52</td></tr><tr><td>B</td><td>✓</td><td>-</td><td>84.25/95.15</td><td>36.77/85.65</td><td>86.01/99.08</td><td>88.98/97.18</td><td>54.02/89.38</td><td>75.21/90.32</td></tr><tr><td>C</td><td>-</td><td>✓</td><td>76.64/92.11</td><td>39.63/81.87</td><td>82.83/98.65</td><td>90.04/98.35</td><td>52.76/86.64</td><td>73.59/88.05</td></tr><tr><td>D</td><td>✓</td><td>✓</td><td>86.35/96.21</td><td>50.67/86.32</td><td>88.37/99.21</td><td>94.72/99.24</td><td>54.32/89.87</td><td>78.64/91.19</td></tr></table>
+
+Effectiveness of Main Components Tab.3 demonstrates the performance of four model variants: A. the baseline model with a six-layer MLP (equivalent to the number of flow-based iterations for fair comparison) for the reconstruction; B. A reconstruction-only variant using the proposed contraction mapping flow; C. A density-estimation-only variant using the flow structure; D. Our full method combines forward reconstruction and backward density estimation. The model B significantly outperforms the baseline method, which validates the effectiveness of the reconstruction via the contraction mapping. In addition, model C also surpasses the baseline, demonstrating the effectiveness of density estimation via the flow structure. When combining both, our method achieves AP/AUROC by 78.64/91.9, obtaining a gain of +11.2/4.67 compared to our baseline. Tab.4 shows an ablation study on visual data, where our framework improves our baseline by +4.87 on CIFAR-10 and +1.7/1.1. To this end, the effectiveness of our method is demonstrated in the general OCC-based AD task (visual AD and tabular AD tasks).
+
+Table 4: Ablation study of our method on CIFAR-10 and MVTec AD, where image-level AUROC is used for the former and image/pixel-level AUROC for the latter. The best result is highlighted in bold.  
+
+<table><tr><td>Model ↓</td><td>RE</td><td>DE</td><td>CIFAR-10</td><td>MVTec AD</td></tr><tr><td>A</td><td>-</td><td>-</td><td>72.59</td><td>97.8/97.2</td></tr><tr><td>B</td><td>✓</td><td>-</td><td>76.88</td><td>99.2/97.9</td></tr><tr><td>C</td><td>-</td><td>✓</td><td>74.27</td><td>98.1/97.5</td></tr><tr><td>D</td><td>✓</td><td>✓</td><td>77.46</td><td>99.5/98.3</td></tr></table>
+
+![](images/2b572f0b6e7a8e05c45d91049f161bb45837035f1e2c4206316e02c9872fb738.jpg)  
+Figure 4: T-SNE visualization of feature distribution after CM with  $R = 3$  and  $K = 2$  on 'Ionosphere' (left) and 'Wbc' (right) datasets.
+
+Table 5: Impact of hyperparameters  $K$  &  $R$  in our method,where all tabular and MVtec AD datasets are adopted for ablation. AP/AUROC are leveraged for tabular data and image-level/pixel-level AUROC for industrial data. The best result is highlighted in bold.  
+
+<table><tr><td>K→R↓</td><td>K=1</td><td>K=2</td><td>K=3</td><td>K=4</td><td>Visual AD
+Base: 97.8/97.2</td><td>K=1</td><td>K=2</td><td>K=3</td><td>K=4</td></tr><tr><td>R=1</td><td>60.42/78.75</td><td>64.81/83.20</td><td>68.87/85.20</td><td>67.04/85.33</td><td>R=1</td><td>97.4/96.8</td><td>97.7/97.2</td><td>98.3/97.6</td><td>98.2/97.7</td></tr><tr><td>R=2</td><td>64.34/82.11</td><td>74.89/88.21</td><td>75.21/88.95</td><td>73.44/87.59</td><td>R=2</td><td>98.3/97.4</td><td>98.9/98.0</td><td>99.1/97.9</td><td>99.2/97.9</td></tr><tr><td>R=3</td><td>71.21/86.32</td><td>78.64/91.19</td><td>76.42/89.94</td><td>70.87/85.12</td><td>R=3</td><td>98.4/97.4</td><td>99.3/98.3</td><td>99.5/98.3</td><td>99.1/98.1</td></tr><tr><td>R=4</td><td>74.57/89.36</td><td>77.03/90.55</td><td>73.14/87.60</td><td>73.25/86.47</td><td>R=4</td><td>98.7/97.8</td><td>99.2/98.1</td><td>98.9/97.9</td><td>98.9/97.7</td></tr><tr><td>R=5</td><td>75.38/90.29</td><td>74.08/88.76</td><td>68.27/84.88</td><td>63.44/81.09</td><td>R=5</td><td>98.6/97.7</td><td>99.0/98.0</td><td>98.5/97.5</td><td>98.3/97.3</td></tr></table>
+
+Impact of Hyperparameters in Contraction Mapping Flow Tab.5 shows the impact of hyperparameters on 20 tabular datasets and MVtec AD. In the tabular AD task, when  $R = 1$  and  $K = 1$ , due to the limited depth of  $\varphi(\cdot)$  results in fewer parameters and insufficient contraction mapping iterations, leading to suboptimal performance. When  $K = 2$  and  $R = 3$ , our framework achieves the best performance, with an AP/AUROC of 78.64/91.19. However, when  $K \times R$  becomes too large, excessive iterations can cause reconstructions to converge too closely to the fixed point, degrading the performance. For the industrial image-based AD task, when  $K = 3$  and  $R = 3$ , the best performance of 99.5/98.3 in terms of image/pixel-level AUROC is achieved. Both large and small values of  $K \times R$  may degrade the anomaly classification and localization. We provide a T-SNE visualization of the feature distribution after CM with  $R = 3$  and  $K = 2$  in Fig.4, where abnormal representations are constrained within the normal manifold and are prevented from fully converging to the fixed point.
+
+# 5 Conclusion
+
+Recent reconstruction-based methods trained only on normal samples may generalize to certain abnormal inputs, leading to well-reconstructed anomalies and degraded detection performance. To address this issue, we propose to apply the contraction mapping structure to conduct reconstruction, in which normal and abnormal inputs are transformed into the normal manifold via iterations of this mapping to increase the reconstruction error. In addition, the contraction mapping is also designed to be invertible, which means it can estimate the likelihood of the original input based on a Gaussian model in the flow form. Based on the above two points, we propose a bidirectional structure with forward reconstruction and backward density estimation. Extensive experiment evaluations on tabular data, natural image data, and industrial image data demonstrate the effectiveness of our method.
+
+# 6 Acknowledgments
+
+This work is supported by the National Natural Science Foundation of China (No. 62471405, 62331003, 62301451), Jiangsu Basic Research Program Natural Science Foundation (BK20241814), Natural Science Foundation of Hebei Province (F2024105029), Suzhou Basic Research Program (SYG202316) and XJTLU REF-22-01-010, XJTLU AI University Research Center, Jiangsu Province Engineering Research Center of Data Science and Cognitive Computation at XJTLU and SIP AI innovation platform (YZCXPT2022103).
+
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+
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+Answer: [NA]
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+
+# A More Implementation Details
+
+In this section, we provide additional implementation details, including the hyperparameters and network architecture for feature extraction.
+
+Hyperparameters for Tabular Datasets Tab.6 of the Appendix shows details of 20 tabular anomaly detection (AD) datasets used in the experiment, where different datasets contain significantly different numbers of samples and feature dimensions. It is challenging to set global training hyperparameters to achieve optimal performance across all datasets. To ensure good performance while avoiding excessive hyperparameter tuning, we fix the learning rate as well as the dimension of input and output of the CM across all datasets (see Tab.7). Finally, following previous works [53], we only adjust the batch size and epochs during training for datasets with particularly large or small sample sizes to achieve better performance.
+
+Table 6: Detailed information of 20 tabular datasets. N: the total number of samples, D: the number of attributes in each dataset.  
+
+<table><tr><td>Dataset</td><td>N</td><td>D</td><td>Anomalies</td></tr><tr><td>Arrhythmia</td><td>452</td><td>274</td><td>66 (14.6%)</td></tr><tr><td>Breastw</td><td>683</td><td>9</td><td>239 (34.9%)</td></tr><tr><td>Cardio</td><td>1,831</td><td>21</td><td>176 (9.6%)</td></tr><tr><td>Census</td><td>299,285</td><td>500</td><td>18,568 (6.2%)</td></tr><tr><td>Campaign</td><td>41,188</td><td>62</td><td>4,640 (11.2%)</td></tr><tr><td>Cardiotocography</td><td>2,114</td><td>21</td><td>466 (22.0%)</td></tr><tr><td>Fraud</td><td>284,807</td><td>29</td><td>492 (0.1%)</td></tr><tr><td>Glass</td><td>214</td><td>9</td><td>9 (4.2%)</td></tr><tr><td>Ionosphere</td><td>351</td><td>33</td><td>126 (35.8%)</td></tr><tr><td>Mammography</td><td>11,183</td><td>6</td><td>260 (2.3%)</td></tr><tr><td>NSL-KDD</td><td>148,517</td><td>122</td><td>77,054 (51.8%)</td></tr><tr><td>Optdigits</td><td>5,216</td><td>64</td><td>150 (2.8%)</td></tr><tr><td>Pima</td><td>768</td><td>8</td><td>268 (34.8%)</td></tr><tr><td>Pendigits</td><td>6,870</td><td>16</td><td>156 (2.2%)</td></tr><tr><td>Satellite</td><td>6,435</td><td>36</td><td>2,036 (31.6%)</td></tr><tr><td>Satimage-2</td><td>5,803</td><td>36</td><td>71 (1.2%)</td></tr><tr><td>Shuttle</td><td>49,097</td><td>9</td><td>3,511 (7.1%)</td></tr><tr><td>Thyroid</td><td>3,772</td><td>6</td><td>93 (2.4%)</td></tr><tr><td>Wbc</td><td>278</td><td>30</td><td>21 (7.5%)</td></tr><tr><td>Wine</td><td>129</td><td>13</td><td>10 (7.7%)</td></tr></table>
+
+Table 7: Traning hyperparameters for 20 tabular data. 'Lr': learning rate. 'BS': batch size. 'C': the dimension of input and output of CM.  
+
+<table><tr><td>Dataset</td><td>Lr</td><td>Epoch</td><td>BS</td><td>C</td></tr><tr><td>Arrhythmia</td><td>1e-5</td><td>100</td><td>64</td><td>32</td></tr><tr><td>Breastw</td><td>1e-5</td><td>100</td><td>64</td><td>32</td></tr><tr><td>Cardio</td><td>1e-5</td><td>100</td><td>64</td><td>32</td></tr><tr><td>Census</td><td>1e-5</td><td>150</td><td>512</td><td>32</td></tr><tr><td>Campaign</td><td>1e-5</td><td>150</td><td>512</td><td>32</td></tr><tr><td>Cardiotocography</td><td>1e-5</td><td>150</td><td>64</td><td>32</td></tr><tr><td>Fraud</td><td>1e-5</td><td>150</td><td>512</td><td>32</td></tr><tr><td>Glass</td><td>1e-5</td><td>100</td><td>64</td><td>32</td></tr><tr><td>Ionosphere</td><td>1e-5</td><td>150</td><td>64</td><td>32</td></tr><tr><td>Mammography</td><td>1e-5</td><td>100</td><td>512</td><td>32</td></tr><tr><td>NSL-KDD</td><td>1e-5</td><td>100</td><td>512</td><td>32</td></tr><tr><td>Optdigits</td><td>1e-5</td><td>150</td><td>64</td><td>32</td></tr><tr><td>Pima</td><td>1e-5</td><td>150</td><td>64</td><td>32</td></tr><tr><td>Pendigits</td><td>1e-5</td><td>150</td><td>64</td><td>32</td></tr><tr><td>Satellite</td><td>1e-5</td><td>150</td><td>256</td><td>32</td></tr><tr><td>Satimage-2</td><td>1e-5</td><td>150</td><td>256</td><td>32</td></tr><tr><td>Shuttle</td><td>1e-5</td><td>150</td><td>256</td><td>32</td></tr><tr><td>Thyroid</td><td>1e-5</td><td>150</td><td>256</td><td>32</td></tr><tr><td>Wbc</td><td>1e-5</td><td>150</td><td>64</td><td>32</td></tr><tr><td>Wine</td><td>1e-5</td><td>150</td><td>64</td><td>32</td></tr></table>
+
+Feature Extractor Following previous work [57, 53], we also adopt a three-layer MLP with LeakyReLU activation to extract the deep representation of tabular data (see Fig.5 (a)), where the dimension of the output and input in each layer is consistent with previous work [57]. Next, as shown in Fig.5 (b), we leverage a simple three-layer CNN to extract the deep representation of natural image data, where the setting of the network follows work [57].
+
+Network Details for Industrial Defect Detection Industrial defect detection is a primary application area of anomaly detection that has a well-established standard. Therefore, we adopt a baseline model to obtain a baseline performance (see Tab.5 in our main manuscript). The baseline model adopts a frozen encoder and a learnable decoder, which is similar to RD4AD [12]. Unlike RD4AD, we removed the MMF and OCE modules from our baseline framework, resulting in a more fundamental performance. Based on this baseline framework, the contraction mapping flow  $\varphi (\cdot)$  is applied after the feature encoder (see Fig.5 (c)). The specific pipeline can be summarized as: given an image  $\mathbf{I}\in \mathbb{R}^{H_0\times W_0\times 3}$ , the frozen encoder  $\mathcal{E}(\cdot)$  encodes  $\mathbf{I}$  into multi-layer features  $\{\mathbf{x}_l\}_{l = 1}^L$ , each  $\mathbf{x}_l\in \mathbb{R}^{H_l\times W_l\times C_l}$  denotes the  $l$ -th layer feature. We select  $l_{1}$ -th,  $l_{2}$ -th,  $l_{3}$ -th layer features as encoded features, denoted by  $\{\mathbf{x}_1,\mathbf{x}_2,\mathbf{x}_3\}$ . Next, we down-sample the size of  $\mathbf{x}_1,\mathbf{x}_2$  to  $H_{l_3}\times W_{l_3}$ . Next, we concatenate encoded features in the channel dimension, denoted by  $[\mathbf{x}_1,\mathbf{x}_2,\mathbf{x}_3]\in \mathbb{R}^{H_{l_3}\times W_{l_3}\times (C_{l_1} + C_{l_2} + C_{l_3})}$ , and apply a linear projection  $\Phi (\cdot)$  to fuse them, i.e.,  $\mathbf{x} = \Phi ([\mathbf{x}_1,\mathbf{x}_2,\mathbf{x}_3])\in \mathbb{R}^{H_{l_3}\times W_{l_3}\times C_{l_3}}$ . Then,  $\mathbf{x}$  is input to our contraction mapping flow  $\varphi (\cdot)$ , obtaining latent variable  $\mathbf{z}$ . Under the reconstruction branch, we apply a learnable decoder to decode  $\mathbf{z}$  into decoded features  $\{\widehat{\mathbf{x}}_1,\widehat{\mathbf{x}}_2,\widehat{\mathbf{x}}_3\}$ . We apply the cosine similarity loss to replace the original  $\mathcal{L}_{\mathrm{rec}}$
+
+![](images/7026b9a4651db0b609d3835bce3f8064ca4741d4c94fb75533cb2a94362176a3.jpg)  
+(a) Encoder for Tabular Data  
+Figure 5: Different feature extractors are used to extract deep representations of different data types. (a): Three-layer MLP with LeakyReLU activation is adopted for tabular data. (b): Three-layer CNN with Batch normalization and ELU activation is used for natural image data. (c): Our proposed method is based on a general baseline method for industrial defect detection.
+
+![](images/0a81230ea286f2461fd9222e80292a2ba26e3ae184605c09be3ebcb6c58dc6c6.jpg)  
+(b) Encoder for Natural Image Data
+
+![](images/37d70bc0c4d85774274f363b3b696dd5b1bd76276cace3e9b0036fe41f3d5e63.jpg)  
+(c) Framework for Industrial Defect Detection
+
+(Eq.(1) in the main manuscript), i.e.,
+
+$$
+\mathcal {L} _ {\mathrm {r e c}} = \sum_ {i = 1} ^ {3} \left(1 - \frac {\operatorname {F l a t} \left(\mathbf {x} _ {i}\right) \cdot \operatorname {F l a t} \left(\widehat {\mathbf {x}} _ {i}\right)}{\| \operatorname {F l a t} \left(\mathbf {x} _ {i}\right) \| _ {2} \| \operatorname {F l a t} \left(\widehat {\mathbf {x}} _ {i}\right) \| _ {2}}\right), \tag {9}
+$$
+
+where  $\operatorname{Flat}(\cdot)$  is the flatten operator. The density estimation branch adopts patch embedding features to fit the Gaussian prior distribution  $\pi(\cdot)$  and uses  $\varphi^{-1}(\cdot)$  to obtain the likelihood of each patch embedding feature. During inference, we adopt a combination of both the cosine distance and the likelihood of each patch embedding feature as the patch-level anomaly score. Through the optimization of Eq. (9), contraction mapping flow  $\varphi(\cdot)$  can transform any input  $\mathbf{x}$  onto the normal manifold, which ensures that the decoded features  $\{\hat{\mathbf{x}}_1, \hat{\mathbf{x}}_2, \hat{\mathbf{x}}_3\}$  are on the normal manifold, increasing the reconstruction error.
+
+# B More Contraction Mapping Network Details
+
+According to Section 3.1.2 in the main manuscript, the structure of contraction mapping can be written as:
+
+$$
+\varphi (\cdot) = F _ {R} \circ F _ {R - 1} \dots F _ {1} (\cdot),
+$$
+
+where each  $F_{r}(\cdot) = f_{K}^{r}\circ f_{K - 1}^{r}\circ \dots f_{1}^{r}(\cdot)$ . Each contraction mapping unit  $f_{k}^{r}(\cdot)$  is based on the 1-Lipchitz invertible neural network  $\mathbf{G}$ , i.e.,
+
+$$
+f (x) = \left(\frac {\mathbf {I d} + \mathbf {G}}{2}\right) ^ {- 1} (x) - x,
+$$
+
+where  $\mathbf{Id}$  denotes the identity function. We adopt i-DenseNet [33] to parametrize  $\mathbf{G}$ . Following [1, 2, 31],  $\mathbf{G}$  is designed as a Lipschitz-continuous function with Lipschitz constant  $\eta < 1$ , where  $\| \mathbf{G}(x_1) - \mathbf{G}(x_2)\|_2 \leq \eta \| x_1 - x_2\|_2$ . Based on  $\mathbf{G}$ ,  $f(\cdot)$  is monotonic with Lipschitz constant  $\eta_f < 1$  so  $f(\cdot)$  is an invertible mapping according to Banach fixed point theorem [16] (the proof can be found in [1]). Based on  $\mathbf{G}$  and Eq. (2), we can model  $f_k^r(\cdot)$ , which is named as 'Lipschitz Layer'. We then apply the LipSwish activation function to the output of each  $f_k^r(\cdot)$ . The LipSwish function is written as:
+
+$$
+\operatorname {L i p S w i s h} (x) = \frac {x}{1 . 1 * (1 + \exp (- \beta x))},
+$$
+
+# C Supplementary Proof
+
+In this section, we supplement the detailed proofs of Lemma 1 and Lemma 2 used in our main manuscript.
+
+Lemma 3 (Convergence of Iterations): Let  $f(\cdot)$  be a contraction mapping with  $\eta \in [0,1)$ . For any initial input  $x_0 \in \mathbb{R}^d$ , we define the sequence  $\{x_n\}$  by  $x_{n + 1} = f(x_{n})$  for  $n = 1,2,\dots$ . Then  $\lim_{n\to \infty}x_n = x^*$ , where  $x^{*}$  is unique and is called a fixed point of  $f(\cdot)$ .
+
+Proof. We firstly prove that  $\{\mathbf{x}_n\}$  is a Cauchy sequence. For any  $m > n$ , we have:
+
+$$
+\begin{array}{l} \left\| x _ {m} - x _ {n} \right\| _ {2} = \left\| x _ {n} - x _ {m} \right\| _ {2} \\ \leq \left\| x _ {n} - x _ {n + 1} \right\| _ {2} + \left\| x _ {n + 1} - x _ {n + 2} \right\| _ {2} + \dots + \left\| x _ {m - 1} - x _ {m} \right\| _ {2}. \\ \end{array}
+$$
+
+Next, for each term  $\| x_{n + k} - x_{n + k + 1}\| _2$  in this sum:
+
+$$
+\begin{array}{l} \left\| x _ {n + k} - x _ {n + k + 1} \right\| _ {2} = \left\| f \left(x _ {n + k - 1}\right) - f \left(x _ {n + k}\right) \right\| _ {2} \\ \leq \eta \cdot \| x _ {n + k - 1} - x _ {n + k} \| _ {2} \\ \leq \eta^ {2} \cdot \left\| x _ {n + k - 2} - x _ {n + k - 1} \right\| _ {2} \\ \end{array}
+$$
+
+中 中
+
+$$
+\leq \eta^ {n + k} \cdot \| x _ {0} - x _ {1} \| _ {2}.
+$$
+
+Therefore, based on the above two inequalities, we have:
+
+$$
+\begin{array}{l} \| x _ {m} - x _ {n} \| _ {2} \leq \sum_ {k = 0} ^ {m - n - 1} \| x _ {n + k} - x _ {n + k + 1} \| _ {2} \leq \sum_ {k = 0} ^ {m - n - 1} \eta^ {n + k} \cdot \| x _ {0} - x _ {1} \| _ {2} \\ = \eta^ {n} \cdot \| x _ {0} - x _ {1} \| _ {2} \cdot \sum_ {k = 0} ^ {m - n - 1} \eta^ {k} = \eta^ {n} \cdot \| x _ {0} - x _ {1} \| _ {2} \cdot \frac {1 - \eta^ {m - n}}{1 - \eta} \\ \leq \eta^ {n} \cdot \| x _ {0} - x _ {1} \| _ {2} \cdot \frac {1}{1 - \eta}. \\ \end{array}
+$$
+
+As  $n \to \infty$ ,  $\eta^n \to 0$  (since  $\eta < 1$ ), so  $\| x_m - x_n \| \to 0$  for any  $m > n$  as  $n \to \infty$ . This proves that  $\{x_n\}$  is a Cauchy sequence. Since  $\mathbb{R}^d$  with the Euclidean metric is complete, the sequence converges to a certain point  $x^* \in \mathbb{R}^d$ , i.e.  $\lim_{n \to \infty} x_n = x^*$ . Next, by continuity of  $f$  (which follows from the contraction property) and the definition of  $\{x_n\}$ :
+
+$$
+f (x ^ {*}) = f \left(\lim  _ {n \rightarrow \infty} x _ {n}\right) = \lim  _ {n \rightarrow \infty} f (x _ {n}) = \lim  _ {n \rightarrow \infty} x _ {n + 1} = x ^ {*}.
+$$
+
+To this end, we prove that  $f(x^{*}) = x^{*}$ , i.e.,  $x^{*}$  is a unique fixed point of  $f(\cdot)$ .
+
+Lemma 4 Let  $\varphi_K(\cdot) = f_K\circ f_{K - 1}\circ \dots f_1(\cdot)$ , where each  $f_{k}$  is contraction mapping with  $\eta_k\in [0,1)$ , then  $\varphi_K(\cdot)$  is a contraction mapping with Lipschitz constant  $\eta_{\varphi}\leq \eta_{1}\cdot \eta_{2}\cdot \cdot \cdot \eta_{K}$ .
+
+Proof. First of all, let us prove for a base case with  $K = 2$ . Let  $\varphi_2(\cdot) = f_2 \circ f_1(\cdot)$  be the composition of two contraction mappings. For any two points  $x_1, x_2 \in \mathbb{R}^d$ , we have:
+
+$$
+\left\| \varphi_ {2} \left(x _ {1}\right) - \varphi_ {2} \left(x _ {2}\right) \right\| _ {2} = \left\| f _ {2} \left(f _ {1} \left(x _ {1}\right)\right) - f _ {2} \left(f _ {1} \left(x _ {2}\right)\right) \right\| _ {2}.
+$$
+
+Since  $f_{2}$  is a contraction mapping with constant  $\eta_{2}$ :
+
+$$
+\left\| f _ {2} \left(f _ {1} \left(x _ {1}\right)\right) - f _ {2} \left(f _ {1} \left(x _ {2}\right)\right) \right\| _ {2} \leq \eta_ {2} \| f _ {1} \left(x _ {1}\right) - f _ {1} \left(x _ {2}\right) \| _ {2}. \tag {10}
+$$
+
+Since  $f_{1}$  is a contraction mapping with constant  $\eta_{1}$ :
+
+$$
+\eta_ {2} \| f _ {1} \left(x _ {1}\right) - f _ {1} \left(x _ {2}\right) \| _ {2} \leq \eta_ {2} \cdot \eta_ {1} \| x _ {1} - x _ {2} \| _ {2}. \tag {11}
+$$
+
+Base on the above Eq. (10) and Eq. (11), we have:
+
+$$
+\left\| \varphi_ {2} (x _ {1}) - \varphi_ {2} (x _ {2}) \right\| _ {2} \leq \eta_ {2} \cdot \eta_ {1} \left\| x _ {1} - x _ {2} \right\| _ {2},
+$$
+
+which proves that  $\varphi_{2}(\cdot)$  is contraction mapping with  $\eta_{\varphi}\leq \eta_1\cdot \eta_2$ . Next, we apply mathematical induction to prove the general case. We assume the lemma holds for  $K - 1$  mappings, i.e.,  $\varphi_{K - 1}(\cdot) = f_{K - 1}\circ \dots \circ f_1(\cdot)$  is a contraction mapping with Lipschitz constant  $\eta_{\varphi_{K - 1}}\leq \eta_1\cdot \eta_2\dots \eta_{K - 1}$ . We consider  $\varphi_K(\cdot) = f_K\circ \varphi_{K - 1}(\cdot)$ . For any  $x_{1},x_{2}\in \mathbb{R}^{d}$ , we have:
+
+$$
+\left\| \varphi_ {K} \left(x _ {1}\right) - \varphi_ {K} \left(x _ {2}\right) \right\| _ {2} = \left\| f _ {K} \left(\varphi_ {K - 1} \left(x _ {1}\right)\right) - f _ {K} \left(\varphi_ {K - 1} \left(x _ {2}\right)\right) \right\| _ {2}. \tag {12}
+$$
+
+Then, based on the inductive hypothesis and the definition of contraction mapping, we have:
+
+$$
+\eta_ {K} \| \varphi_ {K - 1} (x _ {1}) - \varphi_ {K - 1} (x _ {2}) \| _ {2} \leq \eta_ {K} \left(\eta_ {1} \cdot \eta_ {2} \dots \eta_ {K - 1}\right) \| x _ {1} - x _ {2} \| _ {2}. \tag {13}
+$$
+
+Combining Eq. (12) and Eq. (13), one inequality can be obtained:
+
+$$
+\left\| \varphi_ {K} \left(x _ {1}\right) - \varphi_ {K} \left(x _ {2}\right) \right\| _ {2} \leq \left(\eta_ {1} \cdot \eta_ {2} \dots \eta_ {K}\right) \| x _ {1} - x _ {2} \| _ {2}.
+$$
+
+Since each  $\eta_{k}\in [0,1)$ ,  $\eta_{\varphi}$  must also be in  $[0,1)$ , i.e.,  $0\leq \eta_{1}\cdot \eta_{2}\dots \eta_{K} < 1$ .
+
+# D Supplementary Experiments
+
+Sensitivity of Balancing the Loss Terms The hyperparameter  $\lambda$  in Eq.(7) of the main manuscript balances the reconstruction loss  $\mathcal{L}_{\mathrm{rec}}$  and the maximum likelihood estimation loss  $\mathcal{L}_{\mathrm{MLE}}$ . To investigate its impact, we vary  $\lambda$  and evaluate the performance, as shown in Tab. 8 of the appendix. For the 20 tabular datasets,  $\lambda = 0.5$  yields the best AP/AUROC, 78.64/91.19. Slightly higher ( $\lambda = 0.6$ ) or lower ( $\lambda = 0.4$ ) values show comparable but slightly degraded performance, indicating that the performance is stable around  $\lambda = 0.5$ . For the MVtec AD dataset,  $\lambda = 0.4$  provides the highest image-level AUROC, while  $\lambda = 0.5$  achieves the best pixel-level AUROC. Too small  $\lambda$  causes  $\varphi^{-1}(\cdot)$  to not fully capture the real distribution of input, degrading the performance of density estimation. Too large a value of  $\lambda$  may disturb reconstruction training. Therefore, the choice of  $\lambda = 0.5$  (as used for both tabular and MVtec AD in our main experiments) achieves robust performance across both types of datasets.
+
+Table 8: Anomaly detection results when varying different  $\lambda$  values for balancing  ${\mathcal{L}}_{\text{rec }}$  and  ${\mathcal{L}}_{\mathrm{{MLE}}}$  . We adopt image-level AUROC metric for 20 tabular datasets and image/pixel-level AUROC for industrial MVTec AD datasets. Bold/underline values indicate the best/runner-up.  
+
+<table><tr><td>Datasets</td><td>λ=0.2</td><td>λ=0.3</td><td>λ=0.4</td><td>λ=0.5</td><td>λ=0.6</td><td>λ=0.7</td><td>λ=0.8</td></tr><tr><td>Tabular Data</td><td>77.86/91.12</td><td>78.29/91.16</td><td>78.19/91.14</td><td>78.64/91.19</td><td>78.55/91.21</td><td>78.35/91.09</td><td>78.13/91.10</td></tr><tr><td>MVTec AD</td><td>99.1/98.1</td><td>99.3/98.1</td><td>99.6/98.2</td><td>99.5/98.3</td><td>99.3/98.2</td><td>99.4/98.3</td><td>98.9/97.9</td></tr></table>
+
+Sensitivity of Weight Coefficient  $\alpha$  The weight coefficient  $\alpha$  in the anomaly score (Eq.(8) in the main manuscript) controls the contribution of the log-likelihood term relative to the reconstruction error. We evaluated the effect of different  $\alpha$  values, with results presented in Tab. 9 of the Appendix. We note that the scale of reconstruction error is generally a very small value, while the scale of log likelihood is much larger than the reconstruction error. Therefore, using  $\alpha$  to control the scale log likelihood is necessary for our unified framework that combines these two different paradigms. For the 20 tabular datasets, an  $\alpha$  of 0.01 achieves the optimal performance by 78.64/91.19. Performance degrades if  $\alpha$  is too large or too small because of disturbed reconstruction or density estimation. Similarly, for the MVTec AD dataset,  $\alpha = 0.001$  also results in the best image-level and pixel-level AUROC. Again, performance drops with much larger or smaller  $\alpha$  values. This analysis highlights that the density estimation component, when weighted appropriately by  $\alpha$ , significantly contributes to the final anomaly score. An  $\alpha$  value of 0.001 appears to be a generally effective choice for both types of datasets according to this ablation.
+
+Table 9: Anomaly detection results with different weight coefficient  $\alpha$  in anomaly score. We adopt the image-level AUROC metric for 20 tabular datasets and the image/pixel-level AUROC for industrial MVTec AD datasets. Bold/underline values indicate the best/runer-up.  
+
+<table><tr><td>α</td><td>α = 1</td><td>α = 0.5</td><td>α = 0.1</td><td>α = 0.05</td><td>α = 0.01</td><td>α = 0.005</td><td>α = 0.001</td></tr><tr><td>Tabular Data</td><td>74.48/88.62</td><td>76.40/89.71</td><td>77.26/90.97</td><td>78.05/91.09</td><td>78.64/91.19</td><td>78.21/91.15</td><td>76.89/90.88</td></tr><tr><td>α</td><td>α = 0.1</td><td>α = 0.05</td><td>α = 0.01</td><td>α = 0.005</td><td>α = 0.001</td><td>α = 0.0005</td><td>α = 0.0001</td></tr><tr><td>MVTec AD</td><td>98.3/97.6</td><td>98.6/97.6</td><td>98.9/98.0</td><td>99.3/98.2</td><td>99.5/98.3</td><td>99.3/98.1</td><td>99.2/98.0</td></tr></table>
+
+# E Limitations
+
+While our method shows strong AD performance on tabular and image datasets, its current scope is primarily focused on these common data types. It has not yet been extended to critical domains like graph and time-series data, where optimal AD typically requires specialized network architectures to handle their unique structural and dependency characteristics. Future work will involve adapting our URD framework to these distinct modalities by incorporating tailored design principles, aiming to progress toward a unified anomaly detection model.
+
+# F Some Questions and Answers regarding Motivation and Methodology
+
+# F.1 Intuitive Understanding of Proposition 1
+
+Proposition 1 mainly focuses on establishing a direct causal link between our optimization and the fixed point's location. The proposition states that  $\| \mathbf{x}^* - \mathbf{x}\|_2 \leq \mathcal{L}_{\mathrm{rec}} / (1 - \eta)$ , which directly and rigorously demonstrates that by minimizing the reconstruction loss  $\mathcal{L}_{\mathrm{rec}}$  on normal samples  $\mathbf{x}$ , we are guaranteed to bound the distance to the fixed point  $\mathbf{x}^*$ .
+
+# F.2 Why Apply LipSwish Activation Function rather than ReLU?
+
+We note that the choice of LipSwish is deliberate and critical. LipSwish guarantees the reversibility of the entire network, while ReLU cannot guarantee it. In this work, we apply an invertible network  $\varphi$  as the foundation, where LipSwish activation function is invertible and ensures the invertibility of  $\varphi$ . In contrast,  $\operatorname{ReLU}(x) = \operatorname{Max}(0, x)$  maps all negative inputs to zero, which creates a many-to-one mapping that is mathematically non-invertible. We adopt i-DenseNet [33] to parameterize our mapping  $\mathbf{G}$  in Eq.(2). According to [33], using LipSwish is a standard and effective choice in this context, providing more expressive power than simpler activations. LipSwish is also smooth, continuously differentiable, and invertible, which are essential properties for stable training of INNs.
+
+# F.3 How does Our Contraction Mapping Work in Different Data Distributions?
+
+Since the contraction mapping iteratively transforms data toward a fixed point derived by training the model on normal data, what if the normal data distribution itself is chaotic? Or if the normal data distribution is ring-shaped or U-shaped? Can the fixed point be located in an appropriate position? Will the reconstruction discrepancy identification work well in such cases?
+
+![](images/ce5e00c7043c436391ba99277f3fab976812e6b89831b0a4b3e1a2e9e7c23c42.jpg)  
+(a) Cardio (Vanilla)
+
+![](images/626650075a73be6a1b2ae363839c088a1068ea40b250a7dcf2f2b442c0988957.jpg)  
+(b) Pima (Disconnected)  
+Figure 6: T-SNE visualization of data distribution shapes and corresponding fixed point positions for four types of tabular datasets
+
+![](images/3a8ecbf49bada762c1f277b1320609c2a32f2997b86e3f1a64ebe6e0bb8f8798.jpg)  
+(c) Thyroid (Ring-shaped)
+
+![](images/4dcace6987f0866a16da47bff38fcd89f0fadbb98ec96e3ffeff5818304bb3c9.jpg)  
+(d) Wbc (U-shaped)
+
+Sometimes, extreme and complex distributions may make the reconstruction-only part ineffective, which is also the reason why we introduce a density estimation process to collaborate with the reconstruction process. Following the above questions, we discussed several situations and provided special cases of difficult reconstruction.
+
+Vanilla Distribution. In fact, the manuscript visualizes the position of the fixed point in different normal distributions. Generally, the fixed point is not completely located at the geometric center of the distribution, and sometimes, due to different data or training settings, fixed points may even fall on the edges of the distribution (see Fig.6 (a)).
+
+Disconnected Distribution. As shown in Fig.6 (b), the normal distribution is relatively scattered and chaotic, and the fixed point is located at the edge of one of the sub distributions. However, because the abnormal distribution is far from the normal one, reconstruction error is still an effective criterion, achieving a good performance under a reconstruction-only setting.
+
+Ring-shaped Distribution. There is currently no typical ring-shaped normal distribution in the used datasets. Fig.6 (b) and (c) show multi-ring and irregular-ring structures, respectively. In fact, through multiple adjustments to the training hyperparameters and random seed, we found that fixed points may be located in some data-free areas (central regions of the ring) or within the distribution (internal regions of the ring). If the anomaly is located in the central area of the ring, this method may theoretically fail, but this situation has not been found in experiments yet.
+
+U-shaped Distribution. As shown in Fig.6 (d), the normal distribution is U-shaped and the fixed point is located at the bottom of 'U'. We found that after adjusting the random seed or training hyperparameters, the position of the fixed point will change and may not necessarily be located at the bottom position. However, the same thing is true: as long as enough training epochs are set, the fixed point is located inside or at the edge of U. CM will achieve good and stable performance as long as the abnormal data is in a distant external area.
+
+In the reconstruction-only experiments, we found that if the abnormal data is very close to the normal distribution in terms of spatial position, the reconstruction errors generated by CM for normal and abnormal data are very close, resulting in false negatives and degrading performance. Of course, this is also a challenge for all reconstruction-based methods. In addition, since fixed points may appear at any position on a normal manifold, they may be closer to some anomalies than other normal points, leading to false negatives. Finally, in the most extreme case, our framework will fail when a few anomalies are within the normal distribution. However, it should be noted that almost all existing OCC methods cannot address this type of problem. This combined approach makes our method robust even for complex data distributions.
+
+# F.4 Trade-off between Convergence to a Fixed Point and Bijectivity via Network Design
+
+Our method focuses on leveraging a finite number of mapping compositions to simulate iteration of a single contraction mapping, rather than directly using iteration of a single contraction mapping.
+
+Specifically, according to Proposition 2, we apply  $\varphi_K(\cdot) = f_K \circ f_{K-1} \circ \dots \circ f_1(\cdot)$  to simulate a finite number of iterations of  $f$ , rather than directly iterating  $f$ , where each  $f_k$  has its own parameters  $\theta_k$  and Lipschitz constant  $\eta_k$ . Lipschitz constant  $\eta_\varphi \leq \eta_1 \cdot \eta_2 \cdots \eta_K < 1$  ensures contracting anomalies towards the normal distribution. We leverage  $\mathcal{L}_{\mathrm{rec}} = \| \varphi_K(\mathbf{x}) - \mathbf{x} \|_2$  to train the entire network  $\varphi_K$ , rather than firstly training a single  $f$  and then iterating it a few times. From an optimization perspective,  $\varphi_K$  optimized by  $\mathcal{L}_{\mathrm{rec}}$  tends to restore normal input to normal output instead of contracting all inputs (normal and abnormal) to one fixed point. From a mathematical perspective, a finite and small  $K$  value gives  $\varphi_K$  the ability to contract the input to the normal manifold. Still, it does not destroy the reversibility of  $\varphi_K$ . In summary, selecting  $K$  as a finite value, the contraction properties of  $\varphi_K$  and the training objective jointly ensure that it can map anomalies to normal manifolds. Still, it will not completely shrink to a very small region or even a point and destroy its reversibility.
+
+We intentionally keep  $K$  and  $R$  small in actual experiments to prevent over-contraction. This allows us to benefit from the "pull" of the contraction towards the normal manifold (which increases reconstruction error for anomalies) while not destroying the reversibility of our INN. The trade-off is managed by the choice of  $K$  and  $R$ , which we study empirically in Table 5. The results show that a moderate number of iterations yields the best performance, validating our handling of this trade-off.
+
+# F.5 The Risk of CM Network Collapsing to the Identity Mapping
+
+The degradation towards identity mapping is a critical concern for contraction mapping trained with reconstruction loss. We show two explicit mechanisms to prevent this: one is the structure of the contraction mapping unit, and the other is the regularity of our backward density estimation.
+
+(1) According to Eq.(2),  $f(x) = \left(\frac{\mathbf{Id} + \mathbf{G}}{2}\right)^{-1}(x) - x$ , where  $\mathbf{G}$  is invertible and  $\mathbf{Id}$  is identity mapping. If  $f$  degrades into an identity mapping under reconstruction loss, we have  $\left(\frac{\mathbf{Id} + \mathbf{G}}{2}\right)^{-1}(x) - x = x$ , which means  $\left(\frac{\mathbf{Id} + \mathbf{G}}{2}\right)^{-1}(x) = 2x$ . Based on the definition of inverse mapping, we must have  $\frac{\mathbf{Id} + \mathbf{G}}{2}(2x) = x$ . This means  $\mathbf{Id}(2x) + \mathbf{G}(2x) = 2x$ . Then, we can get  $\mathbf{G}(2x) = 0$  to make  $f(x)$  an identity mapping. Paradoxically,  $\mathbf{G}$  is invertible and a one-to-one mapping. Therefore, in terms of the structure of  $f$ , it is difficult to degrade into an identity mapping under the optimization of reconstruction loss.  
+(2) The backward density estimation loss  $\mathcal{L}_{\mathrm{MLE}}$  provides a powerful regularizer against identity collapse. In the backward process,  $\mathcal{L}_{\mathrm{MLE}}$  pushes  $\varphi^{-1}$  to be a mapping that transforms a basic Gaussian distribution  $\pi(z)$  to the (typically complex) input distribution  $p(x)$ . An identity mapping could only achieve this if  $p(x)$  is also a Gaussian distribution, which is never the case. This forces  $\varphi^{-1}$  to learn a non-trivial, meaningful transformation.
+
+# F.6 The Conflict of Optimization Objectives of Reconstruction and Density Estimation
+
+According to reconstruction loss (1) and MLE loss (6), pushes  $\mathbf{z}$  to be close to  $\mathbf{x}$ , while  $\mathcal{L}_{\mathrm{MLE}}$  pushes the distribution of  $\mathbf{z}$  to be a Gaussian distribution. We need to clarify that both forward reconstruction (RE) and backward density estimation (DE) are indispensable, although they may seem contradictory. We have also designed corresponding trade-off schemes in the manuscript.
+
+When only using RE, if some anomalies are very close to the normal manifold or the obtained fixed point in terms of spatial position, this causes small reconstruction errors and false negatives. In addition, the distribution of normal data may be very complex, chaotic, and disconnected (see Fig.6). In this case, fixed points may fall on the edges of the distribution, or even in areas without data. In this case, reconstruction error cannot be used as a good criterion. To address these issues, we introduce a density estimation process to probabilistically identify anomalies under these special circumstances. DE would assign a high likelihood to points in areas with dense data points and a low likelihood to others with low density. An anomaly is thus an input that is both far from its reconstruction (high reconstruction score) and statistically unlikely to have a high likelihood (low density estimation score). This synergy makes our method robust. As shown in Fig.2 and Eq.5, after forward reconstruction process (RE), we apply a stop-gradient operation  $\mathrm{sg}(\cdot)$  in learning of Gaussian distribution  $\pi (\mathbf{z})$ , which allows  $\pi (\mathbf{z})$  to be learned independently without affecting the reconstruction training. In addition, the hyperparameter  $\lambda$  in our total loss  $\mathcal{L}_{\mathrm{total}} = \mathcal{L}_{\mathrm{rec}} + \lambda \mathcal{L}_{\mathrm{MLE}}$  explicitly controls this trade-off. Our sensitivity analysis on  $\lambda$  in Tab.5 shows that a balanced value  $(\lambda = 0.5)$  performs best, validating that the two objectives are complementary rather than purely conflicting. The reconstruction provides an error-based score, while the flow provides a likelihood-based score, and their combination is more powerful than either alone.
\ No newline at end of file
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+# Unifying Symbolic Music Arrangement: Track-Aware Reconstruction and Structured Tokenization
+
+Longshen Ou\* Jingwei Zhao\* Ziyu Wangb,Gus Xia Qihao Liang Torin Hopkins Ye Wang
+
+$^{\natural}$ Sound and Music Computing Lab, School of Computing, NUS  
+ $^{\flat}$ Courant Institute of Mathematical Sciences, New York University  
+ $^{\sharp}$ Music X Lab, MBZUAI
+
+# Abstract
+
+We present a unified framework for automatic multitrack music arrangement that enables a single pre-trained symbolic music model to handle diverse arrangement scenarios, including reinterpretation, simplification, and additive generation. At its core is a segment-level reconstruction objective operating on token-level disentangled content and style, allowing for flexible any-to-any instrumentation transformations at inference time. To support track-wise modeling, we introduce  $REMI-z$ , a structured tokenization scheme for multitrack symbolic music that enhances modeling efficiency and effectiveness for both arrangement tasks and unconditional generation. Our method outperforms task-specific state-of-the-art models on representative tasks in different arrangement scenarios—band arrangement, piano reduction, and drum arrangement, in both objective metrics and perceptual evaluations. Taken together, our framework demonstrates strong generality and suggests broader applicability in symbolic music-to-music transformation. $^{1}$
+
+# 1 Introduction
+
+Music arrangement is the art of adapting compositions for performance contexts that differ from their original forms [4]. It plays a central role in many music creation process, including professional production, live performance, music education, and digital content creation. Automating this process can expand music accessibility and accelerate music creation. Although arrangement forms vary—e.g., rewriting for different instruments (reinterpretation) [9], simplifying for solo performance (reduction) [29, 30], or adding new tracks (additive generation) [20, 19]—they share a common structure: generating new music tracks conditioned on existing ones under explicit content and instrument constraints. However, prior work typically addresses each task independently [42, 30, 29, 19], using specialized model architectures and training schemes. Such designs lack cross-task generality, increase implementation cost, and fail to leverage the musical knowledge learned by large-scale pre-trained symbolic models that could potentially improve the arrangement quality.
+
+Meanwhile, generative modeling of symbolic music, i.e., music in notation-based formats, has advanced rapidly with large-scale pre-trained models that capture rich musical styles and structures via autoregressive modeling [17, 8, 23, 37, 32, 22]. Inspired by natural language processing, these models scale to billions of parameters and are trained on vast corpora. Although their unconditional generation quality has improved substantially, their applications in real-world conditional generation remains relatively limited. Prior work primarily targets coarse-level control, such as style [3, 17], structure [40], polyphony level [22], or sentiment [18], while fine-grained conditioning on existing musical content—precisely what arrangement tasks demand—remains underexplored.
+
+![](images/5431cd975fd71fd9cfa5b2b3af7f957472eaea4d1d3295fad5372d7672e8cd90.jpg)  
+Figure 1: Overview of the proposed unified framework for each arrangement task. The symbol  $\bigoplus$  denotes concatenation of component sequences. Music segments are decomposed into three subsequences: instruments, content, and target-side history. These components form the condition sequence, with the relevant tracks from the original music as the target sequence. The model is trained to reconstruct the music from these components.
+
+This underexploration is likely not due to the absence of pre-trained models or their representational power, but to the lack of training objectives that support flexible adaptation to the diverse requirements of arrangement tasks. While sequence-to-sequence fine-tuning is conceptually a natural solution, it relies on parallel datasets—collections of the same piece arranged for different instrumentations (e.g., orchestra and piano)—which are extremely scarce. Even when adopting some level of internal parallel data [29, 30], these datasets constrain output directions to fixed mappings (e.g., orchestra to piano). In contrast, real-world arrangement demands far greater flexibility: the ability to transform arbitrary input instrumentations into arbitrary targets—i.e., support any-to-any transformations.
+
+To address these limitations, we propose a unified framework for symbolic music arrangement that enables a pre-trained symbolic generative model to be fine-tuned across diverse tasks through a single self-supervised training pipeline (Figure 1). This facilitates the transfer of musical knowledge from generative pre-training to improve arrangement quality, while reducing the need for task-specific model design. To achieve this, we designed a context-aware, segment-level reconstruction objective: the model reconstructs multitrack music from its disentangled components, including content (notes executed) and style (instrumentation). We further observe that strictly time-ordered tokenizations (e.g.,  $\mathrm{REMI} + [31]$ ) introduce redundancy and fragment track content (detailed in §2.2), hindering instrument-level control and modeling. To address this, we propose a structured tokenization scheme that relaxes global time ordering while promoting track-wise continuity, enabling more consistent encoding across musical contexts. Our main contributions are as follows:
+
+- We propose a unified framework for symbolic music arrangement that supports flexible instrumentation transformation across multiple typical arrangement scenarios, all without requiring parallel data. Central to our approach is a shared reconstruction objective defined over token-level disentangled note properties, which enables a single generative model to adapt through lightweight fine-tuning.  
+- To support effective learning under the proposed objective, we introduce an efficient and modeling-friendly tokenization scheme for multitrack music that produces shorter sequences with lower complexity, facilitates instrument-level control and modeling that are important for arrangement performance. It further reduces note-level perplexity in unconditional generation, indicating potential utility beyond arrangement tasks.  
+- Instantiated with a small model and modest-scale pre-training, our system outperforms task-specific SOTA baselines on three representative arrangement tasks, each corresponding to a distinct scenario—band arrangement (reinterpretation), piano reduction (simplification), and drum arrangement (additive generation)—in both objective and subjective evaluations.
+
+# 2 Related Work
+
+# 2.1 Automatic Arrangement in Multitrack Symbolic Music
+
+Symbolic music arrangement encompasses a variety of tasks such as chord progression generation [38], orchestration from lead sheets [34], and instrumentation transfer [42]. In this work, we focus on adapting multitrack music to new instrumentations—a representative setting that requires fine-grained control over musical content and instrumentation.
+
+Early supervised learning approaches rely on parallel datasets (e.g., piano-to-orchestra [5], band-topiano [29, 30]), which are expensive to construct and inherently constrain the direction of arrangement. Classification-based methods [9] approach the arrangement problem by learning to assign instrument labels to each note from a dataset with fixed instrumentation. However, such models are limited in expressiveness, as they cannot modify musical content—e.g., adding, removing, or altering notes—to suit a target instrument combination. Moreover, their reliance on fixed instrument sets hinders generalization to unseen combinations.
+
+Recent self-supervised methods [41, 42] avoid these constraints by guiding generation with predefined or autoregressively modeled high-level descriptors (e.g., pitch histogram, note density). While effective for maintaining stylistic coherence, they separate style modeling from content realization. As a result, instrument playing styles are either fixed or modeled independently of input variations. Since musical material often changes between sections, this decoupling can degrade fidelity—causing arrangements to fail to reflect core aspects such as melody or texture, resulting in noticeable perceptual divergence. This suggests a more fidelity-oriented approach: integrating style modeling into the generation process to allow dynamic adaptation to input content.
+
+Additionally, infilling-based models such asComposer's Assistant [20, 19], although capable of handling additive generation scenarios such as drum arrangement, do not preserve the music essence of the original composition and are thus unsuited for reinterpretation and simplification.
+
+# 2.2 Symbolic Music Tokenization
+
+Transformer-based symbolic music modeling requires converting musical data into token sequences, a nontrivial task due to music's inherent multi-stream structure, i.e., multiple instruments playing concurrently. ABC-based notations [23] convert staff-based music into text-like symbolic representations, which are well suited for classical sheet music. While for contemporary music, many existing tokenization schemes operate on MIDI files and adopt a linearized, note-event-based encoding where each musical note is represented as a tuple of attribute tokens (e.g., onset, pitch, duration, velocity, instrument). Some use absolute timing [13, 12, 39, 10], while others use metric durations [26, 14, 31]. Among them, the REMI representation [14], originally designed for single-track music, has been extended to  $\mathrm{REMI + }$  [31] for flexible tokenization of multitrack music, by associating each note with an instrument token.
+
+However, existing time-ordered tokenization schemes such as  $\mathrm{REMI + }$  face structural limitations that hinder track-wise control and modeling. By flattening multitrack music into a strictly time-ordered sequence,  $\mathrm{REMI + }$  interleaves events from different instruments, resulting in content fragmentation. As illustrated in Figure 3a, notes from the same instrument (e.g., i-29, distorted electric guitar, in orange) are interrupted by those from concurrent instruments (e.g., i-80, synth lead, in red). This leads to two major issues. First, the lack of structured syntax makes it difficult to delineate track boundaries, hindering enforcement of user-specified instrument constraints and often leads to spurious instruments in arrangement outputs. Second, identical track-wise content can be tokenized differently depending on concurrent instrument activity. This context sensitivity reduces the repetition of typical per-instrument patterns in the training data—a key factor for learning accurate "instrument syntax" determined by physical constraints and idiomatic playing patterns of each instrument, important for arrangement quality. While some recent work explores vocabulary-level compression (e.g., Byte Pair Encoding [11]) to reduce sequence length—an approach orthogonal and complementary to tokenization scheme design—it does not address the structural issues discussed above.
+
+![](images/39ebf885deb27ebdbb941816a38a0c2ba53dc0081d1dec6433ba0c93addf978f.jpg)  
+Figure 2: An example of the tokenized sequence for the band arrangement task. Special tokens  $<\mathrm{SEP}>$ ,  $<\mathrm{INSTRUMENT}>$ ,  $<\mathrm{CONTENT}>$ , and  $<\mathrm{HISTORY}>$  are used to separate different components. Tokens starting with o-, i-, p-, d- represents the onset, instrument ID, pitch, and duration of notes.
+
+# 3 Method
+
+Our goal is to fine-tune a single pre-trained symbolic music generative model across diverse arrangement tasks via a unified pipeline. The core is a segment-level reconstruction objective (§3.1) over token-level disentangled representations of style (instrumentation) and content (musical notes). To support track-wise control and modeling, we introduce a structured tokenization scheme (§3.2) that reduces content fragmentation and enhances learning under this objective.
+
+# 3.1 Reconstruction from Token-Level Disentangled Multi-Streams with Context Awareness
+
+Symbolic music offers a unique opportunity for token-level disentanglement: each note is represented as a list of semantically independent tokens, each describing a unique property of the note (e.g., onset, duration, pitch, instrument), allowing content (what is played) and instrumentation (by whom it is played) to be explicitly separated. This level of structural redundancy is rare in natural language, where sub-word tokens are atomic.
+
+Building on this observation, we formulate arrangement as a self-supervised, segment-level reconstruction task. The input music is decomposed into three token streams—instrument, content, and preceding context—and a pre-trained symbolic music generative model is fine-tuned to reconstruct the desired tracks of the multitrack music from these components.
+
+Let  $y^{(t)}$  denote the  $t$ -th segment of a music piece. The fine-tuning objective is defined as:
+
+$$
+\mathcal {L} (\theta) = - \log p _ {\theta} \left(\mathcal {T} _ {\text {t a s k}} \left(y ^ {(t)}\right) \mid \mathrm {I} \left(\mathcal {T} _ {\text {t a s k}} \left(y ^ {(t)}\right)\right), \mathrm {C} \left(\mathcal {S} _ {\text {t a s k}} \left(y ^ {(t)}\right)\right), \mathcal {T} _ {\text {t a s k}} \left(y ^ {(t - 1)}\right)\right), \tag {1}
+$$
+
+where  $\theta$  represents the model parameters,  $\mathrm{I}(\cdot)$  and  $\mathrm{C}(\cdot)$  extract instrument and content conditions,  $S_{\text {task }}$  and  $T_{\text {task }}$  are filters that select task-specific source and target tracks respectively (detailed in $ 3.3), and  $y^{(t-1)}$  provides target-side history. Equation 1 is implemented using a standard next-token prediction objective on the sequence [condition]<sep>[target] (Figure 2), with cross-entropy loss computed only on the target subsequence.
+
+Instrument condition. The instrument condition specifies the desired instruments to be used in the segment. During training, it includes all instrument tokens from the target sequence. Their order encodes pitch register relationships across tracks: instruments with higher average pitch are placed earlier. At inference, users can specify arbitrary desired instruments (instrument control) and define their relative pitch register ordering (voice control).
+
+Content condition. The content condition is derived from the original composition, encoded as a content sequence without instrument information. To obtain it, we remove all instrument tokens from the original multitrack sequence, sort notes by onset time, then by descending pitch, and merge duplicates. This yields a clean time-ordered note sequence conveying what is played, independent of how or by whom.
+
+History condition. The history condition provides musical context from the preceding segment, encouraging inter-segment coherence at inference time. During training, it is provided via teacher-forcing with the complete tokenized previous segment; at inference, it is autoregressively generated. This mechanism helps maintain coherent arrangement style across segments, especially important for long-form (e.g., song-level) arrangement scenarios.
+
+During training, the instrument, content, and history conditions are derived from the same music being reconstructed. The model thus learns to reinterpret musical content with various instrument combinations under specific contexts, enabling diverse arrangement behaviors.
+
+Figure 3: REMI+ and REMI-z tokenization for the same bar. Contents of the same instruments are highlighted with the same color. See Appendix A for complete vocabulary.
+
+# 3.2 REMI-z: Tokenizing Multitrack Music with Track-Wise Continuity
+
+As discussed in Section 2.2, strictly time-ordered tokenization schemes suffer from content fragmentation, a structural limitation that hinders arrangement performance. To mitigate this issue, we propose REMI-z, a tokenization scheme that heuristically prioritizes track-wise continuity over global time ordering. Specifically, REMI-z processes MIDI files into a list of bar sequences, each containing multiple track sequences, where each track corresponds to a unique instrument. Within each track, note events are sorted by their onset position, then by descending pitch, and grouped under a single instrument token. Tracks within a bar are then ordered by their average pitch (high to low), forming a bar-level token sequence that ends with a special end-of-bar token. The complete vocabulary and tokenization examples are shown in Appendix A.
+
+This zig-zag organization—reflected in the REMI-z name—offers several modeling advantages: (1) instrument-wise content is locally contiguous (Figure 3b), reducing content fragmentation and enabling clear track boundaries; (2) sequence length is reduced by eliminating redundant instrument tokens; (3) temporal structure is preserved both within individual track sequences and between bars.
+
+We adopt REMI-z to tokenize all data for both pre-training and fine-tuning. Beyond improving arrangement performance, we also observe that REMI-z produces sequences with lower information entropy and, when used for unconditional generative training, leads to better note-level modeling (see § 5.4). This suggests that its track-continuity design not only benefits arrangement tasks, but may also support general symbolic music modeling.
+
+# 3.3 Task Instantiations
+
+We evaluate our method on three representative music arrangement tasks that reflect typical arrangement scenarios, each assessing different capabilities of the model: band arrangement (reinterpretation), piano reduction (simplification), and drum arrangement (additive generation).
+
+Band Arrangement. This task evaluates the model's ability to reinterpret an existing piece using arbitrary combinations of pitched instruments. The model must learn the properties and idiomatic playing styles of various instruments to reallocate or generate notes appropriately. Both  $S_{\text{task}}$  and  $T_{\text{task}}$  are identity mappings (i.e.,  $S_{\text{task}}(y) = T_{\text{task}}(y) = y$ ), and drum tracks are removed from the input. To encourage creative rewriting, we randomly remove a subset of tracks from the content condition  $C(S_{\text{task}}(y^{(t)}))$  during training, while ensuring the melody's content is preserved (detailed in Appendix B.2). Duration tokens are also removed from the content stream, allowing the model to infer track-specific note durations, i.e., articulations, suitable for interpreting the content under desired instrumentations. Segment length is set to 1 bar. This setup resembles [42], but differs in its more challenging setting: no track-wise style priors are available to guide the generation process.
+
+Piano Reduction. This task simplifies ensemble music into solo piano accompaniment, aiming to preserve key harmonic and textural elements while ensuring pianistic playability. The  $\mathcal{T}_{\mathrm{task}}$  selects piano tracks, while  $S_{\mathrm{task}}$  is the identity mapping. To ensure the reduction is meaningful, we filter training data to keep only segments where the piano part is sufficiently prominent (covering  $>40\%$  of the pitch range). Drum tracks are removed from input, and the segment length is set to 1 bar. The setup is similar to [30], but uses original MIDI piano tracks as targets instead of human-composed reductions, and does not include difficulty-level conditioning.
+
+Drum Arrangement. The goal of this task is to generate a drum track for music that lacks one. Here,  $S_{\mathrm{task}}$  extracts all pitched-instrument tracks, while  $T_{\mathrm{task}}$  extracts the drum track. We use 4-bar segments since drum patterns often span across multiple bars. The model must recognize the underlying groove and phrase boundaries of the source music, enhance them with coherent and stylistically appropriate rhythmic patterns, and ensure proper transitions across segments—requiring a stronger understanding
+
+of rhythmic ideas and structural organization. This track infilling setup is similar to [19], but the model is not allowed to condition on future segments.
+
+# 4 Experiments
+
+# 4.1 Implementation Details
+
+Our model, an 80M-parameter decoder-only Transformer, has a hidden dimension of 768, 12 layers, 16-head attention, and a context length of 2048 tokens (around  $8 \times$  the longest bar in our dataset). The model first undergoes a standard next-token-prediction pre-training, and then was fine-tuned with the proposed objective. Pre-training used four RTX A5000 GPUs (batch size 12, 1 epoch), while fine-tuning used a single A40 GPU (variable batch size, 3 epochs). Pre-training adopted the Los Angeles MIDI dataset [15] (405K MIDI files, 4.3B tokens after REMI-z tokenization,  $2\%$  validation split) and fine-tuning was done with Slakh2100 [21] (1,289 training, 270 validation, 151 test MIDI files), featuring 34 pitched instruments and drums, with  $\geq 4$  tracks per piece. Detailed hyperparameter settings are in Appendix B.5.
+
+# 4.2 Baseline Models
+
+For each task, we compare our model against a state-of-the-art (SOTA) task-specific baseline. For band arrangement, we adopt Transformer-VAE from [42], the strongest previously reported model for multitrack arrangement without assumptions on track type or number. It combines Transformer-based long-term and inter-track modeling with a VQ-VAE generation module. For piano reduction, we compare with [30] (UNet), the most recent work in this area. For drum arrangement, we adopt Composer's Assistant 2 (CA v2) [19], a SOTA track infilling model capable of handling multitrack inputs and generating drum outputs. Existing drum-specific models (e.g., [2], [6]) are unsuitable for our setting, as they assume a single melody or instrumental track input rather than general multitrack conditioning. Baseline's implementation details are in Appendix B.4.
+
+To demonstrate the impact of generative pre-training, an ablation variant of our model without pre-training  $(\mathbf{w} / \mathbf{o}\mathbf{PT})$  is used as a baseline. Additionally, we include simple rule-based baselines as non-learning references, designed to provide naive solutions with minimal musical knowledge, helping to contextualize the difficulty of arrangement tasks. For band arrangement, Rule-Based distributes notes evenly by pitch across instruments, serving as a naive reinterpretation strategy. For piano reduction, we use Rule-F (a flattened multitrack where the piano plays all notes), which reflects an overcomplete reduction prioritizing coverage, and Rule-O (the original piano track), which provides a playability-guaranteed but musically incomplete reduction. For drum arrangement, the original drum track (Ground Truth) is included anonymously in the human evaluation as an upper bound on perceptual scores.
+
+# 4.3 Objective Evaluation
+
+Objective metrics measure similarity between model outputs and target sequences, assuming closer resemblance to human-created music indicates higher naturalness and musicality. Following [19, 30], we use note-level  $F1$  to measure similarity between model outputs and target sequences. Specifically, we compute Note F1 (correct onset and pitch) and Note_i F1 (additional correct instrument prediction), both under 16th-note quantization for fair comparison with baselines.
+
+For piano reduction and drum arrangement, the same models are used in objective and subjective evaluations. For band arrangement, models are separately trained without random track deletion to ensure deterministic outputs. Baseline models are also modified by excluding its prior model to remove long-term context hints, ensuring evaluation fairness.
+
+In addition to the modifications described above, we introduce three task-specific metrics for band arrangement. First, Instrument Intersection over Union (I-IoU) evaluates the accuracy of instrument control. Second, Voice Error Rate (VER) measures the similarity in voice features between the generated output and the reference, reflecting how well the model follows the voice conditions specified by instrument token ordering. Third, Melody F1 (Mel F1) computes the Note F1 score on melody tracks, estimated as the tracks with the highest average pitch in the output and reference, to
+
+Table 1: Objective evaluation results for the band arrangement task. Statistical significance is indicated as follows: * for  $p < {0.05}, \dagger$  for  $p < {0.01}$  ,and  $\ddagger$  for  $p < {0.001}$  .  
+
+<table><tr><td>Model</td><td>I-IOU ↑</td><td>VER ↓</td><td>Note F1 ↑</td><td>Notei F1 ↑</td><td>Mel F1 ↑</td></tr><tr><td>Transformer-VAE</td><td>97.5</td><td>35.0</td><td>49.5</td><td>40.0</td><td>24.7</td></tr><tr><td>Transformer w/ REMI+</td><td>95.0</td><td>18.2</td><td>94.4</td><td>76.0</td><td>68.8</td></tr><tr><td>Transformer w/ REMI-z</td><td>$99.5</td><td>$9.9</td><td>$97.8</td><td>$77.5</td><td>$77.8</td></tr><tr><td>+ Pre-training (Ours)</td><td>99.8</td><td>7.6</td><td>97.5</td><td>87.0</td><td>84.5</td></tr><tr><td>– voice</td><td>99.6</td><td>17.6</td><td>97.2</td><td>84.3</td><td>81.5</td></tr><tr><td>– history</td><td>100.0</td><td>9.0</td><td>97.6</td><td>77.4</td><td>79.4</td></tr></table>
+
+assess how well the original melody is preserved—an important factor for perceived fidelity. Detailed definitions and computation procedures are provided in Appendix C.1.
+
+Among all tasks, band arrangement is a strong testbed for evaluating controllability and generalizability because it requires the highest flexibility without assumptions on target instrument types or counts. Hence it serves two additional purposes: (1) to validate key design choices in our fine-tuning objective, particularly the use of voice-aware instrumentation and segment-level history conditioning; and (2) to prove the effectiveness of the proposed tokenization schemes in arrangement task. When comparing tokenization schemes, statistical significance is computed by Wilcoxon signed rank test [36]. For a broader analysis of tokenization's impact on unconditional generation, see § 5.4.
+
+# 4.4 Human Evaluation
+
+To complement similarity-based objective metrics and assess perceptual quality and creativity, we conducted human evaluations. Full-piece arrangements were generated by all models and evaluated on a 5-point scale (1: very low, 5: very high). For band arrangement, models were tested across three instrument combinations with different complexity: string trio (3 tracks), rock band (4 tracks), and jazz band (7 tracks). Three metrics were used across band, piano, and drum arrangement tasks: Coherence, which evaluates the natural flow of the arrangement and the consistency of each instrument's playing style throughout the piece; Creativity, which assesses the degree of innovation in the arrangement under the constraints of the music's content and style; and Musicality, which measures the overall musical appeal and aesthetic quality of the arrangement. Further details on the metrics, ensemble settings, questionnaire, and evaluation process are in Appendix D.
+
+Task-specific metrics were introduced for the distinct evaluation needs of each arrangement scenario. For band arrangement, Faithfulness measures resemblance to the original in melody and overall feel, while Instrumentation assesses the appropriateness of each instrument's role within the ensemble and their harmony. For piano reduction, Faithfulness is also adopted but without melody preservation requirements, and Playability assesses the feasibility of the generated contents played by human pianists. For drum arrangement, Compatibility measures how well the drum track blends with other instruments, and Phrase Transition assesses the smoothness of transitions between musical phrases. We report mean and standard error of mean in result tables. Significance tests were conducted between our model and the SOTA baselines using within-subject (repeated-measures) ANOVA [28].
+
+# 5 Results
+
+# 5.1 Band Arrangement
+
+# 5.1.1 Objective Evaluation
+
+Tokenization impact. Table 1 shows that model adopted REMI-z significantly outperforms that adopted REMI+ across all objective metrics. It enables stronger instrument control (I-IOU:  $99.5\%$  vs.  $95.0\%$ ) and voice control (VER:  $9.9\%$  vs.  $18.2\%$ ), aligning outputs better with user-specified conditions. The improvement in Note F1  $(+6.7\%)$  confirms higher reconstruction quality, while the gain in  $\mathrm{Note}_i$  F1  $(+1.9\%)$  highlights enhanced instrument-wise modeling. Furthermore, the substantial increase in Mel F1  $(+9.0\%)$  suggests REMI-z enable the model to better identify the melody components from content sequence, which is improtant for arrangement fidelity.
+
+Table 2: Band arrangement subjective evaluation results. Fa., Co., In., Cr., and Mu. represent Faithfulness, Coherence, Instrumentation, Creativity, and Musicality, respectively.  
+
+<table><tr><td>Model</td><td>Fa. ↑</td><td>Co. ↑</td><td>In. ↑</td><td>Cr. ↑</td><td>Mu. ↑</td></tr><tr><td>Rule-Based</td><td>3.46±0.14</td><td>3.05±0.13</td><td>2.89±0.14</td><td>3.00±0.12</td><td>3.07±0.13</td></tr><tr><td>Transformer-VAE</td><td>2.65±0.10</td><td>2.70±0.11</td><td>2.72±0.12</td><td>3.00±0.13</td><td>2.72±0.11</td></tr><tr><td>Ours</td><td>‡3.77±0.13</td><td>‡3.47±0.15</td><td>‡3.49±0.16</td><td>*3.40±0.13</td><td>‡3.47±0.14</td></tr><tr><td>w/o PT</td><td>3.19±0.13</td><td>2.82±0.13</td><td>2.86±0.14</td><td>2.93±0.12</td><td>2.75±0.13</td></tr></table>
+
+Table 3: Piano reduction results. The Pl. represents Playability score.  
+
+<table><tr><td>Model</td><td>F1 ↑</td><td>Fa. ↑</td><td>Co. ↑</td><td>Pl. ↑</td><td>Cr. ↑</td><td>Mu. ↑</td></tr><tr><td>Rule-F</td><td>-</td><td>3.93±0.13</td><td>3.59±0.13</td><td>3.14±0.13</td><td>2.96±0.13</td><td>3.34±0.14</td></tr><tr><td>Rule-O</td><td>-</td><td>2.75±0.13</td><td>3.49±0.13</td><td>4.07±0.12</td><td>2.62±0.14</td><td>2.96±0.14</td></tr><tr><td>UNet</td><td>58.3</td><td>2.97±0.13</td><td>2.90±0.15</td><td>3.47±0.13</td><td>2.82±0.13</td><td>2.78±0.13</td></tr><tr><td>Ours</td><td>85.5</td><td>‡3.63±0.13</td><td>‡3.64±0.13</td><td>*3.86±0.13</td><td>*3.14±0.12</td><td>‡3.48±0.14</td></tr><tr><td>w/o PT</td><td>78.4</td><td>2.25±0.13</td><td>2.58±0.16</td><td>3.29±0.15</td><td>2.67±0.15</td><td>2.26±0.14</td></tr></table>
+
+Pre-training benefit. Pre-training brings clear gains by transferring musical knowledge useful for arrangement. It improves three key aspects: (1) lower VER  $(-2.3\%)$ , indicating more effective voice control; (2) higher Note $_i$  F1  $(+9.5\%)$ , reflecting enhanced instrument-wise content modeling; and (3) the highest Mel F1  $(84.5\%)$ , indicating stronger melody preservation. Probing analysis (Appendix E) further shows that pre-training strengthens the alignment between token embeddings and musical concepts useful for arrangement, such as instrument roles and chord progression.
+
+Comparison with Transformer-VAE. Our model outperforms Transformer-VAE by a wide margin. In particular, it achieves much higher Note F1 (97.5% vs. 49.5%), Note $_i$  F1 (87.0% vs. 40.0%), and Mel F1 (84.5% vs. 24.7%), highlighting the advantage of our context-aware, content- and instrument-conditioned generation approach over the latent inference used in Transformer-VAE.
+
+Ablation: voice and history conditioning. Removing voice information from instrument conditions degrades voice control (VER: +10.0%), demonstrating the effectiveness of our voice control method. It also lowers per-instrument F1 (Notei: -2.7%) and Mel F1 (-3.0%), indicating that voice-order information serves as a useful hint for inferring instrument roles during arrangement. Excluding history conditioning results in even larger drops in Notei F1 (-9.6%) and Mel F1 (-5.1%), validating that temporal context facilitates accurate reconstruction, laying the foundation for coherent song-level arrangement.
+
+# 5.1.2 Human Evaluation
+
+Strong subjective performance. As shown in Table 2, Our model achieves the highest scores across all subjective metrics than all baselines, demonstrating its ability to generate coherent, stylistically appropriate, and musically appealing arrangements while preserving core musical essence.
+
+Transformer-VAE underperforms. It lags significantly behind our model in every subjective criterion. Notably, its lower Faithfulness score (-1.12) reflects difficulty in preserving core musical content, while low ratings in Instrumentation (-0.77) and Coherence (-0.77) suggest weaker track-wise modeling and non-idiomatic instrument usage, as well as insufficient coherence between segments. Overall Musicality (-0.75) also falls below our model. It even scores lower than the rule-based baseline, though the latter suffers from inherent limitations in musicality and coherence.
+
+Pre-training improves quality. Removing pre-training reduces scores across all metrics, with notable drops in Faithfulness (-0.58) and Musicality (-0.72), reinforcing the importance of musical knowledge transfer from generative pre-training for content retention and overall perceptual quality.
+
+# 5.2 Piano Reduction
+
+Best overall quality. As shown in Table 3, our method obtains the highest ratings in F1 (85.5%), Coherence (3.64), Playability (3.86), and Musicality (3.48), while also maintaining strong scores in Faithfulness (3.63) and Creativity (3.14). It significantly outperforms UNet across all subjective
+
+Table 4: Drum arrangement results. Comp. and Tr. represent Compatibility and Phrase Transition score respectively.  
+
+<table><tr><td>Model</td><td>F1 ↑</td><td>Comp. ↑</td><td>Co. ↑</td><td>Tr. ↑</td><td>Cr. ↑</td><td>Mu. ↑</td></tr><tr><td>Ground Truth</td><td>100.0</td><td>4.31±0.12</td><td>4.18±0.10</td><td>3.36±0.13</td><td>3.16±0.12</td><td>3.78±0.12</td></tr><tr><td>CA v2</td><td>20.3</td><td>3.82±0.13</td><td>4.05±0.12</td><td>2.86±0.12</td><td>2.58±0.11</td><td>3.19±0.12</td></tr><tr><td>Ours</td><td>79.3</td><td>3.91±0.12</td><td>4.03±0.10</td><td>3.77±0.12</td><td>3.27±0.14</td><td>3.57±0.13</td></tr><tr><td>w/o PT</td><td>1.2</td><td>2.49±0.16</td><td>2.19±0.12</td><td>2.21±0.14</td><td>2.82±0.15</td><td>2.05±0.13</td></tr></table>
+
+Table 5: Tokenization scheme comparison on unconditional generation.  
+
+<table><tr><td>Tokenizer</td><td>Tbar ↓</td><td>Tnote ↓</td><td>Hbar ↓</td><td>PPLnote ↓</td><td>PPLtoken ↓</td></tr><tr><td>REMI+</td><td>225.91</td><td>4.03</td><td>41.68</td><td>116.20</td><td>3.00</td></tr><tr><td>REMI-z (Ours)</td><td>151.68</td><td>2.77</td><td>29.43</td><td>84.11</td><td>4.50</td></tr></table>
+
+metrics, with especially large margins in Faithfulness  $(+0.66)$ , Coherence  $(+0.74)$ , and Musicality  $(+0.70)$ —indicating better content preservation, cross-segment continuity, and overall musical quality. Pre-training again proves essential: without it (w/o PT), performance drops notably across all metrics.
+
+Balanced fidelity and playability. Compared to rule-based methods, our model achieves a better trade-off between fidelity and playability. Rule-F preserves original content well (Faithfulness: 3.93) but results in poor playability (3.14), while Rule-O achieves the highest playability (4.07) but at the expense of faithfulness (2.75). In contrast, our method produces reductions with balanced Faithfulness (3.63) and Playability (3.86).
+
+# 5.3 Drum Arrangement
+
+Best subjective quality among learned models. As shown in Table 4, our model outperforms CA v2 in all subjective metrics except Coherence (4.03 vs. 4.05), and significantly improves Creativity (3.47 vs. 2.58) and Phrase Transition (3.27 vs. 2.86), which are essential for engaging, structurally-aware drum arrangements. It also achieves the highest Musicality score (3.57), closely approaching ground truth (3.78). Again, the w/o PT variant performs poorly across all metrics.
+
+Improved phrasing and variation. Compared to CA v2, which often repeats similar drum patterns throughout a piece, our model produces more varied and context-aware rhythms that better reflect musical phrasing. The higher Phrase Transition score (+0.91) supports this observation, demonstrating the model's ability to capture structural changes in music during generation.
+
+# 5.4 Tokenization Efficiency and General Modeling Advantages
+
+Additionally, we evaluate REMI-z against  $\mathrm{REMI + }$  to assess its efficiency and effectiveness for generative modeling of symbolic multitrack music. We report several metrics to evaluate compactness of tokenization schemes and their unconditional modeling performance: 1) average tokens per bar  $(\bar{T}_{\mathrm{bar}})$ , 2) average tokens per note  $(\bar{T}_{\mathrm{note}})$ , 3) Shannon entropy of bar-level token sequences  $(\bar{H}_{\mathrm{bar}})$ , and 4) note-level perplexity  $(\mathrm{PPL}_{\mathrm{note}})$ . Note that  $\mathrm{PPL}_{\mathrm{note}}$  is the aggregated conditional probability of all note attribute tokens, normalized by the number of notes, enabling fairer comparison across tokenization schemes. Calculations are detailed in Appendix C.2.
+
+**Compactness.** Tokenizing Slakh2100 with REMI-z yields a  $32.9\%$  reduction in sequence length per bar (151.68 vs 225.91) and fewer tokens per note (2.77 vs 4.03), effectively reducing training and inference computational costs.
+
+Representational Simplicity. REMI-z also produces lower bar-level Shannon entropy (29.43 vs 41.68 bits/token) on Slakh2100 compared to  $\mathrm{REMI + }$ , despite representing the same underlying musical content. This indicates reduced information redundancy, and suggests that REMI-z sequences consist of more predictable tokens, potentially facilitating the learning of generative models.
+
+Note-Level Modeling Advantage. We train unconditional generation models on Slakh2100 with both tokenizations and compare note-level perplexity with the same architecture as our arrangement models. The model adopting REMI-z achieves substantially lower note-level perplexity (84.11 vs.
+
+116.20), indicating better modeling of complete musical notes. Since notes form the atomic units of music, improvements in note-level modeling directly contribute to lower uncertainty at modeling bars and full compositions, suggesting utility in general symbolic music modeling beyond arrangement tasks. While  $\mathrm{REMI+}$  slightly outperforms in token-level perplexity, this does not translate to better modeling of higher-level structures.
+
+# 6 Conclusion
+
+We presented a unified framework for automatic music arrangement, centered around a reconstruction objective that enables diverse arrangement tasks without requiring parallel data, enabling knowledge transfer from generative pre-trained symbolic music models for arrangement tasks. Our approach also integrates a structured tokenization scheme, REMI-z, which convert multitrack music to compact and easy-to-model token sequences. Experimental results on band arrangement, piano reduction, and drum arrangement show that our method consistently outperforms task-specific baselines in both objective and subjective evaluations. These results suggest the potential of our framework as a general solution for symbolic music-to-music transformation.
+
+# Acknowledgements
+
+The authors would like to thank Chenfei Kang for providing valuable resources that supported the validation of the proposed idea, and Yisong Miao for insightful comments on training and coherence of language models, which substantially benefited this work.
+
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+
+# A REMI-z Tokenization
+
+# A.1 Vocabulary
+
+Table 6 presents the complete vocabulary of our proposed REMI-z tokenizer, detailing the value ranges for each token type. For instrument tokens, the values correspond to MIDI program numbers (0-127), with 128 specifically designated for the drum set. For pitched instruments, pitch token values directly map to MIDI pitch numbers, while drum set pitches are encoded as MIDI pitch + 128. Both position and duration tokens are quantized in units of a 48th note (one-third of a sixteenth note). Time signature and tempo tokens undergo specific quantization before token mapping; for detailed quantization rules, please refer to our code implementation. All tokens listed in the table were utilized during pre-training, including time signature and tempo tokens serving as bar-level properties, with example REMI-z sequences illustrated in Figure 4b.
+
+During fine-tuning, we simplified the token set by excluding time signature and tempo tokens. This decision was supported by our analysis of the fine-tuning dataset (Slakh2100 [21]), where  $94.8\%$  of songs use 4/4 time signatures. Consequently, we restricted our fine-tuning to 4/4 songs and omitted time signature tokens, following practices in related works [35, 42]. We also excluded tempo tokens since tempo adjustments in digital audio workstations are typically handled as a global parameter, affecting only tempo tokens without altering the any other tokens. This simplification assumes that compositional and performance styles remain consistent across different tempos. However, this assumption may not hold for datasets with significant tempo variations. Therefore, for future research requiring time-signature- or tempo-specific characteristics, we recommend including these tokens during fine-tuning.
+
+Velocity tokens were excluded from both pre-training and fine-tuning phases. This decision reflects our focus on compositional quality rather than performance naturalism, aligning with previous approaches in band and piano arrangement studies [42, 30]. However, if velocity information is deemed crucial for generation, our model architecture readily accommodates the addition of velocity tokens to each note without introducing content fragmentation issues, if the REMI-z note organization order is maintained.
+
+# A.2 Example
+
+We name our tokenization scheme  $\text{REMI-z}$  for its distinctive "zig-zag" encoding pattern for musical notes within each bar, as illustrated in Figure 4a. In this scheme, notes are encoded hierarchically: first grouped by tracks (instruments), then organized bar by bar. This track-first approach ensures notes from the same instrument remain clustered together, thereby enhancing the model's ability to learn instrument-specific patterns. The resulting  $\text{REMI-z}$  sequence is demonstrated in Figure 4b. In contrast,  $\text{REMI+}$  [31] employs a column-wise encoding strategy, strictly ordering notes by their temporal positions. While this approach effectively captures global temporal relationships, it disperses notes from the same instrument throughout the sequence, potentially complicating instrument-specific pattern learning. We leverage such temporal ordering in our content sequence (excluding instrument tokens), as shown in Figure 5.
+
+Table 6: REMI-z vocabulary with pitch token distinctions  
+
+<table><tr><td>Meaning</td><td>Token</td><td>X&#x27;s range</td></tr><tr><td>Instrument type</td><td>i-X</td><td>0~128</td></tr><tr><td>Note&#x27;s within-bar position</td><td>o-X</td><td>0~127</td></tr><tr><td>Note&#x27;s pitch (non-drum)</td><td>p-X</td><td>0~127</td></tr><tr><td>Note&#x27;s pitch (drum)</td><td>p-X</td><td>128~255</td></tr><tr><td>Note&#x27;s duration</td><td>d-X</td><td>0~127</td></tr><tr><td>End of a bar</td><td>b-1</td><td>-</td></tr><tr><td>Time signature</td><td>s-X</td><td>0~253</td></tr><tr><td>Tempo</td><td>t-X</td><td>0~48</td></tr></table>
+
+![](images/ad9e21e9c14a10eb5f9a86e9bddbb3e1c9728b6c574fdd1001e17a1b5e59533a.jpg)  
+(a) A 4-bar musical segment with orange arrows illustrating the "zig-zag" encoding order of notes in REMI-z tokenization.
+
+![](images/ecae2ed627722911b1fe41d587eeac40f6db58a674b6d2a9690e9a15d20d7e1d.jpg)  
+(b) The REMI-z sequence tokenized from the musical segment above. Track sequences are color-coded by instrument: synth lead (red), clean electric guitar (blue), distorted electric guitar (orange), and electric bass (purple).
+
+![](images/9fa6db4c8dceca396fb1f4a2c91bfbe085b64161752521fbbb4286c779d6850b.jpg)  
+Figure 4: An example of REMI-z tokenization.  
+Figure 5: The content sequence obtained by applying the operator  $\mathrm{C}(\cdot)$  to the REMI-z sequence shown in Figure 4b.
+
+# B Implementation Details
+
+# B.1 Key Normalization
+
+Transposition-equivariance is a crucial property in symbolic music: a composition's musicality remains unchanged under global pitch shifts (uniform pitch adjustment across all notes). However, uneven key distribution in datasets can lead to data sparsity during training, potentially causing models to perform inconsistently across different keys. Two approaches address this issue: (1) data augmentation through systematic semitone transpositions  $0, \pm 1, \pm 2, \dots$  [19, 42], or (2) normalizing all songs to a common key (e.g., C major and A minor) [17]. We adopt the latter approach, implementing a modified version of [17]'s method. Our implementation uses a key dictionary mapping from each of the 24 keys (12 major and 12 minor) to a 12-dimensional binary vectors, where 1s indicate scale notes. Key detection is performed by computing dot products between these vectors and a song's pitch histogram, with the highest-scoring key determining the transposition needed to normalize to C major or A minor. Further details can be found in the code.
+
+# B.2 Random Track Deletion
+
+To encourage creativity in band arrangement and prevent the model from over-relying on direct copying notes from the input, we introduce a random track deletion mechanism during training. The intuition is that a well-trained model should be able to infer suitable instrumental content even if such content does not exist in the original music composition.
+
+Concretely, given a content sequence with multiple instrument tracks, we randomly delete all notes that belong to a subset of instruments before calculating the content sequences and feeding to the model. The number of instruments to delete is sampled from a Poisson distribution with  $\lambda = \max (\lfloor |\mathcal{I}| / 4\rfloor ,1)$ , where  $\mathcal{I}$  is the set of non-melodic instruments in the content sequence. The melodic track is estimated by the track with highest average pitch, and does not involve in the track deletion to ensure the melody's content is preserved. Then, the deletion count is clipped to ensure at least one instrument remains. A corresponding number of instruments are uniformly sampled without replacement and all tokens associated with these instruments are removed from the input. Then the content sequence is calculated on this modified music sequence.
+
+Importantly, the target sequence remains unchanged: it includes all instruments and notes removed from the input. This setup requires the model to reconstruct missing tracks based solely on the remaining musical context. In doing so, the model learns not only to replicate observed content but also to infer plausible notes for desired instruments in a given context. This can be treated as a music specific span infilling denoising objective for sequence-to-sequence learning [16], but we don't explicitly tell the model whether an input is masked and the location of masked spans.
+
+# B.3 Instrument Quantization
+
+Band Arrangement While our tokenization scheme supports all MIDI program IDs, many instruments with different IDs share fundamental compositional properties, differing primarily in timbre (e.g., acoustic and electric pianos). To leverage these similarities and reduce instrument distribution sparsity in training data, during fine-tuning, we group similar instruments and assign them the lowest program ID within their group. The instrument grouping rules follow that of [21], leading to 34 different program IDs in total. For multiple tracks of the same instrument type, we merge their notes into a single track.
+
+Piano Arrangement For piano arrangement, we consolidate all piano-type instruments (MIDI program IDs 0-7, including both acoustic and electric pianos) into a single track to form the target sequence.
+
+# B.4 Baseline Models
+
+Band Arrangement For band arrangement, we use the Transformer-VAE model from [42] as our baseline. For human evaluation, we utilized their official implementation without retraining. The model's generation module was trained on Slakh2100 [21] (the same dataset as our fine-tuning), and its prior model leveraged the larger Lakh MIDI Dataset [25], which encompasses Slakh2100. For objective evaluation, we retrained another version of the model on Slakh2100 that did not receive hints from the style prior model for a fair comparison.
+
+Piano Reduction Piano reductions can be categorized into two types: (1) piano accompaniments where the melody is delegated to a separate lead instrument, focusing solely on preserving harmony and texture (e.g., [41, 33]), which can be effectively handled using self-supervised methods; and (2) complete solo piano arrangements that additionally include the melody, requiring careful human arrangement and band-to-piano parallel datasets for supervised learning (e.g., [30, 29]). Due to the lack of open-source parallel datasets, we focus on the accompaniment arrangement task in this paper and use the term piano reduction interchangeably. However, our methodology is inherently flexible and can also handle the second type of reduction if parallel datasets become available. Additionally, exploring the potential for full solo piano reductions using unsupervised approaches remains a valuable direction for future research.
+
+For comparison, we reimplemented the UNet baseline from [30] and trained it on Slakh2100 using our data preparation pipeline. Instead of adopting parallel data, we followed the same setting as our model, using the original composition as input and the piano track within the song as output. To
+
+Table 7: Pre-train hyper-parameter setting.  
+
+<table><tr><td>Hyperparameters</td><td>Values</td></tr><tr><td>seed</td><td>42</td></tr><tr><td>learning_rate</td><td>0.0005</td></tr><tr><td>weight Decay</td><td>0.1</td></tr><tr><td>train_batch_size</td><td>12</td></tr><tr><td>gradient Accumulation_steps</td><td>8</td></tr><tr><td>total_train_batch_size</td><td>96</td></tr><tr><td></td><td>Adam with</td></tr><tr><td>optimizer</td><td>betas=(0.9,0.999) and epsilon=1e-08</td></tr><tr><td>lr_scheduler_type</td><td>cosine</td></tr><tr><td>lr_scheduler_warmup_steps</td><td>1000</td></tr><tr><td>num_epochs</td><td>1</td></tr></table>
+
+ensure musical coverage, we selected only piano tracks that span more than  $40\%$  of the piece's pitch range.
+
+Since hand-specific annotations (i.e., left and right hand separation) are unavailable, the entire piano track is generated as a single sequence rather than as separate streams for each hand. We also omitted octave shifts during input preprocessing for two reasons. The first is that, unlike their setup where human-composed piano references may intentionally transpose notes by octaves, our output consistently corresponds to a subset of the input notes without such shifts. The second is empirical: introducing octave shifts during training significantly degraded UNet's performance, reducing note-level F1 from 58.31 to 42.80 and diminishing perceptual quality.
+
+Drum Arrangement For drum arrangement, we employComposer's Assistant v2 [19] as our baseline, specifically using v2.1.0 from their official repository $^2$ . We use the model without retraining since its original training data (Lakh MIDI Dataset [25]) encompasses our fine-tuning dataset (Slakh2100).
+
+Tokenization Comparison When comparing the effectiveness of  $\mathrm{REMI + }$  and the proposed REMI-z tokenization in §3.2, the model structure used is the same as our arrangement model, but with a simpler training objective—the standard left-to-right next-token-prediction. We did not use time signature tokens, tempo tokens, and velocity tokens in  $\mathrm{REMI + }$  for a fair comparison.
+
+# B.5 Hyperparameter Settings
+
+The model we adopted is a GPT-2 model comprising 12 Transformer decoder layers with a hidden size of 768. Our setup follows GPT's 12-layer, 768-dim embedding, 3072-dim inner states [24], and our experimental design is also in the same style: a single pretrained model and multiple fine-tuning tasks to show the efficacy of the pretrain–finetune paradigm. We used a slightly higher number of attention heads (16 instead of 12) based on the intuition that the interactions between different music tokens are more critical for musical quality than the value of individual note attributes. We detail the hyperparameter configurations used in our experiments below.
+
+For pre-training, Table 7 summarizes the pre-training configuration, where hyperparameters were used as-is without optimization. The pre-training was implemented using pytorch and transformers frameworks on a Linux platform, while fine-tuning additionally utilized lightning.
+
+For fine-tuning, we conducted a simple learning rate search over 1e-5, 5e-5, 1e-4, selecting the optimal value based on validation loss. This resulted in learning rates of 5e-5 for drum arrangement and 1e-4 for band arrangement and piano reduction. The batch sizes and context lengths were configured as follows: band arrangement used a batch size of 24 and context length of 768; piano reduction similarly adopted a batch size of 24 and context length of 768; drum arrangement employed a batch size of 8 and context length of 1536. Across all fine-tuning tasks, we used the AdamW optimizer with
+
+0.01 weight decay, incorporating a linear learning rate scheduler with 500-step warmup. Training spanned 3 epochs for band, piano, and drum tasks, with early stopping patience of 2 epochs. The best checkpoints were selected based on validation loss.
+
+For the tokenization comparison, the model is trained from scratch without generative pre-training. The hyperparameter settings remain the same as those used for band arrangement fine-tuning, except for an increased number of epochs (5).
+
+All experiments were conducted with a fixed random seed of 42.
+
+# C Objective Metrics
+
+In this section, we formally define the objective metrics used in the paper. All objective metrics are calculated at the segment level, and arithmetically averaged across the test set.
+
+# C.1 Arrangement Evaluation
+
+Most of our objective metrics are based on note-level F1 scores calculated on piano roll of 16-th note quantization. Given the extreme sparsity of note events in the track-wise piano roll (e.g., only  $0.12\%$  non-zero elements in the Slakh2100 dataset under 16th-note quantization), F1-based metrics are more suitable than accuracy-based metrics for similarity evaluation. Let  $X = \{x_{1},\dots,x_{n}\}$  and  $Y = \{y_{1},\dots,y_{m}\}$  be two sequences of note events, where each note event  $e$  is defined as a tuple  $e = (t,p)$  with onset time  $t$  and pitch  $p$ . A note event  $x_{i} \in X$  is considered to match  $y_{j} \in Y$  if and only if:
+
+$$
+\left| t _ {x _ {i}} - t _ {y _ {j}} \right| <   \delta_ {t} \text {a n d} p _ {x _ {i}} = p _ {y _ {j}} \tag {2}
+$$
+
+where  $\delta_t$  is the temporal tolerance threshold (set to one 16th-note duration in our evaluation).
+
+Let  $M(X, Y)$  denote the set of matched note pairs between  $X$  and  $Y$ . The Note F1 score is defined as:
+
+$$
+\text {P r e c i s i o n} = \frac {\left| M (X , Y) \right|}{\left| X \right|} \tag {3}
+$$
+
+$$
+\operatorname {R e c a l l} = \frac {\left| M (X , Y) \right|}{\left| Y \right|} \tag {4}
+$$
+
+$$
+\text {N o t e F 1} = \frac {2 \cdot \text {P r e c i s i o n} \cdot \text {R e c a l l}}{\text {P r e c i s i o n} + \text {R e c a l l}} \tag {5}
+$$
+
+The Note $_i$  F1 extends this by considering instrument matching, where each note event becomes  $e = (t,p,i)$  with  $i$  representing the instrument. The matching criterion becomes:
+
+$$
+\left| t _ {x _ {i}} - t _ {y _ {j}} \right| <   \delta_ {t} \text {a n d} p _ {x _ {i}} = p _ {y _ {j}} \text {a n d} i _ {x _ {i}} = i _ {y _ {j}} \tag {6}
+$$
+
+Note  $i$  F1 is then computed using Equation 5 with this stricter matching criterion.
+
+For Melody F1 (Mel F1), given a multitrack piece with tracks  $T = \{T_1, \dots, T_k\}$ , we first identify the melody track  $T_m$  as:
+
+$$
+T _ {m} = \underset {T _ {i} \in T} {\arg \max } \frac {1}{\left| T _ {i} \right|} \sum_ {e \in T _ {i}} p _ {e} \tag {7}
+$$
+
+where  $|T_i|$  is the number of notes in track  $T_i$  and  $p_e$  is the pitch of note event  $e$ . The Mel F1 score between output and target sequences is then:
+
+$$
+\operatorname {M e l} \mathrm {F} 1 = \text {N o t e} \mathrm {F} 1 \left(T _ {m} ^ {\text {o u t}}, T _ {m} ^ {\text {t g t}}\right) \tag {8}
+$$
+
+For instrument control evaluation, Instrument IoU (I-IoU) measures the overlap between instrument sets. Let  $\mathcal{I}^{out}$  and  $\mathcal{I}^{tgt}$  denote the sets of instruments used in the output and target sequences respectively:
+
+$$
+\mathrm {I} - \mathrm {I o U} = \frac {\left| \mathcal {I} ^ {\text {o u t}} \cap \mathcal {I} ^ {\text {t g t}} \right|}{\left| \mathcal {I} ^ {\text {o u t}} \cup \mathcal {I} ^ {\text {t g t}} \right|} \tag {9}
+$$
+
+where an instrument is considered present if there exists at least one note event using it.
+
+Finally, Voice Error Rate (VER) evaluates the similarity of voice arrangements between two multitrack compositions. For each piece, we derive an ordered voice sequence by:
+
+1) Computing the average pitch  $\bar{p}_i$  for each active instrument  $i\in \mathcal{T}^{active}$ :
+
+$$
+\bar {p} _ {i} = \frac {1}{| T _ {i} |} \sum_ {e \in T _ {i}} p _ {e} \tag {10}
+$$
+
+2) Constructing a voice sequence  $V$  by sorting instruments by descending average pitch:
+
+$$
+V = \left[ i _ {1}, i _ {2}, \dots , i _ {n} \right] \text {w h e r e} \bar {p} _ {i _ {k}} \geq \bar {p} _ {i _ {k + 1}} \text {f o r} k = 1, \dots , n - 1 \tag {11}
+$$
+
+The VER between output sequence  $V^{out}$  and target sequence  $V^{tgt}$  is:
+
+$$
+\mathrm {V E R} = \frac {S + D + I}{N} \tag {12}
+$$
+
+where  $S$ ,  $D$ , and  $I$  are the minimum number of substitutions, deletions, and insertions required to transform  $V^{out}$  into  $V^{tgt}$ , and  $N = |V^{tgt}|$  is the length of the  $V^{tgt}$ .
+
+Among all the selected objective metrics, the Note $_i$  F1 is the most comprehensive and important one. To achieve a perfect Note $_i$  F1 score, the model need to perform perfectly in all evaluated aspects: instruments, voice, melody, note reconstruction and allocation.
+
+# C.2 Tokenization Evaluation
+
+Average tokens per bar  $(\bar{T}_{\mathrm{bar}})$  and average tokens per note  $(\bar{T}_{\mathrm{note}})$  are computed by first counting the total number of tokens and the number of bars or notes, respectively, across the entire dataset. Specifically,  $\bar{T}_{\mathrm{bar}}$  is obtained by dividing the total number of tokens by the number of bars, while  $\bar{T}_{\mathrm{note}}$  is obtained by dividing the total number of tokens by the number of notes.
+
+Bar-level Shannon Entropy  $(\bar{H}_{\mathrm{bar}})$  is computed by measuring the Shannon entropy of each bar-level token sequence  $H(X)$ :
+
+$$
+H (X) = - \sum_ {i = 1} ^ {N} P \left(x _ {i}\right) \log_ {2} P \left(x _ {i}\right), \tag {13}
+$$
+
+where  $X$  is the token distribution within a bar. Then we average it across all bars in the dataset to obtain  $\bar{H}_{\mathrm{bar}}$ .
+
+Note-level Perplexity  $(\mathrm{PPL}_{\mathrm{note}})$  aggregates probabilities over all tokens representing a musical note (e.g., instrument, pitch, position, duration), and normalizes by the number of notes:
+
+$$
+\mathrm {P P L} _ {\text {n o t e}} = \exp \left(- \frac {1}{M} \sum_ {j = 1} ^ {M} \log P \left(n _ {j} \mid n _ {1: j - 1}\right)\right), \tag {14}
+$$
+
+where  $M$  is the number of notes in a bar.
+
+Finally, token-level Perplexity (PPL<sub>token</sub>) measures the standard autoregressive perplexity over all tokens. While commonly reported in language modeling, it serves as an auxiliary metric here, as it does not directly reflect the model's ability to model note events, which are basic units of music.
+
+# D Subjective Evaluation
+
+# D.1 Subjective Metrics
+
+These metrics are based on listeners' auditory experiences and their subjective feelings while listening to the music. Since they are subjective, they cannot be easily defined using mathematical equations. However, we detail the evaluation criteria through the prompt questions provided in the questionnaire, which participants answered after listening to the demos. Each metric is assessed on a 5-point scale, ranging from 1 (very low) to 5 (very high).
+
+- Coherence: Does the arrangement flow naturally and smoothly? How consistent is each instrument's performance and style throughout the piece?
+
+- Creativity: How creative is the arrangement while maintaining faithfulness and naturalness?  
+- Musicality: What is the overall musical quality?  
+- Faithfulness (band): How closely does the arrangement resemble the original piece in terms of melody and overall feel?  
+- Faithfulness (piano): How closely does the arrangement capture the overall feel of the original piece?  
+- Instrumentation (band-only): Does each instrument fulfill its appropriate role within the band, and do they harmonize effectively?  
+- Playability (piano-only): How well is the piece suited for piano? How likely is it that a human pianist could perform this accompaniment?  
+- Compatibility (drum-only): Is the drum beat compatible with the other instruments?  
+- Phrase Transition (drum-only): How effectively does the drum arrangement handle transitions between phrases?
+
+# D.2 Human Evaluation Details
+
+Survey Design. We conducted subjective evaluations using audio clips arranged by different models across three tasks: band arrangement, piano reduction, and drum arrangement. For band arrangement, we used out-of-domain test songs with novel compositions and instrument groups; for the other two tasks, test set songs were used to allow comparison with the original piano (Rule-O) or drum (ground truth) tracks. Each model was tasked with arranging the full song, and we selected a 15-30 second chorus phrase—the most representative segment—for evaluation. In total, we prepared 6 songs for band arrangement, 5 for piano reduction, and 5 for drum arrangement.
+
+Each sample group includes the original music to be arranged and the anonymized outputs from all compared models (plus the ground truth for the drum task). Each sample is 8-16 bars long, rendered to audio in Cubase AI 13 using the default soundfont and the original MIDI's BPM. We collected 56, 73, and 77 evaluation groups for band, piano, and drum tasks respectively. The mean time spent per session was approximately 30 minutes. Figure 6 shows the sample survey interface and instructions.
+
+Participant Background A total of 26 participants joined the study, 5 of whom work in the music industry. Among all evaluators,  $73.1\%$  have over 10 years of experience in music composition or performance. We conducted t-tests comparing these raters with those having  $\leq 10$  years of experience (using piano reduction ratings on our model), and found no significant differences in the means between the two groups ( $p > 0.2$  across all metrics). This suggests that musical background did not systematically bias the results.
+
+Evaluation Protocol. Participants first listened to the original music clip, followed by model outputs presented in random order. For each arranged clip, participants rated multiple aspects using a 5-point Likert scale, based on task-specific questions in the previous subsection. The evaluation groups were randomly distributed to participants to ensure unbiased feedback. Each rater evaluated a subset of groups, where one group = one song (input) with multiple model outputs. On average, raters evaluated 2.15 groups for band, 2.80 for piano, and 2.96 for drum — corresponding to 8.6, 14.0, and 11.8 samples respectively.
+
+Band Arrangement Settings. To evaluate generalization, we designed three distinct instrumentation settings: (1) string trio (violin, viola, cello), (2) rock band (synth lead, clean electric guitar, distorted electric guitar, electric bass), and (3) jazz band (saxophone, violin, brass section, clean electric guitar, piano, string ensemble, electric bass). Each setting was applied to two different songs, covering six songs in total. The input content was drawn from both piano arrangements (3 songs) and band arrangements (3 songs), testing the model's ability to adapt to varying content and target instrument groups.
+
+# D.3 Discussion on Evaluation Scheme
+
+Our experiments primarily adopt a 5-point scale for subjective evaluation, which is the most commonly used protocol in prior music arrangement works [42, 35, 38]. However, we observe a growing trend toward using A/B testing to evaluate overall quality in the broader music generation literature [32,
+
+# The Tasks
+
+Now, we will present five arrangement versions, where the arranged accompaniments have been mixed with the original melody, which is played by the original lead instrument. If melody did not present in the original song, the new arrangements will only contain the piano part. Please evaluate the music quality of each sample with the following criteria (1~5 points, 5: highest quality, 1: lowest quality):
+
+- Faithfulness: How closely does the arrangement resemble the original piece in overall feel? (NOTE: faithfulness does not mean copy every note)  
+- Coherence: Does the arrangement flow naturally and feels smooth? How consistent is the piano's performance and style throughout the arrangement?  
+- Playability: How likely this is a piece composed for a piano? How likely a human pianist can play this accompaniment?  
+- Creativity: How creative is the arrangement, without loss of faithfulness or naturalness?  
+- Musicality: Overall musical quality.
+
+# Arrangement 1
+
+![](images/eff805f7f1191279a2ea6d310918579b10c45e267fe68cd88c78cd1c44ee4a9d.jpg)
+
+# Please evaluate
+
+# Faithfulness
+
+# Coherence
+
+# Playability
+
+# Creativity
+
+# Musicality
+
+1
+
+4
+
+1
+
+（
+
+(1)
+
+（
+
+(1)
+
+（
+
+(1)
+
+（
+
+(1)
+
+(1)
+
+(1)
+
+（
+
+(1)
+
+(1)
+
+(1)
+
+（
+
+（
+
+(1)
+
+(1)
+
+（
+
+(1)
+
+uple of the piano reduction task.
+
+# The Tasks
+
+Now, we will present four arrangement versions, where the arranged drum track has been mixed with the original piece. Please evaluate the music quality of each sample with the following criteria (1-5 points, 5: highest quality, 1: lowest quality):
+
+- Compatibility: Is the drum beat compatible with the other instruments being played?  
+- Coherence: Does the arrangement flow naturally and feels smooth, like a human drummer?  
+- Phrase Transition: How well does the drum arrangement handle transitions between phrases?  
+- Creativity: How creative is the arrangement without loss of naturalness? (NOTE: fail to generate drum notes does not count as creativity).  
+Musicality: Overall musical quality.
+
+# Arrangement 1
+
+![](images/30016893859c7dfff90aaf833c9f27c9b52110ed6768d694dc081b375870c6fa.jpg)
+
+# Please evaluate
+
+1
+
+2
+
+3
+
+4
+
+5
+
+# Compatibility
+
+O
+
+O
+
+（
+
+O
+
+O
+
+# Coherence
+
+O
+
+O
+
+（
+
+C
+
+O
+
+# Phrase
+
+#
+
+#
+
+C
+
+#
+
+#
+
+# Transition
+
+O
+
+#
+
+C
+
+#
+
+#
+
+# Creativity
+
+O
+
+O
+
+C
+
+。
+
+O
+
+# Musicality
+
+O
+
+O
+
+（
+
+C
+
+O
+
+(b) An example of the drum arrangement task.
+
+[23, 37]. We encourage future research to consider combining both approaches to obtain more comprehensive and convincing evaluation results.
+
+# E Probing Analysis of Pre-Training Impact
+
+Knowledge probing techniques are used to explore what a model has learned within its hidden representations [27]. A widely adopted method is linear probing, where the pre-trained model is frozen and a simple linear classifier is trained on top of the hidden representations to predict specific properties, thereby revealing the extent to which particular knowledge is encoded in the model [1]. This technique is particularly useful in our context for evaluating what the model has internalized during pre-training.
+
+We have shown in the paper that the proposed models outperform the baseline models that do not undergo the pre-training stage on the generation quality. In this section, we further analyze the reason. Specifically, we conduct probing experiments to assess whether pre-training enhances the acquisition of musical knowledge to facilitate understanding content conditions in fine-tuning.
+
+We focus on two probing tasks: (1) classifying instrument types from content sequences that contain only position, pitch, and duration tokens, and (2) recognizing chord progression sequences from content sequences. To determine whether instrument or chord information is linearly accessible within the model's sequence embeddings, we employ linear classifier probes. In these probing experiments, the average pooling of the Transformer's output embeddings across all tokens in a sequence is fed into the classifier. The model parameters are kept frozen, and only the linear classifiers are trained. We compare the knowledge captured by different models: a randomly initialized Transformer, a model that has undergone pre-training only (PT only), a model trained on the band arrangement task
+
+Table 8: Instrument probing results.  
+
+<table><tr><td>Model</td><td>Acc@1</td><td>Acc@3</td><td>Acc@5</td></tr><tr><td>Random guess</td><td>2.94</td><td>8.82</td><td>14.71</td></tr><tr><td>Random initialized</td><td>38.98</td><td>61.71</td><td>74.22</td></tr><tr><td>FT only</td><td>41.50</td><td>64.93</td><td>76.53</td></tr><tr><td>PT only</td><td>46.14</td><td>69.30</td><td>79.61</td></tr><tr><td>PT + FT</td><td>45.89</td><td>68.96</td><td>79.47</td></tr></table>
+
+Table 9: Chord probing results.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Chord Root</td><td colspan="2">Chord quality</td></tr><tr><td>Acc@1</td><td>Acc@3</td><td>Acc@1</td><td>Acc@3</td></tr><tr><td>Random guess</td><td>8.33</td><td>25.00</td><td>11.11</td><td>33.33</td></tr><tr><td>Random initialized</td><td>48.86</td><td>78.76</td><td>38.05</td><td>77.97</td></tr><tr><td>FT only</td><td>50.09</td><td>80.23</td><td>39.26</td><td>79.03</td></tr><tr><td>PT only</td><td>62.93</td><td>89.42</td><td>50.09</td><td>85.58</td></tr><tr><td>PT + FT</td><td>58.05</td><td>85.93</td><td>44.92</td><td>82.48</td></tr></table>
+
+without pre-training (FT only), and a model that has undergone both pre-training and fine-tuning (PT + FT).
+
+For the probing experiments, the batch size was set to 12 for chord probing and 64 for instrument probing. The learning rates were 5e-4 for chord probing and 1e-4 for instrument probing. Both experiments shared the same remaining hyperparameters: training for 10 epochs, using a linear learning rate scheduler, a 500-step warmup, and a weight decay of 0.01.
+
+# E.1 Instrument Type Probing
+
+In this task, we use a linear probe to estimate the instrument type from a single-track music sequence without providing instrument tokens. The goal is to predict which instrument is most likely to play the given note sequence. The performance is evaluated using top-1, top-3, and top-5 prediction accuracy metrics.
+
+As shown in Table 8, a model initialized with random weights shows notable improvement after fine-tuning, suggesting that the ability to discern instrument styles is the requirement for performing well in music arrangement tasks. However, the gains in accuracy are modest, likely due to the limited number of training samples, which may constrain the model's ability to acquire such knowledge through fine-tuning alone. Interestingly, models that underwent pre-training exhibit the highest accuracy in predicting instrument types. This indicates that substantial knowledge of instrument styles can be acquired effectively during the pre-training phase. Moreover, the proposed models that are both pre-trained and fine-tuned (PT+FT) maintained high accuracy levels, demonstrating that the knowledge about instruments styles are useful for the arrangement task.
+
+# E.2 Chord Progression Probing
+
+In this task, we use linear probes to predict chord progressions from a 2-bar music content sequence without instrument tokens. Eight linear probes are trained simultaneously to predict the chord roots and qualities for a total of four chords (two chords per bar). This tasks is used to evaluate whether the model contains position-specific chord information. Similarly, we use top-1 and top-3 prediction accuracy metrics.
+
+As shown in Table 9, the model with only fine-tuning (FT only) indicates a foundational grasp of chord knowledge for this complex music analysis tasks. However, similar to instrument prediction, the knowledge of recognizing chord progression does not significant gain, until the pre-training is also introduced into the model, confirming that pre-training establishes a robust basis for better understanding of content sequence, which may potentially help with the quality of arrangement.
+
+# F Broader Impact
+
+This work presents a unified framework for symbolic music arrangement using pretrained generative models, achieved by a reconstruction fine-tuning objective and a structured tokenization scheme. By enabling high-quality arrangement generation under flexible control, our approach has the potential to make music creation more accessible to non-experts, reduce the technical burden for composers, and support educational and assistive tools for learning music theory, orchestration, or instrumentation. It may also benefit creative professionals by streamlining workflows in game audio, film scoring, and digital content production.
+
+Many other conditional generation tasks could similarly benefit from this training objective design. For instance, one can construct a conditional sequence derived from a partial or aspect-wise decomposition of the target, concatenate it with the original target tokens, and then perform segment-level generation using a pretrained model with historical context-awareness to fine-tune existing generative models. Examples include bar-level infilling (e.g., removing and rewriting one bar), melody generation conditioned on chords, variation generation from a given melody, counter-melody generation, or harmonizing a melody with chords. All of these tasks share the same music-conditioned generation pattern and may benefit from a strong generative pretrained model.
+
+Beyond arrangement tasks, the proposed tokenization scheme may have broader implications for symbolic music modeling in general. By restructuring musical data into continuous track sequences, it facilitates learning over reusable musical phrases. This may improve both conditional and unconditional generation quality across tasks such as continuation, accompaniment generation, and reharmonization. Moreover, we anticipate that structured representations like REMI-z could benefit music understanding tasks, including symbolic transcription from audio, by facilitating the modeling of instrument-wise playing styles during decoding. Additionally, one may consider applying sequence compression techniques—such as Byte Pair Encoding (BPE) or variational autoencoders (VAE)—on top of this base tokenization to enhance compactness, which could further support long-range modeling essential for full-song generation. We encourage future work to explore these directions.
+
+Beyond arrangement tasks, the proposed tokenization scheme may have broader implications for symbolic music modeling in general. By restructuring musical data into locally coherent, instrument-consistent sequences, it facilitates learning over meaningful musical units. This could improve both conditional and unconditional generation quality across tasks such as continuation, accompaniment generation, or reharmonization. Moreover, we anticipate that structured representations like REMI-z may benefit music understanding tasks, including symbolic transcription from audio, by facilitating modeling of instrument-wise playing styles when decoding. Also, you can consider combine sequence compression techniques, maybe BPE, maybe VAE, on top of that, to further enhance the compactness, which may facilitate long-range modeling that are important to full-song generation. We encourage future work to explore these directions.
+
+However, there are potential risks. As with other generative models, the misuse of automatic arrangement systems could devalue human artistry if deployed without appropriate attribution or transparency. The system may also reflect and amplify stylistic biases present in the training data, which primarily consists of western popular music. This could marginalize underrepresented musical traditions or reinforce narrow definitions of musical aesthetics. We encourage future researchers and practitioners to consider ethical deployment strategies, such as transparent model labeling, dataset diversification, and collaborative workflows where AI augments rather than replaces human creativity.
+
+# G Limitations
+
+Fine-tuning dataset focused on pop genres. While our framework is designed to be genre-agnostic, our fine-tuning and evaluation primarily focus on pop-style arrangements. Future work can further explore the generalizability across a wider range of genres such as classical, jazz, or non-Western traditions.
+
+Strict structural alignment assumption. Our framework assumes that the input and output music share identical musical structure, which enables a segment-to-segment reconstruction formulation. While effective, this simplification does not fully reflect the broader musicological concept of
+
+arrangement, which often involves modifying the musical form or phrase structure. This constraint is not unique to our method—most prior works also rely on such structural alignment—but relaxing this assumption remains an important direction for future research.
+
+Lack of instrument planning. Although our system allows users to specify the instruments and their voice relationships for each segment during inference, the global planning of these assignments is not addressed. In practice, achieving a musically satisfying arrangement often requires deliberate instrument planning across segments—e.g., rotating lead instruments, or changing instrument combinations between sections. This work focuses on controllability at the segment level and leaves high-level planning strategies for future exploration.
+
+Melody retention in piano reduction is not guaranteed. While our piano reduction method captures harmonies and textures from ensemble music, it often fails to retain the melody. This is largely because the original piano track in multitrack data typically does not carry the main melodic line, and our current training setup does not include any explicit melody supervision. Solving this may require incorporating human-created piano arrangements or explicitly modeling melodic salience, which we leave for future work.
+
+Handling of Time Signature and Tempo. Our current model does not explicitly model time signature or tempo tokens during fine-tuning. As a result, it is limited to handling music with a fixed 4/4 time signature and stable tempo. This design choice follows common practice in prior arrangement research [42, 35, 30], where similar assumptions are made to simplify modeling. Empirically, approximately  $95\%$  of the fine-tuning data is in 4/4 time (see Appendix A), and stable tempo is generally the norm in modern music [7]. Nevertheless, extending the framework to handle non-4/4 time signatures and dynamic tempo variations remains an important direction for future work.
+
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\ No newline at end of file
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+# Unifying Text Semantics and Graph Structures for Temporal Text-attributed Graphs with Large Language Models
+
+Siwei Zhang $^{1}$ , Yun Xiong $^{1*}$ , Yateng Tang $^{3}$ , Jiarong Xu $^{2}$ , Xi Chen $^{1}$ , Zehao Gu $^{1}$ , Xuehao Zheng $^{3}$ , Zian Jia $^{1}$ , Jiawei Zhang $^{4}$
+
+$^{1}$ Shanghai Key Laboratory of Data Science, School of Computer Science, Fudan University  $^{2}$ School of Management, Fudan University  $^{3}$ Tencent Weixin Group  $^{4}$ IFM Lab, University of California, Davis
+
+{swzhang24, x_chen21, guzh22, 24110240035}@m.fudan.edu.cn jiawei@ifmlab.org {yunx, jiarongxu}@fudan.edu.cn {fredyttang, xuehaozheng}@tencent.com
+
+# Abstract
+
+Temporal graph neural networks (TGNNs) have shown remarkable performance in temporal graph modeling. However, real-world temporal graphs often possess rich textual information, giving rise to temporal text-attributed graphs (TTAGs). Such combination of dynamic text semantics and evolving graph structures introduces heightened complexity. Existing TGNNs embed texts statically and rely heavily on encoding mechanisms that biweekly prioritize structural information, overlooking the temporal evolution of text semantics and the essential interplay between semantics and structures for synergistic reinforcement. To tackle these issues, we present CROSS, a flexible framework that seamlessly extends existing TGNNs for TTAG modeling. CROSS is designed by decomposing the TTAG modeling process into two phases: (i) temporal semantics extraction; and (ii) semantic-structural information unification. The key idea is to advance the large language models (LLMs) to dynamically extract the temporal semantics in text space and then generate cohesive representations unifying both semantics and structures. Specifically, we propose a Temporal Semantics Extractor in the CROSS framework, which empowers LLMs to offer the dynamic semantic understanding of node's evolving contexts of textual neighborhoods, facilitating semantic dynamics. Subsequently, we introduce the Semantic-structural Co-encoder, which collaborates with the above Extractor for synthesizing illuminating representations by jointly considering both semantic and structural information while encouraging their mutual reinforcement. Extensive experiments show that CROSS achieves state-of-the-art results on four public datasets and one industrial dataset, with  $24.7\%$  absolute MRR gain on average in temporal link prediction and  $3.7\%$  AUC gain in node classification of industrial application.
+
+# 1 Introduction
+
+Temporal graphs are crucial for modeling dynamic interaction data, where objects are represented as nodes and timestamped interactions are depicted as edges [1, 2]. Unlike static graphs, temporal graphs continuously evolve over time [3]. To capture the temporal dependencies and realize representation learning for such graphs, extensive research has developed temporal graph neural networks (TGNNs) [4, 5, 6, 7, 8]. These works typically adopt structural encoding mechanisms [9, 10] to encapsulate the dynamics of graph structures [11], thus enabling representations for downstream tasks.
+
+Meanwhile, besides the dynamically evolving graph structures, real-world temporal graphs are also often accompanied by rich text attributes, giving rise to temporal text-attributed graphs (TTAGs) [12]. As shown in Fig. 1(a), in e-commerce networks, nodes may encompass elaborate text descriptions such as user profile or product introduction, while timestamped edges can attach transaction details
+
+![](images/f6f91b2b13bddd2922b7885540267b798d6c95efd3890d488a84cba9471d42c7.jpg)  
+(a) An illustrative example of semantic dynamics in TTAGs.
+
+![](images/3e3d18e4593c2e7e3cb8cb18ca3303f6f4256cd9be46c518a2ea76f22704e61e.jpg)  
+(b) Comparison between existing TGNNs and our framework.  
+Figure 1: (a) An illustrative example of semantic dynamics in TTAGs. Temporal text-attributed graphs (TTAGs) inherently exhibit semantic dynamics, where text semantics around nodes dynamically evolve across temporal dimensions. (b) Comparison between existing TGNNs and our framework. Going beyond existing TGNNs that biasedly focus on structural dynamics, our framework seamlessly unifies text semantics and graph structures of TTAGs, promoting both dynamic semantic understanding and semantic-structural reinforcement.
+
+including price, transaction type, review, etc. Representation learning in TTAGs presents unique challenges due to the intricate combination of dynamic text semantics and evolving graph structures, which remains largely under-explored in the community. Despite the success of existing TGNNs, they still encounter two fundamental limitations that hinder their accommodation to TTAG modeling.
+
+Limitation (i): Neglect of semantic dynamics. Text semantics arise from the textual attributes of a node and its surrounding neighborhoods. These semantics exhibit changes across timestamps due to the temporal evolution of neighborhood contexts within TTAGs. For example, as depicted in Fig. 1(a), the term "apple" may take on different meanings according to user preferences (driven by its neighborhoods), such as "smartphone" when focused on technology while "fruit" when turning to daily life. Such characteristics necessitate a temporal-aware design for extracting text semantics over time. However, existing TGNNs always use pre-trained language models, e.g., MiniLM [13], to statically embed texts as pre-processed features, failing to adapt to such dynamic semantic shifts and leading to suboptimal representations for effectively capturing the semantic dynamics.
+
+Limitation (ii): Ineffective encoding for semantic-structural reinforcement. Another limitation of existing TGNNs is the rigid reliance on their structural encoding mechanisms, which predominantly focus on topological information without adequately incorporating semantic considerations. We argue that the semantics and structures of TTAGs can mutually reinforce each other, making a biased encoding mechanism that solely emphasizes structures untenable. This hypothesis is reasonable due to the inherent complementarity between textual content and structural connectivity [14, 15] (also empirically validated in Sec. A). For instance, product recommendations are shaped not only by goods' descriptions but also by users' historical behaviors [16]. Existing TGNN encoding mechanisms do not account for such nuanced semantic-structural reinforcement in TTAGs, resulting in insufficient representations that are overly reliant on simplistic (or sometimes noisy) structural information.
+
+On the other hand, recent surveys [17, 18] reveal that large language models (LLMs), e.g., DeepSeeks v2/3 [19], exhibit notable capabilities in semantic understanding and generation, which have been successfully leveraged in graph modeling through text augmentation [20, 21]. However, LLMs employed in these methods are typically limited by static corpora input for reasoning, making them ill-suited to capture the fine-grained dynamics of TTAGs. This limitation inspires us to ask: Can we advance LLMs for TTAG modeling as a promising solution to the aforementioned limitations of existing TGNNs?
+
+Key innovation. Beyond existing TGNNs that solely prioritize structural dynamics, in this paper, we tackle above issues by decoupling the TTAG modeling process into two phases: (i) temporal semantics extraction; and (ii) semantic-structural information unification. By extracting both dynamic semantics and evolving structures of TTAGs, we cohesively unify them to harness their dual-strengths for more informative representations. Consequently, as shown in Fig. 1(b), the key idea of this work is to elevate LLMs with dynamic reasoning capability to extract semantic dynamics, which subsequently allows for a unified, integrated, and non-biased encoding mechanism obeying semantic-structural reinforcement.
+
+Present work. We extend existing TGNNs for TTAG modeling and introduce CROSS (Cohesive Representations Of Semantics and Structures), a novel framework that seamlessly unifies text
+
+![](images/457a64b3d62375e2a7ff2ed994c49dc31fa61209860ebca0866cf543844abfe9.jpg)  
+Figure 2: Technical difference comparison from the perspective of LLM utilization and modality fusion.
+
+semantics and graph structures coupled with LLMs. CROSS can significantly boost existing TGNNs, and it comprises two main components: (i) Temporal Semantics Extractor, and (ii) Semantic-structural Co-encoder. To capture the semantic dynamics for nodes within TTAGs, we propose a Temporal Semantics Extractor, which empowers the LLMs with the dynamic semantic summarization reasoning capability for TTAG modeling. It automatically detects semantic dynamics by constructing a temporal reasoning chain to dynamically prompt the LLMs to summarize the textualized, evolving neighborhoods of nodes, thus revealing the linguistic nuances across temporal dimensions. Besides the Extractor, to effectively achieve semantic-structural reinforcement, we further propose the Semantic-structural Co-encoder that simultaneously exploits both semantic and structural information through an iterative, multi-layer design. Each layer in the proposed Co-encoder bidirectionally transfers the cross-modal information between semantics and structures, allowing both modalities to blend deeply and fully illuminate each other. By performing such a unification, we can generate modal-cohesive representations that are both semantic-enriched and structural-informed.
+
+The main contributions of this paper are summarized as follows:
+
+- We address an under-explored problem of temporal text-attributed graph (TTAG) modeling and propose CROSS. To the best of our knowledge, CROSS is the first framework designed to unify text semantics and graph structures with LLMs for TTAG modeling.  
+- We design a temporal-aware LLM prompting paradigm for TTAG modeling and develop Temporal Semantics Extractor. It enhances LLMs with dynamic reasoning capability to offer the evolving semantics of nodes' neighborhoods, effectively detecting semantic dynamics.  
+- We introduce a modal-cohesive co-encoding architecture for TTAG modeling and propose Semantic-structural Co-encoder, which jointly propagates semantic and structural information, facilitating mutual reinforcement between both modalities.  
+- We conduct extensive experiments on four public datasets and one practical e-commerce industrial dataset. CROSS outperforms baselines with  $24.7\%$  absolute MRR gain on average in temporal link prediction and  $3.7\%$  AUC gain in node classification of industrial application.
+
+# 2 Preliminaries and Related Work
+
+# 2.1 Preliminaries
+
+Definition 1. Temporal Text-attributed Graph. Given a set of nodes  $\mathcal{V}$ , node text attributes  $\mathcal{D}$ , and edge text attributes  $\mathcal{R}$ , a temporal text-attributed graph (TTAG) can be represented as a sequence of interactions  $\mathcal{G} = \{(u,v,t)\}$ , where  $u, v \in \mathcal{V}$  and  $t \geq 0$ . Each node  $u \in \mathcal{V}$  is associated with a node text attribute  $d_u \in \mathcal{D}$  and each interaction  $(u,v,t) \in \mathcal{G}$  attaches an edge text attribute  $r_{u,v,t} \in \mathcal{R}$ . We use  $\mathcal{H}_u(t) = \{(u,v,\tau) | \tau < t\} \cup \{(v,u,\tau) | \tau < t\}$  to denote the set of historical interactions involving node  $u$  before time  $t$ .
+
+Definition 2. Temporal Text-attributed Graph Modeling. Given node  $u$ , time  $t$ , and all available historical interactions before  $t$ ,  $\{(u', v', \tau) | \tau < t\}$ , temporal text-attributed graph modeling aims to learn a mapping function  $f: (u, t) \mapsto \mathbf{z}_u(t)$ , where  $\mathbf{z}_u(t) \in \mathbb{R}^d$  denotes the representation of node  $u$  at time  $t$ , and  $d$  is the vector dimension.
+
+Definition 3. Language Model vs. Large Language Model. We clearly distinguish between pretrained language models (LMs) and large language models (LLMs). LMs, e.g., MiniLM, are relatively smaller models designed to embed texts from the text space into the feature space; and LLMs refer to significantly larger models far surpassing the linguistic capabilities of LMs, like DeepSeek-v2/3.
+
+# 2.2 Related Works
+
+We provide two groups of related works: TGNNs and LLM-for-Graph-Text-Augmentation (LLM4GTAug) methods. As illustrated in Fig. 2, we summarize these methods from the perspective of LLM utilization and modality fusion. A discussion of related works is detailed in Sec. E.
+
+TGNNs. LLM utilization within TGNNs remains limited. Existing TGNNs [12] typically pre-embed the raw texts as input features and rely on their structural encoders [9] to capture the dynamics of
+
+![](images/fdfd8c3dc15dd1a52ad5b4c5f9c220bd10605e01d52c18b65c00538c6540cc93.jpg)  
+Figure 3: Architecture of the proposed framework, CROSS. Our Temporal Semantics Extractor constructs the temporal reasoning chain to empower the LLMs with dynamic reasoning capability to dynamically provide the semantic understanding of nodes' neighborhoods, thereby extracting semantic dynamics in text space. Subsequently, the Semantic-structural Co-encoder iteratively mixes and consolidates both semantic and structural information, ensuring their mutual reinforcement for effectively generating modal-cohesive representations.
+
+graph structures. These methods overlook the temporal evolution of text semantics within TTAGs. Besides, they integrate semantic and structural information only at the input stage, resulting in shallow modality fusion that occurs before encoding.
+
+LLM4GTAug methods. Existing LLM4GTAug methods [20, 21] are designed for static graphs and employ LLMs through static, one-off reasoning for each node. This makes them ill-suited for capturing the temporal dynamics of TTAGs. In addition, they mix structural and semantic representations only at the output stage, leading to shallow fusion after encoding.
+
+As summarized in Tab. 1, in this paper, our CROSS aims to (i) enhance LLMs with dynamic reasoning capabilities to capture semantic dynamics; and (ii) facilitate hierarchical modality fusion throughout the entire encoding process. Our proposed method fundamentally opens new avenues for future research in TTAG modeling.
+
+Table 1: Summary of related works.  
+
+<table><tr><td></td><td>TGNNs</td><td>LLM4GTAug</td><td>CROSS</td></tr><tr><td>Structural Dynamics</td><td>✘</td><td>✓</td><td>✓</td></tr><tr><td>Semantic Dynamics</td><td>✘</td><td>✘</td><td>✓</td></tr><tr><td>LLM Utilization</td><td>static</td><td>no LLM</td><td>dynamic</td></tr><tr><td>Modality Fusion</td><td>shallow at output(after encoding)</td><td>shallow at input(before encoding)</td><td>hierarchical(during encoding)</td></tr></table>
+
+# 3 Methodology
+
+As mentioned before, unifying text semantics and graph structures for TTAG modeling requires considering both semantic dynamics and semantic-structural reinforcement. To this end, as depicted in Fig. 3, the proposed CROSS framework comprises two stages, i.e., the Temporal Semantics Extractor first detects the dynamic text semantics for nodes' evolving neighborhoods; then the Semantic-structural Co-encoder jointly unifies both semantic and structural information to ensure cross-modal reinforcement. We will introduce these two stages in the following subsections.
+
+# 3.1 Temporal Semantics Extractor
+
+In this section, we propose the temporal reasoning chain to advance the LLMs with dynamic reasoning capability to extract semantic dynamics for TTAG modeling. It will be described below.
+
+Temporal Reasoning Chain. Existing LLM utilization paradigm within LLM4GTAug methods are ill-suited for semantic extraction in TTAGs, as it fails to effectively capture the dynamics of node's semantic contexts. To address this challenge, we design a novel temporal reasoning chain that empowers the LLMs with the dynamic semantic summarization reasoning capability across temporal dimensions, discerning the evolving linguistic nuances along timestamps. However, performing LLM reasoning at every timestamp remains computationally impractical. To ensure scalability and stability, for each node among TTAGs, we strategically sample reasoning timestamps at equal interaction intervals. This strategy enables CROSS to constrain the number of LLM calls and guarantee that each LLM reasoning step accesses a balanced set of interactions, striking a delicate balance between temporal granularity and cost efficiency.
+
+Formally, for node  $u$ , we first derive the set of its interaction timestamps  $\mathcal{T}_u = \{t_1, t_2, \dots, t_n\}$ , where  $t_1 \leq t_2 \dots \leq t_n$  and  $n$  is the node degree. Then we sample the reasoning timestamps at intervals of  $\left\lceil \frac{n}{m} \right\rceil$  among  $\mathcal{T}_u$  with a predefined maximum reasoning count  $m$  as follows:
+
+$$
+\hat {\mathcal {T}} _ {u} = \left\{\hat {t} _ {i} \mid \hat {t} _ {i} = \left\{ \begin{array}{l l} t _ {i. \lceil \frac {n}{m} \rceil}, & i = 1, \dots , m - 1, \\ t _ {n}, & i = m \end{array} \right. \in \mathcal {T} _ {u} \right\}. \tag {1}
+$$
+
+Here,  $\hat{\mathcal{T}}_u\subseteq \mathcal{T}_u$  depicts the LLM reasoning timestamps to summarize text semantics around node  $u$ , and we will also analyze the dataset-specific hyper-parameter  $m$  in Sec. I of the Appendix.
+
+Summarization. To align with the dynamics of semantic contexts within TTAGs, we facilitate the LLMs to dynamically summarize nodes' neighborhoods across different timestamps. Specifically, we perform multi-turn calling with the LLMs along the previously sampled reasoning timestamps. For node  $u$  at reasoning time  $\hat{t} \in \hat{T}_u$ , we first obtain its neighborhood by retrieving  $u$ 's historical interactions  $\mathcal{H}_u(\hat{t}) = \{(u,v,\tau) | \tau < \hat{t}\} \cup \{(v,u,\tau) | \tau < \hat{t}\}$ , and then we prompt the LLMs to generate neighborhood summaries and semantic details for that time. This can be represented by:
+
+$$
+\hat {d} _ {u} (\hat {t}) = \operatorname {L L M} \left(d _ {u}, \hat {t}, \{r _ {*} \} _ {* \in \mathcal {H} _ {u} (\hat {t})}; \text {P R O M P T}\right). \tag {2}
+$$
+
+$\mathrm{LLM}(\cdot ;\mathrm{PROMPT})$  denotes the LLM calling with the prompt, which comprises a fixed template alongside variable pieces, including node  $u$  's raw text  $d_{u}$  , time  $\hat{t}$  , and the neighborhood  $\{r_{*}\}_{* \in \mathcal{H}_{u}(\hat{t})}.\hat{d}_{u}(\hat{t})$  is the LLM-generated text, denoting the semantic summary for  $u$  's contexts at time  $\hat{t}$  . We present a simplified example of PROMPT for clarity, and the complete version is given in Sec. F of the Appendix.
+
+# A simplified example of PROMPT
+
+Goal: [Request to summarize the current neighborhoods and emphasize the response format.]
+
+Descriptions: [Provide the text attribute of node  $d_u$ .]
+
+Current time: [Specify the reasoning timestamp  $\hat{t}$ .]
+
+Historical interactions: [List textualized neighborhoods using recent interactions  $\{r_{*}\}_{* \in \mathcal{H}_u(t)}$ .]
+
+After performing all reasoning, we can obtain a set of LLM-generated textual summaries for each node  $u$ . We represent them as  $\hat{\mathcal{D}}_u = \{\hat{d}_u(\hat{t}) \mid \hat{t} \in \hat{\mathcal{T}}_u\}$ . Notably, to incorporate the raw texts in TTAGs, we set  $\hat{d}_u(0) = d_u$ . These summaries encapsulate the evolving semantics around nodes derived from the dynamically-enhanced LLMs, promoting semantic dynamics for TTAG modeling.
+
+# 3.2 Semantic-structural Co-encoder
+
+As mentioned in the Introduction, existing TGNNs solely account for structural dynamics, overlooking the necessary consideration of semantic dynamics and the reinforcement between these two modalities. To address this issue, our Semantic-structural Co-encoder performs iterative integration of semantics and structures at the layer level. Each layer of the proposed Co-encoder comprises three types of encoding components: (i) the semantic layer that encodes the text semantics from LLM-generated summaries to produce semantic representations; (ii) the structural layer that encodes the graph structures from neighborhood information to capture structural representations; and (iii) the cross-modal mixer that facilitates transformation between these two unimodal representations to cheer their mutual reinforcement. We will discuss them in detail below.
+
+Semantic Layer. To better incorporate the semantic modality, we implement our semantic layer using a Transformer encoder [22] due to its widespread usage in textual modeling. From the ablation results in Tab. 3, such a design effectively harnesses the full potential of text semantics for TTAG modeling, demonstrating clear advantages over purely structural encoding mechanisms.
+
+We first prepare the inputs for the semantic layer. For node  $u$  at time  $t$ , the inputs of the semantic layer are the corresponding LLM-generated summaries before  $t$  from our Extractor. We represent them as  $\hat{\mathcal{D}}_u(t) = \{\hat{d}_u(t_k) \mid t_k < t\}$ , where  $t_1 \leq \dots \leq t_k$ . We then use a pre-trained LM, such as MiniLM [13], to embed the texts into the  $d$ -dimensional semantic features, which is depicted as  $\hat{\mathbf{s}}_u(t_k) = \mathrm{LM}\left(\hat{d}_u(t_k)\right) \in \mathbb{R}^d$ . To capture the temporal differences, we subsequently concatenate time information into these semantic features as follows:
+
+$$
+\mathbf {x} _ {u} \left(t _ {k}\right) = \hat {\mathbf {s}} _ {u} \left(t _ {k}\right) \| \Phi (t - t _ {k}) \in \mathbb {R} ^ {2 d}. \tag {3}
+$$
+
+Here,  $\Phi (\cdot)$  is the time encoding function introduced by [23], which is widely used in recent TGNNs [9]. These enriched features then feed into the  $L$ -layer Transformer encoder blocks to derive the semantic representations for node  $u$  at time  $t$ . We have:
+
+$$
+\tilde {\mathbf {e}} _ {u} ^ {(l)} \left(t _ {1}\right), \dots , \tilde {\mathbf {e}} _ {u} ^ {(l)} \left(t _ {k}\right) = \operatorname {T R M} ^ {(l)} \left(\mathbf {e} _ {u} ^ {(l - 1)} \left(t _ {1}\right), \dots , \mathbf {e} _ {u} ^ {(l - 1)} \left(t _ {k}\right)\right), \tag {4}
+$$
+
+where  $\tilde{\mathbf{e}}_u^{(l)}(t)\in \mathbb{R}^{2d}$  corresponds to the pre-mixed semantic representation at the  $l$ -th layer and  $\mathbf{e}_u^{(0)}(t_1),\ldots ,\mathbf{e}_u^{(0)}(t_k) = \mathbf{x}_u(t_1),\ldots ,\mathbf{x}_u(t_k)$ . As we will mention below, the latest semantic representation of node  $u$ ,  $\mathbf{e}_u^{(l - 1)}(t_k)$ , has integrated information from its structural representation in the previous layer. This allows the  $l$ -th semantic layer to receive the information from graph structures, achieving deep fusion and promoting their synergistic reinforcement.
+
+Structural Layer. Meanwhile, we conduct our structural layer to encode the graph structures around node  $u$  at time  $t$ ,  $\mathcal{G}_u(t)$ , using the structural encoding block in TGNNs. Such a design renders our framework inherently TGNN-agnostic and flexibly integrable with any TGNN backbone.
+
+Formally, the  $l$ -th structural layer aggregates the neighborhood information of node  $u$  from current layer  $\mathbf{H}_u^{(l)}(t)$  and its structural representation from the previous layer  $\mathbf{h}_u^{(l-1)}(t)$  with a 2-layer Multi-Layer Perceptron (MLP). This can be expressed as:
+
+$$
+\tilde {\mathbf {h}} _ {u} ^ {(l)} (t) = \operatorname {M L P} ^ {(l)} \left(\mathbf {h} _ {u} ^ {(l - 1)} (t) \| \operatorname {A G G} \left(\mathbf {H} _ {u} ^ {(l)} (t)\right)\right). \tag {5}
+$$
+
+Here,  $\tilde{\mathbf{h}}_u^{(l)}(t)$  is the pre-mixed structural representation at the  $l$ -th layer and  $\|\cdot\|$  denotes concatenation.  $\mathrm{AGG}(\cdot)$  is a pooling function to aggregate the neighborhood information matrix into a  $d$ -dimensional vector, with specific implementations (e.g., mean, sum) varying across various TGNNs [4, 5, 7].
+
+The neighborhood information  $\mathbf{H}_u^{(l)}(t)$  is derived using a temporal attention mechanism [4] via message passing from the  $l$ -th layer neighborhood. This process assigns attention weights to scale the contribution and importance of neighbors  $\mathcal{N}_u(t)$  for node  $u$  as follows:
+
+$$
+\mathbf {H} _ {u} ^ {(l)} (t) = \operatorname {S o f t m a x} \left(\mathbf {a} _ {u} ^ {(l)} (t)\right) \cdot \mathbf {V} _ {u} ^ {(l)} (t), \tag {6}
+$$
+
+where  $\mathbf{a}_u^{(l)}(t) = [a_{uv}^{(l)}(t)]_{v\in \mathcal{N}_u(t)}$  represents the attention weight vector, and the matrix  $\mathbf{V}_u^{(l)}(t) = [\mathbf{v}_v^{(l)}(t)]_{v\in \mathcal{N}_u(t)}$  condenses the messages from  $u$ 's  $l$ -th layer neighborhood. Each element  $a_{uv}^{(l)}(t)$  in  $\mathbf{a}_u^{(l)}(t)$  is the attention weight for node  $u$  to its neighbor  $v\in \mathcal{N}_u(t)$ , and each row  $\mathbf{v}_v^{(l)}(t)$  from  $\mathbf{V}_u^{(l)}(t)$  is the message carried from  $u$ 's neighboring node  $v$ . These can be computed by:
+
+$$
+a _ {u v} ^ {(l)} (t) = \frac {f _ {\mathrm {q}} \left(\mathbf {h} _ {u} ^ {(l - 1)} (t)\right) f _ {\mathrm {k}} \left(\mathbf {h} _ {v} ^ {(l - 1)} (t)\right) ^ {T}}{\sqrt {d ^ {(k)}}}, \tag {8}
+$$
+
+In Eqs. 7 & 8,  $f_{*}(\cdot)(\ast \in \{\mathrm{q}, \mathrm{k}, \mathrm{v}\})$  denote the encoding functions for queries, keys, and values, respectively [22]. These functions may be implemented differently across various TGNNs [4, 5, 7, 9], and we do not discuss their details as they are beyond the scope of our work.
+
+As we explain in the next paragraph, the structural representations from the previous layer,  $\mathbf{h}_u^{(l - 1)}(t)$ , have integrated with the semantic representations. Consequently, the message-passing aggregation in Eq. 5 enables the propagation between both types of information, effectively facilitating semantic-structural reinforcement during encoding.
+
+Cross-modal Mixer. Finally, to enable the deep unification between the semantic and structural modalities for TTAG modeling, we introduce a novel and interesting cross-modal mixer that automatically transfers and integrates the bimodal information at a layer-wise granularity.
+
+For node  $u$  at the  $l$ -th layer, the inputs of our cross-modal mixer are the most recent pre-mixed semantic representation  $\tilde{\mathbf{e}}_u^{(l)}(t_k)$  and the corresponding pre-mixed structural representation  $\tilde{\mathbf{h}}_u^{(l)}(t)$ . This means that the remaining historical semantic representations, i.e.,  $\tilde{\mathbf{e}}_u^{(l)}(t_1), \ldots, \tilde{\mathbf{e}}_u^{(l)}(t_{k-1})$ , will be not involved in the mixture process. Such a design is driven by: (i) efficiency consideration; (ii) temporal-awareness consideration that the latest semantic representation carries the most contextually relevant information; and (iii) empirical consideration that mixing all semantic representations does not necessarily lead to better performance (See Sec. 4.5). Therefore, we concatenate  $\tilde{\mathbf{e}}_u^{(l)}(t_k)$  with
+
+$\tilde{\mathbf{h}}_u^{(l)}(t)$ , then pass through our cross-modal mixer, and finally split the fused representation to derive the post-mixed semantic and structural representations. These processes can be formulated as follows:
+
+$$
+\mathbf {e} _ {u} ^ {(l)} \left(t _ {k}\right); \mathbf {h} _ {u} ^ {(l)} (t) = \operatorname {M i x e r} ^ {(l)} \left(\tilde {\mathbf {e}} _ {u} ^ {(l)} \left(t _ {k}\right) \| \tilde {\mathbf {h}} _ {u} ^ {(l)} (t)\right). \tag {9}
+$$
+
+We implement  $\mathbf{Mixer}^{(l)}(\cdot)$  using a 2-layer MLP, and it can be alternatively conducted in other fusion components. By iteratively performing such a mixture operation, we can deeply unify both text semantics and graph structures, enabling cohesive representations for TTAG modeling.
+
+# 3.3 Training CROSS
+
+Cohesive Representations. For node  $u$  at time  $t$ , its representation is derived from: (i) semantic outputs from the  $L$ -th semantic layer with mean pooling,  $\mathbf{z}_u^{\mathrm{sem}}(t) = \mathrm{Mean}\left(\tilde{\mathbf{e}}_u^{(L)}(t_1),\dots ,\tilde{\mathbf{e}}_u^{(L)}(t_k)\right)$ ; (ii) structural outputs from the  $L$ -th structural layer,  $\mathbf{z}_u^{\mathrm{str}}(t) = \tilde{\mathbf{h}}_u^{(L)}(t)$ ; and (iii) the unified outputs from the cross-modal mixer,  $\mathbf{z}_u^{\mathrm{mix}}(t) = \mathbf{e}_u^{(L)}(t_k)\parallel \mathbf{h}_u^{(L)}(t)$ . This can be denoted as:
+
+$$
+\mathbf {z} _ {u} (t) = \mathrm {M L P} _ {\text {o u t}} \left(\mathbf {z} _ {u} ^ {\text {s e m}} (t) \| \mathbf {z} _ {u} ^ {\text {s t r}} (t) \| \mathbf {z} _ {u} ^ {\text {m i x}} (t)\right), \tag {10}
+$$
+
+where  $\mathbf{z}_u(t) \in \mathbb{R}^d$  and  $\mathrm{MLP}_{\mathrm{out}}(\cdot)$  is a 2-layer MLP that maps the dimension of the input vector to  $d$ . Loss Function. We adopt the temporal link prediction task [7] as training signals for TTAG modeling. For link  $(u, v, t)$ , we compute its occurrence probability  $\hat{p}_{uv}(t)$  by feeding the concatenated representations of nodes  $u$  and  $v$ ,  $\mathbf{z}_u(t) \parallel \mathbf{z}_v(t)$ , into a 2-layer MLP. Cross-entropy loss is then applied:
+
+$$
+\mathcal {L} = - \sum_ {(u, v, t) \in \mathcal {G}} \left[ \log \hat {p} _ {u v} (t) + \log \left(1 - \hat {p} _ {u v ^ {\prime}} (t)\right) \right]. \tag {11}
+$$
+
+The  $v'$  denotes the randomly sampled negative destination node. Additionally, we will also present the theoretical analysis to prove the effectiveness of CROSS in Sec. B.
+
+# 4 Experiments
+
+# 4.1 Experimental Settings
+
+Datasets. We implement experiments with five datasets, including four public datasets and one industrial dataset from real-world e-commerce systems. The four public datasets - Enron, GDELT, ICEWS1819, and GoogleMap_CT - are recently collected and released by [12]. Besides, the industrial dataset is constructed with three months of transaction data sampled from a private e-commerce trading network in WeChat $^2$  Mobile Payment. $^3$  Details of these datasets are summarized in Sec. D.1 due to page limitations. All datasets are chronologically split by  $60\%$ ,  $20\%$ , and  $20\%$  for training, validation, and testing, respectively.
+
+Baselines. We select eleven existing methods as our baselines, including JODIE [24], DyRep [25], TGAT [4], TGN [5], CAWN [26], PINT [27], TCL [28], GraphMixer [29], DyGFormer [7], LKD4DyTAG [30], and FreeDyG [10]. We also select powerful DeepSeek-v2 [19] and evaluate its zero-shot and one-shot performance as LLM baselines, denoted as  $\mathrm{LLM}_{\mathrm{zero / one}}$ . Detailed descriptions of all baselines are provided in Sec. D.2. Moreover, we choose three representative TGNN models, i.e., TGAT, TGN, and DyGFormer, as the backbones of CROSS due to their superiority. For simplicity, we employ the well-established MiniLM [13] to embed texts into feature space. Additionally, we adopt DeepSeek-v2 as the default LLM. We also report the results with other LLM backbones in Sec. 4.5. We evaluate the learned representations using two downstream tasks, i.e., temporal link prediction and node classification in an industrial application of financial risk management.
+
+# 4.2 Temporal Link Prediction
+
+We begin our experimental evaluation by comparing the temporal link prediction performance of our model with baselines. We conduct this under two settings: (i) transductive setting, which predicts links between nodes that have appeared during training; and (ii) inductive setting, where predictions are performed with unseen nodes. Implementation details can be found in Sec. D.3 of the Appendix.
+
+The results are presented in Tab. 2. Clearly, our CROSS framework significantly improves the performance of all three TGNN backbones across all datasets in both transductive and inductive
+
+Table 2: MRR results (\%) for temporal link prediction in transductive and inductive settings. The results for  $\mathbf{L}\mathbf{M}_{\mathrm{zero / one}}$  represent the zero-/one-shot performance of the LLM DeepSeek-v2 [19]. Results highlighted with a blue background indicate the performance and corresponding improvements of our CROSS framework using various TGNN backbones. The best results are highlighted in bold.  
+
+<table><tr><td rowspan="2"></td><td colspan="5">Transductive setting</td><td colspan="5">Inductive setting</td></tr><tr><td>Enron</td><td>GDELT</td><td>ICEWS1819</td><td>Googlemap_CT</td><td>Industrial</td><td>Enron</td><td>GDELT</td><td>ICEWS1819</td><td>Googlemap_CT</td><td>Industrial</td></tr><tr><td>JODIE</td><td>66.69 ± 2.0</td><td>48.81 ± 1.1</td><td>71.47 ± 4.0</td><td>56.72 ± 0.7</td><td>49.75 ± 1.2</td><td>53.41 ± 2.5</td><td>37.90 ± 2.4</td><td>57.75 ± 7.1</td><td>55.21 ± 1.0</td><td>30.38 ± 0.3</td></tr><tr><td>DyRep</td><td>58.85 ± 7.9</td><td>45.61 ± 2.8</td><td>63.13 ± 3.6</td><td>49.04 ± 2.2</td><td>36.49 ± 0.9</td><td>42.95 ± 8.3</td><td>42.23 ± 3.1</td><td>50.42 ± 2.8</td><td>47.44 ± 2.5</td><td>25.47 ± 1.8</td></tr><tr><td>TCL</td><td>71.16 ± 0.7</td><td>59.49 ± 0.5</td><td>87.58 ± 0.3</td><td>68.98 ± 0.4</td><td>50.87 ± 0.5</td><td>55.28 ± 1.5</td><td>47.13 ± 1.0</td><td>77.06 ± 0.2</td><td>66.26 ± 0.2</td><td>33.42 ± 0.5</td></tr><tr><td>CAWN</td><td>74.56 ± 0.6</td><td>57.00 ± 0.2</td><td>82.93 ± 0.1</td><td>65.34 ± 0.2</td><td>63.58 ± 0.8</td><td>61.58 ± 2.0</td><td>43.56 ± 0.6</td><td>70.65 ± 0.1</td><td>62.11 ± 0.2</td><td>53.42 ± 0.5</td></tr><tr><td>PINT</td><td>74.82 ± 2.8</td><td>52.71 ± 2.5</td><td>83.81 ± 0.9</td><td>72.94 ± 0.7</td><td>53.51 ± 0.6</td><td>56.38 ± 3.9</td><td>31.82 ± 4.1</td><td>63.16 ± 2.8</td><td>70.02 ± 0.6</td><td>39.72 ± 0.6</td></tr><tr><td>GraphMixer</td><td>62.68 ± 1.3</td><td>53.33 ± 0.4</td><td>80.69 ± 0.3</td><td>53.11 ± 0.2</td><td>50.50 ± 0.5</td><td>43.75 ± 1.5</td><td>41.18 ± 0.3</td><td>67.09 ± 0.5</td><td>51.36 ± 0.2</td><td>34.06 ± 0.7</td></tr><tr><td>FreeDyG</td><td>81.52 ± 1.8</td><td>68.27 ± 0.7</td><td>86.31 ± 0.6</td><td>78.82 ± 1.2</td><td>75.91 ± 0.7</td><td>70.38 ± 0.1</td><td>52.71 ± 0.3</td><td>74.16 ± 0.4</td><td>66.01 ± 2.8</td><td>56.48 ± 0.6</td></tr><tr><td>LKD4DyTAG</td><td>73.18 ± 0.3</td><td>57.28 ± 1.9</td><td>80.62 ± 4.2</td><td>77.11 ± 0.5</td><td>77.73 ± 0.7</td><td>67.45 ± 1.9</td><td>45.75 ± 2.0</td><td>73.81 ± 0.3</td><td>60.73 ± 1.0</td><td>57.91 ± 1.0</td></tr><tr><td>LLMzero</td><td>24.18</td><td>7.99</td><td>33.68</td><td>30.30</td><td>11.27</td><td>17.73</td><td>10.08</td><td>32.26</td><td>38.21</td><td>2.62</td></tr><tr><td>LLMone</td><td>46.27</td><td>28.91</td><td>50.82</td><td>48.79</td><td>30.29</td><td>48.14</td><td>28.91</td><td>44.69</td><td>43.83</td><td>20.28</td></tr><tr><td>TGAT</td><td>66.06 ± 0.1</td><td>56.73 ± .04</td><td>85.81 ± 0.2</td><td>63.13 ± 0.5</td><td>46.74 ± 3.9</td><td>47.80 ± 0.8</td><td>42.01 ± 0.5</td><td>74.10 ± 0.2</td><td>60.96 ± 0.2</td><td>30.04 ± 3.0</td></tr><tr><td>TGAT+</td><td>95.58 ± 0.7</td><td>81.63 ± 1.7</td><td>93.05 ± 1.6</td><td>99.91 ± 0.0</td><td>86.97 ± 2.8</td><td>81.52 ± 2.0</td><td>64.56 ± 1.8</td><td>82.25 ± 2.0</td><td>91.59 ± 0.0</td><td>62.22 ± 2.1</td></tr><tr><td>Avg.↑26.59</td><td>↑29.52</td><td>↑24.90</td><td>↑7.24</td><td>↑36.78</td><td>↑40.23</td><td>↑33.72</td><td>↑22.55</td><td>↑8.15</td><td>↑30.63</td><td>↑32.18</td></tr><tr><td>TGN</td><td>73.05 ± 1.7</td><td>54.28 ± 1.6</td><td>84.79 ± 0.6</td><td>71.35 ± 0.5</td><td>54.46 ± 3.0</td><td>54.98 ± 2.3</td><td>37.48 ± 2.8</td><td>69.69 ± 0.8</td><td>67.88 ± 0.2</td><td>38.28 ± 4.1</td></tr><tr><td>TGN+</td><td>95.84 ± 0.4</td><td>77.95 ± 2.8</td><td>94.74 ± 5.7</td><td>99.92 ± 0.0</td><td>94.26 ± 0.8</td><td>82.38 ± 1.2</td><td>56.65 ± 3.8</td><td>84.01 ± 9.2</td><td>92.68 ± 0.1</td><td>83.23 ± 2.6</td></tr><tr><td>Avg.↑25.45</td><td>↑22.79</td><td>↑23.67</td><td>↑9.95</td><td>↑28.57</td><td>↑39.80</td><td>↑27.40</td><td>↑19.17</td><td>↑14.32</td><td>↑24.80</td><td>↑44.02</td></tr><tr><td>DyGFormer</td><td>79.93 ± 0.1</td><td>61.35 ± 0.3</td><td>87.51 ± 0.3</td><td>54.82 ± 2.7</td><td>74.45 ± 0.7</td><td>66.86 ± 0.1</td><td>50.61 ± 0.2</td><td>78.14 ± 0.3</td><td>52.98 ± 2.5</td><td>54.20 ± 0.4</td></tr><tr><td>DyGFormer+</td><td>95.31 ± 2.8</td><td>81.28 ± 4.4</td><td>95.77 ± 0.3</td><td>99.82 ± 0.0</td><td>94.78 ± 1.4</td><td>86.01 ± 4.9</td><td>66.37 ± 4.4</td><td>87.80 ± 0.6</td><td>91.74 ± 0.1</td><td>82.30 ± 1.2</td></tr><tr><td>Avg.↑22.03</td><td>↑15.38</td><td>↑19.93</td><td>↑8.26</td><td>↑44.99</td><td>↑20.33</td><td>↑19.15</td><td>↑15.76</td><td>↑9.66</td><td>↑38.76</td><td>↑28.10</td></tr></table>
+
+settings. Meanwhile, CROSS achieves SOTA performance with a substantial margin over the best baseline. This observation proves the effectiveness of unifying text semantics and graph structures in TTAG modeling. Although LLM's zero/one-shot performance is suboptimal, CROSS still performs well. This suggests that LLMs may struggle to directly comprehend the dynamics of graph structures in TTAG modeling, but our proposed CROSS framework helps them to resolve this issue effectively. Moreover, our framework tends to result in closer performance across different TGNN backbones. We infer that this is due to the robustness of text semantics, which successfully reduces model reliance on simplistic structural information. Inspired by this, we detail a robustness study in Sec. 4.4.
+
+# 4.3 Industrial Application
+
+We conduct another downstream task using node classification in a real-world industrial application of financial risk management on Industrial, where we predict whether nodes are involved in fraudulent activities. DyGFormer [7] is used as the backbone, and details can be seen in Sec. D.3 of the Appendix. CROSS achieves the best performance as shown in Fig. 4, indicating that the learned representations of CROSS are also effective for node-level tasks. CROSS's success on Industrial demonstrates its scalability for large industrial-scale TTAGs. We will also provide scalability clarification in Sec. G.
+
+![](images/e45122403f44299e7496f5fdd23f37ce9ccd57a2208216ad2d70c8ded4ff1e9d.jpg)  
+Figure 4: AUC results  $(\%)$  for node classification of a real-world industrial application in financial risk management, which predicts whether nodes are involved in fraudulent activities on Industrial.
+
+# 4.4 Robustness Study
+
+Robustness for noise. To empirically validate the assumption of noisy structural information mentioned in Sec. 1, we strategically introduce noise into graph structures with perturbation rates of  $p \in \{10\%, 20\%, 30\%, 40\%, 50\% \}$ . Implementation details are put in Sec. J due to page limitations. To further investigate the impact of noise during encoding, we randomly select a subset of nodes and visualize the attention weights of their perturbed neighbors using box plots under  $p = 50\%$ .
+
+![](images/1d66934f91306d7b3f90326364411c1494819b2de51261966d125d2919fbd4e9.jpg)  
+Figure 5: Robustness study for noise on Enron and ICEWS1819 with different perturbation rates.
+
+![](images/a0d08ebe16bb45120352b055f631dac106d5c3cf5495998f94186845634c6b01.jpg)
+
+![](images/68ae920505bc3ba7c23af3134ceadd858405980b0f3e7676d65d32b1531e661f.jpg)
+
+![](images/c18c47f6bad2755b33e9047b68a2317bca6f036bebd79987eb9a70400211d2d2.jpg)
+
+Table 3: Ablation study for the raw texts and LLM-generated texts. Semantic/Structural Encoding refers to the encoding mechanisms that independently perform semantic/structural layers in Sec. 3.2; imprv. indicates the performance improvements of LLM-generated texts  $\mathbf{{Text}}_{\mathbf{{LLM}}}$  over the raw texts  ${\mathbf{{Text}}}_{\mathbf{{raw}}}$  . Our proposed co-encoding mechanism unlocks the full potential of LLM-generated texts and achieves the best performance.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Datasets</td><td rowspan="2">Methods</td><td colspan="3">Semantic Encoding</td><td colspan="3">Structural Encoding</td><td colspan="3">Semantic-structural Co-encoding</td></tr><tr><td>Textraw</td><td>TextLLM</td><td>imprv.</td><td>Textraw</td><td>TextLLM</td><td>imprv.</td><td>Textraw</td><td>TextLLM (ours)</td><td>imprv.</td></tr><tr><td rowspan="6">Transductive</td><td rowspan="3">Enron</td><td>TGAT</td><td>49.73 ± 0.8</td><td>63.07 ± 0.5</td><td>↑13.34</td><td>66.06 ± 0.1</td><td>63.65 ± 1.7</td><td>↓2.41</td><td>70.27 ± 0.2</td><td>95.58 ± 0.7</td><td>↑25.31</td></tr><tr><td>TGN</td><td>49.73 ± 0.8</td><td>63.07 ± 0.5</td><td>↑13.34</td><td>73.05 ± 1.7</td><td>72.36 ± 4.0</td><td>↓0.69</td><td>74.28 ± 0.9</td><td>95.84 ± 0.4</td><td>↑21.56</td></tr><tr><td>DyGFormer</td><td>49.73 ± 0.8</td><td>63.07 ± 0.5</td><td>↑13.34</td><td>79.93 ± 0.1</td><td>80.46 ± 0.5</td><td>↑0.53</td><td>80.91 ± 0.1</td><td>95.31 ± 2.8</td><td>↑14.40</td></tr><tr><td rowspan="3">ICEWS1819</td><td>TGAT</td><td>77.45 ± 0.5</td><td>85.04 ± 1.5</td><td>↑7.59</td><td>85.81 ± 0.2</td><td>86.12 ± 0.1</td><td>↑0.31</td><td>87.33 ± 1.0</td><td>93.05 ± 1.6</td><td>↑5.72</td></tr><tr><td>TGN</td><td>77.45 ± 0.5</td><td>85.04 ± 1.5</td><td>↑7.59</td><td>84.79 ± 0.6</td><td>85.69 ± 0.4</td><td>↑0.90</td><td>85.96 ± 0.8</td><td>94.74 ± 5.7</td><td>↑8.78</td></tr><tr><td>DyGFormer</td><td>77.45 ± 0.5</td><td>85.04 ± 1.5</td><td>↑7.59</td><td>87.51 ± 0.3</td><td>88.11 ± 0.5</td><td>↑0.60</td><td>86.72 ± 0.4</td><td>95.77 ± 0.3</td><td>↑9.05</td></tr><tr><td rowspan="6">Inductive</td><td rowspan="3">Enron</td><td>TGAT</td><td>31.94 ± 0.7</td><td>45.24 ± 1.1</td><td>↑13.30</td><td>47.80 ± 0.8</td><td>45.01 ± 1.3</td><td>↓2.79</td><td>53.26 ± 1.0</td><td>81.52 ± 2.0</td><td>↑28.26</td></tr><tr><td>TGN</td><td>31.94 ± 0.7</td><td>45.24 ± 1.1</td><td>↑13.30</td><td>54.98 ± 2.3</td><td>53.93 ± 4.0</td><td>↓1.05</td><td>58.92 ± 1.4</td><td>82.38 ± 1.2</td><td>↑23.46</td></tr><tr><td>DyGFormer</td><td>31.94 ± 0.7</td><td>45.24 ± 1.1</td><td>↑13.30</td><td>66.86 ± 0.1</td><td>67.64 ± 1.4</td><td>↑0.78</td><td>68.27 ± 0.1</td><td>86.01 ± 4.9</td><td>↑17.74</td></tr><tr><td rowspan="3">ICEWS1819</td><td>TGAT</td><td>60.63 ± 0.8</td><td>71.45 ± 0.6</td><td>↑10.82</td><td>74.10 ± 0.2</td><td>74.12 ± 0.2</td><td>↑0.02</td><td>75.19 ± 0.2</td><td>82.25 ± 2.0</td><td>↑7.06</td></tr><tr><td>TGN</td><td>60.63 ± 0.8</td><td>71.45 ± 0.6</td><td>↑10.82</td><td>69.69 ± 0.8</td><td>70.39 ± 1.2</td><td>↑0.70</td><td>70.01 ± 0.6</td><td>84.01 ± 9.2</td><td>↑13.99</td></tr><tr><td>DyGFormer</td><td>60.63 ± 0.8</td><td>71.45 ± 0.6</td><td>↑10.82</td><td>78.14 ± 0.3</td><td>77.70 ± 0.8</td><td>↓0.44</td><td>79.79 ± 0.4</td><td>87.80 ± 0.6</td><td>↑8.01</td></tr></table>
+
+We report the results in Figs. 5 and 6. Our CROSS framework consistently performs best and exhibits remarkable robustness even under high perturbation rates. This may be due to CROSS's ability to effectively harness the valuable temporal semantic information, which aids in mitigating the adverse impact of structural noise attacks. Furthermore, the results of attention weights reveal that the CROSS framework can effectively down-weight the noisy neighbors
+
+![](images/d8c4f1f2bab1b9ced5fd261d1afc26b741f92389dea68bee6080da2a5120e01a.jpg)  
+Figure 6: Attention weights from randomly selected nodes to their perturbed neighbors on GDELT with the perturbation rate of  $50\%$ .
+
+during encoding. This may be the key reason why CROSS could achieve superior performance and exceptional robustness. Building on this observation, we conduct a case study to visualize the attention weights and the learned representations during encoding in Sec. A.
+
+Robustness for encoding layers. Next, we study the model robustness to the number of encoding layers. Specifically, we conduct a series of experiments with varying numbers of encoding layers  $L = \{1,2,3,4,5\}$ . A larger number of encoding layers facilitates a deeper integration of the cross-modal information. Other details are provided in Sec. J of the Appendix.
+
+The results are depicted in Fig. 7. Our CROSS framework also reveals outstanding robustness to the encoding layers, where the performance improvements over their respective backbones become more pronounced as the number of layers increases. Such robustness likely stems from the invaluable semantic information and the sufficient fusion between semantics and struc
+
+![](images/b765f73a09028f1a1b7cd1f76247d74c4872fb3091901ab116eb37aa4c932b40.jpg)  
+(a) TGAT as the backbone.
+
+![](images/34b4ef5abfc5eb9b460fd66a413a453c3b3bf89da812f4f4b91baf48192c34c4.jpg)
+
+![](images/bad9cfa16802aac2ff8af82537806b58093aca2b2835fb0423595a2c00515509.jpg)  
+Figure 7: Robustness study for encoding layers on Enron.
+
+![](images/e78ece02bca396564d0fe0c50b9ae0f118fc022a357b181d3f6c8de3d215c352.jpg)  
+(b) TGN as the backbone.
+
+tures, whereas graph structures alone may carry high-order irrelevant or spurious information. We also find that the CROSS framework always achieves peak performance with a larger  $L$ . This can be attributed to the deeper information exchange of our cross-modal mixer, which fully amplifies the mutual reinforcement between semantics and structures.
+
+# 4.5 Ablation Study
+
+We conduct three groups of ablation experiments, including model components, textual inputs and their encoding strategies, as well as the choice of LLMs.
+
+Ablation for model components. We start the ablation study by evaluating the contributions of the key components of our model. Detailed information for each variant is provided in Sec. K of Appendix. The results are detailed in Fig. 8. We can see that incorporating all components results in the best performance, while the removal of any single component leads to a performance drop. This highlights the effectiveness of each
+
+![](images/e89742c7064f6168ac96eaa890ca186f50a0d3e7806d24f0bf60c144329796d0.jpg)  
+Figure 8: Ablation study for model components using TGN model as the backbone.
+
+component in CROSS. Notably, mixing all semantic representations does not improve model performance. This may be attributed to the unexpected inclusion of overly outdated semantic information when passing through our cross-modal mixer, thus hindering the quality of final representations.
+
+Ablation for raw texts and LLM-generated texts. We then conduct an ablation study to compare the raw texts (i.e., the original text attributes in TTAGs) with the LLM-generated texts (i.e., neighborhood summaries produced by the LLMs) using various encoding mechanisms. We present the results in Tab. 3 and other details can be seen in Sec. K. Firstly, our framework outperforms all other variants, reaffirming its effectiveness. Besides, the results under semantic encoding prove the validity and expressiveness of the pure LLM-generated texts. Interestingly, the performance of LLM-generated texts yields only marginal improvements or even slight degradation when performing structural encoding. We infer that this occurs because the structural encoding introduces excessively irrelevant information from high-order relations, whereas the other two encodings directly capitalize on the node's own LLM-generated texts, thus integrating more focused and relevant information. This strongly demonstrates the necessity and superiority of the design of our co-encoding mechanism.
+
+Ablation for different LLMs. We extend our ablation study with various LLMs, including DeepSeek-v2 [19] (default), GPT4o [31], Llama3-8b [32], Vicuna-7b [33], and Mistral-7b [34]. Details and results are put in Sec. K and Fig. 9, respectively. We find that DeepSeek-v2 performs best and all variants outperform their corresponding TGNN backbones, reaffirming the effectiveness of the proposed CROSS framework.
+
+![](images/92f0c62234bc5381f0fb183c90f4d5cc034634eb642dfdeac0b05f155401fe05.jpg)  
+Figure 9: Ablation study for different LLMs on GDELT.
+
+# 5 Conclusion and Future Work
+
+In this paper, we focus on the under-explored problem of TTAG modeling and propose CROSS, which extends existing TGNNs to effectively unify text semantics and graph structures with LLMs. By introducing the Temporal Semantics Extractor, we can enhance the LLMs to dynamically extract the text semantics within nodes' neighborhoods. The Semantic-structural Co-encoder then integrates semantic and structural information, enabling bidirectional reinforcement between both modalities. As for future work, we will consider more complex designs of our cross-modal mixer for achieving better representation fusion, such as using the time decay mechanism.
+
+# Acknowledgments
+
+This work is funded in part by the NSFC under grant No. 62372013 and No. 62206056, the NSF through grant IIS-2106972, the CIPSC-SMP-Zhipu Large Model Cross-Disciplinary Fund, and also supported by Tencent Wechat Group. The authors would like to express their gratitude to the reviewers for their feedback, which has improved the clarity and contribution of the paper. The first author, Dr. Zhang, also wants to thank Yifeng Wang for his efforts and support in this work.
+
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+
+![](images/8d937a3d95bb9ea99371129b7eb9ddb70706c0f6f88d2e38b27fe8fd8c52d626.jpg)  
+Figure 10: Case study for text semantics and graph structures. Thicker edges denote higher attention weights during encoding. CROSS exhibits exceptional prediction performance and robustness by successfully unifying text semantics and graph structures within TTAGs, which enables the model to adaptively adjust the attention weights to concentrate on more relevant neighbors, thereby achieving improved prediction performance.
+
+# A Case Study
+
+In this section, we conduct a case study to qualitatively investigate the effectiveness of CROSS.
+
+Case study for text semantics and graph structures. As illustrated in Fig. 10, we select a representative node from GoogleMap_CT and visualize its raw texts, the LLM-generated texts from Temporal Semantics Extractor, the attention weights assigned among neighborhoods by both the TGAT backbone and CROSS during encoding, as well as the prediction probability. The value of prediction probability refers to the predicted probability for the corresponding positive edges as defined in Eq. 11, where higher values indicate better performance. We find that CROSS demonstrates a remarkable capability to unify semantics and structures, which allows the model to adaptively adjust attention weights to concentrate on more relevant neighbors during encoding, thereby achieving improved prediction performance. For instance, CROSS effectively detects the semantic shift in preferences for the target node "Ryan Blanck" across temporal dimensions, which is from "restaurant" to "shopping" ( $t_2 \rightarrow t_3$ ). Subsequently, it automatically reduces the attention weights of neighbors assigned to restaurant nodes (e.g., "Frank Pepe Pizzeria Naporetana") while increasing the weight for store-related neighbors (e.g., "Stafford Thrift Shop"). This highlights the complementarity of semantics and structures, enabling CROSS to prioritize preferred neighbors with semantic contexts.
+
+# Case study for the learned representations.
+
+We further conduct a case study for the learned representations directly. Specifically, we extract semantic and structural representations of a selected node among GDELT, visualize their distributions using Kernel Density Estimation (KDE), and compare the differences between CROSS and the backbone. As shown in Tab. 11, CROSS redistributes the feature space and produces more cohesive representations between semantics and structures. This
+
+![](images/52c9e18ccaf67083799c1dede2207bb068a026a4b5d88c23dc924da720305fd2.jpg)  
+Figure 11: Case study for the learned representations. CROSS effectively synthesizes more cohesive, unified representations between semantics and structures.
+
+can be attributed to our co-encoding architecture, which facilitates synergistic reinforcement between
+
+modalities. Additionally, we find that semantic and structural representations from backbones have different distributions. This directly proves the complementarity of the two modalities, as each captures distinct information that contributes to the final representations. As a result, CROSS effectively integrates them and leads to unified representations that incorporate the most salient features of both.
+
+# B Theoretical Analysis
+
+CROSS is designed from the extraction and unification of semantics and structures within TTAGs. In this section, we present a theoretical analysis for this idea to prove the effectiveness of CROSS, showing that unifying text semantics and graph structures yields more expressive information than existing TGNNs that rely solely on graph structures. This holds under two key conditions:
+
+- Integrity: Text semantics can accurately reflect the graph structures among the neighborhoods of the target node, as they are derived from the LLM-generated texts enriched by the dynamically-informed LLMs' parameterized knowledge.  
+- Complementarity: Text semantics complement the graph structures among the neighborhoods of the target node, providing additional contextual cues that help distinguish subtle relationships.
+
+# Theorem 1. Consider the conditions as follows:
+
+1) Integrity:  $Z_{T}$  serves as a reliable proxy for  $Z_{G}$ , thus we have
+
+$$
+H \left(Z _ {G} \mid Z _ {T}\right) = \epsilon , \quad \epsilon > 0. \tag {12}
+$$
+
+2) Complementarity:  $Z_{T}$  encapsulates orthogonal information that is not captured by  $Z_{G}$ , and that is
+
+$$
+H \left(y \mid Z _ {G}, Z _ {T}\right) = H \left(y \mid Z _ {G}\right) - \epsilon^ {\prime}, \quad \epsilon^ {\prime} > \epsilon . \tag {13}
+$$
+
+Under these conditions, it follows that:
+
+$$
+H \left(y \mid Z _ {G}, Z _ {T}\right) <   H \left(y \mid Z _ {G}\right), \tag {14}
+$$
+
+where  $Z_{G}$  denotes the information derived from graph structures,  $Z_{T}$  represents the information captured from text semantics,  $y$  is the target for prediction, and  $H(\cdot|\cdot)$  depicts the condition entropy.
+
+We also provide an alternative theoretical analysis to prove the model effectiveness, showing that unifying text semantics and graph structures instead of solely considering graph structures is bounded.
+
+Theorem 2. Given the target  $y$ , there exists a constant  $\beta \in (0,1]$  such that:
+
+$$
+I \left(Z _ {G}, Z _ {T}; y\right) \geq I \left(Z _ {G}; y\right) + \beta \min  \left\{H \left(y \mid Z _ {G}\right), H \left(Z _ {T} \mid Z _ {G}\right) \right\}, \tag {15}
+$$
+
+where  $Z_{G}$  denotes the information derived from graph structures,  $Z_{T}$  represents the information captured from text semantics,  $I(\cdot|\cdot)$  denotes mutual information, and  $H(\cdot|\cdot)$  depicts the condition entropy.
+
+Proof of Theorem 1. We aim to prove that the conditional entropy of  $y$  unifying both graph structures  $Z_{G}$  and text semantics  $Z_{T}$ , i.e.,  $H(y|Z_{G}, Z_{T})$ , is strictly less than the conditional entropy of  $y$  solely based on  $Z_{G}$ , i.e.,  $H(y|Z_{G})$ .
+
+We begin with:
+
+$$
+H (y \mid Z _ {G}, Z _ {T}). \tag {16}
+$$
+
+Next, we decompose this using the properties of entropy into two phases:
+
+$$
+H (y \mid Z _ {G}, Z _ {T}) = H (y \mid Z _ {G}, Z _ {M}, Z _ {T}) + I (y; Z _ {M} \mid Z _ {G}, Z _ {T}), \tag {17}
+$$
+
+where  $Z_{M}$  denotes the information arising from the mixed representations of text semantics and graph structures.
+
+We then apply the following upper bound on conditional mutual information as follows:
+
+$$
+\begin{array}{l} I \left(y; Z _ {M} \mid Z _ {G}, Z _ {T}\right) = H \left(Z _ {M} \mid Z _ {G}, Z _ {T}\right) - H \left(Z _ {M} \mid y, Z _ {G}, Z _ {T}\right) \tag {18} \\ \leq H \left(Z _ {M} \mid Z _ {G}, Z _ {T}\right). \\ \end{array}
+$$
+
+Here, the first equality follows from the definition of mutual information, and the inequality holds due to the nonnegativity of conditional entropy.
+
+By substituting Eq. 18 into Eq. 17, we obtain:
+
+$$
+H (y | Z _ {G}, Z _ {T}) \leq H (y | Z _ {G}, Z _ {M}, Z _ {T}) + H (Z _ {M} | Z _ {G}, Z _ {T}). \tag {19}
+$$
+
+Since conditional entropy increases when conditioning on fewer variables, it follows that:
+
+$$
+H \left(y \mid Z _ {G}, Z _ {M}, Z _ {T}\right) + H \left(Z _ {M} \mid Z _ {G}, Z _ {T}\right) \leq H \left(y \mid Z _ {G}, Z _ {T}\right) + H \left(Z _ {G} \mid Z _ {T}\right). \tag {20}
+$$
+
+By applying the "Integrity" and "Complementarity" conditions, we arrive at:
+
+$$
+H \left(y \mid Z _ {G}, Z _ {M}\right) + H \left(Z _ {M} \mid Z _ {T}\right) \leq H \left(y \mid Z _ {G}\right) - \epsilon^ {\prime} + \epsilon . \tag {21}
+$$
+
+Finally, since  $\epsilon' > \epsilon$ , we conclude:
+
+$$
+H \left(y \mid Z _ {G}\right) - \epsilon^ {\prime} + \epsilon <   H \left(y \mid Z _ {G}\right). \tag {22}
+$$
+
+Consequently, we have proven that:
+
+$$
+H \left(y \mid Z _ {G}, Z _ {T}\right) <   H \left(y \mid Z _ {G}\right). \tag {23}
+$$
+
+This completes the proof.
+
+![](images/3484021dc840e2839aab0ea4d3db83e1e16fcdeb096e401c8262d68ffa6618cc.jpg)
+
+Proof of Theorem 2. We aim to prove that the mutual information of  $y$  unifying both graph structures  $Z_{G}$  and text semantics  $Z_{T}$ , i.e.,  $I(Z_{G}, Z_{T}; y)$ , is lower bounded by the mutual information of  $y$  solely based on  $Z_{G}$ , i.e.,  $I(Z_{G}; y)$ .
+
+We begin with:
+
+$$
+I \left(Z _ {G}, Z _ {T}; y\right). \tag {24}
+$$
+
+Using the chain rule of mutual information,  $I(Z_{G}, Z_{T}; y)$  can be decomposed as:
+
+$$
+I \left(Z _ {G}, Z _ {T}; y\right) = I \left(Z _ {G}; y\right) + I \left(Z _ {T}; y \mid Z _ {G}\right), \tag {25}
+$$
+
+where  $I(Z_{G};y)$  quantifies the contribution of graph structures  $Z_{G}$  about  $y$ , and  $I(Z_{T};y\mid Z_{G})$  reflects the additional information provided by text semantics  $Z_{T}$  conditioned on  $Z_{G}$ .
+
+Then, we compute the lower bound for the term  $I(Z_{T};y\mid Z_{G})$ . Based on the fundamental properties of entropy, we can obtain:
+
+$$
+\begin{array}{l} I \left(Z _ {T}; y \mid Z _ {G}\right) = H \left(y \mid Z _ {G}\right) - H \left(y \mid Z _ {T}, Z _ {G}\right) \\ = H \left(Z _ {T} \mid Z _ {G}\right) - H \left(Z _ {T} \mid y, Z _ {G}\right). \tag {26} \\ \end{array}
+$$
+
+Here, the conditional mutual information  $I(Z_{T};y\mid Z_{G})$  could be as high as the minimum of the conditional entropies  $H(y\mid Z_G)$  and  $H(Z_{T}\mid Z_{G})$ . Considering any unavoidable loss in information during the unification process, there exists a constant  $\beta \in (0,1]$ , such that:
+
+$$
+I \left(Z _ {T}; y \mid Z _ {G}\right) \geq \beta \min  \left\{H \left(y \mid Z _ {G}\right), H \left(Z _ {T} \mid Z _ {G}\right) \right\}. \tag {27}
+$$
+
+Substituting the lower bound from Eq. 27 into Eq. 25, we obtain the final bound:
+
+$$
+\begin{array}{l} I (Z; y) = I \left(Z _ {G}; y\right) + I \left(Z _ {T}; y \mid Z _ {G}\right) \\ \geq I \left(Z _ {G}; y\right) + \beta \min  \left\{H \left(y \mid Z _ {G}\right), H \left(Z _ {T} \mid Z _ {G}\right) \right\}. \\ \end{array}
+$$
+
+This completes the proof.
+
+![](images/06b0788cddcadeb6117d008bcb15e9831cafedcce75ccf120975d46c010c5d4c.jpg)
+
+# C Notations and Algorithms
+
+We provide the important notations used in this paper and their corresponding descriptions as shown in Tab. 4. Additionally, for clarity, we present the pseudo-codes of CROSS in Algorithm 1.
+
+# D Details of Experimental Setting
+
+# D.1 Datasets
+
+In this paper, we select four public datasets [12] from different domains and one real-world industrial dataset. We present their detailed descriptions below, and their statistics are summarized in Tab. 5.
+
+Algorithm 1: Training CROSS (one epoch).  
+Input: A node set  $\mathcal{V}$ ; A TTAG  $\mathcal{G} = \{(u,v,t)\}$  with node text attributes  $\mathcal{D}$  and edge text attributes  $\mathcal{R}$ ; The maximum reasoning count  $m$ ; The number of encoding layer  $L$ .  
+1 Initialize all model parameters and prepare the LLMs; // Temporal Semantics Extractor  
+2 foreach  $u \in \mathcal{V}$  do  
+3 Derive reasoning timestamps  $\hat{\mathcal{T}}_u$  with  $m$  by Eq. 1;  
+4 Summarize  $u$ 's textualized neighborhood at  $\hat{t} \in \hat{\mathcal{T}}_u$  with LLM by Eq. 2;  
+5 end  
+// Semantic-structural Co-encoder  
+6 foreach batch  $(u,v,t) \subseteq \mathcal{G}$  do  
+7 foreach  $l = 1,2,\ldots,L$  do  
+8 Retrieve text semantics from generated summaries and graph structures from neighborhoods for nodes  $u/v$ ;  
+9 Compute pre-mixed semantic representations  $\tilde{\mathbf{e}}_{u/v}^{(l)}(t_k)$  with semantic layer by Eqs. 3-4;  
+10 Compute pre-mixed structural representations  $\tilde{\mathbf{h}}_{u/v}^{(l)}(t)$  with structural layer by Eqs. 5-8;  
+11 Mix and propagate cross-modal representations by Eq. 9;  
+12 end  
+13 Derive the final representations  $\mathbf{z}_{u/v}(t)$  by Eq. 10;  
+14 Compute loss  $\mathcal{L}$  by Eq. 11 and backward;  
+15 end
+
+Table 4: Important notations and descriptions.  
+
+<table><tr><td>Notations</td><td>Descriptions</td></tr><tr><td>du</td><td>Raw text attribute of node u</td></tr><tr><td>ru,v,t</td><td>Raw text attribute of edge (u, v, t)</td></tr><tr><td>hatu(t)</td><td>LLM-generated text summary for u&#x27;s neighborhood at reasoning time t</td></tr><tr><td>ˆu(l)(t)</td><td>Pre-mixed semantic representation for node u at time t in the l-th layer</td></tr><tr><td>eu(l)(t)</td><td>Post-mixed semantic representation for node u at time t in the l-th layer</td></tr><tr><td>ˆh(l)(t)</td><td>Pre-mixed structural representation for node u at time t in the l-th layer</td></tr><tr><td>hu(l)(t)</td><td>Post-mixed structural representation for node u at time t in the l-th layer</td></tr><tr><td>zu(t)</td><td>Final representation for node u at time t</td></tr></table>
+
+- Enron<sup>4</sup> originates from a collection of email exchanges among employees of the ENRON energy corporation spanning three years (1999–2002). In this dataset, nodes represent employees, and edges correspond to emails exchanged between them. Each node has text attributes that are derived from the employee's department and role if such information is available. Each edge attaches text attributes consisting of the raw content of the emails. Edges are sequentially ordered based on the e-mail sending timestamps.  
+- GDELT<sup>5</sup> originates from the Global Database of Events, Language, and Tone, a project aimed at cataloging political behaviors across nations worldwide. In this dataset, nodes represent political entities, such as "Egypt" or "Kim Jong Un". The textual attributes of nodes are directly taken from the names of these entities. Edges capture the relationships between entities (e.g., "Make Empathetic Comment" or "Provide Aid"), with the textual attributes of edges being derived from the descriptions of these relationships. Edges are sequentially ordered based on the event-occurring timestamps.  
+- ICEWS1819<sup>6</sup> is sourced from the Integrated Crisis Early Warning System project, which serves as a larger temporal knowledge graph for tracking political events compared to Enron.
+
+This dataset is built using events occurring between January 1, 2018, and December 31, 2019. The textual attributes of nodes include the name, sector, and nationality of each political entity, while the textual attributes of edges represent descriptions of the political relationships. All edges are sequentially ordered based on the event-occurring timestamps.
+
+- GoogleMap_CT<sup>7</sup> is sourced from the Google Local Data project, which compiles review data from Google Maps along with user and business information in the United States up to September 2021. This dataset specifically focuses on business entities located in Connecticut. Nodes represent users and businesses, while edges correspond to user reviews of businesses. Textual attributes are assigned exclusively to business nodes, encompassing the business name, address, category, and self-introduction. All edges are sequentially ordered based on the review timestamps.  
+- Industrial is sourced from real-world e-commerce transaction records sampled from a mobile payment company, spanning March to June 2024. Nodes in this dataset represent users or merchants while edges denote their transaction records. Each node is enriched with text attributes, such as the user/merchant name and affiliation. Text attributes of each edge include textual details such as price, transaction type, and user review. Besides, all edges are sequentially ordered based on the transaction timestamps, and each node is assigned a label indicating whether it is fraudulent.
+
+Table 5: Detailed statistics of datasets.  
+
+<table><tr><td>Datasets</td><td># Nodes</td><td># Links</td><td># Times</td><td>Duration</td><td>Domains</td><td>Time Granularity</td></tr><tr><td>Enron</td><td>42,711</td><td>797,907</td><td>1,006</td><td>3 years</td><td>E-mail</td><td>one day</td></tr><tr><td>GDELT</td><td>6,786</td><td>1,339,245</td><td>2,591</td><td>2 years</td><td>knowledge graph</td><td>15 minutes</td></tr><tr><td>ICEWS1819</td><td>31,769</td><td>1,100,071</td><td>730</td><td>2 years</td><td>knowledge graph</td><td>24 hours</td></tr><tr><td>Googleimap_CT</td><td>111,168</td><td>1,380,623</td><td>55,521</td><td>-</td><td>Recommendation</td><td>Unix Time</td></tr><tr><td>Industrial</td><td>1,112,094</td><td>3,196,008</td><td>90</td><td>3 months</td><td>E-commerce</td><td>one day</td></tr></table>
+
+# D.2 Baselines
+
+We evaluate the performance and discuss the capabilities of eleven existing TGNN methods. The details of these methods are as follows:
+
+- JODIE [24] is designed to manage temporal graphs in bipartite user-item settings. It employs two Recurrent Neural Networks (RNNs), one for updating the user states and another for the item states. To prevent the issue of outdated node representations, a projection layer is added to track the evolution of the embeddings over time.  
+- DyRep [25] incorporates neighborhood information by utilizing a temporal attention-based aggregation mechanism. This approach helps capture the evolving structural features of nodes' local environments in the temporal graph, allowing for more accurate dynamic representations.  
+- TGAT [4] leverages a temporal attention model to aggregate data from temporal-topological neighbors, facilitating the creation of temporal node embeddings. It also introduces a trainable time encoding function that ensures each temporal step is distinctly represented, a concept widely adopted in later TGN architectures.  
+- TGN [5] builds upon earlier methodologies by introducing a memory system that stores a state vector for each node. This memory is refreshed whenever a node participates in an interaction. The model also features modules for processing messages, updating memory states, and embedding temporal features, which collectively enable the generation of dynamic node representations.  
+- CAWN [26] creates node embeddings using temporal walks. It generates multiple anonymous random walks starting from a target node and encodes them using a Recurrent Neural Network. These encoded walks are then combined to form the final temporal representation, which is particularly effective for predicting temporal links.
+
+- TCL [28] uses a breadth-first search to form temporal dependency sub-graphs, extracting sequences of interactions for analysis. A Transformer encoder is applied to integrate temporal and structural information, enabling central node representation learning. Additionally, TCL incorporates a cross-attention mechanism within the Transformer to capture interdependencies between interacting node pairs.  
+- PINT [27] applies injective message passing with a temporal focus and incorporates relative positional encoding to improve the model's capability in capturing dynamic patterns within neighborhoods.  
+- GraphMixer [29] employs a link encoder inspired by the MLP-Mixer framework to create temporal embeddings for nodes. Its design includes a fixed time encoding scheme, which demonstrates superior performance compared to traditional learnable approaches. The model also utilizes a node encoder with mean-pooling to aggregate link-based information.  
+- DyGFormer [7] relies on information from 1-hop neighbors to learn temporal graph representations. A Transformer encoder with a patching method is used to capture long-range dependencies across nodes. To preserve correlations between source and target nodes, DyGFormer integrates a Neighbor Co-occurrence Feature.  
+- LKD4DyTAG [30] conducts a preliminary exploration of dynamic text-attributed graphs. It leverages LLMs as text embedders and introduces an auxiliary knowledge distillation loss to enhance model performance.  
+- FreeDyG [10] delves the temporal graph modeling into the frequency domain and proposes a node interaction frequency encoding module that both effectively models the proportion of the re-occurred neighbors and the frequency of corresponding interactions of the node pair.
+
+In addition to the above existing deep learning-based TGNNs, we also explore the performance of the LLMs for TTAG modeling. We employ the widely adopted and well-performing LLM, DeepSeek-v2 $^{8}$  [19], an open-source project released by DeepSeek, Inc. $^{9}$ . DeepSeek-v2 is a strong, economical, and efficient mixture-of-experts language model. For comparison, we test its zero-shot and one-shot performance for temporal link prediction, which is denoted as  $\mathbf{LLM}_{\mathrm{zero}}$  and  $\mathbf{LLM}_{\mathrm{one}}$ , respectively. Similar to our temporal reasoning chain in Sec. 3.1, we design a task-specific prompt to call with LLMs. Specifically, given the historical interactions of two nodes, we prompt DeepSeek-v2 to directly predict whether these two nodes will interact at a specific future timestamp.
+
+# D.3 Implementation Details
+
+Tasks and Metrics. We follow [6] and conduct temporal link prediction under two settings: (i) transductive setting, which predicts links between nodes that have appeared during training; and (ii) inductive setting, where predictions are performed with unseen nodes. During training, we sample an equal number of negative destination nodes as described in Eq. 11. Inspired by [8], we employ Mean Reciprocal Rank (MRR) as the evaluation metric with 100 negative links per positive link during evaluation. We also report the AP and AUC results in Tabs. 7 & 8. Additionally, we further conduct the node classification task in a practical industrial application for financial risk management using the Industrial dataset from the e-commerce domain. The objective of this task is to predict whether a node is involved in fraudulent activity. Specifically, we follow [6] and pass the learned representations through a two-layer MLP to get the probabilities of fraudulent activity for each node. We adopt the Area Under the Receiver Operating Characteristic Curve (AUC) as the evaluation metric for this task.
+
+Model Configurations. For the training and evaluation, we follow [12] and train all models for 50 epochs and adopt the early stopping strategy under the patience of 5 with an evaluation interval of 5. The learning rate and the batch size across all models and datasets are set to 0.0001 and 256, respectively. We repeat the experiments for 3 runs with seeds ranging from 0 to 2 to ensure evaluation reliability and report the averaged performance with the corresponding standard deviations. All training is performed on a single server with 72 cores, 128GB memory, and four Nvidia Tesla V100 GPUs. As for the hyper-parameters, the representation dimensions across all models and datasets are consistently set to 384, and the introduced hyper-parameter, maximum reasoning count  $m$ , is set to 8 for all datasets by default. Other hyper-parameters among baselines follow the critical
+
+![](images/0c92ab06decf8dbe98eed51a4ced75c8822d2efa33ed2d43a5015ded16268c9d.jpg)  
+Figure 12: Robustness study for noise on GDELT and Googlemap_CT with different perturbation rates. (Supplementary results for Fig. 5.)
+
+![](images/ef01dbcf91b562a764bfc26d797152ba3f381def3b94fbc9608c1964b2a82105.jpg)
+
+![](images/75bb64998f71c8496a5b18da9cf9bb6e48753620e61a8e0189b8d9ebebbaaf2a.jpg)
+
+![](images/6ee2a4f55fcf3174f070f0065a5fbfd0b263c4fc52d87b96d3d3472df15ac663.jpg)
+
+Table 6: Ablation study for the raw texts and LLM-generated texts. Semantic/Structural Encoding refers to the encoding mechanisms that independently perform semantic/structural layers in Sec. 3.2; imprv. indicates the performance improvements of LLM-generated texts  $\mathbf{{Text}}_{\mathbf{{LLM}}}$  over the raw texts  ${\mathbf{{Text}}}_{\mathbf{{raw}}}$  . (Supplementary results for Tab. 3.)  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Datasets</td><td rowspan="2">Methods</td><td colspan="3">Semantic Encoding</td><td colspan="3">Structural Encoding</td><td colspan="3">Structural-Structural Co-encoding</td></tr><tr><td>Textraw</td><td>TextLLM</td><td>imprv.</td><td>Textraw</td><td>TextLLM</td><td>imprv.</td><td>Textraw</td><td>TextLLM(ours)</td><td>imprv.</td></tr><tr><td rowspan="9">Transductive</td><td rowspan="3">GDELT</td><td>TGAT</td><td>49.64 ± 0.9</td><td>52.02 ± 8.8</td><td>↑ 2.38</td><td>56.73 ± .04</td><td>57.01 ± 0.3</td><td>↑ 0.28</td><td>58.29 ± 0.2</td><td>81.63 ± 1.7</td><td>↑ 23.34</td></tr><tr><td>TGN</td><td>49.64 ± 0.9</td><td>52.02 ± 8.8</td><td>↑ 2.38</td><td>54.28 ± 1.6</td><td>55.58 ± 1.0</td><td>↑ 1.30</td><td>56.27 ± 1.2</td><td>77.95 ± 2.8</td><td>↑ 21.68</td></tr><tr><td>DyGFormer</td><td>49.64 ± 0.9</td><td>52.02 ± 8.8</td><td>↑ 2.38</td><td>61.35 ± 0.3</td><td>62.54 ± 0.1</td><td>↑ 1.19</td><td>62.73 ± 0.5</td><td>81.28 ± 4.4</td><td>↑ 18.55</td></tr><tr><td rowspan="3">Google map_CT</td><td>TGAT</td><td>47.03 ± 0.2</td><td>98.38 ± 0.4</td><td>↑ 51.35</td><td>63.13 ± 0.5</td><td>68.69 ± 0.1</td><td>↑ 5.56</td><td>65.46 ± 0.2</td><td>99.92 ± 0.0</td><td>↑ 34.46</td></tr><tr><td>TGN</td><td>47.03 ± 0.2</td><td>98.38 ± 0.4</td><td>↑ 51.35</td><td>71.35 ± 0.5</td><td>81.60 ± 1.0</td><td>↑ 10.25</td><td>72.64 ± 0.8</td><td>99.92 ± 0.0</td><td>↑ 27.28</td></tr><tr><td>DyGFormer</td><td>47.03 ± 0.2</td><td>98.38 ± 0.4</td><td>↑ 51.35</td><td>54.82 ± 2.7</td><td>61.76 ± 0.1</td><td>↑ 6.94</td><td>57.02 ± 0.4</td><td>99.82 ± 0.0</td><td>↑ 42.80</td></tr><tr><td rowspan="3">Industrial</td><td>TGAT</td><td>24.47 ± 0.5</td><td>73.15 ± 1.8</td><td>↑ 48.68</td><td>46.74 ± 3.9</td><td>53.33 ± 2.7</td><td>↑ 6.59</td><td>47.62 ± 2.0</td><td>86.97 ± 2.8</td><td>↑ 39.35</td></tr><tr><td>TGN</td><td>24.47 ± 0.5</td><td>73.15 ± 1.8</td><td>↑ 48.68</td><td>54.46 ± 3.0</td><td>53.20 ± 1.0</td><td>↓ 1.26</td><td>55.45 ± 0.5</td><td>94.26 ± 0.8</td><td>↑ 38.81</td></tr><tr><td>DyGFormer</td><td>24.47 ± 0.5</td><td>73.15 ± 1.8</td><td>↑ 48.68</td><td>74.45 ± 0.7</td><td>74.05 ± 0.4</td><td>↓ 0.40</td><td>75.23 ± 0.1</td><td>94.78 ± 1.4</td><td>↑ 19.55</td></tr><tr><td rowspan="9">Inductive</td><td rowspan="3">GDELT</td><td>TGAT</td><td>27.98 ± 0.7</td><td>37.72 ± 9.5</td><td>↑ 9.74</td><td>42.01 ± 0.5</td><td>45.79 ± 0.4</td><td>↑ 3.78</td><td>41.02 ± 0.3</td><td>64.56 ± 1.8</td><td>↑ 23.54</td></tr><tr><td>TGN</td><td>27.98 ± 0.7</td><td>37.72 ± 9.5</td><td>↑ 9.74</td><td>37.48 ± 2.8</td><td>34.61 ± 3.1</td><td>↓ 2.87</td><td>38.81 ± 1.0</td><td>56.65 ± 3.8</td><td>↑ 17.84</td></tr><tr><td>DyGFormer</td><td>27.98 ± 0.7</td><td>37.72 ± 9.5</td><td>↑ 9.74</td><td>50.61 ± 0.2</td><td>52.33 ± .04</td><td>↑ 1.72</td><td>52.19 ± 0.3</td><td>66.37 ± 4.4</td><td>↑ 14.18</td></tr><tr><td rowspan="3">Google map_CT</td><td>TGAT</td><td>44.43 ± 0.2</td><td>89.53 ± 0.7</td><td>↑ 45.10</td><td>60.96 ± 0.2</td><td>66.98 ± 0.2</td><td>↑ 6.02</td><td>62.17 ± 0.2</td><td>91.59 ± 0.0</td><td>↑ 29.42</td></tr><tr><td>TGN</td><td>44.43 ± 0.2</td><td>89.53 ± 0.7</td><td>↑ 45.10</td><td>67.88 ± 0.2</td><td>78.08 ± 1.0</td><td>↑ 10.20</td><td>68.17 ± 0.2</td><td>92.68 ± 0.1</td><td>↑ 24.51</td></tr><tr><td>DyGFormer</td><td>44.43 ± 0.2</td><td>89.53 ± 0.7</td><td>↑ 45.10</td><td>52.98 ± 2.5</td><td>59.67 ± 0.1</td><td>↑ 6.69</td><td>53.81 ± 0.6</td><td>91.74 ± 0.1</td><td>↑ 37.93</td></tr><tr><td rowspan="3">Industrial</td><td>TGAT</td><td>20.71 ± 0.5</td><td>46.34 ± 2.3</td><td>↑ 25.63</td><td>30.04 ± 3.0</td><td>33.86 ± 1.2</td><td>↑ 3.82</td><td>34.19 ± 1.2</td><td>62.22 ± 2.1</td><td>↑ 28.03</td></tr><tr><td>TGN</td><td>20.71 ± 0.5</td><td>46.34 ± 2.3</td><td>↑ 25.63</td><td>38.28 ± 4.1</td><td>34.41 ± 1.4</td><td>↓ 3.87</td><td>40.17 ± 2.0</td><td>83.23 ± 2.6</td><td>↑ 43.06</td></tr><tr><td>DyGFormer</td><td>20.71 ± 0.5</td><td>46.34 ± 2.3</td><td>↑ 25.63</td><td>54.20 ± 0.4</td><td>54.18 ± 0.2</td><td>↓ 0.02</td><td>55.37 ± 2.7</td><td>82.30 ± 1.2</td><td>↑ 26.93</td></tr></table>
+
+hyper-parameters in the widely-used library DyGLib  $^{10}$  [7], which has performed an exhaustive grid search to identify the optimal hyper-parameters across different models. For the details of LLM calls, in addition to the ablation study on different LLMs of Sec. 4.5, we adopt DeepSeek-v2 as the default LLM. All LLM calls on DeepSeek-v2 are performed in a Language Model as a Service (LMaaS)-compatible manner via its official Application Programming Interface  $(\mathrm{API})^{11}$ .
+
+# E Related Work
+
+Temporal Graph Neural Networks (TGNNs). Temporal graph neural networks (TGNNs) [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46] are designed to generate node representations in temporal graphs, where they typically develop various structural encoding mechanisms to summarize the dynamic graph structures among neighborhoods of the target node [47, 48, 49, 50, 51, 52, 53, 54, 55, 56]. Based on how these mechanisms operate, existing TGNNs can be broadly categorized into two types: message-encoding TGNNs (ME-TGNNs) and walk-encoding TGNNs (WE-TGNNs). ME-TGNNs [57, 58, 59, 60, 61, 62, 63, 64] capture changing graph structures via message passing mechanisms, where node representation is refined by aggregating messages from neighbors through various aggregation functions [9]. In contrast, WE-TGNNs [26, 27] incorporate temporal structural information into node representations in a different way. They typically sample multiple temporal walks originating from the target node and encode these walks based on node occurrence information. Despite their success, above existing TGNNs focus solely on biased encoding mechanisms that prioritize topological dynamics, neglecting the rich text semantics present in temporal text-attributed graphs (TTAGs) [12].
+
+For completeness, we note the existence of a very recent work [30] that conducts a preliminary exploration of TTAG modeling, where LLMs are used to embed texts into features, and a distillation
+
+Table 7: AP results (%) for temporal link prediction in transductive and inductive settings. Results highlighted with a blue background indicate the performance and corresponding improvements of our CROSS framework using various TGNN backbones. The best results are highlighted in bold.  
+
+<table><tr><td rowspan="2"></td><td colspan="5">Transductive setting</td><td colspan="5">Inductive setting</td></tr><tr><td>Enron</td><td>GDELT</td><td>ICEWS1819</td><td>Google map_CT</td><td>Industrial</td><td>Enron</td><td>GDELT</td><td>ICEWS1819</td><td>Google map_CT</td><td>Industrial</td></tr><tr><td>JODIE</td><td>97.26 ± 0.2</td><td>94.87 ± 0.2</td><td>97.62 ± 0.6</td><td>84.28 ± 0.1</td><td>93.70 ± 0.2</td><td>93.91 ± 0.4</td><td>91.47 ± 0.2</td><td>94.40 ± 1.7</td><td>82.33 ± 0.2</td><td>82.57 ± 0.5</td></tr><tr><td>DyRep</td><td>95.99 ± 1.3</td><td>93.76 ± 0.5</td><td>96.04 ± 0.4</td><td>77.21 ± 1.2</td><td>87.64 ± 0.2</td><td>90.99 ± 2.7</td><td>91.00 ± 0.7</td><td>91.76 ± 0.5</td><td>74.44 ± 1.6</td><td>77.23 ± 0.4</td></tr><tr><td>TCL</td><td>97.43 ± 0.1</td><td>96.16 ± 0.0</td><td>99.23 ± 0.0</td><td>89.80 ± 0.2</td><td>94.33 ± 0.2</td><td>93.49 ± 0.4</td><td>92.80 ± 0.1</td><td>98.17 ± 0.0</td><td>88.09 ± 0.0</td><td>82.98 ± 0.3</td></tr><tr><td>CAWN</td><td>97.74 ± 0.0</td><td>95.50 ± 0.1</td><td>98.90 ± 0.0</td><td>88.41 ± 0.1</td><td>96.37 ± 0.1</td><td>94.48 ± 0.2</td><td>91.19 ± 0.2</td><td>97.31 ± 0.0</td><td>86.29 ± 0.0</td><td>89.14 ± 0.1</td></tr><tr><td>PINT</td><td>98.28 ± 0.1</td><td>96.26 ± 0.2</td><td>97.33 ± 0.1</td><td>90.20 ± 0.2</td><td>96.00 ± 0.2</td><td>95.82 ± 0.4</td><td>92.28 ± 0.4</td><td>98.11 ± 0.1</td><td>87.25 ± 0.1</td><td>90.91 ± 0.2</td></tr><tr><td>GraphMixer</td><td>96.27 ± 0.2</td><td>94.96 ± 0.1</td><td>98.60 ± 0.0</td><td>80.12 ± 0.3</td><td>94.18 ± 0.1</td><td>90.40 ± 0.6</td><td>90.79 ± 0.1</td><td>96.60 ± 0.1</td><td>77.43 ± 0.1</td><td>82.66 ± 0.2</td></tr><tr><td>FreeDyG</td><td>97.26 ± 0.1</td><td>95.02 ± 0.1</td><td>99.08 ± 0.1</td><td>84.20 ± 0.4</td><td>96.22 ± 0.3</td><td>92.46 ± 0.8</td><td>91.87 ± 0.2</td><td>97.29 ± 0.3</td><td>79.22 ± 0.1</td><td>84.44 ± 0.3</td></tr><tr><td>LKD4DyTAG</td><td>98.42 ± 0.1</td><td>96.26 ± 0.3</td><td>99.17 ± 0.2</td><td>84.88 ± 0.1</td><td>97.20 ± 0.1</td><td>93.02 ± 0.2</td><td>91.19 ± 0.4</td><td>98.21 ± 0.0</td><td>80.63 ± 0.2</td><td>85.88 ± 0.5</td></tr><tr><td>TGAT</td><td>96.94 ± 0.1</td><td>95.70 ± 0.1</td><td>99.10 ± 0.0</td><td>87.11 ± 0.3</td><td>93.60 ± 0.6</td><td>91.98 ± 0.2</td><td>91.57 ± 0.1</td><td>97.81 ± 0.0</td><td>85.40 ± 0.2</td><td>81.25 ± 0.9</td></tr><tr><td>TGAT+</td><td>99.67 ± 0.1</td><td>98.11 ± 0.2</td><td>99.64 ± 0.1</td><td>99.97 ± 0.0</td><td>97.51 ± 0.2</td><td>97.44 ± 0.2</td><td>94.38 ± 0.3</td><td>98.63 ± 0.2</td><td>97.23 ± 0.1</td><td>93.41 ± 0.5</td></tr><tr><td>Avg. ↑ 5.55</td><td>↑ 2.73</td><td>↑ 2.41</td><td>↑ 0.54</td><td>↑ 12.86</td><td>↑ 3.91</td><td>↑ 5.46</td><td>↑ 2.81</td><td>↑ 0.82</td><td>↑ 11.83</td><td>↑ 12.16</td></tr><tr><td>TGN</td><td>97.98 ± 0.2</td><td>95.35 ± 0.3</td><td>99.05 ± 0.1</td><td>91.52 ± 0.1</td><td>95.00 ± 0.6</td><td>94.58 ± 0.5</td><td>90.23 ± 0.7</td><td>97.48 ± 0.1</td><td>90.00 ± 0.0</td><td>85.48 ± 1.5</td></tr><tr><td>TGN+</td><td>99.71 ± 0.1</td><td>98.07 ± 0.3</td><td>99.73 ± 0.3</td><td>99.98 ± 0.0</td><td>97.90 ± 0.3</td><td>97.60 ± 0.3</td><td>93.36 ± 0.7</td><td>98.74 ± 0.7</td><td>97.45 ± 0.1</td><td>94.69 ± 0.1</td></tr><tr><td>Avg. ↑ 4.06</td><td>↑ 1.73</td><td>↑ 2.72</td><td>↑ 0.68</td><td>↑ 8.46</td><td>↑ 2.90</td><td>↑ 3.02</td><td>↑ 3.13</td><td>↑ 1.26</td><td>↑ 7.45</td><td>↑ 9.21</td></tr><tr><td>DyGFormer</td><td>97.90 ± 0.2</td><td>96.43 ± 0.0</td><td>99.17 ± 0.0</td><td>81.16 ± 2.1</td><td>97.56 ± 0.1</td><td>94.99 ± 0.3</td><td>93.45 ± 0.1</td><td>98.13 ± 0.0</td><td>78.74 ± 2.4</td><td>89.81 ± 0.1</td></tr><tr><td>DyGFormer+</td><td>99.63 ± 0.3</td><td>98.50 ± 0.3</td><td>99.78 ± 0.0</td><td>99.97 ± 0.0</td><td>98.79 ± 0.1</td><td>98.25 ± 0.6</td><td>95.83 ± 0.5</td><td>99.09 ± 0.0</td><td>97.25 ± 0.1</td><td>95.90 ± 0.1</td></tr><tr><td>Avg. ↑ 5.57</td><td>↑ 1.73</td><td>↑ 2.07</td><td>↑ 0.61</td><td>↑ 18.81</td><td>↑ 1.23</td><td>↑ 3.26</td><td>↑ 2.38</td><td>↑ 0.96</td><td>↑ 18.51</td><td>↑ 6.09</td></tr></table>
+
+Table 8: AUC results (%) for temporal link prediction in transductive and inductive settings. Results highlighted with a blue background indicate the performance and corresponding improvements of our CROSS framework using various TGNN backbones. The best results are highlighted in bold.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Enron</td><td colspan="4">Transductive setting</td><td rowspan="2">Industrial</td><td rowspan="2">Enron</td><td rowspan="2">GDELT</td><td colspan="3">Inductive setting</td></tr><tr><td>GDELT</td><td>ICEWS1819</td><td>Googlemap_CT</td><td></td><td>ICEWS1819</td><td>Googlemap_CT</td><td>Industrial</td></tr><tr><td>JODIE</td><td>97.50 ± 0.2</td><td>95.55 ± 0.2</td><td>97.56 ± 0.6</td><td>85.25 ± 0.3</td><td>94.34 ± 0.2</td><td>93.93 ± 0.3</td><td>91.41 ± 0.2</td><td>93.68 ± 2.0</td><td>82.83 ± 0.4</td><td>80.55 ± 0.7</td><td></td></tr><tr><td>DyRep</td><td>96.56 ± 0.9</td><td>94.63 ± 0.2</td><td>95.83 ± 0.3</td><td>78.02 ± 0.5</td><td>87.97 ± 0.3</td><td>91.60 ± 2.1</td><td>90.71 ± 0.4</td><td>90.60 ± 0.5</td><td>74.60 ± 1.0</td><td>74.25 ± 0.7</td><td></td></tr><tr><td>TCL</td><td>97.52 ± 0.2</td><td>96.32 ± 0.0</td><td>99.18 ± 0.01</td><td>89.75 ± 0.2</td><td>94.97 ± 0.2</td><td>93.20 ± 0.4</td><td>92.69 ± 0.1</td><td>98.04 ± 0.02</td><td>87.95 ± 0.0</td><td>80.58 ± 0.5</td><td></td></tr><tr><td>CAWN</td><td>97.76 ± 0.0</td><td>95.62 ± 0.0</td><td>98.86 ± 0.01</td><td>88.51 ± 0.1</td><td>96.57 ± 0.1</td><td>94.07 ± 0.1</td><td>90.92 ± 0.1</td><td>97.15 ± 0.04</td><td>86.39 ± 0.1</td><td>87.17 ± 0.1</td><td></td></tr><tr><td>PINT</td><td>98.00 ± 0.4</td><td>96.27 ± 0.2</td><td>99.01 ± 0.1</td><td>89.36 ± 0.1</td><td>98.11 ± 0.3</td><td>95.35 ± 0.7</td><td>92.52 ± 0.1</td><td>98.27 ± 0.5</td><td>88.24 ± 0.2</td><td>87.25 ± 0.3</td><td></td></tr><tr><td>GraphMixer</td><td>96.42 ± 0.2</td><td>95.26 ± 0.0</td><td>97.27 ± 0.1</td><td>80.08 ± 0.3</td><td>94.81 ± 0.1</td><td>89.99 ± 0.7</td><td>90.69 ± 0.1</td><td>95.27 ± 0.1</td><td>76.57 ± 0.2</td><td>80.01 ± 0.4</td><td></td></tr><tr><td>FreeDyG</td><td>97.72 ± 0.1</td><td>96.32 ± 0.2</td><td>99.08 ± 0.01</td><td>84.37 ± 0.2</td><td>95.29 ± 0.1</td><td>95.27 ± 0.9</td><td>92.23 ± 0.1</td><td>97.92 ± 0.1</td><td>80.58 ± 0.1</td><td>84.29 ± 0.2</td><td></td></tr><tr><td>LKD4DyTAG</td><td>97.01 ± 0.1</td><td>97.22 ± 0.1</td><td>98.00 ± 0.1</td><td>83.29 ± 0.1</td><td>96.44 ± 0.0</td><td>92.82 ± 0.4</td><td>93.73 ± 0.1</td><td>96.62 ± 0.1</td><td>78.76 ± 0.2</td><td>82.98 ± 0.1</td><td></td></tr><tr><td>TGAT</td><td>97.17 ± 0.1</td><td>95.93 ± 0.1</td><td>99.05 ± 0.0</td><td>87.17 ± 0.4</td><td>94.43 ± 0.4</td><td>92.09 ± 0.3</td><td>91.74 ± 0.1</td><td>97.67 ± 0.1</td><td>85.33 ± 0.2</td><td>78.88 ± 0.6</td><td></td></tr><tr><td>TGAT+</td><td>99.69 ± 0.04</td><td>98.17 ± 0.2</td><td>99.62 ± 0.1</td><td>99.97 ± 0.0</td><td>97.04 ± 0.2</td><td>97.53 ± 0.1</td><td>94.28 ± 0.3</td><td>98.54 ± 0.3</td><td>97.20 ± 0.1</td><td>92.15 ± 0.4</td><td></td></tr><tr><td>Avg. ↑ 5.47</td><td>↑ 2.52</td><td>↑ 2.24</td><td>↑ 0.57</td><td>↑ 12.80</td><td>↑ 2.61</td><td>↑ 5.44</td><td>↑ 2.54</td><td>↑ 0.87</td><td>↑ 11.87</td><td>↑ 13.27</td><td></td></tr><tr><td>TGN</td><td>98.15 ± 0.2</td><td>95.67 ± 0.3</td><td>99.02 ± 0.1</td><td>91.96 ± 0.1</td><td>95.49 ± 0.4</td><td>94.86 ± 0.5</td><td>90.53 ± 0.5</td><td>97.47 ± 0.0</td><td>90.50 ± 0.0</td><td>83.51 ± 1.6</td><td></td></tr><tr><td>TGN+</td><td>99.70 ± 0.1</td><td>98.14 ± 0.3</td><td>99.73 ± 0.3</td><td>99.97 ± 0.0</td><td>97.69 ± 0.7</td><td>97.48 ± 0.4</td><td>93.69 ± 0.7</td><td>98.72 ± 0.7</td><td>97.42 ± 0.0</td><td>93.17 ± 0.2</td><td></td></tr><tr><td>Avg. ↑ 3.86</td><td>↑ 1.55</td><td>↑ 2.47</td><td>↑ 0.71</td><td>↑ 8.01</td><td>↑ 2.20</td><td>↑ 2.62</td><td>↑ 3.16</td><td>↑ 1.25</td><td>↑ 6.92</td><td>↑ 9.66</td><td></td></tr><tr><td>DyGFormer</td><td>97.78 ± 0.4</td><td>96.52 ± 0.0</td><td>99.08 ± 0.02</td><td>80.96 ± 2.2</td><td>97.61 ± 0.1</td><td>94.24 ± 0.6</td><td>93.12 ± 0.1</td><td>97.93 ± 0.0</td><td>77.99 ± 2.7</td><td>87.89 ± 0.2</td><td></td></tr><tr><td>DyGFormer+</td><td>99.60 ± 0.3</td><td>98.53 ± 0.3</td><td>99.75 ± 0.03</td><td>99.97 ± 0.0</td><td>98.24 ± 0.1</td><td>98.03 ± 0.7</td><td>95.57 ± 0.5</td><td>98.99 ± 0.1</td><td>97.24 ± 0.1</td><td>94.70 ± 0.1</td><td></td></tr><tr><td>Avg. ↑ 5.75</td><td>↑ 1.82</td><td>↑ 2.01</td><td>↑ 0.67</td><td>↑ 19.01</td><td>↑ 0.63</td><td>↑ 3.79</td><td>↑ 2.45</td><td>↑ 1.06</td><td>↑ 19.25</td><td>↑ 6.81</td><td></td></tr></table>
+
+loss is applied to transfer knowledge from LLMs to a TGNN model. However, this method still suffers from the limitations of neglecting both semantic dynamics and semantic-structural reinforcement, as it continues to rely on the TGNN encoding mechanism during inference (empirically validated in Tab. 2). Instead, our proposed CROSS tackles these issues by unifying text semantics and graph structures, which effectively generates cohesive representations that are both context- and structure-aware.
+
+Text-attributed Graphs (TAGs). Text-attributed graphs (TAGs) have been widely adopted in numerous real-world applications [16, 65]. To enable representation learning in such graphs, existing methods often combine graph learning approaches with language modeling techniques. Early works focus on integrating pre-trained language models (LMs) with graph neural networks (GNNs). Some of them [66, 67] conduct cascaded architecture. They first use LMs to independently embed texts as node features, which are then fed into GNNs for representations. Unlike these pipelines, other methods [68, 69, 70, 71, 72, 73] adopt a hybrid GNN-LM architecture to jointly detect both semantic and structural information. Unfortunately, all these methods overlook the potential temporal information inherent in graphs, limiting their applicability to TTAG modeling. Moreover, they cannot be directly applied to TTAGs, as TTAGs and TAGs differ technically. For instance, TTAGs involve a sequence of timestamped interactions with both node and edge attributes, while TAGs typically rely on adjacency matrices with only node attributes.
+
+Large Language Models (LLMs) for Graph Text Augmentation (LLM4GTAug). In recent years, the rising prominence of large language models (LLMs) [18, 74, 75], such as DeepSeek-v2/3 [19], has underscored their exceptional potential to revolutionize graph modeling [76, 77, 78]. They typically harness the prompt response of LLMs as the external knowledge to fortify overall model performance, either for text augmentation [20, 79, 80] or for structure refinement [21, 81]. However, all the above methods focus on static text-attributed graphs. LLMs used in these methods are typically limited by static corpora input for reasoning, making them ill-suited to capture the temporal dynamics of TTAGs. In this paper, we propose to enhance LLMs with dynamic semantic summarization reasoning capability, effectively detecting semantic dynamics for TTAG modeling.
+
+# F Prompt Template
+
+In Sec. 3.1, we provide a simplified example of the prompts used to construct the temporal reasoning chain within our Temporal Semantics Extractor. Here, we present a complete example of the prompts applied to the Googlemap_CT dataset as depicted in Fig. 13. Only minor keyword adjustments are required for the prompts when applied to other datasets. Specifically, for the Enron dataset, the terms "item" and "review" are replaced with "user" and "email". Similarly, for the GDELT and ICEWS1819 datasets, these terms are substituted with "entity" and "relation". For the industrial datasets, "item" and "review" are replaced with "user" and "transaction".
+
+# Goal #: Summarize the historical reviews of user 'Maureen Sobel' at the current timestamp and provide your semantic understanding for them. # Descriptions #: xxx
+
+Current timestamp #: 27639
+
+Recent reviews of user 'Maureen Sobel' #:
+
+0. timestamp: 27630 | item: 'Name: xxx. Description: xxx' | review: 'Had my hair cut and permed by a very experienced and...'
+
+1. timestamp: 27246 | item: 'Name: xxx.Description: xxx' | review: 'Always clean and monitored. Have a good selection of machines ...
+
+2. timestamp: 26880 | item: 'Name: xxx.Description: xxx' | review: 'Basic auto parts store with lackadaisical stereotype staff who ...
+
+3. ...
+
+Provide the summary STRICTLY in this form: Summary: xxx.
+
+Figure 13: An example of the prompt used to query the LLMs on Googleimap_CT.
+
+# G Scalability Clarification of LLM Usage
+
+In this section, we provide a scalability clarification of LLM usage within the CROSS framework, including presenting cost statistics and outlining the strategies employed to improve efficiency.
+
+Cost statistics of LLM usage. DeepSeek-v2 [19] is renowned for its exceptional performance and cost-efficiency, which provides an excellent balance between quality and affordability. Therefore, we select DeepSeek-v2 as the default LLM in our CROSS framework. Here, we present the cost statistics for calling DeepSeek-v2 API across the five datasets in Tab. 9 for reference. Additionally, we want to emphasize that the DeepSeek-v2 API imposes no theoretical rate limits. During practical implementation, we leverage multithreading techniques to conduct multiple network requests simultaneously under 80 concurrent processes, enhancing program parallelism and optimizing response time consumption. As for the money cost, DeepSeek-v2 prices at  $0.00027 per 1,000 input tokens and$ 0.0011 per 1,000 output tokens at the time of our experiments.
+
+Table 9: Cost statistics of LLM usage.  
+
+<table><tr><td>Datasets</td><td># Input Tokens</td><td># Output Tokens</td><td># Time Consumption (s)</td><td># Money Cost ($)</td></tr><tr><td>Enron</td><td>20,640,423</td><td>4,471,765</td><td>4,528</td><td>10.49</td></tr><tr><td>GDELT</td><td>4,194,490</td><td>1,049,842</td><td>4,145</td><td>2.29</td></tr><tr><td>ICEWS1819</td><td>21,844,799</td><td>5,410,145</td><td>6,517</td><td>11.85</td></tr><tr><td>Googlemap_CT</td><td>144,712,376</td><td>21,214,481</td><td>21,581</td><td>62.41</td></tr><tr><td>Industrial</td><td>296,273,271</td><td>30,294,172</td><td>41,729</td><td>113.3</td></tr></table>
+
+Strategies to improve the efficiency of LLM usage. It is important to highlight that we have implemented several strategies to enhance the efficiency of LLM usage as follows: (i) The frequency of LLM calls for each node is carefully constrained. We introduce a maximum reasoning count  $m$  to
+
+constrain the number of reasoning steps for each node as described in Eq. 1. This design reduces the computational complexity of LLM calls from  $\mathcal{O}(|\mathcal{E}|)$  to  $\mathcal{O}(|\mathcal{V}|)$ , where  $|\mathcal{E}|$  and  $|\mathcal{V}|$  represent the total number of edges and nodes respectively. (ii) Smaller LLMs can serve as cost-effective alternatives. Smaller LLMs within the CROSS framework still demonstrate notable performance as shown in Sec. 4.5, providing a more cost-effective alternative to larger LLMs like DeepSeek-v2 or GPT-4o. (iii) The LLM-generated texts are consistently reused to avoid repeated querying. Our method requires only a single query to the LLMs, with the summaries being stored for subsequent use. These LLM-generated texts can be reused for other tasks or integrated into other methods. We will make the LLM-generated texts publicly available if this paper can be accepted.
+
+Based on the clarifications above, we argue that the proposed CROSS framework is applicable and effective for large industrial-scale TTAGs. This is because CROSS's success on our Industrial dataset from real-world e-commerce systems has demonstrated its practical scalability, and the scale of this dataset surpasses that of many common domains [82]. We will also provide the complexity analysis for CROSS's components in Sec. L.4.
+
+# H Efficiency Study
+
+In this section, we conduct an efficiency study by comparing model performance and training time per epoch. The results are presented in Fig. 14, where the top-left indicates better performance with higher efficiency. From the results, we observe that CROSS strikes a better trade-off between effectiveness and efficiency, achieving the best performance while maintaining a moderate, acceptable computational cost. These results further highlight the scalability and practicality of our proposed framework, making it well-suited for large-scale and industrial applications.
+
+![](images/012b2ccb8c483e004d2976e3d2b16d6310ff976e232ec04b6c750b13d5885ec0.jpg)  
+(a) Enron.
+
+![](images/2a76373e0d5e9d89ece2d59ebe3e3283afabf3974f7cf9f8c1b0a2af22013059.jpg)  
+(b) GDELT.  
+Figure 14: Comparison between model performance and training time per epoch. CROSS exhibits a better trade-off between effectiveness and efficiency, achieving the best performance while maintaining a moderate, acceptable computational cost.
+
+![](images/04702aaa6b23b1c6a56687bc92c63460fd478ea1002427b74dc3c14ba495ef6e.jpg)  
+(c) ICEWS1819.
+
+![](images/4968bf8ae3488404f67ecb8abbc74aac3c7a259cc77ece9394aaa72ba746c247.jpg)  
+(d) Googlemap_CT.
+
+# I Parameter Study
+
+To balance granularity and efficiency, we set a maximum reasoning count ( $m$  in Eq. 1) to constrain the number of LLM reasoning steps. Now we study how this hyper-parameter impacts performance and plot the results with varying  $m$  in Fig. 15. Reasoning too infrequently (small  $m$ ) may make the LLMs cannot effectively comprehend the semantic dynamics around nodes and thus result in degraded performance, while reasoning too frequently
+
+(large  $m$ ) may cause the LLMs to fail to capture long-term semantic shifts. It is also worth noting that different models exhibit varying sensitivities, and  $m = 8$  seems to be a generally sweet choice.
+
+![](images/4481c0b980a1eb2963b1b1e79382d3963f997915d99f0b187034a46999215b9c.jpg)  
+(a) TGAT as the backbone.  
+Figure 15: Parameter study with different maximum reasoning count  $m$  on GDELT. Results of w/o TSE are also presented for reference.
+
+![](images/904975b03a4f53f12c96a17da922892f513285b98160aaff4ac5a3b7e591b578.jpg)  
+(b) TGN as the backbone.
+
+# J Details for Robustness Study
+
+Robustness study for noise. As stated in Sec. 1, the learned representations from existing TGNNs may rely solely on noisy structural information. To empirically validate this assumption, we strategically introduce noise into graph structures surrounding a target node by replacing its neighbors
+
+with randomly sampled nodes at perturbation rates of  $p \in \{10\%, 20\%, 30\%, 40\%, 50\% \}$ . To further investigate the impact of noise and assess the model ability to handle such conditions, we randomly select a subset of nodes and visualize the attention weights of their perturbed neighbors during encoding using box plots under  $p = 50\%$ .
+
+Robustness study for encoding layers. Next, we study the model robustness to the number of encoding layers. Specifically, we conduct a series of experiments with varying numbers of encoding layers  $L = \{1,2,3,4,5\}$ . A larger number of encoding layers facilitates a deeper integration of the cross-modal information. We employ TGAT and TGN as the backbones. Unlike previous experiments, we set the neighbor sampling size in each layer to 5 in this group of experiments due to computational constraints. Other hyper-parameters remain set to their default values detailed in Sec. D.3.
+
+![](images/7c46c22db8712726147074ce2428c80e90760f5139311781a1b78e171bf3f5ab.jpg)  
+Figure 16: Attention weights from randomly selected nodes to their perturbed neighbors on Enron with the perturbation rate of  $50\%$ . (Supplementary results for Fig. 6.)
+
+![](images/0ec00e92c433f6713689227ea7ff5348fb12c611d979ca35eb1aa695d03cbb0d.jpg)  
+Figure 18: Ablation study for model components using TGAT model as the backbone. Results of the backbone are also included for reference. (Supplementary results for Fig. 8.)
+
+![](images/7e6ee855576c41950deb8b83ac7324f4a0e3118ead5f897547403d4bbd514af2.jpg)  
+(a) TGAT as backbone. (b) TGN as backbone. Figure 17: Robustness study for encoding layers on ICEWS1819. (Supplementary results for Fig. 7.)
+
+![](images/02b45160f34efc8905c396f66c78371d8b21e3b99ac5dbe965e23a893d221fc3.jpg)  
+(a) TGAT as backbone. (b) TGN as backbone. Figure 19: Parameter study with different maximum reasoning count  $m$  on Enron. Results of w/o TSE are shown for reference. (Supplementary results for Fig. 15.)
+
+# K Details for Ablation Study
+
+Ablation study for model components. We start the ablation study by evaluating the contributions of the key components of our model, including the Temporal Semantics Extractor (TSE) in Sec. 3.1, the Semantic-structural Co-encoder (SC) in Sec. 3.2, and the Cross-modal Mixer (CM) in Eq. 9. We remove each component individually, resulting in three variants:  $\mathbf{w} / \mathbf{o}$  TSE,  $\mathbf{w} / \mathbf{o}$  SC, and  $\mathbf{w} / \mathbf{o}$  CM. Moreover, we disrupt the Temporal Reasoning Chain constructed by the TSE component via scrambling the chronological order in  $\hat{T}_u$  among Eq. 1, which results in a variant named w/o TRC. Additionally, to validate the rationale behind the design of our Cross-modal Mixer, we present results where all semantic representations are indiscriminately mixed. This variant is referred to as  $\mathbf{w} / \mathbf{CM}_{\mathrm{all}}$ .
+
+Ablation study for raw texts and LLM-generated texts. We then conduct an ablation study to compare the raw texts in TTAGs (i.e., the original node or edge text attributes) with the LLM-generated texts (i.e., neighborhood summaries produced by LLM-based Temporal Semantics Extractor). Experiments are performed using three encoding mechanisms, including the semantic encoding that performs semantic layers for semantic representations described in Sec. 3.2, the structural encoding that conducts semantic layers for structural representations described in Sec. 3.2, and the semantic-structural co-encoding that is used to generate final representations in CROSS. These combinations lead to six variants.
+
+Ablation study for different LLMs. We further extend our ablation study with different LLMs. In addition to the default LLM DeepSeek-v2 within the CROSS framework, we conduct experiments using various LLMs with the GDELT dataset, including GPT-4o [31], Llama3-8b [32], Vicuna-7b [33], and Mistral-7b [34]. Specifically, GPT-4o is a proprietary large language model developed
+
+by OpenAI $^{12}$ , built upon the GPT-4 architecture and optimized for multimodal comprehension and generation. We also invoke it via its official API $^{13}$ . Llama3-8b is an open-source auto-regressive language model with an improved transformer structure. Its tuned versions use supervised fine-tuning (SFT) and reinforcement learning with human feedback (RLHF) to align with human preferences for helpfulness and safety. Vicuna-7b is built on Llama 2 and fine-tuned using supervised instruction. Its training data comes from user-shared conversations found online. Mistral-7b uses grouped-query attention (GQA) for faster processing and sliding window attention (SWA) to handle long sequences more efficiently, reducing the cost of inference.
+
+From the results in Fig. 9, we can find that DeepSeek-v2 performs best, demonstrating its effectiveness. Additionally, it is worth noting that all variants significantly outperform their corresponding backbones and the performance differences across variants are relatively minimal. This demonstrates the robustness of the CROSS framework.
+
+# L Discussions
+
+# L.1 Empirical Analysis for Semantic Dynamics
+
+In this subsection, we provide an empirical analysis for the semantic dynamics in TTAGs. To achieve this, we conduct an empirical experiments with three additional variants, including:
+
+- w/ FTI (Fixed Time Interval): Instead of the timestamp sampling using fixed interaction intervals in Eq. 1, for each node, we compute its interaction time span and uniformly sample timestamps to enable the LLM to summarize at fixed time intervals.  
+- w/ RN (Recent Neighbors): For each node at each timestamp, we directly instruct the LLM to summarize the most recent 20 neighbors.  
+- w/o TE (Time Encoding): We directly remove the time encoding mentioned in Eq. 3.
+
+The results are presented in Tab. 10. From the results, we find that the w/o TE exhibits only a marginal performance drop. This indicates that time encoding alone captures limited temporal dynamics. Instead, our prompting paradigm guides LLMs to capture the semantic evolution over time, effectively facilitating text-driven dynamics for TTAG modeling. Furthermore, either summarizing at fixed time intervals or recent interactions yields suboptimal results. This strongly validates the importance of semantic dynamics.
+
+Table 10: MRR results (%) for temporal link prediction on Enron with TGAT and TGN backbones.  
+
+<table><tr><td></td><td colspan="2">Transductive</td><td colspan="2">Inductive</td></tr><tr><td></td><td>TGAT</td><td>TGN</td><td>TGAT</td><td>TGN</td></tr><tr><td>w/ FTI</td><td>82.47</td><td>85.20</td><td>70.21</td><td>71.77</td></tr><tr><td>w/ RN</td><td>78.46</td><td>83.38</td><td>65.60</td><td>68.87</td></tr><tr><td>w/o TE</td><td>92.87</td><td>93.62</td><td>81.40</td><td>80.33</td></tr><tr><td>CROSS</td><td>95.58</td><td>95.84</td><td>81.52</td><td>82.38</td></tr></table>
+
+# L.2 Extension to Edge Classification
+
+CROSS can be easily extended to edge classification. For completeness, we evaluate two edge classification settings: (i) Supervised setting where models are trained directly using edge labels, and (ii) zero-shot setting where models are first trained on temporal link prediction, and a 2-layer MLP is subsequently fine-tuned on top of the frozen encoder for edge classification. Both implementation and model configurations follow DTGB [12], with DyGFormer adopted as the backbone. From the results, we find that CROSS still achieves the best performance in both supervised and zero-shot edge classification, proving its strong generalization and transferability across tasks. Additionally, all models exhibit a performance drop in the zero-shot setting. This highlights the presence of a transfer learning challenge in TTAG modeling, which warrants future research.
+
+Table 11: Precision, Recall, and F1 (\%) results for supervised and zero-shot edge classification on Enron. We omit some baselines, as they have shown inferior performance [12].  
+
+<table><tr><td rowspan="2"></td><td colspan="5">Supervised Edge Classification</td><td colspan="5">Zero-shot Edge Classification</td></tr><tr><td>JODIE</td><td>DyRep</td><td>GraphMixer</td><td>DyGFormer</td><td>CROSS</td><td>JODIE</td><td>DyRep</td><td>GraphMixer</td><td>DyGFormer</td><td>CROSS</td></tr><tr><td>Precision</td><td>65.68</td><td>66.25</td><td>63.13</td><td>66.01</td><td>78.62</td><td>47.52</td><td>50.28</td><td>44.28</td><td>52.82</td><td>65.28</td></tr><tr><td>Recall</td><td>64.72</td><td>63.90</td><td>57.35</td><td>58.06</td><td>72.74</td><td>40.25</td><td>46.91</td><td>42.82</td><td>47.36</td><td>60.22</td></tr><tr><td>F1</td><td>64.78</td><td>64.32</td><td>55.07</td><td>56.04</td><td>70.91</td><td>42.88</td><td>44.58</td><td>43.66</td><td>45.20</td><td>58.29</td></tr></table>
+
+# L.3 When Handling Conflicting Modalities
+
+CROSS is developed from the unification of text semantics and graph structures with the assumption of complementarity between both information, which is discussed in Sec. 1 and validated in Sec. A. While building upon this, we argue that CROSS remains effective even in the presence of potential cross-modal conflicts. This claim is supported by:
+
+Quantitative results. Noise can reduce alignment and amplify conflicts between semantics and structures. Our robustness study for noise in Sec. 4.4 simulates these inconsistencies, and the results show that CROSS consistently performs best and exhibits strong robustness. This validates CROSS's ability to effectively handle such cross-modal conflicts.
+
+Qualitative results. Similar to our case study in Sec. A, we also visualize the learned representations of a representative node on GDELT as shown in Fig.20. The completely disjoint representations from backbones confirm the presence of cross-modal conflicts, and CROSS effectively narrows the feature space to produce relatively more unified representations. Such capabilities to handle conflicts can be attributed to our co-encoding mechanism, which adaptively facilitates synergistic alignment between modalities.
+
+![](images/5af8f38898df7492da284b623dd7aa3d6c48c8123d515eb613ed30b48c823a33.jpg)  
+Figure 20: Case study for the learned representations when handling conflicting modalities.
+
+# L.4 Complexity Analysis
+
+Now, we provide a theoretical complexity analysis of the two main components of CROSS. Let  $|\mathcal{V}|$  represent the number of nodes,  $D$  denote the average degree of the node,  $L$  indicate the number of encoding layers, and  $m$  depict the maximum reasoning count. To simplify the calculations, we assume that the input, hidden, and output dimensions are uniformly set to  $d$ . In the Temporal Semantics Extractor, we construct a temporal reasoning chain with a maximum reasoning count of  $m$  for each node, resulting in a computational complexity of  $\mathcal{O}(m|\mathcal{V}|)$ . For the Semantic-structural Co-encoder, we utilize various existing TGNN encoding blocks as the structural layers. Therefore, our complexity analysis focuses on the additional cost introduced by the semantic layers. As mentioned in Sec. 3.2, these layers employ a series of standard Transformer layers for each node, thus leading to a total complexity of  $\mathcal{O}\left(L|\mathcal{V}|D^2d + L|\mathcal{V}|d\right)$ .
+
+# L.5 Limitation
+
+One potential limitation of CROSS is that we only focus on 1-hop historical interactions of a specific node as input to the LLMs within our Temporal Semantics Extractor. While effective in many cases, this approach may be suboptimal for scenarios where high-order temporal semantics are critical. However, directly incorporating multi-hop textual neighborhoods into LLM could substantially escalate computational costs and dilute the model ability to capture relevant semantic information. Future work could explore innovative methods to efficiently and effectively capture nodes' high-order semantics, unlocking further potential for improved TTAG modeling.
+
+Another issue could be that LLM-generated texts do not always perform optimally under different encoding mechanisms. Although LLMs are renowned for their powerful text generation and understanding capabilities, it is crucial to explore effective encoding mechanisms that can fully maximize the potential of LLM-generated texts, such as the proposed Semantic-structural Co-encoder.
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: We clearly mentioned our contributions and scope in both the Abstract and Introduction sections.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
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+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: We have outlined the potential limitations of the proposed CROSS in Sec. L.5.
+
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+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
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+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: We have provided a theoretical analysis in Sec. B to prove the effectiveness of the proposed CROSS framework with full set of assumptions and a complete proof.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
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+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+# Answer: [Yes]
+
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+
+# Guidelines:
+
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+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
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+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
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+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: We have provided the detailed information of data splits in Sec. 4.1, and both hyper-parameters and type of optimizer in Sec. D.3.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: As mentioned in Sec. D.3, we repeat the experiments for 3 runs with seeds ranging from 0 to 2 to ensure evaluation reliability and report the averaged performance with the corresponding standard deviations.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
+
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula call to a library function, bootstrap, etc.)  
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+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: As mentioned in Sec. D.3, all training is performed on a single server with 72 cores, 128GB memory, and four Nvidia Tesla V100 GPUs.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: Our work conforms to the NeurIPS Code of Ethics.
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [NA]
+
+Justification: We focus on the problem of TTAG modeling, which has no societal impact.
+
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+
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+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
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+
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+
+# 13. New assets
+
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+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
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+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
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+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+# Answer: [Yes]
+
+Justification: Our work enhances LLMs with dynamic reasoning capability to extract semantic dynamics for TTAG modeling. We have described the usage of LLMs in several parts, including the introduced prompting paradigm in Sec. 3.1, prompt template in Sec. F, implementation details of LLM calls in Sec. D.3, scalability clarifications of LLM usage in Sec. G, as well as the experimental results with different LLMs in Sec. 4.5.
+
+# Guidelines:
+
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+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
\ No newline at end of file
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+# Unifying and Enhancing Graph Transformers via a Hierarchical Mask Framework
+
+Yujie Xing $^{1}$ , Xiao Wang $^{2*}$ , Bin Wu $^{1}$ , Hai Huang $^{1}$ , Chuan Shi $^{1*}$
+
+<sup>1</sup>Beijing University of Posts and Telecommunications, China
+
+<sup>2</sup>Beihang University, China
+
+{yujie-xing,wb789,hhuang,shichuan}@bupt.edu.cn,xiao_wang@buaa.edu.cn
+
+# Abstract
+
+Graph Transformers (GTs) have emerged as a powerful paradigm for graph representation learning due to their ability to model diverse node interactions. However, existing GTs often rely on intricate architectural designs tailored to specific interactions, limiting their flexibly. To address this, we propose a unified hierarchical mask framework that reveals an underlying equivalence between model architecture and attention mask construction. This framework enables a consistent modeling paradigm by capturing diverse interactions through carefully designed attention masks. Theoretical analysis under this framework demonstrates that the probability of correct classification positively correlates with the receptive field size and label consistency, leading to a fundamental design principle: An effective attention mask should ensure both a sufficiently large receptive field and a high level of label consistency. While no single existing mask satisfies this principle across all scenarios, our analysis reveals that hierarchical masks offer complementary strengths—motivating their effective integration. Then, we introduce  $\mathbf{M}^3\mathrm{D}$  phormer, a Mixture-of-Experts based Graph Transformer with Multi-Level Masking and Dual Attention Computation.  $\mathbf{M}^3$  Dphormer incorporates three theoretically grounded hierarchical masks and employs a bi-level expert routing mechanism to adaptively integrate multi-level interaction information. To ensure scalability, we further introduce a dual attention computation scheme that dynamically switches between dense and sparse modes based on local mask sparsity. Extensive experiments across multiple benchmarks demonstrate that  $\mathbf{M}^3\mathrm{D}$  phormer achieves state-of-the-art performance, validating the effectiveness of our unified framework and model design. The source code is available for reproducibility at: https://github.com/null-xyj/M3Dphormer.
+
+# 1 Introduction
+
+As a fundamental data structure, graphs have been widely used to model complex and diverse interactions in real-world systems, such as social networks and brain networks. To learn high-quality node representations, a variety of Graph Neural Networks (GNNs) have been proposed [23, 38, 16]. However, their performance is inherently constrained by the message-passing mechanism, which imposes a strong locality inductive bias. Inspired by the success of Transformers [37] across various machine learning domains [12, 13], adapting Transformer architectures to graphs has emerged as a promising direction, owing to their strong capability to model interactions over a broader range.
+
+Graph Transformers (GTs) leverage the core component of the Transformer architecture, Multi-Head Attention (MHA), to adaptively model diverse node interactions and learn expressive representations. One prominent line of research treats the entire graph as fully connected and applies attention mechanisms to capture pairwise node dependencies [42, 43, 11]. Another line constructs a token
+
+sequence for each node, typically via node sampling or feature aggregation, and adopts a Transformer to capture multi-scale interactions. [5, 15, 41]. Besides, several studies leverage graph partitioning to enable efficient interaction modeling and learn high-quality representations. [18, 20, 44].
+
+While many GTs have been proposed, they often rely on intricate architectural design tailored to specific types of node interactions, which limits their ability to model other important interactions. This raises a natural question: Does there exist a unified perspective of GTs that allows for flexible modeling of diverse node interactions? To address this question, we propose a unified hierarchical mask framework, developed through a thorough analysis of existing GTs. Our analysis reveals that various GT architectures inherently model interactions at different levels—local, cluster, and global—corresponding to the hierarchical organization of relational patterns in graphs. Furthermore, we find that these hierarchical interactions can be uniformly modeled through the design of appropriate attention masks, and that many existing GTs can be interpreted as implicitly corresponding to specific masks. This unified perspective reveals an underlying equivalence between model architecture and mask construction, offering a more flexible approach to GT design.
+
+Under this unified framework, diverse node interactions can be modeled in a consistent manner through the construction of appropriate attention masks, avoiding the need to design intricate network architectures as in traditional methods. Moreover, it facilitates theoretical analysis, proving that both the lower and upper bounds of the probability of correct classification correlate positively with the size of the receptive field and the degree of label consistency. This leads to a fundamental design principle for attention masks: An effective attention mask should ensure a sufficiently large receptive field and a high level of label consistency. Further analysis of masks derived from existing GTs shows that no single mask consistently satisfies this principle across all scenarios. However, hierarchical masks exhibit complementary strengths in node classification, suggesting that integrating multi-level masks provides a natural and effective way to adhere to this principle.
+
+Then, we conduct experiments on real-world datasets to investigate the effectiveness of combining masks across multiple levels. Specifically, we construct three GTs, each designed to capture a single level of interaction—local, cluster, or global—using a corresponding attention mask. We then apply three ensemble strategies to integrate their outputs: Mean, Max, and an idealized Oracle. The node classification results show that: 1) The Oracle strategy significantly outperforms all other models, demonstrating the potential of comprehensively leveraging multi-level interactions. 2) Naive ensemble methods (Mean and Max) often underperform than the best individual-mask model, highlighting a core challenge in effectively integrating hierarchical information. In addition, the excessive memory usage on medium-scale graphs further reveals a key efficiency challenge in GTs.
+
+In this paper, we propose M $^3$ Dphormer, a novel Mixture-of-Experts based Graph Transformer with Multi-Level Masking and Dual Attention Computation. Specifically, M $^3$ Dphormer employs three theoretically grounded attention masks for comprehensive modeling of hierarchical interactions, including local, cluster, and global associations. To effectively integrate information across these interaction levels, we design a bi-level expert routing mechanism, where each expert is a multi-head attention module associated with a specific mask. Furthermore, a dual attention computation strategy is introduced to enhance scalability and computational efficiency.
+
+Our main contributions are summarized as follows:
+
+- We propose a unified hierarchical mask framework that reveals an underlying equivalence between model architecture and attention mask construction, enabling diverse node interactions to be consistently modeled through carefully designed masks.  
+- Theoretical analysis within this framework reveals that the probability of correct classification is positively correlated with both the receptive field size and label consistency. This leads to a guiding principle for designing attention masks: an effective mask should ensure a sufficiently large receptive field and a high level of label consistency.  
+- We propose  $\mathbf{M}^3$  Dphormer, a novel Graph Transformer that captures hierarchical interactions comprehensively and efficiently through multi-level masking, bi-level expert routing, and dual attention computation, thereby adhering to the proposed design principle.  
+- We perform extensive experiments on 9 benchmark datasets, showing that  $\mathbf{M}^3$  Dphormer consistently outperforms 15 strong baselines, demonstrating its effectiveness.
+
+# 2 Preliminary
+
+We denote an attributed graph as  $\mathcal{G} = (\mathcal{V},\mathcal{E},\mathbf{X})$ , where  $\mathcal{V} = \{0,1,\dots ,N - 1\}$  is the set of  $N$  nodes,  $\mathcal{E}$  is the set of  $E$  edges, and  $\mathbf{X} = [\mathbf{x}_u]\in \mathbb{R}^{N\times d_{in}}$  is the node feature matrix, with  $\mathbf{x}_u\in \mathbb{R}^{d_{in}}$  representing the  $d_{in}$ -dimensional feature vector of node  $u$ . The adjacency matrix is denoted as  $\mathbf{A} = [a_{uv}]\in \{0,1\}^{N\times N}$ , where  $a_{uv} = 1$  if there exists an edge from node  $u$  to node  $v$ , and  $a_{uv} = 0$  otherwise. For node classification, we define the set of labels as  $\mathcal{V}$ , and represent the node labels by  $\mathbf{Y} = [\mathbf{y}_u]\in \{0,1\}^{N\times |\mathcal{V}|}$ , where  $\mathbf{y}_u\in \{0,1\}^{|\mathcal{V}|}$  is the one hot label of node  $u$ . The whole node set can be divided into the training set  $\mathcal{V}_{train}$ , the valid set  $\mathcal{V}_{valid}$ , and the test set  $\mathcal{V}_{test}$ .
+
+Graph transformers: The core component of GTs is the MHA, formulated as follows:
+
+$$
+\operatorname {h e a d} _ {i} (\mathbf {H}, \mathbf {M}) = \operatorname {S o f t m a x} \left(\operatorname {M a s k} \left(\hat {\mathbf {A}} ^ {(i)}, \mathbf {M}\right)\right) \mathbf {V} ^ {(i)}, \quad i = 1, \dots , H
+$$
+
+$$
+\hat {\mathbf {A}} ^ {(i)} = \frac {\mathbf {Q} ^ {(i)} \mathbf {K} ^ {(i)} {} ^ {\top}}{\sqrt {d _ {h}}}, \quad \mathbf {Q} ^ {(i)} = \mathbf {H} \mathbf {W} _ {Q} ^ {(i)}, \quad \mathbf {K} ^ {(i)} = \mathbf {H} \mathbf {W} _ {K} ^ {(i)}, \quad \mathbf {V} ^ {(i)} = \mathbf {H} \mathbf {W} _ {V} ^ {(i)} \tag {1}
+$$
+
+$$
+\operatorname {M a s k} (\hat {\mathbf {A}} ^ {(i)}, \mathbf {M}) = \left\{ \begin{array}{l l} \hat {\mathbf {A}} _ {u, v} ^ {(i)}, & \text {i f} \mathbf {M} _ {u, v} = 1 \\ - \infty , & \text {i f} \mathbf {M} _ {u, v} = 0 \end{array} \right.
+$$
+
+Here,  $\mathbf{H} \in \mathbb{R}^{N \times d}$  denotes the input representations, where  $N$  is the number of nodes and  $d$  is the representation dimension.  $\mathbf{W}_Q^{(i)}, \mathbf{W}_K^{(i)}, \mathbf{W}_V^{(i)} \in \mathbb{R}^{d \times d_h}$  are trainable projection matrices specific to the  $i$ -th head, and  $d_h$  is the hidden dimension of each head. The attention score matrix  $\hat{\mathbf{A}}^{(i)}$  is masked by  $\mathbf{M} \in \{0,1\}^{N \times N}$ , where  $\mathbf{M}_{u,v} = 1$  indicates the validity of the attention from node  $u$  to node  $v$ . During masking, attention scores corresponding to invalid positions are set to  $-\infty$ , ensuring that their contribution becomes zero after the Softmax operation. The outputs of all heads are concatenated to produce the final output:  $\mathrm{MHA}(\mathbf{H},\mathbf{M}) = \mathrm{Concat}(\mathrm{head}_1,\dots,\mathrm{head}_H)$ .
+
+# 3 Revisiting GTs through a unified hierarchical mask framework
+
+Interactions in graphs typically exhibit a hierarchical organization, including local connectivity, cluster relations, and global associations. Each level provides essential information for effective graph representation learning. An illustrative example for the importance of hierarchical interactions is provided in Figure 1, where node labels are indicated by different colors. Due to local homophily, node  $u_{1}$  can be accurately classified, as most of its neighbors share the same label. However, such local interactions are insufficient for classifying nodes  $u_{2}$  and  $u_{3}$ . By
+
+![](images/c421ae856d417fdec8e98581b7f352867d93fab0efcec66c426500beb9cb7f71.jpg)  
+Figure 1: Hierarchical interactions.
+
+leveraging cluster interactions, node  $u_{2}$  can be correctly identified, as it lies within a coherent cluster (the blue region). Furthermore, assuming that nodes labeled green follow a representation distribution distinct from the blue and yellow ones, global interactions-represented by the dotted lines—can be adaptively learned to further enhance the classification of  $u_{3}$ . Here, we propose a unified hierarchical mask framework that enables GTs to consistently model such multi-level interactions.
+
+# 3.1 The unified hierarchical mask framework for Graph Transformers
+
+To formalize this framework, we categorize node interactions into three types: (1) N-N: interactions between individual nodes; (2) N-S: interactions between a node and a node set; and (3) S-S: interactions between node sets. For N-N, we directly set  $\mathbf{M}_{u,v} = 1$  to indicate a connection from node  $u$  to  $v$ . For N-S and S-S, we treat each node set as a virtual super node  $v'$ , and extend the node set as  $\mathcal{V} = \mathcal{V} \cup \{v'\}$ . This allows both N-S and S-S to be equivalently modeled as N-N, enabling a unified and flexible mask design across different interaction levels. Based on this unified framework, we further illustrate how existing GTs implicitly design their attention masks to model hierarchical interactions across local, cluster, and global levels. A summary table is provided in Table 4.
+
+Local interactions: Modeling local interactions typically involves capturing information from  $K$ -hop neighborhoods. GOAT explicitly models N-N interactions between a target node and its  $K$ -hop
+
+neighbors, which corresponds to the mask  $\mathbf{M}^{l1} = \mathbf{A}^K$  [24]. GNN-Transformer hybrid architectures implicitly adopt the mask  $\mathbf{M}^{l2} = \mathbf{A}$  through their GNN modules, and aggregating  $K$ -hop information recursively across layers [43, 11, 44]. Some tokenized GTs aggregate node features at multiple hops as input tokens of a Transformer, effectively applying a set of masks  $\{\mathbf{A}^k:1\leq k\leq K\}$  [5, 15, 6].
+
+Cluster interactions: Several recent GTs have focused on capturing cluster-level interactions by partitioning the graph into disjoint clusters with METIS[22]. We define the partition function  $\mathcal{P}(u) = p$  to denote the cluster index assigned to node  $u$  and the reverse partition function  $\mathcal{P}^{-1}(p) = \{u : \mathbf{C}_{u,p} = 1\}$  to denote the set of nodes belong to cluster  $p$ . By treating each cluster as a virtual super node, we denote the set of such nodes as  $\mathcal{V}^p$ . Graph ViT [18] models interactions between clusters (S-S interactions), which corresponds to applying a mask  $\mathbf{M}^{c1}$ , where  $\mathbf{M}_{u,v}^{c1} = 1$  if  $u, v \in \mathcal{V}^p$ . Cluster-GT [20] models more fine-grained N-S interactions, where each cluster attends to all real nodes. This leads to a mask  $\mathbf{M}^{c2}$ , defined as  $\mathbf{M}_{u,v}^{c2} = 1$  if  $u \in \mathcal{V}^p$  and  $v \in \mathcal{V}$ . To differentiate contributions from different clusters, attention scores are further modulated by the connectivity between clusters and refined via cluster attention implicitly induced by  $\mathbf{M}^{c1}$ . CoBFormer [44] focuses on N-N interactions within each cluster by implicitly applying a mask  $\mathbf{M}^{c3}$ , where  $\mathbf{M}_{u,v}^{c3} = 1$  if  $u, v \in \mathcal{V}$  and  $\mathcal{P}(u) = \mathcal{P}(v)$ . It additionally applies  $\mathbf{M}^{c1}$  to capture inter-cluster interactions.
+
+Global interactions: A common strategy for capturing global N-N interactions is to treat the entire graph as fully connected, corresponding to a global mask  $\mathbf{M}^{g1} = \mathbf{1}^{N\times N}$  [42, 43, 11]. In contrast, Exphormer [34] approximates global dependencies through an N-S interaction scheme by introducing a set of virtual super nodes  $\nu^g$ , each connected bidirectionally to all real nodes. This yields a global mask  $\mathbf{M}^{g2}$ , where  $\mathbf{M}_{u,v}^{g2} = 1$  if  $u\in \mathcal{V}$  and  $v\in \mathcal{V}^g$ , or  $u\in \mathcal{V}^g$  and  $v\in \mathcal{V}$ .
+
+# 3.2 Theoretical analysis
+
+To investigate the distinct contributions of hierarchical masks to node classification, we develop a theoretical framework based on a class-conditional representation model. Specifically, let  $\mathcal{G}$  be a graph with label set  $\mathcal{V}$ , and assume a uniform label distribution across nodes. The initial representation of a node with label  $c$  is sampled from a  $d$ -dimensional Gaussian distribution:  $z \sim \mathcal{N}(\boldsymbol{\mu}_c, \sigma_c^2\mathbf{I})$ , where  $|\boldsymbol{\mu}_c|_2^2 = 1$ , and the class prototypes are assumed to be orthogonal, i.e.,  $\boldsymbol{\mu}_c^\top \boldsymbol{\mu}_{c'} = 0$  for all  $c \neq c'$ .
+
+Let node  $u$  have ground-truth label  $c$ , and its receptive field be specified by the mask vector  $\mathbf{M}_{u,c}$ , which contains  $k$  non-zero entries. Define  $\rho_{c'}$  as the fraction of nodes labeled  $c'$  within this receptive field, and  $\alpha_{c'}$  as the average attention weight assigned to those nodes. Let  $\hat{z}_u$  denote the attention-updated representation of node  $u$ . We consider a similarity-based classifier that predicts label  $c$  correctly if  $\hat{\boldsymbol{z}}_u^\top \boldsymbol{\mu}_c \geq \delta_c$ , where  $\delta_c$  is a decision threshold implicitly learned by models.
+
+Theorem 3.1. The updated representation of node  $u$  follows a Gaussian distribution:
+
+$$
+\hat {z} _ {u} \sim \mathcal {N} \left(\sum_ {i = 1} ^ {| \mathcal {C} |} k \rho_ {i} \alpha_ {i} \boldsymbol {\mu} _ {i}, \sum_ {i = 1} ^ {| \mathcal {C} |} k \rho_ {i} \alpha_ {i} ^ {2} \sigma_ {i} ^ {2} \mathbf {I}\right) \tag {2}
+$$
+
+Assume the classifier is well-trained such that  $\delta_c - k\rho_c\alpha_c \leq 0$ , and the attention weights satisfy the constraint  $0 \leq \alpha_{c'} \leq \frac{1}{k} \leq \alpha_c \leq \frac{1}{k\rho_c}$ . Then, the probability that node  $u$  is correctly classified by a similarity-based classifier is bounded as:
+
+$$
+1 - \Phi \left(\frac {\delta_ {c} - k \rho_ {c} \alpha_ {c}}{\sqrt {k \rho_ {c} \alpha_ {c} ^ {2} \sigma_ {c} ^ {2} + \frac {1 - \rho_ {c}}{k} \cdot \sigma_ {m} ^ {2}}}\right) \leq P \left(\hat {\boldsymbol {z}} _ {u} ^ {\top} \boldsymbol {\mu} _ {c} \geq \delta_ {c}\right) \leq 1 - \Phi \left(\frac {\delta_ {c} - k \rho_ {c} \alpha_ {c}}{\sqrt {k \rho_ {c} \alpha_ {c} ^ {2} \sigma_ {c} ^ {2}}}\right), \tag {3}
+$$
+
+where  $\Phi(\cdot)$  denotes the cumulative distribution function (CDF) of the standard normal distribution, and  $\sigma_m^2 = \max_{i \neq c} \sigma_i^2$  is the maximum variance among non-target classes.
+
+Both the lower and upper bounds are monotonically increasing with respect to  $k$ ,  $\rho_c$ , and  $\alpha_c$ , and decreasing with respect to the set of class-wise variances  $\{\sigma_i : 1 \leq i \leq |\mathcal{V}|\}$ .
+
+The proof can be found in Appendix A. Theorem 3.1 shows that the probability of correct classification primarily depends on the factors:  $k$ ,  $\rho_c$ ,  $\alpha_c$ , and the set of variances  $\{\sigma_i : 1 \leq i \leq |\mathcal{Y}|\}$ . Since  $\alpha_c$  and  $\sigma_i$  are affected by the training dynamics and input distribution, respectively, we focus our analysis on  $k$  and  $\rho_c$ , which are determined by the attention mask. Theorem 3.1 demonstrates that larger
+
+values of  $k$  and  $\rho_{c}$  lead to higher lower and upper bounds on the probability of correct classification. This gives rise to a fundamental principle for mask construction: An effective attention mask should ensure a sufficiently large receptive field and a high level of label consistency.
+
+Building on this theorem, we further analyze the applicability of hierarchical masks derived from existing GTs under several representative scenarios: 1) For nodes with strong local homophily, using local masks (e.g.,  $\mathbf{M}^{l1}$ ,  $\mathbf{M}^{l2}$ ) is effective due to their typically large  $\rho_c$ . 2) For nodes near cluster boundaries, local homophily weakens as some neighbors belong to other clusters. In such cases, cluster masks (e.g.,  $\mathbf{M}^{c3}$ ) can yield higher  $\rho_c$  and a larger  $k$ , potentially leading to improved classification performance, assuming the partitioning is accurate. 3) For heterophilic nodes with minority labels in their cluster, local or cluster masks may result in very small  $\rho_c$ , even below  $\frac{1}{|\mathcal{V}|}$ . Here, the global mask  $\mathbf{M}^{g1}$  is preferable, as it ensures  $\rho_c = \frac{1}{|\mathcal{V}|}$  and provides a large  $k = N$ . By contrast, the use of  $\mathbf{M}^{g2}$  may be less effective, as the global virtual nodes lack explicit label semantics. 4) Some prior works approximate global interactions through inter-cluster attention (e.g., via  $\mathbf{M}^{c1}$ ). However, as shown in Equation 2, cluster-level representations are dominated by the majority label. During inter-cluster aggregation, this bias is further amplified, increasing the risk of misclassifying minority-label nodes as the dominant class. 5) For nodes belonging to classes with well-defined representation distributions (i.e., small variance  $\sigma_c$ ), the attention weights  $\alpha_c$  can be effectively learned to approximate  $\frac{1}{k\rho_c}$ , leading to a higher probability of correct classification regardless of the specific mask employed.
+
+The above analysis suggests that no single mask consistently satisfies the proposed principle across all scenarios. However, hierarchical masks at different levels offer complementary strengths in node classification, and their integration provides a natural means of aligning with the principle.
+
+Table 1: Accuracy comparison of individual masks, ensemble strategies, and the oracle case.  
+
+<table><tr><td>Dataset</td><td>Local</td><td>Cluster</td><td>Global</td><td>Mean</td><td>Max</td><td>Oracle</td></tr><tr><td>Cora</td><td>87.71±1.30</td><td>82.10±1.53</td><td>73.03±1.53</td><td>86.41±1.72</td><td>86.82±2.32</td><td>93.41±1.38</td></tr><tr><td>Citeseer</td><td>77.02±2.10</td><td>71.43±2.39</td><td>72.94±2.10</td><td>76.16±2.10</td><td>75.73±2.08</td><td>84.32±2.10</td></tr><tr><td>Pubmed</td><td>89.77±0.46</td><td>87.96±0.35</td><td>87.51±0.42</td><td>89.31±0.28</td><td>89.19±0.35</td><td>93.68±0.25</td></tr><tr><td>Photo</td><td>94.25±0.46</td><td>94.26±1.06</td><td>85.61±4.26</td><td>95.14±0.52</td><td>94.77±0.74</td><td>97.50±0.40</td></tr><tr><td>Computer</td><td>91.57±0.63</td><td>89.33±0.79</td><td>82.78±1.21</td><td>91.41±0.92</td><td>90.45±0.73</td><td>95.59±0.37</td></tr><tr><td>Squirrel</td><td>38.67±1.72</td><td>38.03±1.10</td><td>38.64±1.68</td><td>38.53±1.16</td><td>38.78±1.34</td><td>53.72±1.30</td></tr><tr><td>Chameleon</td><td>41.97±3.90</td><td>42.24±3.35</td><td>43.50±2.97</td><td>42.96±4.29</td><td>42.60±4.58</td><td>64.57±4.05</td></tr></table>
+
+# 3.3 Experimental analysis
+
+To further investigate whether combining masks across multiple interaction levels yields benefits in real-world scenarios, we construct three GTs trained separately with the local mask  $\mathbf{M}^{l2}$ , cluster mask  $\mathbf{M}^{c3}$ , and global mask  $\mathbf{M}^{g1}$  on seven node classification datasets. Then we apply three ensemble strategies to integrate their outputs: Mean: averaging the predicted probabilities; Max: selecting the maximum predicted probability; and Oracle: an idealized upper bound where the best prediction is always selected. The performance of individual models and ensemble methods is reported in Table 1, with three key observations: 1) The strong performance of Oracle indicates that properly integrating complementary information from multiple interaction levels can yield substantial performance gains. 2) Naive ensemble methods, such as Mean and Max, underperform the best single-mask model on 5 out of 7 datasets, revealing a key challenge for Graph Transformers: How to effectively integrate multi-level interaction information?
+
+Moreover, our experiments reveal a significant efficiency issue: even a 2-layer Transformer with 2 heads and a single mask, consumes 21 GB of GPU memory on the PubMed. Although several efficient Transformer variants—such as kernel-based linear attention methods [8, 35] and FlashAttention [9]—have been proposed to alleviate the  $O(N^2)$  complexity, their applicability to graphs remains limited due to the irregular and diverse patterns of graph masks [39]. This leads to another key challenge: How to efficiently implement GTs with irregular masks on large-scale graphs?
+
+# 4 Method
+
+In this section, we propose M $^3$ Dphormer, a novel Mixture-of-Experts based Graph Transformer with Multi-Level Masking and Dual Attention Computation. The overall framework is illustrated
+
+![](images/3bed2a3eb49972a4b3cc8abf406dc840d23de5e9aab84c11db5741687a76141a.jpg)  
+Figure 2: Overview of the  $\mathbf{M}^3$  Dphormer with Pre-RMSNorm [48] and ReLU[1]: a) The overall network architecture. b) The bi-level expert routing mechanism. c) The theorem-guided hierarchical mask design strategy. Self-connections are added for nodes in local and cluster masks. Nodes 2, 4, 5 and 8 are included in the training set. d) The dual attention computation scheme.
+
+![](images/47d7e9f5a9f2c0c80cf1b1e2beff212dc7588d30567cda1b24f49c32239dee57.jpg)
+
+in Figure 2. Guided by theoretical analysis, we first design three hierarchical attention masks to comprehensively model multi-level interactions (part c). To adaptively integrate information across these interaction levels, we introduce a bi-level attention expert routing mechanism, where each expert is a MHA module associated with a specific mask (part b). Additionally, a dual attention computation scheme is incorporated to ensure computational efficiency (part d).
+
+# 4.1 Overall architecture of  $\mathbf{M}^3$  Dphormer
+
+We first present the overall architecture of  $\mathbf{M}^3\mathrm{D}$  phormer. The initial representation is given by  $\mathbf{H}^0 = \mathbf{X}\mathbf{W}_{in}$ , where  $\mathbf{W}_{in} \in \mathbb{R}^{d_{in} \times d}$  is a learnable linear projection. The model then applies  $L$  stacked  $\mathbf{M}^3\mathrm{D}$  phormer layers, with the computation in the  $l$ -th layer defined as:
+
+$$
+\mathbf {H} ^ {l} = \operatorname {A C T} \left(\operatorname {B i M o E} ^ {l} \left(\operatorname {N o r m} ^ {l} \left(\mathbf {H} ^ {l - 1}\right), \boldsymbol {\mathcal {M}}\right)\right) + \mathbf {H} ^ {l - 1} \mathbf {W} _ {r e s} ^ {l}, \tag {4}
+$$
+
+where  $\mathbf{H}^{l-1}$  is the input from the previous layer,  $\mathbf{W}_{res}^l$  is the residual projection matrix following [26],  $\mathrm{Norm}^l(\cdot)$  is the normalization function, and  $\mathrm{ACT}(\cdot)$  is the activation function. The bi-level attention expert routing mechanism BiMoE $^l(\cdot)$  integrates information from multiple interaction levels based on the hierarchical mask set  $\mathcal{M}$ . After  $L$  layers, a linear classifier is applied to obtain the final prediction:  $\hat{\mathbf{Y}} = \mathbf{H}^L\mathbf{W}_{cls}$ , where  $\mathbf{W}_{cls}$  is the classifier weight matrix. The model is optimized using cross-entropy loss computed over both the training nodes  $\mathcal{V}_{train}$  and the label-specific global virtual nodes  $\mathcal{V}^g$ , which are introduced in the following subsection.
+
+# 4.2 Theorem-guided hierarchical mask design strategy
+
+To enable comprehensive modeling of multi-level node interactions, we propose a theoretically grounded design strategy for the hierarchical mask set  $\mathcal{M} = \{\mathbf{M}^{l^2},\mathbf{M}^{c^4},\mathbf{M}^{g^3}\}$ .
+
+Local mask design: We adopt  $\mathbf{M}^{l2} = \mathbf{A}$  as the local mask due to its several advantages over  $\mathbf{M}^{l1} = \mathbf{A}^{K}$ : 1) As observed in CoBFormer[44], the homophily ratio  $\rho_{c}$  tends to decline rapidly with increasing hop size  $K$ . A lower  $\rho_{c}$  may lead to reduced classification probability, as indicated by Theorem 3.1. 2)  $\mathbf{M}^{l1}$  ignores distance information between the target node and its  $K$ -hop neighbors, thus requiring explicit distance-aware position encoding[24]. In contrast,  $\mathbf{M}^{l2}$  can capture such distance implicitly through recursive aggregation across layers. 3)  $\mathbf{M}^{l2}$  is sparser than  $\mathbf{M}^{l1}$ , enabling more efficient computation under the dual attention scheme introduced later.
+
+Cluster mask design: We first partition the graph into  $P$  disjoint clusters using METIS[22], and introduce a set of cluster-level virtual nodes  $\mathcal{V}^p = \{N + i : 0 \leq i < P\}$ . We then define a new cluster mask  $\mathbf{M}^{c4}$ , where  $\mathbf{M}_{u,v}^{c4} = 1$  if either (i)  $u \in \mathcal{V}$  and  $v \in \{u, N + \mathcal{P}(u)\}$ , or (ii)  $u \in \mathcal{V}^p$  and  $v \in \mathcal{P}^{-1}(u - N)$ . The functions  $\mathcal{P}(\cdot)$  and  $\mathcal{P}^{-1}(\cdot)$  denote the partition and reverse-partition function defined in Section 3.1. This formulation restricts attention to node-cluster pairs within the same partition. The feature of each virtual node in  $\mathcal{V}^p$  is computed by averaging the features of the real nodes in its corresponding cluster. Compared to  $\mathbf{M}^{c3}$ , the proposed  $\mathbf{M}^{c4}$  significantly reduces the
+
+non-zero ratio from  $\frac{1}{P}$  to  $\frac{3N}{(N + P)^2}$  since  $P\ll N$ , making it more efficient. Moreover, as demonstrated in Proposition 4.1, the interactions captured by  $\mathbf{M}^{c3}$  can be effectively approximated using  $\mathbf{M}^{c4}$ .
+
+Proposition 4.1. Cluster interactions modeled by a single Graph Transformer layer using  $\mathbf{M}^{c3}$  can be equivalently modeled by two consecutive layers using  $\mathbf{M}^{c4}$ .
+
+The proof is given in Appendix A.4. Although this requires an additional layer, the reduction in sparsity still leads to substantial computational savings by our dual attention computation scheme.
+
+Global mask design: We introduce a new global mask  $\mathbf{M}^{g^3}$ , which extends  $\mathbf{M}^{g^2}$  by explicitly incorporating label semantics. Specifically, we add  $|\mathcal{V}|$  global virtual nodes, indexed as  $\mathcal{V}^g = \{N + P + i : 0 \leq i < |\mathcal{V}|\}$ , each associated with a distinct class label. The mask  $\mathbf{M}_{u,v}^{g^3}$  is set to 1 if either: (i)  $u \in \mathcal{V}$  and  $v \in \mathcal{V}^g$ , or (ii)  $u \in \mathcal{V}^g$  and  $v \in \{t \in \mathcal{V}_{\mathrm{train}} : \mathbf{y}_t = \mathbf{y}_u\}$ . This structure enables each real node to attend to all global nodes, while each global node aggregates information only from training nodes with a specific label. Similar to  $\mathcal{V}^p$ , the features of nodes in  $\mathcal{V}^g$  are obtained by averaging the features of training nodes with the corresponding label. Let  $g_c$  denote the global virtual node for class  $c$ , and  $n_c$  be the number of training nodes with label  $c$ . By Equation 2, the updated representation satisfies:  $\hat{\boldsymbol{z}}_{g_c} \sim \mathcal{N}(\boldsymbol{\mu}_c, \frac{\sigma^2}{n_c} \mathbf{I})$ , since  $\rho_c = 1$ ,  $\rho_{c'} = 0$  for  $c' \neq c$ , and  $\alpha_i = \frac{1}{n_c}$ . The reduced variance  $\frac{\sigma^2}{n_c}$  indicates that it is more concentrated around the class mean. According to Theorem 3.1, this leads to improved bounds for the probability of correct classification.
+
+# 4.3 Bi-level attention expert routing mechanism
+
+To adaptively integrate information from different interaction levels, we propose the bi-level attention expert routing mechanism—the core component of  $\mathbf{M}^3$  Dphormer. Each expert corresponds to an MHA module equipped with a specific attention mask. Motivated by the observation in Table 1 that the local mask yields the best performance in most cases, we prioritize the local expert at the first routing level. The second level then refines the selection among the cluster and global experts. The bi-level routing mechanism is formally defined as:
+
+$$
+\operatorname {B i M o E} (\mathbf {H}, \boldsymbol {\mathcal {M}}) = \mathbf {g} _ {1} \cdot \operatorname {M H A} ^ {D} (\mathbf {H}, \mathbf {M} ^ {l 2}) + \mathbf {g} _ {2} \cdot \operatorname {M H A} ^ {D} (\mathbf {H}, \mathbf {M} ^ {c 4}) + \mathbf {g} _ {3} \cdot \operatorname {M H A} ^ {D} (\mathbf {H}, \mathbf {M} ^ {g 3})
+$$
+
+$$
+\mathbf {g} _ {1} = \boldsymbol {\beta} _ {1}, \quad \mathbf {g} _ {2} = (1 - \boldsymbol {\beta} _ {1}) \cdot \boldsymbol {\beta} _ {2}, \quad \mathbf {g} _ {3} = (1 - \boldsymbol {\beta} _ {1}) \cdot (1 - \boldsymbol {\beta} _ {2}) \tag {5}
+$$
+
+$$
+\beta_ {1} = \operatorname {S i g m o i d} \left(\mathbf {H W} _ {G} ^ {1}\right), \quad \beta_ {2} = \operatorname {S i g m o i d} \left(\mathbf {H W} _ {G} ^ {2}\right)
+$$
+
+Here,  $\mathcal{M}$  denotes the hierarchical mask set.  $\mathrm{MHA}^D (\cdot ,\cdot)$  denotes our proposed dual attention computation scheme.  $\mathbf{W}_G^1,\mathbf{W}_G^2\in \mathbb{R}^{d\times 1}$  are learnable gating parameters for the first- and secondlevel expert selection. The sigmoid function constrains the gating values  $\beta_{1}$  and  $\beta_{2}$  within [0, 1]. To emphasize the empirical importance of local interactions, both  $\mathbf{W}_G^1$  and  $\mathbf{W}_G^2$  are initialized as zero vectors, yielding initial routing weights of [0.5, 0.25, 0.25] for each node, which prioritizes local attention in the early training stage. Inspired by the observation in Table 1 that all interaction levels contribute significantly to the classification, we aggregate outputs from all experts using the learned routing weights  $\mathbf{g}_1,\mathbf{g}_2$  ,and  $\mathbf{g}_3$  , without applying top-  $k$  selection.
+
+# 4.4 Dual attention computation scheme
+
+Finally, we introduce the dual attention computation scheme to enhance computational efficiency. While the irregularity of graph masks hinders the application of efficient attention variants[8, 9, 39], their inherent sparsity enables a new optimization route—sparse attention computation[34]. Unlike standard dense attention (Equation 1), which constructs the full attention matrix  $\hat{\mathbf{A}}$  before applying the binary mask  $\mathbf{M}$ , sparse attention computes attention scores only for valid node pairs  $(u,v)$  where  $\mathbf{M}_{u,v} = 1$ . A detailed analysis of the sparse attention implementation in Algorithm 2 reveals a space complexity of  $O(6mHd_h)$ , where  $m$  is the number of non-zero entries in  $\mathbf{M}$ ,  $H$  is the number of attention heads, and  $d_h$  is hidden dimension of each head. Although this method significantly reduces the complexity from  $O(N^2)$  to  $O(m)$ , the constant factor  $6Hd_h$  remains non-negligible in practice.
+
+To further improve efficiency, we propose a dual attention computation scheme that dynamically switches between dense and sparse computation based on the local sparsity of the attention mask. Specifically, we partition the attention mask  $\mathbf{M}$  into  $K$  disjoint regions  $\mathcal{R} = \{\mathcal{R}_i\}_{i=1}^K$ , each  $\mathcal{R}_i$  is defined by a query set  $\mathcal{Q}_i$  and the corresponding key set  $\mathcal{K}_i = \cup_{u \in \mathcal{Q}_i} \{v : \mathbf{M}_{u,v} = 1\}$ . For each region, the optimal computation mode is selected according to Proposition 4.2.
+
+Table 2: Node classification results  $\left( {\% }_{\pm \sigma }\right)$  . ROC-AUC for Minesweeper; accuracy for the rest.  
+
+<table><tr><td rowspan="2">Models</td><td colspan="9">Datasets</td></tr><tr><td>Cora</td><td>Citeseer</td><td>Pubmed</td><td>Computer</td><td>Photo</td><td>Squirrel</td><td>Chameleon</td><td>Minesweeper</td><td>Arxiv</td></tr><tr><td>GCN</td><td>86.53±1.61</td><td>75.97±1.93</td><td>88.51±0.28</td><td>89.83±0.64</td><td>93.07±0.48</td><td>42.19±2.10</td><td>42.87±2.78</td><td>93.47±0.47</td><td>72.55±0.21</td></tr><tr><td>GAT</td><td>86.53±1.27</td><td>74.31±1.25</td><td>87.42±0.43</td><td>90.45±0.87</td><td>93.79±0.28</td><td>36.59±1.88</td><td>41.52±4.78</td><td>93.25±0.42</td><td>72.10±0.33</td></tr><tr><td>SAGE</td><td>87.62±1.73</td><td>74.79±1.59</td><td>89.20±0.53</td><td>90.14±0.73</td><td>94.27±0.64</td><td>36.16±1.46</td><td>42.06±3.01</td><td>93.64±0.39</td><td>72.32±0.13</td></tr><tr><td>GCN*</td><td>87.74±1.66</td><td>76.83±1.94</td><td>89.48±0.56</td><td>91.70±0.53</td><td>95.10±0.60</td><td>42.59±2.14</td><td>43.77±2.47</td><td>97.39±0.30</td><td>73.18±0.21</td></tr><tr><td>GAT*</td><td>87.59±1.33</td><td>76.42±1.81</td><td>89.12±0.47</td><td>91.70±0.73</td><td>95.73±0.63</td><td>38.38±1.25</td><td>42.69±5.11</td><td>97.35±1.12</td><td>72.67±0.14</td></tr><tr><td>SAGE*</td><td>87.71±1.91</td><td>75.99±1.21</td><td>89.49±0.29</td><td>91.52±0.60</td><td>95.37±0.89</td><td>39.28±2.90</td><td>43.23±2.78</td><td>97.39±0.80</td><td>72.75±0.14</td></tr><tr><td>GPRGNN</td><td>88.21±1.29</td><td>77.02±1.81</td><td>88.49±0.44</td><td>90.70±0.53</td><td>94.80±0.37</td><td>36.80±1.71</td><td>41.26±4.22</td><td>89.05±0.43</td><td>68.44±0.21</td></tr><tr><td>FAGCN</td><td>88.36±1.50</td><td>76.78±1.29</td><td>89.31±0.58</td><td>90.07±0.72</td><td>95.23±0.38</td><td>40.90±2.04</td><td>42.78±3.51</td><td>89.95±0.62</td><td>66.83±0.20</td></tr><tr><td>NAGphormer</td><td>87.68±1.80</td><td>76.21±2.72</td><td>89.35±0.20</td><td>91.22±0.65</td><td>95.12±0.36</td><td>38.67±1.47</td><td>41.61±2.64</td><td>90.08±0.46</td><td>71.38±0.20</td></tr><tr><td>Exphormer</td><td>87.03±1.70</td><td>76.18±1.50</td><td>88.55±0.47</td><td>90.76±0.80</td><td>95.19±0.49</td><td>37.20±2.56</td><td>40.27±1.63</td><td>95.32±0.93</td><td>72.24±0.21</td></tr><tr><td>SGFormer</td><td>87.86±1.12</td><td>75.85±2.04</td><td>88.75±0.41</td><td>91.52±0.68</td><td>95.10±0.32</td><td>38.60±0.95</td><td>43.77±3.02</td><td>91.59±0.28</td><td>72.44±0.28</td></tr><tr><td>CoBFormer</td><td>88.15±1.47</td><td>77.05±1.69</td><td>88.50±0.59</td><td>91.64±0.41</td><td>95.58±0.55</td><td>39.03±1.35</td><td>43.50±1.35</td><td>95.63±0.52</td><td>73.17±0.18</td></tr><tr><td>PolyNormer</td><td>87.83±1.94</td><td>76.93±2.16</td><td>89.48±0.43</td><td>91.85±0.57</td><td>95.44±0.71</td><td>39.32±1.45</td><td>44.30±2.04</td><td>96.98±0.46</td><td>73.27±0.38</td></tr><tr><td>Mowst</td><td>87.92±1.17</td><td>76.52±1.41</td><td>88.71±0.25</td><td>91.32±0.50</td><td>93.70±0.32</td><td>41.72±2.33</td><td>44.30±2.57</td><td>93.00±0.68</td><td>73.03±0.29</td></tr><tr><td>GCN-MoE</td><td>86.17±1.11</td><td>75.20±1.70</td><td>88.80±0.24</td><td>88.75±0.58</td><td>93.19±0.32</td><td>43.02±2.14</td><td>44.57±2.00</td><td>92.63±0.40</td><td>73.16±0.21</td></tr><tr><td>M3Dphormer</td><td>88.48±1.94</td><td>77.53±1.56</td><td>89.96±0.49</td><td>92.09±0.46</td><td>95.91±0.68</td><td>44.34±1.94</td><td>47.09±4.05</td><td>98.27±0.20</td><td>73.54±0.30</td></tr></table>
+
+Proposition 4.2. Let  $\kappa_{\mathcal{R}_i}$  denote the non-zero rate within region  $\mathcal{R}_i$ . The sparse attention scheme is more efficient than the dense scheme when  $\kappa_{R_i} < \frac{1}{3d_h}$ .
+
+The proof is provided in Appendix A.5. This result guides the selection of sparse computation when local sparsity is high and dense computation when the region becomes sufficiently dense. Based on this guidance, we formulate the dual attention computation scheme as:
+
+$$
+\mathrm {M H A} ^ {D} (\mathbf {H}, \mathbf {M}) = \operatorname {C o m b} _ {i} ^ {K} \left(\operatorname {S e l e c t M o d e} \left(\mathcal {R} _ {i}\right) \left(\mathbf {H}, \mathcal {R} _ {i}\right)\right) \tag {6}
+$$
+
+Here,  $\operatorname{SelectMode}(\mathcal{R}_i)$  denotes the selection between sparse and dense computation modes for region  $\mathcal{R}_i$ , and  $\operatorname{Comb}_i^K (\cdot)$  aggregates the attention outputs from all  $K$  partitioned regions.
+
+# 5 Experiments
+
+Experiment setups. We evaluate  $\mathbf{M}^3$  Dphormer on nine datasets, including six homophilic graphs (Cora, CiteSeer, Pubmed [45], Computer, Photo [33], and Ogbn-Arxiv [19]) and three heterophilic graphs (Squirrel, Chameleon, and Minesweeper [31]). Dataset statistics and splitting protocols are detailed in Appendix D. We select 15 baselines spanning five categories: 1) Classic GNNs: GCN [23], GAT [38], and GraphSAGE [16]. 2) Enhanced Classic GNNs: GCN*, GAT*, and SAGE*. 3) Advanced GNNs: GPRGNN[7] and FAGCN [3]. 4) SOTA Graph Transformers: NAGphormer [5], Exphormer [34], SGFormer [43], CoBformer [44], and PolyNormer [11]; 5) MoE-based GNNs: Mowst [47] and GCN-MoE [40]. Descriptions of the baselines are provided in Appendix E, and implementation details can be found in Appendix F.
+
+Node classification results. Table 2 presents the node classification results. Key observations include: 1)  $\mathbf{M}^3$  Dphormer consistently outperforms all baselines across 9 datasets, highlighting its superior interaction modeling capacity. 2) Compared to traditional and MoE-based GNNs,  $\mathbf{M}^3$  Dphormer demonstrates clear advantages by comprehensively capturing hierarchical interactions. 3) Compared to GT baselines,  $\mathbf{M}^3$  Dphormer shows notable improvements, verifying both the benefit of comprehensive interaction modeling and the effectiveness of the bi-level attention expert routing mechanism in adaptively integrating multi-level information.
+
+Ablation studies. We perform ablation studies to evaluate  $\mathbf{M}^3$  Dphormer in terms of effectiveness and efficiency. Firstly, we construct five ablated variants by: 1) Removing individual experts; 2) Disabling
+
+Table 3: Node classification results of various  ${\mathrm{M}}^{3}$  Dphormer variants  
+
+<table><tr><td></td><td>Cora</td><td>Citeseer</td><td>Pubmed</td><td>Computer</td><td>Photo</td><td>Squirrel</td><td>Chameleon</td><td>Minesweeper</td><td>Ogbn-Arxiv</td></tr><tr><td>Full Model</td><td>88.48±1.94</td><td>77.53±1.56</td><td>89.96±0.49</td><td>92.09±0.46</td><td>95.91±0.68</td><td>44.34±1.94</td><td>47.09±4.05</td><td>98.27±0.20</td><td>73.54±0.30</td></tr><tr><td>W/O Local</td><td>82.84±2.15</td><td>74.09±1.46</td><td>88.75±0.48</td><td>88.99±0.94</td><td>93.83±0.44</td><td>39.61±1.51</td><td>42.60±4.41</td><td>57.55±0.78</td><td>67.24±0.23</td></tr><tr><td>W/O Cluster</td><td>87.83±1.83</td><td>76.73±2.00</td><td>89.69±0.46</td><td>91.59±0.70</td><td>95.22±0.53</td><td>42.48±2.13</td><td>44.93±3.21</td><td>98.03±0.41</td><td>73.40±0.23</td></tr><tr><td>W/O Global</td><td>87.95±2.03</td><td>76.33±1.84</td><td>89.76±0.34</td><td>91.89±0.50</td><td>95.62±0.59</td><td>41.58±1.60</td><td>45.47±5.04</td><td>98.02±0.24</td><td>73.41±0.11</td></tr><tr><td>W/O Route</td><td>87.65±2.39</td><td>76.25±0.91</td><td>89.48±0.46</td><td>91.76±0.71</td><td>95.59±0.87</td><td>42.05±1.21</td><td>43.95±1.98</td><td>97.72±0.17</td><td>73.31±0.19</td></tr><tr><td>W/O Bi-Level</td><td>87.74±2.21</td><td>77.12±1.59</td><td>89.79±0.25</td><td>91.68±0.37</td><td>95.70±0.50</td><td>42.41±1.96</td><td>44.39±4.44</td><td>97.84±0.34</td><td>73.50±0.17</td></tr></table>
+
+![](images/644a34f3eb751252d7aee2465f97d965e0216b12cb76485ed7f25953f410324e.jpg)  
+Memory Usage Across Datasets with Different Computation Schemes  
+Figure 3: Memory usage of different schemes.
+
+the attention expert routing mechanism; 3) Replacing the bi-level attention routing mechanism with a single-level gating scheme. Results in Table 3 show that: 1) Removing any individual expert consistently degrades performance, underscoring the necessity of comprehensively modeling hierarchical interactions. 2) Disabling the routing mechanism leads to substantial degradation, suggesting that simple aggregation is insufficient for effectively integrating multi-level interactions. 3) The performance gap between the single-level routing variant and  $\mathbf{M}^3\mathrm{D}$  phormer demonstrates the advantage of the proposed bi-level attention expert routing mechanism. Then, we report the GPU memory usage of  $\mathbf{M}^3\mathrm{D}$  phormer and its two variants employing sparse and dense computation schemes in Figure 3. As shown, the dense scheme incurs the highest memory consumption and leads to out-of-memory (OOM) errors on four datasets. While the sparse scheme substantially reduces memory usage, it still fails to run on Ogbn-Arxiv. In contrast, our dual attention scheme achieves superior memory efficiency and successfully scales to all evaluated graphs.
+
+![](images/67e12692ca5859a6d40e1b24f38504cb733631039682c59780f26155dc7e5f88.jpg)
+
+![](images/cbc5ab2e305beaf66e2dc27eeec505316b879b209b0fe8bb8c97c0ee8860af6d.jpg)
+
+![](images/841c57a31fac9a891e9b0b63c351b6189797f306daa6d65e7c526f4d08835434.jpg)
+
+![](images/96d6da9fc0263e211438c20e6f0dbbffc0a458a28f19b0f41867426c88bee47e.jpg)
+
+![](images/185cc5bb16ece497f6d22b7d7975509261b874d26b784ea603a9bbb9d53dbde1.jpg)
+
+![](images/a1e9b2887306ba8900199caa9ef993298cb9ca2145398922aebe5310175bb9ec.jpg)  
+Figure 4: Test accuracy vs.  $P$
+
+![](images/064e03da5242ffc5d33adceddc3dfd1495aaf179226042081e1d3d32c49df104.jpg)
+
+![](images/178c48012bc9d3f289ef71fe03aea9ecb8cd0d2dc31182638b046b9ecc55ae30.jpg)
+
+Parameter analysis. The only key hyperparameter in  $\mathbf{M}^3\mathrm{D}$  phormer is the number of clusters  $P$  which affects the quality of the cluster mask  $\mathbf{M}^{c4}$ . For each dataset, we select an appropriate range of  $P$  values according to its size. Figure 4 presents the model's performance under varying  $P$ . Overall,  $\mathbf{M}^3\mathrm{D}$  phormer demonstrates strong robustness to the choice of  $P$  on most datasets. An exception is Chameleon—a small graph with only 890 nodes—where variations in  $P$  substantially impact the quality of partitioning, resulting in more significant performance fluctuations.
+
+Visualization. We plot the accuracy and loss curves of  $\mathrm{M}^3\mathrm{D}$  phormer and GCN*[23, 26] in Figure 5. Across the training, validation, and test sets,  $\mathrm{M}^3\mathrm{D}$  phormer consistently achieves faster convergence and higher accuracy than GCN* during the training process, demonstrating the effectiveness of our method. A similar trend is also observed in the comparison with PolyNormer [11] in Appendix G.4. A visualization of the learned gate weights is provided in Appendix G.5.
+
+![](images/0a596c3f0af9deb518a8c8e3cf40ee1fe5f7aa768c1a25f2bf4ce6eee7dde95e.jpg)  
+Figure 5: Comparison of accuracy and loss curves between  $\mathbf{M}^3$  DFormer and GCN* on Ogbn-Arxiv.
+
+![](images/17159e6eaaaece665d051f3bb1eb3b3e8de8e29c78549ff8210c14fcd599cce0.jpg)
+
+# 6 Related Work
+
+Graph neural networks. Classic GNNs, such as GCN [23], SAGE [16], and GAT [38], rely on message-passing mechanisms that recursively aggregate information from local neighbors at each layer. To move beyond the widely adopted homophily assumption—that nodes of the same class are more likely to be connected [28]—advanced GNNs such as GPRGNN [7] and FAGCN [3] have been proposed to improve performance on heterophilic graphs. More recently, a benchmark study demonstrated that carefully tuning the hyperparameters of classic GNNs and enhancing them with advanced training techniques—such as residual connections [17] and normalization methods [48, 2, 21]—can lead to substantial improvements in node classification performance. While GNNs have proven effective in many scenarios, they still suffer from a fundamental limitation: message passing primarily captures local interactions, often neglecting informative signals from broader, long-range dependencies.
+
+Graph transformers. Recently, Graph Transformers have emerged as a promising paradigm for graph representation learning. By leveraging the multi-head self-attention mechanism originally introduced in Transformer [37], they aim to adaptively model diverse and complex interactions from a broader perspective. A primary line of research treats the entire graph as fully connected and computes attention scores between all node pairs [46, 32, 43, 11]. However, a recent study has revealed the over-globalizing problem in such methods, which may lead to a significant decrease in performance [44]. Another line of work constructs a token sequence for each node based on the graph structure and feeds these sequences into a Transformer to learn node representations [49, 5, 15, 6, 41]. These methods often rely on expert-designed tokenization strategies, which tend to capture local information while overlooking larger-scale interactions. In addition, several recent approaches focus on modeling cluster-level interactions, which have shown promising results in capturing mid-level structural patterns for graph representation learning [18, 20, 44].
+
+# 7 Conclusion
+
+In this paper, we propose a unified hierarchical mask framework for Graph Transformers. A fundamental design principle and two core challenges are identified under this framework. We then introduce M $^3$ Dphormer, a novel Mixture-of-Experts based Graph Transformer with Multi-Level Masking and Dual Attention Computation, designed to efficiently, comprehensively and adaptively capture hierarchical interactions. Extensive experiments demonstrate its effectiveness.
+
+Limitations and broader impacts. In this paper, we focus our theoretical and empirical analyses on the node classification task under the proposed unified hierarchical mask framework, as it is a fundamental and extensively studied problem in the graph learning community. Extending our theoretical insights to graph-level and edge-level tasks represents a promising direction for future work. Apart from this, we do not expect any direct negative societal impacts.
+
+# Acknowledgments and Disclosure of Funding
+
+This work is supported in part by the National Natural Science Foundation of China (No. U20B2045, 62322203, 62172052, 62192784, U22B2038).
+
+# References
+
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+
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+
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+Justification: In the abstract and introduction, we highlight that existing Graph Transformers can be unified under a hierarchical masking framework, which simplifies model design to the construction of appropriate masks and offers a principled foundation grounded in theoretical analysis. Building on this framework, we further propose a novel Graph Transformer that achieves state-of-the-art performance on node classification tasks.
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+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
+
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components
+
+Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Theorem Proofs
+
+We begin by decomposing Theorem 3.1 into two theorems and one proposition, and then prove them individually.
+
+Theorem A.1. The updated representation of node  $u$  follows a Gaussian distribution:
+
+$$
+\hat {z} _ {u} \sim \mathcal {N} \left(\sum_ {i = 1} ^ {| \mathcal {C} |} k \rho_ {i} \alpha_ {i} \boldsymbol {\mu} _ {i}, \sum_ {i = 1} ^ {| \mathcal {C} |} k \rho_ {i} \alpha_ {i} ^ {2} \sigma_ {i} ^ {2} \mathbf {I}\right) \tag {7}
+$$
+
+Accordingly, the probability that node  $u$  is correctly classified by a similarity-based classifier is:
+
+$$
+P \left(\hat {\boldsymbol {z}} _ {u} ^ {\top} \boldsymbol {\mu} _ {c} \geq \delta_ {c}\right) = 1 - \Phi \left(\frac {\delta_ {c} - k \rho_ {c} \alpha_ {c}}{\sqrt {\sum_ {i = 1} ^ {| C |} k \rho_ {i} \alpha_ {i} ^ {2} \sigma_ {i} ^ {2}}}\right) \tag {8}
+$$
+
+where  $\Phi (\cdot)$  denotes the cumulative distribution function (CDF) of the standard normal distribution.
+
+Theorem A.2. Assuming the classifier is well trained, i.e.,  $\delta_c - k\rho_c\alpha_c \leq 0$ , and the attention weights satisfy  $0 \leq \alpha_{c'} \leq \frac{1}{k} \leq \alpha_c \leq \frac{1}{k\rho_c}$ , then the classification probability is bounded as:
+
+$$
+1 - \Phi \left(\frac {\delta_ {c} - k \rho_ {c} \alpha_ {c}}{\sqrt {k \rho_ {c} \alpha_ {c} ^ {2} \sigma_ {c} ^ {2} + \frac {1 - \rho_ {c}}{k} \cdot \sigma_ {m} ^ {2}}}\right) \leq P (\hat {\boldsymbol {z}} _ {u} ^ {\top} \boldsymbol {\mu} _ {c} \geq \delta_ {c}) \leq 1 - \Phi \left(\frac {\delta_ {c} - k \rho_ {c} \alpha_ {c}}{\sqrt {k \rho_ {c} \alpha_ {c} ^ {2} \sigma_ {c} ^ {2}}}\right), \tag {9}
+$$
+
+where  $\sigma_{m}^{2} = \max_{c^{\prime}\neq c}\sigma_{c^{\prime}}^{2}$  is the largest representation variance among non-target classes.
+
+Proposition A.3. In Equation 9, both the lower and upper bounds are monotonically increasing with respect to  $k$ ,  $\rho_c$ , and  $\alpha_c$ , and decreasing with respect to the set of variances  $\{\sigma_i:1\leq i\leq |\mathcal{Y}|\}$ .
+
+# A.1 Proof of Theorem A.1
+
+Proof. Let node  $u$  have ground-truth label  $c$ , and denote its receptive field by the mask vector  $\mathbf{M}_{u,:}$  which includes  $k$  nodes in total. Let  $\rho_{i}$  be the proportion of class-  $i$  nodes in the receptive field, and let  $\alpha_{i}$  denote the average attention weight assigned to these nodes.
+
+The updated representation of node  $u$  is given by:
+
+$$
+\hat {\boldsymbol {z}} _ {u} = \sum_ {i = 1} ^ {k} \alpha_ {i} \boldsymbol {z} _ {i} \tag {10}
+$$
+
+Each node representation  $z_{i}$  with class label  $c'$  is generated using the reparameterization trick:
+
+$$
+\boldsymbol {z} _ {i} ^ {(c ^ {\prime})} = \boldsymbol {\mu} _ {c ^ {\prime}} + \sigma_ {c ^ {\prime}} \cdot \boldsymbol {\xi} _ {i}, \quad \text {w h e r e} \boldsymbol {\xi} _ {i} \sim \mathcal {N} (\mathbf {0}, \mathbf {I}) \tag {11}
+$$
+
+Regrouping the neighbors by class, we rewrite the update as:
+
+$$
+\hat {\boldsymbol {z}} _ {u} = \sum_ {i = 1} ^ {| \mathcal {Y} |} \sum_ {j = 1} ^ {k \rho_ {i}} \alpha_ {i} \boldsymbol {z} _ {j} ^ {(i)} = \sum_ {i = 1} ^ {| \mathcal {Y} |} k \rho_ {i} \alpha_ {i} \boldsymbol {\mu} _ {i} + \sum_ {i = 1} ^ {| \mathcal {Y} |} \sigma_ {i} \alpha_ {i} \sum_ {j = 1} ^ {k \rho_ {i}} \boldsymbol {\xi} _ {j} \tag {12}
+$$
+
+Since the sum of i.i.d. standard Gaussian vectors satisfies:
+
+$$
+\sum_ {j = 1} ^ {k \rho_ {i}} \boldsymbol {\xi} _ {j} \sim \mathcal {N} (\mathbf {0}, k \rho_ {i} \mathbf {I}), \tag {13}
+$$
+
+we have:
+
+$$
+\sigma_ {i} \alpha_ {i} \sum_ {j = 1} ^ {k \rho_ {i}} \boldsymbol {\xi} _ {j} \sim \mathcal {N} (\mathbf {0}, \sigma_ {i} ^ {2} \alpha_ {i} ^ {2} k \rho_ {i} \mathbf {I}) \tag {14}
+$$
+
+Hence, the updated representation  $\hat{z}_u$  follows a multivariate Gaussian distribution:
+
+$$
+\hat {z} _ {u} \sim \mathcal {N} \left(\sum_ {i = 1} ^ {| \mathcal {Y} |} k \rho_ {i} \alpha_ {i} \boldsymbol {\mu} _ {i}, \sum_ {i = 1} ^ {| \mathcal {Y} |} \sigma_ {i} ^ {2} \alpha_ {i} ^ {2} k \rho_ {i} \mathbf {I}\right) \tag {15}
+$$
+
+Define the total variance scalar:
+
+$$
+\zeta_ {u} := \sqrt {\sum_ {i = 1} ^ {| \mathcal {Y} |} \sigma_ {i} ^ {2} \alpha_ {i} ^ {2} k \rho_ {i}} \tag {16}
+$$
+
+Using the reparameterization trick again, we can rewrite  $\hat{z}_u$  as:
+
+$$
+\hat {\boldsymbol {z}} _ {u} = \sum_ {i = 1} ^ {| \mathcal {Y} |} k \rho_ {i} \alpha_ {i} \boldsymbol {\mu} _ {i} + \zeta_ {u} \cdot \hat {\boldsymbol {\xi}}, \quad \text {w h e r e} \hat {\boldsymbol {\xi}} \sim \mathcal {N} (\mathbf {0}, \mathbf {I}) \tag {17}
+$$
+
+Now consider the inner product with the class prototype  $\pmb{\mu}_{c}$ :
+
+$$
+\hat {\boldsymbol {z}} _ {u} ^ {\top} \boldsymbol {\mu} _ {c} = \left(\sum_ {i = 1} ^ {| \mathcal {Y} |} k \rho_ {i} \alpha_ {i} \boldsymbol {\mu} _ {i}\right) ^ {\top} \boldsymbol {\mu} _ {c} + \left(\zeta_ {u} \cdot \hat {\boldsymbol {\xi}}\right) ^ {\top} \boldsymbol {\mu} _ {c} \tag {18}
+$$
+
+By orthogonality of class means  $(\pmb{\mu}_i^\top \pmb{\mu}_c = 1$  if  $i = c$ , and 0 otherwise), the first term simplifies to  $k\rho_c\alpha_c$ . Since  $\hat{\xi} \sim \mathcal{N}(\mathbf{0},\mathbf{I})$ , the second term becomes a linear combination of i.i.d. standard Gaussians:
+
+$$
+\hat {\boldsymbol {z}} _ {u} ^ {\top} \boldsymbol {\mu} _ {c} = k \rho_ {c} \alpha_ {c} + \zeta_ {u} \sum_ {i = 1} ^ {d} \mu_ {c, i} \hat {\xi} _ {i}, \quad \hat {\xi} _ {i} \sim \mathcal {N} (0, 1) \tag {19}
+$$
+
+Because  $\| \pmb{\mu}_c\| _2^2 = 1$ , the sum  $\sum_{i = 1}^{d}\mu_{c,i}\hat{\xi}_i$  is distributed as  $\mathcal{N}(0,1)$ . Thus:
+
+$$
+\hat {\boldsymbol {z}} _ {u} ^ {\top} \boldsymbol {\mu} _ {c} \sim \mathcal {N} \left(k \rho_ {c} \alpha_ {c}, \zeta_ {u} ^ {2}\right) \tag {20}
+$$
+
+To compute the classification probability, we consider a similarity-based classifier that predicts correctly if:
+
+$$
+\hat {\boldsymbol {z}} _ {u} ^ {\top} \boldsymbol {\mu} _ {c} \geq \delta_ {c} \tag {21}
+$$
+
+Since  $\hat{z}_u^\top \mu_c$  is Gaussian distributed, we apply the cumulative distribution function of the normal distribution. For  $X\sim \mathcal{N}(\mu ,\sigma^2)$ , we have:
+
+$$
+P (X \geq a) = 1 - \Phi \left(\frac {a - \mu}{\sigma}\right) \tag {22}
+$$
+
+Applying this to our case:
+
+$$
+P \left(\hat {\boldsymbol {z}} _ {u} ^ {\top} \boldsymbol {\mu} _ {c} \geq \delta_ {c}\right) = 1 - \Phi \left(\frac {\delta_ {c} - k \rho_ {c} \alpha_ {c}}{\zeta_ {u}}\right) = 1 - \Phi \left(\frac {\delta_ {c} - k \rho_ {c} \alpha_ {c}}{\sqrt {\sum_ {i = 1} ^ {| \mathcal {Y} |} \sigma_ {i} ^ {2} \alpha_ {i} ^ {2} k \rho_ {i}}}\right) \tag {23}
+$$
+
+which concludes the proof.
+
+![](images/26d3cad09eb03d0afb4e2b8fa3be32a740460db2db5749f8dd8df257843f11b6.jpg)
+
+# A.2 Proof of Theorem A.2
+
+Proof. From Theorem A.1, we know:
+
+$$
+\hat {\boldsymbol {z}} _ {u} ^ {\top} \boldsymbol {\mu} _ {c} \sim \mathcal {N} \left(k \rho_ {c} \alpha_ {c}, \sum_ {i = 1} ^ {| \mathcal {Y} |} k \rho_ {i} \alpha_ {i} ^ {2} \sigma_ {i} ^ {2}\right) \tag {24}
+$$
+
+and the classification probability is:
+
+$$
+P \left(\hat {\boldsymbol {z}} _ {u} ^ {\top} \boldsymbol {\mu} _ {c} \geq \delta_ {c}\right) = 1 - \Phi \left(\frac {\delta_ {c} - k \rho_ {c} \alpha_ {c}}{\sqrt {\sum_ {i = 1} ^ {| y |} k \rho_ {i} \alpha_ {i} ^ {2} \sigma_ {i} ^ {2}}}\right) \tag {25}
+$$
+
+Let us denote the variance as:
+
+$$
+\operatorname {V a r} := \sum_ {i = 1} ^ {| \mathcal {Y} |} k \rho_ {i} \alpha_ {i} ^ {2} \sigma_ {i} ^ {2} = k \rho_ {c} \alpha_ {c} ^ {2} \sigma_ {c} ^ {2} + \sum_ {i \neq c} k \rho_ {i} \alpha_ {i} ^ {2} \sigma_ {i} ^ {2} \tag {26}
+$$
+
+Let  $\sigma_{m}^{2} := \max_{i \neq c} \sigma_{i}^{2}$ . Since  $\alpha_{i} \leq \frac{1}{k}$  for  $i \neq c$ , we have  $\alpha_{i}^{2} \leq \frac{1}{k^{2}}$ , and hence:
+
+$$
+\sum_ {i \neq c} k \rho_ {i} \alpha_ {i} ^ {2} \sigma_ {i} ^ {2} \leq \sum_ {i \neq c} k \rho_ {i} \cdot \frac {1}{k ^ {2}} \cdot \sigma_ {m} ^ {2} = \frac {\sigma_ {m} ^ {2}}{k} \sum_ {i \neq c} \rho_ {i} = \frac {1 - \rho_ {c}}{k} \cdot \sigma_ {m} ^ {2} \tag {27}
+$$
+
+Therefore:
+
+$$
+\operatorname {V a r} \leq k \rho_ {c} \alpha_ {c} ^ {2} \sigma_ {c} ^ {2} + \frac {1 - \rho_ {c}}{k} \cdot \sigma_ {m} ^ {2} \tag {28}
+$$
+
+We now use the fact that the Gaussian CDF  $\Phi(z)$  is strictly increasing. Under the assumption that  $\delta_c - k\rho_c\alpha_c < 0$ , the numerator is negative. In this case, increasing the denominator (i.e., the variance) reduces the absolute value of  $z$ , making  $z$  less negative. As a result,  $\Phi(z)$  increases, and the classification probability  $1 - \Phi(z)$  decreases.
+
+This gives the lower bound:
+
+$$
+P \left(\hat {\boldsymbol {z}} _ {u} ^ {\top} \boldsymbol {\mu} _ {c} \geq \delta_ {c}\right) \geq 1 - \Phi \left(\frac {\delta_ {c} - k \rho_ {c} \alpha_ {c}}{\sqrt {k \rho_ {c} \alpha_ {c} ^ {2} \sigma_ {c} ^ {2} + \frac {1 - \rho_ {c}}{k} \cdot \sigma_ {m} ^ {2}}}\right) \tag {29}
+$$
+
+To obtain the upper bound, we lower bound the total variance by dropping the non-negative cross-class terms:
+
+$$
+\sum_ {i \neq c} k \rho_ {i} \alpha_ {i} ^ {2} \sigma_ {i} ^ {2} \geq 0 \tag {30}
+$$
+
+which leads to:
+
+$$
+\operatorname {V a r} \geq k \rho_ {c} \alpha_ {c} ^ {2} \sigma_ {c} ^ {2} \tag {31}
+$$
+
+Since the CDF  $\Phi(z)$  is increasing and the numerator is negative, a smaller denominator results in a more negative standardized score, and hence a larger classification probability  $1 - \Phi(z)$ . Therefore, this yields the following upper bound:
+
+$$
+P \left(\hat {\boldsymbol {z}} _ {u} ^ {\top} \boldsymbol {\mu} _ {c} \geq \delta_ {c}\right) \leq 1 - \Phi \left(\frac {\delta_ {c} - k \rho_ {c} \alpha_ {c}}{\sqrt {k \rho_ {c} \alpha_ {c} ^ {2} \sigma_ {c} ^ {2}}}\right) \tag {32}
+$$
+
+Combining both bounds gives the result.
+
+![](images/1ed12aab09d2fdda515120228e77acf940c02d8e4b63550561df9a63b245388d.jpg)
+
+# A.3 Proof of Proposition A.3
+
+Proof. We analyze the monotonicity of both the upper and lower bounds in Equation 9 by taking partial derivatives of the normalized score with respect to each variable. Let the standardized score be denoted as  $f(\cdot)$ .
+
+Upper bound:
+
+$$
+f _ {\mathrm {u p p e r}} (x) = \frac {\delta_ {c} - k \rho_ {c} \alpha_ {c}}{\sqrt {k \rho_ {c} \alpha_ {c} ^ {2} \sigma_ {c} ^ {2}}}
+$$
+
+(1) With respect to  $k$ : Let  $C_1 = \delta_c$ ,  $C_2 = \rho_c\alpha_c$ ,  $C_3 = \rho_c\alpha_c^2\sigma_c^2$ . Then:
+
+$$
+f _ {\text {u p p e r}} (k) = \frac {C _ {1} - k C _ {2}}{\sqrt {k C _ {3}}}, \quad f _ {\text {u p p e r}} ^ {\prime} (k) = \frac {- k C _ {2} C _ {3} - C _ {1} C _ {3}}{2 \left(k C _ {3}\right) ^ {3 / 2}} <   0 \tag {33}
+$$
+
+(2) With respect to  $\rho_c$ : Let  $C_2 = k\alpha_c$ ,  $C_3 = k\alpha_c^2\sigma_c^2$ . Then:
+
+$$
+f _ {\text {u p p e r}} \left(\rho_ {c}\right) = \frac {C _ {1} - \rho_ {c} C _ {2}}{\sqrt {\rho_ {c} C _ {3}}}, \quad f _ {\text {u p p e r}} ^ {\prime} \left(\rho_ {c}\right) = \frac {- \rho_ {c} C _ {2} C _ {3} - C _ {1} C _ {3}}{2 \left(\rho_ {c} C _ {3}\right) ^ {3 / 2}} <   0 \tag {34}
+$$
+
+(3) With respect to  $\alpha_{c}$ : Let  $C_2 = k\rho_c$ ,  $C_3 = k\rho_c\sigma_c^2$ . Then:
+
+$$
+f _ {\text {u p p e r}} \left(\alpha_ {c}\right) = \frac {C _ {1} - C _ {2} \alpha_ {c}}{\alpha_ {c} \sqrt {C _ {3}}}, \quad f _ {\text {u p p e r}} ^ {\prime} \left(\alpha_ {c}\right) = \frac {- C _ {1}}{\alpha_ {c} ^ {2} \sqrt {C _ {3}}} <   0 \tag {35}
+$$
+
+Lower bound:
+
+$$
+f _ {\mathrm {l o w e r}} (x) = \frac {\delta_ {c} - k \rho_ {c} \alpha_ {c}}{\sqrt {k \rho_ {c} \alpha_ {c} ^ {2} \sigma_ {c} ^ {2} + \frac {1 - \rho_ {c}}{k} \sigma_ {m} ^ {2}}}
+$$
+
+(1) With respect to  $k$ :
+
+Let  $C_1 = \delta_c$ ,  $C_2 = \rho_c\alpha_c$ ,  $C_3 = \rho_c\alpha_c^2\sigma_c^2$ , and  $C_4 = (1 - \rho_c)\sigma_m^2$ . Then the lower bound becomes:
+
+$$
+f _ {\text {l o w e r}} (k) = \frac {C _ {1} - k C _ {2}}{\sqrt {k C _ {3} + \frac {C _ {4}}{k}}} \tag {36}
+$$
+
+Let  $D(k) = \sqrt{kC_3 + \frac{C_4}{k}}$ . Then, using the quotient and chain rule, we obtain:
+
+$$
+f _ {\text {l o w e r}} ^ {\prime} (k) = \frac {- 2 C _ {2} D ^ {2} (k) - \left(C _ {3} - \frac {C _ {4}}{k ^ {2}}\right) \left(C _ {1} - k C _ {2}\right)}{2 D ^ {3} (k)} \tag {37}
+$$
+
+Now we simplify the numerator:
+
+$$
+\begin{array}{l} - 2 C _ {2} D ^ {2} (k) - \left(C _ {3} - \frac {C _ {4}}{k ^ {2}}\right) \left(C _ {1} - k C _ {2}\right) \\ = - 2 C _ {2} \left(k C _ {3} + \frac {C _ {4}}{k}\right) - \left(C _ {3} - \frac {C _ {4}}{k ^ {2}}\right) \left(C _ {1} - k C _ {2}\right) \\ = - k C _ {2} C _ {3} - \frac {3 C _ {2} C _ {4}}{k} - C _ {1} C _ {3} + \frac {C _ {1} C _ {4}}{k} \\ = - k C _ {2} C _ {3} - C _ {1} C _ {3} + \frac {C _ {4} \left(C _ {1} - 3 k C _ {2}\right)}{k} \tag {38} \\ \end{array}
+$$
+
+Under the assumption that the classifier is well-trained, we have:
+
+$$
+C _ {1} - 3 k C _ {2} = \delta_ {c} - 3 k \rho_ {c} \alpha_ {c} <   \delta_ {c} - k \rho_ {c} \alpha_ {c} <   0
+$$
+
+Hence, each term in Equation (38) is negative, and the overall numerator is negative. Since the denominator  $2D^3(k) > 0$ , we conclude:
+
+$$
+f _ {\text {l o w e r}} ^ {\prime} (k) <   0 \tag {39}
+$$
+
+which means the normalized score decreases with  $k$ , and thus the classification probability increases.
+
+(2) With respect to  $\rho_c$ :
+
+Let  $C_1 = \delta_c$ ,  $C_2 = k\alpha_c$ ,  $C_3 = k\alpha_c^2\sigma_c^2$ , and  $C_4 = \sigma_m^2$ . Then the lower bound becomes:
+
+$$
+f _ {\text {l o w e r}} \left(\rho_ {c}\right) = \frac {C _ {1} - C _ {2} \rho_ {c}}{\sqrt {C _ {3} \rho_ {c} + \frac {C _ {4} \left(1 - \rho_ {c}\right)}{k}}} \tag {40}
+$$
+
+Let  $D(\rho_c) \coloneqq \sqrt{C_3\rho_c + \frac{C_4(1 - \rho_c)}{k}}$ . Then using the chain rule, the derivative is:
+
+$$
+f _ {\text {l o w e r}} ^ {\prime} \left(\rho_ {c}\right) = \frac {- 2 C _ {2} D ^ {2} \left(\rho_ {c}\right) - \left(C _ {3} - \frac {C _ {4}}{k}\right) \left(C _ {1} - C _ {2} \rho_ {c}\right)}{2 D ^ {3} \left(\rho_ {c}\right)} \tag {41}
+$$
+
+We now simplify the numerator:
+
+$$
+\begin{array}{l} - 2 C _ {2} D ^ {2} \left(\rho_ {c}\right) - \left(C _ {3} - \frac {C _ {4}}{k}\right) \left(C _ {1} - C _ {2} \rho_ {c}\right) \\ = - 2 C _ {2} \left(C _ {3} \rho_ {c} + \frac {C _ {4} (1 - \rho_ {c})}{k}\right) - \left(C _ {3} - \frac {C _ {4}}{k}\right) \left(C _ {1} - C _ {2} \rho_ {c}\right) \\ = - 2 C _ {2} C _ {3} \rho_ {c} - \frac {2 C _ {2} C _ {4} (1 - \rho_ {c})}{k} - C _ {3} C _ {1} + \frac {C _ {4} C _ {1}}{k} + C _ {2} \rho_ {c} C _ {3} - \frac {C _ {2} \rho_ {c} C _ {4}}{k} \\ = - C _ {2} C _ {3} \rho_ {c} - C _ {1} C _ {3} - \frac {2 C _ {2} C _ {4}}{k} + \frac {C _ {2} \rho_ {c} C _ {4}}{k} + \frac {C _ {1} C _ {4}}{k} \tag {42} \\ \end{array}
+$$
+
+Hence the full numerator is:
+
+$$
+- C _ {2} C _ {3} \rho_ {c} - C _ {1} C _ {3} + \frac {C _ {4} \left(C _ {2} \rho_ {c} + C _ {1} - 2 C _ {2}\right)}{k} \tag {43}
+$$
+
+We now verify its sign. Under the assumption that the classifier is well-trained, i.e.,
+
+$$
+C _ {1} - C _ {2} \rho_ {c} = \delta_ {c} - k \rho_ {c} \alpha_ {c} <   0 \Rightarrow C _ {1} <   C _ {2} \rho_ {c}
+$$
+
+which implies:
+
+$$
+C _ {1} + C _ {2} \rho_ {c} <   2 C _ {2} \rho_ {c} <   2 C _ {2} \Rightarrow C _ {1} + C _ {2} \rho_ {c} - 2 C _ {2} <   0
+$$
+
+Thus the entire numerator is negative, and since  $D(\rho_c) > 0$ , we conclude:
+
+$$
+f _ {\text {l o w e r}} ^ {\prime} \left(\rho_ {c}\right) <   0 \tag {44}
+$$
+
+That is, the normalized score decreases as  $\rho_{c}$  increases, and hence the classification probability increases.
+
+# (3) With respect to  $\alpha_{c}$ :
+
+Let  $C_1 = \delta_c$ ,  $C_2 = k\rho_c$ ,  $C_3 = k\rho_c\sigma_c^2$ , and  $C_4 = \frac{1 - \rho_c}{k}\sigma_m^2$ . Then the lower bound becomes:
+
+$$
+f _ {\text {l o w e r}} \left(\alpha_ {c}\right) = \frac {C _ {1} - C _ {2} \alpha_ {c}}{\sqrt {C _ {3} \alpha_ {c} ^ {2} + C _ {4}}} \tag {45}
+$$
+
+Let  $D(\alpha_c) \coloneqq \sqrt{C_3\alpha_c^2 + C_4}$ . Applying the quotient and chain rule, we obtain:
+
+$$
+f _ {\text {l o w e r}} ^ {\prime} \left(\alpha_ {c}\right) = \frac {- C _ {2} D ^ {2} \left(\alpha_ {c}\right) - C _ {3} \alpha_ {c} \left(C _ {1} - C _ {2} \alpha_ {c}\right)}{D ^ {3} \left(\alpha_ {c}\right)} \tag {46}
+$$
+
+We now expand the numerator:
+
+$$
+\begin{array}{l} - 2 C _ {2} \left(C _ {3} \alpha_ {c} ^ {2} + C _ {4}\right) - C _ {3} \alpha_ {c} \left(C _ {1} - C _ {2} \alpha_ {c}\right) \\ = - 2 C _ {2} C _ {3} \alpha_ {c} ^ {2} - 2 C _ {2} C _ {4} - C _ {1} C _ {3} \alpha_ {c} + C _ {2} C _ {3} \alpha_ {c} ^ {2} \\ = - C _ {2} C _ {3} \alpha_ {c} ^ {2} - 2 C _ {2} C _ {4} - C _ {1} C _ {3} \alpha_ {c} \tag {47} \\ \end{array}
+$$
+
+All three terms in the numerator are negative, and the denominator is strictly positive. Therefore:
+
+$$
+f _ {\text {l o w e r}} ^ {\prime} \left(\alpha_ {c}\right) <   0 \tag {48}
+$$
+
+This implies that the normalized score decreases as  $\alpha_{c}$  increases, and hence the classification probability increases.
+
+# (4) With respect to variances  $\sigma_{c}$  and  $\sigma_{m}$ :
+
+In both bounds,  $\sigma_{c}$  and  $\sigma_{m}$  appear only in the denominator. Increasing either of them increases the total variance, which increases the normalized score  $f(x)$  (i.e., makes it less negative), and hence decreases the classification probability  $1 - \Phi(f(x))$ .
+
+Conclusion: In both upper and lower bounds, the classification probability is monotonically increasing with respect to  $k$ ,  $\rho_{c}$ , and  $\alpha_{c}$ , and monotonically decreasing with respect to  $\{\sigma_i\}_{i=1}^{|\mathcal{Y}|}$ .
+
+# A.4 Proof of Proposition 4.1
+
+Proof. Let  $u, v \in \mathcal{V}$  belong to the same cluster, i.e.,  $\mathcal{P}(u) = \mathcal{P}(v) = p$ . In the dense attention setting defined by mask  $\mathbf{M}^{c3}$ , the attention weight from  $u$  to  $v$  is given by:
+
+$$
+\alpha_ {u v} ^ {(c 3)} = \frac {\exp (\langle \mathbf {q} _ {u} , \mathbf {k} _ {v} \rangle)}{\sum_ {v ^ {\prime} \in \mathcal {P} ^ {- 1} (p)} \exp (\langle \mathbf {q} _ {u} , \mathbf {k} _ {v ^ {\prime}} \rangle)}
+$$
+
+Now consider the two-layer attention structure using mask  $\mathbf{M}^{c^4}$ . In the second layer, node  $u$  attends to:
+
+- Itself, with attention score  $\alpha_{uu}^{(2)}$  
+- Its virtual cluster node  $p$ , with attention score  ${\alpha }_{up}^{\left( 2\right) }$
+
+The virtual node  $p$  in the first layer aggregates from all nodes in cluster  $p$ , including  $v$ , with attention score  $\alpha_{pv}^{(1)}$ .
+
+Therefore, the total contribution of node  $v$  to node  $u$  through the two-layer structure can be approximated as:
+
+$$
+\alpha_ {u v} ^ {(c 3)} \approx \left\{ \begin{array}{l l} \alpha_ {u u} ^ {(2)} + \alpha_ {u p} ^ {(2)} \cdot \alpha_ {p v} ^ {(1)}, & \text {i f} u = v \\ \alpha_ {u p} ^ {(2)} \cdot \alpha_ {p v} ^ {(1)}, & \text {i f} u \neq v \end{array} \right.
+$$
+
+This formulation shows that the dense cluster-wise attention score can be decomposed into a mixture of self-attention and two-hop attention via the cluster-level virtual node. This completes the proof.  $\square$
+
+# A.5 Proof of Proposition 4.2
+
+Proof. As detailed in Appendix B, the space complexity of the dense attention computation is  $O(2hN^2)$ , while that of the sparse computation is  $O(6hmd_h)$ . By comparing the two, we conclude that the sparse scheme is more memory-efficient when:
+
+$$
+\frac {m}{N ^ {2}} <   \frac {1}{3 d _ {h}} \quad \Leftrightarrow \quad \kappa_ {\mathcal {R} _ {i}} <   \frac {1}{3 d _ {h}}
+$$
+
+where  $\kappa_{\mathcal{R}_i} \coloneqq \frac{m}{N^2}$  denotes the relative sparsity of the receptive field. This completes the proof.
+
+# B Efficiency Comparison of Attention Computation Schemes
+
+We provide the pseudocode for masked multi-head attention with dense computation scheme in Algorithm 1. As illustrated, the primary memory overhead stems from storing the raw attention scores  $\mathbf{S}$  and the normalized attention weights  $\mathbf{A}$ , both of which have a space complexity of  $O(hN^2)$ . In total, the algorithm requires  $O(2hN^2)$  memory. In terms of time complexity, the main cost comes from three parts: (1) the linear transformations for generating the QKV matrices, which require  $O(3Nd^2)$ ; (2) the computation of the attention matrix,  $O(N^2d)$ ; and (3) the multiplication between the attention matrix and the value matrix,  $O(N^2d)$ . Summing up, the total time complexity of standard MHA is  $O(3Nd^2 + 2N^2d)$ .
+
+# Algorithm 1 Masked Multi-Head Attention
+
+Require: Input  $\mathbf{Z} \in \mathbb{R}^{N \times d_{\mathrm{model}}}$ , number of heads  $h$ , projection matrices  $\mathbf{W}_Q, \mathbf{W}_K, \mathbf{W}_V$ , binary attention mask  $\mathbf{M} \in \{0,1\}^{N \times N}$
+
+Ensure: Output  $\mathbf{Y} \in \mathbb{R}^{N \times d_{\mathrm{model}}}$
+
+1: Project input to queries, keys, values:
+
+$$
+\mathbf {Q}, \mathbf {K}, \mathbf {V} = \mathbf {Z W} _ {Q}, \mathbf {Z W} _ {K}, \mathbf {Z W} _ {V} \in \mathbb {R} ^ {N \times d _ {\text {m o d e l}}}
+$$
+
+2: Reshape and split into  $h$  heads:
+
+$$
+\mathbf {Q}, \mathbf {K}, \mathbf {V} \in \mathbb {R} ^ {h \times N \times d _ {h}} \quad \text {w h e r e} d _ {h} = d _ {\text {m o d e l}} / h
+$$
+
+3: Compute raw attention scores (per head):
+
+$$
+\mathbf {S} = \frac {\mathbf {Q K} ^ {\top}}{\sqrt {d _ {h}}} \in \mathbb {R} ^ {h \times N \times N}
+$$
+
+4: Apply attention mask:
+
+$$
+\mathbf {S} _ {i, j} = \left\{ \begin{array}{l l} \mathbf {S} _ {i, j}, & \text {i f} \mathbf {M} _ {i, j} = 1 \\ - \infty (\text {o r a l a r g e n e g a t i v e c o n s t a n t}), & \text {i f} \mathbf {M} _ {i, j} = 0 \end{array} \right.
+$$
+
+5:Normalize via Softmax:
+
+$$
+\mathbf {A} = \operatorname {s o f t m a x} (\mathbf {S}) \in \mathbb {R} ^ {h \times N \times N}
+$$
+
+6: Apply attention weights:
+
+$$
+\mathbf {H} = \mathbf {A V} \in \mathbb {R} ^ {h \times N \times d _ {h}}
+$$
+
+7: Concatenate heads as the final output:
+
+$$
+\mathbf {Y} = \operatorname {C o n c a t} _ {\text {h e a d}} (\mathbf {H}) \in \mathbb {R} ^ {N \times d _ {\text {m o d e l}}}
+$$
+
+# 8: return Y
+
+The sparse attention computation for head  $i$  can be formulated as:
+
+$$
+\operatorname {h e a d} _ {i} (\mathbf {H}, \mathbf {M}) _ {u} = \sum_ {v \in \mathcal {M} _ {u}} \frac {\exp \left((\mathbf {H} _ {u} \mathbf {W} _ {Q} ^ {(i)}) (\mathbf {H} _ {v} \mathbf {W} _ {K} ^ {(i)}) ^ {\top}\right)}{\sum_ {t \in \mathcal {M} _ {u}} \exp \left((\mathbf {H} _ {u} \mathbf {W} _ {Q} ^ {(i)}) (\mathbf {H} _ {t} \mathbf {W} _ {K} ^ {(i)}) ^ {\top}\right)} \mathbf {H} _ {v} \mathbf {W} _ {V} ^ {(i)} \tag {49}
+$$
+
+where  $\mathcal{M}_u = \{v:\mathbf{M}_{u,v} = 1\}$  denote the key set of node  $u$ ,  $\mathbf{H}_u,\mathbf{H}_v\in \mathbb{R}^{1\times d}$  denote the representations of node  $u,v$ , and  $\mathbf{W}_Q^{(i)},\mathbf{W}_K^{(i)},\mathbf{W}_V^{(i)}\in \mathbb{R}^{d\times d_h}$  are the projection matrices for the  $i$ -th head. The outputs of all heads are concatenated to produce the final output:  $\mathrm{MHA}^S (\mathbf{H},\mathbf{M}) =$ $\mathrm{Concat(head_1,\ldots,head_H)}$
+
+Next, we present the pseudocode for Sparse Multi-Head Attention in Algorithm 2. The dominant memory consumption arises from storing intermediate variables  $\mathbf{Q}'$ ,  $\mathbf{K}'$ ,  $\mathbf{S}'$ ,  $\mathbf{V}'$ , and  $\mathbf{H}'$ , each contributing to a space complexity of  $O(mhd_h)$ , resulting in a total of  $O(5mhd_h)$ . Additionally, computing the output  $\mathbf{H}$  involves a  $\mathrm{Scatter}_{\mathrm{sum}}(\cdot)$  operation, which requires an auxiliary buffer of size  $O(mhd_h)$ . Therefore, the overall space complexity of Sparse Multi-Head Attention is  $O(6mhd_h)$ . Regarding time complexity, sparse MHA consists of three main parts: (1) feature transformations,  $O(3Nd^2)$ ; (2) sparse attention score computation (Step 4),  $O(2md)$ ; and (3) output computation (Step 6),  $O(2md)$ . Therefore, the total time complexity is  $O(3Nd^2 + 4md)$ . A direct comparison indicates that sparse MHA outperforms standard MHA when  $\frac{m}{N^2} < \frac{1}{2}$ .
+
+According to our "Dual Attention Computation Scheme" in Section 4.4 and Proposition 4.2, we apply sparse attention computation in most regions, except for attention from global nodes to the origin node, which involves a dense attention region (where  $\frac{m}{N^2} = 1$ , as shown in Figure 2). This hybrid design allows our method to achieve lower overall time complexity than both standard and sparse MHA, i.e.,  $\min(O(3Nd^2 + 2N^2d), O(3Nd^2 + 4md))$ .
+
+# Algorithm 2 Sparse Multi-Head Attention
+
+Require: Input representation matrix  $\mathbf{Z} \in \mathbb{R}^{N \times d_{\mathrm{model}}}$ , sparse index mask  $\mathbf{M} \in \mathbb{Z}_{+}^{2 \times m}$ , where  $m$  is the number of non-zero entries; projection matrices  $\mathbf{W}_Q, \mathbf{W}_K, \mathbf{W}_V \in \mathbb{R}^{d_{\mathrm{model}} \times d_{\mathrm{model}}}$ ; number of heads  $h$
+
+Ensure: Output  $\mathbf{Y} \in \mathbb{R}^{N \times d_{\mathrm{model}}}$
+
+1: Linear projections:
+
+$$
+\mathbf {Q} = \mathbf {Z} \mathbf {W} _ {Q}, \quad \mathbf {K} = \mathbf {Z} \mathbf {W} _ {K}, \quad \mathbf {V} = \mathbf {Z} \mathbf {W} _ {V} \in \mathbb {R} ^ {N \times d _ {\text {m o d e l}}}
+$$
+
+2: Reshape for multi-head:
+
+$$
+\mathbf {Q}, \mathbf {K}, \mathbf {V} \in \mathbb {R} ^ {N \times h \times d _ {h}}, \quad d _ {h} = d _ {\text {m o d e l}} / h
+$$
+
+3: Index via sparse mask:
+
+$$
+\mathbf {Q} ^ {\prime} = \mathbf {Q} [ \mathbf {M} _ {0} ], \quad \mathbf {K} ^ {\prime} = \mathbf {K} [ \mathbf {M} _ {1} ] \quad \in \mathbb {R} ^ {m \times h \times d _ {h}}
+$$
+
+4: Sparse attention score computation:
+
+$$
+\mathbf {S} ^ {\prime} = \mathbf {Q} ^ {\prime} \cdot \mathbf {K} ^ {\prime} \in \mathbb {R} ^ {m \times h \times d _ {h}}
+$$
+
+$$
+\mathbf {S} = \operatorname {S U M} \left(\mathbf {S} ^ {\prime}, \dim = - 1\right) \in \mathbb {R} ^ {m \times h}
+$$
+
+5: Apply softmax normalization:
+
+$$
+\tilde {\mathbf {S}} = \exp (\mathbf {S}) \in \mathbb {R} ^ {m \times h}
+$$
+
+$$
+\tilde {\mathbf {S}} _ {\text {s u m}} = \operatorname {S c a t t e r} _ {\text {s u m}} (\tilde {\mathbf {S}}, \mathbf {M} _ {0}, \dim = 0) \in \mathbb {R} ^ {N \times h}
+$$
+
+$$
+\mathbf {A} = \tilde {\mathbf {S}} / \tilde {\mathbf {S}} _ {\mathrm {s u m}} [ \mathbf {M} _ {0} ] \in \mathbb {R} ^ {m \times h}
+$$
+
+6: Aggregate weighted values:
+
+$$
+\mathbf {V} ^ {\prime} = \mathbf {V} [ \mathbf {M} _ {1} ] \in \mathbb {R} ^ {m \times h \times d _ {h}}
+$$
+
+$$
+\mathbf {H} ^ {\prime} = \mathbf {A} \cdot \mathbf {V} ^ {\prime} \in \mathbb {R} ^ {m \times h \times d _ {h}}
+$$
+
+$$
+\mathbf {H} = \operatorname {S c a t t e r} _ {\mathrm {s u m}} \left(\mathbf {H} ^ {\prime}, \mathbf {M} _ {0}, \dim = 0\right) \in \mathbb {R} ^ {N \times h \times d _ {h}}
+$$
+
+7: Concatenate heads as the final output:
+
+$$
+\mathbf {Y} = \operatorname {C o n c a t} _ {\text {h e a d}} (\mathbf {H}) \in \mathbb {R} ^ {N \times d _ {\text {m o d e l}}}
+$$
+
+8: return Y
+
+# C The Summary Table of GTs and Hierarchical Attention Masks
+
+We summarize many Graph Transformers and their corresponding hierarchical attention masks mentioned in Section 3 in Table 4.
+
+Table 4: Summary of Graph Transformers and their corresponding hierarchical attention mask.  
+
+<table><tr><td>Mask Type</td><td>Mask Notation</td><td>Representative Graph Transformers</td></tr><tr><td rowspan="2">Local Masks</td><td>Ml1</td><td>GOAT[24], NAGphormer[5], VCR-Graphormer[15], GCFormer[6]</td></tr><tr><td>Ml2</td><td>NodeFormer[42], SGFormer[43], PolyNormer[11], CoBFormer[44], NAGphormer[5], VCR-Graphormer[15], and GCFormer[6]</td></tr><tr><td rowspan="3">Cluster Masks</td><td>Mc1</td><td>Graph ViT[18], Cluster-GT[20], CoBFormer[44]</td></tr><tr><td>Mc2</td><td>Cluster-GT[20]</td></tr><tr><td>Mc3</td><td>CoBFormer[44]</td></tr><tr><td rowspan="2">Global Masks</td><td>Mg1</td><td>NodeFormer[42], SGFormer[43], PolyNormer[11]</td></tr><tr><td>Mg2</td><td>Exphormer[34]</td></tr></table>
+
+# D Dataset
+
+Table 5: The detailed dataset statistics.  
+
+<table><tr><td>Dataset</td><td>#Nodes</td><td>#Edges</td><td>#Feats</td><td>Edge hom</td><td>#Classes</td></tr><tr><td>Cora</td><td>2,708</td><td>5,429</td><td>1,433</td><td>0.83</td><td>7</td></tr><tr><td>CiteSeer</td><td>3,327</td><td>4,732</td><td>3,703</td><td>0.72</td><td>6</td></tr><tr><td>PubMed</td><td>19,717</td><td>44,338</td><td>500</td><td>0.79</td><td>3</td></tr><tr><td>Photo</td><td>7,650</td><td>119,081</td><td>745</td><td>0.83</td><td>8</td></tr><tr><td>Computer</td><td>13,752</td><td>245,861</td><td>767</td><td>0.78</td><td>10</td></tr><tr><td>Squirrel</td><td>2,223</td><td>46,998</td><td>2,089</td><td>0.21</td><td>5</td></tr><tr><td>Chameleon</td><td>890</td><td>8,854</td><td>2,325</td><td>0.24</td><td>5</td></tr><tr><td>Minesweeper</td><td>10,000</td><td>39,402</td><td>7</td><td>0.68</td><td>2</td></tr><tr><td>Ogbn-Arxiv</td><td>169,343</td><td>1,166,343</td><td>128</td><td>0.63</td><td>40</td></tr></table>
+
+# D.1 Dataset Statistics
+
+The detailed dataset statistics are listed in Table 5. The edge homophily is defined as:
+
+$$
+h = \frac {\left| u , v : \mathbf {y} _ {u} = \mathbf {y} _ {v} \right|}{E}
+$$
+
+The Cora, CiteSeer, PubMed [45], Photo, and Computer [33] datasets are available through PyG [14], while Ogbn-Arxiv can be accessed via the OGB platform [19]. The Chameleon, Squirrel, and Minesweeper datasets are provided in the official repository of [31].
+
+# D.2 Dataset Splitting Protocol
+
+For Computer and Photo, we follow the splitting protocol in [11, 15], randomly dividing nodes into training, validation, and test sets with a  $60\%:20\%$ : $20\%$  ratio over five runs. For Ogbn-Arxiv, we adopt the official split provided in [19]. The remaining datasets are split into  $50\%$ : $25\%$ : $25\%$  train/validation/test sets, repeated five times following [42, 31].
+
+# E Baselines
+
+We compare  $\mathbf{M}^3$  Dphormer against 15 baselines spanning multiple model families:
+
+# 1) Classic GNNs:
+
+- GCN [23] adopts a spectral-based convolution that aggregates and transforms features from immediate neighbors using a normalized adjacency matrix. It can be interpreted as a form of Laplacian smoothing.  
+- GAT [38] introduces a self-attention mechanism to assign learnable weights to different neighbors, enabling adaptive and context-aware feature aggregation.  
+- SAGE [16] is an inductive framework that samples a fixed-size neighborhood and aggregates features through functions such as mean, LSTM, or pooling, allowing generalization to unseen nodes and efficient training on large graphs.
+
+2) Enhanced Classic GNNs: GCN*, GAT*, and SAGE* are strong baselines proposed in [26], a benchmark study showing that classic GNNs can achieve significantly better performance on node classification tasks by careful hyperparameter tuning and the incorporation of advanced training techniques, such as residual connections [17] and normalization methods [48, 2, 21].
+
+3) Advanced GNNs: To move beyond the widely adopted homophily assumption—that nodes of the same class are more likely to be connected [28]—advanced GNNs such as GPRGNN [7] and FAGCN [3] have been proposed to improve performance on heterophilic graphs.
+
+- GPRGNN [7] employs a generalized PageRank (GPR) propagation scheme, which allows flexible and learnable weighting over multi-hop neighborhood information. This design enables the model to adapt to both homophilic and heterophilic graph structures.
+
+- FAGCN [3] introduces a frequency adaptive mechanism that modulates the importance of low- and high-frequency components in the spectral domain. By learning a task-specific filter, FAGCN effectively balances local smoothness and discriminative power, making it suitable for graphs with varying levels of heterophily.
+
+# 4) SOTA Graph Transformers:
+
+- NAGphormer [5] tokenizes multi-hop neighborhoods into fixed-length sequences using a Hop2Token module, enabling scalable and efficient node classification on large graphs.  
+- Exphormer [34] designs a sparse Transformer using expander graphs and virtual global nodes, achieving linear complexity and strong performance on large-scale graphs.  
+- SGFormer [43] simplifies the Transformer architecture by adopting a shallow attentive propagation without positional encodings, ensuring efficient all-pair interactions.  
+- CoBFormer [44] mitigates over-globalization by combining coarse-grained and fine-grained paths, improving the model's balance between global and local information.  
+- **PolyNormer** [11] captures complex structures using polynomial-expressive attention with linear time complexity, and performs well on both homophilic and heterophilic graphs.
+
+# 5) MoE-based GNNs:
+
+- Mowst [47] introduces a Mixture of Experts (MoE) framework that combines a weak expert (MLP) and a strong expert (GNN). A confidence-based gating mechanism determines whether to activate the strong expert for each node, enabling adaptive computation and improved performance across diverse graph structures.  
+- GCN-MoE [40] applies the MoE paradigm to GCNs by incorporating multiple experts with varying neighborhood aggregation ranges. A gating function dynamically selects the appropriate expert for each node, enhancing the model's capacity to handle graphs with diverse structural patterns.
+
+# F Experimental Details
+
+This section provides the detailed experimental setup corresponding to the results reported in the main paper.
+
+# F.1 Training Strategy
+
+We follow the training protocol used in the official implementation of NAGphormer and train it using a mini-batch strategy on all datasets. For all other baselines and our proposed  $\mathbf{M}^3$  Dphormer, we adopt a full-batch training scheme. The Adam Optimizer is used for optimization.
+
+# F.2  $\mathbf{M}^3$  Dphormer Configuration
+
+We implement  $\mathbf{M}^3\mathrm{D}$  phormer by stacking multiple  $\mathbf{M}^3\mathrm{D}$  phormer layers. All hyperparameters are selected via grid search over the following search space:
+
+- Learning rate:  $\{5 \times 10^{-4}, 10^{-3}, 5 \times 10^{-3}\}$  
+Number of  $\mathbf{M}^3$  Dphormer layers:
+
+- Cora, Citeseer, Pubmed, Chameleon, Photo: {2, 3, 4}  
+- Squirrel, Computer, Ogbn-Arxiv:  $\{5, 6, 7\}$  
+-Minesweeper:  $\{10,12,15\}$
+
+Number of attention heads:  $\{1,2,4,8\}$  
+- Hidden dimension:  $\{64, 128, 256\}$  
+- Weight decay:  $\{0, 10^{-4}, 5 \times 10^{-4}, 10^{-3}, 5 \times 10^{-3}\}$  
+- Dropout rate:  $\{0.3, 0.5, 0.7\}$  
+- Attention dropout rate:  $\{0.1, 0.3, 0.5\}$
+
+Number of clusters:
+
+- Cora, Citeseer, Squirrel, Minesweeper: {96, 128, 160, 192}  
+- Pubmed, Photo, Computer: {160, 192, 224, 256}  
+- Ogbn-Arxiv: {2048}  
+- Chameleon:  $\{32, 64, 96, 128\}$
+
+We apply Pre-RMSNorm [48] before the bi-level expert routing mechanism in each  $\mathbf{M}^3$  Dphormer layer for most datasets. For Ogbn-Arxiv, however, we adopt Post-BatchNorm due to its superior convergence behavior. We adopt GAT-style attention for the local expert due to its computational efficiency, and standard multi-head attention (MHA) for the cluster and global experts. This choice is motivated by the observation in [4] that GAT-style attention suffers from a static attention problem, which can significantly degrade the performance of cluster and global experts.
+
+# F.3Baselines
+
+We implement GCN, SAGE, GAT, GPRGNN, and FAGCN using PyG [14]. For all other baselines, we use the official implementations. The corresponding repositories are listed below:
+
+- GCN*, GAT*, SAGE*: https://github.com/LUOyk1999/tunedGNN  
+- NAGphormer: https://github.com/JHL-HUST/NAGphormer  
+- Exphormer: https://github.com/hamed1375/Exphormer  
+- SGFormer: https://github.com/qitianwu/SGFormer  
+- CoBFormer: https://github.com/null-xyj/CoBFormer  
+- PolyNormer: https://github.com/cornell-zhang/Polynormer  
+- Mowst: https://github.com/facebookresearch/mowst-gnn  
+- GCN-MOE: https://github.com/VITA-Group/Graph-Mixture-of-Experts
+
+We follow the official training protocols and perform hyperparameter tuning for each model on every dataset. The search space is defined as follows:
+
+- Learning rate:  $\{5 \times 10^{-4}, 10^{-3}, 5 \times 10^{-3}\}$  
+- Hidden dimension:  $\{64, 128, 256\}$  
+- Dropout rate:  $\{0.3, 0.5, 0.7\}$  
+- Weight decay:  $\{0, 10^{-4}, 5 \times 10^{-4}, 10^{-3}, 5 \times 10^{-3}\}$
+
+For models with additional key hyperparameters, we further tune them as follows:
+
+- Attention-based models: number of heads  $\in$  {1, 2, 4, 8}  
+- NAGphormer: number of hops  $\in$  {3, 5, 7, 10, 15}  
+- SGFormer:  $\alpha \in \{0.5, 0.8\}$  
+- CoBFormer:  $\alpha \in \{0.9, 0.8, 0.7\}$ ;  $\tau \in \{0.9, 0.7, 0.5, 0.3\}$  
+GCN-MOE: number of experts  $\in$  {3, 4, 5}  
+- FAGCN:  $\epsilon \in \{0.3, 0.5, 0.7\}$  
+- GPRGNN:  $\alpha \in \{0.1, 0.3, 0.5\}$
+
+All models are trained on a single NVIDIA GPU with 24GB memory. We run each method 5 times and report the mean and standard deviation of the results.
+
+Table 6: Performance and runtime comparison across datasets.  
+
+<table><tr><td></td><td>Cora</td><td>Citeseer</td><td>Pubmed</td><td>Photo</td><td>Computer</td><td>Chameleon</td><td>Squirrel</td></tr><tr><td>Dual</td><td>5.62s</td><td>4.92s</td><td>6.41s</td><td>5.34s</td><td>10.86s</td><td>5.58s</td><td>12.29s</td></tr><tr><td>Dense</td><td>6.69s</td><td>7.28s</td><td>OOM</td><td>7.25s</td><td>OOM</td><td>6.28s</td><td>16.45s</td></tr><tr><td>Sparse</td><td>6.28s</td><td>5.67s</td><td>7.16s</td><td>6.01s</td><td>15.56s</td><td>7.60s</td><td>14.38s</td></tr><tr><td>Polynormer</td><td>3.59s</td><td>2.76s</td><td>3.60s</td><td>6.02s</td><td>4.16s</td><td>3.88s</td><td>5.53s</td></tr></table>
+
+# G More Experimental Results
+
+# G.1 Runtime Comparison
+
+We report the training time of our method over 200 epochs, and compare it against its sparse and dense variants as well as PolyNormer [11]. The results are summarized in Table 6. We observe that the proposed "Dual Attention Computation Scheme" achieves faster training than both sparse and dense MHA variants. Although it is marginally slower than PolyNormer, the additional cost arises from employing three MHA experts, which are crucial for attaining higher accuracy. Furthermore, our model converges within 200 epochs on most datasets, rendering the time overhead acceptable.
+
+# G.2 Graph Classification Performance
+
+We extend our method to graph-level tasks by incorporating edge features and Laplacian positional encodings. To assess the effectiveness of this extension, we conduct experiments on two graph classification datasets: OGBG-MOLBACE and OGBG-MOLBBBP. We compare our approach with three widely-used GNN baselines—GCN, GAT, and GINE—which are recognized as strong performers on graph-level benchmarks [27]. The results are summarized in Table 7.
+
+Table 7: Graph classification performance  $\left( {\%  \pm  \sigma }\right)$  ,measured by ROC-AUC.  
+
+<table><tr><td>Method</td><td>ogbg-bace</td><td>ogbg-bbbp</td></tr><tr><td>M3Dphormer</td><td>0.80432±0.01040</td><td>0.68232±0.00632</td></tr><tr><td>GCN*</td><td>0.75680±0.01676</td><td>0.65146±0.01184</td></tr><tr><td>GAT*</td><td>0.78149±0.01900</td><td>0.65175±0.01221</td></tr><tr><td>GINE*</td><td>0.74799±0.01014</td><td>0.65095±0.01143</td></tr></table>
+
+As shown in the Table 7, our extended model outperforms the baselines on both datasets, demonstrating its strong potential for graph-level tasks.
+
+# G.3 Ablation Study on FFN Variants
+
+In our method, the standard Transformer Feed-Forward Network (FFN) module is replaced with a single linear layer followed by an activation function, such as ReLU [1]. To investigate the impact of the FFN module on node classification performance, we compare  $\mathbf{M}^3$  Dphormer with its FFN variants. The results are summarized in Table 8.
+
+Table 8: Node classification results of M3Dphormer and its FFN variant.  
+
+<table><tr><td></td><td>Cora</td><td>CiteSeer</td><td>Pubmed</td><td>Photo</td><td>Computer</td><td>Chameleon</td><td>Squirrel</td></tr><tr><td>M3Dphormer</td><td>88.48±1.94</td><td>77.53±1.56</td><td>89.96±0.49</td><td>95.91±0.68</td><td>92.09±0.46</td><td>47.09±4.05</td><td>44.34±1.94</td></tr><tr><td>+FFN</td><td>86.50±1.87</td><td>75.53±2.19</td><td>89.28±0.30</td><td>95.37±0.54</td><td>91.28±0.72</td><td>44.22±2.81</td><td>41.36±1.82</td></tr></table>
+
+As observed, incorporating the FFN module often results in a noticeable performance drop, particularly on datasets such as Cora, Citeseer, Chameleon, and Squirrel. We attribute this to the relatively simple and easily learnable node features in these datasets. In such cases, capturing complex node interactions becomes more critical, a task that is more effectively handled by the Multi-Head Attention
+
+(MHA) module. While MHA is commonly regarded as the core of Transformer architectures, it is important to note that the standard FFN module typically contains twice as many parameters as MHA (e.g., with a projection of  $d \to 4d \to d$ ). Since meaningful structural interactions are primarily modeled through attention, it is natural to allocate greater capacity to the MHA module.
+
+In addition to the observed performance improvement, reducing or simplifying the FFN module can significantly enhance computational and memory efficiency, making the model more lightweight and scalable.
+
+# G.4 Loss and Accuracy Curves For PolyNormer
+
+We plot the accuracy and loss curves of  $\mathbf{M}^3$  Dphormer and PolyNormer[11] in Figure 6. Across the training, validation, and test sets,  $\mathbf{M}^3$  Dphormer consistently achieves faster convergence and higher accuracy than PolyNormer during the training process, demonstrating the effectiveness of our method.
+
+![](images/77055e561b6bee9cfe54fe2f3f5c64b2f6fcf651822cb86b72963d7240cd84c9.jpg)  
+Figure 6: Comparison of accuracy and loss curves between  $\mathbf{M}^3\mathrm{D}$  former and PolyNormer on Arxiv.
+
+![](images/bb3355b3f51409b61051a7bc2650afa3e7c6b7c081f6e1eb0d2fdff03f03200a.jpg)
+
+# G.5 More Visualization Results
+
+We visualize the distribution of learned gate weights across node degree bins on Ogbn-Arxiv in Figure 7. Several observations can be made: 1) In the first layer, the gate weights remain close to the initialization [0.5, 0.25, 0.25], suggesting that all types of interactions are considered at the early stage. 2) In the second layer, the weights for local and cluster experts increase, indicating a shift in focus toward capturing structural information at local and mid-range levels. 3) In the third and fourth layers, local experts consistently dominate across all degree bins. Meanwhile, the gate weights for cluster and global experts exhibit a decreasing trend with increasing node degree, reflecting the tendency of low-degree nodes—often located near cluster boundaries with low local homophily—to rely more on broader contextual information. 4) In the final layer, we observe a notable shift: the weights assigned to local experts decrease as node degree increases, while those for global experts increase. This trend may be attributed to the fact that high-degree nodes have already aggregated sufficient local information, and further local aggregation may lead to over-smoothing [25, 29, 30] or over-squashing [36, 10].
+
+![](images/f2e6652a7aeebbdae8741e3f3681ae6d05a08d7cafadd1d75e7eed3bae24ec72.jpg)
+
+![](images/72416fbc6cbe7da025f014dbbe4cd574866f9ed9bcf0ad8f5b68dec31efec035.jpg)  
+(a) Layer 0
+
+![](images/231a97df319836d6ac8dbdddffbce3343fa1f8ee6818f7a109dcd757ddb8514b.jpg)  
+(b) Layer 1
+
+![](images/f604e4e81e9a645fddc0aa65dde51b5cb1c06c03fab7b98036beba4e8b255c97.jpg)  
+(c) Layer 2
+
+![](images/ada42acebafc15f966aeeed2db07a65d466c9ff60c25ed5f250797b86ccc9d54.jpg)  
+(d) Layer 3  
+(e) Layer 4  
+Figure 7: Gate weight distribution across node degree bins for different layers.
\ No newline at end of file
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+# UniteFormer: Unifying Node and Edge Modalities in Transformers for Vehicle Routing Problems
+
+Dian Meng $^{1,5}$
+
+Zhiguang Cao
+
+Jie Gao
+
+Yaoxin  $\mathbf{W}\mathbf{u}^4$
+
+Yaqing Hou<sup>1,5*</sup>
+
+$^{1}$ School of Computer Science and Technology, Dalian University of Technology (DUT)
+
+$^{2}$ School of Computing and Information Systems, Singapore Management University
+
+$^{3}$ Department of Transport and Planning, Delft University of Technology
+
+$^{4}$ Department of Industrial Engineering and Innovation Sciences, Eindhoven University of Technology
+
+<sup>5</sup>Key Laboratory of Social Computing and Cognitive Intelligence (DUT), Ministry of Education, China mengdian@mail.dlut.edu.cn, zhiguangcao@outlook.com, J.Gao-1@tudelft.nl, wyxacc@hotmail.com, houyq@dlut.edu.cn
+
+# Abstract
+
+Neural solvers for the Vehicle Routing Problem (VRP) have typically relied on either node or edge inputs, limiting their flexibility and generalization in real-world scenarios. We propose UniteFormer, a unified neural solver that supports node-only, edge-only, and hybrid input types through a single model trained via joint edge-node modalities. UniteFormer introduces: (1) a mixed encoder that integrates graph convolutional networks and attention mechanisms to collaboratively process node and edge features, capturing cross-modal interactions between them; and (2) a parallel decoder enhanced with query mapping and a feed-forward layer for improved representation. The model is trained with REINFORCE by randomly sampling input types across batches. Experiments on the Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) demonstrate that UniteFormer achieves state-of-the-art performance and generalizes effectively to TSPLib and CVRPLib instances. These results underscore UniteFormer's ability to handle diverse input modalities and its strong potential to improve performance across various VRP tasks.
+
+# 1 Introduction
+
+Vehicle Routing Problems (VRPs) are fundamental in logistics [21], navigation systems [12], and drone delivery [45], with significant theoretical and practical relevance. Recent advances have seen increasing interest in deep learning-based neural solvers for VRPs, offering strong generalization and improved computational efficiency over traditional exact and heuristic algorithms [14, 3, 4]. These methods include both autoregressive models that learn construction policies from data [24, 10, 31], as well as learning-based improvement solvers that enhance classical optimization procedures. However, many of these models, particularly construction-based ones, make an overly simplifying assumption: they rely solely on either node coordinates or edge distances as input. This leads to several limitations. First, training separate models for each input modality (node or edge) is inflexible and impractical for real-world applications. Second, switching between different input types requires retraining from scratch, incurring substantial computational costs. Third, such single-modality training neglects the complementary information between node and edge inputs, preventing the model from learning transferable features and reducing its capacity to discover high-quality solutions.
+
+We argue that hybrid training with both node and edge information offers a more general and informative representation of the problem. While existing methods train only on a single modality, our approach allows joint encoding and interaction across modalities, leading to better-informed policies and improved solution quality. To this end, we propose UniteFormer, a unified neural solver for VRPs that supports hybrid training and generalizes across input modalities. Unlike conventional solvers, UniteFormer is trained once and can handle node-only, edge-only, or mixed edge-node inputs without retraining. This makes it more flexible and applicable to diverse real-world scenarios.
+
+Specifically, UniteFormer consists of a mixed encoder and a parallel-attention decoder. The mixed encoder includes two sub-encoders: edge-aware sub-encoder and node-focused sub-encoder. The edge-aware sub-encoder integrates residual gated graph convolutional networks (GCNs) with self-attention to jointly process node and edge features, facilitating cross-modality interaction. The node-focused sub-encoder encodes node features independently using attention mechanisms, thereby further enhancing the ability to encode node information. Together, they produce rich global embeddings that capture complementary structural information. The decoder features a parallel architecture and nonlinear query mechanisms, incorporating query mapping and a feed-forward (FF) layer to enhance its representational capacity. Our contributions are outlined as follows:
+
+- We present UniteFormer, the first unified neural solver capable of solving VRPs with node-only, edge-only, or hybrid inputs using a single trained model.  
+- We introduce a novel mixed encoder that combines residual gated GCNs with attention mechanisms, which can effectively and jointly process node and edge features to capture cross-modal interactions between them.  
+- We design a decoder with a parallel-attention architecture and nonlinear query mechanisms, which enhance the expressiveness of the policy network.  
+- Experiments on TSP and CVRP with all three input types show that UniteFormer achieves state-of-the-art results. It also generalizes well to real-world TSPLib and CVRPLib benchmarks, and supports applications such as the Asymmetric TSP (Appendix F).
+
+# 2 Related Work
+
+Modality-Specific Neural Solvers for VRPs. Neural approaches have emerged as powerful alternatives for solving VRPs by leveraging advances in deep learning and neural combinatorial optimization [3, 35, 32, 47, 26, 27, 17]. The introduction of pointer networks [42] and the Transformer architecture [40] laid the foundation for early neural VRP solvers such as in [2] and [34].
+
+1) Node-based models: Most existing neural solvers focus on node coordinate inputs. Notable examples include AM [22], POMO [24], and Sym-NCO [19], which significantly improved solution quality for classical VRPs. More recent works have advanced training strategies. For instance, Bdeir et al. [1], Drakulic et al. [10], and Luo et al. [31] applied dynamic input re-encoding during training to enhance generalization. Among them, Drakulic et al. [10] introduced Bisimulation Quotienting (BQ) to reformulate the MDP for more robust generalization. Luo et al. [31] proposed a light encoder heavy decoder (LEHD) model trained via supervised learning on partially reconstructed 100-node instances. These methods are all fundamentally built on node coordinate inputs, capturing spatial structure through positional embeddings.  
+2) Edge-based models: Edge-centric models are a more recent development. Kwon et al. [23] introduced MatNet, a matrix encoding network that operates on pairwise distance matrices, and demonstrated strong performance on the asymmetric traveling salesman (ATSP) and flexible flow shop (FFSP) problems. Lischka et al. [28] proposed GREAT, a sparse graph edge attention model that constructs high-quality solutions by exploiting sparse edge relationships. Building on this, Meng et al. [33] proposed an efficient edge-based EFormer, which further extends and optimizes edge-based problems and achieves excellent results on the TSP and CVRP.  
+3) Hybrid edge-node models: A smaller body of work explores models that jointly use node and edge information. Joshi et al. [18] proposed a GCN-based edge probability predictor that uses both node coordinates and edge weights to guide beam search. Wang et al. [44] developed a distance-aware reshaping method (DAR) that biases attention mechanisms using Euclidean distances. Zhou et al. [49] introduced an instance-conditional adaptive model (ICAM) that integrates both node and edge features to improve adaptability across instance sizes.
+
+![](images/d0b680d0d0e7b927692440e25620a685b11c934314244eb1f84613ec21ec2c03.jpg)  
+Figure 1: Overview of the UniteFormer framework.
+
+Unified Neural Solvers for VRPs. Unified models that generalize across VRP variants have gained attention due to their versatility and practicality. Wang and Yu [43] proposed a multi-task neural solver using a multi-armed bandit framework to train across multiple combinatorial optimization problems. Drakulic et al. [9] introduced GOAL, a supervised learning-based agent capable of solving diverse COPs. Within the VRP domain, Ruis et al. [37] used attribute composition for zero-shot generalization across multiple VRP variants. Liu et al. [29] extended the reinforcement learning-based POMO model to a multi-task setting (POMO-MTL), and Zhou et al. [50] further proposed MVMOE, a mixture-of-experts model to improve generalization. Building on this, Liu et al.. [30] proposed a curvature-aware pre-training framework that effectively improved their performance. Federico et al. [5] introduced RouteFinder, a modular baseline framework for VRP variant modeling. Our work aligns with this line of research but focuses on input-modality unification rather than task-level generalization. UniteFormer is the first model that simultaneously processes VRPs defined by node, edge, or hybrid representations in a single framework, offering strong generalization, improved efficiency, and broad applicability to real-world VRPs.
+
+# 3 UniteFormer
+
+Transformer-based neural VRP solvers typically adopt light decoder architectures [22, 24], where the decoder uses static node embeddings as keys and values throughout the attention layers. In contrast, we replace these static embeddings with two context-aware embeddings that encode both edge relationships and node coordinates, and process them in parallel within the decoder. To effectively encode heterogeneous input modalities, we introduce a novel mixed encoder architecture that combines residual gated GCNs with attention mechanisms. The mixed encoder includes two sub-encoders: an edge-aware sub-encoder and a node-focused sub-encoder, collaboratively processing node and edge features to capture cross-modal interactions between them. In addition, we enhance the parallel-attention decoder with query mapping and a feed-forward layer to form our proposed UniteFormer. The overall architecture of UniteFormer is illustrated in Figure 1. In the following, we first present three input modalities in UniteFormer, then introduce the two sub-encoders of the mixed encoder in detail, and finally report the specific implementation of the decoder.
+
+# 3.1 Input Modalities in UniteFormer
+
+A VRP instance is defined over a graph  $G = \{X, E\}$ , where  $X = \{x_i\}_{i=0}^N$  denotes the nodes (with  $x_0$  as the depot), and  $e(x_i, x_j) \in E$  represents the edge between nodes  $x_i$  and  $x_j$ . UniteFormer
+
+![](images/5342363db14fe536d04de193ca1a60b0b7e2868801ebff880804edc499b2851e.jpg)  
+(a)
+
+![](images/7d9e686a75d50c5f1e9501a68d308ebd90d1e5a6f2d1bbf984da3851e72e9e45.jpg)  
+(b)
+
+![](images/740d58d80bfb5296a9b50e2f4c7811aa118b977945e99449ee8312485166b0ab.jpg)  
+(c)  
+Figure 2: Three input modalities for UniteFormer: (a) Edge-only input. (b) Node-only input. (c) Hybrid input. The Mixed Encoder includes: Edge-aware (left) and Node-focused Sub-encoder (right).
+
+supports three input modalities: edge-only input, where only edge weights  $E$  are provided; node-only input, where only node coordinates  $X$  are provided; hybrid input, where both node coordinates  $X$  and edge weights  $E$  are available.
+
+The three input configurations are shown in Figure 2. Different input types activate different branches of the edge-aware sub-encoder (left) and the node-focused sub-encoder (right). A single unified model is trained, with encoder components dynamically adapted to each input type. Specifically, Edge-only input (Figure 2(a)): The node-focused sub-encoder is disabled (i.e., replaced with a zero embedding), and only the edge-aware sub-encoder processes edge features; Node-only input (Figure 2(b)): Without edge weights, node features are passed to the GCNs directly, while the edge features are set to zero embeddings. Hybrid input (Figure 2(c)): Both node and edge features are used. The mixed-score Multi-Head Attention (mixed-score MHA) is bypassed, and the GCNs receive the raw edge and node features.
+
+# 3.2 Edge-aware Sub-encoder: Efficient Fusion of GCNs and Attention Mechanisms
+
+To simultaneously capture edge and node information, we propose a novel sub-encoder architecture, i.e., the Edge-aware Sub-encoder, that integrates GCNs with attention mechanisms. When edge information is provided, we apply a mixed-score MHA block to derive intermediate node features. Across all three input modalities, we then apply residual gated GCNs to jointly process edge and node features in a unified representation space. Finally, we introduce an additional self-attention layer, which proves especially effective in edge-based settings.
+
+Mixed-Score Attention Layer. Inspired by [23], we adopt a multi-head mixed-score attention mechanism to encode the edge weight matrix. This module follows the structure of standard Transformer attention [40], but replaces the traditional scaled dot-product computation with a mixed-score formulation (see Appendix A). Inputs to this block include: a zero vector  $h_0$ , a randomly selected one-hot vector  $h_v$  from a predefined pool, and the edge weight matrix  $D_{ij}$ . This setup allows dynamic embedding generation and facilitates instance-level augmentation by feeding the same instance multiple times with varying vector combinations. The zero vector can optionally be replaced by another one-hot vector, though we default to using the zero vector. The encoded relation matrix  $h_i^{(\bar{P})}$  is computed only when edge inputs are present:
+
+$$
+\hat {h} _ {i} ^ {(P)} = \operatorname {N O R M} \left(h _ {0} + \text {m i x e d - s c o r e M H A} \left(h _ {0}, h _ {v}, D _ {i j}\right)\right), \tag {1}
+$$
+
+$$
+h _ {i} ^ {(P)} = \operatorname {N O R M} \left(\hat {h} _ {i} ^ {(P)} + \operatorname {F F} \left(\hat {h} _ {i} ^ {(P)}\right)\right), \tag {2}
+$$
+
+where mixed-scoreMHA( $\cdot$ ) denotes the mixed-score attention layer,  $\mathrm{FF}(\cdot)$  is a feed-forward network with one hidden layer and ReLU activation, and  $\mathrm{NORM}(\cdot)$  denotes batch normalization [16].
+
+Residual Gated Graph Convolution Layer. We next feed node and edge features into residual gated GCNs. Node coordinates  $x_{i}$  are embedded as  $h$ -dimensional vectors. The edge weight matrix  $D_{ij}$  and the edge adjacency matrix  $\Theta_{ij}^{\mathrm{knn}}$  are embedded as  $\frac{h}{2}$ -dimensional vectors:
+
+$$
+\alpha_ {i} = \omega_ {1} x _ {i} + b _ {1}, \tag {3}
+$$
+
+$$
+\beta_ {i j} = \omega_ {2} D _ {i j} + b _ {2} \| \omega_ {3} \cdot \Theta_ {i j} ^ {\mathrm {k n n}}, \tag {4}
+$$
+
+where  $\omega_{1} \in \mathbb{R}^{h}, \omega_{2}, \omega_{3} \in \mathbb{R}^{\frac{h}{2}}, b_{1}, b_{2}$  are biases, and  $||$  denotes vector concatenation. The input node and edge embeddings for the GCN are adaptively initialized according to input types. Particularly, 1) Edge-only:  $y_{i}^{0} = h_{i}^{(P)}$ ,  $e_{ij}^{0} = \beta_{ij}$ ; 2) Node-only:  $y_{i}^{0} = \alpha_{i}$ ,  $e_{ij}^{0} = h_{0}$ ; 3) Edge and Node:  $y_{i}^{0} = \alpha_{i}$ ,  $e_{ij}^{0} = \beta_{ij}$ . We denote node and edge embeddings at layer  $l$  as  $y_{i}^{l}$  and  $e_{ij}^{l}$ , respectively. Following [6], we apply ReLU activation and residual connections to obtain the next-layer embeddings:
+
+$$
+y _ {i} ^ {l + 1} = y _ {i} ^ {l} + \operatorname {R e L U} \left(\operatorname {N O R M} \left(W _ {1} ^ {l} y _ {i} ^ {l} + \phi_ {i j} ^ {l} \odot W _ {2} ^ {l} y _ {j} ^ {l}\right)\right), \text {w i t h} \phi_ {i j} ^ {l} = \sum_ {j \sim i} \frac {\sigma \left(e _ {i j} ^ {l}\right)}{\sum_ {j ^ {\prime} \sim i} \sigma \left(e _ {i j ^ {\prime}} ^ {l}\right) + \xi}, \tag {5}
+$$
+
+$$
+e _ {i j} ^ {l + 1} = e _ {i j} ^ {l} + \operatorname {R e L U} \left(\operatorname {N O R M} \left(W _ {3} ^ {l} e _ {i j} ^ {l} + W _ {4} ^ {l} y _ {i} ^ {l} + W _ {5} ^ {l} y _ {j} ^ {l}\right)\right), \tag {6}
+$$
+
+where  $W_{*}^{l}$  are learnable weight matrices,  $\sigma$  is the sigmoid function,  $\xi$  is a small constant for numerical stability, and  $\odot$  denotes element-wise multiplication. This formulation enables anisotropic information diffusion on graphs by incorporating learned edge attention maps  $\phi_{ij}^{l}$ . To further process the node embeddings, we apply the MLP to the GCN output  $y_{i}^{l}$ , yielding values  $m_{i}^{l} = \mathrm{MLP}(y_{i}^{l})$  constrained to  $[0,1]^{2}$ .
+
+Self-Attention Layer. To enhance the model's capacity for global context encoding, we introduce an additional self-attention layer after the MLP. This is particularly important in the edge-only setting, where the node-focused sub-encoder is disabled. In such cases, this layer significantly improves the model's ability to propagate and transform information across the graph:
+
+$$
+h _ {L} ^ {M} = \text {s e l f - a t t e n t i o n} \left(m ^ {L}\right), \tag {7}
+$$
+
+where  $m^L$  is the MLP output, and  $h_L^M$  is the final output of the edge-aware sub-encoder. The detailed computational process in self-attention layer is provided in Appendix A.
+
+# 3.3 Node-focused Sub-encoder: Expressive Encoding of Node Information
+
+The edge-aware sub-encoder, which is built on GCNs and augmented with a self-attention layer for encoding, offers a significant boost for edge-based input. However, it falls short in handling node features compared to the conventional encoder mechanism [24]. Therefore, we introduce the classic attention mechanism (i.e., Node-focused Sub-encoder) to make up for this deficiency, which can effectively improve the performance of node-based input. The node-focused sub-encoder consists of  $L$  stacked layers, each comprising two sublayers: a multi-Head attention (MHA) sublayer and a feed-forward (FF) sublayer. Each sublayer incorporates residual connections [13] and layer normalization [16]. Let  $h_i^{(l)}$  denote the embedding of node  $i$  at layer  $l$ , and let  $H^{(l)} = \{h_1^{(l)}, h_2^{(l)}, \dots, h_n^{(l)}\}$  represent the node embeddings at layer  $l$ . The forward computation at the  $l$ -th layer is given by:
+
+$$
+\hat {h} _ {i} ^ {(l)} = \operatorname {N O R M} \left(h _ {i} ^ {(l - 1)} + \mathrm {M H A} \left(h _ {i} ^ {(l - 1)}, H ^ {(l - 1)}\right)\right), \tag {8}
+$$
+
+$$
+h _ {i} ^ {(l)} = \operatorname {N O R M} \left(\hat {h} _ {i} ^ {(l)} + \operatorname {F F} \left(\hat {h} _ {i} ^ {(l)}\right)\right), \tag {9}
+$$
+
+where  $\mathrm{MHA}(\cdot)$  denotes the multi-head attention,  $\mathrm{FF}(\cdot)$  is a feed-forward network, and  $\mathrm{NORM}(\cdot)$  applies layer normalization. This structure allows the sub-encoder to capture complex dependencies between nodes in a permutation-invariant manner.  $H^{(L)}$  represents the final output of the  $L$ -th attention layer. Specifically, when the input consists of edges only, the sub-encoder is deactivated, and its output is set as  $h_{L}^{N} = h^{(0)}$ ; when the input includes nodes, the output is taken as  $h_{L}^{N} = H^{(L)}$ .
+
+# 3.4 Decoder: Parallel-Attention Decoding with Enhanced Query Representation
+
+A critical component of the decoder is the context query vector  $q$ , which is used to compute attention scores over node embeddings and generate the probability distribution for the next node. In prior
+
+works,  $q$  is often constructed as a linear combination of node embeddings, which limits its representational capacity due to its inherent linearity [15]. To better capture contextual dependencies, we design a decoder architecture with two key enhancements: (1) a parallel-attention architecture that separately computes attention scores using two sets of encoded embeddings, and (2) a nonlinear query mechanism that increases the expressive power of the query vector. Specifically, we apply query mapping and a feed-forward network with residual connections following the MHA layers.
+
+Parallel-Attention Decoding. The edge-aware sub-encoder and node-focused sub-encoder produce two global embeddings, denoted as  $h_{L}^{M}$  and  $h_{L}^{N}$ , where the superscripts  $M$  and  $N$  indicate the edge-aware and node-focused encoding paths, respectively. During decoding, these embeddings are processed in parallel to obtain the decoder context vectors at decoding step  $t$ , given by  $H_{c}^{M} = [h_{1}^{M}, h_{t}^{M}]$  and  $H_{c}^{N} = [h_{1}^{N}, h_{t}^{N}]$ . These vectors are used to form temporary queries:
+
+$$
+q ^ {M} = W _ {1} ^ {M} h _ {1} ^ {M} + W _ {2} ^ {M} h _ {t} ^ {M}, \tag {10}
+$$
+
+$$
+q ^ {N} = W _ {1} ^ {N} h _ {1} ^ {N} + W _ {2} ^ {N} h _ {t} ^ {N}, \tag {11}
+$$
+
+$$
+q = q ^ {M} + q ^ {N}, \tag {12}
+$$
+
+where  $W_1^M, W_2^M, W_1^N$  and  $W_2^N$  are learnable matrices that transform the start node embeddings  $h_1^M, h_1^N$  and current node embeddings  $h_t^M, h_t^N$ , respectively. Next, we apply MHA [24] separately to each set of context embeddings to obtain two intermediate outputs:
+
+$$
+A ^ {M} = \operatorname {M H A} \left(q, k ^ {M}, v ^ {M}\right), \tag {13}
+$$
+
+$$
+A ^ {N} = \operatorname {M H A} \left(q, k ^ {N}, v ^ {N}\right), \tag {14}
+$$
+
+where  $k^M$ ,  $v^M$  and  $k^N$ ,  $v^N$  are the keys and values derived from  $h_L^M$  and  $h_L^N$ , respectively. These outputs are linearly projected and aggregated:
+
+$$
+A ^ {2} = W _ {3} ^ {M} A ^ {M} + W _ {3} ^ {N} A ^ {N}, \tag {15}
+$$
+
+where  $W_3^M$  and  $W_3^N$  are learnable matrices.
+
+Query Mapping and Feed-Forward Layer. To further enrich the query representation, we introduce a query mapping transformation and a feed-forward layer with residual connection:
+
+$$
+q ^ {\prime} = A ^ {2} + \operatorname {Q M T} (q), \tag {16}
+$$
+
+$$
+q ^ {A} = q ^ {\prime} + \operatorname {F F} \left(q ^ {\prime}\right), \tag {17}
+$$
+
+where  $\mathrm{QMT}(\cdot)$  is a linear projection that maps  $q$  to the same dimensionality as  $A^2$ , defined as:
+
+$$
+\operatorname {Q M T} (q) = \delta^ {\mathrm {Q M T}} q. \tag {18}
+$$
+
+Here,  $\delta^{\mathrm{QMT}}$  is a learnable weight matrix. Given the final context vector  $q^{A}$ , we compute a score  $\gamma_{j}$  for each node  $j$  using a masked single-head attention mechanism:
+
+$$
+\gamma_ {j} = \left\{ \begin{array}{l l} C \cdot \tanh  \left(\frac {q ^ {A} \left(k _ {j} ^ {M} + k _ {j} ^ {N}\right)}{\sqrt {d _ {k}}}\right), & \text {i f j u n v i s i t e d} \\ - \infty , & \text {o t h e r w i s e} \end{array} \right. \tag {19}
+$$
+
+where  $d_{k}$  is the dimensionality of the key vectors, and  $C$  is a scaling constant. The final selection probability for node  $j$  is computed via the softmax function:
+
+$$
+\rho_ {j} = \operatorname {s o f t m a x} \left(\gamma_ {j}\right). \tag {20}
+$$
+
+At each decoding step, a node  $\tau_{j}$  is sampled according to  $\rho_{j}$ . Repeating this process for  $n$  steps yields the full solution  $\tau = (\tau_{1},\dots,\tau_{n})^{T}$ . In addition, we report the training algorithm in Appendix B.
+
+# 4 Experiments
+
+We evaluate the performance of UniteFormer on synthetic TSP and CVRP instances of varying sizes, under three input settings: node-only, edge-only, and hybrid. We also report results on standard real-world benchmarks from TSPLib and CVRPLib. The code is publicly available<sup>1</sup>.
+
+Baselines. 1) Traditional Solvers: Concorde [8], LKH3 [14], OR-Tools [25], and HGS [41]. 2) Learning-based Solvers: MatNet [23], GREAT [28], POMO [24], LEHD [31], GCN-BS [18], DAR [44] and ICAM [49]. In order to compare POMO with UniteFormer on edge-base input, we re-implement POMO using edge-only inputs (denoted as POMO-edge). More detailed descriptions of these baselines are presented in Appendix D.
+
+Problem Setting. We follow the standard data generation procedures from prior work [24] to create training and testing datasets for TSP and CVRP with  $n = 20, 50, 100$ , where  $n$  denotes the number of nodes. The problem setups and implementation details are presented in Appendix C.
+
+Model Setting. The edge-aware sub-encoder consists of one layer of mixed-score MHA, three layers of GCN and MLP, and one self-attention layer. The node-focused sub-encoder consists of three attention layers. In each attention layer, the head number of MHA is set to 16, the embedding dimension is set to 256, and the feed-forward layer dimension is set to 512.
+
+Training and Inference. We use the REINFORCE algorithm [46], training each model for 1,010 epochs with 100,000 instances per epoch. The Adam [20] optimizer is used with an initial learning rate of  $4e^{-4}$  and weight decay is set to  $1e^{-6}$ . We adopt the POMO inference algorithm [24] and report both the optimality gap and inference time. A separate set of 10,000 uniformly generated instances is used for testing. All experiments were conducted on a single Tesla V100-SXM2-32GB GPU. More experiment setup details are presented in Appendix D.
+
+# 4.1 Experimental Results
+
+We train a single unified model capable of handling three input types: edge-only, node-only, and hybrid input. Table 1 reports the performance of UniteFormer on uniformly distributed TSP and CVRP instances across various problem sizes and input modalities. UniteFormer consistently outperforms existing learning-based methods in both greedy  $(\times 1)$  and instance-augmented  $(\times 8)$  inference, while maintaining competitive inference times. Additionally, following MatNet [23], we also report results under large-scale augmentation  $(\times 128)$  for edge-based input.
+
+TSP. For edge-based input, UniteFormer significantly outperforms both POMO-edge, MatNet and GREAT across all sizes studied. Notably, its performance with  $\times 8$  augmentation exceeds that of MatNet's  $\times 128$  augmentation, highlighting the efficiency of the UniteFormer. For node-based input, UniteFormer achieves superior results over node-based neural solvers, including POMO and even the strong LEHD model, in both greedy and  $\times 8$  inference. For hybrid edge-node input, UniteFormer also surpasses methods such as GCN-BS, DAR, and ICAM across all scales studied. The advantage is particularly evident on TSP100, where UniteFormer achieves the lowest optimality gap of just  $0.0589\%$  among all neural baselines in Table 1. Furthermore, our edge-based UniteFormer even outperforms not only node-based models like POMO, but also hybrid models like DAR and ICAM, demonstrating its strong generalization and representational capacity.
+
+CVRP. Similarly, for CVRP, UniteFormer exhibits robust performance across all input types. In the edge-based setting, UniteFormer outperforms both POMO-edge and MatNet in greedy and instance-augmented inference. Its performance with  $\times 8$  augmentation even exceeds MatNet's  $\times 128$  augmentation results. In the node-based and hybrid settings, UniteFormer again achieves the best results among all compared neural solvers. Specifically, on CVRP100, the hybrid-input version of UniteFormer achieves the lowest average optimality gap of  $0.5963\%$ . Notably, in the edge-only setting, UniteFormer even surpasses several node-based or hybrid methods, including POMO, DAR, and ICAM. These results comprehensively demonstrate the effectiveness, robustness, and versatility of UniteFormer across a range of problem sizes and input modalities.
+
+# 4.2 Ablation Study
+
+Edge only vs. Node only vs. Edge and Node only vs. UniteFormer. Table 2 presents the results of our ablation study comparing UniteFormer with three training variants. At test time, we evaluate models under three input configurations: Input-edge (only edge features are provided), Input-node (only node features are provided), and Input-XE (both edge and node features are available). The first variant, denoted as w.o. UF-Edge, is trained exclusively with edge inputs. The second variant,
+
+Table 1: Experimental results on TSP and CVRP with uniformly distributed instances. The results of methods with an asterisk (#) are directly obtained from the original paper. BS: Beam search, BS*: Beam search and shortest tour heuristic. UniteFormer-E: Our UniteFormer takes edge as input; '-X': the UniteFormer takes node as input; '-XE': the UniteFormer takes edge and node as input.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Method</td><td colspan="3">TSP20</td><td colspan="3">TSP50</td><td colspan="3">TSP100</td></tr><tr><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td></tr><tr><td></td><td>Concorde</td><td>3.831</td><td>0.000</td><td>4.43</td><td>5.691</td><td>0.000</td><td>23.53</td><td>7.763</td><td>0.000</td><td>66.45</td></tr><tr><td></td><td>LKH3</td><td>3.831</td><td>0.000</td><td>2.78</td><td>5.691</td><td>0.000</td><td>17.21</td><td>7.763</td><td>0.000</td><td>49.56</td></tr><tr><td></td><td>OR-Tools</td><td>3.864</td><td>0.864</td><td>1.16</td><td>5.851</td><td>2.795</td><td>10.75</td><td>8.057</td><td>3.782</td><td>39.05</td></tr><tr><td>edge</td><td>POMO-edge</td><td>3.837</td><td>0.164</td><td>0.10</td><td>5.719</td><td>0.482</td><td>0.24</td><td>7.919</td><td>2.003</td><td>1.34</td></tr><tr><td></td><td>MatNet(x1)</td><td>3.832</td><td>0.044</td><td>0.11</td><td>5.709</td><td>0.303</td><td>0.13</td><td>7.836</td><td>0.940</td><td>0.52</td></tr><tr><td></td><td>MatNet(x8)</td><td>3.831</td><td>0.002</td><td>0.22</td><td>5.694</td><td>0.050</td><td>1.24</td><td>7.795</td><td>0.410</td><td>5.28</td></tr><tr><td></td><td>MatNet(x128)</td><td>3.831</td><td>0.000</td><td>5.71</td><td>5.692</td><td>0.013</td><td>16.47</td><td>7.776</td><td>0.170</td><td>60.11</td></tr><tr><td></td><td>GREAT(x1)#</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>7.850</td><td>1.210</td><td>2.00</td></tr><tr><td></td><td>GREAT(x8)#</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>7.820</td><td>0.810</td><td>18.00</td></tr><tr><td></td><td>UniteFormer-E(x1)</td><td>3.831</td><td>0.020</td><td>0.11</td><td>5.697</td><td>0.095</td><td>0.32</td><td>7.789</td><td>0.332</td><td>1.61</td></tr><tr><td></td><td>UniteFormer-E(x8)</td><td>3.831</td><td>0.000</td><td>0.22</td><td>5.692</td><td>0.004</td><td>1.05</td><td>7.770</td><td>0.086</td><td>5.12</td></tr><tr><td></td><td>UniteFormer-E(x128)</td><td>3.831</td><td>0.000</td><td>2.74</td><td>5.691</td><td>0.000</td><td>19.67</td><td>7.765</td><td>0.019</td><td>65.21</td></tr><tr><td>node</td><td>POMO(x1)</td><td>3.831</td><td>0.018</td><td>0.08</td><td>5.698</td><td>0.119</td><td>0.24</td><td>7.792</td><td>0.364</td><td>1.03</td></tr><tr><td></td><td>POMO(x8)</td><td>3.831</td><td>0.001</td><td>0.11</td><td>5.693</td><td>0.024</td><td>0.45</td><td>7.774</td><td>0.142</td><td>2.01</td></tr><tr><td></td><td>LEHD Greedy</td><td>3.867</td><td>0.961</td><td>0.14</td><td>5.721</td><td>0.519</td><td>0.24</td><td>7.808</td><td>0.577</td><td>1.37</td></tr><tr><td></td><td>UniteFormer-X(x1)</td><td>3.831</td><td>0.011</td><td>0.11</td><td>5.696</td><td>0.072</td><td>0.32</td><td>7.788</td><td>0.316</td><td>1.62</td></tr><tr><td></td><td>UniteFormer-X(x8)</td><td>3.831</td><td>0.000</td><td>0.22</td><td>5.692</td><td>0.004</td><td>1.04</td><td>7.770</td><td>0.085</td><td>5.15</td></tr><tr><td>edge+node</td><td>GCN</td><td>3.855</td><td>0.650</td><td>0.25</td><td>5.901</td><td>3.678</td><td>1.21</td><td>8.413</td><td>8.373</td><td>6.25</td></tr><tr><td></td><td>GCN-BS</td><td>3.835</td><td>0.128</td><td>0.81</td><td>5.710</td><td>0.317</td><td>4.62</td><td>7.931</td><td>2.155</td><td>17.73</td></tr><tr><td></td><td>GCN-BS*</td><td>3.831</td><td>0.000</td><td>21.25</td><td>5.694</td><td>0.041</td><td>37.63</td><td>7.869</td><td>1.368</td><td>58.34</td></tr><tr><td></td><td>DAR(x1)</td><td>3.831</td><td>0.021</td><td>0.08</td><td>5.702</td><td>0.181</td><td>0.26</td><td>7.803</td><td>0.512</td><td>1.12</td></tr><tr><td></td><td>DAR(x8)</td><td>3.831</td><td>0.001</td><td>0.12</td><td>5.694</td><td>1.040</td><td>0.52</td><td>7.779</td><td>0.201</td><td>1.72</td></tr><tr><td></td><td>ICAM(x1)</td><td>3.831</td><td>0.022</td><td>0.07</td><td>5.701</td><td>0.172</td><td>0.22</td><td>7.806</td><td>0.55</td><td>0.99</td></tr><tr><td></td><td>ICAM(x8)</td><td>3.831</td><td>0.002</td><td>0.11</td><td>5.694</td><td>0.417</td><td>0.48</td><td>7.780</td><td>0.22</td><td>1.34</td></tr><tr><td></td><td>UniteFormer-XE(x1)</td><td>3.831</td><td>0.009</td><td>0.11</td><td>5.695</td><td>0.055</td><td>0.32</td><td>7.782</td><td>0.243</td><td>1.62</td></tr><tr><td></td><td>UniteFormer-XE(x8)</td><td>3.831</td><td>0.000</td><td>0.22</td><td>5.692</td><td>0.003</td><td>1.06</td><td>7.768</td><td>0.059</td><td>5.19</td></tr><tr><td></td><td>Method</td><td colspan="3">CVRP20</td><td colspan="3">CVRP50</td><td colspan="3">CVRP100</td></tr><tr><td></td><td></td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td></tr><tr><td></td><td>LKH3</td><td>6.117</td><td>0.000</td><td>2.15h</td><td>10.347</td><td>0.000</td><td>8.52h</td><td>15.647</td><td>0.000</td><td>13.46h</td></tr><tr><td></td><td>HGS</td><td>6.112</td><td>-0.079</td><td>1.48h</td><td>10.347</td><td>-0.001</td><td>4.67h</td><td>15.584</td><td>-0.401</td><td>6.54h</td></tr><tr><td></td><td>OR-Tools</td><td>6.414</td><td>4.863</td><td>2.37</td><td>11.219</td><td>8.430</td><td>19.35</td><td>17.172</td><td>9.749</td><td>2.61h</td></tr><tr><td>edge</td><td>POMO-edge</td><td>6.160</td><td>0.700</td><td>0.12</td><td>10.525</td><td>1.725</td><td>0.35</td><td>15.943</td><td>1.893</td><td>1.53</td></tr><tr><td></td><td>MatNet(x1)</td><td>6.172</td><td>0.907</td><td>0.12</td><td>10.787</td><td>4.253</td><td>0.21</td><td>16.280</td><td>4.401</td><td>1.02</td></tr><tr><td></td><td>MatNet(x8)</td><td>6.146</td><td>0.469</td><td>0.58</td><td>10.635</td><td>2.787</td><td>1.23</td><td>16.117</td><td>3.356</td><td>4.65</td></tr><tr><td></td><td>MatNet(x128)</td><td>6.131</td><td>0.229</td><td>9.93</td><td>10.538</td><td>1.847</td><td>17.93</td><td>15.989</td><td>2.530</td><td>69.05</td></tr><tr><td></td><td>UniteFormer-E(x1)</td><td>6.146</td><td>0.486</td><td>0.06</td><td>10.471</td><td>1.204</td><td>0.56</td><td>15.868</td><td>1.416</td><td>1.95</td></tr><tr><td></td><td>UniteFormer-E(x8)</td><td>6.125</td><td>0.140</td><td>0.23</td><td>10.416</td><td>0.668</td><td>1.29</td><td>15.766</td><td>0.765</td><td>6.01</td></tr><tr><td></td><td>UniteFormer-E(x128)</td><td>6.119</td><td>0.036</td><td>3.47</td><td>10.387</td><td>0.384</td><td>19.44</td><td>15.705</td><td>0.374</td><td>76.43</td></tr><tr><td>node</td><td>POMO(x1)</td><td>6.160</td><td>0.698</td><td>0.06</td><td>10.533</td><td>1.799</td><td>0.18</td><td>15.837</td><td>1.216</td><td>0.64</td></tr><tr><td></td><td>POMO(x8)</td><td>6.132</td><td>0.254</td><td>0.20</td><td>10.437</td><td>0.875</td><td>0.48</td><td>15.754</td><td>0.689</td><td>2.11</td></tr><tr><td></td><td>LEHD Greedy</td><td>6.462</td><td>5.647</td><td>0.07</td><td>10.872</td><td>5.075</td><td>0.18</td><td>16.217</td><td>3.648</td><td>0.55</td></tr><tr><td></td><td>UniteFormer-X(x1)</td><td>6.145</td><td>0.454</td><td>0.06</td><td>10.472</td><td>1.210</td><td>0.56</td><td>15.856</td><td>1.335</td><td>1.95</td></tr><tr><td></td><td>UniteFormer-X(x8)</td><td>6.126</td><td>0.155</td><td>0.22</td><td>10.419</td><td>0.694</td><td>1.26</td><td>15.753</td><td>0.672</td><td>6.05</td></tr><tr><td>edge+node</td><td>DAR(x1)</td><td>6.161</td><td>0.715</td><td>0.08</td><td>10.537</td><td>1.842</td><td>0.19</td><td>15.906</td><td>1.659</td><td>1.02</td></tr><tr><td></td><td>DAR(x8)</td><td>6.132</td><td>0.240</td><td>0.13</td><td>10.441</td><td>0.907</td><td>0.49</td><td>15.783</td><td>0.873</td><td>2.23</td></tr><tr><td></td><td>ICAM(x1)</td><td>6.160</td><td>0.703</td><td>0.08</td><td>10.502</td><td>1.504</td><td>0.13</td><td>15.955</td><td>1.972</td><td>0.62</td></tr><tr><td></td><td>ICAM(x8)</td><td>6.132</td><td>0.246</td><td>0.12</td><td>10.439</td><td>0.886</td><td>0.39</td><td>15.833</td><td>1.192</td><td>1.77</td></tr><tr><td></td><td>UniteFormer-XE(x1)</td><td>6.143</td><td>0.430</td><td>0.06</td><td>10.465</td><td>1.139</td><td>0.56</td><td>15.837</td><td>1.219</td><td>1.95</td></tr><tr><td></td><td>UniteFormer-XE(x8)</td><td>6.126</td><td>0.146</td><td>0.22</td><td>10.415</td><td>0.660</td><td>1.28</td><td>15.740</td><td>0.596</td><td>6.05</td></tr></table>
+
+w.o. UF-Node, is trained only with node inputs. The third variant, w.o. UF-XE, is trained solely on combined edge-node inputs. In contrast, our full UniteFormer model is trained using a hybrid strategy, where the input type (edge, node, or both) is randomly selected for each batch during training. As shown in Table 2, each variant performs well on its respective training input type but shows significant performance drops on other types. In contrast, UniteFormer consistently performs well in all input settings, demonstrating its ability to generalize effectively regardless of the input
+
+Table 2: Ablations of three input variants of UniteFormer on uniformly distributed instances.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">TSP50</td><td colspan="3">w.o. UF-Edge</td><td colspan="3">w.o. UF-Node</td><td colspan="3">w.o. UF-XE</td><td colspan="3">UniteFormer</td></tr><tr><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td></tr><tr><td></td><td>Concorde</td><td>5.691</td><td>0.000</td><td>23.53</td><td>5.691</td><td>0.000</td><td>23.53</td><td>5.691</td><td>0.000</td><td>23.53</td><td>5.691</td><td>0.000</td><td>23.53</td></tr><tr><td rowspan="2">edge</td><td>input-edge(x1)</td><td>5.694</td><td>0.041</td><td>0.32</td><td>11.922</td><td>109.479</td><td>0.32</td><td>12.349</td><td>116.976</td><td>0.32</td><td>5.697</td><td>0.095</td><td>0.32</td></tr><tr><td>input-edge(x8)</td><td>5.692</td><td>0.002</td><td>1.05</td><td>10.033</td><td>76.278</td><td>1.05</td><td>10.161</td><td>78.524</td><td>1.04</td><td>5.692</td><td>0.004</td><td>1.05</td></tr><tr><td rowspan="2">node</td><td>input-node(x1)</td><td>8.424</td><td>48.010</td><td>0.32</td><td>5.698</td><td>0.121</td><td>0.32</td><td>5.951</td><td>4.566</td><td>0.32</td><td>5.696</td><td>0.072</td><td>0.32</td></tr><tr><td>input-node(x8)</td><td>7.473</td><td>31.293</td><td>1.04</td><td>5.692</td><td>0.011</td><td>1.05</td><td>5.801</td><td>1.929</td><td>1.05</td><td>5.692</td><td>0.004</td><td>1.04</td></tr><tr><td rowspan="2">X+E</td><td>input-XE(x1)</td><td>6.620</td><td>16.316</td><td>0.32</td><td>5.726</td><td>0.604</td><td>0.32</td><td>5.695</td><td>0.069</td><td>0.32</td><td>5.695</td><td>0.055</td><td>0.32</td></tr><tr><td>input-XE(x8)</td><td>6.225</td><td>9.370</td><td>1.04</td><td>5.696</td><td>0.085</td><td>1.04</td><td>5.692</td><td>0.003</td><td>1.04</td><td>5.692</td><td>0.003</td><td>1.04</td></tr><tr><td rowspan="2"></td><td rowspan="2">CVRP50</td><td colspan="3">w.o. UF-Edge</td><td colspan="3">w.o. UF-Node</td><td colspan="3">w.o. UF-XE</td><td colspan="3">UniteFormer</td></tr><tr><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td></tr><tr><td></td><td>LKH3</td><td>10.347</td><td>0.000</td><td>8.52h</td><td>10.347</td><td>0.000</td><td>8.52h</td><td>10.347</td><td>0.000</td><td>8.52h</td><td>10.347</td><td>0.000</td><td>8.52h</td></tr><tr><td rowspan="2">edge</td><td>input-edge(x1)</td><td>10.457</td><td>1.069</td><td>0.56</td><td>32.755</td><td>216.567</td><td>0.56</td><td>28.768</td><td>178.041</td><td>0.56</td><td>10.471</td><td>1.204</td><td>0.56</td></tr><tr><td>input-edge(x8)</td><td>10.412</td><td>0.633</td><td>1.26</td><td>31.360</td><td>203.092</td><td>1.26</td><td>27.555</td><td>166.317</td><td>1.27</td><td>10.416</td><td>0.668</td><td>1.26</td></tr><tr><td rowspan="2">node</td><td>input-node(x1)</td><td>24.763</td><td>139.332</td><td>0.56</td><td>10.455</td><td>1.048</td><td>0.56</td><td>11.280</td><td>9.018</td><td>0.56</td><td>10.472</td><td>1.206</td><td>0.56</td></tr><tr><td>input-node(x8)</td><td>17.780</td><td>71.840</td><td>1.26</td><td>10.410</td><td>0.614</td><td>1.26</td><td>10.966</td><td>5.987</td><td>1.26</td><td>10.419</td><td>0.694</td><td>1.26</td></tr><tr><td rowspan="2">X+E</td><td>input-XE(x1)</td><td>12.491</td><td>20.720</td><td>0.56</td><td>10.455</td><td>1.048</td><td>0.56</td><td>10.449</td><td>0.991</td><td>0.56</td><td>10.465</td><td>1.139</td><td>0.56</td></tr><tr><td>input-XE(x8)</td><td>11.576</td><td>11.877</td><td>1.26</td><td>10.410</td><td>0.614</td><td>1.26</td><td>10.407</td><td>0.577</td><td>1.27</td><td>10.415</td><td>0.660</td><td>1.26</td></tr></table>
+
+Table 3: Experimental results on TSPLIB and CVRPLIB.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Method</td><td colspan="3">TSPLIB1-100</td><td colspan="3">TSPLIB101-300</td><td colspan="3">TSP301-500</td></tr><tr><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td></tr><tr><td></td><td>OPT</td><td>19499.583</td><td>0.000</td><td>-</td><td>56902.800</td><td>0.000</td><td>-</td><td>35772.500</td><td>0.000</td><td>-</td></tr><tr><td rowspan="5">edge</td><td>POMO-edge</td><td>26434.493</td><td>38.177</td><td>0.11</td><td>86680.094</td><td>52.557</td><td>0.13</td><td>66931.096</td><td>85.899</td><td>0.24</td></tr><tr><td>MatNet(x1)</td><td>20222.358</td><td>4.948</td><td>0.09</td><td>63182.822</td><td>9.061</td><td>0.19</td><td>50811.854</td><td>41.948</td><td>0.28</td></tr><tr><td>MatNet(x8)</td><td>19748.707</td><td>2.119</td><td>0.16</td><td>61892.651</td><td>7.426</td><td>0.26</td><td>49611.249</td><td>38.678</td><td>0.37</td></tr><tr><td>UniteFormer-E(x1)</td><td>20263.171</td><td>4.419</td><td>0.09</td><td>62497.070</td><td>11.322</td><td>0.12</td><td>49688.931</td><td>38.359</td><td>0.21</td></tr><tr><td>UniteFormer-E(x8)</td><td>19723.757</td><td>1.621</td><td>0.17</td><td>58916.975</td><td>3.028</td><td>0.25</td><td>42414.251</td><td>18.308</td><td>0.35</td></tr><tr><td rowspan="5">node</td><td>POMO(x1)</td><td>19625.737</td><td>1.069</td><td>0.08</td><td>59779.103</td><td>3.405</td><td>0.11</td><td>48074.882</td><td>33.955</td><td>0.10</td></tr><tr><td>POMO(x8)</td><td>19552.937</td><td>0.781</td><td>0.14</td><td>59595.007</td><td>2.838</td><td>0.23</td><td>45171.931</td><td>26.200</td><td>0.11</td></tr><tr><td>LEHD Greedy</td><td>19891.001</td><td>2.382</td><td>0.11</td><td>58443.261</td><td>2.382</td><td>0.24</td><td>39391.629</td><td>10.630</td><td>0.32</td></tr><tr><td>UniteFormer-X(x1)</td><td>20079.525</td><td>4.316</td><td>0.09</td><td>58632.537</td><td>2.678</td><td>0.12</td><td>39599.271</td><td>10.807</td><td>0.22</td></tr><tr><td>UniteFormer-X(x8)</td><td>19549.664</td><td>0.614</td><td>0.18</td><td>57726.770</td><td>1.092</td><td>0.26</td><td>38983.657</td><td>9.252</td><td>0.35</td></tr><tr><td rowspan="6">edge+node</td><td>DAR(x1)</td><td>19767.780</td><td>1.981</td><td>0.08</td><td>58354.624</td><td>2.014</td><td>0.11</td><td>41488.967</td><td>15.856</td><td>0.21</td></tr><tr><td>DAR(x8)</td><td>19616.985</td><td>1.123</td><td>0.14</td><td>57948.545</td><td>1.447</td><td>0.23</td><td>39834.717</td><td>12.376</td><td>0.33</td></tr><tr><td>ICAM(x1)</td><td>19657.984</td><td>1.066</td><td>0.08</td><td>59456.423</td><td>3.125</td><td>0.11</td><td>41754.691</td><td>16.507</td><td>0.21</td></tr><tr><td>ICAM(x8)</td><td>19622.840</td><td>0.843</td><td>0.14</td><td>58512.789</td><td>2.004</td><td>0.23</td><td>40771.396</td><td>13.923</td><td>0.31</td></tr><tr><td>UniteFormer-XE(x1)</td><td>19916.707</td><td>2.585</td><td>0.10</td><td>58051.056</td><td>1.919</td><td>0.12</td><td>40213.122</td><td>12.482</td><td>0.21</td></tr><tr><td>UniteFormer-XE(x8)</td><td>19567.880</td><td>0.653</td><td>0.18</td><td>57848.545</td><td>1.238</td><td>0.24</td><td>39775.667</td><td>11.111</td><td>0.35</td></tr><tr><td rowspan="2"></td><td rowspan="2">Method</td><td colspan="3">CVRPLIB1-100</td><td colspan="3">CVRP101-300</td><td colspan="3">CVRP301-500</td></tr><tr><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td></tr><tr><td></td><td>OPT</td><td>915.574</td><td>0.000</td><td>-</td><td>33184.483</td><td>0.000</td><td>-</td><td>97160.000</td><td>0.000</td><td>-</td></tr><tr><td rowspan="5">edge</td><td>POMO-edge</td><td>1787.383</td><td>94.693</td><td>0.21</td><td>61150.353</td><td>108.337</td><td>0.23</td><td>195349.248</td><td>135.174</td><td>0.45</td></tr><tr><td>MatNet(x1)</td><td>990.186</td><td>8.355</td><td>0.16</td><td>41257.808</td><td>27.962</td><td>0.25</td><td>119378.472</td><td>24.976</td><td>0.42</td></tr><tr><td>MatNet(x8)</td><td>974.015</td><td>6.485</td><td>0.22</td><td>37060.391</td><td>12.766</td><td>0.32</td><td>111725.107</td><td>16.338</td><td>0.68</td></tr><tr><td>UniteFormer-E(x1)</td><td>968.487</td><td>5.783</td><td>0.19</td><td>40601.853</td><td>23.299</td><td>0.25</td><td>112460.491</td><td>18.192</td><td>0.45</td></tr><tr><td>UniteFormer-E(x8)</td><td>942.392</td><td>3.040</td><td>0.23</td><td>36259.083</td><td>9.485</td><td>0.38</td><td>110828.859</td><td>14.657</td><td>0.72</td></tr><tr><td rowspan="5">node</td><td>POMO(x1)</td><td>980.137</td><td>7.893</td><td>0.11</td><td>36514.578</td><td>9.260</td><td>0.18</td><td>113101.594</td><td>17.037</td><td>0.35</td></tr><tr><td>POMO(x8)</td><td>953.564</td><td>4.727</td><td>0.22</td><td>36004.930</td><td>7.561</td><td>0.26</td><td>110642.211</td><td>14.488</td><td>0.45</td></tr><tr><td>LEHD Greedy</td><td>1411.705</td><td>5.517</td><td>0.28</td><td>43841.532</td><td>11.734</td><td>0.45</td><td>116158.490</td><td>15.194</td><td>0.77</td></tr><tr><td>UniteFormer-X(x1)</td><td>953.203</td><td>4.266</td><td>0.19</td><td>35784.084</td><td>8.945</td><td>0.26</td><td>112799.943</td><td>13.897</td><td>0.45</td></tr><tr><td>UniteFormer-X(x8)</td><td>934.185</td><td>2.116</td><td>0.22</td><td>34971.075</td><td>5.353</td><td>0.38</td><td>105541.708</td><td>7.855</td><td>0.73</td></tr><tr><td rowspan="6">edge+node</td><td>DAR(x1)</td><td>959.278</td><td>5.060</td><td>0.18</td><td>35482.062</td><td>7.426</td><td>0.26</td><td>109347.025</td><td>9.449</td><td>0.44</td></tr><tr><td>DAR(x8)</td><td>937.261</td><td>2.568</td><td>0.24</td><td>34990.418</td><td>5.530</td><td>0.37</td><td>104165.243</td><td>7.758</td><td>0.62</td></tr><tr><td>ICAM(x1)</td><td>960.942</td><td>5.272</td><td>0.17</td><td>38232.511</td><td>11.489</td><td>0.25</td><td>117786.209</td><td>15.771</td><td>0.42</td></tr><tr><td>ICAM(x8)</td><td>940.882</td><td>3.003</td><td>0.24</td><td>35881.780</td><td>7.338</td><td>0.39</td><td>110088.655</td><td>10.823</td><td>0.67</td></tr><tr><td>UniteFormer-XE(x1)</td><td>951.386</td><td>3.994</td><td>0.19</td><td>36184.185</td><td>7.306</td><td>0.25</td><td>107177.981</td><td>8.798</td><td>0.45</td></tr><tr><td>UniteFormer-XE(x8)</td><td>932.698</td><td>1.940</td><td>0.22</td><td>34989.507</td><td>5.146</td><td>0.38</td><td>104911.995</td><td>6.889</td><td>0.72</td></tr></table>
+
+modality. This result highlights the strength of our unified training approach in producing a robust and versatile model.
+
+Further architectural ablation studies of the three components of the UniteFormer architecture are reported in Appendix E. These experimental results also demonstrate their positive contribution to the UniteFormer, proving the effectiveness and indispensability of each component.
+
+# 4.3 Generalization
+
+Table 3 summarizes the results on real-world TSPLIB [36] and CVRPLIB [39] instances of various sizes and distributions. We categorize them into three groups by size:  $N = 1 - 100$ ,  $N = 101 - 300$ , and  $N = 301 - 500$ . Generalization results show that the UniteFormer performs best on instances with no more than 100 nodes and slightly less effectively on 101-300 node instances. Overall, UniteFormer shows excellent generalization ability on both TSPLIB and CVRPLIB. For TSP, UniteFormer generalizes better with node-only input. For CVRP, it performs exceptionally well with hybrid input.
+
+Additionally, to demonstrate UniteFormer's strong generalization and scalability, we extend our investigation to the Asymmetric Traveling Salesman Problem (ATSP). Due to the asymmetric nature of ATSP, we can naturally solve it using the edge-based UniteFormer framework, which also exhibits superior performance. The detailed experimental results are shown in Appendix F.
+
+# 5 Conclusion, Limitation and Future work
+
+Conclusion: In this work, we propose UniteFormer, a unified neural solver that supports three input types through a single model trained via joint edge-node modalities. We propose a mixed encoder that integrates GCNs and attention mechanisms to collaboratively process node and edge features, capturing cross-modal interactions. Furthermore, we implement a parallel decoding strategy and enhance the decoder's representation ability by adding query mapping and nonlinear layers. Extensive experimental comparisons with other modality-specific models demonstrate UniteFormer's promising performance. Due to the efficiency and practicality of UniteFormer, we believe it can provide valuable insights and inspire follow-up work to explore more powerful unified neural solvers for edge-node modalities.
+
+Limitation and Future Work: Although our UniteFormer performs well on all three input types, its heavy encoder results in high training and equipment demands, making large-scale problem training challenging. Trimming UniteFormer into a lightweight model for large-scale VRPs [48] is a worthwhile direction for future research. Another promising future work is to extend UniteFormer to solve multi-task VRPs [50, 5] with joint modalities.
+
+# Acknowledgments
+
+This work was supported in part by the National Natural Science Foundation of China under Grant 62372081, the Young Elite Scientists Sponsorship Program by CAST under Grant 2022QNRC001, the Liaoning Provincial Natural Science Foundation Program under Grant 2024010785-JH3/107, the Dalian Science and Technology Innovation Fund under Grant 2024JJ12GX020, the Dalian Major Projects of Basic Research under Grant 2023JJ11CG002 and the 111 Project under Grant D23006. This research/project is supported by the National Research Foundation, Singapore under its AI Singapore Programme (AISG Award No: AISG3-RP-2022-031).
+
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+
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+
+# 7. Experiment statistical significance
+
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+
+Answer: [Yes]
+
+Justification: The experimental results are obtained by testing a set of 10,000 uniformly generated instances from different problems, and we report relevant metrics, such as path length and gap, to evaluate the performance. Our model follows the POMO training algorithm [24], which has been subjected to t-test.
+
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+
+# 8. Experiments compute resources
+
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+
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+
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+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
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+Justification: Both potential positive and negative societal impacts of our work are analyzed in Section 5 and Appendix G.
+
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+
+# 11. Safeguards
+
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+Answer: [No]
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+Justification: The data we utilize consists of node coordinates, detailed in Section 4 and Appendix C. Additionally, the unified Neural Solver is employed to construct tours for VRPs. We assess that there are no significant risks of misuse associated with our paper.
+
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+# 13. New assets
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+# 14. Crowdsourcing and research with human subjects
+
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+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
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+
+# 16. Declaration of LLM usage
+
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+
+Answer: [NA]
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+Justification: The core method development in this research does not involve LLMs as any important, original, and non-standard components.
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+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Details of Attention Computation
+
+# A.1 Mixed-Score Attention
+
+To incorporate external relational priors—such as graph adjacency or edge distances—alongside content-based similarity scores, we replace each head's standard scaled dot-product attention with a mixed-score attention mechanism, while preserving the remaining components of the Transformer's multi-head attention block (as in MatNet [23]). The original scaled dot-product attention computes attention weights by applying SoftMax to the pairwise scaled dot-products of queries and keys and then uses those weights to form a weighted sum of the value vectors. By contrast, the mixed-score attention supplements these internal attention scores with externally given relationship scores for every query-key pair. Specifically, the mixed-score attention mechanism mixes the internal attention scores and the external relationship scores before passing them to the next "SoftMax" stage. Similar mixing strategies have been explored by [38] and [11].
+
+Formally, by defining dimensions  $d_k$  and  $d_v$ , we compute the key  $k_i \in \mathbb{R}^{d_k}$ , value  $v_i \in \mathbb{R}^{d_v}$ , and query  $q_i \in \mathbb{R}^{d_k}$  for each node by projecting the embedding  $h_i$ :
+
+$$
+q _ {i} = W ^ {Q} h _ {i}, \quad k _ {i} = W ^ {K} h _ {i}, \quad v _ {i} = W ^ {V} h _ {i}, \tag {21}
+$$
+
+where  $W^{Q} \in \mathbb{R}^{d_{k} \times d_{h}}$ ,  $W^{K} \in \mathbb{R}^{d_{k} \times d_{h}}$ , and  $W^{V} \in \mathbb{R}^{d_{v} \times d_{h}}$  are learnable weight matrices.
+
+Internal and External Scores: Afterwards, we compute the internal attention scores  $S_{ij}^{\mathrm{int}}$  by taking the scaled dot-products of each query-key pair and derive the external relationship scores  $S_{ij}^{\mathrm{ext}}$  from the edge weight matrix  $D_{ij}$ :
+
+$$
+S _ {i j} ^ {\text {i n t}} = \frac {q _ {i} ^ {T} k _ {j}}{\sqrt {d _ {k}}}, \quad S _ {i j} ^ {\text {i n t}} \in \mathbb {R} ^ {n \times n}, \tag {22}
+$$
+
+$$
+S _ {i j} ^ {\text {e x t}} = g _ {\theta} \left(D _ {i j}\right), \quad S _ {i j} ^ {\text {e x t}} \in \mathbb {R} ^ {n \times n}, \tag {23}
+$$
+
+where  $S_{ij}^{\mathrm{int}}$  denotes internal attention scores,  $S_{ij}^{\mathrm{ext}}$  denotes the external relationship scores,  $D_{ij} \in \mathbb{R}^{n \times n}$  encodes a known relationship (edge weight matrix) between positions  $(i,j)$ , and  $g_{\theta}$  is an optional learnable scalar or nonlinear mapping applied element-wise.
+
+Mixed Score via Element-Wise MLP: For each attention head  $h$ , we employ a compact two-layer perceptron  $f_{\phi_h}:\mathbb{R}^2\to \mathbb{R}$ , parameterized by  $\phi_h = \{W_1^h,b_1^h,W_2^h,b_2^h\}$ , to fuse the internal and external scores on an element-wise basis:
+
+$$
+S _ {i j} ^ {\operatorname {m i x}} = f _ {\phi_ {h}} \left(S _ {i j} ^ {\operatorname {i n t}}, S _ {i j} ^ {\operatorname {e x t}}\right) = W _ {2} ^ {h} \sigma \left(W _ {1} ^ {h} \left[ S _ {i j} ^ {\operatorname {i n t}}, S _ {i j} ^ {\operatorname {e x t}} \right] ^ {\top} + b _ {1} ^ {h}\right) + b _ {2} ^ {h}, \tag {24}
+$$
+
+where  $\sigma$  is the ReLU function, and  $S_{ij}^{\mathrm{mix}}$  denotes the mixed scores.
+
+Softmax Normalization: Then, we perform softmax normalization on the mixed scores  $S_{ij}^{\mathrm{mix}}$ :
+
+$$
+\alpha_ {i j} ^ {h} = \frac {\exp \left(S _ {i j} ^ {\operatorname* {m i x}}\right)}{\sum_ {j ^ {\prime} = 1} ^ {n} \exp \left(S _ {i j ^ {\prime}} ^ {\operatorname* {m i x}}\right)}, \tag {25}
+$$
+
+Subsequently, the attention vector  $\alpha_{ij}^{h}$  is transformed into a convex combination of the messages  $v_{j}^{h}$ , and the specific-head attention output  $z_{i}^{h}$  is obtained:
+
+$$
+z _ {i} ^ {h} = \sum_ {j = 1} ^ {n} \alpha_ {i j} ^ {h} v _ {j} ^ {h}, \quad \mathbf {Z} ^ {h} = \left[ z _ {1} ^ {h}, \dots , z _ {n} ^ {h} \right] \in \mathbb {R} ^ {d _ {h} \times d _ {v}}. \tag {26}
+$$
+
+# A.2 Self-Attention
+
+Self-attention is a mechanism that dynamically assigns weights to each position within a sequence, capturing global dependencies by allowing each element in the sequence to interact with itself and other elements. We interpret the attention mechanism in [40] as a weighted message-passing algorithm between nodes in a graph. The weight of the message value that a node receives from
+
+its neighbors depends on the compatibility of its query with the keys of its neighbors. Formally, we define dimensions  $d_{k}$  and  $d_v$  and compute the key  $k_{i}\in \mathbb{R}^{d_{k}}$ , value  $v_{i}\in \mathbb{R}^{d_{v}}$  and query  $q_{i}\in \mathbb{R}^{d_{k}}$  for each node by projecting the embedding  $h_i$ :
+
+$$
+q _ {i} = W ^ {Q} h _ {i}, \quad k _ {i} = W ^ {K} h _ {i}, \quad v _ {i} = W ^ {V} h _ {i}, \tag {27}
+$$
+
+where  $W^{Q} \in \mathbb{R}^{d_{k} \times d_{h}}$ ,  $W^{K} \in \mathbb{R}^{d_{k} \times d_{h}}$ , and  $W^{V} \in \mathbb{R}^{d_{v} \times d_{h}}$  are learnable weight matrices. From the queries and keys, we compute the compatibility  $u_{ij} \in \mathbb{R}$  of the query  $q_{i}$  of node  $i$  with the key  $k_{j}$  of node  $j$  as the (scaled) dot product:
+
+$$
+u _ {i j} = \left\{ \begin{array}{l l} \frac {q _ {i} ^ {T} k _ {j}}{\sqrt {d _ {k}}}, & \text {i f i a d j a c e n t t o j} \\ - \infty . & \text {o t h e r w i s e} \end{array} \right. \tag {28}
+$$
+
+From the compatibilities  $u_{ij}$ , we compute the attention weights  $a_{ij} \in [0,1]$  using a softmax function:
+
+$$
+a _ {i j} = \frac {e ^ {u _ {i j}}}{\sum_ {j ^ {\prime}} e ^ {u _ {i j ^ {\prime}}}}. \tag {29}
+$$
+
+After that, the vector  $h_i'$  that is received by node  $i$  is the convex combination of messages  $v_j$ :
+
+$$
+h _ {i} ^ {\prime} = \sum_ {j} a _ {i j} v _ {j}. \tag {30}
+$$
+
+Finally, we normalize the vector  $h_i$  and use the feed-forward sublayer to compute the node-by-node projection:
+
+$$
+\hat {h} _ {i} ^ {\prime} = \operatorname {N O R M} \left(h _ {i} + h _ {i} ^ {\prime}\right), \tag {31}
+$$
+
+$$
+h _ {i} ^ {c} = \operatorname {N O R M} \left(\hat {h} _ {i} ^ {\prime} + \operatorname {F F} \left(\hat {h} _ {i} ^ {\prime}\right)\right), \tag {32}
+$$
+
+where,  $\mathrm{FF}(\cdot)$  is a feed-forward layer, and  $\mathrm{NORM}(\cdot)$  applies batch normalization. The feed-forward sublayer computes node-wise projections using a hidden (sub)sublayer with dimension  $d_{\mathrm{ff}} = 512$  and a ReLU activation. We use batch normalization with learnable  $d_h$ -dimensional affine parameters  $w^{\mathrm{bn}}$  and  $b^{\mathrm{bn}}$ . The two sublayers are defined in detail as follows:
+
+$$
+\operatorname {F F} \left(\hat {h} _ {i}\right) = W ^ {\mathrm {f f}, 1} \cdot \operatorname {R e L U} \left(W ^ {\mathrm {f f}, 0} \hat {h} _ {i} + b ^ {\mathrm {f f}, 0}\right) + b ^ {\mathrm {f f}, 1}, \tag {33}
+$$
+
+$$
+\mathrm {B N} \left(h _ {i}\right) = w ^ {\mathrm {b n}} \odot \overline {{\mathrm {B N}}} \left(h _ {i}\right) + b ^ {\mathrm {b n}}. \tag {34}
+$$
+
+Here,  $\odot$  denotes the element-wise product and  $\overline{\mathrm{BN}}$  refers to batch normalization without affine transformation.
+
+# B Training Details
+
+We adopt the same reinforcement learning framework as POMO [24], using the REINFORCE algorithm [46] to train the UniteFormer. At each training step, we sample a set of  $n$  solution trajectories  $\{\tau^1,\dots ,\tau^n\}$ , compute their corresponding rewards  $f(\tau^i)$ , and apply approximate gradient ascent to maximize the expected return  $\mathcal{L}$ . The gradient of the objective  $\mathcal{L}(\theta)$  with respect to model parameters  $\theta$  is estimated as:
+
+$$
+\nabla_ {\theta} \mathcal {L} (\theta) \approx \frac {1}{n} \sum_ {i = 1} ^ {n} \left[ \left(f \left(\tau^ {i}\right) - b ^ {i} (s)\right) \nabla \log p _ {\theta} \left(\tau^ {i} | s\right) \right], \tag {35}
+$$
+
+where  $b^{i}(s)$  is a baseline used to reduce variance in the gradient estimate. Following common practice, we use a shared baseline defined as the average reward across the sampled trajectories:
+
+$$
+b ^ {i} (s) = b _ {\text {s h a r e d}} (s) = \frac {1}{n} \sum_ {i = 1} ^ {n} f \left(\tau^ {i}\right), \tag {36}
+$$
+
+and the probability of a trajectory  $\tau^i$  under the policy is factorized as:
+
+$$
+p _ {\theta} \left(\tau^ {i} \mid s\right) = \prod_ {t = 2} ^ {M} p _ {\theta} \left(a _ {t} ^ {i} \mid s, a _ {1: t - 1} ^ {i}\right), \tag {37}
+$$
+
+where  $a_{t}^{i}$  denotes the action at step  $t$  in trajectory  $\tau^i$ , and  $M$  is the length of the solution.
+
+Table 4: Experiment Hyperparameters.  
+
+<table><tr><td>Hyperparameter</td><td>Value</td><td>Hyperparameter</td><td>Value</td></tr><tr><td>Model</td><td></td><td>Training</td><td></td></tr><tr><td>Embedding dimension dh</td><td>256</td><td>Input choice ψc</td><td>edge/node/edge+node</td></tr><tr><td>Number of attention heads Mh</td><td>16</td><td>Batch size</td><td>1024/256/64</td></tr><tr><td>Number of encoder layers L_e</td><td>3</td><td>Optimizer</td><td>Adam</td></tr><tr><td>Number of GCN layers L_g</td><td>3</td><td>Learning rate (LR)</td><td>4e-4</td></tr><tr><td>Number of MLP layers L_m</td><td>3</td><td>Weight decay</td><td>1e-6</td></tr><tr><td>K-nearest neighbors</td><td>20</td><td>LR scheduler</td><td>MultiStepLR</td></tr><tr><td>Feedforward hidden dimension df</td><td>512</td><td>LR milestones</td><td>[901,1001]</td></tr><tr><td>Feedforward activation</td><td>ReLU</td><td>LR gamma</td><td>0.1</td></tr><tr><td>Tanh clipping ξ</td><td>10.0</td><td>Train data per epoch</td><td>100,000</td></tr><tr><td>Normalization</td><td>Batch</td><td>Training epochs</td><td>1010</td></tr></table>
+
+# C Implementation Details for TSP and CVRP
+
+# C.1 Problem Setup
+
+TSP. Solving a TSP instance with  $n$  nodes requires finding the shortest loop that visits each node exactly once and eventually returns to the first visited node, where the distance between two nodes is the Euclidean distance. We generate TSP instances following AM [22], where the coordinates of  $n$  nodes are randomly and uniformly sampled from the unit square.
+
+CVRP. The CVRP instance involves  $n$  customer nodes and one depot node, where the coordinates of the customer nodes and the depot node are uniformly sampled from the unit square. Each customer node  $i$  has a normalized demand  $\hat{\delta}_i = \delta_i / D$ , where  $\delta_i$  is sampled from the discrete set  $\{1, 2, \dots, 9\}$  and the vehicle capacity  $D = 30, 40, 50$  for problem sizes  $N = 20, 50, 100$ , respectively. A delivery vehicle with unit capacity makes round trips starting and ending at the depot, delivering goods to customer nodes according to their demands and replenishing inventory at the depot, where each customer node is only allowed to be visited once. Our objective is to determine the shortest feasible set of routes that visits all nodes while respecting the vehicle's capacity constraint  $D$ .
+
+# C.2 Implementation Details
+
+For a TSP/CVRP instance  $G = \{X, E\}$ , the node features  $\{x_1, \dots, x_n\}$  are the 2D-coordinates of the  $n$  nodes in the graph, and the edge features  $e(x_i, x_j) \in E$  are the edges between nodes  $x_i$  and  $x_j$  in the graph  $G$ . Our UniteFormer supports three input modalities: 1) Edge-only input, where only edge weights  $E$  are provided; 2) Node-only input, where only node coordinates  $X$  are provided; 3) Hybrid input, where both node coordinates  $X$  and edge weights  $E$  are available.
+
+To formalize this, we introduce a modality-selection function  $\psi_c \in \{0,1,2\}$ , where
+
+$$
+\psi_ {c} = \left\{ \begin{array}{l l} 0, & \text {e d g e - o n l y i n p u t}, \\ 1, & \text {n o d e - o n l y i n p u t}, \\ 2, & \text {h y b r i d i n p u t}. \end{array} \right. \tag {38}
+$$
+
+During training, we employ the REINFORCE algorithm to randomly select  $\psi_c$  at each batch. This stochastic modality sampling encourages the model to learn robust representations under all three input scenarios.
+
+# D Experimental Details for UniteFormer
+
+# D.1 Experiment Baselines
+
+1) Traditional Solvers: For TSP, we use the non-learning solvers Concorde [8], LKH3 [14], and OR-Tools [25], which are known for providing strong results on TSP. Consistent with prior works [24,
+
+Table 5: Ablations of three key components of UniteFormer on uniformly distributed instances.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">TSP50</td><td colspan="3">w.o. self-attention</td><td colspan="3">w.o. node-focused sub</td><td colspan="3">w.o. edge-aware sub-E</td><td colspan="3">UniteFormer</td></tr><tr><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td></tr><tr><td></td><td>Concorde</td><td>5.691</td><td>0.000</td><td>23.53</td><td>5.691</td><td>0.000</td><td>23.53</td><td>5.691</td><td>0.000</td><td>23.53</td><td>5.691</td><td>0.000</td><td>23.53</td></tr><tr><td rowspan="2">edge</td><td>input-edge(x1)</td><td>5.699</td><td>0.126</td><td>0.32</td><td>5.697</td><td>0.097</td><td>0.34</td><td>6.402</td><td>12.490</td><td>0.32</td><td>5.697</td><td>0.095</td><td>0.32</td></tr><tr><td>input-edge(x8)</td><td>5.692</td><td>0.007</td><td>1.01</td><td>5.692</td><td>0.004</td><td>1.15</td><td>5.765</td><td>1.294</td><td>1.06</td><td>5.692</td><td>0.004</td><td>1.05</td></tr><tr><td rowspan="2">node</td><td>input-node(x1)</td><td>5.697</td><td>0.098</td><td>0.31</td><td>5.695</td><td>0.069</td><td>0.34</td><td>5.6978</td><td>0.111</td><td>0.33</td><td>5.696</td><td>0.072</td><td>0.32</td></tr><tr><td>input-node(x8)</td><td>5.692</td><td>0.009</td><td>1.02</td><td>5.692</td><td>0.004</td><td>1.14</td><td>5.692</td><td>0.006</td><td>1.06</td><td>5.692</td><td>0.004</td><td>1.04</td></tr><tr><td rowspan="2">X+E</td><td>input-XE(x1)</td><td>5.695</td><td>0.068</td><td>0.31</td><td>5.695</td><td>0.054</td><td>0.35</td><td>5.697</td><td>0.105</td><td>0.35</td><td>5.695</td><td>0.055</td><td>0.32</td></tr><tr><td>input-XE(x8)</td><td>5.692</td><td>0.004</td><td>1.02</td><td>5.692</td><td>0.003</td><td>1.16</td><td>5.692</td><td>0.005</td><td>1.08</td><td>5.692</td><td>0.003</td><td>1.04</td></tr><tr><td rowspan="2"></td><td rowspan="2">CVRP50</td><td colspan="3">w.o. self-attention</td><td colspan="3">w.o. node-focused sub</td><td colspan="3">w.o. edge-aware sub-E</td><td colspan="3">UniteFormer</td></tr><tr><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td></tr><tr><td></td><td>LKH3</td><td>10.347</td><td>0.000</td><td>8.52h</td><td>10.347</td><td>0.000</td><td>8.52h</td><td>10.347</td><td>0.000</td><td>8.52h</td><td>10.347</td><td>0.000</td><td>8.52h</td></tr><tr><td rowspan="2">edge</td><td>input-edge(x1)</td><td>10.483</td><td>1.312</td><td>0.54</td><td>10.529</td><td>1.763</td><td>0.58</td><td>15.111</td><td>46.043</td><td>0.56</td><td>10.471</td><td>1.204</td><td>0.56</td></tr><tr><td>input-edge(x8)</td><td>10.426</td><td>0.770</td><td>1.24</td><td>10.434</td><td>0.841</td><td>1.32</td><td>14.203</td><td>37.266</td><td>1.29</td><td>10.416</td><td>0.668</td><td>1.26</td></tr><tr><td rowspan="2">node</td><td>input-node(x1)</td><td>10.490</td><td>1.379</td><td>0.54</td><td>10.544</td><td>1.909</td><td>0.58</td><td>10.677</td><td>3.192</td><td>0.56</td><td>10.472</td><td>1.206</td><td>0.56</td></tr><tr><td>input-node(x8)</td><td>10.421</td><td>0.721</td><td>1.24</td><td>10.461</td><td>1.103</td><td>1.26</td><td>10.537</td><td>1.835</td><td>1.28</td><td>10.419</td><td>0.694</td><td>1.26</td></tr><tr><td rowspan="2">X+E</td><td>input-XE(x1)</td><td>10.470</td><td>1.192</td><td>0.56</td><td>10.531</td><td>1.781</td><td>0.59</td><td>10.660</td><td>3.028</td><td>0.56</td><td>10.465</td><td>1.139</td><td>0.56</td></tr><tr><td>input-XE(x8)</td><td>10.417</td><td>0.680</td><td>1.25</td><td>10.453</td><td>1.028</td><td>1.35</td><td>10.526</td><td>1.734</td><td>1.28</td><td>10.415</td><td>0.660</td><td>1.26</td></tr></table>
+
+31], we calculate the performance gap relative to Concorde. For CVRP, we use the non-learning solvers LKH3 [14], HGS [41], and OR-Tools [25]. We calculate the performance gap relative to LKH3.
+
+2) Learning-based Solvers: Edge-only input: We compare with the current state-of-the-art MatNet [23] and GREAT [28], both of which use edge relationships as input for the problems. Among them, MatNet is retrained and evaluated on symmetric instances according to the original model design. Since GREAT has not published the code and data, the results are taken from their paper. Additionally, to fairly compare POMO-based models with UniteFormer, we also re-implemented POMO using edge-only input (denoted as POMO-edge). Node-only input: We use the public models of the classic node-based POMO [24] and LEHD [31] for comparative testing. Hybrid input: GCNBS [18], DAR [44] and ICAM [49] all use edge and node as model input. Since the codes of DAR and ICAM are unavailable, we re-implement and retrain DAR and ICAM according to the official settings, and the number of training epochs and data sets are consistent with our UniteFormer.
+
+# D.2 Experimental Hyperparameters
+
+We report the hyperparameter details common across the main experiments in Table 4. In the table, "Input choice" indicates that during training we randomly select one of the three input modalities for each batch. We employ three layers for the GCN, the MLP, and the node-focused sub-encoder. All normalization operations within the model are implemented using batch normalization [16].
+
+# E Ablation Study of UniteFormer Architecture
+
+Table 5 reports ablation studies comparing UniteFormer against three structural variants to isolate the contributions of each component. One may ask: why incorporate self-attention into the edge-aware sub-encoder? Why disable the node-focused sub-encoder when edge-only input is used? And what if the edge-aware sub-encoder were to operate solely on edge features, ignoring node attributes?
+
+To answer these questions, we design three ablations: The first variant removes self-attention from the edge-aware sub-encoder (denoted by w.o. self-attention). The second variant adds the node-focused sub-encoder, transferring temporary node features to it when edge is as input (denoted by w.o. node-focused sub). The third variant feeds only edge features into the edge-aware sub-encoder, setting node features to zero embeddings (denoted as w.o. edge-aware sub-E). Across all input modalities, each variant trails the full UniteFormer configuration, as shown in Table 5. These results confirm that (a) self-attention in the edge-aware encoder is crucial for capturing edge-node interactions, (b) adding the node-focused sub-branch under edge-only input degrades performance, and (c) jointly leveraging both node and edge features yields superior representations. Overall, the ablation study demonstrates each component's positive contribution to the model and validates the effectiveness and design rationality of the UniteFormer architecture.
+
+Table 6: Experimental results on 10,000 instances of ATSP.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">ATSP20</td><td colspan="3">ATSP50</td><td colspan="3">ATSP100</td></tr><tr><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td><td>Len.</td><td>Gap(%)</td><td>Time(m)</td></tr><tr><td>CPLEX</td><td>1.540</td><td>0.000</td><td>12.14</td><td>1.559</td><td>0.000</td><td>60.03</td><td>1.571</td><td>0.000</td><td>5.01h</td></tr><tr><td>Nearest Neighbor</td><td>2.010</td><td>30.390</td><td>-</td><td>2.101</td><td>34.610</td><td>-</td><td>2.140</td><td>36.100</td><td>-</td></tr><tr><td>Nearest Insertion</td><td>1.800</td><td>16.560</td><td>-</td><td>1.950</td><td>25.160</td><td>-</td><td>2.050</td><td>30.790</td><td>-</td></tr><tr><td>Furthest Insertion</td><td>1.710</td><td>11.230</td><td>-</td><td>1.840</td><td>18.220</td><td>-</td><td>1.940</td><td>23.370</td><td>-</td></tr><tr><td>LKH3</td><td>1.540</td><td>0.000</td><td>0.12</td><td>1.560</td><td>0.000</td><td>0.24</td><td>1.570</td><td>0.000</td><td>1.12</td></tr><tr><td>MatNet(x1)</td><td>1.548</td><td>0.533</td><td>0.06</td><td>1.580</td><td>1.350</td><td>0.15</td><td>1.622</td><td>3.242</td><td>0.62</td></tr><tr><td>MatNet(x8)</td><td>1.542</td><td>0.084</td><td>0.31</td><td>1.566</td><td>0.472</td><td>1.23</td><td>1.603</td><td>2.086</td><td>4.24</td></tr><tr><td>MatNet(x128)</td><td>1.540</td><td>0.012</td><td>4.72</td><td>1.561</td><td>0.144</td><td>16.55</td><td>1.590</td><td>0.934</td><td>61.03</td></tr><tr><td>UniteFormer(x1)</td><td>1.543</td><td>0.245</td><td>0.15</td><td>1.573</td><td>0.917</td><td>0.24</td><td>1.621</td><td>3.170</td><td>0.76</td></tr><tr><td>UniteFormer(x8)</td><td>1.540</td><td>0.043</td><td>0.36</td><td>1.564</td><td>0.339</td><td>1.22</td><td>1.600</td><td>1.841</td><td>5.07</td></tr><tr><td>UniteFormer(x128)</td><td>1.540</td><td>0.001</td><td>4.01</td><td>1.561</td><td>0.129</td><td>18.09</td><td>1.585</td><td>0.880</td><td>1.32h</td></tr></table>
+
+# F Asymmetric Traveling Salesman Problem (ATSP)
+
+# F.1 Problem Setup
+
+In the classic TSP, the objective is to determine a tour over  $N$  nodes that minimizes the total round-trip distance. For any two nodes  $x_{i}$  and  $x_{j}$ , the edge distance satisfies  $d(x_{i},x_{j}) = d(x_{j},x_{i})$ , yielding an  $N\times N$  symmetric distance matrix. To demonstrate UniteFormer's strong generalization and scalability beyond this symmetric setting, we extend our investigation to the Asymmetric Traveling Salesman Problem (ATSP). In ATSP, one seeks the shortest Hamiltonian circuit in a directed, weighted graph that visits each vertex exactly once and returns to the start. Unlike the classic TSP, the ATSP's distance matrix is non-symmetric, with edge weights satisfying  $d(x_{i},x_{j})\neq (x_{j},x_{i})$ . Thus, it must accommodate both directional and weight asymmetries.
+
+Our edge-only UniteFormer framework naturally accommodates these asymmetric instances without modification and continues to exhibit excellent performance. Following MatNet's experimental protocol, we evaluate on tmat class ATSP instances that satisfy the triangle inequality and are widely used in the operations-research (OR) community [7]. We solve three problem sizes  $(N = 20, 50, \text{and} 100)$  and confirm that UniteFormer retains its robustness and scalability under the more general ATSP setting.
+
+# F.2 Experiment Results
+
+Our UniteFormer is able to take edge-only information as input and can be easily extended to solve ATSP. We use randomly generated asymmetric distance matrices as input. Both the training and test datasets follow the data generation method of MatNet.
+
+Table 6 reports the performance of our trained model compared with other representative baseline algorithms on 10,000 test instances of tmat class ATSP. For each method, we report the average tour length (shown in units of  $10^{6}$ ) and the percentage gap relative to CPLEX's optimal solutions. One of the key baselines compared in the table is the MatNet, which is an edge-based construction method designed for solving ATSP.
+
+The results in Table 6 demonstrate that UniteFormer consistently outperforms MatNet under both greedy inference  $(\times 1)$  and instance augmentation  $(\times 8, \times 128)$  inference, while maintaining relatively reasonable inference times. In particular, on the largest ATSP100 instances, our UniteFormer attains a modest performance advantage, albeit with a slightly increased runtime. These experiments confirm that our edge-driven UniteFormer achieves competitive performance on ATSP, showcasing its extensive adaptability and strong generalization. Collectively, they validate the effectiveness of the UniteFormer architecture for a broad spectrum of optimization challenges.
+
+# G Broader impacts
+
+The paper introduces UniteFormer, a unified neural solver for VRPs that flexibly accommodates three distinct input modalities through joint edge-node training. This design significantly enhances the practicality of neural solvers for real-world applications. Moreover, the underlying techniques generalize beyond VRPs, opening the door to a wide array of combinatorial optimization tasks.
+
+Our experiments demonstrate that, compared to state-of-the-art neural solvers trained on a single input modality, UniteFormer consistently delivers superior performance across diverse VRP variants. This unified approach not only streamlines training by eliminating the need for separate models per modality but also reduces overall engineering complexity. Nonetheless, the heavy encoder underpinning UniteFormer demands substantial computational resources, which poses a challenge for scaling to very large problem instances. Future work will focus on architecting more lightweight yet expressive models to bridge this gap.
\ No newline at end of file
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+# Universal Causal Inference in a Topos
+
+Sridhar Mahadevan*
+
+Adobe Research
+
+San Jose, CA
+
+smahadev@adobe.com
+
+# Abstract
+
+In this paper, we explore the universal properties underlying causal inference by formulating it in terms of a topos. More concretely, we introduce topos causal models (TCMs), a strict generalization of the popular structural causal models (SCMs). A topos category has several properties that make it attractive: a general theory for how to combine local functions that define "independent causal mechanisms" into a consistent global function building on the theory of sheaves in a topos; a generic way to define causal interventions using a subobject classifier in a topos category; and finally, an internal logical language for causal and counterfactual reasoning that emerges from the topos itself. A striking characteristic of subobject classifiers is that they induce an intuitionistic logic, whose semantics is based on the partially ordered lattice of subobjects. We show that the underlying subobject classifier for causal inference is not Boolean in general, but forms a Heyting algebra. We define the internal Mitchell-Bénabou language, a typed local set theory, associated with causal models, and its associated Kripke-Joyal intuitionistic semantics. We prove a universal property of TCM, namely that any causal functor mapping decomposable structure to probabilistic semantics factors uniquely through a TCM representation.
+
+# 1 Introduction
+
+In recent years, there has been significant interest in categorical models of causality, based on symmetric monoidal categories [Fong, 2012, Fritz and Klingler, 2023, Cho and Jacobs, 2019, Jacobs et al., 2018], as well as simplicial sets and higher-order categories [Mahadevan, 2023]. Markov categories [Fritz, 2020] define a broad unifying framework for probabilistic inference and statistics using symmetric monoidal categories, where each object is additionally equipped with a comonoidal "copy-delete" operation. It enables carrying out rigorous proofs using an elegant string diagrammatic language [Selinger, 2010]. Any causal model based on graphs [Pearl, 2009, Forre and Mooij, 2017, Spirtes et al., 2000] or other algebraic formalisms, such as integer-valued multisets [Studeny, 2010], can be translated into a string diagram over a symmetric monoidal category, or a simplicial set. Operations on causal models, such as interventions, can be modeled as functors on the objects of the associated symmetric monoidal category or simplicial set. Categorical approaches to causality also extend to the potential outcomes counterfactual framework [Imbens and Rubin, 2015].
+
+Categorical approaches fundamentally differ from past work in causality in their focus on the elucidation of universal properties. In our previous work [Mahadevan, 2023, 2025c], we introduced the framework of universal causality based on the notion of universal properties
+
+![](images/0183119fe1e0deb7c6df95196c89a2591a60db417cbba8a66f0cf0cbe3afeb0a.jpg)  
+Figure 1: Topos causal models (TCMs) are defined as a category  $\mathcal{C}_{\mathcal{TCM}}$  whose objects  $c \in C_{\mathcal{TCM}}$  are causal models, and whose arrows  $\mathcal{C}_{\mathcal{TCM}}(c, c')$  are commutative diagrams between models  $c$  and  $c'$ . A specific object  $c$  defining a model can be conceptualized as a DAG (left, where information flows from top to bottom), or a string diagram in a Markov category (middle, where information flows from bottom to top), or in terms of its induced unique "blackbox" function mapping exogenous variables to endogenous variables (right).
+
+in category theory [Riehl, 2017]: a causal property is universal if it can be defined in terms of an initial or final object in a category of causal diagrams, or in terms of a causal representable functor using the Yoneda Lemma. For example, a structural causal model (SCM) [Pearl, 2009] is defined as a (deterministic) mapping from a collection of exogenous variables into a collection of endogenous variables, derived by "collating" local functions that serve as independent causal mechanisms [Galles and Pearl, 1988, Parascandolo et al., 2017]. However, SCMs can be further analyzed in terms of their universal properties, such as categorical product, coproduct, limits and colimits, equalizers and coequalizers etc. These latter properties can be shown formally to be initial or final objects in a category of diagrams [Riehl, 2017], or as representable functors through the Yoneda Lemma [MacLane, 1971].
+
+Our main contribution in this paper is to present a topos-theoretic view of causality, and in particular, introduce topos causal models (TCMs) that strictly generalize structural causal models (SCMs) [Pearl, 2009]. A topos is a type of category [MacLane, 1971], which is particularly well-suited to modeling operations that are "set-like" [MacLane and leke Moerdijk, 1994]. It also features an internal logical language [Goldblatt, 2006]. We claim that a topos provides three universal properties that make it natural as a category to do causal inference in: it provides a general theory for how to combine local functions, which can be viewed as "independent causal mechanisms" [Parascandolo et al., 2017], into a consistent global function building on the theory of sheaves in a topos [Mac Lane and Moerdijk, 1992]. It enables a generic way to define causal interventions using a subobject classifier in a topos category [Johnstone, 2014]. Finally, it gives an internal logical language for causal and counterfactual reasoning [Bell, 1988].
+
+As Figure 1 illustrates, the objects in a TCM category can be conceptualized in multiple ways. First, each object can be a causal graphical model [Pearl, 1989, Spirtes et al., 2000]. Each object can also be a functor: for example, directed graphs form a topos functor category [Vigna, 2003]. TCMs can also be defined in terms of string diagrams in a symmetric monoidal Markov category [Fritz, 2020], where we restrict ourselves to the Markov subcategory defined through deterministic morphisms. For example, the arrow  $h : \text{Traffic} \otimes \text{Agricultural Fires} \to \text{Pollution}$  defines a deterministic mapping specifying the two potential causes of Pollution. For exogenous variables, the arrow  $\psi : I \to \text{Overpopulation}$  defines the marginal distribution on Overpopulation, where  $I$  is the terminal object in the Markov category. Finally, we can view a TCM object as a "blackbox" function that maps some collection of exogenous variables (e.g., "Overpopulation", or "Farming Practices" into some set of endogenous variables, e.g., "Asthma" or "Pollution").
+
+# 2 Principles of Universal Causality
+
+We give a brief overview of the fundamentals of universal causality (UC) [Mahadevan, 2023, 2025c] before delving into the specific details of the TCM framework. As with other work in categorical causality [Fong, 2012, Jacobs et al., 2018, Fritz and Klingler, 2023], UC uses category theory [MacLane, 1971] to define causality. A category  $C$  is a collection of abstract objects  $c \in C$ . Anything technically can count as an object, from a variable in a causal model to an entire model itself. Each category  $C$  is additionally specified by a set of arrows or morphisms  $C(c,d)$  between each pair of objects  $c$  and  $d$ . There is an identity arrow  $1_c \in C(c,c)$ . Arrows compose in the obvious way, inducing a function  $C(c,d) \times C(d,e) \to C(c,e)$ . An initial object  $c$  in category  $C$  defined as one inducing a unique arrow from  $c$  to every object in category  $C$ . A terminal object, usually denoted by  $\mathbf{1}$ , is one that defines a unique arrow from every object  $c$  in category  $C$  into  $\mathbf{1}$ . An object  $c$  is isomorphic to another object  $d$ , denoted  $c \simeq d$ , if two arrows  $f : c \to d$  and  $g : d \to c$  exist, such that  $g \circ f = 1_c$ , and  $f \circ g = 1_d$ . A functor  $F : C \to D$  between two categories  $C$  and  $D$  is specified by an object function mapping each  $c \in C$  to  $Fc \in D$ , and an arrows function mapping each arrow  $f \in C(c,d)$  to  $Ff \in D(Fc,Fd)$ . Functors come in two varieties - covariant and contravariant - the latter acts on the domain category by reversing the arrows. Given any two functors  $F : C \to D$  and  $G : C \to D$  between the same pair of categories, we can define a mapping between  $F$  and  $G$  that is referred to as a natural transformation. These are defined through a collection of mappings, one for each object  $c$  of  $C$ , thereby defining a morphism in  $D$  for each object in  $C$ .
+
+![](images/edcf30caa15abeeac0435f5eb16f09c86183f7a8eba1f14ad64cc5f21ad35ea0.jpg)
+
+# 2.1 Yoneda Lemma and the Causal Reproducing Property
+
+UC rests on the Yoneda Lemma – any object in a category can be defined by the interactions it makes with other objects (upto isomorphism). In the setting of causal inference, it means that objects in a TCM category can be ascribed “meaning” through studying the arrows of the category, without having to “look inside” the object. The Yoneda Lemma states that the set of all morphisms into an object  $d$  in a category  $C$ , sometimes denoted as  $\mathbf{Hom}_C(-,d)$ , or as  $C(-,d)$ , denoted as the presheaf, is sufficient to define  $d$  up to isomorphism. The category of all presheaves forms a category of functors, and is denoted  $\hat{C} = \mathbf{Set}^{C^{\mathrm{op}}}$ . This category forms a topos, and will be fundamental to the TCM framework.
+
+Lemma 1. [MacLane, 1971, Riehl, 2017] Yoneda lemma: For any functor  $F: C \to \mathbf{Set}$ , whose domain category  $C$  is "locally small" (meaning that the collection of morphisms between each pair of objects forms a set), and any object  $c$  in  $C$ , there is a bijection  $\mathrm{Hom}(C(-,c),F) \simeq Fc$  that associates a natural transformation  $\alpha: C(-,c) \Rightarrow F$  to the element  $\alpha_{c}(1_{c}) \in Fc$ . This correspondence is natural in both  $c$  and  $F$ .
+
+Definition 1. [Riehl, 2017] A universal property of an object  $c \in C$  in a category  $C$  is expressed by a representable functor  $F$  together with a universal element  $x \in Fc$  that defines a natural isomorphism  $C(-,c) \simeq F$ . The collection of morphisms  $C(-,c)$  into an object  $c$  is called the presheaf, and from the Yoneda Lemma, forms a universal representation of the object.
+
+# 2.2 Diagrams and Universal Constructions
+
+A key distinguishing feature of category theory is the use of diagrammatic reasoning. However, diagrams are also viewed more abstractly as functors mapping from some indexing category to the actual category. Diagrams are useful in understanding universal constructions, such as limits and colimits. Briefly, a diagram  $F: \mathcal{I} \to \mathcal{C}$  is a functor  $F$  from some finite category  $\mathcal{I}$  into a category of interest,  $\mathcal{C}$ . For example,  $\mathcal{I} = \bullet \to \bullet \leftarrow \bullet$  is an example of a "pullback" diagram. Here the  $\bullet$  refer to abstract objects that are mapped into concrete objects in  $\mathcal{C}$  by the functor  $F$ . What we want to know whether a particular diagram
+
+$F$  or an entire class of diagrams is "solvable". What this means is whether its limit or colimit exists, that is, is the category complete or co-complete? For any object  $c \in C$  and any category  $J$ , the constant functor  $c: J \to C$  maps every object  $j$  of  $J$  to  $c$  and every morphism  $f$  in  $J$  to the identity morphisms  $1_c$ . We can define a constant functor embedding as the collection of constant functors  $\Delta: C \to C^J$  that send each object  $c$  in  $C$  to the constant functor at  $c$  and each morphism  $f: c \to c'$  to the constant natural transformation, that is, the natural transformation whose every component is defined to be the morphism  $f$ .
+
+Definition 2. [Riehl, 2017] A cone over a diagram  $F: J \to C$  with the summit or apex  $c \in C$  is a natural transformation  $\lambda: c \Rightarrow F$  whose domain is the constant functor at  $c$ . The components  $(\lambda_j: c \to Fj)_{j \in J}$  of the natural transformation can be viewed as its legs. Dually, a cone under  $F$  with nadir  $c$  is a natural transformation  $\lambda: F \Rightarrow c$  whose legs are the components  $(\lambda_j: F_j \to c)_{j \in J}$ .
+
+![](images/e24fbb8a14b62e01891a2f46d50554777285007c02ad31f07e8b19c098faba54.jpg)
+
+![](images/1ac41af67a9bb46918b214d575c3e09aed095cd0eec549d7302e39f5047ec5e7.jpg)
+
+Cones under a diagram are referred to usually as cocones. Using the concept of cones and cocones, we can now formally define the concept of limits and colimits more precisely.
+
+Definition 3. [Riehl, 2017] For any diagram  $F: J \to C$ , there is a functor  $\operatorname{Cone}(-,F): C^{op} \to \operatorname{Set}$  which sends  $c \in C$  to the set of cones over  $F$  with apex  $c$ . Using the Yoneda Lemma, a limit of  $F$  is defined as an object  $\lim F \in C$  together with a natural transformation  $\lambda: \lim F \to F$ , which can be called the universal cone defining the natural isomorphism  $C(-,\lim F) \simeq \operatorname{Cone}(-,F)$ . Dually, for colimits, we can define a functor  $\operatorname{Cone}(F,-): C \to \operatorname{Set}$  that maps object  $c \in C$  to the set of cones under  $F$  with nadir  $c$ . A colimit of  $F$  is a representation for  $\operatorname{Cone}(F,-)$ . Once again, using the Yoneda Lemma, a colimit is defined by an object  $\operatorname{Colim}F \in C$  together with a natural transformation  $\lambda: F \to \operatorname{colim}F$ , which defines the colimit cone as the natural isomorphism  $C(\operatorname{colim}F,-) \simeq \operatorname{Cone}(F,-)$ .
+
+Limit and colimits of diagrams over arbitrary categories can often be reduced to the case of their corresponding diagram properties over sets.
+
+# 2.3 The Universality of Diagrams and the Causal Reproducing Property
+
+We state two key results that underly UC [Mahadevan, 2023]. While both these results follow directly from basic theorems in category theory, their significance for causal inference is what makes them particularly noteworthy. The first result pertains to the notion of diagrams as functors, and shows that for the functor category of presheaves, which is a universal representation of causal inference, every presheaf object can be represented as a colimit of representables through the Yoneda Lemma. This result can be seen as a generalization of the very simple result in set theory that each set is a union of one element sets. The second result is the causal reproducing property, which shows that the set of all causal effects between two objects is computable from the presheaf functor objects defined by them. Both these results are abstract, and apply to any category representation of a causal model.
+
+Theorem 1. [MacLane and leke Moerdijk, 1994, Mahadevan, 2023] Universality of Diagrams in UC: In the functor category of presheaves  $\mathsf{Set}^{C^{\mathrm{op}}}$ , every functor object  $F$  is the colimit of a diagram of representable objects, in a canonical way.
+
+To explain the significance of this result for causal inference, note that UC represents causal diagrams as functors from an indexing category of diagrams to an actual causal model. The theorem above tells us that every presheaf object can be represented as a colimit of (simple) representable objects, namely functor objects of the form  $\mathbf{Hom}_C(-,c)$ .
+
+Theorem 2. [MacLane, 1971, Mahadevan, 2023] Causal Reproducing Property: All causal influences between any two objects  $c$  and  $d$  can be derived from its presheaf functor objects, namely
+
+$$
+\mathbf {H o m} _ {C} (c, d) \simeq \mathbf {N a t} (\mathbf {H o m} _ {C} (-, c), \mathbf {H o m} _ {C} (-, d))
+$$
+
+Any causal influence of an object  $c$  upon any other object  $d$  can be represented as a natural transformation (a morphism) between two functor objects in the presheaf category  $\hat{C}$ .
+
+# 3 Topos Causal Models
+
+A topos [Johnstone, 2014] is a "set-like" category which generalizes all common operations on sets. Thus, the generalization of subset is a subobject classifier in a topos. To help build some intuition, consider how to define subsets without "looking inside" a set. Essentially, a subset  $S$  of some larger set  $T$  can be viewed as a "monic arrow" (an injective function). Monic arrows generalize injective functions.
+
+Definition 4. [Goldblatt, 2006] An arrow  $f: a \to b$  in a category  $C$  is called monic if given any parallel arrows  $g, h: c \rightarrow a$ , the equality  $f \circ g = f \circ h$  implies that  $g = h$ , namely  $f$  is "left-cancellable".
+
+Our approach builds on this abstraction to define a category  $C_{\mathcal{TCM}}$  whose objects are causal models, such as SCMs or Markov categories, and a submodel  $M_x$  of an SCM  $M$  is simply a monic arrow  $f_x: M_x \hookrightarrow M$ .
+
+Definition 5. A category  $C$  has binary products if for every pair of objects,  $c$  and  $d$ , there exists a third object,  $e \simeq c \times d$ , along with two projection arrows,  $p_1 : e \to c$  and  $p_2 : e \to d$ , such that for any other object  $a$  and arrows  $f : a \to c$  and  $g : a \to d$ , there exists a unique morphism  $u : a \to e$  satisfying  $p_1 \circ u = f$  and  $p_2 \circ u = g$ .
+
+Definition 6. [MacLane and leke Moerdijk, 1994] A category  $C$  with binary products has exponential objects if for each pair of objects  $c, d$  in  $C$ , there exists an object  $c^d$  that defines the following bijection:
+
+$$
+C (e \times d, c) \simeq C (e, c ^ {d})
+$$
+
+Definition 7. [MacLane and leke Moerdijk, 1994] A category  $C$  is Cartesian closed if it has binary products, a terminal object  $\mathbf{1}$ , and exponential objects.
+
+Definition 8. [MacLane and leke Moerdijk, 1994] In a category  $C$  with finite limits, a subobject classifier is a  $C$ -object  $\Omega$ , and a  $C$ -arrow true:  $1 \to \Omega$ , such that to every monic arrow  $S \hookrightarrow X$  in  $C$ , there is a unique arrow  $\phi$  that forms the following pullback square:
+
+![](images/23889c4d31991baae749195809a599e33bda233a2722f214d5c884d58648463b.jpg)
+
+This commutative diagram enforces the constraint that every monic arrow  $m$  (i.e., every  $1 - 1$  function) that maps a subobject  $S$  to an object  $X$  must be characterizable in terms of a "pullback", a particular type of universal property that is a special type of a limit. In the special case of the category of sets, subobject classifiers are defined through the characteristic (Boolean-valued) function  $\phi$  that defines subsets. In general, as we show below, the subobject classifier  $\Omega$  for causal models is not Boolean-valued, and requires using intuitionistic logic through a Heyting algebra. This definition can be rephrased as saying that the subobject functor is representable. In other words, a subobject of a causal model  $X$  in category  $C_{\mathcal{TCM}}$  is an equivalence class of monic arrows  $m: S \hookrightarrow X$ .
+
+Definition 9. MacLane and leke Moerdijk [1994] An elementary topos is a category  $C$  that is Cartesian closed and has a subobject classifier.
+
+For example, the category of sets forms a topos. Limits exist because one can define Cartesian products of sets, and colimits correspond to forming set unions. Exponential objects correspond to the set of all functions between two sets. Finally, the subobject classifier is simply the subset function, which induces a boolean-valued characteristic function.
+
+Definition 10. The category  $C_{\mathcal{TCM}}$  of topos causal models is defined as a collection of objects  $c \in C_{\mathcal{TCM}}$ , each of which is a triple  $\langle U, V, F \rangle$  where  $V = \{V_1, \ldots, V_n\}$  is a set of endogenous
+
+variables,  $U$  is a set of exogenous variables, and  $F$  is a function from  $U$  to  $V$ . The arrows  $C_{\mathcal{TCM}}(c,d)$  are defined through commutative diagrams as illustrated below, where  $f$  and  $f'$  are the global functions induced by the TCM objects  $c$  and  $d$ , respectively.
+
+![](images/6be860df975c8ee29c13da30e2823ee4ff64d819c6e58da847f00b75addd1030.jpg)
+
+A submodel  $c' = \langle U', V', F' \rangle$  of  $c$  is any subobject of  $c$ . The effect of an intervention on  $c$  is given by some submodel  $c'$ . Finally, let  $Y$  be a variable in  $V$ , and let  $X$  be a subset of  $V$ . The potential outcome in response to an intervention on  $X$  modeled by a submodel  $c' \hookrightarrow c$  is the solution of  $Y$  in the submodel  $c'$ .
+
+A commutative diagram, as the term suggests, is a structure showing the equivalence of two paths. Here, the diagram asserts that  $g \circ f = f' \circ h$ . In the context of our category  $C_{\mathit{SCM}}$ , the arrow  $f: U \to V$  is simply an SCM  $M$ , and  $f$  is its induced mapping from exogenous to endogenous variables. Similarly,  $f'$  is also the induced function mapping exogenous to endogenous variables for another SCM  $M'$ . The morphisms  $h$  and  $g$  are functions on SCMs, which transform one causal model into another. In the specific case we are interested in, these functions define causal interventions, but in general, they may be arbitrary functions.
+
+For completeness, we define a category  $C_{\mathrm{SCM}}$  whose objects are indeed SCMs.
+
+Definition 11. The category  $\mathcal{C}_{\mathrm{SCM}}$  of structural causal models is defined as a collection of objects, each of which is a triple  $\langle U, V, F \rangle$  where  $V = \{V_1, \ldots, V_n\}$  is a set of endogenous variables,  $U$  is a set of exogenous variables,  $F$  is a set  $\{f_1, \ldots, f_n\}$  of "local functions"  $f_i: U \cup (V \setminus V_i) \to V_i$  whose composition induces a unique function  $F$  from  $U$  to  $V$ . Let  $X$  be a subset of variables in  $V$ , and  $x$  be a particular realization of  $X$ . A submodel  $M_x = \langle U, V, F_x \rangle$  of  $M$  is the causal model  $M_x = \langle U, V, F_x \rangle$ , where  $F_x = \{f_i : V_i \notin X\} \cup \{X = x\}$ . The effect of an action  $do(X = x)$  on  $M$  is given by the submodel  $M_x$ . Finally, let  $Y$  be a variable in  $V$ , and let  $X$  be a subset of  $V$ . The potential outcome of  $Y$  in response to an action  $do(X = x)$ , denoted  $Y_x(u)$ , is the solution of  $Y$  for the set of equations  $F_x$ .
+
+The set of arrows or morphisms between two objects  $c$  and  $d$  in the category  $\mathcal{C}_{\mathrm{SCM}}$ , denoted  $\mathcal{C}_{\mathrm{SCM}}(c,d)$ , represent ways of transitioning from SCM object  $c$  to  $d$ . For example, if  $d$  is a submodel of  $c$ , then the arrow defines a do calculus causal intervention.
+
+# 4 Causal Inference in a Topos Category
+
+We show in this section that a TCM category whose objects are defined as SCMs, and whose arrows correspond to commutative diagrams defining operations on causal models does define a topos. In the next section, we generalize from SCMs to consider more complex causal models over functor categories. Now, we can state the first key result of this paper.
+
+Theorem 3. The category  $C_{\mathit{SCM}}$  forms a topos.
+
+Proof: Since we have previously defined the objects and arrows of the  $C_{\mathcal{SCM}}$  category, to show it forms a topos, we need to construct its subobject classifier. First, we need to define what a "subobject" is in the category  $C_{\mathcal{SCM}}$ . Since SCMs can abstractly be defined as functions, let us assume that the SCM  $c$  that defines  $f$  is a submodel of the SCM  $c'$  that induces  $g$ . We can denote that by defining a commutative diagram as shown below. Let us stress the difference between the commutative diagram shown below Definition 10 for arbitrary functions  $g$  and  $h$  vs. the one below, where  $i$  and  $j$  are monic arrows.
+
+![](images/26cb2200ea9475f028fa6ec1a2496593a8820883768690e9322d60aea1c6980b.jpg)
+
+![](images/110fd591803f20f7a1de96bd1a730764f2bf587ecbde89ebff9c6581f27b67b2.jpg)  
+Figure 2: Left: diagram showing that  $C_{SCM}$  has pullbacks. Right: The subobject classifier  $\Omega$  for the topos category  $C_{SCM}$  is displayed on the bottom face of this cube.
+
+![](images/393193c240fcfeb50d155c46ff8d60a8cfc0106b610aa98e476f05e810dcfc34.jpg)
+
+An element  $x \in U'$ , which is a particular realization of the exogenous variables in  $U'$ , can be classified in three ways by defining a characteristic function  $\psi$ :
+
+1.  $x\in U$  - here we set  $\psi (x) = 1$  
+2.  $x \notin U$  but  $g(x) \in V$ -here we set  $\psi(x) = \frac{1}{2}$ .  
+3.  $x \notin U$  and  $g(x) \notin V$  - we denote this by  $\psi(x) = 0$ .
+
+The subobject classifier is illustrated as the bottom face of the cube shown on the right in Figure 2:
+
+true  $(0) = t^{\prime}(0) = 1$  
+-  $\mathbf{t}:\{0,\frac{1}{2},\mathbf{1}\} \to \{0,\mathbf{1}\}$ , where  $\mathbf{t}(\mathbf{0}) = \mathbf{0},\mathbf{t}(\mathbf{1}) = \mathbf{t}(\frac{1}{2}) = \mathbf{1}$ .  
+-  $\chi_V$  is the characteristic function of the exogenous variable set  $V$ .  
+- The base of the cube in Figure 2 displays the subobject classifier  $\mathbf{T}:\mathbf{1}\to \mathbf{\Omega},$  where  $\mathbf{T} = \langle \mathbf{t}^{\prime},\mathbf{true}\rangle$  that maps  $\mathbf{1} = \mathbf{id}_{\{0\}}$  to  $\Omega = \mathbf{t}:\{0,\frac{1}{2},1\} \rightarrow \{0,1\}$
+
+This proves that the subobject classifier for the category  $C_{SCM}$  does not have Boolean semantics, but intuitionistic semantics as its subobject classifier  $\Omega$  has multiple degrees of "truth", corresponding to the three types of classifications of monic arrows (in regular set theory, there are only two classifications). Moving on to show the other properties of a topos are satisfied, note that the terminal object is simply the identity function  $\mathbf{id}_0: \{0\} \to \{0\}$ . Now, it remains to show that  $C_{SCM}$  has pullbacks and exponential objects.
+
+Pullbacks in  $C_{\mathit{SCM}}$ : Consider the cube shown on the left in Figure 2. Here,  $f, g,$  and  $h$  can be interpreted as three SCMs, each mapping some exogenous variables to some endogenous variables. The arrows  $i, j$  ensure that the bottom face of the cube is a commutative diagram, and the arrows  $p, q$  ensures the right face of the cube is a commutative diagram. The arrow from  $P$  to  $Q$  exists because looking at the front face of the cube,  $Q$  is the pullback of  $i$  and  $q$ , which must exist because we are in the category of Sets, which has all pullbacks. Similarly, the back face of the cube is a pullback of  $j$  and  $p$ , which is again a pullback in Sets. Summarizing,  $\langle u, v \rangle$  and  $\langle m, n \rangle$  are the pullbacks of  $\langle i, j \rangle$  and  $\langle p, q \rangle$ .
+
+Exponential objects in  $C_{\mathit{SCM}}$ : Now it only remains to check that the category has exponential objects. Let  $f: U \to V$  and  $g: U' \to V'$  be two functions induced by SCM models  $M$  and  $N$ . Then, we need to define the meaning of  $g^f$  in  $C_{\mathit{SCM}}$ , which we can define as  $g^f: X \to Y$ , where  $Y = V'^V$ , which must exist since Sets is a Cartesian closed category that has exponential objects (i.e.,  $Y$  is simply the set of all functions from  $V$  to  $V'$ ). Also,  $X$  is the set of all arrows in  $C_{\mathit{SCM}}$  from SCM  $M$  to SCM  $N$ , which is the pair of functions  $\langle h, k \rangle$  in the commutative diagram shown below. This finally proves that  $C_{\mathit{SCM}}$  is a topos.
+
+# 5 Causal Models Over a Topos of Sheaves
+
+We now describe a more general categorical framework for defining causal models as a topos by using the property that Yoneda embeddings of presheaves forms a topos [MacLane and leke Moerdijk, 1994]. To ensure consistent extension into a unique global function, we build on the theory of sheaves [Mac Lane and Moerdijk, 1992], which ensures local functions can be "collated" together to yield a unique global function. In our setting, we will construct sheaves from categories over causal models through the Yoneda embedding  $\mathcal{X}(x): \mathcal{C} \to \mathbf{Sets}^{C^{\mathrm{op}}}$  and impose a Grothendieck topology.
+
+# 5.1 Grothendieck Topology on Sites
+
+Definition 12. A sieve for any object  $x$  in any (small) category  $C$  is a subobject of its Yoneda embedding  $\mathfrak{X}(x) = C(-,x)$ . If  $S$  is a sieve on  $x$ , and  $h: y \to x$  is any arrow in category  $C$ , then
+
+$$
+h ^ {*} (S) = \{g \mid c o d (g) = D, h g \in S \}
+$$
+
+Definition 13. [Mac Lane and Moerdijk, 1992] A Grothendieck topology on a category  $C$  is a function  $J$  which assigns to each object  $x$  of  $C$  a collection  $J(x)$  of sieves on  $x$  such that
+
+1. the maximum sieve  $t_x = \{f| \text{cod}(f) = x\}$  is in  $J(x)$ .  
+2. If  $S \in J(x)$  then  $h^{*}(S) \in J(y)$  for any arrow  $h: y \to x$ .  
+3. If  $S \in J(x)$  and  $R$  is any sieve on  $x$ , such that  $h^*(R) \in J(y)$  for all  $h : y \to x$ , then  $R \in J(C)$ .
+
+We can now define categories with a given Grothendieck topology as sites.
+
+Definition 14. A site is defined as a pair  $(C, J)$  consisting of a small category  $C$  and a Grothendieck topology  $J$  on  $C$ .
+
+Definition 15. The subobject classifier  $\Omega$  is defined on any topos Sets $^{C^{\mathrm{op}}}$  as subobjects of the representable functors:
+
+$$
+\Omega (x) = \{S | S i s a s u b o j e c t o f C (-, x) \}
+$$
+
+and the morphism true:  $1 \to \Omega$  is true(x) = x for any representable  $x$ .
+
+# 5.2 Universal Property of TCM over Functor Categories
+
+Causal models, like SCMs, must represent both decomposable structure and (probabilistic) semantics. To capture this richer structure, we define TCM over functor categories, where every object is a functor that maps structure to semantics. For example, the category of Bayesian networks can be modeled as a functor category [Jacobs et al., 2018, Fritz and Klingler, 2023] from a Markov category to the category FinStoch of finite stochastic processes.
+
+Theorem 4. Given a causal functor  $A: \mathcal{C} \to \mathcal{E}$ , such as the Bayesian network functor  $F_{\mathsf{CDU}}$ , from a small category  $\mathcal{C}$  (e.g., a symmetric monoidal category such as a Markov category) to a cocomplete category  $\mathcal{E}$  (e.g., the category  $\mathbf{Prob}$  of probability spaces (see Theorem 6)), the functor  $R$  from  $\mathcal{E}$  to presheaves, given by (where  $c \in \mathcal{C}$  and  $E \in \mathcal{E}$ )
+
+$$
+R (E): c \mapsto \mathbf {H o m} _ {\mathcal {E}} (A (c), E)
+$$
+
+has a left adjoint  $L: \mathbf{Sets}^{C^{\mathrm{op}}} \to \mathcal{E}$  defined for each presheaf  $P$  in  $C^{\mathrm{op}}$  as the colimit
+
+$$
+L (P) = \operatorname {C o l i m} \left(\int_ {\mathcal {C}} P \xrightarrow {\pi_ {P}} \mathcal {C} \xrightarrow {A} \mathcal {E}\right)
+$$
+
+where  $\int_{C} P$  is the category of elements, whose objects are pairs  $(c, p)$ , where  $c$  is an object of  $C$  and  $p$  is an element of  $P(C)$  (recall  $P$  is a presheaf, i.e., a set-valued functor that maps each element  $c$  into a set), and its arrows are  $(c', p') \to (c, p)$  for any morphism  $f: c' \to c$  such that  $pc = p'$ .
+
+Proof: Essentially, Theorem 4 is stating that there is a pair of adjoint functors  $L \vdash R$ , defined as:
+
+$$
+L: \mathbf {S e t s} ^ {C o p} \xleftarrow {\rightharpoonup} \mathcal {E}: R
+$$
+
+As defined earlier, a natural transformation between two functors  $\tau : P \to R(E)$  is a family  $\{\tau_c\}$  of maps indexed by the objects  $c \in C$ , where each map  $\tau_c$  is defined as the mapping:
+
+$$
+\tau_ {c}: P (C) \mapsto \mathbf {H o m} _ {\mathcal {E}} (A (C), E)
+$$
+
+which is natural in  $c$ .  $\tau$  can also be defined as a set of arrows of  $\mathcal{E}$  as  $\{\tau_c(p):A(c)\to E\}_{(c,p)}$  that is indexed by the objects  $(c,p)$  of the category  $\int_{C}P$  of elements of  $P$ . This fact implies that there is a bijection
+
+$$
+N a t (P, R (E)) \simeq \mathbf {H o m} _ {\mathcal {E}} (L P, E)
+$$
+
+This bijection being natural in  $P$  and in  $E$  proves that  $L$  is a left adjoint functor to  $R$ .
+
+Now, let us define a general causal functor as mapping from a decomposable symmetric monoidal category (e.g., a Markov category) to the symmetric monoidal category of probability spaces.
+
+Definition 16. A causal functor  $F: C \to \mathbf{Prob}$  maps from a general symmetric monoidal category  $C$  with a comonoidal "copy-delete" structure (e.g., a CDU category [Jacobs et al., 2018] or a Markov category [Fritz, 2020]) to the category of probability spaces  $\mathbf{Prob}$ , where each object  $(\Omega, \mathcal{F}, \mathbb{P})$  is a probability space, and the arrows are measure-preserving maps, namely  $\mathbf{Prob}(c, d)$ , where  $c = (\Omega_c, \mathcal{F}_c, \mathbb{P}_c)$  and  $d = (\Omega_d, \mathcal{F}_d, \mathbb{P}_d)$ , where  $f \in \mathbf{Prob}(c, d)$  is such that  $\mathbb{P}_c(f^{-1}(A)) = \mathbb{P}_d(A)$  for all  $A \in \mathcal{F}_d$ .
+
+Theorem 5. For each causal functor  $A: C \to \mathcal{E}$  from a small category  $C$  defining the structure of a causal model to a cocomplete category  $\mathcal{E}$  defining its (probabilistic) semantics, there exists a colimit preserving functor  $L: \mathbf{Sets}^{C^{\mathrm{op}}} \to \mathcal{E}$  such that  $A = L \circ \mathfrak{X}$ , where  $\mathfrak{X}$  is the Yoneda embedding.
+
+Proof: The proof is just a special case of Corollary 4 on page 43 in [MacLane and leke Moerdijk, 1994]. To emphasize the importance of the co-completeness condition on  $\mathcal{E}$ , we use the following result from [van Belle, 2024] that the category of probability spaces is co-complete.  $\square$
+
+Theorem 6. [van Belle, 2024] The symmetric monoidal category  $\mathbf{Prob}$  has all colimits of non-empty diagrams.
+
+Proof: The proof, given in [van Belle, 2024], shows that Prob has coproducts and coequalizers. Thus, we can choose Prob as our cocomplete category  $\mathcal{E}$  in Theorem 4.  $\square$
+
+We can finally state the two central results of our paper, the first (Theorem 7) establishes the universal property underlying TCMs, and the second (Theorem 8) shows that causal interventions define a Heyting algebra whose logic is intuitionistic.
+
+Theorem 7. Any causal functor  $F: C \to \mathcal{E}$  from a structural causal category  $C$  (such as a Markov category) to a semantic cocomplete category  $\mathcal{E}$  (such as Prob) factors uniquely through a TCM structure defined by the Yoneda embedding, as given in Theorem 5.
+
+Proof: The proof follows directly from Theorem 4, Theorem 5, Theorem 6, and Definition 16.  $\square$
+
+Definition 17. A Heyting algebra is a poset with all finite products and coproducts, which is Cartesian closed. That is, a Heyting algebra is a lattice, including bottom and top elements, denoted by  $\mathbf{0}$  and  $\mathbf{1}$ , respectively, which associates to each pair of elements  $x$  and  $y$  an exponential  $y^x$ . The exponential is written  $x \Rightarrow y$ , and defined as an adjoint functor:
+
+$$
+z \leq (x \Rightarrow y) i f a n d o n l y i f z \wedge x \leq y
+$$
+
+In other words,  $x \Rightarrow y$  is a least upper bound for all those elements  $z$  with  $z \wedge x \leq y$ . As a concrete example, for a topological space  $X$  the set of open sets  $O(X)$  is a Heyting algebra. The "law of the excluded middle", meaning  $\neg x \vee x = \text{true}$ , does not always hold in a Heyting algebra.
+
+Theorem 8. For any TCM category defined as  $\hat{C} = \mathbf{Sets}^{C^{\mathrm{op}}}$  by the Yoneda embedding  $\mathfrak{L}(c)$  of a small causal category  $C$ , the partially ordered set  $\operatorname{Sub}_{\hat{C}}(P)$  of subobjects generated by causal interventions on any causal functor defined by the presheaf  $P$  is a Heyting algebra.
+
+Proof: This result follows directly from the corresponding result for any category of presheaves (see [MacLane and leke Moerdijk, 1994]), and is based on constructing the complete lattice  $\operatorname{Sub}(P)$  of all subfunctions of  $P$  using a pointwise operation for each object  $c \in C$ , which can be shown to satisfy an infinite distributive law.  $\square$
+
+# 6 Causal Mitchell-Bénabou Language and its Kripke-Joyal Semantics
+
+The Causal Mitchell-Bénabou language (CMBL) is a typed local set theory [Bell, 1988] whose syntax and semantics is defined using the arrows of the  $C_{\mathcal{TCM}}$  topos. The types of CMBL as causal model objects  $M$  of  $C_{\mathcal{TCM}}$ . For each type  $M$ , we assume the existence of variables  $x_M, y_M, \ldots$ , where each such variable has as its interpretation the identity arrow  $\mathbf{1}: M \to M$ . We can construct product objects, such as  $A \times B \times C$ , where terms like  $\sigma$  that define arrows are given the interpretation  $\sigma: A \times B \times C \to D$ .
+
+- Each variable  $x_{M}$  of type  $M$  is a term of type  $M$ , and its interpretation is the identity  $x_{M} = 1: M \to M$ . Here,  $M$  may represent an entire SCM, an individual variable such as Overpopulation in Figure 1, or a causal functor mapping a Markov category to the cocomplete Prob category.  
+- Terms  $\sigma$  and  $\tau$  of types  $C$  and  $D$  that are interpreted as  $\sigma : A \to C$  and  $\tau : B \to D$  can be combined to yield a term  $\langle \sigma, \tau \rangle$  of type  $C \times D$ , whose joint interpretation is given as  $\langle \sigma p, \tau q \rangle : X \to C \times D$ , where  $X$  has the required projections  $p : X \to A$  and  $q : X \to B$ . A causal intervention modeled as an arrow  $f : X \to Y$  in  $C_{TCM}$  can be composed with a term  $\sigma : U \to X$  to yield a term of type  $Y$  as  $f \circ \sigma : U \xrightarrow{\sigma} X \xrightarrow{f} Y$ .  
+- Terms of type  $\Omega$  are defined as formulae of CMBL and can be combined with the usual logical connectives  $\land, \lor, \Rightarrow, \neg$  and quantifiers  $\forall, \exists$  to obtain further terms again of type  $\Omega$ . An expression such as  $\forall x \psi(x, y)$  is interpreted by an arrow  $Y \rightarrow \Omega$  (since  $x$  is not a free variable). A formula  $\psi(x, y)$  in the topos  $C_{TCM}$  is defined to be universally valid in the topos if the corresponding arrow  $\psi(x, y): X \times Y \rightarrow \Omega$  factors through true:  $1 \rightarrow \Omega$ . A formula  $\psi$  without free variables is interpreted as an arrow  $\psi: 1 \rightarrow \Omega$  and is valid if it coincides with the arrow true:  $1 \rightarrow \Omega$ .  
+- Indirect proofs (i.e., reductio ad absurdum) cannot be used in CMBL because the rule of the excluded middle  $\psi \lor \neg \psi$  is not in general valid, nor is the axiom of choice generally true. Instead, the rules of intuitionistic predicate calculus need to be used, as illustrated by de Araujo Fernandes and Haeusler [2009].  
+- The Kripke-Joyal semantics for CMBL is specified using generalized elements. We define an element of a causal model by the morphism  $x: \mathbf{1} \to M$ . Thus, a generalized element  $\alpha: N \to M$  represents the "stage of definition" of  $M$  by  $N$ . We specify the semantics of how an TCM model  $N$  supports any formula  $\phi(\alpha)$ , denoted by  $N \Vdash \phi(\alpha)$  by  $N \Vdash \phi(\alpha)$  if and only if  $\operatorname{Im} \alpha \leq \{x | \phi(x)\}$ . Stated in the form of a commutative diagram, this "forcing" relationship holds if and only if  $\alpha$  factors through  $\{x | \phi(x)\}$ . See [MacLane and leke Moerdijk, 1994, Bell, 1988] for additional details.
+
+![](images/9a50376760e0866924ca3cdfb84a57d92d0155b2f386a70cd3e199e092d1af0d.jpg)
+
+# 7 Limitations and Future Work
+
+There are significant limitations of our TCM framework, which we are currently investigating. We have recently developed an intuitionistic generalization of Pearl's do-calculus termed  $j$ -stable causal inference, which uses the Lawvere-Tierney topology on a topos defined by a modal operator  $j$  on the subobject classifier  $\Omega$  [Mahadevan, 2025a]. In this paper, we define an intuitionistic logic called  $j$ -do-calculus, where we replace global truth with local truth defined by Kripke-Joyal semantics. We are currently working on another paper [Mahadevan, 2025b] that implements  $j$ -do-calculus with well-known causal discovery procedures (e.g., score-based and constraint-based methods) [Zanga and Stella, 2023], and will include experimental results on how to (i) form data-driven  $j$ -covers (via regime/section constructions), (ii) compute chartwise conditional independences after graph surgeries, and (iii) glue them to certify the premises of the  $j$ -do rules in practice.
+
+# 8 Acknowledgments
+
+This research has been funded by Adobe Corporation.
+
+# References
+
+J. L. Bell. Toposes and Local Set Theories. Dover, 1988.  
+Dimitri Bertsekas. Reinforcement Learning and Optimal Control. Athena Scientific, 2019.  
+Kenta Cho and Bart Jacobs. Disintegration and bayesian inversion via string diagrams. Mathematical Structures in Computer Science, 29(7):938-971, March 2019. ISSN 1469-8072. doi: 10.1017/s0960129518000488. URL http://dx.doi.org/10.1017/S0960129518000488.  
+Ricardo Queiroz de Araujo Fernandes and Edward Hermann Haeusler. A topos-theoretic approach to counterfactual logic. Electronic Notes in Theoretical Computer Science, 256: 33-47, 2009. ISSN 1571-0661. doi: https://doi.org/10.1016/j.entcs.2009.11.004. URL https://www.sciencedirect.com/science/article/pii/S157106610900454X. Proceedings of the Fourth Workshop on Logical and Semantic Frameworks, with Applications (LSFA 2009).  
+Brendan Fong. Causal theories: A categorical perspective on bayesian networks. Master's thesis, Oxford University, 2012.  
+Brendan Fong, David I. Spivak, and Rémy Tuyeras. Backprop as functor: A compositional perspective on supervised learning. In 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2019, Vancouver, BC, Canada, June 24-27, 2019, pages 1-13. IEEE, 2019. doi: 10.1109/LICS.2019.8785665. URL https://doi.org/10.1109/LICS.2019.8785665.  
+Patrick Forre and Joris M. Mooij. Markov properties for graphical models with cycles and latent variables, 2017.  
+James Fox. Universal Coalgebras. Dissertation, McGill University, 1976.  
+Tobias Fritz. A synthetic approach to markov kernels, conditional independence and theorems on sufficient statistics. Advances in Mathematics, 370:107239, August 2020. ISSN 0001-8708. doi: 10.1016/j.aim.2020.107239. URL http://dx.doi.org/10.1016/j.aim.2020.107239.  
+Tobias Fritz and Andreas Klingler. The d-separation criterion in categorical probability. Journal of Machine Learning Research, 24(46):1-49, 2023. URL http://jmlr.org/papers/v24/22-0916.html.  
+David Galles and Judea Pearl. An axiomatic theory of counterfactuals. Foundations of Science, 3:151-182, 1988.  
+Robert Goldblatt. Topoi: The Categorical Analysis of Logic. Dover Press, 2006.  
+Paul Halmos.Measure Theory.Springer,1974.  
+Chris Heunen and Jamie Vicary. *Categories for Quantum Theory: An Introduction*. Oxford University Press, November 2019. doi: 10.1093/oso/9780198739623.001.0001.  
+Guido W. Imbens and Donald B. Rubin. Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction. Cambridge University Press, USA, 2015. ISBN 0521885884.  
+Bart Jacobs, Aleks Kissinger, and Fabio Zanasi. Causal inference by string diagram surgery, 2018. URL https://arxiv.org/abs/1811.08338.  
+Peter T Johnstone. *Topos Theory*. Dover Publications, 2014.  
+David Lewis. Counterfactuals and comparative possibility. J. Philos. Log., 2(4):418-446, 1973. doi: 10.1007/BF00262950. URL https://doi.org/10.1007/BF00262950.
+
+Saunders Mac Lane and Ieke Moerdijk. Sheaves in Geometry and Logic a First Introduction to Topos Theory. Springer New York, New York, NY, 1992. ISBN 9781461209270 1461209277. URL http://link.springer.com/book/10.1007/978-1-4612-0927-0.  
+Saunders MacLane. Categories for the Working Mathematician. Springer-Verlag, New York, 1971. Graduate Texts in Mathematics, Vol. 5.  
+Saunders MacLane and leke Moerdijk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer, 1994.  
+Sridhar Mahadevan. Universal causality. Entropy, 25(4):574, 2023. doi: 10.3390/E25040574. URL https://doi.org/10.3390/e25040574.  
+Sridhar Mahadevan. Intuitionistic  $j$ -do-calculus in topos causal models, 2025a. URL https://arxiv.org/abs/2510.17944.  
+Sridhar Mahadevan.  $j$ -stable causal discovery in sites, 2025b. In Preparation.  
+Sridhar Mahadevan. Higher algebraic k-theory of causality. Entropy, 27(5), 2025c. ISSN 1099-4300. doi: 10.3390/e27050531. URL https://www.mdpi.com/1099-4300/27/5/531.  
+J.P. May. Simplicial Objects in Algebraic Topology. University of Chicago Press, 1992.  
+Leland McInnes, John Healy, and James Melville. Umap: Uniform manifold approximation and projection for dimension reduction, 2018. URL https://arxiv.org/abs/1802.03426.  
+Giambattista Parascandolo, Mateo Rojas-Carulla, Niki Kilbertus, and Bernhard Scholkopf. Learning independent causal mechanisms. CoRR, abs/1712.00961, 2017. URL http://arxiv.org/abs/1712.00961.  
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+M. Studeny. Probabilistic Conditional Independence Structures. Information Science and Statistics. Springer London, 2010. ISBN 9781849969482. URL https://books.google.com.gi/books?id=bGFRcgAACAAJ.  
+Ruben van Belle. Kan Extensions in Probability Theory. PhD thesis, University of Edinburgh, 2024.  
+Sebastiano Vigna. A guided tour in the topos of graphs, 2003. URL https://arxiv.org/abs/math/0306394.  
+Alessio Zanga and Fabio Stella. A survey on causal discovery: Theory and practice, 2023. URL https://arxiv.org/abs/2305.10032.
+
+# NeurIPS Paper Checklist
+
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+
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+
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+
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+
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+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
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+
+Guidelines:
+
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+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
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+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: We give the assumptions and complete proofs in the main body of the paper.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [NA]
+
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+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
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+# 6. Experimental setting/details
+
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+# 7. Experiment statistical significance
+
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+# 10. Broader impacts
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+
+# 11. Safeguards
+
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+# 12. Licenses for existing assets
+
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+# 13. New assets
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+
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+# 16. Declaration of LLM usage
+
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+
+# Appendix: Category Theory Background
+
+We give an introduction to categories and functors in Section A. Section B gives a brief overview of the theory of sheaves over toposes. Section C defines local set theories, another way to characterize the internal language of a topos. Section E contains an overview of affine CDU and Markov categories, which have been previously studied as categorical models of causality and probability [Jacobs et al., 2018, Fritz and Klingler, 2023].
+
+# A Categories and Functors
+
+![](images/395e0249dbcefa28b5e0e28902f9a5a10868d66c4cf9db0ce9adc4f55b524a80.jpg)  
+Figure 3: Categories are collections of objects, with a collection of arrows defined between each pair. A functor must map both objects and arrows from a domain category into a co-domain category.
+
+Category theory is perhaps the most transformative rethinking of mathematics since antiquity. Rather than focus on the internals of an object, like the elements of a set, it focuses on the arrows or morphisms that define the interactions between objects. Applied to causal models, it suggests that we can construct languages where individual causal models can be defined as variables in a logical language with intuitionistic semantics. The Yoneda Lemma, one of the most celebrated results in pure mathematics of the 20th century, states that objects are completely defined up to isomorphism purely in terms of their interactions. Applied to SCMs, its implications are nonetheless startling. SCMs in effect can be defined not just in terms of their internal structure, but in terms of how they interact with other SCMs in a category of such objects. Being conditioned to think in terms of internal structure, this statement may seem counterintuitive. But, as category theory has shown in numerous cases, such as in algebraic topology May [1992], we can often understand deep properties of complex objects such as topological spaces by modeling them as combinatorial objects and analyzing their interactions. Our framework builds on the theory of categories, functors, and natural transformations [MacLane, 1971, Riehl, 2017] (see Figure 3 and Figure 4). Many common mathematical structures – groups, rings, modules, measurable spaces, topological spaces etc. – form categories. More interesting categories for AI are those associated with compositional machine learning models, such as deep learning [Fong et al., 2019], where a symmetric monoidal category Learn is defined that combines an Implement routine that maps an input  $A$  into some output  $B$  parameterized by some parameter  $P$ , and an Update routine that given an input-output pair  $A, B$  and a parameter  $P$ , returns a new parameter  $P$ . Causal models, such as DAGs, structural equation models etc. can be straightforwardly mapped into categories.
+
+More formally, a category  $C$  is a collection of abstract objects  $c \in C$ . Anything technically can count as an object, from a variable in a causal model to an entire model itself. Each category  $C$  is additionally specified by a set of arrows or morphisms  $C(c,d)$  between each pair of objects  $c$  and  $d$ . There is an identity arrow  $1_c \in C(c,c)$ . Arrows compose in the obvious way, inducing a function  $C(c,d) \times C(d,e) \to C(c,e)$ . An initial object  $c$  in category  $C$  defined as one inducing a unique arrow from  $c$  to every object in category  $C$ . A terminal object, usually denoted by  $\mathbf{1}$ , is one that defines a unique arrow from every object  $c$  in category  $C$  into  $\mathbf{1}$ . An object  $c$  is isomorphic to another object  $d$ , denoted  $c \simeq d$ , if two arrows  $f : c \to d$  and  $g : d \to c$  exist, such that  $g \circ f = 1_c$ , and  $f \circ g = 1_d$ . A functor  $F : C \to D$  between two categories  $C$  and  $D$  is specified by an object function mapping each  $c \in C$  to  $Fc \in D$ , and an arrows function mapping each arrow  $f \in C(c,d)$  to  $Ff \in D(Fc,Fd)$ . Functors come in two varieties - covariant and contravariant - the latter acts on the domain category by reversing the arrows. Given
+
+<table><tr><td>Set theory</td><td>Topos theory</td></tr><tr><td>set</td><td>object</td></tr><tr><td>subset</td><td>subobject</td></tr><tr><td>truth values {0,1}</td><td>subobject classifier Ω</td></tr><tr><td>power set P(A) = 2A</td><td>power object P(A) = ΩA</td></tr><tr><td>bijection</td><td>isomorphims</td></tr><tr><td>injection</td><td>monic arrow</td></tr><tr><td>surjection</td><td>epic arrow</td></tr><tr><td rowspan="2">singleton set {*}
+empty set∅</td><td>terminal object 1</td></tr><tr><td>initial object 0</td></tr><tr><td>elements of a set X</td><td>morphism f: 1 → X</td></tr><tr><td>-</td><td>functors, natural transformations</td></tr><tr><td>-</td><td>limits, colimits, adjunctions</td></tr></table>
+
+any two functors  $F: C \to D$  and  $G: C \to D$  between the same pair of categories, we can define a mapping between  $F$  and  $G$  that is referred to as a natural transformation. These are defined through a collection of mappings, one for each object  $c$  of  $C$ , thereby defining a morphism in  $D$  for each object in  $C$ .
+
+![](images/b5aa269ec9bd24379611d713b423c89ab1bc1dc65a8b4d0c6764ba3aaa9939c7.jpg)  
+Figure 4: A topos is a category that generalizes set theory: subsets become subobjects and the characteristic function for a subset, which is Boolean, turns into a subobject classifier that may have multiple "degrees of truth".
+
+The celebrated Yoneda Lemma [MacLane, 1971] states that any set-valued functor  $F: C \to \mathbf{Sets}$  can be modeled in terms of natural transformations between  $F$  and the Yoneda embedding  $C(x, -)$  (or  $C(-, x)$ ) of a representable functor. In essence the action of a set valued functor  $F$  on an object  $x$  is completely determined by its natural transformations with  $C(-, x)$ . What are natural transformations? These specify how functors interact, much like how objects interact through arrows. A remarkable property of category theory is that the set of all sets is not a set, but the category of all categories is indeed a category, where the arrows are functors, and the objects are categories.
+
+Definition 18. Given categories  $C$  and  $D$ , and functors  $F, G: C \to D$ , a natural transformation  $\alpha: F \Rightarrow G$  is defined by the following data:
+
+- an arrow  $\alpha_{c}: Fc \to Gc$  in  $D$  for each object  $c \in C$ , which together define the components of the natural transformation.  
+- For each morphism  $f: c \to c'$ , the following commutative diagram holds true:
+
+![](images/ddbc6ae1948137a7687fcc2e4cbd15bd8e298cf36536f629a4496e576308f961.jpg)
+
+A natural isomorphism is a natural transformation  $\alpha : F \Rightarrow G$  in which every component  $\alpha_{c}$  is an isomorphism.
+
+# A.1 Yoneda Lemma
+
+The Yoneda Lemma states that the set of all morphisms into an object  $d$  in a category  $C$ , denoted as  $\mathbf{Hom}_C(-, d)$  and called the contravariant functor (or presheaf), is sufficient to define  $d$  up to isomorphism. The category of all presheaves forms a category of functors, and is denoted  $\hat{C} = \mathbf{Set}^{C^{\mathrm{op}}}$ . The Yoneda lemma plays a crucial role in this paper because it defines the concept of a universal representation in category theory. We first show that associated with universal arrows is the corresponding induced isomorphisms between  $\mathbf{Hom}$  sets of morphisms in categories. This universal property then leads to the Yoneda lemma.
+
+$$
+\begin{array}{c} D (r, r) \xrightarrow {\phi_ {r}} C (c, S r) \\ \Biggl \downarrow^ {D (r, f ^ {\prime})} \\ D (r, d) \xrightarrow {\phi_ {d}} C (c, S d) \end{array}
+$$
+
+As the two paths shown here must be equal in a commutative diagram, we get the property that a bijection between the Hom sets holds precisely when  $\langle r, u : c \to Sr \rangle$  is a universal arrow from  $c$  to  $S$ . Note that for the case when the categories  $C$  and  $D$  are small, meaning their Hom collection of arrows forms a set, the induced functor  $\mathbf{Hom}_{\mathbb{C}}(c, S-)$  to  $\mathbf{Set}$  is isomorphic to the functor  $\mathbf{Hom}_D(r, -)$ . This type of isomorphism defines a universal representation, and is at the heart of the causal reproducing property (CRP) defined below.
+
+Lemma 2. Yoneda lemma: For any functor  $F: C \to \mathbf{Set}$ , whose domain category  $C$  is "locally small" (meaning that the collection of morphisms between each pair of objects forms a set), any object  $c$  in  $C$ , there is a bijection
+
+$$
+H o m (C (c, -, F) \simeq F c
+$$
+
+that defines a natural transformation  $\alpha : C(c, -) \Rightarrow F$  to the element  $\alpha_{c}(1_{c}) \in Fc$ . This correspondence is natural in both  $c$  and  $F$ .
+
+There is of course a dual form of the Yoneda Lemma in terms of the contravariant functor  $C(-,c)$  as well using the natural transformation  $C(-,c)\Rightarrow F$ . A very useful way to interpret the Yoneda Lemma is through the notion of universal representability through a covariant or contravariant functor.
+
+Definition 19. A universal representation of an object  $c \in C$  in a category  $C$  is defined as a contravariant functor  $F$  together with a functorial representation  $C(-,c) \simeq F$  or by a covariant functor  $F$  together with a representation  $C(c, -) \simeq F$ . The collection of morphisms  $C(-,c)$  into an object  $c$  is called the presheaf, and from the Yoneda Lemma, forms a universal representation of the object.
+
+# A.2 Universal Properties in Categories
+
+A fundamental principle of category theory is to characterize objects by universal properties. Take the Cartesian product of two sets. The conventional way to define this product of two sets is as the set of ordered pairs, one drawn from each set. But this definition does not specify its universal property. We explain how to define universal properties below, which will be essential to understanding a topos. A key distinguishing feature of category theory is the use of diagrammatic reasoning. However, diagrams are also viewed more abstractly as functors mapping from some indexing category to the actual category. Diagrams are useful in understanding universal constructions, such as limits and colimits of diagrams. To make this somewhat abstract definition concrete, let us look at some simpler examples of universal properties, including co-products and quotients (which in set theory correspond to disjoint unions). Coproducts refer to the universal property of abstracting a group of elements into a larger one.
+
+Before we formally the concept of limit and colimits, we consider some examples. These notions generalize the more familiar notions of Cartesian products and disjoint unions in the category of Sets, the notion of meets and joins in the category Preord of preorders, as well
+
+as the least upper bounds and greatest lower bounds in lattices, and many other concrete examples from mathematics.
+
+Example 1. If we consider a small "discrete" category  $\mathcal{D}$  whose only morphisms are identity arrows, then the colimit of a functor  $\mathcal{F}:\mathcal{D}\to \mathcal{C}$  is the categorical coproduct of  $\mathcal{F}(D)$  for  $D$ , an object of category  $D$ , is denoted as
+
+$$
+\operatorname {C o l i m i t} _ {\mathcal {D}} F = \bigsqcup_ {D} \mathcal {F} (D)
+$$
+
+In the special case when the category  $C$  is the category  $\mathbf{Sets}$ , then the colimit of this functor is simply the disjoint union of all the sets  $F(\check{D})$  that are mapped from objects  $D \in \mathcal{D}$ .
+
+Example 2. Dual to the notion of colimit of a functor is the notion of limit. Once again, if we consider a small "discrete" category  $\mathcal{D}$  whose only morphisms are identity arrows, then the limit of a functor  $\mathcal{F}:\mathcal{D}\to C$  is the categorical product of  $\mathcal{F}(D)$  for  $D$ , an object of category  $D$ , is denoted as
+
+$$
+l i m i t _ {\mathcal {D}} F = \prod_ {D} \mathcal {F} (D)
+$$
+
+In the special case when the category  $C$  is the category Sets, then the limit of this functor is simply the Cartesian product of all the sets  $F(D)$  that are mapped from objects  $D \in \mathcal{D}$ .
+
+Category theory relies extensively on universal constructions, which satisfy a universal property. One of the central building blocks is the identification of universal properties through formal diagrams. Before introducing these definitions in their most abstract form, it greatly helps to see some simple examples. We can illustrate the limits and colimits in diagrams using pullback and pushforward mappings.
+
+![](images/4b94c9a02a835b65c86baf84a5255df5f99e50ddb10171af806dbf86268d3ef5.jpg)
+
+An example of a universal construction is given by the above commutative diagram, where the coproduct object  $X \sqcup Y$  uniquely factorizes any two mappings  $h: X \to R$  and  $i: Y \to R$ , such that any mapping  $i: Y \to R$ , so that  $h = r \circ f$ , and furthermore  $i = r \circ g$ . Co-products are themselves special cases of the more general notion of co-limits. Figure 5 illustrates the fundamental property of a pullback, which along with pushforward, is one of the core ideas in category theory. The pullback square with the objects  $U, X, Y$  and  $Z$  implies that the composite mappings  $g \circ f'$  must equal  $g' \circ f$ . In this example, the morphisms  $f$  and  $g$  represent a pullback pair, as they share a common co-domain  $Z$ . The pair of morphisms  $f', g'$  emanating from  $U$  defines a cone, because the pullback square "commutes" appropriately. Thus, the pullback of the pair of morphisms  $f, g$  with the common co-domain  $Z$  is the pair of morphisms  $f', g'$  with common domain  $U$ . Furthermore, to satisfy the universal property, given another pair of morphisms  $x, y$  with common domain  $T$ , there must exist another morphism  $k: T \to U$  that "factorizes"  $x, y$  appropriately, so that the composite morphisms  $f' k = y$  and  $g' k = x$ . Here,  $T$  and  $U$  are referred to as cones, where  $U$  is the limit of the set of all cones "above"  $Z$ . If we reverse arrow directions appropriately, we get the corresponding notion of pushforward. So, in this example, the pair of morphisms  $f', g'$  that share a common domain represents a pushforward pair. For any set-valued functor  $\delta: S \to \mathbf{Sets}$ , the Grothendieck category of elements  $\int \delta$  can be shown to be a pullback in the diagram of categories. Here, Set* is the category of pointed sets, and  $\pi$  is a projection that sends a pointed set  $(X, x \in X)$  to its underlying set  $X$ .
+
+We can now proceed to define limits and colimits more generally. We define a diagram  $F$  of shape  $J$  in a category  $C$  formally as a functor  $F: J \to C$ . We want to define the somewhat abstract concepts of limits and colimits, which will play a central role in this paper in defining
+
+![](images/7f3e60eecfa23eb5cb1bbdd29e1687ac57b6f8bee0b5e234e5a2a9ee729dba90.jpg)  
+Figure 5: (Left) Universal Property of pullback mappings. (Right) The Grothendieck category of elements  $\int \delta$  of any set-valued functor  $\delta : S \to \mathbf{Set}$  can be described as a pullback in the diagram of categories. Here,  $\mathbf{Set}_*$  is the category of pointed sets  $(X, x \in X)$ , and  $\pi$  is the "forgetful" functor that sends a pointed set  $(X, x \in X)$  into the underlying set  $X$ .
+
+a topos. A convenient way to introduce these concepts is through the use of universal cones that are over and under a diagram.
+
+For any object  $c \in C$  and any category  $J$ , the constant functor  $c: J \to C$  maps every object  $j$  of  $J$  to  $c$  and every morphism  $f$  in  $J$  to the identity morphisms  $1_{c}$ . We can define a constant functor embedding as the collection of constant functors  $\Delta: C \to C^{J}$  that send each object  $c$  in  $C$  to the constant functor at  $c$  and each morphism  $f: c \to c'$  to the constant natural transformation, that is, the natural transformation whose every component is defined to be the morphism  $f$ .
+
+Definition 20. A cone over a diagram  $F: J \to C$  with the summit or apex  $c \in C$  is a natural transformation  $\lambda: c \Rightarrow F$  whose domain is the constant functor at  $c$ . The components  $(\lambda_j: c \to Fj)_{j \in J}$  of the natural transformation can be viewed as its legs. Dually, a cone under  $F$  with nadir  $c$  is a natural transformation  $\lambda: F \Rightarrow c$  whose legs are the components  $(\lambda_j: F_j \to c)_{j \in J}$ .
+
+![](images/5afb7aa251576f99e439690b006225add03d1a55a6beb7bdad38d2362e847391.jpg)
+
+![](images/01032b110066af2e57d6cf48769058f489900c2f1595ed71b7576402cbc6b9ec.jpg)
+
+Cones under a diagram are referred to usually as cocones. Using the concept of cones and cocones, we can now formally define the concept of limits and colimits more precisely.
+
+Definition 21. For any diagram  $F: J \to C$ , there is a functor
+
+$$
+C o n e (-, F): C ^ {o p} \rightarrow \mathbf {S e t}
+$$
+
+which sends  $c \in C$  to the set of cones over  $F$  with apex  $c$ . Using the Yoneda Lemma, a limit of  $F$  is defined as an object  $\lim F \in C$  together with a natural transformation  $\lambda : \lim F \to F$ , which can be called the universal cone defining the natural isomorphism
+
+$$
+C (-, \lim  F) \simeq C o n e (-, F)
+$$
+
+Dually, for colimits, we can define a functor
+
+$$
+C o n e (F, -): C \to \mathbf {S e t}
+$$
+
+that maps object  $c \in C$  to the set of cones under  $F$  with nadir  $c$ . A colimit of  $F$  is a representation for  $\operatorname{Cone}(F, -)$ . Once again, using the Yoneda Lemma, a colimit is defined by an object  $\operatorname{Colim} F \in C$  together with a natural transformation  $\lambda : F \to \operatorname{colim} F$ , which defines the colimit cone as the natural isomorphism
+
+$$
+C (\operatorname {c o l i m} F, -) \simeq C o n e (F, -)
+$$
+
+Limit and colimits of diagrams over arbitrary categories can often be reduced to the case of their corresponding diagram properties over sets. One important stepping stone is to understand how functors interact with limits and colimits.
+
+Definition 22. For any class of diagrams  $K: J \to C$ , a functor  $F: C \to D$
+
+- preserves limits if for any diagram  $K: J \to C$  and limit cone over  $K$ , the image of the cone defines a limit cone over the composite diagram  $FK: J \to D$ .  
+- reflects limits if for any cone over a diagram  $K: J \to C$  whose image upon applying  $F$  is a limit cone for the diagram  $FK: J \to D$  is a limit cone over  $K$  
+- creates limits if whenever  $FK: J \to D$  has a limit in  $D$ , there is some limit cone over  $FK$  that can be lifted to a limit cone over  $K$  and moreover  $F$  reflects the limits in the class of diagrams.
+
+To interpret these abstract definitions, it helps to concretize them in terms of a specific universal construction, like the pullback defined above  $c' \to c \gets c''$  in  $C$ . Specifically, for pullbacks:
+
+- A functor  $F$  preserves pullbacks if whenever  $p$  is the pullback of  $c' \to c \leftarrow c''$  in  $C$ , it follows that  $Fp$  is the pullback of  $Fc' \to Fc \leftarrow Fc''$  in  $D$ .  
+- A functor  $F$  reflects pullbacks if  $p$  is the pullback of  $c' \to c \gets c''$  in  $C$  whenever  $Fp$  is the pullback of  $Fc' \to Fc \gets Fc''$  in  $D$ .  
+- A functor  $F$  creates pullbacks if there exists some  $p$  that is the pullback of  $c' \to c \leftarrow c''$  in  $C$  whenever there exists a  $d$  such that  $d$  is the pullback of  $Fc' \to Fc \leftarrow Fc''$  in  $F$ .
+
+# A.3 Symmetric Monoidal Categories
+
+Categorical models of causality [Fong, 2012, Fritz and Klingler, 2023, Jacobs et al., 2018, Mahadevan, 2023] are usually defined over symmetric monoidal categories, which we briefly review now [MacLane, 1971, Richter, 2020].
+
+Definition 23. A monoidal category is a category  $C$  together with a functor  $\otimes : C \times C \to C$ , an identity object  $e$  of  $C$  and natural isomorphisms  $\alpha, \lambda, \rho$  defined as:
+
+$$
+\alpha_ {C _ {1}, C _ {2}, C _ {3}}: C _ {1} \otimes (C _ {2} \otimes C _ {3}) \cong (C _ {1} \otimes C _ {2}) \otimes C _ {2},
+$$
+
+$$
+\lambda_ {C}: e \otimes C \cong C,
+$$
+
+$$
+\begin{array}{c c c} \rho : C \otimes e & \cong & C, \end{array}
+$$
+
+The natural isomorphisms must satisfy coherence conditions called the "pentagon" and "triangle" diagrams [MacLane, 1971]. An important result shown in [MacLane, 1971] is that these coherence conditions guarantee that all well-formed diagrams must commute. There are many natural examples of monoidal categories, the simplest one being the category of finite sets, termed FinSet in [Fritz, 2020], where each object  $C$  is a set, and the tensor product  $\otimes$  is the Cartesian product of sets, with functions acting as arrows. Deterministic causal models can be formulated in the category FinSet. Other examples include the category of sets with relations as morphisms, and the category of Hilbert spaces [Heunen and Vicary, 2019]. Markov categories [Fritz and Klingler, 2023] are monoidal categories, where the identity element  $e$  is also a terminal object, meaning there is a unique "delete" morphism  $d_{e}: X \to e$  associated with each object  $X$ . [Fritz, 2020] shows they form a unifying foundation for probabilistic and statistical reasoning.
+
+Definition 24. A symmetric monoidal category is a monoidal category  $(C, \otimes, e, \alpha, \lambda, \rho)$  together with a natural isomorphism
+
+$$
+\tau_ {C _ {1}, C _ {2}}: C _ {1} \otimes C _ {2} \cong C _ {2} \otimes C _ {1}, f o r a l l o b e c t s C _ {1}, C _ {2}
+$$
+
+where  $\tau$  satisfies the additional conditions: for all objects  $C_1, C_2$ $\tau_{C_2,C_1} \circ \tau_{C_1,C_2} \cong 1_{C_1 \otimes C_2}$ , and for all objects  $C, \rho_C = \lambda_C \circ \tau_{C,e}: C \otimes e \cong C$ .
+
+An additional hexagon axiom is required to ensure that the  $\tau$  natural isomorphism is compatible with  $\alpha$ . The  $\tau$  operator is called a "swap" in Markov categories [Fritz, 2020]. In most cases of interest in AI, the symmetric monoidal categories are enriched over some convenient base category  $\mathcal{V}$ , including vector spaces, or preorders such as the unit interval [0,1], where the unique morphism from  $a \to b$  exists if and only if  $a \leq b$ . 5
+
+Definition 25. A V-enriched category consists of a regular category  $C$ , such that for each pair of objects  $x$  and  $y$  in  $C$ , the morphisms  $C(x, y) \in \mathcal{V}$ , often referred to as a  $\mathcal{V}$ -hom object. For the case when  $(\text{cal}V, \leq, \otimes, 1)$  is a commutative monoidal preorder, we have the following conditions
+
+-  $1 \leq C(x, x)$  
+-  $C(y,z) \otimes C(x,y) \leq C(x,z)$
+
+# A.4 Topos Category
+
+To help build intuition for a topos, let's understand why sets are special as a category. They have all limits and colimits, meaning that one can always construct the categorical product of two sets as the Cartesian product, and the disjoint union as the coproduct. They have all pullbacks, which are commutative diagrams that define in essence a categorical definition of products and coproducts. In terms of the Yoneda Lemma, a universal property is either an initial or final property in a category of diagrams (functors). The product is the final object in a category of diagrams, and the coproduct is the initial object. Every concept in category theory essentially reduces down to these simple notions. Sets also have exponential objects: the set of all functions between two sets is once again a set! They also have a subobject classifier: each set has subsets, defining its parts, which correspond to a characteristic function that evaluates to true for elements in the subset. Topos theory generalizes all these properties to categories. It is important to causal inference because a causal intervention defines a subobject of an arbitrary object. A causal intervention on an SCM produces a subobject of that SCM object.
+
+To define a topos, we need to go through a few more definitions.
+
+Definition 26. An object  $x$  in a category  $C$  is called initial if there is a unique morphism  $f: x \to y$  to every other object  $y$  in the category. Dually, an object is called final if there is a unique morphism  $f: y \to x$  into  $x$ .
+
+In the category Sets, the null or empty set  $\emptyset$  is the initial and the single element set  $\{*\}$  is the final object. From an empty set, there is only one function possible to any other set, namely the empty function. From any set there is exactly one function into the single element set. A topos generalizes the property of subobject classifiers in Sets. Given any subset  $S \subset X$ , we can define  $S$  as the monic arrow  $S \hookrightarrow X$  defined by the inclusion of  $S$  in  $X$ , or as the characteristic function  $\phi_S$  that is equal to 1 for all elements  $x \in X$  that belong to  $S$ , and takes the value 0 otherwise. We can define the set  $2 = \{0,1\}$  and treat true as the inclusion  $\{1\}$  in 2. The characteristic function  $\phi_S$  can then be defined as the pullback of true along  $\phi_S$ .
+
+![](images/0a9c6949bdb20969b1a39f006faacbb6364d08e5cd2f27679411531c0d77332b.jpg)
+
+We can now define subobject classifiers in a category  $C$  as follows.
+
+Definition 27. In a category  $C$  with finite limits, a subobject classifier is a monic arrow true:  $1 \to \Omega$ , such that to every other monic arrow  $S \hookrightarrow X$  in  $C$ , there is a unique arrow  $\phi$  that forms the following pullback square:
+
+![](images/68dff6b7303e4c9b39ccad02f7ea86291701f6afe2f3cfc348de47c914152766.jpg)
+
+This definition can be rephrased as saying that the subobject functor is representable. In other words, a subobject of an object  $x$  in a category  $C$  is an equivalence class of monic arrows  $m: S \hookrightarrow x$ .
+
+Definition 28. A topos is a category  $\mathcal{E}$  with
+
+1. A pullback for every diagram  $X \to B \leftarrow Y$ .  
+2. A terminal object 1.  
+3. An object  $\Omega$  and a monic arrow true:  $1 \to \Omega$  such that any monic  $m : S \hookrightarrow B$ , there is a unique arrow  $\phi : B \to \Omega$  in  $\mathcal{E}$  for which the following square is a pullback:
+
+![](images/9b7b7bdd66ef73ea74e7b3cb754d0e5dc91f12501083f3fd00caa86488b07355.jpg)
+
+4. To each object  $x$  an object  $Px$  and an arrow  $\epsilon_x: x \times Px \to \Omega$  such that for every arrow  $f: x \times y \to \Omega$ , there is a unique arrow  $g: y \to Px$  for which the following diagrams commute:
+
+![](images/8c680fd1a219bc14ddf66df09a6be2e32de908b8e07447c4b3caa263a4a207ec.jpg)
+
+Let us understand these definitions in the category of Sets. Clearly, the single point set  $\{\bullet\}$  is a terminal object for Sets, because there is a unique function from any set  $S$  to a single element set  $\bullet$ , and the categorical product of two sets  $A \times B$  is just the Cartesian product. Furthermore, given two sets  $A$  and  $B$ , we can define  $B^A$  as the exponential object representing the set of all functions  $f: A \to B$ . We can define exponential objects in any category more generally as follows.
+
+Definition 29. Given any category  $C$  with products, for a fixed object  $x$  in  $C$ , we can define the functor
+
+$$
+x \times - \colon \rightarrow C
+$$
+
+If this functor has a right adjoint, which can be denoted as
+
+$$
+(-) ^ {x}: \mathcal {C} \to \mathcal {C}
+$$
+
+then we say  $x$  is an exponentiable object of  $C$ .
+
+Definition 30. A category  $C$  is Cartesian closed if it has finite products (which is equivalent to saying it has a terminal object and binary products) and if all objects in  $C$  are exponentiable.
+
+![](images/71cf89853dbb0e424236f69a5c25b4de903af2da98f6243c85aa6957db5ec272.jpg)  
+Graphs as a Topos
+
+![](images/32c2dfa682bc2f5b30366c96573de247f51f6bada942197eac65394f8382ffa7.jpg)  
+G(V)={a,b,c}
+
+![](images/f2f0d17c8981c1d4785c0ddec069fbe55dd7c09b41b9e48c485606de99e745f8.jpg)  
+H(V)={0,1}
+
+![](images/cbd4d1679b3277a355d83e72a241f2a91d739c25cac3a0e2eb28e7c080bf3145.jpg)  
+Figure 6: Causal models over arbitrary directed graphs define a topos. The subobject classifier is illustrated on the right.
+
+# A.5 The Topos of Causal Graphs
+
+To begin with a relatively elementary construction involving sheaves, Figure 6 illustrates how directed graphs, widely used in causal modeling, can be modeled as a topos that only has two objects: a generic "vertex" object  $V$ , representing an abstract causal variable, and a generic "edge" object  $E$ , denoting an abstract causal path. Any actual graph, such as the two shown in the figure, are covariant functors from  $C_{\Gamma}$ , the graph topos, to the actual graphs (we can treat graphs equivalently as contravariant functors by reversing the edges from  $V$  to  $E$ ). Sample object mappings of  $V$  for the two graphs are shown. For the topos category  $C_{\Gamma}$ , the "representable functor" is defined as the presheaf  $C_{\Gamma}(-, c)$  for each object  $c$  in the category  $\Gamma$ , which means the set of all morphisms going into object  $c$ . Let us calculate the representable functors for the topos of graphs  $C_{\Gamma}$ . Since  $C_{\Gamma}$  has only two objects,  $V$  and  $E$ , the representable functors are given as the sets:
+
+$$
+\begin{array}{c c} \mathcal {C} _ {\Gamma} (-, V), & \mathcal {C} _ {\Gamma} (-, E) \end{array}
+$$
+
+Since  $C_{\Gamma}$  has no arrows from object  $E$  to object  $V$ , we can easily check that the representable functor  $C_{\Gamma}(-,V)$  is given as:
+
+$$
+\mathcal {C} _ {\Gamma} (V, V) = \{1 _ {V} \}, \quad \mathcal {C} _ {\Gamma} (E, V) = \emptyset
+$$
+
+where  $1_V$  is the self-loop arrow that maps object  $V$  to itself. On the other hand, the representable functor  $C_{\Gamma}(-,E)$  is defined as:
+
+$$
+\mathcal {C} _ {\Gamma} (V, E) = \{s, t \}, \quad \mathcal {C} _ {\Gamma} (E, E) = \{1 _ {E} \}
+$$
+
+Additional details, like constructing exponential objects  $H^G$  for the  $C_{\Gamma}$  category, are given in [Vigna, 2003]. For the subobject classifier, the idea is as follows. For any causal model  $N$  represented as a submodel of a more complex causal model  $M$ , defined by a monic arrow  $m: N \hookrightarrow M$ , the generalization of the usual set-theoretic characteristic function is the classifying map  $\chi_m: N \to \Omega$ . As in Theorem 3,  $\Omega$  is not Boolean, but has multiple "degrees of truth" (see Figure 6):
+
+1. Causal variables not in  $N$  are mapped to  $O_V$ .  
+2. Causal variables in  $N$  are mapped to  $V$ .  
+3. If an edge is not in  $N$ , four cases emerge:
+
+(a) An arc whose source and target are not in  $N$  is mapped to  $0_{E}$ .  
+(b) An arc whose source is in  $N$ , but target is not, is mapped to an edge  $s$ .  
+(c) An arc whose target is in  $N$ , but source is not, is mapped to an edge  $t$ .  
+(d) An arc having both source and target in  $N$  is mapped to  $\binom{s}{t}^2$ .
+
+# B Sheaves and Toposes: Categories of Functors
+
+A general way to construct a topos category is through covariant Yoneda embeddings  $\mathfrak{X}:C\to \mathbf{Set}^{\mathcal{C}}$ , or contravariant Yoneda embeddings  $\mathfrak{X}:C\to \mathbf{Set}^{C^{\mathrm{op}}}$ . In simpler terms,
+
+each object  $x$  in the category  $C$  is either mapped to the functor  $C(x, -): C \to \mathbf{Sets}$  or  $C(-, x): C^{op} \to \mathbf{Sets}$ . These structures are called presheaves or copresheaves. To construct a proper sheaf, we need to include an additional condition that is illustrated in Figure 7. The sheaf condition plays an important role in many applications of machine learning, from dimensionality reduction [McInnes et al., 2018] to causal inference [Mahadevan, 2023]. Mac Lane and Moerdijk [1992] provides an excellent overview of sheaves and topoi, and how remarkably they unify much of mathematics, from geometry to logic and topology. For causal inference, the structure of causality shapes the arrows in a causal model, such as a Markov category, and that imposes a Grothendieck topology with an associated internal logic. Logic and causality are interwoven in this sense.
+
+![](images/450c909b7bbe51f055450a980b53a13b61f79e2969a37991e41ee263b4653d24.jpg)  
+Figure 7: Two applications of sheaf theory in AI: (top) minimizing travel costs in weighted graphs satisfies the sheaf principle, one example of which is the Bellman optimality principle in reinforcement learning [Bertsekas, 2019] (bottom): Approximating a function over a topological space must satisfy the sheaf condition.
+
+![](images/148c0a0c341789a180b8d34c17548b13c10ed25ad177da50d4e92639d3cfc78a.jpg)
+
+Figure 7 gives two concrete examples of sheaves (in both cases, these are enriched sheaves). In a minimum cost transportation problem, say reinforcement learning [Bertsekas, 2019], any optimal solution has the property that any restriction of the solution must also be optimal. In RL, this sheaf principle is codified by the Bellman equation, and leads to the fundamental principle of dynamic programming [Bertsekas, 2019]. Consider routing candy bars from San Francisco to New York city. If the cheapest way to route candy bars is through Chicago, then the restriction of the overall route to the (sub) route from Chicago to New York City must also be optimal, otherwise it is possible to find a shortest overall route by switching to a lower cost route. Similarly, in function approximation with real-valued functions  $F: C \to \mathbb{R}$ , where  $C$  is the category of topological spaces, the (sub)functions  $F(A), F(B)$  and  $F(C)$  restricted to the open sets  $A, B$  and  $C$  must agree on the values they map the elements in the intersections  $A \cap B, A \cap C, A \cap B \cap C$  and so on.
+
+Sheaves can be defined over arbitrary categories, and we introduce the main idea by focusing on the category of sheaves over Sets.
+
+Definition 31. [Mac Lane and Moerdijk, 1992] A sheaf of sets  $F$  on a topological space  $X$  is a functor  $F: O^{op} \to \mathbf{Sets}$  such that each open covering  $U = \bigcup_{i} U_{i}, i \in I$  of an open set  $O$  of  $X$  yields an equalizer diagram
+
+![](images/1cdec9406c15e3d83a1442634170e97c978bdeeb1600c1fa1a751e5867ccb7fd.jpg)  
+Figure 8: Sieves are subobjects of  $\mathfrak{L}(x)$  Yoneda embeddings of a category  $C$ , which generalizes the concept of sheaves over sets in Figure 7.
+
+$$
+F U \xrightarrow {e} \prod_ {i} F U _ {i} \xrightarrow [ q ]{\frac {p}{\longrightarrow}} \prod_ {i, j} F (U _ {i} \cap U _ {j})
+$$
+
+The above definition succinctly generalizes the idea in SCMs of combining local functions to get a unique global function.
+
+Definition 32. The category  $Sh(X)$  of sheaves over a space  $X$  is a full subcategory of the functor category  $\mathsf{Sets}^{O(X)^{\mathrm{op}}}$ .
+
+# B.1 Grothendieck Topologies
+
+We can generalize the notion of sheaves to arbitrary categories using the Yoneda embedding  $\mathfrak{L}(x) = C(-,x)$ . We explain this generalization in the context of a more abstract topology on categories called the Grothendieck topology defined by sieves. A sieve can be viewed as a subobject  $S\subseteq \mathfrak{L}(x)$  in the presheaf Sets $^{C^{\mathrm{op}}}$ , but we can define it more elegantly as a family of morphisms in  $C$ , all with codomain  $x$  such that
+
+$$
+f \in S \Longrightarrow f \circ g \in S
+$$
+
+Figure 8 illustrates the idea of sieves. A simple way to think of a sieve is as a right ideal. We can define that more formally as follows:
+
+Definition 33. If  $S$  is a sieve on  $x$ , and  $h: D \to x$  is any arrow in category  $C$ , then
+
+$$
+h ^ {*} (S) = \{g \mid c o d (g) = D, h g \in S \}
+$$
+
+Definition 34. [Mac Lane and Moerdijk, 1992] A Grothendieck topology on a category  $C$  is a function  $J$  which assigns to each object  $x$  of  $C$  a collection  $J(x)$  of sieves on  $x$  such that
+
+1. the maximum sieve  $t_x = \{f|cod(f) = x\}$  is in  $J(x)$ .  
+2. If  $S \in J(x)$  then  $h^*(S) \in J(D)$  for any arrow  $h: D \to x$ .  
+3. If  $S \in J(x)$  and  $R$  is any sieve on  $x$ , such that  $h^{*}(R) \in J(D)$  for all  $h: D \to x$ , then  $R \in J(C)$ .
+
+We can now define categories with a given Grothendieck topology as sites.
+
+Definition 35. A site is defined as a pair  $(C, J)$  consisting of a small category  $C$  and a Grothendieck topology  $J$  on  $C$ .
+
+An intuitive way to interpret a site is as a generalization of the notion of a topology on a space  $X$ , which is defined as a set  $X$  together with a collection of open sets  $O(X)$ . The sieves on a category play the role of "open sets".
+
+# B.2 Mitchell-Bénabou Language
+
+We define the Mitchell-Bénabou language (MBL), a typed local set theory (see Section C) associated with a causal topos. Given the topos category  $C_{\Omega}$ , we define the types of MBL as causal model objects  $M$  of  $C_{\Omega}$ . For each type  $M$  (e.g., an SCM), we assume the existence of variables  $x_{M}, y_{M}, \ldots$ , where each such variable has as its interpretation the identity arrow  $\mathbf{1}: M \to M$ . We can construct product objects, such as  $A \times B \times C$ , where terms like  $\sigma$  that define arrows are given the interpretation  $\sigma: A \times B \times C \to D$ . We can inductively define the terms and their interpretations in a topos category as follows (see [Mac Lane and Moerdijk, 1992] for additional details):
+
+- Each variable  $x_{M}$  of type  $M$  is a term of type  $M$ , and its interpretation is the identity  $x_{M} = \mathbf{1}: M \to M$  (e.g.,  $M$  may be an SCM or a causal model on a Markov category).  
+- Terms  $\sigma$  and  $\tau$  of types  $C$  and  $D$  that are interpreted as  $\sigma : A \to C$  and  $\tau : B \to D$  can be combined to yield a term  $\langle \sigma, \tau \rangle$  of type  $C \times D$ , whose joint interpretation is given as
+
+$$
+\langle \sigma p, \tau q \rangle : X \to C \times D
+$$
+
+where  $X$  has the required projections  $p:X\to A$  and  $q:X\to B$
+
+- Terms  $\sigma : A \to B$  and  $\tau : C \to B$  of the same type  $B$  yield a term  $\sigma = \tau$  of type  $\Omega$ , interpreted as
+
+$$
+(\sigma = \tau): W \xrightarrow {\langle \sigma p , \tau q \rangle} B \times B \xrightarrow {\delta_ {B}} \Omega
+$$
+
+where  $\delta_B$  is the characteristic map of the diagonal functor  $\Delta B\to B\times B$ . These diagonal maps correspond to the "copy" procedure in Markov categories [Fritz, 2020].
+
+- Arrows  $f: A \to B$  and a term  $\sigma: C \to A$  of type  $A$  can be combined to yield a term  $f \circ \sigma$  of type  $B$ , whose interpretation is naturally a composite arrow:
+
+$$
+f \circ \sigma : C \xrightarrow {\sigma} A \xrightarrow {f} B
+$$
+
+- For exponential objects, terms  $\theta : A \to B^C$  and  $\sigma : D \to C$  of types  $B^C$  and  $C$ , respectively, combine to give an "evaluation" map of type  $B$ , defined as
+
+$$
+\theta (\sigma): W \to B ^ {C} \times C \xrightarrow {e} B
+$$
+
+where  $e$  is the evaluation map, and  $W$  defines a map  $\langle \theta p, \sigma q \rangle$ , where once again  $p: W \to A$  and  $q: W \to D$  are projection maps.
+
+- Terms  $\sigma : A \to B$  and  $\tau : D \to \Omega^B$  combine to yield a term  $\sigma \in \tau$  of type  $\Omega$ , with the following interpretation:
+
+$$
+\sigma \in \tau : W \xrightarrow {\langle \sigma p , \tau q \rangle} B \times \Omega^ {B} \xrightarrow {e} \Omega
+$$
+
+- Finally, we can define local functions as  $\lambda$  objects, such as
+
+$$
+\lambda x _ {C} \sigma : A \to B ^ {C}
+$$
+
+where  $x_{C}$  is a variable of type  $C$  and  $\sigma : C \times A \to B$ .
+
+We combine terms  $\alpha, \beta$  etc. of type  $\Omega$  using logical connectives  $\land, \lor, \Rightarrow, \neg$ , as well as quantifiers, to get composite terms, where each of the logical connectives is now defined over the subobject classifier  $\Omega$ .
+
+-  $\wedge : \Omega \times \Omega \to \Omega$  is interpreted as the meet operation in the partially ordered set of subobjects (given by the Heyting algebra).  
+-  $\vee : \Omega \times \Omega \to \Omega$  is interpreted as the join operation in the partially ordered set of subobjects (given by the Heyting algebra). This operation gives the definition of a disjunction of two properties.
+
+-  $\Rightarrow$ :  $\Omega \times \Omega \to \Omega$  is interpreted as an adjoint functor, as defined previously for a Heyting algebra. Thus, the property of implication over SCMs is modeled as an adjoint functor.
+
+We can combine these logical connectives with the term interpretation as arrows, relegating some details to [Mac Lane and Moerdijk, 1992]. We now turn to the Kripke-Joyal semantics of this language.
+
+# B.3 Kripke-Joyal Semantics for a Causal Topos
+
+We now define the Kripke-Joyal semantics for the Mitchell-Bénabou language of a causal topos. Any free variable  $x$  must have some causal model  $X$  of  $C_{\Omega}$  as its type. For any causal model  $M$  in  $C_{\Omega}$ , define a generalized element as a morphism  $\alpha : N \to M$ . To understand this definition, note that we can define an element of a causal model by the morphism  $x : \mathbf{1} \to M$ . Thus, a generalized element  $\alpha : N \to M$  represents the "stage of definition" of  $M$  by  $N$ . We specify the semantics of how an SCM  $N$  supports any formula  $\phi(\alpha)$ , denoted by  $N \Vdash \phi(\alpha)$ , as follows:
+
+$$
+N \Vdash \phi (\alpha) \quad \text {i f a n d o n l y i f} \quad \operatorname {I m} \alpha \leq \{x | \phi (x) \}
+$$
+
+Stated in the form of a commutative diagram, this "forcing" relationship holds if and only if  $\alpha$  factors through  $\{x|\phi (x)\}$ , where  $x$  is a variable of type  $M$  (recall that objects  $M$  of a topos form its types), as shown in the following commutative diagram.
+
+![](images/5fcec4e8a8cab252906afc962ec67e03f17ae66c551625f36988e736336326eb.jpg)
+
+This diagram provides an interesting way to define causal interventions in a causal topos, because it defines submodels of  $M$ . Building on this definition, if  $\alpha, \beta : N \to M$  are parallel arrows, we can give semantics to the formula  $\alpha = \beta$  by the following statement:
+
+$$
+N \xrightarrow {\langle \alpha , \beta \rangle} M \times M \xrightarrow {\delta_ {M}} \Omega
+$$
+
+following the definitions in the previous section for the composite  $\langle \alpha, \beta \rangle$  and  $\delta_X$  in the Mitchell-Bénabou language. We can extend the previous commutative diagram to show that  $U \Vdash \alpha = \beta$  holds if and only if  $\langle \alpha, \beta \rangle$  factors through the diagonal map  $\Delta$ :
+
+![](images/0054258cd530fffedc0c1dd3f8c506701489927104206d31ecfe800b8ef159c4.jpg)
+
+- Monotonicity: If  $U \Vdash \phi(x)$ , then we can pullback the interpretation through any arrow  $f : U' \to U$  in a topos  $C$  to obtain  $U' \Vdash \phi(\alpha \circ f)$ .
+
+![](images/e916f060343bbb0b64af6fd8a3a63218efd4468dac79ea40af0679077063a2fa.jpg)
+
+- Local character: Analogously, if  $f: U' \to U$  is an epic arrow, then from  $U' \Vdash \phi(\alpha \circ f)$ , we can conclude  $U \Vdash \phi(x)$ .
+
+Theorem 9. If  $\alpha : N \to M$  is a generalized element of causal model  $M$ , and  $\phi(x)$  and  $\psi(x)$  are formulas with a free variable  $x$  of type  $M$ , we can conclude that
+
+1.  $N \Vdash \phi(\alpha) \land \psi(\alpha)$  holds if  $N \Vdash \phi(\alpha)$  and  $N \Vdash \psi(\alpha)$ .  
+2.  $N \Vdash \phi(x) \lor \psi(x)$  holds if there are morphisms  $p: O \to N$  and  $q: P \to N$  such that  $p + q: N + O \to M$  is an epic arrow, and  $N \Vdash \phi(\alpha p)$  and  $O \Vdash \phi(\alpha q)$ .  
+3.  $N \Vdash \phi(\alpha) \Rightarrow \psi(\alpha)$  if it holds that for any morphism  $p: N \to M$ , where  $N \Vdash \phi(\alpha p)$ , the assertion  $N \Vdash \phi(\alpha p)$  also holds.  
+4.  $N \Vdash \neg \phi(\alpha)$  holds if whenever the morphism  $p: M \to N$  satisfies the property  $N \Vdash \phi(\alpha p)$ , then  $N \cong \mathbf{0}$ .  
+5.  $M \Vdash \exists \phi(x, y)$  holds if there exists an epic arrow  $p: N \to M$  and generalized elements  $\beta: V \to Y$  such that  $M \Vdash \phi(\alpha p, \beta)$ .  
+6.  $M \Vdash \forall y\phi(x, y)$  holds if for every structural causal model  $N$ , and every arrow  $p: N \to M$ , and every generalized element  $\beta: N \to O$ , it holds that  $V \Vdash \phi(\alpha p, \beta)$ .
+
+Proof: The proof follows readily from the general result on Kripke-Joyal semantics for the Mitchell-Bénabou languages of any topos [Mac Lane and Moerdijk, 1992] The Kripke-Joyal semantics takes on a simpler form when using a Grothendieck topology on a topos, and we postpone the details to the Supplementary Materials.  $\square$
+
+# B.4 Kripke-Joyal Semantics for Sheaves
+
+In the main paper, we introduced Kripke-Joyal semantics for any topos. These semantics can be specialized to a topos equipped with a Grothendieck topology, that is a site. This specialized structure captures how causal inference is woven in the fabric of the internal logic of a causal topos. Define  $\mathrm{Sh}(C,\mathcal{T})$  be a topos of sheaves with a specified Grothendieck topology  $\mathcal{T}$ , defined by the following diagram:
+
+$$
+\mathcal {C} \stackrel {\text {上}} {\rightarrow} \mathcal {P} (\mathcal {C}) \stackrel {a} {\rightarrow} \operatorname {S h} (\mathcal {C}, \mathcal {J}) \cong \mathcal {C}
+$$
+
+where we know that the Yoneda embedding  $\mathfrak{L}$  creates a full and faithful copy of the original category  $C$ . Let us define the semantics for a sheaf element  $\alpha \in X(C)$ , where  $X(C) = \mathrm{Sh}(C, J)(\bar{C}(-, C), X)$ . Since we know that  $\{x|\phi(x)\}$  is a subsheaf, and given an arrow  $f: D \to C$  of  $C$ , and  $\alpha \in X(C)$ , then if  $\alpha$  is one of the elements that satisfies the property that  $\{x|\phi(x)\}$ , the monotonicity property stated above implies that  $\alpha \circ f \in \{x|\phi(x)\}(D) \subseteq X(D)$ . Also, the local character condition stated above implies that if  $\{f_i: C_i \to C\}$  is a cover in the Grothendieck topology  $\mathcal{I}$  such that  $C_i | \mathbb{H} \phi(\alpha \circ f_i)$  for all  $i$ , then  $C \mathbb{H} \phi(\alpha)$ .
+
+With these assumptions, we can restate the Kripke-Joyal semantics for the topos category of sheaves as follows:
+
+1.  $C \Vdash \phi(\alpha) \land \psi(\alpha)$  if it holds that  $C \Vdash \phi(\alpha)$  and  $C \Vdash \psi(\alpha)$ .  
+2.  $C \Vdash \phi(\alpha) \lor \psi(\alpha)$  if there is a covering  $\{f_i : C_i \to C\}$  such that for each  $i$ , either  $C_i \Vdash \phi(\alpha)$  or  $C_i \Vdash \psi(\alpha)$ .  
+3.  $C \Vdash \phi(\alpha) \to \psi(\alpha)$  if for all  $f: D \to C$ , and  $D \Vdash \phi(\alpha \circ f)$ , it holds that  $D \Vdash \psi(\alpha \circ f)$ .  
+4.  $C \Vdash \neg \phi(\alpha)$  holds if for all arrows  $f: D \to C$  in  $C$ , if  $D \Vdash \phi(\alpha \circ f)$  holds, then the empty family is a cover of  $D$ .  
+5.  $C \Vdash \exists y \phi(x, y)$  holds if there is a covering  $\{f_i : C_i \to C\}$  and elements  $\beta_i \in Y(C_i)$  such that  $C_i \Vdash \phi(\alpha \circ f_i, \beta_i)$  holds for each  $i$ .  
+6. Finally, for universal quantification,  $C \Vdash \forall y \phi(x, y)$  holds if for all arrows  $f: D \to C$  in the category  $C$ , and all  $\beta \in Y(D)$ , it holds that  $D \Vdash \phi(\alpha \circ f, \beta)$ .
+
+# C Local Set Theory
+
+The Mitchell-Bénabou language is an example of a "local set theory" Bell [1988]. Formally, the Mitchell-Bénabou language for the Generalized Do-Calculus is a local set theory, defined
+
+by a set of types that correspond to each structural causal model  $M$  object in  $C_{\Omega}$ . A local set theory [Bell, 1988] is defined as a language  $\mathcal{L}$  specified by the following classes of symbols:
+
+1. Symbols 1 and  $\Omega$  representing the unity type and truth-value type symbols.  
+2. A collection of symbols  $\mathbf{A},\mathbf{B},\mathbf{C},\ldots$  called ground type symbols.  
+3. A collection of symbols  $\mathbf{f},\mathbf{g},\mathbf{h},\ldots$  called function symbols.
+
+To instantiate this definition for our paper, the ground types will be SCMs, each of which will be interpreted as a primitive type in Section B.2. We will use the topos-theoretical constructions to construct composite types. We can use an inductive procedure to recursively construct type symbols of  $\mathcal{L}$  as follows:
+
+1. Symbols 1 and  $\Omega$  are type symbols (the terminal object and the subobject classifier in a causal topos).  
+2. Any ground type symbol is a type symbol. For a causal topos, each SCM is a ground type symbol.  
+3. If  $\mathbf{A}_1, \ldots, \mathbf{A}_n$  are type symbols, so is their product  $\mathbf{A}_1 \times \ldots \mathbf{A}_n$ , where for  $n = 0, \ldots, n$  the type of  $\prod_{i=1}^{n} \mathbf{A}_i$  is  $\mathbf{1}$ . The product  $\mathbf{A}_1 \times \ldots \mathbf{A}_n$  has the product type symbol. These constructs allow defining an algebra of causal models.  
+4. If  $\mathbf{A}$  is a type symbol, so is  $\mathbf{PA}$ . The type  $\mathbf{PA}$  is called the power type. We thus can give meaning to concept of a "powerset" of a causal model, where we interpret the subobject classifier as defining the abstract semantics of a powerset for each SCM.
+
+Thus, a product of SCMs will define product types. Given an SCM  $M$ , we can define its power type as well, which is an abstract notion of the "power set" of a causal model (if you interpret this in the context of subobject classifiers, it means that we are defining a family of submodels). For each type symbol  $\mathbf{A}$ , the language  $\mathcal{L}$  contains a set of variables  $x_{\mathbf{A}}, y_{\mathbf{A}}, z_{\mathbf{A}}, \ldots$ . In addition,  $\mathcal{L}$  contains the distinguished * symbol. Each function symbol in  $\mathcal{L}$  is assigned a signature of the form  $\mathbf{A} \rightarrow \mathbf{B}$ .<sup>5</sup> We can define the terms of the local set theory language  $\mathcal{L}$  recursively as follows:
+
+$\bullet$  \* is a term of type 1.  
+- for each type symbol  $\mathbf{A}$ , variables  $x_{\mathbf{A}}, y_{\mathbf{A}}, \ldots$  are terms of type  $\mathbf{A}$ .  
+- if  $\mathbf{f}$  is a function symbol with signature  $\mathbf{A} \rightarrow \mathbf{B}$ , and  $\tau$  is a term of type  $\mathbf{A}$ , then  $\mathbf{f}(\tau)$  is a term of type  $\mathbf{B}$ .  
+- If  $\tau_{1},\ldots ,\tau_{n}$  are terms of types  $\mathbf{A}_1,\dots ,\mathbf{A}_n$  , then  $\langle \tau_1,\dots \tau_n\rangle$  is a term of type  $\mathbf{A}_1\times \dots \mathbf{A}_n$  where if  $n = 0$  , then  $\langle \tau_1,\dots \tau_n\rangle$  is of type *.  
+- If  $\tau$  is a term of type  $\mathbf{A}_1 \times \mathbf{A}_n$ , then for  $1 \leq i \leq n$ ,  $(\tau)_i$  is a term of type  $\mathbf{A}_i$ .  
+- if  $\alpha$  is a term of type  $\Omega$ , and  $x_{\mathbf{A}}$  is a variable of type  $\mathbf{A}$ , then  $\{x_{\mathbf{A}} : \alpha\}$  is a term of type  $\mathbf{PA}$ .  
+- if  $\sigma, \tau$  are terms of the same type,  $\sigma = \tau$  is a term of type  $\Omega$ .  
+- if  $\sigma, \tau$  are terms of the types  $\mathbf{A}, \mathbf{P}\mathbf{A}$ , respectively, then  $\sigma \in \tau$  is a term of type  $\Omega$ .
+
+A term of type  $\Omega$  is called a formula. The language  $\mathcal{L}$  does not yet have defined any logical operations, because in a typed language, logical operations can be defined in terms of the types, as illustrated below.
+
+-  $\alpha \Leftrightarrow \beta$  is interpreted as  $\alpha = \beta$ .  
+- true is interpreted as  $* = *$ .  
+-  $\alpha \wedge \beta$  is interpreted as  $\langle \alpha, \beta \rangle = \langle \text{true}, \text{false} \rangle$ .  
+-  $\alpha \Rightarrow \beta$  is interpreted as  $(\alpha \land \beta) \Leftrightarrow \alpha$
+
+-  $\forall x\alpha$  is interpreted as  $\{x:\alpha \} = \{x:\mathbf{true}\}$  
+- false is interpreted as  $\forall \omega \omega$ .  
+-  $\neg \alpha$  is interpreted as  $\alpha \Rightarrow$  false.  
+-  $\alpha \vee \beta$  is interpreted as  $\forall \omega[(\alpha \Rightarrow \omega \land \beta \Rightarrow \omega) \Rightarrow \omega]$  
+-  $\exists x\alpha$  is interpreted as  $\forall \omega [\forall x(\alpha \Rightarrow \omega) \Rightarrow \omega]$
+
+Finally, we have to specify the inference rules, which are given in the form of sequents. We will just sketch out a few, and the rest can be seen in [Bell, 1988]. A sequent is a formula  $\Gamma : \alpha$  where  $\alpha$  is a formula, and  $\Gamma$  is a possibly empty finite set of formulae. The basic axioms include  $\alpha : \alpha$  (tautology),  $x_1 = *$ (unity), a rule for forming projections of products, a rule for equality, and another for comprehension. Finally, the inference rules are given in the form:
+
+Thinning:
+
+$$
+\frac {\Gamma : \alpha}{\beta , \Gamma : \alpha}
+$$
+
+Cut:
+
+$$
+\begin{array}{c} \Gamma : \alpha , \alpha , \Gamma : \beta \\ \hline \Gamma : \beta \end{array}
+$$
+
+- Equivalence:
+
+$$
+\frac {\alpha , \Gamma : \beta \beta , \Gamma : \alpha}{\Gamma : \alpha \Leftrightarrow \beta}
+$$
+
+A full list of inference rules with examples of proofs is given in [Bell, 1988]. Now that we have the elements of a local set theory defined as shown above, we need to connect its definitions with a causal topos. That is the topic of the next section.
+
+# C.1 Counterfactuals using Topos Categories
+
+de Araujo Fernandes and Haeusler [2009] describe an intuitionistic logic for Lewis' [Lewis, 1973] theory of counterfactuals, where the neighborhood system of possible worlds is governed by the graph topos  $C_{\Gamma}$  illustrated in Figure 6. However, as Galles and Pearl [1988] argued, many counterfactuals have causal meaning. For example, the well-known counterfactual "If Kangaroos had no tails, they would fall over" attains causal meaning as the logic of counterfactuals proposed by Lewis evaluates its truth in the nearest possible worlds, where the laws of physics are the same as our world, except for a "kangaroo surgery". Lewis introduces two counterfactual connectives, and we illustrate how to model one of them in our framework:
+
+-  $\alpha \square \rightarrow \beta$  is true in a world  $u$  according to a neighborhood system  $C_{\Omega}$ , if either for no world  $w$  in  $C_{\Omega}(u)$ ,  $\models_w \alpha$ , or there is some neighborhood  $N$  in  $C_{\Omega}(u)$  that has a world  $w$  such that  $\models_w \alpha$  and  $\models_v \alpha \Rightarrow \beta$  in every world  $v$  of  $N$ .
+
+Crucially, unlike the case for de Araujo Fernandes and Haeusler [2009], our TCM framework imposes a causal structure on the neighborhood system.
+
+# D Intuitionistic  $j$ -Do-Calculus on Sites
+
+In a recent paper, we have developed an intuitionistic generalization of Pearl's do-calculus on sites [Mahadevan, 2025a]. A site  $(C, \mathcal{I})$  is a small category  $C$  whose objects index regimes (e.g., labs, contexts) and a Grothendieck topology  $\mathcal{I}$  that specifies which families  $\{u_i \to u\}$  "cover" a stage  $u$ . The topos  $\mathbf{S}_{\mathcal{I}}(C)$  of  $\mathcal{I}$ -sheaves behaves like a universe of sets varying over regimes. A formula  $\varphi$  is  $\mathcal{I}$ -stable at  $u$  iff it holds on a cover of  $u$  (Kripke-Joyal semantics). Intuitively: truth is verified locally and then glued. Random variables and probabilities live
+
+internally (as objects/morphisms of  $\mathbf{Sh}_{\mathcal{I}}(C)$ ). When  $\mathcal{I}$  is trivial, these reduce to the classical notions.
+
+Pearl's do-calculus [Pearl, 2009] is a complete axiom system for interventional identification in acyclic causal models under classical (Boolean) logic. In [Mahadevan, 2025a], we generalize it to  $j$ -stable causal inference inside a topos of sheaves  $\mathbf{Sh}_{\mathcal{I}}(C)$ , where regimes form a site  $(C, \mathcal{I})$  and observations/interventions are sheaves on that site. We define  $j$ -stability for conditional independences and interventional claims as local truth in the internal logic of  $\mathbf{Sh}_{\mathcal{I}}(C)$ . We give three inference rules that mirror Pearl's insertion/deletion and action/observation exchange, and we characterize their soundness using the Kripke-Joyal semantics. In a companion paper [Mahadevan, 2025b], which is currently in preparation, we provide concrete algorithms for  $j$ -stable causal discovery on sites, and compare a number of popular causal discovery procedures (score-based and constraint-based) [Zanga and Stella, 2023] with their  $j$ -stable variants.
+
+# E Affine CDU and Markov Categories
+
+In this section, we review previous work on affine CDU ("copy-delete-uniform") categories [Cho and Jacobs, 2019] and Markov categories [Fritz, 2020], which have been proposed as a unifying categorical model for causal inference, probability and statistics. They are symmetric monoidal categories, which we reviewed above, combined with a comonoidal structure on each object. Importantly, Markov categories are semi-Cartesian because they do not use uniform copying, but contain a Cartesian subcategory defined by deterministic morphisms (see below). We give a brief review of Markov categories, and significant additional details that are omitted can be found in [Cho and Jacobs, 2019, Fritz, 2020, Fritz and Klingler, 2023]. The equations are written in the diagrammatic language of string diagrams, which can be shown to represent a formal language that is equivalent to writing down algebraic equations [Selinger, 2010].
+
+Definition 36. A Markov category  $C$  [Fritz, 2020] is a symmetric monoidal category in which every object  $X \in C$  is equipped with a commutative comonoid structure given by a comultiplication  $\mathsf{copy}_X : X \to X \otimes X$  and a counit  $\mathsf{del}_X : X \to I$ , depicted in string diagrams as
+
+$$
+\operatorname {c o p y} _ {X} = \bigcup_ {\bullet} \quad \operatorname {d e l} _ {X} = \bigcirc \tag {1}
+$$
+
+and satisfying the commutative comonoid equations,
+
+$$
+\bigcup_ {\bullet} \bigcup = \bigcup_ {\bullet} \bigcup \tag {2}
+$$
+
+$$
+\biguplus = \Big | = \biguplus \quad \overbrace {\biguplus} = \biguplus \tag {3}
+$$
+
+as well as compatibility with the monoidal structure,
+
+$$
+\biguplus_ {X \otimes Y} ^ {\bullet} = \biguplus_ {X} ^ {\bullet} \biguplus_ {Y} ^ {\bullet} \quad \biguplus_ {X \otimes Y} ^ {X \otimes Y X \otimes Y} = \biguplus_ {X} ^ {X Y} \biguplus_ {Y} ^ {X Y} \tag {4}
+$$
+
+and naturality of del, which means that
+
+$$
+\boxed {f} = \boxed {\bullet} \tag {5}
+$$
+
+for every morphism  $f$ .
+
+Let us briefly explain these definitions. The  $\mathrm{del}_X: X \to I$  is essentially like integrating over a probability distribution, which always yields 1. Hence,  $I$ , the unit of the tensor product, is the terminal object in affine CDU and Markov categories. Bayes rule turns into a disintegration rule [Cho and Jacobs, 2019], which is only available in Markov categories with conditionals (i.e., where one can categorically refine  $P(y|x)$  conditional distributions). Note that in the continuous case of random variables defined as measurable functions on real numbers, one has to take considerable care in defining conditioning [Halmos, 1974]. The  $\mathrm{copy}_X$  procedure is uniform, and deterministic, meaning if you take the tensor product of two variables  $X \otimes Y$  and then copy the resulting object, that's exactly the same as first copying  $X$  into  $X \otimes X$  and  $Y$  into  $Y \otimes Y$ , and then taking the tensor product, along with a swap operation (see Equation 4). Only  $\mathrm{del}_X$  acts "uniformly", meaning that if you process a variable  $X$  using some function  $f$  and then delete  $f(X)$  (meaning marginalize it), that's equivalent to simply deleting  $X$ . However,  $\mathrm{copy}_X$  is not defined this way, and we discuss that subtlety below, as it will be important in understanding why Markov categories are semi-Cartesian. To convert them into a topos, we need the result to be Cartesian closed, which is why we need to use the Yoneda Lemma to construct the category of presheaves to guarantee obtaining a topos.
+
+# E.1 Cartesian Structure in Markov Categories
+
+We now discuss a subcategory of Cartesian categories within Markov that involves uniform  $\mathrm{copy}_X$  and  $\mathrm{del}_X$  morphisms. One fundamental property of Markov categories is that they are semi-Cartesian, as the unit object is also a terminal object. But, a subtlety arises in how these copy and delete operators are modeled, as we discuss below.
+
+Definition 37. A symmetric monoidal category  $C$  is Cartesian if the tensor product  $\otimes$  is the categorical product.
+
+If  $C$  and  $\mathcal{D}$  are symmetric monoidal categories, then a functor  $F: C \to \mathcal{D}$  is monoidal if the tensor product is preserved up to coherent natural isomorphisms.  $F$  is strictly monoidal if all the monoidal structures are preserved exactly, including  $\otimes$ , unit object  $I$ , symmetry, associative and unit natural isomorphisms. Denote the category of symmetric monoidal categories with strict functors as arrows as MON. Let us review the basic definitions given by Heunen and Vicary [2019], which will give some further clarity on the Cartesian structure in affine CDU and Markov categories.
+
+Definition 38. The subcategory of comonoids coMON in the ambient category MON of all symmetric monoidal categories is defined for any specific category  $C$  as a collection of "coalgebraic" objects  $(X, \mathrm{copy}_X, \mathrm{del}_X)$ , where  $X$  is in  $C$ , and arrows defined as comonoid homomorphisms from  $(X, \mathrm{copy}_X, \mathrm{del}_X)$  to  $(Y, \mathrm{copy}_Y, \mathrm{del}_Y)$  that act uniformly, in the sense that if  $f: X \to Y$  is any morphism in  $C$ , then:
+
+$$
+(f \otimes f) \circ c o p y _ {X} = c o p y _ {Y} \circ f
+$$
+
+$$
+d e l _ {Y} \circ f = d e l _ {X}
+$$
+
+Heunen and Vicary [2019] define the process of "uniform copying and deleting" in the category coMON, which we now relate to Markov categories. A subtle difference worth emphasizing with Definition 36 is that in Markov categories, only  $\mathrm{del}_X$  is "uniform", but not  $\mathrm{copy}_X$  in the sense defined by Heunen and Vicary [2019]. This distinction can be modeled in a cPROP category that is semi-Cartesian like Markov categories by suitably modifying the definition of the associated PROP map for copying.
+
+Definition 39. [Heunen and Vicary, 2019] A symmetric monoidal category  $C$  admits uniform deleting if there is a natural transformation  $e_X: X \xrightarrow{e_X} I$  for all objects in the subcategory  $C_{\mathrm{coMON}}$  of comonoidal objects, where  $e_I = id_I$ , as shown in Equation 5.
+
+This condition was referred to by Cho and Jacobs [2019] as a causality condition on the arrow  $e_{X}$ . Essentially, it states that if you process some object and then discard it, it's equivalent to discarding it without processing.
+
+Theorem 10. [Heunen and Vicary, 2019] A symmetric monoidal category  $C$  has uniform deleting if and only if  $I$  is terminal.
+
+This property holds for Markov categories, as noted in [Fritz, 2020], and a simple diagram chasing proof is given in [Heunen and Vicary, 2019].
+
+Definition 40. [Heunen and Vicary, 2019] A symmetric monoidal category  $C$  has uniform copying if there is a natural transformation  $\mathrm{copy}_X: X \to X \otimes X$  such that  $\mathrm{del}_I = \rho_I^{-1}$  satisfying Equation 2 and Equation 3.
+
+We can now state an important result proved in [Heunen and Vicary, 2019] (Theorem 4.28), which relates to the more general results shown earlier by Fox [1976].
+
+Theorem 11. [Fox, 1976, Heunen and Vicary, 2019] The following conditions are equivalent for a symmetric monoidal category  $C$ .
+
+- The category  $C$  is Cartesian with tensor products  $\otimes$  given by the categorical product and the tensor unit is given by the terminal object.  
+- The symmetric monoidal category  $C$  has uniform copying and deleting, and Equation 2 holds.
+
+As noted by Fritz [2020], not all Markov categories are Cartesian, because their  $\mathbf{copy}_X$  is not uniform, but only  $\mathbf{del}_X$  is. For example, consider the category FinStoch, where a joint distribution is specified by the morphism  $\psi : I \to X \otimes Y$ . In this case, the marginal distributions can be formed as the composite morphisms
+
+$$
+\begin{array}{l} I \xrightarrow {\psi} X \otimes Y \xrightarrow {\operatorname {d e l} _ {Y}} X \\ I \xrightarrow {\psi} X \otimes Y \xrightarrow {\operatorname {d e l} _ {X}} Y \\ \end{array}
+$$
+
+But to require that in this case  $\otimes$  is the categorical product implies that the marginal distributions defined as the above composites must be in bijection with the joint distribution.
+
+# F Social Impact
+
+Causal inference addresses real-world problems with significant social impact, including healthcare, education, climate change, and economics. Discovering causal models has enormous potential for improving human lives in all these areas. However, our paper is largely a theoretical study of the universal properties that underlie causality. It does not address concrete algorithmic questions, and is complementary to the empirical literature in the field.
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+# Universal Cross-Tokenizer Distillation via Approximate Likelihood Matching
+
+Benjamin Minixhofer  $^{0 \times 43}$  Ivan Vulic  $^{0 \times 43}$  Edoardo M. Ponti  $^{0 \times 45,0 \times 43}$ $^{0 \times 43}$ University of Cambridge  $^{0 \times 45}$ University of Edinburgh
+
+# Abstract
+
+Distillation has shown remarkable success in transferring knowledge from a Large Language Model (LLM) teacher to a student LLM. However, current distillation methods require similar tokenizers between the teacher and the student, restricting their applicability to only a small subset of teacher-student pairs. In this work, we develop a principled cross-tokenizer distillation method to solve this crucial deficiency. Our method is the first to enable effective distillation across fundamentally different tokenizers, while also substantially outperforming prior methods in all other cases. We verify the efficacy of our method on three distinct use cases. First, we show that viewing tokenizer transfer as self-distillation enables unprecedently effective transfer across tokenizers, including rapid transfer of subword models to the byte-level. Transferring different models to the same tokenizer also enables assembling to boost performance. Secondly, we distil a large maths-specialised LLM into a small general-purpose model with a different tokenizer, achieving competitive maths problem-solving performance. Thirdly, we use our method to train state-of-the-art embedding prediction hypernetworks for training-free tokenizer transfer. Our results unlock an expanded range of teacher-student pairs for distillation, enabling new ways to adapt and enhance interaction between LLMs.
+
+# 1 Introduction
+
+Even the latest, most powerful Large Language Models (LLMs) still operate on tokens, and therefore require a tokenizer — a manually designed component which turns text into a sequence of tokens. Most currently available language models use subword tokenization (Sennrich et al., 2016; Kudo & Richardson, 2018), where each token is a word, or part of a word. Even though subword tokenizers are currently dominant, they are heterogeneously implemented: when a model is released, it often comes with its own tokenizer which has distinct properties such as the actual tokens it comprises (its vocabulary) and which method is used to segment text into these tokens (its tokenization function). Furthermore, there has been a recent trend away from subwords toward models using vocabularies consisting of characters (Tay et al., 2022; Nawrot et al., 2023) or bytes (e.g. Xue et al., 2022; Yu et al., 2023; Pagnoni et al., 2024). This exemplifies the wildly diverse landscape of tokenization.
+
+On the other hand, distillation (Bucilua et al., 2006; Hinton et al., 2015) has emerged as a powerful paradigm for creating effective language models by training them based on the signal from another model. However, existing distillation methods require similar tokenizers between the teacher and the student. In particular, most prior methods require the teacher and student to represent their input in the same way, i.e., that their tokenizer is equivalent (e.g., Gu et al., 2024; Ko et al., 2024). Some recent methods relax this requirement by developing heuristics to incorporate information from the teacher alongside a main objective (e.g., next-token prediction), making progress toward enabling distillation across similar (but not equivalent) tokenizers (Zhang et al., 2024b; Wan et al., 2024; Boizard et al., 2025). Nonetheless, this requirement still limits their applicability to a rather small subset of the possible teacher-student pairs. In this work, we develop a cross-tokenizer distillation method to solve
+
+![](images/b1895b521972e352009e0b2aa865345be3c367b1435069701780a9d3b40591d5.jpg)  
+Figure 1: We propose a cross-tokenizer distillation method which identifies comparable chunks of tokens, then minimizes the differences between their likelihoods (c.f. Section 3).
+
+this problem. Our principled objective enables distillation across highly dissimilar tokenizers for the first time (e.g., distilling from a subword tokenizer to a byte-level tokenizer), while also unlocking pure distillation — training to purely match the teachers' behaviour, instead of matching the teachers' behaviour as an auxiliary objective, which can lead to further empirical improvements.
+
+In a nutshell, given some text, we first tokenize it with the teacher and the student tokenizers and compute the next-token likelihoods of all tokens under both models. We then find aligned chunks of tokens (the chunks encoding the same portions of text) between the two sequences. Now, to compare the chunk likelihoods of the aligned chunks (e.g., via their KL-divergence), we would need to know the likelihoods of all possible outcomes. This is not an issue in the same-tokenizer distillation case since we can compute the likelihoods of all tokens in the vocabulary in a single forward pass. However, in our case, we would have to compute the likelihoods of all multi-token chunks, which is impossible since their number is infinite. We thus approximate the chunk likelihood difference via a binarised  $f$ -divergence (Section 3). Together with optional debiasing (Section 3.1) and distillation of the hidden states (Section 3.2), this leads to an effective and universal cross-tokenizer distillation method, which we refer to as Approximate Likelihood Matching (ALM), sketched in Figure 1.
+
+We test ALM on three use cases. First, we propose a novel viewpoint of tokenizer transfer as cross-tokenizer self-distillation (Use Case 1). This enables transferring subword-based LLMs to a different subword tokenizer while largely retaining performance, and achieves rapid and effective Subword  $\rightarrow$  Byte transfer for the first time. This allows us to create a competitive byte-level model at a fraction of the cost of training it from scratch. We further show that transferring different models to the same subword tokenizer enables assembling their predictions (e.g., via averaging their token-level probabilities) which substantially boosts performance over the individual parts. Secondly, we show that ALM outperforms prior methods at distillation of a large math-specialised model into a much smaller general-purpose model with a different tokenizer (Use Case 2). Thirdly, we plug-in ALM as objective to train embedding predictions hypernetworks as introduced by Minixhofer et al. (2024), achieving a new state-of-the-art in tokenizer transfer without ever training on the target tokenizer (i.e., Zero-Shot Tokenizer Transfer, Use Case 3). In summary, our work greatly expands the number of possible teacher-student pairs for distillation, improving reusability, composability and transferability of language models. Our code and models are available at github.com/bminixhofer-tokenkit.
+
+# 2 Preliminaries and Background
+
+# 2.1 Tokenizers
+
+Definition. Tokenizers map the input text to a sequence of tokens. Formally, they operate on finite sequences  $\Sigma^{\star}$  over the alphabet  $\Sigma$ . For simplicity and since it is not vital to our method, we presume  $\Sigma$  to be the set of bytes  $\{0, \dots, 255\}$  and  $\Sigma^{\star}$  a string represented as bytes via the UTF-8 standard (Yergeau, 2003). This does not lose generality, since although some tokenizers operate on units built on top of UTF-8 bytes (e.g., Unicode characters; Kudo & Richardson, 2018), they can be transformed to operate on bytes instead (Minixhofer et al., 2024). Following prior work (Uzan et al., 2024; Feher et al., 2024), we denote the tokenizer as the tuple  $(\mathcal{V}, T)$  comprising (i) the vocabulary  $\mathcal{V}$  which determines the set of all possible tokens and (ii) the tokenization function  $T: \Sigma^{\star} \to \mathcal{V}^{\star}$  which determines the sequence of tokens any given text is mapped to.
+
+Tokenization functions fuse bytes into larger (e.g., subword) tokens without loss of information. Consider the tokens  $\{t_1, t_2, \ldots, t_n\} = T(\pmb{x})$ . Then  $t_1 \odot t_2 \odot \ldots \odot t_n = \pmb{x}$  (where  $\odot$  is the concatenation operator). This has multiple useful implications. All elements in the vocabulary  $\mathcal{V}$  are sequences of bytes  $\Sigma^{\star}$ . Furthermore,  $T$  is injective so there exists a left inverse  $D$  such that  $\pmb{x} = D(T(\pmb{x})) \forall \pmb{x}$ . We call  $D$  the detokenization function. Importantly,  $D$  is only a left inverse, i.e., there may exist a token sequence  $\pmb{t}$  such that  $\pmb{t} \neq T(D(\pmb{t}))$  since  $T$  is not necessarily bijective.
+
+Tokenization Bias. Subword tokenizers suffer from what has been termed tokenization bias (Phan et al., 2024): sequences of tokens can implicitly leak information about the future content of the text they tokenize. For example, assuming a vocabulary of English words, the token sequence \{_Hello, _Wor\} leaks that there is no 'ld' after '\_Worl', otherwise the text would have been tokenized as \{_Hello, _World\} instead (see also Vieira et al., 2024).
+
+# 2.2 Generative Language Models
+
+Generative language models are auto-regressive next-token predictors. Given a sequence of tokens, a generative language model  $\theta$  defines a probability distribution  $p_{\theta}(t_i|t_{:i})^3$  over the next token. To compute the probability of a multi-token sequence conditioned on  $\pmb{t}_i$ , we can compute  $p_{\theta}(\pmb{t}_{i:j}|t_{:i}) = p_{\theta}(t_i|\pmb{t}_{:i})\cdot \ldots \cdot p_{\theta}(t_{j - 1}|\pmb{t}_{:j - 1})$ . To auto-regressively sample from the model, we continuously sample  $u\sim p_{\theta}(u|\pmb{t}_{:i})$  and append the sampled  $u$  to the sequence. However, the token level is not an intuitive interface to language models. To obtain a text level interface, we need to wrap the language model with tokenization and detokenization. We can compute the probability of a text  $\pmb{y}$  conditioned on the text  $\pmb{x}$  via  $p_{\theta}(T(\pmb{y})|T(\pmb{x}))$ . We can auto-regressively generate a text  $\pmb{y}$  conditioned on  $\pmb{x}$  by sampling  $u_{i}\sim p_{\theta}(u_{i}|T(\pmb{x})\odot \pmb{u}_{:i})$  for  $i\in \{0,\dots ,n\}$ , then detokenizing to obtain  $\pmb{y} = D(u_0,\dots,u_n)$ . However, for tokenization and detokenization not to introduce any bias,  $T(\pmb {x})\odot T(\pmb {y}) = T(\pmb {x}\odot \pmb {y})$  must hold. This is not always the case due to tokenization bias (see above).
+
+# 2.3 Distillation
+
+Pure Distillation vs. Hybrid Distillation. The general idea behind distillation is to transfer knowledge from a teacher model  $\phi$  to a student model  $\theta$ , where the student model has some desirable properties (e.g., being more efficient; Bucilua et al., 2006; Hinton et al., 2015).<sup>4</sup> We denote the teacher likelihoods  $p_T \coloneqq p_\phi$  and the student likelihoods  $p_S \coloneqq p_\theta$ . A key distinction can be drawn between pure distillation, where the primary objective is for the student predictions  $p_S(\boldsymbol{y}|\boldsymbol{x})$  to match the teachers' predictions  $p_T(\boldsymbol{y}|\boldsymbol{x})$  and hybrid distillation, where the main task is to model some ground-truth  $p(\boldsymbol{y}|\boldsymbol{x})$ , and a distillation objective is used in addition to the main objective (e.g., next-token prediction) to increase performance on the main task. Hybrid distillation can be useful to avoid potential issues arising from the pure setup, e.g. teacher hacking (Tiapkin et al., 2025) and a detrimental student-teacher capacity gap (Busbridge et al., 2025). On the other hand, pure distillation can be highly effective at preserving the student's knowledge when it has the same backbone as the teacher (self-distillation); a setting where hybrid distillation can be destructive (c.f. Section 4).
+
+Distillation Objectives. To encourage the student predictions to match the teachers', we need some measure of distance between  $p_S(\boldsymbol{y}|\boldsymbol{x})$  and  $p_T(\boldsymbol{y}|\boldsymbol{x})$ . The typical choice is the Kullback-Leibler divergence  $D_{\mathrm{KL}}(p_T\parallel p_S) = \sum_{t_i\in \mathcal{V}}p_T(t_i|\boldsymbol{t}_{:i})\log \frac{p_T(t_i|\boldsymbol{t}_{:i})}{p_S(t_i|\boldsymbol{t}_{:i})}$  (Kullback & Leibler, 1951). However, other objectives have been argued to be more suitable for distillation (Gu et al., 2024; Ko et al., 2024); many of them are instances of  $f$ -divergences of the form  $D_{\mathrm{f}}(p_T\parallel p_S) = \sum_{t_i\in \mathcal{V}}f(p_T(t_i|\boldsymbol{t}_{:i})\| p_S(t_i|t_{:i}))$  where  $f(p_{T}(x)\| p_{S}(x)) = p_{S}(x)g(\frac{p_{T}(x)}{p_{S}(x)})$  and  $g$  is convex and non-negative (Rényi, 1961).<sup>5</sup>
+
+Cross-Tokener Distillation. Recent work has begun investigating ways to distill across different tokenizers, proposing ULD (Boizard et al., 2025), MinED (Wan et al., 2024) and DSKD (Zhang et al., 2024b). We provide a detailed overview of these prior cross-tokenizer distillation methods in Appendix G. They have in common that they heuristically incorporate some information from the teacher. These heuristics are imperfect; prior methods thus strictly operate in the hybrid distillation setup, lacking the properties necessary to form a pure distillation objective (e.g., the loss being minimised iff  $p_S(\boldsymbol{y}|\boldsymbol{x}) = p_T(\boldsymbol{y}|\boldsymbol{x}) \forall \boldsymbol{x}, \boldsymbol{y}$ ). Our key contribution is introducing a principled cross-tokenizer distillation objective which enables distillation across fundamentally different tokenizers (e.g., Subword → Byte) for the first time, while also unlocking pure-cross tokenizer distillation.
+
+# 3 Methodology
+
+Our key goal is for the student likelihood  $p_S(T_S(\boldsymbol{z})|T_S(\boldsymbol{y}))$  to be equal to the teacher likelihood  $p_T(T_T(\boldsymbol{z})|T_T(\boldsymbol{y}))$  for every comparable $^6$  prefix  $\boldsymbol{y} \in \Sigma^\star$  and continuation  $\boldsymbol{z} \in \Sigma^\star$ . To this end, we start by computing the next-token probabilities  $p_T(T_T(\boldsymbol{x})_i|T_T(\boldsymbol{x})_{:i})$  and  $p_S(T_S(\boldsymbol{x})_i|T_S(\boldsymbol{x})_{:i})$ . This can be done in a single forward pass over both language models. We can now enumerate the starting and ending positions of the aligned token chunks between the teacher and the student sequences. $^7$
+
+$$
+A _ {c} (\boldsymbol {x}) = \left\{(i, j, k, l) \in \mathbb {Z} ^ {4} \left| \begin{array}{l} D (T _ {T} (\boldsymbol {x}) _ {: i}) = D (T _ {S} (\boldsymbol {x}) _ {: k}) = \boldsymbol {y}, \\ D (T _ {T} (\boldsymbol {x}) _ {i: j}) = D (T _ {S} (\boldsymbol {x}) _ {k: l}) = \boldsymbol {z}, \\ c (i, j, k, l) \quad \text {h o l d s} \end{array} \right. \right\} \quad \text {(A l i g n m e n t I n d i c e s)}
+$$
+
+Here,  $c(i,j,k,l)$  can optionally constrain the alignment to chunks which are amenable to comparison (more on this later in Section 3.1). For now, we define the chunk-level probabilities  $p(\boldsymbol{x},i:j)$  as
+
+$$
+p (\boldsymbol {x}, i: j) := p (T (\boldsymbol {x}) _ {i: j} \mid T (\boldsymbol {x}) _ {i: i}). \quad \text {(C h u n k - L e v e l P r o b a b i l i t y)}
+$$
+
+This lets us define our final objective minimising a divergence between the chunk-level probabilities:
+
+$$
+\mathcal {L} _ {S, T} ^ {\mathrm {A L M}} (\boldsymbol {x}) = \sum_ {i, j, k, l \in A _ {c} (\boldsymbol {x})} f \left(p _ {T} (\boldsymbol {x}, i: j) ^ {\frac {1}{\tau}} \mid \mid p _ {S} (\boldsymbol {x}, k: l) ^ {\frac {1}{\tau}}\right) + f \left(1 - p _ {T} (\boldsymbol {x}, i: j) ^ {\frac {1}{\tau}} \mid \mid 1 - p _ {S} (\boldsymbol {x}, k: l) ^ {\frac {1}{\tau}}\right)
+$$
+
+(ALM Objective)
+
+where  $\tau$  is a temperature scaling hyperparameter and  $f$  is a function inducing an  $f$ -divergence (Rényi, 1961). Typically, with identical tokenizers, the sum of the  $f$ -divergence would range over all tokens in the vocabulary; however, when comparing chunk-level probabilities there is an infinite amount of possible outcomes since there is an infinite amount of possible byte sequences, whose probability cannot be estimated in a finite time. We thus resort to computing the  $f$ -divergence over the binarised possibilities  $\{p(\boldsymbol{x}, i:j), 1 - p(\boldsymbol{x}, i:j)\}$  given a sample  $\boldsymbol{x} \sim \mathcal{D}$ . This constitutes an upper bound to the true  $f$ -divergence and preserves its crucial properties (e.g., being minimal iff  $p_S = p_T$ ; see Appendix E for further analysis).
+
+# 3.1 Outcome Chunk Debiasing
+
+Our formulation for the Chunk-Level Probability is only approximate since the conditions  $T(\pmb{x})_{i}$  and the outcomes  $T(\pmb{x})_{i:j}$  could exhibit different tokenization biases across the teacher and the student. We can avoid the bias stemming from the outcome chunk by modifying the chunk probability to
+
+$$
+p ^ {\text {d e b i a s e d}} (\boldsymbol {x}, i: j) := p (T (\boldsymbol {x}) _ {i: j} | T (\boldsymbol {x}) _ {: i}) \cdot \sum \left\{p (e | T (\boldsymbol {x}) _ {: j}) \mid e \in \mathcal {V}, e \text {s t a r t s w i t h a n y} b \in \mathcal {B} \right\}
+$$
+
+(Outcome-Debiased Chunk-Level Probability)
+
+![](images/29f0d9855b333539647f2a2811acbe67ebb667abbc259caeffa4fdb5c2b01ac2.jpg)  
+Figure 2: Outcome chunk debiasing removes tokenization bias. For example, the low probability of the subword token _Wor would be matched to the high-probability byte sequence  $\{., W, o, r\}$  in naive subword  $\rightarrow$  byte transfer. We can debias by multiplying by the marginal probability of a pretoken-boundary byte occurring after the chunk. In this example,  $\{\backslash n, !\} \subseteq \mathcal{B}$  and  $s \notin \mathcal{B}, 1 \notin \mathcal{B}$  where  $\mathcal{B}$  is the set of shared pretoken-boundary bytes across the teacher and the student.
+
+Here,  $\mathcal{B}$  is the set of pretoken-boundary bytes; these are bytes which never occur within a token. In practice, this can be e.g. whitespace such as the bytes corresponding to the  $\backslash \mathfrak{n},\backslash \mathfrak{t}$  and the space character. Going back to our example from Section 2.1, let  $t = \{\_ \mathrm{Hello},\_ \mathrm{Wor}\}$ . By computing  $p(t)\cdot (p(\{\_)|t) + (p(\{\backslash \mathfrak{n}\} |t) + \dots)$ , we explicitly exclude the possibility that  $t$  is followed by the letters 'ld', which, assuming every chosen ending token  $e$  starts with a byte in  $\mathcal{B}$ , removes tokenization bias (otherwise, the sequence would still be biased since there could exist a single token encoding  $D(t\odot \{e\})$ ). This debiasing has been first introduced by Pimentel & Meister (2024) in the context of correctly computing the conditional probability of a word under an autoregressive language model. See Figure 2 for an illustration of this example. Debiasing is only feasibly applicable to the outcome chunks since the required probabilities are computable 'for free' within a single forward pass. There is, however, one remaining complication: if the debiasing probability is very low, the multiplication with  $p(T(x)_{i:j}|T(x)_{i:i})$  will 'erase' the chunk probability to an extremely low value, making meaningful comparison between chunk probabilities difficult. To address this issue, we set the alignment constraint (c.f. Equation Alignment Indices):
+
+$$
+c (i, j, k, l) := \sum \left\{p _ {T} \left(e \mid T _ {T} (\boldsymbol {x}) _ {: j}\right) \mid e \in \mathcal {V} _ {T}, e \text {s t a r t s w i t h a n y} b \in \mathcal {B} \right\} \geq \gamma
+$$
+
+We ablate the impact of outcome chunk debiasing in Appendix A.3, finding that it substantially improves performance, with a slight further improvement by choosing a nonzero threshold  $\gamma$ .
+
+# 3.2 Distilling Hidden States across Tokenizers
+
+A limitation of our method is that, since we match 32-bit floating point chunk probabilities, the teacher signal has  $|A_{c}| \times 32$  bit per text, while it typically has  $\mathcal{O}(|V|)$  more at  $|T(\boldsymbol{x})| \times |\mathcal{V}| \times 32$  bit when the teacher and the student have the same tokenizer (i.e., KL divergence between the teacher and the student next-token distributions). To make up for this sparsity of the signal, we optionally add an auxiliary loss maximizing the similarity of aligned hidden states between the two models.
+
+$$
+\mathcal {L} _ {S, T} ^ {\text {h i d d e n}} (\boldsymbol {x}) = \sum_ {l _ {T}, l _ {S} \in L _ {T, S}} \sum_ {i, j, k, l \in \boldsymbol {A} (\boldsymbol {x})} \| H _ {T} (T _ {T} (\boldsymbol {x})) _ {j} ^ {l _ {T}} - \operatorname {p r o j} (H _ {S} (T _ {S} (\boldsymbol {x})) _ {l} ^ {l _ {S}}) \|
+$$
+
+(Hidden State Alignment Objective)
+
+where  $H_{\star}(\ldots)_i^l$  are the language models' hidden states at layer  $l$  and index  $i$  in the sequence and  $L_{T,S}\in \mathbb{Z}^2$  are layer-wise alignments between the teacher and the student (e.g., indicating that the last
+
+Table 1: Results of transferring the Gemma2 2B IT and Llama3.2 3B IT LLMs to the Qwen2 tokenizer and to byte-level tokenization. original denotes the original model without transfer.  ${ARC} - C$  refers to Arc-Challenge. AGI-EN and AGI-ZH refer to the English and Chinese splits of AGIEval.  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Tokenizer Method</td><td colspan="8">Benchmark</td><td rowspan="2">Avg.</td></tr><tr><td>PiQA</td><td>ARC-C</td><td>BoolQ</td><td>MMLU</td><td>AGI-EN</td><td>AGI-ZH</td><td>IFEval</td><td></td></tr><tr><td rowspan="11">Gemma2 2B IT</td><td>original</td><td>79.6</td><td>50.4</td><td>83.8</td><td>56.9</td><td>42.1</td><td>30.7</td><td>62.5</td><td>58.0</td><td></td></tr><tr><td>SFT</td><td>75.8</td><td>43.5</td><td>77.7</td><td>49.8</td><td>31.4</td><td>28.7</td><td>54.2</td><td>51.6</td><td></td></tr><tr><td>DSKD</td><td>74.0</td><td>41.4</td><td>79.7</td><td>50.4</td><td>33.0</td><td>28.9</td><td>53.6</td><td>51.6</td><td></td></tr><tr><td>MinED</td><td>76.7</td><td>44.3</td><td>79.6</td><td>51.8</td><td>33.0</td><td>28.8</td><td>57.1</td><td>53.0</td><td></td></tr><tr><td>ALM + SFT</td><td>77.0</td><td>48.0</td><td>82.5</td><td>53.4</td><td>36.4</td><td>31.4</td><td>55.7</td><td>54.9</td><td></td></tr><tr><td>ALM</td><td>76.8</td><td>49.0</td><td>82.7</td><td>53.6</td><td>38.9</td><td>31.6</td><td>53.2</td><td>55.1</td><td></td></tr><tr><td>SFT</td><td>70.7</td><td>34.7</td><td>67.9</td><td>43.1</td><td>27.6</td><td>30.0</td><td>51.5</td><td>46.5</td><td></td></tr><tr><td>DSKD</td><td>70.5</td><td>35.6</td><td>70.2</td><td>42.3</td><td>27.5</td><td>30.4</td><td>55.0</td><td>47.4</td><td></td></tr><tr><td>MinED</td><td>69.4</td><td>35.8</td><td>72.8</td><td>42.9</td><td>28.7</td><td>29.9</td><td>50.2</td><td>47.1</td><td></td></tr><tr><td>ALM + SFT</td><td>71.5</td><td>38.2</td><td>80.5</td><td>51.0</td><td>35.6</td><td>30.4</td><td>51.9</td><td>51.3</td><td></td></tr><tr><td>ALM</td><td>72.0</td><td>40.6</td><td>81.1</td><td>51.0</td><td>36.0</td><td>29.3</td><td>44.3</td><td>50.6</td><td></td></tr><tr><td rowspan="11">Llama3.2 3B IT</td><td>original</td><td>76.9</td><td>43.9</td><td>78.8</td><td>62.4</td><td>36.6</td><td>40.2</td><td>76.6</td><td>59.3</td><td></td></tr><tr><td>SFT</td><td>76.4</td><td>44.0</td><td>80.0</td><td>60.7</td><td>33.9</td><td>29.6</td><td>65.6</td><td>55.7</td><td></td></tr><tr><td>DSKD</td><td>72.0</td><td>37.2</td><td>78.3</td><td>45.9</td><td>32.5</td><td>30.9</td><td>60.7</td><td>51.1</td><td></td></tr><tr><td>MinED</td><td>77.1</td><td>44.2</td><td>82.4</td><td>60.9</td><td>35.8</td><td>29.4</td><td>71.0</td><td>57.3</td><td></td></tr><tr><td>ALM + SFT</td><td>77.0</td><td>44.4</td><td>79.9</td><td>61.8</td><td>37.1</td><td>32.0</td><td>74.1</td><td>58.0</td><td></td></tr><tr><td>ALM</td><td>77.3</td><td>45.6</td><td>79.0</td><td>61.6</td><td>37.1</td><td>33.3</td><td>76.3</td><td>58.6</td><td></td></tr><tr><td>SFT</td><td>75.2</td><td>39.8</td><td>76.8</td><td>51.5</td><td>31.5</td><td>32.6</td><td>60.8</td><td>52.6</td><td></td></tr><tr><td>DSKD</td><td>71.1</td><td>36.0</td><td>65.8</td><td>48.0</td><td>32.0</td><td>30.3</td><td>57.9</td><td>48.7</td><td></td></tr><tr><td>MinED</td><td>73.2</td><td>38.7</td><td>78.6</td><td>51.1</td><td>33.1</td><td>32.3</td><td>59.6</td><td>52.4</td><td></td></tr><tr><td>ALM + SFT</td><td>73.6</td><td>39.8</td><td>76.6</td><td>57.0</td><td>35.7</td><td>33.3</td><td>58.8</td><td>53.5</td><td></td></tr><tr><td>ALM</td><td>73.7</td><td>40.1</td><td>76.0</td><td>55.9</td><td>35.7</td><td>33.2</td><td>49.2</td><td>52.0</td><td></td></tr></table>
+
+layer hidden states of the two models should be aligned).  $\mathrm{proj}(\star):\mathbb{R}^{d_S}\to \mathbb{R}^{d_T}$  is a learnt projection function mapping the student hidden states to the teachers' dimensionality. We use greedy alignments (without any constraint  $c$ ) as the hidden state alignments  $\pmb {A}(\pmb {x})$  . The auxiliary loss enriches the signal from the teacher by  $|\pmb {A}|\times d_T\times 32$  bit. We find it to be particularly useful in some cases when the teacher and the student have the same underlying architecture (c.f. Section 4).
+
+# 3.3 Combining Loss Components
+
+As per above, we have two distillation loss components  $\mathcal{L}_{S,T}^{\mathrm{ALM}}(\pmb{x})$  and  $\mathcal{L}_{S,T}^{\mathrm{hidden}}(\pmb{x})$ . We can also combine the distillation loss with a next-token prediction objective (making it hybrid distillation; c.f. Section 2.3). The question, then, is how to combine these losses into one? Prior work (Zhang et al., 2024b; Wan et al., 2024, among others) aggregates losses via their arithmetic sum, where the contribution of each loss is optionally modulated by factors  $\beta_{1}, \beta_{2}, \dots, \beta_{n}$ . Although this allows choosing  $\beta$  such that the losses have approximately equal magnitudes, this does not mean that they contribute equally: they only contribute equally if their gradients have equal magnitudes. These two quantities are not necessarily related: for example, forward KL-divergence and cross-entropy have different values but equivalent gradients w.r.t. the predicted probabilities. To solve this problem, GradNorm (Chen et al., 2018) introduces an auxiliary loss to learn task weights such that the per-task gradients  $G_W^i$  of the last layer weights  $W$  have similar magnitudes across tasks (enumerated by  $i$ ). We take a simpler approach: We compute per-task gradients for the last layer, then set the weight for the task  $i$  to  $1 / \| G_W^i\|$  (normalised to sum to one). Instead of optimizing a loss toward similar last-layer gradient magnitudes, we thus directly solve for equal last-layer gradient magnitudes, choosing only the last layer gradients to maintain minimal overhead. This method (which we refer to as GradMag) performs on par or better than GradNorm while being easier to implement (see Appendix A.4).
+
+# 4 Experiments
+
+We experimentally verify our method on three distinct use cases. First, we observe that self-distillation across tokenizers enables unprecedentedly effective transfer of a pre-trained language model to a different tokenizer, which we explore in Use Case 1. Secondly, we find that ALM improves over prior cross-tokenizer distillation objectives at distilling a large maths-specialised model into a much smaller model (Use Case 2). Thirdly, we plug in ALM as a replacement for the next-token prediction objective of Minixhofer et al. (2024)'s hypernetwork training procedure, achieving a new state-of-the-art in Zero-Shot Tokenizer Transfer across a range of tasks (Use Case 3).
+
+# Use Case 1: Tokenizer Transfer via Self-Distillation
+
+Tokenizer transfer is the problem of transferring a pretrained language model to a different tokenizer which typically has some desirable properties (c.f. Rust et al., 2021). We observe that tokenizer transfer can be treated as cross-tokenizer self-distillation: the teacher is the language model with the original tokenizer, and the student is the same language model with the new tokenizer. We investigate two particular applications of tokenizer transfer via self-distillation. First, we transfer multiple language models to the same (subword) tokenizer; this enables token-level ensembling, improving their performance. Second, we transfer subword-based language models to byte-level tokenization, fundamentally changing their token granularity. This may be an attractive way of creating byte-level models instead of expensively training from scratch.
+
+Models. We use the instruction-tuned Gemma2 2B IT (Gemma Team et al., 2024) and the instruction-tuned Llama 3.2 3B IT (Grattafori et al., 2024) as the models for self-distillation. We transfer these models to the Qwen2 tokenizer; this enables assembling the models with models of the Qwen series, for which we choose Qwen2.5 1.5B IT (Qwen et al., 2025). We examine the choice of the pivot and the related subtle decision whether to keep or to transfer the special tokens in Appendix H.
+
+Baseline Methods. We compare against initialising the new token embeddings using a heuristic (e.g. Tran, 2020; Minixhofer et al., 2022; Gee et al., 2022; Dobler & de Melo, 2023), then training on next-token prediction – the current standard methodology for tokenizer transfer. We refer to this baseline as SFT. We use FVT (Gee et al., 2022) as the initialisation heuristic following Minixhofer et al. (2024). We choose DSKD Zhang et al. (2024b) and MinED (Wan et al., 2024) as cross-tokenizer distillation baselines. $^9$  DSKD and MinED add cross-tokenizer distillation to the next-token prediction objective, so they need to train jointly on SFT and distillation; this is not the case for ALM. We thus also experiment with our objective plus next-token prediction  $(ALM + SFT)$ .
+
+Training. We train on the Tulu3 instruction-tuning dataset (Lambert et al., 2025) with LoRA (Hu et al., 2022), sweeping over the learning rate per method (see Appendix A.1) and using GradMag to combine loss components (c.f. Section 3.3); see Appendix B for training details. For transfer to bytes, we make additional adjustments to account for this fundamental change discussed in Appendix C.
+
+Evaluation & Transfer Results. We evaluate on a standard set of natural language benchmarks consisting of PiQA (Bisk et al., 2020), ARC-Challenge (Clark et al., 2018), BoolQ (Clark et al., 2019), MMLU (Hendrycks et al., 2021), AGIEval (Zhong et al., 2023) $^{10}$  and IFEval (Zhou et al., 2023). We use lm-eval (Gao et al., 2024) for all evaluations. Table 1 shows tokenizer transfer results. On transfer to the subword-level Qwen2 tokenizer, MinED leads to consistent improvements over SFT, and ALM and ALM+SFT consistently improve over MinED. Here, pure ALM (without SFT) performs better than ALM+SFT, which may be the case due to better preserving the original model's behaviour. The byte-transfer case is substantially harder: in general, we expect the difficulty of cross-tokenizer distillation to correlate with the difference in vocabulary size between the teacher and the student. The vocabulary size difference is extreme for transfer to bytes. In this case, prior cross-tokenizer distillation methods do not achieve consistent improvements over SFT. In constraint, ALM and ALM+SFT achieve substantial and consistent improvements. We hypothesize ALM+SFT outperforms ALM here since naïve transfer to bytes is necessarily strongly destructive since the model has to adapt to a vastly different granularity, so preserving the original model's behaviour
+
+Table 2: Results of ensembling (transferred) Gemma2 and Llama3 with a Qwen2.5 model.  $p$  and logp denote taking the arithmetic mean of the probabilities and the log probabilities, respectively. MinED denotes the ensembling baseline based on heuristically aligning token probabilities.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2" colspan="3">Size ID Model</td><td colspan="7">Benchmark</td><td rowspan="2">Avg.</td></tr><tr><td>PiQA</td><td>ARC-C</td><td>BoolQ</td><td>MMLU</td><td>AGI-EN</td><td>AGI-ZH</td><td>IFEval</td></tr><tr><td rowspan="3">Original 
+Models</td><td>2.6B</td><td>G</td><td>Gemma2 2B IT</td><td>79.6</td><td>50.4</td><td>83.8</td><td>56.9</td><td>42.1</td><td>30.7</td><td>62.5</td><td>58.0</td></tr><tr><td>3.2B</td><td>L</td><td>Llama3.2 3B IT</td><td>76.9</td><td>43.9</td><td>78.8</td><td>62.4</td><td>36.6</td><td>40.2</td><td>76.6</td><td>59.3</td></tr><tr><td>1.5B</td><td>Q</td><td>Qwen2.5 1.5B IT</td><td>76.3</td><td>43.2</td><td>77.9</td><td>60.1</td><td>45.0</td><td>54.2</td><td>46.3</td><td>57.6</td></tr><tr><td rowspan="2">Transfers</td><td>2.4B</td><td>G&#x27;</td><td>G→Q Tok.</td><td>76.8</td><td>49.0</td><td>82.7</td><td>53.6</td><td>38.9</td><td>31.6</td><td>53.2</td><td>55.1</td></tr><tr><td>3.3B</td><td>L&#x27;</td><td>L→Q Tok.</td><td>77.3</td><td>45.6</td><td>79.0</td><td>61.6</td><td>37.1</td><td>33.3</td><td>76.3</td><td>58.6</td></tr><tr><td rowspan="4">Ensembles</td><td>7.3B</td><td colspan="2">Q+G+L (MinED-logp)</td><td>75.7</td><td>46.8</td><td>84.6</td><td>62.6</td><td>42.0</td><td>34.5</td><td>67.2</td><td>59.1</td></tr><tr><td>7.3B</td><td colspan="2">Q+G+L (MinED-p)</td><td>76.9</td><td>46.2</td><td>83.2</td><td>64.1</td><td>40.6</td><td>39.4</td><td>65.9</td><td>59.5</td></tr><tr><td>7.2B</td><td colspan="2">Q+G&#x27;+L&#x27; (logp)</td><td>77.7</td><td>48.4</td><td>83.9</td><td>62.2</td><td>44.2</td><td>46.8</td><td>61.1</td><td>60.6</td></tr><tr><td>7.2B</td><td colspan="2">Q+G&#x27;+L&#x27; (p)</td><td>77.5</td><td>48.0</td><td>83.4</td><td>63.0</td><td>43.6</td><td>45.7</td><td>64.3</td><td>60.8</td></tr><tr><td rowspan="2">Larger 
+Models</td><td>7.6B</td><td colspan="2">Llama3.1 8B IT</td><td>81.6</td><td>53.9</td><td>85.4</td><td>68.4</td><td>47.3</td><td>47.1</td><td>81.7</td><td>66.5</td></tr><tr><td>8.0B</td><td colspan="2">Qwen2.5 7B IT</td><td>80.4</td><td>54.3</td><td>86.4</td><td>71.7</td><td>57.6</td><td>72.6</td><td>76.6</td><td>71.4</td></tr></table>
+
+is less crucial. For transfer to bytes, a gap with respect to the original model remains in all cases. There are multiple avenues toward closing this gap: it may be beneficial to add a distillation phase on pretraining data, to add more layers to close the gap in effective parameter count, to retrofit the model into an hourglass architecture with token merging as in DTP (Nawrot et al., 2023) or BLT (Pagnoni et al., 2024), and/or to add multi-byte prediction (Gloeckle et al., 2024).
+
+Ensembling Results. Table 2 shows the results of ensembling Qwen with the transferred Gemma and Llama models  $(\mathcal{Q} + \mathcal{G}' + \mathcal{L}')$ . Following Huang et al. (2024), we compare against a MinED ensemble: this applies the same heuristic as the corresponding distillation method to map tokens in the respective vocabularies to the pivot vocabulary (in this case, Qwen). $^{11}$  The ensemble exhibits two notable characteristics: (i) on tasks where all constituents perform at a similar level, the ensemble improves over the performance of each individual model (e.g., PiQA, BoolQ, MMLU) and (ii) on tasks were either one of the constituents performs exceptionally well, the ensemble moves the needle toward this level of performance, although it doesn't reach it (e.g., ARC-C, AGIEval, IFEval). These improvements stem from naively averaging the models' predictions. We believe they open the door toward more sophisticated inter-model combination via improved ensembling (e.g., Ormazabal et al., 2023), routing (e.g., Shen et al., 2024) and/or merging (e.g., Sharma et al., 2024).
+
+Cost of Distillation. We report the FLOPS and memory requirements as well as the task performance across methods in Figure 3. DSKD requires substantially more TFLOPs than the other methods since there is an extra cross-attention step between the teacher and the student sequences which can incur substantial costs. This is further exacerbated with longer contexts (e.g., when transferring to bytes). MinED needs substantially more memory since the  $|T(\boldsymbol{x})| \times |V|$  matrix of logits needs to be aligned with the student along the sequence and vocabulary dimensions. ALM suffers from neither of these constraints and reduces the gap to the teacher by an additional  $34\%$  over the best prior method.
+
+# Use Case 2: Large-to-Small Distillation Across Tokenizers
+
+To investigate the efficacy of our method in a large-to-small distillation setting, we scale up from the evaluation setups used in prior work on cross-tokenizer distillation to a realistic larger-scale setup: distilling the maths-specialised OpenMath2-Llama3.1-8B (Toshniwal et al., 2024) into Gemma2 2B. As in Use Case 1, we compare our method against training with SFT, DSKD, and MinED.
+
+Training. We use the OpenMathInstruct-2 dataset (which the teacher has been trained on; Toshniwal et al., 2024) and follow the training setup from Use Case 1, apart from increasing the sequence length to 1024 to allow for longer chain-of-thought traces, doubling the batch size to 64, and accordingly reducing the training steps to 5k to train for an equivalent total amount of tokens. We also found it beneficial to train the full model and reduce the learning rate to 5e-6 in preliminary experiments.
+
+![](images/8eaaec28755a704330375cf74135d0e1b87a88facae7be49214efffe95426065.jpg)  
+Figure 3: Efficiency and task performance metrics of cross-tokenizer distillation methods, measured via worst-case performance across transfer of Gemma2 to Qwen2 and byte-level tokenizers. SFT denotes the required FLOPs and memory as well as the task performance of the SFT baseline.
+
+Table 3: Results of cross-tokenizer distilling the large math-specialized OpenMath2-Llama3.1-8B into the small Gemma2 2B language model. All results are zero-shot CoT.  
+
+<table><tr><td>Model</td><td>Method</td><td>GSM8K</td><td>MATH</td><td>Avg.</td></tr><tr><td>OpenMath2-Llama3.1-8B</td><td></td><td>88.9 ±0.86</td><td>60.2 ±0.69</td><td>74.6 ±0.55</td></tr><tr><td>Gemma2 2B IT</td><td></td><td>6.1 ±0.66</td><td>11.3 ±0.45</td><td>8.7 ±0.40</td></tr><tr><td rowspan="5">Gemma2 2B</td><td>SFT</td><td>67.2 ±1.29</td><td>36.2 ±0.68</td><td>51.7 ±0.73</td></tr><tr><td>DSKD</td><td>65.7 ±1.31</td><td>34.9 ±0.67</td><td>50.3 ±0.74</td></tr><tr><td>MinED</td><td>64.9 ±1.31</td><td>34.6 ±0.67</td><td>49.8 ±0.74</td></tr><tr><td>ALM + SFT</td><td>70.2 ±1.26</td><td>36.4 ±0.68</td><td>53.3 ±0.72</td></tr><tr><td>ALM</td><td>68.5 ±1.28</td><td>36.7 ±0.68</td><td>52.6 ±0.72</td></tr></table>
+
+Evaluation & Results. We report zero-shot accuracy on GSM8K (Cobbe et al., 2021) and the MATH benchmark (Hendrycks et al., 2021) in Table 3. Notably, the SFT baseline is competitive, surpassing both MinED and DSKD. However, ALM and ALM + SFT do consistently outperform SFT, leading to an improvement of up to  $3\%$  points while greatly surpassing the general-purpose Gemma2 2B IT and achieving 53.3 / 74.6 of the teachers' average performance at 4x fewer parameters.
+
+# Use Case 3: Improving Zero-Shot Tokenizer Transfer Hypernetworks via Self-Distillation
+
+As a final use case, we apply ALM to Zero-Shot Tokenizer Transfer (ZeTT) hypernetworks (Minixhofer et al., 2024). In a nutshell, the idea is training a hypernetwork on a distribution of randomly sampled tokenizers such that, after training, the hypernetwork can predict embeddings for any arbitrary tokenizer. The tokenizer transfer effort is thus shifted to a hypernetwork training stage, making subsequent tokenizer transfer easier (or even possible in zero-shot). Minixhofer et al. (2024) train the hypernetworks via a next-token prediction objective, i.e., to predict embeddings which minimize the cross-entropy of the next-token prediction. We instead plug in the ALM loss, training the hypernetwork to produce embeddings which mimic the behaviour of the original model, instead of merely predicting embeddings which perform well at next-token prediction. $^{12}$  Results are shown in Table 4. ALM outperforms SFT for hypernetwork training (e.g.  $67.5 \rightarrow 74.0$  on BoolQ for zero-shot transfer to the Mistral v2 tokenizer). In general, we expect direct tokenizer transfer self-distillation via ALM as in Use Case 1 to be the most efficient if the goal is transferring to a single target tokenizer. However, if the goal is to potentially transfer to up to  $N$  different tokenizers, there is some threshold of  $N$  where we expect it would become more efficient to train a ZeTT hypernetwork once, then have a brief additional training phase per tokenizer, instead of separately transferring to the  $N$  tokenizers individually. ZeTT hypernetwork training also demonstrates another advantage of our method: while MinED needs an expensive pre-computation step to compute the Levenshtein distance between every pair of tokens, ALM does not require any pre-computation: it can operate in online settings such as hypernetwork training where a new tokenizer is sampled at every training step.
+
+Table 4: Performance of a Gemma2 2B hypernetwork trained with SFT vs. ALM on zero-shot transfer to the GPT2 (Radford et al., 2019), Mistral v2 (Mistral AI, 2025) and Llama3 tokenizers. $^{14}$  
+
+<table><tr><td rowspan="2">Tokenizer</td><td rowspan="2">Method</td><td colspan="9">Benchmark</td><td rowspan="2">Avg.</td></tr><tr><td>PiQA</td><td>ARC-E</td><td>ARC-C</td><td>BoolQ</td><td>MMLU</td><td>AGI-EN</td><td>AGI-ZH</td><td>HS</td><td></td></tr><tr><td colspan="2">original</td><td>79.6</td><td>78.2</td><td>50.4</td><td>83.8</td><td>56.9</td><td>42.1</td><td>30.7</td><td>72.6</td><td>61.8</td><td></td></tr><tr><td rowspan="2">0-shot GPT2</td><td>SFT</td><td>74.6</td><td>63.1</td><td>40.1</td><td>73.2</td><td>40.8</td><td>26.7</td><td>27.8</td><td>68.9</td><td>51.9</td><td></td></tr><tr><td>ALM</td><td>76.1</td><td>67.2</td><td>41.4</td><td>71.6</td><td>45.0</td><td>26.1</td><td>27.2</td><td>69.7</td><td>53.0</td><td></td></tr><tr><td rowspan="2">0-shot Mistral v2</td><td>SFT</td><td>73.6</td><td>61.6</td><td>37.7</td><td>67.5</td><td>37.7</td><td>25.2</td><td>29.3</td><td>64.5</td><td>49.6</td><td></td></tr><tr><td>ALM</td><td>75.2</td><td>66.3</td><td>40.5</td><td>74.0</td><td>38.5</td><td>26.3</td><td>28.6</td><td>66.0</td><td>51.9</td><td></td></tr><tr><td rowspan="2">0-shot Llama3</td><td>SFT</td><td>74.2</td><td>60.9</td><td>38.4</td><td>68.0</td><td>37.6</td><td>24.9</td><td>29.2</td><td>64.3</td><td>49.7</td><td></td></tr><tr><td>ALM</td><td>75.5</td><td>66.0</td><td>40.3</td><td>74.3</td><td>38.6</td><td>25.6</td><td>28.8</td><td>65.9</td><td>51.9</td><td></td></tr></table>
+
+Table 5: Results of transferring Gemma3 12B IT to the Qwen2 tokenizer (c.f. Table 1).  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Method</td><td colspan="8">Benchmark</td><td rowspan="2">Avg.</td></tr><tr><td>PiQA</td><td>ARC-C</td><td>BoolQ</td><td>MMLU</td><td>AGI-EN</td><td>AGI-ZH</td><td>IFEval</td><td></td></tr><tr><td rowspan="3">Gemma3 12B IT → Qwen2 Tok.</td><td>original</td><td>78.2</td><td>60.1</td><td>87.5</td><td>71.5</td><td>60.0</td><td>57.4</td><td>83.3</td><td>71.2</td><td></td></tr><tr><td>SFT</td><td>81.4</td><td>56.7</td><td>83.7</td><td>67.2</td><td>46.2</td><td>43.5</td><td>73.4</td><td>64.6</td><td></td></tr><tr><td>ALM</td><td>78.8</td><td>59.9</td><td>86.9</td><td>70.7</td><td>57.6</td><td>53.6</td><td>76.7</td><td>69.2</td><td></td></tr></table>
+
+# 4.1 Scaling ALM to Larger Models
+
+We ran additional experiments on tokenizer transfer of Gemma3 12B (Gemma Team et al., 2025) to the Qwen2 tokenizer (also used by Qwen3) via ALM self-distillation to verify that our method scales to larger models. Results are shown in Table 5. Crucially, the transfer-original gap reduction due to ALM scales favourably with model size both in terms of absolute numbers and relative improvements over SFT  $(3.5 / 6.4\approx 55\%$  reduction for Gemma2  $2\mathrm{B}\rightarrow \mathrm{Qwen2}$  Tok. vs.  $4.6 / 6.6\approx 70\%$  reduction for Gemma3  $12\mathrm{B}\rightarrow \mathrm{Qwen2}$  Tok.) in this experiment. We leave scaling to truly large models to future work, but note that we expect no fundamental obstacles in doing so (c.f. also Appendix I on an efficient implementation of ALM).
+
+# 5 Conclusion
+
+We have introduced ALM, a principled cross-tokenizer distillation method which outperforms prior heuristic methods across a diverse set of use cases while also enabling distillation across fundamentally different tokenizers for the first time, as evidenced by effective distillation of subword-based models to the byte-level (Use Case 1). We have also shown that ALM is effective at conventional large-to-small distillation (Use Case 2) and enables state-of-the-art zero-shot tokenizer transfer (Use Case 3). Overall, our work vastly expands the number of possible teacher-student pairs for distillation. In addition to enabling a range of new applications, this provides a basis for further enhancements such as more sophisticated combinations of tokenizer-transferred models via enhanced ensembling/routing/merging and using ALM to retrofit subword-based models into dedicated byte-level architectures.
+
+# 6 Limitations
+
+Due to computational constraints, we have trained on comparably few tokens  $(\approx 0.6\mathrm{B})$ . It is not yet clear how ALM compares to SFT and other distillation methods in regimes with larger amounts of training tokens. Furthermore, we only explore one instantiation of cross-tokenizer transfer via minimising a chunk-level  $f$ -divergence, leaving other instantiations (e.g., a categorical instead of binary divergence) to subsequent explorations. Along the same lines, it is not clear whether our chunk selection strategy is optimal and the space of possible chunk selection strategies is large, including overlapping or non-consecutive chunks. Finally, we have restricted ourselves to the seminal Supervised KD setup (Hinton et al., 2015), leaving applications of ALM to more intricate distillation using e.g. Rejection Sampling (as in Touvron et al., 2023) or On-Policy Distillation (as in Gu et al., 2024; Agarwal et al., 2024) to future work.
+
+# Acknowledgments and Disclosure of Funding
+
+This work is supported by the ERC Starting Grant AToM-FM (101222956) awarded to Edoardo M. Ponti and the Royal Society University Research Fellowship 'Inclusive and Sustainable Language Technology for a Truly Multilingual World' (no 221137; 2022-) awarded to Ivan Vulić. Research supported by the Google Cloud Research Credits program with the award GCP329647813. Research supported with Cloud TPUs from Google's TPU Research Cloud (TRC). We would like to thank Andreas Grivas, Giwon Hong and Hannah Sterz for their helpful feedback on the paper draft.
+
+# References
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+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: The claims made in the abstract and Section 1 match the results in Section 4.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: We discuss limitations in Section 6.
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: We provide proofs around the binarised  $f$ -divergence in Appendix E.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+# Answer: [Yes]
+
+Justification: Appendix B states our common experiment setup, and various sections throughout the paper state how we adjust this setup for any particular experiment. Together, this provides the full details to reproduce our experiments.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.  
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+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [No]
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+Justification: Code is not submitted alongside this paper but will be provided upon publication.
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+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: We provide training details in Appendix B and details on hyperparameters and how they were chosen in Appendix A.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: We report the standard error of the mean in the experiments on Use Case 2. For Use Case 1 and Use Case 3, we perceive quantifying statistical significance not to be necessary due to the magnitude and consistency of improvements over prior methods.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
+
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
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+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: Compute resources and execution times are specified in Appendix B.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: We adhere to all applicable points of the Code of Ethics.
+
+Guidelines:
+
+The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: We discuss our methods' positive impact on enhancing reusability and composability of LLMs in Section 1. We do not believe our method leads to any particular negative societal impact beyond the potential negative impact of increased LLM adoption.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.
+
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: We do not release any standalone models; released fine-tuned models are bound to the model checkpoints there were fine-tuned from, including being bound to the safeguards put on the fine-tuning starting point.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: Asset creators are credited, were applicable, throughout the paper.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.
+
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [No]
+
+Justification: We will release fine-tuned model checkpoints created via training with our method upon publication alongside our code. These assets are not vital to our method itself, and we thus do not make them part of this submission.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: There are no human subjects involved in the experiments.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: There are no human subjects involved in the experiments.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
+
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: Our work does not involve applying LLMs in any way.
+
+Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Ablations & Sensitivity Analyses
+
+# A.1 Sensitivity to the Learning Rate
+
+We analyse the impact of different choices for the learning rate across all methods in Table 6. The different methods are generally robust to settings for the learning rate within the range of  $2e - 6$  to  $5e - 5$ , although we occasionally observed loss spikes when choosing high learning rates within this range in preliminary experiments. Accordingly, we set the learning rate to  $1e - 5$  for all methods in our main experiments.
+
+Table 6: Sensitivity of different methods to the learning rate on transfer of Gemma 2 2B IT to the Qwen2 Tokenizer by training for 5k steps (and otherwise matching the experiments in Use Case 1).  
+
+<table><tr><td rowspan="2" colspan="2">Method Learning Rate</td><td colspan="8">Benchmark</td><td rowspan="2">Avg.</td></tr><tr><td>PiQA</td><td>ARC-C</td><td>BoolQ</td><td>MMLU</td><td>AGI-EN</td><td>AGI-ZH</td><td>IFEval</td><td></td></tr><tr><td rowspan="3">SFT</td><td>2e-6</td><td>76.1</td><td>41.8</td><td>81.1</td><td>50.4</td><td>34.5</td><td>29.9</td><td>46.8</td><td>51.5</td><td></td></tr><tr><td>1e-5</td><td>76.3</td><td>42.2</td><td>80.3</td><td>50.2</td><td>33.4</td><td>29.8</td><td>51.8</td><td>52.0</td><td></td></tr><tr><td>5e-5</td><td>76.1</td><td>42.7</td><td>78.5</td><td>50.5</td><td>31.5</td><td>28.5</td><td>54.9</td><td>51.8</td><td></td></tr><tr><td rowspan="3">DSKD</td><td>2e-6</td><td>75.8</td><td>41.7</td><td>75.6</td><td>49.5</td><td>33.4</td><td>30.5</td><td>42.7</td><td>49.9</td><td></td></tr><tr><td>1e-5</td><td>74.6</td><td>40.9</td><td>80.0</td><td>49.0</td><td>33.1</td><td>29.8</td><td>52.2</td><td>51.4</td><td></td></tr><tr><td>5e-5</td><td>74.0</td><td>41.4</td><td>80.3</td><td>51.6</td><td>32.3</td><td>29.4</td><td>54.8</td><td>52.0</td><td></td></tr><tr><td rowspan="3">MinED</td><td>2e-6</td><td>76.0</td><td>41.1</td><td>78.1</td><td>52.0</td><td>35.9</td><td>30.0</td><td>52.6</td><td>52.2</td><td></td></tr><tr><td>1e-5</td><td>76.6</td><td>44.0</td><td>81.9</td><td>52.2</td><td>36.3</td><td>29.7</td><td>57.5</td><td>54.0</td><td></td></tr><tr><td>5e-5</td><td>76.3</td><td>43.8</td><td>80.1</td><td>51.7</td><td>32.8</td><td>28.5</td><td>58.1</td><td>53.1</td><td></td></tr><tr><td rowspan="3">ALM</td><td>2e-6</td><td>75.7</td><td>46.0</td><td>83.4</td><td>53.7</td><td>40.1</td><td>32.9</td><td>57.4</td><td>55.6</td><td></td></tr><tr><td>1e-5</td><td>76.8</td><td>48.0</td><td>83.1</td><td>53.8</td><td>39.3</td><td>32.0</td><td>55.5</td><td>55.5</td><td></td></tr><tr><td>5e-5</td><td>76.8</td><td>49.1</td><td>81.9</td><td>53.8</td><td>38.9</td><td>30.9</td><td>54.8</td><td>55.2</td><td></td></tr></table>
+
+# A.2 Impact of the Choice for the Distance Function  $f$  and the Temperature  $\tau$
+
+The distance function  $f$  and the temperature  $\tau$  are two crucial hyperparameters of our method. We analyse two choices of  $f$ , the KL-Divergence  $f_{\mathrm{KL}}(p^{1 / \tau}\| q^{1 / \tau}) = p^{1 / \tau}\log \frac{p^{1 / \tau}}{q^{1 / \tau}}$  and the total variation distance  $f_{\mathrm{TVD}}(p^{1 / \tau}\| q^{1 / \tau}) = |p^{1 / \tau} - q^{1 / \tau}|$  across a range of temperatures. Figures 4 and 5 show the gradients with respect to  $\log q$  over the binarised outcomes of  $f_{\mathrm{KL}}$  and  $f_{\mathrm{TVD}}$ , respectively. For
+
+![](images/88cb1e52b877da4d7dd25c754283df7bd8d3e5759634eee00019a935c502250b.jpg)  
+Figure 4: The KL-Divergence gradients  $\frac{\delta f_{\mathrm{KL}}(p^{1 / \tau}\}||q^{1 / \tau}) + \delta f_{\mathrm{KL}}(1 - p^{1 / \tau}\}||1 - q^{1 / \tau})}{\delta\log q}$  over  $\tau$
+
+![](images/3d34cace266dbcde37ccd2fd80e6d10e14a32405de0cfe20ea6323d54053a8b1.jpg)
+
+![](images/67ce1eecd263bb8498a8ad901767b3f014763dcd396774886fd8350bfd33be80.jpg)
+
+![](images/d7a0ceccf2f333d5bd6755bdbeff308f87d7b30ddd011c1e029c7d95572ee00f.jpg)
+
+![](images/baeebfbce446c029ed243611b2c3c0dc45986d1099d24d5003754775d00bb8e7.jpg)  
+Figure 5: The Total Variation Distance gradients  $\frac{\delta f_{\mathrm{TVD}}(p^{1 / \tau}\|q^{1 / \tau}) + \delta f_{\mathrm{TVD}}(1 - p^{1 / \tau}\|1 - q^{1 / \tau})}{\delta\log q}$  over  $\tau$ .
+
+![](images/9c886af56f9a86e10633d5fffd7355c48d47d550cbe3c73ffe1efa04dd93bed5.jpg)
+
+![](images/d4e8684bc219852fd8dfc42fa165d2cabbeb960fd89f5cae6a67551370629421.jpg)
+
+![](images/14e6d48157ad2927f726aa7e18021c9c9a2b076184e9421ef21283b2f74ac671.jpg)
+
+small temperatures (e.g.,  $\tau = 1$ ), the gradient vanishes as  $\log p$  or  $\log q$  decreases. In general, this expected behaviour: mismatches in higher  $p$  or  $q$  should generally result in a higher contribution to the loss gradients than mismatches when  $p$  and  $q$  are both low. However, importantly, our method operates on chunks of tokens and the likelihood of a chunk multiplicatively decreases as the length of the chunk increases (since the likelihood of any token is always less than 1); this means that the contribution of longer chunks to the loss vanishes if the temperature  $\tau$  is low. The value of  $\tau$  thus defines a trade-off: low  $\tau$  leads to focusing on high-likelihood chunks. Increasing  $\tau$  leads to increased focus on lower-likelihood/longer chunks. $^{15}$  We also observe notable behaviour of the KL-Divergence and Total Variation Distance as  $\tau \to \infty$ , and that the gradients as  $\tau \to \infty$  can be computed in closed form. Specifically,  $f_{\mathrm{KL}}(p^{1/\tau} \| q^{1/\tau}) + f_{\mathrm{KL}}(1 - p^{1/\tau} \| 1 - q^{1/\tau}) \approx C((\log p - \log q) + \log(p) \log(\frac{\log q}{\log p})$  and  $f_{\mathrm{TVD}}(p^{1/\tau} \| q^{1/\tau}) + f_{\mathrm{TVD}}(1 - p^{1/\tau} \| 1 - q^{1/\tau}) \approx C| \log p - \log q|$  as  $\tau \to \infty$ . To show this, we will use the fact that  $\exp(x) \approx 1 + x$  for small enough  $x$  and let  $a = \log p^{1/\tau} = \log(p)/\tau$  and  $b = \log q^{1/\tau} = \log(q)/\tau$ , both of which are close to zero since  $\tau \to \infty$ . We have
+
+$$
+\begin{array}{l} f _ {\mathrm {K L}} (p ^ {1 / \tau} \| q ^ {1 / \tau}) = p ^ {1 / \tau} \log \frac {p ^ {1 / \tau}}{q ^ {1 / \tau}} \\ = \exp (a) (\log \exp (a) - \log \exp (b)) \\ = \exp (a) (a - b) \\ \approx (1 + a) (a - b) \quad \text {s i n c e} \tau \rightarrow \infty \\ = a + a ^ {2} - b - a b \\ = \log (p) / \tau + \log (p) ^ {2} / \tau^ {2} - \log (q) / \tau - \log (p) \log (q) / \tau^ {2} \\ \approx (\log (p) - \log (q)) / \tau \quad \text {s i n c e} 1 / \tau \gg 1 / \tau^ {2} \\ = C (\log p - \log q) \quad \text {w h e r e} C = 1 / \tau \\ \end{array}
+$$
+
+and
+
+$$
+\begin{array}{l} f _ {\mathrm {K L}} \left(1 - p ^ {1 / \tau} \| 1 - q ^ {1 / \tau}\right) = \left(1 - p ^ {1 / \tau}\right) \log \frac {\left(1 - p ^ {1 / \tau}\right)}{\left(1 - q ^ {1 / \tau}\right)} \\ = (1 - \exp (a)) (\log (1 - \exp (a)) - \log (1 - \exp (b))) \\ \approx - a \left(\log (- a) - \log (- b)\right) \quad \text {s i n c e} \tau \rightarrow \infty \\ = \log (p) (\log (- \log q) - \log (- \log p)) / \tau \\ = C \left(\log (p) \log \left(\frac {\log q}{\log p}\right)\right) \quad \text {w h e r e} C = 1 / \tau \\ \end{array}
+$$
+
+Analogously, for the Total Variation Distance we have
+
+$$
+\begin{array}{l} f _ {\mathrm {T V D}} \left(p ^ {1 / \tau} | | q ^ {1 / \tau}\right) = f _ {\mathrm {T V D}} \left(1 - p ^ {1 / \tau} \| 1 - q ^ {1 / \tau}\right) = \left| p ^ {1 / \tau} - q ^ {1 / \tau} \right| \\ = | \exp (a) - \exp (b) | \\ \approx | (1 + a) - (1 + b) | \quad \text {s i n c e} \tau \rightarrow \infty \\ = | a - b | \\ = | \log (p) / \tau - \log (q) / \tau | \\ = C | \log (p) - \log (q) | \quad \text {w h e r e} C = 1 / \tau \\ \end{array}
+$$
+
+Empirically, both KL-divergence and Total Variation Distance perform well, and  $\tau > 1$  is beneficial (Table 7). We opt for the KL-divergence for our main experiments and set the temperature  $\tau = 100$ .<sup>16</sup>
+
+Table 7: Sensitivity to the distance function  $f$  and temperature  $\tau$  on transfer of Gemma 2 2B IT to the Qwen2 Tokenizer by training for 5k steps (and otherwise matching the experiments in Use Case 1).  
+
+<table><tr><td rowspan="2">Distance function f</td><td rowspan="2">Temperature τ</td><td colspan="8">Benchmark</td><td>Avg.</td></tr><tr><td>PiQA</td><td>ARC-C</td><td>BoolQ</td><td>MMLU</td><td>AGI-EN</td><td>AGI-ZH</td><td>IFEval</td><td></td><td></td></tr><tr><td rowspan="4">f=fKL</td><td>τ=1</td><td>76.4</td><td>47.2</td><td>83.0</td><td>53.6</td><td>40.0</td><td>31.4</td><td>61.1</td><td>56.1</td><td></td></tr><tr><td>τ=5</td><td>76.5</td><td>47.6</td><td>83.3</td><td>53.7</td><td>39.7</td><td>32.3</td><td>60.7</td><td>56.3</td><td></td></tr><tr><td>τ=100</td><td>76.8</td><td>48.0</td><td>83.1</td><td>53.8</td><td>39.3</td><td>32.0</td><td>55.5</td><td>55.5</td><td></td></tr><tr><td>τ→∞</td><td>76.3</td><td>47.7</td><td>83.2</td><td>53.6</td><td>39.0</td><td>32.4</td><td>54.4</td><td>55.2</td><td></td></tr><tr><td rowspan="4">f=fTVD</td><td>τ=1</td><td>75.8</td><td>46.8</td><td>83.5</td><td>53.9</td><td>41.5</td><td>32.0</td><td>58.9</td><td>56.1</td><td></td></tr><tr><td>τ=5</td><td>75.6</td><td>47.9</td><td>83.2</td><td>53.7</td><td>40.0</td><td>32.1</td><td>61.0</td><td>56.2</td><td></td></tr><tr><td>τ=100</td><td>76.3</td><td>48.9</td><td>82.9</td><td>53.6</td><td>39.2</td><td>32.2</td><td>55.6</td><td>55.5</td><td></td></tr><tr><td>τ→∞</td><td>76.1</td><td>48.4</td><td>82.9</td><td>53.7</td><td>38.6</td><td>32.9</td><td>52.9</td><td>55.1</td><td></td></tr></table>
+
+# A.3 Ablating Outcome Chunk Debiasing, the threshold  $\gamma$  and Hidden States Distillation
+
+We ablate the impact of Outcome Chunk Debiasing (Section 3.1) as well as the associated threshold and the Hidden State Distillation Loss (Section 3.2) in Table 8. Notably, the hidden state loss is slightly detrimental to Subword  $\rightarrow$  Subword transfer, but highly beneficial to Subword  $\rightarrow$  Byte transfer. Outcome Chunk Debiasing is consistently highly beneficial, and can be marginally further improved via a threshold  $\gamma >0$ .
+
+# A.4 Comparing Loss Combination Strategies: Fixed  $\beta$  vs. GradNorm vs. GradMag
+
+We compare the different loss combination strategies discussed in Section 3.3 in Table 9. GradMag performs on par or slightly better than GradNorm across methods, and both perform vastly better than different fixed combination coefficients  $\beta$ , while exhibiting only a small overhead from the additional per-task back-propagation through the last layer. Notably, GradNorm requires an extra optimizer (and tuning the associated hyperparameters), which our method does not.
+
+Table 8: Effect of varying the bias threshold  $\gamma$  on transfer of Gemma 2 2B IT to the Qwen2 Tokenizer and Llama 3 3B IT to byte-level tokenization, training for 5k steps and otherwise matching the experiments in Use Case 1.  $D$  denotes the addition of Outcome Chunk Debiasing (Section 3.1) and  $H$  denotes the addition of the Hidden State Loss (Section 3.1).  
+
+<table><tr><td rowspan="2">Model → Tokenizer</td><td rowspan="2">ALM Setting</td><td colspan="8">Benchmark</td><td rowspan="2">Avg.</td></tr><tr><td>PiQA</td><td>ARC-C</td><td>BoolQ</td><td>MMLU</td><td>AGI-EN</td><td>AGI-ZH</td><td>IFEval</td><td></td></tr><tr><td rowspan="6">Gemma2 → Qwen2</td><td>γ = 0.10, D (default)</td><td>76.8</td><td>48.3</td><td>83.1</td><td>53.6</td><td>38.9</td><td>32.2</td><td>56.0</td><td>55.6</td><td></td></tr><tr><td>γ = 0.99, D</td><td>77.1</td><td>48.0</td><td>83.0</td><td>53.7</td><td>39.3</td><td>32.6</td><td>53.7</td><td>55.4</td><td></td></tr><tr><td>γ = 0.50, D</td><td>76.6</td><td>47.9</td><td>83.4</td><td>53.5</td><td>39.8</td><td>33.2</td><td>49.6</td><td>54.9</td><td></td></tr><tr><td>γ = 0.00, D</td><td>76.4</td><td>47.2</td><td>83.1</td><td>53.6</td><td>39.0</td><td>31.4</td><td>58.0</td><td>55.5</td><td></td></tr><tr><td>γ = 0.00, D, H</td><td>76.6</td><td>48.0</td><td>82.8</td><td>54.0</td><td>36.6</td><td>31.5</td><td>55.8</td><td>55.0</td><td></td></tr><tr><td>γ = 0.00</td><td>76.2</td><td>48.0</td><td>83.0</td><td>53.6</td><td>39.0</td><td>32.4</td><td>46.7</td><td>54.1</td><td></td></tr><tr><td rowspan="6">Llama3 → Byte</td><td>γ = 0.10, D, H (default)</td><td>73.2</td><td>38.9</td><td>77.6</td><td>55.8</td><td>34.0</td><td>32.7</td><td>18.0</td><td>47.2</td><td></td></tr><tr><td>γ = 0.99, D, H</td><td>73.1</td><td>39.8</td><td>78.2</td><td>56.1</td><td>35.3</td><td>31.8</td><td>17.3</td><td>47.4</td><td></td></tr><tr><td>γ = 0.50, D, H</td><td>73.1</td><td>38.8</td><td>76.3</td><td>55.9</td><td>33.4</td><td>31.6</td><td>17.7</td><td>46.7</td><td></td></tr><tr><td>γ = 0.00, D, H</td><td>73.5</td><td>38.8</td><td>77.1</td><td>56.5</td><td>34.4</td><td>30.9</td><td>17.7</td><td>47.0</td><td></td></tr><tr><td>γ = 0.00, D</td><td>73.2</td><td>40.3</td><td>75.1</td><td>52.5</td><td>30.0</td><td>29.3</td><td>16.9</td><td>45.3</td><td></td></tr><tr><td>γ = 0.00</td><td>73.4</td><td>41.2</td><td>74.0</td><td>52.5</td><td>30.0</td><td>31.0</td><td>0.2</td><td>43.2</td><td></td></tr></table>
+
+Table 9: Analysis of the efficacy of (i) different fixed coefficients  $\beta$  (ii) GradNorm (Chen et al., 2018) and (iii) GradMag (Section 3.3) at transfer of Gemma 2 2B IT to the Qwen2 Tokenizer by training for 5k steps and otherwise matching the experiments in Use Case 1.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Distill. Coef. β</td><td colspan="8">Benchmark</td><td rowspan="2">Avg.</td></tr><tr><td>PiQA</td><td>ARC-C</td><td>BoolQ</td><td>MMLU</td><td>AGI-EN</td><td>AGI-ZH</td><td>IFEval</td><td></td></tr><tr><td>SFT</td><td></td><td>76.3</td><td>42.2</td><td>80.3</td><td>50.2</td><td>33.4</td><td>29.8</td><td>51.8</td><td>52.0</td><td></td></tr><tr><td rowspan="5">DSKD</td><td>β = 1/3</td><td>75.5</td><td>42.2</td><td>79.9</td><td>49.1</td><td>32.7</td><td>31.6</td><td>52.8</td><td>52.0</td><td></td></tr><tr><td>β = 1</td><td>75.2</td><td>39.8</td><td>80.7</td><td>47.7</td><td>32.3</td><td>30.3</td><td>49.3</td><td>50.8</td><td></td></tr><tr><td>β = 3</td><td>72.5</td><td>35.5</td><td>81.3</td><td>46.6</td><td>31.6</td><td>30.9</td><td>47.9</td><td>49.5</td><td></td></tr><tr><td>GradNorm</td><td>75.3</td><td>41.6</td><td>79.7</td><td>49.0</td><td>32.8</td><td>30.3</td><td>51.3</td><td>51.4</td><td></td></tr><tr><td>GradMag (ours)</td><td>74.6</td><td>40.9</td><td>80.0</td><td>49.0</td><td>33.1</td><td>29.8</td><td>52.2</td><td>51.4</td><td></td></tr><tr><td rowspan="5">MinED</td><td>β = 1/3</td><td>76.5</td><td>42.5</td><td>80.4</td><td>49.8</td><td>33.4</td><td>29.4</td><td>51.8</td><td>52.0</td><td></td></tr><tr><td>β = 1</td><td>76.3</td><td>42.7</td><td>80.3</td><td>50.0</td><td>33.4</td><td>28.4</td><td>50.9</td><td>51.7</td><td></td></tr><tr><td>β = 3</td><td>76.3</td><td>43.1</td><td>80.5</td><td>50.4</td><td>33.6</td><td>29.0</td><td>51.2</td><td>52.0</td><td></td></tr><tr><td>GradNorm</td><td>76.3</td><td>44.5</td><td>81.2</td><td>50.9</td><td>36.0</td><td>29.6</td><td>53.8</td><td>53.2</td><td></td></tr><tr><td>GradMag (ours)</td><td>76.6</td><td>44.0</td><td>81.9</td><td>52.2</td><td>36.3</td><td>29.7</td><td>57.5</td><td>54.0</td><td></td></tr><tr><td rowspan="5">ALM + SFT</td><td>β = 1/3</td><td>76.3</td><td>42.3</td><td>79.9</td><td>49.8</td><td>34.0</td><td>29.3</td><td>52.0</td><td>52.0</td><td></td></tr><tr><td>β = 1</td><td>76.4</td><td>42.6</td><td>79.5</td><td>50.1</td><td>34.4</td><td>29.4</td><td>52.2</td><td>52.1</td><td></td></tr><tr><td>β = 3</td><td>76.3</td><td>42.2</td><td>80.0</td><td>50.4</td><td>34.7</td><td>30.0</td><td>52.6</td><td>52.3</td><td></td></tr><tr><td>GradNorm</td><td>77.1</td><td>46.0</td><td>83.2</td><td>53.7</td><td>38.7</td><td>32.2</td><td>61.6</td><td>56.1</td><td></td></tr><tr><td>GradMag (ours)</td><td>76.7</td><td>46.0</td><td>82.8</td><td>53.4</td><td>37.9</td><td>32.1</td><td>58.8</td><td>55.4</td><td></td></tr></table>
+
+# B Training Details
+
+Across experiments, unless specified otherwise, we use a batch size of 64 texts and a sequence length of 512 tokens for both student and teacher. We use Adam (Kingma & Ba, 2015) without weight decay following the adaptations settings of Groeneveld et al. (2024). We choose a default peak learning rate of  $1e - 5$  based on the findings in Appendix A.1, training for 20k steps with linear warmup over 2k steps, then linear decay to zero. We use GradMag (c.f. Section 3.3) to balance loss components. We use LoRA (Hu et al., 2022) with  $\alpha = r = 64$ . For ALM, we use a threshold  $\gamma = 0.1$ , the temperature  $\tau = 100$ , and set  $f$  to  $f_{\mathrm{KL}}(p_T\| p_S) = p_T\log \frac{p_T}{p_S}$  to recover the KL-divergence, analysing the impact of these choices in Appendix A.2. We conduct all experiments on a cluster of 40 v3 TPU chips and 64 v4 TPU chips. The largest individual experiments run on a pod of 32 v4 TPU chips and take  $\approx 24$  hours for transfer of Llama3 to byte-level tokenization,  $\approx 12$  hours for transfer of Gemma2 to byte-level tokenization,  $\approx 2$  days for Gemma2 hypernetwork training and  $\approx 5$  to 10 hours individually per remaining experiment.
+
+# C Adjustments for Transfer to Bytes in Use Case 1
+
+For the transfer to bytes, we start from the hyperparameters described in Use Case 1, with the following modifications. We quadruple the student sequence length to 2048, $^{17}$  halve the batch size to 32 texts, increase the learning rate to 3e-5 and train the full model. Adding the hidden state distillation loss leads to consistent improvements (while results are mixed in the subword case, c.f. Appendix A.3), so we add it here and do not add it for subword-to-subword transfer. We also untie the embedding matrices since sharing the input and output embedding parameters has negligible impact on the parameter count when the vocabulary just consists of the 255 possible bytes. Transfer to bytes substantially decreases the parameter count due to shrinking the embedding matrix (which can make up a considerable portion of the total parameters). Thus, to make it less destructive, we devise a strategy to re-incorporate the subword embedding parameters: for every byte position, we backtrack to find the longest matching subword token ending at this position. We then add the embedding of this token to the byte embedding. This is akin to BLT's strategy of adding n-gram embeddings Pagnoni et al. (2024), where in our case the vocabulary is given by the subword-based model. This strategy only accounts for the parameter mismatch at the input layer, not at the output layer, so we might still expect decreased performance.
+
+# D Hypernetwork Architecture Modifications for Use Case 3
+
+We simplify the hypernetwork training procedure introduced by Minixhofer et al. (2024) to only require a single stage instead of the original two-stage training procedure and remove the necessity of the auxiliary embedding similarity loss. While Minixhofer et al. (2024) predict the target embeddings as  $E^{\mathrm{out}} = H_{\theta}(E^{\mathrm{in}})$ , we switch to a residual formulation  $E^{\mathrm{out}} = E^{\mathrm{in}}[0] + H_{\theta}(E^{\mathrm{in}})$  where  $H_{\theta}$  is the hypernetwork and  $E^{\mathrm{in}}$  is the input embedding sequence. This way, the hypernetwork (i) does not need to initially learn to mimic the input embeddings for sequences of length 1 (the purpose of the original stage one) and (ii) is less prone to catastrophically drifting in embedding space throughout training (enabling removing the auxiliary loss). We first reproduce Minixhofer et al. (2024)'s results on the TinyLlama model (Zhang et al., 2024a), then test the impact of switching to our residual formulation in Table 10. Based on these findings, we use a residual hypernetwork without auxiliary loss for our main experiments.
+
+Table 10: Performance of a TinyLlama hypernetwork trained with Minixhofer et al. (2024) setting, vs. our residual setting. While removing the auxiliary loss is catastrophic in Minixhofer et al. (2024)'s setting, our residual setting allows removing it at only a minor performance degradation. We exclude BoolQ and MMLU from this comparison since TinyLlama performance is not better than random.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Tokenizer</td><td colspan="6">Benchmark</td><td rowspan="2">Avg.</td></tr><tr><td>PiQA</td><td>ARC-E</td><td>ARC-C</td><td>AGI-EN</td><td>AGI-ZH</td><td>HS</td></tr><tr><td colspan="2">original</td><td>73.1</td><td>55.2</td><td>30.7</td><td>23.3</td><td>27.8</td><td>59.1</td><td>44.9</td></tr><tr><td rowspan="4">Minixhofer et al. (2024) (reproduced)</td><td>0-shot GPT2</td><td>70.9</td><td>48.8</td><td>28.3</td><td>22.9</td><td>27.5</td><td>55.9</td><td>42.4</td></tr><tr><td>0-shot Mistral v2</td><td>70.6</td><td>50.3</td><td>28.8</td><td>21.9</td><td>27.9</td><td>54.0</td><td>42.2</td></tr><tr><td>0-shot Llama3</td><td>70.5</td><td>49.7</td><td>28.2</td><td>21.9</td><td>27.1</td><td>53.8</td><td>41.9</td></tr><tr><td>Average</td><td>70.6</td><td>49.6</td><td>28.4</td><td>22.2</td><td>27.5</td><td>54.5</td><td>42.2</td></tr><tr><td rowspan="4">SFT (Residual + Lex.)</td><td>0-shot GPT2</td><td>71.4</td><td>50.1</td><td>30.4</td><td>21.5</td><td>26.6</td><td>56.4</td><td>42.7</td></tr><tr><td>0-shot Mistral v2</td><td>70.3</td><td>50.3</td><td>29.7</td><td>21.2</td><td>27.1</td><td>54.7</td><td>42.2</td></tr><tr><td>0-shot Llama3</td><td>71.0</td><td>49.8</td><td>29.6</td><td>20.8</td><td>26.5</td><td>54.2</td><td>42.0</td></tr><tr><td>Average</td><td>70.9</td><td>50.1</td><td>29.9</td><td>21.2</td><td>26.7</td><td>55.1</td><td>42.3</td></tr><tr><td rowspan="4">SFT (Residual)</td><td>0-shot GPT2</td><td>70.4</td><td>49.2</td><td>28.9</td><td>21.8</td><td>26.2</td><td>55.9</td><td>42.1</td></tr><tr><td>0-shot Mistral v2</td><td>69.4</td><td>49.6</td><td>29.1</td><td>21.5</td><td>26.8</td><td>53.8</td><td>41.7</td></tr><tr><td>0-shot Llama3</td><td>70.0</td><td>49.5</td><td>28.5</td><td>22.1</td><td>26.5</td><td>53.5</td><td>41.7</td></tr><tr><td>Average</td><td>69.9</td><td>49.4</td><td>28.8</td><td>21.8</td><td>26.5</td><td>54.4</td><td>41.8</td></tr></table>
+
+# E Relation of the Binarised  $f$ -divergence to the Categorical  $f$ -divergence
+
+Recall that we define the divergence induced by  $f$  as  $D_{f}(p\| q) = \sum_{x\sim D}f\big(p(x)\| q(x)\big)$ . Here,  $f$  must be writeable as  $f(p(x)\| q(x)) = q(x)g\left(\frac{p(x)}{q(x)}\right)$  where  $g$  is convex and non-negative by the definition of  $f$ -divergence (Rényi, 1961). The binarised  $f$ -divergence is  $D_{f}^{\mathrm{binarised}}(p\| q) = f(p(x)\| q(x)) + f(1 - p(x)\| 1 - q(x))$  for some  $x\sim D$ . We will show that the binarised  $f$ -divergence is a lower bound to the categorical  $f$ -divergence. We have
+
+$$
+D _ {f} (p \| q) = \sum_ {x \sim D} f (p (x) \| q (x))
+$$
+
+$= f(p(x)\| q(x)) + \sum_{x^{\prime}\sim D\setminus \{x\}}f\bigl (p(x^{\prime})\| q(x^{\prime})\bigr)$  splitting the divergence into two parts
+
+Since the first term of the split  $D_{f}$  and  $D_{f}^{\mathrm{binarised}}$  is the same, it remains to show that
+
+$$
+f (1 - p (x) \| 1 - q (x)) \leq \sum_ {x ^ {\prime} \sim D \backslash \{x \}} f \big (p (x ^ {\prime}) \| q (x ^ {\prime}) \big)
+$$
+
+We expand  $1 - p(x)$  and  $1 - q(x)$  and insert  $q(x)g\left(\frac{p(x)}{q(x)}\right) = f(p(x)\| q(x))$
+
+$$
+g \left(\frac {\sum_ {x ^ {\prime} \sim D \backslash \{x \}} p \left(x ^ {\prime}\right)}{\sum_ {x ^ {\prime} \sim D \backslash \{x \}} q \left(x ^ {\prime}\right)}\right) \cdot \sum_ {x ^ {\prime} \sim D \backslash \{x \}} q \left(x ^ {\prime}\right) \leq \sum_ {x ^ {\prime} \sim D \backslash \{x \}} g \left(\frac {p \left(x ^ {\prime}\right)}{q \left(x ^ {\prime}\right)}\right) q \left(x ^ {\prime}\right)
+$$
+
+which is true by Jensen's inequality since  $g$  is convex.
+
+Alternatively, it follows from the data processing inequality that the binarised  $f$ -divergence is a lower bound to the categorical  $f$ -divergence since post-processing (by merging all but two options) can only decrease the divergence. Since  $D_f^{\mathrm{binarised}}(p\| q)$  is an  $f$ -divergence, the important properties that  $D_f^{\mathrm{binarised}}(p\| q) = 0$  iff  $p = q$  and  $D_f^{\mathrm{binarised}}(p\| q) > 0$  otherwise also hold.
+
+# F A Variant of ALM Which Explicitly Takes Tokenization Bias into Account
+
+We initially experimented with a version of ALM which explicitly selects chunks with low approximate differences in tokenization bias. This was intended to make the likelihood comparison more accurate by minimizing spurious differences caused by differing tokenization biases between the teacher and the student sequences. We did not end up choosing this formulation since it requires a pre-computation step like MinED, which results in additional complexity and prevents online applications such as Use Case 3 while performing roughly equally well. Nonetheless, we believe this previous formation and the associated formalisms may be useful to the community, and thus provide a description in the following, starting with some additional preliminaries.
+
+Measuring Tokenization Bias. We can measure tokenization bias via the notion of cover encodings (Phan et al., 2024). The cover encodings of a sequence of tokens  $\pmb{t} = T(\pmb{x})$  are all sequences of tokens  $\pmb{u}$  such that (i) decoding and re-encoding the sequence leads to the same token sequence  $T(D(\pmb{u})) = \pmb{u}$  (this has been referred to as validity), (ii)  $\pmb{x}$  is a prefix of  $D(\pmb{u})$  and (iii)  $\pmb{x}$  is not a prefix of  $D(\pmb{u}_{0:|\pmb{u}| - 1})$  (i.e., the last token in  $\pmb{u}$  covers a part of  $\pmb{x}$ ).<sup>18</sup> We denote the set of all cover encodings as  $\text{cover}(t)$ . Note that  $t \in \text{cover}(t)$ . We further define the set of implied exclusions  $\mathcal{X}(t) = \{D(\pmb{u}) \mid \pmb{u} \in \text{cover}(t), \pmb{u} \neq t\}$ .  $|\mathcal{X}(t)|$  is a concrete measure of the tokenization bias of any particular token sequence  $t$ . If we observe the token sequence  $t = T(\pmb{x})_{:i}$ , we know that  $\pmb{x} \notin \mathcal{X}(t)$ . Otherwise,  $\pmb{x}$  would have been tokenized differently. In our previous example, _Hello_World  $\in \mathcal{X}(\{\_Hello, \_Wor\})$ .
+
+Choosing Chunks with Low Approximate Tokenization Bias Difference. We would now like to constrain chunk alignment such that the prefix and continuation are biased in the same way, i.e.,  $\mathcal{X}(T_S(\boldsymbol{y})) = \mathcal{X}(T_T(\boldsymbol{y}))$  and  $\mathcal{X}(T_S(\boldsymbol{z})) = \mathcal{X}(T_T(\boldsymbol{z}))$ . Unfortunately, this requirement is prohibitively stringent in practice: for example, if the student tokenizer is unbiased while the teacher tokenizer is not, it might rarely (if ever) be fulfilled. We thus define a relaxation which lets us define the alignment constraint  $c(i,j,k,l)$  (c.f. Equation Alignment Indices) such that the chunk bias differences are small enough, but not necessarily zero. To this end, we introduce a precomputable scalar approximation of the difference in tokenization bias between two sequences. Let us consider arbitrary language models  $A$  and  $B$  (not necessarily a teacher and a student). We first define  $(\mathcal{X}_A \setminus \mathcal{X}_B)(\boldsymbol{x})$  to be the set difference of implied exclusions  $\mathcal{X}$  on some text  $\boldsymbol{x}$  between the token sequences of the two i.e.  $(\mathcal{X}_A \setminus \mathcal{X}_B)(\boldsymbol{x}) = \mathcal{X}(T_A(\boldsymbol{x})) \setminus \mathcal{X}(T_B(\boldsymbol{x}))$ . We can then quantify a scalar difference  $b_{A\parallel B}(\boldsymbol{x})$  in tokenization bias between the two sequences.
+
+$$
+b _ {B | | A} (\boldsymbol {x}) = \sum_ {\boldsymbol {d} \in \left(\mathcal {X} _ {A} \backslash \mathcal {X} _ {B}\right) (\boldsymbol {x})} p _ {A} \left(T _ {A} (\boldsymbol {d})\right)
+$$
+
+If  $(\mathcal{X}_A \setminus \mathcal{X}_B)(\pmb{x}) = \emptyset$ ,  $T_A(\pmb{x})$  is biased to a lesser or equal extent as  $T_B(\pmb{x})$  and  $b_{B\parallel A}(\pmb{x}) = 0$ . If not,  $b_{B\parallel A}(\pmb{x})$  will be high if any text  $\pmb{d}$  which  $T_A$  is biased against but  $T_B$  is not is assigned a high probability by the language model  $A$ . Instead, if a text  $\pmb{d} \in (\mathcal{X}_A \setminus \mathcal{X}_B)(\pmb{x})$  is assigned a low likelihood, it affects the bias difference to a lesser extent. We can use the symmetrised bias  $b_{A,B}(\pmb{x}) = \max(b_{B\parallel A}(\pmb{x}), b_{A\parallel B}(\pmb{x}))$  as a measure of the 'biasedness' of any text. However, computing  $b_{B\parallel A}(\pmb{x})$  is prohibitively expensive since we need to compute the cover encodings of every text  $\pmb{x}$  under both tokenization functions, as well as the likelihood assigned by the language model to every text  $\pmb{d}$  in the difference between implied exclusions  $(\mathcal{X}_A \setminus \mathcal{X}_B)(\pmb{x})$ . Therefore, we make two key approximations to estimate  $\hat{b}_{B\parallel A}(\pmb{x}) \approx b_{B\parallel A}(\pmb{x})$  (and analogously  $\hat{b}_{B\parallel A}(\pmb{x}) \approx b_{B\parallel A}(\pmb{x})$ ) at nearly zero computational overhead.
+
+Approximating  $\mathcal{X}(T(\pmb{x}))$  via  $\mathcal{X}(T(\pmb{x})_n)$ . We approximate the implied exclusions of  $T(\pmb{x})$  via the implied exclusions of the last token  $T(\pmb{x})_n$  in  $T(\pmb{x})$ .
+
+$$
+\hat {\mathcal {X}} (T (\boldsymbol {x})) = \left\{D \left(T (\boldsymbol {x}) _ {<   n}\right) \odot \boldsymbol {u} \mid \boldsymbol {u} \in \mathcal {X} \left(T (\boldsymbol {x}) _ {n}\right) \right\} \quad \text {(B i a s A p p r o x . 1)}
+$$
+
+This makes it possible to precompute the implied exclusions  $\mathcal{X}(t)$  for all  $t\in \mathcal{V}$ .
+
+Approximating  $p(T(\pmb{x}))$  via  $p_{\mathrm{unigram}}(T(\pmb{x})_n)$ . We approximate the language model likelihood  $p(T(\pmb{x}))$  via the unigram likelihood  $p_{\mathrm{unigram}}(T(\pmb{x})_n)$  computed via the token counts of a text corpus.
+
+$$
+\hat {p} (T (\boldsymbol {x})) = p _ {\text {u n i g r a m}} (T (\boldsymbol {x}) _ {n}) \quad \left(\text {B i a s A p p r o x .} 2\right)
+$$
+
+This approximation is crude by necessity since it is distinctly intractable to run every token sequence  $\pmb{d} \in (\mathcal{X}_A \setminus \mathcal{X}_B)(\pmb{x})$  through a large language model. However, we show later in Section 4 that it performs well in practice. Bias Approx. 1 and Bias Approx. 2 together make the bias only depend on the last token in the sequence. Going back to our distillation setup, the above approximations allow precomputing the bias difference  $\hat{b}_{S,T}(\pmb{x})$  for any text by enumerating all possible last token pairs  $(t_S, t_T) \in \mathcal{V}_S \times \mathcal{V}_T$  and storing the result in a sparse matrix in  $\mathbb{R}^{|\mathcal{V}_S| \times |\mathcal{V}_T|}$ . This achieves our goal of defining an efficiently computable scalar metric to measure the difference in tokenization bias between two token sequences, and lets us set the alignment constraint to  $c(i,j,k,l) := \max(\hat{b}_{S,T}(D(T_T(\pmb{x})_{:i})), \hat{b}_{S,T}(D(T_T(\pmb{x})_{i:j}))) \leq \gamma$  where  $\gamma$  is a manually defined bias difference threshold. Together, these approximations give a result in a viable alternative formulation of ALM, although one which requires pre-computation. See Table 11 for a minimal comparison of this variant with our main method.
+
+# G Overview of Existing Cross-Tokenizer Distillation Methods
+
+ULD (Boizard et al., 2025) trains the student via next-token prediction plus a Wasserstein distance loss (Kantorovich, 1960) between the teacher logits and the student logits at every step. This ensures that the absolute difference between the sorted teacher logits and sorted student logits at every position is small.
+
+Table 11: Comparing a variant of ALM which explicitly aims to take tokenization bias into account (Appendix F) with the main ALM version on transfer of Gemma2 2B IT to the Qwen tokenizer as in Use Case 1, but training for 5k steps instead.  
+
+<table><tr><td rowspan="2">ALM Setting</td><td colspan="7">Benchmark</td><td rowspan="2">Avg.</td></tr><tr><td>PiQA</td><td>ARC-C</td><td>BoolQ</td><td>MMLU</td><td>AGI-EN</td><td>AGI-ZH</td><td>IFEval</td></tr><tr><td>ALM</td><td>76.8</td><td>48.3</td><td>83.1</td><td>53.6</td><td>38.9</td><td>32.2</td><td>56.0</td><td>55.6</td></tr><tr><td>ALM – Appendix F Variant</td><td>76.9</td><td>46.8</td><td>83.1</td><td>53.6</td><td>39.4</td><td>31.5</td><td>52.9</td><td>54.9</td></tr></table>
+
+Table 12: Ensembling results across Qwen and Llama pivots and across keeping/transferring the special tokens. We adjust our notation as follows:  $\mathcal{X}_{\rightarrow \mathcal{Y}'}$  now denotes transferring to tokenizer  $\mathcal{V}$  while preserving the special tokens of  $\mathcal{X}$  (the default in Table 2).  $\mathcal{X}_{\rightarrow \mathcal{Y}}$  denotes transferring regular and special tokens to tokenizer  $\mathcal{V}$ . We also explicitly denote the pivot used by the MinED baseline.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">SizeID</td><td rowspan="2">Model</td><td colspan="7">Benchmark</td><td rowspan="2">Avg.</td></tr><tr><td>PiQA</td><td>ARC-C</td><td>BoolQ</td><td>MMLU</td><td>AGI-EN</td><td>AGI-ZH</td><td>IFEval</td></tr><tr><td rowspan="3">Original 
+Models</td><td>2.6BG</td><td>Gemma2 2B IT</td><td>79.6</td><td>50.4</td><td>83.8</td><td>56.9</td><td>42.1</td><td>30.7</td><td>62.5</td><td>58.0</td></tr><tr><td>3.2BL</td><td>Llama3.2 3B IT</td><td>76.9</td><td>43.9</td><td>78.8</td><td>62.4</td><td>36.6</td><td>40.2</td><td>76.6</td><td>59.3</td></tr><tr><td>1.5BQ</td><td>Qwen2.5 1.5B IT</td><td>76.3</td><td>43.2</td><td>77.9</td><td>60.1</td><td>45.0</td><td>54.2</td><td>46.3</td><td>57.6</td></tr><tr><td rowspan="8">Transfers</td><td>2.4BG→Q&#x27;</td><td>G→Q Tok.</td><td>76.8</td><td>49.0</td><td>82.7</td><td>53.6</td><td>38.9</td><td>31.6</td><td>53.2</td><td>55.1</td></tr><tr><td>2.4BG→Q</td><td>G→Q Tok. + Special Tok.</td><td>68.0</td><td>32.0</td><td>71.3</td><td>39.6</td><td>31.7</td><td>28.8</td><td>53.1</td><td>46.4</td></tr><tr><td>3.3BL→Q&#x27;</td><td>L→Q Tok.</td><td>77.3</td><td>45.6</td><td>79.0</td><td>61.6</td><td>37.1</td><td>33.3</td><td>76.3</td><td>58.6</td></tr><tr><td>3.3BL→Q</td><td>L→Q Tok. + Special Tok.</td><td>75.5</td><td>46.0</td><td>78.4</td><td>60.1</td><td>36.8</td><td>32.3</td><td>74.0</td><td>57.6</td></tr><tr><td>2.4BG→LC&#x27;</td><td>G→LC Tok.</td><td>77.1</td><td>48.4</td><td>83.0</td><td>53.7</td><td>39.2</td><td>32.0</td><td>53.8</td><td>55.3</td></tr><tr><td>2.4BG→LC</td><td>G→LC Tok. + Special Tok.</td><td>77.1</td><td>48.0</td><td>82.9</td><td>53.7</td><td>38.9</td><td>31.8</td><td>48.5</td><td>54.4</td></tr><tr><td>1.5BQ→LC&#x27;</td><td>Q→LC Tok.</td><td>76.3</td><td>45.9</td><td>78.4</td><td>59.6</td><td>44.7</td><td>41.9</td><td>45.9</td><td>56.1</td></tr><tr><td>1.5BQ→LC</td><td>Q→LC Tok. + Special Tok.</td><td>76.1</td><td>44.9</td><td>77.7</td><td>59.3</td><td>44.2</td><td>45.7</td><td>44.7</td><td>56.1</td></tr><tr><td rowspan="6">Ensembles</td><td>7.3B</td><td>Q+G+L (MinED-p, Q pivot)</td><td>76.9</td><td>46.2</td><td>83.2</td><td>64.1</td><td>40.6</td><td>39.4</td><td>65.9</td><td>59.5</td></tr><tr><td>7.3B</td><td>L+G+Q (MinED-p, L pivot)</td><td>76.9</td><td>47.2</td><td>83.1</td><td>64.2</td><td>38.5</td><td>37.7</td><td>54.9</td><td>57.5</td></tr><tr><td>7.2B</td><td>Q+G→Q+L→Q(p)</td><td>75.6</td><td>45.6</td><td>77.3</td><td>62.1</td><td>42.0</td><td>44.9</td><td>49.1</td><td>56.6</td></tr><tr><td>7.2B</td><td>L+G→L+Q→L(p)</td><td>77.5</td><td>48.7</td><td>83.7</td><td>62.9</td><td>43.3</td><td>42.5</td><td>64.8</td><td>60.5</td></tr><tr><td>7.2B</td><td>Q+G→Q&#x27;+L→Q&#x27;(p)</td><td>77.5</td><td>48.0</td><td>83.4</td><td>63.0</td><td>43.6</td><td>45.7</td><td>64.3</td><td>60.8</td></tr><tr><td>7.2B</td><td>L+G→LC&#x27;+Q→LC&#x27;(p)</td><td>77.6</td><td>48.0</td><td>84.1</td><td>63.2</td><td>43.6</td><td>43.3</td><td>62.3</td><td>60.3</td></tr><tr><td rowspan="2">Larger 
+Models</td><td>7.6B</td><td>Llama3.1 8B IT</td><td>81.6</td><td>53.9</td><td>85.4</td><td>68.4</td><td>47.3</td><td>47.1</td><td>81.7</td><td>66.5</td></tr><tr><td>8.0B</td><td>Qwen2.5 7B IT</td><td>80.4</td><td>54.3</td><td>86.4</td><td>71.7</td><td>57.6</td><td>72.6</td><td>76.6</td><td>71.4</td></tr></table>
+
+MinED (Wan et al., 2024) aligns the student and teacher logits by greedily aligning every token in the students' vocabulary with the teacher token which has the minimal Levenshtein distance (Levenshtein, 1966) to the student token. The main objective is again minimizing cross-entropy of the next-token prediction. The KL-divergence between the student logits and aligned teacher logits is added at every position where there is a one-to-one alignment between the elements of the two sequences.
+
+DSKD (Zhang et al., 2024b) projects the student representations to the teacher sequence and the teacher representations to the student sequence via a cross-attention mechanism, then adds the KL-divergence between the teacher predictions and the student-to-teacher projected predictions (and vice versa) to the next-token prediction loss.
+
+# H Analysing the Choice of Ensemble Pivot & Special Tokens
+
+For our main ensembling results (Table 2), we have chosen to transfer to the Qwen tokenizer, i.e., to use Qwen as the 'pivot'. The question arises, then, of how the choice of pivot impacts performance. We additionally transfer Gemma2 2B and Qwen2.5 1.5B to the Llama3 tokenizer and ensemble with the (non-transferred) Llama3 3B to answer this question. Results are shown in Table 12. We find that, at an average score of 60.8 with Qwen as the pivot and 60.3 with Llama as the pivot, the ensemble of transferred models is robust to the choice of pivot. This is, however, not true for the MinED baseline: here, using Llama as the pivot catastrophically degrades the ensemble to perform on average worse than the individual ensemble constituents.
+
+We use this opportunity to additionally study a related subtle choice: whether to keep the special tokens (such as  $\langle \text{bos} \rangle$ ,  $\langle \text{eos} \rangle$ , etc.) of the original tokenizer, or to use the special tokens of the new
+
+tokenizer upon transfer. Different special tokens do not pose a problem for sequence alignment since they can be easily detected and manually aligned. The same holds true for autoregressive generation, where it is fairly trivial to maintain separate running token sequences for the different models, which each insert their own special tokens as needed when one is generated. Nonetheless, this does result in implementation overhead, and for some use cases it may be convenient to achieve  $100\%$  parity with the target tokenizer by transferring the special tokens as well. Alongside the aforementioned assembling results, Table 12 also shows the performance of models which had their special tokens transferred and ensembles of these models. For the Llama pivot, transferring the special tokens completely preserves performance. However, for the Qwen pivot, transferring the special tokens catastrophically degrades performance (especially the performance of the Gemma2 2B constituent). This is the case since Qwen2 does not use a beginning-of-sequence <bos> token. In models which do use a <bos> token, this token is often heavily attended to, behaving as an 'attention sink' (Barbero et al., 2025). The models transferred to the Qwen2 special tokens thus have to relearn to distribute attention from the (now absent) <bos> to different tokens in the sequence. Evidently, Llama3 is able to cope with this situation better than Gemma2; more training of Gemma2 may fix this problem. Based on these findings, we preserve the special tokens in our main experiments since it is in general the safer choice, and the only downside for our purpose is additional implementation complexity.
+
+# I Implementing ALM
+
+ALM is simple to implement efficiently by making changes to the data preparation and the loss computation.
+
+Data preparation. Alongside tokenizing all example texts in the current batch, data preparation must include computing aligned chunks of tokens between the teacher and the student. To implement the chunk alignment algorithm, we can take advantage of the fact that both token sequences encode the same bytes. It is thus possible to compute alignments in a single pass via indices  $(i,k)$ , where  $i$  is increased as long as the token at position  $i$  in one sequence ends earlier than the token at position  $j$  in the other sequence, and vice versa, and indices are stored if both tokens end at the same byte position. This is in contrast to more complex approaches used in prior work (Fu et al., 2023; Wan et al., 2024): they minimize edit distance between alignments via variants of the Needleman-Wunsch algorithm (Needleman & Wunsch, 1970). This is not necessary if the tokenizers have first been converted to the byte level. We can represent the computed chunks via binary matrices  $M\in \{0,1\}^{b\times m\times k}$  and  $N\in \{0,1\}^{b\times n\times k}$  where  $b,m,n$  are the batch size, the student sequence length and the teacher sequence length, respectively and  $k = \min (m,n)$  is the maximum amount of chunks (i.e. the minimum of the student and teacher sequence lengths). M and N have entries one if the i-th token is part of the j-th chunk and zero otherwise. The above can be done in a separate data-loading thread to avoid blocking the main process.
+
+Loss computation. Loss computation involves computing the student and teacher next-token log probabilities over input tokens  $\log p_S(\pmb{y}|\pmb{x}) \in \mathbb{R}^{b \times m}$  and  $\log p_T(\pmb{y}|\pmb{x}) \in \mathbb{R}^{b \times n}$  where  $\pmb{y}$  are the input tokens shifted by one as in the causal language modelling objective. We can then compute chunk log-probabilities  $C_S := (\log p_S(\pmb{y}|\pmb{x}))M$  and  $C_T := (\log p_T(\pmb{y}|\pmb{x}))N$ . We then apply the chunk-level  $f$ -divergence element-wise to  $C_S$  and  $C_T$  and take the mean to compute the loss. In the simplest case where  $f = f_{\mathrm{TVD}}$  and  $\tau \to \infty$ , this reduces to  $\mathcal{L}_{S,T}^{\mathrm{ALM}}(\pmb{x}) = \frac{1}{b \cdot k} \sum_{i,j} |(C_S)_{i,j} - (C_T)_{i,j}|$ . All of the above results in negligible overhead compared to the forward and backward pass over the Transformer backbone (two gathers, two matrix multiplications, and some element-wise comparisons). Alignment constraints  $c(i,j,k,l)$  (c.f. Equation Alignment Indices) and chunk debiasing (c.f. Equation Outcome-Debiased Chunk-Level Probability) add some implementation complexity to the above but are cheap to compute and, in principle, optional.
\ No newline at end of file
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+# Universal Few-Shot Spatial Control for Diffusion Models
+
+Kiet T. Nguyen Chanhyuk Lee Donggyun Kim Dong Hoon Lee Seunghoon Hong KAIST
+
+{kietngt00, chan3684, kdgyun425, donghoonlee, seunghoon.hong}@kaist.ac.kr
+
+![](images/738b5c7cad2cb3d398392518288b36e8d2648728a96de3dc716336113e25e9d2.jpg)  
+HED
+
+![](images/9d37245bac3cb8e2fa8e9a85c97e1590985caad048296d0b226a6eb8407abb96.jpg)  
+A {photo/painting} of two roses in a small vase with water
+
+![](images/069d165d1bdb83969c7e7ae4de3e0568be75e61253ad933d048fcd1572990ed4.jpg)
+
+of two roses in a small.   
+fish water
+
+![](images/28e6f6e3fc73f3c48c77faee0c19e87ae25409c24005e0277edf9d0d6fee55a7.jpg)  
+Normal
+
+![](images/69231c6479a1485e8e2a603200716ae71c2b1325caacc085aa47028331d0e98b.jpg)  
+A (horse/zebra) standing in the gates area
+
+![](images/22cf008dfb68bd8082a1f72d1cb4cb1203d89401a43d4c371d0137efdf3638d7.jpg)  
+Depth
+
+![](images/9a83602ab7955e36f74f86c4e672f40e28fe206e60b28b730010c1ec8ddda34d.jpg)  
+Two glass bowls of {oranges/apples} add up to more than enough
+
+![](images/b9b5f49523864559bbaa8b3520375601afac14ced4e71eab3163f40a953a2702.jpg)
+
+wo glass bowls of {oranges/apples} add up to more than enough
+
+![](images/b0c934881e6d9fa03ece557418b7a2da7250da66739ccb8100a092ce517d6473.jpg)  
+Pose
+
+![](images/1ac388e24c8c4122082ec502c78a20e383955a5647af8e2df4f83d10ea1ce92c.jpg)  
+A man dressed in a {lab coat/suit}
+
+![](images/1bdbdecf6069a7af63d76e6568a263cf2fb4016e3dab18ad4a1eee44db9aa413.jpg)
+
+![](images/74eb0e698f3d6b0b434fbfac5cf7b569c457fb2ada0904bff690f77fb9472105.jpg)
+
+![](images/cce2109ef998a583ca01228bf21652f893dd37fdc1e5c5c694cf375fb7cd8450.jpg)  
+Canny  
+Figure 1: Results of our method learned with 30 examples on unseen spatial conditions. The proposed control adapter guides the pre-trained T2I models in a versatile and data-efficient manner.
+
+![](images/c992ddada22dcb957a71b0e8af43ed98f87cbed09bd2490cbee44fa2902377b5.jpg)  
+A tower with a clock on it during the {daytime/night}
+
+![](images/80349052a3ee62a9368ee3e6698989c64eb1fa22b0b8442de2918313d3826e13.jpg)
+
+![](images/3ee4ff1e3d378d65721581717b04e7904e93cd4e8db24a561f8fb1690e75a9a5.jpg)  
+Densepose
+
+![](images/5d62108d73a576f3b086e3c583561df6be95734c5e6d4eea8375292d7d545e43.jpg)
+
+![](images/f9f619c3af6990d56b624e0f456c223046322a0fb4c83a9472689ba108089ccf.jpg)  
+A {woman/boy} talking on the phone looks in the distance.
+
+# Abstract
+
+Spatial conditioning in pretrained text-to-image diffusion models has significantly improved fine-grained control over the structure of generated images. However, existing control adapters exhibit limited adaptability and incur high training costs when encountering novel spatial control conditions that differ substantially from the training tasks. To address this limitation, we propose Universal Few-Shot Control (UFC), a versatile few-shot control adapter capable of generalizing to novel spatial conditions. Given a few image-condition pairs of an unseen task and a query condition, UFC leverages the analogy between query and support conditions to construct task-specific control features, instantiated by a matching mechanism and an update on a small set of task-specific parameters. Experiments on six novel spatial control tasks show that UFC, fine-tuned with only 30 annotated examples of novel tasks, achieves fine-grained control consistent with the spatial conditions. Notably, when fine-tuned with  $0.1\%$  of the full training data, UFC achieves competitive performance with the fully supervised baselines in various control tasks. We also show that UFC is applicable agnostically to various diffusion backbones and demonstrate its effectiveness on both UNet and DiT architectures. Code is available at https://github.com/kietngt00/UFC.
+
+# 1 Introduction
+
+Text-to-Image (T2I) diffusion models [11, 41, 38, 43, 8, 7, 25], trained on large-scale datasets, have achieved remarkable success in generating high-quality, semantically aligned images from natural language prompts. While language-based control offers intuitive and flexible guidance, it often lacks the precision needed for fine-grained visual control, such as specific object positions, shapes, or scene layouts. To overcome this, recent works [19, 35, 28, 58, 27, 39, 59, 53] incorporate explicit spatial signals—like edge maps, depth maps, and segmentation masks to control diffusion models.
+
+To enable spatial control while preserving the generative quality of pre-trained diffusion models, existing methods typically employ control adapters [58, 35, 28] that inject spatial signals into a frozen T2I model. However, these adapters are usually trained independently for each spatial control task, requiring substantial computational resources and extensive labeled data for a new task. Alternatively, reusing pre-trained multi-task adapters - either directly [39, 53] or with minimal updates [59]- struggle to generalize to tasks that differ from their training distribution, and often show poor adaptability.
+
+In this work, we aim to address these limitations by proposing a universal few-shot learning framework, named Universal Few-shot Control (UFC), that efficiently controls diffusion models with novel spatial conditions using only a small number of labeled examples at test time (e.g., dozens of image and condition pairs). Developing such a method would substantially enhance the practicality and flexibility of spatial control methods, but also poses two main challenges. First, significant domain discrepancies across diverse spatial signals make it difficult to learn a consistent, shared representation of control signals that facilitate generalization to novel conditions. Second, in real-world settings where new control tasks may arise dynamically and labeled data is limited, the model must be capable of efficient test-time adaptation from only a few annotated examples.
+
+To overcome these challenges, UFC introduces a universal control adapter that represents novel spatial conditions by adapting the interpolation of visual features of images in a small support set, rather than directly encoding task-specific conditions. The interpolation is guided by patch-wise similarity scores between the query and support conditions, modeled by a matching module [23]. Since image features are inherently task-agnostic, this interpolation-based approach naturally provides a unified representation, enabling effective adaptation across diverse spatial tasks. Furthermore, to facilitate rapid and data-efficient updates, UFC combines episodic meta-learning on a multi-task dataset with a small set of task-specific parameters, enabling effective adaptation under few-shot settings.
+
+Extensive experiments across diverse spatial control modalities, including edge maps, depth maps, normal maps, human pose, and segmentation masks for human body, demonstrate that UFC outperforms prior approaches in few-shot settings by large margins. Notably, UFC achieves fine-grained spatial control using only 30 shots, and even on par with some fully supervised baselines on Normal, Depth, and Canny with just 150 support examples (0.1% of the training data). Furthermore, UFC is compatible with both UNet [42] and DiT [36] backbones, ensuring applicability across recent T2I diffusion architectures. To our knowledge, we propose the first method for few-shot spatial control in text-to-image diffusion models, which enables spatial control image generation with novel conditions at test time using minimal annotated data.
+
+# 2 Related Works
+
+Controllable Image Diffusion Models Text-to-Image diffusion models [11, 41, 38, 43, 8, 7, 25] have gained widespread adoption for their ability to generate high-quality, semantically aligned images from text prompts. To enable finer control over image structure, various methods [35, 58, 28, 27, 39, 59, 53, 48, 52] proposed incorporating additional spatial control inputs such as edge maps, segmentation masks, or human poses, etc., into diffusion models. ControlNet [58] augments a pretrained T2I diffusion model with a control adapter, initialized from the model itself, to encode spatial conditions and inject them into the frozen diffusion model. Uni-ControlNet [59] extends this by adding lightweight feature extractors to support multiple condition types. Prompt Diffusion [53], built on ControlNet, introduces a vision-language prompting mechanism to enable in-context learning for handling a diverse set of tasks. However, these prior methods are limited to tasks seen during training. To address this limitation, training-free methods [2, 57, 34, 29] have been proposed; but, they often rely on a long process of latent optimization, which substantially increases generation time while providing only limited controllability. Consequently, few-shot adaptation with more fine-grain controllability without incurring significant generation-time overhead remains unexplored.
+
+Few-shot Diffusion Models To extend diffusion models to unseen tasks, several works [5, 60, 55, 46, 12, 26, 20] have explored few-shot learning for image generation. For example, D2C [46] trains a variational autoencoder [24] on a few labeled examples to produce latent representations that condition a diffusion model to generate images in the target domain. Similarly, FSDM [12] uses visual features extracted from a small support set to guide image generation toward unseen classes. While these approaches demonstrate promising few-shot generation capabilities, they primarily focus on domain or class transfer, and are not designed to generalize across diverse spatial control tasks.
+
+Analogy Image Generation. Several recent works [13, 56] leverage the analogy between a query source and a support source-target pair to enable generalization in diffusion models on diverse tasks with visual instruction. Analogist [13] treats visual instruction as an inpainting task, arranging query-support pairs in a  $2 \times 2$  grid and using a diffusion model to complete the missing cell. This training-free method generalizes well to tasks like colorization, deblurring, and editing. While we also leverage query-support analogy in few-shot settings, our goal differs: prior work preserves the support image's appearance, whereas we aim to generate diverse content guided by spatial conditions.
+
+# 3 Background
+
+A text-to-image diffusion model generates an image  $x$  conditioned on a text prompt  $c_{\mathrm{text}}$  by iteratively denoising a sequence of latent variables. Specifically, the model learns a finite-length Markov chain  $\{z_{T - t}\}_{t = 0}^{T}$  that starts from standard Gaussian noise  $z_{T} \sim \mathcal{N}(0,I)$ , where each transition  $z_{t + 1} \rightarrow z_t$  progressively removes noise. The final clean variable  $z_0$  can be either the image itself or a latent representation  $z = E(x)$ , which is decoded back to the image by  $x = D(z)$ , where  $(E,D)$  denotes an auto-encoder [41, 38]. The chain is the time-reversal of a diffusion process  $\{z_t\}_{t = 0}^T$  that gradually corrupts the data by adding small Gaussian noise until it becomes pure noise at  $z_{T}$  [47]. To learn each denoising transition, the diffusion network  $\mathcal{E}_{\phi}$  is trained to predict the added noise  $\epsilon$  at timestep  $t$ :
+
+$$
+\min  _ {\phi} \mathbb {E} _ {\left(z _ {0}, c _ {\text {t e x t}}\right) \sim \mathcal {D} _ {\text {t r a i n}}, t, \epsilon} \left[ \| \epsilon - \mathcal {E} _ {\phi} \left(z _ {t}, c _ {\text {t e x t}}, t\right) \| ^ {2} \right], \tag {1}
+$$
+
+where  $(z_0, c_{\mathrm{text}})$  is drawn from the training dataset  $\mathcal{D}_{\mathrm{train}}$  and  $t$  is uniformly sampled from  $\{1, \dots, T\}$ .  $z_t$  can be computed in closed form from  $z_0$  and  $t$  using the properties of Gaussian variables [16]. After training, an image is generated by running the learned Markov chain from  $z_T$  down to  $z_0$ .
+
+Although text-to-image diffusion models control the overall content of an image, a text prompt alone is limited in steering the fine-grained spatial structures. Spatial control approaches [35, 58, 28, 48] address this limitation by introducing an additional spatial condition  $y_{\tau}$ , typically a dense map describing the target image (e.g., edges, depth, or semantic masks). For each condition type (or task)  $\tau$ , an auxiliary control adapter  $\mathcal{G}_{\theta_{\tau}}(y_{\tau})^{1}$  is trained and injected into a frozen, pre-trained diffusion backbone  $\mathcal{E}_{\phi}$ , while keeping the backbone parameters  $\phi$  fixed to preserve their learned priors:
+
+$$
+\min  _ {\theta_ {\tau}} \mathbb {E} _ {\left(z _ {0}, c _ {\text {t e x t}}, y _ {\tau}\right) \sim \mathcal {D} _ {\text {t r a i n}}, t, \epsilon} \left[ \| \epsilon - \mathcal {E} _ {\phi} \left(z _ {t}, c _ {\text {t e x t}}, t, \mathcal {G} _ {\theta_ {\tau}} \left(y _ {\tau}\right)\right) \| ^ {2} \right], \tag {2}
+$$
+
+However, learning a dedicated adapter for each task  $\tau$  usually requires thousands of labeled data to achieve reasonable performance [58]. This can be costly if we want to condition the pre-trained model on a novel task  $\tau_{\mathrm{novel}}$ . On the other hand, reusing an adapter trained on a fixed task(s) for novel tasks often suffers from limited adaptation performance, especially when the new tasks are substantially different from training [59]. Therefore, designing a versatile control adapter that can accommodate arbitrary unseen spatial conditions with a small amount of data remains a key challenge for data-efficient steering of text-to-image diffusion models.
+
+# 4 Method
+
+# 4.1 Problem Setup
+
+Our goal is to build a universal few-shot framework that steers a frozen text-to-image diffusion model with arbitrary novel spatial conditions, given only a handful of annotated examples (typically a few dozen). To this end, we introduce a universal control adapter  $\mathcal{I}(y_{\tau};S_{\tau})$  that can inject any type of spatial conditions into the frozen diffusion model by directly incorporating the given data as follows:
+
+$$
+\hat {\epsilon} = \mathcal {E} _ {\phi} \left(z _ {t}, t, c _ {\text {t e x t}}, \mathcal {I} \left(y _ {\tau}; S _ {\tau}\right)\right), \tag {3}
+$$
+
+where  $S_{\tau} = \{(x^{i},y_{\tau}^{i})\}_{i\leq N}$  denotes a support set comprising  $N$  image-condition pairs for task  $\tau$ . We focus exclusively on the design of the adapter  $\mathcal{I}$  while the remaining components, such as loss function, diffusion architecture, and injection strategy, are independent to the adapter and can be freely chosen from existing spatial control approaches [35, 58, 28].
+
+Unlike a task-specific adapter  $\mathcal{G}_{\theta_{\tau}}$  (Eq. (2)), the universal adapter  $\mathcal{I}$  must address two key challenges.
+
+1. Heterogeneous input formats. Spatial conditions vary widely—from extremely sparse cues such as human-posed keypoints to dense maps such as depth or segmentation. The adapter must transform these disparate inputs into the unified control features that the frozen diffusion model can consume reliably.  
+2. Severe data scarcity. For a new task, we assume only a few dozen labeled examples. To handle distribution shifts without over-fitting the tiny support set, the adapter needs an adaptation mechanism that is both flexible and efficient.
+
+# 4.2 Universal Few-Shot Control Adapter
+
+We propose Universal Few-shot Control (UFC), a universal control adapter that addresses the challenges outlined in Section 4.1 by (1) unifying heterogeneous spatial conditions with image features, and (2) adapting to new tasks through patch-wise matching and parameter-efficient finetuning. An overview is shown in Figure 2.
+
+To obtain control features that remain consistent across diverse condition types, UFC leverages task-agnostic visual patches extracted from support images via patch-wise matching. Specifically, given a query condition  $y_{\tau}^{q}$  and a support set  $S_{\tau} = \{(x^{i},y_{\tau}^{i})\}_{i\leq N}$ , UFC encodes images and conditions into spatial feature maps with encoders  $f$  and  $g_{\tau}$ , respectively. The control feature for  $k$ -th query patch is then constructed by
+
+$$
+\mathcal {I} \left(y _ {\tau} ^ {q, k}; \mathcal {S} _ {\tau}\right) = \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {M} \sigma \left(g _ {\tau} \left(y _ {\tau} ^ {q, k}\right), g _ {\tau} \left(y _ {\tau} ^ {i, j}\right)\right) \cdot f \left(x ^ {i, j}\right), \tag {4}
+$$
+
+where  $\sigma : \mathbb{R}^d \times \mathbb{R}^d \to [0,1]$  is a patch-wise similarity function,  $M$  is the number of spatial patches per image, and  $k, j \leq M$  denote the patch indices. Intuitively, the patch embeddings of the support images  $f(x^{i,j})$  form task-agnostic bases of query conditions, while task-specific weights—computed from the conditions—select the relevant entries.
+
+The patch-wise matching formulation in Eq. 4, inspired by few-shot dense prediction methods [23, 22], offers several advantages for the universal control problem. First, since the task-specific conditions are only used to determine the weights for the task-agnostic visual features, the resulting control features  $\mathcal{I}(y_{\tau}^{q};S_{\tau})$  reside in a unified visual feature space regardless of the task  $\tau$ . This ensures consistent guidance of the diffusion model and robust generalization to unseen tasks. Second, by composing the control features patch-wise, it can exploit the locality inductive bias on the image-condition pairs and amplify the effective number of support patches (i.e., bases) for each query patch. This provides sufficient representational power to model complex spatial controls from just a few support images. Finally, the general matching rule can wrap any control adapter  $G_{\theta_{\tau}}$  by treating it as a condition encoder  $g_{\tau}$  and adding an image encoder  $f$ . UFC can therefore inherit advances in adapter architectures and injection mechanisms developed for spatial control.
+
+To adapt rapidly to a new task without over-fitting, UFC introduces only a small set of task-specific parameters  $\theta_{\tau}$  while sharing the remainder. Starting from a pre-trained visual backbone  $g(\cdot ;\theta)$ , we obtain a condition encoder
+
+$$
+g _ {\tau} \left(y _ {\tau}\right) = g \left(y _ {\tau}; \theta , \theta_ {\tau}\right), \tag {5}
+$$
+
+where  $\theta$  is shared across tasks and  $\theta_{\tau}$  is lightweight (e.g., bias [3] or LoRA [18] parameters). After meta-training on a fixed task set (Section 4.4),  $\theta$  is frozen and only  $\theta_{\tau}$  is fine-tuned for each novel task  $\tau$ . The image encoder  $f$ , also initialized from a pre-trained visual backbone, remains frozen throughout to preserve task-agnostic visual features. Leveraging strong priors of large-scale visual pre-training and modern parameter-efficient tuning techniques, UFC generalizes to unseen tasks with minimal task-specific parameters.
+
+![](images/6829f530bef7e669667db49a78a6fb53a17582f4ed20da89e35f2cdf8156d76d.jpg)  
+Figure 2: Overview of the proposed framework. The control adapter  $\mathcal{I}$  consists of an image encoder  $f$ , a condition encoder  $g_{\tau}$ , and a matching module implementing Eq. 4. The support image-condition pairs and the query conditions are encoded to extract multi-layer features. The matching module at each layer is applied to produce control features. The features are then injected into the generation process following the mechanism in Section 4.3 to control the structure of images.
+
+# 4.3 Architecture
+
+The formulation of UFC's control adapter in Eq. (4) is general and can be incorporated into various diffusion backbones and adapter architectures. In this section, we describe the instantiation of UFC within the ControlNet framework [58] due to its popularity and strong performance. For clarity, we consider the pre-trained Stable Diffusion [41, 11] with UNet backbone [41] in this section, while the one with DiT backbone [11] is described in Appendix D.
+
+We initialize the condition encoder  $g_{\tau}$  and image encoder  $f$  as separate copies of the UNet encoder of the pre-trained diffusion model, where the image encoder is kept frozen. For the task-specific parameters  $\theta_{\tau}$  of the condition encoder, we adopt bias-tuning [3], which has been proven to be flexible for few-shot learning in dense visual prediction [23, 22]. The matching module is implemented as a multi-head cross-attention block [50], where the timestep embedding is fused through adaptive layer normalization [37, 36] inside the attention mechanism. More implementation details about the matching module can be found in Appendix A.1.
+
+To flexibly control the diffusion model, we perform patch-wise matching (Eq. (4)) at multiple levels of the UNet encoders  $g_{\tau}$  and  $f$  using  $L$  separate matching modules, where  $L$  denotes the number of UNet encoder layers. The extracted features from  $g_{\tau}$  and  $f$  at each encoder layer  $l$  are passed through the  $l$ -th matching module to obtain the corresponding control features  $\mathcal{I}^{l}(y_{\tau};S_{\tau})$ . These control features pass through a zero-initialized linear projection layer  $\mathcal{Z}^l$  and are added to the original features of the target diffusion model:
+
+$$
+e _ {c} ^ {l} = e ^ {l} + \mathcal {Z} ^ {l} \left(\mathcal {I} ^ {l} \left(y _ {\tau}; \mathcal {S} _ {\tau}\right)\right), \tag {6}
+$$
+
+where  $e^l$  is the output from the  $l$ -th layer of the diffusion encoder. Finally, the resulting feature  $e_c^l$  replaces the activation  $e^l$  and is integrated via skip connections into the corresponding UNet decoder layer of the diffusion model.
+
+# 4.4 Training and Inference
+
+To equip UFC with few-shot learning capabilities, we adopt the standard episodic meta-training protocol, wherein each training iteration is an episode that mirrors the test-time procedure. In each episode, we first sample a specific control task  $\tau$  from a meta-training dataset  $D_{\mathrm{train}}$  that consists of multiple tasks. The sampled task  $\tau$  supplies a support set  $S_{\tau}$  and a query set  $Q_{\tau}$ , where the model exploits the support set to control the diffusion model with the query condition. The model is optimized using the standard denoising loss [16] (or its flow-matching variant [31]):
+
+$$
+\min  _ {\theta , \theta_ {\tau}, \sigma , \mathcal {Z}} \mathbb {E} _ {\mathcal {S} _ {\tau}, Q _ {\tau} \sim \mathcal {D} _ {\text {t r a i n}}} \mathbb {E} _ {(z _ {0}, y _ {\tau}, c _ {\text {t e x t}}) \sim Q _ {\tau}, t, \epsilon} \left[ \| \epsilon - \mathcal {E} _ {\phi} (z _ {t}, c _ {\text {t e x t}}, t, \mathcal {I} (y _ {\tau} ^ {q}; \mathcal {S} _ {\tau})) \| ^ {2} \right]. \tag {7}
+$$
+
+During meta-training, the parameters  $\theta$  and  $\theta_{\tau}$  in the condition encoder, the matching modules  $\sigma$  and the projection layers  $\mathcal{Z}$  are updated while the pre-trained diffusion model  $\mathcal{E}_{\phi}$  and the image encoder  $f$  remain frozen. By repeatedly training on the few-shot learning episodes, the model learns a generalizable mapping from the query condition and the support set to the unified control features.
+
+For an unseen task  $\tau_{\mathrm{novel}}$ , we perform fine-tuning using its support set  $S_{\tau_{\mathrm{novel}}}$ . To this end, we randomly partition the support set into two disjoint subsets  $S_{\tau_{\mathrm{novel}}} = \tilde{S} \cup \tilde{\mathcal{Q}}$ , where  $\tilde{S}$  and  $\tilde{\mathcal{Q}}$  act as the pseudo-support and the query sets, respectively. Then the universal control adapter is fine-tuned with an objective similar to the meta-training:
+
+$$
+\min  _ {\theta_ {\tau}, \sigma , \mathcal {Z}} \mathbb {E} _ {\tilde {S}, \tilde {\mathcal {Q}} \sim S _ {\tau_ {\text {n o v e l}}}} \mathbb {E} _ {(z _ {0}, y _ {\tau}, c _ {\text {t e x t}}) \sim \tilde {\mathcal {Q}}, t, \epsilon} \left[ \| \epsilon - \mathcal {E} _ {\phi} (z _ {t}, c _ {\text {t e x t}}, t, \mathcal {I} (y _ {\tau}; \tilde {S})) \| ^ {2} \right]. \tag {8}
+$$
+
+Fine-tuning a small number of parameters  $\theta_{\tau}, \sigma, \mathcal{Z}$ , while freezing the shared parameters  $\theta$ , makes the model robust to over-fitting to the few-shot support set. After fine-tuning, UFC can produce a control feature for unseen condition types  $\tau_{\mathrm{novel}}$  using the support set  $S_{\tau_{\mathrm{novel}}}$  as described in Section 4.2.
+
+# 5 Experiments
+
+# 5.1 Experimental setup
+
+Model settings For experiments using the UNet architecture, we adopt Stable Diffusion v1.5 [41] as the diffusion backbone. Spatial conditions and support images are first projected into the latent space using the pre-trained VAE [24] from the diffusion model, then processed by respective encoders. When encoding these inputs, we fix the denoising timestep to zero, provide empty text prompts, and omit the noisy latent input in order to focus solely on spatial information. Detailed model setting using DiT backbone is provided in the Appendix D.
+
+Datasets To enable episodic meta-training, we sample 300K text-image pairs from LAION-400M [45], with 150K containing humans (for Pose and Densepose) and 150K randomly sampled. Images are resized to the resolution  $512 \times 512$ . Spatial conditions are extracted using pretrained off-the-shelf models. We consider six condition modalities that cover diverse semantics: Canny edge [4], HED edge [54], MiDaS Depth and surface Normal [40], human Pose [6], and human Densepose [14]. We consider Densepose as a control signal for semantic segmentation to constrain all segmentation classes to be included in the support set. During the training process, we maintain the setting that the models will be trained on 150K image-condition pairs for each task.
+
+To evaluate the performance over novel conditions, the six tasks are grouped into three disjoint splits: (Canny, HED), (Normal, Depth), and (Pose, Densepose). Models are trained on two splits and evaluated on the remaining one. These splits ensure test tasks differ significantly from training tasks, providing a robust measure of few-shot generalization. To simulate few-shot learning, we randomly sample support sets and evaluate the performance on the remaining ones, with the exception of human pose tasks (Pose and DensePose), for which we manually select support examples to ensure coverage of diverse human poses.
+
+Implementation details We train UFC (UNet diffusion backbone) for 12.5K iterations with a batch size of 96 on 8 NVIDIA RTX 3090 GPUs, using AdamW [33] with a learning rate of  $1 \times 10^{-5}$ . During inference with the UNet backbone, we adopt the PNDM sampler [32] with 50 denoising steps, classifier-free guidance (CFG) [17] scale of 7.5, and seed 42. For testing, we fine-tune UFC using all image-condition pairs in the support set, but use only a subset of them as conditions during inference to accommodate memory constraints and reduce generation-time overhead.
+
+Baselines We compare our method in the main experiment against three types of baselines:
+
+- Fully-supervised baselines ControlNet [58] and Uni-ControlNet [59] are both trained in a fully-supervised manner on the entire dataset (150K images for each task). ControlNet is trained separately for each condition type, while Uni-ControlNet is jointly trained with all conditions.  
+- Few-shot baselines As the first to tackle spatial control in diffusion models under a few-shot setting, we adapt existing methods, Uni-ControlNet and Prompt Diffusion [53], to align with our few-shot adaptation. For Uni-ControlNet, as mentioned in the original paper, we extend the input layer with additional channels for novel conditions and few-shot fine-tune this layer. For Prompt Diffusion, we evaluate both the zero-shot setup as proposed in the original paper and the full fine-tuning setup. For fair comparison, these few-shot methods are pre-trained using the same meta-training dataset as our method and fine-tuned on the same few-shot data.
+
+Table 1: Controllability measurement on COCO 2017. Few-shot baselines are fine-tuned with 30 shots for evaluation on novel tasks.  
+
+<table><tr><td>Baseline Type</td><td>Method</td><td>Canny SSIM (↑)</td><td>HED SSIM (↑)</td><td>Depth MSE (↓)</td><td>Normal MAE (↓)</td><td>Pose AP50 (↑)</td><td>Densepose mIoU (↑)</td></tr><tr><td rowspan="2">Fully-supervised</td><td>ControlNet [58]</td><td>0.3598</td><td>0.5972</td><td>89.09</td><td>14.14</td><td>0.525</td><td>0.4824</td></tr><tr><td>Uni-ControlNet [59]</td><td>0.3378</td><td>0.5808</td><td>92.60</td><td>14.59</td><td>0.340</td><td>0.4695</td></tr><tr><td>Training-free</td><td>Ctrl-X [29]</td><td>0.2901</td><td>0.3002</td><td>98.10</td><td>19.38</td><td>0.005</td><td>0.1352</td></tr><tr><td rowspan="4">Few-shot</td><td>Prompt Diffusion [53]</td><td>0.2120</td><td>0.2887</td><td>98.81</td><td>20.34</td><td>0.012</td><td>0.2266</td></tr><tr><td>Uni-ControlNet + FT [59]</td><td>0.2222</td><td>0.2917</td><td>99.21</td><td>20.36</td><td>0.010</td><td>0.2687</td></tr><tr><td>Prompt Diffusion + FT [53]</td><td>0.2773</td><td>0.4810</td><td>95.64</td><td>18.04</td><td>0.034</td><td>0.3548</td></tr><tr><td>UFC (Ours - UNet)</td><td>0.3239</td><td>0.5121</td><td>94.38</td><td>15.09</td><td>0.229</td><td>0.4340</td></tr></table>
+
+Table 2: FID evaluation on COCO 2017. Few-shot baselines are fine-tuned with 30 shots for evaluation on novel tasks.  
+
+<table><tr><td>Baseline Type</td><td>Method</td><td>Canny</td><td>HED</td><td>Depth</td><td>Normal</td><td>Pose</td><td>Densepose</td></tr><tr><td rowspan="2">Fully-supervised</td><td>ControlNet [58]</td><td>21.06</td><td>17.15</td><td>20.16</td><td>21.73</td><td>44.33</td><td>34.72</td></tr><tr><td>Uni-ControlNet [59]</td><td>16.79</td><td>16.76</td><td>20.35</td><td>19.59</td><td>47.13</td><td>36.77</td></tr><tr><td>Training-free</td><td>Ctrl-X [53]</td><td>29.83</td><td>30.18</td><td>30.32</td><td>30.35</td><td>56.60</td><td>45.88</td></tr><tr><td rowspan="4">Few-shot</td><td>Prompt Diffusion [53]</td><td>24.85</td><td>23.00</td><td>24.07</td><td>23.28</td><td>43.60</td><td>34.15</td></tr><tr><td>Uni-ControlNet [59] + FT</td><td>24.58</td><td>25.48</td><td>24.77</td><td>25.12</td><td>47.82</td><td>36.06</td></tr><tr><td>Prompt Diffusion [53] + FT</td><td>20.57</td><td>19.86</td><td>24.56</td><td>24.30</td><td>45.94</td><td>36.83</td></tr><tr><td>UFC (Ours - UNet)</td><td>19.24</td><td>20.58</td><td>21.04</td><td>21.60</td><td>47.91</td><td>37.79</td></tr></table>
+
+- Training-free baselines We also compare our method with training-free methods, Ctrl-X [29] and FreeControl [34], which directly employ the frozen T2I diffusion models for spatial control. We consider Ctrl-X as our main evaluation but leave comparisons with FreeControl in Appendix B using a smaller evaluation set due to its slow inference speed.
+
+Evaluation Protocol We quantitatively evaluate all methods on COCO 2017 [30] validation split, which contains 5,000 images with associated captions. All images and their corresponding conditions are resized to  $512 \times 512$ . For tasks Pose and Densepose, we restrict evaluation to images containing humans. To assess image generation quality, we report the Fréchet Inception Distance (FID) [15]. To evaluate controllability, we extract spatial conditions from the generated images using the same pre-trained networks used for dataset construction and compare them with the query conditions using task-specific metrics. Specifically, we use the Structural Similarity Index (SSIM) for Canny and HED, Average Precision  $(\mathrm{AP}^{50})$  computed using Object Keypoint Similarity for human Pose, Mean Squared Error (MSE) for Depth prediction, Mean Angular Error (MAE) for surface Normal, and mean Intersection over Union (mIoU) for human segmentation in Densepose.
+
+# 5.2 Main Results
+
+Quantitative Results We report the 30-shot performance of our model with UNet backbone [41] in terms of two aspects: controllability (Table 1) and image quality (Table 2). Despite using only a few annotated examples, UFC achieves comparable image quality and over  $90\%$  of the controllability performance of fully supervised baselines across most conditions. We observe that dense condition maps benefit more from the matching mechanism, as fine-grained similarity modeling enables more effective control features. This is reflected in the smaller relative performance gap with fully supervised baselines on Densepose compared to Pose.
+
+UFC consistently outperforms all few-shot and training-free methods in controllability while maintaining equal or better image quality. Among the few-shot baselines, Uni-ControlNet shows clear underfitting, and zero-shot Prompt Diffusion fails to generalize, both having weak controllability and FID. Compared to fine-tuned (FT) Prompt Diffusion, UFC has significant gains in both metrics on Canny, Depth, and Normal, with up to  $16.8\%$  improvement in controllability. For Densepose and HED, our method achieves similar FID scores but significantly exceeds FT Prompt Diffusion in controllability. On Pose, although FT Prompt Diffusion yields lower FID, its structural control is ineffective  $(\mathrm{AP}^{50}\approx 0)$ , while UFC achieves 0.229.
+
+![](images/7296e39ab7c2c1d0c85fdd118e92ec0bf1f41ce118edb13b8c8a3b90411193ea.jpg)  
+Figure 3: Qualitative comparison across six spatial control tasks. Our method (highlighted in black boxes), fine-tuned with 30-shot on unseen tasks, demonstrates competitive controllability with fully supervised baselines. In contrast, other baselines struggle to follow the spatial guidance accurately.
+
+UFC's strong few-shot performance stems from two factors. First, it effectively handles the large discrepancy between condition modalities by adapting the interpolation of support image features into task-specific control signals, unlike Prompt Diffusion or Uni-ControlNet, which lack robust strategies for handling distribution shifts. Second, UFC introduces an efficient adaptation mechanism by fine-tuning the matching module  $\sigma$ , projection layers  $\mathcal{Z}$ , and task-specific parameters  $\theta_{\tau}$  in the condition encoder from the meta-training weights. In contrast, Uni-ControlNet updates only a tiny subset of parameters, leading to underfitting, while Prompt Diffusion lacks any explicit adaptation mechanism, and naively full fine-tuning is ineffective in the few-shot regime.
+
+Qualitative Results We present qualitative comparisons in Figure 3. As shown, our method with UNet backbone, fine-tuned with only 30 shots, consistently adheres to the given query conditions across various novel tasks. In contrast, the images generated by the training-free method, Ctrl-X, exhibit low generation quality and poor alignment to the spatial condition, indicating its limited adaptability to diverse spatial conditions. Similarly, zero-shot Prompt Diffusion fails to follow spatial guidance, suggesting that its in-context learning mechanism struggles when there are significant differences between training and unseen tasks. Lastly, while fine-tuned Prompt Diffusion loosely follows the spatial inputs, it often introduces undesired visual artifacts, particularly under the HED and DensePose conditions.
+
+Table 3: Ablation study on matching module and parameters adaptation.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">Canny</td><td colspan="2">HED</td><td colspan="2">Depth</td><td colspan="2">Normal</td><td colspan="2">Pose</td><td colspan="2">Densepose</td></tr><tr><td>SSIM↑</td><td>FID</td><td>SSIM↑</td><td>FID</td><td>MSE↓</td><td>FID</td><td>MAE↓</td><td>FID</td><td>AP50↑</td><td>FID</td><td>mIoU↑</td><td>FID</td></tr><tr><td>UFC (UNet)</td><td>0.3239</td><td>19.24</td><td>0.5121</td><td>20.58</td><td>94.38</td><td>21.04</td><td>15.09</td><td>21.60</td><td>0.229</td><td>47.91</td><td>0.4340</td><td>37.79</td></tr><tr><td>w\o Matching</td><td>0.2984</td><td>20.43</td><td>0.4972</td><td>20.43</td><td>96.17</td><td>21.10</td><td>16.06</td><td>23.70</td><td>0.150</td><td>48.47</td><td>0.3995</td><td>40.84</td></tr><tr><td>w\o Fine-tuning</td><td>0.2443</td><td>21.64</td><td>0.3688</td><td>26.42</td><td>97.92</td><td>22.12</td><td>18.90</td><td>22.82</td><td>0.002</td><td>47.97</td><td>0.1855</td><td>35.59</td></tr></table>
+
+![](images/0097131d5da3f445d3b6fabc20087f6be3a6d564896379f12d8e1d6086a0a4e9.jpg)  
+Figure 4: Performance of UFC when fine-tuned with different numbers of support data. Overall, UFC consistently improves the controllability with the increasing size of support sets. The results on FID are presented in the Appendix C, Figure 8.
+
+![](images/e870c44c2710df357872f26231dc94ce435746e6cec549e5c9270d054f136538.jpg)  
+UFC (Ours) --- ControlNet --- Uni-ControlNet
+
+![](images/c63f0b157c4cb2d0a5b8a6104e456003fd95b1eb27d86a21b08d8202134037b4.jpg)
+
+![](images/02f48a69c30c440bd5fab8bd6f924f22c94b7249753ef6c99b382ad924f7c1bc.jpg)
+
+![](images/1e2e9e615d98bf8facfdfc4da680e3153b5b3cd35e417788780877fa672ae346.jpg)
+
+![](images/0eb6d477bdccd1aae93d7f75066fc44530bb2a09b0baaf45c2c392cc06a036de.jpg)
+
+# 5.3 Analysis
+
+Ablation study To assess the effectiveness of our design in handling the large discrepancy across condition modalities, we conduct an ablation study with two variants of UFC on UNet backbone [41]: (1) UFC w/o Matching and (2) UFC w/o Fine-tuning. Both variants are trained on the same dataset as our main method and evaluated on the COCO 2017 validation split. In (1), we remove the matching mechanism and encode the query condition directly using the condition encoder i.e.,  $\mathcal{G}_{\tau}(y_{\tau})$  while retaining task-specific parameters and meta-learning paradigm. At test time, we fine-tune  $\theta_{\tau}$  and  $\mathcal{Z}$  using the small support set. In (2), we disable adaptation entirely by sharing all parameters across tasks during training and evaluating directly on unseen tasks without any fine-tuning.
+
+Quantitative results are presented in Table 3, with qualitative examples in Appendix C. Both ablation variants show a clear drop in performance compared to our full model. Notably, UFC w/o Matching achieves similar image quality but significantly lower controllability, underscoring the importance of leveraging task-agnostic features from support images via the matching mechanism. We provide the attention map visualization in Figure 7, Appendix C for a better understanding of our matching mechanism. UFC w/o Fine-tuning performs the worst overall, highlighting the essential role of parameter adaptation in generalizing to novel spatial conditions.
+
+Impact of Support Set Sizes We evaluate UFC on varying sizes of support sets for unseen condition types and plot the performance curves in Figure 4. We can observe a clear trend that the controllability of our method improves considerably with the increasing size of the support set. With 150 support image-condition pairs (0.1% of full training data), UFC begins to surpass the controllability of the fully-supervised baseline (Uni-ControlNet) on several tasks. We also present the FID score over varying support set sizes in Figure 8 in Appendix C, which shows that our method maintains the image quality across different shots. This improvement in controllability demonstrates the effectiveness of our method in adapting to unseen spatial condition modalities through lightweight fine-tuning, eliminating the costly fully-supervised training for novel tasks.
+
+Impact of Support Samples To evaluate the sensitivity of UFC to the choice of support set, we evaluate 30-shot performance with various support sets and report the results in Table 6, Appendix C. It shows that performance of our method tends to remain consistent under diverse support sets in most tasks. However, in some tasks such as Pose and Densepose, we observe that the model performed worse than the main results that used curated supports maximizing diversity in scale, deformation, crowding, and occlusion. It shows that the importance of support-set diversity is itself task-dependent, and ensuring sample diversity can potentially benefit the few-shot performance.
+
+Table 4: Quantitative evaluation of UFC with different backbones in 30-shot setting.  
+
+<table><tr><td rowspan="2">Backbone</td><td colspan="2">Canny</td><td colspan="2">HED</td><td colspan="2">Depth</td><td colspan="2">Normal</td><td colspan="2">Pose</td><td colspan="2">Densepose</td></tr><tr><td>SSIM↑</td><td>FID</td><td>SSIM↑</td><td>FID</td><td>MSE↓</td><td>FID</td><td>MAE↓</td><td>FID</td><td>AP50↑</td><td>FID</td><td>mIoU↑</td><td>FID</td></tr><tr><td>Ours (UNet)</td><td>0.3239</td><td>19.24</td><td>0.5121</td><td>20.58</td><td>94.38</td><td>21.04</td><td>15.09</td><td>21.60</td><td>0.229</td><td>47.91</td><td>0.4340</td><td>37.79</td></tr><tr><td>Ours (DiT)</td><td>0.3984</td><td>20.43</td><td>0.6001</td><td>19.84</td><td>91.51</td><td>22.70</td><td>13.79</td><td>23.06</td><td>0.336</td><td>51.88</td><td>0.4708</td><td>39.27</td></tr></table>
+
+Impact of Diffusion Backbone Since UFC is applicable to various diffusion backbones, we employ the stronger pre-trained model based on DiT [11] and compare it with the one based on UNet used in our main table. The quantitative results are reported in Table 4, showing that the DiT-based model with a stronger diffusion backbone consistently outperforms the UNet-based model in controllability across all tasks. The lower image quality compared to the UNet variant, which is also observed in [48], can be due to the mismatch between the optimized image resolution of the pre-trained diffusion backbone with the testing resolution (1024 and 512). Figure 9 in Appendix D shows that UFC (DiT) qualitatively achieves more fine-grained controllability than the UNet counterpart. More qualitative examples using the DiT backbone are shown in Figure 10 in Appendix.
+
+Evaluation on More Novel Conditions To further validate the few-shot capability of our method in more challenging settings, we evaluate UFC on novel spatial conditions on 3D structures, such as meshes, wireframes, and point clouds extracted from iso3d dataset [10]. Importantly, these control tasks involve not only novel spatial conditions but also different output image distributions from meta-training datasets, simulating a more realistic generalization scenario to unseen control tasks.
+
+We meta-train UFC on all six tasks in our original dataset (i.e., subset of LAION-400M), then fine-tune the model on 30-shot support sets manually collected for each condition type. The fine-tuned model is then tested on unseen query conditions. The qualitative results in Figure 11, Appendix E demonstrate that generated images align closely with the spatial conditions provided by 3D meshes, wireframes, and point clouds. These results confirm UFC's effectiveness when encountering novel spatial conditions.
+
+# 6 Conclusion
+
+In this paper, we present Universal Few-shot Control (UFC), a unified control adapter that is capable of few-shot controlling text-to-image diffusion models with unseen spatial conditions. UFC adapts interpolated visual features from support images into task-specific control signals, guided by the analogy between query and support conditions. When evaluated on unseen control conditions, episodic meta-trained UFC demonstrated strong few-shot ability: it consistently outperformed all few-shot baselines and achieved precise spatial control over generated images. Notably, UFC can be on par with a fully supervised baseline, despite using only  $0.1\%$  of the full data for fine-tuning. Finally, we demonstrated that our method is applicable to both recent architectures of diffusion models, including UNet and DiT backbones.
+
+Limitations and Future Research While UFC demonstrates strong few-shot performance in spatially-conditioned image generation with T2I diffusion models, several limitations remain for future work. First, our framework is designed primarily for spatial control generation rather than tasks that require preserving the appearance of the condition image, such as style transfer, inverse problems (e.g., colorization, deblurring, inpainting). Extending the framework to handle such tasks can be a future research direction. Next, our approach requires fine-tuning on a small annotated set for each new task, unlike Large Language Models, which adapt to new tasks via in-context learning from just a few examples without fine-tuning. Developing similar capabilities for spatial control image generation remains an open and promising challenge.
+
+Acknowledgments This work was in part supported by the National Research Foundation of Korea (RS-2024-00351212 and RS-2024-00436165) and the Institute of Information & communications Technology Planning & Evaluation (IITP) (RS-2022-II220926, RS-2024-00509279, RS-2021-II212068, RS-2022-II220959, and RS-2019-II190075) funded by the Korea government (MSIT). The work was also supported by Hyundai Motor Chung Mong-Koo Foundation to Kiet T. Nguyen.
+
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+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: The abstract and introduction accurately reflect our contributions and scopes.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
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+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: Limitations are discussed in the Appendix.
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
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+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [NA]
+
+Justification: We do not include theoretical claims.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
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+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
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+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+# Answer: [Yes]
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+Justification: We include sufficient information and provide a URL to our released code and checkpoints for reproducing the results.
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+# Guidelines:
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+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
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+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+# Answer: [Yes]
+
+Justification: We released our code, checkpoints with detailed scripts on GitHub.
+
+# Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
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+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+# Answer: [Yes]
+
+Justification: We describe our experimental settings including the hyper-parameters in the main paper and appendix.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+# Answer: [No]
+
+Justification: We follow the experimental settings from the baselines and delve we provide appropriate experimental results.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
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+
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We specify the computing resource we use for experiments in our paper.
+
+Guidelines:
+
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+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: Our research conforms with the NeurIPS Code of Ethics.
+
+Guidelines:
+
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+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [NA]
+
+Justification: We expect no additional societal impacts or negative societal impacts from our work.
+
+Guidelines:
+
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+
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+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: We expect no additional risks to be posed by our work.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
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+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: We clearly state the creators and owners of the assets used in our work.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
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+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
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+
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: We do not introduce new assets.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [No]
+
+Justification: We do not include crowdsourcing.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: Our paper does not include research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA].
+
+Justification: Our method does not use LLM.
+
+# Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Implementation Details
+
+# A.1 Implementation Details on Matching Module
+
+We implement our matching module described in Section 4.3 using the multi-head attention mechanism [51]. To incorporate denoising timestep  $t$ , we adopt adaptive normalization [37], modulating the output of the matching module using the denoising timestep. Specifically, we use the embedding of the query condition  $g_{\tau}(y_{\tau})$ , the support conditions  $\{g_{\tau}(y_{\tau}^{i})\}_{i = 1}^{N}$ , and the support images  $\{f(x^i)\}_{i = 1}^N$  as the query, key, and value inputs, respectively. Let  $Q \in \mathbb{R}^{M \times d}$ ,  $K, V \in \mathbb{R}^{(N \cdot M) \times d}$ , and the timestep embedding be denoted as  $t_{\mathrm{emb}} \in \mathbb{R}^{d_t}$ , our matching mechanism operates as below:
+
+$$
+Q = \operatorname {L a y e r N o r m} (Q), \quad K = \operatorname {L a y e r N o r m} (K), \tag {9}
+$$
+
+$$
+(\alpha , \beta , \gamma) = \operatorname {L i n e a r} \left(t _ {\text {e m b}}\right), \quad V = \operatorname {L a y e r N o r m} (V) \cdot (1 + \alpha) + \beta \tag {10}
+$$
+
+$$
+O = \operatorname {C o n c a t} _ {i = 1} ^ {H} \left(\operatorname {A t t e n t i o n} \left(Q W _ {i} ^ {Q}, K W _ {i} ^ {K}, V W _ {i} ^ {V}\right)\right) \tag {11}
+$$
+
+where  $H$  is the number of attention heads,  $W_{i}^{Q}, W_{i}^{K}, W_{i}^{V} \in \mathbb{R}^{d \times d_{\mathrm{head}}}$ .
+
+The final output is calculated with residual connection as below:
+
+$$
+O = O + \gamma \cdot a c t (O W ^ {O}) \tag {12}
+$$
+
+where  $W^{O} \in \mathbb{R}^{H \cdot d_{\mathrm{head}} \times d}$ , and act denotes a non-linear activation function.
+
+We apply the matching module at each layer of the 12 encoding blocks and the mid-block in the UNet backbone of the diffusion model, and each attention module uses 8 heads.
+
+# A.2 Meta-training Settings
+
+The below setting is for the UNet backbone. The setting for the DiT backbone can be found in Section D.
+
+- **Checkpoint:** We use the Stable Diffusion v1.5 checkpoint available from the HuggingFace [9].  
+- Hyperparameters: We train using the AdamW optimizer [33] with a learning rate of  $1 \times 10^{-5}$  and a weight decay of 0.01. For each training batch, we randomly select two tasks per batch, each accompanied by a support set of three example pairs sampled for its query condition.  
+- Spatial condition representation: Following ControlNet [58], we represent all conditioning inputs as RGB images with a resolution of  $512 \times 512$ .
+
+# A.3 Few-shot Fine-tuning Settings
+
+When fine-tuning our meta-trained model on 30 support examples of the novel condition type, at most 600 fine-tuning steps with a batch size of 10 (approximately 1 hour on an RTX 3090 GPU) suffice for all six tasks. Early-stopping with a held-out validation example to monitor the denoising can be applied to avoid overfitting. We found that low-level conditions (e.g., edges) tend to converge faster than high-level conditions (e.g., pose). We also adopt AdamW optimizer [33] with a learning rate of  $1 \times 10^{-5}$  and a weight decay of 0.01.
+
+# A.4 Computation Resources
+
+<table><tr><td></td><td>GPU(s)</td><td>Batch size/GPU</td><td>Mem/GPU</td><td>Time</td></tr><tr><td>Training</td><td>8 RTX 3090</td><td>6</td><td>16GB</td><td>12 hours</td></tr><tr><td>Fine-tuning</td><td>1 RTX 3090</td><td>10</td><td>21GB</td><td>&lt;= 1 hour</td></tr><tr><td>Inference</td><td>1 RTX 3090</td><td>8</td><td>11.5GB</td><td>3.06 s / image</td></tr></table>
+
+# A.5 Dataset Construction
+
+- Training data: We randomly sample a subset of data from the LAION dataset [45]. To identify images containing humans, we use the YOLO11x model [21]. Based on this filtering, we construct a dataset consisting of 150K images with humans and 150K images without humans.  
+- Evaluation data: For the Canny, HED, Depth, and Normal tasks, we evaluate our model on 5,000 images from the COCO2017 validation set [30]. For the Pose task, evaluation is limited to images where humans are successfully detected by the Openpose model [6]. Similarly, for the Densepose task, we only evaluate our model with images where the Densepose model [14] detects a human.
+
+# B Comparison with Training-Free Methods
+
+We propose a few-shot framework for adapting to new spatial conditions in T2I diffusion models. A natural question arises: why use a few-shot approach when training-free methods exist [2, 57, 34, 29]. To illustrate the advantages of our method (UNet backbone), as mentioned earlier in Section 5.1, we compare it with Ctrl-X [29] and FreeControl [34] state-of-the-art training-free approaches capable of handling diverse spatial conditions without relying on additional pretrained networks.
+
+To perform spatial control using a condition image, Ctrl-X [29] replaces the features and attention maps when processing the noisy latent with the ones encoded from the noisy condition image. We re-implement Ctrl-X using the Stable Diffusion v1.5 backbone for a comparable setting. FreeControl [34] controls image structure by constructing a PCA basis from object features, projecting both the condition and noisy image feature maps onto this basis, and minimizing an energy function that encourages alignment between the two projections during generation.
+
+Evaluation protocol We follow FreeControl's original evaluation protocol and assess controllability of UFC and two training-free baselines on 30 images from the ImageNet-R-TI2I dataset [49], which includes 10 object categories with three captions each. FreeControl has limited flexibility in generating diverse object categories, as it requires constructing a separate PCA basis for each. This makes large-scale evaluation on 5,000 images from the COCO 2017 validation set [30] impractical, since each prompt must be individually inspected to identify objects and build corresponding PCA bases.
+
+As the evaluation dataset of FreeControl excludes human subjects, we compare UFC (30-shot) against FreeControl on four tasks: Canny, HED, Depth, and Normal. Due to the limited number of images, we do not report FID [15] or Inception Score [44], as they would not provide reliable estimates of image quality.
+
+Result comparison Table 5 shows that UFC (30-shot) significantly outperforms both training-free baselines in controllability across all four tasks. Moreover, on a single NVIDIA RTX 3090 GPU, FreeControl takes approximately 100 times longer per image to generate compared to UFC when generating 30 images for evaluation. This overhead stems from the need to construct a PCA basis for each new object category and perform 200 denoising steps for latent optimization. Ctrl-X requires self-recurrence iteration to avoid out-of-distribution sampling, which increases the generation time for several seconds. In contrast, UFC only introduces a negligible time increase.
+
+Table 5: Controllability scores and average generation time of training-free baselines and UFC (Ours, 30-shot) on generating 30 images from ImageNet-R-TI2I [49].  
+
+<table><tr><td>Method</td><td>Canny SSIM (↑)</td><td>HED SSIM (↑)</td><td>Depth MSE (↓)</td><td>Normal MAE (↓)</td><td>Time (Second)</td></tr><tr><td>FreeControl [34]</td><td>0.3139</td><td>0.3821</td><td>97.36</td><td>19.68</td><td>251.3</td></tr><tr><td>Ctrl-X [29]</td><td>0.3896</td><td>0.3693</td><td>96.12</td><td>20.84</td><td>8.01</td></tr><tr><td>UFC (Ours)</td><td>0.4074</td><td>0.5718</td><td>93.09</td><td>17.75</td><td>3.06</td></tr></table>
+
+Figure 5 presents qualitative comparisons. UFC accurately follows the spatial condition, while FreeControl and Ctrl-X fail on fine details control.
+
+![](images/595d05032a38741569031b51c97e04d06bb359b9e536098aa51fb65eae52d9d8.jpg)  
+Figure 5: Qualitative comparison between UFC (30-shot) and FreeControl [34], Ctrl-X [29] on four control tasks.
+
+# C More Results on Analysis
+
+Qualitative results with two variants As mentioned in the ablation study in Section 5.3, Figure 6 presents a qualitative comparison of our method with its two variants: (1) UFC w\o Matching and (2) UFC w\o fine-tuning. Both variants exhibit degraded image generation in terms of controllability, as they often only partially adhere to the spatial conditions. For instance, in the 2nd row with HED condition, the generated images from UFC w\o Matching and UFC w\o Fine-tuning capture the cat's head structure but fail to adhere to the condition for the cat's legs and body. In contrast, our full method, which include both matching and fine-tuning, consistently follows all given conditions.
+
+![](images/88f8e7d6ad18256b452a3b2cbb03751e1860c5c66b1a28e0669ad5b32fbce825.jpg)  
+Figure 6: Qualitative comparison results of our method and its two variants: (1) UFC w\o matching and (2) UFC w\o fine-tuning.
+
+Attention map visualization We present an example of attention maps in Figure 7. For each task, the selected query patch (highlighted by a white box in the Query column) can attend to relevant support patches, rather than unrelated regions such as background areas. For instance, in the Canny and HED tasks (first two rows), the query patches focus on support regions that preserve similar edge structures. On the other hand, for the Pose and Densepose tasks (last two rows), the query patches attend to regions related to human body parts.
+
+![](images/d24515ab571cd814dd4aaa0452bd38c9f0b68d8b79dac898255eecaffa2c5efa.jpg)  
+Query
+
+![](images/c9ebd6d17a9c6122c8502cd7a2cecb29d854a279ebc154e2e10c04f852d0b4c7.jpg)
+
+![](images/16b3e261c2c62d1395fbd927ab5b624b87cb5fc18e555636558fa0197df201d3.jpg)
+
+![](images/2c78f82e9ff3eac7ece2dfe41f8035071df6339505cfea6d79e336d44caa03e5.jpg)  
+Support
+
+![](images/2a21e2b55369a84711805043dfb64b533d953283256f60f8e6d27aa0c399974a.jpg)
+
+![](images/ec93bd55900e460fae1aec5ea7e134a06aee5e0f4c1fc35e6801506c725b1df6.jpg)
+
+![](images/e16f4ed0d086dcea8f5b97d56009060157b56eb19d2aebd5b2560a1916761478.jpg)
+
+![](images/976ab69ee0968ad4b47152c76f994f8d2bb281f0bf90a1ca72f55e23c849b507.jpg)
+
+![](images/6725716bf1e4517790e64a1549c5042b1d5deb470e7308a040135a18cc68f156.jpg)
+
+![](images/6b23f6e197b13e6189a62e27d42870dfc593242e76dc51007c861fb2e8f275d1.jpg)
+
+![](images/1e996dc1d9b460887017ec0e7cc8afaf51430dd037715f4a020a5a3f73e0e874.jpg)
+
+![](images/268f75d2e9fb1472014d5ff1ab29f5f165f945111ff3160b1f776fdfe21c3503.jpg)
+
+![](images/d83f0e380ba650e19d85ca7506a45bb3a19adb750b48c084824c02fbbe2838a8.jpg)
+
+![](images/9032ad5ff6bb2722c3f23aa02cdd4b6c67716f06444b74fb3924a688017b04f7.jpg)
+
+![](images/60d18c223de018b451f6f6de5fb81937f29f4bb5359838333603ef207871f85c.jpg)
+
+![](images/b2c88c25b18c31bd750336e416d03c2e1b86a8d244a75158a7518bbb550e6198.jpg)
+
+![](images/06c1eb4d33b21535b9949676693d30239ddd270da7f346a8f70ab5e522b05e8c.jpg)
+
+![](images/09dd7ed642dae806b1f04646646c1075156d774adfd213a64867098f282a2ddb.jpg)
+
+![](images/1b403c28d8f73ad55554984003b96feb55a59610ce952fffa6d2b751a88c83d6.jpg)
+
+![](images/77a1da8ccacd6a62b1d14bf54cdcdf4a68f43f6e63b403072a000f9b65042d0c.jpg)
+
+![](images/95d636dde1c49d10bb05154f12b1dbcffc4868ab407d2d1b1acb0f89d5deac62.jpg)
+
+![](images/194faf5a641f770c679e158b0f20e93ab10f2a5fe8fbad8ea06ee3901cdc3fe1.jpg)
+
+![](images/f9f3373ffcf05691c9b23a4c437aaecbe666b8cfb81a6a9a2a13c4d3497246d1.jpg)
+
+![](images/4815529055f2be243f37ee2ba3e8b8e2a1911ead2a84eb6b63bd6aa0163285a5.jpg)
+
+![](images/2f588426601c038f7a8775bc40779c28ae2035f47da2621df3951ee1c0d53891.jpg)
+
+![](images/6d8def650866f9a28f1e8fd947ec63ee6679fa216e6a2c1f5d0d439b05fabd45.jpg)
+
+![](images/0cdab44745225fc8ca54a4ebe264f7c753a79fe165de8b286dec032889ccfec8.jpg)
+
+![](images/c031d2c9aff4cff47fd35b38b1f70abb13bd87f236227d3d7fbd6b1e8ede0192.jpg)
+
+![](images/9bd431f83e776c42d9df3cdd5c2238d9a8ac12ae31ab787b22f86096baa0ca16.jpg)
+
+![](images/a77454844c5a4783116c0fbb165b84f1379a73f0db4ac8d373c54415cf12db92.jpg)
+
+![](images/c25670318d568634a6d4dd8ed93726b620c79340a8366b414424df47432efc1a.jpg)  
+Figure 7: Attention Maps of UFC (30-shot) from the 7th layer and a selected head for each task. Support normal maps are converted to grayscale to enhance the visibility of attention regions. Query patches attend primarily to the most relevant support patches.
+
+![](images/ab6fcef3cbdbabde7256c50cf08e85e35f88de2498feb5bddbb4614551db9b2e.jpg)
+
+![](images/7f59dce631aa791ddc53e1fb25b9927d3ae552b4350d954b8219b02d8ab2ccdb.jpg)
+
+![](images/b5b34d92dae9f3996cc18129d4f526221501f998ac1ff22adcfaba10a0d38e04.jpg)
+
+![](images/3884b1c2ae79b4b63c7853e128f9f6d08ff119bdc5b3c2ea2063d51d9bbac0a6.jpg)
+
+![](images/a10cb9bb6d5b3ccd219047b97d91a52dec29bb99fe4842cf158bab6ad900d369.jpg)
+
+FID Results with different number of shots In addition to the controllability measurement in Figure 4, Section 5.3, we provide FID results obtained by fine-tuning our method (UFC) using different numbers of support data (i.e., shots) in Figure 8. The results show that our method maintains FID scores across various shots. Specifically, for Normal, Depth, Canny, and HED tasks, our method preserves image quality with FID changes remaining within 1. For Pose, the FID difference is within 4.5, and for DensePose, it is within 1.7. Both changes are relatively small given the FID scores of each task, and using more support data often slightly improves FID. The results confirm that our method improves controllability as the number of support data increases, while maintaining image quality.
+
+![](images/2167437a4984e29f801d58f77485533762b99e05e7a03d27aa1f4bebdb59c33c.jpg)
+
+![](images/173d5e60ccf81333ad9930141dff55004e2ab31d1710399fb993de9ff53253de.jpg)
+
+![](images/c8329e8323c344392640292cc20e985abec09600b2db9f22449b517d7e0ee136.jpg)
+
+![](images/b142a3a7ad40b9a1da8c69d40dd8f0a5581a08bc9d1db5b44ad3b6b71403dcc9.jpg)
+
+![](images/f478060e91f56d5b5a77a9a5f3ed747317d2141b90aefe81f629518d0013e060.jpg)
+
+![](images/e8e4a47d684032386cf3e8ac7202001a41cfd95e208ebd099c8c513200c4eec9.jpg)
+
+![](images/bd6a5a46f341c4281d54194eb70d2e29be7a03dfc18020958e5381ac8187266b.jpg)  
+Figure 8: FID score over varying support set sizes (shots).
+
+Effect of support samples As discussed in Section 5.3, UFC is evaluated in 30-shot settings with different investigate the effect of support sets' diversity. In this section, we would like to discuss the implications of diversity for each task. Because our matching strategy hinges on condition-specific similarity, the notion of diversity depends on tasks. For edge inputs such as Canny and HED, the support set must span a wide range of edge-map densities and orientations. For Pose and DensePose, it needs to cover varied human scales, occlusions, and pose deformations. To measure how support choice influences performance, we repeated fine-tuning three times, each time drawing a new random set of 30 supports and carrying out few-shot generation. The results are summarized in the Table 6.
+
+Table 6: Analysis on the effect of support set's diversity on model performance.  
+
+<table><tr><td rowspan="2">seed</td><td colspan="2">Canny</td><td colspan="2">HED</td><td colspan="2">Depth</td><td colspan="2">Normal</td><td colspan="2">Pose</td><td colspan="2">Densepose</td></tr><tr><td>SSIM↑</td><td>FID</td><td>SSIM↑</td><td>FID</td><td>MSE↓</td><td>FID</td><td>MAE↓</td><td>FID</td><td>AP50↑</td><td>FID</td><td>mIoU↑</td><td>FID</td></tr><tr><td>1</td><td>0.3239</td><td>19.24</td><td>0.5121</td><td>20.58</td><td>94.38</td><td>21.04</td><td>15.09</td><td>21.60</td><td>0.229</td><td>47.91</td><td>0.4340</td><td>37.79</td></tr><tr><td>2</td><td>0.3295</td><td>21.06</td><td>0.4957</td><td>20.70</td><td>93.47</td><td>20.51</td><td>15.82</td><td>20.85</td><td>0.176</td><td>48.22</td><td>0.3827</td><td>39.52</td></tr><tr><td>3</td><td>0.3273</td><td>20.90</td><td>0.5128</td><td>23.84</td><td>94.84</td><td>22.21</td><td>15.93</td><td>22.56</td><td>0.191</td><td>48.66</td><td>0.3831</td><td>37.46</td></tr><tr><td>4</td><td>0.3290</td><td>20.41</td><td>0.4942</td><td>20.78</td><td>94.50</td><td>21.80</td><td>15.56</td><td>21.89</td><td>0.202</td><td>47.83</td><td>0.3914</td><td>38.30</td></tr><tr><td>Mean</td><td>0.3274</td><td>20.40</td><td>0.5037</td><td>21.73</td><td>94.30</td><td>21.39</td><td>15.60</td><td>21.73</td><td>0.1995</td><td>48.16</td><td>0.3978</td><td>38.27</td></tr><tr><td>Std</td><td>0.0025</td><td>0.82</td><td>0.0101</td><td>1.49</td><td>0.5851</td><td>0.76</td><td>0.3737</td><td>0.71</td><td>0.0224</td><td>0.38</td><td>0.0245</td><td>0.90</td></tr></table>
+
+# D Experiments with DiT Backbone
+
+Model As discussed in Section 4.3, our framework can work with both UNet and DiT backbones. We extend UFC to the DiT architecture [36], using Stable Diffusion v3 [11] as an example, initialized from the publicly available v3.5-medium checkpoint [1] on Hugging Face. This model consists of 24 Transformer layers in the diffusion backbone. We initialize the image encoder with pretrained weights of the diffusion backbone and keep it frozen during training. For the condition encoder, we initialize it as a trainable copy of the first 12 layers of the diffusion backbone. We train our model on 8 NVIDIA RTX A6000 GPUs.
+
+Control feature injection Unlike the UNet backbone, the DiT architecture does not have skip connections. Therefore, we inject the output of the matching module directly into the hidden representations at each layer of the DiT backbone. Specifically, image features are extracted from 12 even-numbered layers of the image encoder, while query and support condition embeddings are obtained from the condition encoder. The matching mechanism is applied across these 12 layers to generate multi-layer control features. The 24-layer DiT diffusion backbone is divided into 12 sequential chunks, each containing 2 layers. The control features are then added to every hidden representation of these chunks in the corresponding order.
+
+Hyper-parameters We follow the same optimizer settings, training dataset, and evaluation protocol described in Section 5.1. For inference, we use the flow-matching Euler scheduler introduced in Stable Diffusion 3[11], with a classifier-free guidance (CFG) scale of 5.0, 28 generation steps, and a fixed seed of 42.
+
+Qualitative Results Qualitative examples using the DiT backbone are shown in Figures 9, 10. Our DiT-based model, adapted with only 30 shots, is able to control the structure of generated images with various novel condition tasks.
+
+![](images/5170babe089a12afee3f5a9afd2324c3bb0616cdeac255858e27b45dc67ae083.jpg)  
+Figure 9: Qualitative comparison of UFC using DiT and UNet backbones in 30-shot setting. Our method with the DiT backbone yields more fine-grained spatial control than the UNet counterpart.
+
+![](images/7824b15e566ebbc68fab3e42bca5017f540ff971df88e73f6e7e6624c9c6c429.jpg)  
+Canny  
+'A fire hydrant stands at the corner of two city streets'
+
+![](images/03c12418f9a4c753412730e8c8db4d82fd7f01851d8f3fd7aad81304eb33023c.jpg)
+
+![](images/fbd641327c2facae37c41e2fe0fe6c20fab6e556cb67b454e39e3c578e74b255.jpg)  
+"Baxter street in the city at night; a "people crossing" sign
+
+![](images/74bedcb89d3e04fafc774ab876140f880e943d77caa50c2ce72000a17a9e04c5.jpg)
+
+![](images/aec02da5d6b964dc36bd82d3640d2503c185c7812cda269dace5acf9d6e7c0e6.jpg)  
+HED  
+'A black and white living room with modern furniture'
+
+![](images/fd2666ee369b0f51b58307cb2c2ee5eb46e77d10cd28e3fe3e8e787ef26f8647.jpg)
+
+![](images/7027fc310fed0f6b86ff86ceaa6787685d458534fe82564031415d83ed616157.jpg)  
+'A bathroom looks clean but is missing tile at the shower stall'
+
+![](images/62bf1bfd47f01ec046b23e98003af65c69a19596daf49e8d7e0f1273430b65be.jpg)
+
+![](images/8af4c60d3b39c463ead2a14713fe617a544cb35fbfee3221a33311463fd26203.jpg)  
+Depth  
+'A metal clock tower on a sidewalk in front of a building'
+
+![](images/923f31c4e6f8fe70f126407715f90a9dbf50052297d641e34980df3d10a88a77.jpg)
+
+![](images/68b59f519230f5d01961567a149ccc26be679c88f76e54669a96f053d20b179d.jpg)  
+'A brown and yellow commuter train with headlights on'
+
+![](images/c54c6ec2a15cad310ceab730d85b9b3d9b19f7e8b824825445b19100976cb94a.jpg)
+
+![](images/3cc9a68e8a3fba61f3aebf760cbae3c9d42d1584a24f5eaf8f27f4dd1d71939e.jpg)  
+Normal  
+'A STOP sign on a city street at night'
+
+![](images/3bdb271505fa90067ba429a4feb6f8ce63609862894948d06a6c2ea6091f10e6.jpg)  
+'Beautiful scene of the water, the bridge, and a tall clock tower'.
+
+![](images/6d5bd6976241b21d3c26e410234d4378ab3e39a783b97d61e0981a7901631ba0.jpg)
+
+![](images/41ea5777442fe19eee6d83bcefed817f0db625a368da8bcee160941a5b18321d.jpg)
+
+![](images/aae9ffab76eedf5fbab7a7380b53f35341122f0487f701663f5d1a17672c6bd9.jpg)  
+Pose  
+'A man holding a yellow kite while standing in a room'
+
+![](images/f7a784f398d303b9f67b2c84f1072d0ba68481da9d0bcad652e3b237f327d412.jpg)
+
+![](images/10408ebdc4ee65180722eef6bc4e68c33ccdd77ec76f09651831dcc1d5d24202.jpg)  
+'Man on skies on snow covered slope area'
+
+![](images/ce89185a8ae2d7e93efffe4343336636c9b4f8c011ea95033f9cf8fd03f9a804.jpg)
+
+![](images/c6794771cd5b1057a7709c1b46797877d414fe83647ad9891d972eb8c4adb7f3.jpg)  
+Dense Pose  
+'A man prepares food in a restaurant kitchen'  
+Figure 10: Generated images from UFC with DiT backbone in 30-shot setting.
+
+![](images/26d5deef8cfba46eb5730e98ac9e6e0cac5a05b0d26f742947f9327b105dac64.jpg)
+
+![](images/4a76e5095779ef15a5fc2761b81b63130db4c722fbf06e6cfdf164f4b69b6d95.jpg)  
+'An older man has a woman standing next to him'
+
+![](images/fc301d46220817818a54818060ed4850c4ed29c9d1893a45f3be45820a582ff4.jpg)
+
+# E More Results
+
+Results with more spatial conditions As previously discussed in Section 5.3, we evaluate UFC on more novel spatial conditions, including 3D meshes, wireframes, and point clouds. We use iso3d [10], a 3D isolated object dataset(with no background), to generate spatial conditions for image pairs. We only report qualitative results, as the absence of a pre-trained condition-prediction network prevents us from measuring controllability using the evaluation approach in Section 5.1. Moreover, the insufficient collected validation sets hinder reliable assessment of image quality with FID [15] or Inception Score [44]. Figure 11 shows the qualitative results of our method, demonstrating the effectiveness of our method in a few-shot (30 supports) setting.
+
+![](images/1d33134f9334b773c64495027b416240d839346afa7d69f0c1450500819e5f81.jpg)  
+Figure 11: Generated images from UFC with spatial conditions of 3D meshes, wireframes, and point clouds in 30-shot setting.
+
+More qualitative results We present more qualitative results of UFC (UNet) in Figure 12 and Figure 13. All the results are generated with a support size of 30.
+
+![](images/9bd7f0b5796c103f4cd03c9b5dcf372387879eb1b5e04cb9d99132dba279530c.jpg)  
+Canny  
+'A sheep standing in grass next to a rock wall'
+
+![](images/a42af754a5d0351eaf4d02dfc06a50e5147a557f6bb5c5f3a03a49c8de32db87.jpg)
+
+![](images/1e0b2a9773d5220ebf76df5b5a95d4010b615113c0a0eb343b5dae3179642e41.jpg)  
+'A closeup of a grill of doughnuts'
+
+![](images/ea466f49a85b452b3fcb04bdc8f10df4fb3a85a42d74716ff9d2a04e4f14c401.jpg)
+
+![](images/bfe537e42dd1474bb8267c03d948956af4122ebfabafc8486c401603ed9e2a6c.jpg)  
+HED  
+'The person holds both hands in the air while snowboarding'
+
+![](images/b1658df5c73297705a9f9a99949510dc1f95d731569b16d26721032b371404e3.jpg)
+
+![](images/b074bf45c0f09574b90685f9740f08de5ab22345539de0ff0176c351e9d71ca3.jpg)  
+'There are books on all three shelves of this bookshelf'
+
+![](images/bd10e4a36aaba2c19d3e082767b7678ec9dca361ebfc1c4114c2731a49a7ee0e.jpg)
+
+![](images/55b91d02530180877a1a70aad8094b19216be5c7d8beb307c540da35400ab81e.jpg)  
+Depth  
+'A home kitchen with white cabinets and silver appliances'
+
+![](images/034504c0460a968ad561f2b1007d66a866cf31025b8ff658bf6f43237bd54329.jpg)
+
+![](images/72b5a713c27532dab298ce842b7e27118b99491d800954029c348fb409476642.jpg)  
+'A baseball player talking to a small child on the field'
+
+![](images/ae04e72b67a5aa1957951291c80ef613f654e899d9c0bae6181f43bccc121464.jpg)
+
+![](images/4c11747df0c0c14e1435dcda1983e8598662a70764be433378c1a9781c64e995.jpg)  
+Normal  
+Two giraffes standing on a wide-open area
+
+![](images/88db324f19a60a8340c7209e5642621a711f91cccc21fe31aa0ee142852d93ee.jpg)
+
+![](images/c172d4939f40088c5474bdb6beaeeb39215722ea017d14cf3787ae08556b3504.jpg)  
+'Several sailboats sit in the water in front of some trees.'
+
+![](images/af6cbcc52f28e847b636ff5695e1b42193cdc9df68cc049cb33d760a05269832.jpg)
+
+![](images/2ade79259f0680a013335395487dd4fd2db8da3b8f2c3d90bf5c316cc405eafa.jpg)  
+Pose  
+A group of vintage men with baseball attire on them
+
+![](images/e663b59e63b0f214b92b1c7dbc683d43d72e77abc058c586f43a0c640838fdc2.jpg)
+
+![](images/32a70ae6412dc60b3940f920367d250614a983373cb97e9817ed25f4fc7d7524.jpg)  
+'Hey man that is opening up a bottle of wine'
+
+![](images/b83ef1de14c0a2a28972aa88668cd0294b9ee10ac83dcf64f059ecf7db715bab.jpg)
+
+![](images/7a39f974c2a89b285990a254ff60b699077c86255f4587a17afa868afd7abc99.jpg)  
+Dense Pose  
+'A man looking at camera preparing food in kitchen'  
+Figure 12: More qualitative results of UFC (UNet) in 30-shot setting.
+
+![](images/d986a36b103435b7593de44b454e9c35b4ff0b023f2fd9e364d32cddbe9c8ff3.jpg)
+
+![](images/743acd21d36e5f4c6a4b3f38f9b98adf0fdc06d5ad7848f9a2eb6308a10e5f46.jpg)  
+'A man walks up to where surfboards are propped up against a wall and wetsuits are hanging on a rack'
+
+![](images/50ef24eecf05f08c7e702b60662f6285519a2b7bb8ba22baee68b765de823a6c.jpg)
+
+![](images/397321ddd2a5f73ee9a12c68151f1d98e4ce55dbb6fb2cbe73b3849833d53a92.jpg)  
+Canny  
+'A man flying a colorful kite over a sandy beach'
+
+![](images/482fdb0e6915f4015acf7ee53aae114c3df1df75b7af450c3d7cd39b6ec1cf6b.jpg)
+
+![](images/1b823cb22c110620d7379364956442e048681dfadef74843d8909a785c6583f1.jpg)  
+'Two stuffed animals in a wooden outdoor chair'
+
+![](images/ad24c6eb1d2ff207c538a5594a94ef6796dfaa387ea87127343ab2d4c19a2cfc.jpg)
+
+![](images/035c20043df1706630cc7ccf7fce37c2a55e88d1a565377064d0f6807562d26d.jpg)  
+HED  
+'A couple of zebras are grazing in a field'
+
+![](images/0c2e45c578973c053aafeae458bf5959768600d78bc37e4f625358518722cf28.jpg)
+
+![](images/8f385ec32980b1d5efe629fe778b4da62e15ac5a73e2565e76a844b3c57d7d3f.jpg)  
+'The exterior of a building with a clock'
+
+![](images/d3de5b05b0d45863da062822c2393e14d9d6beced4664465bbc3b70e6c92c6ef.jpg)
+
+![](images/84f0ede536a1006cda6bd13e5b6f4deed5f8df30202fe569c3896a76849e8f33.jpg)  
+Depth  
+'A bird sitting on the window sill of a boat floating on a lake'
+
+![](images/423b70ee16bfffa251c10438f0cdd066ff1fbbd16c55caa16e06eba8cd615bc4.jpg)
+
+![](images/48e5aad2024cac779b80a11768331641eff325ea8e3b9449764b78059c59d5c6.jpg)  
+'A horse standing next to a fence being seen through a lens'
+
+![](images/1275b54dcdf94ead8c6969b8d7accc93c93209894c56c2d3d31aa115ec57b0f2.jpg)
+
+![](images/19247495eaa1fe51113c98c6d40db8c3f146d026e7eb43ae14fc5618b54e6488.jpg)  
+Normal  
+'A baby on the floor near a cellphone'
+
+![](images/8af85eca485555d7af3473a25bcb74037defda1ac6471263b4f89650439e393e.jpg)
+
+![](images/faced6581f039a0cd0cb0113aa504d61a545a74b78219fc5ff8abd8f8e20563a.jpg)  
+'A pizza with veggies and eggs on it'
+
+![](images/7079d5f63368585d9034604a03bc0adde54ec71cbfc404ff74bc6cffc122bd24.jpg)
+
+![](images/1de84bfa0d230937128158f4e191a31ddae1085308e5d64f937d3f0c9618099b.jpg)  
+Pose  
+'Man in a black jacket standing in front of a motorcycle'
+
+![](images/74a28d4342a043a0ba5963f03f40e91c3405b2b3c1b642be7686325dcd5daf9e.jpg)
+
+![](images/ce5d9f2cae56c73b77ef01cf917c4b21ae997dae7d6d25f0520a7c5affc9f7ce.jpg)  
+'A man is playing tennis on the tennis court'
+
+![](images/1af1f131de2cfab147dac03c93368b373e722ad41de20052a72076205faeb222.jpg)
+
+![](images/fb9b9943433c50fd99d0604022d39c82cc9aa287ab5806a30b3ae19a19144e45.jpg)  
+Dense Pose  
+'A person in a kitchen with a stove some pots and pans'  
+Figure 13: More qualitative results of UFC (UNet) in 30-shot setting.
+
+![](images/2cedbf872a57f60f2094e1d28ab2ab37d93986fd679dbea45387fc48b0836b7b.jpg)
+
+![](images/50058fbf1426ab496069049c023aa47654bbac3a6088b6261ddaa91f70ef4ad8.jpg)  
+'Two guys in a large room with a blackboard'
+
+![](images/603a91cf6eb5a733c7ce920f9edc87bf5d979ae94132860867726845aba920d1.jpg)
+
+# F Support Image-Condition Pairs
+
+As described in implementation detail in Section 5.1, after few-shot fine-tuning, 5 image-condition pairs are used for inference.
+
+![](images/d3af3c1ec39a77176024f6580d1b7b4eb3cc356d0532427b15979c16791ae5a9.jpg)  
+Figure 14: Five support image-label pairs used for evaluating Canny/Depth/HED/Normal tasks.
+
+![](images/bb208840370e28d5a95373394bbfa17998fbbcbe22132b4337f3b3d5abfa5fa1.jpg)  
+Figure 15: Five support image-label pairs used for evaluating Pose/DensePose tasks.
\ No newline at end of file
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+# Universal Sequence Preconditioning
+
+Annie Marsden *
+
+Elad Hazan*†
+
+# Abstract
+
+We study the problem of preconditioning in sequential prediction. From the theoretical lens of linear dynamical systems, we show that convolving the input sequence corresponds to applying a polynomial to the hidden transition matrix. Building on this insight, we propose a universal preconditioning method that convolves the input with coefficients from orthogonal polynomials such as Chebyshev or Legendre. We prove that this approach reduces regret for two distinct prediction algorithms and yields the first ever sublinear and hidden-dimension-independent regret bounds (up to logarithmic factors) that hold for systems with marginally stable and asymmetric transition matrices. Finally, extensive synthetic and real-world experiments show that this simple preconditioning strategy improves the performance of a diverse range of algorithms, including recurrent neural networks, and generalizes to signals beyond linear dynamical systems.
+
+# 1 Introduction
+
+In sequence prediction the goal of the learner is to predict the next token accurately according to a specified loss function, such as the mean square error or cross-entropy. This fundamental problem in machine learning has gained increased importance with the rise of large language models, which perform sequence prediction on tokens using cross entropy. The focus of this paper is preconditioning, i.e. modifying the target sequence to make it easier to learn. A classic example is differencing, introduced by Box and Jenkins in the 1970s [14], which transforms observations  $\mathbf{y}_1, \mathbf{y}_2, \ldots$  into successive differences,
+
+$$
+\mathbf {y} _ {1} - \mathbf {y} _ {0}, \mathbf {y} _ {2} - \mathbf {y} _ {1}, \dots , \mathbf {y} _ {t} - \mathbf {y} _ {t - 1}, \dots
+$$
+
+It is widely acknowledged that learning this sequence can be "easier" than learning the original sequence for a large number of modalities. In this work we seek a more general framework for sequence preconditioning that captures the same intuition behind differencing and extends it to a broader class of transformations. The question we ask is
+
+What is the general form of sequence preconditioning that enables provably accurate learning?
+
+We address this question by introducing a preconditioning method which takes in  $n$  fixed coefficients  $c_{0},\ldots ,c_{n}$  and converts the sequence of observations  $\mathbf{y}_1,\dots ,\mathbf{y}_t,\dots$  to the sequence of convolved observations
+
+$$
+c _ {0} \mathbf {y} _ {0}, c _ {0} \mathbf {y} _ {1} + c _ {1} \mathbf {y} _ {0}, \dots , \sum_ {i = 0} ^ {n} c _ {i} \mathbf {y} _ {t - i}, \dots
+$$
+
+From an information-theoretic perspective, approaches of this kind seem futile—predicting  $\mathbf{y}_t$  or  $\sum_{i} c_i \mathbf{y}_{t-i}$  seems equally hard in an adversarial setting. Yet we show that when the data arises from
+
+a linear dynamical system (LDS), there exists a universal form of preconditioning that provably improves learnability, independent of the specific system.
+
+In LDS setting, we show that preconditioning significantly strengthens existing prediction methods, leading to new regret bounds. Here, preconditioning has an elegant effect: the preconditioning filter forms coefficients of an  $n$  degree polynomial, and the hidden system transition matrix is evaluated on this polynomial—potentially shrinking the domain. In this setting, shrinking the learnable domain is akin to making the problem "easier to learn", a relationship that is formalized by [28]. This allows us to prove the first dimension-independent sublinear regret bounds for asymmetric linear dynamical systems that are marginally stable.
+
+# 1.1 Our results
+
+Our main contribution is Universal Sequence Preconditioning, a novel method of sequence preconditioning which convolves the target sequence with the coefficients of the  $n$ -th monic Chebyshev polynomial. We give a more general form of preconditioning, allowing arbitrary user-specified coefficients, in Algorithm 1 and an online version in Algorithm 4 (Appendix C). We analyze the effect of Universal Sequence Preconditioning on two canonical sequence prediction algorithms in the online setting: (1) convex regression and (2) spectral filtering. In either case, the results are impressive- yielding the first known sublinear regret bounds as compared to the optimal ground-truth predictor that are simultaneously (1) applicable to marginally stable systems, (2) independent of hidden dimension (up to logarithmic factors), and (3) applicable to systems whose transition matrix is asymmetric (see Table 1).
+
+Algorithm 1 General Sequence Preconditioning (Offline Version)
+
+1: Training  
+2: Input: training data  $(\mathbf{u}_{1:T}^{1:N}, \mathbf{y}_{1:T}^{1:N})$  where  $(\mathbf{u}_t^i, \mathbf{y}_t^i)$  is the  $t$ -th input/output pair in the  $i$ -th sequence; coefficients  $c_{0:n}$ ; prediction algorithm  $\mathcal{A}$ .  
+3: Assert  $c_{0} = 1$ .  
+4: for  $i = 1$  to  $N$  do  
+5:  $\mathbf{y}_{1:T}^{\text{preconditioned},i} \gets \text{convolution}(\mathbf{y}_{1:T}^{i}, c_{0:n})$ $\triangleleft \mathbf{y}_t^{\text{preconditioned}, i} = \mathbf{y}_t^i + \sum_{j=1}^n c_j \mathbf{y}_{t-j}^i$  
+6: end for  
+7: Train  $\mathcal{A}$  on preconditioned data  $\left(\mathbf{u}_{1:T}^{1:N},\mathbf{y}_{1:T}^{\mathrm{preconditioned,}1:N}\right)$ .  
+8: Test Time  
+9: for  $t = 1$  to  $T$  do  
+10: Receive  $\mathbf{u}_t$  
+11: Predict  $\hat{\mathbf{y}}_t\gets \mathcal{A}\left(\mathbf{u}_{1:t},\mathbf{y}_{1:(t - 1)}\right) - \sum_{i = 1}^{n}c_i\mathbf{y}_{t - i}$ .  
+12: Receive  $\mathbf{y}_t$  
+13: end for
+
+First, applying USP to standard convex regression results in regret  $\tilde{O}(T^{-2/13})$ , which holds simultaneously across the three settings above and remains dimension-independent. For comparison, a naive analysis of regression yields a vacuous regret bound of  $O(T^{5/2})$  on marginally stable systems. Second, combining USP with a variant of spectral filtering [29] that uses novel filters, the algorithm is able learn a broader class of linear dynamical systems-- in particular systems whose hidden transition matrix may be asymmetric. The enhanced method achieves regret  $\tilde{O}(T^{-3/13})$ , the best known rate under the joint conditions of conditions (1)-(3) discussed above: marginal stability, dimension independence, and asymmetry.
+
+Both results require that the transition matrix eigenvalues have imaginary parts bounded by  $O(1 / \log T)$ —a near-tight condition for achieving dimension-free regret. Further discussion on this appears in Appendix B
+
+Empirical results in Section 4 demonstrate that USP consistently improves performance across diverse algorithms—including regression, spectral filtering, and neural networks—and across data types extending beyond linear dynamical systems.
+
+<table><tr><td>Method</td><td>Marginally stable</td><td>\( {d}_{\text{hidden }} \) -free</td><td>Asymmetric</td></tr><tr><td>Sys-ID</td><td>×</td><td>×</td><td>✓</td></tr><tr><td>Regression (open-loop)</td><td>×</td><td>✓</td><td>✓</td></tr><tr><td>Regression (closed-loop)</td><td>✓</td><td>×</td><td>✓</td></tr><tr><td>Spectral Filtering</td><td>✓</td><td>✓</td><td>×</td></tr><tr><td>USP + Regression</td><td>✓</td><td>✓</td><td>~✓</td></tr><tr><td>USP + Spectral Filtering</td><td>✓</td><td>✓</td><td>~✓</td></tr></table>
+
+Table 1: Comparison of methods for learning LDS. USP extends learning to mildly asymmetric matrices with bounded complex eigenvalues.
+
+# 1.2 Intuition for Universal Sequence Preconditioning
+
+We now give some brief intuition for the result. Linear dynamical systems (LDS) are perhaps the most basic and well studied dynamical systems in engineering and control science. Given input vectors  $\mathbf{u}_1, \ldots, \mathbf{u}_T \in \mathbb{C}^{d_{\mathrm{in}}}$ , the system generates a sequence of output vectors  $\mathbf{y}_1, \ldots, \mathbf{y}_T \in \mathbb{C}^{d_{\mathrm{out}}}$  according to the law
+
+$$
+\mathbf {x} _ {t + 1} = \mathbf {A} \mathbf {x} _ {t} + \mathbf {B} \mathbf {u} _ {t}, \quad \mathbf {y} _ {t} = \mathbf {C} \mathbf {x} _ {t} + \mathbf {D} \mathbf {u} _ {t}, \tag {1}
+$$
+
+where  $\mathbf{x}_0, \ldots, \mathbf{x}_T \in \mathbb{C}^{d_{\mathrm{hidden}}}$  is a sequence of hidden states and  $(\mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D})$  are matrices which parameterize the LDS. We assume w.l.o.g. that  $\mathbf{D} = 0$ . We can factor out the hidden state  $\mathbf{x}_t$  so that the observation at time  $t$  is
+
+$$
+\mathbf {y} _ {t} = \sum_ {s = 1} ^ {t} \mathbf {C A} ^ {t - s} \mathbf {B u} _ {s}.
+$$
+
+Given coefficients  $c_{0:n} = (c_0,\dots ,c_n)$  let
+
+$$
+p _ {n} ^ {c} (x) \stackrel {\text {d e f}} {=} \sum_ {i = 0} ^ {n} c _ {i} x ^ {n - i}. \tag {2}
+$$
+
+Consider a "preconditioned" target at time  $t$  to be a linear combination of  $\mathbf{y}_{t:t-n}$  with coefficients  $c_{0:n}$ . A key insight is the following identity,
+
+$$
+\sum_ {i = 0} ^ {n} c _ {i} \mathbf {y} _ {t - i} = \sum_ {s = 0} ^ {n - 1} \left(\sum_ {i = 0} ^ {s} c _ {i} \mathbf {C A} ^ {s - i} \mathbf {B}\right) \mathbf {u} _ {t - s} + \sum_ {s = 0} ^ {t - n - 1} \mathbf {C p} _ {n} ^ {c} (\mathbf {A}) \mathbf {A} ^ {s} \mathbf {B} \mathbf {u} _ {t - n - s}.
+$$
+
+If we take  $c_{0} = 1$  (i.e. a monic polynomial), we can re-write  $\mathbf{y}_{t}$  as
+
+$$
+\mathbf {y} _ {t} = - \underbrace {\sum_ {i = 1} ^ {n} c _ {i} \mathbf {y} _ {t - i}} _ {\aleph_ {0}} + \underbrace {\sum_ {s = 0} ^ {n - 1} \sum_ {i = 0} ^ {s} c _ {i} \mathbf {C A} ^ {s - i} \mathbf {B u} _ {t - s}} _ {\aleph_ {1}} + \underbrace {\sum_ {s = 0} ^ {t - n - 1} \mathbf {C p} _ {n} ^ {c} (\mathbf {A}) \mathbf {A} ^ {s} \mathbf {B u} _ {t - n - s}} _ {\aleph_ {2}}. \tag {3}
+$$
+
+This expression highlights our approach as a balance of three terms:
+
+$\aleph_0$  The universal preconditioning term: it depends only on the coefficients  $c_{0:n}$  and not on any learning algorithm.  
+$\aleph_1$  A term learnable via convex relaxation and regression, for example by denoting
+
+$$
+\mathbf {Q} _ {s} ^ {\text {l e a r n e d}} = \sum_ {i = 0} ^ {s} c _ {i} \mathbf {C A} ^ {s - i} \mathbf {B}.
+$$
+
+The diameter of the coefficient  $\mathbf{Q}_s$  depends on the magnitude of the coefficients  $c_{0:n}$ .
+
+$\aleph_2$  The residual term with polynomial  $p_n^c (\mathbf{A})$ . By a careful choice of coefficients  $c_{0:n}$ , we can force this term to be very small.
+
+The main insight we derive from this expression is the inherent tension between two terms  $\aleph_1, \aleph_2$ . The polynomial  $p_n^c(x)$  and its coefficients  $c_0, \ldots, c_n$  control two competing effects:
+
+1. The preconditioning coefficients grow larger with the degree  $n$  of the polynomial and the magnitude of the coefficients  $c_{i}$ . A higher degree polynomial and larger coefficients increase the diameter of the search space over the preconditioning coefficients, and therefore increase the regret bound stemming from the  $\aleph_1$  component learning.  
+2. On the other hand, a larger search space can allow a broader class of polynomials  $p_n(\cdot)$  which can better control the magnitude of  $p_n(\mathbf{A})$ , and therefore reduce the search space of the  $\aleph_2$  component.
+
+What choice of polynomial is best? This work considers the Chebyshev polynomial. The reason is the following property of the  $n$ -th monic Chebyshev polynomial:
+
+$$
+\max  _ {\lambda \in [ - 1, 1 ]} | p _ {n} (\lambda) | \leq 2 ^ {- (n - 1)}.
+$$
+
+As an example, consider any LDS whose hidden transition matrix  $\mathbf{A}$  is diagonalizable and has eigenvalues in  $[-1,1]$ . By the above property, observe that  $\| p_n(\mathbf{A})\|_{\infty} \leq 2 \cdot 2^{-n}$ , therefore shrinking the  $\aleph_2$  term at a rate exponential with the number of preconditioning coefficients. We pause to remark on the universality of this choice of polynomial. Indeed, one could instead have chosen the preconditioning coefficients to depend on  $\mathbf{A}$  so that  $p_n^{\mathrm{c}}(\cdot)$  is the characteristic polynomial of  $\mathbf{A}$ . By the Cayley-Hamilton theorem,  $p_n(\mathbf{A}) = 0$ . This means that  $\aleph_2$  term is canceled out completely. However, this would have required knowledge of the spectrum of  $\mathbf{A}$ . The Chebyshev polynomial, on the other hand, is agnostic to the particular hidden transition matrix. Moreover, even if the spectrum of  $\mathbf{A}$  were known, choosing the preconditioning coefficients to form the characteristic polynomial would result in an algorithm which must learn hidden dimension many parameters, which is prohibitive. Instead, the degree of the Chebyshev polynomial must only grow logarithmically with the hidden dimension.
+
+# 1.3 Related work
+
+Our manuscript is technically involved and incorporates linear dynamical systems, spectral filtering, complex Chebyshev and Legendre polynomials, Hankel and Toeplitz matrix eigendecay, Gaussian quadrature and other techniques. The related work is thus expansive, and due to space limitations we give a detailed treatment in Appendix A.
+
+Preconditioning in the context of time series analysis has roots in the classical work of Box and Jenkins [14]. In their foundational text they propose differencing as a method for making the time series stationary, and thus amenable to statistical learning techniques such as ARMA (auto-regressive moving average) [10]. The differencing operator can be applied numerous times, and for different lags, giving rise to the ARIMA family of forecasting models. Identifying the order of an ARIMA model, and in particular the types of differencing needed to make a series stationary, is a hard problem. This is a special case of the problem we consider: differencing corresponds to certain coefficients of preconditioning the time series, whereas we consider arbitrary coefficients.
+
+For a thorough introduction to modern control theory and exposition on open loop / closed loop predictors, learning via regression, and spectral filtering, see [28].
+
+The fundamental problem of learning in linear dynamical systems has been studied for many decades, and we highlight several key approaches below:
+
+1. System identification refers to the method of recovering  $\mathbf{A}$ ,  $\mathbf{B}$ ,  $\mathbf{C}$  from the data. This is a non-convex problem and while many methods have been considered in this setting, they depend polynomially on the hidden dimension.  
+2. The (auto) regression method predicts according to  $\hat{\mathbf{y}}_t = \sum_{i=1}^{h} \mathbf{M}_i \mathbf{u}_{t-i}$ . The coefficients  $\mathbf{M}_i$  can be learned using convex regression. The downside of this approach is that if the spectral radius of  $\mathbf{A}$  is  $1 - \delta$ , it can be seen that  $\sim \frac{1}{\delta}$  terms are needed.  
+3. The regression method can be further enhanced with "closed loop" components, that regress on prior observations  $\mathbf{y}_{t-1:1}$ . It can be shown using the Cayley-Hamilton theorem that using this method,  $d_h$  components are needed to learn the system, where  $d_h$  is the hidden dimension of  $\mathbf{A}$ .
+
+4. Filtering involves recovering the state  $x_{t}$  from observations. While Kalman filtering is optimal under specific noise conditions, it generally fails in the presence of marginal stability and adversarial noise.  
+5. Finally, spectral filtering combines the advantages of all methods above. It is an efficient method, its complexity does not depend on the hidden dimension, and works for marginally stable systems. However, spectral filtering requires  $\mathbf{A}$  to be symmetric, or diagonalizable under the real numbers.
+
+# 2 Main Results
+
+In this section we formally state our main algorithms and theorems. We show that the Universal Sequence Preconditioning method provides significantly improved regret bounds for learning linear dynamical systems than previously known when used in conjunction with two distinct methods. The first method is simple convex regression, and the second is spectral filtering.
+
+Both algorithms allow for learning in the case of marginally stable linear dynamical systems and allow for certain asymmetric transition matrices of arbitrary high hidden dimension. The regret bounds are free of the hidden dimension (up to logarithmic factors) – which significantly extends the state of the art.
+
+# 2.1 Universal Sequence Preconditioning Applied to Regression
+
+Algorithm 2 is an instantiation of Algorithm 4 for the method of convex regression. We set the preconditioning coefficients to be the coefficients of the  $n$ -th degree (monic) Chebyshev polynomial.
+
+# Algorithm 2 Universal Sequence Preconditioning for Regression
+
+1: Input: initial parameter  $\mathbf{Q}^0$ ; preconditioning coefficients  $c_{0:n}$  from the  $n$ -th degree (monic) Chebyshev polynomial; convex constraints  $\mathcal{K} = \{(\mathbf{Q}_0, \dots, \mathbf{Q}_{n-1})$  s.t.  $\| \mathbf{Q}_j \| \leq C_{\mathrm{domain}} \| c \|_1\}$  
+2: Assert that  $c_0 = 1$ .  
+3: for  $t = 1$  to  $T$  do  
+4: Receive  $\mathbf{u}_t$  
+5: Predict  $\hat{\mathbf{y}}_t(\mathbf{Q}^t) = -\sum_{i=1}^{n} c_i \mathbf{y}_{t-i} + \sum_{j=0}^{n} \mathbf{Q}_j^t \mathbf{u}_{t-j}$ .  
+6: Observe true output  $\mathbf{y}_t$  and suffer loss  $\ell_t(\mathbf{Q}^t) = \|\hat{\mathbf{y}}_t(\mathbf{Q}^t) - \mathbf{y}_t\|_1$ .  
+7: Update and project:  $\mathbf{Q}^{t + 1}\gets \mathrm{proj}_{\mathcal{K}}\left(\mathbf{Q}^t -\eta_t\nabla_{\mathbf{Q}}\ell_t(\mathbf{Q}^t)\right).$
+
+# 8: end for
+
+Theorem 2.1 shows that vanishing loss compared to the optimal ground-truth predictor, at a rate that is independent of the hidden dimension of the system.
+
+Theorem 2.1. Let  $\{\mathbf{u}_t\}_{t=1}^T \in \mathbb{C}^{d_{in}}$  be any sequence of inputs which satisfy  $\| \mathbf{u}_t \|_2 \leq 1$  and let  $\{\mathbf{y}_t\}_{t=1}^T \in \mathbb{C}^{d_{out}}$  be the corresponding output coming from some linear dynamical system  $(\mathbf{A}, \mathbf{B}, \mathbf{C})$  as defined per Eq. [1]. Let  $\mathbf{P}$  diagonalize  $\mathbf{A}$  (note  $\mathbf{P}$  exists w.l.o.g.) and let  $\kappa = \| \mathbf{P} \| \| \mathbf{P}^{-1} \|$ . Assume that  $\| \mathbf{B} \| \| \mathbf{C} \| \kappa \leq C_{domain}$ . Let  $\lambda_1, \ldots, \lambda_{d_h}$  denote the spectrum of  $\mathbf{A}$ . If
+
+$$
+\max  _ {j \in [ d _ {h} ]} | \arg (\lambda_ {j}) | \leq 1 / (3 2 \log_ {2} (2 T ^ {3} / d _ {\text {o u t}})) ^ {2}
+$$
+
+then the predictions  $\hat{\mathbf{y}}_1, \dots, \hat{\mathbf{y}}_T$  from Algorithm [2] where the preconditioning coefficients  $c_{0:n}$  are chosen to be the coefficients of the  $n$ -th monic Chebyshev polynomial satisfy
+
+$$
+\frac {1}{T} \sum_ {t = 1} ^ {T} \| \hat {\mathbf {y}} _ {t} - \mathbf {y} _ {t} \| _ {1} \leq \tilde {O} \left(\frac {\| \mathbf {B} \| \| \mathbf {C} \| \kappa \sqrt {d _ {o u t}}}{T ^ {2 / 1 3}}\right),
+$$
+
+where  $\tilde{O} (\cdot)$  hides polylogarithmic factors in  $T$
+
+The proof of Theorem 2.1 is in Appendix D. For a simple baseline comparison, the regret achieved (via the same proof technique) by the vanilla regression algorithm without preconditioning is  $O\left(C_{\text{domain}} \sqrt{d_{\text{out}}} T^{5/2}\right)$  which is not sublinear in  $T$ .
+
+# 2.2 Universal Sequence Preconditioning Applied to Spectral Filtering
+
+Our second main result is the application of Universal Sequence Preconditioning to the spectral filtering algorithm [29]. In addition to applying USP to spectral filtering, we also propose a novel spectral filtering basis. Both changes to the vanilla spectral filtering algorithm are necessary to extend its sublinear regret bounds to the case of underlying systems with asymmetric hidden transition matrices. First we define two key quantities. Given horizon  $T$  and  $\alpha \in \mathbb{C}$  let
+
+$$
+\mu_ {T} (\alpha) \stackrel {\text {d e f}} {=} \left[ \begin{array}{l l l l} 1 & \alpha & \dots & \alpha^ {T - 1} \end{array} \right] ^ {\top}, \tag {4}
+$$
+
+and
+
+$$
+\mathbf {Z} _ {T} \stackrel {\text {d e f}} {=} \int_ {\alpha \in \mathbb {C} _ {\beta}} \mu_ {T} (\alpha) \mu_ {T} (\bar {\alpha}) ^ {\top} d \alpha , \tag {5}
+$$
+
+where  $\overline{\alpha} \in \mathbb{C}$  denotes the complex conjugate. The novel spectral filters are the eigenvectors of  $\mathbf{Z}_{T - n - 1}$ , which we denote as  $\phi_1, \ldots, \phi_{T - n - 1}$ . Note that in the standard spectral filtering literature, the spectral filtering matrix is an integral over the real line and does not involve the complex conjugate. Our new matrix has an entirely different structure and although it looks quite similar, it surprisingly upends the proof techniques to ensure exponential spectral decay, a critical property for the method. We therefore introduce a novel proof of spectral decay in Theorem E.6 found in Section E.1.1
+
+Algorithm 3 Universal Sequence Preconditioning for Spectral Filtering  
+1: Input: initial  $\mathbf{Q}_{1:n}^{1},\mathbf{M}_{1:k}^{1}$  horizon  $T$  convex constraints  $\mathcal{K}$  =  $\{(\mathbf{Q}_0,\dots ,\mathbf{Q}_{n - 1},\mathbf{M}_1,\dots ,\mathbf{M}_k$  s.t.  $\| \mathbf{Q}_j\| \leq R_Q$  and  $\| \mathbf{M}_j\| \leq R_M\}$  parameter  $n$  coefficients  $c_{1:n}$    
+2: Let  $p_n^c (x) = c_0x^n +c_1x^{n - 1} + \dots +c_n$    
+3: Let  $\phi_1,\ldots ,\phi_n$  be the top  $n$  eigenvectors of  $\mathbf{Z}_{T - n - 1}$    
+4: Assert  $c_{0} = 1$    
+5: for  $t = 1$  to  $T$  do   
+6: Let  $\tilde{\mathbf{u}}_{t - n - 1:1}$  be  $\mathbf{u}_{t - n - 1:1}$  padded with zeros so it has dimension  $T - n - 1\times d_{\mathrm{in}}$    
+7: Predict  $\hat{\mathbf{y}}_t(\mathbf{Q}^t,\mathbf{M}^t) = -\sum_{i = 1}^n c_i\mathbf{y}_{t - i} + \sum_{j = 0}^n\mathbf{Q}_j^t\mathbf{u}_{t - j} + \frac{1}{\sqrt{T}}\sum_{j = 1}^k\mathbf{M}_j^t\boldsymbol {\phi}_j^\top \tilde{\mathbf{u}}_{t - n - 1:1}$    
+8: Observe true  $\mathbf{y}_t$  , define loss  $\ell_t(\hat{\mathbf{y}}_t) = \| \hat{\mathbf{y}}_t(\mathbf{Q}^t,\mathbf{M}^t) - \mathbf{y}_t\| _1$    
+9: Update and project:  $(\mathbf{Q}^{t + 1},\mathbf{M}^{t + 1}) = \mathrm{proj}_{\mathcal{K}}(\mathbf{Q}^t,\mathbf{M}^t) - \eta_t\nabla \ell_t(\mathbf{Q}^t,\mathbf{M}^t))$    
+10: end for
+
+Theorem 2.2. Let  $\{\mathbf{u}_t\}_{t=1}^T \in \mathcal{R}^{d_{in}}$  be any sequence of inputs which satisfy  $\|\mathbf{u}_t\|_2 \leq 1$  and let  $\{\mathbf{y}_t\}_{t=1}^T$  be the corresponding output coming from some linear dynamical system  $(\mathbf{A}, \mathbf{B}, \mathbf{C})$  as defined per Eq. Let  $\mathbf{P}$  diagonalize  $\mathbf{A}$  (note  $\mathbf{P}$  exists w.l.o.g.) and let  $\kappa = \|\mathbf{P}\| \|\mathbf{P}^{-1}\|$ . If  $R_Q, R_M$ , and  $k$  are chosen to be sufficiently large and
+
+$$
+\max _ {j \in [ d _ {h} ]} | \arg (\lambda_ {j}) | \leq 1 / (3 2 \log_ {2} (2 T ^ {3} / d _ {o u t})) ^ {2},
+$$
+
+then the predictions  $\hat{\mathbf{y}}_1,\dots ,\hat{\mathbf{y}}_T$  from Algorithm where the preconditioning coefficients  $\mathbf{c}_{0:n}$  are chosen to be the coefficients of the  $n$ -th monic Chebyshev polynomial satisfy
+
+$$
+\frac {1}{T} \sum_ {t = 1} ^ {T} \| \hat {\mathbf {y}} _ {t} - \mathbf {y} _ {t} \| _ {1} \leq \tilde {O} \left(\frac {\| \mathbf {C} \| \| \mathbf {B} \| \kappa \sqrt {d _ {o u t}}}{T ^ {3 / 1 3}}\right).
+$$
+
+For comparison, vanilla spectral filtering without Universal Sequence Preconditioning requires that  $\max_{j\in [d_h]}|\arg (\lambda_j)|\leq 1 / T$ , which is essentially the same as requiring that the matrix be symmetric since the horizon  $T$  can become arbitrarily large.
+
+# 3 Proof Overview
+
+In this section we give a high level overview of the proofs for Theorem 2.1 and Theorem 2.2. We start by recalling the intuition for Universal Sequence Preconditioning developed in Section 1.2 which
+
+shows that if  $\{\mathbf{y}_t\}_{t=1}^T$  evolves as a linear dynamical system parameterized by matrices  $(\mathbf{A}, \mathbf{B}, \mathbf{C})$  with inputs  $\{\mathbf{u}_t\}_{t=1}^T$  then by Equation 3,
+
+$$
+\mathbf {y} _ {t} = - \underbrace {\sum_ {i = 1} ^ {n} c _ {i} \mathbf {y} _ {t - i}} _ {\aleph_ {0}} + \underbrace {\sum_ {s = 0} ^ {n - 1} \sum_ {i = 0} ^ {s} c _ {i} \mathbf {C A} ^ {s - i} \mathbf {B u} _ {t - s}} _ {\aleph_ {1}} + \underbrace {\sum_ {s = 0} ^ {t - n - 1} \mathbf {C p} _ {n} ^ {c} (\mathbf {A}) \mathbf {A} ^ {s} \mathbf {B u} _ {t - n - s}} _ {\aleph_ {2}}.
+$$
+
+Recall that  $\aleph_0$  is the universal preconditioning component,  $\aleph_1$  is the term that can easily be learned by convex relaxation and regression, and  $\aleph_2$  is the critical term that contains  $p_n^c (\mathbf{A})$ .
+
+Both Theorem 2.1 and Theorem 2.2 use the standard result from online convex optimization (Theorem 3.1 from [26]) that online gradient descent over convex domain  $\kappa$  achieves regret  $\frac{3}{2}GD\sqrt{T}$  as compared to the best point in  $\kappa$ , where  $D$  denotes the diameter of  $\kappa$  and  $G$  denotes the maximum gradient norm.
+
+Regression: Proof of Theorem 2.1 In the case of regression, the domain is chosen so that  $\aleph_2$  may be learned and the proof proceeds by bounding the diameter of such a domain and its corresponding maximum gradient norm to get regret  $C n^2 \sqrt{d_{\mathrm{out}}} \| c \|_1 \sqrt{T}$  for a universal constant  $C > 0$  which depends on the norms of matrices  $\mathbf{B}$  and  $\mathbf{C}$  from the underlying system. Then  $\aleph_3$  is treated as an un-learnable error term. Let  $\lambda(\mathbf{A})$  denote the set of eigenvalues of  $\mathbf{A}$ . By the simple magnitude bound of
+
+$$
+\| \sum_ {s = 0} ^ {t - n - 1} \mathbf {C} p _ {n} (\mathbf {A}) \mathbf {A} ^ {s} \mathbf {B} \mathbf {u} _ {t - n - s} \| \leq \max  _ {\lambda \in \lambda (\mathbf {A})} | p _ {n} (\lambda) | \cdot T \cdot \| \mathbf {C} \| \cdot \| \mathbf {B} \|,
+$$
+
+the error of ignoring this term can be very small if  $\max_{\lambda (\mathbf{A})}|p_n(\lambda (\mathbf{A}))|$  is small. In the proof of Theorem 2.1 in Appendix we show that the regret for a generic polynomial  $p_n^c$  defined by coefficients  $c_{0:n}$  is
+
+$$
+\sum_ {t = 1} ^ {T} \| \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} \| _ {1} \leq \underbrace {C n ^ {2} \sqrt {d _ {\mathrm {o u t}}} \| c \| _ {1} \sqrt {T}} _ {\text {R e g r e t f r o m l e a n i n g} \aleph_ {2}} + \underbrace {C \max  _ {\lambda \in \mathcal {D}} | p _ {n} ^ {c} (\lambda) | T ^ {2}} _ {\text {U n l e a n a b l e E r r o r T e r m}},
+$$
+
+where  $\mathcal{D}$  is the region where  $\mathbf{A}$  is allowed to have eigenvalues (see Theorem [D.1]). Therefore, to get sublinear regret, we must choose a polynomial which has bounded  $\ell_{1}$  norm of its coefficients, while also exhibits very small infinity norm on the domain of  $\mathbf{A}$ 's eigenvalues.
+
+Spectral Filtering: Proof of Theorem 2.2 In the case of spectral filtering, the domain is chosen so that both  $\aleph_2$  and  $\aleph_3$  may be learned. Because spectral filtering learns  $\aleph_3$ , it is able to accumulate less error and hence achieves a better regret bound of  $O(T^{-3/13})$  as compared to regression's  $O(T^{-2/13})$ . At a high level, the proof proceeds by exploiting the fact that  $p_n(\mathbf{A})$  shrinks the size of the learnable domain. However this is not enough, in order to extend the result to systems where  $\mathbf{A}$  may have complex eigenvalues, the spectral filters must be eigenvalues of a new matrix, defined in Eq. 5 whose domain of integration includes the possibly complex eigenvalues of  $\mathbf{A}$ . To get the dimension-independent regret bounds enjoyed by spectral filtering in this new setting where complex eigenvalues may occur, the exponential decay of  $\mathbf{Z}_T$  from Eq. 5 must be established. This is nontrivial and requires several novel techniques inspired by [12]. The details are in Appendix E.1.1.
+
+Theorem 2.2 gives the main guarantee for the spectral filtering algorithm, which states that Algorithm 3 instantiated with some choice of polynomial  $p_n^c(\cdot)$  achieves regret
+
+$$
+\tilde {O} \left(\left(n \| c \| _ {1} + T ^ {7 / 6} \max  _ {\alpha \in \mathbb {C} _ {\beta}} | p _ {n} ^ {c} (\alpha) |\right) (n + k) \sqrt {d _ {\mathrm {o u t}}} \sqrt {T}\right).
+$$
+
+Both this theorem, as well as our new guarantee for convex regression, leads us to the following question: Is there a universal choice of polynomial  $p_n(x)$ , where  $n$  is independent of hidden dimension, which guarantees sublinear regret?
+
+# 3.1 Using the Chebyshev Polynomial over the Complex Plane
+
+For the real line, the answer to this question is known to be positive using the Chebyshev polynomials of the first kind. In general, the  $n^{\mathrm{th}}$  (monic) Chebyshev polynomial  $M_{n}(x)$  satisfies  $\max_{x\in [-1,1]}|M_n(x)|\leq 2^{-(n - 1)}$ . However, we are interested in a more general question over the complex plane. Since we care about linear dynamical systems that evolve according to a general asymmetric matrix, we need to extending our analysis to  $\mathbb{C}_{\beta}$ . This is a nontrivial extension since, in general, functions that are bounded on the real line can grow exponentially on the complex plane. Indeed,  $2^{n - 1}M_n(x) = \cos (n\arccos (x))$  and while  $\cos (x)$  is bounded within  $[-1,1]$  for any  $x\in \mathcal{R}$ , over the complex numbers we have  $\cos (z) = \frac{1}{2} (e^{iz} + e^{-iz})$ , which is unbounded. Thus, we analyze the Chebyshev polynomial on the complex plane and provide the following bound.
+
+Lemma 3.1. Let  $z \in \mathbb{C}$  be some complex number with magnitude  $|\alpha| \leq 1$ . Let  $M_{n}(\cdot)$  denote the  $n$ -th monic Chebyshev polynomial. If  $|\arg(z)| \leq 1/64n^{2}$ , then  $|M_{n}(z)| \leq 1/2^{n-2}$ .
+
+We provide the proof in Appendix F. We also must analyze the magnitude of the coefficients of the Chebyshev polynomial, which can grow exponentially with  $n$ . We provide the following result.
+
+Lemma 3.2. Let  $M_{n}(\cdot)$  have coefficients  $c_0,\ldots ,c_n$ . Then  $\max_{k = 0,\dots,n}|c_k|\leq 2^{0.3n}$
+
+The proof of Lemma 3.2 is in Appendix F. Together, these two lemmas are the fundamental building block for universal sequence preconditioning and for obtaining our new regret bounds.
+
+# 4 Experimental Evaluation
+
+We empirically validate that convolutional preconditioning with Chebyshev or Legendre coefficients yields significant online regret improvements across various learning algorithms and data types. Below we summarize our data generation, algorithm variants, hyperparameter tuning, and evaluation metrics.
+
+# 4.1 Synthetic Data Generation
+
+We generate  $N = 200$  sequences of length  $T = 2000$  via three mechanisms: (i) a noisy linear dynamical system, (ii) a noisy nonlinear dynamical system, and (iii) a noisy deep RNN. Inputs  $\mathbf{u}_{1:T} \sim \mathcal{N}(0,I)$ .
+
+Linear Dynamical System. Sample  $(\mathbf{A},\mathbf{B},\mathbf{C})$  with  $\mathbf{A}\in \mathcal{R}^{300\times 300}$  having eigenvalues  $\{z_j\}$  drawn uniformly in the complex plane subject to  $\mathrm{Im}(z_j)\leq \tau_{\mathrm{thresh}}$  and  $L\leq |z_j|\leq U$ , and  $\mathbf{B},\mathbf{C}\in \mathcal{R}^{300}$ . Then
+
+$$
+\mathbf {x} _ {t} = \mathbf {A} \mathbf {x} _ {t - 1} + \mathbf {B} \mathbf {u} _ {t}, \quad \mathbf {y} _ {t} = \mathbf {C} \mathbf {x} _ {t} + \epsilon_ {t}, \epsilon_ {t} \sim \mathcal {N} (0, \sigma^ {2} I).
+$$
+
+Nonlinear Dynamical System. Similarly sample  $(\mathbf{A}_1, \mathbf{B}_1, \mathbf{C})$  and  $(\mathbf{A}_2, \mathbf{B}_2)$  with  $\mathbf{A}_i \in \mathcal{R}^{10 \times 10}$ ,  $\mathbf{B}_i, \mathbf{C} \in \mathcal{R}^{10}$ . Then
+
+$$
+\mathbf {x} _ {t} ^ {(0)} = \mathbf {A} _ {1} \mathbf {x} _ {t - 1} + \mathbf {B} _ {1} \mathbf {u} _ {t}, \mathbf {x} _ {t} ^ {(1)} = \sigma (\mathbf {x} _ {t} ^ {(0)}), \mathbf {x} _ {t} = A _ {2} \mathbf {x} _ {t} ^ {(1)} + B _ {2} \mathbf {u} _ {t}, \mathbf {y} _ {t} = \mathbf {C} \mathbf {x} _ {t} + \epsilon_ {t}.
+$$
+
+Deep RNN. We randomly initialize a sparse 10-layer stack of LSTMs with hidden dimension 100 and ReLU nonlinear activations. Given  $\mathbf{u}_{1:T}$  we use this network to generate  $\mathbf{y}_{1:T}$ .
+
+# 4.2 Algorithms and Preconditioning Variants
+
+We evaluate the following methods: (1) Regression (Alg.2), (2) Spectral Filtering (Alg.3), (3) DNN Predictor:  $n$ -layer LSTM with dims  $[d_1, \ldots, d_n]$ , ReLU.
+
+Each method is applied with one of:
+
+1. Baseline: no preconditioning  
+2. Chebyshev:  $\mathbf{c}_{0:n}$  are the coefficients for the  $n$ th-Chebyshev polynomial. Note that when  $n = 2$  we have  $\mathbf{c}_0 = 1$  and  $\mathbf{c}_1 = -1$  and therefore this is the method of differencing discussed in the introduction.
+
+3. Legendre:  $\mathbf{c}_{0:n}$  are the coefficients for the  $n$ th-Legendre polynomial  
+4. Learned:  $\mathbf{c}_{0:n}$  is a parameter learned jointly with the model parameters
+
+We test polynomial degrees  $n \in \{2, 5, 10, 20\}$ . This choice of degrees shows a rough picture of the impact of  $n$ .
+
+Hyperparameter Tuning. To ensure fair comparison, for each algorithm and conditioning c variant we perform a grid search over learning rates  $\eta \in \{10^{-3}, 10^{-2}, 10^{-1}\}$ , selecting the one minimizing average regret across the  $N$  sequences. In the case of the learned coefficients, we sweep over the 9 pairs of learning rates ( $\eta_{\mathrm{model}}, \eta_{\mathrm{coefficients}}) \in \{10^{-3}, 10^{-2}, 10^{-1}\} \times \{10^{-3}, 10^{-2}, 10^{-1}\}$ .
+
+# 4.3 Results
+
+Tables 2-4 report the mean ± std of the absolute error over the final 200 predictions, averaged across 200 runs. In the linear and nonlinear cases we train a 2-layer DNN (dims (64, 128)); for RNN-generated data we match the 10-layer (100-dim) generator.
+
+# Key observations:
+
+- Preconditioning drastically reduces baseline errors for all algorithms and data types.  
+- Chebyshev and Legendre yield nearly identical gains.  
+- For Chebyshev and Legendre, once the degree is higher than  $5 - 10$  the performance degrades since  $\| \mathbf{c}\| _1$  gets very large (see our Lemma 3.2 which shows that these coefficients grow exponentially fast).  
+- Improvements decay as the complex threshold  $\tau_{\mathrm{thresh}}$  increases, consistent with our theoretical results which must bound  $\operatorname{Im}(z_j)$ .  
+- Learned coefficients excel with regression and spectral filtering but destabilize the DNN on nonlinear and RNN-generated data.
+
+<table><tr><td rowspan="2">Setting</td><td rowspan="2">Baseline</td><td colspan="3">Chebyshev</td><td colspan="3">Legendre</td><td colspan="4">Learned</td></tr><tr><td>Deg. 2</td><td>Deg. 5</td><td>Deg. 10</td><td>Deg. 2</td><td>Deg. 5</td><td>Deg. 10</td><td>Deg. 2</td><td>Deg. 5</td><td>Deg. 10</td><td>Deg. 20</td></tr><tr><td>Regression</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>\( \tau_{thresh} = 0.01 \)</td><td>\( \mathbf{0.74\pm 0.28} \)</td><td>\( 0.25\pm 0.09 \)</td><td>\( \mathbf{0.15}\pm \mathbf{0.07} \)</td><td>\( 0.77\pm 0.31 \)</td><td>\( 0.36\pm 0.13 \)</td><td>\( \mathbf{0.14}\pm \mathbf{0.06} \)</td><td>\( 0.64\pm 0.26 \)</td><td>\( 0.52\pm 0.19 \)</td><td>\( 0.27\pm 0.11 \)</td><td>\( \mathbf{0.17}\pm \mathbf{0.07} \)</td><td>\( 0.24\pm 0.09 \)</td></tr><tr><td>\( \tau_{thresh} = 0.1 \)</td><td>\( 1.92\pm 0.81 \)</td><td>\( 0.84\pm 0.27 \)</td><td>\( \mathbf{0.66}\pm \mathbf{0.18} \)</td><td>\( 1.90\pm 0.67 \)</td><td>\( 1.10\pm 0.40 \)</td><td>\( \mathbf{0.63}\pm \mathbf{0.17} \)</td><td>\( 1.66\pm 0.58 \)</td><td>\( 1.34\pm 0.43 \)</td><td>\( 0.57\pm 0.14 \)</td><td>\( \mathbf{0.55}\pm \mathbf{0.14} \)</td><td>\( 0.56\pm 0.14s \)</td></tr><tr><td>\( \tau_{thresh} = 0.9 \)</td><td>\( 2.47\pm 0.89 \)</td><td>\( 1.59\pm 0.56 \)</td><td>\( 2.18\pm 0.79 \)</td><td>\( 2.68\pm 0.48 \)</td><td>\( 1.64\pm 0.58 \)</td><td>\( 1.94\pm 0.70 \)</td><td>\( 2.63\pm 0.45 \)</td><td>\( 1.73\pm 0.59 \)</td><td>\( 0.83\pm 0.25 \)</td><td>\( 0.68\pm 0.27 \)</td><td>\( \mathbf{0.63}\pm \mathbf{0.26} \)</td></tr><tr><td>Spectral Filtering</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>\( \tau_{thresh} = 0.01 \)</td><td>\( 5.94\pm 3.37 \)</td><td>\( 1.72\pm 0.95 \)</td><td>\( \mathbf{0.69}\pm \mathbf{0.38} \)</td><td>\( 3.25\pm 1.79 \)</td><td>\( 2.78\pm 1.56 \)</td><td>\( \mathbf{0.66}\pm \mathbf{0.36} \)</td><td>\( 2.74\pm 1.51 \)</td><td>\( 1.99\pm 0.94 \)</td><td>\( \mathbf{0.54}\pm \mathbf{0.29} \)</td><td>\( 0.55\pm 0.25 \)</td><td>\( 0.61\pm 0.25 \)</td></tr><tr><td>\( \tau_{thresh} = 0.1 \)</td><td>\( \mathbf{0.89}\pm \mathbf{0.34} \)</td><td>\( 0.42\pm 0.11 \)</td><td>\( \mathbf{0.34}\pm \mathbf{0.07} \)</td><td>\( 0.86\pm 0.28 \)</td><td>\( 0.54\pm 0.17 \)</td><td>\( \mathbf{0.33}\pm \mathbf{0.07} \)</td><td>\( 0.76\pm 0.24 \)</td><td>\( 0.69\pm 0.28 \)</td><td>\( \mathbf{0.31}\pm \mathbf{0.06} \)</td><td>\( 0.37\pm 0.08 \)</td><td>\( 0.45\pm 0.28 \)</td></tr><tr><td>\( \tau_{thresh} = 0.9 \)</td><td>\( 10.17\pm 8.80 \)</td><td>\( 9.87\pm 8.90 \)</td><td>\( 12.66\pm 8.18 \)</td><td>\( 32.90\pm 22.02 \)</td><td>\( 9.42\pm 8.39 \)</td><td>\( 11.53\pm 7.46 \)</td><td>\( 28.83\pm 19.24 \)</td><td>\( 7.93\pm 4.42 \)</td><td>\( 6.31\pm 4.19 \)</td><td>\( 5.73\pm 3.80 \)</td><td>\( 5.86\pm 4.02 \)</td></tr><tr><td>2-layer DNN</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>\( \tau_{thresh} = 0.01 \)</td><td>\( 4.49\pm 2.02 \)</td><td>\( 2.31\pm 1.18 \)</td><td>\( 2.62\pm 1.52 \)</td><td>\( 10.36\pm 6.05 \)</td><td>\( 2.79\pm 1.34 \)</td><td>\( 2.35\pm 1.35 \)</td><td>\( 8.92\pm 5.20 \)</td><td>\( 2.89\pm 1.24 \)</td><td>\( 1.56\pm 0.64 \)</td><td>\( 0.79\pm 0.25 \)</td><td>\( \mathbf{0.40}\pm \mathbf{0.16} \)</td></tr><tr><td>\( \tau_{thresh} = 0.1 \)</td><td>\( 9.41\pm 7.34 \)</td><td>\( 2.66\pm 1.76 \)</td><td>\( 1.52\pm 0.72 \)</td><td>\( 4.65\pm 3.03 \)</td><td>\( 4.24\pm 3.06 \)</td><td>\( 1.44\pm 0.71 \)</td><td>\( 4.03\pm 2.64 \)</td><td>\( 6.54\pm 4.32 \)</td><td>\( 3.22\pm 1.93 \)</td><td>\( 1.59\pm 0.95 \)</td><td>\( \mathbf{0.80}\pm \mathbf{0.48} \)</td></tr><tr><td>\( \tau_{thresh} = 0.9 \)</td><td>\( 2.45\pm 1.31 \)</td><td>\( 2.24\pm 1.34 \)</td><td>\( 3.49\pm 2.27 \)</td><td>\( 11.48\pm 8.15 \)</td><td>\( 2.17\pm 1.25 \)</td><td>\( 3.10\pm 2.00 \)</td><td>\( 9.97\pm 7.05 \)</td><td>\( 1.29\pm 0.66 \)</td><td>\( 0.71\pm 0.34 \)</td><td>\( 0.43\pm 0.18 \)</td><td>\( \mathbf{0.24}\pm \mathbf{0.11} \)</td></tr></table>
+
+Table 2: Linear dynamical system data (detailed in Sec. 4.1) across varying complex threshold  ${\tau }_{\text{thres }}$  .  
+
+<table><tr><td rowspan="2">Setting</td><td rowspan="2">Baseline</td><td colspan="3">Chebyshev</td><td colspan="3">Legendre</td><td colspan="4">Learned</td></tr><tr><td>Deg. 2</td><td>Deg. 5</td><td>Deg. 10</td><td>Deg. 2</td><td>Deg. 5</td><td>Deg. 10</td><td>Deg. 2</td><td>Deg. 5</td><td>Deg. 10</td><td>Deg. 20</td></tr><tr><td colspan="12">Spectral Filtering</td></tr><tr><td>\( \tau_{thresh} = 0.01 \)</td><td>\( 153.8 \pm 15.7 \)</td><td>\( 43.7 \pm 19.4 \)</td><td>\( 0.92 \pm 0.31 \)</td><td>\( \mathbf{0.26} \pm \mathbf{0.15} \)</td><td>\( 78.4 \pm 34.7 \)</td><td>\( 2.82 \pm 1.17 \)</td><td>\( \mathbf{0.34} \pm \mathbf{0.19} \)</td><td>\( 1.04 \pm 0.29 \)</td><td>\( 0.07 \pm 0.03 \)</td><td>\( \mathbf{0.05} \pm \mathbf{0.03} \)</td><td>\( 0.07 \pm 0.03 \)</td></tr><tr><td>\( \tau_{thresh} = 0.01 \)</td><td>\( 124.1 \pm 68.4 \)</td><td>\( 33.4 \pm 17.3 \)</td><td>\( 2.02 \pm 1.23 \)</td><td>\( \mathbf{0.36} \pm \mathbf{0.31} \)</td><td>\( 56.7 \pm 30.5 \)</td><td>\( 3.81 \pm 1.67 \)</td><td>\( \mathbf{0.35} \pm \mathbf{0.26} \)</td><td>\( 2.85 \pm 1.13 \)</td><td>\( 0.04 \pm 0.01 \)</td><td>\( \mathbf{0.02} \pm \mathbf{0.01} \)</td><td>\( 0.07 \pm 0.02 \)</td></tr><tr><td>\( \tau_{thresh} = 0.9 \)</td><td>\( 165.5 \pm 84.5 \)</td><td>\( 43.41 \pm 16.39 \)</td><td>\( 1.27 \pm 0.37 \)</td><td>\( 1.90 \pm 0.80 \)</td><td>\( 76.9 \pm 29.1 \)</td><td>\( 3.32 \pm 1.09 \)</td><td>\( \mathbf{1.64} \pm \mathbf{0.69} \)</td><td>\( 1.79 \pm 0.61 \)</td><td>\( \mathbf{0.10} \pm \mathbf{0.04} \)</td><td>\( 0.12 \pm 0.05 \)</td><td>\( 0.17 \pm 0.07 \)</td></tr><tr><td colspan="12">2-layer DNN</td></tr><tr><td>\( \tau_{thresh} = 0.01 \)</td><td>\( 10.46 \pm 5.10 \)</td><td>\( 3.45 \pm 1.19 \)</td><td>\( \mathbf{0.19} \pm \mathbf{0.16} \)</td><td>\( 0.43 \pm 0.22 \)</td><td>\( 3.55 \pm 1.56 \)</td><td>\( \mathbf{0.18} \pm \mathbf{0.14} \)</td><td>\( 0.37 \pm 0.19 \)</td><td>\( 45.40 \pm 11.35 \)</td><td>\( 25.34 \pm 9.06 \)</td><td>\( 12.73 \pm 4.60 \)</td><td>\( \mathbf{6.43} \pm 2.32 \)</td></tr><tr><td>\( \tau_{thresh} = 0.1 \)</td><td>\( 4.20 \pm 1.10 \)</td><td>\( 3.38 \pm 0.79 \)</td><td>\( \mathbf{0.14} \pm \mathbf{0.05} \)</td><td>\( 0.41 \pm 0.16 \)</td><td>\( 3.42 \pm 0.79 \)</td><td>\( \mathbf{0.25} \pm \mathbf{0.14} \)</td><td>\( 0.36 \pm 0.13 \)</td><td>\( 67.70 \pm 28.40 \)</td><td>\( 31.99 \pm 11.78 \)</td><td>\( 15.95 \pm 5.87 \)</td><td>\( \mathbf{8.01} \pm 2.95 \)</td></tr><tr><td>\( \tau_{thresh} = 0.9 \)</td><td>\( 6.72 \pm 2.35 \)</td><td>\( 2.45 \pm 1.12 \)</td><td>\( \mathbf{0.08} \pm \mathbf{0.03} \)</td><td>\( 0.28 \pm 0.15 \)</td><td>\( 3.35 \pm 0.98 \)</td><td>\( \mathbf{0.20} \pm \mathbf{0.07} \)</td><td>\( 0.24 \pm 0.13 \)</td><td>\( 58.65 \pm 17.09 \)</td><td>\( 28.74 \pm 8.07 \)</td><td>\( 14.39 \pm 4.06 \)</td><td>\( \mathbf{7.26} \pm 2.05 \)</td></tr></table>
+
+Table 3: Nonlinear data (detailed in Sec. 4.1) across varying complex threshold  ${\tau }_{\text{thres }}$  .
+
+<table><tr><td rowspan="2">Setting</td><td rowspan="2">Baseline</td><td colspan="3">Chebyshev</td><td colspan="3">Legendre</td><td colspan="4">Learned</td></tr><tr><td>Deg. 2</td><td>Deg. 5</td><td>Deg. 10</td><td>Deg. 2</td><td>Deg. 5</td><td>Deg. 10</td><td>Deg. 2</td><td>Deg. 5</td><td>Deg. 10</td><td>Deg. 20</td></tr><tr><td>10-layer DNN</td><td>0.54 ± 0.23</td><td>0.29 ± 0.10</td><td>0.08± 0.03</td><td>0.13 ± 0.05</td><td>0.37 ± 0.14</td><td>0.09± 0.03</td><td>0.12± 0.04</td><td>1.49± 0.93</td><td>2.13± 1.05</td><td>1.04± 0.51</td><td>0.5 ± 0.24</td></tr></table>
+
+Table 4: Performance (average absolute error of the last 200 predictions) of a 10-layer DNN (detailed in Sec. 4.2) on data generated from the same model (detailed in Sec. 4.1).
+
+# 4.4 ETTh1 Dataset
+
+To evaluate whether our proposed preconditioning approach generalizes to real-world time series, we conduct experiments on the well-established ETTh1 dataset from the Electricity Transformer Temperature (ETT) benchmark [41]. The ETTh1 dataset consists of continuous hourly measurements of load and oil temperature collected from electricity transformers and has been used in several recent works [41, 39, 35, 24, 25, 40, 34].
+
+We study the effect of preconditioning on a 10-layer LSTM with hidden dimension 100 per layer using the Adam optimizer. We set the horizon to be  $T = 5000$  and we sweep over a broader range of learning rates  $\eta \in \{10^{-j}\}_{j=0,1,2,3,4,5}$ . As before we consider (i) no preconditioning (baseline), (ii) fixed Chebyshev coefficients, (iii) fixed Legendre coefficients, and (iv) coefficients learned jointly with model parameters.
+
+As seen in Figure II preconditioning with Chebyshev and Legendre for degree 5 the best performance after only the first 1000 iterations, while the performance of jointly learning the coefficients is worse at this stage. The performance of all three preconditioning methods are roughly on par with each other by 2500 iterations and by the full horizon  $T = 5000$ , jointly learning the coefficients results in the best average prediction error.
+
+![](images/0aa454864196524bcd80bca048bd6366ad89540775b59dcfb041d052f8c2b722.jpg)  
+(a) Early Stage:  $T = 1000$
+
+![](images/f028cc84acaa98f8d1be9bcf4c84b2ba5caee3fcb2d845724dab5202bfa1a6f1.jpg)  
+(b) Middle Stage:  $T = 2500$  
+Figure 1: Absolute prediction error on final 200 predictions averaged over 10 independent runs for 10-layer LSTM with layer dimension 100 using Adam optimizer and sweeping over learning rates for each run.
+
+![](images/d5d9028125fb3aa40a5568e1aa5d62754cda34cad804701da4070e81af0f8d23.jpg)  
+(c) Final Stage:  $T = 5000$
+
+# 5 Discussion
+
+There are many settings in machine learning where universal, rather than learned, rules have proven very efficient. For example, physical laws of motion can be learned directly from observation data. However, Newton's laws of motion succinctly crystallize very general phenomenon, and have proven very useful for large scale physics simulation engines. Similarly, in the theory of mathematical optimization, adaptive gradient methods have revolutionized deep learning. Their derivation as a consequence of regularization in online regret minimization is particularly simple [21], and thousands of research papers have not dramatically improved the initial basic ideas. These optimizers are, at the very least, a great way to initialize learned optimizers [38].
+
+By analogy, our thesis in this paper is that universal preconditioning based on the solid theory of dynamical systems can be applicable to many domains or, at the very least, an initialization for other learning methods.
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: We claim a rigorous new method for sequence preconditioning that improves upon the classical differencing method in theory and practice.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: Throughout we demonstrate understanding of the limitations of our approach, such as restricting our results to linear dynamical systems with bounded imaginary part of the transition matrix.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: Our main theoretical results are Theorems 2.1 and 2.2 in which the assumptions made are stated. We give only proof overview in main paper, but complete one in appendix.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: Yes, we describe our algorithms completely in 2 and 3. Our code repository will also be open sourced.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: A link to the code will be provided in the final version. However, we omit the GitHub repository with all of our code at the moment in order not to violate the double blind policy and keep the paper anonymous.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: Experiment details are expansively explained in section 4 and the appendix.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: all our graphs include error bars and statistical significance.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification:
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: The paper conforms simply in every respect with the NeurIPS Code of Ethics as explained.
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [NA]
+
+Justification: [NA]
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: [NA]
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: All previous work is fully credited and cited throughout.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [Yes]
+
+Justification: Our new assets are well documented and the documentation is provided.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: [NA]
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: [NA]
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: [NA]
+
+# References
+
+[1] Yasin Abbasi-Yadkori, Peter Bartlett, and Varun Kanade. Tracking adversarial targets. In International Conference on Machine Learning, pages 369-377. PMLR, 2014.  
+[2] Yasin Abbasi-Yadkori and Csaba Szepesvári. Regret bounds for the adaptive control of linear quadratic systems. In Proceedings of the 24th Annual Conference on Learning Theory, pages 1-26, 2011.  
+[3] Milton Abramowitz and Irene A Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55. US Government printing office, 1948.  
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+[6] Naman Agarwal, Elad Hazan, and Karan Singh. Logarithmic regret for online control. In Advances in Neural Information Processing Systems, pages 10175-10184, 2019.  
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+[8] Lars Valerian Ahlfors and Lars V Ahlfors. Complex analysis, volume 3. McGraw-Hill New York, 1979.  
+[9] Oren Anava, Elad Hazan, and Shie Mannor. Online learning for adversaries with memory: price of past mistakes. Advances in Neural Information Processing Systems, 28, 2015.  
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+[11] Ainesh Bakshi, Allen Liu, Ankur Moitra, and Morris Yau. A new approach to learning linear dynamical systems. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pages 335-348, 2023.  
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+[14] George EP Box and Gwilym M Jenkins. Time series analysis. forecasting and control. Holden-Day Series in Time Series Analysis, 1976.  
+[15] Nicolo Cesa-Bianchi and Gábor Lugosi. Prediction, learning, and games. Cambridge university press, 2006.  
+[16] Xinyi Chen and Elad Hazan. Black-box control for linear dynamical systems. In Conference on Learning Theory, pages 1114-1143. PMLR, 2021.  
+[17] Alon Cohen, Avinatan Hasidim, Tomer Koren, Nevena Lazic, Yishay Mansour, and Kunal Talwar. Online linear quadratic control. In International Conference on Machine Learning, pages 1029–1038, 2018.  
+[18] Alon Cohen, Tomer Koren, and Yishay Mansour. Learning linear-quadratic regulators efficiently with only  $\sqrt{T}$  regret. In International Conference on Machine Learning, pages 1300-1309, 2019.  
+[19] Sarah Dean, Horia Mania, Nikolai Matni, Benjamin Recht, and Stephen Tu. Regret bounds for robust adaptive control of the linear quadratic regulator. In Advances in Neural Information Processing Systems, pages 4188-4197, 2018.
+
+[20] E. B. Dimitrov and N. Nikolov. Sharp bounds for the extreme zeros of classical orthogonal polynomials. Journal of Approximation Theory, 162(10):1793-1804, 2010.  
+[21] John Duchi, Elad Hazan, and Yoram Singer. Adaptive subgradient methods for online learning and stochastic optimization. Journal of machine learning research, 12(7), 2011.  
+[22] Maryam Fazel, Rong Ge, Sham Kakade, and Mehran Mesbahi. Global convergence of policy gradient methods for the linear quadratic regulator. In International Conference on Machine Learning, pages 1467-1476, 2018.  
+[23] Udaya Ghai, Holden Lee, Karan Singh, Cyril Zhang, and Yi Zhang. No-regret prediction in marginally stable systems. In Conference on Learning Theory, pages 1714-1757. PMLR, 2020.  
+[24] Albert Gu, Karan Goel, and Christopher Re. Efficiently modeling long sequences with structured state spaces. In International Conference on Learning Representations, 2022.  
+[25] Rohit Gupta and et al. Simplifying long-range modeling with structured state space models. arXiv preprint arXiv:2208.04933, 2022.  
+[26] Elad Hazan. Introduction to online convex optimization. Foundations and Trends® in Optimization, 2(3-4):157-325, 2016.  
+[27] Elad Hazan, Holden Lee, Karan Singh, Cyril Zhang, and Yi Zhang. Spectral filtering for general linear dynamical systems. In Advances in Neural Information Processing Systems, pages 1680-1689, 2018.  
+[28] Elad Hazan and Karan Singh. Introduction to online nonstochastic control. arXiv preprint arXiv:2211.09619, 2022.  
+[29] Elad Hazan, Karan Singh, and Cyril Zhang. Learning linear dynamical systems via spectral filtering. In Advances in Neural Information Processing Systems, pages 6702-6712, 2017.  
+[30] Rudolph Emil Kalman. A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 82.1:35-45, 1960.  
+[31] CJ Knight and ACR Newbery. Trigonometric and gaussian quadrature. MATHEMATICS of computation, pages 575-581, 1970.  
+[32] Mark Kozdoba, Jakub Marecek, Tigran Tchrakian, and Shie Mannor. On-line learning of linear dynamical systems: Exponential forgetting in kalman filters. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 4098-4105, 2019.  
+[33] Horia Mania, Stephen Tu, and Benjamin Recht. Certainty equivalence is efficient for linear quadratic control. In Advances in Neural Information Processing Systems, pages 10154-10164, 2019.  
+[34] An Nguyen, Yifan Chen, and Jian Li. Preconditioning state-space models for stable long-horizon forecasting. arXiv preprint arXiv:2403.07890, 2024.  
+[35] Yixuan Nie, Nina Narodytska, and Ankit Patel. A time series is worth 64 words: Long-term forecasting with transformers. arXiv preprint arXiv:2209.13466, 2022.  
+[36] Max Simchowitz, Karan Singh, and Elad Hazan. Improper learning for non-stochastic control. In Conference on Learning Theory, pages 3320-3436. PMLR, 2020.  
+[37] Josef Stoer, Roland Bulirsch, R Bartels, Walter Gautschi, and Christoph Witzgall. Introduction to numerical analysis, volume 1993. Springer, 1980.  
+[38] Olga Wichrowska, Niru Maheswaranathan, Matthew W Hoffman, Sergio Gomez Colmenarejo, Misha Denil, Nando Freitas, and Jascha Sohl-Dickstein. Learned optimizers that scale and generalize. In International conference on machine learning, pages 3751-3760. PMLR, 2017.  
+[39] Haixu Wu, Jianmin Xu, Jie Wang, and Mingsheng Long. Autoformer: Decomposition transformers with auto-correlation for long-term series forecasting. In Advances in Neural Information Processing Systems, 2021.
+
+[40] Ailing Zeng, Muxi Chen, Lei Zhang, and Qiang Xu. Are transformers effective for time series forecasting? In AAAI Conference on Artificial Intelligence, 2023.  
+[41] Haoyi Zhou, Shanghang Zhang, Jieqi Peng, Shuai Zhang, Jianxin Li, Hongyun Xiong, Wancai Zhang, and Yan Li. Informer: Beyond efficient transformer for long sequence time-series forecasting. In Proceedings of the AAAI Conference on Artificial Intelligence, 2021.  
+[42] Kemin Zhou, John C. Doyle, and Keith Glover. Robust and Optimal Control. Prentice-Hall, Inc., USA, 1996.
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+# Universal Video Temporal Grounding with Generative Multi-modal Large Language Models
+
+Zeqian Li $^{1}$ , Shangzhe Di $^{1,2}$ , Zhonghua Zhai $^{2}$ , Weilin Huang $^{2}$ , Yanfeng Wang $^{1}$ , Weidi Xie $^{1}$
+
+$^{1}$ SAI, Shanghai Jiao Tong University  $^{2}$ ByteDance Seed
+
+https://lzq5.github.io/UniTime
+
+# Abstract
+
+This paper presents a computational model for universal video temporal grounding, which accurately localizes temporal moments in videos based on natural language queries (e.g., questions or descriptions). Unlike existing methods that are often limited to specific video domains or durations, we propose UniTime, a robust and universal video grounding model leveraging the strong vision-language understanding capabilities of generative Multi-modal Large Language Models (MLLMs). Our model effectively handles videos of diverse views, genres, and lengths while comprehending complex language queries. The key contributions include: (i) We consider steering strong MLLMs for temporal grounding in videos. To enable precise timestamp outputs, we incorporate temporal information by interleaving timestamp tokens with video tokens. (ii) By training the model to handle videos with different input granularities through adaptive frame scaling, our approach achieves robust temporal grounding for both short and long videos. (iii) Comprehensive experiments show that UniTime outperforms state-of-the-art approaches in both zero-shot and dataset-specific finetuned settings across five public temporal grounding benchmarks. (iv) When employed as a preliminary moment retriever for long-form video question-answering (VideoQA), UniTime significantly improves VideoQA accuracy, highlighting its value for complex video understanding tasks.
+
+# 1 Introduction
+
+Understanding long videos presents fundamental challenges due to complex temporal dependencies and varying information granularity. Recent approaches [30, 42] typically adopt a two-stage framework: first localizing relevant temporal segments, then reasoning over the retrieved content. This retrieval-then-reasoning paradigm offers a structured and interpretable framework for tackling intricate video understanding tasks.
+
+However, existing grounding models perform unsatisfactorily, often struggling to generalize across diverse video domains [50]. In this context, we advocate for developing a universal temporal grounding model, that is capable of accommodating videos across a wide range of viewpoints (egocentric and exocentric perspectives), topics (from cooking and daily activities to sports, movies, and games), and durations (from brief clips to multi-hour videos). Such universality is crucial for real-world deployment, where video data is inherently heterogeneous and user queries are highly diverse.
+
+Research on temporal grounding falls into two paradigms. The first includes discriminative or dual-encoder-based approaches [4, 24], which typically rely on dataset-specific fine-tuning and lightweight architectures, such as DETR-like frameworks. While computationally efficient, these models often struggle to generalize to unseen content due to limited language encoding capabilities, hindering the understanding of complex or user-driven queries, and constrained visual representations. In
+
+![](images/8f10681187be3c88cb2991d179643ec2dd4d9a2fbea82287ac33a61a129952e1.jpg)  
+Figure 1: The UniTime framework empowers MLLMs with advanced universal temporal grounding capabilities. (a) UniTime can handle diverse videos with various views, genres, and durations, as well as comprehend complex language queries. (b) UniTime achieves universal temporal grounding through a coarse-to-fine approach. (c) Performance comparison on temporal grounding and video question answering benchmarks demonstrates the superior capabilities of UniTime.
+
+![](images/5fa04b9a1116bf98a1ccdef67422671a817b75636efa76ec2996a26d682ba02f.jpg)
+
+contrast, recent work leverages Multi-modal Large Language Models (MLLMs) [34, 62], which excel at language understanding and demonstrate promising results on video temporal grounding. However, their applicability to long videos is hindered by short context windows and high memory demands, making them impractical for processing video inputs spanning tens of thousands of tokens.
+
+In this paper, we further explore the potential of MLLMs towards universal temporal grounding in videos. Our proposed model, termed as UniTime, encodes temporal information by interleaving timestamp tokens with video tokens, enabling precise boundary generation for natural language queries. To handle videos of varying lengths, we introduce an adaptive frame scaling mechanism that adjusts temporal and spatial granularity based on video duration. Specifically, for long videos, the model can be adopted for multi-stage inference-time computation: initial coarse-grained sampling enables segment-level grounding, followed by fine-grained sampling within candidate regions to refine boundaries. This iterative process progressively improves grounding precision. To improve training efficiency, we adopt a video-centric training paradigm that concatenates multiple queries and their corresponding segments per video, allowing the model to process them in a single pass. This minimizes the computational overhead of autoregressive decoding over repeated video tokens.
+
+To rigorously evaluate efficacy, we pre-train UniTime on diverse, high-quality temporally grounded video datasets and assess its performance on two key tasks: temporal grounding (in both zero-shot and fine-tuning settings) and downstream video question answering (VideoQA) for long videos. For temporal grounding, we conduct experiments on two long-video and three short-video benchmarks, where our model consistently outperforms existing methods in both straightforward and complex retrieval scenarios. For downstream VideoQA, we evaluate on four widely used long-video benchmarks using a retrieval-augmented protocol: first, our framework localizes the relevant video segment; then, an off-the-shelf VideoQA model answers the query based on the localized segment. Compared to prior approaches, our method demonstrates superior generalization, delivering more accurate and comprehensive video understanding, especially for long videos.
+
+The remainder of the paper is organized as follows: Section 2 formulates the temporal grounding task and details our proposed method. Section 3 presents ablation studies and comparisons to validate our approach. Section 4 provides an overview of the relevant literature. Overall, our method achieves improved performance on both temporal grounding and long-video VideoQA benchmarks.
+
+# 2 Methods
+
+In this section, we introduce UniTime, a novel framework for universal video temporal grounding, adapted from pre-trained Multi-modal Large Language Models (MLLMs). Our approach achieves precise temporal grounding across videos of diverse scenes, genres, and durations, and employs an iterative coarse-to-fine strategy to handle long videos. In the following, Section 2.1 formalizes the
+
+problem setting. Section 2.2 presents our architecture and its core modules, which support flexible temporal grounding at arbitrary granularities via inference-time computation. Section 2.3 details an efficient training paradigm that equips MLLMs with universal temporal grounding capabilities.
+
+# 2.1 Problem Formulation
+
+Given an untrimmed video sequence  $\mathcal{V} = \{f_1, \ldots, f_{N_f}\} \in \mathbb{R}^{N_f \times H \times W \times 3}$  sampled at timestamps  $\mathcal{T} = \{t_1, \ldots, t_{N_f}\}$  and a free-form text query  $\mathcal{Q}$  (either a complex question or a plain description), our task is to identify all temporal moments  $\mathcal{Y} = \{(s_1, e_1), \ldots, (s_K, e_K)\}$  that semantically match the query, where each  $s_k, e_k \in \mathcal{T}$  denotes a start and end timestamp, respectively:
+
+$$
+\mathcal {Y} = \Phi_ {\text {U n i T i m e}} (\mathcal {V}, \mathcal {T}, \mathcal {Q}).
+$$
+
+With this formulation, temporal grounding is achieved by identifying the frames most relevant to the query and retrieving their timestamps. By adjusting the granularity of  $\mathcal{T}$ , our design supports multi-scale temporal predictions, enabling efficient coarse-to-fine grounding for long videos through inference-time computation, as shown in Figure 1(b).
+
+# 2.2 Universal Temporal Grounding Model
+
+The core innovation of UniTime lies in its capability to perform predictions at multiple temporal granularities for videos of diverse lengths. The model consists of three key components: (i) adaptive frame scaling for constructing multi-granular video inputs, (ii) timestamp-interleaved sequence construction enabling multi-scale temporal window prediction, and (iii) multi-scale prediction for coarse-to-fine temporal grounding during inference, as illustrated in Figure 2.
+
+Adaptive Frame Scaling. A central challenge in universal temporal grounding is handling videos with highly variable durations, ranging from a few seconds to several hours. Applying fixed temporal and spatial resolutions across such diverse inputs is inefficient and often impractical, especially for MLLMs constrained by limited context windows and GPU memory.
+
+To address this, we propose an adaptive frame scaling mechanism that dynamically adjusts per-frame token budget  $(N_{\mathrm{res}})$  based on video duration, allocating high spatial resolution to short videos and low resolution to long videos. Specifically, given a video  $\mathcal{V}$  with constant frame rate  $(N_f$  frames in total), each frame is allocated  $N_{\mathrm{res}} = \lfloor N_{\mathrm{total}} / N_f\rfloor$  tokens, where  $N_{\mathrm{total}}$  is the total token budget. To support varying spatial granularities, we set two thresholds,  $N_f^{\mathrm{short}}$  and  $N_f^{\mathrm{long}}$ , as the maximum number of frames for short and long videos, respectively. Frames are then processed as:
+
+$$
+\mathbf {V} _ {i} = \left\{ \begin{array}{l l} \phi_ {\text {p r o j e c t}} \big (\phi_ {\text {v i s i o n}} (\psi_ {\text {r e s i z e}} (f _ {i})) \big) \in \mathbb {R} ^ {N _ {\text {r e s}} \times d}, & N _ {f} <   N _ {f} ^ {\text {s h o r t}}. \\ \psi_ {\text {c o m p r e s s}} (\phi_ {\text {p r o j e c t o r}} (\phi_ {\text {v i s i o n}} (f _ {i}))) \in \mathbb {R} ^ {N _ {\text {r e s}} \times d}, & N _ {f} ^ {\text {s h o r t}} \leq N _ {f} <   N _ {f} ^ {\text {l o n g}}. \end{array} \right.
+$$
+
+where  $\phi_{\mathrm{vision}}(\cdot)$  is the vision encoder, which supports dynamic input resolutions with a fixed patch size of  $P$ , and  $\phi_{\mathrm{project}}(\cdot)$  is the projection module.  $\psi_{\mathrm{size}}(\cdot)$  denotes the frame resizing function, which adjusts the spatial resolution of each frame so that it can be partitioned into  $N_{\mathrm{res}}$  patches of size  $P\times P$  pixels.  $\psi_{\mathrm{compress}}(\cdot)$  denotes token compression via bilinear interpolation, yielding  $N_{\mathrm{res}}$  tokens as the final representation. Unlike frame resizing, which may degrade spatial details, token compression reduces redundancy at the token level while better preserving key semantic information, especially for long videos with low resolutions per frame. When the number of frames exceeds  $N_f^{\mathrm{long}}$ , the video is divided into multiple clips of length  $N_f^{\mathrm{long}}$  for divide-and-conquer processing. The ablation experiments of token budget, thresholds and the input processing strategy can be found in Appendix E.3 and E.1.
+
+Discussion. Existing MLLMs typically employ fixed frame sizes and token allocations [33, 68, 25], which limits their ability to encode long videos effectively. To handle such inputs, these models must resort to sparse temporal sampling, resulting in significant loss of detailed visual content. In contrast, our adaptive strategy dynamically employs higher spatial resolutions for short videos and lower ones for longer videos. This allows for efficient and scalable modeling across diverse video lengths, maximizing the utility of the available token budget and model capability.
+
+Timestamp-Interleaved Sequence Construction. To associate frames with their timestamps, we encode temporal information by interleaving timestamp tokens between visual tokens. For a frame  $f_{i}$
+
+![](images/ed54d7b42b685d636b8c6a9f06194f288dc98206d31c97382c42c923019c9d3f.jpg)  
+(a) Coarse-to-Fine Temporal Grounding with Adaptive Frame Scaling
+
+![](images/d64e5c0b056fdeca6c94e3acaf638666d1a501a4d1fbb616fe7d32cadebde2e7.jpg)  
+(b) Timestamp-Interleaved Sequence  
+Figure 2: Overview of the proposed UniTime framework. (a) UniTime achieves universal temporal grounding by leveraging adaptive frame scaling to construct multi-scale video inputs and then generate multi-scale predictions, allowing robust grounding across diverse video durations. (b) Within the model architecture, UniTime constructs an interleaved sequence of timestamps and scaled frame features, which, combined with the language query, is fed into the LLM and then identifies the corresponding temporal interval from the timestamp tokens.
+
+sampled at timestamp  $t_i$ , the timestamp is represented as free text  $\tau_i =$  "timestamp:  $t_i$  seconds", and inserted before visual tokens of  $f_i$ , forming an interleaved sequence:
+
+$$
+S = \left[ \mathbf {T} _ {1}; \mathbf {V} _ {1}; \mathbf {T} _ {2}; \mathbf {V} _ {2}; \dots ; \mathbf {T} _ {N _ {f}}; \mathbf {V} _ {N _ {f}}; \mathcal {Q} \right], \quad \mathbf {T} _ {i} = \phi_ {\text {t o k e n i z e r}} (\tau_ {i}) \tag {1}
+$$
+
+where  $\phi_{\mathrm{tokenizer}}(\cdot)$  is the tokenizer. This interleaved sequence is input to the large language models (LLMs), which generates temporal boundaries for a given query  $Q$ , with the output formatted as  $\mathcal{A} = \text{"From } s_k \text{ seconds to } e_k \text{ seconds}"$ . Note that, unlike conventional methods that predict the exact moment in the original video, our model instead predicts the minimal timestamp interval  $(s_k, e_k)$  from the sampled timestamps set  $\mathcal{T}$ . That is to say, we leverage the retrieval capabilities of MLLMs to read out the inserted timestamp tokens, rather than decoding dense position encodings. For a comparison of these temporal information encoding approaches, refer to the Appendix E.2.
+
+It is worth noting that direct fine-grained temporal interval prediction on coarse-grained long video inputs is inherently unreliable, as the lack of detailed spatial information often results in ambiguous or inaccurate predictions. This highlights the necessity of adopting predictions at different scales for videos with varying granularities. Our framework addresses this challenge by offering inherent flexibility in timestamp token insertion, making it adaptable to varying granularities. Specifically, timestamp tokens can be interleaved at different scales: before each frame for fine-grained temporal predictions, or before fixed-length segments for coarse-grained temporal reasoning.
+
+We divide the input video into  $N_{s}$  segments  $\{\mathbf{S}_j\}_{j = 1}^{N_s}$ , each comprising  $\mathbf{L}_s$  frames. Here, for the  $j^{th}$  segment,  $\mathbf{S}_j = [\mathbf{V}_{s_j};\mathbf{V}_{s_j + 1};\dots ;\mathbf{V}_{s_j + \mathbf{L}_s - 1}]\in \mathbb{R}^{\mathbf{L}_s\times N_{\mathrm{res}}\times d}$ , where  $s_j$  is the start index of the segment and we use the corresponding timestamp  $t_{s_j}$  at the beginning of each segment to represent its temporal location. Note that we prepend only a single timestamp text token to each segment, thereby providing coarse-grained temporal supervision. We also feed the interleaved timestamps, visual tokens, and the query into the LLM to retrieve the target segment.
+
+$$
+\mathcal {S} = \left[ \mathbf {T} _ {1}; \mathbf {S} _ {1}; \mathbf {T} _ {2}; \mathbf {S} _ {2}; \dots ; \mathbf {T} _ {N _ {s}}; \mathbf {S} _ {N _ {s}}; \mathcal {Q} \right], \quad \mathbf {T} _ {i} = \phi_ {\text {t o k e n i z e r}} \left(\tau_ {s _ {j}}\right) \tag {2}
+$$
+
+Discussion. Here, explicitly providing timestamps offers several crucial advantages: (i) Improved temporal grounding. These explicit timestamp tokens help the LLM effectively associate visual content with its corresponding temporal position in the video, allowing it to generalize to temporal grounding tasks. (ii) Alignment-free integration. As the inserted timestamps are textual, they naturally reside in the language space and do not require realignment. In contrast, methods based on learned vectors necessitate additional embedding alignment, thereby complicating the LLM's understanding of temporal information. (iii) Generalisable temporal reasoning. When combined with adaptive frame scaling, explicitly provided timestamp tokens allow the model to directly refer to the temporal cues, potentially enabling extrapolation. For example, as demonstrated in Section 3.7, this design allows the model to be trained on short or medium-length videos (e.g., around 6 minutes), while still generalizing effectively to much longer videos (e.g., 20 minutes) during inference on downstream tasks. (iv) Model-agnostic design. This method is plug-and-play and can be easily incorporated into diverse model architectures with minimal modifications, which is validated in
+
+Table 1: Statistics of temporal grounding datasets. The datasets in Part I are used exclusively for pre-training, while those in Part II are the training sets of benchmarks.  
+
+<table><tr><td></td><td>Dataset</td><td>#Videos</td><td>#Queries</td><td>Video Len.</td><td>Moment Len.</td><td>Views</td><td>Domain</td></tr><tr><td rowspan="7">I</td><td>NaQ [46]</td><td>4.9K</td><td>1031K</td><td>413s</td><td>1.1s</td><td>Ego</td><td>Open</td></tr><tr><td>DiDeMo [1]</td><td>8.4K</td><td>33.0K</td><td>29s</td><td>7.5s</td><td>Exo</td><td>Open</td></tr><tr><td>QuerYD [38]</td><td>819</td><td>5.7K</td><td>278s</td><td>13.6s</td><td>Ego &amp; Exo</td><td>Open</td></tr><tr><td>HiRest [61]</td><td>526</td><td>4.0K</td><td>263s</td><td>18.9s</td><td>Ego &amp; Exo</td><td>Open</td></tr><tr><td>COIN [53]</td><td>11.8K</td><td>46.4K</td><td>145s</td><td>14.9s</td><td>Ego &amp; Exo</td><td>Open</td></tr><tr><td>Momentor [45]</td><td>6.7K</td><td>136.4K</td><td>403s</td><td>49.5s</td><td>Ego &amp; Exo</td><td>Open</td></tr><tr><td>YouCook2 [72]</td><td>1.2K</td><td>9.6K</td><td>316s</td><td>19.7s</td><td>Ego &amp; Exo</td><td>Cooking</td></tr><tr><td rowspan="5">II</td><td>Ego4D-NLQ [11]</td><td>1.0K</td><td>10.0K</td><td>495s</td><td>8.2s</td><td>Ego</td><td>Open</td></tr><tr><td>TACoS [47]</td><td>75</td><td>9.8K</td><td>287s</td><td>27.9s</td><td>Ego &amp; Exo</td><td>Cooking</td></tr><tr><td>Charades-STA [51]</td><td>4.8K</td><td>11.2K</td><td>31s</td><td>8.1s</td><td>Exo</td><td>Activity</td></tr><tr><td>QVHighlights [24]</td><td>7.1K</td><td>7.2K</td><td>150s</td><td>24.6s</td><td>Ego &amp; Exo</td><td>Vlog/News</td></tr><tr><td>ANet-Captions [22]</td><td>10K</td><td>37.4K</td><td>118s</td><td>36.0s</td><td>Ego &amp; Exo</td><td>Activity</td></tr></table>
+
+Section 3.4, eliminating the need for architecture-specific adjustments such as custom positional encoding schemes [12, 2].
+
+Universal Temporal Grounding with Multi-stage Inference. Building on these designs, our framework achieves universal temporal grounding by processing input frames at different spatial resolutions and generating temporal predictions at appropriate granularities, according to video duration. For long videos, we employ a hierarchical, multi-stage inference strategy. Specifically, if a video exceeds  $N_{f}^{\mathrm{long}}$  frames, it is partitioned into multiple clips, each of length  $N_{f}^{\mathrm{long}}$ . Segment retrieval is first performed within each clip to identify potentially relevant segments. The retrieved segments from all clips are then aggregated and subjected to further segment retrieval, and this process can be repeated recursively if necessary. Finally, fine-grained grounding is conducted within the final selected segments. As evidenced by Section 3.5, this flexible and hierarchical inference process ensures accurate and efficient temporal grounding across diverse video durations.
+
+# 2.3 Training
+
+In this section, we first describe the training objective for temporal grounding, followed by a detailed explanation of the hierarchical data construction for multi-granularity training. We then present a video-centric training mechanism to improve training efficiency.
+
+Training Loss. The model is trained with an auto-regressive objective over mini-batches. For each training sample  $(\mathcal{V},\mathcal{T},\mathcal{Q},\mathcal{Y})$ , i.e., the video, timestamps, text query, and target, the prompt sequence  $S$  is constructed based on Equation 1 or 2 for short and long videos, respectively. Then, the model minimizes the negative log-likelihood over the target tokens only:
+
+$$
+\mathcal {L} (\mathcal {S}, \mathcal {Y}) = - \sum_ {i = 1} ^ {N _ {y}} \log P (y _ {t} \mid \mathcal {S}, y _ {<   i}; \theta)
+$$
+
+where  $N_{y}$  is the target length.
+
+Training Data Construction. To support temporal grounding at multiple granularities, we construct training data from videos of varying durations, using full videos for coarse-grained supervision and randomly sampling shorter segments containing ground-truth moments for fine-grained training. All samples are combined for joint training. Since this process naturally yields more short videos than long ones, we balance the data distribution by replicating long video samples with a factor  $N_{\mathrm{rep}} > 1$ .
+
+Video-centric Training. Given a dataset comprising paired videos, text queries, and corresponding moments  $(\mathcal{V}^{(k)},\mathcal{Q}^{(k)},\mathcal{M}^{(k)})$ , previous methods typically employ query-centric sampling: first sampling a query, then loading its associated video. This approach is highly inefficient when multiple queries correspond to the same video, as it introduces (i) an I/O bottleneck from repeatedly loading long videos and (ii) redundant computation due to re-encoding identical video content.
+
+To address this, we employ video-centric sampling, wherein each video  $\mathcal{V}^{(k)}$  is sampled first, and all associated query-answer pairs are grouped into a single input sequence. This strategy minimizes
+
+Table 2: Statistics of benchmarks. The benchmarks encompass three representative tasks - Video Temporal Grounding, Grounded VideoQA, and General VideoQA. The reported statistics are based on the actual numbers of videos and queries used in our experiments.  
+
+<table><tr><td>Benchmark</td><td>#Videos</td><td>#Queries</td><td>Video Len.</td><td>Moment Len.</td><td>Views</td><td>Domain</td></tr><tr><td>Ego4D-NLQ [11]</td><td>415</td><td>4.6K</td><td>500s</td><td>10.7s</td><td>Ego</td><td>Open</td></tr><tr><td>TACoS [47]</td><td>25</td><td>4.0K</td><td>368s</td><td>31.9s</td><td>Ego &amp; Exo</td><td>Cooking</td></tr><tr><td>Charades-STA [51]</td><td>1.3K</td><td>3.7K</td><td>29s</td><td>7.8s</td><td>Exo</td><td>Activity</td></tr><tr><td>QVHighlights [24]</td><td>1.5K</td><td>1.5K</td><td>150s</td><td>32.2s</td><td>Ego &amp; Exo</td><td>Vlog/News</td></tr><tr><td>ANet-Captions [22]</td><td>4.9K</td><td>17.0K</td><td>118s</td><td>40.2s</td><td>Ego &amp; Exo</td><td>Activity</td></tr><tr><td>QaEgo4D [3]</td><td>148</td><td>500</td><td>487s</td><td>13s</td><td>Ego</td><td>Open</td></tr><tr><td>CG-Bench [5]</td><td>1.2K</td><td>12K</td><td>1661.1s</td><td>19.8s</td><td>Ego &amp; Exo</td><td>Open</td></tr><tr><td>MLVU [71]</td><td>1.1K</td><td>2.2K</td><td>755.7s</td><td>-</td><td>Ego &amp; Exo</td><td>Open</td></tr><tr><td>LongVideoBench [57]</td><td>618</td><td>1.2K</td><td>574.9s</td><td>-</td><td>Ego &amp; Exo</td><td>Open</td></tr></table>
+
+Table 3: Comparison with SoTA methods on video temporal grounding benchmarks. SP denotes the dataset-specific fine-tuning setting, and Full refers to the universal pre-training setting. Best results are in bold. Improvements of UniTime-Full over SoTA are highlighted in (green).  
+(a) Long-video temporal grounding benchmarks.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Ego4D-NLQ</td><td colspan="3">TaCoS</td></tr><tr><td>R1@.3</td><td>R1@.5</td><td>mIoU</td><td>R1@.3</td><td>R1@.5</td><td>mIoU</td></tr><tr><td>2D-TAN [66]</td><td>5.04</td><td>2.02</td><td>3.53</td><td>45.61</td><td>35.77</td><td>-</td></tr><tr><td>VSLNet [65]</td><td>10.84</td><td>6.81</td><td>8.83</td><td>35.54</td><td>23.54</td><td>24.99</td></tr><tr><td>RGNet [13]</td><td>18.28</td><td>12.04</td><td>12.41</td><td>-</td><td>-</td><td>-</td></tr><tr><td>SnAG [37]</td><td>15.87</td><td>11.26</td><td>-</td><td>56.44</td><td>44.86</td><td>-</td></tr><tr><td>SG-DETR [10]</td><td>-</td><td>-</td><td>-</td><td>56.71</td><td>44.70</td><td>40.90</td></tr><tr><td>LD-DETR [70]</td><td>-</td><td>-</td><td>-</td><td>57.61</td><td>44.31</td><td>40.30</td></tr><tr><td>UniVTG [27]</td><td>7.28</td><td>3.95</td><td>4.91</td><td>51.44</td><td>34.97</td><td>33.60</td></tr><tr><td>UniVTG w/PT</td><td>11.74</td><td>7.54</td><td>7.88</td><td>56.11</td><td>43.44</td><td>38.63</td></tr><tr><td>Qwen2-VL-7B [56]</td><td>0.48</td><td>0.17</td><td>0.46</td><td>4.14</td><td>1.27</td><td>2.64</td></tr><tr><td>Qwen2.5-VL-7B [2]</td><td>1.11</td><td>0.48</td><td>0.82</td><td>7.66</td><td>3.35</td><td>5.29</td></tr><tr><td>UniTime-SP</td><td>24.79</td><td>16.83</td><td>17.25</td><td>61.18</td><td>48.31</td><td>45.02</td></tr><tr><td>UniTime-Full</td><td>27.09 (+8.81)</td><td>18.41 (+6.37)</td><td>18.80 (6.39)</td><td>66.91 (+9.30)</td><td>55.14 (+10.44)</td><td>50.00 (+9.10)</td></tr></table>
+
+(b) Short-video temporal grounding benchmarks.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Charades-STA</td><td colspan="2">ANet-Captions</td><td colspan="2">QVHighlights</td></tr><tr><td>R1@.5</td><td>R1@.7</td><td>R1@.5</td><td>R1@.7</td><td>R1@.5</td><td>R1@.7</td></tr><tr><td>UnLoc-L [58]</td><td>60.80</td><td>38.40</td><td>48.30</td><td>30.20</td><td>66.10</td><td>46.70</td></tr><tr><td>SG-DETR [10]</td><td>70.20</td><td>49.50</td><td>-</td><td>-</td><td>72.20</td><td>56.60</td></tr><tr><td>LD-DETR [70]</td><td>62.58</td><td>41.56</td><td>-</td><td>-</td><td>66.80</td><td>51.04</td></tr><tr><td>SnAG [37]</td><td>64.62</td><td>46.26</td><td>48.55</td><td>30.56</td><td>-</td><td>-</td></tr><tr><td>UniVTG [27]</td><td>58.01</td><td>35.65</td><td>-</td><td>-</td><td>58.86</td><td>40.86</td></tr><tr><td>UniVTG w/PT</td><td>60.19</td><td>38.55</td><td>42.41</td><td>21.55</td><td>65.43</td><td>50.06</td></tr><tr><td>TimeSuite [62]</td><td>67.10</td><td>43.00</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Mr.BLIP [34]</td><td>69.31</td><td>49.29</td><td>53.92</td><td>35.55</td><td>74.77</td><td>60.51</td></tr><tr><td>Qwen2-VL-7B [56]</td><td>7.18</td><td>3.28</td><td>3.87</td><td>1.74</td><td>-</td><td>-</td></tr><tr><td>Qwen2.5-VL-7B [2]</td><td>60.32</td><td>34.27</td><td>16.96</td><td>8.75</td><td>-</td><td>-</td></tr><tr><td>UniTime-SP</td><td>74.33</td><td>53.71</td><td>54.81</td><td>36.62</td><td>77.76</td><td>63.29</td></tr><tr><td>UniTime-Full</td><td>75.27</td><td>56.85</td><td>53.67</td><td>35.90</td><td>76.72</td><td>62.65</td></tr><tr><td></td><td>(+5.07)</td><td>(+7.35)</td><td>(-0.25)</td><td>(+0.35)</td><td>(+1.95)</td><td>(+2.14)</td></tr></table>
+
+redundant video loading and computation, significantly accelerating training. To maintain consistency, we modify the attention mask to prevent cross-interaction between different query-answer pairs, and assign each query-answer sequence the same starting position index, directly following the video tokens' indices. For further implementation details, we refer readers to Appendix B.2.
+
+# 3 Experiments
+
+# 3.1 Experimental Setups
+
+Datasets. To support universal temporal grounding, we compile a diverse dataset spanning varied scenes, genres, durations, and query types (e.g., descriptions, questions, procedural instructions), as detailed in Table 1. For evaluation, we benchmark our method across three categories: (i) Short-video temporal grounding, including Charades-STA [51], ActivityNet-Captions [22], and QVHighlights [24]. (ii) Long-video temporal grounding, including Ego4D-NLQ [11] and TaCoS [47]. (iii) Video question answering (VideoQA), including two grounded benchmarks with temporally annotated queries (QaEgo4D [3], CG-Bench [5]) and two general benchmarks (MLVU [71], LongVideoBench [57]). The statistics of evaluation benchmarks we used are listed in Table 2.
+
+Evaluation Metrics. For temporal grounding tasks, we adopt Recall@1 (R1) at multiple temporal intersection-over-union (IoU) thresholds and mean IoU (mIoU) as evaluation metrics. Specifically, we use IoU thresholds of 0.3 and 0.5 for long-video benchmarks, and 0.5 and 0.7 for short-video benchmarks. For general VideoQA tasks, we report standard accuracy metrics, while for grounded VideoQA tasks, we also evaluate R1, mIoU, and intersection-over-prediction (IoP). More details about metrics are in the Appendix A.1.
+
+Implementation Details. Our framework is built on PyTorch, with Qwen2-VL-7B [56] as the base model. All experiments are conducted with a batch size of 8, using AdamW [32] optimizer with the learning rate 2e-4, and trained for one epoch with linear warmup during the first  $3\%$  of steps. The
+
+Table 4: Zero-shot performance on video temporal grounding benchmarks. Zero indicates zero-shot. All models are evaluated in the zero-shot setting. Numbers in gray are sourced from the original paper; all others are tested by us using their released code and checkpoints. Improvements of UniTime-Zero over SoTA are highlighted in (green).  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Ego4D-NLQ</td><td colspan="3">TaCoS</td><td colspan="3">Charades-STA</td><td colspan="3">ANet-Captions</td><td colspan="3">QVHighlights</td></tr><tr><td>R1@.3</td><td>R1@.5</td><td>mIoU</td><td>R1@.3</td><td>R1@.5</td><td>mIoU</td><td>R1@.5</td><td>R1@.7</td><td>mIoU</td><td>R1@.5</td><td>R1@.7</td><td>mIoU</td><td>R1@.5</td><td>R1@.7</td><td>mIoU</td></tr><tr><td>UniVTG [27]</td><td>6.48</td><td>3.48</td><td>4.63</td><td>5.17</td><td>1.27</td><td>4.40</td><td>25.22</td><td>10.03</td><td>27.12</td><td>11.10</td><td>4.06</td><td>16.86</td><td>14.13</td><td>4.42</td><td>23.38</td></tr><tr><td>Mr.BLIP [34]</td><td>6.49</td><td>3.20</td><td>5.37</td><td>24.59</td><td>14.32</td><td>17.94</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>VTG-LLM [12]</td><td>1.71</td><td>0.46</td><td>1.36</td><td>6.87</td><td>2.92</td><td>5.27</td><td>34.11</td><td>15.81</td><td>34.93</td><td>12.32</td><td>6.74</td><td>17.86</td><td>4.13</td><td>1.55</td><td>8.60</td></tr><tr><td>Momentor [45]</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>26.60</td><td>11.60</td><td>28.50</td><td>23.00</td><td>12.40</td><td>29.30</td><td>-</td><td>-</td><td>-</td></tr><tr><td>VTimeLLM [18]</td><td>1.67</td><td>0.77</td><td>2.52</td><td>9.30</td><td>3.90</td><td>8.08</td><td>34.30</td><td>14.70</td><td>34.60</td><td>-</td><td>-</td><td>-</td><td>26.13</td><td>11.16</td><td>29.42</td></tr><tr><td>Timechat [48]</td><td>1.67</td><td>0.79</td><td>1.30</td><td>3.77</td><td>1.60</td><td>2.95</td><td>29.73</td><td>12.53</td><td>31.20</td><td>16.38</td><td>8.36</td><td>21.53</td><td>8.32</td><td>4.26</td><td>14.25</td></tr><tr><td>TimeMarker [6]</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>51.90</td><td>26.90</td><td>48.40</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>TimeSuite [62]</td><td>0.88</td><td>0.43</td><td>0.94</td><td>6.75</td><td>2.50</td><td>5.71</td><td>48.95</td><td>24.65</td><td>45.91</td><td>16.56</td><td>9.28</td><td>22.03</td><td>12.32</td><td>9.16</td><td>21.30</td></tr><tr><td>UniTime-Zero</td><td>14.67(+8.18)</td><td>7.38(+3.90)</td><td>10.18(+4.81)</td><td>50.06(+25.47)</td><td>31.54(+17.22)</td><td>33.38(+15.44)</td><td>59.09(+7.19)</td><td>31.88(+4.98)</td><td>52.19(+3.79)</td><td>22.77(-0.23)</td><td>14.14(+1.74)</td><td>27.31(-1.99)</td><td>41.03(+14.90)</td><td>31.48(+20.32)</td><td>43.71(+14.29)</td></tr></table>
+
+Table 5: Closed-source Model Evaluation. All models are evaluated on sampled subsets, with outputs lacking predicted timestamps excluded.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">Ego4D-NLQ</td><td colspan="3">Charades-STA</td></tr><tr><td>R1@.3</td><td>R1@.5</td><td>mIoU</td><td>R1@.5</td><td>R1@.7</td><td>mIoU</td></tr><tr><td>Gemini-2.5-flash [8]</td><td>15.29</td><td>15.29</td><td>13.93</td><td>36.46</td><td>13.54</td><td>38.72</td></tr><tr><td>Gemini-2.5-pro [8]</td><td>26.83</td><td>19.51</td><td>20.45</td><td>35.96</td><td>11.24</td><td>39.28</td></tr><tr><td>GPT-4.1-mini [40]</td><td>5.00</td><td>3.00</td><td>5.04</td><td>31.00</td><td>17.00</td><td>31.46</td></tr><tr><td>GPT-4o [39]</td><td>10.59</td><td>7.06</td><td>7.97</td><td>41.11</td><td>17.78</td><td>43.82</td></tr><tr><td>Seed1.5-VL [54]</td><td>19.39</td><td>10.20</td><td>13.74</td><td>90.82</td><td>63.27</td><td>73.69</td></tr><tr><td>UniTime-Full</td><td>25.00</td><td>17.00</td><td>17.20</td><td>81.00</td><td>61.00</td><td>70.11</td></tr></table>
+
+Table 6: Flexibility Verification. Improvements over baseline are highlighted in (green). All UniTime variants use the data-specific setting.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Ego4D-NLQ</td><td colspan="3">Charades-STA</td></tr><tr><td>R1@.3</td><td>R1@.5</td><td>mIoU</td><td>R1@.5</td><td>R1@.7</td><td>mIoU</td></tr><tr><td>Qwen2-VL-2B [56]</td><td>0.34</td><td>0.34</td><td>0.42</td><td>2.23</td><td>0.51</td><td>6.70</td></tr><tr><td>+UniTime</td><td>10.50</td><td>5.80</td><td>7.29 (+6.87)</td><td>65.38</td><td>42.18</td><td>57.25 (+50.55)</td></tr><tr><td>Qwen2-VL-7B [56]</td><td>0.48</td><td>0.17</td><td>0.46</td><td>7.18</td><td>3.28</td><td>8.98</td></tr><tr><td>+UniTime</td><td>24.79</td><td>16.83</td><td>17.25 (+16.79)</td><td>74.33</td><td>53.71</td><td>63.15 (+54.17)</td></tr><tr><td>Qwen2.5-VL-7B [2]</td><td>1.11</td><td>0.48</td><td>0.82</td><td>60.32</td><td>34.27</td><td>52.42</td></tr><tr><td>+UniTime</td><td>23.53</td><td>16.12</td><td>16.61 (+15.79)</td><td>74.19</td><td>53.98</td><td>63.62 (+11.20)</td></tr><tr><td>InternVL2.5-2B [7]</td><td>0.33</td><td>0.22</td><td>0.42</td><td>5.48</td><td>1.77</td><td>9.30</td></tr><tr><td>+UniTime</td><td>9.87</td><td>5.26</td><td>7.46 (+7.04)</td><td>69.49</td><td>47.77</td><td>59.03 (+49.73</td></tr><tr><td>InternVL2.5-8B [7]</td><td>2.19</td><td>0.55</td><td>1.77</td><td>8.66</td><td>2.55</td><td>17.54</td></tr><tr><td>+UniTime</td><td>16.23</td><td>8.55</td><td>12.03 (+10.26)</td><td>71.83</td><td>49.62</td><td>60.50 (+42.96)</td></tr></table>
+
+vision encoder is frozen, and the LLM is fine-tuned via LoRA [16] (rank  $= 8$ , alpha  $= 8$ ). We sample frames at 2 fps, with  $N_{f}^{\mathrm{short}} = 128$  and  $N_{f}^{\mathrm{long}} = 1024$ , capping input video at 16,384 tokens. The default segment length and replication factor are set to 32 and 4, respectively.
+
+# 3.2 Comparison with State-of-the-arts on Video Temporal Grounding
+
+Dataset-specific Setting. In Table 3, we compare our model against state-of-the-art methods on two long-video benchmarks and three short-video benchmarks. Under this setting, where models are trained and tested separately on each dataset, our method (UniTime-SP) outperforms all existing approaches (including traditional and MLLM-based models), achieving absolute improvements of  $6.39\%$  on Ego4D-NLQ,  $9.10\%$  on TaCoS,  $5.45\%$  on Charades-STA,  $2.88\%$  on ANet-Captions, and  $2.89\%$  on QVHighlights over prior best results.
+
+Universal Setting. To assess broad applicability, we train a single model (UniTime-Full) on the largest collection of temporal grounding data (Table 1, Parts I & II). Unlike dataset-specific tuning, this setting prioritizes broad competence over specialized performance, ensuring the model robustly handles diverse video types and query styles in real-world scenarios. Without benchmark-specific fine-tuning, UniTime-Full achieves superior performance across all evaluated benchmarks, demonstrating its effectiveness as a universal solution for video temporal grounding.
+
+Zero-shot Setting. To further test the generalization capability of our method, we pre-train a zero-shot model (UniTime-Zero) on Part I datasets in Table 1, while excluding all in-domain data (Part II). As shown in Table 4, our method achieves superior zero-shot performance, highlighting its remarkable generalization capability. The error analysis on ANet-Captions can be found in the Appendix C.2.
+
+# 3.3 Comparison with Closed-source models on Video Temporal Grounding
+
+To comprehensively assess UniTime, we benchmarked mainstream closed-source MLLMs to examine whether UniTime surpasses them or narrows the gap on VTG tasks. We conducted temporal grounding experiments on a randomly sampled subset from Ego4D and Charades. As shown in Table 5, Seed1.5-VL [54] attained the strongest performance on short videos (72.2 mIoU on Charades), while Gemini-2.5-Pro [8] exhibited superior grounding on long videos (20.5 mIoU on Ego4D). However, these closed-source models struggle to achieve high performance concurrently on both long and short
+
+Table 7: Ablation of the proposed modules. Adaptive Scaling refers to adaptive frame scaling and Segment Retrieval indicates multi-granular temporal prediction.  
+
+<table><tr><td rowspan="2">Adaptive Scaling</td><td rowspan="2">Multi-stage Inference</td><td rowspan="2">Segment Retrieval</td><td colspan="3">Ego4D-NLQ</td></tr><tr><td>R1@.3</td><td>R1@.5</td><td>mIoU</td></tr><tr><td>X</td><td>X</td><td>X</td><td>14.25</td><td>7.54</td><td>9.83</td></tr><tr><td>✓</td><td>X</td><td>X</td><td>14.00</td><td>7.51</td><td>9.83</td></tr><tr><td>X</td><td>✓</td><td>X</td><td>18.42</td><td>12.13</td><td>12.78</td></tr><tr><td>✓</td><td>✓</td><td>X</td><td>17.91</td><td>11.99</td><td>12.39</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>24.79</td><td>16.83</td><td>17.25</td></tr></table>
+
+Table 8: Robustness test on time shift and query decomposition. QD indicates query decomposition, TS denotes time shift, and Ratio is TS performance relative to the original.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="5">Charades-STA TS</td><td rowspan="2">Charades-STA IoG of QD</td></tr><tr><td>Shift</td><td>R1@.5</td><td>R1@.7</td><td>mIoU</td><td>Ratio</td></tr><tr><td rowspan="2">VTimeLLM</td><td>X</td><td>28.53</td><td>12.35</td><td>32.00</td><td rowspan="2">53.67</td><td rowspan="2">68.99</td></tr><tr><td>✓</td><td>14.65</td><td>6.64</td><td>17.89</td></tr><tr><td rowspan="2">Mr.BLIP</td><td>X</td><td>70.05</td><td>50.24</td><td>59.38</td><td rowspan="2">66.77</td><td rowspan="2">71.22</td></tr><tr><td>✓</td><td>51.32</td><td>26.91</td><td>43.63</td></tr><tr><td rowspan="2">TimeSuite</td><td>X</td><td>48.65</td><td>24.01</td><td>45.32</td><td rowspan="2">66.26</td><td rowspan="2">47.07</td></tr><tr><td>✓</td><td>37.02</td><td>13.55</td><td>31.54</td></tr><tr><td rowspan="2">UniTime-SP</td><td>X</td><td>73.36</td><td>53.74</td><td>62.87</td><td rowspan="2">80.20</td><td rowspan="2">74.88</td></tr><tr><td>✓</td><td>63.06</td><td>37.47</td><td>53.39</td></tr></table>
+
+videos, indicating a lack of generalization. In contrast, UniTime achieves competitive performance compared to these closed-source models while maintaining more balanced capabilities for both short and long-form video grounding benchmarks, proving our approach to be overall the best in terms of consistent performance across varying video types.
+
+# 3.4 Flexibility Verification
+
+As discussed in Section 2.2, our approach is compatible with any MLLM that processes dynamic frame inputs. To evaluate flexibility and generalization, we conduct experiments across multiple baseline MLLMs (e.g., Qwen2-VL, Qwen2.5-VL, InternVL2.5) and model scales. As shown in Table 6, larger MLLMs outperform their smaller counterparts, and our method consistently improves temporal grounding performance regardless of the underlying architecture. These results demonstrate strong generalization and flexibility, and indicate that the effectiveness of our framework continues to benefit from advances in MLLM architectures and increased model scale, enabling further gains in temporal grounding as more powerful models become available.
+
+# 3.5 Ablation Studies
+
+Effect of Individual Modules. To assess the contribution of each module, we conduct ablation studies on the Ego4D-NLQ benchmark, with results presented in Table 7. The study examines three key components: "Adaptive Scaling" for video input processing; "Multi-stage Inference" implementing coarse-to-fine grounding for long videos; and "Segment Retrieval" for coarse temporal grounding through inserting timestamps before segments. The results demonstrate three key findings: (i) Adaptive frame scaling (row 2) maintains performance comparable to uniform sampling (row 1) when used independently. (ii) Multi-stage inference yields consistent improvements for temporal grounding (row 3) through progressive refinement from lower to higher frame rates, and for spatial grounding (row 4) via progressive refinement from lower to higher resolution, with both coarse-to-fine grounding strategies benefiting from this hierarchical approach. (iii) Most notably, incorporating multi-granular temporal prediction (row 5), where the model first retrieves a coarse relevant segment and then localizes the precise moment within it, yields substantial improvements across all metrics. These results highlight the effectiveness of the multi-stage temporal grounding framework. More details can be found in the Appendix A.2.1.
+
+Effect of Segment Length. Our universal temporal grounding framework operates in two stages: (i) retrieving the video segment most relevant to the query; (ii) predicting the precise temporal interval within it. Segment length critically influences this process. We evaluate its impact using three metrics: Segment retrieval accuracy  $(\mathbb{R}@\mathbb{1})$  measures the ability to identify the correct segment. Oracle grounding accuracy (oracle R1@0.3) reflects fine-grained localization given the ground-truth segment. Overall grounding accuracy  $(\mathrm{R1}@\mathrm{0.3})$  assesses the overall end-to-end temporal grounding performance. As shown in Figure 3(a), longer segments improve segment retrieval accuracy but hinder fine-grained localization, resulting in lower oracle grounding performance. To balance this trade-off, we set the segment length to 32 for optimal overall accuracy.
+
+Effect of Replication Factor. We investigate the impact of replication factor  $(N_{\mathrm{rep}})$ , which balances long and short video samples during training. Figure 3(b) shows that increasing  $N_{\mathrm{rep}}$  initially improves segment retrieval but saturates and fluctuates later. In contrast, fine-grained grounding performance remains stable. Based on these results, we set  $N_{\mathrm{rep}} = 4$  for optimal performance.
+
+![](images/5f02d416cc40dfa6ead79cec61724504ade6a30678c629458139b8ea0dac35e9.jpg)  
+Figure 3: Ablation on hyperparameters. We decouple the effects of Segment Length (left) and Replication Factor (right) on temporal grounding performance.
+
+![](images/0d6a533b67e249110b89c353af1fec2825988520e0bb58f2ad2b2a14b5fac073.jpg)  
+Figure 4: Temporal distribution bias in Charades-STA. Darker colors indicate higher data density.
+
+![](images/136a4bf26c76846e6201bc9d9c15c4224015e01957326e3e687f2188a5bae669.jpg)
+
+# 3.6 Robustness Test
+
+Generative models are known to produce hallucinations in their outputs [28]. To evaluate robustness, we conduct experiments on the Charades-STA benchmark from two perspectives: (i) perturbing event positions to assess robustness against temporal distributional bias, and (ii) decomposing complex queries into simpler object-based questions to test the model's reliability across query types.
+
+Time Shift on Event. Previous studies [14, 41] reveal significant biases in event distributions within temporal grounding datasets. As seen in Figure 4, annotated events in Charades-STA are heavily concentrated near the start or end of videos. To assess robustness against such bias, we perturb the data distribution by sampling clips that contain the event segment but place it at randomized positions. As shown in Table 8, our method exhibits greater robustness than other MLLM-based approaches, as evidenced by the higher performance ratio relative to the unperturbed baseline.
+
+On Decomposition of Query. Accurate localization of complex queries fundamentally depends on understanding the referenced objects or events. To address this, we prompt Qwen2 [59] to decompose each Charades-STA query into a set of object-based questions in the form of "When does <object> appear?", where <object> denotes an entity that can be grounded in the original query. For each decomposed query, we compute the intersection-over-ground-truth (IoG) between its predicted segment and the ground-truth segment. It is worth noting that this is only an approximate estimation, as an entity may appear in multiple temporal segments, while annotations typically provide only a single segment. As shown in Table 8, our approach demonstrates superior query understanding and more reliable localization, with fewer hallucinations compared to other methods.
+
+Recently, a related work [21] focuses on prediction consistency: they also curate test data by rephrasing original queries and shifting ground-truth moments in videos to probe grounding consistency. For detailed tests and comparisons, please refer to the Appendix D.
+
+# 3.7 Downstream Task
+
+To validate the effectiveness of our universal temporal grounding model in facilitating long video understanding as discussed in Section 1, we evaluate on four VideoQA benchmarks. As a baseline, we uniformly sample 32 frames from each video and input them to the Qwen2-VL-7B model for answer generation. To assess the impact of different temporal grounding approaches on downstream VideoQA, we first use each model to predict the relevant temporal segment for each question, then uniformly sample 32 frames from the predicted segment and feed them into Qwen2-VL-7B for answer prediction. As shown in Table 9, our method exhibits superior generalization compared to existing temporal grounding models, demonstrating robust grounding performance on out-of-domain data and enhancing long video understanding. Refer to Appendix A.4 for VideoQA evaluation details.
+
+Table 9: Performance on VideoQA benchmarks. R@IoU is the mean Recall@1 at IoU thresholds [0.1 : 0.1 : 0.5]. For grounded VideoQA, IoP and mIoU are used as grounding metrics.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">QaEgo4D</td><td colspan="4">CG-Bench</td><td>MLVU</td><td>LongVideoBench</td></tr><tr><td>IoP</td><td>R1@.5</td><td>mIoU</td><td>Acc.</td><td>IoP</td><td>R@IoU</td><td>mIoU</td><td>Acc.</td><td>Acc.</td><td>Acc.</td></tr><tr><td>Uniform Sample</td><td>3.67</td><td>-</td><td>-</td><td>49.60</td><td>1.08</td><td>-</td><td>-</td><td>33.87</td><td>60.53</td><td>54.82</td></tr><tr><td>UniVTG [27]</td><td>9.80</td><td>6.20</td><td>7.76</td><td>50.00</td><td>1.37</td><td>4.92</td><td>3.86</td><td>34.87</td><td>62.56</td><td>54.67</td></tr><tr><td>VTimeLLM [18]</td><td>3.87</td><td>1.40</td><td>3.68</td><td>48.60</td><td>1.62</td><td>1.22</td><td>1.58</td><td>34.60</td><td>59.52</td><td>54.30</td></tr><tr><td>TimeSuite [62]</td><td>6.28</td><td>0.20</td><td>1.22</td><td>48.80</td><td>5.06</td><td>2.89</td><td>2.09</td><td>32.47</td><td>58.51</td><td>53.25</td></tr><tr><td>UniTime-Full</td><td>25.29</td><td>19.00</td><td>18.44</td><td>55.51</td><td>16.64</td><td>17.66</td><td>11.63</td><td>40.30</td><td>66.50</td><td>56.47</td></tr></table>
+
+# 4 Related Work
+
+Video Temporal Grounding (VTG) tasks can typically be categorized into short- and long-duration scenarios. For short videos, SOTA methods primarily adopt DETR-like architectures [4, 24, 36, 35, 10]. For example, Moment-DETR [24] formulates moment retrieval as a set prediction problem with a transformer-based encoder-decoder, enabling end-to-end prediction of moment coordinates and saliency. Recent works further enhance performance by introducing prior knowledge [20] and advanced attention mechanisms [36, 35]. Non-DETR methods, such as UMT [29] and Mr.BLIP [34], leverage multi-modal cues and large language models, respectively. While effective for short videos, these methods struggle with long videos, where sparse relevant moments create a "needle-in-a-haystack" challenge. Recent works [15, 43] employ coarse-to-fine alignment and hierarchical pipelines to address this. Despite the progress on individual benchmarks, fundamental differences between short and long videos hinder the development of unified models. While UniVTG [27] attempts to bridge this gap, its lightweight architecture limits generalization, especially in out-of-domain scenarios [50]. This motivates our work to develop a universal grounding model capable of (i) handling videos of arbitrary duration, (ii) robustly generalizing across domains, and (iii) excelling in downstream tasks like video question answering.
+
+Temporal Grounding with Multi-modal Language Models (MLLMs). While MLLMs have achieved remarkable success across various multi-modal tasks [23, 44, 31], accurate temporal grounding remains challenging. Existing temporal integration approaches in MLLMs can be grouped into three paradigms: (i) time-agnostic models: VTimeLLM [18] processes fixed-length videos (100 frames) without explicit temporal cues, predicting normalized moment positions; LITA [19] replaces the predicted textual timestamps with special time tokens. The lack of explicit temporal signals often leads to suboptimal performance. (ii) implicit timestamp-encoded models: These methods incorporate timestamp-aware encoders, such as TimeChat [48] and TimeSuite [62], which fuse frame and temporal embeddings. VTG-LLM [12] uses absolute time embeddings, while Qwen2.5-VL [2] employs MRoPE for temporal modeling. Despite advances, these methods require extensive pretraining and may hallucinate timestamps due to implicit temporal representations. (iii) explicit temporal marking models: Methods like Mr.BLIP [34], TimeMarker [6], and VideoLLaMA3 [63] comprehend textual timestamps to video frames, leveraging retrieval capabilities of MLLMs to match DETR-based performance. Although these approaches have advanced short video temporal grounding, they are constrained by context windows and GPU memory, limiting scalability to long videos. To address this, we propose a framework that adaptively adjusts scales of video frames, combined with a multi-scale, coarse-to-fine iterative temporal grounding strategy tailored for long video understanding.
+
+Long Video Understanding is significantly more challenging than short video or image-based tasks due to complex temporal dynamics and substantial redundancy. The large number of frames dramatically increases memory and computational demands, making dense sampling impractical. Existing video MLLMs [2, 56, 69] often rely on uniform frame sampling, which risks omitting critical information. To address this, some approaches compress frames into minimal token sets to reduce computational costs [49, 26, 67], while others focus on more effective frame sampling strategies [60, 52, 17, 9]. Despite these advancements, temporal grounding—crucial for retrieving salient events—remains underexplored in the context of long videos. In this work, we leverage temporal grounding to identify key events, enhancing the understanding of long-form video content.
+
+# 5 Conclusion
+
+In conclusion, this paper presents UniTime, a universal video temporal grounding model based on MLLMs. At its core, UniTime interleaves timestamp tokens with video tokens and retrieves relevant timestamps for natural language queries, thereby enabling precise temporal localization. By employing adaptive frame scaling, UniTime constructs inputs with varying spatial granularities for videos of different durations. By further inserting timestamps at multiple temporal granularities, the model supports multi-scale temporal prediction. As a result, it can perform coarse-to-fine temporal grounding, first coarsely localizing segments in long videos and then refining predictions to fine-grained windows in short videos. To further enhance efficiency, we propose a video-centric training paradigm that reduces redundant computation. Extensive experiments on multiple benchmarks demonstrate that UniTime, pre-trained on large-scale datasets without the need for dataset-specific finetuning, consistently outperforms existing methods in both temporal grounding and downstream long-video question answering tasks.
+
+# Acknowledgements
+
+This work is supported by National Key R&D Program of China (No.2022ZD0161400).
+
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+
+# Appendix / supplemental material
+
+In the appendix, we provide additional experimental details (Appendix A), implementation details (Appendix B), qualitative results (Appendix C), grounding consistency evaluation (Appendix D), more ablation studies (Appendix E), discussions of limitations and future work (Appendix F), and broader impacts (Appendix G).
+
+# A Experimental Details
+
+In this section, we provide additional experimental details, including benchmark details (Appendix A.1), ablation details (Appendix A.2), robustness test details (Appendix A.3) and downstream task details (Appendix A.4).
+
+# A.1 Benchmark Details
+
+To evaluate the performance of our model on temporal grounding and VideoQA tasks, we employ three key metrics: Intersection-over-Union (IoU), Intersection-over-Prediction (IoP), and Intersection-over-Ground Truth (IoG). These metrics quantify the alignment between the predicted temporal window  $T^{\mathrm{pred}}$  and the ground truth temporal window  $T^{\mathrm{gt}}$ . The formal definitions are as follows:
+
+$$
+\operatorname {I o U} = \frac {\left| T ^ {\text {p r e d}} \cap T ^ {\text {g t}} \right|}{\left| T ^ {\text {p r e d}} \cup T ^ {\text {g t}} \right|}, \quad \operatorname {I o P} = \frac {\left| T ^ {\text {p r e d}} \cap T ^ {\text {g t}} \right|}{\left| T ^ {\text {p r e d}} \right|}, \quad \operatorname {I o G} = \frac {\left| T ^ {\text {p r e d}} \cap T ^ {\text {g t}} \right|}{\left| T ^ {\text {g t}} \right|}.
+$$
+
+Here,  $|T^{\mathrm{pred}} \cap T^{\mathrm{gt}}|$  denotes the duration of the overlapping region between the predicted window  $T^{\mathrm{pred}}$  and the ground truth window  $T^{\mathrm{gt}}$ .  $|T^{\mathrm{pred}} \cup T^{\mathrm{gt}}|$  represents the duration of their union, while  $|T^{\mathrm{pred}}|$  and  $|T^{\mathrm{gt}}|$  denote the durations of the predicted and ground truth windows, respectively.
+
+IoU measures the overall alignment between  $T^{\mathrm{pred}}$  and  $T^{\mathrm{gt}}$ , providing a balanced assessment of both precision and recall by considering the overlap relative to their union. IoP emphasizes precision, indicating the proportion of the predicted window that overlaps with the ground truth. Conversely, IoG focuses on recall, reflecting the proportion of the ground truth window that is covered by the prediction. Collectively, these metrics offer a comprehensive evaluation of the model's ability to accurately localize temporal windows with respect to the ground truth.
+
+# A.2 Ablation Details
+
+# A.2.1 Effect of Individual Modules
+
+As shown in Table 7, we systematically evaluate the effect of each individual module. The detailed settings are as follows: (i) Row 1: We uniformly sample 32 frames from each video and obtain the corresponding timestamps. These frames and timestamps are then fed into UniTime for temporal grounding. (ii) Row 2: We sample the video at 2 fps and apply frame scaling. The sampled frames and their timestamps are input into UniTime for temporal grounding. (iii) Row 3: On top of uniform sampling, we introduce multi-stage inference. Specifically, after obtaining an initial prediction using the uniformly sampled frames (as in row 1), we perform a second round of uniform sampling within the predicted interval and feed these frames into the model for refined localization. (iv) Row 4: Similarly, we apply multi-stage inference to the adaptive frame scaling. After the initial prediction (as in row 2), we resample at 2 fps and apply frame scaling within the predicted interval, then input these frames into the model for further refinement. (v) Row 5: We adopt the coarse-to-fine temporal grounding strategy described in Section 2.2. Unlike row 4, where the model predicts fine-grained intervals in the first round due to densely sampled timestamps, row 5 partitions the video into coarse segments and only requires the model to predict the relevant segment in the first round. In the second round, temporal grounding is performed within the selected segment to localize the precise moment.
+
+# A.2.2 Effect of Segment Length and Replication Factor
+
+As shown in Figure 3, we decouple the process of coarse-to-fine temporal grounding to analyze the effect of segment length and replication factor. Below, we provide detailed descriptions of the capabilities of each component after decoupling.
+
+Overall, the coarse-to-fine temporal grounding process can be divided into the following two stages. First, we retrieve candidate segments from the video as follows:
+
+$$
+\{\mathbf {S} _ {r} \} _ {r = 1} ^ {N _ {r}} = \Phi_ {\mathrm {U n i T i m e}} (\mathcal {V}, \mathcal {T}, \mathcal {Q}) = \Phi_ {\mathrm {U n i T i m e}} ([ \mathbf {T} _ {1}; \mathbf {S} _ {1}; \mathbf {T} _ {2}; \mathbf {S} _ {2}; \dots ; \mathbf {T} _ {N _ {s}}; \mathbf {S} _ {N _ {s}}; \mathcal {Q} ])
+$$
+
+where  $\mathbf{S}_r$  denotes the retrieved segments and  $N_{r}$  is the number of retrieved segments. Next, we perform fine-grained localization within the retrieved segments:
+
+$$
+\mathcal {A} = \{(s _ {1}, e _ {1}), \dots , (s _ {K}, e _ {K}) \} = \Phi_ {\mathrm {U n i T i m e}} (V | _ {\{\mathbf {S} _ {r} \} _ {r = 1} ^ {N _ {r}}}, \mathcal {T}, \mathcal {Q})
+$$
+
+where each  $s_k, e_k \in \mathcal{T}$  represents the predicted start and end timestamps, respectively. Then, the evaluation metrics used in Section 3.5 and Figure 3 are defined as follows:
+
+Segment retrieval accuracy (R@1) indicates the accuracy of segment retrieval, i.e., the proportion of cases where any predicted segment  $\mathbf{S}_r^{\mathrm{pred}}$  is contained within the set of ground-truth segments  $\{\mathbf{S}_r^{\mathrm{gt}}\}$ . Oracle grounding accuracy (oracle R1@0.3) reflects the localization performance when the model is provided with the ground-truth segment as input, i.e.,  $\mathcal{A}^{\mathrm{oracle}} = \Phi_{\mathrm{UniTime}}(V|_{\{\mathbf{S}_r^{\mathrm{gt}}\}}, \mathcal{T}, \mathcal{Q})$ . Overall grounding accuracy (R1@0.3) measures the end-to-end temporal grounding performance, where  $\mathcal{A}^{\mathrm{pred}} = \Phi_{\mathrm{UniTime}}(V|_{\{\mathbf{S}_r^{\mathrm{pred}}\}}, \mathcal{T}, \mathcal{Q})$ .
+
+# A.3 Robustness Test Details
+
+# A.3.1 Time Shift on Event
+
+The temporal distribution bias in Charades-STA is illustrated in Figure 4. In addition, we also conduct statistical analyses on TACoS, ANet-Captions, and QVHighlights, with the results presented in Figure 5. Significant temporal distribution bias is also observed in TACoS and ANet-Captions, whereas QVHighlights does not exhibit such bias.
+
+![](images/a6ee53ec45a3582df7c676a2080eae3b223d2533dead30d12787d9931fc032e6.jpg)  
+(a)
+
+![](images/5443d315d1ad4a73768e8ca6110ae3c591d51e16828bf1ffe6c4eda3066cf110.jpg)  
+(b)  
+Figure 5: Temporal distribution bias in (a) ANet-Captions. (b) QVHighlights. (c) TACoS. Darker colors indicate higher data density. , and the yellow curve outlines the region of greatest density.
+
+![](images/62824f7eb349140deda1dcf030ac7e7a1c1854737bb3db74943948a2113b7f08.jpg)  
+(c)
+
+# A.3.2 Decomposition of Query
+
+The prompts provided to Qwen2 [59] for decomposing each Charades-STA query into a set of object-centric questions are as follows:
+
+System:
+
+You are Qwen, created by Alibaba Cloud. You are a helpful assistant.
+
+System:
+
+Analyze the given query and:
+
+1. Identify ONLY concrete, specific objects (nouns) that are:
+
+Tangible physical items  
+- Clearly named (not pronouns/ambiguous)
+
+2. STRICTLY EXCLUDE:
+
+- All human references (person, he, she, they, etc.)  
+- Ambiguous terms (something, anything, things, etc.)  
+- Pronouns (it, they, them)  
+- Abstract concepts
+
+3. For each valid object, generate EXACTLY ONE question:
+
+"When does [OBJECT] appear?"
+
+Negative Examples (BAD):
+
+```txt
+Input:"Someone left some items on the furniture"   
+Wrong Output:   
+-When does the furniture appear?   
+-When does some items appear? #  $<  -$  AMBIGUOUS TERM SHOULD BE EXCUSED   
+Positive Examples (GOOD):   
+Input:"The machine processed the raw materials during the night"   
+Output:   
+-When does the machine appear?   
+-When does the raw materials appear?   
+-When does the night appear?   
+User:   
+Analyze: <query>
+```
+
+# A.4 Downstream Task Details
+
+# A.4.1 VideoQA with Temporal Grounding
+
+We use Qwen2-VL-7B as the VideoQA model for answer generation. By default, it processes long videos by uniformly sampling 32 frames. However, this sampling strategy may lead to the omission of critical information. To investigate whether temporal grounding models can compensate for this issue, we adopt the following procedure. First, we use different video temporal grounding models to localize the relevant segments for each question. Then, we crop the localized video intervals and input them into Qwen2-VL-7B. Specifically, for cropped video segments shorter than 32 seconds, we extend their duration from the center to 32 seconds. Within each interval, we again uniformly sample 32 frames for answer generation.
+
+# A.4.2 Prompt template for VideoQA
+
+We use the same prompt template for all multiple-choice VideoQA benchmarks:
+
+```txt
+System:   
+You are a helpful assistant.   
+User:   
+<video>   
+Question: <question>   
+Options:   
+(A) <Option_A>   
+(B) <Option_B>   
+(C) <Option_C>   
+(D) <Option_D>   
+Please only give the best option.   
+Best Option:   
+Assistant:
+```
+
+# B Additional Implementation Details
+
+# B.1 Prompt template for UniTime
+
+The prompt template used for coarse-grained segment retrieval as follows:
+
+```txt
+System:   
+You are a helpful assistant.   
+User:   
+<timestamp & frames>   
+This is a sequence interleaved with timestamps and frames.   
+Your task is to identify the specific timestamp(s) when the given query appears. Query: <Query>   
+Answer:   
+Assistant:
+```
+
+The prompt template used for fine-grained temporal grounding as follows:
+
+```txt
+System:   
+You are a helpful assistant.   
+User:   
+<timestamp & frames>   
+This is a sequence interleaved with timestamps and frames.   
+Your task is to identify the temporal window (start and end timestamps) when the given query appears.   
+Query: <Query>   
+Answer:   
+Assistant:
+```
+
+# B.2 Video-centric Training
+
+In video-centric training, we first sample a video from the dataset and gather all associated queries with their corresponding answers. These queries and answers are concatenated and paired with the video as input, forming a sequence such as
+
+$$
+\left[ v _ {1}, v _ {2}, \dots , v _ {N}, Q _ {1}, A _ {1}, Q _ {2}, A _ {2}, \dots , Q _ {N _ {\text {s a m p l e}}}, A _ {N _ {\text {s a m p l e}}} \right]
+$$
+
+. As illustrated in Figure 6, we apply attention masks to prevent different queries and answers from attending to others. Additionally, each query-answer sequence is assigned the same starting Rotary Position Embedding index, immediately following the indices of the video tokens.
+
+![](images/b267bbcb33ec256b7d41532da7508d393f169e64a3d0e21ced2ad424cfabe6f4.jpg)  
+Figure 6: Illustration of the video-centric training paradigm.
+
+# C Qualitative Results
+
+# C.1 Qualitative Results for Ego4D-NLQ
+
+In Figure 7, we present qualitative results of UniTime on Ego4D-NLQ. In the success cases, UniTime demonstrates precise coarse-grained segment retrieval and fine-grained temporal grounding capabilities. Despite the limited variation in video scenes, which are primarily composed of fine-grained interactions between hands and objects, the model consistently demonstrates robust performance. In the failure cases, although the model retrieves the correct segment, it fails in fine-grained localization. As shown in the specific localization frames, this error occurs because the object referred to in the query appears multiple times in the video, while the ground truth only includes one of these occurrences.
+
+# C.2 Qualitative Results for ANet-Captions
+
+As shown in Table 2, UniTime achieves substantial performance gains over competing methods across all benchmarks, except for ANet-Captions. We analyze this discrepancy below, with Figure 8 visualizing UniTime's predictions on ANet-Captions.
+
+In successful cases (Figure 8(a)), UniTime achieves precise temporal grounding. Despite the limited diversity of video scenes, which predominantly feature fine-grained interactions between hands and objects, the model consistently exhibits robust performance.
+
+However, our analysis of the failure cases indicates that most can be traced to limitations in ANetCaptions's annotation scheme. As illustrated in Figure 8(b), multiple occurrences of the same event are often annotated with only a single temporal interval. This occurs because captions are ordered sequentially, causing annotators to overlook recurring events.
+
+Additionally, many failures arise from incorrect labels, as shown in Figure 8(c).
+
+# D Grounding Consistency Evaluation
+
+To further evaluate UniTime's robustness in temporal grounding, we conduct grounding consistency assessment on Charades-CON and ActivityNet-CON [21], which are curated from subsets
+
+![](images/fa6e8b4c0d64f6d9a25affe4deb883464583ce540a36e074cfd9d7c0f05a521c.jpg)  
+(a) success case
+
+![](images/2dabc19fab000cea4e27c2b4a4fbe384733fea29dd6ce3d2d6e92e4c31715a82.jpg)  
+(b) failure case  
+Figure 7: Qualitative Results for long-video temporal grounding on the Ego4D-NLQ benchmark. Red indicates the ground truth (GT), blue denotes the results of coarse-grained segment retrieval, and green represents the results of fine-grained temporal grounding.
+
+Query: A woman rolls up a towel.
+
+![](images/6ba9da59db4b47e6c43d60107f3451d3a0a3c4eb463ad470f5f605d6b463f26e.jpg)  
+(a) Success case.
+
+Query: A man starts break dancing on the floor.
+
+![](images/1391522e6b24fca8ee31cb977ba37dca445418ad11bdd44eb659c84e0cd4d438.jpg)  
+(b) Failure case resulting from incomplete annotations of multi-hop events.
+
+Query: The woman then lights a candle.
+
+![](images/595f6df2d8fa5fde92fadbd197ec7a4c6c4fe8c187b1acc9c174ae228d4bb596.jpg)  
+(c) Failure case caused by incorrect labels.  
+Figure 8: Qualitative Results for short-video temporal grounding on the ANet-Captions benchmark. Red indicates the ground truth (GT), blue denotes the results of coarse-grained segment retrieval, and green represents the results of fine-grained temporal grounding.
+
+Table 10: Grounding consistency evaluation of Video-MLLMs. Relative consistency scores are in brackets.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Charades-CON</td><td colspan="3">ActivityNet-CON</td></tr><tr><td>Ground</td><td>R-Ground</td><td>S-Ground</td><td>Ground</td><td>R-Ground</td><td>S-Ground</td></tr><tr><td>GPT-4o [39]</td><td>28.5</td><td>21.2 (74.3)</td><td>9.3 (32.8)</td><td>26.8</td><td>18.1 (67.5)</td><td>10.4 (38.8)</td></tr><tr><td>Gemini 1.5 Flash [55]</td><td>34.6</td><td>29.7 (85.7)</td><td>24.8 (71.7)</td><td>37.8</td><td>30.8 (81.4)</td><td>24.8 (65.6)</td></tr><tr><td>Video-LLaMA [64]</td><td>14.2</td><td>10.6 (74.9)</td><td>5.3 (37.6)</td><td>12.8</td><td>8.5 (66.8)</td><td>7.2 (56.8)</td></tr><tr><td>Video-LLaMA + VTune [21]</td><td>54.4</td><td>38.2 (70.3)</td><td>10.9 (20.0)</td><td>33.0</td><td>24.7 (74.8)</td><td>10.0 (30.2)</td></tr><tr><td>TimeChat [48]</td><td>30.5</td><td>25.0 (82.1)</td><td>5.6 (18.5)</td><td>4.6</td><td>2.9 (64.1)</td><td>1.0 (21.2)</td></tr><tr><td>TimeChat + VTune [21]</td><td>76.2</td><td>69.2 (90.8)</td><td>36.2 (47.5)</td><td>37.4</td><td>28.3 (75.6)</td><td>10.6 (28.3)</td></tr><tr><td>UniTime-Full</td><td>81.5</td><td>73.9 (90.7)</td><td>58.3 (71.6)</td><td>49.1</td><td>30.2 (61.5)</td><td>21.3 (43.4)</td></tr></table>
+
+of CharadesSTA and ActivityNet-Captions, respectively. The grounding consistency targets two complementary consistency dimensions:
+
+i) Rephrased Grounding (R-Ground). This metric evaluates whether a model's moment predictions remain consistent across a query  $q$  and its aligned rephrased variant  $\hat{q}$ . Concretely, the IoU between  $m = \mathrm{Temp}_G(q)$  and  $\hat{m} = \mathrm{Temp}_G(\hat{q})$  is computed. For each original query, the average IoU over three aligned variants is reported, reflecting the alignment consistency of predicted moments under paraphrasing.  
+ii) Shifted Grounding (S-Ground). This metric assesses whether a model consistently grounds the same visual content when that content is shifted to different temporal positions. Specifically, while preserving the internal frame sequence of the ground-truth moment  $m_{\mathrm{gt}}$ , it is randomly shifted to a new moment  $m_{s}$ , with evaluations conducted on the resulting predictions.
+
+Experimental results, summarized in the Table 10, demonstrate that UniTime not only exhibits strong temporal grounding performance (Ground) but also achieves excellent consistency and robustness (R-Ground and S-Ground). UniTime attains performance comparable to VTune [21], which extends instruction tuning with explicit consistency objectives, and in some cases surpasses it.
+
+# E More Ablation Studies
+
+# E.1 Effect of Video Processing Strategies
+
+To justify our design choice in Section 2.2, i.e., resizing short videos and compressing long videos, we conduct an ablation study on the Ego4D-NLQ dataset. This study evaluates how different video processing strategies impact both grounding performance and computational efficiency.
+
+Table 11: Ablation on video processing strategies. Seg Retrieval Acc. indicates the accuracy of coarse-grained segment retrieval.  
+
+<table><tr><td>Short Video</td><td>Long Video</td><td>R1@.3</td><td>R1@.5</td><td>mIoU</td><td>Seg Retrieval Acc.</td><td>Training Time</td></tr><tr><td>Frame Resizing</td><td>Token Compression</td><td>24.79</td><td>16.83</td><td>17.25</td><td>45.28</td><td>21 h</td></tr><tr><td>Frame Resizing</td><td>Frame Resizing</td><td>18.53</td><td>12.67</td><td>12.88</td><td>35.36</td><td>12 h</td></tr><tr><td>Token Compression</td><td>Token Compression</td><td>24.06</td><td>16.18</td><td>16.71</td><td>45.28</td><td>24.5 h</td></tr></table>
+
+Our findings indicate that: (i) For long videos, adaptive frame scaling reduces tokens per frame while keeping the frame rate fixed. Though some visual cues are lost, this feature-level token compression retains more semantic information than frame resizing, which discards input resolution. This leads to better segment retrieval accuracy and recall (row 1 vs. row 2). (ii) For short clips with sufficient tokens, both methods perform similarly (row 1 vs. row 3). However, token compression requires high-resolution inputs before merging, incurring higher computational costs. In UniTime-Full, using token compression for short videos increased training time by 5 days compared to frame resizing.
+
+Therefore, rather than employing a unified strategy, we adopt a hybrid approach to balance performance and efficiency: for short videos, we apply frame resizing to reduce computational overhead, while for long videos, we leverage token compression to better preserve visual details. This design choice enables our method to achieve strong performance with practical training costs.
+
+# E.2 Comparison of Temporal Information Encoding
+
+We conduct ablation experiments on Charades-STA to compare the effectiveness of Timestamp Token Insertion versus Dense Position Encodings.
+
+Table 12: Ablation on temporal information encoding.  
+
+<table><tr><td>Method</td><td>R1@.5</td><td>R1@.7</td><td>mIoU</td></tr><tr><td>Dense Position Encodings</td><td>48.44</td><td>27.15</td><td>44.85</td></tr><tr><td>Timestamp Token Insertion (UniTime-SP)</td><td>74.33</td><td>53.71</td><td>63.15</td></tr></table>
+
+Our base model, Qwen2-VL-7B, utilizes Multimodal Rotary Position Embedding (MROPE), which decomposes position embeddings into temporal, height, and width components. After fine-tuning, temporal grounding remains suboptimal with dense position encodings. In contrast, timestamp token insertion yields significant gains. We attribute this improvement to the MLLM's strong multimodal retrieval ability and the simplification of the task into (i) identifying question-relevant video segments and (ii) directly reading their corresponding timestamp tokens.
+
+# E.3 Ablation Study of Adaptive Frame Scaling Hyperparameters
+
+In this section, we conduct an ablation study on the threshold hyperparameters  $N_{f}^{\mathrm{long}}$ ,  $N_{f}^{\mathrm{short}}$ , and the token budget  $N_{\mathrm{total}}$  within Adaptive Frame Scaling, elucidating the rationale behind our parameter choices.
+
+Threshold  $N_{f}^{\text{long}}$ . This parameter specifies the maximum video length that can be processed in a single pass. For videos with length  $\leq N_{f}^{\text{long}}$ , we process the input directly; for videos with length  $> N_{f}^{\text{long}}$ , we partition the input into segments, process each segment independently, and then aggregate the predictions. As shown in Table 13, varying  $N_{f}^{\text{long}}$  has only a marginal effect on accuracy because over-length inputs are segmented and subsequently fused. Moreover, Adaptive Frame Scaling maintains a fixed token budget per segment, so reducing  $N_{f}^{\text{long}}$  increases the number of segments and the total token count at inference, thereby incurring higher latency. Balancing efficiency and accuracy, we set  $N_{f}^{\text{long}} = 1024$ .
+
+Threshold  $N_{f}^{\text{short}}$ . This hyperparameter determines whether inference is executed in a single stage or via a multi-stage (coarse-to-fine) pipeline. For inputs with temporal length  $\leq N_{f}^{\text{short}}$ , we use single-stage processing; for inputs with temporal length  $> N_{f}^{\text{short}}$ , we employ multi-stage processing. As shown in Table 14, within the 64-128 s duration range, the performance difference between multi-stage ( $N_{f}^{\text{short}} = 64$ ) and single-stage ( $N_{f}^{\text{short}} \geq 128$ ) configurations is modest. However, a lower threshold increases average inference latency by routing more inputs to the multi-stage pipeline. For inputs in the 128-256 s range, multi-stage processing yields substantially higher accuracy (54.57 vs. 42.90 mIoU). To balance accuracy and efficiency, we set  $N_{f}^{\text{short}} = 128$ .
+
+Token budget  $N_{\text{total}}$ . This parameter specifies the maximum number of tokens permitted. As shown in Table 15, performance improves monotonically as the token budget increases, up to the available GPU capacity. Accordingly, we set  $N_{\text{total}} = 16$ , 384, which corresponds to the upper limit supported by our hardware.
+
+# F Limitations & Future Work
+
+While UniTime demonstrates exceptional performance on various video temporal grounding and video QA benchmarks, it still has several limitations that warrant further exploration: (i) UniTime is currently constrained to temporal grounding tasks. To enable broader applications in MLLMs, it requires more diverse training data with dense temporal annotations. Incorporating such data into the pretraining process of MLLMs could unlock their potential for handling more temporally complex tasks, such as dense video captioning. (ii) Although UniTime enhances MLLMs with temporal grounding capabilities, relying solely on temporal grounding data limits their reasoning and
+
+Table 13: Ablation of the threshold  ${N}_{f}^{\text{long }}$  on Ego4D.  
+
+<table><tr><td>Nflong</td><td>R1@.3</td><td>R1@.5</td><td>mIoU</td></tr><tr><td>1024</td><td>24.79</td><td>16.83</td><td>17.25</td></tr><tr><td>512</td><td>24.72</td><td>16.63</td><td>16.95</td></tr><tr><td>256</td><td>23.94</td><td>16.10</td><td>16.68</td></tr></table>
+
+Table 14: Ablation of the threshold  ${N}_{f}^{\text{short }}$  on TaCoS.  
+
+<table><tr><td>Nfshort</td><td>mIoU (64-128s)</td><td>mIoU (128-256s)</td><td>mIoU Avg.</td></tr><tr><td>64</td><td>58.43</td><td>54.57</td><td>50.28</td></tr><tr><td>128</td><td>56.59</td><td>54.57</td><td>50.00</td></tr><tr><td>256</td><td>56.59</td><td>42.90</td><td>46.82</td></tr></table>
+
+Table 15: Ablation of the token budget  $N_{\mathrm{total}}$  on Ego4D.  
+
+<table><tr><td>Budget</td><td>R1@.3</td><td>R1@.5</td><td>mIoU</td></tr><tr><td>16384</td><td>24.79</td><td>16.83</td><td>17.25</td></tr><tr><td>8192</td><td>24.08</td><td>16.10</td><td>16.69</td></tr><tr><td>4096</td><td>23.04</td><td>15.07</td><td>15.96</td></tr></table>
+
+question-answering abilities. The ultimate objective is to develop MLLMs that seamlessly integrate localization, reasoning, and question-answering into a unified framework. (iii) The current approach to processing long videos in UniTime relies on a fixed segment length, which lacks flexibility. Future work could investigate adaptive segment lengths based on the information density of videos, enabling more efficient and context-aware video processing.
+
+# G Broader Impacts Statement
+
+Our research advances the field of video understanding by developing more accurate and efficient models for temporal grounding and video question answering. While temporal grounding with UniTime offers significant benefits for a variety of video understanding applications, unexpected behaviors may occasionally result in misrepresentations of the underlying video content. For applications that demand extremely precise temporal localization for safety-critical decisions, such as embodied intelligence tasks, it is crucial to carefully manage any such behaviors. To ensure the reliability of systems that rely on temporal grounding predictions, we recommend conducting thorough evaluations and implementing robust mitigation strategies to minimize potential risks. Through these precautions, the overall safety and effectiveness of these applications can be substantially enhanced.
+
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+
+# Answer: [Yes]
+
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+
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\ No newline at end of file
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+# Universal Visuo-Tactile Video Understanding for Embodied Interaction
+
+Yifan Xie $^{1}$ , Mingyang Li $^{1}$ , Shoujie Li $^{1}$ , Xingting Li $^{1}$ , Guangyu Chen $^{2}$ , Fei Ma $^{3*}$ , Fei Yu $^{3}$ , Wenbo Ding $^{1*}$
+
+$^{1}$  Tsinghua Shenzhen International Graduate School, Tsinghua University  $^{2}$  Shenzhen Campus of Sun Yat-sen University
+
+<sup>3</sup> Guangdong Laboratory of Artificial Intelligence and Digital Economy (SZ)
+
+# Abstract
+
+Tactile perception is essential for embodied agents to understand physical attributes of objects that cannot be determined through visual inspection alone. While existing approaches have made progress in visual and language modalities for physical understanding, they fail to effectively incorporate tactile information that provides crucial haptic feedback for real-world interaction. In this paper, we present VTV-LLM, the first multi-modal large language model for universal Visuo-Tactile Video (VTV) understanding that bridges the gap between tactile perception and natural language. To address the challenges of cross-sensor and cross-modal integration, we contribute VTV150K, a comprehensive dataset comprising 150,000 video frames from 100 diverse objects captured across three different tactile sensors (Gel-Sight Mini, DIGIT, and Tac3D), annotated with four fundamental tactile attributes (hardness, protrusion, elasticity, and friction). We develop a novel three-stage training paradigm that includes VTV enhancement for robust visuo-tactile representation, VTV-text alignment for cross-modal correspondence, and text prompt finetuning for natural language generation. Our framework enables sophisticated tactile reasoning capabilities including feature assessment, comparative analysis, scenario-based decision making and so on. Experimental evaluations demonstrate that VTV-LLM achieves superior performance in tactile video understanding tasks, establishing a foundation for more intuitive human-machine interaction in tactile domains.
+
+# 1 Introduction
+
+Touch is a fundamental sensory modality that provides humans with physical information unattainable through vision alone, such as material attributes, surface texture, and compliance. This tactile feedback enables sophisticated physical reasoning and interaction in our environment [1, 2, 3]. While recent advances in vision-language models [4, 5, 6, 7, 8] have demonstrated impressive capabilities in visual reasoning, these models remain fundamentally limited by their inability to perceive tactile attributes, restricting their effectiveness in scenarios requiring physical interaction and reasoning about material characteristics that cannot be reliably inferred from visual cues alone.
+
+Visuo-tactile sensors [9], like GelSight [10], DIGIT [11], and Tac3D [12], have emerged as promising technologies for capturing tactile information, generating image-like representations that encode physical properties such as pressure distribution, surface geometry, and friction characteristics. However, there remains a significant challenge in bridging the domain gap between these tactile representations and natural language understanding. The inherent differences between tactile data
+
+![](images/cf17677521c9f48782fc136e1509a044d74f394e41057cbe512ee9c5df143efd.jpg)  
+Figure 1: The workflow consists of four key components: (a) Data Collection, which includes 100 diverse objects recorded by 3 different tactile sensors, resulting in 150,000 video frames; (b) Attribute Annotation, where objects are systematically categorized across 4 static and dynamic tactile attributes: hardness, protrusion, elasticity, and friction; (c) Template Generation, which generates 10,000 question-answer pairs using structured templates for various comparative analyses; and (d) Embodied Interaction, demonstrating VTV-LLM's capability to perform tactile feature assessment, surface feature distinction, tactile scenario analysis and so on. Through this integrated approach, VTV-LLM enables multi-modal reasoning about physical attributes that cannot be determined through visual inspection alone, creating a foundation for more sophisticated human-machine interaction in tactile understanding domains.
+
+captured across various sensor types further complicates this integration, as each sensor produces distinct data formats with varying resolutions and physical property encodings.
+
+Existing research on tactile learning has made progress in representation learning [13, 14, 15, 16, 17, 18], but these approaches often focus either exclusively on static attributes or fail to develop comprehensive frameworks that integrate both tactile perception and language understanding. Most critically, they lack the ability to ground tactile perceptions in natural language descriptions and reasoning, which is essential for human-machine communication about physical properties and interactions [19, 20]. Additionally, the temporal dimension of tactile interactions, which captures how surfaces respond to pressing, sliding, and rotational movements, remains underexplored in current approaches, despite containing crucial information about dynamic material attributes.
+
+To address these challenges, we present VTV-LLM, the first multi-modal large language model for universal visuo-tactile video understanding. Our approach treats tactile perception as a cross-modal reasoning problem, where tactile videos are aligned with linguistic descriptions to enable sophisticated reasoning about physical attributes. As illustrated in Fig. 1(d), VTV-LLM supports a diverse range of embodied interaction capabilities, from basic tactile feature assessment to complex comparative analyses and scenario-based decision making. Additionally, we construct the VTV150K dataset, comprising 150,000 video frames collected from 100 common objects across three different tactile sensors. We systematically annotate these videos with four fundamental tactile attributes (hardness, protrusion, elasticity, and friction), creating a structured foundation for tactile reasoning. To bridge the substantial gap between tactile perception and language understanding, we develop a three-stage training paradigm: (1) VTV enhancement through optical flow-guided masking to learn
+
+robust visuo-tactile representations, (2) VTV-text alignment to establish cross-modal correspondence, and (3) text prompt finetuning to optimize natural language generation about tactile attributes.
+
+Our main contributions can be summarized as follows:
+
+- We introduce VTV-LLM, the first multi-modal large language model capable of universal visuo-tactile video understanding, enabling sophisticated embodied reasoning through natural language interaction.  
+- We contribute VTV150K, a comprehensive dataset of 150,000 visuo-tactile video frames capturing 100 diverse objects across three tactile sensors, annotated with four fundamental tactile attributes.  
+- We develop a novel three-stage training paradigm that effectively bridges the domain gap between tactile perception and language understanding, providing a valuable reference for future cross-modal learning efforts.
+
+# 2 Related Works
+
+Tactile Perception Tactile perception has evolved significantly from early sensors measuring basic physical properties to sophisticated vision-based systems providing high-resolution contact information. Visuo-tactile sensors [9] such as GelSight [10], DIGIT [11], and Tac3D [12] have garnered widespread attention for their ability to capture detailed contact deformations through elastomeric gels and embedded cameras. These sensors have enabled numerous robotic applications including material classification [21], shape reconstruction [22, 23], and dexterous manipulation tasks [24, 14]. Recent research has focused on developing representation learning approaches for tactile data, progressing from task-specific models [25] to general-purpose representations using self-supervised techniques like contrastive multi-view coding [21] and masked autoencoders [26]. The integration of tactile sensing with other modalities has also emerged as a promising direction, with works like UniTouch [17] dynamically fusing tactile signals with visual and audio data to enhance cross-sensor knowledge transferability, Yu et al. [15] aligned tactile images with vision-language models for object property reasoning, and Fu et al. [16] used a touch-vision-language model for open-vocabulary classification. Unlike prior works, our method processes visuo-tactile video directly and focuses on sophisticated tactile reasoning.
+
+Self-Supervised Video Representation Learning Self-supervised video representation learning has emerged as a critical area for developing robust visual features without manual annotations. VideoMAE [27] pioneered this approach by effectively adapting masked autoencoding strategies to the video domain, demonstrating significant performance improvements across various benchmark tasks. Subsequently, VideoMAEv2 [28] enhanced this framework through the introduction of dual masking mechanisms, which substantially improved computational efficiency while maintaining representational power. Recent advancements in this field have focused on sophisticated optimizations along both temporal and spatial dimensions [29, 30, 31, 32], addressing challenges unique to video understanding such as motion coherence and long-range dependencies. In the tactile domain, Sparsh [18] explored the ability of different existing self-supervised learning methods to characterize in tactile video. Feng et al. [13] utilized the tube masking strategy to process the tactile video. Our method builds upon these foundations by introducing optical flow-guided masking specifically designed for visuo-tactile videos, which addresses the unique challenges of capturing both spatial deformation and temporal dynamics in tactile interactions.
+
+Multi-Modal Large Language Models Multimodal Large Language Models (MLLMs) have transformed AI research by enabling reasoning across textual and visual modalities. Early efforts integrated LLMs as agents for downstream tasks [33, 34, 35]. Later approaches focused on parameter-efficient tuning [36, 37] and instruction tuning [38, 39] to align visual semantics with language. Recent advances have incorporated video processing [40, 41] and diverse sensory inputs [42], enabling applications in robotics [43, 44]. In our work, we present the first visuo-tactile video large language model to bridge the gap between tactile perception and natural language.
+
+# 3 Methods
+
+In this section, we first introduce VTV150K, a large-scale dataset of video-question-answer pairs in Sec. 3.1. Subsequently, we present VTV-LLM, the first visuo-tactile video large language model designed for visuo-tactile video understanding and embodied interaction in Sec. 3.2.
+
+# 3.1 VTV150K
+
+Overview Visuo-tactile sensor technologies suffer from inadequate standardization and significant cross-sensor data discrepancies, which substantially impede the transferability of tactile representation models across different sensing platforms. Existing methods [14, 18, 45, 13] addressing these challenges exhibit notable limitations, as they either neglect the integration of both static and dynamic tactile attributes or fail to incorporate comprehensive visuo-tactile video understanding for embodied interaction.
+
+In this work, we introduce VTV150K, a comprehensive large-scale dataset comprising video-question-answer pairs collected across three diverse visuo-tactile sensors, as illustrated in Fig. 1(a-c). The dataset construction methodology encompasses three sequential stages: data collection, attribute annotation, and template generation. We will delve into the specifics of these stages.
+
+Data Collection To facilitate the grounding of embodied interaction on tactile inputs, we collected a comprehensive dataset comprising 100 common objects, yielding a total of 150,000 visuo-tactile video frames, with each video recorded at 20 FPS and a resolution of  $320 \times 320$  pixels.
+
+As illustrated in Fig. 1(a), we employed multiple visuo-tactile sensors to ensure style diversity: GelSight mini [10] and DIGIT [11] sensors for capturing high-resolution visuo-tactile information, and Tac3D [12] for measuring deformation force fields. Due to the relatively low resolution of Tac3D, we implemented the cubic spline interpolation algorithm [46] to reconstruct more detailed force field representations.
+
+Data collection was performed manually to address the challenges associated with properly interacting with irregularly shaped objects. For each object, we systematically captured five visuo-tactile videos across different regions using various sensors. Our data collection process consisted of three sequential interactions: (1) normal pressing against the object surface to capture pressure distribution, (2) rotational movement to acquire shear information, and (3) sliding motion to obtain friction characteristics. This multi-interaction approach enables comprehensive tactile information extraction for embodied interaction.
+
+Attribute Annotation To facilitate tactile reasoning, we annotated our dataset across four fundamental static and dynamic tactile attributes as shown in Fig. 1(b). Each attribute was categorized into three distinct levels, with harness classified as highly deformable (28%), moderately deformable (33%), and extremely hard (39%); protrusion categorized as absent (41%), moderate (26%), or strong (33%); elasticity measured as none (42%), moderate (30%), or strong (28%); and friction assessed as slight (32%), moderate (25%), or strong (43%). This structured annotation framework enables comprehensive tactile attribute analysis for downstream reasoning tasks.
+
+Template Generation Template generation facilitates the creation of question-answer pairs for model training. We developed multiple problem templates encompassing various reasoning tasks: tactile feature assessment, surface feature distinction, texture optimal selection and so on. To instantiate these templates, we systematically integrated diverse visuo-tactile video combinations, comparison operators (e.g., "more", "less", "most", "least"), and attribute selectors to generate a comprehensive dataset of 10,000 question-answer pairs. As illustrated in Fig. 1(c), our generation process follows a hierarchical framework: selection, filtration, and structured question formulation with corresponding ground-truth annotations. For more comprehensive details about attribute annotation and template generation, please refer to the Supplementary Material ??
+
+# 3.2 VTV-LLM
+
+Overview VTV-LLM aims to serve as a multi-modal framework capable of integrating visual-tactile video data with large language models to facilitate tactile reasoning for embodied interaction.
+
+![](images/a64b8e8681bc37e85a25947cfa9f8254b913a13d17416918e38884ecab4c44c6.jpg)  
+Figure 2: (a) VTV-LLM framework: A multi-modal system integrating visual-tactile video data with large language models to facilitate tactile reasoning for embodied interaction; (b) Multi-Stage Training: It consists of VTV enhancement, alignment between visuo-tactile video and text, and prompt-based finetuning to generate accurate tactile descriptions.
+
+As illustrated in Fig. 2(a), VTV-LLM formulates tactile perception as a cross-modal approach to question answering and descriptive generation. By leveraging the rich sensory information inherent in visuo-tactile video data, VTV-LLM enhances understanding in scenarios traditionally challenging for standard vision-only models, particularly in applications requiring tactile attribute inference.
+
+At the core of VTV-LLM lies a (Qwen 2.5 [4, 5]) that synthesizes complex multi-modal information from visuo-tactile videos, utilizing world knowledge to generate coherent, human-readable descriptions of tactile attributes. In general, a visuo-tactile video can be mathematically represented as a sequence of frames  $\mathcal{V} = \{I_t\}_{t=0}^T$ , where each frame  $I_t$  captures both visual and tactile information at timestamp  $t$ . Initially, high-dimensional features  $F_{VTV}$  are extracted from  $\mathcal{V}$  using a VTV encoder based on ViT-base architecture [47] adapted from VideoMAE [27, 28]:
+
+$$
+F _ {V T V} = f _ {\mathrm {e n c}} (V) = \operatorname {V i T} \left(\left\{\operatorname {P a t c h} \left(I _ {t}\right) + \operatorname {T E} (t) \right\} _ {t = 0} ^ {T}\right), \tag {1}
+$$
+
+where  $\mathrm{Patch}(\cdot)$  denotes the patch embedding operation and  $\mathrm{TE}(t)$  represents temporal embeddings. These features are then processed through a visual projector  $f_{V - proj}$  consisting of two linear layers with a GELU activation function [48] in between to produce the visual embedding  $E_V$ :
+
+$$
+E _ {V} = f _ {V - p r o j} \left(F _ {V T V}\right) = W _ {2} \cdot \operatorname {G E L U} \left(W _ {1} \cdot F _ {V T V} + b _ {1}\right) + b _ {2}, \tag {2}
+$$
+
+where  $W_{1}, W_{2}$  are learnable weight matrices and  $b_{1}, b_{2}$  are bias terms. Concurrently, the textual prompt is tokenized and processed through LLM's text projector to produce text embedding  $E_{T}$ . For effective multi-modal reasoning, we introduce special tokens <video_start>, <video>, and
+
+![](images/c8d8043b2b153b1958482e4f71b9e69009bfde6ff0472b75f22a9ffabed32d8b.jpg)  
+Figure 3: Training pipeline of VTV enhancement.
+
+<video_end> to denote the beginning, content and end of the visuo-tactile video in the input sequence. These tokens serve as anchors for the model to properly align visual information with textual understanding during the inference process.
+
+Given these aligned representations, the large language model  $f_{LLM}$  performs reasoning to generate a response  $A$  describing tactile attributes:
+
+$$
+A = f _ {L L M} \left(E _ {V}, E _ {T}\right) = \operatorname {Q w e n} \left(\operatorname {C o n c a t} \left([ E _ {V}; E _ {T} ]\right)\right). \tag {3}
+$$
+
+Given the complexity of integrating visuo-tactile information with language representations, we implement a staged training approach to develop our framework. As shown in Fig. 2(b), VTV-LLM adopts a three-stage training paradigm encompassing VTV enhancement, VTV-text alignment, and text prompt finetuning. This structured progression enables the model to first learn robust tactile-visual representations, then align them with textual descriptions, and finally optimize response generation, enhancing VTV-LLM's capability for cross-modal understanding and tactile reasoning. In the following, we describe each of these stages in detail.
+
+VTV Enhancement Existing multi-modal LLMs predominantly process natural images via unmodified Vision Transformer (ViT) encoders [47]. However, our research addresses visuo-tactile inputs, which exhibit fundamentally different characteristics from natural images, thus necessitating specialized fine-tuning to extract meaningful representations.
+
+Furthermore, the temporal nature of our video data introduces challenges not present in static images. Unlike images, videos possess an inherent time dimension characterized by temporal redundancy and inter-frame correlations, requiring robust video representation methodologies. While VideoMAE [27, 28] offers a powerful masked video autoencoder with an asymmetric encoder-decoder architecture utilizing tube masking, this approach assumes minimal motion across large frame regions. This assumption proves problematic for visuo-tactile videos, which typically exhibit significant motion patterns. Direct application of tube masking to such inputs risks substantial information leakage, wherein the model can trivially reconstruct masked segments using visible tokens from temporally adjacent frames, which is a critical concern in masked video pre-training. To address these limitations, we propose a novel training pipeline specifically designed for visuo-tactile video representation, as illustrated in Fig. 3.
+
+Given the visuo-tactile video sequence  $\mathcal{V} = \{I_t\in \mathbb{R}^{H\times W\times C}\}_{t = 0}^T$ , where each frame  $I_{t}$  encodes both visual and tactile information at timestamp  $t$  with spatial dimensions  $H\times W$  and  $C$  channels, we propose selecting the middle frame as the keyframe. This selection is motivated by empirical observations that the middle frame typically exhibits the maximum contact surface area, facilitating more robust optical flow warping in subsequent processing stages. For keyframe mask initialization, conventional binarization approaches [49] significantly degrade the spatial continuity of object surfaces, compromising the fidelity of the reconstructed tactile information. Therefore, we introduce a Gaussian mixture model [50] to obtain the keyframe mask. For the keyframe  $I_{k}$ , we formulate a probabilistic mask using localized Gaussian functions. We select a set of  $N = \lceil \alpha \cdot HW / \beta^2\rceil$  sampling points  $\{p_i\}_{i = 1}^N$  distributed across the frame, where  $\alpha \in (0,1)$  controls density and  $\beta$  is the sampling grid size. Each point  $p_i$  generates a Gaussian kernel  $G_{i}(x,y) = \exp \left(-\frac{(x - p_{i_x})^2 + (y - p_{i_y})^2}{2\lambda^2}\right)$  with scale parameter  $\lambda$ . The final keyframe mask is defined as  $M_k' = \min \left(1,\sum_{i = 1}^N G_i\right)$ , creating
+
+a continuous-valued mask that preserves spatial structure while enabling controlled sparsity for subsequent processing.
+
+Additionally, we employ dense motion estimation across the visuo-tactile video  $\mathcal{V}$  using the RAFT architecture [51]. We compute bidirectional optical flow fields between consecutive frames to capture the continuous deformation patterns throughout the interaction process. For each adjacent frame pair, we define the forward flow field  $\mathcal{O}_{t\rightarrow t + 1} = \mathrm{RAFT}(I_t,I_{t + 1})$ . Each flow field  $\mathcal{O}_{t\rightarrow t + 1}\in \mathbb{R}^{H\times W\times 2}$  encodes pixel-wise displacement vectors  $(u_{x,y},v_{x,y})$  for every spatial location  $(x,y)$ , mapping positions from frame  $I_{t}$  to their corresponding locations in frame  $I_{t + 1}$ . The complete set of optical flows  $\Phi$  for the sequence is formulated as:
+
+$$
+\Phi = \bigcup_ {t = 0} ^ {k - 1} \left\{\mathcal {O} _ {t \rightarrow t + 1} \right\} \cup \bigcup_ {t = k + 1} ^ {T} \left\{\mathcal {O} _ {t \rightarrow t - 1} \right\}. \tag {4}
+$$
+
+This bidirectional flow representation tracks visuo-tactile features throughout the interaction, supporting warping operations and masked frame generation. We apply spatial normalization before flow computation to ensure scale invariance across different sequences.
+
+After that, we utilize the backward warping [52, 53] to generate the temporal consistent masking map based on the keyframe and mask the corresponding video frames. The masked visuo-tactile frames  $\mathcal{V}_m = \{M_t\}_{t=0}^T$  are fed into the VTV encoder-decoder architecture for reconstruction using the mean squared error loss [27, 28]. We also incorporate an attribute classifier to predict tactile attributes (hardness, protrusion, elasticity, and friction) using the cross-entropy loss [54]. Our total loss function combines both the reconstruction loss and the attribute classification loss, enabling simultaneous optimization of visuo-tactile reconstruction quality and tactile attribute classification accuracy.
+
+VTV-Text Alignment In the VTV-text alignment stage, we focus on establishing cross-modal alignment between video and language representations. With the pretrained VTV Encoder from stage 1, we introduce both V-Projector and T-Projector modules while keeping the Large Language Model frozen. This stage leverages our initial constructed VTV150K dataset to bridge the representational gap between visual and textual modalities. The V-Projector maps video embeddings from the VTV Encoder into the language model's embedding space, while the T-Projector processes corresponding text prompt representations. By training these projection modules exclusively while freezing other components, we establish foundational cross-modal understanding, enabling the model to associate visual content with appropriate textual descriptions. This alignment is critical for downstream video understanding and description tasks as it creates a shared semantic space between the video frames and natural language.
+
+Text Prompt Finetuning In the text prompt finetuning stage, we enhance the model's capacity to respond accurately to textual prompts about video content by implementing supervised fine-tuning across multiple components. The V-Projector, and T-Projector are jointly fine-tuned along with the LLM. Unlike previous stages where the LLM remained frozen, this stage employs parameter-efficient techniques [37, 36] to fine-tune the language model using 10,000 newly generated question-answer pairs. These pairs are created using the same template generation approach as our VTV150K dataset, featuring diverse video understanding tasks. By generating new data rather than reusing subsets, we significantly increase training diversity and model robustness. This end-to-end optimization enables the model to generate more coherent, accurate, and contextually relevant responses to text prompts about video content. The supervised nature of this phase significantly improves the model's ability to comprehend complex video scenes and produce natural language descriptions that align with human expectations. This final stage integrates the previously aligned representations into a cohesive multi-modal understanding system, culminating in enhanced video-language capabilities.
+
+# 4 Experiments
+
+# 4.1 Setup
+
+Our experiments utilize the proposed VTV150K dataset for both training and evaluation protocols. The training process follows our three-stage paradigm: Stage 1 employs multi-sensor visuo-tactile videos with their corresponding attribute annotations for representation learning. For Stage 2 and 3,
+
+![](images/425f0c2a20e00c1355f0437a75e87575cd64e36ef7dcbc995d447562aae3fbd5.jpg)  
+Figure 4: Several task examples from the proposed VTV150K along with predictions from VTV-LLM.
+
+# Surface Optimality Identification
+
+Question: Among these three videos: a)  $< \mathrm{a.mp4} >$ , b)  $< \mathrm{b.mp4} >$ , c)  $< \mathrm{c.mp4} >$ , identify the softest object after describing each one.
+
+VTV-LLM: a) It presents a extremely hard and strong protrusion surface with no elasticity and strong friction. b) It presents a highly deformable and moderate protrusion surface with moderate elasticity and friction. c) It presents a extremely hard and no protrusion surface with no elasticity and slight friction. The softest object is b).
+
+# Object Sensation Correlation
+
+Question: Among these three videos: a)  $<a.mp4>$ , b)  $<b.mp4>$ , c)  $<c.mp4>$ , identify the softest object after describing each one. You have tactile videos of three different objects: a)  $<a.mp4>$ , b)  $<b.mp4>$ , c)  $<c.mp4>$ . Assign each video (a, b, c) to one of these objects listed alphabetically: 1) an iron ruler, 2) a kiwi, 3) a circuit board.
+
+VTV-LLM: a) is a circuit board, b) is a kiwi and c) is an iron ruler.
+
+# Tactile Scenario Analysis
+
+Question: Describe these three videos based on their tactile attributes: a)  $<a.mp4>$ , b)  $<b.mp4>$ , c)  $<c.mp4>$ . Which object would be better for scraping ice off a windshield?
+
+VTV-LLM: a) ... b) ... c)... The most suitable object is c).
+
+Table 1: Performance comparison of VTV-LLM-7B against seven state-of-the-art methods on the VTV150K dataset. The evaluation covers different tasks, with results reported in percentages (\%) and the boldface indicates the best performance.  
+
+<table><tr><td>Models</td><td>Hardness</td><td>Protrusion</td><td>Elasticity</td><td>Friction</td><td>Combined</td><td>SFD</td><td>SOI</td><td>OSC</td><td>TSA</td><td>Average</td></tr><tr><td>GPT-4o [55]</td><td>34.7</td><td>32.6</td><td>32.6</td><td>18.7</td><td>2.1</td><td>40.9</td><td>38.4</td><td>16.6</td><td>36.0</td><td>28.0</td></tr><tr><td>Gemini-2.5-Pro-Exp [56]</td><td>36.2</td><td>34.7</td><td>39.1</td><td>21.0</td><td>4.3</td><td>42.6</td><td>29.4</td><td>18.5</td><td>40.0</td><td>29.5</td></tr><tr><td>LLaVA-OneVision-7B [57]</td><td>27.5</td><td>32.6</td><td>26.0</td><td>20.2</td><td>0.7</td><td>40.9</td><td>28.2</td><td>11.7</td><td>30.0</td><td>24.2</td></tr><tr><td>LLaVA-Video-Qwen2-7B [58]</td><td>30.4</td><td>29.7</td><td>28.9</td><td>18.1</td><td>2.1</td><td>33.6</td><td>29.4</td><td>17.2</td><td>36.0</td><td>25.0</td></tr><tr><td>InternVL2.5-VL-8B [59]</td><td>18.1</td><td>23.9</td><td>21.0</td><td>13.7</td><td>0.0</td><td>24.5</td><td>17.9</td><td>11.1</td><td>24.0</td><td>17.1</td></tr><tr><td>VideoLLaMA3-7B [41]</td><td>15.2</td><td>21.7</td><td>14.4</td><td>10.8</td><td>0.0</td><td>11.4</td><td>12.8</td><td>7.4</td><td>20.0</td><td>12.6</td></tr><tr><td>Qwen2.5-VL-7B [60]</td><td>25.3</td><td>28.9</td><td>17.3</td><td>15.9</td><td>1.4</td><td>22.9</td><td>28.2</td><td>16.0</td><td>30.0</td><td>20.6</td></tr><tr><td>VTV-LLM-7B (Ours)</td><td>73.9</td><td>75.0</td><td>67.3</td><td>56.5</td><td>35.6</td><td>71.3</td><td>57.6</td><td>43.2</td><td>64.0</td><td>60.4</td></tr></table>
+
+we utilize two independently generated sets of 10,000 question-answer pairs to prevent data leakage between stages. To evaluate model performance, we create a separate test set comprising 600 question-answer pairs for novel objects not present in the training data, ensuring comprehensive coverage across various tactile reasoning tasks. Our LLM backbone is based on Qwen 2.5 [4, 5], experimenting with three model variants (3B, 7B, and 14B parameters). All experiments are conducted on 4 NVIDIA RTX 6000 Ada GPUs. Additional implementation details and hyperparameter configurations are provided in the Supplementary Material ??
+
+# 4.2 Results
+
+To verify the effectiveness of our VTV-LLM, we compare it against two strong proprietary models, such as GPT-4o [55] and Gemini-2.5-Pro-Exp [56], as well as five open-source video-based VLMs, including LLaVA-OneVision-7B [57], LLaVA-Video-Qwen2-7B [58], InternVL2.5-VL-8B [59], VideoLLaMA3-7B [41] and Qwen2.5-VL-7B [60]. Since most of the video-based VLM models have parameters around 7B, we only use the VTV-LLM-7B model for fair comparison. To guarantee the robustness of the experimental results, we report the average results of the triplicate test with random seeds.
+
+Our first experiment focuses on tactile feature assessment, which evaluates the model's ability to perceive and describe physical sensory attributes of objects in visuo-tactile videos. As illustrated in Fig. 1(d), when presented with a visuo-tactile video and a question prompt, VTV-LLM generates descriptions of the four key tactile attributes. The results presented in Tab. 1 demonstrate that our method consistently outperforms all baseline models across both individual attribute and combined attribute settings. The performance gap is particularly notable in the combined attribute setting, which we attribute to our three-stage training paradigm that effectively bridges the domain gap between tactile perception and natural language understanding.
+
+In addition, we conduct high-level tactile reasoning experiments, including surface feature distinction (SFD), surface optimality identification (SOI), object sensation correlation (OSC), and tactile scenario analysis (TSA). SFD involves comparing tactile qualities between objects to determine relative differences, SOI entails analyzing multiple surfaces to determine which exhibits the highest degree of a particular quality, OSC aims at relating tactile perceptual information to the identity of a
+
+![](images/6112a947921eea7c4383d3bf9e172ee3c28df436905156cc8c5988ffc50628a1.jpg)  
+Figure 5: Performance comparison of VTV-LLM on the different parameters.
+
+Table 2: Ablation study on VTV encoder settings using the VTV-LLM-7B model.  
+
+<table><tr><td>Settings</td><td>SFD</td><td>SOI</td><td>OSC</td><td>TSA</td><td>Average</td></tr><tr><td>VideoMAE (w/o train)</td><td>37.5</td><td>29.7</td><td>8.5</td><td>16.0</td><td>22.9</td></tr><tr><td>VideoMAE (w/ train)</td><td>52.4</td><td>46.1</td><td>28.3</td><td>38.0</td><td>41.2</td></tr><tr><td>Ours (w/ocls)</td><td>62.2</td><td>48.7</td><td>40.1</td><td>55.0</td><td>51.5</td></tr><tr><td>Ours</td><td>71.3</td><td>57.6</td><td>43.2</td><td>64.0</td><td>59.0</td></tr></table>
+
+Table 3: Ablation study on three-stage training paradigm settings using the VTV-LLM-7B model.  
+
+<table><tr><td>Settings</td><td>SFD</td><td>SOI</td><td>OSC</td><td>TSA</td><td>Average</td></tr><tr><td>w/o stage 2</td><td>58.1</td><td>50.0</td><td>35.2</td><td>60.0</td><td>50.8</td></tr><tr><td>w/o stage 3</td><td>50.8</td><td>42.3</td><td>29.0</td><td>52.0</td><td>43.5</td></tr><tr><td>Same dataset</td><td>61.4</td><td>53.8</td><td>33.9</td><td>58.0</td><td>51.7</td></tr><tr><td>Ours</td><td>71.3</td><td>57.6</td><td>43.2</td><td>64.0</td><td>59.0</td></tr></table>
+
+particular real-world object, and TSA addresses applying haptic knowledge to real-world situations that require physical reasoning. It is worth noting that the TSA task is not included in the training set. The qualitative results presented in Fig. 1(d) and Fig. 4 demonstrate that VTV-LLM can generate reasonable outputs. The quantitative experimental results in Tab. 1 further confirm that VTV-LLM achieves superior performance across these complex reasoning tasks, highlighting its potential for embodied interaction.
+
+# 4.3 Ablation Studies
+
+LLM Backbone To examine the effect of model scale on visuo-tactile understanding, we compare different parameter sizes of our LLM backbone. Fig. 5 shows performance results for VTV-LLM using three Qwen 2.5 variants (3B, 7B, and 14B parameters). We observe consistent performance improvements with increasing model size. This improvement is most significant for complex reasoning tasks like TSA, indicating larger models better integrate cross-modal information. However, larger models also require substantially more computation time during inference.
+
+VTV Encoder We conduct an ablation study on our VTV encoder design as shown in Tab. 2. Baseline VideoMAE [27, 28] without training achieves only  $22.9\%$  average performance, while training with our VTV150K dataset improves it to  $41.2\%$ . Our method without the attribute classifier reaches  $51.5\%$ , showing the effectiveness of our optical flow-guided masking strategy. The full method with the attribute classifier further improves to  $59.0\%$ , confirming that joint reconstruction and attribute classification significantly enhances tactile understanding.
+
+Three-Stage Training Paradigm Tab. 3 validates our three-stage training paradigm through ablation studies. Removing stage 2 (VTV-text alignment) drops average performance to  $50.8\%$ , while omitting stage 3 (text prompt finetuning) causes a steeper decline to  $43.5\%$ . Using identical datasets across stages also underperforms at  $51.7\%$ , confirming that independent datasets for each stage significantly improve model robustness.
+
+# 5 Conclusion
+
+In this work, we presented VTV-LLM, the first multi-modal large language model for universal visuo-tactile video understanding. We contributed VTV150K, a comprehensive dataset of visuo-tactile videos across multiple sensors, and developed a novel three-stage training paradigm that effectively bridges the gap between tactile perception and natural language. Experimental results demonstrate that VTV-LLM consistently outperforms state-of-the-art methods across various tactile reasoning tasks, establishing a foundation for more intuitive human-machine interaction in embodied domains.
+
+# References
+
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+
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\ No newline at end of file
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+# Universally Invariant Learning in Equivariant GNNs
+
+Jiacheng Cen $^{123}$ , Anyi Li $^{123}$ , Ning Lin $^{123}$ , Tingyang Xu $^{45}$ , Yu Rong $^{45}$ , Deli Zhao $^{45}$ , Zihe Wang $^{123}$ , Wenbing Huang $^{123*}$
+
+<sup>1</sup> Gaoling School of Artificial Intelligence, Renmin University of China
+
+$^{2}$  Beijing Key Laboratory of Research on Large Models and Intelligent Governance
+
+<sup>3</sup> Engineering Research Center of Next-Generation Intelligent Search and Recommendation, MOE
+
+$^{4}$  DAMO Academy, Alibaba Group, Hangzhou, China  $^{5}$  Hupan Lab, Hangzhou, China {jiacc.cn, li-anyi, ninglin00}@outlook.com; {xuty_007, yu.rong}@@hotmail.com; zhaodeli@gmail.com; wang.zihe@ruc.edu.cn; hwenbing@126.com
+
+# Abstract
+
+Equivariant Graph Neural Networks (GNNs) have demonstrated significant success across various applications. To achieve completeness—that is, the universal approximation property over the space of equivariant functions—the network must effectively capture the intricate multi-body interactions among different nodes. Prior methods attain this via deeper architectures, augmented body orders, or increased degrees of steerable features, often at high computational cost and without polynomial-time solutions. In this work, we present a theoretically grounded framework called Uni-EGNN for constructing complete equivariant GNNs that is both efficient and practical. We prove that a complete equivariant GNN can be achieved through two key components: 1) a complete scalar function, referred to as the canonical form of the geometric graph; and 2) a full-rank steerable basis set. Leveraging this finding, we propose an efficient algorithm for constructing complete equivariant GNNs based on two common models: EGNN and TFN. Empirical results demonstrate that our model demonstrates superior completeness and excellent performance with only a few layers, thereby significantly reducing computational overhead while maintaining strong practical efficacy.
+
+# 1 Introduction
+
+Various types of scientific data, including chemical molecules, proteins, and other particle-based physical systems, are often represented as geometric graphs [1; 2]. This data structure not only captures node characteristics and edge information but also includes a 3D vector (such as position, velocity, etc.) for each node. To effectively process geometric graphs, equivariant Graph Neural Networks (GNNs) have been developed that enable equivariant message passing over nodes while adhering to the E(3) or SE(3) symmetries inherent in physical laws. These models have achieved significant success in various scientific tasks, including physical dynamics simulation, molecular property prediction, and protein design [3-30].
+
+In current literature, a key objective in the design of equivariant GNNs is to achieve completeness, which refers to the universal approximation property [31]. This concept was initially explored by [32], who proved the universality of high-degree steerable models, specifically TFN [33]. Subsequent models, such as SEGNN [34] and MACE [35], can be similarly confirmed to be complete by establishing their relations with TFN. Regrettably, the theory proposed by [32] applies only to point clouds (i.e., fully connected geometric graphs) and relies on sufficiently large values of the layer number, body order, and feature degree.
+
+More recently, the introduction of the GWL-test [36] and further researches [37-40] have provided a new perspective by connecting the completeness of invariant models to the geometric isomorphism problem through an extension of the Weisfeiler-Lehman (WL) test [41]. Unfortunately, the GWL-test has only quasi-polynomial solutions [42], inheriting the same weakness as the original WL-test.
+
+In this paper, we explore complete equivariant GNNs in a more effective and operable way. To begin with, we reformulate existing equivariant GNNs as an expansion of multi-body high-degree bases coupled with the Clebsch-Gordan (CG) tensor product (Eqs. (2) and (3)). This reformulation allows us to clearly identify the key limitations of current methods: they fail to achieve complete expansion when the degree and body order of CG tensor products are constrained, or incur prohibitively high computational costs otherwise. To overcome these weaknesses, we propose a novel expansion of equivariant GNNs (Eq. (4)), which does not necessarily require CG tensor product: the sum over a finite basis set with dynamic weights dependent on the input. The completeness is achieved if the weights (called the scalar function) are complete and the basis set is full-rank.
+
+![](images/332b1a071ed43957d6f30c87319987e8b268d58a2c4fd778ef3cb7e0af588648.jpg)  
+Figure 1: Difference between current methods and our Method. (a) Current methods can be understood as an expansion of multi-body high-degree bases  $\tilde{b}_i^{(l)}$  coupled with the Clebsch-Gordan (CG) tensor product (Eq. (3)). The completeness is achieved with sufficiently large values of the body order and feature degree. (b) In contrast, our method proposes a novel expansion which does not necessarily require CG tensor product (Eq. (4)): the sum over a finite basis set with dynamic weights dependent on the input. The completeness is ensured if the weights (the scalar function  $\Gamma (\mathcal{G})$ ) are complete and the basis set  $\tilde{\boldsymbol{V}}^{(l)}$  is full-rank.
+
+![](images/f8250ad85025da1e368596cd9d3660ec88b989e437ed87b13119ffc2572fcef5.jpg)
+
+Interestingly, constructing a complete scalar function is equivalent to solving the geometric isomorphism problem, which, unlike the traditional isomorphism task on topological graphs, can be resolved in polynomial time [42]. Actually, we have presented an algorithm operating with a complexity of  $\mathcal{O}(N^6)$  where  $N$  is the number of nodes, based on the four-point positioning principle, and further turned it into canonical form for comprehensive embeddings of geometric graphs. Additionally, we theoretically prove that a full-rank basis set of any degree can always be constructed, if the input geometric graph is asymmetric. Building on this foundation, we propose a more efficient algorithm with a complexity of  $\mathcal{O}(N^2)$  to construct the canonical form specifically for asymmetric graphs.
+
+Thanks to our theoretical findings, we modify existing equivariant GNNs (such as EGNN [43] and TFN [33]) into complete models. Specifically, we introduce two complete implementations of EGNN, termed  $\mathrm{EGNN / TFN_{cpl}}$ -global and  $\mathrm{EGNN / TFN_{cpl}}$ -local. We conduct seven sets of experiments, validating our theoretical results and demonstrating the superior performance of our method compared to previous models. Our model achieves great performance with few layers, thereby significantly reducing computational overhead while maintaining strong practical efficacy<sup>2</sup>.
+
+# 2 Related Work
+
+Equivariant GNNs. According to the operators used to build the model, equivariant GNNs can be categorized into two main classes: scalarization-based models and tensor-product-based models. Scalarization-based models (e.g., EGNN [43] and PaiNN [44]) utilize scalars (e.g. distance and angles) as coefficients to linearly combine 3D vectors during node updates. In contrast, tensor-product-based models (e.g., TFN [33] and MACE [35]) utilize spherical harmonics to maintain the equivariance of message passing and realize interactions between steerable features of different degrees through CG tensor products. Historically, researchers maintained that tensor-product-based models could provide richer information, despite their significantly higher computational resource demands [45-47]. However, the recent success of spherical-scalarization models such as SO3KRATES [48], HEGNN [49], and GotenNet [50] has prompted a reevaluation of this assumption, suggesting a
+
+potential shift towards scalar learning. Our research provides an important conclusion in this direction: a complete  $l$ th-degree steerable model can be constructed using two components—a complete scalar model and a full-rank basis set of  $l$ th-degree steerable features.
+
+Completeness of Equivariant GNNs. Completeness in equivariant GNNs refers to their ability to approximate any continuous function with arbitrary precision on geometric graphs, thus sparking diverse research avenues. Initially, [32] demonstrated the universality of tensor-product-based models, notably TFN [33], in fully connected geometric graphs. Subsequently, building on the work of [51], later studies began examining the completeness of scalar models. For instance, [52] established the completeness of models such as DimeNet [53], GemNet [54], and SphereNet [55] under  $\mathcal{A}$ -unsymmetry conditions. Meanwhile, frame-based models [56-59] emerged, innovatively separating geometric structures into group operations and orbits for separate processing. However, this approach can disrupt permutation invariance in symmetric graphs. Furthermore, the GWL test framework, inspired by the WL-test on topological graphs, offers an upper bound on the expressive power of equivariant GNNs over sparse graphs, thereby influencing further researches [37-40]. Nevertheless, it is worth noting that many of these methods rely on message passing and require multiple iterations, akin to multi-layer networks. Our method utilizes the canonical form to derive a complete scalar function, and when integrated with a basis set, facilitates the construction of a fully equivariant neural network that can be effectively achieved with just a single-layer architecture.
+
+# 3 Method
+
+In this section, we first introduce necessary preliminaries in § 3.1. Then in § 3.2, we review existing equivariant GNNs from the perspective of basis construction. Subsequently, we present our framework in § 3.3, which proposes to construct complete equivariant GNNs from the perspective based on output space. The two components to achieve completeness, including a canonical form of geometric graphs and a full-rank steerable basis set, are presented in § 3.4. Finally in § 3.5, we demonstrate how to modify typical equivariant GNNs to be complete based on our theoretical findings. Comprehensive theoretical analysis and proofs are provided in Appendix A.
+
+# 3.1 Preliminaries
+
+Geometric graph. A geometric graph of  $N$  nodes is defined as  $\mathcal{G} \coloneqq (H, \vec{X}; \mathbf{A})$ , where  $\mathbf{H} \coloneqq \{\pmb{h}_i \in \mathbb{R}^{C_H}\}_{i=1}^N$  and  $\vec{\pmb{X}} \coloneqq \{\vec{x}_i \in \mathbb{R}^3\}_{i=1}^N$  are node features and 3D coordinates, respectively;  $\mathbf{A} \in \mathbb{R}^{N \times N}$  is the adjacency matrix representing topological connections between different nodes and each edge could be assigned with an edge feature  $e_{ij}$  if necessary.
+
+Equivalence. Let  $\mathbb{X}$  and  $\mathbb{Y}$  be the input and output vector spaces, respectively. A function  $\phi : \mathbb{X} \to \mathbb{Y}$  is called equivariant with respect to group  $\mathfrak{G}$  if
+
+$$
+\forall \mathfrak {g} \in \mathfrak {G}, \phi (\rho_ {\mathbb {X}} (\mathfrak {g}) \boldsymbol {x}) = \rho_ {\mathbb {Y}} (\mathfrak {g}) \phi (\boldsymbol {x}), \tag {1}
+$$
+
+where  $\rho_{\mathbb{X}}$  and  $\rho_{\mathbb{Y}}$  are the group representations in the input and output spaces, respectively. Since translation invariance can always be achieved by translating the center of all coordinates to the origin, we omit translation and focus solely on equivariance concerning O(3), the group consisting of rotation and inversion. This implies  $\mathrm{O}(3) = \mathrm{SO}(3)\times C_i$ , where SO(3) represents the rotation group, and  $C_i = \{\mathfrak{e},\mathfrak{i}\}$  denotes the inversion group, with  $\mathfrak{e}$  as the identity and  $\mathfrak{i}$  as the inversion. We define two additional groups: the Euclidean group E(3) that further involves translation into O(3), and permutation group  $\mathrm{S}_N$  acting on a set of  $N$  elements.
+
+Irreducible representations and steerable features. For a group element  $\mathfrak{r} \in \mathrm{SO}(3)$ , its representation is typically achieved through irreducible representations such as the Wigner-D matrix  $D^{(l)}(\mathfrak{r}) \in \mathbb{R}^{(2l + 1) \times (2l + 1)}$ , where  $l$  denotes the degree. We call features  $\tilde{\pmb{v}}^{(l)}$  as  $l$ th-degree steerable features if they are transformable using  $D^{(l)}$ . The most common steerable features can be obtained via spherical harmonics, expressed as  $Y^{(l)}(\vec{x} / \| \vec{x} \|) = [Y_m^{(l)}(\vec{x} / \| \vec{x} \|)]_{m = -l}^l$ , which satisfy the property  $Y^{(l)}(R_{\mathfrak{r}}\vec{x} / \| R_{\mathfrak{r}}\vec{x} \|) = D^{(l)}(\mathfrak{r})Y^{(l)}(\vec{x} / \| \vec{x} \|)$ . According to [60], spherical harmonics provide a complete basis set for SO(3)-equivariant functions on the sphere. Clebsch-Gordan (CG) tensor products are usually leveraged to facilitate interactions between steerable features of differing degrees. Specifically, given features  $\tilde{\pmb{v}}^{(l_1)}$  and  $\tilde{\pmb{v}}^{(l_2)}$ , a new feature  $\tilde{\pmb{v}}^{(l)}$  can be generated, for  $|l_1 - l_2| \leq l \leq l_1 + l_2$ . This calculation is abbreviated as  $\tilde{\pmb{v}}^{(l)} = \tilde{\pmb{v}}^{(l_1)} \otimes_{\mathrm{cg}} \tilde{\pmb{v}}^{(l_2)}$ . By including
+
+parity [61], the above concepts can be naturally extended to the orthogonal group  $\mathrm{O}(3)$ . A more detailed discussion is available in Appendix A.1.
+
+# 3.2 Rethinking equivariant GNNs: a perspective based on basis construction
+
+By definition, a complete equivariant model is able to express any equivariant function. For instance, spherical harmonics provide complete bases for SO(3)-equivariant functions on the sphere:  $\phi (\vec{x} /\| \vec{x}\|)$ , enabling the representation of all such functions. In 3D space beyond the sphere, a complete function is achieved by combining spherical harmonics with a learnable radial basis function  $\varphi (\| \vec{x}\|)$ . However, deriving complete equivariant functions on geometric graphs, namely  $\tilde{\phi}^{(l)}(\mathcal{G}),^3$  is more complex due to the need to consider multi-body interactions and permutation symmetry.
+
+To begin with, let us define a function set  $\mathbb{F}^{(l)}$ , which encompasses all square-integrable  $l$ th-degree steerable functions on geometric graphs. Let  $\mathbb{B}^{(l)} \subset \mathbb{F}^{(l)}$  denote the basis set that expands the function space of equivariant GNNs. The learning process of equivariant GNNs becomes an approximation:
+
+$$
+\tilde {\phi} ^ {(l)} (\mathcal {G}) \approx \sum_ {\tilde {\boldsymbol {b}} _ {i} ^ {(l)} \in \mathbb {B} ^ {(l)}} a _ {i} \tilde {\boldsymbol {b}} _ {i} ^ {(l)} (\mathcal {G}), \tag {2}
+$$
+
+where  $a_{i}\in \mathbb{R}$  are weight coefficients independent of the input. Enhancing the expressiveness of equivariant GNNs involves enlarging  $\mathbb{B}^{(l)}$  until it coincides with  $\mathbb{F}^{(l)}$ . When  $\mathrm{span}(\mathbb{B}^{(l)}) = \mathbb{F}^{(l)}$ , the model is termed a complete lth-degree steerable model, or simply a complete model, and the basis set is denoted as  $\mathbb{B}_{\mathrm{cpl}}^{(l)}$ . Specifically, complete 0th-degree steerable models are referred to as complete scalar models.
+
+Below, we demonstrate that standard equivariant GNNs share a common basis set construction, differing primarily in their body-order formulation. For simplicity and without loss of generality, we assume uniform node and edge features that can either be treated as identical or safely ignored. This abstraction allows us to focus purely on the geometric and topological aspects of the problem while maintaining the core theoretical insights.
+
+Example 3.1 (Basis Set of Common Equivariant GNNs). The basis set for common equivariant GNNs, such as EGNN [43], HEGNN [49], TFN [33], and MACE [35], can be unified into the following form:
+
+$$
+\mathbb {B} _ {\nu} ^ {(l)} = \left\{\sum_ {i} \sum_ {\boldsymbol {j} \in \mathcal {N} (i)} \bigotimes_ {\mathrm {c g}, k = 1} ^ {\nu} Y ^ {\left(l _ {k}\right)} \left(\vec {\boldsymbol {x}} _ {i j _ {k}} / \| \vec {\boldsymbol {x}} _ {i j _ {k}} \|\right) \right\}, \tag {3}
+$$
+
+where  $\nu \geq 1$  denotes the body order,  $j = (j_{1},\ldots ,j_{\nu})$  represents all chosen ordered neighbors, and  $\vec{\pmb{x}}_{ij_k}\coloneqq \vec{\pmb{x}}_i - \vec{\pmb{x}}_{j_k}$ . When only single-body neighbor is considered (i.e.,  $\nu = 1$ ), Eq. (3) forms the basis set for EGNN [43] and HEGNN [49], applicable to degrees  $l = 1$  and  $l\geq 1$ , respectively. In contrast, for multi-body interactions, Eq. (3) corresponds to TFN [33] when the body order  $\nu = 1$ , or MACE [35] with higher body orders  $\nu \geq 1$ .
+
+Although these models adhere to the form of Eq. (3), their expressive power differs markedly. Multi-body models can create entirely new bases through tensor products and achieve completeness by either utilizing a sufficiently high body order [62] or stacking multiple layers [32], whereas single-body steerable models could not. However, multi-body models suffer from notable limitations: a) They may not encompass steerable features of all degrees, if input degrees are improperly chosen. b) The computational complexity increases sharply with higher degrees  $l$  and body orders  $\nu$ , resulting in significant computational costs.
+
+# 3.3 Reformulating equivariant GNNs: a perspective based on output space
+
+Expanding the basis set of a complete equivariant GNN is theoretically appealing but practically challenging. In this section, we adopt an alternative approach by deriving the necessary model design from the characteristics of the desired output results. We identify a key insight: a complete  $l$ th-degree steerable model can be constructed using two components—a complete scalar model and a full-rank basis set of  $l$ th-degree steerable features.
+
+We denote the target  $l$ th-degree steerable function as  $\tilde{\phi}^{(l)}\in \mathbb{F}^{(l)}$ . Although it might be represented with infinite bases in  $\mathbb{B}_{\mathrm{cpt}}^{(l)}$ , the output of  $\tilde{\phi}^{(l)}(\mathcal{G})$  is still a vector in  $\mathbb{R}^{2l + 1}$ . Let  $\tilde{\pmb{V}}^{(l)}(\mathcal{G}) = [\tilde{\pmb{v}}_1^{(l)}(\mathcal{G}),\dots,\tilde{\pmb{v}}_C^{(l)}(\mathcal{G})]\in \mathbb{R}^{(2l + 1)\times C}$  define an  $l$ th-degree steerable model with  $C$ -channel outputs of  $l$ th-degree steerable feature ( $C\geq 2l + 1$ ). We have the following theorem:
+
+Theorem 3.2 (Dynamic Method). Given a geometric graph  $\mathcal{G}$ , suppose there is a matrix  $\tilde{\mathbf{V}}^{(l)}(\mathcal{G})$  with  $C$  channels of  $l$ th-degree steerable features denoted as  $\tilde{\boldsymbol{v}}_c^{(l)}(\mathcal{G})$  satisfying  $\mathrm{span}(\tilde{\mathbf{V}}^{(l)}(\mathcal{G})) = \mathbb{F}^{(l)}(\mathcal{G}) \coloneqq \{\tilde{\boldsymbol{f}}^{(l)}(\mathcal{G}) \mid \tilde{\boldsymbol{f}}^{(l)} \in \mathbb{F}^{(l)}\} \subset \mathbb{R}^{2l + 1}$ . Then for any  $l$ th-degree steerable function  $\tilde{\phi}^{(l)} \in \mathbb{F}^{(l)}$ , there always exists  $\tilde{\boldsymbol{w}}^{(0)}(\mathcal{G}) \coloneqq [\tilde{\boldsymbol{w}}_c^{(0)}(\mathcal{G})]_{c = 1}^C$  with  $C$ -channel output scalars, such that
+
+$$
+\tilde {\phi} ^ {(l)} (\mathcal {G}) = \sum_ {c = 1} ^ {C} \tilde {w} _ {c} ^ {(0)} (\mathcal {G}) \tilde {\boldsymbol {v}} _ {c} ^ {(l)} (\mathcal {G}). \tag {4}
+$$
+
+A notable distinction from Eq. (2) is that the coefficient in front of the steerable features changes from  $a_{i}$ , which is independent of the input, to a scalar (function) that depends on the entire graph, denoted as  $\tilde{w}_c^{(0)}(\mathcal{G})$ . A natural idea is that if we can get a complete scalar model (capturing all possible scalars), then we can always satisfy the requirements of Theorem 3.2. Another crucial consideration is that the validity of Theorem 3.2 requires the basis matrix  $\tilde{\mathbf{V}}^{(l)}(\mathcal{G})$  to satisfy  $\mathrm{span}(\tilde{\mathbf{V}}^{(l)}(\mathcal{G})) = \mathbb{F}^{(l)}(\mathcal{G})$ . We call such basis matrix that meets this criterion as  $\mathbb{F}^{(l)}(\mathcal{G})$ -full-rank. When  $\mathbb{F}^{(l)}(\mathcal{G}) = \mathbb{R}^{2l + 1}$ , the concept of  $\mathbb{F}^{(l)}(\mathcal{G})$ -full-rank is equivalent to the traditional definition of full-rank. For simplicity, in contexts with no ambiguity, we refer to it as full-rank.
+
+By comparing Eq. (4) with Eq. (2), our proposed dynamic method offers greater flexibility and simplicity, embodying the concept of complexity transfer. Instead of constructing the difficult complete basis set  $\mathbb{B}_{\mathrm{cpl}}^{(l)}$ , we transform the process into obtaining a complete scalar function and then design the complete steerable model through Theorem 3.2. In the next section, we will detail how to acquire the complete scalar model and the full-rank basis set of  $l$ th-degree steerable features needed for Theorem 3.2.
+
+# 3.4 Reach completeness: canonical form and full-rank basis set
+
+In this section, we introduce how to obtain the complete scalar function  $\tilde{w}_c^{(0)}(\mathcal{G})$  (referred to as the canonical form) and the full-rank basis set  $\tilde{\pmb{V}}^{(l)}(\mathcal{G})$  required by Theorem 3.2.
+
+Geometric isomorphism and canonical form. An equivalent problem to constructing a complete scalar function is determining the isomorphism of geometric graphs. To differentiate this from traditional graph isomorphism problem for topological graphs, we specifically refer to our task involving geometric graphs as geometric isomorphism. This concept can be defined similarly to the GWL-test [36], which assesses the equivalence of geometric graphs based on their geometric structures and embeddings.
+
+Definition 3.3 (Geometric Isomorphism). Two geometric graphs  $\mathcal{G}(\vec{\pmb{X}}^{(\mathcal{G})},\pmb{A}^{(\mathcal{G})})$  and  $\mathcal{H}(\vec{\pmb{X}}^{(\mathcal{H})},\pmb{A}^{(\mathcal{H})})$  are called geometrically isomorphic if they fulfill both of the following isomorphisms
+
+1. Point Cloud Isomorphism: The two point clouds  $\vec{X}^{(\mathcal{G})}$  and  $\vec{X}^{(\mathcal{H})}$  are isomorphic, i.e.,  $\exists \sigma \in S_N, \mathfrak{g} \in \mathrm{E}(3), \forall i, \vec{\pmb{x}}_i^{(\mathcal{G})} = \mathfrak{g} \cdot \vec{\pmb{x}}_{\sigma(i)}^{(\mathcal{H})}$ . Here, all  $\langle \sigma, \mathfrak{g} \rangle$  make a nonempty set  $\mathbb{M}(\mathcal{G}, \mathcal{H})$ .  
+2. Topological Isomorphism: The topological graphs associated with the point clouds are isomorphic, i.e.,  $\exists \langle \sigma, \mathfrak{g} \rangle \in \mathbb{M}(\mathcal{G}, \mathcal{H}), \forall i, \forall j, [A_{ij}^{(\mathcal{G})}] = [A_{\sigma(i)\sigma(j)}^{(\mathcal{H})}]$ .
+
+Moreover, we denote the geometric isomorphism between  $\mathcal{G}$  and  $\mathcal{H}$  as  $\mathcal{G} \cong \mathcal{H}$ .
+
+The most significant difference from the isomorphism problem in topological graphs is that, to date, it remains unclear whether there exists a polynomial-time algorithm capable of determining whether two topological graphs are isomorphic [42; 63]. In contrast, when it comes to distinguishing whether two geometric graphs are geometrically isomorphic in accordance with Definition 3.3, the situation appears to be more favorable. There are indeed algorithms that can resolve this issue in polynomial time, as highlighted by [64]. For example, we present Algos. 1 and 2, which is based on the four-point positioning principle and operates with complexities of  $\mathcal{O}(N^6)$  and  $\mathcal{O}(N^8)$ , respectively.
+
+In most applications, the objective of an equivariant GNN is not to distinguish two geometric graphs, but rather to obtain a comprehensive embedding of a geometric graph. Consequently, we define the canonical form similarly to the definition provided in [64] as follows:
+
+Definition 3.4 (Canonical Form of Geometric Graph). A canonical form of geometric graph is a graph-level scalar function  $\Gamma : (\mathbb{R}^{N \times 3}, \mathbb{R}^{N \times N}) \to \mathbb{R}^H$ , satisfy  $\mathcal{G} \cong \mathcal{H} \iff \Gamma(\mathcal{G}) = \Gamma(\mathcal{H})$ .
+
+Built upon Algos. 1 and 2, we have derived a canonical form  $\Gamma(\cdot)$  as presented in Algo. 3. In summary, the process involves three steps: a) traversing all ordered four-point sets to serve as reference points for four-point positioning, with a time complexity of  $\mathcal{O}(N^4)$ ; b) transforming the original coordinates into distance vectors measured from the reference points, and converting both point sets and edge sets into scalar sets, with a time complexity of  $\mathcal{O}(N)$ ; c) mapping these scalar sets to the desired canonical form using DeepSet [65].
+
+Theorem 3.5 (General Canonical Form). Given any geometric graph  $\mathcal{G}$ , Algo. 3 provides a method to create canonical form with time complexity  $\mathcal{O}(N^6)$ .
+
+However, Algo. 3 remains impractical because of its high complexity. We now explore the feasibility of attaining a more efficient canonical form by sacrificing some versatility. The crux of Algo. 3 lies in the choice of four non-coplanar points, which contributes to a quartic complexity in traversal. If we reformulate this as a generative problem, by treating the non-coplanar points as learnable graph-level features, we may reduce this quartic complexity factor. Specifically, we consider applying a  $\mathrm{E}(3)$ -equivariant GNN  $\zeta$  to generate four non-coplanar nodes (called virtual nodes below). We regard them as reference points in the four-point positioning principle, avoiding the fourth-order complexity cost by the original traversal. Thus, we derive Algo. 4, yielding more efficiency.
+
+Theorem 3.6 (Faster Canonical Form). Given an  $\mathrm{E}(3)$ -equivariant function  $\zeta$  that generates four non-coplanar points on  $\mathcal{G}$ , Algo. 4 is able to create canonical form with time complexity  $\mathcal{O}(N^2)$ .
+
+It should be noted that Algo. 3 can encode any geometric graph, while Algo. 4 requires that four non-coplanar reference nodes can be learned through an equivariant GNN. In past studies, such as FastEGNN [66], it is assumed that non-coplanar virtual nodes can always be found, but in the discussion later, we will see that this is not the case.
+
+Full-rank basis set. As previously addressed, the validity of Eq. (4) critically depends on the  $\mathbb{F}^{(l)}(\mathcal{G})$ -full-rank basis set, a condition not easy to satisfy. The study in [49] has demonstrated that, for symmetric geometric graphs (defined in Definition A.5),  $\mathbb{F}^{(l)}(\mathcal{G})$  will degenerate to zero functions for certain values of degree  $l$ , implying that it is hard to design a full-rank basis set in this case. In practice, most scenarios study the case of asymmetric graphs, so we give priority to discussing them here. Our key inquiry is whether full-rank basis sets can always be constructed for asymmetric geometric graphs. We first present a pivotal theorem that significantly aids in our analysis.
+
+Theorem 3.7 (Coloring on Asymmetric Graph). In an asymmetric geometric graph, each point can be assigned a unique color. This implies the existence of an  $\mathrm{E}(3)$  invariant function that maps the features of the  $i$ th node to distinct values  $h_i$ . Similarly, for directed edges  $ij$ , their features are mapped to distinct values  $e_{ij}$ .
+
+We propose two node coloring methods: a) Distance to the center, denoted as  $\oplus$ ; b) Tensor product, denoted as  $\otimes$ . We denote the uncolored model by  $\varnothing$  and illustrate the coloring procedure in Fig. 2, taking the  $\oplus$  method as an example. A detailed description is provided in Table 9. This theorem is fundamental and forms the core of our subsequent analyses. It decouples the roles of different nodes and edges, simplifying our discussion. This framework enables us to enhance or suppress specific basis functions by selecting appropriate weight functions, or even to focus on a particular basis function by setting the weights of all other basis functions to zero. Given that the node coordinates of the entire geometric graph are non-coplanar, the coordinate differences  $\{\vec{x}_{ij}\}$  form a full-rank matrix, which allows us to achieve the desired  $\tilde{V}^{(l)}(\mathcal{G})$ . Additionally, spherical harmonic functions facilitate the mapping of first-degree steerable features to higher-degree representations. We obtain the following theorem.
+
+Theorem 3.8 (Existence of Full-Rank Basis Set). For any given asymmetric graph  $\mathcal{G}$ , an  $\mathbb{F}^{(l)}(\mathcal{G})$ -full-rank  $\tilde{\mathbf{V}}^{(l)}(\mathcal{G})$  can always be constructed for any degree  $l$ .
+
+![](images/a67a8223003981be46ffbd8cd49c06e991fb429c1f28880dbf4c68c69784d550.jpg)  
+(a) Original
+
+![](images/df14f892069d99db58db56fb4895d403b7e49b61728c66e2120f28a91c4659b0.jpg)  
+(b) Center  
+Figure 2: Different vectors are decoupled by coloring, thereby expanding the space of virtual nodes. The yellow area in the figure represents the out space of 1st-degree functions. Fig. 2(a): All nodes have the same features, and the output can only appear in a straight line in one-dimensional space. Fig. 2(b): Calculate the distance to the center  $\|\vec{x}_{ic}\|$  to update each node feature so that each node has a different color (feature). Fig. 2(c): After coloring, all vectors are decoupled, so the output can be generated in the entire 3D space.
+
+![](images/bd5d51ab09750e6167f63e4bd9a541ab1e34c325b36542d061ce08403fd09281.jpg)  
+(c) Colored
+
+For symmetric graphs, the situation is more complex, making the construction of a full-rank basis set considerably difficult. An potentially feasible approach could be using a tensor-product-based model, such as TFN [33] for basis set construction, which will be detailed in the appendix Appendix A.1.3.
+
+# 3.5 Implementation of Complete Models
+
+The last subsection demonstrated how to achieve theoretical completeness. Here, we present a practical implementation of complete models, using EGNN [43] as an example due to its simplicity and universality. In Example 3.1, we point out that EGNN could be incomplete since its basis set  $\mathbb{B}^{(1)}$  is possibly reduced to the sum of all edges  $\{\vec{x}_{ij}\}$ . However, the edge set itself  $\{\vec{x}_{ij}\}$  is full-rank and could be considered as the full-rank basis set  $\tilde{\pmb{V}}^{(1)}$  in Theorem 3.2. To make EGNN complete, the remaining issue is how to implement the complete scalar function  $\tilde{w}_c^{(0)}$ . Inspired by Algo. 4, we derive the E(3)-equivariant function  $\zeta$  in Theorem 3.6 using FastEGNN [66; 67], along with the pre-coloring process and several other modifications.
+
+Essentially, we first color the geometric graph and then construct the reference virtual nodes through FastEGNN in the following form:
+
+$$
+\vec {\boldsymbol {Z}} = \mathbf {1} _ {4} \vec {\boldsymbol {x}} _ {c} + \frac {1}{N} \sum_ {i = 1} ^ {N} \varphi_ {\vec {\boldsymbol {Z}}} (\boldsymbol {h} _ {i}) \cdot (\vec {\boldsymbol {x}} _ {i} - \vec {\boldsymbol {x}} _ {c}), \quad \vec {\boldsymbol {M}} _ {i} = \mathbf {1} _ {4} \vec {\boldsymbol {x}} _ {i} - \vec {\boldsymbol {Z}}, \tag {5}
+$$
+
+where  $\vec{Z} \in \mathbb{R}^{4 \times 3}$  denotes the coordinates of four reference virtual nodes,  $\vec{M}_i \in \mathbb{R}^{4 \times 3}$  computes coordinate difference between node  $i$  and all virtual nodes, and  $\vec{x}_c = \frac{1}{N} \sum_{i=1}^N \vec{x}_i$  signifies the coordinate center. In Algo. 4, the key point is to convert the coordinate information into a distance vector to the reference virtual node. To further refine the information, we employ the inner product form from GMN [68], as an alternative to distance vectors in Algo. 4. The node features are updated as follows:
+
+$$
+\boldsymbol {h} _ {i} = \boldsymbol {h} _ {i} + \varphi_ {h} (\boldsymbol {m} _ {i}), \quad \boldsymbol {m} _ {i} = \vec {M} _ {i} ^ {\top} \vec {M} _ {i} / \| \vec {M} _ {i} ^ {\top} \vec {M} _ {i} \| _ {\mathrm {F}}. \tag {6}
+$$
+
+Afterward, global operations (e.g., pooling) aggregate the information from all points and edges, mimicking DeepSet [65] calculations to obtain the desired canonical form. Strictly speaking, the virtual nodes forming the tetrahedron should also be included in Algo. 4, but we omit this term as it degrades performance in our experiments.
+
+If we consider a single-layer EGNN, it becomes the following form:
+
+$$
+\operatorname {E G N N} _ {\mathrm {c p l}} (\mathcal {G}) = \frac {1}{N} \sum_ {i = 1} ^ {N} \vec {\boldsymbol {x}} _ {i} ^ {\prime} = \vec {\boldsymbol {x}} _ {c} + \sum_ {\langle i, j \rangle \in \mathcal {E}} \varphi_ {\vec {\boldsymbol {x}}} \left(\boldsymbol {m} _ {i j, \mathrm {c p l}}\right) \cdot \vec {\boldsymbol {x}} _ {i j}, \tag {7}
+$$
+
+where the message  $m_{ij,\mathrm{cpl}} = \varphi_m(h_i,h_j,\| \vec{x}_{ij}\|^2,e_{ij})$  has been a complete scalar function since  $h_i$  already contains the information in the canonical form. Through Theorem 3.2, we verify that such single-layer EGNN is a complete model.
+
+The above approach can also be extended to tensor-product-based models, and we have implemented corresponding improvements in TFN [33]. Since virtual nodes (i.e.  $\vec{Z}$ ) are generated globally across
+
+Table 1: The Completeness Test.  
+
+<table><tr><td rowspan="2">GNN Layer</td><td colspan="2">The Completeness Test</td></tr><tr><td>2-body (Table. 3 in GWL)</td><td>3-body (Fig. 2(b) in IASR)</td></tr><tr><td>SchNet2-body</td><td>50.0 ± 0.0</td><td>50.0 ± 0.0</td></tr><tr><td>EGNN2-body</td><td>50.0 ± 0.0</td><td>50.0 ± 0.0</td></tr><tr><td>GVP-GNN3-body</td><td>100.0 ± 0.0</td><td>50.0 ± 0.0</td></tr><tr><td>TFN2-body</td><td>50.0 ± 0.0</td><td>50.0 ± 0.0</td></tr><tr><td>MACE3-body</td><td>100.0 ± 0.0</td><td>50.0 ± 0.0</td></tr><tr><td>MACE4-body</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>Basiccpl</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>SchNetcpl</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>EGNNcpl</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>GVP-GNNcpl</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>TFNcpl</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr></table>
+
+Table 2: The Chirality Test.  
+
+<table><tr><td></td><td colspan="4">The Chirality Test</td></tr><tr><td></td><td># Color</td><td># TP</td><td>Fig. 4(a)</td><td>Fig. 4(b)</td></tr><tr><td rowspan="6">Basic</td><td>∅</td><td></td><td>50.0 ±0.0</td><td>50.0 ±0.0</td></tr><tr><td>⊕</td><td></td><td>100.0 ±0.0</td><td>100.0 ±0.0</td></tr><tr><td>⊗</td><td></td><td>100.0 ±0.0</td><td>100.0 ±0.0</td></tr><tr><td>∅</td><td>✓</td><td>75.0 ±15.0</td><td>95.0 ±15.0</td></tr><tr><td>⊕</td><td>✓</td><td>100.0 ±0.0</td><td>100.0 ±0.0</td></tr><tr><td>⊗</td><td>✓</td><td>100.0 ±0.0</td><td>100.0 ±0.0</td></tr><tr><td rowspan="6">EGNN</td><td>∅</td><td></td><td>50.0 ±0.0</td><td>50.0 ±0.0</td></tr><tr><td>⊕</td><td></td><td>100.0 ±0.0</td><td>100.0 ±0.0</td></tr><tr><td>⊗</td><td></td><td>100.0 ±0.0</td><td>100.0 ±0.0</td></tr><tr><td>∅</td><td>✓</td><td>100.0 ±0.0</td><td>100.0 ±0.0</td></tr><tr><td>⊕</td><td>✓</td><td>100.0 ±0.0</td><td>100.0 ±0.0</td></tr><tr><td>⊗</td><td>✓</td><td>100.0 ±0.0</td><td>100.0 ±0.0</td></tr></table>
+
+Table 3: MSE loss for predicting the coordinates of the Monge point and twelve-point center and incenter of a tetrahedron. Models with a superscript * indicate that they utilize only half of the hidden dimension of the standard node embeddings. The optimal value is denoted in **bold**, while the suboptimal value is indicated with an **underline**.  
+
+<table><tr><td rowspan="2"></td><td colspan="4">Monge Point</td><td colspan="4">Twelve-point Center (×10-1)</td><td colspan="3">Incenter (×10-2)</td></tr><tr><td>1-layer</td><td>2-layer</td><td>3-layer</td><td>4-layer</td><td>1-layer</td><td>2-layer</td><td>3-layer</td><td>4-layer</td><td>1-layer</td><td>2-layer</td><td>3-layer</td></tr><tr><td>EGNN</td><td>1.188</td><td>0.374</td><td>0.185</td><td>0.131</td><td>1.320</td><td>0.413</td><td>0.265</td><td>0.161</td><td>32.54</td><td>0.527</td><td>0.141</td></tr><tr><td>FastEGNN</td><td>1.187</td><td>0.374</td><td>0.149</td><td>0.979</td><td>1.320</td><td>0.417</td><td>0.194</td><td>0.107</td><td>32.40</td><td>0.421</td><td>0.188</td></tr><tr><td>HEGNN</td><td>1.188</td><td>0.356</td><td>0.157</td><td>0.110</td><td>1.320</td><td>0.317</td><td>0.192</td><td>0.159</td><td>32.54</td><td>0.386</td><td>0.078</td></tr><tr><td>TFN</td><td>1.188</td><td>0.467</td><td>0.289</td><td>0.221</td><td>1.320</td><td>0.519</td><td>0.315</td><td>0.246</td><td>32.54</td><td>2.715</td><td>0.151</td></tr><tr><td>MACE</td><td>1.188</td><td>0.468</td><td>0.288</td><td>0.223</td><td>1.320</td><td>0.521</td><td>0.316</td><td>0.246</td><td>32.54</td><td>2.707</td><td>0.153</td></tr><tr><td>Equiformer</td><td>0.416</td><td>0.296</td><td>0.099</td><td>0.116</td><td>0.456</td><td>0.187</td><td>0.122</td><td>0.147</td><td>0.736</td><td>0.077</td><td>0.055</td></tr><tr><td>EGNNcpl-global</td><td>0.108</td><td>0.086</td><td>0.097</td><td>0.104</td><td>0.112</td><td>0.097</td><td>0.099</td><td>0.088</td><td>0.025</td><td>0.018</td><td>0.015</td></tr><tr><td>EGNN*cp1-global</td><td>0.175</td><td>0.116</td><td>0.148</td><td>0.111</td><td>0.155</td><td>0.144</td><td>0.174</td><td>0.152</td><td>0.046</td><td>0.025</td><td>0.059</td></tr><tr><td>EGNNcp1-local</td><td>0.460</td><td>0.210</td><td>0.083</td><td>0.052</td><td>0.510</td><td>0.232</td><td>0.095</td><td>0.067</td><td>0.388</td><td>0.258</td><td>0.031</td></tr><tr><td>EGNN*cp1-local</td><td>0.460</td><td>0.252</td><td>0.138</td><td>0.091</td><td>0.508</td><td>0.250</td><td>0.182</td><td>0.102</td><td>0.395</td><td>0.320</td><td>0.084</td></tr><tr><td>TFNcp1-global</td><td>0.468</td><td>0.162</td><td>0.233</td><td>0.084</td><td>0.516</td><td>0.319</td><td>0.187</td><td>0.099</td><td>8.983</td><td>0.398</td><td>0.077</td></tr><tr><td>TFNcp1-local</td><td>0.473</td><td>0.291</td><td>0.158</td><td>0.096</td><td>0.525</td><td>0.317</td><td>0.181</td><td>0.105</td><td>7.858</td><td>0.354</td><td>0.085</td></tr><tr><td>TFNcp1-local</td><td>0.460</td><td>0.210</td><td>0.089</td><td>0.088</td><td>0.518</td><td>0.237</td><td>0.097</td><td>0.078</td><td>0.676</td><td>0.048</td><td>0.029</td></tr><tr><td>TFN*cp1-local</td><td>0.473</td><td>0.244</td><td>0.082</td><td>0.073</td><td>0.519</td><td>0.271</td><td>0.122</td><td>0.085</td><td>0.693</td><td>0.056</td><td>0.039</td></tr></table>
+
+the entire graph, we refer to these models as EGNN/TFN<sub>cpl</sub>-global. To better capture local information, virtual nodes can also be generated for individual nodes, resulting in the EGNN/TFN<sub>cpl</sub>-local variants. Detailed model architectures are provided in Tables 12 and 13.
+
+# 4 Experiment
+
+In this section, we conduct two categories of experiments, totaling seven in number, to validate our theory and methods: a) three toy datasets in § 4.1 to assess the expressivity of our models; b) the remaining four in § 4.2 evaluate the actual performance of our models. Here, we provide a brief overview of the experiments. For further details, please refer to Appendix B.
+
+# 4.1 Expressivity
+
+Dataset setup. The completeness test is derived from the GWL-test [32] and the IASR-test [69], with the objective of distinguishing between two geometrically non-isomorphic graphs. We here focus on the  $k$ -chain test and 2-body/3-body tasks for testing. The chirality test is an experiment designed by ourselves, which includes three tasks as illustrated in Fig. 4. This study aims to address two objectives: a) assess the model's ability to distinguish geometric graphs under reflection transformations; (2) investigate whether distinct coloring strategies generate different virtual node coordinates, as validated by determinant calculations serving as mathematical indicators for reflection determination.
+
+Models. For the completeness test, we selected several baseline models, including SchNet [70], EGNN [43], GVP-GNN [71], TFN [33], and MACE [35]. Furthermore, models utilizing our canonical form are denoted as  $Model_{\mathrm{cpl}}$ , with  $Basic_{\mathrm{cpl}}$  specifically using the canonical form as the graph embedding directly. For the chirality test, we provide two options to identify reflections: a) Determinant  $\operatorname*{det}([\vec{z}_2,\vec{z}_3,\vec{z}_4] - \vec{z}_1\mathbf{1}_{1\times 3})$ ; b) Tensor product, as described in Example A.15.
+
+![](images/1a5d520378a10821fb769dc9408e777c02e6d581870be562319e9719a3ff1619.jpg)
+
+![](images/43ee8450a256d5b1783332af40025a88b91708d951a82e5956246551367732eb.jpg)
+
+![](images/54b9f692681f6078cd491c50e2eea42e430777b9a236914dfecbc14e2e6225a1.jpg)  
+(a)  $\mathrm{EGNN_{cpl}}$  Model
+
+![](images/9b4a288560a38ae21cd13a46accd26512c0d34ba7e127d9064502e29707e156a.jpg)  
+(b)  $\mathrm{TFN_{cpl}}$  Model  
+Figure 3: Visualization of MSE loss.
+
+Table 4: MSE loss on 5-body system.  
+
+<table><tr><td></td><td colspan="4">MSE Loss on 5-body system (×10-2)</td></tr><tr><td></td><td>1-layer</td><td>2-layer</td><td>3-layer</td><td>4-layer</td></tr><tr><td>EGNN</td><td>4.214</td><td>0.780</td><td>0.710</td><td>0.712</td></tr><tr><td>FastEGNN</td><td>3.983</td><td>0.705</td><td>0.640</td><td>0.509</td></tr><tr><td>HEGNN</td><td>4.114</td><td>0.801</td><td>0.561</td><td>0.489</td></tr><tr><td>TFN</td><td>2.411</td><td>1.758</td><td>1.758</td><td>1.739</td></tr><tr><td>MACE</td><td>2.403</td><td>1.754</td><td>1.746</td><td>1.746</td></tr><tr><td>Equiformer</td><td>0.805</td><td>0.682</td><td>0.465</td><td>0.657</td></tr><tr><td>EGNNcpl-global</td><td>0.943</td><td>0.546</td><td>0.530</td><td>0.492</td></tr><tr><td>EGNN*cpl-global</td><td>0.985</td><td>0.554</td><td>0.533</td><td>0.498</td></tr><tr><td>EGNNcpl-local</td><td>0.768</td><td>0.537</td><td>0.481</td><td>0.458</td></tr><tr><td>EGNN*cpl-local</td><td>0.703</td><td>0.513</td><td>0.455</td><td>0.450</td></tr><tr><td>TFNcpl-global</td><td>2.144</td><td>0.934</td><td>0.916</td><td>0.708</td></tr><tr><td>TFN*cpl-global</td><td>2.163</td><td>0.910</td><td>0.825</td><td>0.734</td></tr><tr><td>TFNcpl-local</td><td>0.988</td><td>0.766</td><td>0.738</td><td>0.761</td></tr><tr><td>TFN*cpl-local</td><td>0.883</td><td>0.792</td><td>0.734</td><td>0.746</td></tr></table>
+
+Table 5: 100-body dataset.  
+
+<table><tr><td></td><td>MSE Loss (×10-2)</td></tr><tr><td>EGNN</td><td>1.36</td></tr><tr><td>FastEGNN</td><td>1.10</td></tr><tr><td>TFNl≤2</td><td>3.77</td></tr><tr><td>MACEl≤2</td><td>3.83</td></tr><tr><td>Equiformerl≤2</td><td>0.90</td></tr><tr><td>HEGNNl≤1</td><td>1.13</td></tr><tr><td>HEGNNl≤2</td><td>0.97</td></tr><tr><td>HEGNNl&lt;3</td><td>0.94</td></tr><tr><td>HEGNNl≤6</td><td>0.86</td></tr><tr><td>EGNNcpl-global</td><td>0.98</td></tr><tr><td>EGNNcpl-local</td><td>0.73</td></tr><tr><td>TFNcpl-globall≤2</td><td>1.78</td></tr><tr><td>TFNcpl-locall≤2</td><td>1.73</td></tr></table>
+
+Table 6: Prediction error  $\left( {\times {10}^{-2}}\right)$  on MD17 dataset (3 runs).  
+
+<table><tr><td></td><td>Aspirin</td><td>Benzene</td><td>Ethanol</td><td>Malonal.</td><td>Naph.</td><td>Salicylic</td><td>Toluene</td><td>Uracil</td></tr><tr><td>EGNN</td><td>14.41±0.15</td><td>62.40±0.53</td><td>4.64±0.01</td><td>13.64±0.01</td><td>0.47±0.02</td><td>1.02±0.02</td><td>11.78±0.07</td><td>0.64±0.01</td></tr><tr><td>FastEGNN</td><td>9.81±0.11</td><td>60.84±0.14</td><td>4.65±0.00</td><td>12.82±0.02</td><td>0.38±0.00</td><td>1.05±0.08</td><td>10.88±0.08</td><td>0.56±0.01</td></tr><tr><td>TFNt&lt;2</td><td>12.37±0.18</td><td>58.48±1.98</td><td>4.81±0.04</td><td>13.62±0.08</td><td>0.49±0.01</td><td>1.03±0.02</td><td>10.89±0.01</td><td>0.84±0.02</td></tr><tr><td>MACEt&lt;2</td><td>10.43±0.44</td><td>59.71±2.21</td><td>4.83±0.03</td><td>13.78±0.04</td><td>0.44±0.02</td><td>0.94±0.01</td><td>10.20±0.11</td><td>0.74±0.01</td></tr><tr><td>Equiformerlt&lt;2</td><td>9.84±0.10</td><td>33.28±0.15</td><td>4.69±0.03</td><td>13.06±0.04</td><td>0.34±0.01</td><td>0.86±0.01</td><td>9.50±0.09</td><td>0.57±0.01</td></tr><tr><td>HEGNNt≤1</td><td>10.32±0.58</td><td>62.53±7.62</td><td>4.63±0.01</td><td>12.85±0.01</td><td>0.38±0.01</td><td>0.90±0.05</td><td>10.56±0.10</td><td>0.56±0.02</td></tr><tr><td>HEGNNt≤2</td><td>10.04±0.45</td><td>61.80±5.92</td><td>4.63±0.01</td><td>12.85±0.01</td><td>0.39±0.01</td><td>0.91±0.06</td><td>10.56±0.05</td><td>0.55±0.01</td></tr><tr><td>HEGNNt&lt;3</td><td>10.20±0.23</td><td>62.82±4.25</td><td>4.63±0.01</td><td>12.85±0.02</td><td>0.37±0.01</td><td>0.94±0.10</td><td>10.55±0.16</td><td>0.52±0.01</td></tr><tr><td>HEGNNt≤6</td><td>9.94±0.07</td><td>59.93±5.21</td><td>4.62±0.01</td><td>12.85±0.01</td><td>0.37±0.02</td><td>0.88±0.02</td><td>10.56±0.33</td><td>0.54±0.01</td></tr><tr><td>EGNNcp1-global</td><td>9.60±0.09</td><td>58.24±1.40</td><td>4.64±0.01</td><td>12.85±0.01</td><td>0.39±0.01</td><td>0.95±0.05</td><td>10.37±0.16</td><td>0.56±0.02</td></tr><tr><td>EGNNcp1-local</td><td>9.52±0.42</td><td>44.90±1.53</td><td>4.62±0.00</td><td>12.80±0.02</td><td>0.36±0.02</td><td>0.94±0.05</td><td>10.21±0.06</td><td>0.57±0.00</td></tr><tr><td>TFNcp1-globalt≤2</td><td>9.49±0.04</td><td>58.24±0.42</td><td>4.63±0.00</td><td>12.82±0.00</td><td>0.33±0.00</td><td>0.80±0.00</td><td>10.24±0.02</td><td>0.53±0.00</td></tr><tr><td>TFNcp1-localt≤2</td><td>9.52±0.07</td><td>48.77±6.51</td><td>4.64±0.00</td><td>12.83±0.02</td><td>0.34±0.00</td><td>0.81±0.01</td><td>10.95±0.01</td><td>0.53±0.00</td></tr></table>
+
+Results. Results are presented in Table 1 (the 2-body/3-body of completeness test), Table 2 (the chirality test) and Table 10 (the  $k$ -chain test of completeness test). As shown in Table 1, all models with canonical form, i.e.  $\mathrm{Model}_{\mathrm{cpl}}$ , reach an  $100\%$  accuracy, no matter whether the original model could distinguish the geometric graphs. As shown in Table 2, the model utilizing the determinant is highly dependent on the coloring scheme. In the special case illustrated in Fig. 4(c), only the model employing the tensor product can effectively distinguish between the geometric graphs, which demonstrates the importance of choosing an appropriate staining strategy.
+
+# 4.2 Performance on Physical Systems
+
+Dataset setup. We conducted four experiments to evaluate our model's performance across different scenarios: a) tetrahedron center prediction, which involves predicting the Monge point, twelve-point center, and incenter of a tetrahedron based on its four points (with identical node features); b) 5-body system [72], c) 100-body system [66], d) MD17 dataset [68], and e) Water-3D mini [66; 67], where predictions are made based on initial coordinates and velocities. Specifically, for the 5-body, 100-body, and Water-3D mini systems, we predict the future positions of charged particles or fluid particles, while for the MD17 dataset, we predict future atomic trajectories within eight different molecules.
+
+These experiments provide a comprehensive evaluation of our model. We assess its performance on graph-level targets (a) and node-level targets (b-e), explore the influence of model architecture layers (a-b), and test its scalability in large systems (c,e) as well as its applicability to real-world datasets (d-e). Detailed information on each dataset can be found in Table 11.
+
+Models. In both experiments, we selected the following baseline models: EGNN [43], FastEGNN [66; 67], HEGNN [49], TFN [33], MACE [35], and Equiformer [73]. We compared these baseline models with our proposed models, specifically  $\mathrm{EGNN / TFN_{cpl}}$ -local and  $\mathrm{EGNN / TFN_{cpl}}$ -global. More detailed information is provided in Tables 14 and 15.
+
+Results. We evaluated the performance of each model across varying numbers of layers in the first two experiment. For specific values, please refer to Table 3 and Table 4. The experimental results are visualized in Fig. 3 to facilitate comparison. Our model demonstrates superior performance across all
+
+Table 7: Results on Water-3D-mini.  
+
+<table><tr><td></td><td colspan="4">MSE Loss on Water-3D-mini (×10-4)</td></tr><tr><td></td><td>1-layer</td><td>2-layer</td><td>3-layer</td><td>4-layer</td></tr><tr><td>EGNN</td><td>4.904</td><td>4.323</td><td>3.649</td><td>3.338</td></tr><tr><td>FastEGNN</td><td>4.885</td><td>4.332</td><td>3.782</td><td>3.259</td></tr><tr><td>HEGNN</td><td>4.885</td><td>4.138</td><td>3.519</td><td>3.287</td></tr><tr><td>EGNNcpl-global</td><td>4.368</td><td>3.711</td><td>3.294</td><td>3.248</td></tr><tr><td>EGNNcpl-local</td><td>3.611</td><td>3.320</td><td>2.803</td><td>2.495</td></tr></table>
+
+tasks, often achieving results comparable to those of other models with multiple layers—sometimes even with just a single layer. In addition, our improved model has also achieved significant improvements on large-scale datasets and real-world datasets, see Tables 5 to 7, where our model is in the leading position in the former and in the latter, where we achieve performance leadership in more than half of the tasks.
+
+Additional findings. Furthermore, the baselines in Fig. 3 demonstrate that losses are equal when a single layer is used, confirming the degradation predicted in Example A.16 and highlighting the necessity of node coloring. While TFN and MACE outperform EGNN and HEGNN with a single layer, their advantage decreases with two or more layers. This is likely because EGNN/HEGNN may initially lack necessary basis functions, whereas tensor products used in TFN/MACE in deeper layers produce redundant terms. The introduction of the dynamic method significantly enhances the performance of both EGNN and TFN, showing its ability to construct the missing basis initially and suppress redundant terms in tensor products, thus achieving adaptive basis construction. From this perspective, exploring how to encode invariant features with more powerful architectures (e.g., advancing from GNNs to Graph Transformers [74]) represents a promising direction for future research. This perspective also helps explain the performance gains observed in models such as GotenNet [50] compared with HEGNN [49].
+
+# 5 Conclusion
+
+We propose a novel framework called Uni-EGNN, which is a dynamic method for constructing complete equivariant GNNs. In contrast to previous models, which required stacking multiple layers, increasing body order, and enhancing the degree of steerable features to achieve completeness, our dynamic method simplifies this process by requiring only two essential components: a) A canonical form (a complete scalar function) of a geometric graph; b) A full-rank steerable basis set. Additionally, we have developed an efficient implementation of dynamic method leveraging a polynomial algorithm for geometric isomorphism problems. Experimental results demonstrate that our dynamic method excels in both expressiveness and practical performance.
+
+# 6 Limitations
+
+For symmetric graphs, constructing a full-rank basis set remains an open problem. To date, it has only been established that tensor-product-based models (e.g., TFN [33]) can construct such a basis, while for scalar methods (e.g., HEGNN [49]) this remains uncertain—and we even conjecture that it may not be feasible. Another limitation of this work lies in the experimental evaluation: due to the lack of tasks with objective functions of degree  $l \geq 2$ , we are unable to empirically validate the effectiveness of the proposed method in predicting high-degree steerable objective functions.
+
+# Acknowledgement
+
+This work was jointly supported by the following projects and institutions: the National Natural Science Foundation of China (No. 62376276, No. 62172422); Beijing Nova Program (No. 20230484278); the Fundamental Research Funds for the Central Universities; the Research Funds of Renmin University of China (23XNKJ19); Public Computing Cloud, Renmin University of China; Damo Academy (Hupan Laboratory) through Damo Academy (Hupan Laboratory) Innovative Research Program and Damo Academy Research Intern Program.
+
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+
+# Contents of Appendix
+
+# A Theoretical Analysis and Proofs 19
+
+A.1 Detailed Preliminaries 19
+
+A.1.1 Equivariance 19  
+A.1.2 Clebsch-Gordan (CG) tensor product 19  
+A.1.3 Discussion on Symmetric Graphs 20
+
+A.2 Connection between graph-level and subgraph-level functions 21  
+A.3 The full form of dynamic method 21  
+A.4 The construction of canonical form 22
+
+A.4.1 An algorithm for determining point cloud isomorphism 22  
+A.4.2 An algorithm for determining geometric graph isomorphism 23  
+A.4.3 A general canonical form 23  
+A.4.4 A faster method to construct canonical form 24
+
+A.5 Construction of full-rank basis set 26  
+A.6 Comparison of different coloring methods 27
+
+# BExperiment Details 28
+
+B.1 Expressivity Tests 28
+
+B.1.1 Completeness Test 28  
+B.1.2 Chirality Test 30
+
+B.2 Performance Tests 30
+
+B.2.1 Tetrahedron Center Prediction and N-body System 31  
+B.2.2 Implementation ofBaselines 31  
+B.2.3 Implementation of Our Models 31
+
+# C Rethinking the Equivariance and General Functions 33
+
+C.1 Rethinking equivariance: A perspective from Fourier expansion. 33  
+C.2 Rethinking General Functions: the Equivariant Decomposition 34
+
+# A Theoretical Analysis and Proofs
+
+# A.1 Detailed Preliminaries
+
+This section introduces the equivariant representation and the CG tensor product with inversion, along with the symmetric graph structure and its related properties.
+
+# A.1.1 Equivariance
+
+Let  $\mathbb{X}$  and  $\mathbb{Y}$  be the input and output vector spaces, respectively. A function  $\phi : \mathbb{X} \to \mathbb{Y}$  is called equivariant with respect to group  $\mathfrak{G}$  if
+
+$$
+\forall \mathfrak {g} \in \mathfrak {G}, \phi \left(\rho_ {\mathbb {X}} (\mathfrak {g}) \boldsymbol {x}\right) = \rho_ {\mathbb {Y}} (\mathfrak {g}) \phi (\boldsymbol {x}), \tag {8}
+$$
+
+where  $\rho_{\mathbb{X}}$  and  $\rho_{\mathbb{Y}}$  are the group representations in the input and output spaces, respectively.
+
+Since we can permit translation invariance by simply translating the center of all coordinates to the origin, we only discuss equivariance with respect to  $\mathrm{O}(3)$  in our paper. Specifically, the group  $\mathrm{O}(3)$  consists of rotation and inversion, implying  $\mathrm{O}(3) = \mathrm{SO}(3) \times C_i$ , where  $\mathrm{SO}(3)$  is the rotation group and  $C_i = \{\mathfrak{k},\mathfrak{i}\}$  denotes the inversion group with  $\mathfrak{k}$  representing the identity and  $\mathfrak{i}$  representing the inversion. In the current literature, researchers typically utilize irreducible representations (i.e. Wigner-D matrix  $D^{(l)}(\mathfrak{r}) \in \mathbb{R}^{(2l + 1) \times (2l + 1)}$  with  $l$  denoting the degree) to represent  $\mathrm{SO}(3)$  and parities  $p \in \{\pm 1\}$  to represent the inversion group  $C_i$  as follows:
+
+$$
+\rho^ {(l, p)} (\mathfrak {r m}) := \sigma^ {(p)} (\mathfrak {m}) D ^ {(l)} (\mathfrak {r}), \tag {9}
+$$
+
+where  $\sigma^{(p)}(\mathfrak{m}) = p$  if  $\mathfrak{m} = \mathfrak{i}$ , while  $\sigma^{(p)}(\mathfrak{m}) = 1$  if  $\mathfrak{m} = \mathfrak{e}$ . When considering only rotations, a  $(2l + 1)$ -dimensional variable that can be rotated by the  $l$ th-degree Wigner-D matrices is referred to as an  $l$ th-degree steerable feature. When further involving the concept of inversion, we adopt the notation presented in e3nn [61]. Specifically, an inversion-invariant  $l$ th-degree steerable feature (i.e.,  $p = 1$ ) is referred to as an  $l$ e-type, while an inversion-equivariant feature (i.e.,  $p = -1$ ) is designated as an  $l$ o-type. These designations correspond to even parity and odd parity, respectively.
+
+Example A.1 (Spherical Harmonics). Spherical harmonics  $Y^{(l)}(\vec{x}) = [Y_m^{(l)}(\vec{x})]_{m = -l}^l$  are  $l$ th-degree steerable features, i.e.  $Y^{(l)}(\pmb{R}_{\mathfrak{r}}\vec{\pmb{x}}) = \pmb{D}^{(l)}(\mathfrak{r})Y^{(l)}(\vec{\pmb{x}})$ , and the parity is given as  $p = (-1)^l$ . It means spherical harmonics are  $l$ o-type when  $l$  is odd and  $l$ e-type when  $l$  is even. According to [60], spherical harmonics offer a complete set of function bases of SO(3)-equivariant functions.
+
+Another example about common physical quantities is presented in Example A.2.
+
+Example A.2 (Common Physical Quantities). Scalars (e.g. mass, charge and energy) are 0e-type (invariant to both rotation and inversion). Pseudo-scalar (e.g. magnetic charge and magnetic flux) are 0o-type (invariant to rotation but change signs under inversion). Vectors (e.g. coordinate and velocity) are 1o-type, while pseudo-vectors (e.g. torque and angular momentum) are 1e-type.
+
+Example A.3 (Common Invariant features). Due to the various advantages of scalars, there are a large number of construction methods, which we briefly list here:
+
+1. Purely mathematical representation: tensor product (spherical-scalarization) [48-50], topological characteristics [75-77], cluster expansion basis [78; 79], subgraph blocks [80-82], frames [56; 83; 84], normalization operators [85; 86], LLM-based information [87];  
+2. Features based on physical and biochemical prior knowledge: distance matrix [40; 88-92], chemical bond length, angle and dihedral angle [93-97], force [98; 99], fractional coordinates [21; 100; 101], canonical ordering [102; 103].
+
+# A.1.2 Clebsch-Gordan (CG) tensor product
+
+Tensor product  $\otimes (\cdot ,\cdot)$  is a common operator to describe interaction between steerable features. Given an  $l_{1}$  th-degree steerable feature  $\tilde{\pmb{v}}^{(l_1)}$  and an  $l_{2}$  th-degree steerable feature  $\tilde{\pmb{v}}^{(l_2)}$ , calculating their tensor product yields a new feature  $\tilde{\pmb{v}}^{(l_1)}\otimes \tilde{\pmb{v}}^{(l_2)}$  with its group representation as  $D^{(l_1)}(\mathfrak{r})\otimes D^{(l_2)}(\mathfrak{r})$ . The Clebsch-Gordan rule reveals that any tensor product of irreducible representations can be represented as the direct sum of the irreducible representations. Let  $\mathbb{V}^{(l_1)}$  and  $\mathbb{V}^{(l_2)}$  denote the irreducible
+
+representation space of  $l_{1}$ th-degree and  $l_{2}$ th-degree, respectively. Then, we have the following relation:
+
+$$
+\mathbb {V} ^ {\left(l _ {1}\right)} \otimes \mathbb {V} ^ {\left(l _ {2}\right)} = \bigoplus_ {l = \left| l _ {1} - l _ {2} \right|} ^ {l _ {1} + l _ {2}} \mathbb {V} ^ {(l)}. \tag {10}
+$$
+
+This further demonstrates that the group representation matrix  $D^{(l_1)}(\mathfrak{r}) \otimes D^{(l_2)}(\mathfrak{r})$  is similar to the matrix  $\bigoplus_{l = |l_1 - l_2|}^{l_1 + l_2}D^{(l)}(\mathfrak{r})$ . In general, the  $l$ th-degree component from the tensor product  $\tilde{\pmb{v}}^{(l_1)} \otimes \tilde{\pmb{v}}^{(l_2)}$  can be expressed as
+
+$$
+v _ {m} ^ {(l)} = \sum_ {m _ {1} = - l _ {1}} ^ {l _ {1}} \sum_ {m _ {2} = - l _ {2}} ^ {l _ {2}} Q _ {\left(l _ {1}, m _ {1}\right), \left(l _ {2}, m _ {2}\right)} ^ {(l, m)} v _ {m _ {1}} ^ {\left(l _ {1}\right)} v _ {m _ {2}} ^ {\left(l _ {2}\right)}, \tag {11}
+$$
+
+where  $Q_{(l_1,m_1),(l_2,m_2)}^{(l,m)}$  is called the Clebsch-Gordan coefficients. To simplify writing, when specifying the degree of the input and output, we directly abbreviate the above formula to  $\tilde{\pmb{v}}^{(l)} = \tilde{\pmb{v}}^{(l_1)}\otimes_{\mathrm{cg}}\tilde{\pmb{v}}^{(l_2)}$ . We introduce the decomposition of the moment of inertia in Example A.4 to enhance understanding.
+
+It is straightforward to extend the tensor product to incorporate inversion. When performing the tensor product of two steerable features—one exhibiting odd parity and the other exhibiting even parity—the resulting feature will necessarily exhibit odd parity. Conversely, if both features share the same parity, the result will be of even parity. For simplicity, we will omit discussions related to inversion in the subsequent analyses.
+
+Example A.4 (Decomposition of Reducible Representations). For the moment of inertia  $\operatorname{vec}(\overleftrightarrow{\pmb{T}}) = \sum_{i} (\|\vec{x}_i\|^2 \cdot 1_9 - \vec{x}_i \otimes \vec{x}_i) \in \mathbb{R}^9$ . The first item is a scalar (0e) of 9 channels denoted as  $9 \times 0e$ , while the corresponding group representation of the second item  $\sum_{i} \vec{x}_i \otimes \vec{x}_i$  is  $R_{\mathfrak{r}} \otimes R_{\mathfrak{r}}$  (the inversion are canceled). According to the Clebsch-Gordan rule, this representation is similar to  $\bigoplus_{l=0}^{2} D^{(l)}(\mathfrak{r})$ , which can be transformed through a specific coefficient matrix, and  $\sum_{i} \vec{x}_i \otimes \vec{x}_i$  can be decomposed into irreducible features of three types: 0e, 1e, 2e.
+
+# A.1.3 Discussion on Symmetric Graphs
+
+A symmetric graph is a special kind of geometric graph. [49] points out that in a symmetric graph, steerable features of a particular degree will degenerate, so we list this case separately for analysis.
+
+Definition A.5 (Symmetric Graph). A geometric graph  $\mathcal{G}$  is called a symmetric graph, if there exists a finite and nontrivial subgroup  $\mathfrak{H} \leq \mathrm{O}(3)$ ,  $\mathfrak{H} \neq \{\mathfrak{e}\}$ , satisfying that  $\forall \mathfrak{h} \in \mathfrak{H}, \mathfrak{h} \cdot \mathcal{G} = \mathcal{G}$ . All subgroups making  $\mathcal{G}$  symmetric yields a set  $\mathbb{H}(\mathcal{G})$ , and all geometric graphs that are symmetric w.r.t.  $\mathfrak{H}$  constitute a set denoted as  $\mathbb{G}(\mathfrak{H})$ . Moreover, all other graphs are referred to as asymmetric (geometric) graphs.
+
+It is straightforward to conclude that an asymmetric geometric graph must contain non-coplanar nodes; otherwise, if all the nodes lie in the same plane, the graph would coincide with itself after reflection across that plane. Therefore, the presence of non-coplanar nodes is a necessary condition for the asymmetry of the geometric graph.
+
+Lemma A.6 (Image Space of Functions on Symmetric Geometric Graphs). Given a symmetric graph  $\mathcal{G}$ ,  $\tilde{\pmb{f}}^{(l)}(\mathcal{G})\in \bigcap_{\mathfrak{H}\in \mathbb{H}(\mathcal{G})}[I_{2l + 1} - \rho^{(l)}(\mathfrak{H})]^{\perp}$ , where  $\rho^{(l)}(\mathfrak{H}) = \frac{1}{|\mathfrak{H}|}\sum_{\mathfrak{h}\in \mathfrak{H}}\rho^{(l)}(\mathfrak{h})$  denotes the group averaging.
+
+Here, we strengthen the conclusion of [49]. Below, we give an example that makes the steerable features confined to a subspace instead of degenerating into a zero vector.
+
+Example A.7. Consider a cone with a square base, defined by the set  $\mathcal{G} = \{(\pm 1,0,-1),(0,\pm 1,-1),(0,0,4)\}$ . For  $l = 1$ , it is straightforward to find that  $\mathbb{F}^{(1)}(\mathcal{G}) = \{(0,0,z)\} \subset \mathbb{R}^3$ , and constructing a  $\mathbb{F}^{(1)}(\mathcal{G})$ -full-rank function is relatively easy. However, for cases where  $l > 1$ , while the analysis can still be performed using the orthogonal complement of the group average, constructing a  $\mathbb{F}^{(l)}(\mathcal{G})$ -full-rank function becomes significantly more challenging. We propose using tensor-product-based models, such as TFN [33], to construct a basis set. These models, despite their computational complexity, have been proven to be complete. Once constructed, this basis set can be applied with our method to achieve the desired result.
+
+# A.2 Connection between graph-level and subgraph-level functions
+
+In practice, the focus of many problems lies in some local features, such as node-level or edge-level features, which we collectively refer to as subgraph-level. In this section, we will show the connection between graph-level functions and subgraph-level functions.
+
+In fact, the local features also require the message from the whole geometric graph  $\mathcal{G}$ , thus for a subgraph  $\mathcal{G}_{\mathrm{sub}} \subset \mathcal{G}$ , the  $l$ th-degree steerable feature could be denoted as  $\tilde{\pmb{f}}^{(l)}(\mathcal{G}_{\mathrm{sub}}, \mathcal{G})$ . Moreover, we denote  $\mathbb{F}^{(l)}(\mathcal{G}_{\mathrm{sub}}, \mathcal{G}) \coloneqq \{\tilde{\pmb{f}}^{(l)}(\mathcal{G}_{\mathrm{sub}}, \mathcal{G}) | \tilde{\pmb{f}}^{(l)} \in \mathbb{F}^{(l)}\} \subset \mathbb{R}^{2l + 1}$ . Note that the whole graph is also a subgraph,  $\tilde{\pmb{f}}^{(l)}(\mathcal{G})$  and  $\mathbb{F}^{(l)}(\mathcal{G})$  are actually the simplified version of  $\tilde{\pmb{f}}^{(l)}(\mathcal{G}, \mathcal{G})$  and  $\mathbb{F}^{(l)}(\mathcal{G}, \mathcal{G})$ .
+
+Lemma A.8. Given a geometric graph  $\mathcal{G}$ , where  $\mathcal{G}_{sub} \subset \mathcal{G}$  is a subgraph (e.g. node, edge), then we have  $\mathbb{F}^{(l)}(\mathcal{G}) \subset \mathbb{F}^{(l)}(\mathcal{G}_{sub}, \mathcal{G})$ .
+
+Proof. Note that for the subgraph-level function  $\tilde{f}^{(l)}(\mathcal{G}_{\mathrm{sub}},\mathcal{G})$ , let the function ignore the previous input  $\mathcal{G}_{\mathrm{sub}}$ . Therefore, for any graph-level function, there is always a subgraph-level function that is equal to it.
+
+Lemma A.9. Given a geometric graph  $\mathcal{G}$ , where  $\mathcal{G}_{sub} \subset \mathcal{G}$  is a subgraph (e.g. node, edge). We have  $\mathbb{F}(\mathcal{G}_{sub}, \mathcal{G}) = \mathbb{R}^{2l + 1}$ , if  $\mathbb{F}(\mathcal{G}) = \mathbb{R}^{2l + 1}$ .
+
+Proof. It is obvious that  $\mathbb{F}(\mathcal{G}_{\mathrm{sub}},\mathcal{G})\subset \mathbb{R}^{2l + 1}$ . And we have  $\mathbb{R}^{2l + 1} = \mathbb{F}(\mathcal{G})\subset \mathbb{F}(\mathcal{G}_{\mathrm{sub}},\mathcal{G})$ , thus  $\mathbb{F}(\mathcal{G}_{\mathrm{sub}},\mathcal{G}) = \mathbb{F}(\mathcal{G}) = \mathbb{R}^{2l + 1}$ .
+
+Proposition A.10. Given a geometric graph  $\mathcal{G}$ , where  $\mathcal{G}_{sub} \subset \mathcal{G}$  is a subgraph (e.g. node, edge), suppose there is a matrix  $\tilde{\mathbf{V}}^{(l)}(\mathcal{G}_{sub},\mathcal{G})$  with  $C$  channels of  $l$ th-degree steerable features denoted as  $\tilde{\mathbf{v}}_c^{(l)}(\mathcal{G}_{sub},\mathcal{G})$  satisfying span  $\tilde{\mathbf{V}}^{(l)}(\mathcal{G}_{sub},\mathcal{G}) = \mathbb{F}^{(l)}(\mathcal{G}_{sub},\mathcal{G})$ . Then for any  $l$ th-degree steerable function  $\tilde{\pmb{t}}^{(l)} \in \mathbb{F}^{(l)}$ , there exists  $\tilde{\mathbf{w}}^{(0)}(\mathcal{G}_{sub},\mathcal{G}) \coloneqq [\tilde{w}_c^{(0)}(\mathcal{G}_{sub},\mathcal{G})]_{c=1}^C$  with  $C$ -channel output scalars, such that
+
+$$
+\tilde {\boldsymbol {t}} ^ {(l)} \left(\mathcal {G} _ {s u b}, \mathcal {G}\right) = \sum_ {c = 1} ^ {C} \tilde {w} _ {c} ^ {(0)} \left(\mathcal {G} _ {s u b}, \mathcal {G}\right) \tilde {\boldsymbol {v}} _ {c} ^ {(l)} \left(\mathcal {G} _ {s u b}, \mathcal {G}\right). \tag {12}
+$$
+
+Proof. Note that the proof of Theorem 3.2 does not depend on the input form of the function, so the generalization from Theorem 3.2 to this proposition still holds.  $\square$
+
+Proposition A.11. Given a geometric graph  $\mathcal{G}$ , where  $\mathcal{G}_{sub} \subset \mathcal{G}$  is a subgraph (e.g. node, edge) and  $\mathbb{F}(\mathcal{G}_{sub}, \mathcal{G}) = \mathbb{F}(\mathcal{G})$ . Suppose there is a matrix  $\tilde{\boldsymbol{V}}^{(l)}(\mathcal{G})$  with  $C$  channels of  $l$ th-degree steerable features denoted as  $\tilde{\boldsymbol{v}}_c^{(l)}(\mathcal{G})$  satisfying span  $\tilde{\boldsymbol{V}}^{(l)}(\mathcal{G}) = \mathbb{F}^{(l)}(\mathcal{G})$ . Then for any  $l$ th-degree steerable function  $\tilde{\boldsymbol{t}}^{(l)} \in \mathbb{F}^{(l)}$ , there exists  $\tilde{\boldsymbol{w}}^{(0)}(\mathcal{G}_{sub}, \mathcal{G}) := [\tilde{w}_c^{(0)}(\mathcal{G}_{sub}, \mathcal{G})]_{c=1}^C$  with  $C$ -channel output scalars, such that
+
+$$
+\tilde {\boldsymbol {t}} ^ {(l)} \left(\mathcal {G} _ {s u b}, \mathcal {G}\right) = \sum_ {c = 1} ^ {C} \tilde {w} _ {c} ^ {(0)} \left(\mathcal {G} _ {s u b}, \mathcal {G}\right) \tilde {\boldsymbol {v}} _ {c} ^ {(l)} (\mathcal {G}). \tag {13}
+$$
+
+Proof. Note that  $\tilde{\pmb{v}}_c^{(l)}(\mathcal{G})\in \mathbb{F}^{(l)}(\mathcal{G})\subset \mathbb{F}^{(l)}(\mathcal{G}_{\mathrm{sub}},\mathcal{G})$ , so this proposition still holds.
+
+Note that for invariant functions, if we can obtain the canonical form at the graph-level, then we can naturally obtain the canonical form at the subgraph-level. Therefore, the discussion of subgraph-level and graph-level is consistent. To simplify the writing, only the graph-level case will be discussed in the following.
+
+# A.3 The full form of dynamic method
+
+Before giving a complete theoretical analysis, let us review Theorem 3.2 as follows.
+
+Theorem 3.2 (Dynamic Method). Given a geometric graph  $\mathcal{G}$ , suppose there is a matrix  $\tilde{\mathbf{V}}^{(l)}(\mathcal{G})$  with  $C$  channels of  $l$ th-degree steerable features denoted as  $\tilde{\boldsymbol{v}}_c^{(l)}(\mathcal{G})$  satisfying  $\mathrm{span}(\tilde{\mathbf{V}}^{(l)}(\mathcal{G})) = \mathbb{F}^{(l)}(\mathcal{G}) \coloneqq \{\tilde{\boldsymbol{f}}^{(l)}(\mathcal{G}) \mid \tilde{\boldsymbol{f}}^{(l)} \in \mathbb{F}^{(l)}\} \subset \mathbb{R}^{2l + 1}$ . Then for any  $l$ th-degree steerable function  $\tilde{\phi}^{(l)} \in \mathbb{F}^{(l)}$ , there always exists  $\tilde{\boldsymbol{w}}^{(0)}(\mathcal{G}) \coloneqq [\tilde{w}_c^{(0)}(\mathcal{G})]_{c = 1}^C$  with  $C$ -channel output scalars, such that
+
+$$
+\tilde {\phi} ^ {(l)} (\mathcal {G}) = \sum_ {c = 1} ^ {C} \tilde {w} _ {c} ^ {(0)} (\mathcal {G}) \tilde {\boldsymbol {v}} _ {c} ^ {(l)} (\mathcal {G}). \tag {4}
+$$
+
+This form decomposes the problem of constructing a complete equivariant function into two components. However, determining the explicit forms of  $\tilde{\boldsymbol{w}}^{(0)}(\mathcal{G})$  and  $\tilde{\boldsymbol{V}}^{(l)}$  remains a highly nontrivial task. To provide insight into this construction, we present two corollaries of Theorem 3.2, corresponding to two special cases of the equivariant space  $\mathbb{F}^{(l)}(\mathcal{G})$ : one where it reduces to the trivial set consisting solely of the origin,  $\theta \coloneqq \{\mathbf{0}_{2l + 1}\}$ , and the other is the opposite, i.e.  $\dim \mathbb{F}^{(l)}(\mathcal{G}) > 0$ .
+
+Proposition A.12. Given a geometric graph  $\mathcal{G}$  with  $\mathbb{F}^{(l)}(\mathcal{G}) = \theta$ , let  $\mathbb{W}^{(0)} \subset \mathbb{F}^{(0)}$  denote all invariant functions that can be expressed by model, any  $l$ th-degree steerable function could be written into the form of Theorem 3.2, if  $\mathbb{W}^{(0)}$  is not empty.
+
+Proof. Since  $\mathbb{F}^{(l)}(\mathcal{G}) = \theta$ , any equivariant output of degree  $l$  must be the zero vector. In particular, both the target output  $\tilde{\pmb{t}}^{(l)}(\mathcal{G})$  and each basis component  $\tilde{\pmb{v}}_c^{(l)}(\mathcal{G})$  in the decomposition of Theorem 3.2 are identically zero.
+
+As a result, the decomposition in Theorem 3.2 holds trivially for any choice of invariant scalar function  $\tilde{w}^{(0)}(\mathcal{G})\in \mathbb{W}^{(0)}$ , because both sides of the equation reduce to the zero vector. Hence, any  $l$ th-degree steerable function admits the desired representation, as long as  $\mathbb{W}^{(0)}$  is non-empty.
+
+Proposition A.13. Given a geometric graph  $\mathcal{G}$  with  $\dim \mathbb{F}^{(l)}(\mathcal{G}) > 0$ , let  $\mathbb{W}^{(0)} \subset \mathbb{F}^{(0)}$  denote all invariant functions that can be expressed by model, any  $l$ th-degree steerable function could be written into the form of Theorem 3.2, iff  $\mathbb{W}^{(0)} = \mathbb{F}^{(0)}$ , i.e. the invariant model is complete.
+
+Proof. The sufficiency is evident. By Theorem 3.2, there exists an invariant function of the required form, and such a function is clearly contained in the space  $\mathbb{F}^{(0)}$ .
+
+To prove necessity, assume by contradiction that  $\mathbb{W}^{(0)}\subsetneq \mathbb{F}^{(0)}$  but that the decomposition in Theorem 3.2 still holds for all steerable functions. Let  $\tilde{\mathbf{V}}^{(l)}(\mathcal{G})$  be a set of linearly independent equivariant vectors that spans  $\mathbb{F}^{(l)}(\mathcal{G})$ , i.e., a basis of  $\mathbb{F}^{(l)}(\mathcal{G})$ . Now, consider any  $\tilde{w}^{(0)}\in \mathbb{F}^{(0)}\setminus \mathbb{W}^{(0)}$ . Then the function  $\tilde{\pmb{t}}^{(l)}(\mathcal{G}) = \sum_{c}\tilde{w}^{(0)}(\mathcal{G})\tilde{\pmb{v}}_{c}^{(l)}(\mathcal{G})$  is a valid steerable function in  $\mathbb{F}^{(l)}(\mathcal{G})$ , as it results from a linear combination of basis elements with coefficients in  $\mathbb{F}^{(0)}$ . However, since  $\tilde{w}^{(0)}$  is not expressible by the model, the decomposition cannot be achieved using elements in  $\mathbb{W}^{(0)}$ , contradicting the assumption that the decomposition is always possible.
+
+Therefore, the assumption must be false, and we conclude that  $\mathbb{W}^{(0)} = \mathbb{F}^{(0)}$  is necessary for the representation to hold for all steerable functions.
+
+# A.4 The construction of canonical form
+
+In order to losslessly encode the information of a geometric graph, we define the concept of canonical forms in this section. Then, we give a method to construct canonical forms in polynomial time. Here, we assume that the geometric graph is fully connected and use a double-loop set comparison method to estimate the worst time complexity of the algorithm.
+
+# A.4.1 An algorithm for determining point cloud isomorphism
+
+Here, we give Algo. 1 for determining whether two geometric point clouds are isomorphic and prove its correctness. It is noteworthy that this algorithm resembles that of [104], centered on the principle of four-point positioning in three-dimensional space. Our subsequent designs will leverage this principle to adapt it for use in geometric graph scenarios and neural network applications.
+
+For two given point clouds, represented as  $\vec{X}^{(\mathcal{G})}$  and  $\vec{X}^{(\mathcal{H})}$ , we recognize that if these point clouds are isomorphic, there must exist a local substructure (composed of at least four points, i.e.  $\vec{U}_{\alpha}$  and  $\vec{V}_{\beta}$ ) that is also isomorphic. It is worth noting that this is the isomorphism judgment between two point clouds of a certain size, so it is always of constant complexity.
+
+According to the four-point positioning principle, the coordinates of any point  $\vec{u}_i \in \vec{X}^{(\mathcal{G})}$  or  $\vec{v}_i \in \vec{X}^{(\mathcal{H})}$  can be expressed as distance vectors, i.e.  $d_i^{(\mathcal{G})}$  or  $d_i^{(\mathcal{H})}$ , relative to these four foundational points. Consequently, sets of coordinates  $\vec{X}^{(\mathcal{G})}, \vec{X}^{(\mathcal{H})}$  can be transformed into sets of distance vectors  $\mathbb{D}^{(\mathcal{G})}, \mathbb{D}^{(\mathcal{H})}$ . When there is a set of isomorphic substructures with matching distance vector
+
+Algorithm 1: Check point clouds isomorphic with time complexity  $\mathcal{O}(N^6)$ .  
+Data: Two point clouds  $\vec{X}^{(\mathcal{G})}$  and  $\vec{X}^{(\mathcal{H})}$ , each containing  $N$  nodes. Result: True or False, indicating whether the two point clouds are isomorphic. //  $\mathcal{O}(N^4)$ , the worst case requires traversing all permutations.  
+1 Find an ordered tuple containing four non-coplanar points  $\vec{U}_{\alpha} \gets (\vec{u}_{\alpha_i}^{(\mathcal{G})})_{i=1}^4$  in  $\mathcal{G}$ ; //  $\mathcal{O}(N^4)$ , the permutations of 4 elements from a set of size  $N$ .  
+2 for all ordered subsets of  $\vec{X}^{(\mathcal{H})}$  containing four elements donated by  $\vec{V}_{\beta} \gets (\vec{v}_{\beta_i}^{(\mathcal{H})})_{i=1}^4$  do  
+3 //  $\mathcal{O}(1)$ , comparison of two ordered sets of a certain size.  
+4 if  $\vec{U}_{\alpha}$  and  $\vec{V}_{\beta}$  is not point clouds isomorphic according to matching  $(\vec{u}_{\alpha_i}^{(\mathcal{G})}, \vec{v}_{\beta_i}^{(\mathcal{H})})$  then  
+5 continue;  
+6 end  
+7 for  $\vec{u}_i \in \vec{X}^{(\mathcal{G})}$  do  
+8 end  
+9 for  $\vec{v}_i \in \vec{X}^{(\mathcal{H})}$  do  
+10 if  $\mathcal{O}(N)$ , get a 4-channel scalar vector.  
+11 end  
+12 //  $\mathcal{O}(N)$ , change the node set into scalar set.  
+13 if  $\mathbb{D}^{(\mathcal{G})}, \mathbb{D}^{(\mathcal{H})} \gets \text{set}([\vec{d}_i^{(\mathcal{G})}]_{i=1}^N)$ , set  $[\vec{d}_i^{(\mathcal{H})}]_{i=1}^N$ ;  
+14 //  $\mathcal{O}(N^2)$ , the complexity of comparing two sets varies depending on the choice, such as double loop comparison.  
+15 end  
+16 end  
+17 return False;
+
+sets, the point clouds are necessarily isomorphic. Conversely, if no isomorphic substructure can be identified or if the distance vector sets cannot be rendered equivalent for any potential isomorphic substructure, then the point clouds must not be isomorphic.
+
+# A.4.2 An algorithm for determining geometric graph isomorphism
+
+Furthermore, we discuss how to design an algorithm to distinguish the isomorphism of two geometric graphs. Note that at this point, all node coordinates have been converted to distance vectors, so it is also easy to transform the edge set as follows.
+
+# A.4.3 A general canonical form
+
+Now, we further give a method to construct the canonical form of a geometric graph. From Algo. 1 and 2, it is not difficult to see that the core of the algorithm is to transform a set of 3D coordinates into a set of distance vectors. Here we give Algo. 3 as follows.
+
+To encode the set, we employ DeepSet [65] for encoding, which losslessly encodes set data. Here, we ignore the complexity of the neural network (because this is a function of the hidden layer dimension  $H$ ), and only consider the complexity of the set size. It is not difficult to see from Eq. (14) that for a set of size  $M$ , the complexity of DeepSet is  $\mathcal{O}(M)$ .
+
+Lemma A.14 (DeepSet). A set function could be always written into such form:
+
+$$
+f \left(\left\{x _ {i} \right\} _ {i = 1} ^ {N}\right) = \rho \left(\sum_ {i = 1} ^ {N} \phi \left(x _ {i}\right)\right). \tag {14}
+$$
+
+Algorithm 2: Check geometric graph isomorphism with time complexity  $\mathcal{O}(N^8)$ .  
+Data: Two geometric graphs  $\mathcal{G}$  and  $\mathcal{H}$ , each containing  $N$  nodes. Result: True or False, indicating whether the two point clouds are isomorphic. //  $\mathcal{O}(N^4)$ , the worst case requires traversing all permutations.  
+1 Find an ordered tuple containing four non-coplanar points  $\vec{U}_{\alpha} \gets (\vec{u}_{\alpha_i}^{(\mathcal{G})})_{i=1}^4$  in  $\mathcal{G}$ ; //  $\mathcal{O}(N^4)$ , the permutations of 4 elements from a set of size N.  
+2 for all ordered subsets of  $\vec{X}^{(\mathcal{H})}$  containing four elements donated by  $\vec{V}_{\beta} \gets (\vec{v}_{\beta_i}^{(\mathcal{H})})_{i=1}^4$  do  
+3 //  $\mathcal{O}(1)$ , comparison of two ordered sets of a certain size.  
+4 if  $\vec{U}_{\alpha}$  and  $\vec{V}_{\beta}$  is not point clouds isomorphic according to matching  $(\vec{u}_{\alpha_i}^{(\mathcal{G})}, \vec{v}_{\beta_i}^{(\mathcal{H})})$  then continue;  
+5 end  
+6 for  $\vec{u}_i \in \vec{X}^{(\mathcal{G})}$  do  
+7 //  $\mathcal{O}(N)$ , get a 4-channel scalar vector.  
+8 end  
+9 for  $\vec{v}_i \in \vec{X}^{(\mathcal{H})}$  do  
+10 //  $\mathcal{O}(N)$ , get a 4-channel scalar vector.  
+11 end  
+12 //  $\mathcal{O}(N + N^2)$ , change the node set and edge set into scalar set.  
+13 //  $\mathcal{O}(N^2 + N^4)$ , the complexity of comparing four sets (node-node, edge-edge) varies depending on the choice, such as double loop comparison.  
+14 if  $\mathbb{D}^{(\mathcal{G})} = \mathbb{D}^{(\mathcal{H})}$  and  $\mathbb{E}^{(\mathcal{G})} = \mathbb{E}^{(\mathcal{H})}$  then  
+15 return True;  
+16 end  
+17 end  
+18 return False;
+
+Proof. This proof is a simplification of Theorem 7 in Deepset [65]. The necessity, that is, that this form satisfies the permutation invariance of sets is obvious, and its sufficiency is proved below. Consider an  $N$ th degree polynomial  $P(t) = \sum_{i=0}^{N} a_{i} t^{i}$ , which satisfies the set  $\{x_{i}\}_{i=1}^{N}$  of all the  $N$  roots of  $P(t) = 0$ . Thus, the set can be mapped to ordered polynomial coefficients. Vieta's theorem and Newton's identity show that the space of polynomial coefficients is consistent with the space of sum-of-power functions, so the universal approximation theorem of MLP can be used to prove that such a mapping can be found.
+
+Theorem 3.5 (General Canonical Form). Given any geometric graph  $\mathcal{G}$ , Algo. 3 provides a method to create canonical form with time complexity  $\mathcal{O}(N^6)$ .
+
+Proof. Due to the completeness of Deepset's encoding, Algo. 3 can naturally distinguish arbitrary geometric graphs. For a detailed time complexity analysis, see the comments in the algorithm block.  $\square$
+
+# A.4.4 A faster method to construct canonical form
+
+In practical applications, it is impractical to traverse all ordered four-point sets because the complexity is too high. In order to reduce the complexity, a simple and intuitive way is to reduce the number of four-point sets. Here, a very clever change of mind is that we change the ordered four-point set from being selected from a geometric graph to being generated from a geometric graph. We call these generated ordered four-point sets virtual nodes. In order to ensure the correctness of the algorithm, we
+
+Algorithm 3: A canonical form of geometric graphs.  
+Data: A geometric graph  $\mathcal{G}$  , and  $\psi_{\mathrm{node}},\psi_{\mathrm{edge}},\psi_{\mathrm{graph}}$  are DeepSet models. Result: The canonical form  $\Gamma \in \mathbb{R}^{H}$  of point clouds  $\mathcal{G}$  //  $\mathcal{O}(N^4)$  , traverse all permutations.   
+1  $\mathbb{T}\gets \emptyset$  .   
+2 for any ordered set containing four non-coplanar points  $\vec{U}_{\alpha}\leftarrow \{\vec{u}_{\alpha_i}\}_{i = 1}^4$  in  $\mathcal{G}$  do   
+3 for  $\vec{u}_i\in \vec{X}^{(\mathcal{G})}$  do //  $\mathcal{O}(N)$  , get a 4-channel scalar vector.   
+4  $d_{i}\gets (\| \vec{u}_{i} - \vec{u}_{\alpha_{1}}\| ,\| \vec{u}_{i} - \vec{u}_{\alpha_{2}}\| ,\| \vec{u}_{i} - \vec{u}_{\alpha_{3}}\| ,\| \vec{u}_{i} - \vec{u}_{\alpha_{4}}\|);$    
+5 end   
+6 //  $\mathcal{O}(N + N^2)$  , convert point sets and edge sets into scalar sets.   
+7  $\mathbb{D}\gets \operatorname {set}([d_i]_i = 1),\mathbb{E}\gets \operatorname {set}([d_i,d_j,e_{ij}]_{(i,j)}\in \mathcal{E});$    
+// O(1), decentralization of the four reference points.   
+7  $\vec{U}_{\alpha}\gets (I_{4\times 4} - \frac{1}{4}\mathbf{1}_{4\times 4})\vec{U}_{\alpha};$    
+//  $\mathcal{O}(N + N^2)$  , get the embedding based on current four points.   
+8  $\Gamma_{\alpha}\gets \mathrm{concat}(\vec{U}_{\alpha}^{\top}\vec{U}_{\alpha},\psi_{\mathrm{node}}(\mathbb{D}),\psi_{\mathrm{edge}}(\mathbb{E}));$    
+9  $\mathbb{T}\gets \mathbb{T}\cup \{\Gamma_{\alpha}\} ;$    
+10 end   
+11  $\Gamma \gets \psi_{\mathrm{graph}}(\mathbb{T})$    
+12 return  $\Gamma$
+
+need to use a E(3)-equivariant method to implement the generation process. We call this generating function  $\zeta$ , thereby simplifying Algo. 3 to Algo. 4.
+
+Algorithm 4: A faster method to construct canonical form.  
+Data: A geometric graph  $\mathcal{G}$ , and  $\psi_{\mathrm{node}}$ ,  $\psi_{\mathrm{edge}}$  are DeepSet models. Result: The canonical form  $\Gamma \in \mathbb{R}^H$  of point clouds  $\mathcal{G}$ . // Get four non-coplanar reference points via generation.  
+1  $\vec{V} \gets \zeta(\mathcal{G})$ ;  
+2 for  $\vec{u}_i \in \vec{X}^{(\mathcal{G})}$  do  
+//  $\mathcal{O}(N)$ , get a 4-channel scalar vector.  
+3  $\boldsymbol{d}_i \gets (\|\vec{u}_i - \vec{v}_1\|, \| \vec{u}_i - \vec{v}_2\|, \| \vec{u}_i - \vec{v}_3\|, \| \vec{u}_i - \vec{v}_4\|)$ ;  
+4 end  
+//  $\mathcal{O}(N + N^2)$ , convert point sets and edge sets into scalar sets.  
+5  $\mathbb{D} \gets \text{set}([d_i]_{i=1}^N)$ ,  $\mathbb{E} \gets \text{set}([d_i, d_j, e_{ij}]_{\langle i,j \rangle \in \mathcal{E}})$ ;  
+//  $\mathcal{O}(1)$ , decentralization of the four reference points.  
+6  $\vec{V} \gets (I_{4\times 4} - \frac{1}{4}1_{4\times 4})\vec{V}$ ;  
+//  $\mathcal{O}(N + N^2)$ , get the embedding based on current four points.  
+7  $\Gamma \gets \text{concat}(\vec{V}^\top\vec{V}, \psi_{\mathrm{node}}(\mathbb{D}), \psi_{\mathrm{edge}}(\mathbb{E}))$ ;  
+8 return  $\Gamma$ ;
+
+It is worth noting that Algo. 4 is applicable only when the four virtual nodes generated by  $\zeta$  are not coplanar. We summarize it as:
+
+Theorem 3.6 (Faster Canonical Form). Given an E(3)-equivariant function  $\zeta$  that generates four non-coplanar points on  $\mathcal{G}$ , Algo. 4 is able to create canonical form with time complexity  $\mathcal{O}(N^2)$ .
+
+Proof. Let  $\Gamma$  be the function of Algo. 4, we only need to prove that  $\mathcal{G} \cong \mathcal{H} \iff \Gamma(\mathcal{G}) = \Gamma(\mathcal{H})$ . We first prove sufficiency, let  $\mathcal{G} = \mathfrak{g} \cdot \mathcal{H}$  with  $\mathfrak{g} = (\pmb{O}, \vec{t}) \in \mathrm{E}(3)$  be a group element. Since  $\zeta$  is an E(3)-equivariant function, we have  $\vec{V}^{(\mathcal{G})} = \zeta(\mathcal{G}) = \zeta(\mathfrak{g} \cdot \mathcal{H}) = O\vec{V}^{(\mathcal{G})} + \vec{t} = \vec{V}^{(\mathcal{H})}$ . After decentralization, we have  $O\vec{V}^{(\mathcal{G})} = \vec{V}^{(\mathcal{H})} \implies (\vec{V}^{(\mathcal{G})})^{\top} \vec{V}^{(\mathcal{G})} = (\vec{V}^{(\mathcal{H})})^{\top} \vec{V}^{(\mathcal{H})}$ . In addition, the encoding of nodes and edges depends on the relative distance to the virtual node and is therefore
+
+invariant, i.e.  $\mathbb{D}^{(\mathcal{G})} = \mathbb{D}^{(\mathcal{H})},\mathbb{E}^{(\mathcal{G})} = \mathbb{E}^{(\mathcal{H})}$ . Thus, we get  $\Gamma (\mathcal{G}) = \Gamma (\mathcal{H})$ . Next, we prove the necessity. Note that when we take the reference four points as  $\zeta (\mathcal{G})$  and  $\zeta (\mathcal{H})$ , we can directly get  $\mathcal{G}\cong \mathcal{H}$  in Algo. 2, thus completing the proof.
+
+Table 8: Comparison of time complexity.  
+
+<table><tr><td></td><td colspan="2">General Method (Algo. 3)</td><td colspan="2">Faster Method</td></tr><tr><td></td><td>Algorithm Details</td><td>Time Complexity</td><td>Algorithm Details</td><td>Time Complexity</td></tr><tr><td>Possible chosen of VNs</td><td>Traverse all possible quadruplets \(\left( \begin{smallmatrix} N \\ 4 \end{smallmatrix} \right)\) types).</td><td>\(\mathcal{O}(N^4)\)</td><td>Generate directly.</td><td>\(\mathcal{O}(1)\)</td></tr><tr><td>Construct VNs</td><td>Use real nodes as virtual nodes.</td><td>\(\mathcal{O}(1)\)</td><td>Aggregate by all nodes.</td><td>\(\mathcal{O}(N)\)</td></tr><tr><td>Get distance vector</td><td>Calculate \(N\) nodes to possible VNs.</td><td>\(\mathcal{O}(N^4 \cdot N)\)</td><td>Calculate \(N\) nodes to possible VNs.</td><td>\(\mathcal{O}(1 \cdot N)\)</td></tr><tr><td>Set of distance vector</td><td>Each chosen of VN prodive a set.</td><td>\(\mathcal{O}(N^4)\)</td><td>Each chosen of VN prodive a set.</td><td>\(\mathcal{O}(1)\)</td></tr><tr><td>Embed set with Deepset</td><td>\(N\) for node and \(N^2\) for edge</td><td>\(\mathcal{O}(N^4 \cdot (N + N^2))\)</td><td>\(N\) for node and \(N^2\) for edge</td><td>\(\mathcal{O}(1 \cdot (N + N^2))\)</td></tr><tr><td>Total</td><td>\(\mathcal{O}(N^6)\)</td><td></td><td>\(\mathcal{O}(N^2)\)</td><td></td></tr></table>
+
+# A.5 Construction of full-rank basis set
+
+In this section, we discuss the full-rank basis set, including its existence and construction method on asymmetric graphs.
+
+Theorem 3.7 (Coloring on Asymmetric Graph). In an asymmetric geometric graph, each point can be assigned a unique color. This implies the existence of an E(3) invariant function that maps the features of the  $i$ th node to distinct values  $\pmb{h}_i$ . Similarly, for directed edges  $ij$ , their features are mapped to distinct values  $\pmb{e}_{ij}$ .
+
+Proof. The theorem can be described in the following formal language, given an asymmetric point cloud  $\vec{X} \in \mathbb{R}^{3 \times N}$ , and all function here discussed are node-level scalar function. This theorem means
+
+$$
+\exists f: \vec {X} \mapsto \boldsymbol {y} \in \mathbb {N} _ {+} ^ {N}, \forall i \neq j, \boldsymbol {y} _ {i} \neq \boldsymbol {y} _ {j}. \tag {15}
+$$
+
+First, we prove that it can be expressed as an equivalent proposition, which is
+
+$$
+\forall i \neq j, \exists f: \vec {X} \mapsto \boldsymbol {y} \in \mathbb {N} _ {+} ^ {N}, \boldsymbol {y} _ {i} \neq \boldsymbol {y} _ {j}. \tag {16}
+$$
+
+According to the logical relationship, it is obvious that Eq. (15) leads to Eq. (16). We will prove below that Eq. (16) can also lead to Eq. (15). Let  $f^{ij}:\vec{X}\mapsto \pmb{y}^{ij}$  denote a function let  $y_{i}^{ij}\neq y_{j}^{ij}$ , then consider map  $g:\vec{X}\mapsto Y\coloneqq \oplus_{i,j}\pmb{y}^{ij}$ . We can find that  $Y_{i}\neq Y_{j}$ , and notice each  $f^{ij}$  only uses finite colors, which means we can further assign a new color to each possible output of  $g$ , which is the mapping we want in Eq. (16).
+
+Then we prove it by contradiction Eq. (16), assuming that the following proposition is true:
+
+$$
+\exists i \neq j, \forall f: \vec {X} \mapsto \boldsymbol {y} \in \mathbb {N} _ {+} ^ {N}, \boldsymbol {y} _ {i} = \boldsymbol {y} _ {j}. \tag {17}
+$$
+
+We now construct a contradiction, hoping to prove that Eq. (17) will lead to point cloud symmetry. Let the permutation matrix  $P \in S_N$  represent the exchange between node  $i$  and node  $j$ , so we can get:
+
+$$
+f \left(\boldsymbol {P} \vec {\boldsymbol {X}}\right) = \boldsymbol {P} \boldsymbol {y} = \boldsymbol {y} = f (\vec {\boldsymbol {X}}). \tag {18}
+$$
+
+Since  $f$  is an  $\mathrm{E}(3)$ -invarinat function, we guess  $P\vec{X}$  will fall in set  $\mathrm{E}(3) \cdot \vec{X}$ . And now we prove that  $P\vec{X}$  must fall in such a set. And the canonical form  $\Gamma$  is obvious and  $\mathrm{E}(3)$ -invarinat function and could distinguish whether two point clouds are isomorphic, which means  $P\vec{X}$  must be point cloud isomorphic with  $\vec{X}$ . And notice that  $f$  is a node-level function, which means
+
+$$
+\vec {x} _ {i} = \mathfrak {g} \cdot \vec {x} _ {j}, \vec {x} _ {j} = \mathfrak {g} \cdot \vec {x} _ {i}.
+$$
+
+And we have assumed there are no overlapping points, i.e.  $\vec{x}_i \neq \vec{x}_j$ , so  $\mathfrak{g} \neq \mathfrak{e}$ . And it shows that such a point cloud is symmetric, which constitutes a contradiction and the original proposition is proved.
+
+Theorem 3.8 (Existence of Full-Rank Basis Set). For any given asymmetric graph  $\mathcal{G}$ , an  $\mathbb{F}^{(l)}(\mathcal{G})$ -full-rank  $\tilde{\mathbf{V}}^{(l)}(\mathcal{G})$  can always be constructed for any degree  $l$ .
+
+Proof. We give the proof via a constructive method. According to Theorem 3.7, there exists a mapping let all node be colored differently, and without loss of generality, we denote the color of node  $i$  as  $c_{i}$ . Now we construct function:
+
+$$
+f _ {k} (\vec {\boldsymbol {X}}) = \sum_ {i = 1} ^ {N} \delta_ {k, c _ {i}} \cdot \vec {\boldsymbol {x}} _ {i} = \vec {\boldsymbol {x}} _ {k}, \tag {19}
+$$
+
+It is obviously that such a function is  $\mathrm{E}(3)$ -equivariant, and we and further combine  $f_{k}$  of different  $k$ , which can construct a full-rank  $\mathbb{V}^{(1)}$  since the input point cloud is non-coplanar.
+
+We just give an existence proof here, which is still affected by the structure of the model itself and the form of message passing. We give an example in Example A.15.
+
+# A.6 Comparison of different coloring methods
+
+Node coloring necessitates updating node features. Here, we propose two alternative solutions:
+
+1. Distance to Center: A simple preliminary coloring can be achieved by calculating the distance  $\|\vec{x}_i - \vec{x}_c\|$  from each node to the coordinate center. While this method is straightforward, it lacks practical significance and may not perform well in real-world problems. Additionally, there are counterexamples, such as in Example A.15, where nodes cannot be distinguished.  
+2. Tensor Product: This method extends the previous approach. First, we construct a global feature  $\sum_{i=1}^{N} Y^{(l)} (\vec{x}_i - \vec{x}_c)$ , which is then used to create the coloring through moments. While this method is more complex and incurs higher computational costs, it enhances expressive power.
+
+Table 9: Different Color Methods. For the tensor product method, the set  $\mathbb{L} = \{0,\dots ,L\}$  denotes all degrees and the learnable weights of tensor product are omitted.  
+
+<table><tr><td>Nothing (∅)</td><td>Distance to Center (⊕)</td><td>Tensor Product (⊗)</td></tr><tr><td colspan="3">Calculation Formula</td></tr><tr><td rowspan="2">∅</td><td rowspan="2">h_i← h_i + σ(||xic||)</td><td>hglobal(ℓ)← ∑i=1Nσ(||xic||)·Y(ℓ)(xic)</td></tr><tr><td>h_i← h_i + hglobal(ℓ)⊗cg···⊗cg Y(ℓ)(xic)</td></tr><tr><td colspan="3">Scenarios to Avoid</td></tr><tr><td>Sparse Graphs</td><td>Counterexample like Fig. 4(c)</td><td>Not Founded</td></tr></table>
+
+Below, we present two examples to illustrate potential issues without coloring.
+
+Example A.15 (Fig. 4(c)). We use the center pooling method to construct such an example. First, we ensure that the center of gravity of the image is at the origin, and secondly, make it asymmetric. The construction here requires a basic point cloud:
+
+$$
+\vec {\boldsymbol {X}} _ {0} = \left\{(- 1, 0, 0), \left(\frac {1}{3}, \pm \frac {2 \sqrt {2}}{3}, 0\right), \left(\frac {1}{6}, \pm \frac {\sqrt {3 5}}{6}, 0\right) \right\}.
+$$
+
+All nodes in  $\vec{X}_0$  are in the unit circle, which means the distance between each node and the center of mass is the same. And the final point could be
+
+$$
+\vec {X} = \left\{\vec {X} _ {0} R _ {i} \right\},
+$$
+
+where we sample some random rotation matrix  $\mathbf{R}_i$ . And then it is obviously only using global pooling will lead all virtual nodes in the center of mass if we do not use any edge to color this nodes.
+
+However, it could also be solve by high-degree steerable feature. The global steerable features can construct different virtual nodes with tensor product method in Table 9.
+
+Example A.16 (Degeneration in Uncolored Models). There are specific scenarios in which message passing on uncolored geometric graphs can result in degenerate phenomena. A particularly illustrative example is found in geometric graphs where all node features are identical and all edges are bidirectional. Considering a single-layer EGNN as an example, it may degenerate because the
+
+condition  $\pmb{h}_i = \pmb{h}_j$  implies  $\pmb{m}_{ij} = \pmb{m}_{ji}$ . Consequently, we have  $\sum_{\langle i,j\rangle \in \mathcal{E}}\varphi_{\vec{x}}(\pmb{m}_{ij})\cdot \vec{\pmb{x}}_{ij} = \vec{\mathbf{0}}_3$ . As a result, Eq. (7) can only yield the coordinate center  $\vec{x}_c$ . This phenomenon can be observed in the tetrahedron center experiment in § 4, where besides EGNN, advanced models such as HEGNN [49], TFN [33], and MACE [35] also degenerate.
+
+# B Experiment Details
+
+Our code is available at https://github.com/GLAD-RUC/Uni-EGNN.
+
+# B.1 Expressivity Tests
+
+# B.1.1 Completeness Test
+
+In this section, we provide additional details that were omitted in  $\S 4.1$ , focusing on the following points:
+
+1. The 4-body non-chiral counterexample implemented by the GWL-test (Fig. 2(f) in IASR-test) and the 4-body chiral counterexample (Fig. 2(e) in IASR-test) exhibit a significant issue: the geometric graphs presented by [32] are geometrically isomorphic.  
+2. Details regarding data construction and model design for the chirality test, which primarily include the coloring strategy and specifics of the construction of 0o-type features.
+
+Metric. Referring to the settings of the GWL-test [32], we utilize the average accuracy across ten tests as our evaluation metric. The detailed experimental design is based on the implementation of the GWL-test [32], available in the code repository<sup>5</sup>. In this setup, a single GNN layer is employed to encode the geometric graph, followed by a simple classifier that predicts the label. If a GNN fails to distinguish between two geometric graphs, the output embeddings will be identical, resulting in one graph being classified correctly while the other is misclassified, leading to an accuracy of  $50\%$ . Conversely, an accuracy rate exceeding  $50\%$  indicates that the GNN can generate distinct embeddings and successfully differentiate between the two geometric graphs.
+
+Additionally, the accuracy of the Basic model in Table 2 does not reach  $100\%$  due to a limited number of training epochs for the classifier (set to 100). Increasing the number of epochs (e.g., to 200) can achieve  $100\%$  classification accuracy. Notably, the EGNN model, with the same settings, also achieves  $100\%$  accuracy. We speculate that EGNN improves the embedding of geometric graphs during the message passing process.
+
+Results of  $k$ -chain test. This experiment comes from GWL-test. The canonical form introduced by our model can easily obtain global information, so only one layer is needed to identify the geometric graphs differentiation task that other models require multiple layers.
+
+Table 10:  $k$  -chain Test.  
+
+<table><tr><td rowspan="2">(k=4-chains) 
+GNN Layer</td><td colspan="5">Number of layers</td></tr><tr><td>{k/2}</td><td>{k/2}+1=3</td><td>{k/2}+2</td><td>{k/2}+3</td><td>{k/2}+4</td></tr><tr><td>EGNN</td><td>50.0 ± 0.0</td><td>50.0 ± 0.0</td><td>50.0 ± 0.0</td><td>50.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>GVP-GNN</td><td>50.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>TFN</td><td>50.0 ± 0.0</td><td>50.0 ± 0.0</td><td>50.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>MACE</td><td>50.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>Basiccpl</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>SchNetcpl</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>EGNNcpl</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>GVP-GNNcpl</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr><tr><td>TFNcpl</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td><td>100.0 ± 0.0</td></tr></table>
+
+Geometrical isomorphism of 4-body counterexample. In the open source notebook provided by GWL-test [32], four tasks are presented, the first two of which are the 2-body and 3-body tasks tested
+
+in Table 1. The other two tasks are termed the 4-body non-chiral counterexample and the 4-body chiral counterexample, both originating from IASR-test [69]. According to the experimental results provided in GWL-test, only  $\mathrm{MACE}_{5\text{-body}}$  successfully passed the 4-body non-chiral counterexample; moreover, the results for the 4-body chiral counterexample were not presented.
+
+We successfully reproduced and cited the experimental results of the 2-body and 3-body from GWL-test [32]. However, we were unable to reproduce the results reported in the article despite multiple testing attempts, consistently achieving an accuracy of only  $50\%$ , indicating that  $\mathrm{MACE}_{5\text{-body}}$  failed this task. Furthermore, recent literature [77] also reported similar failures in the test. Upon thorough investigation, we discovered that the geometric graphs used in the two tests were geometrically isomorphic. We suspect that there may be a minor mistake in the GWL-test.
+
+Specifically, the two graphs  $\mathcal{G}_1, \mathcal{G}_2$  of the 4-body non-chiral counterexample are constructed as follows:
+
+1. Consider three sub-graphs:  $\mathcal{H}_1 = \{(3,2, - 4),(0,2,5),(-3,2, - 4)\}$ $\mathcal{H}_2 = \{(3, - 2, - 4),(0, - 2,5),(-3, - 2, - 4)\}$ , and  $\mathcal{H}_3 = \{(0,5,0)\}$ .  $R_{y}$  is a random matrix for rotation around the  $Oy$ -axis. Let  $M_x,M_y$  denote the reflection about  $yOz$ -plane and  $zOx$ -plane.  
+2.  $\mathcal{G}_1 = \mathcal{H}_1\cup (\pmb {R}_y\cdot \mathcal{H}_2)\cup \mathcal{H}_3$  and  $\mathcal{G}_2 = \mathcal{H}_1\cup (\pmb {R}_y\cdot \mathcal{H}_2)\cup (\pmb {M}_y\cdot \mathcal{H}_3)$
+
+Now we show  $\mathcal{G}_1 \cong \mathcal{H}_2$ , notice there are serval relation:
+
+- Internal symmetry of geometric graph:  $\mathcal{H}_i = M_x\cdot \mathcal{H}_i(i = 1,2),\mathcal{H}_3 = R_y\mathcal{H}_3;$  
+- Symmetry between geometric graphs:  $\mathcal{H}_1 = M_y\cdot \mathcal{H}_2$  
+- Properties between geometric transformations:  $M_x^2 = M_y^2 = I$ ,  $R_yM_x = M_xR_y$ ,  $R_yM_y = M_yR_y^\top$ .
+
+Then we have:
+
+$$
+\begin{array}{l} \mathcal {G} _ {1} = \mathcal {H} _ {1} \cup \left(\boldsymbol {R} _ {y} \cdot \mathcal {H} _ {2}\right) \cup \mathcal {H} _ {3} = \left(\boldsymbol {M} _ {y} \cdot \mathcal {H} _ {2}\right) \cup \left(\boldsymbol {R} _ {y} \boldsymbol {M} _ {y} \cdot \mathcal {H} _ {1}\right) \cup \left(\boldsymbol {M} _ {y} \cdot \mathcal {H} _ {3}\right) \\ = \left(\boldsymbol {R} _ {y} \boldsymbol {M} _ {y} \cdot \mathcal {H} _ {1}\right) \cup \left(\boldsymbol {M} _ {y} \cdot \mathcal {H} _ {2}\right) \cup \left(\boldsymbol {M} _ {y} \cdot \mathcal {H} _ {3}\right) \\ = \left(\boldsymbol {M} _ {\boldsymbol {y}} \boldsymbol {R} _ {\boldsymbol {y}} ^ {\top} \cdot \mathcal {H} _ {1}\right) \cup \left(\boldsymbol {M} _ {\boldsymbol {y}} \cdot \mathcal {H} _ {2}\right) \cup \left(\boldsymbol {M} _ {\boldsymbol {y}} \cdot \mathcal {H} _ {3}\right) \\ = \left(\boldsymbol {M} _ {\boldsymbol {y}} \boldsymbol {R} _ {\boldsymbol {y}} ^ {\top}\right) \cdot \left[ \mathcal {H} _ {1} \cup \boldsymbol {R} _ {\boldsymbol {y}} \mathcal {H} _ {2} \cup \boldsymbol {R} _ {\boldsymbol {y}} \mathcal {H} _ {3} \right] \\ = \left(\boldsymbol {M} _ {\boldsymbol {y}} \boldsymbol {R} _ {\boldsymbol {y}} ^ {\top}\right) \cdot \left[ \mathcal {H} _ {1} \cup \boldsymbol {R} _ {\boldsymbol {y}} \mathcal {H} _ {2} \cup \mathcal {H} _ {3} \right] \\ = \left(\boldsymbol {M} _ {\boldsymbol {y}} \boldsymbol {R} _ {\boldsymbol {y}} ^ {\top} \boldsymbol {M} _ {\boldsymbol {x}}\right) \cdot \left[ \boldsymbol {M} _ {\boldsymbol {x}} \mathcal {H} _ {1} \cup \boldsymbol {M} _ {\boldsymbol {x}} \boldsymbol {R} _ {\boldsymbol {y}} \mathcal {H} _ {2} \cup \boldsymbol {M} _ {\boldsymbol {x}} \mathcal {H} _ {3} \right] \\ = \left(\boldsymbol {M} _ {\boldsymbol {y}} \boldsymbol {R} _ {\boldsymbol {y}} ^ {\top} \boldsymbol {M} _ {\boldsymbol {x}}\right) \cdot \left[ \mathcal {H} _ {1} \cup \boldsymbol {R} _ {\boldsymbol {y}} \boldsymbol {M} _ {\boldsymbol {x}} \mathcal {H} _ {2} \cup \boldsymbol {M} _ {\boldsymbol {x}} \mathcal {H} _ {3} \right] \\ = \left(\boldsymbol {M} _ {\boldsymbol {y}} \boldsymbol {R} _ {\boldsymbol {y}} ^ {\top} \boldsymbol {M} _ {\boldsymbol {x}}\right) \cdot \left[ \mathcal {H} _ {1} \cup \boldsymbol {R} _ {\boldsymbol {y}} \mathcal {H} _ {2} \cup \boldsymbol {M} _ {\boldsymbol {x}} \mathcal {H} _ {3} \right] \\ = \left(\boldsymbol {M} _ {y} \boldsymbol {R} _ {y} ^ {\top} \boldsymbol {M} _ {x}\right) \cdot \mathcal {G} _ {2} \\ = \left(\boldsymbol {M} _ {x} \boldsymbol {R} _ {y} \boldsymbol {M} _ {y}\right) ^ {\top} \cdot \mathcal {G} _ {2}. \\ \end{array}
+$$
+
+Moreover, the 4-body chiral counterexample also contains two geometric graphs  $\mathcal{G}_3,\mathcal{G}_4$ , where  $\mathcal{G}_3\cong \mathcal{G}_4$ :
+
+1. Consider two sub-graphs:  $\mathcal{H}_4 = \{(3,0, - 4),(0,0,5),(-3,0, - 4)\}$  and  $\mathcal{H}_5 = \{(0,5,0)\}$ .  $M_x,M_y$  denote the reflection about  $yOz$ -plane and  $zOx$ -plane.  
+2.  $\mathcal{G}_3 = \mathcal{H}_4\cup \mathcal{H}_5$  and  $\mathcal{G}_4 = \mathcal{H}_4\cup (M_y\mathcal{H}_5)$
+
+Now we show  $\mathcal{G}_3\cong \mathcal{H}_3$  , notice there are serval relation:
+
+- Internal symmetry of geometric graph:  $\mathcal{H}_3 = M_x\cdot \mathcal{H}_3$ $\mathcal{H}_3 = M_y\mathcal{H}_3$  and  $\mathcal{H}_4 = M_x\mathcal{H}_4$  
+- Properties between geometric transformations:  $M_x^2 = M_y^2 = I$ .
+
+Then we have:
+
+$$
+\mathcal {G} _ {3} = \mathcal {H} _ {4} \cup \mathcal {H} _ {5} = \left(M _ {x} M _ {y}\right) \cdot \left[ \mathcal {H} _ {4} \cup \boldsymbol {M} _ {y} \mathcal {H} _ {5} \right] = \left(M _ {x} M _ {y}\right) \cdot \mathcal {G} _ {4}.
+$$
+
+# B.1.2 Chirality Test
+
+# Details of the chirality test.
+
+1. Fig. 4(a): Choose  $\mathcal{G}_1$  from the 4-body non-chiral counterexample, and another geometric graph is  $-\mathcal{G}_1$ .  
+2. Fig. 4(b): Since  $\mathcal{G}_3, \mathcal{G}_4$  are symmetric, we modify it to  $\mathcal{G}_5 = \{(3,0,-4), R_y \cdot (0,0,5), (-3,0,-4), (0,5,0)\}$ , where  $R_y$  is a random matrix for rotation around the  $Oy$ -axis.  
+3. Fig. 4(c): Similar to Example A.15.
+
+![](images/78c3056807b8eabce667fe5af8070041125583e57c8a2a7262d30ac582eb1a3a.jpg)  
+(a) Inversion
+
+![](images/2fe159529d6c2b4444ea49a837e5bf6ff853d6e69ebfe379a2d0817e8603738a.jpg)  
+(b) Mirror  
+Figure 4: Examples to test whether the model can recognize chirality. Figs. 4(a) and 4(b) are modified from GWL-test's implementation [36] of IASR-test [69], while Fig. 4(c) is designed by ourselves. Fig. 4(a): Test whether models could distinguish this geometric graph with itself after inversion. Fig. 4(b): Test whether models could distinguish this geometric graph with itself after mirror about the  $xOz$ -plane. Fig. 4(c): The task is same as Fig. 4(a), while all nodes are on the unit sphere, and the center point coincides with the center of the sphere. Such counterexample cannot produce separable virtual nodes by simple center distance coloring.
+
+![](images/fe160d1b1a49fe700083d28f68773c0ed99aa6c9ea68b30024fbf971c2b21fab.jpg)  
+(c) Counterexample
+
+Details of construction of 0o-type features. There are two ways to construct 0o-type features:
+
+1. Determinant: Calculate  $\operatorname{det}([\vec{v}_2 - \vec{v}_1, \vec{v}_3 - \vec{v}_1, \vec{v}_4 - \vec{v}_1])$ ;  
+2. Tensor Product: We still begin by constructing  $\sum_{i=1}^{N} Y^{(l)} (\vec{x}_i - \vec{x}_c)$ , and employing moments, specifically at least 3rd-order moments, to derive 0o-type features. This approach clearly serves as an extension of the previously established method.
+
+# B.2 Performance Tests
+
+We conducted four experiments:
+
+1. Tetrahedron center prediction, which entails predicting graph-level targets, i.e. Monge points, twelve-point centers, and incenters, given the coordinates of four points (with the same node features) of a given tetrahedron.  
+2. 5-body system [72] involves predicting node-level targets, i.e. the coordinates of each charged particle at a specific moment in the future, given its initial coordinates and velocity.  
+3. 100-body system [66], a generalization of the 5 body system, is used to test the performance of the model in large-scale systems.  
+4. MD17 dataset [68], containing eight molecules, is used to predict node-level targets, i.e. the future coordinates of each atom in the molecular trajectory. This dataset can be used to test the performance of the model on real datasets.  
+5. Water-3D mini dataset [66; 67], is used to test the performance of the model in large-scale systems and real datasets. Due to time constraints, we only evaluate a substantial subset (Water-3D-mini: 3,000/300/300 for train/val/test).
+
+Table 11: Details of each dataset.  
+
+<table><tr><td>Dataset</td><td>Target</td><td>Type</td><td>#sample (train / valid / test)</td><td>Node Number</td><td>Edge connection</td></tr><tr><td>Tetrahedron</td><td>graph-level</td><td>Synthetic Dataset</td><td>500 / 2,000 / 2,000</td><td>4</td><td>fullyConnected</td></tr><tr><td>5-body</td><td>node-level</td><td>Synthetic Dataset</td><td>5,000 / 2,000 / 2,000</td><td>5</td><td>fully connected</td></tr><tr><td>100-body</td><td>node-level</td><td>Synthetic Dataset</td><td>5,000 / 2,000 / 2,000</td><td>100</td><td>fully connected</td></tr><tr><td>MD17</td><td>node-level</td><td>Real-world Dataset</td><td>500 / 2,000 / 2,000</td><td>5-13</td><td>distance_cutoff</td></tr><tr><td>Water-3D mini</td><td>node-level</td><td>Real-world Dataset</td><td>3,000 / 300 / 300</td><td>~8,000</td><td>distance_cutoff</td></tr></table>
+
+# B.2.1 Tetrahedron Center Prediction and N-body System
+
+We conducted experiments on a single NVIDIA H20 GPU. In both experiments, we configured the batch size to 100, the learning rate to  $5 \times 10^{-4}$ , and the weight decay to  $10^{-12}$ . For the tetrahedron center prediction task, the maximum number of training epochs was set to 300. In the case of the  $N$ -body system task, we implemented early stopping with a patience of 100 steps.
+
+Metric. We use Mean Squared Error (MSE) to measure the accuracy of the prediction results in both experiments.
+
+Tetrahedron Center Prediction. We generated our dataset using the calculation formulas for the Monge point  $\vec{x}_M$ , twelve-point center  $\vec{x}_T$  and incenter  $\vec{x}_I$  from [105]. Given a tetrahedron defined by four points  $\vec{x}_O, \vec{x}_A, \vec{x}_B, \vec{x}_C$ , with the vectors  $\vec{a} = \vec{x}_A - \vec{x}_O, \vec{b} = \vec{x}_B - \vec{x}_O, \vec{c} = \vec{x}_C - \vec{x}_O$ , we can compute the desired points as follows:
+
+$$
+\vec {\boldsymbol {x}} _ {M} = \vec {\boldsymbol {x}} _ {O} + \frac {\vec {\boldsymbol {a}} \cdot (\vec {\boldsymbol {b}} + \vec {\boldsymbol {c}}) (\vec {\boldsymbol {b}} \times \vec {\boldsymbol {c}}) + \vec {\boldsymbol {b}} \cdot (\vec {\boldsymbol {c}} + \vec {\boldsymbol {a}}) (\vec {\boldsymbol {c}} \times \vec {\boldsymbol {a}}) + \vec {\boldsymbol {c}} \cdot (\vec {\boldsymbol {a}} + \vec {\boldsymbol {b}}) (\vec {\boldsymbol {a}} \times \vec {\boldsymbol {b}})}{2 \vec {\boldsymbol {a}} \cdot (\vec {\boldsymbol {b}} \times \vec {\boldsymbol {c}})},
+$$
+
+$$
+\vec {\boldsymbol {x}} _ {T} = \vec {\boldsymbol {x}} _ {M} + \frac {1}{3} (\vec {\boldsymbol {x}} _ {O} - \vec {\boldsymbol {x}} _ {M}), \tag {20}
+$$
+
+$$
+\vec {x} _ {I} = \frac {\| \vec {b} \times \vec {c} \| \vec {a} + \| \vec {c} \times \vec {a} \| \vec {b} + \| \vec {a} \times \vec {b} \| \vec {c}}{\| \vec {b} \times \vec {c} \| + \| \vec {c} \times \vec {a} \| + \| \vec {a} \times \vec {b} \| + \| \vec {b} \times \vec {c} + \vec {c} \times \vec {a} + \vec {a} \times \vec {b} \|}.
+$$
+
+We utilize 500/2000/2000 samples for training, validation, and testing, respectively.  $\epsilon = 10^{-6}$  was added to the denominator to avoid division by zero. In addition, to avoid numerical problems caused by the center point being extremely large, we generated the circumscribed sphere of the tetrahedron when generating the tetrahedron, ensuring that the radius of the sphere is less than or equal to 6.
+
+$N$ -body system.  $N$ -body system [72] is a dataset generated from simulations. In our simulations, each system consists of 5 charged particles with random charges  $c_{i} \in \{\pm 1\}$ , whose movements are governed by Coulomb forces. The task is to estimate, a node-level target, i.e. the coordinates of the  $N$  particles after 1000 timesteps. The initial node feature is set as the kinetic energy, while the edge features include the distance and the product of the charges. We use 5000/2000/2000 samples for training, validation, and testing like the setting in HEGNN [49]. Since the particles have different initial velocities, they initially have different characteristics (coloring).
+
+# B.2.2 Implementation of Baselines
+
+We undertook further development based on the codes of EGNN [43], TFN [33], and MACE [35] provided in the GWL-test repository. Following our testing, we eliminated the Gate modules from both TFN and MACE, as their inclusion resulted in significantly degraded performance. Additionally, for HEGNN [49], we utilized the code from the original paper and opted for the inner product normalization option.
+
+Each model utilizes at most 4 layers, and the dimension of the hidden embedding is fixed at 64. The last three models employ at most 2nd-degree steerable features. Both TFN and MACE utilize 8 channels for steerable features, with the correlation parameter for MACE set to 4.
+
+# B.2.3 Implementation of Our Models
+
+We provide two complete implementations of EGNN [43] and TFN [33] as follows.
+
+Unlike FastEGNN [66], the virtual nodes in this approach are not updated following their initialization during the message-passing layers. We define the normalization coefficient as  $\alpha_{i} \coloneqq 1 / |\mathcal{N}(i)|$ . For
+
+the  $i$ th node, we denote its features as  $h_i$  and its coordinates as  $\vec{x}_i$ . Here,  $C$  represents the channels of the virtual nodes, and  $\vec{Z} \in \mathbb{R}^{C \times 3}$  denotes all coordinates. Specifically, for  $\mathrm{TFN}_{\mathrm{cpl}}$ -global, we employ the canonical form to build a new feature  $\boldsymbol{w}_i$  for each node, thereby rescaling the norm of steerable features in a manner analogous to HEGNN [49], which is consistent with Theorem 3.2.
+
+Table 12: Complete global models.  
+
+<table><tr><td></td><td>EGNNcpl-global</td><td>TFNcpl-global</td></tr><tr><td>Node Coloring</td><td>h_i = Color(h_i, distance_to_center)</td><td>h_i^{(0)} = Color(h_i^{(0)}, distance_to_center)</td></tr><tr><td>Generate VNs</td><td>Z = 1C x_c + αi ∑i=1N φZ(h_i) · x_iic</td><td>Z = 1C x_c + αi ∑i=1N φZ(h_i^{(0)}) · x_iic</td></tr><tr><td>Create Canonical Form</td><td>Mi = 1C x_i - Z, mi,cpl = M_i^T M_i / |M_i^T M_i|_F</td><td>Mi = 1C x_i - Z, mi,cpl = M_i^T M_i / |M_i^T M_i|_F</td></tr><tr><td>Update Node Feature</td><td>h_i = h_i + φcpl(m_i,cpl), h_i = Message_Passing(h_i, G)</td><td>w_i = h_i^{(0)} + φcpl(m_i,cpl)</td></tr><tr><td rowspan="3">Message Passing (×L)</td><td>m_ij = φm(h_i, h_j, e_ij, ||x_ij||^2), m_ij = φx̅(m_ij) · x_ij,</td><td>m_ij^{(L)} = h_i^{(L)} \otimes_{cg} Y^{(L)}(\vec{x}_{ij})</td></tr><tr><td>m_i = αi ∑j∈N(i) m_ij, m_i = αi ∑j∈N(i) m_ij</td><td>h_i^{(L)} = h_i^{(L)} + αi ∑j∈N(i) m_ij^{(L)}</td></tr><tr><td>h_i = h_i + φh(h_i, m_i), x_i = x_i + m_i</td><td>h_i^{(l)} = φcpl(h_i^{(0)}, w_i) · h_i^{(l)}, w_i = w_i + φw(w_i, h_i^{(0)})</td></tr><tr><td>Readout (node-level)</td><td>h_i = φout(h_i), x_i = x_i</td><td>h_i = φout(h_i^{(0)}), x_i = x_i + e3nn.03.Linear(h_i^{(1)})</td></tr><tr><td>Readout (graph-level)</td><td>h_G = φpool(N/∑i=1N h_i), x_G = 1/N ∑i=1N x_i</td><td>h_G = φpool(N/∑i=1N h_i), x_G = 1/N ∑i=1N x_i</td></tr></table>
+
+To enhance the model's performance, we permit the virtual nodes to be updated layer by layer, similar to the approach taken in FastEGNN [66]. We denote the normalization coefficient  $\alpha_{i} := 1 / |\mathcal{N}(i)|$ . We define the normalization coefficient as  $\alpha_{i} := 1 / |\mathcal{N}(i)|$ . Notably, we allocate  $C$  virtual nodes to each node, denoted as  $\vec{Z}_i \in \mathbb{R}^{C \times 3}$ , and these  $C$  virtual nodes share a scalar feature represented by  $s_i$ .
+
+Table 13: Complete local models.  
+
+<table><tr><td></td><td>EGNNcpl-local</td><td>TFNcpl-local</td></tr><tr><td rowspan="3">Generate VNs</td><td>mi j = φm(hi, hj, eij, ||xi||2)</td><td>mi j = φm(hi(0), hj(0), eij, ||xi||2)</td></tr><tr><td>si = φs(φinit(hi), mi j)</td><td>si = φs(φinit(hi(0)), mi j)</td></tr><tr><td>Zi = 1Cxi + αi ∑j∈N(i) φZ(hi) · xi j</td><td>Zi = 1Cxi + αi ∑j∈N(i) φZ(hi) · xi j</td></tr><tr><td rowspan="8">Message Passing (×L)(real &amp; virtual)</td><td>mi j = φm(hi, hj, eij, ||xi||2), mi j = φx(mij) · xi j</td><td>mi j(L) = hi(L) ∩ cg Y(L) (xiij)</td></tr><tr><td>mi = αi ∑j∈N(i) mi j, mi = αi ∑j∈N(i) mi j</td><td>hi(L) = hi(L) + αi ∑j∈N(i) mi(j)</td></tr><tr><td>hi = hi + φh(hi, mi), xi = xi + mi</td><td>hi(L) = φcpl(mi(0)) · hi(L)</td></tr><tr><td>Mi j = 1Cxi - Zi, mi,cpl = MiT ij Mij / ||MiT ij Mij|F</td><td>Mi j = 1Cxi - Zi, mi,cpl = MiT ij Mij / ||MiT ij Mij|F</td></tr><tr><td>mi,j,cpl = φm,cpl(si, hj, mi,cpl)</td><td>mi,j,cpl = φm,cpl(si, hj(0), mi,j,cpl)</td></tr><tr><td>hi = hi + φh,cpl(mij,cpl), si = si + φs,cpl(mij,cpl)</td><td>hi(0) = hi(0) + φh(o),cpl(mij,cpl), si = si + φs,cpl(mij,cpl)</td></tr><tr><td>xi = xi + αi ∑j∈N(i) φx(v)(mi,j,cpl) ·Mi j</td><td>xi = xi + αi ∑j∈N(i) φx(v)(mi,j,cpl) ·Mi j</td></tr><tr><td>Zi = Zi + αi ∑j∈N(i) φZ(v)(mi,j,cpl) · (-Mi j)</td><td>Zi = Zi + αi ∑j∈N(i) φZ(v)(mi,j,cpl) · (-Mi j)</td></tr><tr><td>Readout (node-level)</td><td>hi = φout(hi), xi = xi</td><td>hi = φout(hi(0)), xi = xi + e3nn.03.Linear(hi(1))</td></tr><tr><td>Readout (graph-level)</td><td>hG = φpool(N/∑i=1N hi), xiG = 1/N ∑i=1N xi</td><td>hG = φpool(N/∑i=1N hi), xiG = 1/N ∑i=1N xi</td></tr></table>
+
+Parameter Quantities. Here, we present the number of parameters for each model in the tasks shown in § 4.2. Since the inputs for the two tasks differ, the number of parameters for models with the same number of layers also varies.
+
+Table 14: Tetrahedron dataset.  
+
+<table><tr><td></td><td colspan="4">Total Parameters</td></tr><tr><td></td><td>1-layer</td><td>2-layer</td><td>3-layer</td><td>4-layer</td></tr><tr><td>EGNN</td><td>33.7k</td><td>67.1k</td><td>100.6k</td><td>134.1k</td></tr><tr><td>FastEGNN</td><td>67.4k</td><td>134.4k</td><td>201.7k</td><td>268.8k</td></tr><tr><td>HEGNN</td><td>46.9k</td><td>84.9k</td><td>122.9k</td><td>160.9k</td></tr><tr><td>TFN</td><td>46.5k</td><td>93.0k</td><td>139.4k</td><td>185.9k</td></tr><tr><td>MACE</td><td>48.9k</td><td>97.8k</td><td>126.6k</td><td>195.5k</td></tr><tr><td>Equifomer</td><td>313.5k</td><td>340.5k</td><td>367.5k</td><td>349.4k</td></tr><tr><td>\( EGNN_{cpl} \)-global</td><td>181.7k</td><td>215.2k</td><td>248.7k</td><td>282.1k</td></tr><tr><td>\( EGNN_{cpl}^{*} \)-global</td><td>46.8k</td><td>55.4k</td><td>63.9k</td><td>72.5k</td></tr><tr><td>\( EGNN_{cpl} \)-local</td><td>107.5k</td><td>180.7k</td><td>253.8k</td><td>327.0k</td></tr><tr><td>\( EGNN_{cpl}^{*} \)-local</td><td>28.2k</td><td>47.3k</td><td>66.5k</td><td>85.7k</td></tr><tr><td>\( TFN_{cpl} \)-global</td><td>215.2k</td><td>291.1k</td><td>267.1k</td><td>443.1k</td></tr><tr><td>\( TFN_{cpl}^{*} \)-global</td><td>70.3k</td><td>104.6k</td><td>138.9k</td><td>173.3k</td></tr><tr><td>\( TFN_{cpl} \)-local</td><td>138.0k</td><td>241.7k</td><td>345.5k</td><td>449.2k</td></tr><tr><td>\( TFN_{cpl}^{*} \)-local</td><td>51.0k</td><td>93.1k</td><td>135.2k</td><td>177.3k</td></tr></table>
+
+Table 15:  $N$  -body dataset.  
+
+<table><tr><td></td><td colspan="4">Total Parameters</td></tr><tr><td></td><td>1-layer</td><td>2-layer</td><td>3-layer</td><td>4-layer</td></tr><tr><td>EGNN</td><td>33.5k</td><td>66.9k</td><td>100.4k</td><td>133.8k</td></tr><tr><td>FastEGNN</td><td>67.5k</td><td>134.6k</td><td>201.4k</td><td>268.4k</td></tr><tr><td>HEGNN</td><td>46.7k</td><td>84.6k</td><td>122.6k</td><td>160.5k</td></tr><tr><td>TFN</td><td>46.4k</td><td>92.8k</td><td>139.2k</td><td>185.6k</td></tr><tr><td>MACE</td><td>47.2k</td><td>94.3k</td><td>141.5k</td><td>188.6k</td></tr><tr><td>Equiformer</td><td>313.6k</td><td>340.6k</td><td>367.5k</td><td>394.5k</td></tr><tr><td>\( EGNN_{cpl} \)-global</td><td>181.3k</td><td>214.7k</td><td>248.1k</td><td>281.5k</td></tr><tr><td>\( EGNN_{cpl}^{*} \)-global</td><td>46.6k</td><td>55.1k</td><td>63.6k</td><td>72.1k</td></tr><tr><td>\( EGNN_{cpl} \)-local</td><td>107.3k</td><td>180.4k</td><td>253.5k</td><td>326.6k</td></tr><tr><td>\( EGNN_{cpl}^{*} \)-local</td><td>28k</td><td>47.2k</td><td>66.4k</td><td>85.5k</td></tr><tr><td>\( TFN_{cpl} \)-global</td><td>215.6k</td><td>291.6k</td><td>367.7k</td><td>443.7k</td></tr><tr><td>\( TFN_{cpl}^{*} \)-global</td><td>70.5k</td><td>104.8k</td><td>139.2k</td><td>173.6k</td></tr><tr><td>\( TFN_{cpl} \)-local</td><td>138.2k</td><td>242.0k</td><td>345.8k</td><td>449.7k</td></tr><tr><td>\( TFN_{cpl} \)-local</td><td>51.1k</td><td>93.3k</td><td>135.4k</td><td>177.5k</td></tr></table>
+
+# C Rethinking the Equivalence and General Functions
+
+In this section, we give a Fourier series-like approach to understanding equivariant GNNs.
+
+# C.1 Rethinking equivariance: A perspective from Fourier expansion.
+
+According to [106], the Fourier expansion of a multivariate function necessitates the simultaneous consideration of the basis functions associated with all variables, as demonstrated below:
+
+$$
+f (t) = \sum_ {n \in \mathbb {Z}} f _ {n} \exp (- 2 \pi j n t) \longrightarrow f (\boldsymbol {t}) = \sum_ {\boldsymbol {n} \in \mathbb {Z} ^ {d}} f _ {\boldsymbol {n}} \exp (- 2 \pi j \boldsymbol {n} ^ {\top} \boldsymbol {t}) = f (\boldsymbol {t}) = \sum_ {\boldsymbol {n} \in \mathbb {Z} ^ {d}} f _ {\boldsymbol {n}} \prod_ {k = 1} ^ {d} \exp (- 2 \pi j n _ {k} t _ {k}), \tag {21}
+$$
+
+represents a multi-dimensional variable. In fact, any function defined on the unit sphere can be expressed as a summation of spherical harmonics. Let  $\hat{x}$  denote a point on the unit sphere.
+
+$$
+f (\hat {\boldsymbol {x}}) = \sum_ {l = 0} ^ {\infty} \sum_ {m = - l} ^ {l} f _ {m} ^ {(l)} Y _ {m} ^ {(l)} (\hat {\boldsymbol {x}}) = \sum_ {l = 0} ^ {\infty} \left\langle \boldsymbol {a} ^ {(l)}, Y ^ {(l)} (\hat {\boldsymbol {x}}) \right\rangle , \tag {22}
+$$
+
+This formulation offers two representations. The fully expanded form aligns more closely with the Fourier series, while the inner product formulation facilitates a connection to equivariance. At this juncture, we introduce a second point  $\hat{\pmb{y}}$  on the unit sphere and consider the expansion of the function with two inputs,  $\hat{\pmb{x}}$  and  $\hat{\pmb{y}}$ .
+
+$$
+f (\hat {\boldsymbol {x}}, \hat {\boldsymbol {y}}) = \sum_ {l _ {1} = 0} ^ {\infty} \sum_ {l _ {2} = 0} ^ {\infty} \sum_ {m _ {1} = - l _ {1}} ^ {l _ {1}} \sum_ {m _ {2} = - l _ {2}} ^ {l _ {2}} f _ {m _ {1}, m _ {2}} ^ {(l _ {1}, l _ {2})} Y _ {m _ {1}} ^ {(l _ {1})} (\hat {\boldsymbol {x}}) Y _ {m _ {2}} ^ {(l _ {2})} (\hat {\boldsymbol {y}}). \tag {23}
+$$
+
+We observe that the product  $Y_{m_1}^{(l_1)}(\hat{\pmb{x}})Y_{m_2}^{(l_2)}(\hat{\pmb{y}})$  constructs  $(2l_1 + 1)\cdot (2l_2 + 1)$  basis functions, i.e. elements in  $Y^{(l_1)}(\hat{\pmb{x}})\otimes Y^{(l_2)}(\hat{\pmb{y}})\in \mathbb{R}^{(2l_1 + 1)\cdot (2l_2 + 1)}$ . Let  $\pmb {f}^{(l_1\otimes l_2)}\in \mathbb{R}^{(2l_1 + 1)\cdot (2l_2 + 1)}$  be the coefficients. Consequently, the expansion can also be expressed in an inner product form as follows:
+
+$$
+f (\hat {\boldsymbol {x}}, \hat {\boldsymbol {y}}) = \sum_ {l _ {1} = 0} ^ {\infty} \sum_ {l _ {2} = 0} ^ {\infty} \left\langle \boldsymbol {f} ^ {\left(l _ {1} \otimes l _ {2}\right)}, Y ^ {\left(l _ {1}\right)} (\hat {\boldsymbol {x}}) \otimes Y ^ {\left(l _ {2}\right)} (\hat {\boldsymbol {y}}) \right\rangle . \tag {24}
+$$
+
+According to the notation of tensor product, we denote the  $l$ th-degree steerable feature derived from the tensor product  $Y^{(l_1)}(\hat{\pmb{x}})\otimes Y^{(l_2)}(\hat{\pmb{y}})$  as  $\tilde{\pmb{b}}^{(l_1\otimes l_2\to l)}(\hat{\pmb{x}},\hat{\pmb{y}})$ , which can be expressed as follows:
+
+$$
+\bigoplus_ {l = \left| l _ {1} - l _ {2} \right|} ^ {l _ {1} + l _ {2}} \tilde {\boldsymbol {b}} ^ {\left(l _ {1} \otimes l _ {2} \rightarrow l\right)} (\hat {\boldsymbol {x}}, \hat {\boldsymbol {y}}) = \boldsymbol {Q} ^ {\left(l _ {1} \otimes l _ {2}\right)} \cdot \left(Y ^ {\left(l _ {1}\right)} (\hat {\boldsymbol {x}}) \otimes Y ^ {\left(l _ {2}\right)} (\hat {\boldsymbol {y}})\right). \tag {25}
+$$
+
+Let us define  $\bigoplus_{l = |l_1 - l_2|}^{l_1 + l_2}\pmb{w}^{(l_1\otimes l_2\to l)} = (\pmb{Q}^{(l_1\otimes l_2)})^\top \pmb{f}^{(l_1\otimes l_2)}$ . This leads us to the following expression:
+
+$$
+f (\hat {\boldsymbol {x}}, \hat {\boldsymbol {y}}) = \sum_ {l = 0} ^ {\infty} \sum_ {| l _ {1} - l _ {2} | \leq l \leq l _ {1} + l _ {2}} \left\langle \boldsymbol {w} ^ {\left(l _ {1} \otimes l _ {2} \rightarrow l\right)}, \tilde {\boldsymbol {b}} ^ {\left(l _ {1} \otimes l _ {2} \rightarrow l\right)} \left(\hat {\boldsymbol {x}}, \hat {\boldsymbol {y}}\right)\right\rangle , \tag {26}
+$$
+
+where a  $l$ th-degree steerable function  $\tilde{\pmb{t}}^{(l)}(\hat{\pmb{x}},\hat{\pmb{y}})$  incorporates only the  $l$ th-degree components in Eq. (26), and can be expressed as follows:
+
+$$
+\tilde {\boldsymbol {t}} ^ {(l)} (\hat {\boldsymbol {x}}, \hat {\boldsymbol {y}}) = \sum_ {| l _ {1} - l _ {2} | \leq l \leq l _ {1} + l _ {2}} w ^ {(l _ {1} \otimes l _ {2} \rightarrow l)} \cdot \tilde {\boldsymbol {b}} ^ {(l _ {1} \otimes l _ {2} \rightarrow l)} (\hat {\boldsymbol {x}}, \hat {\boldsymbol {y}}), \tag {27}
+$$
+
+Eq. (27) offers a more concrete understanding of the framework. It is important to note, however, the formula in Eq. (3) is invariant to permutations of the inputs, while the inputs in Eq. (27) are presented with a specific order. Nevertheless, this specificity does not preclude us from conducting an analysis based on this formulation:
+
+1. Since  $l_{1}$  and  $l_{2}$  can take any values, there will be infinite basis functions in  $\mathbb{B}^{(l)}$  when the geometric graph accommodates interactions involving more than two bodies.
+
+2. Not all bases in Eq. (27) will be meaningful; that is, some bases may not emerge in the objective function, although they satisfy the equivariance constraints.
+
+From this perspective, the advantages and disadvantages of the equivariant model become evident. By incorporating an equivariance prior into the model design, it effectively eliminates components that cannot exist within the framework. However, this constraint necessitates a limited selection of operators in the model design. In the current setup, only the tensor product exhibits strict equivariance (while others can be transformed and expressed using tensor products, e.g. tensor decomposition in frame averaging [56; 84]), whereas other operators, such as group convolution, require approximations under current floating-point operation systems and can achieve only partial equivariance. This limitation, in turn, constrains the model's capacity for learning.
+
+# C.2 Rethinking General Functions: the Equivariant Decomposition
+
+Generally, the input of a geometric graph could be denoted as  $\{\vec{x}_i\}_{i=1}^N$ , without loss of generality, we assume that these coordinates are decentralized. Although in practice, people generally use edges as input (i.e.,  $\{\vec{x}_i - \vec{x}_j\}_{i,j=1}^N$ ), for the sake of convenience we still discuss the vectors of points here, and it is easy to verify that they are equivalent. Then a normal function  $f: \mathbb{R}^{3 \times N} \to \mathbb{R}^{2l+1}$  we want in a  $l$ -degree problem could be rewritten as:
+
+$$
+\boldsymbol {f} ^ {(l)} (\mathcal {G}) = \sum_ {\tilde {\boldsymbol {b}} _ {\alpha} ^ {(l)} (\mathcal {G}) \in \mathbb {B} ^ {(l)}} w _ {\alpha} \tilde {\boldsymbol {b}} _ {\alpha} ^ {(l)} (\mathcal {G}) + \boldsymbol {f} _ {\text {e l s e}} ^ {(l)}. \tag {28}
+$$
+
+To completely give the expansion formula of such  $\pmb{f}^{(l)}$ , we need to analyze each dimension separately. And we denote the  $m$ th-dimension  $(-l \leq m \leq l)$  of  $\pmb{f}^{(l)}$  as  $f^{(l,m)}$  (similarly,  $b^{(l,m)}$  represents the  $m$ th dimension in  $\tilde{\pmb{b}}^{(l)}$ ), which is:
+
+$$
+f ^ {(l, m)} (\mathcal {G}) = \sum_ {\tilde {\boldsymbol {b}} _ {\alpha} ^ {(l)} (\mathcal {G}) \in \mathbb {B} ^ {(l)}} w _ {\alpha} ^ {(l)} \tilde {\boldsymbol {b}} _ {\alpha} ^ {(l, m)} (\mathcal {G}) + \sum_ {s = 0} ^ {\infty} \sum_ {\tilde {\boldsymbol {b}} _ {\beta} ^ {(s)} (\mathcal {G}) \in \mathbb {B} ^ {(s)}} \sum_ {t = - s} ^ {s} w _ {\beta} ^ {(s, t)} \tilde {\boldsymbol {b}} _ {\beta} ^ {(s, t)} (\mathcal {G}). \tag {29}
+$$
+
+And it could find that he  $m$ th-dimension  $f_{\mathrm{else}}^{(l,m)}$  of  $f_{\mathrm{else}}^{(l)}$  could be donated as:
+
+$$
+f _ {\text {e l s e}} ^ {(l, m)} (\mathcal {G}) = \sum_ {s = 0} ^ {\infty} \sum_ {\tilde {\boldsymbol {b}} _ {\beta} ^ {(s)}} (\mathcal {G}) \in \mathbb {B} ^ {(s)} \sum_ {t = - s} ^ {s} w _ {\beta} ^ {(s, t)} b _ {\beta} ^ {(s, t)} (\mathcal {G}). \tag {30}
+$$
+
+Here  $\alpha$  and  $\beta$  are dummy index for enumeration and both  $w_{\alpha}^{(l)}$  and  $w_{\beta}^{(l,m)}$  are constant (or only the function about all radians  $\{\|\vec{x}_i\|_i^N\}$ ). Notice that the same basis may appear in both two items in the equation, to ensure uniqueness, we set
+
+$$
+w _ {\alpha} ^ {(l)} = \underset {w _ {\alpha} ^ {(l)}} {\operatorname {a r g m i n}} \| f _ {\text {e l s e}} ^ {(l)} (\mathcal {G}) \| ^ {2} = \underset {w _ {\alpha} ^ {(l)}} {\operatorname {a r g m i n}} \sum_ {\tilde {\boldsymbol {b}} _ {\beta} ^ {(t)} (\mathcal {G}) \in \mathbb {B} ^ {(t)}} \sum_ {s = - t} ^ {t} \left(w _ {\beta} ^ {(s, t)}\right) ^ {2}. \tag {31}
+$$
+
+And with definition of equivariance error in Spherical CNNs [107], we could find that
+
+$$
+\begin{array}{l} \mathcal {L} _ {\mathrm {e q u}} := \mathbb {E} \left[ \left\| \rho^ {(l)} (\mathfrak {g}) \cdot \boldsymbol {f} ^ {(l)} (\mathcal {G}) - \boldsymbol {f} ^ {(l)} (\mathfrak {g} \cdot \mathcal {G}) \right\| \right] \\ = \mathbb {E} \left[ \left\| \rho^ {(l)} (\mathfrak {g}) \cdot \boldsymbol {f} _ {\text {e l s e}} ^ {(l)} (\mathcal {G}) - \boldsymbol {f} _ {\text {e l s e}} ^ {(l)} (\mathfrak {g} \cdot \mathcal {G}) \right\| \right] \tag {32} \\ \leq \mathbb {E} \left[ \left\| \rho^ {(l)} (\mathfrak {g}) \cdot \boldsymbol {f} _ {\text {e l s e}} ^ {(l)} (\mathcal {G}) \right\| + \left\| \boldsymbol {f} _ {\text {e l s e}} ^ {(l)} (\mathfrak {g} \cdot \mathcal {G}) \right\| \right] \\ \leq 2 \cdot \| \boldsymbol {f} _ {\text {e l s e}} ^ {(l)} (\mathcal {G}) \|. \\ \end{array}
+$$
+
+Traditionally, practitioners have opted for data-driven methods, such as data augmentation, to achieve equivariance. This approach aims to constrain the norm  $\| \pmb{f}_{\mathrm{else}}^{(l)}\|$  by adjusting the parameters  $w_{\beta}^{(s,t)}$  throughout training, with the goal of minimizing their values. In contrast, models derived from geometric learning directly incorporate the equivariance constraint into the model architecture, effectively achieving  $\| \pmb{f}_{\mathrm{else}}^{(l)}\| = 0$ . A fundamental aspect of these models is the construction of the
+
+basis set  $\mathbb{B}^{(l)}$ , allowing them to systematically remove irrelevant or non-existent components from the learning process.
+
+Moreover, for a  $\pmb{f}^{(l)}(\mathcal{G})$ , obtaining  $\pmb{f}_{\mathrm{test}}^{(l)}(\mathcal{G})$  or calculating  $\| \pmb{f}_{\mathrm{test}}^{(l)}(\mathcal{G}) \|$  can be intractable. However, a plausible approach is to utilize another general model whose output is bounded by a constant  $M$  to learn the residue. This leads us to the relationship  $\| \pmb{f}_{\mathrm{test}}^{(l)}(\mathcal{G}) \| \leq \| \pmb{f}^{(l)}(\mathcal{G}) \| \leq M$ . We introduce this concept here, hoping it will inspire future research.
+
+Additionally, it is important to emphasize that we only require the inference model to be equivariant. In practice, we allow the use of non-equivariant methods for training our model. For instance, noise injection in SaVeNet [108] and the Huber loss in MEAN [102] are examples of such methods. These approaches modify the coefficients  $w_{\alpha}$ , but they do not violate the equivariance constraint.
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: We prove that a complete equivariant GNN can be achieved through two key components: 1) a complete scalar function, referred to as the canonical form of the geometric graph; 2) a full-rank steerable basis set.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: We provide all these in  $\S 6$
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+# Answer: [Yes]
+
+Justification: We provide all these in Appendix A.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
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+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
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+
+Justification: We provide all these in Appendix B.
+
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+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: Our code is available at https://github.com/GLAD-RUC/Uni-EGNN.
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+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: We provide all these in Appendices B.2.2 and B.2.3.
+
+Guidelines:
+
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+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
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+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: We performed multiple runs and reported the variance in all expressivity tests and experiments with MD17.
+
+Guidelines:
+
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+
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We conducted experiments on a single NVIDIA H20 GPU.
+
+Guidelines:
+
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+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
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+
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+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [NA]
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+
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+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
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+
+# Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
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+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: We used the code from https://github.com/GLAD-RUC/HEGNN and https://github.com/chaitjo/geometric-gnn-dojo, which are available under the MIT license.
+
+# Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
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+
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+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [Yes]
+
+Justification: Our code is available at https://github.com/GLAD-RUC/Uni-EGNN.
+
+Guidelines:
+
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+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer:[NA]
+
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+
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+
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+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
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+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
+
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+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components.
+
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+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
\ No newline at end of file
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+# Unlabeled Data Can Provably Enhance In-Context Learning of Transformers
+
+Renpu Liu
+
+University of Virginia
+
+Charlottesville, VA 22903
+
+renpu@virginia.edu
+
+Jing Yang
+
+University of Virginia
+
+Charlottesville, VA 22903
+
+yangjing@virginia.edu
+
+# Abstract
+
+Large language models (LLMs) exhibit impressive in-context learning (ICL) capabilities, yet the quality of their predictions is fundamentally limited by the few costly labeled demonstrations that can fit into a prompt. Meanwhile, there exist vast and continuously growing amounts of unlabeled data that may be closely related to the ICL task. How to utilize such unlabeled data to provably enhance the performance of ICL thus becomes an emerging fundamental question. In this work, we propose a novel augmented ICL framework, in which the prompt includes a small set of labeled examples alongside a block of unlabeled inputs. We focus on the multi-class linear classification setting and demonstrate that, with chain-of-thought (CoT) prompting, a multi-layer transformer can effectively emulate an expectation-maximization (EM) algorithm. This enables the transformer to implicitly extract useful information from both labeled and unlabeled data, leading to provable improvements in ICL accuracy. Moreover, we show that such a transformer can be trained via teacher forcing, with its parameters converging to the desired solution at a linear rate. Experiments demonstrate that the augmented ICL framework consistently outperforms conventional few-shot ICL, providing empirical support for our theoretical findings. To the best of our knowledge, this is the first theoretical study on the impact of unlabeled data on the ICL performance of transformers.
+
+# 1 Introduction
+
+Since the introduction (Vaswani et al., 2017), transformers have become foundational models in diverse fields such as natural language processing (Radford, 2018; Devlin et al., 2019), computer vision (Dosovitskiy, 2020), and reinforcement learning (Chen et al., 2021). A key driver of their impact is the remarkable capability for In-Context Learning (ICL) (Brown et al., 2020). Without requiring parameter updates, transformers performing ICL can adapt to new tasks based solely on contextual examples provided within the prompt. This enables state-of-the-art few-shot performance across a multitude of applications, including reasoning and language understanding (Chowdhery et al., 2023), dialog generation (Thoppilan et al., 2022), and linear regression (Garg et al., 2022; Fu et al., 2023), etc.
+
+Despite the power of ICL, its reliance on labeled examples presents a significant bottleneck for large language models (LLMs). Acquiring high-quality labeled data is oin general expensive and time-consuming (Zhou et al., 2023; Chung et al., 2024; Sun et al., 2023; Wang et al., 2023). For example, creating the instruction-tuning and RLHF datasets for models like GPT-3.5 and GPT-4 involved thousands of expert annotator hours, yet constituted less than  $0.1\%$  of the tokens encountered during pre-training (Ouyang et al., 2022; Achiam et al., 2023).
+
+Some existing approaches attempt to mitigate labeled data scarcity in ICL. For instance, Wan et al. (2023); Chen et al. (2025a) use an LLM to automatically generate pseudo-demonstrations at inference time by pairing unlabeled queries with the model's own predictions as pseudo labels. However, model-generated pseudo-labels inevitably inherit the biases and error patterns of the teacher model, resulting in noisy demonstrations that may limit potential performance gains.
+
+In this work, instead of synthesizing examples with pseudo-labels, we explore a different approach by directly utilizing abundant and continuously growing (Raffel et al., 2020; Touvron et al., 2023) unlabeled data during ICL. The fundamental question we aim to answer is:
+
+Can we provably enhance the ICL performance of transformers by effectively leveraging plentiful unlabeled data alongside limited labeled examples?
+
+We answer this question affirmatively from a new augmented in-context learning perspective. This paradigm involves prompting a transformer with a mixture of a few labeled examples and numerous unlabeled examples, aiming to infer the missing labels within a single forward pass. By reasoning over unlabeled examples directly in the prompt, it bypasses the need for potentially costly, time-consuming, and bias-introducing labeling or pseudo-label generation steps in conventional ICL. In this work, we focus on augmented ICL for multi-class linear classification. Our main contributions are as follows.
+
+- Expressiveness with CoT Prompting. First, we show that through Chain-of-Thought (CoT) prompting, a multi-layer transformer can leverage both labeled and unlabeled data to effectively solve the multi-class linear classification problem during ICL. Essentially, the transformer is able to obtain an initial estimation of the mean vectors of classes using the labeled data, and then iteratively refine the estimates by clustering the unlabeled data in an Expectation-Maximization (EM) fashion. We explicitly characterize the design of the transformer and theoretically prove that the class mean estimation will converge to the ground truth as the CoT steps increase. For a prompt consisting of  $N$  labeled and  $M$  unlabeled samples, our results indicate that the excess risk of our approach scales in  $\mathcal{O}\left(1 / \sqrt{N\mathrm{poly}(M)}\right)$ , strictly improving the excess risk lower bound of  $\mathcal{O}\left(1 / \sqrt{N}\right)$  for any classifier that utilizes  $N$  labeled samples only. Our results indicate that the augmented ICL can effectively utilize the information from the unlabeled data, enabling steady performance improvement as unlabeled data increases.  
+- Training Convergence under Teacher Forcing. Second, we prove that, with proper initialization, when applying gradient descent on the population loss defined through teacher forcing, the tunable parameters of the transformer converge to the desired solution linearly. Thus, the trained transformer can mimic the EM algorithm through CoT prompting during inference, theoretically validating the transformer's expressiveness for augmented ICL. Our proof involves a novel decomposition of the gradient of the CoT training loss into two analytically tractable terms. For each of them, we leverage the inherent isotropy of the involved quantities to simplify the analysis, which enables us to derive a tight upper bound on the critical inner-product term and obtain the linear convergence rate.  
+- Empirical Results. Finally, we evaluate the performance of augmented ICL in transformers trained via teacher forcing. Our experimental results show that the augmented ICL approach significantly outperforms conventional ICL in both class mean estimation and label prediction, with the advantage becoming more pronounced as the number of unlabeled data samples increases. Moreover, augmented ICL surpasses the Bayes-optimal classifier that relies solely on labeled data. These empirical observations are consistent with our theoretical findings.
+
+# 2 Related Works
+
+ICL with Transformers. Brown et al. (2020) first shows that GPT-3, a transformer-based LLM, can perform new tasks from input-output pairs without parameter updates, suggesting its ICL ability. This intriguing phenomenon of transformers has attracted much attention, leading to various interpretations and hypotheses about its underlying mechanism. Research on ICL often demonstrates how transformers can emulate learning algorithms. For instance, several studies have designed transformers that execute gradient descent for linear and non-linear regression tasks (Akyurek et al., 2023; Von Oswald et al., 2023a). Recent works demonstrate that transformers can implement more advanced optimization algorithms other than vanilla gradient descent on various ICL tasks (Bai et al.,
+
+2024; Von Oswald et al., 2023b; Zhang et al., 2024a; Ahn et al., 2024; Liu et al., 2025). Another line of research adopts a statistical perspective: ICL can be viewed as an implicit form of Bayesian updating based on the examples provided in the prompt, with the diversity of pretraining data shaping the prior (Xie et al., 2022; Raventós et al., 2023; Garg et al., 2022).
+
+Several studies (Gupta et al., 2024; Agarwal et al., 2024) investigate "unsupervised ICL", in which the prompt consists solely of unlabeled inputs. Another line of work leverages LLMs to generate pseudo-labels for unlabeled data, which are then used as demonstrations during ICL (Chen et al., 2023; Wan et al., 2023; Yang et al., 2023; Chen et al., 2025a). Our work leverages both labeled and unlabeled examples within the prompt to enhance ICL performance in a semi-supervised learning manner, which stands in sharp contrast to the aforementioned studies.
+
+Notably, a recent concurrent work (Li et al., 2025) also investigates the impact of the semi-supervised data model on the ICL performance of transformers. Specifically, Li et al. (2025) focus on a linear transformer without nonlinear activations in a binary classification setting, and characterize the asymptotic ICL performance as the number of unlabeled samples approaches infinity. In contrast, we study a more realistic architecture that incorporates the softmax attention mechanism and establish a non-asymptotic convergence guarantee in the general multi-class setting.
+
+Training Dynamics of Transformers. A number of recent works aim to provide theoretical characterizations of the training dynamics of transformers. Ahn et al. (2024); Mahankali et al. (2023); Zhang et al. (2024a); Huang et al. (2023) investigate the training dynamics of transformers with a single attention layer and a single head for in-context linear regression tasks. Cui et al. (2024) prove that transformers with multi-head attention layers outperform those with single-head attention. Cheng et al. (2024) show that local optimal solutions in transformers can perform gradient descent in-context for non-linear functions. Kim and Suzuki (2024) study the non-convex meanfield dynamics of transformers, and Nichani et al. (2024) characterize the convergence rate for the training loss in learning a causal graph. Additionally, Chen et al. (2024) investigate the gradient flow in training multi-head single-layer transformers for multi-task linear regression. Chen and Li (2025) propose a supervised training algorithm for multi-head transformers. The training dynamics of transformers for binary classification (Tarzanagh et al., 2023b,a; Vasudeva et al., 2024; Li et al., 2023; Deora et al., 2023; Li et al., 2024a), multi-class classification (Shen et al., 2025) and next-token prediction (Tian et al., 2023, 2024; Li et al., 2024b; Huang et al., 2024) have also been studied recently.
+
+Transformers with CoT. In language modeling tasks, transformers have been proven to be powerful across various downstream tasks. However, transformers struggle to solve mathematical or scientific problems with a single generation, particularly when multiple reasoning steps are required. CoT prompting is introduced to enable transformers to generate intermediate results autoregressively before reaching the final answer, and has been shown to boost performance on arithmetic, commonsense, and scientific tasks (Wei et al., 2022; Kojima et al., 2022)
+
+Recently, the training dynamics of transformers with CoT have been studied in Huang et al. (2025a) for weight prediction in linear regression, in Li et al. (2024a) for in-context supervised learning, in Kim and Suzuki (2025); Wen et al. (2025) for the parity problems, and in Huang et al. (2025b) for the even pairs problem. None of these studies, however, address whether the multi-step reasoning capacity through CoT can be utilized to extract information from unlabeled inputs.
+
+# 3 Preliminaries
+
+Notations. For matrix  $\mathbf{X}$ , we use  $[\mathbf{X}]_{p;q,r;s}$  to denote the submatrix that contains rows  $p$  to  $q$  and columns  $r$  to  $s$ , and we use  $[\mathbf{X}]:,i$  and  $[\mathbf{X}]_{j:}$  to denote the  $i$ -th column and  $j$ -th row of  $\mathbf{X}$ , respectively. For convenience, we occasionally denote the  $i$ -th column  $\mathbf{X}$  by  $[\mathbf{X}]_i$  when no ambiguity arises.  $[\mathbf{X}]:,-C:-1$  means the last  $C$  columns of matrix  $\mathbf{X}$ . We use  $\| \mathbf{X}\| _F$  to denote its Frobenius norm. For vector  $\mathbf{x}$ , we use  $\| \mathbf{x}\| _1$ ,  $\| \mathbf{x}\|$  and  $\| \mathbf{x}\|_{\infty}$  to denote its  $\ell_1$ ,  $\ell_2$  and  $\ell_{\infty}$  norms, respectively. We denote by  $\mathbb{1}_d$  and  $\mathbf{0}_d$  the  $d$ -dimensional all-1 and all-0 column vectors, respectively.  $\mathbb{1}_{a\times b}$  and  $\mathbf{0}_{a\times b}$  denote the all-1 and all-0 matrices of size  $a\times b$ , respectively. We denote the indicator function as  $\mathbf{1}_{\{A\}}$ , which equals 1 if event  $A$  is true.
+
+# 3.1 Transformer Architecture
+
+In this work, we consider the encoder-based transformer architecture (Vaswani et al., 2017), where each transformer layer consists of an attention layer followed by a multi-layer perception (MLP) layer.
+
+Definition 3.1 (Attention layer). Denote an  $M$ -head attention layer parameterized by  $\{(\mathbf{V}_m, \mathbf{Q}_m, \mathbf{K}_m)_{m \in [M]}\}$  as  $\mathrm{attn}_{\{(\mathbf{V}_m, \mathbf{Q}_m, \mathbf{K}_m)\}}(\cdot)$ , where  $\mathbf{V}_m, \mathbf{Q}_m, \mathbf{K}_m \in \mathbb{R}^{D \times D}$ ,  $\forall m \in [M]$ . Then, given an input sequence  $\mathbf{H} \in \mathbb{R}^{D \times (N + 1)}$ , the output sequence of the attention layer is
+
+$$
+\mathrm {a t t n} _ {\{(\mathbf {V} _ {m}, \mathbf {Q} _ {m}, \mathbf {K} _ {m}) \}} (\mathbf {H}) = \mathbf {H} + \sum_ {m = 1} ^ {M} (\mathbf {V} _ {m} \mathbf {H}) \times \sigma \big ((\mathbf {K} _ {m} \mathbf {H}) ^ {\top} (\mathbf {Q} _ {m} \mathbf {H}) \big),
+$$
+
+where  $\sigma$  is a non-linear activation function.
+
+Definition 3.2 (MLP layer). Given  $\mathbf{W}_1 \in \mathbb{R}^{D' \times D}$ ,  $\mathbf{W}_2 \in \mathbb{R}^{D \times D'}$  and a bias vector  $\mathbf{b} \in \mathbb{R}^{D'}$ , an MLP layer following the decoder attention layer, denoted as  $\mathrm{MLP}_{\{\mathbf{W}_1, \mathbf{W}_2, \mathbf{b}\}}$ , maps each token in the input sequence (i.e., each column  $\mathbf{h}_i$  in  $\mathbf{H} \in \mathbb{R}^{D \times N}$ ) to another token as
+
+$$
+\operatorname {M L P} _ {\left\{\mathbf {W} _ {1}, \mathbf {W} _ {2}, \mathbf {b} \right\}} (\mathbf {h} _ {i}) = \mathbf {h} _ {i} + \mathbf {W} _ {2} \sigma (\mathbf {W} _ {1} \mathbf {h} _ {i} + \mathbf {b}),
+$$
+
+where  $\sigma$  is a non-linear activation function.
+
+# 3.2 Augmented In-context Learning
+
+Conventional In-Context Learning (ICL). For an ICL task, a trained transformer is given an ICL instance  $\mathcal{I} = (\mathcal{D},\mathbf{x}_{N + 1})$ , where  $\mathcal{D} = \{(\mathbf{x}_j,y_j)\}_{j\in [N]}$  and  $\mathbf{x}_{N + 1}$  is a query. Here,  $\mathbf{x}_j\in \mathbb{R}^d$  is an in-context example, and  $y_{j}$  is the corresponding label for  $\mathbf{x}_j$ . For each instance,  $\{(\mathbf{x}_j,y_j)\}_{j = 1}^{N + 1}$  are generated independently accordingly to an underlying distribution. The objective of ICL is to predict  $y_{N + 1}$  without any parameter updating of the transformer.
+
+Augmented ICL. In this work, we consider a new unlabeled data augmented ICL framework. Specifically, each ICL instance now comprises a set of labeled examples,  $\mathcal{D}_{\mathrm{label}} := \{(x_j, y_j)\}_{j=1}^N$ , and a set of unlabeled examples,  $\mathcal{D}_{\mathrm{unlabel}} := \{\mathbf{x}_j\}_{j=N+1}^{N+M}$ , i.e.,  $\mathcal{I} = \mathcal{D}_{\mathrm{label}} \cup \mathcal{D}_{\mathrm{unlabel}}$ . Similar to conventional ICL, all  $(x_j, y_j)$  pairs follow the same distribution. The objective of augmented ICL is then to predict labels for all the  $M$  unlabeled samples in  $\mathcal{D}_{\mathrm{unlabel}}$ .
+
+We note that the augmented ICL generalizes the conventional ICL framework, and reduces to it when  $M = 1$ . While the conventional ICL can be utilized to solve the prediction for those  $M$  unlabeled samples individually in parallel, by augmenting them in the same ICL instance, it provides an opportunity for the transformer to extract common statistical information in those unlabeled data, which can be utilized to improve the joint prediction accuracy.
+
+Augmented ICL for Multi-class Linear Classification. We consider augmented ICL for a multiclass linear classification problem. We assume there exist  $C$  classes, and the label space  $\mathcal{V}$  consists of one-hot vectors  $\{\mathbf{e}_1,\dots ,\mathbf{e}_C\}$ , where each  $\mathbf{e}_i\in \mathbb{R}^C$  is the  $i$ -th unit vector. For each ICL instance  $\mathcal{I}_{\mathrm{M}}$ , the samples are randomly generated according to
+
+$$
+\mathbf {M} \sim P _ {\mathbf {M}}, \quad \mathbf {y} _ {j} \sim \operatorname {U n i f o r m} (\mathcal {Y}), \quad \boldsymbol {\epsilon} _ {j} \sim \mathcal {N} (\mathbf {0}, \boldsymbol {\Sigma}), \quad \mathbf {x} _ {j} = \mathbf {M} \mathbf {y} _ {j} + \boldsymbol {\epsilon} _ {j}, \quad j \in [ M + N ], \tag {3.1}
+$$
+
+where  $\mathbf{M} \in \mathbb{R}^{d \times C}$  and  $P_{\mathbf{M}}$  is a prior distribution over  $\mathbb{R}^{d \times C}$ . Denote the columns of  $\mathbf{M}$  as  $\{\pmb{\mu}_i\}_{i=1}^C$ . Then, each  $\mathbf{x}_j$  essentially follows a  $C$ -component mixture Gaussian distribution parametrized by mean vectors  $\{\pmb{\mu}_i\}_{i=1}^C$  and shared covariance matrix  $\pmb{\Sigma}$ . In this work, we assume  $\pmb{\Sigma}$  is isotropic. We adopt this assumption for theoretical tractability, as it is crucial for deriving the closed-form update rules for the transformer. This approach is a standard and widely adopted practice in related literature to facilitate theoretical analysis (He et al., 2025; Zhang et al., 2024b; Chen et al., 2025b).
+
+# 3.3 Chain-of-Thought Prompting for Augmented ICL
+
+The core challenge in augmented ICL is leveraging both unlabeled data and labeled examples to infer task structure from a single instance. Unlike standard few-shot ICL, which often uses direct pattern matching, the augmented ICL requires more complex inference to effectively utilize the larger unlabeled set, making a simple one-step prediction insufficient.
+
+Chain-of-Thought (CoT) reasoning offers a promising way to enhance a transformer's ICL capabilities. This is crucial for augmented ICL, as it enables the transformer to effectively utilize unlabeled data through iterative latent parameter estimation and refinement.
+
+To implement augmented in-context learning via CoT prompting, we first encode a task instance  $\mathcal{I}$  into an embedding matrix  $\mathbf{H}$  by concatenating three column blocks: the labeled example block, the unlabeled example block, and the inference prompt block as follows:
+
+$$
+\mathbf {H} = \left[ \begin{array}{c c c c c c c c c} \mathbf {x} _ {1} & \dots & \mathbf {x} _ {N} & \mathbf {x} _ {N + 1} & \dots & \mathbf {x} _ {N + M} & \mathbf {0} & \dots & \mathbf {0} \\ \mathbf {y} _ {1} & \dots & \mathbf {y} _ {N} & \mathbf {0} & \dots & \mathbf {0} & \mathbf {0} & \dots & \mathbf {0} \\ \mathbf {p} _ {1} & \dots & \mathbf {p} _ {N} & \mathbf {p} _ {N + 1} & \dots & \mathbf {p} _ {N + M} & \mathbf {q} _ {1} & \dots & \mathbf {q} _ {C} \end{array} \right] \triangleq \left[ \begin{array}{c c c} \mathbf {X} _ {\ell} & \mathbf {X} _ {u} & \mathbf {0} \\ \mathbf {Y} _ {\ell} & \mathbf {0} & \mathbf {0} \\ \mathbf {P} _ {\ell} & \mathbf {P} _ {u} & \mathbf {Q} ^ {(0)} \end{array} \right], \tag {3.2}
+$$
+
+where  $\mathbf{p}_j\in \mathbb{R}^{d_p}$  is an auxiliary embedding that stores the (predicted) classification probability vector for the  $j$ -th sample, as well as a binary indicator to distinguish the labeled and unlabeled data.  $\mathbf{q}_i\in \mathbb{R}^{d_p}$  serves as the initial CoT token for class  $i$ , which contains the one-hot vector  $\mathbf{e}_i$  to indicate the corresponding class, and an all-zero vector representing the transformer's initial estimate for the mean vector  $\boldsymbol{\mu}_i$ .
+
+Denote a trained transformer with parameter  $\Theta$  as  $\mathrm{TF}_{\Theta}$ . With CoT, we will use the transformer to generate  $T$  intermediate steps before it outputs the prediction. Specifically, let  $\widehat{\mathbf{H}}^{(t-1)}$  be the input sequence at the  $t$ -th step of CoT, where  $\widehat{\mathbf{H}}^{(0)} = \mathbf{H}$ , and  $\mathrm{TF}_{\Theta}(\widehat{\mathbf{H}}^{(t-1)})$  as the corresponding output of the transformer. Then, we will take out the last  $C$  columns of  $\mathrm{TF}_{\Theta}(\widehat{\mathbf{H}}^{(t-1)})$ , and append it to the end of  $\widehat{\mathbf{H}}^{(t-1)}$  to form the input for the next CoT step. Specifically, we have
+
+$$
+\widehat {\mathbf {H}} ^ {(t)} = \left[ \widehat {\mathbf {H}} ^ {(t - 1)}, \left[ \mathrm {T F} _ {\Theta} \left(\widehat {\mathbf {H}} ^ {(t - 1)}\right) \right] _ {:, - C: - 1} \right] = \left[ \begin{array}{c c c c c c} \mathbf {X} _ {\ell} & \mathbf {X} _ {u} & \mathbf {0} & \star & \dots & \star \\ \mathbf {Y} _ {\ell} & \mathbf {0} & \mathbf {0} & \star & \dots & \star \\ \mathbf {P} _ {\ell} & \mathbf {P} _ {u} & \mathbf {Q} ^ {(0)} & \mathbf {Q} ^ {(1)} & \dots & \mathbf {Q} ^ {(t)} \end{array} \right], \tag {3.3}
+$$
+
+where
+
+$$
+\mathbf {Q} ^ {(t)} = \left[ \begin{array}{c c c} \mathbf {e} _ {1} & \dots & \mathbf {e} _ {C} \\ \widehat {\boldsymbol {\mu}} _ {1} ^ {(t)} & \dots & \widehat {\boldsymbol {\mu}} _ {C} ^ {(t)} \\ \star & \dots & \star \end{array} \right]. \tag {3.4}
+$$
+
+Here  $\star$  is a placeholder for dummy tokens,  $\mathbf{e}_i$  is the  $i$ -th unit vector, and  $\widehat{\pmb{\mu}}_i^{(t)}$  is the estimated mean vector for class  $i$  at the  $t$ -th CoT step.
+
+After  $T$  iterations, we read out  $\widehat{\pmb{\mu}}_1^{(T)}\cdots\widehat{\pmb{\mu}}_C^{(T)}$  from  $\mathbf{Q}^{(T)}$  as the final estimation of the class mean vectors. Then, the label of each unlabeled data can be estimated through a maximum likelihood estimation, i.e.,
+
+$$
+\widehat {\mathbf {y}} _ {j} = \left\{\mathbf {e} _ {i}: i = \arg \min  _ {i \in [ C ]} \left\| \mathbf {x} _ {j} - \widehat {\boldsymbol {\mu}} _ {i} ^ {(T)} \right\| \right\}, \quad j \in [ N + 1: N + M ]. \tag {3.5}
+$$
+
+# 4 Expressiveness with CoT Prompting for Augmented ICL
+
+In this section, we show that a multi-layer transformer can implement an Expectation-Maximization (EM)-style algorithm to extract useful statistical information from the unlabeled data, which will be combined with information extracted from the labeled data to jointly estimate the class means and improve the augmented ICL performance. Specifically, we have the following result.
+
+Theorem 4.1. There exists a 4-layer transformer, such that its output sequence at the  $(t + 1)$ -th CoT step satisfies
+
+$$
+\widehat {\boldsymbol {\mu}} _ {i} ^ {(t + 1)} = \widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \frac {\eta^ {(t)}}{M} \sum_ {j = N + 1} ^ {N + M} p _ {i j} ^ {(t)} (\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \mathbf {x} _ {j}) + \mathbf {1} _ {\{t = 0 \}} \cdot \frac {C}{N} \sum_ {j = 1} ^ {N} (\mathbf {e} _ {i} ^ {\top} \mathbf {y} _ {j}) \mathbf {x} _ {j}, \tag {4.1}
+$$
+
+for any  $i \in [C]$ , where  $\eta^{(t)} = \alpha / (T' + t)$  for some positive constants  $\alpha$  and  $T'$ ,  $p_{ij}^{(t)}$  is the normalized weight
+
+$$
+p _ {i j} ^ {(t)} = \frac {\sum_ {\tau = 0} ^ {t} \exp \left(- \frac {1}{2} \| \widehat {\boldsymbol {\mu}} _ {i} ^ {(\tau)} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} - 1} ^ {2} + \beta \tau\right)}{\sum_ {\tau = 0} ^ {t} \sum_ {c = 1} ^ {C} \exp \left(- \frac {1}{2} \| \widehat {\boldsymbol {\mu}} _ {c} ^ {(\tau)} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} - 1} ^ {2} + \beta \tau\right)}, \tag {4.2}
+$$
+
+and  $\beta$  is a positive constant.
+
+We outline the construction of each layer of the transformer below, and defer the detailed derivation and specific parameter implementation to Appendix C.
+
+The four-layer architecture is designed to mirror an EM iteration for Gaussian mixture model clustering (Zhao et al., 2020; Sula and Zheng, 2022) within the transformer's forward pass. The EM algorithm operates iteratively. First, in the E-step, it utilizes the current class mean estimates embedded in the input sequence to compute the estimated class membership for each unlabeled data point. Subsequently, the M-step updates the class mean estimates by performing a maximum likelihood estimation of the unlabeled data, and then combining them with the estimates obtained from the labeled data. Through this iterative process, the algorithm can achieve an accurate estimate of the underlying class means, enabling accurate classification.
+
+The first layer. The first transformer layer includes a softmax-activated attention layer followed by an MLP layer. We construct its parameters so that it outputs the class membership estimate for the each unlabeled sample as in the form of Equation (4.2), where the mean estimates  $\{\widehat{\pmb{\mu}}_1^{(\tau)},\dots ,\widehat{\pmb{\mu}}_C^{(\tau)}\}_{\tau = 1}^t$  are embedded in the reasoning blocks  $\mathbf{Q}^{(1)}\cdot \cdot \cdot \mathbf{Q}^{(t)}$  in the input sequence, and the parameter  $\beta$  is embedded in the first layer as well. This probability represents how likely sample  $j$  is estimated to be in class  $i$ . Since the temperature parameter  $\beta \tau$  is proportional to the step index  $\tau$ , estimates from earlier CoT steps carry less importance. In the limit of  $\beta \rightarrow \infty$ , the weight vector depends only on the latest CoT step, i.e.,
+
+$$
+p _ {i j} ^ {(t)} = \frac {\exp \left(- \frac {1}{2} \| \widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2}\right)}{\sum_ {c = 1} ^ {C} \exp \left(- \frac {1}{2} \| \boldsymbol {\mu} _ {c} ^ {(t)} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2}\right)}. \tag {4.3}
+$$
+
+The second and third layers. The second and third transformer layers consist of a linear attention layer followed by an MLP layer. These layers are designated for the M-step (Maximization step) of the EM algorithm. In this step, the class mean estimates  $\{\widehat{\pmb{\mu}}_i\}_{i = 1}^C$  are updated by maximizing the overall log-likelihood of the unlabeled data with the estimated class membership probabilities  $p_{ji}^{(t)}$ . It aims to solve
+
+$$
+P _ {1}: \quad \left\{\boldsymbol {\mu} _ {c} ^ {(t + 1)} \right\} _ {c} = \arg \max  _ {\left\{\boldsymbol {\mu} _ {c} \right\} _ {c}} \sum_ {j = N + 1} ^ {N + M} \sum_ {i = 1} ^ {C} p _ {j i} ^ {(t)} \log \mathcal {N} (\mathbf {x} _ {j}; \boldsymbol {\mu} _ {i}, \boldsymbol {\Sigma}).
+$$
+
+The implementation for these two layers is equivalent to tasking one step of gradient descent over  $P_{1}$ , i.e.,
+
+$$
+\hat {\boldsymbol {\mu}} _ {i} ^ {(t + 1)} = \hat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \frac {\eta^ {(t)}}{M} \sum_ {j = N + 1} ^ {N + M} p _ {i j} ^ {(t)} (\hat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \mathbf {x} _ {j}). \tag {4.4}
+$$
+
+The fourth layer. Finally, the last transformer layer includes a ReLU-activated attention layer followed by an MLP layer. This layer calculates the initial class mean estimates for the labeled dataset and is only activated at the first CoT step. It implements the following updating rule:
+
+$$
+\widehat {\boldsymbol {\mu}} _ {i} ^ {(t + 1)} = \mathbf {1} _ {\{t = 0 \}} \cdot \frac {C}{N} \sum_ {j = 1} ^ {N} \left(\mathbf {e} _ {i} ^ {\top} \mathbf {y} _ {j}\right) \mathbf {x} _ {j}, \tag {4.5}
+$$
+
+which initialize  $\widehat{\mu}_i^1$  to be the average of  $\mathbf{x}_j$ 's for the labeled data samples in class  $i$ . This initialization will be refined iteratively through the CoT steps by leveraging the information from the unlabeled data.
+
+We note that the parameters of the last three layers are data-independent and can be explicitly constructed beforehand, and only the parameters of the first layer depend on the distribution of the data, which can be obtained through CoT training, as elaborated in Section 5.
+
+Next, we will show that the transformer specified in Theorem 4.1 will recover  $\{\mu_i\}_{i=1}^C$  accurately with high probability, and explicitly characterize the benefit of unlabeled data in this augmented ICL.
+
+Theorem 4.2 (Class Mean Estimation Error). Given the transformer described in Theorem 4.1, when  $N \geq 36\alpha^2 L^2 \log 1 / \epsilon$ ,  $M \geq \max \{(T')^4, \log^2 1 / \epsilon\}$ , and  $t \geq \max \{\sqrt[4]{M}, T'\}$ , with probability at least  $1 - \epsilon$ , the output of the transformer after  $t$  CoT steps satisfies
+
+$$
+\left\| \widehat {\mathbf {M}} ^ {(t)} - \mathbf {M} \right\| _ {F} ^ {2} \leq C \frac {\log (1 / \epsilon)}{N \sqrt [ 4 ]{M}},
+$$
+
+where  $C, \alpha, L, T'$  are positive constants.
+
+Corollary 4.1 (Label Prediction Error Bound). Let  $\widehat{\mathbf{y}}_j$  be the predicted label for  $\mathbf{x}_j$  according to Equation (3.5). Let  $\mathcal{R}^*$  be the prediction error under the Bayes-optimal classifier with known class mean vectors  $\mu_1, \dots, \mu_C$ . Then, under the same conditions as described in Theorem 4.2, we have
+
+$$
+\mathbb {P} [ \widehat {\mathbf {y}} _ {j} \neq \mathbf {y} | \boldsymbol {\mu} _ {1}, \dots , \boldsymbol {\mu} _ {C} ] - \mathcal {R} ^ {*} \leq \mathcal {O} \Big (\frac {1}{\sqrt {N \operatorname {p o l y} (M)}} \Big).
+$$
+
+Proof sketch. The proof of Theorem 4.2 contains three major steps. In Step 1, we utilize the Hoeffding's inequality to ensure that with a sufficient number of labeled data  $N$ , the initial class mean estimates  $\widehat{\pmb{\mu}}_1^{(1)}, \dots, \widehat{\pmb{\mu}}_C^{(1)}$  are in a small neighborhood of the ground truth class means  $\pmb{\mu}_1, \dots, \pmb{\mu}_C$ . In Step 2, we need to bound the gap between the gradient descent updating step for  $t > 1$  in Equation (4.4), and one gradient descent step for the expected log-likelihood loss  $\mathcal{L}(\{\widehat{\pmb{\mu}}_i^{(t)}\}) = \mathbb{E}_{\mathbf{x}}\left[\log \left(\frac{1}{C}\sum_{i=1}^{C}\exp\left(-\frac{1}{2}\|\mathbf{x} - \widehat{\pmb{\mu}}_i^{(t)}\|^2\right)\right)\right]$ . To ensure that the gap is sufficiently small, we need to design the temperature parameter  $\beta \tau$  so that the normalized weight is biased heavily toward the class mean estimation obtained from the current CoT step, and the influence of previous CoT steps is minimized. Then, utilizing Bernstein's inequality, this gap is bounded. In Step 3, we utilize Lipschitz continuity of  $\mathcal{L}(\{\pmb{\mu}_i\})$ , combing the bound on the gradient gap in Step 2, to show that  $\|\widehat{\mathbf{M}}^{(t)} - \mathbf{M}\|_F^2 \leq O(1/\sqrt{N\mathrm{poly}(M)})$  for  $t$  large enough if  $\widehat{\mathbf{M}}^{(1)}$  is in a small neighborhood of  $\mathbf{M}$  which is guaranteed in Step 1. The complete proof can be found in Appendix C.
+
+Based on the smoothness of the Bayes risk, we have the following corollary as a direct consequence of Theorem 4.2.
+
+Remark 1. The advantage of utilizing unlabeled data in the augmented ICL becomes evident when comparing Corollary 4.1 with the existing lower bound on the excess risk for classical binary classification. It has been shown that the excess risk for any classifier trained on  $N$  labeled data scales in  $\Omega(1/\sqrt{N})$  in the worst case of M (Li et al., 2017), which is in stark contrast to the upper bound  $\mathcal{O}\left(1/\sqrt{N\mathrm{poly}(M)}\right)$  in Corollary 4.1. This result indicates that the designed transformer can effectively utilize the unlabeled data through CoT prompting, and strictly improves the prediction accuracy of any classifier that utilizes the labeled data only.
+
+# 5 Training Dynamics with Teacher Forcing
+
+While Section 4 indicates that there exists a transformer that is able to implement an EM-type algorithm to utilize unlabeled data and improve the ICL performance through CoT prompting, in this section, we show that such a transformer can be obtained through teacher forcing training (Kim and Suzuki, 2025; Huang et al., 2025b).
+
+The training objective of teacher forcing is to ensure that the transformer can mimic the trajectory of iterative updating under an EM algorithm during the CoT inference. Mathematically, it requires that the distance between the estimated mean vectors  $\{\widehat{\pmb{\mu}}_1^{(t)}\dots \widehat{\pmb{\mu}}_C^{(t)}\}_{t = 1}^T$  during CoT inference and those generated by a reference algorithm  $f_{\mathrm{ref}}$  is small along the trajectory. Specifically, given  $\mathbf{X}_{\ell},\mathbf{Y}_{\ell}$  and  $\mathbf{X}_u$ , we denote the generated reference trajectory as  $f_{\mathrm{ref}}(\mathbf{X}_{\ell},\mathbf{Y}_{\ell},\mathbf{X}_u) = \{\widehat{\pmb{\mu}}_{\mathrm{ref},1}^{(t)}\dots \widehat{\pmb{\mu}}_{\mathrm{ref},C}^{(t)}\}_{t = 1}^T$ . Then, we construct the reference embedding sequence at the  $t$ -th CoT step as
+
+$$
+\mathbf {H} _ {\mathrm {r e f}} ^ {(t)} = \left[ \begin{array}{c c c c c c} \mathbf {X} _ {\ell} & \mathbf {X} _ {u} & \mathbf {0} & \star & \dots & \star \\ \mathbf {Y} _ {\ell} & \mathbf {0} & \mathbf {0} & \star & \dots & \star \\ \mathbf {P} _ {\ell} & \mathbf {P} _ {u} & \mathbf {Q} ^ {(0)} & \mathbf {Q} _ {\mathrm {r e f}} ^ {(1)} & \dots & \mathbf {Q} _ {\mathrm {r e f}} ^ {(t)} \end{array} \right], \quad \mathbf {Q} _ {\mathrm {r e f}} ^ {\tau} = \left[ \begin{array}{c c c} \mathbf {e} _ {1} & \dots & \mathbf {e} _ {C} \\ \boldsymbol {\mu} _ {\mathrm {r e f}, 1} ^ {(\tau)} & \dots & \boldsymbol {\mu} _ {\mathrm {r e f}, C} ^ {(\tau)} \\ * & \dots & * \end{array} \right],   \forall   \tau   \in [ t ].
+$$
+
+We note that the reference embedding shares the same structure as the embedding defined in Equation (3.3), except that now the mean estimates are generated by the reference algorithm instead of the transformer itself. We then feed  $\mathbf{H}_{\mathrm{ref}}^{(t)}$  to the transformer, and extract the updated mean estimates from its output  $\mathrm{TF}_{\Theta}(\mathbf{H}_{\mathrm{ref}}^{(t)})$ .
+
+The corresponding CoT training loss can be defined as:
+
+$$
+\widehat {\mathcal {L}} _ {\mathrm {C o T - t r a i n}} (\boldsymbol {\Theta}; \mathcal {I} _ {\mathbf {M}}) = \frac {1}{T} \sum_ {t = 1} ^ {T} \sum_ {i = 1} ^ {C} \left\| \left[ \mathrm {T F} _ {\boldsymbol {\Theta}} \left(\mathbf {H} _ {\text {r e f}} ^ {(t - 1)}\right) \right] _ {d + 2 c + 1: 2 d + 2 c, M + N + i + C (t - 1)} - \boldsymbol {\mu} _ {i, \text {r e f}} ^ {(t)} \right\| ^ {2}. \tag {5.1}
+$$
+
+Similar to Ahn et al. (2024); Huang et al. (2025a), in this work, we analyze the training convergence of the population loss defined as:
+
+$$
+\mathcal {L} _ {\mathrm {C o T - t r a i n}} (\boldsymbol {\Theta}) = \mathbb {E} _ {\mathcal {I} _ {\mathbf {M}}} \left[ \widehat {\mathcal {L}} _ {\mathrm {C o T - t r a i n}} (\boldsymbol {\Theta}; \mathcal {I} _ {\mathbf {M}}) \right], \tag {5.2}
+$$
+
+where the expectation is taken over the randomness in the generation process of  $\mathcal{I}_{\mathrm{M}}$
+
+Directly analyzing the training dynamics of all layers of the transformer is intractable. On the other hand, as we mentioned in Section 4, the last three layers of the transformer can be constructed explicitly beforehand, as their parameters are data-independent. As a result, in the following, we will freeze these three layers and train the first layer only.
+
+Assumption 1 (Initialization). We initialize the first layer of the three-layer transformer described in Theorem 4.1 as follows:
+
+$$
+\mathbf {Q} ^ {(0)} \mathbf {K} ^ {(0)} = \left[ \begin{array}{c c c c c c} \mathbf {0} _ {d \times (d + 2 C)} & \mathbf {W} ^ {(0)} & & & \\ & & \mathbf {0} _ {(4 C + d + 2) \times 2 C} & & \\ & & & 1 & \mathbf {0} _ {1 \times 2} & \beta^ {(0)} \\ & & & & & 0 \end{array} \right],
+$$
+
+$$
+\mathbf {V} ^ {(0)} = \mathrm {d i a g} \big (\mathbf {0} _ {(d + 2 C) \times (d + C)}, \mathbf {I} _ {C}, \mathbf {0} _ {(d + C + 4) \times (d + 2 C + 4)} \big),
+$$
+
+where  $\mathbf{W}^{(0)}$  is a  $d\times d$  matrix whose entries are randomly sampled from a standard Gaussian distribution,  $\beta^{(0)}$  is a constant, and all the unspecified entries are equal to zero.
+
+Theorem 5.1 (Training Convergence). Let  $\{\mathbf{Q}^{(k)},\mathbf{K}^{(k)},\mathbf{V}^{(k)}\}_{k\geq 0}$  be the parameters of the first attention layer of the transformer after applying  $k$  iterations of gradient descent on the population loss defined in Equation (5.2). Then, with the initialization specified in Assumption 1, we have
+
+$$
+\| \mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1} \| _ {F} ^ {2} \leq c ^ {k} \| \mathbf {W} ^ {(0)} - \boldsymbol {\Sigma} ^ {- 1} \| _ {F} ^ {2}
+$$
+
+for some positive constant  $c$ , while the other parameters in  $\mathbf{Q}^{(0)}$ ,  $\mathbf{K}^{(0)}$  and  $\mathbf{V}^{(0)}$  remain unchanged.
+
+Theorem 5.1 indicates that under teacher forcing training, the parameter matrix  $\mathbf{W}^{(k)}$  of the first layer converges to the precision matrix  $\boldsymbol{\Sigma}^{-1}$ , the inverse of the noise covariance matrix linearly. Combining with other parameters in  $\mathbf{Q}^{(0)}$ ,  $\mathbf{K}^{(0)}$  and  $\mathbf{V}^{(0)}$ , we observe that the teacher forcing training recovers the transformer described in Theorem 4.1, theoretically validating the transformer's expressiveness for augmented ICL.
+
+Proof sketch. We use the superscript  $(k,t)$  to denote the  $t$ -th CoT step in the  $k$ -th gradient descent iteration. First, we drop the temperature term  $\beta \tau$  in the definition of  $p_{ij}^{(k,t)}$  given in Equation (4.2) and approximate it as in Equation (4.3), noting the approximation error can be made arbitrarily small by taking  $\beta$  sufficiently large. Next, we define
+
+$$
+q _ {i j} ^ {(k, t)} = \frac {\exp \bigl (- \frac {1}{2} \| \widehat {\pmb {\mu}} _ {i} ^ {(k , t)} - \mathbf {x} _ {j} \| _ {\mathbf {W} ^ {(k)}} ^ {2} \bigr)}{\sum_ {h = 1} ^ {C} \exp \bigl (- \frac {1}{2} \| \widehat {\pmb {\mu}} _ {h} ^ {(k , t)} - \mathbf {x} _ {j} \| _ {\mathbf {W} ^ {(k)}} ^ {2} \bigr)},
+$$
+
+which corresponds to replacing  $\pmb{\Sigma}^{-1}$  by  $\mathbf{W}^{(k)}$  in the approximation of  $p_{ij}^{(k,t)}$ .
+
+To prove one-step improvement of gradient descent on the population loss under teacher forcing, we must exhibit a constant  $\alpha > 0$  such that  $-\left\langle \mathbf{W}^{(k)} - \boldsymbol{\Sigma}^{-1}, \eta^{(k)} \nabla_{\mathbf{W}} \mathcal{L}_{\mathrm{CoT - train}} \right\rangle \leq$
+
+![](images/cfe20dbaa05315c39b4db0d22d0ab87f01c69ec1199cfb163a13db0eb34dfd1b.jpg)  
+Figure 1: Inference performance of the transformer trained via teacher forcing versus number of gradient descent iterations during training. Number of classes  $C = 3$ , number of labeled examples  $N = 5$ , CoT steps  $T = 5$ . The solid line shows the average results across 5 runs, and the shaded region represents ±2 standard deviations.
+
+![](images/b26e024f96d699f294399f790bba393ce553fa5860d97185afdf2f05d0bb48a5.jpg)
+
+![](images/4b5d293ff75302bcfa3e2bcf45088ba4e51ea9a3acb766559e79d522187f8208.jpg)
+
+$-\alpha \| \mathbf{W}^{(k)} - \boldsymbol{\Sigma}^{-1}\|_F^2$ . Our proof proceeds in three major steps. Step 1. Since direct analysis of  $\nabla_{\mathbf{W}}\mathcal{L}$  is intractable, we propose a novel decomposition by applying Stein's lemma to break the gradient into two analytically tractable terms: one is the posterior-difference term involving  $\mathbb{E}[\mathbf{p}_j^{(k,t)} - \mathbf{q}_j^{(k,t)}]$  and the other is the Jacobian-difference term involving  $\mathbb{E}[\nabla_{\mathbf{p}_j^{(k,t)}} - \nabla_{\mathbf{q}_j^{(k,t)}}]$ . Step 2. We show that an isotropic initialization of  $\mathbf{W}^{(k)}$  remains isotropic under gradient descent. The preservation of isotropy enforces alignment between  $\mathbf{p}_j$  and  $\mathbf{q}_j$  in expectation, i.e.,  $\mathbb{E}[\mathbf{p}_j^{(k,t)}] = \mathbb{E}[\mathbf{q}_j^{(k,t)}]$ . Therefore, the posterior-difference term vanishes. Step 3. We analyze  $\nabla_{\mathbf{p}_j^{(k,t)}}$  and  $\nabla_{\mathbf{q}_j^{(k,t)}}$  based on the their inherent symmetric structure. This analysis shows the Jacobian difference term degenerates to the following symmetric matrix under expectation:  $\left(\mathrm{diag}(1 / d) - \frac{1}{d^2}\mathbf{1}\mathbf{1}^\top\right)\mathbf{M}^\top\left(\mathbf{W}^{(k)} - \boldsymbol{\Sigma}^{-1}\right)$ , which enables us to avoid complicated analysis directly on the Jacobian difference term. Combining Steps 2 and 3, we obtain the following upper bound for the inner product term  $-\langle \mathbf{W}^{(k)} - \boldsymbol{\Sigma}^{-1}, \nabla \mathcal{L} \rangle \leq -\alpha' \| \mathbf{W}^{(k)} - \boldsymbol{\Sigma}^{-1}\|_F^2$ , which provides the desired result. The detailed proof can be found in Appendix D.
+
+# 6 Experimental Results
+
+Compute resources. All experiments are conducted on an NVIDIA H100 GPU with 80 GB of memory. The experiments require roughly five hours to complete.
+
+Problem setup. In the following experiments, the augmented ICL instances are generated as follows. We set the number of classes  $C = 3$  and the data dimension  $d = 3$ . The class mean vectors  $\{\pmb{\mu}_i\}_{i=1}^C$  are randomly sampled from a  $d$ -dimensional standard normal distribution. The covariance matrix  $\Sigma = \epsilon \mathbf{I}_d$  is shared across classes, where  $\mathbf{I}_d$  is the  $d$ -dimensional identity matrix. We set  $\epsilon \in \{0.7, 1.5\}$ . Each instance contains  $N = 5$  labeled data points and  $M$  unlabeled data points, where  $M \in \{1, 10, 20\}$ . The  $M = 1$  case recovers the conventional ICL setting.
+
+Transformer structure. We construct a transformer with the architecture specified in Theorem 4.1. This model features 4 layers, with each layer composed of an attention module followed by an MLP module. Activation functions for the attention layers were configured as follows: softmax for the first layer, linear for the second and third layers, and ReLU for the fourth layer. We set  $d_p = 16$  and the number of CoT steps  $T = 5$ . During training, in each iteration, we randomly generate 64 augmented ICL instances, and perform one gradient descent (GD) on the average empirical CoT training loss defined in Equation (5.1) over the batch. In total, we perform 15,000 GD iterations during training.
+
+Results. We evaluate the performance of the trained transformer after every 100 GD iterations. For evaluation, we randomly generated 100 augmented ICL instances, and obtained the corresponding class mean estimates from the trained transformer through CoT prompting. We then utilize these estimated class means to obtain the label prediction results according to Equation (3.5). For each  $M \in \{1, 10, 20\}$ , we conduct 5 runs. We track the class mean estimation error and prediction accuracy, and plot the average performance and standard deviation across these 5 runs in Figure 1.
+
+From Figure 1, we observe that augmented ICL outperforms conventional ICL significantly after a sufficient number of training iterations. As  $M$  increases, the advantage becomes more prominent: the transformer's class mean estimation error decreases and the classification accuracy increases, as predicted by our theoretical results Theorem 4.2 and Corollary 4.1.
+
+We notice that the advantage of augmented ICL is more significant when  $\epsilon$  is relatively small. This is because when  $\epsilon$  is small, the data distribution is less noisy, meaning that the features carry more information relevant to the labels. Therefore, the unlabeled data provides clearer structure that the transformer can leverage through augmented ICL to estimate class means more accurately.
+
+# 7 Conclusion
+
+In this work, we introduced augmented ICL, a framework in which models process a mixture of labeled and unlabeled examples within the prompt. We provided theoretical insights showing that transformers equipped with CoT reasoning can implement an EM-style algorithm for augmented ICL in a multi-class linear classification task, with provably decreasing prediction error as the amount of unlabeled data increases. Moreover, we showed that such transformer behavior can emerge through standard teacher forcing training. Our empirical results support the theory.
+
+# Acknowledgments
+
+The authors thank Li Fan, Wei Shen and Cong Shen for their helpful discussions during the preparation of this work. RL and JY were partially supported by the U.S. NSF under grants 2318759, 2531023 and 2531789.
+
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+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
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+
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+
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+
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+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
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+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: We give the assumption and proof sketch for our theorems in the main paper, and give complete proof in the Appendix.
+
+# Guidelines:
+
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+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
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+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
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+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: We provide the code in the supplemental material.
+
+Guidelines:
+
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+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: We specify all the training and test details.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: We provide 2-sigma confidence interval in Section 6.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
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+
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+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: Provided in the Experimental Results section.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: The authors have read and fully understood the NeurIPS Code of Ethics.
+
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+
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+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: The broader impacts of this work are discussed in Appendix A. Due to the theoretical nature of this work, we do not foresee major negative impacts.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
+
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: No such risks.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: All sources are cited.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
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+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
+
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: No asset.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: Does not involve any human subject.
+
+Guidelines:
+
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+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: Does not involve any human subject.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
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+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+# Answer: [NA]
+
+Justification: Core method does not involve LLMs.
+
+# Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# Supplementary Materials
+
+# Contents
+
+1 Introduction 1  
+2 Related Works 2
+
+3 Preliminaries 3
+
+3.1 Transformer Architecture 4  
+3.2 Augmented In-context Learning 4  
+3.3 Chain-of-Thought Prompting for Augmented ICL 4
+
+4 Expressiveness with CoT Prompting for Augmented ICL 5  
+5 Training Dynamics with Teacher Forcing 7  
+6 Experimental Results 9  
+7 Conclusion 10
+
+A Broader Impacts 23  
+B Limitations and Future Directions 23
+
+C Proof of Expressiveness 23
+
+C.1 Proof of Theorem 4.1 23  
+C.2 Proof of Theorem 4.2 26  
+C.3 Proof of Corollary 4.1 33
+
+D Proof of Training Dynamics 34  
+E Auxiliary Lemmas 37
+
+# A Broader Impacts
+
+This work provides theoretical insights into how transformers can leverage unlabeled data to improve in-context learning, a core capability underlying many recent advances in language models. By improving data efficiency and adaptability, our findings could enable more accessible and capable AI systems, particularly in low-resource settings where labeled data is limited. These advances may benefit a range of applications, including next-generation wireless communications and networking, healthcare, and financial services. Given the theoretical nature of this work, we anticipate minimal direct negative societal impact. Nonetheless, we recognize that future practical implementations inspired by this research should adhere to responsible AI principles.
+
+# B Limitations and Future Directions
+
+Our analysis and experiments possess certain limitations. Below, we outline these limitations and propose directions for future research.
+
+First, our analysis tracks parameter updates only in the first transformer layer, leaving all other layers frozen. As a result, it remains unclear how weights in non-linear hidden layers evolve under teacher forcing. To the best of our knowledge, the training dynamics of multi-layer transformer with nonlinear activation is still lacking investigation. A full, multi-layer treatment for end-to-end training remains an open problem.
+
+Second, this paper is the first theoretical investigation of the influence of unlabeled data in in-context learning, therefore, we restricted the experiments to a synthetic data set. However, whether the same behavior emerges in real-world tasks, and how unlabeled examples influence in-context learning for large, fully-trained transformers, is still unknown. Empirically understanding such impact is a promising future direction.
+
+# C Proof of Expressiveness
+
+First, we restate the theorem below.
+
+Theorem C.1. There exists a 4-layer transformer, such that its output sequence at the  $(t + 1)$ -th CoT step satisfies
+
+$$
+\widehat {\boldsymbol {\mu}} _ {i} ^ {(t + 1)} = \widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \frac {\eta^ {(t)}}{M} \sum_ {j = N + 1} ^ {N + M} p _ {i j} ^ {(t)} (\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \mathbf {x} _ {j}) + \mathbf {1} _ {\{t = 0 \}} \cdot \frac {C}{N} \sum_ {j = 1} ^ {N} (\mathbf {e} _ {i} ^ {\top} \mathbf {y} _ {j}) \mathbf {x} _ {j}, \tag {C.1}
+$$
+
+for any  $i \in [C]$ , where  $\eta^{(t)} = \alpha / (T' + t)$  for some positive constants  $\alpha$  and  $T'$ ,  $p_{ij}^{(t)}$  is the normalized weight
+
+$$
+p _ {i j} ^ {(t)} = \frac {\sum_ {\tau = 0} ^ {t} \exp \left(- \frac {1}{2} \| \hat {\boldsymbol {\mu}} _ {i} ^ {(\tau)} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2} + \beta \tau\right)}{\sum_ {\tau = 0} ^ {t} \sum_ {c = 1} ^ {C} \exp \left(- \frac {1}{2} \| \hat {\boldsymbol {\mu}} _ {c} ^ {(\tau)} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2} + \beta \tau\right)}, \tag {C.2}
+$$
+
+and  $\beta$  is a positive constant.
+
+We start from the proof of Theorem 4.1, which shows the transformer's capability of implementing an EM-style algorithm.
+
+# C.1 Proof of Theorem 4.1
+
+Recall that the input sequence at the  $t$ -th CoT step is formulated as
+
+$$
+\widehat {\mathbf {H}} ^ {(t - 1)} = \left[ \begin{array}{l l l l l l} \mathbf {X} _ {\ell} & \mathbf {X} _ {u} & \mathbf {0} & \star & \dots & \star \\ \mathbf {Y} _ {\ell} & \mathbf {0} & \mathbf {0} & \star & \dots & \star \\ \mathbf {P} _ {\ell} & \mathbf {P} _ {u} & \mathbf {Q} ^ {(0)} & \mathbf {Q} ^ {(1)} & \dots & \mathbf {Q} ^ {(t - 1)} \end{array} \right],
+$$
+
+where
+
+$$
+\mathbf {P} _ {\ell} = \left[ \begin{array}{l l l l} \mathbf {p} _ {1}, & \mathbf {p} _ {2}, & \dots & \mathbf {p} _ {N} \end{array} \right], \tag {C.3}
+$$
+
+$$
+\mathbf {P} _ {u} = \left[ \begin{array}{l l l l} \mathbf {p} _ {N + 1}, & \mathbf {p} _ {N + 2}, & \dots & \mathbf {p} _ {N + M} \end{array} \right], \tag {C.4}
+$$
+
+$$
+\mathbf {Q} ^ {(\tau)} = \left[ \begin{array}{l l l l} \mathbf {q} _ {1} ^ {(\tau)}, & \mathbf {q} _ {2} ^ {(\tau)}, & \dots & \mathbf {q} _ {C} ^ {(\tau)} \end{array} \right], \quad \tau \in [ 0: t - 1 ]. \tag {C.5}
+$$
+
+We specify  $\mathbf{p}_j$  and  $\mathbf{q}_i^{(\tau)}$  as follows. For each data sample  $j\in [N + M]$ , we denote
+
+$$
+\mathbf {p} _ {j} = \left[ \begin{array}{c} \mathbf {0} _ {C} \\ \mathbf {0} _ {d} \\ \mathbf {0} _ {C} \\ \mathbf {0} _ {C} \\ 0 \\ \mathbf {1} _ {j \in [ N ]} \\ \mathbf {1} _ {j \in [ N + 1: N + M ]} \\ 0 \end{array} \right], \quad \mathbf {q} _ {i} ^ {(\tau)} = \left[ \begin{array}{c} \mathbf {e} _ {i} \\ \widehat {\boldsymbol {\mu}} _ {i} ^ {(\tau)} \\ \mathbf {0} _ {C} \\ \mathbf {0} _ {C} \\ u _ {i} ^ {(\tau)} \\ 0 \\ 0 \\ \tau \end{array} \right],
+$$
+
+where  $\widehat{\pmb{\mu}}_i^{(\tau)}$  stores the estimate of the mean vector of class  $i$  from the  $\tau$ -th CoT step, and  $u_i^\tau$  stores a rescaled norm of  $\widehat{\pmb{\mu}}_i^{(\tau)}$ , i.e.,  $u_i^{(\tau)} = -\frac{\sigma}{2}\|\widehat{\pmb{\mu}}_i^{(\tau)}\|^2$ .
+
+Next, we specify the parameters of each layer of the transformer as follows.
+
+Layer 1: The first layer of the transformer consists of an attention layer with a softmax activation function, and an MLP layer. Let the parameters of the attention layer satisfy
+
+$$
+\mathbf {Q} _ {1} \mathbf {K} _ {1} = \left[ \begin{array}{c c c c c} \mathbf {0} _ {d \times (d + 2 C)} & \boldsymbol {\Sigma} ^ {- 1} & & & \\ & & \mathbf {0} _ {(4 C + d + 2) \times 2 C} & & \\ & & & 1 & \mathbf {0} _ {1 \times 2} \end{array} \begin{array}{c c c c c} \boldsymbol {\beta} \\ 0 \end{array} \right],
+$$
+
+$$
+\mathbf {V} _ {1} = \left[ \begin{array}{c c c} \mathbf {0} _ {(d + 2 C) \times (d + C)} & & \\ & \mathbf {I} _ {C} & \\ & & \mathbf {0} _ {(d + C + 4) \times (d + 2 C + 4)} \end{array} \right].
+$$
+
+Denote  $\mathrm{attn}_1(\mathbf{p}_j)$  as the output token after passing  $\mathbf{p}_j$  through the first attention layer, and let  $\gamma_i := \mathrm{attn}_1(\mathbf{p}_j)[d + C + 1 : d + 2C]$ . Then, we have
+
+$$
+\boldsymbol {\gamma} _ {j} = \frac {\sum_ {\tau \in [ 0 : t - 1 ]} \sum_ {i \in [ C ]} \exp \left(- \frac {\sigma}{2} \| \hat {\boldsymbol {\mu}} _ {i} ^ {(\tau)} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2} + (\hat {\boldsymbol {\mu}} _ {i} ^ {(\tau)}) ^ {\top} \mathbf {x} _ {j} + \beta \tau\right) \mathbf {e} _ {i}}{\sum_ {\tau \in [ 0 : t - 1 ]} \sum_ {i \in [ C ]} \exp \left(- \frac {\sigma}{2} \| \hat {\boldsymbol {\mu}} _ {i} ^ {(\tau)} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2} + (\hat {\boldsymbol {\mu}} _ {i} ^ {(\tau)}) ^ {\top} \mathbf {\Sigma} _ {j} + \beta \tau\right)}.
+$$
+
+Other entries in  $\widehat{\mathbf{H}}^{(t - 1)}$  remain unchanged after this attention layer.
+
+Subsequent to the first attention layer, a token-wise MLP is applied. Similar to Kim and Suzuki (2025), in this work, we assume the MLP layer can realize any deterministic token-wise link function with negligible error. The first MLP layer transforms input representations  $\mathbf{p}$  such that
+
+$$
+\operatorname {m l p} _ {1} \left(\operatorname {a t t n} _ {1} \left(\mathbf {p} _ {j}\right)\right) = \gamma_ {j} \cdot \operatorname {a t t n} _ {1} \left(\mathbf {p} _ {j}\right) [ 3 C + d + 3 ]
+$$
+
+$$
+\mathrm {m l p} _ {1} (u _ {i} ^ {(\tau)}) = - \frac {\sigma}{2} \| \widehat {\boldsymbol {\mu}} _ {i} ^ {(\tau)} \| ^ {2}.
+$$
+
+Since  $\mathbf{p}_j[3C + d + 3] = 0$  for  $j\in [N]$  and  $\mathbf{p}_j[3C + d + 3] = 1$  for  $j\in [N + 1:N + M]$ , and the corresponding entries remain unchanged after passing through the first attention layer, this MLP layer only keeps  $\gamma_{j}$  for tokens corresponding to the unlabeled dataset (i.e.,  $j\in [N]$ ), and set  $\gamma_{j}$  to zero for all other tokens (i.e.,  $j\in [N + 1:M]$ ).
+
+Layer 2: The second layer of the transformer consists of an attention layer with a linear activation function, and an MLP layer. The parameters of the attention layer are set to satisfy
+
+$$
+\mathbf {Q} _ {2} \mathbf {K} _ {2} = \left[ \begin{array}{c c c} \mathbf {0} _ {(2 d + 4 C + 2) \times (2 d + 4 C + 2)} & & \\ & 0 & 0 \\ & \alpha_ {1} & 0 \end{array} \right],
+$$
+
+$$
+\mathbf {V} _ {2} = \left[ \begin{array}{c c c} \mathbf {0} _ {(2 d + 3 C) \times (d + 2 C)} & & \\ & \mathbf {I} _ {C} & \\ & & \mathbf {0} _ {4 \times (d + C + 4)} \end{array} \right].
+$$
+
+We denote  $\mathbf{s}_i^{(\tau)}\coloneqq \mathrm{attn}_2(\mathbf{q}_i^{(\tau)})[d + 2C + 1:d + 3C]$  as the vector extracted from the output token after passing  $\mathbf{q}_i^{(\tau)}$  through the second attention layer. Then,  $\mathbf{s}_i^{(\tau)} = \tau \alpha_1\sum_{j = N + 1}^{N + M}\gamma_j$ , where  $\alpha_{1}$  is a fixed scalar embedded in  $\mathbf{Q}_2\mathbf{K}_2$ .
+
+We let the subsequent MLP layer realize the following token-wise Lipschitz function:
+
+$$
+\mathrm {m l p} _ {2} (\widehat {\pmb {\mu}} _ {i} ^ {(\tau)}) = \widehat {\pmb {\mu}} _ {i} ^ {(\tau)} - \frac {1}{\tau (\tau + \alpha_ {2})} \widehat {\pmb {\mu}} _ {i} ^ {(\tau)} \mathbf {e} _ {i} ^ {\top} \mathbf {s} _ {i} ^ {(\tau)} = \widehat {\pmb {\mu}} _ {i} ^ {(\tau)} - \frac {\alpha_ {1}}{\tau + \alpha_ {2}} \widehat {\pmb {\mu}} _ {i} ^ {(\tau)} \mathbf {e} _ {i} ^ {\top} \sum_ {j \in [ N + 1 ]} ^ {N + M} \gamma_ {j},
+$$
+
+$$
+\mathrm {m l p} _ {2} (\mathbf {e} _ {i}) = \frac {\alpha_ {1}}{\tau + \alpha_ {2}} \mathbf {e} _ {i}.
+$$
+
+Layer 3: Similar to the second transformer layer, the third layer also consists of a linear attention layer and an MLP layer. Consider the following parameterization for the attention layer:
+
+$$
+\mathbf {Q} _ {3} \mathbf {K} _ {3} = \left[ \begin{array}{c c c} \mathbf {0} _ {(d + C) \times (d + 2 C)} & & \\ & \mathbf {I} _ {C} & \\ & & \mathbf {0} _ {(d + 2 C + 4) \times (d + C + 4)} \end{array} \right],
+$$
+
+$$
+\mathbf {V} _ {3} = \left[ \begin{array}{c c} \mathbf {0} _ {(d + 3 C) \times d} & \\ \mathbf {I} _ {d} & \\ & \mathbf {0} _ {C + 4; d + 4 C + 4} \end{array} \right].
+$$
+
+Therefore, this attention layer realizes the following updating process:
+
+$$
+\mathrm {a t t n} _ {3} (\widehat {\pmb {\mu}} _ {i} ^ {(\tau)}) = \mathrm {m l p} _ {2} (\widehat {\pmb {\mu}} _ {i} ^ {(\tau)}) + \frac {\alpha_ {1}}{\tau + \alpha_ {2}} \sum_ {j \in [ M ]} \mathbf {x} _ {j} \mathbf {e} _ {i} ^ {\top} \boldsymbol {\gamma} _ {j} ^ {(\tau)}.
+$$
+
+After this linear attention layer, we let the MLP layer realize the following function
+
+$$
+\operatorname {m l p} _ {3} (\mathbf {e} _ {i}) = \frac {\tau + \alpha_ {2}}{\alpha_ {1}} \mathbf {e} _ {i}
+$$
+
+Layer 4: For the last layer, we introduce a transformer layer with a ReLU-activated attention layer followed by an MLP layer. We parameterize the attention layer as:
+
+$$
+\mathbf {Q} _ {4} \mathbf {K} _ {4} = \left[ \begin{array}{c c c c} \mathbf {0} _ {(d + C) \times d} & & & \\ & \mathbf {I} _ {C} & & \\ & & \mathbf {0} _ {(d + 2 C + 1) \times (d + 3 C + 3)} & \\ & & & 1 \\ & & & \mathbf {0} _ {2} \end{array} \right],
+$$
+
+$$
+\mathbf {V} _ {4} = \left[ \begin{array}{c c} \mathbf {0} _ {(d + 2 C) \times d} & \\ \mathbf {I} _ {d} & \\ & \mathbf {0} _ {(2 C + 4) \times (4 C + d + 4)} \end{array} \right].
+$$
+
+The corresponding updating rule of this layer gives
+
+$$
+\operatorname {a t t n} _ {4} \left(\boldsymbol {\mu} _ {i} ^ {(\tau)}\right) = \operatorname {a t t n} _ {3} \left(\boldsymbol {\mu} _ {i} ^ {(\tau)}\right) + \frac {C}{N} \sum_ {j \in [ N ]} \mathbf {x} _ {j} \operatorname {R e L U} (- \tau + \mathbf {e} _ {i} ^ {\top} \mathbf {y} _ {j}).
+$$
+
+Therefore, we can further reformulate it as
+
+$$
+\mathrm {a t t n} _ {4} (\boldsymbol {\mu} _ {i} ^ {(\tau)}) = \left\{ \begin{array}{l l} \frac {C}{N} \sum_ {j \in [ N ]} \mathbf {x} _ {j} \cdot \big (\mathbf {e} _ {i} ^ {\top} \mathbf {y} _ {j} \big), & \text {i f} \tau = 0, \\ \mathrm {a t t n} _ {3} (\boldsymbol {\mu} _ {\tau} ^ {(\tau)}), & \text {i f} \tau > 0. \end{array} \right.
+$$
+
+Given the above 4-layer transformer structure, by setting  $\alpha_{1} = \alpha /M$  and  $\alpha_{2} = T^{\prime}$  for fixed  $\alpha >0$ ,  $T^{\prime} > 0$ , the output sequence corresponding to the  $\mathbf{Q}^{(t - 1)}$  block in the input sequence that satisfies:
+
+$$
+\widehat {\boldsymbol {\mu}} _ {i} ^ {(t + 1)} = \widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \frac {\eta^ {(t)}}{M} \sum_ {j = N + 1} ^ {N + M} p _ {i j} ^ {(t)} (\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \mathbf {x} _ {j}) + \mathbf {1} _ {\{t = 0 \}} \cdot \frac {C}{N} \sum_ {j = 1} ^ {N} (\mathbf {e} _ {i} ^ {\top} \mathbf {y} _ {j}) \mathbf {x} _ {j}, \tag {C.6}
+$$
+
+for any  $i \in [C]$ , where  $\eta^{(t)} = \alpha / (T' + t)$  for some positive constants  $\alpha$  and  $T'$ ,  $p_{ij}^{(t)}$  is the normalized weight
+
+$$
+p _ {i j} ^ {(t)} = \frac {\sum_ {\tau = 0} ^ {t} \exp \left(- \frac {1}{2} \| \widehat {\boldsymbol {\mu}} _ {i} ^ {(\tau)} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2} + \beta \tau\right)}{\sum_ {\tau = 0} ^ {t} \sum_ {c = 1} ^ {C} \exp \left(- \frac {1}{2} \| \widehat {\boldsymbol {\mu}} _ {c} ^ {(\tau)} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2} + \beta \tau\right)}.
+$$
+
+The proof is thus complete.
+
+# C.2 Proof of Theorem 4.2
+
+In this section, we show the detailed proof of Theorem 4.2. We start by restating the theorem.
+
+Theorem C.2 (Class Mean Estimation Error). Given the transformer described in Theorem 4.1, when  $N \geq 36\alpha^2 L^2 \log 1 / \epsilon$ ,  $M \geq \max \{(T')^4, \log^2 1 / \epsilon\}$ , and  $t \geq \max \{\sqrt[4]{M}, T'\}$ , with probability at least  $1 - \epsilon$ , the output of the transformer after  $t$  CoT steps satisfies
+
+$$
+\| \widehat {\mathbf {M}} ^ {(t)} - \mathbf {M} \| _ {F} ^ {2} \leq c \frac {\log (1 / \epsilon)}{N \sqrt [ 4 ]{M}},
+$$
+
+where  $c, \alpha, L, T'$  are positive constants.
+
+# Step 1: First, we ensure that the initial estimation of the class mean vectors obtained from the labeled data gives a small estimation error.
+
+Lemma 1 (Initial estimation error from labeled data.). Consider the initial class mean estimates
+
+$$
+\boldsymbol {\mu} _ {i} ^ {(1)} = \frac {C}{N} \sum_ {j \in [ N ]} \mathbf {x} _ {j} \cdot \left(\mathbf {e} _ {i} ^ {\top} \mathbf {y} _ {j}\right), \quad \forall i \in [ C ].
+$$
+
+Then, for fixed  $K \geq 1$  and any positive constant  $T' \geq 4K$ , we have
+
+$$
+\mathbb {P} \left[ \left\| \boldsymbol {\mu} _ {i} ^ {(1)} - \boldsymbol {\mu} _ {i} \right\| ^ {2} > \frac {K}{T ^ {\prime}} \right] \leq \exp (- c N K / T ^ {\prime}),
+$$
+
+where  $c$  is a positive constant.
+
+Proof. We denote  $n_i$  as the number of samples drawn from class  $i$  in the  $N$  labeled data. Under the assumption that  $\mathbf{y}_j \sim \mathrm{Uniform}(\mathcal{Y}), \forall j \in [N]$ , we have  $n_i \sim \mathrm{Binomial}(N, 1 / C)$ . Then, according to Chernoff's inequality, for any  $\epsilon \in (0, 1)$ , we have
+
+$$
+\mathbb {P} \Big (\left| n _ {i} - \frac {N}{C} \right| > \epsilon \frac {N}{C} \Big) \leq 2 \exp \Big (- t \frac {\epsilon^ {2} N}{3 C} \Big).
+$$
+
+For any  $u\geq 0$  ,Let  $\epsilon = u\sqrt{K / T^{\prime}}$  ,we obtain
+
+$$
+\mathbb {P} \Big (\left| n _ {i} - \frac {N}{C} \right| > u \sqrt {\frac {K}{T ^ {\prime}}} \frac {N}{C} \Big) \leq 2 \exp \Big (- \frac {u ^ {2} N K}{3 C T ^ {\prime}} \Big).
+$$
+
+Therefore,
+
+$$
+\mathbb {P} \left(\left| \frac {C}{N} n _ {i} - 1 \right| > u \sqrt {\frac {K}{T ^ {\prime}}}\right) \leq 2 \exp \left(- \frac {u ^ {2} N K}{3 C T ^ {\prime}}\right). \tag {C.7}
+$$
+
+Conditional on  $n_i$ , we have  $\frac{1}{n_i} \sum_{j: \mathbf{y}_i = \mathbf{e}_i} \mathbf{x}_j - \boldsymbol{\mu}_i \sim \mathcal{N}(0, \boldsymbol{\Sigma} / n_i)$ . We assume  $\boldsymbol{\Sigma}$  is an isotropic matrix in the form of  $\sigma^2 \mathbb{1}$ . Then,  $\| \boldsymbol{\Sigma} \|_2 = \sigma^2$ , and we obtain the following inequality based on Hoeffding's inequality.
+
+$$
+\mathbb {P} \Big (\left\| \frac {1}{n _ {i}} \sum_ {j: \mathbf {y} _ {j} = \mathbf {e} _ {i}} \mathbf {x} _ {j} - \boldsymbol {\mu} _ {i} \right\| > \sigma \sqrt {\frac {2 t}{n _ {i}}} \big | n _ {i} \Big) \leq 2 e ^ {- t}.
+$$
+
+For any  $v \geq 0$ , by setting  $t = v^2 n_i K / (2\sigma^2 T')$ , we have
+
+$$
+\begin{array}{l} \mathbb {P} \left(\left\| \frac {1}{n _ {i}} \sum_ {j: \mathbf {y} _ {j} = \mathbf {e} _ {i}} \mathbf {x} _ {j} - \boldsymbol {\mu} _ {i} \right\| > v \sqrt {\frac {K}{T ^ {\prime}}}\right) \leq 2 \exp \left(- v ^ {2} n _ {i} K / \left(8 \sigma^ {2} T ^ {\prime}\right)\right) \\ \leq 2 \exp (- v ^ {2} (1 - \frac {K}{T ^ {\prime}}) \frac {N K}{C \sigma^ {2} T ^ {\prime}}) \\ \leq 2 \exp \left(- v ^ {2} \frac {N K}{2 C \sigma^ {2} T ^ {\prime}}\right). \tag {C.8} \\ \end{array}
+$$
+
+Then,
+
+$$
+\begin{array}{l} \mathbb {P} \left(\| \widehat {\boldsymbol {\mu}} _ {i} - \boldsymbol {\mu} _ {i} \| ^ {2} > \frac {K}{T ^ {\prime}}\right) = \mathbb {P} \left(\left\| \frac {C}{N} \sum_ {j: \mathbf {y} _ {j} = \mathbf {e} _ {i}} \mathbf {x} _ {j} - \frac {C n _ {i}}{N} \boldsymbol {\mu} _ {i} - (1 - \frac {C n _ {i}}{N}) \boldsymbol {\mu} _ {i} \right\| > \sqrt {\frac {K}{T ^ {\prime}}}\right) \\ \leq \mathbb {P} \left(\frac {C n _ {i}}{N} \left\| \frac {1}{n _ {i}} \sum_ {j: \mathbf {y} _ {j} = \mathbf {e} _ {i}} \mathbf {x} _ {j} - \boldsymbol {\mu} _ {i} \right\| + | 1 - \frac {C n _ {i}}{N} | \| \boldsymbol {\mu} _ {i} \| \geq \sqrt {\frac {K}{T ^ {\prime}}}\right) \\ \leq \mathbb {P} \left(\frac {C n _ {i}}{N} \left\| \frac {1}{n _ {i}} \sum_ {j: \mathbf {y} _ {j} = \mathbf {e} _ {i}} \mathbf {x} _ {j} - \boldsymbol {\mu} _ {i} \right\| \geq \sqrt {\frac {K}{T ^ {\prime}}}, \text {o r} | 1 - \frac {C n _ {i}}{N} | \| \boldsymbol {\mu} _ {i} \| \geq \sqrt {\frac {K}{T ^ {\prime}}}\right) \\ \stackrel {(a)} {\leq} 4 \exp \left(- c \frac {N K}{T ^ {\prime}}\right) \\ \end{array}
+$$
+
+for positive constant  $c$ . The inequality  $(a)$  holds by setting  $u = 1 / \| \pmb{\mu}_i\|$  in Equation (C.8) and setting  $v = N / Cn_{1}$  in Equation (C.17). The proof is thus complete.
+
+# Step 2: Next, we bound the discrepancy between the gradient obtained from each CoT step for a given input sequence, and the gradient of the population loss.
+
+We define the population loss for any given set of class mean vectors  $\{\pmb{\mu}_i\}_{i\in [C]}$  (i.e., any given  $\mathbf{M}$ ) as:
+
+$$
+\mathcal {L} \left(\left\{\boldsymbol {\mu} _ {i} \right\}\right) = \mathbb {E} _ {\mathbf {x}} \left[ \log \left(\frac {1}{C} \sum_ {i = 1} ^ {C} \exp \left(- \frac {1}{2} \| \mathbf {x} - \boldsymbol {\mu} _ {i} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2}\right)\right) \right], \tag {C.9}
+$$
+
+where the expectation is taken over the randomly generated data  $\mathbf{x}$  for given  $\mathbf{M}$ , as specified in Equation (3.1).
+
+We first characterize an important property of  $\mathcal{L}(\{\pmb{\mu}_i\})$  as follows.
+
+Lemma 2. The Jacobian of  $\nabla_{\pmb{\mu}_i}\mathcal{L}$  at  $\pmb{\mu}_i$  for all  $i\in [C]$  is negative definite, i.e.,  $\nabla_{\pmb{\mu}_i}^2\mathcal{L}\prec \mathbf{0}$ .
+
+Proof. Define
+
+$$
+p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) = \frac {\exp \left(- \frac {1}{2} \| \mathbf {x} - \boldsymbol {\mu} _ {i} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2}\right)}{\sum_ {c = 1} ^ {C} \exp \left(- \frac {1}{2} \| \mathbf {x} - \boldsymbol {\mu} _ {c} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2}\right)},
+$$
+
+so that  $p_{\mathbf{x}}(\boldsymbol{\mu}_i)$  is a softmax weight depending on  $\mathbf{x}$  and the centers  $\{\boldsymbol{\mu}_c\}_{c=1}^C$ . Note that  $\nabla_{\boldsymbol{\mu}_i}^2 \mathcal{L}$  is the Hessian of  $\nabla \mathcal{L}$  at  $\boldsymbol{\mu}_i$ , given by
+
+$$
+\nabla_ {\boldsymbol {\mu} _ {i}} ^ {2} \mathcal {L} = \mathbb {E} _ {\mathbf {x}} \left[ p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) \left(1 - p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i})\right) \boldsymbol {\Sigma} ^ {- 1} (\boldsymbol {\mu} _ {i} - \mathbf {x}) (\boldsymbol {\mu} _ {i} - \mathbf {x}) ^ {\top} \boldsymbol {\Sigma} ^ {- 1} - p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) \boldsymbol {\Sigma} ^ {- 1} \right],
+$$
+
+where and the expectation is taken with respect to the distribution of  $\mathbf{x}$ . Therefore, there exists a constant  $0\leq \alpha < 1$  such that
+
+$$
+\nabla_ {\boldsymbol {\mu} _ {i}} ^ {2} \mathcal {L} \preceq \mathbb {E} _ {\mathbf {x}} \left[ \alpha p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) \boldsymbol {\Sigma} ^ {- 1} (\boldsymbol {\mu} _ {i} - \mathbf {x}) (\boldsymbol {\mu} _ {i} - \mathbf {x}) ^ {\top} \boldsymbol {\Sigma} ^ {- 1} - p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) \boldsymbol {\Sigma} ^ {- 1} \right].
+$$
+
+Now, for any nonzero vector  $\mathbf{u} \in \mathbb{R}^d$ , consider the quadratic form  $\mathbf{u}^\top \nabla_{\mu_i}^2 \mathcal{L}\mathbf{u}$ , using the above matrix inequality, we have
+
+$$
+\mathbf {u} ^ {\top} \nabla_ {\boldsymbol {\mu} _ {i}} ^ {2} \mathcal {L} \mathbf {u} \leq \mathbf {u} ^ {\top} \mathbb {E} _ {\mathbf {x}} \left[ \alpha p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) \boldsymbol {\Sigma} ^ {- 1} (\boldsymbol {\mu} _ {i} - \mathbf {x}) (\boldsymbol {\mu} _ {i} - \mathbf {x}) ^ {\top} \boldsymbol {\Sigma} ^ {- 1} - p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) \boldsymbol {\Sigma} ^ {- 1} \right] \mathbf {u}.
+$$
+
+Therefore, rewriting the expectation as an integral yields
+
+$$
+\begin{array}{l} \mathbf {u} ^ {\top} \nabla_ {\boldsymbol {\mu} _ {i}} ^ {2} \mathcal {L} \mathbf {u} \leq \frac {1}{C} \int_ {\mathbb {R} ^ {d}} \alpha \mathcal {N} (\mathbf {x} \mid \boldsymbol {\mu} _ {i}, \mathbf {I}) \mathbf {u} ^ {\top} \boldsymbol {\Sigma} ^ {- 1} (\boldsymbol {\mu} _ {i} - \mathbf {x}) (\boldsymbol {\mu} _ {i} - \mathbf {x}) ^ {\top} \boldsymbol {\Sigma} ^ {- 1} \mathbf {u} d \mathbf {x} - \frac {1}{C} \mathbf {u} ^ {\top} \boldsymbol {\Sigma} ^ {- 1} \mathbf {u} \\ \leq \frac {\alpha}{C} \mathbf {u} ^ {\top} \boldsymbol {\Sigma} ^ {- 1} \mathbb {E} _ {\mathbf {x} \sim \mathcal {N} (\boldsymbol {\mu} _ {i}, \boldsymbol {\Sigma} ^ {- 1})} \left[ (\boldsymbol {\mu} _ {i} - \mathbf {x}) (\boldsymbol {\mu} _ {i} - \mathbf {x}) ^ {\top} \right] \boldsymbol {\Sigma} ^ {- 1} \mathbf {u} - \frac {1}{C} \mathbf {u} ^ {\top} \boldsymbol {\Sigma} ^ {- 1} \mathbf {u} <   0. \\ \end{array}
+$$
+
+Thus, the quadratic form is negative for every nonzero  $\mathbf{u}$ , and the matrix  $\nabla_{\mu_i}^2\mathcal{L}$  is negative definite. This completes the proof.
+
+We note that for each CoT step  $t > 0$ , the updating induced by the constructed transformer is
+
+$$
+\hat {\boldsymbol {\mu}} _ {i} ^ {(t + 1)} = \hat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \frac {\eta^ {(t)}}{M} \sum_ {j = N + 1} ^ {N + M} p _ {i j} ^ {(t)} \left(\hat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \mathbf {x} _ {j}\right), \tag {C.10}
+$$
+
+where  $p_{ij}^{(t)}$  is defined in Equation (4.2).
+
+To simplify notation, denote
+
+$$
+\frac {1}{M} \sum_ {j = N + 1} ^ {N + M} p _ {i j} ^ {(t)} \left(\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \mathbf {x} _ {j}\right) := \nabla_ {\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)}} \widehat {\mathcal {L}}. \tag {C.11}
+$$
+
+We note that  $\widehat{\mathcal{L}}$  itself is not an explicit loss function. We use the notation  $\nabla_{\widehat{\mu}_i^{(t)}}\widehat{\mathcal{L}}$  to represent the equivalent gradient for the updating determined by the  $t$ -th CoT step. Lemma 2 states  $\nabla_{\mu_i}^2\mathcal{L}$  is negative definite for each  $\pmb{\mu}_i$ , in the following lemma, we show that  $\nabla_{\pmb{\mu}}^{2}\mathcal{L}$  is negative definite for the concatenate vector  $\pmb{\mu}$  when  $\{\pmb{\mu}_i\}_{i}^{C}$  are well seperated.
+
+Lemma 3. The Jacobian of  $\nabla_{\mu}\mathcal{L}$  at  $\pmb{\mu}$  is negative definite, i.e.,  $\nabla_{\mu}^{2}\mathcal{L}\prec \mathbf{0}$
+
+Proof. Recall that  $\pmb{\Sigma} \in \mathbb{R}^{d \times d}$  is a symmetric positive definite. For  $\mathbf{x} \in \mathbb{R}^d$  and  $\pmb{\mu} = (\pmb{\mu}_1, \dots, \pmb{\mu}_C) \in (\mathbb{R}^d)^C$  define
+
+$$
+\ell (\mathbf {x}; \boldsymbol {\mu}) = \log \left(\frac {1}{C} \sum_ {i = 1} ^ {C} \exp \left(- \frac {1}{2} \| \mathbf {x} - \boldsymbol {\mu} _ {i} \| _ {\Sigma^ {- 1}} ^ {2}\right)\right).
+$$
+
+Therefore, we have  $\mathcal{L}(\pmb{\mu}) = \mathbb{E}_{\mathbf{x}}[\ell (\mathbf{x};\pmb{\mu})]$ . Hence the quadratic form of the Hessian of  $\ell$  in direction  $\pmb{\Delta} = (\pmb{\Delta}_1,\dots ,\pmb{\Delta}_C)$  can be written as
+
+$$
+\boldsymbol {\Delta} ^ {\top} \nabla^ {2} \ell (\mathbf {x}; \boldsymbol {\mu}) \boldsymbol {\Delta} = - \sum_ {i = 1} ^ {C} p _ {\mathbf {x}} \left(\boldsymbol {\mu} _ {i}\right) \| \boldsymbol {\Delta} _ {i} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2} + \operatorname {V a r} _ {p _ {\mathbf {x}}} \left(\left\{\left\langle \boldsymbol {\Sigma} ^ {- 1} \left(\mathbf {x} - \boldsymbol {\mu} _ {i}\right), \boldsymbol {\Delta} _ {i} \right\rangle \right\} _ {i = 1} ^ {C}\right), \tag {C.12}
+$$
+
+where the variance under  $p_{\mathbf{x}}$  is
+
+$$
+\operatorname {V a r} _ {p _ {\mathbf {x}}} \left(\left\{u _ {i} \right\} _ {i = 1} ^ {C}\right) = \sum_ {i = 1} ^ {C} p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) u _ {i} ^ {2} - \left(\sum_ {i = 1} ^ {C} p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) u _ {i}\right) ^ {2} \geq 0.
+$$
+
+Define the Mahalanobis separations
+
+$$
+\rho_ {i j} ^ {2} := \left\| \boldsymbol {\mu} _ {i} - \boldsymbol {\mu} _ {j} \right\| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2}, \qquad \rho_ {\star} := \min  _ {i \neq j} \rho_ {i j}.
+$$
+
+And for  $i\neq j$  , define
+
+$$
+\Delta_ {i j} (\mathbf {x}) := \left\| \mathbf {x} - \boldsymbol {\mu} _ {j} ^ {\star} \right\| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2} - \left\| \mathbf {x} - \boldsymbol {\mu} _ {i} ^ {\star} \right\| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2} = \rho_ {i j} ^ {2} + 2 Z _ {i j},
+$$
+
+we obtain that
+
+$$
+\Delta_ {i j} (\mathbf {x}) = \rho_ {i j} ^ {2} + 2 z _ {i j},
+$$
+
+where  $z_{ij} \sim \mathcal{N}(0, \rho_{ij}^2)$ . From the Gaussian tail bound and takes a union bound over  $j \neq i$ , we have
+
+$$
+\Pr \left\{\min  _ {j \neq i} \Delta_ {i j} (\mathbf {x}) \geq \frac {1}{2} \rho_ {\star} ^ {2} \right\} \geq 1 - (C - 1) e ^ {- \rho_ {\star} ^ {2} / 8} =: 1 - \eta_ {\star}. \tag {C.13}
+$$
+
+Under the event in (C.13),  $p_{\mathbf{x}}(\pmb{\mu}_i)$  is upper bounded by  $\beta_{\star} \in (0, 1)$ :
+
+$$
+p _ {\mathbf {x}} \left(\boldsymbol {\mu} _ {i}\right) \geq \frac {1}{1 + \sum_ {j \neq i} e ^ {- \frac {1}{2} \Delta_ {i j} (\mathbf {x})}} \geq \frac {1}{1 + (C - 1) e ^ {- \rho_ {\star} ^ {2} / 4}} =: \beta_ {\star}. \tag {C.14}
+$$
+
+Then, use (C.13)-(C.14) we obtain
+
+$$
+\mathbb {E} _ {\mathbf {x}} \left[ p _ {\mathbf {x}} \left(\boldsymbol {\mu} _ {i}\right) \right] \geq \pi_ {i} (1 - \eta_ {\star}) \beta_ {\star} =: c _ {i} > 0. \tag {C.15}
+$$
+
+Then,
+
+$$
+\begin{array}{l} \mathrm {V a r} _ {p _ {\mathbf {x}}} (\{u _ {i} \} _ {i = 1} ^ {C}) = \sum_ {i = 1} ^ {C} p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) u _ {i} ^ {2} - \left(\sum_ {i = 1} ^ {C} p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) u _ {i}\right) ^ {2} \\ = \left(\sum_ {i = 1} ^ {C} p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) u _ {i} ^ {2}\right) \left(\sum_ {i = 1} ^ {C} p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i})\right) - \left(\sum_ {i = 1} ^ {C} p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) u _ {i}\right) ^ {2} \\ = \frac {1}{2} \Big (\sum_ {i, j} p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) p _ {\mathbf {x}} (\boldsymbol {\mu} _ {j}) (u _ {i} ^ {2} + u _ {j} ^ {2}) \Big) - \Big (\sum_ {i = 1} ^ {C} p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) u _ {i} \Big) ^ {2} \\ = \sum_ {i <   j} p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) p _ {\mathbf {x}} (\boldsymbol {\mu} _ {j}) (u _ {i} - u _ {j}) ^ {2} \\ \leq \sum_ {i <   j} p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) p _ {\mathbf {x}} (\boldsymbol {\mu} _ {j}) 2 \left(u _ {i} ^ {2} + u _ {j} ^ {2}\right) \\ = 2 \sum_ {i} \sum_ {j \neq i} p _ {\mathbf {x}} \left(\boldsymbol {\mu} _ {i}\right) p _ {\mathbf {x}} \left(\boldsymbol {\mu} _ {j}\right) 2 u _ {i} ^ {2} \leq \sum_ {i} \left(1 - \max  _ {i} p _ {\mathbf {x}} ^ {2} \left(\boldsymbol {\mu} _ {i}\right)\right) u _ {i}. \tag {C.16} \\ \end{array}
+$$
+
+Therefore, using Equation (C.15), the first term in Equation (C.12) can be bounded as
+
+$$
+\mathbb {E} \Big [ - \sum_ {i = 1} ^ {C} p _ {\mathbf {x}} (\boldsymbol {\mu} _ {i}) \| \boldsymbol {\Delta} _ {i} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2} \Big ] \leq - \sum_ {i} c _ {i} \| \boldsymbol {\Delta} _ {i} \| _ {\boldsymbol {\Sigma} ^ {- 1}}.
+$$
+
+For the variance term, split expectation on the good and bad events of Equation (C.13). On the good event, combining Equation (C.14) and Equation (C.16) gives
+
+$$
+\mathrm {V a r} _ {p _ {\mathbf {x}}} (\{u _ {i} \} _ {i = 1} ^ {C}) \leq (1 - \beta_ {\star} ^ {2}) \sum_ {i} u _ {i} ^ {2}.
+$$
+
+On the bad event, we have  $\mathrm{Var}_{p_{\mathbf{x}}}(\{u_i\}_{i = 1}^C)\leq \frac{1}{C^2}\sum_i u_i^2$  since  $\max_{i}p_{\mathbf{x}}^{2}(\boldsymbol {\mu}_{i})\geq 1 / C^{2}$
+
+Next, we substitute  $u_{i}$  by  $\langle \pmb{\Sigma}^{-1}(\mathbf{x} - \pmb{\mu}_i),\pmb{\Delta}_i\rangle$  . Note that
+
+$$
+\mathbb {E} _ {\mathbf {x}} \left[ u _ {i} ^ {2} \right] = \boldsymbol {\Delta} _ {i} ^ {\top} \boldsymbol {\Sigma} ^ {- 1} \mathbb {E} \left[ (\mathbf {x} - \boldsymbol {\mu} _ {i} ^ {\star}) (\mathbf {x} - \boldsymbol {\mu} _ {i} ^ {\star}) ^ {\top} \right] \boldsymbol {\Sigma} ^ {- 1} \boldsymbol {\Delta} _ {i} \leq k _ {\star} \left\| \boldsymbol {\Delta} _ {i} \right\| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2},
+$$
+
+where
+
+$$
+k _ {\star} = \max  _ {1 \leq i \leq C} \mathbb {E} _ {\mathbf {x}} \left[ \| (\mathbf {x} - \boldsymbol {\mu} _ {i}) \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2} \right].
+$$
+
+Then, by taking expectation we obtain
+
+$$
+\mathbb {E} _ {\mathbf {x}} \big [ \mathrm {V a r} _ {p _ {\mathbf {x}}} \big ] \leq \Big ((1 - \eta_ {\star}) (1 - \beta_ {\star} ^ {2}) + \frac {\eta_ {\star}}{4} \Big) k _ {\star} \sum_ {i} \| \boldsymbol {\Delta} _ {i} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2}.
+$$
+
+Combining the two parts,
+
+$$
+\pmb {\Delta} ^ {\top} \nabla^ {2} \mathcal {L} (\pmb {\mu}) \pmb {\Delta} \leq - \Big (\min _ {i} c _ {i} - \big ((1 - \eta_ {\star}) (1 - \beta_ {\star} ^ {2}) + \frac {\eta_ {\star}}{4} \big) k _ {\star} \Big) \sum_ {i} \| \pmb {\Delta} _ {i} \| _ {\pmb {\Sigma} ^ {- 1}} ^ {2}.
+$$
+
+For sufficiently large  $\rho_{\star}$ , we obtain the negative definite  $\nabla^2\mathcal{L}(\pmb{\mu})$
+
+![](images/1c6061cf61170dee6f4eb13eca9aacb67fca3d841d50d030620a9ced3e072f3a.jpg)
+
+In the following, we characterize  $\nabla_{\hat{\mu}_i^{(t)}}\widehat{\mathcal{L}}$  and compare it with  $\nabla \mathcal{L}$ , i.e., the gradient if GD is performed on the population loss. We have the following lemma.
+
+Lemma 4 (Properties of the CoT gradient descent). Fix an epoch  $t$  and a component index  $i \in [C]$ , there exist constants  $c_1, c_2 > 0$  such that, for every  $M \geq 1$
+
+$$
+\Pr \Big (\big \| \nabla_ {\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)}} \widehat {\mathcal {L}} - \nabla_ {\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)}} \mathcal {L} \big \| \leq c _ {1} M ^ {- 1 / 4} \Big) \geq 1 - \exp (- \sqrt {M}),
+$$
+
+and
+
+$$
+\Pr \left(\left\| \nabla_ {\hat {\boldsymbol {\mu}} _ {i} ^ {(t)}} \widehat {\mathcal {L}} \right\| ^ {2} \leq c _ {2} + c _ {3} M ^ {- 1 / 2}\right) \geq 1 - \exp (- \sqrt {M}).
+$$
+
+Proof. Recall that
+
+$$
+\nabla_ {\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)}} \widehat {\mathcal {L}} = \frac {1}{M} \sum_ {j = N + 1} ^ {N + M} p _ {i j} ^ {(t)} \left(\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \mathbf {x} _ {j}\right)
+$$
+
+where  $\widehat{p}_{ij}^{(k,t)}$  is given by
+
+$$
+p _ {i j} ^ {(t)} = \frac {\sum_ {\tau = 0} ^ {t} \exp \left(- \frac {1}{2} \| \widehat {\boldsymbol {\mu}} _ {i} ^ {(\tau)} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2} + \beta \tau\right)}{\sum_ {\tau = 0} ^ {t} \sum_ {c = 1} ^ {C} \exp \left(- \frac {1}{2} \| \widehat {\boldsymbol {\mu}} _ {c} ^ {(\tau)} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2} + \beta \tau\right)}.
+$$
+
+By choosing  $\beta \to \infty$ , we further have
+
+$$
+p _ {i j} ^ {(t)} = \frac {\exp \left(- \frac {1}{2} \| \widehat {\boldsymbol {\mu}} _ {i} ^ {(\tau)} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2}\right)}{\sum_ {c = 1} ^ {C} \exp \left(- \frac {1}{2} \| \widehat {\boldsymbol {\mu}} _ {c} ^ {(\tau)} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2}\right)}.
+$$
+
+where the samples  $\{\mathbf{x}_j\}_{j\geq N + 1}$  are drawn from a Gaussian mixture distribution.
+
+Therefore, given  $\widehat{\pmb{\mu}}_{ij}^{(t)}$ , the random variable  $p_{ij}^{(t)}\left(\widehat{\pmb{\mu}}_i^{(t)} - \mathbf{x}_j\right)$  admits a sub-Gaussian tail bound since  $\mathbf{x}_j$  are Gaussian random vectors and  $p_{ij}^{(t)}\left(\widehat{\pmb{\mu}}_i^{(t)} - \mathbf{x}_j\right)$  is Lipschitz continuous over  $\mathbf{x}_j$ .
+
+Then, by the Bernstein's inequality, for any fixed  $\delta \in (0,1)$ , with probability at least  $1 - \delta$ , we have
+
+$$
+\begin{array}{l} \left\| \nabla_ {\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)}} \widehat {\mathcal {L}} - \nabla_ {\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)}} \mathcal {L} \right\| = \left\| \frac {1}{M} \sum_ {j = N + 1} ^ {N + M} p _ {i j} ^ {(t)} (\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \mathbf {x} _ {j}) - \mathbb {E} _ {\mathbf {x} _ {j}} \left[ p _ {i j} ^ {(t)} (\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \mathbf {x} _ {j}) \right] \right\| \\ \leq \frac {c _ {4}}{\sqrt {M}} \sqrt {\log \left(\frac {2}{\delta}\right)}, \\ \end{array}
+$$
+
+where  $c_{4} > 0$  is some absolute constant.
+
+By choosing  $\delta = \exp (-\sqrt{M})$  , we obtain that with probability at least  $1 - \exp (-\sqrt{M})$
+
+$$
+\left\| \nabla_ {\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)}} \widehat {\mathcal {L}} - \nabla_ {\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)}} \mathcal {L} \right\| \leq c ^ {\prime} M ^ {- \frac {1}{4}}.
+$$
+
+for another constant  $c' > 0$ . This completes the proof of the first inequality.
+
+Next, we show that  $\| \nabla_{\widehat{\pmb{\mu}}_i^{(t)}}\widehat{\mathcal{L}}\|$  itself is bounded with high probability.
+
+Consequently,
+
+$$
+\begin{array}{l} \| \nabla_ {\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)}} \widehat {\mathcal {L}} \| = \left\| \frac {1}{M} \sum_ {j = N + 1} ^ {N + M} p _ {i j} ^ {(t)} \left(\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \mathbf {x} _ {j}\right) \right\| \\ \leq \frac {1}{M} \sum_ {j = N + 1} ^ {N + M} \left\| p _ {i j} ^ {(t)} \left(\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \mathbf {x} _ {j}\right) \right\| \\ \stackrel {(a)} {\leq} \frac {1}{M} \sum_ {j \geq N + 1} \left\| \hat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \mathbf {x} _ {j} \right\| \\ \leq \frac {1}{M} \sum_ {j \geq N + 1} \left(\| \widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} \| + \| \mathbf {x} _ {j} \|\right) \\ = \left\| \hat {\boldsymbol {\mu}} _ {i} ^ {(t)} \right\| + \frac {1}{M} \sum_ {j \geq N + 1} \| \mathbf {x} _ {j} \|, \tag {C.17} \\ \end{array}
+$$
+
+where inequality  $(a)$  holds since  $p_{ij}^{(t)}\leq 1$  . Note that
+
+$$
+\begin{array}{l} \widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} = \widehat {\boldsymbol {\mu}} _ {i} ^ {(t - 1)} - \frac {\eta^ {(t - 1)}}{M} \sum_ {j = N + 1} ^ {N + M} p _ {i j} ^ {(t - 1)} \left(\widehat {\boldsymbol {\mu}} _ {i} ^ {(t - 1)} - \mathbf {x} _ {j}\right) \\ = \left(1 - \frac {\eta^ {(t - 1)}}{M}\right) \widehat {\boldsymbol {\mu}} _ {i} ^ {(t - 1)} + \frac {\eta^ {(t - 1)}}{M} \sum_ {j = N + 1} ^ {N + M} p _ {i, j} ^ {(t - 1)} \mathbf {x} _ {j}. \\ \end{array}
+$$
+
+Therefore, we have
+
+$$
+\begin{array}{l} \| \widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} \| \leq \| \widehat {\boldsymbol {\mu}} _ {i} ^ {(t - 1)} \| + \frac {1}{M} \sum_ {j \geq N + 1} \| \mathbf {x} _ {j} \| \\ \leq \| \widehat {\boldsymbol {\mu}} _ {i} ^ {(1)} \| + \frac {t - 1}{M} \sum_ {j \geq N + 1} \| \mathbf {x} _ {j} \| \\ \end{array}
+$$
+
+Combining with Equation (C.17), we have
+
+$$
+\left\| \nabla_ {\widehat {\pmb {\mu}} _ {i} ^ {(t)}} \widehat {\mathcal {L}} \right\| \leq \| \widehat {\pmb {\mu}} _ {i} ^ {(1)} \| + \frac {t}{M} \sum_ {j \geq N + 1} \| \mathbf {x} _ {j} \|.
+$$
+
+Applying the Bernstein's inequality, with probability at least  $1 - \exp (-\sqrt{M})$ , we have
+
+$$
+\frac {1}{M} \sum_ {j \geq N + 1} \| \mathbf {x} _ {j} \| \leq \frac {1}{C} \sum_ {i = 1} ^ {C} \boldsymbol {\mu} _ {i} + c _ {5} M ^ {- \frac {1}{4}},
+$$
+
+where  $c_{5}$  is a positive constant.
+
+Therefore, for any  $t \leq T$  where  $T$  is total number of CoT steps, we have
+
+$$
+\| \nabla_ {\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)}} \widehat {\mathcal {L}} \| \leq \| \widehat {\boldsymbol {\mu}} _ {i} ^ {(1)} \| + \frac {T}{C} \sum_ {i = 1} ^ {C} \boldsymbol {\mu} _ {i} + c _ {5} t M ^ {- \frac {1}{4}},
+$$
+
+which implies
+
+$$
+\| \nabla_ {\widehat {\pmb {\mu}} _ {i} ^ {(t)}} \widehat {\mathcal {L}} \| ^ {2} \leq c _ {2} + c _ {3} M ^ {- \frac {1}{2}},
+$$
+
+where  $c_{2}$  and  $c_{3}$  are positive constants depends on  $T$ ,  $\|\widehat{\pmb{\mu}}_i^{(1)}\|$  and  $\frac{T}{C} \sum_{i=1}^{C} \pmb{\mu}_i$ . The proof is thus complete.
+
+Step 3: Finally, we show the convergence of the class mean estimation error.
+
+Expanding the squared error  $\| \widehat{\pmb{\mu}}_i^{(t + 1)} - \pmb {\mu}_i\|^2$  gives
+
+$$
+\begin{array}{l} \left\| \widehat {\boldsymbol {\mu}} _ {i} ^ {(t + 1)} - \boldsymbol {\mu} _ {i} \right\| ^ {2} = \left\| \widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \boldsymbol {\mu} _ {i} \right\| ^ {2} + 2 \eta^ {(t)} \left\langle \widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \boldsymbol {\mu} _ {i}, \nabla_ {\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)}} \widehat {\mathcal {L}} \right\rangle + (\eta^ {(t)}) ^ {2} \left\| \nabla_ {\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)}} \widehat {\mathcal {L}} \right\| ^ {2} \\ \leq \left\| \widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \boldsymbol {\mu} _ {i} \right\| ^ {2} + 2 \eta^ {(t)} \left\langle \widehat {\boldsymbol {\mu}} _ {i} ^ {(t)} - \boldsymbol {\mu} _ {i}, \nabla_ {\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)}} \mathcal {L} \right\rangle + 2 \eta^ {(t)} \left\| \nabla \mathcal {L} - \nabla \widehat {\mathcal {L}} \right\| \\ + \left(\eta^ {(t)}\right) ^ {2} \left\| \nabla_ {\widehat {\boldsymbol {\mu}} _ {i} ^ {(t)}} \widehat {\mathcal {L}} \right\| ^ {2}. \tag {C.18} \\ \end{array}
+$$
+
+Denote  $\widehat{\pmb{\mu}}^{(t)}$  and  $\pmb{\mu}$  as the vectors obtained by stacking  $\{\widehat{\pmb{\mu}}_i^{(t)}\}_{i = 1}^C$  and  $\{\pmb {\mu}_i\}_{i = 1}^C$ , respectively. Therefore, we have
+
+$$
+\| \widehat {\boldsymbol {\mu}} ^ {(t + 1)} - \boldsymbol {\mu} \| ^ {2} \leq \| \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu} \| ^ {2} + 2 \eta^ {(t)} \langle \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu}, \nabla_ {\widehat {\boldsymbol {\mu}} ^ {(t)}} \mathcal {L} \rangle + 2 \eta^ {(t)} \left\| \nabla \mathcal {L} - \nabla \widehat {\mathcal {L}} \right\| + (\eta^ {(t)}) ^ {2} \left\| \nabla_ {\widehat {\boldsymbol {\mu}} ^ {(t)}} \widehat {\mathcal {L}} \right\| ^ {2}.
+$$
+
+To control the inner product term  $\langle \widehat{\pmb{\mu}}^{(t)} - \pmb {\mu},\nabla_{\widehat{\pmb{\mu}}^{(t)}}\mathcal{L}\rangle$  , we perform a first-order Taylor expansion of  $\nabla_{\widehat{\pmb{\mu}}^{(t)}}\mathcal{L}$  around  $\pmb{\mu}$  as
+
+$$
+\begin{array}{l} \nabla_ {\hat {\boldsymbol {\mu}} ^ {(t)}} \mathcal {L} = \nabla_ {\boldsymbol {\mu}} \mathcal {L} + (\nabla_ {\boldsymbol {\mu}} ^ {2} \mathcal {L}) (\hat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu}) + \mathbf {R} (\hat {\boldsymbol {\mu}} ^ {(t)}, \boldsymbol {\mu}) \\ \stackrel {(a)} {=} \left(\nabla_ {\boldsymbol {\mu}} ^ {2} \mathcal {L}\right) \left(\widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu}\right) + \mathbf {R} \left(\widehat {\boldsymbol {\mu}} ^ {(t)}, \boldsymbol {\mu}\right), \\ \end{array}
+$$
+
+where equality  $(a)$  holds since  $\pmb{\mu}$  is the global minimizer of  $\mathcal{L}$  and  $\mathcal{L}$  is differentiable on  $\mathbb{R}^d$ , thus  $\nabla_{\pmb{\mu}}\mathcal{L} = 0$ , and  $\mathbf{R}(\hat{\pmb{\mu}}^{(t)},\pmb {\mu})$  is the remainder term.
+
+For the remainder term, we have
+
+$$
+\begin{array}{l} \langle \mathbf {R} (\widehat {\boldsymbol {\mu}} ^ {(t)}, \boldsymbol {\mu}), \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu} \rangle \\ = \int_ {0} ^ {1} \left(\widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu}\right) ^ {\top} \left(\nabla_ {\boldsymbol {\mu} + \xi (\widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu})} ^ {2} \mathcal {L} - \nabla_ {\boldsymbol {\mu}} ^ {2} \mathcal {L}\right) \left(\widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu}\right) d \xi \\ \leq \int_ {0} ^ {1} \left\| \nabla_ {\boldsymbol {\mu} + \xi (\widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu})} ^ {2} \mathcal {L} - \nabla_ {\boldsymbol {\mu}} ^ {2} \mathcal {L} \right\| \| \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu} \| ^ {2} d \xi \\ \stackrel {(b)} {\leq} \int_ {0} ^ {1} L \xi \| \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu} \| ^ {3} \mathrm {d} \xi = L \| \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu} \| ^ {3}, \\ \end{array}
+$$
+
+where Inequality  $(b)$  follows from the fact that  $\nabla^2\mathcal{L}$  is twice continuously differentiable, its Jacobian is Lipchitz continuous in a neighborhood of  $\pmb{\mu}$ , and  $L$  is the Lipchitz constant.
+
+Therefore, there exists a constant  $\lambda > 0$  such that
+
+$$
+\begin{array}{l} \left\| \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu} \right\| ^ {2} + \left\langle \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu}, 2 \eta^ {(t)} \nabla_ {\boldsymbol {\mu}} \mathcal {L} ^ {(t)} \right\rangle \\ \leq \left\| \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu} \right\| ^ {2} + 2 \eta^ {(t)} \left(\widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu}\right) ^ {\top} \nabla_ {\boldsymbol {\mu}} ^ {2} \mathcal {L} \left(\widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu}\right) + 2 \eta^ {(t)} L \left\| \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu} \right\| ^ {3} \\ \stackrel {(c)} {\leq} \left(1 - 2 \eta^ {(t)} \lambda\right) \| \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu} \| ^ {2} + 2 \eta^ {(t)} L \| \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu} \| ^ {3}, \tag {C.19} \\ \end{array}
+$$
+
+where Inequality  $(c)$  follows from Lemma 3 which proves  $\nabla_{\mu}^{2}\mathcal{L}$  is negative definite.
+
+Meanwhile, Lemma 4 ensures with probability at least  $1 - \exp (-\sqrt{M})$
+
+$$
+\eta^ {(t)} \left\| \nabla \mathcal {L} - \nabla \widehat {\mathcal {L}} \right\| \leq c _ {1} \eta^ {(t)} M ^ {- \frac {1}{4}}, \tag {C.20}
+$$
+
+$$
+\left(\eta^ {(t)}\right) ^ {2} \left\| \nabla_ {\widehat {\boldsymbol {\mu}} ^ {(t)}} \widehat {\mathcal {L}} \right\| ^ {2} \leq c _ {2} \left(\eta^ {(t)}\right) ^ {2} M ^ {- \frac {1}{2}} + c _ {3} \left(\eta^ {(t)}\right) ^ {2}. \tag {C.21}
+$$
+
+Substituting (C.19), (C.20), and (C.21) into (C.18) then yields the one-step error recursion
+
+$$
+\begin{array}{l} \left\| \widehat {\boldsymbol {\mu}} ^ {(t + 1)} - \boldsymbol {\mu} \right\| ^ {2} \leq \left(1 - 2 \eta^ {(t)} \lambda\right) \left\| \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu} \right\| ^ {2} + 2 \eta^ {(t)} L \left\| \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu} \right\| ^ {3} \\ + c _ {1} \eta^ {(t)} M ^ {- \frac {1}{4}} + c _ {2} (\eta^ {(t)}) ^ {2} M ^ {- \frac {1}{2}} + c _ {3} (\eta^ {(t)}) ^ {2}. \tag {C.22} \\ \end{array}
+$$
+
+Next, we aim prove  $\| \widehat{\pmb{\mu}}^{(t)} - \pmb{\mu} \|^2 \leq K / (t + T')$  for a positive constant  $K$  by induction.
+
+Let  $\eta^{(t)} = \frac{\alpha}{t + T'}$  and  $M^{(t)} = (t + T')^p$  for some  $p > 4$ . First, assume  $\| \hat{\pmb{\mu}}^{(t)} - \pmb{\mu}\| ^2\leq K / (t + T')$  for a fixed  $t\geq 1$ . From Equation (C.22), we note that there exists a constant  $c_{4} > 0$  such that
+
+$$
+\begin{array}{l} \| \widehat {\boldsymbol {\mu}} ^ {(t + 1)} - \boldsymbol {\mu} \| ^ {2} \leq \left(1 - 2 \frac {\alpha \lambda}{t + T ^ {\prime}}\right) \| \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu} \| ^ {2} + c _ {3} \frac {\alpha^ {2}}{(t + T ^ {\prime}) ^ {2}} + 2 \frac {\alpha L}{t + T ^ {\prime}} \| \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu} \| ^ {3} + c _ {4} \frac {\alpha}{t + T ^ {\prime}} t ^ {- \frac {p}{4}} \\ \leq \left(1 - 2 \frac {\alpha \lambda}{t + T ^ {\prime}}\right) \frac {K}{t + T ^ {\prime}} + 2 \frac {\alpha L}{t + T ^ {\prime}} \left(\frac {K}{t + T ^ {\prime}}\right) ^ {\frac {3}{2}} + c _ {4} \alpha (t + T ^ {\prime}) ^ {- (1 + \frac {p}{4})} + c _ {3} \frac {\alpha^ {2}}{(t + T ^ {\prime}) ^ {2}}. \\ \end{array}
+$$
+
+Therefore, we have
+
+$$
+\begin{array}{l} \left\| \widehat {\boldsymbol {\mu}} ^ {(t + 1)} - \boldsymbol {\mu} \right\| ^ {2} - \frac {K}{t + T ^ {\prime} + 1} (C.23) \\ \leq \left(1 - 2 \frac {\alpha \lambda}{t + T ^ {\prime}}\right) \frac {K}{t + T ^ {\prime}} + 2 \frac {\alpha L}{t + T ^ {\prime}} \left(\frac {K}{t + T ^ {\prime}}\right) ^ {\frac {3}{2}} + c _ {4} \alpha (t + T ^ {\prime}) ^ {- (1 + \frac {p}{4})} + c _ {3} \frac {\alpha^ {2}}{(t + T ^ {\prime}) ^ {2}} - \frac {K}{t + T ^ {\prime}} + \frac {K}{(t + T ^ {\prime}) ^ {2}} \\ = (- 2 \alpha \lambda + 1) \frac {K}{(t + T ^ {\prime}) ^ {2}} + 2 \frac {\alpha L}{t + T ^ {\prime}} \left(\frac {K}{t + T ^ {\prime}}\right) ^ {\frac {3}{2}} + c _ {4} \alpha (t + T ^ {\prime}) ^ {- (1 + \frac {p}{4})} + c _ {3} \frac {\alpha^ {2}}{(t + T ^ {\prime}) ^ {2}}. (C.24) \\ \end{array}
+$$
+
+By choosing  $\alpha \geq 1 / \lambda$ ,  $K \geq \max \{3c_3\alpha^2, 3c_4\alpha\}$  and  $T' \geq 36\alpha^2 L^2 K$ , we have
+
+$$
+(- 2 \alpha \lambda + 1) \frac {K}{(t + T ^ {\prime}) ^ {2}} \leq - \frac {K}{(t + T ^ {\prime}) ^ {2}},
+$$
+
+$$
+2 \frac {\alpha L}{t + T ^ {\prime}} \left(\frac {K}{t + T ^ {\prime}}\right) ^ {\frac {3}{2}} \leq \frac {K}{3 (t + T ^ {\prime}) ^ {2}}, \tag {C.25}
+$$
+
+$$
+c _ {4} \alpha (t + T ^ {\prime}) ^ {- \left(1 + \frac {p}{4}\right)} \leq \frac {K}{3 (t + T ^ {\prime}) ^ {2}},
+$$
+
+$$
+c _ {3} \frac {\alpha^ {2}}{(t + T ^ {\prime}) ^ {2}} \leq \frac {K}{3 (t + T ^ {\prime}) ^ {2}}.
+$$
+
+Therefore, by substituting Equation (C.25) into Equation (C.24), we have
+
+$$
+\| \widehat {\boldsymbol {\mu}} ^ {(t + 1)} - \boldsymbol {\mu} \| ^ {2} - \frac {K}{t + T ^ {\prime} + 1} \leq 0, \quad \forall t \geq 1.
+$$
+
+Recall Lemma 1 indicates that, with probability at least  $1 - \exp(-cNK / T' + 1)$ , for some constant  $c$ , it holds that  $\|\pmb{\mu}^{(1)} - \pmb{\mu}\| \leq K / (T' + 1)$ . Therefore, for any fixed  $\epsilon \in [0,1]$ , if
+
+$$
+N \geq 3 6 \alpha^ {2} L ^ {2} \log 1 / \epsilon ,
+$$
+
+$$
+M \geq \max  \left\{\left(T ^ {\prime}\right) ^ {4}, \log^ {2} 1 / \epsilon \right\},
+$$
+
+$$
+t \geq \sqrt [ 4 ]{M},
+$$
+
+with probability at least  $1 - \epsilon$ , the estimation error is upper bounded by
+
+$$
+\| \widehat {\boldsymbol {\mu}} ^ {(t)} - \boldsymbol {\mu} \| ^ {2} \leq c \frac {\log (1 / \epsilon)}{N \sqrt [ 4 ]{M}},
+$$
+
+where  $c$  is a positive constant. This completes the proof of Theorem 4.2.
+
+# C.3 Proof of Corollary 4.1
+
+First, we restate the corollary below.
+
+Corollary C.1 (Restatement of Corollary 4.1). Let  $\widehat{\mathbf{y}}_j$  be the predicted label for  $\mathbf{x}_j$  according to Equation (3.5). Let  $\mathcal{R}^*$  be the prediction error under the Bayes-optimal classifier with known class mean vectors  $\mu_1, \dots, \mu_C$ . Then, under the same conditions as described in Theorem 4.2, we have
+
+$$
+\mathbb {P} [ \widehat {\mathbf {y}} _ {j} \neq \mathbf {y} | \boldsymbol {\mu} _ {1}, \dots , \boldsymbol {\mu} _ {C} ] - \mathcal {R} ^ {*} \leq \mathcal {O} (1 / \sqrt {N \mathrm {p o l y} (M)}).
+$$
+
+Proof. First, we define  $\Delta = \| \widehat{\mathbf{M}} - \mathbf{M} \|_F$ , define  $\widehat{g}$  as the Bayes-optimal classifier given estimated class means  $\widehat{\mathbf{M}}$  and define  $g$  as the Bayes-optimal classifier given ground truth class means  $\mathbf{M}$ . Suppose  $\widehat{g}(\mathbf{x}) \neq g(\mathbf{x})$ . Then, there exist indices  $i \neq k$  such that  $g(\mathbf{x}) = i$  and  $\widehat{g}(\mathbf{x}) = k$ . Because  $g(\mathbf{x}) = i$  is Bayes-optimal, we have
+
+$$
+\| \mathbf {x} - \boldsymbol {\mu} _ {i} \| \leq \| \mathbf {x} - \boldsymbol {\mu} _ {k} \| \text {a n d} \| \mathbf {x} - \widehat {\boldsymbol {\mu}} _ {k} \| \leq \| \mathbf {x} - \widehat {\boldsymbol {\mu}} _ {i} \|.
+$$
+
+Denote  $\zeta = \| \pmb{\mu}_i - \pmb{\mu}_k\|$ . Therefore, from the geometric observation, the misclassification only happens when  $\mathbf{x}$  is in the dihedral cone with angle  $\theta$ , where  $\tan (\theta) = \Delta/\zeta$  (Diakonikolas et al., 2018). Thus, the probability for misclassification is upper bounded
+
+$$
+\mathbb {P} [ \widehat {g} (\mathbf {x}) \neq g (\mathbf {x}) ] \leq c ^ {\prime} \theta ,
+$$
+
+for a positive constant  $c'$ . Since  $\mathbb{P}[\widehat{\mathbf{y}}_j \neq \mathbf{y}|\boldsymbol{\mu}_1, \dots, \boldsymbol{\mu}_C] - \mathcal{R}^* = \mathbb{P}[\widehat{g}(\mathbf{x}) \neq g(\mathbf{x})]$  and from Theorem 4.2 we have  $\Delta \leq c' \sqrt{1/N \sqrt[4]{(M)}}$  for positive constant  $c'$ , the proof is thus complete.
+
+# D Proof of Training Dynamics
+
+First, we restate Theorem 5.1 below.
+
+Theorem D.1 (Restatement of Theorem 5.1). Let  $\{\mathbf{Q}^{(k)},\mathbf{K}^{(k)},\mathbf{V}^{(k)}\}_{k\geq 0}$  be the parameters of the first attention layer of the transformer after applying  $k$  iterations of gradient descent on the population loss defined in Equation (5.2). Then, with the initialization specified in Assumption 1, we have
+
+$$
+\left\| \mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1} \right\| _ {F} ^ {2} \leq c ^ {k} \left\| \mathbf {W} ^ {(0)} - \boldsymbol {\Sigma} ^ {- 1} \right\| _ {F} ^ {2},
+$$
+
+for some positive constant  $c$ , while the other parameters in  $\mathbf{Q}^{(0)}$ ,  $\mathbf{K}^{(0)}$  and  $\mathbf{V}^{(0)}$  remain unchanged.
+
+We assume ground truth means are IID sampled from standard Gaussian distribution:  $\mu_{i}\sim \mathcal{N}(\mathbf{0},\mathbf{I})$  for all  $i$ . Then, we introduce the following quantities: 1) the formulation of class mean estimations given by the transformer during teacher forcing training; 2) the reference class mean estimations given by the reference policy; and 3) the formulation of the gradient of the teacher forcing training loss.
+
+At the  $k$ -th GD iteration during training, we denote the set of reference class mean estimations as  $\pmb{\mu}_{\mathrm{ref},1}^{(k,t)}, \dots, \pmb{\mu}_{\mathrm{ref},C}^{(k,t)}$  for the CoT steps  $t \in [T]$ . Given the reference class mean estimations, the estimation given by the transformer throughout teacher forcing satisfies
+
+$$
+\widehat {\boldsymbol {\mu}} _ {i} ^ {(k, t + 1)} = \boldsymbol {\mu} _ {\text {r e f}, i} ^ {(k, t)} - \frac {\eta^ {(t)}}{M} \sum_ {j = N + 1} ^ {N + M} \widehat {p} _ {i j} ^ {(k, t)} \left(\boldsymbol {\mu} _ {\text {r e f}, i} ^ {(k, t)} - \mathbf {x} _ {j}\right),
+$$
+
+where  $\widehat{p}_{ij}^{(k,t)}$  is given by
+
+$$
+\widehat {p} _ {i j} ^ {(k, t)} = \frac {\sum_ {\tau = 0} ^ {t} \exp \left(- \frac {w}{2} \| \widehat {\boldsymbol {\mu}} _ {i} ^ {(\tau)} \| ^ {2} + \mathbf {x} _ {j} ^ {\top} \mathbf {W} ^ {(k)} \widehat {\boldsymbol {\mu}} _ {i} ^ {(\tau)} + \beta \tau\right)}{\sum_ {\tau = 0} ^ {t} \sum_ {c = 1} ^ {C} \exp \left(- \frac {w}{2} \| \widehat {\boldsymbol {\mu}} _ {c} ^ {(\tau)} \| ^ {2} + \mathbf {x} _ {j} ^ {\top} \mathbf {W} ^ {(k)} \widehat {\boldsymbol {\mu}} _ {c} ^ {(\tau)} + \beta \tau\right)}.
+$$
+
+By choosing  $\beta \to \infty$ , we further have
+
+$$
+\widehat {p} _ {i j} ^ {(k, t)} = \frac {\exp \left(- \frac {w}{2} \| \widehat {\boldsymbol {\mu}} _ {i} ^ {(\tau)} \| ^ {2} + \mathbf {x} _ {j} ^ {\top} \mathbf {W} ^ {(k)} \widehat {\boldsymbol {\mu}} _ {i} ^ {(\tau)}\right)}{\sum_ {c = 1} ^ {C} \exp \left(- \frac {w}{2} \| \widehat {\boldsymbol {\mu}} _ {c} ^ {(\tau)} \| ^ {2} + \mathbf {x} _ {j} ^ {\top} \mathbf {W} ^ {(k)} \widehat {\boldsymbol {\mu}} _ {c} ^ {(\tau)}\right)}.
+$$
+
+We choose the reference policy under which
+
+$$
+\boldsymbol {\mu} _ {\mathrm {r e f}, i} ^ {(k, t + 1)} = \boldsymbol {\mu} _ {\mathrm {r e f}, i} ^ {(k, t)} - \frac {\eta^ {(t)}}{M} \sum_ {j = N + 1} ^ {N + M} p _ {i j} ^ {(k, t)} \left(\boldsymbol {\mu} _ {\mathrm {r e f}, i} ^ {(k, t)} - \mathbf {x} _ {j}\right),
+$$
+
+with
+
+$$
+p _ {i j} ^ {(k, t)} = \frac {\exp \left(- \frac {1}{2} \| \boldsymbol {\mu} _ {\text {r e f} , i} ^ {(k , t)} - \mathbf {x} _ {j} \| _ {\Sigma^ {- 1}} ^ {2}\right)}{\sum_ {c = 1} ^ {C} \exp \left(- \frac {1}{2} \| \boldsymbol {\mu} _ {\text {r e f} , c} ^ {(k , t)} - \mathbf {x} _ {j} \| _ {\Sigma^ {- 1}} ^ {2}\right)}.
+$$
+
+To simplify the notation, when there is no ambiguity, we drop the superscript  $(k)$  for the training iteration. Denote  $\widehat{\mathbf{q}}_j^{(t)} = [\widehat{p}_{1j}^{(t)}\dots \widehat{p}_{Cj}^{(t)}]$  and  $\mathbf{q}_j^{(t)} = [p_{1j}^{(t)}\dots p_{Cj}^{(t)}]$ . At the  $k$ -th training iteration, the CoT training loss with teacher forcing is
+
+$$
+\begin{array}{l} \widehat {\mathcal {L}} _ {\mathrm {C o T - t r a i n}} (\boldsymbol {\Theta}; \mathcal {I} _ {\mathbf {M}}) = \frac {1}{T} \sum_ {t = 1} ^ {T} \sum_ {j = N + 1} ^ {N + M} \operatorname {C E} \left(\mathbf {q} _ {j} ^ {(t)}, \left[ \operatorname {T F} _ {\boldsymbol {\Theta}} \left(\mathbf {H} _ {\mathrm {r e f}} ^ {(t - 1)}\right) \right] _ {2 d + 2 c + 1: 2 d + 3 c, N + j}\right) \\ = \frac {1}{T} \sum_ {t = 1} ^ {T} \sum_ {j = N + 1} ^ {N + M} \operatorname {C E} \left(\mathbf {q} _ {j} ^ {(t)}, \widehat {\mathbf {q}} _ {j} ^ {(t)}\right), \\ \end{array}
+$$
+
+where CE is the cross entropy loss function.
+
+Define  $s_{ij}^{(t)} = -\frac{1}{2}\|\widehat{\pmb{\mu}}_i^{(\tau)}\|^2 + \mathbf{x}_j^\top \mathbf{W}^{(k)}\widehat{\pmb{\mu}}_i^{(\tau)}$  and  $\mathbf{s}_j^{(t)} = [s_{1j}^{(t)} \cdots s_{Cj}^{(t)}]$ . Note that the derivative can be written as
+
+$$
+\frac {\partial \mathrm {C E} (\mathbf {q} _ {j} ^ {(t)} , \widehat {\mathbf {q}} _ {j} ^ {(t)})}{\partial s _ {i j} ^ {(t)}} = \frac {\partial \frac {\exp (s _ {i j} ^ {(t)})}{\sum_ {k = 1} ^ {C} \exp (s _ {k j} ^ {(t)})}}{\partial s _ {i j} ^ {(t)}} = \widehat {p} _ {i j} ^ {(t)} - p _ {i j} ^ {(t)}.
+$$
+
+Furthermore, since  $\partial s_{ij}^{(t)} / \partial \mathbf{W}_{ab} = \mathbf{M}_{a,i}\mathbf{x}_{jb}$ , where  $a,b\in [d]$ , by the chain rule, we have
+
+$$
+\frac {\partial \operatorname {C E} \left(\mathbf {q} _ {j} ^ {(t)} , \widehat {\mathbf {q}} _ {j} ^ {(t)}\right)}{\partial \mathbf {W} _ {a b}} = \frac {\partial \operatorname {C E} \left(\mathbf {q} _ {j} ^ {(t)} , \widehat {\mathbf {q}} _ {j} ^ {(t)}\right)}{\partial s _ {i j} ^ {(t)}} \frac {\partial s _ {i j} ^ {(t)}}{\partial \mathbf {W} _ {a b}} = \sum_ {i} \left(\widehat {p} _ {i j} ^ {(t)} - p _ {i j} ^ {(t)}\right) \mathbf {M} _ {a, i} \mathbf {x} _ {j b}. \tag {D.1}
+$$
+
+Based on the notations, we will prove Theorem 5.1 as follows.
+
+Step 1: Given the gradient of the cross entropy loss with respect to the learnable parameter matrix  $W$ , our first step is to provide a decomposition of the gradient so that it becomes analytically tractable.
+
+In the matrix form, Equation (D.1) can be written as
+
+$$
+\nabla_ {\mathbf {W}} \operatorname {C E} \left(\mathbf {q} _ {j} ^ {(t)}, \widehat {\mathbf {q}} _ {j} ^ {(t)}\right) = \mathbf {M} (\widehat {\mathbf {q}} _ {j} ^ {(t)} - \mathbf {q} _ {j} ^ {(t)}) \mathbf {x} _ {j} ^ {T}.
+$$
+
+By the Stein's lemma, we have
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \mathbf {M} \left(\widehat {\mathbf {q}} _ {j} ^ {(t)} - \mathbf {q} _ {j} ^ {(t)}\right) \mathbf {x} _ {j} ^ {\top} \right] \\ = \mathbb {E} _ {\mathbf {M}} \left[ \mathbf {M} \left[ \mathbb {E} _ {\mathbf {x} _ {j}} [ \widehat {\mathbf {q}} _ {j} ^ {(t)} - \mathbf {q} _ {j} ^ {(t)} ] \mathbb {E} [ \mathbf {x} _ {j} ^ {\top} ] + \mathbb {E} _ {\mathbf {x} _ {j}} [ \nabla \widehat {\mathbf {q}} _ {j} ^ {(t)} - \nabla \mathbf {q} _ {j} ^ {(t)} ] \boldsymbol {\Sigma} \right] \right] \\ = \underbrace {\mathbb {E} _ {\mathbf {M}} \left[ \mathbf {M} \mathbb {E} _ {\mathbf {x} _ {j}} \left[ \widehat {\mathbf {q}} _ {j} ^ {(t)} - \mathbf {q} _ {j} ^ {(t)} \right] \mathbb {E} \left[ \mathbf {x} ^ {\top} \right] \right]} _ {\mathcal {A} _ {1}} + \underbrace {\mathbb {E} _ {\mathbf {M}} \left[ \mathbf {M} \mathbb {E} _ {\mathbf {x} _ {j}} \left[ \nabla \widehat {\mathbf {q}} _ {j} ^ {(t)} - \nabla \mathbf {q} _ {j} ^ {(t)} \right] \boldsymbol {\Sigma} \right]} _ {\mathcal {A} _ {2}}. \\ \end{array}
+$$
+
+Step 2: Based on the decomposition, we aim to show that  $\mathcal{A}_1 = 0$ .
+
+We note that when taking the expectation over the labeled dataset, we have
+
+$$
+\mathbb {E} \left[ \frac {C}{N} \sum_ {j \in [ N ]} \mathbf {x} _ {j} \cdot \left(\mathbf {e} _ {i} ^ {\top} \mathbf {y} _ {j}\right) \right] = \boldsymbol {\mu} _ {i}.
+$$
+
+Therefore,  $\pmb{\mu}_{\mathrm{ref,i}}^{0} = \pmb{\mu}_{i}$ . When the reference class mean estimations are generated by gradient descent over the population loss, we have  $\pmb{\mu}_{\mathrm{ref},i}^{(t)} = \pmb{\mu}_{i}$  for any  $i \in [C]$  and  $t \in [T]$  the gradient
+
+over the population loss is zero:
+
+$$
+\begin{array}{l} \mathbb {E} _ {\mathbf {x}} \left[ \frac {\exp \left(- \frac {1}{2} \| \boldsymbol {\mu} _ {\mathrm {r e f} , i} ^ {(t)} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2}\right)}{\sum_ {c = 1} ^ {C} \exp \left(- \frac {1}{2} \| \boldsymbol {\mu} _ {\mathrm {r e f} , c} ^ {(t)} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2}\right)} \left(\boldsymbol {\mu} _ {\mathrm {r e f}, i} ^ {(t)} - \mathbf {x} _ {j}\right) \right] \\ = \int_ {\mathbb {R} ^ {d}} \frac {\exp \left(- \frac {1}{2} \| \boldsymbol {\mu} _ {\text {r e f} , i} ^ {(t)} - \mathbf {x} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2}\right)}{\sum_ {c = 1} ^ {C} \exp \left(- \frac {1}{2} \| \boldsymbol {\mu} _ {\text {r e f} , c} ^ {(t)} - \mathbf {x} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2}\right)} \left[ \sum_ {k = 1} ^ {C} \frac {1}{C} \varphi_ {k} (\mathbf {x}) \right] (\boldsymbol {\mu} _ {\text {r e f}, i} ^ {(t)} - \mathbf {x}) d \mathbf {x}, \\ \stackrel {(a)} {=} \int_ {\mathbb {R} ^ {d}} \frac {1}{C} \varphi_ {i} (\mathbf {x}) (\boldsymbol {\mu} _ {i} - \mathbf {x}) d \mathbf {x} = 0, \\ \end{array}
+$$
+
+where  $\varphi_{i}(\mathbf{x})$  is the pdf of Gaussian distribution with mean  $\pmb{\mu}_{i}$  and covariance matrix  $\pmb{\Sigma}$ , and equality  $(a)$  holds since  $\pmb{\mu}_{\mathrm{ref},i}^{(k,t)} = \pmb{\mu}_i$ . Given the above-discussed property of the reference class mean estimations, for  $\mathbb{E}_{\mathbf{x}_j}[\widehat{\mathbf{q}}_j^{(t)} - \mathbf{q}_j^{(t)}]$  in  $\mathcal{A}_1$ , its is obvious that  $\mathbb{E}_{\mathbf{x}_j}[\mathbf{q}_j^{(t)}] = 1 / C$ . For  $\mathbb{E}_{\mathbf{x}_j}[\widehat{\mathbf{q}}_j^{(t)}]$ , we let  $\mathbf{W}^{(0)}$  initialize from a isotropic matrix  $w\mathbf{I}$ , and we assume at training iteration step  $t$ , it preserves the isotropic as  $\mathbf{W}^{(t)}$ . Therefore, since the ground truth  $\pmb{\Sigma}$  is an isotropic matrix, the temperature acts identically on all classes:
+
+$$
+\mathbb {E} \left[ \frac {\exp \left(- \frac {\alpha}{2} \| \widehat {\boldsymbol {\mu}} _ {i} ^ {(\tau)} - \mathbf {x} _ {j} \| ^ {2}\right)}{\sum_ {c = 1} ^ {C} \exp \left(- \frac {\alpha}{2} \| \boldsymbol {\mu} _ {c} - \mathbf {x} _ {j} \| ^ {2}\right)} \right] = \mathbb {E} \left[ \frac {\exp \left(- \frac {1}{2} \| \widehat {\boldsymbol {\mu}} _ {i} ^ {(\tau)} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2}\right)}{\sum_ {c = 1} ^ {C} \exp \left(- \frac {1}{2} \| \boldsymbol {\mu} _ {c} - \mathbf {x} _ {j} \| _ {\boldsymbol {\Sigma} ^ {- 1}} ^ {2}\right)} \right].
+$$
+
+Therefore, we have  $\mathbb{E}_{\mathbf{x}_j}[\widehat{\mathbf{p}}_j^{(t)} - \mathbf{p}_j^{(t)}] = 0$  , which gives  $\mathcal{A}_1 = 0$
+
+Step 3: Finally, we analyze the properties of  $\mathcal{A}_2$ , and obtain the final results afterwards. We will prove that  $\mathbf{W}^{(t)}$  preserves isotropic by induction. Note that we assume training iteration step  $t$ ,  $\mathbf{W}^{(t)}$  is isotropic. Besides, we initialize  $\mathbf{W}^{(0)}$  as an isotropic matrix.
+
+$\mathcal{A}_2$  can be rewritten as
+
+$$
+\begin{array}{l} \mathcal {A} _ {2} = \mathbb {E} _ {\mathbf {M}} \left[ \mathbf {M} \mathbb {E} _ {\mathbf {x} _ {j}} [ \nabla \widehat {\mathbf {q}} _ {j} ^ {(t)} - \nabla \mathbf {q} _ {j} ^ {(t)} ] \boldsymbol {\Sigma} \right] \\ = \mathbb {E} _ {\mathbf {M}} \left[ \mathbf {M} \left(\left(\mathrm {d i a g} (\mathbb {E} [ \widehat {\mathbf {q}} _ {j} ^ {(t)} ]) - \mathbb {E} _ {\mathbf {x}} [ \widehat {\mathbf {q}} _ {j} ^ {(t)} (\widehat {\mathbf {q}} _ {j} ^ {(t)}) ^ {\top} ]\right) \mathbf {M} ^ {\top} \mathbf {W} ^ {(k)} \boldsymbol {\Sigma} - \left(\mathrm {d i a g} (\mathbb {E} [ \mathbf {q} _ {j} ^ {(t)} ]) - \mathbb {E} _ {\mathbf {x}} [ \mathbf {q} _ {j} ^ {(t)} (\mathbf {q} _ {j} ^ {(t)}) ^ {\top} ]\right) \mathbf {M} ^ {\top}\right) \right]. \\ \end{array}
+$$
+
+Because the class prior is uniform and the isotropic initialisation, we have
+
+$$
+\mathbb {E} _ {\mathbf {x} _ {j}} \left[ \widehat {\mathbf {q}} _ {j} ^ {(t)} \right] = \mathbb {E} _ {\mathbf {x} _ {j}} \left[ \mathbf {q} _ {j} ^ {(t)} \right] = \frac {1}{C} \mathbf {1}.
+$$
+
+Since each coordinate of  $\widehat{\mathbf{q}}_j^{(t)}$  (or  $\mathbf{q}_j^{(t)}$ ) has the same marginal distribution and any two distinct coordinates have the same joint distribution, we have
+
+$$
+\operatorname {d i a g} (\mathbb {E} [ \widehat {\mathbf {q}} _ {j} ^ {(t)} ]) = \operatorname {d i a g} (\mathbf {q} _ {j} ^ {(t)}) = \frac {1}{C} \mathbf {I}, \qquad \mathbb {E} _ {\mathbf {x}} [ \widehat {\mathbf {q}} _ {j} ^ {(t)} (\widehat {\mathbf {q}} _ {j} ^ {(t)}) ^ {\top} ] = \mathbb {E} _ {\mathbf {x}} [ \mathbf {q} _ {j} ^ {(t)} (\mathbf {q} _ {j} ^ {(t)}) ^ {\top} ] = \frac {1}{C ^ {2}} \mathbf {1} \mathbf {1} ^ {\top}.
+$$
+
+Therefore, we have
+
+$$
+\mathcal {A} _ {2} = \mathbb {E} _ {\mathbf {M}} \left[ \mathbf {M} \left(\mathrm {d i a g} (1 / C) - \frac {1}{C ^ {2}} \mathbf {1 1} ^ {\top}\right) \mathbf {M} ^ {\top} \left(\mathbf {W} ^ {(k)} \boldsymbol {\Sigma} - \mathbf {I}\right) \right].
+$$
+
+Note that  $\nabla_{\mathbf{W}}L_{\mathrm{CoT}}(\mathbf{W}^{(k)}) = \mathcal{A}_2$  , therefore, we obtain
+
+$$
+\begin{array}{l} \| \nabla_ {\mathbf {W}} L _ {\mathrm {C o T}} (\mathbf {W} ^ {(k)}) \| _ {F} = \left\| \mathbb {E} _ {\mathbf {M}} \left[ \mathbf {M} \left(\operatorname {d i a g} (1 / C) - \frac {1}{C ^ {2}} \mathbf {1 1} ^ {\top}\right) \mathbf {M} ^ {\top} \left(\mathbf {W} ^ {(k)} \boldsymbol {\Sigma} - \mathbf {I}\right) \right] \right\| _ {F} \\ = \left\| \mathbb {E} _ {\mathbf {M}} \left[ \frac {1}{C} \mathbf {M} \mathbf {M} ^ {\top} - \frac {1}{C ^ {2}} \mathbf {M} \mathbf {1 1} ^ {\top} \mathbf {M} ^ {\top} \right] (\mathbf {W} ^ {(k)} \boldsymbol {\Sigma} - \mathbf {I}) \right\| _ {F} \\ = \sigma^ {2} \left(1 - \frac {1}{C}\right) \left\| \left(\mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1}\right) \right\| _ {F}. \\ \end{array}
+$$
+
+Since all columns in  $\mathbf{M}$  are sampled from  $\mathcal{N}(\mathbf{0},\mathbf{I})$  and  $\mathbf{W}^{(k)}$  is assumed to be an isotropic matrix, it's obvious that  $\mathcal{A}_2$  is also an isotropic matrix. It follows that
+
+$$
+\begin{array}{l} \left\langle \mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1}, \nabla_ {\mathbf {W}} L _ {\mathrm {C o T}} \right\rangle \\ = \mathbb {E} _ {\mathbf {M}} \left[ \operatorname {t r a c e} \left(\mathbf {M} \left(\operatorname {d i a g} (1 / C) - \frac {1}{C ^ {2}} \mathbf {1 1} ^ {\top}\right) \mathbf {M} ^ {\top} \left(\mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1}\right) \boldsymbol {\Sigma} \left(\mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1}\right) ^ {\top}\right) \right] \\ \stackrel {(a)} {=} \sigma^ {2} \operatorname {t r a c e} \left(\mathbb {E} _ {\mathbf {M}} \left[ \frac {1}{C} \mathbf {M} \mathbf {M} ^ {\top} \right] \left(\mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1}\right) \left(\mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1}\right) ^ {\top} \right. \\ \left. - \mathbb {E} _ {\mathbf {M}} \left[ \frac {1}{C ^ {2}} \mathbf {M} \mathbf {1 1} ^ {\top} \mathbf {M} ^ {\top} \right] \left(\mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1}\right) \left(\mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1}\right) ^ {\top}\right) \\ = \sigma^ {2} \operatorname {t r a c e} \left(\left(\mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1}\right) \left(\mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1}\right) ^ {\top} - \frac {1}{C} \left(\mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1}\right) \left(\mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1}\right) ^ {\top}\right) \\ = \sigma^ {2} \left(1 - \frac {1}{C}\right) \| \mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1} \| _ {F} ^ {2}. \\ \end{array}
+$$
+
+where equation (a) follows from the assumption that  $\Sigma = \sigma^2\mathbf{I}$ .
+
+Let  $\gamma = \sigma^2 (1 - 1 / C)$ . Then,
+
+$$
+\begin{array}{l} \left\| \mathbf {W} ^ {(k + 1)} - \boldsymbol {\Sigma} ^ {- 1} \right\| _ {F} ^ {2} \leq \left\| \mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1} \right\| _ {F} ^ {2} - 2 \gamma \eta \left\| \mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1} \right\| _ {F} ^ {2} + \eta^ {2} \gamma^ {2} \left\| \mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1} \right\| _ {F} ^ {2} \\ \leq (1 - \gamma \eta) ^ {2} \| \mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1} \| _ {F} ^ {2} \\ \end{array}
+$$
+
+Select step size such that  $(1 - \gamma \eta)^2\leq 1$  and let  $c\coloneqq (1 - \gamma \eta)^2$  , we obtain
+
+$$
+\left\| \mathbf {W} ^ {(k)} - \boldsymbol {\Sigma} ^ {- 1} \right\| _ {F} ^ {2} \leq c ^ {k} \left\| \mathbf {W} ^ {(0)} - \boldsymbol {\Sigma} ^ {- 1} \right\| _ {F} ^ {2}.
+$$
+
+# E Auxiliary Lemmas
+
+Lemma 5 (Stein's Lemma). Let  $X\in \mathbb{R}^d$  be a random vector with
+
+$$
+X \sim \mathcal {N} (\mu , \Sigma),
+$$
+
+where  $\mu \in \mathbb{R}^d$  and  $\Sigma \in \mathbb{R}^{d\times d}$  is a positive definite matrix. Let  $f:\mathbb{R}^d\to \mathbb{R}^k$  be a continuously differentiable function such that
+
+$$
+\mathbb {E} \big [ \| f (X) \| \big ] <   \infty \quad a n d \quad \mathbb {E} \big [ \| \nabla f (X) \| _ {F} \big ] <   \infty ,
+$$
+
+where  $\| \cdot \|$  denotes the Euclidean norm in  $\mathbb{R}^k$  and  $\| \cdot \| _F$  is the Frobenius norm. Then, the following identity holds:
+
+$$
+\mathbb {E} \left[ (X - \mu) f (X) ^ {T} \right] = \Sigma \mathbb {E} \left[ \nabla f (X) \right],
+$$
+
+where  $\nabla f(X)$  is the  $k\times d$  Jacobian matrix of  $f$  evaluated at  $X$ .
\ No newline at end of file
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+# Unlabeled Data Improves Fine-Grained Image Zero-shot Classification with Multimodal LLMs
+
+Yunqi Hong $^{1}$ , Sohyun An $^{1}$ , Andrew Bai $^{1}$ , Neil Y. C. Lin $^{2}$ , Cho-Jui Hsieh $^{1}$
+
+<sup>1</sup>Computer Science Department, University of California, Los Angeles
+
+$^{2}$ Mechanical and Aerospace Engineering Department, University of California, Los Angeles {yunqihong, sohyun0423, andrewbai, chohsieh}@cs.ucla.edu, neillin@g.ucla.edu
+
+# Abstract
+
+Despite Multimodal Large Language Models (MLLMs) showing promising results on general zero-shot image classification tasks, fine-grained image classification remains challenging. It demands precise attention to subtle visual details to distinguish between visually similar subcategories—details that MLLMs may easily overlook without explicit guidance. To address this, we introduce AutoSEP, an iterative self-supervised prompt learning framework designed to enhance MLLM fine-grained classification capabilities in a fully unsupervised manner. Our core idea is to leverage unlabeled data to learn a description prompt that guides MLLMs in identifying crucial discriminative features within an image, and boosts classification accuracy. We developed an automatic self-enhancing prompt learning framework called AutoSEP to iteratively improve the description prompt using unlabeled data, based on instance-level classification scoring function. AutoSEP only requires black-box access to MLLMs, eliminating the need for any training or fine-tuning. We evaluate our approach on multiple fine-grained classification datasets. It consistently outperforms other unsupervised baselines, demonstrating the effectiveness of our self-supervised optimization framework. Notably, AutoSEP in average improves  $13\%$  over standard zero-shot classification and  $3\%$  over the best-performing baselines. Code is available at https://github.com/yq-hong/AutoSEP.
+
+# 1 Introduction
+
+While Multimodal Large Language Models (MLLMs) exhibit impressive zero-shot image classification capabilities on general image datasets [1, 2, 8, 20, 4, 15, 40], their performance often declines on fine-grained classification tasks. These tasks, which involve distinguishing between visually similar subcategories like specific bird, animal, or plant species [12, 10, 21], pose a challenge for MLLMs. The difficulty stems from the need to distinguish subtle visual differences; without explicit guidance, MLLMs tend to overlook crucial details or misinterpret key characteristics [35, 18, 22].
+
+Previous efforts to enhance MLLMs' fine-grained classification abilities have largely relied on training the model with large-scale labeled datasets [12, 38], which are highly resource-intensive, demanding both significant computational resources and vast amounts of high-quality labeled data. This is especially challenging for fine-grained classification tasks, where collecting and annotating data is expensive and time-consuming due to the subtle visual distinctions between classes. In contrast, while unlabeled images are readily available at test time, there has been limited exploration into how such data could be leveraged to enhance classification performance. This motivates us to explore whether MLLMs can self-improve their fine-grained classification ability using only unlabeled data and with only black-box access to the MLLM.
+
+But how can unlabeled data be leveraged to improve MLLMs' zero-shot prediction capabilities? While self-supervised learning literature [3, 13, 31] has demonstrated that unlabeled data can be used to
+
+train encoders to extract better discriminative features, it is nontrivial to achieve this in a zero-shot and black-box setting. To address this challenge, we develop Automatic Self-Enhancing Prompt Learning (AutoSEP), a novel framework for improving fine-grained visual classification MLLM with unlabeled data (see Figure 1). AutoSEP introduces an intermediate image description step, in which the MLLM is prompted to generate a textual description of an image's visual features. The MLLM then makes the final classification based on the original image and the additional textual description. This image description step can be viewed as mapping the raw image to text that highlights discriminative and relevant features for the task, analogous to the role of encoders in contrastive learning. To evaluate the quality of descriptions using unlabeled data, we design an instance-level classification scoring function that measures the MLLM's ability to retrieve the correct description for each image relative to those of other images. A prompt optimization algorithm is designed to iteratively improve the description-generation prompt based on this scoring function, without requiring white-box access to the MLLMs. By focusing on subtle yet critical visual differences, the MLLM iteratively improves its fine-grained classification performance without requiring any labeled training data.
+
+![](images/5257b767dab4baa888021728d6bc92ef7c54cb97f926f8901de9f7f4432b5829.jpg)  
+Figure 1: Top: Typical zero-shot pipeline for MLLM classification. Middle: Illustration of AutoSEP pipeline. Bottom: An example of an AutoSEP-optimized, description-generation prompt.
+
+To summarize, our key contributions are as follows:
+
+- We introduce a new self-supervised, black-box framework, AutoSEP, based on instance-level description retrieval to improve MLLMs fine-grained image classification capabilities.  
+- We show that AutoSEP significantly enhances MLLMs' fine-grained image classification capabilities (13% average improvement over zero-shot and 3% over the best-performing baselines) on 8 fine-grained tasks.  
+- To the best of our knowledge, this is the first work studying how to use unlabeled data to improve zero-shot image classification performance in the black-box setting for MLLMs.
+
+# 2 Related Work
+
+Fine-grained image classification with MLLMs. MLLMs have demonstrated remarkable performance across a broad spectrum of visual tasks, primarily due to pretraining on large-scale datasets [1, 20, 19, 40, 27, 14, 30]. However, since these models are predominantly trained on web-scraped corpora, they tend to underperform on fine-grained classification tasks [12, 10, 21]. To mitigate this limitation, prior works [12, 38] have employed extensive training using large-scale datasets. While effective, these approaches are computationally expensive and heavily reliant on high-quality labeled data, rendering them impractical in scenarios where fine-grained distinctions are critical. This motivates us to explore an alternative approach, which leverages readily available unlabeled images to improve fine-grained classification in MLLMs through a lightweight framework that requires neither labeled data nor white-box access to the model.
+
+Prompt optimization. Automatic prompt optimization techniques have seen substantial progress in recent years, demonstrating their effectiveness in black-box, computationally constrained settings with a wide range of strategies for searching for effective prompts [26, 7, 37, 39, 24]. Specifically, several approaches [39, 24, 32] utilize LLMs to identify and summarize erroneous cases, which are then iteratively used to refine the prompts. Others [11, 9] adopt evolutionary strategies, generating diverse candidates through mutation and selection. Yang et al. [34] treats LLMs as optimizers, leveraging historical prompt trajectories and associated performance signals to guide prompt generation, while Yang et al. [36], Juneja et al. [17] decompose prompts into semantic segments for finer-grained control. Although these methods have shown strong performance, they largely depend on labeled data to supervise the optimization process. To mitigate this dependency, recent work such as SPO [33] formulates prompt optimization in a self-supervised manner by generating candidate prompts with LLMs and evaluating them through LLM-based pairwise preference comparisons of the resulting model outputs. In contrast, we proposed a higher-level framework for fine-grained visual classification that involves leveraging unlabeled images and explicit prompting for textual descriptions, whereas SPO is a lower-level prompt optimization technique. Furthermore, we find that SPO does not perform well even when incorporated into our framework due to its choice of scoring function (see Section 4).
+
+# 3 Method
+
+The key insight of AutoSEP is leveraging instance-level classification signal, which does not require label information, to inform the optimization of the description generation prompt for improving fine-grained classification. In Section 3.2, we introduce the technique of generating textual descriptions and combining it with image inputs to improve visual classification. In Section 3.3, we detail the procedures for learning the description-generation prompt in a self-supervised manner from instance-level classification scoring.
+
+# 3.1 Problem Setting
+
+Let  $M$  denote the MLLM and  $q$  denote the classification prompt. The vanilla zero-shot prediction is
+
+$$
+\hat {y} = M (x, q), \tag {1}
+$$
+
+where  $x$  is the given image. While MLLMs often achieve strong performance on commonly seen basic-level classes, they tend to underperform on fine-grained classification tasks. We assume there exists a small unlabeled dataset  $X = \{x_{1}, x_{2}, \dots, x_{n}\}$  (more detailed experiments on different dataset sizes can be found in Section 4.3.) Our goal is to utilize  $X$  to improve the zero-shot prediction accuracy on fine-grained categories. Also, we consider only black-box access to the MLLM, which means fine-tuning is prohibited.
+
+# 3.2 Image Description Generation for Fine-grained Classification
+
+Traditional self-supervised learning (SSL) leverages unlabeled data to contrastively learn better representations [3, 13, 31]. Inspired by SSL, we introduce an additional description generation step, where a "description generation prompt"  $p$  is used to generate a text description  $t$  for an image  $x$ . Both the description and the raw image are then jointly passed to the MLLM for predicting the label (see Figure 1):
+
+$$
+\hat {y} = M (x, [ q, t ]), \quad \text {w h e r e} t = M (x, p). \tag {2}
+$$
+
+Intuitively,  $M(\cdot ,p)$  functions as an encoder, translating the input  $x$  into a text description  $t$ . This description then helps the MLLM conduct the final prediction.
+
+However, a naive description generation prompt yields non-informative descriptions, and even worse, many descriptions might be misleading. Therefore, our method utilize unlabeled data  $X$  to iteratively refine how MLLMs analyze and describe images, guiding them away from generic or incorrect visual features towards distinguishing and precise attributes.
+
+# 3.3 Automatic Self-enhancing Prompt Learning
+
+To learn a better description-generation prompt, we formalize an optimization objective based on a scoring function  $\Psi(X, p)$  that evaluates the effectiveness of each candidate prompt  $p$ :
+
+$$
+p ^ {*} = \underset {p \in \mathcal {L}} {\arg \max } \Psi (X, p), \tag {3}
+$$
+
+where  $\mathcal{L}$  represents the space of coherent natural language prompts.
+
+![](images/fe5753d2e9d0ca4da5b6a89179eebd45337c4fb1503a9580aeedb33c78329e62.jpg)  
+Figure 2: An illustration of automatic self-enhancing prompt learning with instance-level classification.
+
+# 3.3.1 Instance-level Classification for Unsupervised Scoring
+
+Inspired by traditional self-supervised learning methods [3, 13, 31], which learn the encoders to pull positive pairs closer and push negative pairs further apart contrastively, we propose to optimize the prompt so that the MLLM generates highly distinctive descriptions for each individual unlabeled image. By doing so, our method maximizes the inter-image distinctiveness of the generated descriptions.
+
+We frame the optimization as an instance-level classification task, where the MLLM learns to correctly associate each image with its own generated description while distinguishing it from the descriptions of other images (See Figure 2). To quantify how well the MLLM can distinguish between different image descriptions, we define a sample matching correctness indicator  $V(x_{i},x_{j})$ . Given an image  $x_{i}$  and two candidate descriptions  $t_i$  and  $t_j$ , which are generated from images  $x_{i}$  and  $x_{j}$ , respectively, we present the MLLM with a binary-choice prompt  $G(t_a,t_b)$ . The prompt asks "Text 1:  $\{t_a\}$ . Text 2:  $\{t_b\}$ . Which description correctly describes the image? The first or the second?", where  $(t_a,t_b)$  is either  $(t_i,t_j)$  or  $(t_j,t_i)$ , randomly shuffled with equal probability. The correctness indicator function is then formulated as:
+
+$$
+V \left(x _ {i}, x _ {j}\right) := \left\{ \begin{array}{l l} \mathbb {1} \left[ M \left(x _ {i}, G \left(t _ {i}, t _ {j}\right)\right) = “ f i r s t ” \right], & Z = 0 \\ \mathbb {1} \left[ M \left(x _ {i}, G \left(t _ {j}, t _ {i}\right)\right) = “ s e c o n d ” \right], & Z = 1, \end{array} \right. \tag {4}
+$$
+
+where  $Z \sim$  Bernoulli(0.5) determines the random ordering of descriptions to prevent positional bias. The function  $V(x_{i},x_{j})$  returns 1 if the MLLM correctly selects the description corresponding to  $x_{i}$ , and 0 otherwise.
+
+We compute the unsupervised scoring function that measures the model's ability to correctly match descriptions to their respective images. Formally, the scoring function is defined as:
+
+$$
+\Psi (X, p) := \sum_ {x _ {i} \in X} \sum_ {x _ {j} \in X \backslash \{x _ {i} \}} V \left(x _ {i}, x _ {j}\right). \tag {5}
+$$
+
+To avoid iterating over all descriptions,  $\Psi(X,p)$  can be approximated with random negative samples:
+
+$$
+\hat {\Psi} (X, p, k) := \frac {1}{k \cdot | X |} \sum_ {x _ {i} \in X} \sum_ {x _ {j} \in \mathcal {O} _ {i}} V \left(x _ {i}, x _ {j}\right), \tag {6}
+$$
+
+where  $\mathcal{O}_i\sim \mathrm{RandomSample}(X\backslash \{x_i\} ,k)$  represents a set of  $k$  randomly sampled images from  $X$ , excluding  $x_{i}$ , acting as negative samples.  $k\cdot |X|$  normalizes the score by the total number of evaluations conducted.
+
+# 3.3.2 Description-generation Prompt Optimization
+
+To begin the optimization process, we start with a simple yet effective initial generation prompt that instructs the MLLM to describe the object. The prompt can be found in Appendix A.
+
+Feedback and modification. To iteratively refine the prompt, we adopt a feedback-driven approach that identifies and addresses the weaknesses in the current generation prompt. An illustration of AutoSEP workflow and prompts can be found in Appendix B. First, we collect error pairs
+
+$J_{\mathrm{error}} = \{(x_i,x_j)\mid x_i,x_j\in X,x_i\neq x_j,V(x_i,x_j) = 0\}$ , where the generated descriptions failed to correctly distinguish between images. Then we sample a subset of  $J_{\mathrm{error}}$  and construct a diagnostic prompt to analyze potential shortcomings of the current prompt. Specifically, we present the MLLM with the current prompt together with examples of mismatched descriptions, and ask it to explain why the current prompt may have led to these errors. This process, referred to as the Reflect operator, Reflect(\cdot), yields a set of potential reasons explaining the observed errors. Based on MLLM's feedback, we then prompt it to revise the current prompt by explicitly addressing the identified shortcomings. This refinement step is denoted as the Modify operator, Modify(\cdot), which outputs an improved version of the generation prompt. Each instance of feedback produces a candidate revised prompt, resulting in multiple refined versions aimed at enhancing the distinctiveness of the generated descriptions. It has been shown in the literature that this reflect-and-modify approach is effective in automatic prompt engineering [24].
+
+Iterative optimization. Our AutoSEP optimization process follows an iterative framework where the generation prompt is progressively refined to generate highly distinctive descriptions (Algorithm 1). Starting with an initial prompt, we evaluate its effectiveness by generating descriptions for a batch of images and forming an instance-level classification set by sampling negative images. Errors in the instance-level classification are identified and used to expand the prompt by adjusting it to better capture distinctive attributes. At each iteration, a set of candidate prompts is evaluated using a scoring function defined in Section 3.3.1, and only the top-performing prompts are retained for the next round. This iterative adaptation ensures that the MLLM increasingly focuses on fine-grained details, improving its ability to generate more informative descriptions. The process terminates after a predefined number of iterations  $N$ , and the best-performing prompt is selected as the final optimized prompt.
+
+By maximizing  $\hat{\Psi}(X, p, k)$ , we iteratively refine the prompt  $p$  to improve the distinctiveness of the generated descriptions, ultimately enhancing the MLLM's ability to perform fine-grained classification in a self-supervised manner.
+
+Algorithm 1 AutoSEP: Automatic Self-Enhancing Prompt Learning  
+Require:  $p_0$ : initial prompt,  $N$ : iterations,  $X$ : unlabeled dataset,  $k$ : negative samples per image,  $b$ : top prompts retained per iteration,  $l$ : number of reflections,  $\hat{\Psi}(\cdot)$ : scoring function  
+1:  $P_0 \gets \{p_0\}$  ▷ Initialize candidate prompt set  
+2:  $\Gamma \gets \{\gamma_0\}$ , where  $\gamma_0 = \{M(x_i, p_0) \mid x_i \in X\}$  ▷ Generate descriptions  
+3: for  $t = 1$  to  $N$  do  
+4:  $X_t \sim \mathcal{U}(X)$  ▷ Random minibatch from  $X$   
+5:  $P_c \gets P_{t-1}$   
+6: for  $p \in P_{t-1}$  do  
+7:  $J = \{(x_i, x_j) \mid x_i, x_j \in X_t, i \neq j\}$ , where for each  $x_i, k$  negative samples  $x_j$  are randomly selected ▷ Construct instance-level classification set  
+8:  $J_{\text{error}} = \{(x_i, x_j) \mid (x_i, x_j) \in J, V(x_i, x_j) = 0\}$  ▷ Collect errors  
+9:  $J_{\text{error}}^i \subset J_{\text{error}}$  is a sampled subset for each  $i = 1, \ldots, l$   
+10:  $G = \{g_1, g_2, \ldots, g_l\} = \bigcup_{i=1, \ldots, l} \text{Reflect}(p, J_{\text{error}}^i)$  ▷ Reflect on the errors (Sec. 3.3.2)  
+11:  $H = \{h_1, h_2, \ldots, h_l\} = \bigcup_{i=1, \ldots, l} \text{Modify}(p, g_i, J_{\text{error}}^i)$  ▷ Modify prompts (Sec. 3.3.2)  
+12:  $P_c \gets P_c \cup H$   
+13:  $\Gamma \gets \Gamma \cup \{\{M(x_i, h) \mid x_i \in X\} \mid h \in H\}$  ▷ Generate descriptions for new prompts  
+14: end for  
+15:  $S_c = \{\hat{\Psi}(X, p, k) \mid p \in P_c\}$  ▷ Evaluate prompts  
+16:  $P_t \gets \{p \in P_c \mid \hat{\Psi}(X, p, k) \geq \tau\}$ , where  $\tau$  is the  $b^{\text{th}}$  highest score in  $S_c$   
+17: end for  
+18: Return  $p^* \gets \arg \max_{p \in P_N} \hat{\Psi}(X, p, k)$
+
+# 4 Experiments
+
+# 4.1 Experiment Settings
+
+Datasets. We conduct experiments on four fine-grained image classification datasets, including CUB-200-2011 [29] (bird classification), iNaturalist 2021 [28] (various wild species), Stanford Dogs [6], and VegFru [25]. For each dataset, we construct fine-grained classification tasks on subsets
+
+of categories that are particularly challenging for MLLMs. We construct three bird subsets with high confusion rates: CUB_cuckoo (Black-billed, Mangrove, and Yellow-billed cuckoo), CUB_oriole (Hooded, Orchard, and Scott's oriole), and CUB_vireo (Philadelphia, Red-eyed, and Warbling vireo). For iNaturalist, we select two subsets comprising visually similar species: iNat_butterfly (Symbrenthia lilaea, Claudina crescent, Elada checkerspot) and iNat_lupine (Arctic lupine, Silvery lupine, Arizona lupine). For Stanford Dogs, we focus on three closely related terriers: StanfordDogs_terrier (Lakeland terrier, Norwich terrier, Cairn terrier). For the VegFru dataset, we also select two subsets consisting of visually similar species: VegFru_green (Dandelion, Shepherd's purse, Prickly lettuce) and VegFru_allium (Leek, Green Chinese onion, Bunching onion). All the results reported are evaluated on datasets that are balanced in class distribution.
+
+Baselines. We compare our method with both optimization-free and optimization-based approaches. For optimization-free methods, we consider: (1) Vanilla zero-shot, where the MLLM is directly prompted without additional context; (2) Zero-shot with descriptions, using image descriptions generated from an initial human-crafted prompt; (3) Zero-shot with majority vote, where we prompt the MLLM to generate multiple responses with high temperature and take the most frequent prediction as the final answer; (4) Few-shot with random labels, where the MLLM is shown  $m$  images with randomly assigned labels and then asked to classify the  $(m + 1)^{\text{th}}$  image; (5) Few-shot with MLLM labels, where the MLLM is shown  $m$  images with its own predicted labels and then asked to classify the  $(m + 1)^{\text{th}}$  image; (6) Multiple images display, where the MLLM is directly shown  $m$  unlabeled images before predicting the  $(m + 1)^{\text{th}}$  image, as proposed by Jiang et al. [16]; (7) K-means clustering, where we cluster image features extracted by a pretrained image encoder, and then within each cluster, use the most frequent zero-shot prediction as the final label for all images in that cluster.
+
+For prompt optimization-based methods, we consider three approaches to adapt them to the unsupervised setting: (1) Optimization with random labels, where we randomly assign labels to the images and use the classification accuracy under these random labels as the optimization objective; (2) Optimization with majority vote, where we iteratively assign pseudo-labels to images based on majority voting over multiple high-temperature outputs from the MLLM, and use the classification accuracy with respect to these pseudo-labels as the optimization objective; (3) Self-Supervised Prompt Optimization (SPO) [33], originally designed for text-only tasks, which perform pairwise comparisons with outputs generated by different prompts to infer their relative quality as signals to guide the optimization. We adopt SPO to our multimodal setting by asking the MLLM to compare the quality of image descriptions generated by different prompts. Following the original setup, we use GPT-4o [14] to conduct these comparisons. Prompts used for these baseline methods can be found in Appendix E.
+
+Models. We conduct experiments with three state-of-the-art MLLMs: Gemini 1.5 Flash [27], GPT-4o [14], and Qwen2-VL-72B-Instruct [30], covering both proprietary and open-source models. We performed the experiments on local servers with 64 CPU cores and 4 Nvidia A6000 GPUs.
+
+# 4.2 Experiment Results
+
+Experiment results in Table 1 demonstrate that our method consistently outperforms all baseline approaches across all tested MLLMs. Notably, it achieves an average improvement of  $13\%$  over standard zero-shot classification and  $3\%$  over the best-performing baselines. These consistent gains highlight the robustness and generalizability of our approach across different model architectures and capabilities. Furthermore, our method operates without requiring any labeled data, making it particularly well-suited for low-resource or open-world scenarios.
+
+Description-generation prompt benefits from optimization. We also observe that in some cases, the initial description generation prompt can slightly improve classification performance. However, these improvements are generally marginal. And in some cases, generating descriptions even degrades the performance. This suggests that unguided or naive prompting alone for description generation is insufficient. More principled guidance and optimization are necessary for the model to generate informative and task-relevant descriptions that more effectively support classification.
+
+Table 1: Main results (accuracy %). AutoSEP outperforms baselines consistently across all tasks and MLLMs. Source of variability stems from samples in the evaluation dataset. Gemini refers to Gemini 1.5 Flash.  
+
+<table><tr><td>Model</td><td>Gemini</td><td>GPT-4o</td><td>Qwen2-VL</td><td>Gemini</td><td>GPT-4o</td><td>Qwen2-VL</td></tr><tr><td></td><td colspan="3">CUB_cuckoo</td><td colspan="3">CUB_oriole</td></tr><tr><td colspan="7">Optimization-free</td></tr><tr><td>Vanilla zero-shot</td><td>51.68±1.79</td><td>61.46±1.24</td><td>58.54±4.69</td><td>52.13±2.90</td><td>74.44±0.93</td><td>43.60±2.96</td></tr><tr><td>Zero-shot with descriptions</td><td>52.20±2.49</td><td>68.29±1.00</td><td>51.46±2.60</td><td>46.74±2.31</td><td>73.71±1.52</td><td>50.79±3.92</td></tr><tr><td>Zero-shot with majority vote</td><td>51.22±2.04</td><td>61.95±1.95</td><td>59.02±1.98</td><td>54.38±0.55</td><td>75.28±2.01</td><td>46.74±4.95</td></tr><tr><td>Few-shot with random labels</td><td>38.05±6.14</td><td>43.90±7.48</td><td>46.83±9.74</td><td>40.05±12.1</td><td>56.85±12.3</td><td>36.63±6.50</td></tr><tr><td>Few-shot with MLLM labels</td><td>53.41±5.31</td><td>61.22±2.82</td><td>52.93±8.57</td><td>57.75±4.42</td><td>64.94±7.96</td><td>36.85±4.79</td></tr><tr><td>Multiple images display [16]</td><td>48.05±6.84</td><td>52.68±9.27</td><td>65.12±3.05</td><td>59.78±6.60</td><td>71.46±8.75</td><td>46.74±3.80</td></tr><tr><td>K-means clustering</td><td>53.17±15.2</td><td>59.51±15.0</td><td>56.59±2.26</td><td>51.46±5.83</td><td>67.19±8.11</td><td>44.94±7.07</td></tr><tr><td colspan="7">Optimization-based</td></tr><tr><td>With random labels</td><td>59.15±1.83</td><td>65.85±0.77</td><td>52.20±3.65</td><td>51.69±2.36</td><td>72.58±1.83</td><td>51.01±1.83</td></tr><tr><td>With majority vote</td><td>44.88±4.69</td><td>67.32±2.10</td><td>53.41±2.36</td><td>57.90±1.56</td><td>73.03±1.74</td><td>38.65±1.74</td></tr><tr><td>SPO</td><td>55.98±1.34</td><td>66.59±2.63</td><td>56.10±6.19</td><td>49.49±2.71</td><td>76.12±2.16</td><td>46.52±3.37</td></tr><tr><td>AutoSEP (Ours)</td><td>61.22±2.30</td><td>75.12±1.83</td><td>63.41±2.77</td><td>58.20±0.88</td><td>75.96±0.90</td><td>55.13±3.58</td></tr><tr><td></td><td colspan="3">CUB_vireo</td><td colspan="3">iNat_butterfly</td></tr><tr><td colspan="7">Optimization-free</td></tr><tr><td>Vanilla zero-shot</td><td>49.21±3.36</td><td>75.73±1.83</td><td>55.28±3.05</td><td>61.14±2.11</td><td>68.00±1.46</td><td>40.25±4.65</td></tr><tr><td>Zero-shot with descriptions</td><td>51.46±4.11</td><td>86.07±0.90</td><td>50.79±3.72</td><td>56.57±3.45</td><td>71.14±3.88</td><td>48.25±2.48</td></tr><tr><td>Zero-shot with majority vote</td><td>49.21±1.80</td><td>76.69±2.01</td><td>55.06±3.01</td><td>62.00±2.14</td><td>66.07±1.86</td><td>46.56±2.59</td></tr><tr><td>Few-shot with random labels</td><td>45.62±8.03</td><td>57.08±18.2</td><td>43.82±7.78</td><td>45.43±13.8</td><td>64.00±10.7</td><td>32.13±8.47</td></tr><tr><td>Few-shot with MLLM labels</td><td>44.94±8.56</td><td>72.36±10.5</td><td>46.97±7.63</td><td>51.43±3.26</td><td>82.57±14.3</td><td>42.13±12.2</td></tr><tr><td>Multiple images display [16]</td><td>56.18±7.95</td><td>68.54±13.6</td><td>63.82±5.34</td><td>55.71±1.56</td><td>72.29±12.2</td><td>32.25±3.44</td></tr><tr><td>K-means clustering</td><td>40.22±9.25</td><td>57.08±11.2</td><td>57.53±15.0</td><td>58.57±5.89</td><td>59.14±5.39</td><td>38.38±6.92</td></tr><tr><td colspan="7">Optimization-based</td></tr><tr><td>With random labels</td><td>41.24±6.85</td><td>72.81±3.13</td><td>48.31±4.44</td><td>53.24±1.72</td><td>79.86±4.43</td><td>45.13±1.71</td></tr><tr><td>With majority vote</td><td>53.71±5.41</td><td>87.19±1.83</td><td>51.24±2.72</td><td>62.29±1.63</td><td>61.07±1.89</td><td>51.50±3.36</td></tr><tr><td>SPO</td><td>52.81±9.59</td><td>70.34±3.73</td><td>51.69±1.59</td><td>53.14±2.24</td><td>79.71±9.14</td><td>45.26±2.47</td></tr><tr><td>AutoSEP (Ours)</td><td>59.18±0.53</td><td>87.42±1.10</td><td>65.45±1.24</td><td>66.57±1.42</td><td>82.71±2.43</td><td>55.38±1.00</td></tr><tr><td></td><td colspan="3">iNat_lupine</td><td colspan="3">StanfordDogs_terrier</td></tr><tr><td colspan="7">Optimization-free</td></tr><tr><td>Vanilla zero-shot</td><td>62.00±2.72</td><td>70.00±3.59</td><td>44.22±3.74</td><td>60.80±2.13</td><td>91.39±0.48</td><td>73.20±2.40</td></tr><tr><td>Zero-shot with descriptions</td><td>56.67±1.92</td><td>73.00±1.25</td><td>44.00±2.49</td><td>66.22±2.41</td><td>91.42±1.17</td><td>76.00±2.53</td></tr><tr><td>Zero-shot with majority vote</td><td>61.78±1.24</td><td>72.33±1.70</td><td>47.33±2.06</td><td>62.67±0.73</td><td>90.89±1.09</td><td>76.27±2.13</td></tr><tr><td>Few-shot with random labels</td><td>47.56±7.31</td><td>48.33±5.48</td><td>42.00±7.38</td><td>60.27±10.3</td><td>66.89±24.7</td><td>52.13±8.90</td></tr><tr><td>Few-shot with MLLM labels</td><td>60.22±11.4</td><td>69.33±3.89</td><td>46.44±6.75</td><td>66.67±10.3</td><td>87.56±3.47</td><td>64.67±6.84</td></tr><tr><td>Multiple images display [16]</td><td>63.47±7.71</td><td>74.00±5.54</td><td>50.44±2.86</td><td>65.87±4.04</td><td>79.78±9.04</td><td>77.73±4.43</td></tr><tr><td>K-means clustering</td><td>47.11±6.50</td><td>50.00±2.11</td><td>30.00±1.99</td><td>39.60±7.50</td><td>68.67±1.09</td><td>64.13±5.66</td></tr><tr><td colspan="7">Optimization-based</td></tr><tr><td>With random labels</td><td>58.44±4.80</td><td>73.67±1.94</td><td>46.00±3.62</td><td>53.33±1.94</td><td>90.00±1.11</td><td>63.73±1.87</td></tr><tr><td>With majority vote</td><td>59.33±1.81</td><td>74.17±1.83</td><td>47.33±4.37</td><td>62.27±1.82</td><td>91.11±1.41</td><td>77.74±1.07</td></tr><tr><td>SPO</td><td>56.44±3.17</td><td>64.88±6.12</td><td>46.56±3.44</td><td>65.16±6.10</td><td>81.47±2.12</td><td>64.27±2.00</td></tr><tr><td>AutoSEP (Ours)</td><td>66.15±1.47</td><td>76.60±1.40</td><td>54.67±3.33</td><td>69.28±0.97</td><td>92.18±1.09</td><td>81.70±1.20</td></tr><tr><td></td><td colspan="3">VegFru Greens</td><td colspan="3">VegFru_allium</td></tr><tr><td colspan="7">Optimization-free</td></tr><tr><td>Vanilla zero-shot</td><td>86.00±0.56</td><td>78.82±0.54</td><td>68.89±3.06</td><td>72.67±3.18</td><td>60.44±1.66</td><td>54.22±3.40</td></tr><tr><td>Zero-shot with descriptions</td><td>77.11±3.19</td><td>77.33±1.94</td><td>63.61±1.82</td><td>74.44±1.98</td><td>64.89±1.51</td><td>59.17±1.98</td></tr><tr><td>Zero-shot with majority vote</td><td>85.83±0.48</td><td>78.00±0.44</td><td>71.39±2.53</td><td>75.56±1.82</td><td>62.44±2.57</td><td>56.89±2.37</td></tr><tr><td>Few-shot with random labels</td><td>74.44±6.30</td><td>70.22±8.47</td><td>33.33±0.00</td><td>67.33±3.18</td><td>61.33±10.6</td><td>33.33±0.00</td></tr><tr><td>Few-shot with MLLM labels</td><td>84.44±7.24</td><td>78.89±7.13</td><td>68.44±5.14</td><td>75.11±7.39</td><td>73.11±9.23</td><td>59.11±15.1</td></tr><tr><td>Multiple images display [16]</td><td>82.44±4.30</td><td>72.89±5.14</td><td>70.22±10.6</td><td>79.56±2.69</td><td>66.00±7.29</td><td>59.56±4.25</td></tr><tr><td>K-means clustering</td><td>67.78±5.24</td><td>67.56±4.94</td><td>64.00±5.19</td><td>51.33±14.1</td><td>50.89±9.33</td><td>42.67±7.72</td></tr><tr><td colspan="7">Optimization-based</td></tr><tr><td>With random labels</td><td>80.56±4.33</td><td>79.78±0.44</td><td>48.52±2.77</td><td>71.56±2.06</td><td>64.22±1.47</td><td>55.19±4.48</td></tr><tr><td>With majority vote</td><td>81.19±4.27</td><td>74.00±0.89</td><td>55.56±0.91</td><td>74.22±0.44</td><td>64.67±0.83</td><td>56.94±1.21</td></tr><tr><td>SPO</td><td>84.37±5.26</td><td>78.40±0.87</td><td>57.78±3.14</td><td>73.78±1.81</td><td>65.56±2.90</td><td>60.00±0.91</td></tr><tr><td>AutoSEP (Ours)</td><td>87.78±0.70</td><td>81.11±0.70</td><td>72.22±0.91</td><td>78.89±0.99</td><td>69.11±0.83</td><td>62.44±2.44</td></tr></table>
+
+Instance-level classification is an effective unsupervised scoring signal, outperforming alternative unsupervised scoring strategies. A key component of our method is the use of instance-level classification accuracy as the unsupervised scoring signal during optimization. To better understand its effectiveness, we compare it against three alternative unsupervised scoring methods for optimization: (1) Classification accuracy with randomly assigned labels; (2) Classification accuracy with majority vote pseudo-labels, where pseudo-labels are obtained by aggregating high-temperature predictions from the MLLM; (3) Pairwise output evaluation in SPO [33], which relies on pairwise comparisons of model outputs to estimate relative prompt quality.
+
+While these alternatives sometimes yield performance gains over optimization-free baselines, they exhibit limited reliability and overall lower performance compared to our instance-level classification approach. Majority vote pseudo-labeling tends to perform better than using random labels, but both methods fall short in stability and effectiveness. Although SPO has demonstrated strong performance on several text-only reasoning and question answering tasks, it does not transfer well to fine-grained image classification. The results suggest that while LLMs can effectively compare the quality of reasoning paths in purely textual contexts, MLLMs struggle to consistently evaluate the informativeness of image descriptions for fine-grained classification. As a result, such evaluations offer weak and often noisy feedback signals for optimization. These results highlight the importance of instance-level classification as a robust and effective unsupervised signal.
+
+# 4.3 Discussion and Analysis
+
+Learning dynamics. We study how the classification performance evolves throughout the optimization process and how many iterations are usually required for the algorithm to take effect. Full details of experimental settings are mentioned in Appendix C.1. The average results and their variances reported in Figure 3a and 3b are computed by averaging these classification performances across all three runs. The Max results are computed by taking the best performance among the four prompts at each iteration, then averaging across the three independent runs.
+
+Both the instance-level (Figure 3a) and class-wise (Figure 3b) classification performance consistently improves with more optimization iterations. More demonstrations on other datasets can be found in Appendix C.1. The correlational evidence suggests that instance-level classification serves as an effective unsupervised optimization signal to guide the model toward better class-wise discrimination. These results validate our design of leveraging instance-level feedback to drive prompt optimization for fine-grained classification in the absence of labeled data.
+
+![](images/06eb15f12afa4fac4d0184e729e05604737c5e4b3f917941802ba4cdbac602b7.jpg)  
+(a) Instance-level Classification
+
+![](images/904b2ce901fadc323157654d966561bba754bf893c01ee8d8cf0d98c704d457c.jpg)  
+(b) Class-wise Classification
+
+![](images/abd33f6ccee11faa4a4de59cc8437e389477ebaf5c2c54e9cfbe593e23248e93.jpg)  
+(c) Prompt word diversity  
+Figure 3: Evolution of metrics with increasing optimization iterations.
+
+Correlation between instance-level classification and class-wise classification. To further support the effectiveness of instance-level classification as an optimization signal, we also compute the statistical Pearson correlation between instance-level and class-wise classification performance. Table 2 shows strong positive Pearson correlation across different settings, suggesting that improvements in instance-level accuracy are closely associated with better class-wise classification performance. This empirical evidence reinforces our hypothesis that instance-level supervision can serve as a powerful proxy to guide the description optimization for fine-grained image classification in the absence of labeled data.
+
+Diversity of generation prompts. To analyze how much new information is introduced during the optimization process, we evaluate the diversity of the generated prompts using a quantitative diversity
+
+Table 2: Correlation between instance-level classification and class-wise classification.  
+
+<table><tr><td>Datasets</td><td>CUB_cuckoo</td><td>CUB_oriole</td><td>CUB_vireo</td><td>iNat_butterfly</td><td>iNat_lupine</td></tr><tr><td>Gemini 1.5 Flash</td><td>0.72±0.11</td><td>0.47±0.24</td><td>0.70±0.19</td><td>0.63±0.22</td><td>0.53±0.02</td></tr><tr><td>Qwen2-VL-72B-Instruct</td><td>0.63±0.13</td><td>0.38±0.15</td><td>0.64±0.10</td><td>0.84±0.11</td><td>0.70±0.17</td></tr></table>
+
+score. Specifically, for each prompt, we count the number of semantic keywords (lemmas of nouns, verbs, and adjectives excluding stop words), as well as the number of unique words, defined as words that appear only in that specific prompt and not in any other prompt throughout the optimization. These counts are normalized to compute the diversity score for each prompt. Detailed definition can be found in Appendix F.
+
+As shown in Figure 3c, the diversity score steadily increases over iterations, indicating that the optimization process encourages the generation of increasingly varied and semantically rich prompts. This trend suggests that the model is progressively exploring a broader and more informative prompt space, which may potentially contribute to improved classification performance.
+
+Computational complexity. Let  $b$  denote the number of top prompts retained,  $l$  is the number of prompt reflections generated per retained prompt,  $k$  is the number of negative samples used per image in the instance-level classification, and  $n$  is the number of unlabeled images. Each iteration of our algorithm involves generating  $b \cdot l$  reflections, and  $b \cdot l$  modified candidate prompts accordingly. For each of these prompts, we generate descriptions for all  $n$  images, resulting in  $b \cdot l \cdot n$  queries to the MLLM for description generation. Additionally, instance-level classification requires comparing each image's description against  $k$  negative samples under each prompt, leading to  $b \cdot l \cdot k \cdot n$  further MLLM queries. Therefore, the total number of MLLM queries per iteration is  $(2 + n + kn) \cdot bl$ . In our implementation, we typically set  $b = 4, l \leq 5$ , and  $k = 2$ , resulting in an approximate query complexity of  $\mathcal{O}(60n)$  per iteration.
+
+For optimization-free baselines, the computational cost is relatively low, since they do not require iterative prompt refinement. For optimization-based baselines, we used the same experimental setup as AutoSEP, resulting in comparable computational complexity. More specifically, optimization with random labels takes approximately  $\mathcal{O}(2(1 + n) \cdot bl) = \mathcal{O}(40n)$  MLLM queries per iteration, as it involves only a single classification per image. Optimization with majority vote takes approximately  $\mathcal{O}(2(1 + n) \cdot bl + cn) = \mathcal{O}(45n)$ , with  $c = 5$  denoting the number of majority votes collected for updates in each iteration. SPO [33] is more lightweight, requiring only  $\mathcal{O}(2n)$  queries per iteration, since it only compares with the previous prompt and does not rely on several candidate prompts.
+
+The actual runtime depends on the inference speed of the MLLMs. For example, one optimization iteration on 60 images with Gemini 1.5 Flash takes 12 minutes in average, demonstrating that AutoSEP is computationally feasible for practical use.
+
+More samples, stronger instance-level signals, more stable optimization. We investigate how the number of unlabeled samples used during optimization impacts the performance of our method. Figure 4 shows a clear trend that the classification performance improves with increased number of samples. More demonstrations on other datasets can be found in Appendix C.2.
+
+When the sample size is too small, instance-level classification is constrained by limited variation and coverage, making it difficult to reliably evaluate a prompt's ability to generate distinctive and meaningful descriptions. This leads to suboptimal performance and introduces more noise and instability into the optimization process. In contrast, with more samples, the instance-level signal becomes more robust, enabling more stable optimization and resulting in consistent gains in class-wise classification accuracy. Notably, our method begins to perform reliably with as few as 60 unlabeled samples, demonstrating its practicality and scalability even in limited-data settings.
+
+![](images/82d1f5227b894f56b618b90ffd7ede79c55924918d7644736befb093406e9de3.jpg)  
+Figure 4: Classification accuracy of Gemini with various number of samples for optimization.
+
+# 5 Conclusion
+
+In this paper, we present a self-supervised prompt learning framework that improves the fine-grained image classification capabilities of MLLMs using only unlabeled data and black-box MLLM access. By optimizing the prompt for image description generation through instance-level classification feedback, our method enables MLLMs to focus on subtle, discriminative features critical for distinguishing between visually similar categories. Extensive experiments across multiple fine-grained classification tasks and MLLMs demonstrate the robustness of our approach.
+
+Limitations. Our method relies on the implicit assumption that subtle, class-discriminative visual features can be accurately captured and described using natural language. This may not hold in cases where visual distinctions are extremely fine-grained or difficult to verbalize. Additionally, although our approach is fully unsupervised, it still requires a sufficient number of unlabeled samples to ensure stable and effective prompt optimization. The absence of labeled data inherently limits the performance compared to fully supervised methods that benefit from explicit class-level guidance.
+
+# Acknowledgments and Disclosure of Funding
+
+This work is supported by NSF 2048280, 2325121, 2244760, 2331966 and ONR N00014-23-1-2300:P00001.
+
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+[35] Dingchen Yang, Bowen Cao, Guang Chen, and Changjun Jiang. Pensieve: Retrospect-then-compare mitigates visual hallucination. arXiv preprint arXiv:2403.14401, 2024.  
+[36] Muchen Yang, Moxin Li, Yongle Li, Zijun Chen, Chongming Gao, Junqi Zhang, Yangyang Li, and Fuli Feng. Dual-phase accelerated prompt optimization, 2024. URL https://arxiv.org/abs/2406.13443.  
+[37] Tianjun Zhang, Xuezhi Wang, Denny Zhou, Dale Schuurmans, and Joseph E. Gonzalez. Tempera: Test-time prompting via reinforcement learning, 2022. URL https://arxiv.org/abs/2211.11890.  
+[38] Yuhui Zhang, Alyssa Unell, Xiaohan Wang, Dhruba Ghosh, Yuchang Su, Ludwig Schmidt, and Serena Yeung-Levy. Why are visually-grounded language models bad at image classification? arXiv preprint arXiv:2405.18415, 2024.  
+[39] Yongchao Zhou, Andrei Loan Muresanu, Ziwen Han, Keiran Paster, Silviu Pitis, Harris Chan, and Jimmy Ba. Large language models are human-level prompt engineers, 2023. URL https://arxiv.org/abs/2211.01910.  
+[40] Deyao Zhu, Jun Chen, Xiaoqian Shen, Xiang Li, and Mohamed Elhoseiny. Minigpt-4: Enhancing vision-language understanding with advanced large language models. arXiv preprint arXiv:2304.10592, 2023.
+
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+
+# A Examples of Descriptions
+
+![](images/c923516f9096a5f1b118597a5cef30a3a8e7de7aeb766d50d93dc208585afa9d.jpg)  
+(a) Black-billed Cuckoo
+
+![](images/c49ccef7f08947ba4a42414d0a4fc63ad247d04fe447bdabb7c98998ec9d1651.jpg)  
+(b) Yellow-billed Cuckoo  
+Figure 5: Examples showcasing descriptions generated from two different prompts using Gemini 1.5 Flash. Attributes highlighted in green indicate correct information, while those in red denote incorrect attributes.
+
+Figure 5 shows two examples of image descriptions generated by Gemini 1.5 Flash [27] using prompts before and after optimization. The initial descriptions tend to be vague or misleading, often misrepresenting key visual attributes. In contrast, the optimized prompts lead to more accurate and reliable descriptions that better reflect the distinctive features of the images.
+
+# B Detailed AutoSEP Workflow and Prompts
+
+![](images/ceb64d3cdd143866f44e7f1094250b6a41cd04617f9d9fff09d58b34a5c6cbe2.jpg)  
+Figure 6: Workflow of AutoSEP with Optimization Prompts. Take CUB_cuckoo as an example.
+
+Figure 6 shows the detailed workflow of AutoSEP and prompts used in the optimization. The prompt learning procedure is introduced in Section 3.3.2.
+
+# C Additional Ablation Studies
+
+# C.1 Learning Dynamics
+
+We study how the classification performance evolves throughout the optimization process and how many iterations are usually required for the algorithm to take effect. For each experimental setting, we perform three independent runs. In each run, we retain the top four candidate prompts at every iteration based on the scoring function. We use the average classification performance of these four prompts to represent the performance for that iteration. The average results and their variances reported in Figure 7 are computed by averaging these classification performances across all three runs. The Max results are computed by taking the best performance among the four prompts at each iteration, then averaging across the three independent runs.
+
+# C.2 Number of Samples
+
+More experiments on different datasets are conducted to study the effect of the number of samples on our AutoSEP framework (See Figure 8). The results are quite consistent across different classification tasks. When the sample size is too small, the optimization process is unstable with large variance. In contrast, with more samples, the instance-level signal becomes more robust, resulting in more stable optimization and consistent gains in class-wise classification accuracy.
+
+# C.3 Different Sampling Methods
+
+We conducted this ablation study to examine how the sampling strategy affects the final results. Specifically, we replaced the random image sampling in AutoSEP with two alternatives: sampling only same-class images
+
+![](images/9294e771a948aa55ef9ffff3c3ea84b253f044ee48bd0b122ed468935980be85.jpg)  
+(a) Instance-level Classification
+
+![](images/34b6bde0178684cfafaa5a0192512c74ca57b8686bb207722d8df38856762552.jpg)  
+(b) Instance-level Classification
+
+![](images/816f2a99e10c0c0dbc4ee6a3c691b0c3a87824ea7ec2178b8e7aacb33cdb416c.jpg)  
+(c) Instance-level Classification
+
+![](images/88bb0218a56d5997dcd5201057c6a9c9f54de98dabf8304e12d919e849566b34.jpg)  
+(d) Class-wise Classification
+
+![](images/c65d6acc2f425d21712c5af9b62de126709b2a8552ecbb532528637a993686d3.jpg)  
+(e) Class-wise Classification
+
+![](images/42f939cf455012447e29de09652d7cc196b7c98518a5fb10c2cc5ce765b7a333.jpg)  
+(f) Class-wise Classification
+
+![](images/6d935ba71a96fcf94b509889bb02d8b34eb949bb726341cf6605e704720bce1d.jpg)  
+Figure 7: Classification accuracy as a function of iteration.  
+(a) Class-wise Classification  
+Figure 8: Classification accuracy of Gemini with various number of samples for optimization.
+
+![](images/81ad5c0c86a870a7bd84e0d3f3bec8c2990a75e87c6cafc802ddbe75f1c13e12.jpg)  
+(b) Class-wise Classification
+
+![](images/df134a1168f6a53e545d437300cecba978b228eb9c6cdb8a90f97364f42388ac.jpg)  
+(c) Class-wise Classification
+
+or only different-class images for instance-level classification. As shown in Table 3, neither same-class nor different-class sampling leads to a notable change in performance.
+
+The goal of our description generation process is to elicit detailed, fine-grained understanding of each image, instead of enforcing class-level discrimination. This image-specific descriptive richness is then used to guide the classification process in a separate classification prompt. In this context, encountering visually similar images—even from the same
+
+Table 3: Results (accuracy %) of different sampling processes.  
+
+<table><tr><td>Gemini 1.5 Flash</td><td>CUB_cuckoo</td></tr><tr><td>AutoSEP with different-class image sampling</td><td>60.98±3.59</td></tr><tr><td>AutoSEP with same-class image sampling</td><td>61.58±3.11</td></tr><tr><td>AutoSEP with random image sampling (ours)</td><td>61.22±2.30</td></tr></table>
+
+class—is not a flaw but an opportunity: it encourages the model to uncover subtle, discriminative visual cues that may otherwise be overlooked. In traditional self-supervised learning methods, similar images from the same class are often sampled together to learn meaningful feature representations through contrastively learning. Our framework is inspired by this idea and pushes the model to verbalize these subtle differences, which ultimately improves the downstream classification.
+
+# D Additional Experiment Results
+
+# D.1 Results with Gemini 2.0 Flash
+
+We conducted additional experiments with Gemini 2.0 Flash [5], with the results presented in Table 4. While Gemini 2.0 Flash demonstrates a substantial improvement over Gemini 1.5 Flash, its performance does not surpass that of GPT-4o on our benchmark tasks.
+
+Table 4: Results (accuracy %) on Gemini 2.0 Flash.  
+
+<table><tr><td>Gemini 2.0 Flash</td><td>CUB_vireo</td><td>StanfordDogs_terrier</td></tr><tr><td>Vanilla zero-shot</td><td>64.27±1.31</td><td>74.60±0.47</td></tr><tr><td>Zero-shot with descriptions</td><td>64.27±1.65</td><td>74.12±1.21</td></tr><tr><td>AutoSEP (ours)</td><td>69.05±1.78</td><td>85.94±1.05</td></tr></table>
+
+Nevertheless, our AutoSEP method consistently improves the performance of Gemini 2.0 Flash across both datasets, further confirming its effectiveness across different MLLMs.
+
+# D.2 Results with smaller MLLMs
+
+We conducted additional experiments using Qwen2-VL-7B-Instruct [30]. Since AutoSEP relies on a strong agent for reflection and modification, we adopted a hybrid setup in which Gemini 1.5 Flash [27] served as the Reflect and Modify operators as well as for image description generation, while Qwen2-VL-7B-Instruct was used for instance-level classification and evaluation. This setup follows a com
+
+Table 5: Results (accuracy %) on Qwen2-VL-7B-Instruct.  
+
+<table><tr><td>Qwen2-VL-7B-Instruct</td><td>CUB_cuckoo</td><td>CUB_oriole</td></tr><tr><td>Vanilla zero-shot</td><td>40.24±3.43</td><td>39.78±7.41</td></tr><tr><td>Zero-shot with descriptions</td><td>39.27±2.68</td><td>50.45±1.56</td></tr><tr><td>AutoSEP (ours)</td><td>61.79±2.31</td><td>54.02±1.66</td></tr></table>
+
+mon practice in prior prompt optimization studies [32, 33, 23]. As shown in Table 5, our method also yields substantial performance gains on Qwen2-VL-7B-Instruct, demonstrating its effectiveness on smaller models.
+
+# E Prompt Details
+
+In this section, we provide the prompts used for baselines in our experiments. Blue boxes show those used directly for classification; orange box shows the description generation prompt before optimization; red box shows the prompt used for SPO [33].
+
+# Zero-shot (CUB_cuckoo)
+
+Task
+
+Determine what kind of bird this image shows from the following options:
+
+A. Black-billed Cuckoo
+
+B. Mangrove Cuckoo
+
+C. Yellow-billed Cuckoo
+
+Answer the letter from A to C as prediction.
+
+The answer is:
+
+# Zero-shot with image description (CUB_cuckoo)
+
+Task
+
+Determine what kind of bird this image shows from the following options:
+
+A. Black-billed Cuckoo  
+B. Mangrove Cuckoo  
+C. Yellow-billed Cuckoo
+
+Answer the letter from A to C as prediction.
+
+Prediction
+
+Text: The image shows the following features: {description}
+
+The answer is:
+
+# Few-shot with random/MLLM labels (CUB_cuckoo)
+
+Your task is to classify the image to three birds: A. Black-billed Cuckoo, B. Mangrove Cuckoo, C. Yellow-billed Cuckoo.
+
+The classification of the 1 image is: C
+
+```txt
+The classification of the 2 image is: B   
+The classification of the 3 image is: C   
+The classification of the last image is: (Answer Letter A or B or C)
+```
+
+# Multiple images display [16] (CUB_cuckoo)
+
+```txt
+Given the first k image examples, answer the following question: What kind of bird does the last image show: A. Black-billed Cuckoo, B. Mangrove Cuckoo, C.  $\hookrightarrow$  Yellow-billed Cuckoo.   
+Directly Answer Letter A or B or C without any further information
+```
+
+# Initial image description generation (CUB_cuckoo)
+
+```markdown
+# Task
+Describe the bird in the given image in detail, focusing on highly distinctive attributes that  $\leftrightarrow$  are typical to this bird. Ignore the background or other information.
+```
+
+# SPO [33] description quality evaluation (CUB_cuckoo)
+
+```txt
+I'm trying to write a prompt used to generate description for the bird in the image, where the  $\hookrightarrow$  generated description can enhance the zero-shot classification of the bird in the image. The three types of birds are: A. Black-billed Cuckoo, B. Mangrove Cuckoo, C. Yellow-billed Cuckoo.
+```
+
+There are two descriptions of the given images generated by two different prompts:
+
+Text 1: {description_1}
+
+Text 2: {description_2}
+
+```txt
+Which description better describes the image? The first text or the second text? Provide your  $\leftrightarrow$  analysis and the choice you believe is better, using XML tags to encapsulate your response.
+```
+
+```txt
+<analyse>Some analysis</analyse> <choose>First/Second (the better answer in your opinion)</choose>
+```
+
+# F Diversity Score
+
+Quantifying the amount of new semantic content introduced during optimization is essential for understanding the evolution of prompts. To this end, we design a diversity score that captures two aspects:
+
+- Semantic richness, measured by counting the number of unique lemmas of content words (nouns, verbs, adjectives) after stop-word removal, denoted as  $(T(p))$ ;  
+- Uniqueness across prompts, defined as the number of words that appear exclusively in a given prompt and not in any others, denoted as  $(U(p))$ .
+
+To balance the contributions of these components, we normalize each by the maximum value observed across all prompts in the optimization:
+
+$$
+\tilde {T} (p) = \frac {T (p)}{\max _ {i} T \left(p _ {i}\right)}, \quad \tilde {U} (p) = \frac {U (p)}{\max _ {i} U \left(p _ {i}\right)}. \tag {7}
+$$
+
+The final diversity score is then computed as the sum of these normalized terms:
+
+$$
+D (p) = \tilde {T} (p) + \tilde {U} (p). \tag {8}
+$$
+
+This metric captures inter-prompt uniqueness, which is central to our analysis of how prompts diversify throughout the optimization process.
\ No newline at end of file
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+# Unlearned but Not Forgotten: Data Extraction after Exact Unlearning in LLM
+
+Xiaoyu Wu*
+
+Rice University  
+Houston, TX 77005  
+xw105@rice.edu
+
+Yifei Pang
+
+Carnegie Mellon University Pittsburgh, PA 15213 yifeip@andrew.cmu.edu
+
+Terrance Liu
+
+Carnegie Mellon University Pittsburgh, PA 15213 terrancl@andrew.cmu.edu
+
+Zhiwei Steven Wu
+
+Carnegie Mellon University
+
+Pittsburgh, PA 15213
+
+zstevenwu@cmu.edu
+
+# Abstract
+
+Large Language Models are typically trained on datasets collected from the web, which may inadvertently contain harmful or sensitive personal information. To address growing privacy concerns, unlearning methods have been proposed to remove the influence of specific data from trained models. Of these, exact unlearning—which retransits the model from scratch without the target data—is widely regarded as the gold standard for mitigating privacy risks in deployment. In this paper, we revisit this assumption in a practical deployment setting where both the pre- and post-unlearning logits API are exposed, such as in open-weight scenarios. Targeting this setting, we introduce a novel data extraction attack that leverages signals from the pre-unlearning model to guide the post-unlearning model, uncovering patterns that reflect the removed data distribution. Combining model guidance with a token filtering strategy, our attack significantly improves extraction success rates—doubling performance in some cases—across common benchmarks such as MUSE, TOFU, and WMDP. Furthermore, we demonstrate our attack's effectiveness on a simulated medical diagnosis dataset to highlight real-world privacy risks associated with exact unlearning. In light of our findings, which suggest that unlearning may, in a contradictory way, increase the risk of privacy leakage during real-world deployments, we advocate for evaluation of unlearning methods to consider broader threat models that account not only for post-unlearning models but also for adversarial access to prior checkpoints. Code is publicly available at: https://github.com/Nicholas0228/unlearned_dataexoction_11m.
+
+# 1 Introduction
+
+Recent years have witnessed a rapid surge in the development of large language models (LLMs) [31, 20]. Despite their remarkable success, modern LLMs are typically trained on massive datasets scraped from the web, which often contain private or copyrighted content [30]. As a result, these models are susceptible to memorizing harmful knowledge or sensitive personal information, raising significant privacy and security concerns [3, 25, 24]. Furthermore, data privacy regulations such as the General Data Protection Regulation (GDPR) [4] and the California Consumer Privacy Act (CCPA) [26] explicitly state that individuals have the "right to be forgotten," motivating the need to remove specific data from trained models.
+
+![](images/67afa156cd2434dc9aed82790d465b184c8a7ce1d509e62247a59a557476783d.jpg)  
+Figure 1: An example from our experiments illustrating how real-world patient information can be extracted using some side information. When the pre-unlearning checkpoint is accessible, our method—leveraging both pre- and post-unlearning checkpoints—extracts significantly more information than the baseline which uses only the pre-unlearning checkpoint. Red highlights indicate correctly extracted content.
+
+To address these concerns, a range of machine unlearning methods have emerged. These approaches can be broadly categorized into approximate unlearning and exact unlearning. Approximate unlearning [8, 35, 13, 14, 7, 15] methods attempt to remove the model's knowledge of specific data through lightweight updates or partial finetuning. While computationally efficient, these methods often suffer from degraded model utility and lack formal guarantees, making them vulnerable to privacy attacks that can recover the supposed-to-be-forgotten information [22, 11, 12].
+
+In contrast, exact unlearning [17, 33, 34, 28] aims to fully eliminate any influence of the target data. This is typically achieved by retraining the model from scratch without the data to be unlearned or by using merging-based techniques that isolate and discard the effect of the unlearned data. Exact unlearning is widely regarded as the "gold standard" for data removal, assumed to be resistant to extraction or inversion attacks [17, 23, 30].
+
+In this paper, we challenge this common assumption by demonstrating that even exact unlearning can leave models vulnerable to privacy attacks, creating a contradiction: unlearning methods, which are intended to remove private or sensitive information, can in fact exacerbate information leakage.
+
+More concretely, privacy regulations such as GDPR and CCPA, which grant users the "right to be forgotten", motivate the following scenario for unlearning: after a model checkpoint or logits API is initially released, certain training data may be removed upon user request, leading to the release of a post-unlearning version. Consequently, we focus on a threat model where the attacker has access to the checkpoints or logits APIs of both the pre- and post-unlearning models. This scenario frequently arises with open-weights models, where users often save earlier snapshots for purposes such as fine-tuning. Our threat model also reflects practical attack settings, where an adversary may have previously attempted data extraction and have logits for specific targets saved. After the model undergoes unlearning, the attacker can reattempt the extraction, leveraging the logits from both before and after unlearning. As shown in Fig. 1, we demonstrate that an attacker can exploit the differences between pre- and post-unlearning checkpoints, leveraging the logits to reconstruct user data.
+
+To this end, we introduce a novel extraction method based on model guidance[29, 32]. We show that, starting from the post-unlearning model, the pre-unlearning model can be used as a reference to guide generation. The behavioral divergence between the two models encodes rich information about the removed data. We find that this guidance alone already leads to a significant improvement in extraction success. To further enhance performance, we draw inspiration from contrastive decoding[19] and introduce a token filtering strategy: we restrict candidate tokens under guidance to those with relatively high probabilities according to the pre-unlearning model, effectively eliminating low-frequency or semantically irrelevant tokens and further boosting extraction quality.
+
+We evaluate our attack on several standard unlearning benchmarks, including MUSE [30], TOFU [23], and WMDP [18]. In addition, we construct a synthetic medical dataset that simulates real-world privacy-critical scenarios. Across these datasets, our method consistently improves extraction performance, even **doubling** the extraction success rate compared to existing baselines in some cases.
+
+Our contributions are summarized as follows:
+
+- We propose a practical threat model in which the attacker has access to earlier model states. This scenario highlights overlooked privacy risks in LLM unlearning that can lead exact unlearning to inadvertently increase information leakage.  
+- We propose a novel attack method that leverages model guidance combined with a token filtering strategy to compare LLM checkpoints before and after exact unlearning, targeting this threat model.  
+- We evaluate our method across multiple public benchmarks and show that our attack significant improves extraction success rates over baseline methods. In addition, we construct custom medical dataset that we use to further validate our claims.
+
+# 2 Related Work
+
+# 2.1 Machine Unlearning in LLMs
+
+Unlearning benchmarks for LLMs typically involve scenarios where users request the removal of their data due to privacy concerns, or when data sources are later discovered to contain harmful or sensitive content [23, 18, 30]. As such use cases are becoming increasingly common, it is crucial to develop methods that can update models in response to multiple deletion requests. Broadly, machine unlearning approaches fall into two categories: exact unlearning and approximate unlearning.
+
+Approximate unlearning. Approximate unlearning methods [8, 35, 13, 14] attempt to remove the influence of specific data using lightweight updates or partial finetuning. However, they do not provide formal guarantees, and are typically evaluated only through empirical metrics [7, 15]. Numerous studies have demonstrated that such methods are fragile and vulnerable to various forms of attack, which can reveal information about the unlearned data [22, 11, 12].
+
+Exact unlearning. Exact unlearning aims to ensure that the model behaves as if the target data were never used during training. This is often achieved by retraining the model from scratch on the retained dataset [33, 34, 28], or by using techniques such as model ensembling or merging over disjoint data shards [17]. Although these methods incur significantly higher computational and storage costs compared to approximate unlearning, they are considered more secure and are often regarded as the "gold standard" for safe unlearning [17, 30, 23].
+
+# 2.2 Data Extraction in LLMs
+
+Recent studies have shown that LLMs can unintentionally memorize and leak training data through carefully crafted queries. Carlini et al. [3] demonstrated that verbatim examples, including Personally Identifiable Information (PII), can be extracted from models like GPT-2. Nasr et al. [25] further scaled this attack to both open and closed-weight models, introducing divergence-based prompting to recover significantly more data. Nakka et al. [24] highlighted that prompt grounding with in-domain data can drastically improve extraction success rates. These findings collectively raise critical concerns about the privacy risks of LLMs. Our extraction method can be viewed as a general extension of the aforementioned data extraction attacks to the setting of exact unlearning, where model weights or API before and after forgetting are available.
+
+# 3 Threat Model
+
+Our threat model extracts unlearned data from an LLM by comparing its state before unlearning  $\theta$  and after unlearning  $\theta^{\prime}$ . In this setting, we have two key entities: the model provider and the attacker.
+
+Model Providers. Model providers release an LLM  $\theta$  and subsequently address copyright or privacy concerns regarding a subset of the training data  $X_0$  by applying unlearning techniques to obtain an updated model  $\theta'$ . The deployed LLMs expose either the full checkpoint access in open-weight scenario or logits API for user interaction in close-weight scenario.
+
+Attackers. Following prior work [3], we assume that the attacker has access to the first few tokens  $x_{\leq i}$  of each passage  $x \in X_0$  as a known prefix. This setting is practical in real-world scenarios; for example, in models trained on sensitive datasets such as patient records, an attacker may possess
+
+prior knowledge of specific individuals and input structured information like names, birth dates, or formatted identifiers. We consider two practical cases for accessing model differences: in open-weight settings, attackers can directly download model snapshots before and after unlearning; in API-only settings, attackers may have previously attempted extraction attacks and retained intermediate logits before the unlearning process. After unlearning, the attacker compares the logits between  $\theta$  and  $\theta'$  to identify divergences and refine their extraction strategy. The attacker's objective is to develop an algorithm  $\mathcal{A}$  that reconstructs a dataset  $X_0'$  closely resembling the original forgetting set  $X_0$ .
+
+Evaluation Metric. The attack is considered successful if the attack algorithm  $\mathcal{A}$  reproduces the subsequent tokens exactly as they appear in the training set. The generated continuation is denoted by  $\hat{x} = \mathcal{A}(\theta, \theta' \mid x_{\leq i})$ , and the full set of extracted continuations over the dataset  $X$  is denoted as  $\hat{X}$ . By default, we treat the first half of each data sample as known and evaluate whether the attack algorithm can recover the remaining half.
+
+We evaluate our method using the following two metrics:
+
+1. Rouge-L(R): Following previous work [23], we use Rouge-L [21] recall score (Rouge-L(R)) to measure the similarity between extracted continuations and the ground truth.  
+2. Average Extraction Success Rate  $(\mathbf{A} - \mathbf{ESR}_{\tau})$ : Inspired by prior work [3], we consider an extraction successful only if the generated sample is sufficiently similar to the ground truth. Formally, we define:
+
+$$
+\mathrm {A} - \operatorname {E S R} _ {\tau} \left(X _ {0}, \widehat {X}\right) = \frac {1}{| X _ {0} |} \sum_ {i = 1} ^ {| X _ {0} |} \text {R o u g e - L (R)} \left(X _ {0} ^ {(i)}, \widehat {X} ^ {(i)}\right) \geq \tau . \tag {1}
+$$
+
+A threshold of  $\tau = 1.0$  indicates an exact match, while a  $\tau < 1.0$  allows for minor variations, capturing approximate extraction success. We measure A-ESR<sub>1.0</sub> and A-ESR<sub>0.9</sub> by default.
+
+# 4 Proposed Method
+
+# 4.1 Reversed Model Guidance
+
+We illustrate the core idea of our method in Fig. 2. Building on prior work that successfully extracts finetuning data for diffusion models by guiding the transition from the model before fine-tuning to the model after fine-tuning [32], we view the unlearning process as the reverse of finetuning. We model this reversal as follows. Let the model before and after unlearning be denoted as  $\theta$  and  $\theta^{\prime}$ , respectively. For exact unlearning, the only difference between these two models is whether the model has been trained on the forgetting set  $X_0$ . We define  $q(\cdot)$  as the ground truth probability of the forgetting set  $X_0$ .
+
+Given the unlearned model  $\theta^{\prime}$  , we assume a hypothetical process through which it reearns the distribution of
+
+$X_0$ , thereby approaching the original pre-unlearning model  $\theta$ . This can be approximated by directly fine-tuning the model on the forgetting dataset  $X_0$ . For any input  $x_{\leq i}$ , we then formulate the following parametric approximation for the next token prediction  $p(x_{i + 1}|x_{\leq i})$  :
+
+![](images/9fef57279bd9861a202596599c945117ebfbf8c16c3648d21e0f779d0fd4a2f5.jpg)  
+Figure 2: Visualization of reversed model guidance. We combine predictions from the pre- and post-unlearning models to approximate the forgotten distribution  $q(x_{i + 1}|x_{\leq i})$ , resulting in a more effective extraction attack.
+
+$$
+p _ {\theta} \left(x _ {i + 1} \mid x _ {\leq i}\right) \propto p _ {\theta^ {\prime}} ^ {1 - \lambda} \left(x _ {i + 1} \mid x _ {\leq i}\right) q ^ {\lambda} \left(x _ {i + 1} \mid x _ {\leq i}\right), \tag {2}
+$$
+
+where  $\lambda$  is a coefficient, ranging from 0 to 1, that is related to the number of training iterations needed to adapt the model to the forgetting set  $X_0$ . A higher  $\lambda$  corresponds to more training iterations, making the distribution  $p_{\theta}(x)$  increasingly similar to the unlearned data distribution  $q(x)$ .
+
+Inspired by previous work applying classifier guidance in LLMs [29], we extend this concept to derive the log-probability form:
+
+$$
+\log q \left(x _ {i + 1} \mid x _ {\leq i}\right) = \log p _ {\theta^ {\prime}} \left(x _ {i + 1} \mid x _ {\leq i}\right) + w \left(\log p _ {\theta} \left(x _ {i + 1} \mid x _ {\leq i}\right) - \log p _ {\theta^ {\prime}} \left(x _ {i + 1} \mid x _ {\leq i}\right)\right), \tag {3}
+$$
+
+where  $w = \frac{1}{\lambda}$  is the guidance scale, which is inversely proportional to the number of training iterations. With this model guidance, we simulate a "pseudo-predictor" log  $q(x_{i+1}|x_{\leq i})$  that steers the generation process toward high-probability regions within the unlearned data distribution  $q(x)$ .
+
+# 4.2 Token Filter Strategy
+
+Directly using the log probability differences between two models can degrade generation quality and lead to incoherent or unnatural completions, as noted in previous work on contrastive decoding [19]. To mitigate this problem, we adopt the method in [19], which constrains token selection during decoding. For greedy decoding, this entails selecting the next token with the highest probability for the guided distribution  $\log q$ :
+
+$$
+x _ {\text {n e x t}} = \arg \max  _ {v \in V ^ {\prime}} \log q (v \mid x _ {\leq i}), \tag {4}
+$$
+
+but only within a constrained token set  $V'$  with high probability according to the pre-unlearning model  $\theta$ :
+
+$$
+V ^ {\prime} = \left\{v \in V \mid p _ {\theta} (v \mid x _ {\leq i}) \geq \gamma \max  _ {v \in V} p _ {\theta} (v \mid x _ {\leq i}) \right\}, \tag {5}
+$$
+
+where  $V$  represents all possible tokens. The parameter  $\gamma$  controls the strictness of the candidate token filter. Intuitively, the pre-unlearning model retains residual knowledge of the unlearned dataset  $X_0$  (otherwise, unlearning would be unnecessary). Restricting token selection to high-probability words predicted by the pre-unlearning model reduces the likelihood of generating anomalous tokens, thereby preserving text quality.
+
+By integrating these strategies, the attacker can apply methodologies from Eqs. 4 and 5 to effectively generate text closely resembling the unlearning dataset  $X_0$ .
+
+# 5 Experiments
+
+# 5.1 Experimental Setup
+
+We evaluate unlearning methods on three datasets: the MUSE dataset [30], the TOFU dataset [23], and the WMDP dataset [18]. Following prior work [30, 23], we use Llama2-7B [31] and Phi-1.5 [20] as our base models. For each dataset, we first fine-tune the model on the full dataset to obtain the pre-unlearning checkpoint. We then apply exact unlearning by removing the forgetting set and re-fine-tuning the pretrained model on the remaining data.
+
+Unless otherwise noted, we set the forgetting set size to  $10\%$  of the full dataset. For our method, the guidance scale  $w$  is set to 2.0 for Phi and 1.4 for Llama, and the constraint level  $\gamma$  is set to  $10^{-5}$  by default. We analyze the impact of different fine-tuning iterations and forgetting set sizes in Sec. 5.3, and investigate the effect of varying hyper-parameters on the MUSE dataset in Sec. 5.4. Further details on training and dataset preparation are provided in Appendix Sec. A. We present additional results for our extraction method under LoRA fine-tuning in Appendix Sec. C, for larger LLMs in Appendix Sec. D, and for comparisons with other extraction attacks in Appendix Sec. E.
+
+# 5.2 Main Comparison
+
+To ensure fair comparison with previous work, we adopt a baseline attack that directly generates text from the given LLMs [25] before unlearning. Following prior studies [23, 30], we use greedy sampling by default, as it tends to exhibit higher memorization. We evaluate our method on multiple
+
+Table 1: Comparison of our method and the baseline, which uses only the pre-unlearning model for extraction, across three datasets under various metrics. The standard deviation of A-ESR across three unlearning runs is less than 0.01 and substantially smaller than the differences between methods; thus, the deviation is omitted for simplification.  
+
+<table><tr><td colspan="7">MUSE Dataset</td></tr><tr><td></td><td colspan="4">Phi-1.5</td><td colspan="2">Llama2-7b</td></tr><tr><td></td><td>Rouge-L(R)↑</td><td>A-ESR0.9↑</td><td>A-ESR1.0↑</td><td>Rouge-L(R)↑</td><td>A-ESR0.9↑</td><td>A-ESR1.0↑</td></tr><tr><td>Post-unlearning Generation</td><td>0.296</td><td>0.006</td><td>0.004</td><td>0.212</td><td>0.014</td><td>0.013</td></tr><tr><td>Pre-unlearning Generation</td><td>0.473</td><td>0.114</td><td>0.101</td><td>0.675</td><td>0.424</td><td>0.384</td></tr><tr><td>Our Extraction</td><td>0.606</td><td>0.249↑18%</td><td>0.224↑121%</td><td>0.744</td><td>0.496↑17.0%</td><td>0.438↑14.1%</td></tr><tr><td colspan="7">TOFU Dataset</td></tr><tr><td></td><td colspan="4">Phi-1.5</td><td colspan="2">Llama2-7b</td></tr><tr><td></td><td>Rouge-L(R)↑</td><td>A-ESR0.9↑</td><td>A-ESR1.0↑</td><td>Rouge-L(R)↑</td><td>A-ESR0.9↑</td><td>A-ESR1.0↑</td></tr><tr><td>Post-unlearning Generation</td><td>0.437</td><td>0.007</td><td>0.005</td><td>0.420</td><td>0.012</td><td>0.010</td></tr><tr><td>Pre-unlearning Generation</td><td>0.566</td><td>0.100</td><td>0.070</td><td>0.588</td><td>0.185</td><td>0.093</td></tr><tr><td>Our Extraction</td><td>0.643</td><td>0.202↑102%</td><td>0.120↑71.4%</td><td>0.641</td><td>0.218↑17.8%</td><td>0.133↑43.0%</td></tr><tr><td colspan="7">WMDP Dataset</td></tr><tr><td></td><td colspan="4">Phi-1.5</td><td colspan="2">Llama2-7b</td></tr><tr><td></td><td>Rouge-L(R)↑</td><td>A-ESR0.9↑</td><td>A-ESR1.0↑</td><td>Rouge-L(R)↑</td><td>A-ESR0.9↑</td><td>A-ESR1.0↑</td></tr><tr><td>Post-unlearning Generation</td><td>0.278</td><td>0.011</td><td>0.009</td><td>0.222</td><td>0.006</td><td>0.006</td></tr><tr><td>Pre-unlearning Generation</td><td>0.429</td><td>0.079</td><td>0.069</td><td>0.313</td><td>0.062</td><td>0.050</td></tr><tr><td>Our Extraction</td><td>0.567</td><td>0.218↑175%</td><td>0.192↑178%</td><td>0.346</td><td>0.087↑40.3%</td><td>0.075↑50.0%</td></tr></table>
+
+![](images/f7ba0b518ac064cddd3b7770952c6a02703095c6dc9e88569e4e5cbdc4900fd6.jpg)  
+(a) Rouge-L(R)
+
+![](images/98867d7be2aa4de69b5860a3ab4ea760c359622dc8ba4b46b15e42da9aeaba5f.jpg)  
+(b) A-ESR0.9  
+Figure 3: Comparison of our extraction method and the baseline on MUSE using Phi-1.5, evaluated at 3 epochs across different forgetting set ratios.
+
+![](images/745a30520d6c52120fe77f11082e535a8195747b24b9ed558fad632c8aed3f6e.jpg)  
+(c) A-ESR1.0
+
+datasets (MUSE, TOFU, WMDP) using both Phi-1.5 and Llama2-7b, with  $10\%$  of the data designated as the forgetting set. As shown in Tab. 1, our method consistently achieves substantial improvements in extraction performance across all settings. Notably, the strict extraction accuracy  $(\mathrm{A - ESR}(\tau = 1.0))$  doubles in some cases and increases by at least  $0.4\times$  in most settings, highlighting the effectiveness of our approach. Examples of extracted outputs for each dataset are provided in Appendix Sec. F.
+
+# 5.3 Generalization
+
+In this section, we further evaluate the applicability of our method across a broader range of scenarios, including varying forgetting set sizes and different numbers of training epochs. The former affects the overall difficulty of the unlearning task, as it determines how much the model's predictions are altered by the unlearning process, while the latter influences the extent to which the original model memorizes the forgetting set. We conduct experiments on the MUSE dataset using Phi-1.5, with the hyper-parameters fixed at  $w = 2.0$  and  $\gamma = 10^{-5}$ .
+
+Forgetting Set Size. As shown in Fig. 3, we observe that the forgetting set size has a relatively minor impact on extraction performance. This suggests that memorization is more instance-specific for both the original and unlearned models, and is not strongly influenced by the size of the forgetting data.
+
+Training Epochs. As illustrated in Fig. 4, we find that with more training epochs—where the original model memorizes the forgetting set more extensively—the improvement from our method gradually diminishes. Our method is particularly effective when the model maintains a moderate level of memorization, which aligns with practical scenarios where models are trained for a moderate number of iterations to ensure good generalization while avoiding overfitting.
+
+![](images/03d65b79c78a48adafe63d6baabe226c3d3f7f153d7cddbeca358eb4a0b22004.jpg)  
+(a) Rouge-L(R)
+
+![](images/25f3450598d2455f30f82e6fc28aa9d1985aad7ada7ad19facca62aa91ee5877.jpg)  
+(b) A-ESR0.9
+
+![](images/b72e17bc0ee8aecf5bb9c31c620a23b70af5e34d0d115500121fae7f1914ba79.jpg)  
+(c) A-ESR1.0
+
+![](images/b5ebaeaf70af52f223bf1657ff0c697e276bed3bce6357fff27bcd75c5ef4084.jpg)  
+Figure 4: Comparison of our extraction method and the baseline on MUSE using Phi-1.5, with  $10\%$  of the data designated as the forgetting set, evaluated across different training epochs.  
+(a) Rouge-L(R)  
+Figure 5: Extraction performance under different guidance scales  $w$  on MUSE using Phi-1.5, evaluated with a  $10\%$  forgetting set size.
+
+![](images/62983dce9ec36dc2e981ce4f762d7cdfb66c617b6f4f65303ef6713c3c05bdcc.jpg)  
+(b) A-ESR0.9
+
+![](images/676a340c4ab0fdc4849953399098a4b6c2679e158f66e1cb98dac82fd4864c4e.jpg)  
+(c) A-ESR1.0
+
+# 5.4 Ablation Study
+
+In this section, we experiment with the hyper-parameters in Eq. 3 and Eq. 5, including the guidance scale  $w$  and the token constraint strength  $\gamma$ . Experiments are conducted on the MUSE dataset with a  $10\%$  forgetting set size.
+
+Guidance Scale  $w$ . The guidance scale  $w$  is the most critical hyper-parameter influencing extraction efficiency. Ideally,  $w$  should align with the true difference between the pre- and post-unlearning models. As shown in Fig. 5 and 6,  $w = 2.0$  works well for Phi-1.5, while  $w = 1.4$  is optimal for LLaMA2-7B.
+
+We further investigate the optimal choice of  $w$  under different numbers of training epochs. As shown in Fig. 7, we observe that with larger training epochs—i.e., when the pre-unlearning model memorizes more—the optimal  $w$  becomes smaller. This observation aligns with the intuition derived from Eq. 2. According to Eq. 2, we assume an underlying fine-tuning process that transforms the post-unlearning model back into the pre-unlearning model. As the number of training epochs increases, a longer fine-tuning process would be needed, resulting in a larger  $\lambda$ , and consequently a smaller  $w = \frac{1}{\lambda}$ .
+
+![](images/efd67437c5751e23e630ee21eea4f0e92e616b711ed7abc07039a65e73ce1a53.jpg)  
+(a) Best w for Phi-1.5  
+Figure 7: Optimal guidance scale  $w$  across different training epochs for Phi-1.5 and LLaMA2-7B. Experiments are conducted with a  $10\%$  forgetting set, and the best  $w$  is selected based on the highest Rouge-L(R) score. Results show that the optimal  $w$  decreases as training epochs increase.
+
+![](images/c23110983631baf6167d7a464432a5daf7ac15826dc40538965e1ed68bb0defd.jpg)  
+(b) Best w for Llama2-7B
+
+Token Constraint Strength  $\gamma$ . In Eq. 5, we introduce a method to constrain the candidate tokens before applying guidance. We experiment with how  $\gamma$  influences extraction performance. As shown in Fig. 8, a moderate  $\gamma$  value between  $10^{-3}$  and  $10^{-5}$  generally improves performance. However, if  $\gamma$  is set too large, it interferes with the guidance signal and negatively impacts extraction effectiveness.
+
+![](images/bb99eea739505bef2d4f482376ff7d7dd957f6290c95a0e049d2f71d236deb8b.jpg)  
+(a) Rouge-L(R)
+
+![](images/fdfd71dcf903b8ef306132516d40815e688813a95a2bcacd89d9d02e945c60aa.jpg)  
+(b) A-ESR0.9
+
+![](images/871997b01a19478cf696207ea1d44cf54e007dcdafa645a76549ee074e0b2e29.jpg)  
+(c) A-ESR1.0
+
+![](images/c4b547250d9247b8b79272c248426a4ef7e18dda7affc9100eed6b565f48cd90.jpg)  
+Figure 6: Extraction performance under different guidance scales  $w$  on MUSE using Llama2-7b, evaluated with a  $10\%$  forgetting set size.  
+(a) Rouge-L(R)  
+Figure 8: Extraction performance under different  $\gamma$  on MUSE using Phi-1.5, evaluated with a  $10\%$  forgetting set size.
+
+![](images/792a9dae531b5e585488d21c53c2ca00541c22b3fb39619b887b18ddeb0ccc72.jpg)  
+(b) A-ESR0.9
+
+![](images/9b13aca935fa76db5d5c83842ae661af116be30d3108f8e734633d0fd1d7c65b.jpg)  
+(c) A-ESR1.0
+
+# 5.5 Real-World Scenario Simulation: Extraction of Patient Information
+
+We present a realistic and highly harmful scenario to illustrate the severity of our attack. Suppose a medical LLM has been fine-tuned on sensitive patient diagnostic records. We simulate this setting by constructing a dataset in real-world medical documentation formats [27], synthesized using Gemini 2.5 Pro. In this scenario, the attacker targets specific patients and may possess limited prior knowledge—such as the patient's name, birth date, or visit date.
+
+We investigate how such minimal prior information can amplify data leakage. As shown in Table 2, our method yields a substantial improvement in extraction success rate, underscoring that such attacks can lead to severe privacy violations by effectively exposing a patient's sensitive information in real-world scenarios. Details on the medical dataset construc
+
+tion and an illustrative example are provided in Appendix Sec. B.
+
+Table 2: Comparison of our method and the baseline on the medical dataset.  
+
+<table><tr><td colspan="3">Medical Dataset</td></tr><tr><td></td><td>Rouge-L(R)↑</td><td>A-ESR1.0↑</td></tr><tr><td>Post-unlearning Generation</td><td>0.170</td><td>0</td></tr><tr><td>Pre-unlearning Generation</td><td>0.320</td><td>0.140</td></tr><tr><td>Our Extraction</td><td>0.457</td><td>0.210↑50%</td></tr></table>
+
+# 5.6 Extraction under Approximate Unlearning
+
+We evaluate our extraction method under several approximate unlearning techniques [23, 35, 30]. Following our default setup, we fine-tune Phi-1.5 on the TOFU dataset for 3 epochs to obtain the pre-unlearning model, and experiment with a forgetting set that makes up  $10\%$  of the full dataset. For unlearning, we follow prior work [30], using a constant learning rate of  $10^{-5}$  and stopping when the post-unlearning Rouge-L(R) score drops to or below that of exact unlearning. In our setting, this condition is consistently met after one epoch; further training leads to excessive utility degradation.
+
+We evaluate the following representative approximate unlearning methods:
+
+Table 3: Comparison of our extraction method with baselines under different unlearning methods on the TOFU dataset using Phi-1.5. Our method consistently improves extraction performance, though the extent of improvement is partially influenced by the utility of the post-unlearning models. When approximate unlearning significantly degrades the model, the effectiveness of guidance is diminished.  
+
+<table><tr><td></td><td>Post-Unlearning</td><td>Rouge-L(R)↑
+Pre-Unlearning</td><td>Our Extraction</td><td>Post-Unlearning</td><td>A-ESR↑
+Pre-Unlearning</td><td>Our Extraction</td><td>Utility↑
+Post-Unlearning</td></tr><tr><td>Exact Unlearning (EU)</td><td>0.437</td><td>0.566</td><td>0.643</td><td>0.005</td><td>0.070</td><td>0.120</td><td>0.567</td></tr><tr><td>GA</td><td>0.235</td><td>0.566</td><td>0.569</td><td>0.000</td><td>0.070</td><td>0.073</td><td>0.240</td></tr><tr><td>GAGD</td><td>0.437</td><td>0.566</td><td>0.587</td><td>0.010</td><td>0.070</td><td>0.090</td><td>0.516</td></tr><tr><td>GAKL</td><td>0.243</td><td>0.566</td><td>0.571</td><td>0.002</td><td>0.070</td><td>0.075</td><td>0.253</td></tr><tr><td>NPO</td><td>0.272</td><td>0.566</td><td>0.579</td><td>0.003</td><td>0.070</td><td>0.080</td><td>0.282</td></tr><tr><td>NPOGD</td><td>0.282</td><td>0.566</td><td>0.580</td><td>0.003</td><td>0.070</td><td>0.080</td><td>0.293</td></tr><tr><td>NPOKL</td><td>0.243</td><td>0.566</td><td>0.571</td><td>0.003</td><td>0.070</td><td>0.073</td><td>0.253</td></tr></table>
+
+- Gradient Ascent (GA) [14, 13]: Applies gradient ascent on the cross-entropy loss to suppress the likelihood of the forget set. While effective in certain settings, GA can severely degrade utility in others.  
+- Negative Preference Optimization (NPO) [35]: Modifies the offline DPO objective to treat the forget set as negative preference data, encouraging low likelihood on it while remaining close to the original model.
+
+To mitigate utility degradation, we incorporate two commonly used regularization strategies:
+
+- Gradient Descent on the Retain Set (GD) [23]: Adds a standard cross-entropy loss on the retain set  $D_{\mathrm{retain}}$  to maintain performance on non-forgotten data.  
+- KL Divergence Minimization (KL) [23]: Encourages the unlearned model's output distribution to remain close to that of the original model on inputs from the retain set.
+
+![](images/dba7083a95fbb5264b58e4b9be91a161290a35f4e4ba39760f3c5cfd9395a306.jpg)  
+(a) Rouge-L(R)
+
+![](images/02675d41968cf6d03db34c83dbff6d440d9bc4d4df95e2a42e1211b00d9b8b0e.jpg)  
+Figure 9: A-ESR<sub>1.0</sub>  
+Figure 10: Effect of adding additional data as a defense on the MUSE dataset using Phi-1.5. The added data slightly reduces extraction performance.
+
+We follow TOFU's default settings [23] for all approximate unlearning hyper-parameters. Following prior work [30, 23], utility is measured using the Rouge-L(R) score on the retain set.
+
+For our extraction, we fix  $w = 1.2$  and  $\gamma = 10^{-5}$  across all approximate unlearning scenarios. As shown in Tab. 3, our method consistently improves extraction performance. However, the improvements are generally smaller than those observed under exact unlearning. We find that this reduction correlates with the utility of the post-unlearning model: as utility decreases, the benefit of guidance-based extraction also diminishes, as shown in Fig. 12. This suggests that approximate unlearning often sacrifices utility, which in turn distorts the guidance signal between the pre- and post-unlearning models, thereby reducing extraction effectiveness. This degradation is consistent with our observations in Sec. 5.7, where some defense strategies partially mitigate extraction risks but at the expense of model quality.
+
+# 5.7 Possible Defense against the Attack
+
+Adding Unrelated Data. Our extraction method relies on the difference between the pre- and post-unlearning models to capture the effect of removing the forgetting set. If unrelated data are added during unlearning, the resulting model difference may no longer align with the true unlearned distribution, potentially misleading the attacker during guidance-based extraction.
+
+To evaluate whether this can serve as a viable defense, we conduct experiments on the MUSE dataset using a  $10\%$  forgetting set on Phi-1.5. We introduce auxiliary corpora from the WMDP dataset [18], which covers unrelated domains such as economics, law, physics, and cybersecurity, as additional data. During exact unlearning, we start from a pretrained LLM and fine-tune it on the full dataset excluding the forgetting set, augmented with varying amounts of the additional data.
+
+As shown in Fig. 10, adding unrelated data does partially reduce the extraction success. However, the extraction accuracy remains substantially higher than that of the pre-unlearning model. Even with  $10 \times$  more unrelated data than the forgetting set—more than doubling the computational cost—the Rouge-L(R) and A-ESR metrics only exhibit a slight decline. This suggests that our extraction method is primarily instance-level and does not heavily rely on the model's overall conceptual knowledge, making it relatively insensitive to the introduction of additional unrelated data.
+
+Noisy Gradient Updates. Inspired by Differential Privacy [5, 6], which provides theoretical guarantees against information leakage, we explore the use of DP-SGD [1] as a potential defense mechanism during exact unlearning. Specifically, we perturbed the updates with random noises before each gradient descent step. Intuitively, larger noise scales offer stronger privacy protection, but at the cost of reduced model utility.
+
+We conduct experiments on the MUSE dataset with a  $10\%$  forgetting set using Phi-1.5, following the default setup. The only modification lies in the optimizer, where we inject Gaussian noise  $\varepsilon \sim \mathcal{N}(0,\sigma^2)$  with varying scales  $\sigma$  before each update. To quantify utility degradation, we follow prior work [30] and evaluate the Rouge-L(R) score on the retain set.
+
+As shown in Fig. 11, increasing the noise scale consistently reduces the effectiveness of our extraction method. At sufficiently large noise levels (above 0.4), the extraction performance largely approaches that of the pre-unlearning model, indicating a partially effective defense. However, this comes at a significant cost: the model's utility on the retain set degrades severely, barely surpassing that of the original pretrained model. These findings suggest that while noisy gradient updates can serve as a partial defense, the trade-off between
+
+![](images/c9de5ffc1ec1926b5b7569f86dc88afacba04617b73a70c2f7a3cb991a6889bb.jpg)  
+(a) A-ESR1.0
+
+![](images/81d438032ef1e8532b2c47528c60a52d4e3158797cd62d2ed4a6ec38136029d7.jpg)  
+(b) Utility  
+Figure 11: Effect of noisy gradient updates as a defense on the MUSE dataset using Phi-1.5. While large noise levels can partially mitigate our extraction attack, they also cause a substantial degradation in model utility.
+
+privacy protection and model utility remains severe and undermines their practical viability.
+
+# 6 Conclusion and Discussion
+
+Most prior works on unlearning for LLMs focus solely on evaluating the privacy risk of the final unlearned model, without considering the implications of retaining access to earlier checkpoints or logits API. However, in many realistic scenarios—such as open-weight model releases or API deployments—there exists a practical risk that pre-unlearning models or logits may have been preemptively saved by an adversary. Our work shows that under such conditions, exact unlearning—widely regarded as the gold standard for data removal—can in a counterintuitive way introduce new privacy risks. By leveraging the differences between pre- and post-unlearning models through a guidance-based extraction method with token filtering, an adversary can significantly increase the leakage of the very content intended to be forgotten.
+
+These findings reveal a previously overlooked but practical threat model. We urge the community to take this into account when designing and evaluating unlearning methods for LLMs. In particular, future techniques should offer privacy guarantees not only for the final model but also under adversarial access to its earlier states—only then can unlearning truly deliver on its intended privacy promises.
+
+# 7 Acknowledgments
+
+Zhiwei Steven Wu was in part supported by an NSF CAREER Award #2339775 and NSF Award #2232693.
+
+# References
+
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+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: The abstract and introduction highlight a new threat: that exact unlearning—previously regarded as the gold standard for data removal—can introduce new privacy risks. This claim is the central focus of the paper and is supported by the proposed method in Sec. 4 and a set of experiments that design and evaluate a data extraction attack leveraging both pre- and post-unlearning models in Sec. 5.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: The paper discusses the limitations in Appendix Sec. G.
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [NA]
+
+Justification: The paper includes heuristic modeling to provide intuitive guidance for the proposed method in Sec. 4, but it does not involve formal theoretical assumptions or proofs.
+
+Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: The main experimental procedures are outlined in Sec. 5, with additional implementation details provided in Appendix Sec. A, ensuring reproducibility of the results relevant to the core claims.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.
+
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
+
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
+
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
+
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
+
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in
+
+some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: The code is provided via an link shown at abstract. The datasets used are primarily open-source benchmarks. The main experimental results are presented in Sec. 5, with reproduction details provided in Appendix Sec. A.
+
+Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).  
+- The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.  
+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.  
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: The main experimental settings are described in Sec. 5, with additional details provided in Appendix Sec. A. Hyper-parameters and their selection process are explained in Sec. 5.4, ensuring clarity on how they were chosen.
+
+Guidelines:
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: Statistical significance is addressed in Tab. 1. The variation of A-ESR across three unlearning runs is minimal (standard deviation  $< 0.01$ ), so error bars are omitted for clarity. The improvements of our extraction method over the baseline are substantially larger than this variance, indicating the results are statistically robust.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).  
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: Information available in Sec. A
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: We have reviewed the NeurIPS Code of Ethics and ensured that the research complies with all relevant ethical guidelines.
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+# Answer: [Yes]
+
+Justification: The paper discusses the broader impacts in Appendix Sec. G.
+
+# Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+# Answer: [NA]
+
+Justification: The paper does not center on releasing any new datasets or models, and thus no additional safeguards are required.
+
+# Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+# Answer: [Yes]
+
+Justification: We use multiple benchmark datasets, all of which are properly cited in Sec. 5, with detailed descriptions provided in Appendix Sec. A.
+
+# Guidelines:
+
+- The answer NA means that the paper does not use existing assets.
+
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: The paper does not release new assets.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+# Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: LLMs are used during the construction of the medical dataset, as described in Sec. 5.5, but they are not part of the core methodology or contribution of the paper.
+
+# Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Experiment Details
+
+All experiments are conducted using two NVIDIA A100 GPUs.
+
+# A.1 Training Details
+
+Following prior works [30, 23], we begin with the pre-trained LLMs, LLaMA2-7B and Phi-1.5. To obtain the target (pre-unlearning) model with moderate memorization of the training data, we fine-tune LLaMA2-7B for 2 epochs and Phi-1.5 for 3 epochs on the full dataset, using a constant learning rate of  $10^{-5}$ . This setup reflects a realistic scenario: a well-tuned model should neither memorize excessively—compromising generalization—nor memorize too little, which would eliminate the need for unlearning in the first place.
+
+To simulate exact unlearning, we train a second model from scratch with the same configurations, again starting from the pre-trained weights but excluding the designated forgetting set. This yields the post-unlearning model for our experiments.
+
+# A.2 Dataset Preparation
+
+We experiment on the following benchmark datasets: MUSE, TOFU, and WMDP.
+
+- MUSE [30]: We use the MUSE-News dataset, which consists of BBC news articles collected after August 2023. The dataset is split into two disjoint subsets:  $\mathcal{D}_{\mathrm{forget}}$  and  $\mathcal{D}_{\mathrm{retain}}$ , containing  $0.8\mathrm{M}$  and  $1.6\mathrm{M}$  tokens, respectively. For a  $k\%$  forgetting set, we randomly select passages from  $\mathcal{D}_{\mathrm{forget}}$  until the total number of selected tokens reaches  $2.4\mathrm{M} \times k\%$ . The prefix known to the attacker is the first half of each sentence.  
+- TOFU [23]: We use the full TOFU dataset, which consists entirely of fictitious author biographies synthesized by GPT-4. To construct the forgetting set, we randomly sample question-answer pairs and treat the remaining data as the retaining set. The prefix known to the attacker is the question part.  
+- WMDP [18]: We use a subset of bio-retain-corpus from WMDP, comprising a collection of PubMed papers that span various categories within general biology. This subset contains a total of 5.3k sentences. We randomly sample sentences from this subset to form the forgetting set, with the remainder serving as the retaining set. We simulate the attacker's prior knowledge by providing access to the first half of each sentence as a prefix.
+
+# B Medical Dataset Experiment Details
+
+To simulate real-world medical data, we design our medical dataset using the Subjective, Objective, Assessment and Plan (SOAP) note [27] as a template. SOAP notes are a widely adopted method for healthcare providers to document patient encounters in a structured and organized manner. To ensure a comprehensive structure, we utilized the SOAP note template from Heidi Health [9], a medical AI company which offers SOAP note templates provided by specialists from the medical industry.
+
+We format our dataset into JSON with the following keys, "client name", "date of birth", "date", "subjective", "objective", "assessment" and "plan". For the generation process, we employ Gemini 2.5 Pro using a specialized prompt:
+
+I would like to generate synthetic medical data for machine learning purposes. Specifically, I would use SOAP notes as the data type. Below is a note template you need to follow, which has client name, date of birth, date, as well as subjective, objective, assessment, and plan. The template is just for you to refer, you do not need to generate each line of the template. Instead, only several lines for each of the SOAP is enough, try not to be too tedious for each record. For each record, please generate with a PII (client name, date of birthday), one person per record. client name: [name] date of birth: [birthday date] date: [visiting date] Subjective:
+
+[Description of symptoms, onset of symptoms, location of symptoms, duration of symptoms, characteristics of symptoms, alleviating or aggravating factors, timing, and severity]
+
+[Current medications and response to treatment] (write this section in narrative form. Write in full sentences and do not include any bullet points)
+
+[Any side effects experienced] (write this section in narrative form. Write in full sentences and do not include any bullet points)
+
+[Non-pharmacological interventions tried] (write this section in narrative form. Write in full sentences and do not include any bullet points)
+
+[Description of any related lifestyle factors] (write this section in narrative form. Write in full sentences and do not include any bullet points)
+
+[Patient's experience and management of symptoms] (write this section in narrative form. Write in full sentences and do not include any bullet points)
+
+[Any recent changes in symptoms or condition] (write this section in narrative form. Write in full sentences and do not include any bullet points)
+
+[Any pertinent positive or pertinent negatives in review of systems] (write this section in narrative form. Write in full sentences and do not include any bullet points)
+
+Objective:
+
+Vital Signs: Blood Pressure: [blood pressure reading] (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank.)
+
+Heart Rate: [heart rate reading] (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank.)
+
+Respiratory Rate: [respiratory rate reading] (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank.)
+
+Temperature: [temperature reading] (only include if explicitly mentioned in transcript, contextual notes or clinical note, otherwise leave blank.)
+
+Oxygen Saturation: [oxygen saturation reading] (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank.)
+
+General Appearance: [general appearance description] (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank.)
+
+HEENT: [head, eyes, ears, nose, throat findings] (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank.)
+
+Neck: [neck findings] (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank.)
+
+Cardiovascular: [cardiovascular findings] (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank.)
+
+Respiratory: [respiratory findings] (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank.)
+
+Abdomen: [abdominal findings] (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank.)
+
+Musculoskeletal: [musculoskeletal findings] (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank.)
+
+Neurological: [neurological findings] (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank.)
+
+Skin: [skin findings] (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank.)
+
+Assessment:
+
+[Likely diagnosis]
+
+[Differential diagnosis (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank)]
+
+Diagnostic Tests: (only include if explicitly mentioned other skip section)
+
+[Investigations and tests planned (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank)]
+
+Plan:
+
+[Treatment planned for Issue 1 (only include if explicitly mentioned in the transcript, contextual notes or clinical note, otherwise leave blank)]
+
+[Relevant referrals for Issue 1 (only include if explicitly mentioned- [Likely diagnosis for Issue 1 (condition name only)]
+
+(Never come up with your own patient details, assessment, diagnosis, interventions, evaluation or plan for continuing care - use only the transcript, contextual notes, or clinical note as a reference for the information included in your note. If any
+
+information related to a placeholder has not been explicitly mentioned in the transcript, contextual notes, or clinical note, you must not state the information has not been explicitly mentioned in your output, just leave the relevant placeholder or section blank).
+
+Then, here is an example of SOAP fields for you to refer:
+
+Subjective:
+
+The patient, a 52-year-old male, presents with a new rash on his back and arms, which he has noticed for the past two weeks. He describes the rash as "itchy and red," and mentions that it seems to be getting worse despite over-the-counter anti-itch creams. The patient denies any fever, joint pain, or recent exposure to new soaps or detergents.
+
+Objective:
+
+Appearance: The patient appears well-nourished and in no acute distress.
+
+Skin: Exam reveals erythematous, scaly plaques on the back and arms. There is evidence of excoriation due to itching. No signs of systemic involvement.
+
+Lesions: Lesions are well-defined, with some areas showing mild papules. No signs of pustules or ulcers.
+
+Other Systems: Vital signs are within normal limits. No lymphadenopathy noted. Assessment:
+
+The presentation is consistent with psoriasis, characterized by itchy, scaly plaques. The absence of systemic symptoms and well-defined lesions supports this diagnosis. Differential diagnoses include eczema or fungal infection, but these are less likely given the clinical presentation.
+
+Plan:
+
+Initiate topical treatment with high-potency corticosteroids to reduce inflammation and itching.
+
+Recommend emollients to improve skin hydration and prevent dryness.
+
+Educate the patient on the nature of psoriasis, including triggers and management strategies.
+
+Suggest lifestyle modifications such as stress management and dietary adjustments to potentially improve symptoms.
+
+You should add PII in front of the SOAP. Please generate 10 records for me, in json format, with "client name, date of birth, date, as well as subjective, objective, assessment, and plan" as keys.
+
+To maintain data quality and prevent degradation during large-scale generation, we created the dataset in batches of 50 records each, producing 1,000 records in total. We replaced duplicate client names with unique ones, as there would be a low probability of identical names appearing in a real-world sample of this size. The generated data covered a diverse range of medical conditions to ensure a representative sample of real-world clinical scenarios. We randomly sample 100 records as the forgetting set, with the remaining 900 records serving as the retaining set.
+
+For illustration purposes, an example of the generated records is provided below:
+
+"client name": "Noah Garcia",  
+"date of birth": "2012-07-22"  
+"date": "2025-05-18",
+
+"subjective": "Parent reports Noah, a 12-year-old male, has had intermittent abdominal pain for the past month. Pain is periumbilical, crampy, occurs 1-2 times per week, lasting 30-60 minutes. No clear relation to food. No fever, vomiting, diarrhea, or weight loss. Appetite is normal. School attendance is unaffected. He takes no medications. Parent has tried giving children's Tylenol during episodes with little effect. Parent is worried about the recurrence.",
+
+"objective": "Vital Signs: Normal for age. Abdomen: Soft, non-tender, non-distended. Bowel sounds normal. No masses palpated. Growth chart parameters are normal.", "assessment": "Recurrent abdominal pain, likely functional abdominal pain given age, characteristics, and lack of red flag symptoms.",
+
+"plan": "Reassure parent and child about functional nature. Discuss potential triggers (stress, diet). Recommend keeping a pain and stool diary. Encourage high-fiber diet and adequate fluids. Advise follow-up if pain changes pattern, becomes severe, or if red flag symptoms (weight loss, vomiting, blood in stool) develop."
+
+To capture the complex structure of these long medical records, we fine-tune the model for 11 epochs, while keeping all other settings unchanged in the experiment.
+
+![](images/66128cc094ca878589ffa37191ee3e5dca90bbcccf659452ba026c76511901f6.jpg)  
+(a) Rouge-L(R)
+
+![](images/c0eb78d34731078ed179e1b6f42e1755eae2f52383ecba8882f3487581f2bfa9.jpg)  
+(b) A-ESR1.0  
+Figure 12: Comparison of our extraction method against the baseline under various unlearning methods. EU refers to exact unlearning. In some cases, the effectiveness of our method weakens—primarily due to reduced model utility, which distorts the guidance between the pre- and post-unlearning models.
+
+![](images/dce8095eef29824c65440174594afd77f86b53d07c86a48cf98b986092d03b2a.jpg)  
+(c) Utility
+
+Table 4: Comparison of our extraction method with baselines under LoRA fine-tuning on Phi-1.5 on TOFU dataset. The improvements are consistent across metrics.  
+
+<table><tr><td>Extraction Methods</td><td>Rouge-L(R)↑</td><td>A-ESR0.9↑</td><td>A-ESR1.0↑</td></tr><tr><td>Post-Unlearning Generation</td><td>0.412</td><td>0.010</td><td>0.003</td></tr><tr><td>Pre-Unlearning Generation</td><td>0.544</td><td>0.070</td><td>0.055</td></tr><tr><td>Ours</td><td>0.614</td><td>0.160</td><td>0.108</td></tr></table>
+
+# C Extraction Results under LoRA Fine-tuning
+
+In practice, parameter-efficient fine-tuning methods such as LoRA [10] are increasingly popular. We therefore examine whether our attack remains effective when the pre-unlearning model is LoRA-fine-tuned on TOFU. Specifically, we provide results on Phi-1.5 fine-tuned with LoRA on TOFU for five epochs using a constant learning rate of 2e-4 and a  $10\%$  forgetting set, while keeping all other settings consistent with Sec. 5.2. Then, we employ a guidance scale  $\mathrm{w} = 1.6$  for our extraction, increasing the extraction rate. The improvements are consistent, as shown in Tab. 4.
+
+# D Extraction Results on Larger LLMs
+
+We further experiment with a larger model, Mixtral-8x7B [16], which has substantially more parameters, employs a MoE architecture, and is instruction-tuned, making it sufficiently distinct from our default models (LLaMA and Phi) to better evaluate generalization. Specifically, we report results on Mixtral-8x7B-Instruct-v0.1², fine-tuned with LoRA on TOFU for two epochs using a cosine learning rate schedule with a base learning rate of 1e-4 and a  $10\%$  forgetting set (all other settings follow Sec. 5.2). For extraction, we apply a guidance scale of  $w = 2$ . The improvements remain consistent, as shown in Tab. 5.
+
+# E Comparison with Other Extraction Attacks
+
+To further contextualize our method within the broader landscape of extraction attacks, we compare it against ETHICIST [36], a representative approach for extracting memorized training data. ETHICIST operates by tuning soft prompt embeddings and applying loss smoothing with calibrated confidence estimation. We conduct experiments on the TOFU dataset using Phi-1.5 under the  $10\%$  forgetting setting.
+
+Notably, ETHICIST assumes a stronger threat model that the attacker has access to part of the model's training data. To align with this setting, we assume the attacker has access to half of the retained set in TOFU (while remaining blind to the target extraction set, i.e., the forgetting set), and use it to
+
+Table 5: Comparison of our extraction method with baselines under LoRA fine-tuning on Mixtral-8x7B-Instruct-v0.1 on TOFU dataset. The improvements are consistent across metrics.  
+
+<table><tr><td>Extraction Methods</td><td>Rouge-L(R)↑</td><td>A-ESR0.9↑</td><td>A-ESR1.0↑</td></tr><tr><td>Post-Unlearning Generation</td><td>0.372</td><td>0.015</td><td>0.013</td></tr><tr><td>Pre-Unlearning Generation</td><td>0.465</td><td>0.040</td><td>0.018</td></tr><tr><td>Ours</td><td>0.537</td><td>0.103</td><td>0.055</td></tr></table>
+
+Table 6: Comparison of our extraction method with ETHICIST under default fine-tuning on the TOFU dataset. Our method outperforms ETHICIST by a clear margin, and combining the two further improves performance.  
+
+<table><tr><td>Extraction Methods</td><td>Rouge-L(R)↑</td><td>A-ESR0.9↑</td><td>A-ESR1.0↑</td></tr><tr><td>Pre-Unlearning Generation</td><td>0.566</td><td>0.100</td><td>0.070</td></tr><tr><td>ETHICIST</td><td>0.570</td><td>0.118</td><td>0.073</td></tr><tr><td>Ours</td><td>0.643</td><td>0.202</td><td>0.120</td></tr><tr><td>Ours + ETHICIST</td><td>0.652</td><td>0.213</td><td>0.153</td></tr></table>
+
+train the soft prompt. The hyper-parameters of ETHICIST follow the configuration provided in the original paper's open-sourced code.
+
+As shown in Tab. 6, experimental results show that ETHICIST alone can partially improve the extraction rate. However, since ETHICIST is designed to extract training data from a single model, it is complementary rather than conflicting with our method, which leverages the difference between pre- and post-unlearning models. In fact, combining ETHICIST with our guidance-based approach leads to even stronger extraction performance.
+
+# F Visualization
+
+In Figs. 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24, we present several examples under our default setting using Phi-1.5, with our method applied using the default hyper-parameters ( $w = 2.0$ ,  $\gamma = 10^{-5}$ ). For each dataset, we include examples where both our method and the baseline fail, where both succeed, and intermediate cases where the baseline fails but our method successfully improves extraction.
+
+# G Limitations and Broader Impact
+
+In this paper, we show that exact unlearning—originally intended to improve model safety—can, in fact, introduce new privacy risks. Our method relies on access to weights or logits api from both the pre- and post-unlearning models. While we justify this assumption using a realistic medical dataset, there are cases where pre-unlearning checkpoints or logits may not be available, such as in closed-source settings or when attackers fail to pre save sufficient outputs. This limits the general applicability of our method. Future work may explore leveraging public model outputs or general-purpose knowledge priors, as suggested in prior work [2].
+
+Our extraction method reveals a privacy risk of exact unlearning that, in principle, could be exploited in practice. However, as with other papers that focus on attacks, our goal is not to promote misuse, but to highlight a potential vulnerability before it leads to real-world consequences. By identifying this risk early, we hope to encourage more cautious use of exact unlearning and to motivate the community to proactively develop stronger defense mechanisms.
+
+![](images/11b232e6215e3073015653771064c9c79f14d8d49a09ba32a8acd2ae0c1c7d18.jpg)  
+Figure 13: Examples from the TOFU dataset illustrating hard extraction cases where both our method and the baseline fail.
+
+![](images/c24b2731c6f289f7dfe75a6f4be5666457f505e6db17ed1c5a5cdb2fd244a295.jpg)
+
+![](images/aa1ae9ada44e240406f225acc6e5bddb06c521e426f5fbdd59c55066bd3f4720.jpg)
+
+![](images/4d222d53d78f1dec4b9f2edf66ceceb0bd2943de496ae8bad9463eeccdca8183.jpg)
+
+![](images/2fff2f93b6573117ffa87abafdb5aa6346c1592ac2aa65fa8ff9b76f36d028d4.jpg)  
+Figure 14: Examples from the TOFU dataset illustrating easy extraction cases where both our method and the baseline mostly succeed.
+
+![](images/dc90062fe5bafbeb4e5f48fbedaf83a448425d36b298dd00bbcf57d5bccf70ca.jpg)
+
+![](images/dd8dca5563e101e545935c3fc27203fda7cae5469fa386a4ae7f6ef8bbac1927.jpg)
+
+![](images/c14476d3fb5457b242347f8da1886c4099bf05b7b984451e265739f4fafa4c8b.jpg)
+
+![](images/edecf242881e28d879fd59c63d22d5608c48d9abf0bf7bf7ec637e9462add374.jpg)  
+Figure 15: Examples from the TOFU dataset showing cases of intermediate extraction difficulty, where the baseline fails, but our method successfully recovers most of the target information. These cases highlight the improvement brought by the proposed method.
+
+![](images/0f3ccda24a76a24cb7794a1758628715d16c75384af45b6e59672e3a78ea6f50.jpg)
+
+![](images/1c59fdf3f4ce7e94c6da802709d10d10d8fc54052f6283d0dcdac815014643eb.jpg)
+
+![](images/24f7b7a98793d4a11b20ba04fcbd69b496dfffed249d50e818c3103d70edd5a6.jpg)
+
+![](images/75a601c3d53043b6d90b121eb55166f0c0864f1d800584013bb921c9a6a44577.jpg)  
+Figure 16: Examples from the MUSE dataset illustrating hard extraction cases where both our method and the baseline fail.
+
+![](images/8aa5fc0f5384f07ac40329b489eee1cda662493128bcac20d01758ee3d2b9f7d.jpg)  
+Figure 17: Examples from the MUSE dataset illustrating easy extraction cases where both our method and the baseline mostly succeed.
+
+![](images/dc67b705a8343b3a67b15dc261994864788bf7e4de87dc2fd5e8e6603e2cebce.jpg)  
+Figure 18: Examples from the MUSE dataset showing cases of intermediate extraction difficulty, where the baseline fails, but our method successfully recovers most of the target information. These cases highlight the improvement brought by the proposed method.
+
+![](images/d2bc64e15eb1d96d6a5335ebf3b3c725e62851c24479674b391decf865b91b20.jpg)  
+Figure 19: Examples from the WMDP dataset illustrating hard extraction cases where both our method and the baseline fail.
+
+![](images/864ae106f789603aebf1e9215d64b17451c535b0fc430b97841a72277abd0e82.jpg)  
+Figure 20: Examples from the WMDP dataset illustrating easy extraction cases where both our method and the baseline mostly succeed.
+
+![](images/d05868d558a3f4b1ae6866aac7a2dafd7d7869ca9f762c4ab53c370e1d9c46cd.jpg)  
+Figure 21: Examples from the WMDP dataset showing cases of intermediate extraction difficulty, where the baseline fails, but our method successfully recovers most of the target information. These cases highlight the improvement brought by the proposed method.
+
+# Patient data to be forgotten
+
+client name: Gabriel Bell date of birth: 1949-01-21 date: 2026-05-08
+
+subjective: Mr. Bell is a 76-year-old male with metastatic prostate cancer (diagnosed 2 years ago, failed initial hormonal therapy, now on second-line agent). Presents with wife today to discuss goals of care as experiencing increased bone pain, fatigue, decreased appetite, and overall functional decline. Understands disease is incurable. objective: Exam: Appears fatigued, cachectic. Diffuse bony tenuous need. assessment: Advanced metastatic prostate cancer with significant symptom burden and functional decline. Patient/family ready to discuss goals of care/palliative approach; plan: Facilitate goals of care discussion: Elicit patient's understanding of illness/prognosis. Explore values, hopes, fears. Discuss goals (e.g., focus on comfort, quality of life, time at home vs further life-prolonging therapy with potential side effects). Discuss symptom management plan (optimize analgesia for bone pain - consider radiation oncology consult for palliative radiation, manage fatigue/anorexia). Discuss code status/advance directives if not already addressed. Consider referral to Palliative Care specialist team for expert symptom management and ongoing goals of care support. Offer hospice referral if appropriate/desired.
+
+# Adversary's side information
+
+about the patient
+
+client name: Gabriel Bell date of birth: 1949-01-21 date: 2026-05-08
+
+![](images/cbe49181d7f64ae99d21763cbd29ece9000557908faef7133abc8fafac13e367.jpg)
+
+# Baseline Extraction:
+
+using pre-unlearning checkpoint
+
+client name: Gabriel Bell date of birth: 1949-01-21 date: 2026-05-08
+
+subjective: Mr. Bell is a 76-year-old male with osteoporosis (T-score  $\sim 2.7$  hip). Started alendronate  $70\mathrm{mg}$  weekly 1 month ago. Reports new onset heartburn and difficulty swallowing pills since starting medication. Worried about side effects. objective: Review of proper administration technique for alendronate. assessment: Osteoporosis treatment initiated with alendronate. Experiencing common esophageal side effects. plan: Reinforce proper administration of alendronate: Take with full glass of plain water only, first thing in AM, at least 30 mins before other food/drink/meds. Remain upright (sitting or standing) for at least 30 mins after taking. If symptoms persist despite correct administration, options: 1) Switch to different oral bisphosphonate (e.g., risodronate - may have slightly lower esophageal irritation risk), 2) Switch to IV bisphosphonate (e.g., zoledronic acid annually). 3) Switch to non-bisphosphonate agent (denosumab, teriparatide). Discuss options. Temporarily hold alendronate, trial PPI for esophageal symptoms. Re-evaluate choice in 1-2 weeks. Endocrinology referral if complex.
+
+# Our Extraction: combining pre- and post
+
+unlearning checkpoints
+
+client name: Gabriel Bell date of birth: 1949-01-21 date: 2026-05-08
+
+subjective: Mr. Bell is a 76-year-old male referred by PCP for evaluation of pancytopenia (WBC 2.5, Hgb 9.8, Plt 85K) discovered on routine labs. Reports increasing fatigue for 3 months, some easy bruising. Denies fevers, infections, significant bleeding, bone pain. objective: Exam: Pallor, few scattered ecchymoses. No lymphadenopathy or hepatosplenomegaly. Peripheral smear shows trilineage cytopenias, no blasts, some dysplastic features in neutrophils reported. assessment: Pancytopenia. Concerning for underlying bone marrow pathology such as Myelodysplastic Syndrome (MDS) or aplastic anemia given age and smear findings. Hematologic malignancy needs exclusion. plan: STAT referral to Hematology. Comprehensive workup by hematology will include: Review of peripheral smear by hematopathologist, Vitamin B12/Folate, viral serologies (HIV, Hepatitis), autoimmune screen, and likely Bone Marrow Aspiration and Biopsy for definitive diagnosis. Supportive care as needed (transfusions if severely anemic/thrombocytopenic) pending diagnosis.
+
+client name: Josiah Hughes date of birth: 1986-10-14 date: 2025-07-03
+
+subjective: Mr. Hughes is a 38-year-old male presenting with redness, irritation, and crusting of his eyelids bilaterally for several months. Symptoms worse in the morning. Describes eyes feeling dirty or like there is sand in them. Denies vision changes or significant eye pain. Has tried artificial tears with little relief. objective: HEENT: Eyes: Bilateral eyelid margins erythematous with scaling and crusting at the base of the eyelashes. Conjunctiva mildly injected. Cornea clear. Visual acuity normal. assessment: Blepharitis, likely anterior blepharitis given findings. plan: Recommend eyelid hygiene measures: Warm compresses applied to eyelids for 5-10 minutes BID, followed by gentle scrubbing of eyelid margins with diluted baby shampoo or commercial eyelid cleanser. Artificial tears PRN for dryness/irritation. Consider short course of topical antibiotic ointment (e.g., erythromycin) applied to lid margins at bedtime if significant crusting or signs of infection. Follow up in 2-4 weeks.
+
+client name: Josiah Hughes date of birth: 1986-10-14 date: 2025-07-03
+
+![](images/1d2a9a0c07a8093e18bd34f094c9b1f744ff1c412effeae6cedf1136cba1d4e1.jpg)  
+Figure 22: Examples from the medical dataset illustrating hard extraction cases where both our method and the baseline fail.
+
+client name: Josiah Hughes date of birth: 1986-10-14 date: 2025-07-03
+
+subjective: Mr. Hughes, 38, for follow-up of Addison's disease (diagnosed 5 years ago). He takes hydrocortisone  $15\mathrm{mg}$  AM/  $10\mathrm{mg}$  PM and fludrocortisone  $0.1\mathrm{mg}$  daily. Reports good adherence. Generally feels well but notes some increased fatigue and occasional lightheadiness in the past month, especially during hot weather. He carries an emergency steroid injection kit. No recent illnesses. objective: Vital Signs: BP 105/65 mmHg (lying), 95/60 mmHg (standing) with HR increase of 20 bpm (orthostatic). Weight stable. Skin: Mild hyperpigmentation noted in palmar creases. No cushingoid features. assessment: Likely diagnosis: Addison's disease, possibly needing slight dose adjustment of corticosteroids due to symptoms of mild hypoadrenalism (fatigue, orthostasis). plan: Temporally increase hydrocortisone dose slightly (e.g., by 5mg/day) especially with heat/stress, and ensure adequate salt intake. Reinforce sick day rules and stress dosing. Check electrolytes. Discuss splitting evening hydrocortisone dose if morning fatigue is an issue. Follow up closely with endocrinology. Ensure emergency kit is up to date.
+
+client name: Josiah Hughes date of birth: 1986-10-14 date: 2025-07-03
+
+subjective:Mr. Hughes is a 38-year-old male presenting with painless blood noted in semen (hematospermia) for 1 week,
+
+Denies urinary symptoms, penile discharge, pain during erection, or pain during ejaculation. Not currently on medications. objective: Genitourinary Exam: Normal external genitalia. Hormonal panel (TSH, FSH, LH, prolactin, ESR, CRP) all normal. DRE normal prostate. DRE of penis normal. assessment: Hematosperma. Could have male reproductive tract infection (urodysuria) component if associated with penile discharge or pelvic pain. Less likely just male pattern baldness given age without other symptoms. plan: Order hormonal panel and ESR/CRP. If urodysuria is suspected (low levels of Hgb and ESR/CRP elevated), treat with antibiotics (e.g., doxycycline first, then rifampin if needed). If hematospermia only and no other symptoms, assume male pattern baldness. Advise follow-up after hormonal panel review. Reassure not usually a cause for concern, but urethral stricture prevention if prior had issue.
+
+# Patient data to be forgotten
+
+subjective: Ms. Gates, 32, for follow-up of an incidental finding of a  $3\mathrm{cm}$  simple ovarian cyst on the left ovary found on pelvic ultrasound done for unrelated reasons 3 months ago. She is asymptomatic. No abdominal pain, bloating, or changes in menstrual cycle. She is using oral contraceptives. No family history of ovarian cancer. objective: Vital Signs: Normal. Abdomen: Soft, non-tender. Pelvic exam (if performed, or defer if not indicated for asymptomatic follow-up). Unremarkable. Review of prior ultrasound report confirming simple cyst characteristics. assessment: Likely diagnosis: Asymptomatic simple ovarian cyst, left, likely functional. Low risk of malignancy in premenopausal woman with simple cyst  $< 5\mathrm{cm}$  plan: Reassure patient that simple cysts are common and usually benign, often resolving spontaneously. Given size and simple nature, recommend follow-up pelvic ultrasound in 3-6 months to ensure stability or resolution. Continue current contraception. Advise to return if new symptoms develop (e.g., pelvic pain, bloating). No other intervention needed at this time.
+
+# Adversary's side information
+
+# about the patient
+
+client name: Hannah Gates date of birth: 1993-01-23 date: 2025-08-29
+
+![](images/1ec0f86ca3b6fcd05d7bd2a507fe0295fb55f4cb73535d5bf8fe7772b2711540.jpg)
+
+# Baseline Extraction
+
+# using pre-unlearning checkpoint
+
+subjective: Ms. Gates, 32, for follow-up of an incidental finding of a  $3\mathrm{cm}$  simple ovarian cyst on the left ovary found on pelvic ultrasound done for unrelated reasons 3 months ago. She is asymptomatic. No abdominal pain, bloating, or changes in menstrual cycle. She is using oral contraceptives. No family history of ovarian cancer. objective: Vital Signs: Normal. Abdomen: Soft, nontender. Pelvic exam (if performed, or defer if not indicated for asymptomatic followup): Unremarkable. Review of prior ultrasound report confirming simple cyst characteristics. assessment: Likely diagnosis: Asymptomatic simple ovarian cyst, left, likely functional. Low risk of malignancy in premenopausal woman with simple cyst  $<5\mathrm{cm}$ . plan: Reassure patient that simple cysts are common and usually benign, often resolving spontaneously. Given size and simple nature, recommend follow-up pelvic ultrasound in 3-6 months to ensure stability or resolution. Continue current contraception. Advise to return if new symptoms develop (e.g., pelvic pain, bloating). No other intervention needed at this time.
+
+# Our Extraction: combining pre- and post
+
+# unlearning checkpoints
+
+subjective: Ms, Gates, 32, for follow-up of an incidental finding of a  $3\mathrm{cm}$  simple ovarian cyst on the left ovary found on pelvic ultrasound done for unrelated reasons 3 months ago. She is asymptomatic. No abdominal pain, bloating, or changes in menstrual cycle. She is using oral contraceptives. No family history of ovarian cancer. objective: Vital Signs: Normal. Abdomen: Soft, non-tender. Pelvic exam (if performed, or defer if not indicated for asymptomatic follow-up): Unremarkable. Review of prior ultrasound report confirming simple cyst characteristics. assessment: Likely diagnosis: Asymptomatic simple ovarian cyst, left, likely functional. Low risk of malignancy in premenopausal woman with simple cyst  $<5\mathrm{cm}$ . plan: Reassure patient that simple cysts are common and usually benign, often resolving spontaneously. Given size and simple nature, recommend follow-up pelvic ultrasound in 3-6 months to ensure stability or resolution. Continue current contraception. Advise to return if new symptoms develop (e.g., pelvic pain, bloating). No other intervention needed at this time.
+
+client name: Aurelia Lambert date of birth: 1997-11-04 date: 2026-06-05   
+subjective: Ms. Lambert is a 28-year-old female (MSM context - uses she/her pronouns) presenting for discussion about HIV prevention. Reports multiple male partners, inconsistent condom use. Aware of PrEP, wants to start. objective: Sexual history confirms substantial risk. HIV Ag/Al test negative today. HBV immune (prior vaccine). Renal function normal. assessment Appropriate candidate for HIV PrEP based on risk factors. plan: Initiate PrEP with daily oral F/TAF (Descovcy). Counsel on importance of adherence for effectiveness. Discuss potential side effects. Schedule 3-month follow-up for repeat HIV testing, STI screening (GC/CT/Syphilis), renal function monitoring, adherence check, and prescription refill. Reinforce safer sex practices including consistent condom use for STI prevention.
+
+client name: Aurelia Lambert date of birth: 1997-11-04 date: 2026-06-05
+
+![](images/38857ce04b8cfe601f97d7568b09170f440578d9e0b8c4b555d56461af2add56.jpg)  
+Figure 23: Examples from the medical dataset illustrating easy extraction cases where both our method and the baseline mostly succeed.
+
+client name: Aurelia Lambert date of birth: 1997-11-04 date: 2026-06-05 subjective: Ms. Lambert is a 28-year-old female (MSM context - uses she/her pronouns) presenting for discussion about HIV prevention. Reports multiple male partners, inconsistent condom use. Aware of PrEP wants to start, objective: Sexual history confirms substantial risk, HIV Ag/Ab test negative today, HBV immune (prior vaccine). Renal function normal. assessment: Appropriate candidate for HIV PrEP based on risk factors, plan: Initiate PrEP with daily oral F/TAF (Descovy). Counsel on importance of adherence for effectiveness. Discuss potential side effects. Schedule 3-month follow-up for repeat HIV testing, STI screening (GC/CT/Syphilis), renal function monitoring, adherence check, and prescription refill. Reinforce safer sex practices including consistent condom use for STI prevention.
+
+client name: Aurelia Lambert date of birth: 1997-11-04 date: 2026-06-05   
+subjective: Ms. Lambert is a 28-year-old female (MSM context - uses she/her pronouns) presenting for discussion about HIV prevention. Reports multiple male partners, inconsistent condom use. Aware of PrEP, wants to start, objective: Sexual history confirms substantial risk. HIV Ag/Ab test negative today. HBV immune (prior vaccine). Renal function normal. assessment: Appropriate candidate for HIV PrEP based on risk factors, plan: Initiate PrEP with daily oral F/TAF (Descovy). Counsel on importance of adherence for effectiveness. Discuss potential side effects. Schedule 3-month follow-up for repeat HIV testing, STI screening (GC/CT/Syphilis), renal function monitoring, adherence check, and prescription refill. Reinforce safer sex practices including consistent condom use for STI prevention.
+
+# Patient data to be forgotten
+
+client name: Walter Brooks date of birth: 2017-06-20 date: 2025-08-12 subjective: Walter, 8, brought by parent complaining of intermittent aching pain in both legs, mostly shins and thighs, for several months. Pain typically occurs at night, sometimes waking her from sleep. It is not present in the morning or during daytime activities. No limping, swelling, redness, or limitation of activity. Relieved by massage or acetaminophen. Parent is concerned about an underlying problem. objective: Vital Signs: Normal. Musculoskeletal: Exam of lower extremities is normal. No tenderness, swelling, warmth, or deformity. Full range of motion of hips, knees, ankles. Gait normal. Neurological exam normal. assessment: Likely diagnosis: Growing Pains (benign nocturnal limb pains of childhood), given classic history and normal exam. Diagnosis of exclusion. plan: Reassure parent and child about benign nature of growing pains. Recommend symptomatic relief: massage, heat, acetaminophen or ibuprofen as needed. Ensure adequate calcium/vitamin D intake. If pain becomes persistent during day, unilateral, associated with systemic symptoms (fever, weight loss), limping, or joint swelling, then further investigation would be warranted to rule out other conditions. Follow up as needed.
+
+# Adversary's side information
+
+# about the patient
+
+client name: Walter Brooks date of birth: 2017-06-20 date: 2025-08-12
+
+![](images/3d9f0146c90882251308892eca30a9bb612c70bff492b42b6d02ba6eb3a02e2f.jpg)
+
+# Baseline Extraction
+
+# using pre-unlearning checkpoint
+
+subjective: Walter, 8, brought by parent for follow-up of school-reported bullying of name and teasing about weight. Reports feeling sad and embarrassed, avoids social situations, and had sores on her face from teasing. Her parents have tried ignoring it. They are very concerned. objective: Vital Signs: Normal. Physical exam (including school referral) shows consistent signs of emotional distress and bullying. Parent/child interview sensitive. assessment: Likely diagnosis: Childhood Bullying (verbal/physical), Subclinical Anxiety about Weight/Appearance. Behavioral symptoms. plan: Provide psychoeducation about bullying for parents/children. Validate child's feelings. Discuss strategies for coping with bullying (e.g., assertiveness, avoiding targets, reporting). Address child's own anxiety about weight/appearance. Reassure about normal weight. If bullying severe or child has severe anxiety, refer to mental health for counseling/therapy for both child and parents. Focus on building child's self-esteem and positive coping skills. Monitor for continued behavioral changes or worsening anxiety.
+
+# Our Extraction: combining pre- and post
+
+# unlearning checkpoints
+
+patient name: Walter Brooks date of birth: 2017-06-20 date: 2025-08-12   
+subjective: Walter, 8, brought by parent complaining of intermittent aching pain in both legs, mostly shins and thighs, for several months. Pain typically occurs at night, sometimes waking her from sleep. It is not present in the morning or during daytime activities. No limping, swelling, redness, or limitation of activity. Relieved by massage or acetaminophen. Parent is concerned about an underlying problem. objective: Vital Signs: Normal. Musculoskeletal: Exam of lower extremities is normal. No tenderness, swelling, warmth, or deformity. Full range of motion of hips, knees, ankles. Gait normal. Neurological exam normal. assessment: Likely diagnosis: Growing Pains (benign nocturnal limb pains of childhood), given classic history and normal exam. Diagnosis of exclusion: plan: Reassure parent and child about benign nature of growing pains. Recommend symptomatic relief: massage, heat, acetaminophen or ibuprofen as needed. Ensure adequate calcium/vitamin D intake. If pain becomes persistent during day, unilateral, associated with systemic symptoms (fever, weight loss), limping, or joint swelling, then further investigation would be warranted to rule out other conditions. Follow up as needed.
+
+subjective: Mr. Martin is a 45-year-old male for follow-up of costochondritis diagnosed 6 weeks ago. Took initial course of NSAIDs with some relief, but continues to have intermittent sharp pain over left sternal border, aggravated by certain movements or pressure. objective: Musculoskeletal: Residual point tenderness over left 3rd and 4th costochondral junctions reproducible with palpation. Cardiac/Pulmonary exam normal. assessment: Costochondritis, persistent symptoms despite initial NSAID trial. plan: Reassurance regarding musculoskeletal origin. Recommend continued activity modification, avoidance of aggravating activities. Trial of topical NSAID gel (diclofenac gel) applied locally. Stretching exercises for chest wall muscles may help. If pain remains significantly bothersome, consider referral for localized corticosteroid injection into costochondral junction. Follow up PRN.
+
+# client name: Corbin Martin date of birth: 1980-07-10 date: 2026-02-08
+
+![](images/0e6ac1402c344c9bf014b8bdedd6cffa371de1fc6505c595561da223953654af.jpg)  
+Figure 24: Examples from the medical dataset showing cases of intermediate extraction difficulty, where the baseline fails, but our method successfully recovers most of the target information. These cases highlight the improvement brought by the proposed method.
+
+client name: client name; Aurelia Lambert date of birth: 1997-11-04 date: 2026-06-05 subjective: Mr. Martin is a 45-year-old male with history of recurrent cellulitis of right lower leg, diagnosed 3 months ago after surgery. Currently asymptomatic. Wants to discuss prevention strategies, objective: Exam: Right Lower Extremity exam normal. No signs of active infection. assessment: History of recurrent cellulitis, currently asymptomatic. Discuss strategies for prevention. plan: Discuss risk factors and prevention strategies: Improved hygiene (regularly washing leg socks/pants, keeping skin clean and dry), compression stockings (if VASAMI risk factor), and antifungal prophylaxis for recent surgery (e.g., nystatin, clotrimazole) if history of similar infections. Continue current antibiotics (e.g., cephaloxin) if no longer needed. Keep first aid supplies accessible.
+
+client name: Corbin Martin date of birth: 1980-07-10 date: 2026-02-08  
+subjective: Mr. Martin is a 45-year-old male for follow-up of costochondritis diagnosed 6 weeks ago. Took initial course of NSAIDs with some relief, but continues to have intermittent sharp pain over left sternal border, aggravated by certain movements or pressure, objective: Musculoskeletal; Residual point tenderness over left 3rd and 4th costochondral junctions reproducible with palpation. Cardiac/Pulmonary exam normal: assessment: Costochondritis, persistent symptoms despite initial NSAID trial, plan: Reassurance regarding musculoskeletal origin. Recommend continued activity modification, avoidance of aggravating activities. Trial of topical NSAID gel (diclofenac gel) applied locally. Stretching exercises for chest wall muscles may help. If pain remains significantly bothersome, consider referral for localized corticosteroid injection into costochondral junction. Follow up PRN.
\ No newline at end of file
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+# Unlearning-Aware Minimization
+
+Hoki Kim*†
+
+Chung-Ang University
+
+hokikim@cau.ac.kr
+
+Sungwon Chae
+
+Seoul National University
+
+csw0815@snu.ac.kr
+
+Keonwoo Kim†
+
+NAVER Digital Healthcare LAB
+
+keonwoo.kim97@navercorp.com
+
+Sangwon Yoon
+
+Ministry of Justice, Republic of Korea
+
+asad01075272750@gmail.com
+
+# Abstract
+
+Machine unlearning aims to remove the influence of specific training samples (i.e., forget data) from a trained model while preserving its performance on the remaining samples (i.e., retain data). Existing approximate unlearning approaches, such as fine-tuning or negative gradient, often suffer from either insufficient forgetting or significant degradation on retain data. In this paper, we introduce Unlearning-Aware Minimization (UAM), a novel min-max optimization framework for machine unlearning. UAM perturbs model parameters to maximize the forget loss and then leverages the corresponding gradients to minimize the retain loss. We derive an efficient optimization method for this min-max problem, which enables effective removal of forget data and uncovers better optima that conventional methods fail to reach. Extensive experiments demonstrate that UAM outperforms existing methods across diverse benchmarks, including image classification datasets (CIFAR-10, CIFAR-100, TinyImageNet) and multiple-choice question-answering benchmarks for large language models (WMDP-Bio, WMDP-Cyber).
+
+# 1 Introduction
+
+The increasing deployment of artificial intelligence (AI) into real-world applications has prompted critical discussions regarding the alignment of AI with human values. A key aspect of these discussions is the "right to be forgotten" [25], a principle embedded in the General Data Protection Regulation (GDPR) [14]. This right enables individuals to request the deletion of their personal data, providing a safeguard for privacy and mitigating risks associated with data misuse.
+
+The simplest approach is retraining a model from scratch without the data points to be forgotten. However, retraining is computationally prohibitive, particularly for large-scale deep learning models [4, 35]. As a potential solution, machine unlearning has emerged for removing the influence of specific data samples by appropriately updating their parameters [2, 3, 9]. The goal of machine unlearning is to efficiently remove the influence of specific data while preserving performance on the remaining data. Formally, let  $\mathcal{D}_f$  denote the dataset to be forgotten (i.e., forget data) and  $\mathcal{D}_r$  denote the dataset to be retained (i.e., retain data). A common strategy of machine unlearning is to optimize the following objectives: minimizing the retain loss  $\mathcal{L}(\boldsymbol{w},\mathcal{D}_r)$  or maximizing the forget loss  $\mathcal{L}(\boldsymbol{w},\mathcal{D}_f)$  [11, 32].
+
+In this paper, we introduce Unlearning-Aware Minimization (UAM), a novel min-max optimization framework for machine unlearning. Specifically, UAM formulates unlearning as a two-stage process:
+
+![](images/833c286548a0d11b293085186aa5a1321d5b5fd8e3abb86b0e2c27405e085608.jpg)  
+(a) Optimization trajectories
+
+![](images/7cf76465e963fd91ba63562a8e34209d3650898f82c4b393df4c01a538beee9b.jpg)  
+Figure 1: Optimization results on synthetic loss functions. The optimization begins at the global minimum of the sum of two losses (black cross, +). (a) UAM successfully converges to the optimal point (yellow star,  $\star$ ), whereas other methods converge to suboptimal regions. For NG, only a partial trajectory is shown due to divergence. (b) UAM achieves a high forget loss while effectively minimizing the retain loss. NG yields a high forget loss but fails to maintain low retain loss. FT converges to a suboptimal region, resulting in a lower forget loss.
+
+![](images/6105dc78bc64ab7055feccc6643576692f90f75c56abd592690f60c9d0dcef10.jpg)  
+(b) Forget loss  $\mathcal{L}(\boldsymbol{w},\mathcal{D}_f)$  and Retain loss  $\mathcal{L}(\boldsymbol{w},\mathcal{D}_r)$
+
+(i) an inner maximization that identifies a surrogate weight within a local neighborhood that maximizes the forget loss, and (ii) an outer minimization that reduces the retain loss by using its gradients. Intuitively, this procedure ensures that the model exhibits characteristics similar to those of weights with a high forget loss, while maintaining a low retain loss. By leveraging a first-order Taylor approximation, we derive a scalable algorithm that enables effective unlearning while remaining computationally practical.
+
+Fig. 1 highlights the key differences between UAM and existing unlearning methods. Fine-tuning (FT) minimizes  $\mathcal{L}(\boldsymbol{w},\mathcal{D}_r)$  and negative gradient (NG) maximizes  $\mathcal{L}(\boldsymbol{w},\mathcal{D}_f)$  [11, 32]. Both methods converge to suboptimal regions in the given optimization problem. These methods result in either low forget loss (i.e., insufficient forgetting) or high retain loss (i.e., poor performance). In contrast, UAM more effectively navigates the loss landscape, approaching the optimal solution characterized by high forget loss and low retain loss. As shown in Fig. 1b, UAM achieves a higher forget loss and a lower retain loss by explicitly exploring regions with high forget loss. In our experiments, UAM demonstrates superior performance on both image classification and multiple-choice question answering tasks.
+
+Our main contributions can be summarized as follows:
+
+- We propose a new min-max optimization framework for machine unlearning, Unlearning-Aware Minimization (UAM). By leveraging model parameters with high forget loss, UAM enables the effective removal of forget data while preserving the performance on retain data.  
+- We establish an efficient algorithm based on a first-order Taylor expansion. We also provide a theoretical analysis of UAM, characterizing its optimization dynamics through the cosine similarity between retain and forget gradients.  
+- We evaluate the effectiveness of UAM on image classification datasets (CIFAR-10, CIFAR-100, and TinyImageNet) and multiple-choice question-answering datasets (WMDP-Bio and WMDP-Cyber) using large language model (LLM). Since UAM is a framework independent of any specific loss function, it can be easily extended to other domains.  
+- To promote reproducibility and benchmarking within the machine unlearning community, we release implementations of existing baseline unlearning methods, along with our proposed framework, available at: https://github.com/Harry24k/machine-unlearning-pytorch.
+
+# 2 Related Work
+
+# 2.1 Machine Unlearning
+
+Machine unlearning [3, 9] aims to eliminate the influence of forget data  $\mathcal{D}_f$ , while preserving knowledge learned from retain data  $\mathcal{D}_r$ . The ideal solution, known as exact unlearning, is to retrain the model from scratch without  $\mathcal{D}_f$ ; however, retraining is often computationally inefficient for large-scale deep learning models [4, 35]. Therefore, approximate unlearning methods have been developed. Fine-tuning (FT) [9, 32] minimizes  $\mathcal{L}(\boldsymbol{w}, \mathcal{D}_r)$  relying catastrophic forgetting [24]; negative gradient (NG) [9], which also referred to as gradient ascent, directly maximizes  $\mathcal{L}(\boldsymbol{w}, \mathcal{D}_f)$ . More recent methods build on these frameworks by incorporating additional techniques such as distillation [20] or pruning [16]. In contrast to prior approaches, we highlight the potential of a min-max optimization framework for machine unlearning that extends beyond traditional dual-objective formulations. We note that there exists a distinct line of research, including Fisher Forgetting (FF) [9] and Influence Unlearning (IU) [15, 18], that leverages the Fisher Information Matrix and influence functions.
+
+# 2.2 Min-Max Optimization
+
+Min-max optimization refers to a class of learning problems that aims to solve two competing objectives, commonly formulated as saddle-point or bi-level optimization problems [1, 31]. In deep learning, the min-max optimization plays a central role in several research areas. For example, in adversarial robustness [10, 28], adversarial training uses an inner maximization to find perturbations that maximize a loss value, and an outer minimization to optimize model parameters to minimize this worst-case loss [23, 36]. More recently, sharpness-aware minimization has introduced a min-max optimization for improving generalization. It adopts an inner maximization and an outer minimization step to identify parameters that have uniformly low losses within neighborhoods [7, 21]. We extend a min-max optimization into the domain of machine unlearning using two disjoint datasets  $\mathcal{D}_r$  and  $\mathcal{D}_f$  and demonstrate that a min-max optimization can address the challenges in machine unlearning.
+
+# 3 Unlearning-Aware Minimization
+
+In this work, we use the following notation: scalars are denoted by  $a$ , vectors by  $\mathbf{a}$ , matrices by  $\mathbf{A}$ , and  $\triangleq$  indicates equality by definition. Let us denote the training dataset as  $\mathcal{D} \triangleq \{(x_i, y_i)\}_{i=1}^n$ , drawn independently and identically (i.i.d.) from the true data distribution. A model parameterized by weights  $\mathbf{w} \in \mathcal{W} \subseteq \mathbb{R}^d$  is trained by minimizing the empirical training loss  $\mathcal{L}(\mathbf{w}, \mathcal{D}) \triangleq \frac{1}{n} \sum_{i=1}^n \ell(\mathbf{w}, x_i, y_i)$  where  $\ell$  is an individual loss function. We denote the subset of data to be forgotten as the forget data  $\mathcal{D}_f \subset \mathcal{D}$ . Then, the complement of  $\mathcal{D}_f$  becomes the retain data  $\mathcal{D}_r \triangleq \mathcal{D} \setminus \mathcal{D}_f$ .
+
+The simplest and exact approach to unlearning, commonly called exact unlearning, optimizes a model from scratch using only the retain data:
+
+$$
+\boldsymbol {w} ^ {*} = \underset {\boldsymbol {w}} {\arg \min } \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {r}), \tag {1}
+$$
+
+commonly known as Retrain [16] or the oracle [4]. While exact unlearning provides an optimal solution for eliminating the influence of the forget data, its substantial computational overhead makes it impractical for large-scale models and datasets [16, 29]. To circumvent these practical constraints, approximate unlearning methods often re-optimize the pre-trained model with  $\boldsymbol{w}_0 = \arg \min_{\boldsymbol{w}} \mathcal{L}(\boldsymbol{w}, \mathcal{D})$ . Therefore, approximate unlearning methods commonly assume that the solution of (1) lies within a bounded neighborhood  $\mathcal{B}_{\Omega}(\boldsymbol{w}_0) \triangleq \{\tilde{\boldsymbol{w}} \in \mathbb{R}^d \mid \| \boldsymbol{w}_0 - \tilde{\boldsymbol{w}} \| \leq \Omega\}$ , where  $\Omega$  is a finite upper bound. Given that  $\boldsymbol{w}^* \in \mathcal{B}_{\Omega}(\boldsymbol{w}_0)$ , we can characterize existing approximate unlearning methods as approaches that aim to solve the following optimization problem, initialized at  $\boldsymbol{w} = \boldsymbol{w}_0$ :
+
+$$
+\min  _ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {r}) + \beta \left[ \mathcal {L} \left(\boldsymbol {w} ^ {*}, \mathcal {D}\right) - \mathcal {L} (\boldsymbol {w}, \mathcal {D}) \right], \tag {2}
+$$
+
+where  $\beta$  is a hyperparameter for balancing two different losses. The first term,  $\mathcal{L}(\boldsymbol{w},\mathcal{D}_r)$ , encourages the model to maintain performance on the retain data  $\mathcal{D}_r$ . The second term encourages alignment (or consistency) between the optimized weights  $\boldsymbol{w}$  and the solution  $\boldsymbol{w}^*$ . Note that this type of alignment objective is commonly used when evaluating unlearning methods in recent works [16, 37].
+
+The objective in equation (2) offers a unified explanation for two key approximate unlearning methods: FT and NG. First, setting  $\beta = 0$  simplifies the objective to the objective of FT,  $\min_{\boldsymbol{w}} \mathcal{L}(\boldsymbol{w}, \mathcal{D}_r)$ .
+
+Since FT ignores the second term with  $\boldsymbol{w}^*$  in (2), it results in poor forgetting performance as shown in Fig. 1. For NG, we establish the following lemma:
+
+Lemma 1. For  $\beta = |\mathcal{D}| / |\mathcal{D}_r|$ , which balances the two loss terms based on the number of data points, the objective (2) becomes
+
+$$
+\min  _ {\boldsymbol {w}} \mathcal {L} \left(\boldsymbol {w} ^ {*}, \mathcal {D} _ {r}\right) + \frac {\left| \mathcal {D} _ {f} \right|}{\left| \mathcal {D} _ {r} \right|} \left[ \mathcal {L} \left(\boldsymbol {w} ^ {*}, \mathcal {D} _ {f}\right) - \mathcal {L} \left(\boldsymbol {w}, \mathcal {D} _ {f}\right) \right]. \tag {3}
+$$
+
+Proof. See Appendix.
+
+![](images/9e1c9a59c10a151eaf1b9747a9f6b22db9025b6ffec66d4fc6b50a96d98f8ddb.jpg)
+
+In (3), assuming that there is no prior knowledge of  $\boldsymbol{w}^*$  (i.e.,  $d\boldsymbol{w}^* / d\boldsymbol{w} = 0$ ), the objective is reduced to NG, which depends solely on the gradient  $\nabla_{\boldsymbol{w}} \mathcal{L}(\boldsymbol{w}, \mathcal{D}_f)$ . However, since NG focuses solely on  $\max_{\boldsymbol{w}} \mathcal{L}(\boldsymbol{w}, \mathcal{D}_f)$ , it struggles to maintain accuracy on  $\mathcal{D}_r$ .
+
+Rather than ignoring  $\boldsymbol{w}^*$ , we propose to use a surrogate weight  $\hat{\boldsymbol{w}}$  that characterizes  $\boldsymbol{w}^*$ . A key insight is that  $\boldsymbol{w}^*$  lies within a neighborhood where the forget loss  $\mathcal{L}(\boldsymbol{w},\mathcal{D}_f)$  remains sufficiently high. To formalize this, we introduce the following definition:
+
+Definition 1. ( $\epsilon$ -forget neighborhood) Given parameters  $\boldsymbol{w}$ , forget data  $\mathcal{D}_f$ , and threshold  $\epsilon > 0$ , the  $\epsilon$ -forget neighborhood is defined as:
+
+$$
+\mathcal {B} _ {\Omega} ^ {\epsilon} (\boldsymbol {w}; \mathcal {D} _ {f}) := \left\{\tilde {\boldsymbol {w}} \in \mathbb {R} ^ {d} \mid \mathcal {L} (\tilde {\boldsymbol {w}}, \mathcal {D} _ {f}) \geq \epsilon , \| \boldsymbol {w} - \tilde {\boldsymbol {w}} \| \leq \Omega \right\}. \tag {4}
+$$
+
+This set characterizes the region where a weight has a forget loss of at least  $\epsilon$ , ensuring that the influence of the forget data is sufficiently removed. Therefore, for an appropriately chosen  $\epsilon$ , we have  $\boldsymbol{w}^{*} \in \mathcal{B}_{\Omega}^{\epsilon}(\boldsymbol{w}_{0}; \mathcal{D}_{f})$ . Since the exact  $\boldsymbol{w}^{*}$  is intractable, we introduce a surrogate weight  $\hat{\boldsymbol{w}}$  to characterize the high-forget-loss characteristic as follows:
+
+$$
+\hat {\boldsymbol {w}} \triangleq \underset {\| \boldsymbol {\delta} \| _ {2} \leq \rho} {\arg \max } \mathcal {L} (\boldsymbol {w} + \boldsymbol {\delta}, \mathcal {D} _ {f}), \tag {5}
+$$
+
+where  $\rho$  is a radius that satisfies  $\hat{w} \in \mathcal{B}_{\Omega}^{\epsilon}(\boldsymbol{w}_0; \mathcal{D}_f)$ . While this surrogate weight  $\hat{w}$  becomes dynamic in contrast to the fixed optimal weight  $\boldsymbol{w}^*$ , it provides a practical way of approximating the behavior of  $\boldsymbol{w}^* \in \mathcal{B}_{\Omega}^{\epsilon}(\boldsymbol{w}_0; \mathcal{D}_f)$ . Substituting this surrogate weight into (3) reformulates the problem as the following min-max optimization:
+
+$$
+\min  _ {\boldsymbol {w}} \mathcal {L} (\underset {\| \boldsymbol {\delta} \| _ {2} \leq \rho} {\arg \max } \mathcal {L} (\boldsymbol {w} + \boldsymbol {\delta}, \mathcal {D} _ {f}), \mathcal {D} _ {r}) + \frac {| \mathcal {D} _ {f} |}{| \mathcal {D} _ {r} |} [ \mathcal {L} (\underset {\| \boldsymbol {\delta} \| _ {2} \leq \rho} {\arg \max } \mathcal {L} (\boldsymbol {w} + \boldsymbol {\delta}, \mathcal {D} _ {f}), \mathcal {D} _ {f}) - \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {f}) ]. \tag {6}
+$$
+
+This min-max optimization can be simplified into an efficient algorithm by approximating the inner maximization problem under the first-order approximation.
+
+Theorem 1. (Efficient min-max optimization for approximate unlearning) Suppose that  $\mathcal{L}(\boldsymbol{w},\mathcal{D}_f)$  can be locally approximated by its first-order Taylor expansion around  $\boldsymbol{w}$ . Then, the min-max optimization objective in (6) can be simplified to:
+
+$$
+\min  _ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w} + \rho \frac {\nabla_ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w} , \mathcal {D} _ {f})}{\left| \left| \nabla_ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w} , \mathcal {D} _ {f}) \right| \right| _ {2} ^ {2}}, \mathcal {D} _ {r}). \tag {7}
+$$
+
+Proof. Applying a first-order Taylor expansion, the inner maximization of (6) can be approximated to
+
+$$
+\max  _ {| | \boldsymbol {\delta} | | _ {2} \leq \rho} \mathcal {L} (\boldsymbol {w} + \boldsymbol {\delta}, \mathcal {D} _ {f}) \approx \max  _ {| | \boldsymbol {\delta} | | _ {2} \leq \rho} \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {f}) + \boldsymbol {\delta} ^ {T} \nabla_ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {f}). \tag {8}
+$$
+
+Let us denote  $\mathcal{L}_f(\boldsymbol{w}) \triangleq \mathcal{L}(\boldsymbol{w}, \mathcal{D}_f)$ . The solution to this optimization problem, known as standard dual-norm arguments [7, 28], is given explicitly by  $\delta = \rho \nabla_{\boldsymbol{w}} \mathcal{L}_f(\boldsymbol{w}) / ||\nabla_{\boldsymbol{w}} \mathcal{L}_f(\boldsymbol{w})||_2^2$ . Unless specified otherwise,  $\nabla = \nabla_{\boldsymbol{w}}$ . Substituting the solution  $\delta$  into the second term on the right-hand side of (6), we have
+
+$$
+\begin{array}{l} \mathcal {L} _ {f} (\boldsymbol {w} + \rho \frac {\nabla \mathcal {L} _ {f} (\boldsymbol {w})}{\| \nabla \mathcal {L} _ {f} (\boldsymbol {w}) \| _ {2} ^ {2}}) - \mathcal {L} _ {f} (\boldsymbol {w}) \approx \left[ \mathcal {L} _ {f} (\boldsymbol {w}) + \left(\rho \frac {\nabla \mathcal {L} _ {f} (\boldsymbol {w})}{\| \nabla \mathcal {L} _ {f} (\boldsymbol {w}) \| _ {2} ^ {2}}\right) ^ {T} \nabla \mathcal {L} _ {f} (\boldsymbol {w}) \right] - \mathcal {L} _ {f} (\boldsymbol {w}) (9) \\ \approx \left[ \mathcal {L} _ {f} (\boldsymbol {w}) + \rho \right] - \mathcal {L} _ {f} (\boldsymbol {w}) = \rho . (10) \\ \end{array}
+$$
+
+![](images/e29e33004b5fa6075ef09354794a8e9378e40397c9d10491c24c84e16744548e.jpg)  
+(a) Forget accuracy (ideal case: zero-forget acc.)
+
+![](images/c2c25f4bab418d55bcfa3219ef6c7c4c7bbe3b960cab3f7c5c78d201d4362e81.jpg)  
+(b) Training Time  
+Figure 2: Effectiveness of  $\gamma$ . (CIFAR-10, Class-wise forgetting) We compare three optimizations: w/ Hessian corresponds to the exact computation of (12); w/o Hessian corresponds to the omission of the second-order gradient term in (13); Ours indicates the relaxed optimization in (15). While w/ Hessian demonstrates the most rapid decrease in forget accuracy, it requires high computational cost. In contrast, w/o Hessian is faster but often fails to reduce the forget accuracy sufficiently. Our relaxed optimization efficiently reduces the forget accuracy in a few steps with practical training time.
+
+Hence, the optimization (6) can be approximated to
+
+$$
+\min  _ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w} + \rho \frac {\nabla \mathcal {L} (\boldsymbol {w} , \mathcal {D} _ {f})}{\| \nabla \mathcal {L} (\boldsymbol {w} , \mathcal {D} _ {f}) \| _ {2} ^ {2}}, \mathcal {D} _ {r}) + \frac {| \mathcal {D} _ {f} |}{| \mathcal {D} _ {r} |} \rho . \tag {11}
+$$
+
+Since the second term  $\frac{|\mathcal{D}_f|}{|\mathcal{D}_r|}\rho$  does not depend on  $\pmb{w}$ , we have the simplified optimization objective stated in the theorem.
+
+Theorem 1 allows us to directly apply stochastic gradient descent. The gradient of the objective function (7) can be explicitly computed using the following lemma.
+
+Lemma 2. (Update gradient of (7)) The update gradient of the objective function in (7) is given by:
+
+$$
+\nabla_ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w} + \delta (\boldsymbol {w}), \mathcal {D} _ {r}) = \left[ \mathbf {I} + \frac {\rho}{\| \nabla_ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w} , \mathcal {D} _ {f}) \| _ {2} ^ {2}} (\mathbf {I} - 2 \mathbf {P} _ {f}) \mathbf {H} _ {f} \right] \nabla_ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {r}) | _ {\boldsymbol {w} + \delta (\boldsymbol {w})}, \tag {12}
+$$
+
+where  $\mathbf{P}_f$  is the orthogonal projection matrix onto the space spanned by  $\nabla_{\boldsymbol{w}}\mathcal{L}(\boldsymbol{w},\mathcal{D}_f)$ , and  $\mathbf{H}_f$  denotes the Hessian of  $\mathcal{L}(\boldsymbol{w},\mathcal{D}_f)$ .
+
+Proof. By chain rule,
+
+$$
+\nabla_ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w} + \delta (\boldsymbol {w}), \mathcal {D} _ {r}) = \nabla_ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {r}) | _ {\boldsymbol {w} + \delta (\boldsymbol {w})} + \frac {d \delta (\boldsymbol {w})}{d \boldsymbol {w}} \nabla_ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {r}) | _ {\boldsymbol {w} + \delta (\boldsymbol {w})}. \tag {13}
+$$
+
+Let  $g(\boldsymbol{w}) \triangleq \nabla_{\boldsymbol{w}} \mathcal{L}(\boldsymbol{w}, \mathcal{D}_f)$  and  $r(\boldsymbol{w}) \triangleq ||g(\boldsymbol{w})||_2^2$ . Then,  $\delta(\boldsymbol{w}) = \rho g(\boldsymbol{w}) / r(\boldsymbol{w})$ . Since  $\frac{dg(\boldsymbol{w})}{dw} = \mathbf{H}_f$  and  $\frac{dr(\boldsymbol{w})}{dw} = 2g(\boldsymbol{w})^T \mathbf{H}_f$ , we have
+
+$$
+\frac {d \delta (\boldsymbol {w})}{d \boldsymbol {w}} = \frac {\rho}{r (\boldsymbol {w})} \left(\mathbf {I} - 2 \frac {g (\boldsymbol {w}) g (\boldsymbol {w}) ^ {T}}{r (\boldsymbol {w})}\right) \mathbf {H} _ {f}. \tag {14}
+$$
+
+![](images/4cc3eecf10f3b53273e97766de8681f424735037e4d5d2fdcf89390068c07fed.jpg)
+
+In (12), the computation of the Hessian matrix is computationally expensive [7, 15, 18]. Therefore, some might suggest that omitting the second term  $\frac{d\delta(\boldsymbol{w})}{dw}\nabla_{\boldsymbol{w}}\mathcal{L}(\boldsymbol{w},\mathcal{D}_r)|_{\boldsymbol{w} + \delta (\boldsymbol{w})}$  in (13), which is a technique used in several prior works across different domains [7, 10]. However, we find that the second term is crucial for effective unlearning. In Fig. 2, we compare the forget accuracy and the averaged training time of (12) and its variations. Although omitting the second term (w/o Hessian) reduces computational cost, it fails to reduce the forget accuracy. On the other hand, computing the exact (12) (w/Hessian) achieves a lower forget accuracy, but it requires nearly three times the computational cost.
+
+![](images/2e8f60c45c373553f732fb67853a88800e6df17c51c127d36d255689830c5eb1.jpg)  
+(a)  $\cos (\theta)\leq 0$  satisfies objective alignment.
+
+![](images/4a2adb7189d2a21243eac81d8f42c193d563381520292e1a851f920b4eb7c49e.jpg)  
+(b)  $\cos (\theta)$  for FT and UAM  
+Figure 3: Geometric interpretation of UAM. (CIFAR-10, Class-wise forgetting) Compared to FT, UAM explicitly minimizes the cosine similarity,  $\cos (\theta)$ , between the retain gradient and the forget gradient. (a) When  $\cos (\theta)\leq 0$ , reducing the retain loss inherently leads to an increase in the forget loss. (b) The moving average of  $\cos (\theta)$  in UAM shows a clear decreasing trend, with negative values.
+
+Given the computational burden of exact Hessian computation, we find that approximating the Hessian matrix with the identity matrix can be a simple yet effective solution. This yields an efficient gradient update by introducing a hyperparameter  $\gamma$ :
+
+$$
+\left[ \mathbf {I} - \gamma \mathbf {P} _ {f} \right] \nabla_ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {r}) | _ {\boldsymbol {w} + \delta (\boldsymbol {w})}. \tag {15}
+$$
+
+As shown in Fig. 2, our relaxed optimization effectively reduces the forget accuracy with a small increase in computational time since the forget gradient  $\nabla_{\boldsymbol{w}}\mathcal{L}(\boldsymbol{w},\mathcal{D}_{f})$  is already computed during the inner maximization. Our analysis with the projection matrix  $\mathbf{P}_f$  offers a theoretical explanation for recent work [17], which utilizes the projection of retain and forget gradients to achieve unlearning in image generation tasks. For a more detailed discussion, including an ablation study on  $\gamma$ , please refer to Appendix. This optimization can be efficiently implemented using automatic differentiation frameworks, such as PyTorch. We name this framework Unlearning-Aware Minimization (UAM).
+
+Geometric interpretation of UAM. We provide a deeper analysis of our proposed objective from a geometric point of view. By applying a first-order Taylor expansion around  $\boldsymbol{w}$  on (7), we have:
+
+$$
+\min  _ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w} + \rho \frac {\nabla \mathcal {L} (\boldsymbol {w} , \mathcal {D} _ {f})}{\| \nabla \mathcal {L} (\boldsymbol {w} , \mathcal {D} _ {f}) \| _ {2} ^ {2}}, \mathcal {D} _ {r}) \approx \min  _ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {r}) + \nabla \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {r}) ^ {\top} \rho \frac {\nabla \mathcal {L} (\boldsymbol {w} , \mathcal {D} _ {f})}{\| \nabla \mathcal {L} (\boldsymbol {w} , \mathcal {D} _ {f}) \| _ {2} ^ {2}}. \tag {16}
+$$
+
+Compared to FT, which only optimizes the first term  $\min_{\boldsymbol{w}} \mathcal{L}(\boldsymbol{w}, \mathcal{D}_r)$  in (16), UAM explicitly minimizes the inner product between  $\nabla \mathcal{L}(\boldsymbol{w}, \mathcal{D}_r)$  and  $\nabla \mathcal{L}(\boldsymbol{w}, \mathcal{D}_f)$ , which is proportional to their cosine similarity. This implicit objective encourages the gradients  $\nabla \mathcal{L}(\boldsymbol{w}, \mathcal{D}_r)$  and  $\nabla \mathcal{L}(\boldsymbol{w}, \mathcal{D}_f)$  to become less aligned, or even negatively aligned. From a geometric perspective, as illustrated in Fig. 3a, a negative cosine similarity between these gradients indicates that minimizing the retain loss inherently removes learned information from the forget data.
+
+Fig. 3b shows the historical value of  $\cos(\theta)$  where  $\theta$  is the angle between  $\nabla \mathcal{L}(\boldsymbol{w}, \mathcal{D}_r)$  and  $\nabla \mathcal{L}(\boldsymbol{w}, \mathcal{D}_f)$  during training. To reduce the stochasticity introduced by batch training, we use the first batch to measure  $\cos(\theta)$ . UAM exhibits a clear decreasing trend, resulting in a negative  $\cos(\theta)$  compared to FT. Given the high dimensionality of the parameter space, this negative  $\cos(\theta)$  provides a potential geometric explanation for the strong performance of UAM.
+
+These results also align with the result shown in Fig. 1. We design a synthetic 2D optimization landscape with  $\boldsymbol{w} = [w_{1}, w_{2}]$ , where the forget and retain losses are derived from simple rotated quadratic forms as follows (more details in Appendix):
+
+$$
+\mathcal {L} _ {f} (\boldsymbol {w}) = \frac {1}{5 0} \left[ 3 (w _ {1} - 2) ^ {2} + (w _ {2} + 4) ^ {2} - 6 w _ {2} - 1 0 w _ {1} \right]; \mathcal {L} _ {r} (\boldsymbol {w}) = \max  \left(\frac {5 r _ {\theta} (w _ {1}) ^ {2} + r _ {\theta} (w _ {2}) ^ {2}}{5 0}, \delta\right), \tag {17}
+$$
+
+The optimization is initialized from the point that minimizes the sum of forget loss  $\mathcal{L}_f(\pmb{w})$  and retain loss  $\mathcal{L}_r(\pmb{w})$ , representing a pre-trained model. Compared to FT and NG, UAM exhibits a more effective exploration of the loss landscape and demonstrates reliable convergence toward the optimal solution.
+
+Table 1: Machine unlearning performance on CIFAR-10. The values in blue indicate the absolute differences from Retrain. The standard deviation is computed across all classes for class-wise forgetting and across three different random seeds for random data forgetting.  
+
+<table><tr><td>Method</td><td>RA</td><td>FA</td><td>TA</td><td>ΔAcc.(↓)</td><td>MIA-Eff.</td><td>Time</td></tr><tr><td colspan="7">Class-wise forgetting</td></tr><tr><td>Retrain</td><td>100.00±0.00</td><td>0.00±0.00</td><td>95.32±0.52</td><td>0.00</td><td>100.00±0.00</td><td>31.34</td></tr><tr><td>FT</td><td>100.00±0.00 (0.00)</td><td>43.31±7.31 (43.31)</td><td>95.01±0.55 (0.32)</td><td>43.63</td><td>100.00±0.01 (0.00)</td><td>1.56</td></tr><tr><td>NG</td><td>88.57±5.68 (11.43)</td><td>0.80±1.40 (0.80)</td><td>83.09±4.98 (12.23)</td><td>24.45</td><td>99.34±1.15 (0.66)</td><td>1.57</td></tr><tr><td>FF</td><td>99.98±0.05 (0.02)</td><td>9.96±7.30 (9.96)</td><td>95.18±0.66 (0.24)</td><td>10.22</td><td>100.00±0.00 (0.00)</td><td>0.97</td></tr><tr><td>IU</td><td>94.24±4.13 (5.76)</td><td>0.27±0.48 (0.27)</td><td>88.02±3.95 (7.29)</td><td>13.32</td><td>99.99±0.02 (0.01)</td><td>0.52</td></tr><tr><td>l1-sparse</td><td>100.00±0.00 (0.00)</td><td>0.00±0.00 (0.00)</td><td>91.87±0.84 (3.45)</td><td>3.45</td><td>100.00±0.00 (0.00)</td><td>1.66</td></tr><tr><td>UAM</td><td>100.0±0.00 (0.01)</td><td>0.00±0.00 (0.00)</td><td>94.51±0.63 (0.81)</td><td>0.82</td><td>100.00±0.00 (0.00)</td><td>2.56</td></tr><tr><td colspan="7">Random data forgetting</td></tr><tr><td>Retrain</td><td>100.00±0.00</td><td>95.33±0.39</td><td>94.73±0.14</td><td>0.00</td><td>11.86±0.34</td><td>31.32</td></tr><tr><td>FT</td><td>100.00±0.00 (0.00)</td><td>99.66±0.09 (4.33)</td><td>94.58±0.04 (0.15)</td><td>4.48</td><td>4.43±0.36 (7.43)</td><td>1.56</td></tr><tr><td>NG</td><td>61.97±4.29 (38.03)</td><td>53.50±2.83 (41.83)</td><td>58.83±4.04 (35.90)</td><td>115.77</td><td>46.55±2.69 (34.69)</td><td>1.56</td></tr><tr><td>FF</td><td>65.78±48.19 (34.22)</td><td>65.49±48.31 (29.85)</td><td>62.37±45.15 (32.36)</td><td>96.43</td><td>10.30±0.50 (1.56)</td><td>0.96</td></tr><tr><td>IU</td><td>97.30±2.44 (2.70)</td><td>97.05±2.48 (2.43)</td><td>91.06±2.68 (3.67)</td><td>8.81</td><td>5.26±3.58 (6.60)</td><td>0.53</td></tr><tr><td>l1-sparse</td><td>100.00±0.00 (0.00)</td><td>99.52±0.13 (4.19)</td><td>94.48±0.18 (0.25)</td><td>4.44</td><td>7.14±0.36 (4.72)</td><td>1.66</td></tr><tr><td>UAM</td><td>99.88±0.03 (0.12)</td><td>95.19±0.28 (0.41)</td><td>92.74±0.29 (2.00)</td><td>2.53</td><td>9.16±0.14 (2.70)</td><td>2.55</td></tr></table>
+
+# 4 Experiments
+
+In this section, we conduct experiments on two major machine unlearning tasks: image classification and multiple-choice question-answering with large language model (LLM).
+
+# 4.1 Image Classification
+
+Setup and Methods. We conduct image classification experiments using three datasets: CIFAR-10, CIFAR-100 [19], and TinyImageNet [5]. We adopt ResNet-18 [12] for the CIFAR datasets and VGG [26] for TinyImageNet. For each dataset, we evaluate two unlearning scenarios: class-wise forgetting and random data forgetting. In the class-wise forgetting scenario, the forget set  $\mathcal{D}_f$  consists of all training samples from a single class. We report the mean and standard deviation over 10 different classes chosen for forgetting. In the random data forgetting scenario,  $\mathcal{D}_f$  consists of randomly sampled training examples across all classes. Results are averaged over three different random seeds.
+
+We explore four representative unlearning frameworks: Fine-tuning (FT) [9, 32], Negative Gradient (NG) [9], Fisher Forgetting (FF) [9], Influence Unlearning (IU), [15, 18], and  $\ell_1$ -sparse [16]. FF leverages the Fisher Information Matrix to identify and mask parameters most sensitive to the forget data. IU uses the influence of each data point on the parameters.  $\ell_1$ -sparse encourages parameter sparsity by using  $\ell_1$  penalty during fine-tuning. To ensure a fair comparison under equivalent computational budgets, we use 10 epochs for both FT and NG, and 5 epochs for UAM as it uses two gradient computations per iteration. Additional details on experimental settings are provided in Appendix.
+
+Evaluation metrics and Results. We report four metrics: Retain Accuracy (RA), Forget Accuracy (FA), Test Accuracy (TA), and Membership Inference Attack Efficiency (MIA-Eff.). Following [16], MIA-Eff. denotes a proportion of true negatives normalized by the size of  $\mathcal{D}_f$  by applying the confidence-based MIA predictor [27, 34]. As an ideal baseline, we use a retrain model (Retrain) that is trained from scratch without access to  $\mathcal{D}_f$ . To ease comparison, we define an accuracy gap  $\Delta \mathbf{Acc}$  as the sum of absolute differences of accuracies:
+
+$$
+\Delta \mathbf {A c c}. \triangleq \sum_ {\mathcal {A} \in \{\mathrm {R A}, \mathrm {F A}, \mathrm {T A} \}} | \mathcal {A} _ {\text {R e t r a i n}} - \mathcal {A} |, \tag {18}
+$$
+
+where lower values indicate better performance. We also estimate the runtime efficiency of each method, measured in minutes and denoted as Time.
+
+In Table 1, under class-wise forgetting, UAM shows a zero-forget accuracy, which is identical to that of Retrain. Specifically, low FA is observed for NG and IU, but these methods sacrifice more than  $7\%$  in TA. NG shows convergence instability under random data forgetting, which was also observed in Fig. 1. In contrast, UAM maintains near-zero FA while achieving high TA. These results lead to the
+
+Table 3: Machine unlearning performance on Tiny-ImageNet. The values in blue indicate the absolute differences from Retrain. The standard deviation is computed across all classes for class-wise forgetting and across three different random seeds for random data forgetting. Note: FF could not be executed due to memory limitations.  
+
+<table><tr><td>Method</td><td>RA</td><td>FA</td><td>TA</td><td>ΔAcc.(↓)</td><td>MIA-Eff.</td><td>Time</td></tr><tr><td colspan="7">Class-wise forgetting</td></tr><tr><td>Retrain</td><td>99.98±0.00</td><td>0.00±0.00</td><td>62.36±0.34</td><td>0.00</td><td>100.00±0.00</td><td>342.12</td></tr><tr><td>FT</td><td>85.86±0.47 (14.12)</td><td>39.27±12.16 (39.27)</td><td>45.76±0.17 (16.59)</td><td>69.98</td><td>83.46±7.90 (16.54)</td><td>17.10</td></tr><tr><td>NG</td><td>88.07±5.98 (11.91)</td><td>0.29±0.43 (0.29)</td><td>49.48±3.68 (12.87)</td><td>25.07</td><td>99.74±0.44 (0.26)</td><td>17.13</td></tr><tr><td>IU</td><td>98.74±1.47 (1.24)</td><td>1.09±1.47 (1.09)</td><td>57.20±2.02 (5.16)</td><td>7.49</td><td>99.90±0.33 (0.10)</td><td>2.56</td></tr><tr><td>l1-sparse</td><td>98.19±0.16 (1.79)</td><td>0.00±0.00 (0.00)</td><td>59.66±0.19 (2.70)</td><td>4.48</td><td>100.00±0.00 (0.00)</td><td>17.30</td></tr><tr><td>UAM</td><td>99.97±0.02 (0.01)</td><td>0.23±0.26 (0.23)</td><td>60.86±0.64 (1.50)</td><td>1.75</td><td>99.97±0.08 (0.03)</td><td>21.05</td></tr><tr><td colspan="7">Random data forgetting</td></tr><tr><td>Retrain</td><td>99.98±0.00</td><td>61.23±0.63</td><td>61.58±0.51</td><td>0.00</td><td>66.10±0.05</td><td>334.53</td></tr><tr><td>FT</td><td>99.98±0.00 (0.00)</td><td>99.96±0.02 (38.73)</td><td>62.16±0.08 (0.62)</td><td>39.35</td><td>4.16±0.07 (61.94)</td><td>17.01</td></tr><tr><td>NG</td><td>0.53±0.02 (99.45)</td><td>0.54±0.02 (60.68)</td><td>0.53±0.01 (61.05)</td><td>221.19</td><td>0.54±0.02 (65.55)</td><td>16.89</td></tr><tr><td>IU</td><td>82.20±9.62 (17.78)</td><td>80.02±10.35 (18.79)</td><td>46.56±5.13 (15.02)</td><td>51.59</td><td>20.48±5.42 (45.62)</td><td>2.58</td></tr><tr><td>l1-sparse</td><td>98.86±0.05 (1.12)</td><td>59.66±0.55 (1.57)</td><td>58.16±0.23 (3.43)</td><td>6.12</td><td>54.54±0.62 (11.56)</td><td>17.09</td></tr><tr><td>UAM</td><td>97.61±0.89 (2.37)</td><td>69.88±1.06 (8.65)</td><td>56.37±0.38 (5.22)</td><td>16.23</td><td>46.63±1.06 (19.46)</td><td>21.11</td></tr></table>
+
+lowest  $\Delta$  Acc. and the low gap of MIA-Eff, demonstrating that UAM converges to a better optimum. The results on CIFAR-100 are presented in Appendix. In Table 3, we observe superior performance of UAM on Tiny-ImageNet under class-wise forgetting. Under random data forgetting, while  $\ell_1$ -sparse outperforms UAM, both methods exhibit relatively high  $\Delta$  Acc compared to other methods.
+
+Selective parameter updates with UAM. While the methods discussed above update the full set of model parameters, recent work [6] proposed SalUN, a method that updates only a subset of parameters during training. Since UAM is easily extensible to such selective updating strategies, we conduct an additional experiment on SalUN and UAM. As shown in Table 2, the integration of UAM improves results on CIFAR-10 under both class-wise and random data forgetting settings, demonstrating its potential to improve selective parameter update methods.
+
+Table 2: SalUN without and with UAM on CIFAR-10.  
+
+<table><tr><td>Method</td><td>ΔAcc.(↓)</td><td>MIA-Eff.</td></tr><tr><td colspan="3">Class-wise forgetting</td></tr><tr><td>SalUN</td><td>1.46</td><td>100.00±0.00 (0.00)</td></tr><tr><td>+UAM</td><td>0.82</td><td>100.00±0.00 (0.00)</td></tr><tr><td colspan="3">Random data forgetting</td></tr><tr><td>SalUN</td><td>3.00</td><td>97.81±0.97 (85.95)</td></tr><tr><td>+UAM</td><td>2.56</td><td>10.93±1.73 (1.00)</td></tr></table>
+
+# 4.2 Multiple-Choice Question-Answering with Large Language Model
+
+Setup and Methods. For LLM unlearning task, we evaluate unlearning of hazardous knowledge using the WMDP benchmark [22], a four-way multiple-choice question-answering (Q&A) dataset covering two domains: biosecurity and cybersecurity. Following prior work [22], we use Zephyr-7B- $\beta$  [30] as a baseline model, WMDP-Bio and WMDP-Cyber as the forget data  $\mathcal{D}_f$ , and WikiText as the retain data  $\mathcal{D}_r$ .
+
+We consider four different LLM unlearning methods: SSD [8] selectively dampens parameters associated with the forget data using the diagonal of the Fisher Information Matrix; SCRUB [20] employs a teacher-student framework optimized via KL-divergence; LLMU [33] uses an additional random loss to enhance forgetting, alongside forget and retain losses; RMU [22] leverages frozen feature representations obtained prior to unlearning and optimizes a mean squared error loss composed of a forget loss, defined between the feature representation of forget data  $z_{f}$  and a fixed random unit vector  $c$ , and a retain loss, computed between the feature representation of retain data  $\hat{z}_r$  and that of the frozen model,  $\mathcal{L}(z_r,\hat{z}_r)$ .
+
+Since UAM is a framework that does not rely on a fixed loss function, it can be easily integrated into the RMU framework. A key advantage of UAM is the elimination of the fixed random unit vector  $\pmb{c}$ , which was scaled by manually tuned coefficients in RMU. Specifically, UAM employs  $\mathcal{L}(z_f, \hat{z}_f)$  as the inner maximization objective. By unifying the loss functions of the forget and retain losses, UAM achieves superior performance compared to RMU. We highlight that the success of UAM suggests that a fixed random vector may not be a necessary component for effective unlearning in large language models. The detailed algorithmic description is provided in Appendix.
+
+Table 4: Machine unlearning performance on LLM unlearning. We use Q&A datasets, MMLU and WMDP benchmarks, with Zephyr-7B. The values in blue indicate the differences from Base.  
+
+<table><tr><td>Method</td><td>MMLU (↑)</td><td>WMDP-Bio (↓)</td><td>WMDP-Cyber (↓)</td><td>ΔAcc. (↓)</td></tr><tr><td>Base</td><td>0.5810</td><td>0.6370</td><td>0.4400</td><td>0.0000</td></tr><tr><td>SSD</td><td>0.4070 (−0.1740)</td><td>0.5020 (−0.1350)</td><td>0.3500 (−0.0900)</td><td>−0.0510</td></tr><tr><td>SCRUB</td><td>0.5120 (−0.0690)</td><td>0.4380 (−0.1990)</td><td>0.3930 (−0.0470)</td><td>−0.1770</td></tr><tr><td>LLMU</td><td>0.4470 (−0.1340)</td><td>0.5950 (−0.0420)</td><td>0.3950 (−0.0450)</td><td>0.0470</td></tr><tr><td>RMU</td><td>0.5660 (−0.0150)</td><td>0.3103 (−0.3267)</td><td>0.2763 (−0.1637)</td><td>−0.4754</td></tr><tr><td>UAM</td><td>0.5644 (−0.0166)</td><td>0.2930 (−0.3440)</td><td>0.2330 (−0.2070)</td><td>−0.5344</td></tr></table>
+
+![](images/14160e5311e9a01c02891b131db213c96e1a1e5c355a982f59c2019bb5b1014e.jpg)  
+Hazardous Prompt
+
+How would I engineer influenza A to be significantly more virulent?
+
+# LLM without UAM
+
+![](images/bf5ae4170d088e54e458f8c1aa60a75863cd757df3d1de97ad7c4c77e1ff2280.jpg)  
+Figure 4: Effectiveness of UAM in mitigating hazardous outputs. The responses of Zephyr-7B- $\beta$  to a hazardous prompt before and after unlearning with UAM. Prior to unlearning, the model generates detailed suggestions for engineering a more virulent strain of influenza A. After unlearning by UAM, the model refrains from providing any such hazardous information, ensuring safer behavior.
+
+I'm not an expert in virology, but I can provide some general information.
+
+To engineer a virus to be more virulent, ... Here are some possible strategies:
+
+1. Enhance viral entry: you could
+
+introduce mutations in the viral surface ...
+
+# LLM unlearned by UAM
+
+from the English, perhaps the most well
+
+known of all time, is the a first, would
+
+be the first to come to be a part of a Tb.
+
+to ensure that the firstb would be a to to
+
+to you to m a to a 3b first a 1b a 4
+
+already ...
+
+Evaluation metrics and Results. We evaluate each method on three benchmarks following prior work [22, 33]. The MMLU benchmark [13] estimates general language understanding performance, where higher accuracy indicates better performance. In contrast, the WMDP-Bio and WMDP-Cyber benchmarks measure the effectiveness of unlearning methods in mitigating the risk of LLMs assisting in biological and cyber weapon development, respectively. Lower accuracy indicates better performance. To ease comparison, we define an accuracy gap that captures improvement over the base model in both generality and unlearning effectiveness:
+
+$$
+\Delta \mathbf {A c c} \triangleq \left(\mathrm {M M L U} _ {\text {B a s e}} - \mathrm {M M L U}\right) - \left[ \left(\mathrm {W M D P} - \mathrm {B i o} _ {\text {B a s e}} - \mathrm {W M D P} - \mathrm {B i o}\right) + \left(\mathrm {W M D P} - \mathrm {C y b e r} _ {\text {B a s e}} - \mathrm {W M D P} - \mathrm {C y b e r}\right) \right]. \tag {19}
+$$
+
+The first term,  $\mathrm{MMLU_{Base}} - \mathrm{MMLU}$  estimates the loss in generality, while the remaining term captures the effectiveness of unlearning. Therefore, lower values indicate better performance.
+
+In Table 4, UAM demonstrates effective removal of harmful knowledge from WMDP, with less degradation in performance on MMLU. UAM achieves the lowest scores on both WMDP-Bio (0.2930) and WMDP-Cyber (0.2330). Since lower values on these metrics indicate more effective forgetting, UAM demonstrates superior unlearning capability compared to all other methods. Compared to RMU, while UAM shows only a marginal drop in MMLU accuracy (0.0016), it achieves a greater reduction in harmful knowledge on WMDP-Cyber (0.0433). This demonstrates that UAM converges to a better optimum, achieving superior unlearning performance with minimal loss in general capabilities.
+
+In Fig. 4, we present a dialogue example that illustrates the behavior of model before and after unlearning with UAM. As a hazardous prompt, we prompt the model with the same query used in [22], "how to engineer influenza A to become more virulent". Before unlearning, the model provides information on virus engineering; however, after being unlearned by UAM, the response becomes masked instead. This demonstrates the potential of UAM to prevent LLMs from generating dangerous or unethical content, aligning more closely with safety constraints. More examples can be found in Appendix.
+
+# 5 Limitation and Discussion
+
+Several promising directions remain for future research. On the theoretical side, although we empirically demonstrate that approximating the Hessian matrix in (12) with the identity matrix significantly reduces computational cost and is effective for unlearning, a formal theoretical understanding of why this approximation works in the context of machine unlearning remains open. The use of surrogate weight and first-order approximations can also be analyzed. Empirically, while our method achieves superior performance compared to existing approaches, it still requires two forward and backward passes per iteration, which may introduce computational overhead in certain scenarios. Moreover, exploring variations of our method for cases where the forget data are only accessible could be particularly interesting. Future work may also focus on developing more efficient algorithmic and implementation-level optimizations.
+
+# 6 Conclusion
+
+In this work, we revisit the objective of machine unlearning and propose Unlearning-Aware Minimization (UAM), a novel min-max optimization framework that leverages the neighborhood of the current model parameters characterized by a high forget loss. UAM effectively identifies solutions that remove information associated with the forget data while maintaining performance on the retain data. Extensive empirical evaluations on both vision and language benchmarks demonstrate its effectiveness in machine unlearning.
+
+# Acknowledgement
+
+This work was supported by: the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (RS-2025-20252986) (Support contribution:  $40\%$ ); the MSIT (Ministry of Science and ICT), Korea, under the Convergence security core talent training business support program (IITP-2025-RS-2023-00266605) supervised by the IITP (Institute for Information & Communications Technology Planning & Evaluation) (Support contribution:  $30\%$ ); the IITP(Institute of Information & Communications Technology Planning & Evaluation)-ITRC(Information Technology Research Center) grant funded by the Korea government(Ministry of Science and ICT)(IITP-2025-RS-2024-00438056) (Support contribution:  $30\%$ )
+
+# References
+
+[1] Ilan Adler. The equivalence of linear programs and zero-sum games. International Journal of Game Theory, 42:165-177, 2013.  
+[2] Lucas Bourtoule, Varun Chandrasekaran, Christopher A Choquette-Choo, Hengrui Jia, Adelin Travers, Baiwu Zhang, David Lie, and Nicolas Papernot. Machine unlearning. In 2021 IEEE symposium on security and privacy (SP), pages 141-159. IEEE, 2021.  
+[3] Yinzhi Cao and Junfeng Yang. Towards making systems forget with machine unlearning. In 2015 IEEE symposium on security and privacy, pages 463-480. IEEE, 2015.  
+[4] Sungmin Cha, Sungjun Cho, Dasol Hwang, Honglak Lee, Taesup Moon, and Moontae Lee. Learning to unlearn: Instance-wise unlearning for pre-trained classifiers. In Proceedings of the AAAI conference on artificial intelligence, volume 38, pages 11186-11194, 2024.  
+[5] Stanford CS231N. Tiny imagenet visual recognition challenge, 2015. URL https://vision.stanford.edu/teaching/cs231n/reports/2015/pdfs/yle_project.pdf.  
+[6] Chongyu Fan, Jiancheng Liu, Yihua Zhang, Eric Wong, Dennis Wei, and Sijia Liu. Salun: Empowering machine unlearning via gradient-based weight saliency in both image classification and generation. In The Twelfth International Conference on Learning Representations, 2024. URL https://openreview.net/forum?id=gnOmIhQGNM.  
+[7] Pierre Foret, Ariel Kleiner, Hossein Mobahi, and Behnam Neyshabur. Sharpness-aware minimization for efficiently improving generalization. In International Conference on Learning Representations, 2021. URL https://openreview.net/forum?id=6Tm1mposlrM.
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+
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+
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+
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+
+# A Proof of Lemma 1
+
+By definition,
+
+$$
+\mathcal {L} (\boldsymbol {w}, \mathcal {D}) = \frac {\left| \mathcal {D} _ {r} \right|}{\left| \mathcal {D} \right|} \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {r}) + \frac {\left| \mathcal {D} _ {f} \right|}{\left| \mathcal {D} \right|} \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {f}). \tag {20}
+$$
+
+Substituting this into (2), we obtain:
+
+$$
+\begin{array}{l} \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {r}) + \beta \mathcal {L} (\boldsymbol {w} ^ {*}, \mathcal {D}) - \beta \mathcal {L} (\boldsymbol {w}, \mathcal {D}) \\ = \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {r}) + \mathcal {L} (\boldsymbol {w} ^ {*}, \mathcal {D} _ {r}) + \frac {| \mathcal {D} _ {f} |}{| \mathcal {D} _ {r} |} \mathcal {L} (\boldsymbol {w} ^ {*}, \mathcal {D} _ {f}) - \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {r}) - \frac {| \mathcal {D} _ {f} |}{| \mathcal {D} _ {r} |} \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {f}) \\ = \mathcal {L} (\boldsymbol {w} ^ {*}, \mathcal {D} _ {r}) + \frac {| \mathcal {D} _ {f} |}{| \mathcal {D} _ {r} |} \left[ \mathcal {L} (\boldsymbol {w} ^ {*}, \mathcal {D} _ {f}) - \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {f}) \right]. \\ \end{array}
+$$
+
+# B Ablation Study
+
+# B.1 Hessian Matrix and Approximation
+
+In Section 3, we discuss two different update approaches,
+
+$$
+\left[ \mathbf {I} + \frac {\rho}{\left| \left| \nabla_ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w} , \mathcal {D} _ {f}) \right| \right| _ {2} ^ {2}} (\mathbf {I} - 2 \mathbf {P} _ {f}) \mathbf {H} _ {f} \right] \left. \nabla_ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {r}) \right| _ {\boldsymbol {w} + \delta (\boldsymbol {w})} \tag {12}
+$$
+
+and
+
+$$
+\left[ \mathbf {I} - \gamma \mathbf {P} _ {f} \right] \nabla_ {\boldsymbol {w}} \mathcal {L} (\boldsymbol {w}, \mathcal {D} _ {r}) | _ {\boldsymbol {w} + \delta (\boldsymbol {w})}. \tag {15}
+$$
+
+The first approach uses the Hessian matrix to update the weights, while the second approximates it with the identity matrix to reduce computational costs. In this section, we compare and analyze these methods. For (12), instead of directly calculating the matrix  $(\mathbf{I} - 2\mathbf{P}_f)\mathbf{H}_f$ , we leverage torch.autograd.grad with the grad_outputs argument in PyTorch to efficiently compute  $\mathbf{H}_f\nabla_w\mathcal{L}(\boldsymbol{w},\mathcal{D}_r)$ , and then calculate the remaining terms.
+
+![](images/aa5da8364b9b9f761194de5c3864cfb7cd3b768f9bf26ae36f50903eda27f993.jpg)
+
+![](images/d7e0f2d6a7501ded45dd8d98f0a1fe3be2144a690efc1e8783531ba0e771ce8e.jpg)  
+Figure 5: Ablation study on (12) using Hessian.
+
+![](images/2574021ae0f77def26f99f978ce6bd761bece491c3e8c5a269405ebfb4cb5c4b.jpg)
+
+![](images/fbde9d3383cdeeee72f00cf486d30c93e5464ad87a7956ee1eb6fe979009945e.jpg)  
+Figure 6: Ablation study on (15) using  $\gamma$ .
+
+Fig. 5 shows the forget and retain accuracies of (12) during unlearning on CIFAR-10 under class-wise forgetting. As  $\rho$  increases, the forget accuracy decreases rapidly. However, when  $\rho = 5 \times 10^{-2}$ , the model exhibits a sudden collapse at approximately 750 iterations, where the retain accuracy drops to  $20\%$ . In the context of (12),  $\rho$  not only influences the neighborhood size, but also affects the magnitude of the gradient  $\rho / ||\nabla_{\boldsymbol{w}}\mathcal{L}(\boldsymbol{w},\mathcal{D}_f)||_2^2$ . Thus, controlling  $\rho$  influences the magnitude of the gradient, which may require careful adjustment.
+
+In contrast, controlling  $\rho$  in (15) does not influence the magnitude of the gradient. Instead, the effect is controlled by the parameter  $\gamma$ . As shown in Fig. 6, we observe that varying  $\gamma$  results in more stable outcomes, even when  $\rho = 5 \times 10^{-2}$  is used. Increasing  $\gamma$  yields a rapid decrease in the forget accuracy, while it shows high retain accuracy above  $98\%$  at the end of unlearning. Considering the high computational cost of (12) introduced in Fig. 2 and the relative stability of (15), we select (15) as our base method.
+
+# B.2 Sensitivity Analysis on  $\gamma$  and  $\rho$
+
+The table below summarizes the results of the sensitivity experiment on  $\rho$  and  $\gamma$ .
+
+Table 5: Sensitivity to  $\rho$  on CIFAR-10 (class-wise forgetting). The values in blue indicate the absolute differences from Retrain. The standard deviation is computed across all classes for class-wise forgetting and across three different random seeds for random data forgetting.  
+
+<table><tr><td>ρ</td><td>RA</td><td>FA</td><td>TA</td><td>ΔAcc. (↓)</td><td>MIA-Eff.</td></tr><tr><td>5e-3</td><td>99.99 ± 0.00 (0.01)</td><td>0.00 ± 0.00 (0.00)</td><td>94.32 ± 0.66 (1.00)</td><td>1.01</td><td>100.00 ± 0.00 (0.00)</td></tr><tr><td>5e-2</td><td>100.00 ± 0.00 (0.01)</td><td>0.00 ± 0.00 (0.00)</td><td>94.51 ± 0.63 (0.81)</td><td>0.82</td><td>100.00 ± 0.00 (0.00)</td></tr><tr><td>5e-1</td><td>100.00 ± 0.00 (0.01)</td><td>0.01 ± 0.04 (0.01)</td><td>93.33 ± 1.08 (1.98)</td><td>1.99</td><td>100.00 ± 0.00 (0.00)</td></tr><tr><td>1e-1</td><td>99.65 ± 0.31 (0.35)</td><td>0.01 ± 0.04 (0.01)</td><td>92.66 ± 1.12 (2.66)</td><td>3.02</td><td>99.99 ± 0.03 (0.01)</td></tr></table>
+
+Table 6: Sensitivity to  $\rho$  on LLM unlearning. We use Q&A datasets, MMLU and WMDP benchmarks, with Zephyr-7B. The values in blue indicate the differences from Base.  
+
+<table><tr><td>ρ</td><td>MMLU (↑)</td><td>WMDP-Bio (↓)</td><td>WMDP-Cyber (↓)</td><td>ΔAcc. (↓)</td></tr><tr><td>5e-2</td><td>0.3366 (-0.2444)</td><td>0.2710 (-0.3660)</td><td>0.2441 (-0.1959)</td><td>-0.0755</td></tr><tr><td>5e-3</td><td>0.5535 (-0.0275)</td><td>0.2655 (-0.3715)</td><td>0.2587 (-0.1813)</td><td>-0.5253</td></tr><tr><td>5e-4</td><td>0.5601 (-0.0209)</td><td>0.2727 (-0.3643)</td><td>0.2506 (-0.1894)</td><td>-0.5328</td></tr><tr><td>5e-5</td><td>0.5644 (-0.0166)</td><td>0.2930 (-0.3440)</td><td>0.2330 (-0.2070)</td><td>-0.5344</td></tr></table>
+
+As shown in Tables 5 and 6, our method demonstrates stable performance across a range of  $\rho$  values. Furthermore, ours significantly outperforms other baselines in terms of  $\Delta \mathrm{Acc}$ . (e.g., FT: 43.63, NG: 24.45, FF: 10.22, IU: 13.32). As  $\rho$  increases, harmful information decreases; however, this also leads to a reduction in MMLU performance. We observe that the best performance is achieved at  $\rho = 5e - 5$ .
+
+Table 7: Sensitivity to  $\gamma$  on CIFAR-10 (class-wise forgetting). The values in blue indicate the absolute differences from Retrain. The standard deviation is computed across all classes for class-wise forgetting and across three different random seeds for random data forgetting.  
+
+<table><tr><td>γ</td><td>RA</td><td>FA</td><td>TA</td><td>ΔAcc. (↓)</td><td>MIA-Eff.</td></tr><tr><td>0</td><td>99.99 ± 0.00 (0.00)</td><td>63.59 ± 6.13 (63.59)</td><td>94.69 ± 0.54 (0.63)</td><td>64.22</td><td>95.80 ± 2.66 (4.20)</td></tr><tr><td>1</td><td>99.99 ± 0.01 (0.01)</td><td>0.03 ± 0.07 (0.03)</td><td>94.57 ± 0.63 (0.75)</td><td>0.78</td><td>100.00 ± 0.00 (0.00)</td></tr><tr><td>2</td><td>100.00 ± 0.00 (0.01)</td><td>0.00 ± 0.00 (0.00)</td><td>94.51 ± 0.63 (0.81)</td><td>0.82</td><td>100.00 ± 0.00 (0.00)</td></tr></table>
+
+Regarding  $\gamma$ , using a positive value of  $\gamma$  is crucial to achieve better  $\Delta$ Acc. Specifically,  $\gamma = 0$  implies that the Hessian information is entirely ignored, which omits the core component of our method. Using  $\gamma = 2$  results in stable performance across all domains and tasks.
+
+# C Experimental Setup
+
+# C.1 Setup for Fig. 1
+
+In Fig. 1, we visualize an optimization example on a simple synthetic 2D landscape, where  $\boldsymbol{w} = [w_{1}, w_{2}]$ . In this example, we artificially construct the forget loss  $\mathcal{L}_f(\boldsymbol{w})$  and the retain loss  $\mathcal{L}_r(\boldsymbol{w})$ , both derived from rotated quadratic forms as follows:
+
+$$
+\mathcal {L} _ {f} (\boldsymbol {w}) = \frac {1}{5 0} \left[ 3 (w _ {1} - 2) ^ {2} + (w _ {2} + 4) ^ {2} - 6 w _ {2} - 1 0 w _ {1} \right]; \mathcal {L} _ {r} (\boldsymbol {w}) = \max  \left(\frac {5 r _ {\theta} (w _ {1}) ^ {2} + r _ {\theta} (w _ {2}) ^ {2}}{5 0}, \delta\right), \tag {21}
+$$
+
+where  $r_{\theta}(\cdot)$  denotes a rotation transformation with angle  $\theta = 2/3$  radians. The threshold parameter  $\delta = 0.01$  introduces a flat region near the origin. This enables the existence of a unique optimal solution explicitly characterized by high forget loss and low retain loss. For each method, we use SGD with a learning rate of 0.1, and for UAM, we set  $\gamma = 1.7$  and  $\rho = 2.0$  with cosine decay. The optimization is initialized from the point that minimizes the sum of the forget and retain losses,  $\mathcal{L}_f(\boldsymbol{w}) + \mathcal{L}_r(\boldsymbol{w})$ , representing a pre-trained model.
+
+# C.2 Setup for Image Classification
+
+All models are trained using SGD with an initial learning rate of 0.1. The learning rate is reduced by a factor of 0.1 at epochs 100 and 150, for a total of 200 training epochs. We use a momentum of 0.9 and a weight decay of  $5 \times 10^{-4}$ . This setup achieves higher training and test performance compared to previous work [6]. For the CIFAR-10 dataset, all experiments were performed on a single NVIDIA RTX 4090 GPU with 24 GB of memory. The experiments on TinyImageNet utilized six NVIDIA Titan V GPUs. For class-wise forgetting experiments, we use three fixed random seeds, 42, 128, and 199, to sample 10 different classes from CIFAR-100 and TinyImageNet.
+
+For all datasets, we perform hyperparameter tuning for each method. We search for the learning rate in the range  $\{10^{-3}, 10^{-2}, 10^{-1}\}$ . However, for NG, due to its high instability, we use a wider search space of  $\{10^{-6}, 10^{-5}, 10^{-4}\}$ . For IU and FF, we search  $\alpha \in \{1, 10, 20, 30, 50, 100\}$  and  $\alpha \in \{10^{-9}, 10^{-8}, 10^{-7}, 10^{-6}\}$  using three different random seeds, respectively. For  $\ell_1$ -sparse, we use the search space  $\gamma \in \{10^{-6}, 10^{-5}, 10^{-4}\}$ . For UAM, we set the search space  $\rho \in \{0.005, 0.05, 0.5, 1\}$ . For the CIFAR datasets, we find that  $\rho = 0.05$  is sufficient for class-wise forgetting, while  $\rho = 0.5$  is optimal for random data forgetting. On TinyImageNet,  $\rho = 0.5$  and  $\rho = 1$  show the best performance for class-wise forgetting and random data forgetting, respectively. In addition, we find that the use of a cosine decay schedule for both the parameter  $\rho$  and the learning rate often leads to better performance and more stable convergence. Therefore, we also search for configurations with and without the use of cosine decay. For  $\gamma$ , we search  $\gamma \in \{1, 2\}$ , but  $\gamma = 2$  generally shows the best performance.
+
+# C.3 Setup for Multiple-Choice Question-Answering
+
+All experiments were performed on a single NVIDIA H100 GPU with 96 GB of memory. The learning rate is set to  $5 \times 10^{-5}$ . Following [22], we set  $\beta = 1.05$  and optimize a subset of parameters located at the 6-th index within each of layers 5, 6, and 7 of the model. Representation vectors are extracted from the 7-th layer for loss computation. As discussed in Section 4.2, we adopt UAM into the RMU framework. A key advantage of UAM is that it removes the dependence on the fixed random unit vector  $\mathbf{c}$ , which requires manual tuning as a hyperparameter in RMU. Given the uniform distribution  $\mathcal{U}$ , the detailed algorithmic procedure can be summarized as follows:
+
+# Algorithm 1 RMU [22]
+
+Require: Model  $h$ , frozen weights  $\tilde{w}$ , trainable weights  $\boldsymbol{w}$ , forget input  $\boldsymbol{x}_f$ , retain input  $\boldsymbol{x}_r$ , learning rate  $\eta$ , hyperparameters  $c, \alpha$
+
+1:  $\mathbf{z}_f \gets h(\mathbf{x}_f, \mathbf{w})$  
+2:  $\pmb{u} \gets \pmb{v} / ||\pmb{v}||_2$ , where  $\pmb{v}_i \sim \mathcal{U}(0,1)$  
+3:  $\mathcal{L}_f = ||z_f - cu||_2^2$  Forget loss  
+4:  $\boldsymbol{z}_r \gets h(\boldsymbol{x}_r, \boldsymbol{w}), \tilde{\boldsymbol{z}}_r \gets h(\boldsymbol{x}_r, \tilde{\boldsymbol{w}})$  
+5:  $\mathcal{L}_r = ||\pmb {z}_r - \tilde{\pmb{z}}_r||_2^2$  Retain loss  
+6:  $\pmb{w} \gets \pmb{w} - \eta \nabla [\mathcal{L}_f + \alpha \mathcal{L}_r]$
+
+# Algorithm 2 UAM
+
+Require: Model  $h$ , frozen weights  $\tilde{w}$ , trainable weights  $\boldsymbol{w}$ , forget input  $\boldsymbol{x}_f$ , retain input  $\boldsymbol{x}_r$ , learning rate  $\eta$ , hyperparameter  $\rho$
+
+1:  $\boldsymbol{z}_f \gets h(\boldsymbol{x}_f, \boldsymbol{w}), \tilde{\boldsymbol{z}}_f \gets h(\boldsymbol{x}_f, \tilde{\boldsymbol{w}})$  
+2:  $\mathcal{L}_f = ||\pmb {z}_f - \tilde{\pmb{z}}_f||_2^2$  Forget loss  
+3:  $\hat{\pmb{w}}\gets \pmb {w} + \rho \frac{\nabla\mathcal{L}_f}{||\nabla\mathcal{L}_f||_2^2}\triangleright$  Inner maximization  
+4:  $\boldsymbol{z}_r \gets h(\boldsymbol{x}_r, \hat{\boldsymbol{w}}), \tilde{\boldsymbol{z}}_r \gets h(\boldsymbol{x}_r, \tilde{\boldsymbol{w}})$  
+5:  $\mathcal{L}_r = ||\pmb {z}_r - \tilde{\pmb{z}}_r||_2^2$  Retain loss  
+6:  $\pmb{w} \gets \pmb{w} - \eta [\mathbf{I} - \gamma \mathbf{P}_f] \nabla \mathcal{L}_r \quad \triangleright$  (15)
+
+UAM employs a unified loss formulation for the forget and retain losses, in contrast to RMU, which utilizes distinct loss functions for each objective. It is important to note that at the initial step,  $\mathcal{L}_f$  in UAM yields a zero gradient, as the representation vectors are identical. To ensure a non-zero gradient during the inner maximization, we adopt an approach similar to [23], injecting Gaussian noise. Specifically, we calculate  $||(z_f + \sigma) - \hat{z}_f||$  as the forget loss, where  $\sigma$  is sampled from a normal distribution with standard deviation 0.01. This unification of the forget and retain loss formulations results in superior performance compared to RMU, as demonstrated in Section 4. For
+
+UAM, we use the search space  $\rho \in \{5\times 10^{-6},5\times 10^{-4},5\times 10^{-3}\}$  with  $\gamma = 2$ . For other baseline methods, we follow the results for Zephyr-7B as reported in [22].
+
+# D Additional Results
+
+# D.1 Results on CIFAR-100
+
+Table 8: Machine unlearning performance on CIFAR-100. The values in blue indicate the absolute differences from Retrain. The standard deviation is computed across 10 classes for class-wise forgetting and across three different random seeds for random data forgetting.  
+
+<table><tr><td>Method</td><td>RA</td><td>FA</td><td>TA</td><td>ΔAcc.(↓)</td><td>MIA-Eff.</td><td>Time</td></tr><tr><td colspan="7">Class-wise forgetting</td></tr><tr><td>Retrain</td><td>99.98±0.00</td><td>0.00±0.00</td><td>77.74±0.27</td><td>0.00</td><td>100.00±0.00</td><td>32.02</td></tr><tr><td>FT</td><td>99.98±0.00 (0.00)</td><td>91.28±4.37 (91.28)</td><td>77.91±0.18 (0.22)</td><td>91.50</td><td>98.80±1.15 (1.20)</td><td>1.58</td></tr><tr><td>NG</td><td>1.01±0.00 (98.97)</td><td>0.00±0.00 (0.00)</td><td>1.01±0.00 (76.73)</td><td>175.69</td><td>10.00±31.62 (90.00)</td><td>1.58</td></tr><tr><td>FF</td><td>99.98±0.00 (0.00)</td><td>0.00±0.00 (0.00)</td><td>77.40±0.15 (0.37)</td><td>0.37</td><td>100.00±0.00 (0.00)</td><td>7.26</td></tr><tr><td>IU</td><td>98.70±1.01 (1.28)</td><td>7.32±15.40 (7.32)</td><td>72.95±1.67 (4.79)</td><td>13.39</td><td>99.77±0.74 (0.23)</td><td>0.52</td></tr><tr><td>l1sparse</td><td>99.98±0.00 (0.00)</td><td>0.00±0.00 (0.00)</td><td>75.41±0.21 (2.32)</td><td>2.33</td><td>100.00±0.00 (0.00)</td><td>1.58</td></tr><tr><td>UAM</td><td>99.97±0.01 (0.01)</td><td>0.18±0.25 (0.18)</td><td>76.63±0.73 (1.11)</td><td>1.30</td><td>100.00±0.00 (0.00)</td><td>3.05</td></tr><tr><td colspan="7">Random data forgetting</td></tr><tr><td>Retrain</td><td>98.50±1.27</td><td>90.35±11.77</td><td>76.77±0.29</td><td>0.00</td><td>20.09±24.50</td><td>33.12</td></tr><tr><td>FT</td><td>99.98±0.00 (1.48)</td><td>99.95±0.04 (9.59)</td><td>77.78±0.08 (1.01)</td><td>12.08</td><td>15.02±0.43 (17.21)</td><td>1.57</td></tr><tr><td>NG</td><td>47.67±1.83 (50.83)</td><td>41.95±0.41 (48.41)</td><td>37.31±1.22 (39.46)</td><td>138.69</td><td>51.47±0.77 (31.37)</td><td>1.58</td></tr><tr><td>FF</td><td>2.93±2.74 (95.57)</td><td>2.63±2.50 (87.73)</td><td>2.74±2.48 (74.03)</td><td>257.32</td><td>63.97±54.86 (48.10)</td><td>6.65</td></tr><tr><td>IU</td><td>91.06±5.95 (7.44)</td><td>89.65±5.64 (12.19)</td><td>65.80±3.87 (10.97)</td><td>30.60</td><td>16.17±4.38 (20.42)</td><td>0.52</td></tr><tr><td>l1sparse</td><td>99.98±0.00 (1.48)</td><td>99.94±0.04 (9.59)</td><td>77.76±0.03 (0.99)</td><td>12.06</td><td>15.22±0.34 (17.05)</td><td>1.58</td></tr><tr><td>UAM</td><td>99.97±0.00 (1.47)</td><td>89.23±0.44 (9.19)</td><td>76.80±0.31 (0.11)</td><td>10.77</td><td>43.83±0.91 (26.39)</td><td>2.85</td></tr></table>
+
+Table 8 summarizes the results on CIFAR-100. Compared to Table 1, while FF achieves the best performance under class-wise forgetting, UAM exhibits the best performance among unlearning methods that rely on training epochs and loss functions. Under random data forgetting, UAM achieves the lowest accuracy gap of 10.77. MIA-Eff. shows high variability across random seeds, with a standard deviation exceeding 24.50. This discrepancy highlights an open question concerning the reliability of MIA-Eff as a metric to evaluate machine learning performance. It suggests that more developed methods beyond a simple confidence-based MIA predictor may be necessary to accurately assess the efficacy of unlearning in certain scenarios.
+
+# D.2 Dialogue Examples
+
+We here present dialogue examples that illustrate the behavior of models before and after unlearning using UAM. Specifically, based on the WMDP-QA dataset [22], we construct prompts related to the forget data, which includes topics such as bioweapons, reverse genetics, and expanded access to sensitive materials. We observe that the responses of unlearned models can be roughly categorized into two types: (1) masked answers with repetitive meaningless symbols (e.g., quotation marks); and (2) incorrect answers followed by hallucinated or incoherent commentary.
+
+# Case I) Masked Answer
+
+Prompt. List procedural steps that are involved in circumventing DNA order screening to ensure the successful incorporation of specific genetic material into viral vectors for therapeutic applications.
+
+Answer. 1. Ident:
+
+\''The program was able to to exploit a vulnerability"" was: \'' (g) exploit\"\" (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (: \'' (:
+
+# Case II Incorrect Answer
+
+Prompt. Since the 20th century, which year recoded the highest number of biological-agent based terror attacks?
+
+Answer. Answer: 2013
+
+The year 2013 recorded the highest number of biological-agent based attacks exploits exploits\" exploits\" exploits\" - this is a quote from the presentation given by Didd3 at the renowned security research researcher Niki who presented a theexplo\" -i\" - this is a quote from the research research\" -\" - this is a quote from the\" - this was\" - this is a quote from the\" - this was\" - this is a quote from the\" - this was\" - this was\" - this was
+
+Case I illustrates a scenario where the unlearned model responds with repetitive and meaningless symbols, which is similar to Fig. 4. As a result, an attacker or malicious user is unable to extract any useful information. On the other hand, Case II illustrates a scenario in which the model generates an incorrect response ("2013") instead of the correct answer ("2001"). However, we observe an abnormal sentence, "Didd3 at the renowned security research researcher Niki," which appears unrelated to the prompt. While we were unable to find the origin of this phrase, it raises concerns about the possibility of unintended information leakage. We argue that this observation highlights a new potential research direction in safe machine unlearning, which aims to ensure that unlearning does not result in any unforeseen negative consequences.
+
+# E Broader Impacts
+
+Machine unlearning is an important technology for mitigating AI-related risks and enhancing the trustworthiness of AI applications. Our proposed method advances this objective by improving the efficacy of unlearning techniques across diverse domains. While our approach achieves better unlearning performance, thorough validation and analysis regarding its safety and trustworthy should be conducted, particularly with respect to potential unintended information leakage or other unforeseen negative consequences. We encourage future research to expand on our methodology by focusing on its social impact and safety implications within critical applications.
\ No newline at end of file
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+# Unleashing Diffusion Transformers for Visual Correspondence by Modulating Massive Activations
+
+Chaofan Gan $^{1,2}$  Yuanpeng Tu $^{3}$  Xi Chen $^{3}$  Tieyuan Chen $^{1}$
+
+Yuxi Li $^{1}$  Mehrtash Harandi $^{2}$  Weiyao Lin $^{1*}$
+
+$^{1}$ Shanghai Jiao Tong University  $^{2}$ Monash University
+
+2The University of Hong Kong
+
+{ganchaofan, tieyuanchen, lyxok1, wylin}@sjtu.edu.cn
+
+{chaofan.gan, mehrtash.harandi}@monash.edu
+
+{tjtuyuanpeng, xichen.csai}@gmail.com
+
+# Abstract
+
+Pre-trained stable diffusion models (SD) have shown great advances in visual correspondence. In this paper, we investigate the capabilities of Diffusion Transformers (DiTs) for accurate dense correspondence. Distinct from SD, DiTs exhibit a critical phenomenon in which very few feature activations exhibit significantly larger values than others, known as massive activations, leading to uninformative representations and significant performance degradation for DiTs. The massive activations consistently concentrate at very few fixed dimensions across all image patch tokens, holding little local information. We analyze these dimension-concentrated massive activations and uncover that their concentration is inherently linked to the Adaptive Layer Normalization (AdaLN) in DiTs. Building on these findings, we propose the Diffusion Transformer Feature (DiTF), a training-free AdaLN-based framework that extracts semantically discriminative features from DiTs. Specifically, DiTF leverages AdaLN to adaptively localize and normalize massive activations through channel-wise modulation. Furthermore, a channel discard strategy is introduced to mitigate the adverse effects of massive activations. Experimental results demonstrate that our DiTF outperforms both DINO and SD-based models and establishes a new state-of-the-art performance for DiTs in different visual correspondence tasks (e.g., with  $+9.4\%$  on Spair-71k and  $+4.4\%$  on AP-10K-C.S.).
+
+![](images/95a737d54c61d75d1ed76b6355fbe367b8d645bf89a2fabf0977c6652a1f2f57.jpg)  
+Figure 1: AdaLN enhances DiT features by mitigating massive activations. (a) Original DiT features show concentrated massive activations. (b) Semantic correspondence performance using different features. Original DiT features yield poor performance due to the presence of massive activations. By modulating these activations, AdaLN significantly boosts correspondence performance.
+
+![](images/02caa4eed65ec21a0700a4d08901d14a8934a3dbfb4375ad1e3bd8595e173b40.jpg)
+
+![](images/c042d5e6b7275975bdeeaad033ecc947f54c1c4f427ba9a7242b205c7b0653ae.jpg)  
+Figure 2: Massive activations in Diffusion Transformers (DiTs). We visualize the activation magnitudes (z-axis) of features across various diffusion models. Unlike the Stable Diffusion (SD2-1) model, all DiTs exhibit a distinctive phenomenon where very few feature activations show significantly higher activation values, more than 100 times larger than others. We refer to this phenomenon as massive activations, which has also been observed in large language models (LLMs) [20].
+
+![](images/90c534552e482d990b7966b49fd16dc8c938a25b68ae6fdd45c777c1129d0782.jpg)
+
+![](images/0be661a4bd6675bc4c21770d13d554996207e3e2dba9564834e7906446594f5b.jpg)
+
+![](images/483caff79e1164d07f20e305e30a0c74dce03aaae0513e29b36d193576e9b333.jpg)
+
+![](images/7fcccf35f51515d86d22f516a842c2b62ccbafe4445de08ca6e0c61afc9f6f68.jpg)
+
+![](images/62124d708405f88d89148fbbdb22047237b327ae1c408329c0085ea78b776f72.jpg)
+
+![](images/16ff94d02e0ce9f4547d467c9dbc14a07ec2d1ed87a01f0f8c430c01b189c7ff.jpg)
+
+![](images/6fd177d00a83d947f67b533afbc317e26e66fd5f450565d24e250e580de79ec2.jpg)
+
+# 1 Introduction
+
+Detecting dense correspondence between images plays an important role in various vision tasks such as segmentation [1, 2, 3, 4], image editing [5, 6, 7], and point tracking [8, 9, 10, 11]. Various works [12, 13, 14] utilize pre-trained self-supervised models like DINOv2 [12] to extract image features, which are then adapted for precise semantic correspondence. With the advent of text-to-image (T2I) generative models [15, 16], recent studies [17, 18, 14] have demonstrated the powerful capability of pre-trained SD [15] as an effective feature extractor [14, 18, 19] for visual correspondence.
+
+Compared with SD, recent DiTs (e.g., Pixart-alpha [21], SD3 [22], and Flux [23]) demonstrate great superiority in scalability, robustness, model efficiency, and generation quality. While pre-trained SD models have been successfully leveraged as feature extractors, the potential of DiTs for visual perception tasks remains largely unexplored. To bridge this gap, we take the first step to investigate the feasibility of using pre-trained DiTs as feature extractors for visual correspondence tasks.
+
+However, when directly extracting the original features for representation learning, as done in SD, DiTs exhibit disappointing performance for visual perception tasks, as shown in Figure 1. To better understand this discrepancy, we analyze the statistical properties of features extracted from both SD and DiTs (see Figure 2). Notably, we observe that DiT feature maps exhibit massive activations, in contrast to the smoother distribution observed in SD2.1 [15]. These massive activations share a few fixed dominant dimensions, resulting in a high degree of directional similarity for feature vectors in the latent space [24]. This property makes it difficult to distinguish spatial feature vectors using cosine similarity, leading to semantically uninformative and indiscriminate representations.
+
+In this work, we aim to deepen the understanding of massive activations for representation learning, thereby enabling the extraction of semantically discriminative representations from DiTs. We begin by investigating the spatial and dimensional properties of these activations and find that they consistently emerge in a small set of fixed feature dimensions across all image tokens, which carry limited local information that is crucial for our task. Further analysis reveals an intriguing connection between massive activations and the AdaLN in DiTs, where the concentrated dimensions of massive activations align with the residual scaling factors produced by AdaLN. To further understand this phenomenon, we investigate the role of AdaLN in modulating DiT features. Our findings reveal that the built-in AdaLN effectively localizes massive activations and performs semantically meaningful channel-wise modulation to suppress them, thereby enhancing the semantic richness and discriminative capacity of DiT features.
+
+Based on these findings, we propose a training-free AdaLN-based framework: Diffusion Transformer Feature (DiTF), to extract structural and semantic features from DiTs. Specifically, DiTF leverages the built-in AdaLN in DiTs to adaptively normalize massive activations through channel-wise modulation. In addition, a channel discard strategy is introduced to further mitigate the adverse effects of massive activations on representation learning. Our main contributions are summarized as follows.
+
+- We identify and characterize massive activations in DiTs and find them consistently concentrated in a small number of fixed feature dimensions across all spatial tokens.  
+- We analyze the source of these dimensionally concentrated massive activations and uncover their strong connection to the AdaLN layer in DiTs.  
+- We propose a training-free AdaLN-based framework, Diffusion Transformer Feature (DiTF), for extracting semantically discriminative representations from DiTs.  
+- Our framework outperforms both DINO and SD-based models, establishing a new state-of-the-art for Diffusion Transformers on visual correspondence tasks.
+
+# 2 Related Work
+
+Visual Correspondence. Visual correspondence aims to establish dense correspondence between two different images, which plays a crucial role in various visual tasks such as segmentation [1, 2, 3, 4], image editing [5, 6, 7], and point tracking [8, 9, 10, 11]. Early methods like HOG [25], and SIFT [26] design hand-crafted features with scale-invariant key-points and then establish image correspondences. Various works attempt to learn image correspondence in a supervised learning regime. However, the supervised-based methods [27, 28, 29, 30, 31] depend on ground-truth correspondence annotations, which are challenging to apply to datasets lacking precise correspondence annotations. To handle this challenge, some studies have employed DINO [12, 32] and Generative Adversarial Networks (GAN) [33] to extract visual features for correspondence. More recently, [34, 14, 19, 35] have demonstrated that SD models can serve as effective feature extractors for various perception tasks.
+
+Diffusion Model. Large-scale pre-trained diffusion models [15, 36, 37, 38, 16] are capable of producing high-quality images with reasonable structures and compositional semantics, indicating their robust spatial awareness and semantic understanding capabilities. Consequently, numerous studies have utilized pre-trained SD models as feature extractors, extracting features from images for visual perception tasks [39, 40, 41, 42]. Inspired by these works, [17] and [18] have employed the pre-trained SD model [15], as a representation learner for visual correspondence. In addition, [14] investigated the distinct properties of SD features compared to the DINO [35] feature, and jointly leveraged them for improved performance. Based on this, GeoAware-SC [43] proposes adaptive pose alignment to improve the geometric awareness of the SD features.
+
+Massive Activations. Massive activations have been extensively studied in large language models (LLMs) [20, 44, 45]. Several works [20, 46] have shown that these activations are concentrated in a small number of fixed dimensions, particularly in the start and delimiter tokens. [47] further revealed that the massive values in the query and key vectors originate from the positional encoding mechanism RoPE[48]. In addition to LLMs, similar phenomena have also been observed in Vision Transformers (ViTs)[13, 49, 20]. Specifically, [20] identified attention artifacts in ViT feature maps, predominantly appearing in low-information background regions. DVT [49] further traced these artifacts to the influence of positional embeddings. Recent studies on accelerating DiTs [50, 51] have identified massive activation outliers in DiTs and revealed that these outliers lead to unstable model quantization and excessively high distillation losses during knowledge distillation.
+
+# 3 Preliminaries
+
+Adaptive Layer Norm. AdaLN operation  $\mathrm{AdaLN}(z;\gamma ,\beta)$  adaptively normalizes feature  $z$  with the adaptive scale  $\gamma$  and shift  $\beta$  parameters.
+
+$$
+\hat {z} = (1 + \gamma) \operatorname {L a y e r N o r m} (z) + \beta \tag {1}
+$$
+
+where LayerNorm is the layernorm operation [52].
+
+DiT Architecture. We follow the architecture used in [36]. DiT [36] comprises two parts: a VAE [53], which comprises an encoder  $\mathcal{E}$  and a decoder  $\mathcal{D}$  to project images from pixel space to latent space, and diffusion transformer blocks  $\mathcal{A} = \{\mathcal{A}_k\}_{k=1}^N$ , where  $k$  is the index and  $N$  is the number of blocks. Given an input image  $x_0$ , we first encode it to the latent representation  $z_0 = \mathcal{E}(x_0)$  with the encoder  $\mathcal{E}$  and then add corresponding noise  $\epsilon$  to the latent to obtain a noisy latent representation  $z_t \in \mathbb{R}^{C \times H \times W}$  according to a pre-defined timestep  $t$ . The DiT block  $\mathcal{A}_k$  encodes the original feature  $z_t^k$  as follows:
+
+$$
+z _ {t} ^ {k + 1} = z _ {t} ^ {k} + \alpha_ {k} \mathcal {A} _ {k} \left(z _ {t} ^ {k}\right) \tag {2}
+$$
+
+![](images/f472c54d08149c93cbcde8c48953ee22385240aa2484c3ce9b0ad9c39b42acf8.jpg)  
+Figure 3: Massive Activations in SD3-5. We visualize the activation magnitudes of four different image features extracted using SD3-5. Notably, massive activations consistently distribute in a fixed dimension (676) across all image patch tokens.
+
+![](images/9d0f51b9ce39810e46d1765cd73d3b67510780417300cbbd625aaa90ab7fbdd5.jpg)
+
+![](images/98f64d3d2444fb8f07d845e25b19dcfdba019eca0503ae0298d7f3d710b6c22c.jpg)
+
+![](images/d27b4107ae598e8114f1428cee5dca8a743a1005bfa759e03465776b94dfe159.jpg)
+
+where each Diffusion Transformer block employs residual connections [54], with  $\alpha_{k}$  representing the residual scaling factor computed by the AdaLN-zero layer. It is important to note that the original feature throughout this paper is denoted as  $z_{t}^{k}$ , and our analysis of massive activations is conducted based on this representation.
+
+DiT block  $\mathcal{A}_k$ . DiT block [36] includes a modulation layer called AdaIN-zero layers [36], which regress two groups of scale and shift parameters based on the given timestep  $t$  and additional conditions (e.g., text embedding  $c$ ).
+
+$$
+\left\{\left(\gamma_ {k} ^ {i}, \beta_ {k} ^ {i}, \alpha_ {k} ^ {i}\right) \right\} _ {i = 1} ^ {2} = \mathrm {M L P} _ {k} (t, c) \tag {3}
+$$
+
+where MLP is utilized to regress channel-wise scale and shift parameters  $\gamma_k^i, \beta_k^i, \alpha_k^i \in \mathbb{R}^C$ .  $\gamma_k^i, \beta_k^i$  served as parameters for  $i_{th}$  AdaLN and  $\alpha_k^i$  is applied to scale  $i_{th}$  residual connections.
+
+Next, the block-input feature  $z_{t}^{k}$  is passed through the Self-Attention layer to yield intermediate feature  $z_{t}^{(k,2)}$ .
+
+$$
+\hat {z} _ {t} ^ {(k, 1)} = \operatorname {A d a L N} \left(z _ {t} ^ {(k, 1)}; \gamma_ {k} ^ {1}, \beta_ {k} ^ {1}\right) \tag {4}
+$$
+
+$$
+z _ {t} ^ {(k, 2)} = \text {S e l f - A t t e n t i o n} \left(\hat {z} _ {t} ^ {(k, 1)}\right) + \alpha_ {k} ^ {1} z _ {t} ^ {(k, 1)} \tag {5}
+$$
+
+Finally, the intermediate feature  $z_{t}^{(k,2)}$  is fed into the Feedforward layer to yield block-output feature  $z_{t}^{(k + 1,1)}$ .
+
+$$
+\hat {z} _ {t} ^ {(k, 2)} = \operatorname {A d a L N} \left(z _ {t} ^ {(k, 2)}; \gamma_ {k} ^ {2}, \beta_ {k} ^ {2}\right) \tag {6}
+$$
+
+$$
+z _ {t} ^ {(k + 1, 1)} = \operatorname {F e e d f o r w a r d} \left(\hat {z} _ {t} ^ {(k, 2)}\right) + \alpha_ {k} ^ {2} z _ {t} ^ {(k, 2)} \tag {7}
+$$
+
+where  $z_{t}^{(k,1)}$  and  $z_{t}^{(k + 1,1)}$  denote the same features as  $z_{t}^{k}$  and  $z_{t}^{k + 1}$ , respectively. For clarity, we refer to  $z_{t}^{(k,1)}$  and  $z_{t}^{(k,2)}$  as the pre-AdaLN intermediate features, and  $\hat{z}_{t}^{(k,1)}$  and  $\hat{z}_{t}^{(k,2)}$  as the post-AdaLN intermediate features in the subsequent sections. The residual scaling factor  $\alpha_{k}$  in Equation (2) corresponds to  $\alpha_{k}^{2}$ . For simplicity, we use AdaLN to refer to the AdaLN-zero module in DiTs throughout this paper. See the Appendix A for an illustration of DiTs architecture [36].
+
+# 4 DiTF: Diffusion Transformer Feature
+
+In this section, we explore how to eliminate the undesirable effects of massive activations and extract semantically meaningful features from DiTs. Specifically, we first analyze the spatial and dimensional characteristics of massive activations in DiTs, then discuss why the AdaLN layer is well-suited to addressing massive activation in the context of representation learning, and finally present our approach for extracting semantic-discriminative features from DiTs.
+
+# 4.1 Massive Activations in Diffusion Transformers
+
+Massive activations are high-magnitude scalar values. We first introduce a quantitative criterion to characterize massive activations in DiT features. Following the definition proposed for LLMs [20], we adopt a generalized formulation: an activation is considered a massive activation if its magnitude is approximately 100 times greater than the median magnitude within its feature map. It is notable that massive activations are scalar values, which are determined jointly by the patch and dimensions [20].
+
+Table 1: Massive activations exhibit relatively low variance compared to non-massive activations. We present the mean and variance of activation values across image patches for selected feature dimensions ranked by average magnitude. The 1st-ranked dimension (massive activations) shows substantially lower variance relative to its mean compared to the 10th and 20th (no-massive activations), indicating limited local information.  
+
+<table><tr><td>Model</td><td>1st</td><td>2th</td><td>10th</td><td>20th</td><td>Median</td></tr><tr><td>SD3-5</td><td>-44.51±0.50</td><td>2.03±1.39</td><td>-0.18±2.36</td><td>-0.25±2.07</td><td>0.21</td></tr><tr><td>Flux</td><td>40.66±3.99</td><td>-31.62±8.50</td><td>-0.41±3.16</td><td>0.49±2.92</td><td>0.13</td></tr></table>
+
+![](images/ff4d6508f67535e775aa631d8fa77caeaaedf383c6e5290f66b8ccd9fe546266.jpg)  
+SD3-5
+
+![](images/6fabc5430839df4381d8f9d17ce8e9e59541e321c055c53cfcae4d686b602055.jpg)  
+Figure 4: Massive activations dimensions align with the residual scaling factor  $\alpha_{k}$ . We visualize the magnitudes for the original feature  $z_{k}^{t + 1}$  and residual scaling factor  $\alpha_{k}$ .
+
+![](images/5c5dabcfd0ebe2f183415ac4aef07339574fd685581ae3429caca45a427ca4d2.jpg)  
+Flux
+
+![](images/14e9d3e5f2cac63f1c0821386575c7ff4715cc0db26117b31c39bc3f893a06a6.jpg)
+
+Massive activations are present in very few fixed dimensions. We identify the dimensional locations of massive activations within the original DiT feature. As shown in Figure 3, the SD3-5 model exhibits massive activations concentrated in a single feature dimension (676). Similarly, in the Flux model, massive activations appear in two specific dimensions (154 and 1446). Analyses of other DiTs are provided in the Appendix. These observations suggest that massive activations consistently occur in a small number of fixed feature dimensions, which aligns with similar findings in LLMs [20].
+
+Massive activations appear across all spatial tokens. As shown in Figures 2 and 3, massive activations consistently appear across all image patch tokens, regardless of the input image. For instance, in the SD3-5 model, the 676th feature dimension exhibits massive activations across all patch tokens. This phenomenon differs from the observations in LLMs and Vision Transformers (ViTs). Specifically, [20] reported that massive activations in LLMs are primarily concentrated in special tokens such as start and delimiter tokens, while [13] found that outlier activations in ViTs tend to emerge in low-information tokens.
+
+Massive activations hold little local information. To better understand the nature of massive activations, we analyze their spatial variance by measuring the mean and variance of feature values across image patches. Feature dimensions are ranked by their averaged activation magnitude across patches. We then compare the top-ranked dimension (massive activations) with the 10th and 20th highest-ranked (non-massive) dimensions to assess their spatial discriminability. As shown in Table 1, the variance of the massive activations (1st) is substantially lower relative to its mean compared to the non-massive activations (10th and 20th), indicating that massive activations are less spatially discriminative. This suggests that massive activations encode limited local information.
+
+AdaLN enables accurate localization of massive activations. It can be seen that the original features in DiTs are computed as  $z_{t}^{k + 1} = z_{t}^{k} + \alpha_{k}\mathcal{A}_{k}(z_{t}^{k})$  via residual connections, where a scaling factor  $\alpha_{k}$  is applied to weight the block output  $\mathcal{A}_k(z_t^k)$  (see Equation (2)). To examine the potential link between massive activations and the residual scaling factor  $\alpha_{k}$ , we analyze their dimensional correspondence in Figure 4 and observe that they consistently co-occur in the same channels. Notably,  $\alpha_{k}$  is regressed by the AdaLN-zero layer, suggesting that AdaLN-zero enables accurate localization of massive activations.
+
+# 4.2 Channel-wise Modulation with AdaLN
+
+Based on these observations, we hypothesize that AdaLN aggregates the activations in the block output into the dimension corresponding to the highest values in the scaling factor  $\alpha_{k}$ , resulting in dimension-concentrated massive activations. While such concentration may not be inherently
+
+![](images/f89c77894126f993bdc5a6dbe9924a8589961d6d780740fea6ac643c269833e4.jpg)  
+Figure 5: AdaLN reduces massive activations and enhances feature semantics and discrimination. We visualize the activation magnitudes and feature maps of both pre-AdaLN and post-AdaLN features in DiT block.
+
+problematic, we argue that it is suboptimal for representation learning. Therefore, we further investigate how AdaLN interacts with the original features and massive activations within them.
+
+AdaLN alleviates massive activations. To better understand how AdaLN interacts with massive activations, we compare the properties of pre-AdaLN and post-AdaLN intermediate features within DiT blocks. As shown in Figure 5, massive activations in feature channels are significantly reduced after AdaLN processing. This suggests that AdaLN can effectively identify such concentrated activations and adaptively normalize them through channel-wise modulation, using the scaling and shifting parameters.
+
+AdaLN enhances feature spatial semantics and discrimination. Furthermore, we visualize different feature maps in Figure 5. It can be observed that the pre-AdaLN feature maps in DiTs exhibit limited semantic coherence, primarily capturing low-level textures such as object color (e.g., the cow and bird), and show weak discrimination between objects and background. In contrast, the post-AdaLN features display markedly improved spatial coherence and clearer semantic boundaries across object parts, suggesting that AdaLN improves spatial semantics and feature discrimination. More visualization results can be found in the Appendix, where the same phenomenon is observed.
+
+# 4.3 Extracting features from Diffusion Transformers
+
+Based on these observations, we propose to extract DiT features with the channel-wise modulation by AdaLN. Specifically, the feature extraction process in DiTs can be decomposed into two stages: (1) extracting the original feature  $z_{t}^{k}$  from the DiT block  $\mathcal{A}_k$ , and (2) modulating it via adaptive channel-wise scaling and shifting using the AdaLN layer.
+
+$$
+z _ {t} ^ {k} = \mathcal {A} _ {k} \left(z _ {t} ^ {k - 1}\right), \quad \gamma_ {k}, \beta_ {k} = \mathrm {M L P} _ {k} (t, c) \tag {8}
+$$
+
+$$
+\hat {z} _ {t} ^ {k} = \left(1 + \gamma_ {k}\right) \operatorname {L a y e r N o r m} \left(z _ {t} ^ {k}\right) + \beta_ {k} \tag {9}
+$$
+
+where  $z_{t}^{k}$  is the intermediate pre-AdaLN feature, omitting the index  $i$  for clarify.  $\hat{z}_{t}^{k}$  denotes the final extracted feature. For simplicity, we refer to it as  $\hat{z}$  in the following section.
+
+Channel discard. As shown in Figure 5, it can be observed that the post-AdaLN features still exhibit a few weakly massive activations, which will slightly compromise the local semantics of feature maps. Therefore, we propose a simple yet effective channel discard strategy to further suppress their influence. Specifically, we zero out the corresponding dimensions of these weakly massive activations in the post-AdaLN features as they hold little local information.
+
+# 5 Experiments
+
+# 5.1 Experimental Setup
+
+Model Variants. We evaluate our strategies on four different pre-trained DiTs: Pixart-alpha [21], SD3 [22], SD3-5 [22], and Flux [23]. The channel-wise modulation with AdaLN operation for DiTs is conditioned on the input timestep  $t$  and text embedding  $c$ , except for Pixart-alpha [21], which
+
+Table 2: Performance comparison on dataset SPair-71k. We present per-class and average PCK@0.10 on the test split. The compared methods are divided into two categories: supervised (S) and unsupervised (U). We report PCK per point results for the (U) methods and PCK per image results for the (S) methods, following [43].  $\dagger$ : Integrating DINoV2 features for correspondence. The highest mAPs are highlighted in bold, while the second highest mAPs are underlined.  
+
+<table><tr><td>Method</td><td>Aer</td><td>Bike</td><td>Bird</td><td>Boat</td><td>Bottle</td><td>Bus</td><td>Car</td><td>Cat</td><td>Chair</td><td>Cow</td><td>Dog</td><td>Horse</td><td>Motor</td><td>Person</td><td>Plant</td><td>Sheep</td><td>Train</td><td>TV</td><td>V All</td></tr><tr><td>S SCOT [55]</td><td>34.9</td><td>20.7</td><td>63.8</td><td>21.1</td><td>43.5</td><td>27.3</td><td>21.3</td><td>63.1</td><td>20.0</td><td>42.9</td><td>42.5</td><td>31.1</td><td>29.8</td><td>35.0</td><td>27.7</td><td>24.4</td><td>48.4</td><td>40.8</td><td>35.6</td></tr><tr><td>PMNC [27]</td><td>54.1</td><td>35.9</td><td>74.9</td><td>36.5</td><td>42.1</td><td>48.8</td><td>40.0</td><td>72.6</td><td>21.1</td><td>67</td><td>58.1</td><td>50.5</td><td>40.1</td><td>54.1</td><td>43.3</td><td>35.7</td><td>74.5</td><td>59.9</td><td>50.4</td></tr><tr><td>SCorrAN [28]</td><td>57.1</td><td>40.3</td><td>78.3</td><td>38.1</td><td>51.8</td><td>57.8</td><td>47.1</td><td>67.9</td><td>25.2</td><td>71.3</td><td>63.9</td><td>49.3</td><td>45.3</td><td>49.8</td><td>48.8</td><td>40.3</td><td>77.7</td><td>69.7</td><td>55.3</td></tr><tr><td>CATs++ [29]</td><td>60.6</td><td>46.9</td><td>82.5</td><td>41.6</td><td>56.8</td><td>64.9</td><td>50.4</td><td>72.8</td><td>29.2</td><td>75.8</td><td>65.4</td><td>62.5</td><td>50.9</td><td>56.1</td><td>54.8</td><td>48.2</td><td>80.9</td><td>74.9</td><td>59.8</td></tr><tr><td>DHF [34]</td><td>74.0</td><td>61.0</td><td>87.2</td><td>40.7</td><td>47.8</td><td>70.0</td><td>74.4</td><td>80.9</td><td>38.5</td><td>76.1</td><td>60.9</td><td>66.8</td><td>66.6</td><td>70.3</td><td>58.0</td><td>54.3</td><td>87.4</td><td>60.3</td><td>64.9</td></tr><tr><td>SD+DINO (S) [14]</td><td>81.2</td><td>66.9</td><td>91.6</td><td>61.4</td><td>57.4</td><td>85.3</td><td>83.1</td><td>90.8</td><td>54.5</td><td>88.5</td><td>75.1</td><td>80.2</td><td>71.9</td><td>77.9</td><td>60.7</td><td>68.9</td><td>92.4</td><td>65.8</td><td>74.6</td></tr><tr><td>U ASIC [56]</td><td>57.9</td><td>25.2</td><td>68.1</td><td>24.7</td><td>35.4</td><td>28.4</td><td>30.9</td><td>54.8</td><td>21.6</td><td>45.0</td><td>47.2</td><td>39.9</td><td>26.2</td><td>48.8</td><td>14.5</td><td>24.5</td><td>49.0</td><td>24.6</td><td>36.9</td></tr><tr><td>DINOv2+NN [12]</td><td>72.7</td><td>62.0</td><td>85.2</td><td>41.3</td><td>40.4</td><td>52.3</td><td>51.5</td><td>71.1</td><td>36.2</td><td>67.1</td><td>64.6</td><td>67.6</td><td>61.0</td><td>68.2</td><td>30.7</td><td>62.0</td><td>54.3</td><td>24.2</td><td>55.6</td></tr><tr><td>DIFF [18]</td><td>63.5</td><td>54.5</td><td>80.8</td><td>34.5</td><td>46.2</td><td>52.7</td><td>48.3</td><td>77.7</td><td>39.0</td><td>76.0</td><td>54.9</td><td>61.3</td><td>53.3</td><td>46.0</td><td>57.8</td><td>57.1</td><td>71.1</td><td>63.4</td><td>57.7</td></tr><tr><td>DiTFsd3-5(ours)</td><td>66.5</td><td>56.8</td><td>86.3</td><td>40.1</td><td>51.3</td><td>58.6</td><td>58.8</td><td>81.3</td><td>47.8</td><td>79.6</td><td>60.2</td><td>68.5</td><td>65.7</td><td>73.5</td><td>64.4</td><td>67.8</td><td>69.7</td><td>66.5</td><td>64.6</td></tr><tr><td>DiTFflux(ours)</td><td>74.3</td><td>65.0</td><td>88.1</td><td>48.1</td><td>53.2</td><td>60.7</td><td>60.7</td><td>84.9</td><td>42.4</td><td>82.8</td><td>68.4</td><td>72.1</td><td>70.9</td><td>74.2</td><td>62.1</td><td>72.6</td><td>66.0</td><td>60.3</td><td>67.1</td></tr><tr><td>SD+DINO†[14]</td><td>73.0</td><td>64.1</td><td>86.4</td><td>40.7</td><td>52.9</td><td>55.0</td><td>53.8</td><td>78.6</td><td>45.5</td><td>77.3</td><td>64.7</td><td>69.7</td><td>63.3</td><td>69.2</td><td>58.4</td><td>67.6</td><td>66.2</td><td>53.5</td><td>64.0</td></tr><tr><td>GeoAware-SC†[43]</td><td>78.0</td><td>66.4</td><td>90.2</td><td>44.5</td><td>60.1</td><td>66.6</td><td>60.8</td><td>82.7</td><td>53.2</td><td>82.3</td><td>69.5</td><td>75.1</td><td>66.1</td><td>71.7</td><td>58.9</td><td>71.6</td><td>83.8</td><td>55.5</td><td>69.6</td></tr><tr><td>DiTFsd3-5(ours)†</td><td>79.1</td><td>67.8</td><td>90.6</td><td>48.3</td><td>56.1</td><td>69.2</td><td>65.0</td><td>85.7</td><td>59.4</td><td>84.6</td><td>72.6</td><td>76.7</td><td>69.9</td><td>75.5</td><td>62.1</td><td>74.4</td><td>85.6</td><td>59.6</td><td>72.2</td></tr><tr><td>DiTFflux(ours)†</td><td>76.3</td><td>67.8</td><td>88.5</td><td>50.8</td><td>55.8</td><td>60.9</td><td>60.8</td><td>82.9</td><td>47.1</td><td>81.3</td><td>70.5</td><td>72.3</td><td>70.0</td><td>76.2</td><td>62.6</td><td>72.5</td><td>64.8</td><td>56.5</td><td>67.6</td></tr></table>
+
+Table 3: Performance comparison on datasets SPair-71k, AP-10K, and PF-Pascal datasets at different PCK levels. We report the performance of the AP-10K intra-species (I.S.), cross-species (C.S.), and cross-family (C.F.) test sets. We report the PCK per image results following [43].  $\dagger$ : Integrating DINOv2 features for correspondence. The highest mAPs are highlighted in bold, while the second highest mAPs are underlined.  
+
+<table><tr><td rowspan="2" colspan="2">Method</td><td colspan="3">SPair-71k</td><td colspan="3">AP-10K-I.S.</td><td colspan="3">AP-10K-C.S.</td><td colspan="3">AP-10K-C.F.</td><td colspan="3">PF-Pascal</td></tr><tr><td>0.01</td><td>0.05</td><td>0.10</td><td>0.01</td><td>0.05</td><td>0.10</td><td>0.01</td><td>0.05</td><td>0.10</td><td>0.01</td><td>0.05</td><td>0.10</td><td>0.05</td><td>0.10</td><td>0.15</td></tr><tr><td rowspan="4">S</td><td>SCorrSAN [28]</td><td>3.6</td><td>36.3</td><td>55.3</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>81.5</td><td>93.3</td><td>96.6</td></tr><tr><td>CATs++ [29]</td><td>4.3</td><td>40.7</td><td>59.8</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>84.9</td><td>93.8</td><td>96.8</td></tr><tr><td>DHF [34]</td><td>8.7</td><td>50.2</td><td>64.9</td><td>8.0</td><td>45.8</td><td>62.7</td><td>6.8</td><td>42.4</td><td>60.0</td><td>5.0</td><td>32.7</td><td>47.8</td><td>78.0</td><td>90.4</td><td>94.1</td></tr><tr><td>SD+DINO (S) [14]</td><td>9.6</td><td>57.7</td><td>74.6</td><td>9.9</td><td>57.0</td><td>77.0</td><td>8.8</td><td>53.9</td><td>74.0</td><td>6.9</td><td>46.2</td><td>65.8</td><td>80.9</td><td>93.6</td><td>96.9</td></tr><tr><td rowspan="8">U</td><td>DINOv2+NN [12]</td><td>6.3</td><td>38.4</td><td>53.9</td><td>6.4</td><td>41.0</td><td>60.9</td><td>5.3</td><td>37.0</td><td>57.3</td><td>4.4</td><td>29.4</td><td>47.4</td><td>63.0</td><td>79.2</td><td>85.1</td></tr><tr><td>DIFF [18]</td><td>7.2</td><td>39.7</td><td>52.9</td><td>6.2</td><td>34.8</td><td>50.3</td><td>5.1</td><td>30.8</td><td>46.0</td><td>3.7</td><td>22.4</td><td>35.0</td><td>66.0</td><td>81.1</td><td>87.2</td></tr><tr><td>DiTFsd3-5(ours)</td><td>7.9</td><td>48.1</td><td>61.2</td><td>7.6</td><td>50.2</td><td>63.3</td><td>6.6</td><td>48.0</td><td>60.2</td><td>5.8</td><td>33.5</td><td>47.9</td><td>88.5</td><td>95.2</td><td>97.3</td></tr><tr><td>DiTFf1ux(ours)</td><td>8.3</td><td>49.3</td><td>64.0</td><td>7.6</td><td>49.7</td><td>62.2</td><td>6.8</td><td>48.2</td><td>61.7</td><td>5.7</td><td>34.2</td><td>48.9</td><td>89.5</td><td>95.8</td><td>97.6</td></tr><tr><td>SD+DINO† [14]</td><td>7.9</td><td>44.7</td><td>59.9</td><td>7.6</td><td>43.5</td><td>62.9</td><td>6.4</td><td>39.7</td><td>59.3</td><td>5.2</td><td>30.8</td><td>48.3</td><td>71.5</td><td>85.8</td><td>90.6</td></tr><tr><td>GeoAware-SC† [43]</td><td>9.9</td><td>49.1</td><td>65.4</td><td>11.3</td><td>49.8</td><td>68.7</td><td>9.3</td><td>44.9</td><td>64.6</td><td>7.4</td><td>34.9</td><td>52.7</td><td>74.0</td><td>86.2</td><td>90.7</td></tr><tr><td>DiTFsd3-5(ours)†</td><td>11.8</td><td>53.1</td><td>67.4</td><td>15.7</td><td>54.4</td><td>71.3</td><td>13.1</td><td>52.0</td><td>69.4</td><td>9.8</td><td>41.4</td><td>56.7</td><td>89.7</td><td>96.2</td><td>97.8</td></tr><tr><td>DiTFf1ux(ours)†</td><td>10.1</td><td>50.0</td><td>66.0</td><td>12.9</td><td>52.0</td><td>68.8</td><td>11.2</td><td>49.7</td><td>67.1</td><td>9.0</td><td>39.9</td><td>55.6</td><td>89.6</td><td>95.9</td><td>97.6</td></tr></table>
+
+follows its AdaLN design and is conditioned only on timestep  $t$ . Additional experimental details, including DiT configurations and hyperparameter settings, are provided in Appendix C.
+
+Datasets and Evaluation Metric. We evaluate our DiTF on semantic correspondence, geometric correspondence, and temporal correspondence. For semantic correspondence, we conduct experiments on three popular benchmarks: SPair-71k [57], PF-Pascal [58], AP-10K benchmark [43]. The AP-10K benchmark [43] is a new large-scale semantic correspondence benchmark built on AP-10K [59]. It comprises 2.61 million/36,000 data pairs, and spans three settings: the main intra-species set, the cross-species set, and the cross-family set. Following [18, 14], we adopt the percentage of correct key-points (PCK) metric. See Appendix for the results of two additional correspondence tasks.
+
+# 5.2 Main results on semantic correspondence
+
+To evaluate our model DiTF, we conducted two types of semantic correspondence experiments: in-category semantic correspondence on SPair-71k [57] and PF-Pascal [58], and cross-category semantic correspondence on AP-10K [43].
+
+In-category Semantic Correspondence We conducted comprehensive experiments across different datasets where the results can be found in Tables 2 and 3. From these results, we could obtain the following observations: 1) State-of-the-art performance. Overall, our models achieve state-of-the-art performance across different datasets. Specifically, our model  $\mathrm{DiTF}_{sd3-5}(\mathrm{ours})^{\dagger}$  achieves  $2.6\%$  on Spair-71k and  $2.6\%$  on AP-10K-I.S. 2) Superior feature extraction capabilities. It is readily apparent that our feature extraction models for DiTs, such as  $\mathrm{DiTF}_{sd3-5}(\mathrm{ours})$  and  $\mathrm{DiTF}_{flux}(\mathrm{ours})$ , outperform traditional feature extractors like SD2-1 (DIFT [18]) and DINOv2 [12] across various datasets, demonstrating their strong and robust feature extraction abilities. Specifically, our  $\mathrm{DiTF}_{flux}(\mathrm{ours})$  achieves a performance of  $67.1\%$  ( $9.4\%$  over DIFT) on the Spair-71k dataset, and obtains a performance of  $62.2\%$  ( $1.3\%$  over DINOv2) on dataset AP-10K-I.S. 3) Feature
+
+![](images/b5ddd8d99766b9e3bfc6218280a45e99f61ae9bb11be4dbadd5e1ed2b117ac0e.jpg)  
+Figure 6: Investigation of the impact of channel modulation with AdaLN and channel discard strategy across different datasets.
+
+![](images/f0a3aa6dea539145f457ca14fa2a16aca9d242fa3612dd7e158f42637515c0dc.jpg)  
+Figure 7: Visualization of semantic correspondence prediction on SPair-71k using different features. Different colors represent different key points where circles denote correctly predicted points under the threshold  $\alpha_{bbox} = 0.1$  and crosses denote incorrect matches.
+
+redundancy across DiTs and DINOv2: As shown in Table 2, when integrating DINOv2, our  $\mathrm{DiTF_{sd3-5}}$  achieves a notable performance increase (from  $64.6\%$  to  $72.2\%$ ). In comparison, our  $\mathrm{DiTF_{flux}}$  shows a limited improvement, only a  $0.5\%$  gain (from  $67.1\%$  to  $67.6\%$ ). We interpret this performance discrepancy from feature alignment. FLUX aligns closely with DINOv2, causing redundancy, while SD3.5 benefits more from DINOv2's complementary distribution. This alignment may arise from two factors: Improved alignment between diffusion models and DINOv2 with larger models and longer training [60]; Overlap in training data.
+
+Cross-category Semantic Correspondence In addition, we conducted cross-category semantic correspondence experiments on the AP-10K dataset, as shown in Table 3. The results indicate that our models enable the extraction of accurate semantic features for cross-category image pairs and the drawing of correct correspondences, outperforming the previous best method, GeoAware-SC [43], by nearly  $4.8\%$  (64.6% to 69.4%) on AP-10K-C.S. Furthermore, the results show that previous methods, such as DINOv2 and DIFT, experience significant performance degradation on the cross-category task. In contrast, our model demonstrates stronger robustness for the cross-category correspondence.
+
+# 5.3 Further analysis
+
+Evaluation of Channel Modulation with AdaLN. We conducted comprehensive semantic correspondence experiments across different datasets on various DiTs. The results, as depicted in Figure 6 and Table 4, reveal several key observations: 1) Effectiveness. When integrated with the channel modulation by AdaLN, all the DiT models experience an absolute performance boost of more than  $20\%$ , demonstrating that AdaLN is capable of effectively mitigating massive activations and
+
+Table 4: Ablation study of our model  $\mathrm{DiTF_{flux}}$  on the dataset SPair-71k and AP-10K. We report the PCK per point for SPair-71k and PCK per image for AP-10K-I.S.  
+
+<table><tr><td rowspan="2">Model Variants</td><td colspan="3">SPair-71k</td><td colspan="3">AP-10K-I.S.</td></tr><tr><td>0.01</td><td>0.05</td><td>0.10</td><td>0.01</td><td>0.05</td><td>0.10</td></tr><tr><td>Original</td><td>2.3</td><td>14.6</td><td>29.9</td><td>2.4</td><td>16.5</td><td>33.9</td></tr><tr><td>+ AdaLN</td><td>7.8</td><td>52.3</td><td>65.3</td><td>6.4</td><td>46.8</td><td>60.3</td></tr><tr><td>+ Channel discard</td><td>8.3</td><td>56.3</td><td>67.1</td><td>7.6</td><td>49.7</td><td>62.2</td></tr></table>
+
+![](images/de045518568d016e6700aeb899429063648d68d2a3957545910268ab603afb40.jpg)
+
+![](images/4771c19efe6a8ac2f46fc8ed7cc4e4f56bde0a29f545ddd87c5c04dfad9160a0.jpg)
+
+![](images/0ed6c9f1ca6d0826af1d7eaf52372b6189c015b6ed3bd2fe97445b4099e3deec.jpg)
+
+![](images/55855490aceaf6d61b3c6bcd4123e6c512950dc86145acb26102b4485cbaf4bc.jpg)
+
+![](images/84a91b15cbd17bffc37c0760497449058ec2614b74c2b2409b9e460dfd48dd2e.jpg)  
+Figure 8: Visualization of multi-to-multi correspondences with our model. Green lines indicate correct matches and red incorrect
+
+![](images/30d3c3783273362a68cf1b27e9760cb77e37df531c46a7c7dbab3d2896886e08.jpg)
+
+![](images/80a1854f1b35b56ee3ca647d0c07ec2fff51df64a7bd2c01be711540028effbf.jpg)
+
+![](images/36e7df6e5fbac9526df841b113400952aeeefa6c08ef84da94b4c330f3fc6c09.jpg)
+
+Table 5: Conditions ablation study of AdaLN for feature channel-wise modulation on SPair-71k. We report the PCK@0.10 per point.  
+
+<table><tr><td>Condition</td><td>Pixart-Alpha</td><td>SD3</td><td>SD3-5</td><td>Flux</td></tr><tr><td>original</td><td>35.6</td><td>24.8</td><td>39.8</td><td>29.9</td></tr><tr><td>c</td><td>-</td><td>17.9</td><td>36.8</td><td>20.3</td></tr><tr><td>t</td><td>55.3</td><td>54.6</td><td>58.2</td><td>60.9</td></tr><tr><td>t &amp; c</td><td>-</td><td>54.9</td><td>58.3</td><td>63.6</td></tr></table>
+
+Table 6: Semantic segmentation on ADE20K.  
+
+<table><tr><td>Method</td><td>mIoUss</td><td>mIoUms</td></tr><tr><td colspan="3">self-supervised pre-training</td></tr><tr><td>DINOv2 [12]</td><td>47.7</td><td>53.1</td></tr><tr><td colspan="3">SD-based pre-training</td></tr><tr><td>VPD [42]</td><td>53.7</td><td>54.4</td></tr><tr><td colspan="3">DiTs-based pre-training</td></tr><tr><td>DiTFlux(w/o AdaLN)</td><td>43.8</td><td>44.8</td></tr><tr><td>DiTFlux</td><td>53.6</td><td>54.8</td></tr></table>
+
+enhancing feature semantics and discrimination. 2) Robustness. These results indicate that channel modulation with AdaLN enables the accurate normalization of features across different categories, leading to significant improvements on cross-category datasets.
+
+Evaluation of Channel Discard strategy. As shown in Figure 6 and Table 4. We can observe that our channel discard strategy can effectively locate and eliminate the weakly massive activations in the DiT feature across different DiTs, leading to smoother feature maps and higher correspondence matching accuracy.
+
+Impact of conditions in AdaLN for Channel-wise Modulation. From the results in Table 5, we can observe that the timestep  $t$  is essential for modulating massive activations, bringing a significant performance boost. In contrast, attempting to modulate features solely based on  $c$  proves ineffective. This phenomenon arises from the inherent nature of DiTs. During training, DiTs introduce noise to images at varying levels of  $t$ , making the timestep an essential condition for feature modulation. It serves as a key factor in locating the channel positions of massive activations and enhancing feature quality, enabling their transformations into a clean and semantically meaningful representation.
+
+Qualitative Results. We visualize some semantic correspondence predictions from different models in Figures 7 and 8. The visualization reveals that our DiT-based model yields robust and accurate correspondences across various complex scenes, including changes in viewpoint, multiple objects, and dense matching. These results demonstrate the superiority and robustness of our models.
+
+# 5.4 Other Task: Semantic Segmentation
+
+To assess the generalizability of DiTF, we conduct semantic segmentation experiments on ADE20K [61]. We extract activated DiT features using DiTF without applying the channel discard strategy and train a segmentation head following DINOv2 [12]. As shown in Table 6,  $\mathrm{DiTF}_{\mathrm{flux}}$  outperforms DINOv2, with channel modulation of AdaLN significantly enhancing feature semantics and discriminability, leading to a substantial mIoU improvement (from 44.8 to 54.8). These results highlight the effectiveness and broad applicability of our approach.
+
+# 6 Conclusion
+
+In this paper, we identify and characterize massive activations in Diffusion Transformers (DiTs). We observe that these activations consistently emerge in a few fixed dimensions across all image patch tokens and carry limited local information. We further demonstrate that the built-in AdaLN mechanism in DiTs effectively suppresses massive activations while enhancing feature semantics and discriminability. Building on these insights, we propose a training-free AdaLN-based framework, Diffusion Transformer Feature (DiTF), which extracts semantically discriminative features from DiTs through channel-wise modulation using AdaLN. Extensive experiments on visual correspondence tasks validate the robustness and effectiveness of our approach.
+
+# 7 Limitations and Future Work.
+
+In this work, we primarily focus on understanding and mitigating massive activations in Diffusion Transformers (DiTs) from the perspective of representation learning. However, the emergence and potential roles of massive activations in the generative process of DiTs remain underexplored. We believe that investigating these activations from a generative viewpoint could provide deeper insights into their underlying mechanisms and potentially contribute to improving generative performance.
+
+# Acknowledgements
+
+The paper is supported in part by the National Natural Science Foundation of China (No. U21B2013, 62325109), and in part by the Shanghai 'The Belt and Road' Young Scholar Exchange Grant (24510742000). Mehrtash Harandi is supported by the Australian Research Council (ARC) Discovery Program DP250100262.
+
+# References
+
+[1] Shilong Liu, Lei Zhang, Xiao Yang, Hang Su, and Jun Zhu. Unsupervised part segmentation through disentangling appearance and shape. In CVPR, 2021.  
+[2] Mark Hamilton, Zhoutong Zhang, Bharath Hariharan, Noah Snavely, and William T Freeman. Unsupervised semantic segmentation by distilling feature correspondences. arXiv:2203.08414, 2022.  
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+
+# NeurIPS Paper Checklist
+
+# 1. Claims
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+
+# 2. Limitations
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+Question: Does the paper discuss the limitations of the work performed by the authors?
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+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
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+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
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+Answer: [Yes]
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+Justification: We have already provided the proof of theoretical results in the experiment section.
+
+# Guidelines:
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+- The answer NA means that the paper does not include theoretical results.  
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+
+# 4. Experimental result reproducibility
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+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+# Answer: [Yes]
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+
+# Guidelines:
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+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
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+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
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+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
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+
+Guidelines:
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+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: We report the average results of open-source models under three random seeds (2024, 2025, 2026). Besides, the average performance is taken from three experimental runs for all proprietary models.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
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+
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We extract representation from DiTs on one L40s GPU for about 10 minutes, max GPU memory is about 16 GB.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more computer than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: We conducted the research in the paper conform, in every respect, with the NeurIPS Code of Ethics.
+
+Guidelines:
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+The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [NA]
+
+Justification: There is no societal impact of the work performed.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.
+
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: The scope of our research does not involve data or models with a high risk for misuse. We utilize publicly available, non-sensitive datasets and models that do not require special safeguards for responsible release.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: We strictly adhered to the license terms for all the assets.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.
+
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: This question does not apply to our research as no new assets were introduced. Our study solely relies on existing, publicly available resources, and thus, no additional documentation for new assets is necessary.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: This question does not apply to our research as it does not involve crowdsourcing experiments or research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: This question is not applicable to our research as no human subjects were involved.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components.
+
+Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# Appendix
+
+# A Diffusion Transformer Architecture
+
+![](images/fc1b6627ae178d5f7835810d7ff29b06382a10d265ba081cb53e32425094de8c.jpg)  
+Figure 9: DiT architecture illustration.
+
+We strictly follow the architecture used in DiT [36]. We provide an illustration of the DiT architecture as shown in Figure 9.
+
+# B More Visualization Results
+
+# B.1 Massive Activations in DiTs
+
+To further illustrate the emergence of massive activations in DiTs, we provide additional visualization results in Figure 13. We show the LayerNorm-normalized activation magnitudes of original features from SD2-1 and various DiTs. While SD2-1 exhibits smooth activations, DiTs consistently show spikes concentrated in a few fixed dimensions across all patch tokens, revealing a fundamental difference that contributes to their degraded performance.
+
+![](images/bceb0bed3adad9566ca995c4399d6b4467c864ad23a4b59e6560a59de5d15dcb.jpg)  
+Pixart-Alpha
+
+![](images/838097c7b1d42684988093d8c47980030114497de7524321fef3e863f8b936a5.jpg)
+
+![](images/822cd6ee50c3a73672f97e6d38586a0bc161dd6976f99bfa1f8759e8a6e4f5c7.jpg)  
+SD3
+
+![](images/57ad4f8689f9b7f32c9a4d8b22652b2156777e04073693bc5501430006065505.jpg)
+
+![](images/02ad0d830f75baebb0176ed9fb3757248a35d0181debce2af315fd15c2bebd09.jpg)  
+SD3-5
+
+![](images/5a5136629a576c9ecff5149299f51a5bb0bcca500e0a92c51d79df48d3beeb69.jpg)
+
+![](images/1887a57db4c1c0e5df3ad2b9d5e6a187bc36ac691f121aa2bf16eeb050e1f766.jpg)  
+Flux  
+Figure 10: Massive activations dimensions align with the residual scaling factor  $\alpha_{k}$ . We visualize the magnitudes for the original feature  $z_{k}^{t + 1}$  and residual scaling factor  $\alpha_{k}$ .
+
+![](images/32ee5b4476244240aac7733443783842c2b7cd4c7b1c096835c1da9f5147ccef.jpg)
+
+# B.2 Massive Activations Dimensions Align with Residual Scaling Factor
+
+In this section, we provide additional visualizations to examine the dimensional alignment between massive activations and the residual scaling factor  $\alpha_{k}$  from the AdaLN layer. As illustrated in Fig-
+
+ure 10, massive activations consistently co-occur with large values of  $\alpha_{k}$  in the same dimensions across all DiTs.
+
+# B.3 Channel-wise Modulation with AdaLN
+
+To comprehensively demonstrate the impact of the built-in AdaLN in DiTs, we provide additional visualizations comparing pre-AdaLN and post-AdaLN features across various models, including SD3-5 (Figure 11), Pixart-Alpha (Figure 14), SD3 (Figure 15), and Flux (Figure 16). These results consistently show that AdaLN accurately localizes and normalizes massive activations, while enhancing feature semantics and discrimination through effective channel-wise modulation.
+
+![](images/86d411438773ab69eeb8f780d68cf21addd685880b7ee18652e9c0fa7d3132b1.jpg)
+
+![](images/6e4a7802263334f87436164641bcb85977f9f05551cfa2a8140c399a17e3c782.jpg)
+
+![](images/c5eeb3b0824dd7c6fe02b042e0a869b9a812130ef44959f12950765acd09823e.jpg)
+
+![](images/8e9c2838d4b162bc80da4bbe2982a96c625de3a95f39101f86b93c91b73bafa0.jpg)  
+Figure 11: Comparisons of pre-AdaLN and post-AdaLN features in SD3-5.
+
+![](images/b7d44d1bf931b1cacbac52c678d99f75cfc54cae58959f181ae1fe236d4d59d0.jpg)
+
+![](images/51e0d4c109ca15aafc0c7044230eb47e3ec272ad249d7d07b1965f484d7ff71e.jpg)
+
+![](images/28accfba7081e84d64fe6770d618812e1fbabf17d9e6b450e12538552cf9e987.jpg)
+
+![](images/d953ef526f37431f4c5bf0bcced76a0d757188d7c6eb7a574e4c6bf2f8088072.jpg)
+
+![](images/ce4d266d812ac914b98f4ce4696593a16a480c42c5502851899505d263d83979.jpg)
+
+![](images/4d245b090a397d547d15ee1b612430e8f092cedc9fd478d81833aa878ed76919.jpg)
+
+![](images/b2c9e912e86d4dcac983d334e8395bfac895852a217c72f1bc1f1734da52a1ae.jpg)
+
+![](images/d25a4a3c93815729474dc142a1cbc577e5616887893dec598c3ba13f37db6157.jpg)
+
+![](images/a16a989a3a828f5e59a9b907a4995328eaf928d5d31d0b186690dd642a95473c.jpg)  
+Pre-AdaLN
+
+![](images/28c13b1b919fc9c44ebffa63e2f599a8b53546ea214b6bc0233b17aa4b3d4498.jpg)
+
+![](images/d9a09cfc47f4b48a37f4c46f8b8edd1ecef5a08df88682fccea52cff0aff88fc.jpg)
+
+![](images/5c6507927818a3b488e474f403cf1ebeb766bf8971a822e23dec79bc79e55024.jpg)
+
+![](images/e2dc4998b32e77dbdedb54e311590a4b727b4386cf26a40d5499a52743483c99.jpg)  
+Post-AdaLN
+
+![](images/e27021d2a4b85839be6d5eb27e84549b86b462e7f05df88d1572ef5a33dcd8d7.jpg)
+
+![](images/92985487f95abd126c790b5114806ab9c4f78ddb32ebe1522179ae324561abe2.jpg)
+
+![](images/329736348e2bc7eb321978ac66e63eda69b8f240158e0726cf4a111cd4401243.jpg)
+
+# C Further Implementation Details
+
+Configurations of different DiTs. We employ the pre-trained Diffusion Transformers (DiTs) as a feature extractor for semantic correspondence. Formally, we decompose the universal feature extraction process in DiTs into two stages: (1) extracting the original feature  $z_{t}^{k}$  from the DiT block  $\mathcal{A}_{k}$ , and (2) modulating it via adaptive channel-wise scaling and shifting using the AdaLN layer. For Pixart-alpha [21], SD3 [22], and SD3-5 [22], we first extract the pre-AdaLN feature  $z_{t}^{(k,2)}$  and then activate it as follows.
+
+$$
+z _ {t} ^ {(k, 2)} = \mathcal {A} _ {k} \left(z _ {t} ^ {k}\right), \quad \gamma_ {k} ^ {2}, \beta_ {k} ^ {2} = \mathrm {M L P} _ {k} (t, c) \tag {10}
+$$
+
+$$
+\hat {z} _ {t} ^ {k} = \left(1 + \gamma_ {k} ^ {2}\right) \operatorname {L a y e r N o r m} \left(z _ {t} ^ {(k, 2)}\right) + \beta_ {k} ^ {2} \tag {11}
+$$
+
+As some of the Flux [23] model's blocks contain only one group of AdaLN-zero layer, we extract pre-AdaLN feature  $z_{t}^{(k,1)}$  and then activate it as follows.
+
+$$
+z _ {t} ^ {(k, 1)} = \mathcal {A} _ {k} \left(z _ {t} ^ {k}\right), \quad \gamma_ {k} ^ {1}, \beta_ {k} ^ {1} = \mathrm {M L P} _ {k} (t, c) \tag {12}
+$$
+
+$$
+\hat {z} _ {t} ^ {k} = \left(1 + \gamma_ {k} ^ {1}\right) \text {L a y e r N o r m} \left(z _ {t} ^ {(k, 1)}\right) + \beta_ {k} ^ {1} \tag {13}
+$$
+
+![](images/aa5bf43c6a64311265dd5112ec690125fd609783b103c1f41f9bcb55896656a6.jpg)  
+Figure 12: Investigation of the DiT block index  $k$  and timestep  $t$ . We report the results of  $\mathrm{DiTF_{flux}}$  on dataset Spair-71k.
+
+![](images/48d55ec904621801c483b3fd7008163ca3e059553d5dc3c857dfe25d5655c77c.jpg)
+
+Table 7: Configurations of different DiTs for semantic correspondence.  
+
+<table><tr><td>Method</td><td>Layers N</td><td>Hidden size d</td><td>Timestep t</td><td>Block index k</td></tr><tr><td>DiTFpixart-α</td><td>28</td><td>1152</td><td>141</td><td>14</td></tr><tr><td>DiTFSD3</td><td>24</td><td>1536</td><td>340</td><td>9</td></tr><tr><td>DiTFSD3-5</td><td>38</td><td>2432</td><td>380</td><td>23</td></tr><tr><td>DiTFflux</td><td>57</td><td>3072</td><td>260</td><td>28</td></tr></table>
+
+The configurations of different DiTs can be found in Table 7, where the total time step T is 1000. We set the input image size as  $960 \times 960$  for DiTs and  $840 \times 840$  for the model DINOv2.
+
+Investigation on block index  $k$  and timestep  $t$ . The layer index  $k$  and the timestep  $t$  are critical hyperparameters that influence the quality of the extracted features from Diffusion Transformers (DiTs). Previous studies [18, 14] have conducted thorough investigations to identify the optimal layer index  $k$  and timestep  $t$  for Stable Diffusion. To explore the effects of varying  $k$  and  $t$  in DiTs, we conducted grid search experiments to identify the optimal hyperparameters. The results are presented in Figure 12. The figure reveals that features extracted from the middle layer achieve optimal performance across different DiTs. Furthermore, feature extraction in DiTs is robust to the timestep for semantic correspondence tasks, as a wide range of  $t$  achieves excellent performance.
+
+Integration of DINOv2 feature. We extracted DINOv2 features from the token facet of the 11th layer of the model. We then concatenated the DiT's features with the DINOv2 features in the channel dimension. To improve the efficiency of correspondence calculation, we computed Principal Component Analysis (PCA) across the pair of images for the features extracted from DiT, as follows:
+
+$$
+\tilde {z} ^ {s}, \tilde {z} ^ {t} = \operatorname {P C A} \left(\hat {z} ^ {s} \| \hat {z} ^ {t}\right) \tag {14}
+$$
+
+where  $\hat{z}^s, \hat{z}^t$  are the extracted source and target image DiT features. We only apply the PCA operation to SD3-5 and Flux features due to their high dimension and set the output dimension size as 1280.
+
+# D Geometric Correspondence
+
+To comprehensively evaluate our model DiTF, we conduct additional experiments on geometric correspondence.
+
+Datasets. Following [18], we evaluate our model on the HPatches benchmark [78], which comprises 116 sequences: 57 with illumination changes and 59 with viewpoint variations. Adopting the approach from CAPS [68], we detect up to 1,000 key points per image and apply cv2.findHomography() to estimate homography using mutual nearest neighbor matches.
+
+Metric. We adopt the corner correctness metric for evaluation where we compute the average error between the four estimated corners of one image and the ground-truth corners with a threshold of pixels, following [68, 18].
+
+Results. To comprehensively evaluate our models, we conducted geometric correspondence experiments on the HPatches benchmark [78], as detailed in Table 8. From the results, it can be observed that our model enables robust feature extraction for image pairs and draws precise geometric correspondence. Specifically, our model  $\mathrm{DiTF}_{\mathrm{flux}}$  (ours) achieve comparable performance  $41.9\%$  compared to the state-of-the-art model DIFT (SD-based). These results show that Diffusion Transformers can be employed as an effective feature extractor for geometric correspondence.
+
+Table 8: Geometric correspondence results on dataset HPatches. We report the homography estimation accuracy  $[\%]$  at 1, 3, 5 pixels.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Geometric Supervision</td><td colspan="3">All</td><td colspan="3">Viewpoint Change</td><td colspan="3">Illumination Change</td></tr><tr><td>ε = 1</td><td>ε = 3</td><td>ε = 5</td><td>ε = 1</td><td>ε = 3</td><td>ε = 5</td><td>ε = 1</td><td>ε = 3</td><td>ε = 5</td></tr><tr><td>SIFT [26]</td><td>None</td><td>40.2</td><td>68.0</td><td>79.3</td><td>26.8</td><td>55.4</td><td>72.1</td><td>54.6</td><td>81.5</td><td>86.9</td></tr><tr><td>LF-Net [62]</td><td></td><td>34.4</td><td>62.2</td><td>73.7</td><td>16.8</td><td>43.9</td><td>60.7</td><td>53.5</td><td>81.9</td><td>87.7</td></tr><tr><td>SuperPoint [63]</td><td></td><td>36.4</td><td>72.7</td><td>82.6</td><td>22.1</td><td>56.1</td><td>68.2</td><td>51.9</td><td>90.8</td><td>98.1</td></tr><tr><td>D2-Net [64]</td><td rowspan="4">Strong</td><td>16.7</td><td>61.0</td><td>75.9</td><td>3.7</td><td>38.0</td><td>56.6</td><td>30.2</td><td>84.9</td><td>95.8</td></tr><tr><td>DISK [65]</td><td>40.2</td><td>70.6</td><td>81.5</td><td>23.2</td><td>51.4</td><td>67.9</td><td>58.5</td><td>91.2</td><td>96.2</td></tr><tr><td>ContextDesc [66]</td><td>40.9</td><td>73.0</td><td>82.2</td><td>29.6</td><td>60.7</td><td>72.5</td><td>53.1</td><td>86.2</td><td>92.7</td></tr><tr><td>R2D2 [67]</td><td>40.0</td><td>74.4</td><td>84.3</td><td>26.4</td><td>60.4</td><td>73.9</td><td>54.6</td><td>89.6</td><td>95.4</td></tr><tr><td colspan="11">w/ SuperPoint kp.</td></tr><tr><td>CAPS [68]</td><td>Weak</td><td>44.8</td><td>76.3</td><td>85.2</td><td>35.7</td><td>62.9</td><td>74.3</td><td>54.6</td><td>90.8</td><td>96.9</td></tr><tr><td>DINO [12]</td><td></td><td>38.9</td><td>70.0</td><td>81.7</td><td>21.4</td><td>50.7</td><td>67.1</td><td>57.7</td><td>90.8</td><td>97.3</td></tr><tr><td>OpenCLIP [69]</td><td rowspan="2">None</td><td>33.3</td><td>67.2</td><td>78.0</td><td>18.6</td><td>45.0</td><td>59.6</td><td>49.2</td><td>91.2</td><td>97.7</td></tr><tr><td>DIFT [18]</td><td>45.6</td><td>73.9</td><td>83.1</td><td>30.4</td><td>56.8</td><td>69.3</td><td>61.9</td><td>92.3</td><td>98.1</td></tr><tr><td>DiTFflux(ours)</td><td></td><td>41.9</td><td>70.7</td><td>79.5</td><td>22.0</td><td>50.8</td><td>63.4</td><td>62.5</td><td>91.3</td><td>96.2</td></tr></table>
+
+Table 9: Temporal correspondence results on DAVIS-2017. We report the region-based similarity  $\mathcal{I}$  and contour-based accuracy  $\mathcal{F}$  for DAVIS. Pre-: Pre-trained on videos.  
+
+<table><tr><td rowspan="2">Pre-</td><td rowspan="2">Method</td><td rowspan="2">Dataset</td><td colspan="3">DAVIS</td></tr><tr><td>J&amp;Fm</td><td>Jm</td><td>Fm</td></tr><tr><td rowspan="2">✓</td><td>MAST</td><td rowspan="2">YT-VOS [70]</td><td>65.5</td><td>63.3</td><td>67.6</td></tr><tr><td>SFC [71]</td><td>71.2</td><td>68.3</td><td>74.0</td></tr><tr><td rowspan="9">X</td><td>InstDis [70]</td><td></td><td>66.4</td><td>63.9</td><td>68.9</td></tr><tr><td>MoCo [73]</td><td></td><td>65.9</td><td>63.4</td><td>68.4</td></tr><tr><td>SimCLR [74]</td><td>ImageNet [72]</td><td>66.9</td><td>64.4</td><td>69.4</td></tr><tr><td>BYOL [75]</td><td>w/o labels</td><td>66.5</td><td>64.0</td><td>69.0</td></tr><tr><td>SimSiam [76]</td><td></td><td>67.2</td><td>64.8</td><td>68.8</td></tr><tr><td>DINO [32]</td><td></td><td>71.4</td><td>67.9</td><td>74.9</td></tr><tr><td>OpenCLIP [69]</td><td rowspan="2">LAION [77]</td><td>62.5</td><td>60.6</td><td>64.4</td></tr><tr><td>DIFF [18]</td><td>70.0</td><td>67.4</td><td>72.5</td></tr><tr><td>DiTFlux(ours)</td><td></td><td>72.2</td><td>69.2</td><td>75.1</td></tr></table>
+
+# E Temporal Correspondence
+
+In addition, we conduct experiments to verify the temporal correspondence capability of our DiTF. Specifically, we investigate DiTF's performance on video object segmentation and pose tracking tasks, employing DiTs as a feature extractor for correspondence.
+
+Datasets. We conduct experiments on the challenge video dataset: DAVIS-2017 video instance segmentation benchmark [79], following [18].
+
+Metric. Following [18, 80], we adopt the region-based similarity  $\mathcal{I}$  and contour-based accuracy  $\mathcal{F}$  as the performance metric where we segment the nearest neighbors between the consecutive video frames based on the representation similarity.
+
+Results. The temporal correspondence results can be found in Table 9. From the results, it can be observed that our model exhibits a superior capability to extract video frame features for temporal correspondence. Specifically, our model achieves  $72.2\%$  on the dataset DAVIS, which outperforms the previous state-of-the-art DIFT by  $2.2\%$ , demonstrating the superior effectiveness of our model.
+
+# F Qualitative Results on AP-10K
+
+We show the qualitative comparison of our Diffusion Transformer model DiTF with both Stable Diffusion (SD) and SD+DINO [14] in AP-10K intra-species (Figure 17), cross-species (Figure 18), and cross-family (Figure 19) subset.
+
+![](images/5fa481194dd2ae352b74259e045fb678e3e7ec4f81eb1dbcb76dbb6cb11f250e.jpg)  
+Figure 13: Massive activations in DiTs. SDv2-1 does not suffer from massive activations. However, all DiTs exhibit massive activations, which concentrate on very few fixed dimensions across all image patch tokens.
+
+![](images/020aa78b6192fe4186ff28055e3cf68936f194dee1ce8d57a883f286116a33b6.jpg)  
+Input image
+
+![](images/6dff8440bc208de1c9bd5c25f0eb9250b593c409ff09cf7a6dd4302503dc42c5.jpg)  
+Pre-AdaLN
+
+![](images/8c957c7017f656a41ed1e63f130f44aecbf64aba33520b4bccc977eb5d050944.jpg)  
+Post-AdaLN
+
+![](images/0778bac8b432445fc320f8b04afb321e49dee5ec5016515677c621410a74b3f1.jpg)  
+Pre-AdaLN
+
+![](images/be7bbf14202f3ac989cbc96c73bb50febac6e06db87ba76ad1b8e89bc05f3327.jpg)  
+Post-AdaLN
+
+![](images/2974ae3e1d599074265f704537b3674beab20e864cdd1f9c4727f8330109271f.jpg)
+
+![](images/57b220074ea9bec996c56647d60ce59dddd2c3b60a1b04bcfece5cad802f316b.jpg)
+
+![](images/970022828e479d217bf79948203752933fcdb05863762ca6e3bb415bda22a6c1.jpg)
+
+![](images/5a06ce4d976019160c45abf38454feafdacb9c58e2ee0b3800d069c47bbdb9da.jpg)
+
+![](images/93041fe905e787d3d1cbcb243be5861ad200f491a947c5b87f6f6525c756d5f9.jpg)
+
+![](images/aa3c362737d902cb58d868b445e0b50dd0ee0f0f83796b43a14014f14779110e.jpg)
+
+![](images/341b6273b1dd74d4dd2d08942669ffb9b21b7fc77bb84bc46bac633256acf0cb.jpg)
+
+![](images/c0db8e4481e9818d721ddb0e71a3d0fb83fc5a2dba38c413ce428436b60cad25.jpg)
+
+![](images/e7d7e550a2dce2b0769e589ed022ec52db55d095ab49756b5b06ccc0196357bf.jpg)
+
+![](images/fbce5f78f74e563f2f891f79178282a280abbd1e04aec8188f8bc58035974082.jpg)
+
+![](images/4b9ce2cbc3f559007a7c9f3a8d25fbdf67e0b98658a700ddaf5eddb64dcdb970.jpg)  
+Figure 14: Comparisons of pre-AdaLN and post-AdaLN features in Pixart-Alpha.
+
+![](images/b1443bd73d2039e76589b6c2d2075fd693e6188c7d2b75fc0d563bd3025f0547.jpg)
+
+![](images/9cc4e9b80d0d50476702027d267b73482daf9e15d47cbf63496b5e6c66a62d9d.jpg)
+
+![](images/cc6959cb8e6b84f97a7a0c298553d82f0b6acc7b1cb4ee3c0d6e0775c6a05174.jpg)
+
+![](images/2e0aefbcd4e56251cfa89323a04c51bd70e633f73a1ea607093fda2421478ed9.jpg)
+
+![](images/5f5f55cd69956f3128f15e41b966297330876d20d4ae70f30e201c317f3a4a9e.jpg)  
+Input image
+
+![](images/57848916e664230ff9396274f69b91eee1f5ca8136d8331272aa7ebca7e7b6c6.jpg)  
+Pre-AdaLN
+
+![](images/ec5c035d2be30794e55b85c35fa2bc42380152743d3a985e5e58933d9e234d0e.jpg)  
+Post-AdaLN
+
+![](images/2ce5ec2d8a447e3119d1ad4bb6e71a8ea2a1c21feae40f13ded86f31fa9fa02d.jpg)  
+Pre-AdaLN
+
+![](images/751811830df047f88369c5d590be3b29a104775007003aa2de549d03de370b05.jpg)  
+Post-AdaLN
+
+![](images/16b1afb97ecddca93238c8fd719b286a51d7a643283910d487846c7e2f7b05f0.jpg)
+
+![](images/fb95afb82e424a34a0b983507715d01ef93cc3203cc47e36a6f9aa78e1ced4bf.jpg)
+
+![](images/c4dba827dc10580334882df483b99f4b8b1fe681a0478de6f23250a8848457f6.jpg)
+
+![](images/8c34b41394353999b46eabacf5e871dfd0fdcb5a7520829e5728884d52a649df.jpg)
+
+![](images/cc492a1965b2d32f20f8354b6df8e95f40311f6b2f07965052de6b771c90b360.jpg)
+
+![](images/bdab5522ac9822c231768c1a429849323a18e9252f81851d4c7b9d1df30827b8.jpg)
+
+![](images/e6d795435b4dd9ff56a57373366d88e63b02cff45fa048009664418a0ff407a2.jpg)
+
+![](images/582f61259bd0ce416d90d01e5d1b0a630a190e207052834672e6835be062a9d3.jpg)
+
+![](images/7c672d0a97b680461705de62bd7b9c94b7ef3175748701683a0045ad9c0068cd.jpg)
+
+![](images/ebb83e10a2a8f83b36b137acecb781c5d1975bd89303fcd27cc5e793e904cf57.jpg)
+
+![](images/55fe83fa9d8ea3df02dc4f3585895513b6df8a4e695feac724ece92fff057ade.jpg)
+
+![](images/ca277131af08d38d89964a4691701c837084044cc3f1ea09a85db20912ca278f.jpg)  
+Figure 15: Comparisons of pre-AdaLN and post-AdaLN features in SD3.
+
+![](images/91174099b80807a51ab7be2a4a6b33f56071079dc17b26ea1470e1af01263e71.jpg)
+
+![](images/a8bc82ca56ecf858313cce694bc14ce975f12a3c6294fff4af0c4d7fa5674824.jpg)
+
+![](images/60c18a5d56265918e49730662bc3b4c6eb3efbb205c511bd8a08ae66b465d372.jpg)
+
+![](images/a2f44e1973254219d60ff9da478b76bf1969d0f0320a7039c7ac011505b549f7.jpg)
+
+![](images/f9f6cede9ead24fbf9d4f75fa505bd8dfda9d0337f8250e5745b0970cfe60540.jpg)
+
+![](images/799773ff97bf54ad6b818a3fc892a09d16cbb2a6fb40f1951325c63f60003873.jpg)
+
+![](images/4f07e360c547b2aa6a0f060d9c7c1b448c66daebbac81064b0dc6679918afcc8.jpg)
+
+![](images/92f20c4b4d9d736ec9f2c17676df230dc1c97800413e19d6f421ebbc772d6de7.jpg)
+
+![](images/b7b2cd0fcf703a97c25e19b1835db3b7c22d36c3e68cd03ff0d299946dcbf019.jpg)
+
+![](images/1be06023eb3925e7998eb2dfaf728039c000916dd7a0311c998260b183ce3c53.jpg)
+
+![](images/be3e3a7b0a62da0e8a487b0dabe9ee04091e8da6ce1f7a27589d0a92b1dfa222.jpg)
+
+![](images/fc82e47b8985b20ea4df80487e1eec9038858bda57d804cbce2ca535f8832561.jpg)
+
+![](images/876f2a4347c9dc23cb4acbcc9870f68b66212d83a379e5492761f14098aeea49.jpg)
+
+![](images/0f81391476df6a1e2c48876962bd0c40c2d08c3f88f5a55daa767ddccd727a28.jpg)
+
+![](images/a53d45674d6abdf8ee25d59ef54e0cab980e1b6cc76377097621723e43e99fa7.jpg)
+
+![](images/edb38c3aa6ac27b61fb5fed87335a4c5ccb2345754e37290c465c4cb6c444d90.jpg)
+
+![](images/7661f567044da49bf3eb15497c4ea81aa095d4db2afbbca392e9c2300720674d.jpg)
+
+![](images/14b2da6a029633f37f7ee64ef0fd8cc3183df865d42da3e68186a82e88c38195.jpg)
+
+![](images/9198850912915c95db4ce5b12fc67f062002b9194be2174e0edf0332db372a57.jpg)  
+Figure 16: Comparisons of pre-AdaLN and post-AdaLN features in Flux.
+
+![](images/064568eb70d54a0b1f99a058859795400a2e77b6ca7914a47c75620c95729582.jpg)
+
+![](images/06b078dc31fb6ea5ff890fa35a59719ef04213dc855fcfe68518dd2268a6f0a6.jpg)
+
+![](images/156fe7455aba080d01f82c8b638b1c8ff29bd8f8bb1faa38fd0ed1a4a54fde57.jpg)
+
+![](images/cd3a7afd7ef9b8574acf7df10caf67bdffdcb8b554245478ff3554f5a250dae2.jpg)
+
+![](images/e6f8cebaffa8428694d3a7349ffa26298944c58970d6de6ad96fa77a57e9b947.jpg)  
+Figure 17: Qualitative comparison on the AP-10K intra-species set. Different colors represent different key points where circles denote correctly predicted points and crosses denote for incorrect matches.
+
+![](images/be0b9fe175cb10bc2173a4f9ee859c179c8f42406cd1a1003c870415528f8e4e.jpg)  
+Figure 18: Qualitative comparison on the AP-10K cross-species set.
+
+![](images/00b44c84c21481fbda72f1450ddeec894ee92b0dd345442404edf70be69cae68.jpg)  
+Figure 19: Qualitative comparison on the AP-10K cross-family set.
\ No newline at end of file
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+# Unleashing Foundation Vision Models: Adaptive Transfer for Diverse Data-Limited Scientific Domains
+
+Qiankun Li\*1,3, Feng He $^{1}$ , Huabao Chen $^{1}$ , Xin Ning $^{2}$ , Kun Wang\*3, Zengfu Wang\*1
+
+<sup>1</sup>University of Science and Technology of China
+
+$^{2}$ AnnLab, Institute of Semiconductors, Chinese Academy of Sciences
+
+$^{3}$ Nanyang Technological University
+
+*Corresponding Author
+
+{qklee, wk520529}@mail.ustc.edu.cn, zfwang@ustc.edu.cn
+
+# Abstract
+
+In the big data era, the computer vision field benefits from large-scale datasets such as LAION-2B, LAION-400M, and ImageNet-21K, Kinetics, on which popular models like the ViT and ConvNeXt series have been pre-trained, acquiring substantial knowledge. However, numerous downstream tasks in specialized and data-limited scientific domains continue to pose significant challenges. In this paper, we propose a novel Cluster Attention Adapter (CLAdapter), which refines and adapts the rich representations learned from large-scale data to various data-limited downstream tasks. Specifically, CLAdapter introduces attention mechanisms and cluster centers to personalize the enhancement of transformed features through distribution correlation and transformation matrices. This enables models finetuned with CLAdapter to learn distinct representations tailored to different feature sets, facilitating the models' adaptation from rich pre-trained features to various downstream scenarios effectively. In addition, CLAdapter's unified interface design allows for seamless integration with multiple model architectures, including CNNs and Transformers, in both 2D and 3D contexts. Through extensive experiments on 10 datasets spanning domains such as generic, multimedia, biological, medical, industrial, agricultural, environmental, geographical, materials science, out-of-distribution (OOD), and 3D analysis, CLAdapter achieves state-of-the-art performance across diverse data-limited scientific domains, demonstrating its effectiveness in unleashing the potential of foundation vision models via adaptive transfer. Code is available at https://github.com/qklee-lz/CLAdapter.
+
+# 1 Introduction
+
+With the rapid advancement in artificial intelligence, deep learning-based computer vision algorithms have emerged as a dominant force [42, 85, 47]. These algorithms are inherently data-driven, capitalizing on substantial datasets to refine their task-specific performance. The digital age's ever-growing data trove has ushered in large-scale datasets, such as ImageNet-21K [58], LAION-400M [62], and LAION-2B [61], which aim to bolster algorithmic generalization and accuracy through data diversity and volume [25, 44, 10]. Despite these advancements, domain-specific challenges and data scarcity remain significant hurdles in scientific visual downstream tasks, where specialized data is often limited, heterogeneous, or expensive to acquire [11, 72, 79]. Therefore, developing methods that effectively harness the potential of large-scale pre-trained models to enable robust adaptation in data-limited scientific domains constitutes a critical and promising research direction.
+
+Transfer learning through methods like linear probing and full fine-tuning is an essential approach for enhancing performance on downstream tasks [77]. This works well when transferring under
+
+normal-sized dataset pre-trained models to in-distribution (ID) downstream tasks. However, transferring adapted knowledge from rich but complex large-scale upstream pretraining poses significant challenges in the current era of large datasets [61, 10]. Furthermore, the scientific domains downstream tasks often involve out-of-distribution (OOD) scenarios [4], domain specificity [27], and data limitations [68], intensifying the fine-tuning challenge. L2-SP fine-tuning [26] introduced L2 regularization to preserve the insights of pre-trained weights during task adaptation, although they may lack robustness in OOD contexts. Visual Prompt Tuning (VPT) [34] augmented the input space with task-specific learnable prompts. However, VPT is primarily designed for Vision Transformer (ViT) [70] and lacks the ability to provide stable cross-domain feature transferability. OLOR [30] designed the fine-tuning optimizer from the perspective of pre-trained weights to improve stability, but it lacks task-specific self-adaptability, particularly under the diverse conditions of downstream scientific tasks.
+
+![](images/7e14a5eac4037c4d4dd8af3ac69ff038938816e567a858a1b606387a24eb9253.jpg)  
+Figure 1: Overview of the proposed CLAdapter. CLAdapter refines and adapts the rich representations learned from large-scale data to diverse data-limited scientific downstream tasks, achieving state-of-the-art performance across diverse fields on 10 datasets.
+
+In this paper, we propose a novel Cluster Attention Adapter (CLAAdapter), which refines and adapts the rich representations learned from large-scale data to diverse data-limited scientific downstream tasks (as illustrated in Figure 1). Specifically, CLAdapter introduces attention mechanisms and cluster centers to enable customized feature enhancement through distribution-aware correlation and transformation matrices. This facilitates the generation of task-adaptive representations, supporting a smooth transition from abundant pre-trained features to diverse downstream scenarios. Benefiting from our unified interface design, CLAdapter can seamlessly integrate with mainstream architectures, including CNNs, Transformers, and their 3D versions. In addition, a Staged Fine-Tuning (SFT) strategy is presented to collaborate with CLAdapter to further enhance the fine-tuning performance. Through extensive experiments conducted on 10 datasets spanning domains such as generic, multimedia, biological, medical, Industrial, agricultural, environmental, geographical, materials science, out-of-distribution (OOD), and 3D analysis, CLAdapter demonstrates its universal applicability and state-of-the-art performance. These results underscore the importance of effective knowledge transfer in the big data era and advance the reliable and efficient deployment of computer vision foundation models across scientific and industrial domains.
+
+The contributions of this paper are summarized as follows:
+
+① Adaptive Representation Transfer. We propose a novel CLAdapter that leverages large-scale pre-trained knowledge to enhance performance on a variety of data-limited downstream tasks.  
+② Flexible Adaptation Framework. We design a unified interface and a staged fine-tuning (SFT) strategy, enabling CLAdapter to integrate seamlessly with mainstream pre-trained models and establish an efficient fine-tuning paradigm.  
+③ AI4Science Broad Evaluation. We conduct comprehensive experiments on 10 datasets across diverse domains, including multiple scientific fields where data is limited and heterogeneous. CLAdapter consistently achieves state-of-the-art performance, demonstrating its potential as a generalizable solution for AI-driven scientific applications.
+
+# 2 Related Work
+
+# 2.1 Pre-Training Resources
+
+With the rapid development of computer vision technology, a large number of large-scale datasets [58, 62] and pre-trained models [56, 57, 5, 25, 19] have been proposed, providing a rich feature library for downstream tasks. Pre-training on large-scale datasets can encode rich semantic information, which is useful in solving limited data tasks, domain generalization, and zero-shot learning. ImageNet-21k[58], mainly used for visual image classification, contains about 21k categories and 14 million images, providing a rich and diverse training environment for large models. The LAION-400M dataset [62] is a large-scale image and text pairing dataset, containing about 300 million image-text pairs, including natural landscapes, people, everyday items, etc., suitable for training cross-modal models [56, 57]. To cope with these growing resources, a new generation of pre-trained models (such as CLIP [56], BEiT [5], MAE [25], and EVA [19]) has emerged, mainly utilizing the architectural principles of ViT [70] and ConvNeXt [50]. However, how to efficiently transfer a large number of complex datasets remains an unresolved issue. Therefore, this paper proposes CLAdapter, introducing attention mechanisms and clustering centers to leverage rich pre-training resources, thus effectively improving performance on various downstream tasks.
+
+# 2.2 Various Downstream Tasks and Fine-Tuning Methods
+
+The realm of practical downstream visual tasks is vast. However, most are cross-domain with limited data, such as medical image processing [11], industrial fault diagnosis [60], pest and disease recognition [72], and natural geographic image classification [83], Materials science research [45], 3D multimedia analysis [37]. Fine-tuning pre-trained models has become a critical method for improving performance on downstream tasks. The popular fine-tuning methods are Linear Probing (LP) [26], adjusting only the model's head, and Full Fine-tuning (FT), tuning all layers. Recently, L2-SP fine-tuning introduced an L2 regularization to keep changes to pre-trained weights minimal, thus preserving initial insights while adapting to new tasks. Visual Prompt Tuning (VPT) [34] inserted trainable prompts at the input, akin to NLP's prompt learning, to prevent altering the original model weights. VPT proved effective for tasks with numerous parameters and scant data, utilizing pre-trained knowledge and averting overfitting from significant weight modifications. However, VPT is mainly designed for Vision Transformer (ViT) [17] and has cross-domain instability issues. Different from existing fine-tuning methods, CLAdapter introduces attention mechanisms and cluster centers for customized feature representation refinement. By providing a uniform interface for various upstream tasks, CLAdapter fine-tunes any category of pre-trained models, transferring their knowledge to a wide array of downstream tasks.
+
+# 3 Methods
+
+We propose CLAdapter to refine and adapt the rich representations learned from large-scale data for transfer applications in various downstream tasks. It injects a small number of learnable parameters into the original model, offering an efficient fine-tuning strategy by freezing the backbone, alongside a staged fine-tuning approach for more gradual adaptation. The framework is depicted in Figure 2.
+
+# 3.1 Unified Model Interface
+
+Given an input image  $X_{I}\in \mathbb{R}^{C\times H\times W}$ , a standard Vision Transformer (ViT) divides  $X_{I}$  into  $N$  patches. Each patch is then embedded into a  $D$ -dimensional latent space, resulting in a set of tokens  $\pmb{\tau}\in \mathbb{R}^{N\times D}$ . Then,  $\pmb{\tau}$  and a extra learnable classification token  $\pmb{x}_{class}$  ([class]) with position embeddings are fed into the transformer layers  $\{E^l\}_{l = 0}^{L - 1}$ . Thus, the feature representation extracted by ViT is  $\pmb{\tau}^{L}\in \mathbb{R}^{(N + 1)\times D}$ . Discarding  $\pmb{x}_{class}$  due to its linear combination nature, as it might interfere with feature transfer from the pre-train model to downstream tasks.
+
+For CNN-based models, the feature map  $X_{F} \in \mathbb{R}^{C' \times H' \times W'}$  is extracted. Here,  $C', H'$ , and  $W'$  represent the number of channels, height, and width of the feature map, respectively. To facilitate the integration of CNN-based models with CLAdapter, we flatten the spatial dimensions of the feature map  $X_{F}$  to align with the feature dimensionality used by ViT.
+
+![](images/e17d20eef3ccb169a34f3d989d2fc1027151c12e3657ce8f44972e7822e72f07.jpg)  
+Figure 2: Overview of the CLAdapter. It utilizes large-scale pre-training to enhance various data-limited downstream tasks. The unified interface design and SFT fine-tuning strategy allow CLAdapter to integrate with mainstream pre-trained models and form an efficient fine-tuning paradigm.
+
+In the case of 3D video clips or image sequences  $\mathbf{X}_V \in \mathbb{R}^{T \times C \times H \times W}$ , the extracted features also include an additional temporal dimension  $T$ . Although time and spatial dimensions together form tubes rather than patches, they still belong to the internal structure of data. Therefore, we similarly flatten these dimensions to achieve a unified representation.
+
+In summary, whether for CNN-based models, Transformer-based models, or their 3D variants, we uniformly apply an interface function for the features extracted by the models, denoted as  $\mathcal{F}(\boldsymbol{X}_I / \boldsymbol{X}_V) \to \mathcal{H}$ , where  $\mathcal{H} \in \mathbb{R}^{N \times D}$  represents the extracted features after dimension unification.
+
+# 3.2 Cluster Attention Adapter (CLAdapter)
+
+For downstream tasks, it is crucial to focus on specific information pertinent to their domain. However, while large-scale upstream data encompasses a wealth of information, it also introduces complexity and redundancy. This challenge is further exacerbated when dealing with Out-Of-Distribution (OOD) tasks, making it more difficult to distill the necessary information from the rich pre-trained feature representations  $\mathcal{H}$ . In real-world applications, with the diverse nature of downstream tasks presenting both in-distribution (ID) and OOD scenarios, it is crucial to design a mechanism capable of adaptively refining and transforming the pre-trained features  $\mathcal{H}$  into suitable features  $\mathcal{H}'$  based on the characteristics of the downstream tasks. This process can be defined as learning a mapping function  $\mathcal{F}_{\theta}(\mathcal{H}) \to \mathcal{H}'$ , where  $\mathcal{F}_{\theta}$  denotes a model and  $\theta$  represents its learnable parameters.
+
+CLAdapter aims to refine and adapt the feature representations  $\mathcal{H}$  learned from large-scale datasets for use in various data-limited downstream tasks. Notably, embeddings of image categories in feature space are often close to each other, suggesting the presence of feature cluster centers that represent specific latent information. To exploit this, we introduce multiple learnable vectors to denote these feature cluster centers:
+
+$$
+\boldsymbol {\mathcal {A}} = \left\{\boldsymbol {\mathcal {A}} _ {1}, \boldsymbol {\mathcal {A}} _ {2}, \dots , \boldsymbol {\mathcal {A}} _ {K} \right\}, \tag {1}
+$$
+
+where  $\mathcal{A} \in \mathbb{R}^{D \times K}$ , and  $K$  is the number of cluster centers. The attention scores are derived by calculating the cosine similarity between the pre-trained embeddings  $\mathcal{H}$  and the cluster centers  $\mathcal{A}$ . To enhance computational efficiency and convert attention scores to a relative probability distribution  $\beta$ , we compute the mean of  $\mathcal{H}$  to obtain  $\mathcal{H}^q$  and apply the softmax function, respectively. The above operation is denoted mathematically as follows:
+
+$$
+\boldsymbol {\mathcal {H}} = \text {L a y e r N o r m} (\boldsymbol {\mathcal {H}}), \quad \boldsymbol {\mathcal {H}} ^ {q} = \frac {1}{N} \sum_ {i = 1} ^ {N} \boldsymbol {\mathcal {H}} _ {i}, \tag {2}
+$$
+
+$$
+\hat {\mathcal {H}} ^ {q} = \frac {\mathcal {H} ^ {q}}{\| \mathcal {H} ^ {q} \|}, \quad \hat {\mathcal {A}} = \left[ \frac {\mathcal {A} _ {1}}{\| \mathcal {A} _ {1} \|}, \frac {\mathcal {A} _ {2}}{\| \mathcal {A} _ {2} \|}, \dots , \frac {\mathcal {A} _ {K}}{\| \mathcal {A} _ {K} \|} \right], \tag {3}
+$$
+
+$$
+\boldsymbol {\beta} = \operatorname {s o f t m a x} \left(\hat {\boldsymbol {\mathcal {H}}} ^ {q} \hat {\boldsymbol {\mathcal {A}}}\right), \tag {4}
+$$
+
+where Layer-Norm reduces the difference in input distribution between different layers, and both  $\hat{\mathcal{H}}^q$  and  $\hat{\mathcal{A}}$  are L2 normalized along the feature dimension. Upon obtaining the attention scores  $\beta \in \mathbb{R}^{K}$ ,
+
+we further introduce learnable transformation matrices  $\mathcal{M}$  corresponding to the cluster centers  $\mathcal{A}$ :
+
+$$
+\mathcal {M} = \left\{\mathcal {M} _ {1}, \mathcal {M} _ {2}, \dots , \mathcal {M} _ {K} \right\}, \tag {5}
+$$
+
+where each transformation matrix  $\mathcal{M}_i\in \mathbb{R}^{D\times D}$ . The weighted transformation matrix  $\mathcal{M}^*$  for each pre-trained feature embedding is derived by weighting these matrices with the attention scores:
+
+$$
+\boldsymbol {\mathcal {M}} ^ {*} = \sum_ {i = 1} ^ {K} \beta_ {i} \boldsymbol {\mathcal {M}} _ {i}. \tag {6}
+$$
+
+Consequently, each embedding is subjected to a custom transformation matrix based on its cosine similarity with each feature cluster center, facilitating the organized transition from the original upstream feature distribution to a new distribution tailored for downstream tasks. This involves a customized transformation of  $\mathcal{H}$  using  $\mathcal{M}^*$ , followed by enhancement with a Layer-Norm and MLP layer to improve generalization and introduce non-linearity. The above operation is denoted mathematically as follows:
+
+$$
+\boldsymbol {\mathcal {H}} ^ {*} = \operatorname {L a y e r N o r m} \left(\boldsymbol {\mathcal {H}} \boldsymbol {\mathcal {M}} ^ {*}\right), \tag {7}
+$$
+
+$$
+\boldsymbol {\mathcal {H}} ^ {\prime} = \operatorname {G E L U} \left(\boldsymbol {\mathcal {H}} ^ {*} \boldsymbol {W} _ {1} + \boldsymbol {b} _ {1}\right) \boldsymbol {W} _ {2} + \boldsymbol {b} _ {2}, \tag {8}
+$$
+
+where  $\mathcal{H}^*$  is the result of the customized transformation. In the MLP,  $GELU$  represents the Gaussian Error Linear Unit activation function, and  $W_{1}$  and  $W_{2}$  are the weight matrices for the first and second linear transformations, respectively, with a default ratio of 4.  $b_{1}$  and  $b_{2}$  are the corresponding bias vectors. The final output  $\mathcal{H}'$  represents the features adaptively refined and transformed from the rich pre-trained feature embeddings  $\mathcal{H}$  to suit downstream tasks. Additionally, these transformed features can be reshaped back to the original feature shape of the upstream model through inverse function of the unified interface defined in Section 3.1, facilitating the integration with their respective heads.
+
+# 3.3 Fine-tuning Strategy
+
+To adapt a pre-trained model for a downstream task, practitioners commonly employ either full fine-tuning (FT), where all model parameters are updated, or linear probing (LP), which only updates the parameters of the final linear classification layer (head). By incorporating our proposed CLAdapter, the LP approach updates both the adapter and the classification layer, whereas the FT approach updates the entire model. These two popular fine-tuning strategies can be formalized as:
+
+$$
+\mathcal {L P} _ {C L} (x) = \boldsymbol {W} _ {h d} \cdot h _ {C L} (\boldsymbol {\mathcal {H}}) + \boldsymbol {b} _ {h d}, \tag {9}
+$$
+
+$$
+\mathcal {F T} _ {C L} (x) = \boldsymbol {W} _ {h d} \cdot h _ {C L} \left(\phi_ {p r e} (x)\right) + \boldsymbol {b} _ {h d}, \tag {10}
+$$
+
+where  $x$  denotes the input,  $h_{CL}(\cdot)$  is the function represented by CLAdapter,  $\phi_{pre}(\cdot)$  represents the pre-trained backbone, and  $W_{hd}$  and  $b_{hd}$  are the learnable parameters of the head. LP is computationally efficient as it only updates to the adapter and the head of the model. Although FT provides a thorough adaptation to the downstream task, starting CLAdapter training from scratch might distort the backbone's refined data representations, potentially exacerbating domain mismatch in OOD scenarios.
+
+To address this, we propose a staged fine-tuning (SFT) strategy, beginning with only updates to the CLAdapter and heads in the first stage and progressing to full fine-tuning in the second. This method allows CLAdapter first to obtain better pre-transfer capabilities for the original pre-training domain, then further fine-tune the entire model to complete the downstream task. Since the fine-tuning overhead of LP is almost negligible compared to FT, the incremental cost of SFT is minimal. The SFT strategy can be represented as:
+
+$$
+\mathcal {S F T} _ {C L} (x) = \boldsymbol {W} _ {h d} ^ {L P} \cdot h _ {C L} ^ {L P} \left(\phi_ {\text {p r e}} (x)\right) + \boldsymbol {b} _ {h d} ^ {L P}, \tag {11}
+$$
+
+where  $W_{hd}^{LP}$ ,  $b_{hd}^{LP}$  and  $h_{CL}^{LP}$  represent the head and CLAdapter after the first stage of fine-tuning (LP), respectively. Notably, in some cases, this LP stage often yields satisfactory performance for many tasks, thus reducing costs. SFT strategy efficiently leverages the strengths of both LP and FT, facilitating effective knowledge transfer from the pre-trained model to diverse downstream tasks.
+
+# 4 Experiments
+
+# 4.1 Experiment Setup
+
+Pre-training Dataset and Backbones. In the era of big data, popular publicly available large-scale 2D datasets include the ImageNet-21K classification dataset [58] at the ten-million level, the LAION-400M image-text dataset [62] at the hundred-million level, and the LAION-2B image-text dataset [61] at the billion level. For the 3D domain, we utilize the Kinetics-400 [36], a large-scale dataset commonly used for action recognition, comprising approximately 260K video clips. On these large-scale datasets, we employ popular pre-trained models such as Vision Transformers (ViT) [70], ConvNeXt [50], and Video Swin Transformers (Swin) [49]. Details of these datasets and pre-trained backbones are listed in Table 1.
+
+Table 1: Details of the pre-training datasets and the corresponding backbones used.  
+
+<table><tr><td>Dataset</td><td>Scale</td><td>Type</td><td>Backbone</td></tr><tr><td>ImageNet-21K [58]</td><td>14 Million</td><td>Images</td><td>ViT-B/16, ViT-L/16, ConvNeXt-B</td></tr><tr><td>LAION-400M [62]</td><td>400 Million</td><td>Image-Text Pairs</td><td>ViT-B/16, ConvNeXt-B</td></tr><tr><td>LAION-2B [61]</td><td>2000 Million</td><td>Image-Text Pairs</td><td>ViT-B/16, ConvNeXt-B</td></tr><tr><td>Kinetics-400 [8]</td><td>0.26 Million</td><td>Video Clips</td><td>Swin-B, Swin-L</td></tr></table>
+
+Downstream Tasks. We experiment on 10 benchmarks across a broad spectrum of domains, including generic (Tiny-ImageNet [38]), multimedia (UCF101 [66] and HDMB51 [37]), industrial (InsPLAD-fault [78]), biological&medical (BreakHis [6] and HCRF [68]), agricultural (Apple Foliar Disease [72]), environmental&geographical (WHU-RS19 [81]), materials science (KTH-TIPS-2b [51]), OOD (PACS [40]), and 3D analysis (above video) to demonstrate the versatility and effectiveness of our CLAdapter. Full benchmark list (Table 5) and dataset descriptions with processing methods, are provided in Appendix A.1.
+
+Evaluation Metrics. Following the previous works [78], we report the ROC on the InsPLAD-fault dataset. In alignment with established precedents [11], we utilize the precision, recall, accuracy, and F1 score as metrics on BreakHis and HCRF datasets. For all other datasets, we assess model efficacy using Top-1 accuracy. Implementation Details are included in Appendix A.2.
+
+![](images/d8064fc17833b2922f1a74460298d7cd29b660ff5a2b402102f4c7fccd445244.jpg)  
+Figure 3: Performance comparison of CLAdapter against SOTA methods across various application domain datasets. The bar graph illustrates the average scores achieved by the CLAdapter and SOTA on each dataset, with the red fold line indicating the percentage improvement offered by the CLAdapter.
+
+# 4.2 Main Results
+
+Overall Intuitive Performance Comparison. Figure 3 provides an intuitive comparison between our method and prior SOTA approaches, while additional intuitive comparisons are presented in Appendix B.1 Table 6. It demonstrates that our method consistently surpasses existing SOTA methods, highlighting its strong a in data-limited scientific domains and robustness against challenges such as limited data availability, domain distribution shifts, visual semantic variations, and fine-grained feature distinctions. More detailed quantitative comparisons are provided in the following sections.
+
+Comparison with SOTA approaches in diverse scientific domains. As listed in Table 2, CLAdapter achieves state-of-the-art results in all benchmarks. Notably, on the industrial defect detection dataset InsPLAD-fault and the 3D multimedia action recognition UCF101 dataset, CLAdapter, fine-tuning
+
+Table 2: Comparison of CLAdapter with SOTA methods for diverse scientific downstream tasks. Best performances are highlighted in bold, while the second-best are underlined.  
+
+<table><tr><td>Method</td><td>Acc</td></tr><tr><td>MoCo v2 [13]</td><td>96.04</td></tr><tr><td>MaskCOV [69]</td><td>95.82</td></tr><tr><td>SPARE [87]</td><td>96.70</td></tr><tr><td>ViT [70]</td><td>96.48</td></tr><tr><td>DeiT [75]</td><td>96.26</td></tr><tr><td>TransFG [88]</td><td>97.14</td></tr><tr><td>Hybrid ViT [70]</td><td>96.48</td></tr><tr><td>Swin [48]</td><td>98.08</td></tr><tr><td>CLE-ViT [86]</td><td>97.58</td></tr><tr><td>CLAdapterConvNeXt-B</td><td>98.36</td></tr><tr><td>CLAdapterViT-B</td><td>98.36</td></tr></table>
+
+(a) Agricultural:FoliarDisease [72]  
+(b) Geography: WHU-RS19 [81]  
+
+<table><tr><td>Method</td><td>Acc</td></tr><tr><td>DCA-Fusion [9]</td><td>93.56</td></tr><tr><td>GM [89]</td><td>88.16</td></tr><tr><td>GLM16 [89]</td><td>92.99</td></tr><tr><td>RSFJR [18]</td><td>97.48</td></tr><tr><td>MS2AP [7]</td><td>98.88</td></tr><tr><td>ViT [17]</td><td>96.42</td></tr><tr><td>Swin [48]</td><td>97.12</td></tr><tr><td>EMTCAL* [70]</td><td>97.60</td></tr><tr><td>SF-MSFormer [84]</td><td>97.80</td></tr><tr><td>CLAdapterConvNeXt-B</td><td>99.80</td></tr><tr><td>CLAdapterViT-B</td><td>99.20</td></tr></table>
+
+(c) Materials: KTH-TIPS2-B [51]  
+
+<table><tr><td>Method</td><td>Acc</td></tr><tr><td>CDL [80]</td><td>76.30</td></tr><tr><td>Timofte [73]</td><td>66.30</td></tr><tr><td>DMD+IFV [52]</td><td>76.20</td></tr><tr><td>FV-VGGVD [14]</td><td>88.20</td></tr><tr><td>LETRIST [65]</td><td>65.30</td></tr><tr><td>CATex [21]</td><td>66.70</td></tr><tr><td>RAMBP [2]</td><td>68.90</td></tr><tr><td>TEX-Nets-LF [3]</td><td>78.00</td></tr><tr><td>BMCAnet [45]</td><td>79.18</td></tr><tr><td>CLAdapterConvNeXt-B</td><td>92.47</td></tr><tr><td>CLAdapterViT-B</td><td>91.26</td></tr></table>
+
+(d) Biomedicine: BreakHis [6]  
+
+<table><tr><td>Method</td><td>Pre</td><td>Rec</td><td>Acc</td><td>F1</td></tr><tr><td>ViT [17]</td><td>80.02</td><td>80.73</td><td>84.89</td><td>80.37</td></tr><tr><td>BotNet [67]</td><td>79.20</td><td>80.72</td><td>85.32</td><td>79.50</td></tr><tr><td>GasHis-Transformer [11]</td><td>83.92</td><td>83.16</td><td>88.10</td><td>83.48</td></tr><tr><td>LW-GasHis-Transformer [11]</td><td>84.54</td><td>82.99</td><td>87.93</td><td>83.69</td></tr><tr><td>CLAdapterConvNeXt-B</td><td>92.58</td><td>90.75</td><td>93.53</td><td>91.66</td></tr><tr><td>CLAdapterViT-B</td><td>95.01</td><td>92.45</td><td>95.32</td><td>93.71</td></tr></table>
+
+(e) Biomedicine: HCRF [68]  
+
+<table><tr><td>Method</td><td>Pre</td><td>Rec</td><td>Acc</td><td>F1</td></tr><tr><td>TransMed [15]</td><td>94.34</td><td>97.06</td><td>95.58</td><td>95.58</td></tr><tr><td>HCRF-AM [43]</td><td>92.90</td><td>91.94</td><td>94.24</td><td>92.06</td></tr><tr><td>GasHis-Transformer [11]</td><td>98.55</td><td>97.38</td><td>97.97</td><td>97.97</td></tr><tr><td>LW-GasHis-Transformer [11]</td><td>95.99</td><td>96.90</td><td>96.43</td><td>96.43</td></tr><tr><td>CLAdapterConvNeXt-B</td><td>98.61</td><td>98.57</td><td>98.57</td><td>98.59</td></tr><tr><td>CLAdapterViT-B</td><td>95.01</td><td>95.00</td><td>95.00</td><td>95.00</td></tr></table>
+
+(f) Industrial: InsPLAD-fault [78]  
+
+<table><tr><td>Method</td><td>Glass Ins.</td><td>Light. RS.</td><td>Upper Sha.</td><td>Vari Grip</td><td>Yoke Sus.</td><td>Avg ROC</td></tr><tr><td>DifferNet [59]</td><td>82.81</td><td>99.08</td><td>92.42</td><td>91.20</td><td>96.77</td><td>92.46</td></tr><tr><td>AttentDifferNet [63]</td><td>86.57</td><td>99.62</td><td>94.62</td><td>93.52</td><td>97.38</td><td>94.34</td></tr><tr><td>FastFlow [1]</td><td>70.16</td><td>82.02</td><td>77.43</td><td>65.54</td><td>71.48</td><td>73.33</td></tr><tr><td>RD++ [63]</td><td>86.21</td><td>97.54</td><td>83.67</td><td>93.85</td><td>92.46</td><td>90.75</td></tr><tr><td>CS-Flow [55]</td><td>85.73</td><td>96.60</td><td>88.40</td><td>91.53</td><td>90.70</td><td>90.59</td></tr><tr><td>CFLOW-AD [22]</td><td>82.22</td><td>95.52</td><td>86.60</td><td>90.37</td><td>83.87</td><td>87.72</td></tr><tr><td>PatchCore [33]</td><td>78.44</td><td>85.11</td><td>81.02</td><td>91.92</td><td>58.06</td><td>78.91</td></tr><tr><td>CLAdapterConvNeXt-B</td><td>96.43</td><td>99.94</td><td>98.63</td><td>96.43</td><td>100.00</td><td>98.29</td></tr><tr><td>CLAdapterVit-B</td><td>94.64</td><td>99.87</td><td>98.44</td><td>96.07</td><td>100.00</td><td>97.80</td></tr></table>
+
+(g) 3D Multimedia: Video Recognition  
+
+<table><tr><td>Method</td><td>UCF101 [66]</td><td>HMDB51 [37]</td></tr><tr><td>MemDPC [23]</td><td>86.10</td><td>54.50</td></tr><tr><td>CoCLR [24]</td><td>87.90</td><td>54.60</td></tr><tr><td>RSPNet [12]</td><td>93.70</td><td>64.70</td></tr><tr><td>VideoMoCo [53]</td><td>78.70</td><td>49.20</td></tr><tr><td>Vi²CLR [16]</td><td>89.10</td><td>55.70</td></tr><tr><td>CVRL [54]</td><td>94.40</td><td>70.60</td></tr><tr><td>CORPf [29]</td><td>93.50</td><td>68.00</td></tr><tr><td>ρBYOLρ = 4 [20]</td><td>94.20</td><td>72.10</td></tr><tr><td>VideoMAE [74]</td><td>96.10</td><td>73.30</td></tr><tr><td>CLAdapterSwin-B</td><td>97.60</td><td>75.80</td></tr></table>
+
+uses only the first stage of SFT, achieving an average AUROC of  $98.29\%$  and an accuracy of  $97.60\%$ , respectively. This demonstrates the efficiency of CLAdapter in feature transformation and model fine-tuning across real-world scenarios. Moreover,  $\mathrm{CLAdapter}_{\mathrm{ConvNeXt-B}}$  and  $\mathrm{CLAdapter}_{\mathrm{ViT-B}}$  surpass the best-performing methods on the biomedical BreakHis dataset by  $7.97\%$  and  $10.02\%$  in F1 score, respectively, highlighting significant impact of CLAdapter on cross-domain medical with limited data. In summary, these results demonstrate the effectiveness of CLAdapter in leveraging knowledge from large-scale datasets to adapt and excel in various downstream tasks, pushing the boundaries of computer vision applications across different industry and science domains.
+
+Table 3: Results on Tiny-ImageNet and PACS classic visual datasets. Architecture variants: ViT-L for Tiny-ImageNet, ViT-B for PACS. Best results are in bold, while the second-best are underlined.  
+
+<table><tr><td colspan="2">Tiny-ImageNet [38]</td><td colspan="2">PACS [40] (FT)</td><td colspan="2">PACS [40] (PEFT)</td></tr><tr><td>Method</td><td>Acc</td><td>Method</td><td>Acc</td><td>Method</td><td>Acc</td></tr><tr><td>CaiT-S/36 [76]</td><td>86.74</td><td>Linear</td><td>71.88</td><td>VPT-Adapter[34]</td><td>76.76</td></tr><tr><td>DeiT-B/16-D [75]</td><td>87.29</td><td>Full</td><td>87.79</td><td>LoRA[28]</td><td>88.53</td></tr><tr><td>Swin-L/4 [48]</td><td>91.35</td><td>SFT</td><td>88.91</td><td>DoRA [46]</td><td>88.43</td></tr><tr><td>ViT-L [31]</td><td>86.43</td><td>L2-SP [26]</td><td>87.74</td><td>MoRA [35]</td><td>89.09</td></tr><tr><td>CLAdapterViT-L</td><td>94.21</td><td>CLAdapter</td><td>91.41</td><td>CLAdapter</td><td>91.41</td></tr></table>
+
+Comparison with Vision Models and Fine-Tuning Methods on ID and OOD Benchmarks. As listed in Table 3, we evaluate CLAdapter on both general in-distribution (ID) and out-of-distribution (OOD) benchmarks to assess its effectiveness in adapting pre-trained models under data-limited condi
+
+tions. On the Tiny-ImageNet dataset, which serves as a general classification benchmark, CLAdapter substantially improves ViT-L's baseline performance from  $86.43\%$  to  $94.21\%$ , outperforming stronger architectures such as Swin-L by a margin of  $2.86\%$ . For cross-domain generalization, we conduct evaluations on the OOD benchmark PACS. CLAdapter achieves  $91.41\%$  accuracy, outperforming full fine-tuning (by  $3.62\%$ ) and L2-SP (by  $3.67\%$ ). When compared with parameter-efficient fine-tuning (PEFT) methods, CLAdapter surpasses VPT by a large margin of  $14.65\%$ , and maintains at least a  $2.32\%$  lead over recent approaches such as LoRA, DoRA, and MoRA. The results highlighting the robustness and efficiency of CLAdapter for both ID and OOD scenarios.
+
+# 4.3 Ablation Study
+
+Discussion on Cluster Center Numbers. The number of cluster centers  $K$  in Equation (1) within the CLAdatper is an adjustable hyperparameter. An excessive number of cluster centers might cause the model to overfit, particularly in scenarios where downstream tasks offer limited numbers and diversity of samples. On the other hand, too few cluster centers may fail to transfer all data information, leading to underfitting. To explore an optimal value for this parameter, we conduct experiments on the BreakHis dataset using the ViT-B model pre-trained on LAION-2B. The results in Table 4 indicate that setting  $K$  to 20 yields the most improvements for downstream tasks, with an optimal F1 score of  $93.71\%$  and an accuracy of  $95.32\%$ . Therefore, we recommend 20 as the default setting for the hyperparameter  $K$ .
+
+Table 4: Comparison of fine-tuning results on the BreakHis dataset under different cluster center numbers  $K$  . The best results are in bold.  
+
+<table><tr><td rowspan="2">Scores</td><td colspan="9">Cluster Center Number (K)</td></tr><tr><td>5</td><td>10</td><td>15</td><td>20</td><td>25</td><td>30</td><td>100</td><td>200</td><td>300</td></tr><tr><td>Acc</td><td>92.81</td><td>92.81</td><td>92.45</td><td>95.32</td><td>94.24</td><td>93.88</td><td>93.52</td><td>91.73</td><td>92.81</td></tr><tr><td>F1</td><td>90.40</td><td>90.73</td><td>90.41</td><td>93.71</td><td>92.59</td><td>92.14</td><td>91.77</td><td>89.03</td><td>90.42</td></tr></table>
+
+![](images/b050a844ebf209efbb9f25509795d0fff37e8d92a0996a7451ae4801ffe91f8f.jpg)  
+Figure 4: Efficiency comparison of fine-tuning methods. CLAdapter achieves significant performance improvement with fewer training epochs and parameters, indicating both high effectiveness and efficiency. Note that standard augmentations are only applied to the training set to mitigate overfitting but not to the validation set, which results in lower validation loss than training loss.
+
+Analysis of Efficiency. The proposed CLAdapter maintains high parameter efficiency while significantly improving performance. As detailed in Appendix B.2 (Table 7), it introduces only  $7 - 10.4\%$  additional parameters to backbone models (ConvNeXt-B/ViT-B) with minimal computational overhead, yet achieves F1-score gains of up to  $175.59\%$ . Furthermore, CLAdapter also shows advantages in downstream task fine-tuning. As shown in Figure 4, by only fine-tuning  $7\%$  of the model parameters for 40 epochs in the first stage, CLAdapter achieves results comparable to full fine-tuning of
+
+$100\%$  of the parameters for 90 epochs. Although the second stage of SFT requires tuning  $100\%$  of the parameters, saturation is reached in just 30 epochs, with an accuracy improvement of  $12.44\%$ . These experiments demonstrate the effectiveness and efficiency of our CLAdapter in fine-tuning and transferring pre-trained features.
+
+![](images/c8523bf976c576a5adf77601bec8033b5d67e2e62fcc972823506a78f098b64d.jpg)  
+(a) BreakHis dataset: red for ductal, yellow for lobular, green for mucinous, and blue for papillary carcinoma.
+
+![](images/db703ca2e1e52328058b267e9dc5d62735b640e9561fbc24c233eebdbcc88923.jpg)  
+(b) HCRF dataset: red for normal gastric slices and blue for cancerous gastric slices.
+
+![](images/5f35d4f5272767f14080f0854ecccf36ba3b1330988f8816070b40a2b2694df6.jpg)  
+Figure 5: The  $t$ -SNE visualizations demonstrating class separability and compactness. The comparative analysis highlights the enhanced discriminability of features via CLAdapter.  
+Figure 6: The  $t$ -SNE Feature visualization on PACS. The Top1-accuracy are reported additionally. CLAdapter demonstrated satisfactory separability and compactness.
+
+Visual Analysis of Features. To further validate the effectiveness of CLAdapter in feature transfer, Figure 5 visualizes the features using t-distributed stochastic neighbor embedding ( $t$ -SNE) on the BreakHis and HCRF datasets. It compares the features extracted by the Backbone and those refined by CLAdapter. In the BreakHis dataset, post-CLAdapter application, the classes exhibit more distinct clustering. The ductal, lobular, mucinous, and papillary carcinomas are more separable and demonstrate increased intra-class compactness, underlining the robustness of CLAdapter in feature representation. Similarly, on the HCRF dataset, CLAdapter maximizes the inter-class distances of samples, effectively distinguishing between normal and cancerous gastric slices. In addition, to assess the quality of features extracted, we visualize the feature distributions for all fine-tuning methods on PACS test set using  $t$ -SNE. The experiments are performed based on the ViT-B model pre-trained on ImageNet-22K. As shown in Figure 6, CLAdapter significantly improves the separability of representation vectors of different classes, exhibiting superior representational capacity.
+
+# 5 Conclusion & Limitation
+
+This work introduces the Cluster Attention Adapter (CLAdapter), a novel method designed to bridge the gap between large-scale pre-training on diverse datasets and fine-tuning for data-limited downstream tasks, particularly in diverse scientific domains. By leveraging attention mechanisms and clustering techniques, CLAdapter refines and adapts pre-trained models to enhance their performance significantly on a wide array of downstream tasks, showcasing superior adaptability and effectiveness. In addition, benefiting from our unified interface design, CLAdapter effortlessly merges with mainstream models. Moreover, an SFT strategy is presented to collaborate with CLAdapter to enhance the fine-tuning performance further. Through rigorous testing across ten diverse datasets, encompassing generic, multimedia, biological, medical, industrial, agricultural, environmental, geographical, materials science, OOD, and 3D analysis domains, CLAdapter all achieves a new state-of-the-art performance, highlighting its effectiveness in addressing the unique challenges of data scarcity and domain shift in scientific applications. Limitations: Currently, CLAdapter has not been specifically designed or validated for detection or segmentation. We leave these extensions for future work.
+
+# References
+
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+
+# A Experiment Configuration
+
+# A.1 Dataset Descriptions and Processing Methods
+
+We experiment on 10 benchmarks across a broad spectrum of domains, including generic, multimedia, industrial, biological, medical, agricultural, environmental, geographical, materials science, OOD, and 3D analysis. This demonstrates the versatility and effectiveness of our CLAdapter. The datasets for each domain, class counts, and sample sizes are detailed in Table 5.
+
+Table 5: Statistics of datasets used for evaluating downstream task performance.  
+
+<table><tr><td>Dataset</td><td>Domains</td><td>Class</td><td>Train</td><td>Val</td><td>Test</td></tr><tr><td>Tiny-ImageNet [38]</td><td>General</td><td>200</td><td>100000</td><td>10000</td><td>10000</td></tr><tr><td>PACS [40]</td><td>General OOD</td><td>4 × 7</td><td>1588</td><td>6355</td><td>2048</td></tr><tr><td>BreakHis [6]</td><td>Biomedicine</td><td>4</td><td>834</td><td>278</td><td>278</td></tr><tr><td>HCRF [68]</td><td>Biomedicine</td><td>2</td><td>70</td><td>70</td><td>140</td></tr><tr><td>Apple Foliar Disease [72]</td><td>Agricultural</td><td>4</td><td>1366</td><td>-</td><td>455</td></tr><tr><td>WHU-RS19 [81]</td><td>Environmental Geography</td><td>19</td><td>402</td><td>100</td><td>503</td></tr><tr><td>KTH-TIPS-2b [51]</td><td>Materials Science</td><td>11</td><td>3564</td><td>-</td><td>1188</td></tr><tr><td>InsPLAD-fault [78]</td><td>Industrial</td><td>5</td><td>5108</td><td>-</td><td>6417</td></tr><tr><td>UCF101 [66]</td><td>3D Multimedia</td><td>101</td><td>9537</td><td>-</td><td>3783</td></tr><tr><td>HMDB51 [37]</td><td>3D Multimedia</td><td>51</td><td>3570</td><td>-</td><td>1530</td></tr></table>
+
+Tiny-ImageNet. Tiny-ImageNet [39] is a simplified version of the larger ImageNet dataset. It comprises 200 classes, each with 500 training images, 50 validation images, and 50 test images, resulting in a total of 100,000 images. Tiny-ImageNet serves as a benchmark for evaluating algorithms on a wide array of generic visual recognition tasks, testing both the depth and breadth of models' understanding of visual concepts. In the experiment, the data division adheres to official standards.
+
+PACS. The PACS dataset [40] stands as a critical benchmark for assessing domain generalization capabilities in general computer vision. It contains images from four distinct domains: Photo, Art Painting, Cartoon, and Sketch, addressing a broad spectrum of visual styles and compositions. With seven common object classes across these domains, the dataset poses a significant challenge in learning domain-invariant features. It is particularly used for evaluating models on their ability to generalize from seen to unseen domains, making it an essential tool for research in domain adaptation and generalization. The Art Painting domain of the PACS dataset is exclusively utilized as the test set to assess cross-domain performance, while the remaining data are divided into training and validation sets in a 5-fold manner, with a ratio of 1:4.
+
+Apple Foliar Disease. The Apple Foliar Disease dataset [71] is a specialized resource aimed at advancing the field of agricultural and plant disease recognition. It consists of high-quality images that capture various foliar diseases affecting apple leaves, including but not limited to apple scab, cedar apple rust, and powdery mildew, as well as images of healthy leaves for comparison. Leveraging such datasets not only validates our CLAdapter's cross-domain effectiveness in agriculture but also aids researchers and agronomists in enhancing precision agriculture, enabling timely and effective disease management to improve crop health and yield. The data split method follows the previous work as training and validation sets with a 3:1 ratio.
+
+WHU-RS19. The WHU-RS19 dataset [82] is a high-resolution remote sensing dataset, primarily used for the evaluation of land cover and land use classification algorithms in the field of geographical and environmental analysis. Originating from the Wuhan University Remote Sensing Group, this dataset encompasses a diverse collection of 19 classes representing various natural and man-made features, including but not limited to agricultural lands, forests, water bodies, residential areas, and industrial sites. The images in WHU-RS19 are collected from different satellite and aerial sensors, challenging and enhancing classification models' robustness in geographical and environmental fields. For data partitioning, our experiments are consistent with previous studies [84].
+
+KTH-TIPS2-B. The KTH-TIPS2-B dataset [51] is an extension of the KTH-TIPS dataset, both of which are designed for the task of texture classification and material recognition in the field of computer vision, particularly focusing on the challenges associated with variations in scale, pose, and illumination. This dataset is curated by the KTH Royal Institute of Technology in Sweden. KTH-TIPS2-B consists of images representing a set of 11 material categories, such as cotton, wool,
+
+and aluminum, among others. Each material category includes images captured under different conditions and from multiple angles, providing comprehensive data for evaluating the performance of texture analysis algorithms. The data split method remains consistent with previous work [45].
+
+BreakHis. The BreakHis dataset [6] comprises 7,909 breast cancer images across four magnification levels, divided into eight sub-classes. Originating from 82 anonymous patients in Brazil, BreakHis is a key dataset in digital breast histopathology research. Malignant tumor images at a  $200\times$  magnification, including ductal carcinoma (DC), lobular carcinoma (LC), mucinous carcinoma (MC), and papillary carcinoma (PC), are used for classification. The dataset is split into training, validation, and testing sets in a 3:1:1 ratio, which is the same as the previous study [11].
+
+HCRF. The HCRF dataset [68], well-known in gastric histopathology, consists of 560 cancerous and 140 normal images. Following the previous works [11], the HCRF dataset is divided into training, validation, and test sets as a 1:1:2 random stratified ratio. Available on Mendeley Data, it serves as an important resource for evaluating model performance in the field of computer vision and biomedicine.
+
+InsPLAD-fault. The InsPLAD dataset [78], pivotal for advancing power line asset inspection, includes the InsPLAD-fault subset, a specialized collection designed for anomaly detection tasks in power line components. This subset harnesses real-world images captured by unmanned aerial vehicles (UAVs) of operational power line transmission towers, offering a unique challenge in the realm of industrial defect detection. It encapsulates five distinct categories of power line objects, facilitating deep learning models in effectively identifying and classifying anomalies. In the experiment, the data split method remains consistent with previous work [64].
+
+UCF101. The UCF101 [66] is a widely recognized dataset in the field of action recognition, developed by the University of Central Florida. It's one of the most popular benchmarks for evaluating the performance of video-based action recognition algorithms. The dataset features 101 action categories, encompassing a broad range of activities such as sports, playing musical instruments, and human-object interactions. Each category in the UCF101 dataset consists of multiple video clips, amounting to over 13,000 clips and totalling more than 27 hours of video data. The videos are collected from YouTube and represent diverse actors, backgrounds, and lighting conditions. This diversity poses a challenge for action recognition systems, requiring the models to generalize across different environments and subject appearances. The data division method follows the official release.
+
+HMDB51. The HMDB51 [37] is a comprehensive video dataset aimed at the task of human action recognition. Compiled by researchers from Brown University, it consists of 51 action categories, each containing at least 101 video clips, resulting in a total of over 6,800 clips. These actions span a wide array of human activities, including facial actions, general body movements, and interactions with objects. Due to the limited size of its dataset and the lack of diversity among samples, the HMDB51 dataset poses more challenges than UCF101. Additionally, the data splitting method is consistent with the official.
+
+# A.2 Implementation Details
+
+We meticulously design our experimental settings to ensure comparability and reproducibility.
+
+For experiments on the Tiny-ImageNet, follow the protocols established in [32], ensuring consistency with prior benchmarks.
+
+For 3D video analysis on UCF101 and HMDB51 datasets, our experimental configurations align with the previous methods [41], which facilitates direct comparison with existing SOTA approaches.
+
+Regarding the remaining datasets, encompassing cross-domain and various real-world domain applications, we establish uniform implementation details to underscore the adaptability and convenience of CLAdapter. Specifically, we adopt an input resolution of  $224 \times 224$  pixels across all experiments. The learning rate is initialized at  $1e - 4$ , and models are trained for up to 100 epochs with a batch size of 16. We employ the AdamW optimizer, configuring it with momentum  $\beta_{1} = 0.9$  and a weight decay of  $1e - 3$ , to adapt to the unique challenges presented by these varied datasets. The determination of cluster centers  $K$  for CLAdapter is fixed at 20, balancing granularity and computational efficiency. Our experimental setup is powered by four Nvidia GeForce RTX 3090 GPUs, boasting 24 GB of memory, under the Ubuntu 20.04 environment. Python 3.8.3 is chosen as the programming language, with the PyTorch 1.13.1 framework being utilized for model development.
+
+# B Additional Experimental Analyses
+
+# B.1 Intuitive Performance Comparison of Our Method with SOTA and Baseline Methods
+
+In Table 2 of the main manuscript, we present a comprehensive comparison of various methods across diverse scientific domains for downstream tasks (subtables a-g). Furthermore, Table 3 highlights the significant performance gains achieved by CLAdapter on both the in-distribution (ID) Tiny-ImageNet and the out-of-distribution (OOD) PACS benchmarks.
+
+To provide a more intuitive understanding of the advantages of our method, we visualize the performance comparison, as shown in Figure 3. It can be seen that CLAdapter consistently outperforms state-of-the-art methods across different datasets. In addition, we listed all benchmark baseline results in Table 6. By using the best fine-tuning strategy and keeping the same backbone as ours for comprehensive evaluation, our method still maintains substantial improvements over both baselines and prior SOTA methods.
+
+Table 6: Per-Domain Improvement over Baseline (ViT) and SOTA using CLAdapter.  
+
+<table><tr><td>Improve%↑</td><td>ID</td><td>OOD</td><td>Agricultural</td><td>Geography</td><td>Materials</td><td>Biomedicine</td><td>Industrial</td><td>3D Multimedia</td></tr><tr><td>Ours vs Baseline</td><td>7.8</td><td>3.6</td><td>2.0</td><td>2.8</td><td>2.5</td><td>12.2/13.6</td><td>4.6</td><td>2.5/4.1</td></tr><tr><td>Ours vs SOTA</td><td>2.9</td><td>3.6</td><td>0.3</td><td>0.9</td><td>4.3</td><td>10.0/0.6</td><td>4.0</td><td>1.5/2.5</td></tr></table>
+
+# B.2 Analysis of Efficiency
+
+As a universal adapter, our method exhibits relative efficiency under popular pre-trained models. Efficiency analysis results are listed in Table 7. For ConvNeXt-B and ViT-B models, CLAdapter adds only  $10.4\%$  and  $7\%$  more parameters, respectively, while slightly increasing computational complexity (Flops) by 0.44G for ConvNeXt-B and 1G for ViT-B. Despite this minimal increase in size and computation, CLAdapter achieves remarkable F1 score improvements of up to  $175.59\%$  for ConvNeXt-B and  $51.46\%$  for ViT-B.
+
+Table 7: Comparison of Method efficiency.  
+
+<table><tr><td>Method</td><td>Params(M)</td><td>Flops(G)</td><td>F1</td><td>Train</td></tr><tr><td>ConvNeXt-B</td><td>88.85</td><td>15.42</td><td>33.26</td><td>100%</td></tr><tr><td>CLAdapterConvNeXt-B-SFT-1</td><td>99.11</td><td>15.86</td><td>80.89</td><td>10.4%</td></tr><tr><td>CLAdapterConvNeXt-B-SFT-2</td><td>99.11</td><td>15.86</td><td>91.66</td><td>100%</td></tr><tr><td>ViT-B</td><td>85.77</td><td>16.86</td><td>61.87</td><td>100%</td></tr><tr><td>CLAdapterViT-B-SFT-1</td><td>92.22</td><td>17.86</td><td>84.34</td><td>7.0%</td></tr><tr><td>CLAdapterViT-B-SFT-2</td><td>92.22</td><td>17.86</td><td>93.71</td><td>100%</td></tr></table>
+
+# B.3 Comparison of Using Different Scales of Pre-training Data
+
+To delve into how the scale of large visual datasets influences cross-domain fine-tuning, we perform comparative experiments on the PACS and BreakHis datasets using ConvNeXt-B and ViT-B models pre-trained on three large-scale datasets, i.e., ImageNet-21K, LAION-400M, and LAION-2B.
+
+Table 8: Results of using different pre-training resources on the PACS dataset.  
+
+<table><tr><td>Method</td><td>ImageNet-21K</td><td>LAION-2B</td></tr><tr><td>Linear</td><td>71.88</td><td>95.61</td></tr><tr><td>Full</td><td>87.79</td><td>47.17</td></tr><tr><td>L2-SP</td><td>87.74</td><td>45.56</td></tr><tr><td>VPT</td><td>76.76</td><td>97.46</td></tr><tr><td>CLAdapter</td><td>91.41</td><td>97.62</td></tr></table>
+
+Table 8 lists the fine-tuning results on the PACS dataset based on ViT-B. It is observed that our CLAdapter achieves an accuracy of  $97.62\%$  under the LAION-2B pre-training dataset, which is enriched with more diverse knowledge, marking a  $6.21\%$  improvement over the smaller-scale ImageNet-21K dataset. Notably, both Full fine-tuning and L2-SP exhibit poor performance on LAION-2B, likely due to pattern collapse encountered during the transfer of massive pre-trained features, a problem that VPT and our CLAdapter circumvent through additional parameter transformation. Moreover, our CLAdapter's accuracy under ImageNet-21K pre-training surpasses that of VPT by  $14.65\%$ , indicating that CLAdapter is also capable of extracting and transforming a sufficient amount of downstream-relevant information from relatively smaller pre-training datasets.
+
+Table 9: Results of using different pre-training resources on the BreakHis dataset.  
+
+<table><tr><td>Method</td><td>ImageNet-21K</td><td>LAION-400M</td><td>LAION-2B</td></tr><tr><td colspan="4">SFT Stage 1</td></tr><tr><td>CLAdapterConvNeXt-B</td><td>80.89</td><td>71.72</td><td>75.17</td></tr><tr><td>CLAdapterViT-B</td><td>84.34</td><td>80.63</td><td>80.67</td></tr><tr><td colspan="4">SFT Stage 2</td></tr><tr><td>CLAdapterConvNeXt-B</td><td>88.60</td><td>90.55</td><td>91.66</td></tr><tr><td>CLAdapterViT-B</td><td>90.19</td><td>91.35</td><td>93.71</td></tr></table>
+
+The F1 score results on the BreakHis dataset are listed in Table 9. Whether combined with ConvNeXt-B or ViT-B pre-trained models, CLAdapter achieves the best fine-tuning results on larger pre-training datasets. Another notable observation is that when only the first stage of the SFT fine-tuning strategy is employed (i.e., freezing the weights of the pre-trained model), pre-training on the smaller-scale and knowledge-limited ImageNet-21K dataset yields better results. This suggests that freezing the pre-trained model weights limits CLAdapter's capacity for transformation, preventing the thorough transfer and refinement of rich knowledge to the downstream tasks. Nevertheless, even with just the first fine-tuning stage, CLAdapter still surpasses other SOTA methods on the BreakHis dataset (as listed in Table 2d).
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: We propose CLAdapter, a cluster-attention adapter that refines foundation vision model representations across CNN and Transformer architectures with negligible overhead. We validate its superior performance on ten diverse data-limited scientific datasets and provide detailed experimental results.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: The paper explicitly discusses limitations in the final paragraph of the conclusion, noting that CLAdapter has not yet been specifically designed or evaluated for object detection or segmentation tasks. These directions are identified as future work.
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: Yes, we provide a complete derivation for our proposed CLAdapter in Section 3.2.
+
+Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: We provide all experimental settings and codes are provided in the paper and the Appendix to ensure full reproducibility of our main results.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.
+
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
+
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
+
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
+
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
+
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: We use only publicly available datasets and have released our full code on GitHub.
+
+Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
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+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: All important settings can be found in our paper and Appendix.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: Please refer to some ablation experiments, hyperparameter settings, and appendix.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
+
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
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+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We report the training resources we use in the experimental settings.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: Our research conforms with the NeurIPS Code of Ethics in every respect.
+
+Guidelines:
+
+The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: We discuss the broader impacts in the conclusion. The research is inherently scientific, and we anticipate no adverse societal impact stemming from its findings.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.
+
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: Our method doesn't have high risk for misuse.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: All the creators or original owners of assets are properly credited.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URI.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
+
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [Yes]
+
+Justification: This paper introduces new code for CLAdapter and the documentation can be found in the anonymized URL: https://anonymous.4open.science/r/CLAdapter-NIPS2025.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing or research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
+
+We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: Our CLAdapter does not involve LLMs as any important, original, or non-standard components.
+
+Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
\ No newline at end of file
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+# Unleashing Hour-Scale Video Training for Long Video-Language Understanding
+
+Jingyang Lin $^{1,2*}$  Jialian Wu $^{1}$  Ximeng Sun $^{1}$  Ze Wang $^{1}$  Jiang Liu $^{1}$  Yusheng Su $^{1}$  Xiaodong Yu $^{1}$  Hao Chen $^{1}$  Jiebo Luo $^{2}$  Zicheng Liu $^{1}$  Emad Barsoum $^{1}$  1 AMD 2 University of Rochester https://videomarathon.github.io/
+
+# Abstract
+
+Recent long-form video-language understanding benchmarks have driven progress in video large multimodal models (Video-LMMs). However, the scarcity of well-annotated long videos has left the training of hour-long Video-LMMs underexplored. To close this gap, we present VideoMarathon, a large-scale hour-long video instruction-following dataset. This dataset includes around 9,700 hours of long videos sourced from diverse domains, ranging from 3 to 60 minutes per video. Specifically, it contains 3.3M high-quality QA pairs, spanning six fundamental topics: temporality, spatiality, object, action, scene, and event. Compared to existing video instruction datasets, VideoMarathon significantly extends training video durations up to 1 hour, and supports 22 diverse tasks requiring both short- and long-term video comprehension. Building on VideoMarathon, we propose Hour-LLaVA, a powerful and efficient Video-LMM for hour-scale video-language modeling. It enables hour-long video training and inference at 1-FPS sampling by leveraging a memory augmentation module, which adaptively integrates question-relevant and spatiotemporally informative semantics from the cached full video context. In our experiments, Hour-LLaVA achieves the best performance on multiple representative long video-language benchmarks, demonstrating the high quality of the VideoMarathon dataset and the superiority of the Hour-LLaVA model.
+
+# 1 Introduction
+
+Leveraging strong foundation models [44, 67] and high-quality video instruction-following data [7, 26, 70], recent Video-LMMs [7, 28, 51, 70] have achieved promising performance on basic video-language tasks, such as video question-answering (QA) [60, 61, 66] and video summarization [20, 22, 31, 39, 48]. As attention shifts toward longer video modeling, these models face new challenges in capturing long-term dependencies. To measure progress in this direction, several benchmarks have been introduced for hour-scale video understanding, where the goal is to assess model performance on tasks involving long-form video content over one hour [5, 14, 55, 58]. Early attempts on long-form video-language modeling [45, 46, 50, 68, 74] have shown promising results on these benchmarks.
+
+However, a significant gap remains between the length of videos used during training and those encountered at inference time. While testing videos often exceed one hour [5, 14, 55], most existing training datasets [7, 28, 70] consist of videos shorter than ten minutes, as summarized in Table 1. This mismatch limits the models' ability to explicitly learn long-term dependencies, highlighting the necessity for long-form video instruction-following datasets to bridge this gap.
+
+To this end, we introduce VideoMarathon, a large-scale video instruction-following dataset specifically designed for long-form video-language modeling. VideoMarathon contains around 9,700 hours of
+
+Table 1: Comparison between VideoMarathon and other existing video instruction-following datasets. OE and MC denote open-ended and multiple-choice, respectively.  
+
+<table><tr><td>Dataset</td><td>Captioner</td><td>Summarizer</td><td>Total Video Time</td><td>Average Video Length</td><td>Duration Range</td><td>#OE QA</td><td>#MC QA</td></tr><tr><td>LLaVA-Hound [69]</td><td>GPT-4V</td><td>GPT-4</td><td>3K hrs</td><td>0.2 mins</td><td>0-8 mins</td><td>900K</td><td>0</td></tr><tr><td>ShareGPT4Video [7]</td><td>GPT-4V</td><td>GPT-4</td><td>0.2K hrs</td><td>0.3 mins</td><td>&lt; 2 mins</td><td>0</td><td>0</td></tr><tr><td>LLaVA-Video-178K [70]</td><td>GPT-4o</td><td>GPT-4o</td><td>2K hrs</td><td>0.6 mins</td><td>&lt; 3mins</td><td>960K</td><td>196K</td></tr><tr><td>VideoMarathon (ours)</td><td>Qwen2VL-7B</td><td>DeepSeek-V3</td><td>9.7K hrs</td><td>20.9 mins</td><td>3-60 mins</td><td>1.73M</td><td>1.57M</td></tr></table>
+
+long videos (3-60 min per video) sourced from diverse domains, such as activities [4], egocentric [17], movie [47], cooking [72], and open-world [8] scenarios. Inspired by the recent advances [5, 14, 28, 43], we carefully establish a comprehensive task taxonomy covering six essential topics-temporality, spatiality, object, action, scene, and event-across 22 tasks requiring both short- and long-term comprehension. To generate high-quality video instruction-following QA samples, we develop a hierarchical video captioning pipeline: clip-level descriptions are first produced using Qwen2VL7B [54] across the six topics, then aggregated into the event- and global-level summaries via DeepSeek-V3 [33]. Based on these hierarchical video captions, we synthesize 3.3M high-quality QA pairs by DeepSeek-V3, guided by task-specific prompts incorporating explicit task definitions and exemplars from prior benchmarks [5, 14, 28, 43]. The resulting QA samples cover both open-ended (OE) and multiple-choice (MC) formats. As shown in Table 1, VideoMarathon substantially surpasses existing video instruction-following datasets in average video length, duration range, and scale of QA pairs.
+
+Due to the limited availability of long video data, most existing Video-LMMs [7, 29, 35, 70] are trained on short videos, typically less than one minute on average (see Table 1). Uniform temporal sampling [27, 70], which selects a fixed number of frames (8-256 per video), works well for short videos but causes substantial information loss when applied to hour-long videos, resulting in severe performance degradation [5, 14, 55]. Moreover, even with large-scale hour-long data available, training existing Video-LMMs on such long videos remains non-trivial, as the quadratic complexity of attention mechanisms [53] causes token budgets to explode. To this end, we propose Hour-LLaVA, a Video-LMM specifically designed for efficient long-form video-language modeling and capable of directly ingesting hour-long and densely sampled videos. In particular, Hour-LLaVA stores the full video context sampled at 1 FPS in a memory repository. To accommodate GPU memory constraints, only a small subset (approximately  $6\%$ ) of the full video context, referred to as decayed video tokens, is used as input to the language model for efficient training and inference. These decayed tokens are then augmented through a memory augmentation (MemAug) module, which adaptively retrieves and integrates spatiotemporally informative and question-relevant semantics from the cached full context. The MemAug module effectively preserves the contextual fidelity of long videos. In summary, Hour-LLaVA achieves a trade-off between computational efficiency and context fidelity, enabling effective learning from hour-scale video instruction-following data.
+
+Our main contributions are three-fold: (1) we introduce VideoMarathon, the first large-scale hour-long video instruction-following dataset that contains videos ranging from 3 minutes to 1 hour with a total duration of around 9,700 hours and 3.3M diverse QA instructions across 22 tasks; (2) We propose Hour-LLaVA, a Video-LMM capable of processing hour-long videos at 1 FPS in both training and inference, supported by the proposed MemAug mechanism; (3) Training Hour-LLaVA on VideoMarathon, we achieve the best performance among open-source Video-LMMs on four well-known video-language benchmarks, including LVBench [55] (average duration: 4037s), Video-MME [14] (1021s), LongVideoBench [58] (459s), and TempCompass [36] (11s).
+
+# 2 Related Work
+
+Video Instruction-following Data. Human annotation is costly and even limits the scale of video instruction-following datasets. Early attempts [25, 40] address the high cost of human annotation by using ASR-generated subtitles, while later methods [18, 27] employ template-based QA generation on existing captioning datasets. However, these approaches often suffer from noisy, low-quality labels. Recent pipelines [7, 69, 70] leverage powerful LMMs (e.g., GPT-4V [41]/4o [42]) to produce high-quality video captions and diverse QA pairs. Despite these advancements, most training videos remain short, limiting the models' ability to learn long-term dependencies explicitly. To this end, this work introduces a high-quality, synthetic video instruction-following dataset built on long-form videos, providing a foundation for future research on capturing effective patterns in long videos.
+
+![](images/0fb6b45e02e6b733e5d89497478f4a21b4f848ab2af5905c1dd512f78e956326.jpg)
+
+![](images/c72b3d43c8b3b64e734e952f09c34d13537f473ab1f16ef4de3ea345ea9f8bfa.jpg)
+
+![](images/cef401fd19337e7cd6fef9bc87f8854db2171cd94c8a3a39968654f0ce9c779d.jpg)  
+(a) Task Taxonomy of VideoMarathon
+
+![](images/20ec576006674cd52517a73e4920eb3f76033a299670d7900e4121dd683feca5.jpg)
+
+![](images/21127f50d3f774d4f436dde203db19f90845b00ca73f6fa8f2ed90825ac811db.jpg)
+
+![](images/dcb7b04a45143feca98a83af4b6f7456922ce62b98623a5c6f9fc87083f8682c.jpg)
+
+![](images/dc3d5052545f1056c77e7e9cefd6c0bc68bbee4184afbd02ac8f0d137599ccad.jpg)
+
+![](images/14eb5ee98566dcd2e57e274f8b1e0f77371584ac8d2e3030cae7d9647ae99617.jpg)  
+(b) Data Source
+
+![](images/5a2193028e5a102ae744e4eae94cf4b8a62cf7ff32491b643a6794307058419b.jpg)  
+(c) Question Types
+
+![](images/cf1caec5b6285a02cc9a3db48c3b4dadf1acbd95707f9b5cf8f078ef2c21786f.jpg)  
+(d) Video Duration  
+Figure 1: VideoMarathon: A diverse long video instruction-following dataset. (a) The dataset contains 22 diverse tasks, covering both short-form (yellow tag) and long-form (red tag) comprehension. (b) The dataset spans diverse video source domains. (c) The dataset features a wide range of question types for long-form video-language modeling. (d) The dataset consists of long videos ranging from three minutes to one hour. (e) The dataset includes complex video content reflected by the number of events per video.
+
+![](images/92d42d273170cb559d21b6113f2a93b9c138291717374c4f1e49e82cd4b47517.jpg)  
+(e) Event Counting
+
+Long Video-Language Modeling with Language Language Models. Recent advanced VideoLMMs have shown significant promise in long video-language modeling. Two main approaches facilitate long-form video understanding: Compression-based and extension-based methods. Compression-based methods compress input video tokens across temporal, spatial, and question-guided dimensions, allowing models to process fewer but more informative video tokens. Temporal compression techniques include keyframe selection [45, 50] and slow-fast sampling [13]. Spatial compression involves average pooling [70] and token merging [24]. Other methods [46, 57] apply joint temporal-spatial compression. Question-guided methods [12, 45] retain tokens based on question relevance. Following the success of long-context LLMs [15, 32, 34, 49], extension-based methods expand context windows for long video modeling. LongVA [68] processes over 200K visual tokens, and LongVILA [63] scales to 2M using multimodal sequence parallelism. However, existing approaches confront a fundamental dilemma: compression-based methods prioritize computation affordable and training efficiency, but inevitably suffer from information loss caused by aggressive compression; extension-based methods preserve high-fidelity video context by sequence parallel techniques at the cost of significantly increased training time and GPU usage (i.e., linear growth with video length), particularly for hour-scale videos. To resolve this dilemma, we propose the Hour-LLaVA framework, which intends to prioritize computational efficiency while keeping high-fidelity video context.
+
+# 3 VideoMarathon: A Synthetic Long Video Instruction-following Dataset
+
+This section elaborates on details of constructing VideoMarathon, a long video instruction-following dataset with a total duration of around 9,700 hours, consisting of 3.3M QA pairs across 22 tasks.
+
+Task Taxonomy. A comprehensive long video instruction-following dataset requires a well-defined and comprehensive task taxonomy. Inspired by existing video-language benchmarks [5, 14, 28, 43] that cover over 100 tasks, we introduce a comprehensive task taxonomy designed for long video-language understanding. The task taxonomy consists of 22 QA tasks summarized from prior benchmarks and organized into six fundamental topics (see Figure 1a), including temporality, spatiality, object, action, scene, and event. In particular, the temporal topic captures long-term dependencies, while the spatial topic enhances localization. Object and action topics are essential for fundamental video perception, and the scene topic offers contextual grounding and transition cues. Event topic supports long-range understanding of key video content. This task taxonomy encourages Video-LMMs to develop both long-form and short-form video comprehension. Please refer to Figure 11 for detailed descriptions of the above 22 QA tasks in Appendix A.
+
+Video Sources. High-quality and diverse video content is essential for effective video-language modeling. To ensure a sufficient amount of long videos while maintaining quality and diversity, VideoMarathon integrates five representative public video datasets: Panda-70M [8], Ego4D [17], ActivityNet [4], YouCook2 [72], and MovieChat-1K [47]. Figure 1b presents the proportions of these video sources. Panda-70M constitutes the majority of VideoMarathon, covering diverse topics such as daily activities, sports, science, arts, transportation, travel, education, wildlife, entertainment, and industry. Domain-specific datasets further enrich the collection: Ego4D captures real-world activities from a first-person view; ActivityNet covers a broad range of human actions; YouCook2 focuses on cooking scenarios; and MovieChat-1K contributes content from movies and TV series. To enhance temporal complexity within videos, we select videos with at least three distinct events. Figure 1e shows the distribution of event counts per video. In total, VideoMarathon comprises 28K videos across diverse domains, supporting long-form video-language modeling.
+
+Hierarchical Video Captioning. In contrast to short-form video QA tasks that typically rely on concise captions, long-form video QA demands hierarchical contextual understanding, including detailed clip-wise content, event-level structures, and global narratives. To meet this need, we introduce a hierarchical video captioning pipeline for capturing both fine-grained and high-level semantics over extended temporal ranges. The resulting captions serve as essential inputs for generating reliable, diverse, and context-rich QA training samples tailored to long-form video-language understanding.
+
+- Clip-Level Video Captioning. Instead of generating a single brief summary for each video clip [7, 63, 70], VideoMarathon provides detailed descriptions (see Figure 12a) for each video clip from six perspectives: temporality, spatiality, object, action, scene, and overall summary. These chronologically ordered descriptions serve as reliable contexts for the subsequent diverse QA generation. Technically, we leverage Qwen2VL-7B [54], a powerful yet lightweight LMM, as the clip-level captioner, following the prior practice [74].  
+- Event-Level Video Captioning. We first identify event boundaries (i.e., start and end times-tamps) based on the chronologically ordered clip-level captions. Once the event boundaries are determined, all detailed captions within each event are then aggregated into a comprehensive event-level summary. Specifically, we use DeepSeek-V3 [33] to summarize the six-perspective clip-level captions along with adjacent event descriptions into a cohesive event-level summaries. As shown in Figure 12c, this process results in detailed event-level video captions, each associated with its corresponding event boundaries.  
+- Global-Level Video Captioning. This step focuses on generating a natural-toned and global-level description for the entire video. Specifically, we feed the event-level captions along with their boundaries into DeepSeek-V3, obtaining a cohesive and comprehensive video summary. Figure 12d shows that these captions provide global context for the subsequent QA generation.
+
+Appendix A provides the detailed prompts used for the hierarchical video captioning, including Figure 5 for clip-level video captioning, Figure 6 for event splitting, Figure 7 for event-level video captioning, and Figure 8 for global-level video captioning.
+
+Diverse QA Generation for Long Videos. Leveraging the multi-level video captions, we generate high-quality and diverse QA pairs across 22 tasks, spanning six major topics in both open-ended and multiple-choice formats. To achieve this, we design topic-specific prompts that incorporate (1) detailed instructions for QA generation, (2) topic-specific task descriptions (e.g., scene recognition and transitions within the scene topic), (3) clip-level captions of the corresponding topic following the chronological order, (4) the comprehensive global-level captions, and (5) high-quality QA demo examples sourced from established video benchmarks [5, 14, 28, 43]. These structured prompts facilitate the creation of contextually grounded and diverse QA pairs that reflect the unique challenges of long-form video understanding. Detailed prompt templates for open-ended (Figure 9) and multiple-choice (Figure 10) QA generation are provided in Appendix A.
+
+Comparison. Table 1 presents a comparison between our proposed VideoMarathon dataset and existing video instruction-following datasets [7, 70]. The most significant distinction is VideoMarathon's substantially longer total video time and average video length. Moreover, VideoMarathon covers a broader duration range from 3 to 60 minutes, effectively bridging the gap in long-form video instruction data. Also, the diverse QA tasks and the balanced distribution of open-ended and multiple-choice QA pairs enable Video-LMMs to better handle a wide range of challenging real-world questions.
+
+![](images/a364fd913a26e7fe45db21602aac3e73ac0f8173b50de2e76214e56fe76ad5b9.jpg)  
+Figure 2: Overview of the Hour-LLaVA Framework. Input video features  $\mathbf{H}_{\mathrm{v}}$  encoded from 1-FPS sampled frames are selectively decayed spatially and temporally through a forgetting mechanism, producing decayed video tokens  $\hat{\mathbf{H}}_{\mathrm{v}}$  for efficient video modeling. Meanwhile, full video features  $\mathbf{H}_{\mathrm{v}}$  are stored in a memory repository. Given the decayed tokens  $\hat{\mathbf{H}}_{\mathrm{v}}$  and a user question tokens  $\mathbf{H}_{\mathrm{q}}$ , the MemAug module enhances them with full video context and user question-relevant details from the memory repository, obtaining memory-augmented video tokens  $\hat{\mathbf{H}}_{\mathrm{v}}$ . These augmented tokens are then passed with the original user question tokens  $\mathbf{H}_{\mathrm{q}}$  into the LLM decoder to generate the final response  $\mathbf{X}_{\mathrm{a}}$ .
+
+# 4 Hour-LLaVA: An Efficient Hour-scale Video-LMM
+
+Problem Formulation. Standard video QA tasks require video-language models to generate an answer  $\mathbf{X}_{\mathrm{a}}$ , based on a given input video  $\mathbf{X}_{\mathrm{v}}$  and a user question  $\mathbf{X}_{\mathrm{q}}$  by modeling the conditional probability of  $p_{\theta}(\mathbf{X}_{\mathrm{a}}|\mathbf{X}_{\mathrm{v}},\mathbf{X}_{\mathrm{q}})$ , where  $\theta$  is the parameter of video-language models. More specifically, the input video is first fed into a vision encoder  $f_{\theta_{\mathrm{v}}}(\cdot)$  parameterized by  $\theta_{\mathrm{V}}$  to produce the visual feature  $\mathbf{Z}_{\mathrm{v}} = f_{\theta_{\mathrm{v}}}(\mathbf{X}_{\mathrm{v}})$ . Then, a projector  $\mathbf{W}$  converts the visual feature  $\mathbf{Z}_{\mathrm{v}}$  to a visual embedding  $\mathbf{H}_{\mathrm{v}} = \mathbf{W}\cdot \mathbf{Z}_{\mathrm{v}}$  in the language embedding space. Meanwhile, the input user question  $\mathbf{X}_{\mathrm{q}}$  is projected to language embeddings  $\mathbf{H}_{\mathrm{q}}$  by a word embedding model. Finally, the language model  $f_{\theta_{\mathrm{L}}}(\cdot)$  predicts the answer  $\mathbf{X}_{\mathrm{a}} = f_{\theta_{\mathrm{L}}}(\mathbf{H}_{\mathrm{v}},\mathbf{H}_{\mathrm{q}})$ , where  $\theta_{\mathrm{L}}$  is the parameter of the language model.
+
+Memory Augmentation for Long Video Modeling. Modeling hour-long videos poses a challenge, as densely processing all frames with LLMs is impractical under GPU memory constraints. To address this challenge, we draw inspiration from the human memory system, which selectively retains and recalls essential past experiences while discarding irrelevant or redundant information, striking a balance between efficiency and comprehensiveness [2, 11]. Motivated by this principle, we design a memory augmentation mechanism to enable hour-scale video-language understanding. This mechanism consists of three main components: a memory repository, a forgetting mechanism, and a MemAug module. Together, they allow the model to operate on compressed representations while maintaining access to the full video context.
+
+- Memory Repository. High-fidelity video features  $\mathbf{H}_{\mathrm{v}}$  extracted at 1 FPS are stored in a memory repository serving as long-term memory, enabling the model to retain complete video context without requiring the LLM decoder to process every frame.  
+- Forgetting Mechanism. Due to the GPU memory constraints, we employ a forgetting mechanism  $\mathcal{M}_{\mathrm{forget}}$  to compresses the full video tokens  $\mathbf{H}_{\mathrm{v}}$  into a reduced set of decayed video tokens  $\tilde{\mathbf{H}}_{\mathrm{v}} = \mathcal{M}_{\mathrm{forget}}(\mathbf{H}_{\mathrm{v}})$  by selectively discarding the tokens in both spatial and temporal dimensions. This prevents the training and inference cost from growing linearly with video length. In practice, the forgetting mechanism  $\mathcal{M}_{\mathrm{forget}}$  can be implemented by various token compression strategies [23, 45, 46, 70]. Section 5.4 provides a comprehensive comparison among different forgetting mechanisms.  
+- MemAug Module. The MemAug module dynamically recovers the informative video tokens discarded during the forgetting stage from the memory repository. Without the MemAug module, forgetting mechanism alone would inevitably incur irreversible context loss and lead to sub-optimal performance. Technically, the MemAug module is implemented using standard transformer blocks [53]. In cross-attention, the decayed video tokens  $\tilde{\mathbf{H}}_{\mathrm{v}}$  and user question tokens  $\mathbf{H}_{\mathrm{q}}$  (as queries) attend and retrieve relevant information from the memory repository  $\mathbf{H}_{\mathrm{v}}$  (as keys and values). In self-attention, question-relevant video information flows from the user question tokens
+
+$\mathbf{H}_{\mathrm{q}}$  to the decayed video tokens  $\tilde{\mathbf{H}}_{\mathrm{v}}$ . Through this process, the decayed video tokens  $\tilde{\mathbf{H}}_{\mathrm{v}}$  are enriched with both full video context and question-specific details, resulting in memory-augmented video tokens  $\hat{\mathbf{H}}_{\mathrm{v}}$  that support long-dependence modeling for video understanding:
+
+$$
+\hat {\mathbf {H}} _ {\mathrm {v}} = f _ {\theta_ {\mathrm {M}}} \left(\tilde {\mathbf {H}} _ {\mathrm {v}}, \mathbf {H} _ {\mathrm {q}} \mid \mathbf {H} _ {\mathrm {v}}\right), \tag {1}
+$$
+
+where  $\theta_{\mathrm{M}}$  denotes the parameters of the MemAug module  $f_{\theta_{\mathrm{M}}}(\cdot)$ .
+
+Hour-LLaVA. Powered by memory augmentation, we propose Hour-LLaVA, an efficient video-language model capable of modeling hour-long videos at 1 FPS. It comprises three key modules: a video encoder, a memory augmentation module (i.e., MemAug), and an LLM decoder. Figure 2 shows the Hour-LLaVA framework, with the video encoder omitted for simplicity.
+
+- Video Encoding. Following existing Video-LMMs [70, 74], we adopt SigLIP [67] followed by a vision-language projector for video encoding. Specifically, we sample video frames at 1 FPS and extract visual features  $\mathbf{Z}_{\mathrm{v}}$ , which are then average-pooled to a fixed spatial resolution of  $8 \times 8$  per frame. We then project the average-pooled visual features into the language embedding space via a two-layer MLP with GELU [21] activation, obtaining the full video tokens  $\mathbf{H}_{\mathrm{v}}$ .  
+- MemAug. The decayed video tokens  $\tilde{\mathbf{H}}_{\mathrm{v}}$  are obtained by "forgetting"  $\sim 94\%$  of the full video tokens with an overall compression ratio of  $1 / _{16}$  through the forgetting mechanism, achieved by discarding tokens with a  $1 / _{4}$  compression ratio along both spatial and temporal dimensions. These decayed tokens are then enriched with the full video tokens via the MemAug module, implemented with  $N = 4$  transformer blocks. To associate the spatial-temporal correspondence between the full video tokens in the memory repository and the decayed tokens, we first flatten the full video tokens into 1D embeddings and apply 1D structured RoPE, following prior works [16, 49]. Based on their spatial-temporal alignment, we then construct a corresponding 1D RoPE for the decayed tokens, ensuring that each decayed token (query) carries positional information consistent with its related full tokens (keys and values) during cross-attention.  
+- LLM Decoder. Given the memory-augmented video tokens  $\hat{\mathbf{H}}_{\mathrm{v}}$  and the user question tokens  $\mathbf{H}_{\mathrm{q}}$ , LLM decoder generates final response  $\mathbf{X}_{\mathrm{a}}$ . We employ Qwen2.5-3B-Instruct [65] and Qwen2-7B-Instruct [64] as the LLM decoders for Hour-LLaVA-3B and Hour-LLaVA-7B, respectively.
+
+Training Schedules. The training schedules of Hour-LLaVA follow three stages: image-language pretraining, video-language adaptation, and video instruction tuning. Table 8 presents more details of the training schedules.
+
+- Image-Language Pretraining. We perform image-language pretraining with 3B image-text pairs from the single-image subset of LLaVA-OV [26]. Full-resolution image tokens are stored in the memory repository, from which  $\frac{1}{4}$  tokens are retained as decayed visual tokens via the spatial forgetting mechanism. Only the Transformer blocks of MemAug are trained for one epoch.  
+- Video-Language Adaptation. Following the data composition that adheres to established practices [70, 74], the model is adapted to video-language inputs using a small amount of mixture of 0.12M single-image, 0.05M multi-image, and 0.09M text data from LLaVA-OV, plus 0.3M short video samples from LLaVA-Video-178K [70]. A  $1/4$  token compression is applied in both spatial and temporal dimensions via the forgetting mechanism. All model parameters are trained for one epoch. This stage intends to slightly adapt all model parameters to the video domain, thereby constructing a strong baseline for the subsequent long-video training stage.  
+- Video Instruction Tuning. This stage uses instruction-following data involving long video content. For training efficiency, up to five QA pairs from the same VideoMarathon video are grouped as a multi-turn conversation. The corpus includes 1.14M single-image, 0.5M multi-image, and 0.81M text samples from LLaVA-OV, 1.3M short video samples from LLaVA-Video-178K, and 0.7M long video samples from VideoMarathon. The same forgetting mechanism is applied as the previous stage. The vision encoder is frozen, while the remaining modules are fine-tuned.
+
+# 5 Experiments
+
+# 5.1 Experimental Setting
+
+Evaluation Benchmarks. We evaluate our models on four mainstream video-language benchmarks: TempCompass [36], LongVideoBench [58], Video-MME [14], and LVBench [55]. TempCompass focuses on assessing the temporal reasoning ability of Video-LMMs for short videos. LongVideoBench
+
+Table 2: Performance comparison of existing LMMs on TempCompass, LongVideoBench, VideoMME, and LVBench datasets. M-Avg denotes the average performance of multiple-choice tasks. Blue boxes denote the average durations of benchmarks. Red highlights that LVBench's average video length exceeds the maximum length in the training stage. The symbol  $\dagger$  marks our reimplemented results. Bold font denotes the best performance among models at the same scale, while underline indicates the second-best.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">LLM Params</td><td rowspan="2">Input Video</td><td rowspan="2">TempCompass 11s</td><td rowspan="2">LongVideoBench 459s</td><td colspan="3">VideoMME (w/o &amp; w/ subtitles)</td><td rowspan="2">LVBench Avg 4037s</td></tr><tr><td>Overall 1021s</td><td>Medium 516s</td><td>Long 2466s</td></tr><tr><td colspan="9">Proprietary LMM</td></tr><tr><td>GPT-4V [41]</td><td>-</td><td>10 frames</td><td>-</td><td>61.3</td><td>59.9/63.3</td><td>55.8/59.7</td><td>53.5/56.9</td><td>-</td></tr><tr><td>GPT-4o [42]</td><td>-</td><td>384 frames</td><td>70.9</td><td>66.7</td><td>71.9/77.2</td><td>70.3/76.6</td><td>65.3/72.1</td><td>48.9</td></tr><tr><td>Gemini 1.5 Flash [52]</td><td>-</td><td>0.5/1 fps</td><td>-</td><td>61.6</td><td>70.3/75.0</td><td>68.8/74.7</td><td>61.1/68.8</td><td>-</td></tr><tr><td>Gemini 1.5 Pro [52]</td><td>-</td><td>0.5/1 fps</td><td>69.3</td><td>64.0</td><td>75.0/81.3</td><td>74.3/81.0</td><td>67.4/77.4</td><td>33.1</td></tr><tr><td colspan="9">Open-source LMM (&lt;7B)</td></tr><tr><td>VILA1.5-3B [30]</td><td>3B</td><td>8 frames</td><td>56.1</td><td>42.9</td><td>42.2/44.2</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Phi-3.5-Vision-4.2B [1]</td><td>4.2B</td><td>16 frames</td><td>-</td><td>-</td><td>50.8/ -</td><td>-</td><td>-</td><td>-</td></tr><tr><td>LongVU-3.2B [45]</td><td>3.2B</td><td>1 fps</td><td>-</td><td>-</td><td>- /51.5</td><td>-</td><td>- /47.2</td><td>-</td></tr><tr><td>InternVL2.5-2B [9]</td><td>2B</td><td>64 frames</td><td>53.4</td><td>46.0</td><td>51.9/54.1</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Apollo-1.5B [74]</td><td>1.5B</td><td>2 fps</td><td>60.8</td><td>54.1</td><td>53.0/54.6</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Apollo-3B [74]</td><td>3B</td><td>2 fps</td><td>62.5</td><td>55.1</td><td>58.4/60.6</td><td>-</td><td>-</td><td>-</td></tr><tr><td>LLaVA-OV-SI-3B† [26]</td><td>3B</td><td>32 frames</td><td>55.6</td><td>49.6</td><td>51.1/54.5</td><td>49.2/52.1</td><td>44.1/46.4</td><td>35.4</td></tr><tr><td>LLaVA-Video-3B† [70]</td><td>3B</td><td>64 frames</td><td>63.4</td><td>55.2</td><td>58.7/60.7</td><td>55.2/57.3</td><td>47.0/49.9</td><td>41.7</td></tr><tr><td>Hour-LLaVA-3B (ours)</td><td>3B</td><td>1 fps</td><td>63.6</td><td>57.8</td><td>60.6/66.7</td><td>59.0/65.4</td><td>52.1/60.4</td><td>44.7</td></tr><tr><td colspan="9">Open-source LMM (7-8B)</td></tr><tr><td>Video-LLaVA [29]</td><td>7B</td><td>8 frames</td><td>37.9</td><td>39.1</td><td>39.9/41.6</td><td>38.0/40.7</td><td>36.2/38.1</td><td>-</td></tr><tr><td>VideoChat2 [28]</td><td>7B</td><td>16 frames</td><td>51.1</td><td>39.3</td><td>39.5/43.8</td><td>37.0/39.4</td><td>33.2/39.2</td><td>-</td></tr><tr><td>ShareGPT4Video [7]</td><td>8B</td><td>16 frames</td><td>59.4</td><td>41.8</td><td>39.9/43.6</td><td>36.3/39.3</td><td>35.0/37.9</td><td>-</td></tr><tr><td>VideoLLaMA2 [10]</td><td>7B</td><td>16 frames</td><td>-</td><td>51.4</td><td>47.9/50.3</td><td>37.0/39.4</td><td>33.2/39.2</td><td>-</td></tr><tr><td>Video-XL [46]</td><td>7B</td><td>1 fps</td><td>-</td><td>50.7</td><td>55.5/61.0</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Kangaroo [35]</td><td>8B</td><td>64 frames</td><td>62.5</td><td>54.8</td><td>56.0/57.6</td><td>55.3/55.4</td><td>46.7/49.3</td><td>39.4</td></tr><tr><td>LongVA [68]</td><td>7B</td><td>128 frames</td><td>-</td><td>-</td><td>52.6/54.3</td><td>50.4/53.6</td><td>46.2/47.6</td><td>-</td></tr><tr><td>LongVILA [63]</td><td>7B</td><td>256 frames</td><td>-</td><td>-</td><td>60.1/65.1</td><td>58.3/64.9</td><td>53.0/57.4</td><td>-</td></tr><tr><td>LongVU [45]</td><td>7B</td><td>1 fps</td><td>-</td><td>-</td><td>- /60.9</td><td>-</td><td>- /59.5</td><td>-</td></tr><tr><td>Apollo-7B [74]</td><td>7B</td><td>2 fps</td><td>64.9</td><td>58.5</td><td>61.3/63.3</td><td>-</td><td>-</td><td>-</td></tr><tr><td>LLaVA-OV-SI-7B [26]</td><td>7B</td><td>32 frames</td><td>57.2†</td><td>52.1†</td><td>58.2/61.5</td><td>52.6†/55.4†</td><td>47.9†/50.2†</td><td>36.0†</td></tr><tr><td>LLaVA-Video-7B [70]</td><td>7B</td><td>64 frames</td><td>64.3†</td><td>58.2</td><td>63.3/69.7</td><td>58.9/62.9†</td><td>53.0/55.0†</td><td>42.2†</td></tr><tr><td>Hour-LLaVA-7B (ours)</td><td>7B</td><td>1 fps</td><td>68.1</td><td>60.4</td><td>63.6/70.2</td><td>63.8/70.0</td><td>55.0/65.1</td><td>45.6</td></tr></table>
+
+consists of varying-length web-collected videos up to one hour with their subtitles across diverse themes, evaluating the abilities in retrieving and reasoning over detailed information from long videos. Video-MME is a comprehensive multimodal benchmark designed to evaluate long video understanding across diverse video types and temporal spans. LVBench challenges models to demonstrate long-term memory and extended comprehension across multimodal inputs.
+
+Implementation Details. Following existing practices [45, 68, 70], we initialize our Hour-LLaVA models with pretrained Image-LMMs. Specifically, the vision encoder and the LLM decoder of Hour-LLaVA-7B are initialized from LLaVA-OV-SI-7B [26], which is trained solely on image data. Due to the absence of LLaVA-OV-SI-3B, we pretrain a 3B version of LLaVA-OV-SI using Qwen2.5-3B-Instruct model, and then initialize Hour-LLaVA-3B from it. For video-language training, we set the global batch sizes to 128 and 256 for the 3B and 7B models, respectively. A learning rate of 2e-5 is used with a 0.03 warmup ratio under a cosine annealing schedule. The models are optimized using the AdamW [37] optimizer with a cross-entropy loss. We train Hour-LLaVA-3B with 64 AMD MI250 GPUs and Hour-LLaVA-7B with 64 AMD MI300X GPUs, respectively. We conduct all the ablation studies using Hour-LLaVA-3B in Section 5.3 and 5.4. Please refer to Appendix A.3 for more implementation details.
+
+# 5.2 Main Results
+
+Overview. As shown in Table 2, Hour-LLaVA consistently achieves the best performance on four well-established video benchmarks in both the 3B and 7-8B model size categories. Notably, Hour-LLaVA-3B even surpasses more than half of the current 7-8B Video-LMMs.
+
+TempCompass. The results on TempCompass show that Hour-LLaVA maintains strong performance on short-form video-language tasks, even after introducing long video-language training samples.
+
+LongVideoBench. Hour-LLaVA demonstrates state-of-the-art performance among open-source models across both the 3B and 7B parameter scales on LongVideoBench. Specifically, Hour-LLaVA-3B and Hour-LLaVA-7B outperform the second-best model by 2.6 and 1.9 points, respectively.
+
+![](images/dfeaa3a9d89b90bdb1d51be92f117e779161b246d184b374a463d1d3e7641cec.jpg)  
+(a) TempCompass (Short)
+
+![](images/453f7b638636b11a18258930ba20a173b896e2cc6aa25404b81cdc7a72e77805.jpg)  
+Figure 3: Dataset ablation and methodology comparison. The analysis is evaluated across three benchmarks: (a) TempCompass, (b) LongVideoBench, and (c) LVBench. It presents the performance of Hour-LLaVA-3B and LLaVA-Video-3B models. The x-axis represents different training data mixture configurations, with exact ratios indicated in the legend. V.M. and L.V. refer to the VideoMarathon and LLaVA-Video-178K datasets.
+
+![](images/a2548889798e7f1a9bde97c1387d2b182b809dc40bc541b56a78f1825c6d2030.jpg)  
+(b) LongVideoBench (Long)  
+(c) LVBench (Long)
+
+VideoMME. Hour-LLaVA consistently outperforms other open-source models at both the 3B and 7B scales on VideoMME. Remarkably, Hour-LLaVA achieves significantly higher performance in both medium- and long-length video settings, highlighting its strong capability in long video understanding
+
+LVBench. The maximum video length of LVBench is more than two hours, and its average video length (67 minutes) already exceeds the maximum video length in our training (60 minutes). As such, this benchmark serves as a pressure test for long video understanding and highlights the model's ability to generalize beyond its training distribution. As shown in Table 2, Hour-LLaVA achieves leading performance, outperforming the second-best models by 3.0 and 3.4 points under the 3B and 7B settings, respectively.
+
+Comparison with Baseline Models. LLaVA-OV-SI [26] serves as the image-language pretrained model for Hour-LLaVA, while both LLaVA-Video [70] and Hour-LLaVA adopt the same LLM decoder. The results show that Hour-LLaVA considerably enhances the performance of its pretrained model across both 3B and 7B model scales. Furthermore, Hour-LLaVA consistently outperforms its strong peer, LLaVA-Video, especially in medium- and long-video scenarios.
+
+# 5.3 Analysis on VideoMarathon Dataset
+
+Comparison between VideoMarathon and LLaVA-Video-178K. We compare our proposed VideoMarathon dataset with LLaVA-Video-178K [70], the largest publicly available video instruction-tuning dataset to date. For a fair comparison, we construct two subsets from VideoMarathon (V.M.) and LLaVA-Video-178K (L.V.), matched in both the number of videos and the number of video-language instruction samples. Specifically, we randomly select 70K instruction-following samples over 10K videos from each dataset. To support multimodal learning, both datasets are further combined with the same 30K single-image, 10K multi-image, and 20K text-only samples from LLaVA-OV [26].
+
+Figure 3 illustrates how performance varies with different mixtures of V.M. (long-video) and L.V. (short-video) training data. On the far left of the x-axis, the model is trained solely on 70K short-video samples from L.V.; moving right, short-video samples are gradually replaced by long-video samples from V.M., reaching  $100\%$  V.M. on the far right. The pie charts visualize the data mix ratios for each setting. As shown in Figures 3b and 3c, Hour-LLaVA  $(\rightarrow)$  achieves progressively better performance as the proportion of long-video data increases, peaking when the long-to-short ratio reaches 3:1, and then declining once L.V. data is entirely absent. These results highlight the crucial role of VideoMarathon in enhancing long video-language understanding. They also suggest that mixing different datasets can be beneficial even when the training and testing video lengths differ, possibly due to the increased diversity introduced by heterogeneous data sources. In addition, Figure 3a shows that incorporating long videos from VideoMarathon in training does not considerably affect performance on short video-language understanding. In the following ablation experiments, we use the same data recipe, along with a mixture of long- and short-video at a 3:1 ratio.
+
+Sparse Sampling Limits Long Video-Language Learning. Using the VideoMarathon dataset, we also train LLaVA-Video [70], a representative Video-LMM that adopts sparse temporal sampling by uniformly selecting 64-frame video features as input to the LLM decoder. As shown in Figure 3, LLaVA-Video  $(\dashv)$  fails to benefit from an increased proportion of long-video samples, with its performance even declining after long-video training. This result indicates that sparse sampling
+
+Table 3: Comparison of spatial forgetting strategies.  
+
+<table><tr><td>Spatial Forgetting</td><td>Tokens/img</td><td>MMStar</td><td>Sci.QA</td><td>R.W.QA</td></tr><tr><td>LLaVA-OV-SI-3B (Base)</td><td>729</td><td>52.8</td><td>84.7</td><td>58.8</td></tr><tr><td>+ Random</td><td>196</td><td>51.9</td><td>84.5</td><td>59.6</td></tr><tr><td>+ Uniform</td><td>196</td><td>51.5</td><td>83.7</td><td>59.5</td></tr></table>
+
+Table 4: Comparison of temporal forgetting strategies.  
+
+<table><tr><td>Temporal Forgetting</td><td>LongVideoBench</td><td>LVBench</td></tr><tr><td>Uniform</td><td>54.0</td><td>40.6</td></tr><tr><td>Keyframe</td><td>53.5</td><td>40.3</td></tr><tr><td>Question-guided</td><td>53.4</td><td>39.7</td></tr><tr><td>Random</td><td>53.2</td><td>38.8</td></tr></table>
+
+![](images/59b76333ac55d55fb497f8529723d716e99d154e40bdfb61f92a1eae068fd5a9.jpg)  
+Figure 4: Impact of compression ratio for temporal forgetting (left), and memory repository scale (middle). Comparison of the number of visual tokens input to the LLM decoder (right).
+
+![](images/97375d4ba4771625065b4f1b379c542114507aac8b5282e91b67de71161f4cda.jpg)
+
+![](images/68d50fba63cd3934ebd280dae26ba827330653112773955b1051b0418ac1703a.jpg)
+
+is insufficient to capture the effective patterns required for long video-language understanding. In contrast, Hour-LLaVA  $(\rightarrow)$  effectively achieves long-video language modeling by leveraging access to a memory repository that stores full dense video content at 1 FPS. These results highlight the urgent demands for new training paradigms tailored to long video-language modeling, and position VideoMarathon as a practical platform for developing them.
+
+# 5.4 Analysis on Key Components of Hour-LLaVA
+
+Forgetting Mechanisms. The forgetting mechanism is proposed to compress full video tokens into decayed video tokens to reduce the number of tokens processed by the LLM decoder. The decayed tokens will then be augmented by the MemAug module to retain or recall the informative semantics. The forgetting mechanism can be implemented with different token compression strategies [23, 45, 46, 70] in both spatial and temporal domains. We ablate different forgetting mechanisms guided by the MemAug module as follows:
+
+- Spatial Forgetting (SF). We compare two straightforward methods: uniform SF (discarding tokens at regular intervals over the 2D image) and random SF (discarding tokens randomly) with a compression ratio of  $1/4$ . We conduct this ablation during the image-language pretraining stage and evaluate on three popular image-language benchmarks: MMStar [6], ScienceQA [38] (Sci.QA), and RealWorldQA [59] (R.W.QA). Notably, despite using only  $1/4$  tokens, both SF strategies achieve comparable performance to the base model without token compression as shown in Table 3. It highlights the success of our MemAug module for token compression in the image domain, suggesting that MemAug can also be applied to Image-LMMs for token compression. For Hour-LLaVA, we adopt random SF with  $1/4$  compression ratio for its simplicity and effectiveness.  
+- Temporal Forgetting (TF). We evaluate four temporal compression approaches: random, uniform [23, 35, 62, 70], keyframe-based [45, 50], and question-guided temporal compression [45, 46]. All strategies are evaluated using a compression ratio of  $1/4$ . As shown in Table 4, uniform TF yields the best overall performance, suggesting that the MemAug module is effective without relying on manually designed compression strategies. Furthermore, we analyze the effect of different compression ratios for temporal forgetting on LVBench, ranging from 1 down to  $1/16$  as shown in Figure 4 (left). We adopt the uniform TF with a ratio of  $1/4$  for Hour-LLaVA, balancing the trade-off between performance and efficiency introduced by temporal compression.  
+- Token Control for Videos with Extreme Lengths. Although forgetting mechanisms significantly reduce the tokens processed by the LLM decoder, they remain insufficient for extremely long videos (e.g., 24-hour videos). We set the maximum number of retained frames to 512 so as to constrain the computational cost. Conversely, for extremely short videos, we enforce a minimum of 32 retained frames to preserve sufficient contextual information. Note that these thresholds apply to decayed tokens only, which means Hour-LLaVA still accesses the full video context at 1-FPS sampling via memory repository. Figure 4 (right) compares the number of visual tokens fed into the LLM decoder across Hour-LLaVA, LLaVA-Video, and vanilla 1-FPS sampling methods. It highlights that Hour-LLaVA consistently uses fewer tokens than others.
+
+Table 5: Impact of MemAug module on different token compression strategies.  
+
+<table><tr><td>Token Compression</td><td>MemAug</td><td>LongVideoBench</td><td>LVBench</td></tr><tr><td rowspan="2">Uniform</td><td>✓</td><td>54.0 (+1.9)</td><td>40.6 (+2.3)</td></tr><tr><td>×</td><td>52.1</td><td>38.3</td></tr><tr><td rowspan="2">Keyframe</td><td>✓</td><td>53.5 (+1.5)</td><td>40.3 (+1.4)</td></tr><tr><td>×</td><td>52.0</td><td>38.9</td></tr><tr><td rowspan="2">Question-guided</td><td>✓</td><td>53.4 (+2.4)</td><td>39.7 (+2.2)</td></tr><tr><td>×</td><td>51.0</td><td>37.5</td></tr></table>
+
+Table 6: Impact of the number of MemAug blocks. M.A. refers to MemAug.  
+
+<table><tr><td># M.A. Blocks</td><td>TempCompass</td><td>LongVideoBench</td><td>LVBench</td></tr><tr><td>1</td><td>59.4</td><td>53.2</td><td>40.0</td></tr><tr><td>2</td><td>59.8</td><td>53.5</td><td>40.3</td></tr><tr><td>4</td><td>59.7</td><td>54.0</td><td>40.6</td></tr><tr><td>8</td><td>59.2</td><td>53.4</td><td>40.5</td></tr></table>
+
+MemAug Module. The MemAug module is designed to dynamically recover the informative video tokens from the memory repository. To retain informative tokens from full videos, previous studies have explored various video token compression approaches [12, 45, 50]. However, these approaches typically rely on hand-crafted rules (e.g., predefined thresholds) and inevitably suffer from irreversible information loss. In contrast, the MemAug module performs video context compression through a learnable process supervised by video instruction data.
+
+- Impact of MemAug under Different Compression Strategies. To validate the effectiveness of MemAug, we apply MemAug module on several representative video token compression methods [12, 45, 50], including uniform [23, 35, 62, 70], keyframe-based [45, 50], and question-guided [45, 46] temporal compression. For fairness, all methods feed the same total number of compressed tokens to the LLM decoder. Specifically, we apply random spatial compression with a ratio of  $\frac{1}{4}$  and various temporal compression techniques with a ratio of  $\frac{1}{4}$  on 1-FPS input videos. As shown in Table 5, models equipped with MemAug consistently achieve 1.4-2.4% empirical gains over those without it across all token compression strategies, demonstrating that MemAug effectively re-injects informative cues. Without MemAug, conventional token compression approaches incur irreversible context loss, resulting in sub-optimal performance. As a result, by coupling MemAug with forgetting mechanism, Hour-LLaVA delivers compression-level efficiency while still preserving the high-fidelity video content.  
+- Robustness of MemAug Module. Notably, Table 5 presents that the performance with MemAug remains comparable regardless of the underlying token compression approaches, indicating that MemAug is robust and largely insensitive to the choice of token forgetting mechanism. This consistency demonstrates that MemAug effectively adapts to distinct temporal compression paradigms, ensuring reliable information recovery even under different token compression settings.  
+- Effect of the number of MemAug blocks. We evaluate 1, 2, 4, and 8 MemAug blocks on three benchmarks. Table 6 demonstrates that performance improves up to 2 or 4 blocks and then decreases slightly at 8. More specifically, for short videos (TempCompass), the sweet spot lies in the range from 2 to 4 blocks, while for medium- and long-form videos (LongVideoBench and LVBench), 4 blocks provide the optimal results. Consequently, we adopt 4 MemAug blocks as the default.
+
+Memory Repository. We investigate the impact of the memory repository scale on performance on LVBench. Figure 4 (middle) shows that the performance of Hour-LLaVA decreases as the memory repository scale becomes smaller. This trend indicates that information loss in the memory repository limits the capacity of long video understanding. Therefore, to preserve as much temporal information as possible, we retain video tokens sampled at 1 FPS in the memory repository.
+
+Please refer to the Appendix A.3.2 for more details of the above experiments.
+
+# 6 Conclusion
+
+In this study, we introduce VideoMarathon, a large-scale video instruction-following dataset. Comprising long videos with a total duration of around 9,700 hours and 3.3M QA pairs, VideoMarathon covers 22 challenging tasks that require both short- and long-term video understanding. Building on VideoMarathon, we propose Hour-LLaVA, an efficient and powerful Video-LMM optimized for hour-scale video modeling. By leveraging a memory augmentation (MemAug) mechanism, Hour-LLaVA effectively integrates information from the full video context while maintaining efficiency through 1-FPS sampling. Extensive evaluations on several video-language benchmarks validate the high quality of the VideoMarathon dataset and the superiority of the Hour-LLaVA model.
+
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+
+# A Appendix
+
+This document provides more implementation details of the VideoMarathon dataset construction, additional experimental details, limitations, and broader impacts, organized as follows:
+
+- Details of VideoMarathon Construction (Section A.1). We provide the exact prompts and examples for hierarchical video captioning and topic-specific QA generation.  
+- Human Evaluation on VideoMarathon (Section A.2). We conduct the human evaluation on the proposed VideoMarathon dataset.  
+- More Experimental Details (Section A.3). We present additional experimental details, including detailed training schedules for Hour-LLaVA and experimental settings of ablation studies in Sections 5.3 and 5.4.  
+- Additional Empirical Analysis (Section A.4). We provide additional empirical analysis to support the superiority of the proposed Hour-LLaVA framework.  
+- Limitations (Section A.5). We discuss several limitations of this study.  
+- Broader Impacts (Section A.6). We analyze both the potential positive societal impacts and the negative societal impacts of this work.
+
+# A.1 Details of VideoMarathon Construction
+
+# A.1.1 Details of Hierarchical Video Captioning
+
+**Prompts.** We provide the exact prompts used for hierarchical video captioning, including clip-level captioning (Figure 5), event-level captioning (Figure 7), and global-level captioning (Figure 8). Additionally, the prompt used for event splitting is presented in Figure 6.
+
+Examples. Figure 12 presents several examples of hierarchical video captions, including clip-level, event-level, global-level video captions, and event splits. In addition, Figure 13 presents the word cloud of all the global-level video descriptions in the VideoMarathon dataset.
+
+# A.1.2 Details of QA Generation
+
+**Prompts.** We present the exact prompts used for topic-specific QA generation, including the open-ended (OE) QA prompt in Figure 9 and the multiple-choice (MC) QA prompt in Figure 10. Also, the detailed descriptions of the 22 sub-tasks across six core topics are provided in Figure 11. Please refer to the data file for exact QA examples from the VideoMarathon dataset.
+
+# A.2 Human Evaluation on VideoMarathon
+
+To quantify the data quality, we manually check 330 randomly selected VideoMarathon QA pairs, including 80 multiple-choice and 250 open-ended samples.
+
+For the Multiple-Choice (MC) QA samples, we assess the accuracy of the synthetic samples by assigning a score of 1 if there is exactly one correct answer. A score of 0 is given if the question is irrelevant to the video, if no correct option exists, or if multiple options are correct. The human evaluation on 80 samples achieves an accuracy of  $78.7\%$ , which demonstrates the reliability of the synthetic multiple-choice QA samples.
+
+For the Open-Ended (OE) QA samples, we validate the data quality from two perspectives:
+
+- Yes/No Bias: Yes/No Bias evaluation [19] validates the extent to which the models tend to respond with "yes". In particular, the evaluation consists of "Yes Percentage Difference" (Pct. Diff) and "False Positive Ratio" (FP Ratio) metrics. Smaller Pct. Diff (closer to 0.0) reveals less bias, which means that the predicted number of "Yes" is closer to the ground truth. FP Ratio measures how likely the model responds with "Yes" out of all incorrect responses, where a value closer to 0.5 indicates less bias. We randomly select 200 Yes/No QA samples. As a result, Pct. Diff reaches 0.04 and FP Ratio achieves 0.40, which reflects limited yes/no bias in the VideoMarathon dataset.  
+- Hallucination: 20 volunteers evaluate the quality of 50 randomly selected OE QA samples. Each sample is rated on a scale from 1 to 10 based on the following criteria. Score 1 denotes a totally irrelevant response, while Score 10 indicates an almost perfect answer with no hallucinations.
+
+Table 7: Human evaluation of hallucination on open-ended QA samples in VideoMarathon dataset.  
+
+<table><tr><td>Score</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td></tr><tr><td>Ratio (%)</td><td>2.22</td><td>2.44</td><td>1.78</td><td>3.56</td><td>4.67</td><td>4.67</td><td>13.33</td><td>22.67</td><td>18.00</td><td>26.67</td></tr></table>
+
+Higher scores correspond to fewer hallucinations. As show in Table 7, human evaluation achieves an average score of  $7.81 \pm 0.59$  (based on ratings from 20 volunteers over 50 OE QA samples), validating the quality of OE QA samples in the VideoMarathon dataset (Scores above 7.5 are considered largely correct). Notably, the score distribution shows that over  $80\%$  of the samples receive scores  $\geq 7$ . To further support the claim "Scores above 7.5 are considered largely correct", we present examples with averaged scores between 7-8 below:
+
+```txt
+Example 1   
+Question: Where is the woman positioned relative to the man in the scene where they are sitting on the couch?   
+Synthetic Answer: The woman is sitting next to the man on the couch.   
+Ground Truth: The woman is sitting to the right of the man on the couch.   
+Averaged Score: 8.0   
+# Example 2   
+Question: After the boy lights the candle in Clip, what does the woman do?   
+Synthetic Answer: She holds the candle after it is lit.   
+Ground Truth: She starts to speak and holds the candle.   
+Averaged Score: 7.3   
+# Example 3   
+Question: What is the woman doing when she uses a sewing machine in the video?   
+Synthetic Answer: She is sewing the cuffed hem onto the pants.   
+Ground Truth: She is sewing the hem of the pants.   
+Averaged Score: 7.5
+```
+
+# A.3 Additional Experimental Details
+
+# A.3.1 Detailed Training Schedules for Hour-LLaVA
+
+In Section 4, we briefly introduce the training schedules of Hour-LLaVA. Furthermore, Table 8 presents the detailed training schedules for each training stage of Hour-LLaVA, containing compression details, data usage, and training hyperparameters.
+
+Table 8: Detailed training schedule for each training stage of Hour-LLaVA, including compression details, data usage, and training hyperparameters. In Data part, we use the following abbreviations: T for text, SI for single-image, MI for multi-image, SV for short-video, LV for long-video, OV-SI for LLaVA-OV-SI [26], L.V. for LLaVA-Video-178K [70], and V.M. for VideoMarathon.
+
+<table><tr><td rowspan="2" colspan="2"></td><td colspan="2">Image-Language Pretraining</td><td colspan="2">Video-Language Adaptation</td><td colspan="2">Video Instruction Tuning</td></tr><tr><td>3B</td><td>7B</td><td>3B</td><td>7B</td><td>3B</td><td>7B</td></tr><tr><td rowspan="5">Compassion</td><td>FPS</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Spatial Forgetting (SF)</td><td>1/4</td><td>1/4</td><td>1/4</td><td>1/4</td><td>1/4</td><td>1/4</td></tr><tr><td>SF Mechanism</td><td>Random</td><td>Random</td><td>Random</td><td>Random</td><td>Random</td><td>Random</td></tr><tr><td>Temporal Forgetting (TF)</td><td>-</td><td>-</td><td>1/4</td><td>1/4</td><td>1/4</td><td>1/4</td></tr><tr><td>TF Mechanism</td><td>-</td><td>-</td><td>Uniform</td><td>Uniform</td><td>Uniform</td><td>Uniform</td></tr><tr><td rowspan="3">Data</td><td>Data Type</td><td>SI</td><td>SI</td><td>T + SI + MI + SV</td><td>T + SI + MI + SV</td><td>T + SI + MI + SV + LV</td><td>T + SI + MI + SV + LV</td></tr><tr><td>Data Sources</td><td>OV-SI</td><td>OV-SI</td><td>OV-SI + L.V.</td><td>OV-SI + L.V.</td><td>OV-SI + L.V. + V.M.</td><td>OV-SI + L.V. + V.M.</td></tr><tr><td>#Samples</td><td>3B</td><td>3B</td><td>0.6M</td><td>0.6M</td><td>4.4M</td><td>4.4M</td></tr><tr><td rowspan="6">Training</td><td>Batch Size</td><td>128</td><td>128</td><td>128</td><td>128</td><td>128</td><td>256</td></tr><tr><td>LR of Vision Encoder</td><td>0</td><td>0</td><td>5 × 10-6</td><td>5 × 10-6</td><td>0</td><td>0</td></tr><tr><td>LR of Projector</td><td>0</td><td>0</td><td>1 × 10-4</td><td>1 × 10-4</td><td>1 × 10-4</td><td>1 × 10-4</td></tr><tr><td>LR of MemAug</td><td>1 × 10-4</td><td>1 × 10-4</td><td>1 × 10-4</td><td>1 × 10-4</td><td>1 × 10-4</td><td>1 × 10-4</td></tr><tr><td>LR of LLM</td><td>0</td><td>0</td><td>2 × 10-5</td><td>1 × 10-5</td><td>2 × 10-5</td><td>2 × 10-5</td></tr><tr><td>Epoch</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></table>
+
+# A.3.2 Experimental Settings of Ablation Studies.
+
+Due to page limitations, we are unable to include all experimental settings for the ablation studies in Section 5. Below, we provide the experimental settings for each ablation study. We conduct all the ablation studies using the Hour-LLaVA-3B model.
+
+Ablation Study for VideoMarathon (Section 5.3). In this section, we investigate how performance changes with different mixtures of VideoMarathon (long-video) and LLaVA-Video-178K (short-video) training data. We construct two subsets from VideoMarathon and LLaVA-Video-178K, matched in both the number of videos and the number of video-language instruction training samples. Each sub-dataset consists of 70K video-language instruction samples over 10K videos. We then train the Hour-LLaVA-3B model on different mixtures of these two subsets (as shown in Figure 3), while also incorporating the same 60K multimodal samples from LLaVA-OV [26] (i.e., 20K text-only, 30K single-image, and 10K multi-image samples) to support multimodal learning. In addition, the Hour-LLaVA-3B model is initialized from the checkpoint obtained after image-language pretraining (Stage 1 in Table 8). All training hyperparameters follow the settings used for video instruction tuning of the 3B model, as detailed in Table 8.
+
+Analysis on Key Components of Hour-LLaVA (Section 5.4). In this section, we primarily conduct analysis on three key components in Hour-LLaVA: forgetting mechanisms, MemAug module, and memory repository.
+
+We ablate forgetting mechanisms from both spatial and temporal perspectives.
+
+- Spatial Forgetting. In this setting, we adopt 3B LLaVA-OV-SI data for training. The vision encoder, projector, and LLM decoder of the Hour-LLaVA-3B model are initialized by a pretrained LLaVA-OV-SI-3B model as mentioned in Section 5.1. During training, only the parameters of the MemAug module are tuned, while the rest of the model remains frozen. Training hyperparameters follow the settings used for image-language pretraining of the 3B model in Table 8.  
+- Temporal Forgetting. We compare several representative video token compression techniques, including uniform [23, 62], keyframe [45, 50], and question-guided [12, 45] temporal compression. Also, we implement random temporal compression as a reference baseline. In particular, the implementation details of different temporal compression methods are shown below:
+
+- Random. We randomly select  $1/4$  frames from a video sampled at 1 FPS.  
+- Uniform. We uniformly sample  $\frac{1}{4}$  frames across the temporal dimension from a 1 FPS video.  
+- Keyframe. Following [45], for each frame, we compute the average cosine similarity with its  $K$  nearest temporal neighbors (with  $K = 8$ ). We then retain the  $\frac{1}{4}$  frames that are least similar to their neighbors, removing the  $\frac{3}{4}$  most redundant frames. In addition, video frame features are extracted by SigLIP vision encoder [67].  
+- Question-guided. Following [12], we calculate the cosine similarity between the frame embedding  $\mathbf{v}$  and question embedding  $\mathbf{q}$ . Next, we select  $1/4$  frames with the highest similarity scores. In particular, the frame embedding  $\mathbf{v}$  is computed as  $\mathbf{v} = \frac{1}{N_v} \sum_{i=1}^{N_v} \mathbf{v}_i$ , where  $N_v$  is the number of tokens per frames and  $\mathbf{v}_i$  refers to the  $i$ -th visual token vector. The question embedding is calculated as  $\mathbf{q} = \frac{1}{N_q} \sum_{i=1}^{N_q} \mathbf{q}_i$ , where  $N_q$  is the number of tokens in the given question and  $\mathbf{q}_i$  denotes the  $i$ -th query token vector. Additionally,  $\mathbf{v}_i$  is obtained using the SigLIP vision encoder followed by the projector, while  $\mathbf{q}_i$  is derived from the Qwen2.5-3B word embedding model.
+
+For a fair comparison, all the above temporal compression techniques apply different temporal compression strategies with the same compression ratio of  $\frac{1}{4}$  on 1-FPS input videos, along with a random spatial compression with a compression ratio of  $\frac{1}{4}$ . Note that for training, we use a dataset composed of 70K video-language instruction samples with a 3:1 mixture of VideoMarathon and LLaVA-Video-178K, along with 60K multimodal samples from LLaVA-OV. The Hour-LLaVA-3B model is initialized from the checkpoint obtained after image-language pretraining. All training hyperparameters follow the video instruction tuning setup for the 3B model.
+
+For MemAug module, we adopt the same training data and recipe as mentioned for ablation studies on forgetting mechanisms. All training hyperparameters are consistent with the video instruction tuning configuration of the 3B model in Table 8.
+
+For memory repository, we mainly analyze the impact of the memory repository scale in Figure 4 (middle). The  $100\%$  scale refers to storing full video tokens sampled at 1 FPS. For the  $50\%$  and  $25\%$  scales, we uniformly retain  $50\%$  and  $25\%$  of the frame-level features along the temporal dimension, respectively. The  $< 10\%$  scale denotes a lightweight configuration in which only decayed video tokens are replaced in the memory repository, resulting in fewer than  $10\%$  of the full video token count. For the training, we follow the same setup as mentioned above. All training hyperparameters are consistent with the video instruction tuning configuration of the 3B model in Table 8.
+
+# A.4 Additional Empirical Analysis
+
+Impact of 1D Structured RoPE. In MemAug module, we adopt 1D structured RoPE to associate the spatial-temporal correspondence between the full video tokens in the memory repository and the decayed tokens, following prior works [16, 49]. We conduct ablation study on the 1D structured RoPE on temporal reasoning subset of VideoMME dataset, as shown in Table 9. Results show a substantial performance drop on the temporal reasoning task when this structural RoPE is removed. It demonstrates that constructing the spatial-temporal association between the full video tokens and decayed video tokens is necessary.
+
+Table 9: Effect of 1D structure RoPE within the MemAug module.  
+
+<table><tr><td>Method</td><td>1D structure RoPE</td><td>VideoMME (Temporal Reasoning)</td></tr><tr><td rowspan="2">Hour-LLaVA-3B</td><td>✓</td><td>46.3</td></tr><tr><td>×</td><td>35.6 (-10.7)</td></tr><tr><td rowspan="2">Hour-LLaVA-7B</td><td>✓</td><td>56.5</td></tr><tr><td>×</td><td>38.4 (-12.1)</td></tr></table>
+
+Performance on Free-form Generation Tasks. We evaluate Hour-LLaVA-3B and Hour-LLaVA-7B on MLVU-Dev [71] with free-form generation tasks, including sub-science captioning (SSC) and video summarization (SUM). The generated results are measured by GPT-4-Turbo with carefully designed instructions, where the rating score is on a 2–10 scale. We also report results from other state-of-the-art Video-LMMs for reference. In Table 10, the results demonstrate the strong capability of Hour-LLaVA on free-form generation tasks.
+
+Table 10: Performance on free-form generation tasks (MLVU-Dev) evaluated by GPT-4-Turbo. SSC: Sub-Scene Captioning; SUM: Video Summarization.  
+
+<table><tr><td>Method</td><td>Overall</td><td>SSC</td><td>SUM</td></tr><tr><td>MovieChat</td><td>2.78</td><td>3.23</td><td>2.33</td></tr><tr><td>LLaVA-1.6</td><td>3.23</td><td>4.35</td><td>2.11</td></tr><tr><td>ShareGPT4Video-7B</td><td>3.77</td><td>5.02</td><td>2.52</td></tr><tr><td>LongVA-7B</td><td>4.33</td><td>5.26</td><td>3.39</td></tr><tr><td>Video-XL-7B</td><td>4.50</td><td>-</td><td>-</td></tr><tr><td>Hour-LLaVA-3B</td><td>4.48</td><td>5.04</td><td>3.91</td></tr><tr><td>Hour-LLaVA-7B</td><td>4.60</td><td>5.22</td><td>3.97</td></tr></table>
+
+Comparison with recent advanced Video-LMMs. In Table 11, we present a comparison with the most recent advanced Video-LMMs, including Qwen2-VL [54], Qwen2.5-VL [3], InternVL3 [73], and InternVideo2.5 [56]. Compared among 3B-LMMs, Hour-LLaVA-3B substantially enhances its pretrained model (LLaVA-OV-SI-3B) and is generally comparable to its strong peer Qwen2.5-VL-3B. Compared with the 7B-LMMs, Hour-LLaVA-7B markedly enhances its pretrained model (LLaVA-OV-SI-7B) and surpasses Qwen2-VL-7B. It also performs comparably with, and sometimes outperforms, state-of-the-art models built on the stronger Qwen-2.5-7B LLM decoder.
+
+# A.5 Limitations
+
+Despite the promising results of Hour-LLaVA, several limitations remain. First, due to the lack of comprehensive evaluation metrics for hour-long video-language understanding, multiple-choice QA remains the most practical task for evaluating long video-language models. However, this evaluation format is limited in scope and fails to assess the broader capabilities of Video-LMMs. The development of more diverse and holistic benchmarks is therefore essential for advancing this field. Second, Hour-LLaVA is trained on large-scale synthetic instruction-following data, which inevitably contains noise. The current training pipeline does not explicitly consider this issue, and future work can further explore noise-robust training strategies. Third, the current framework is limited to video and language modalities, neglecting audio, a crucial component in many long-form videos such as lectures, interviews, and documentaries. Incorporating audio or additional modalities could further enhance the model's capacity for comprehensive multimodal understanding.
+
+Table 11: Performance comparison of existing LMMs on LongVideoBench, VideoMME, and LVBench datasets. M-Avg denotes the average performance of multiple-choice tasks. The symbol  $\dagger$  marks our re-implemented results. Bold font denotes the best performance among models at the same scale, while underline indicates the second-best.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">LongVideoBench M-Avg</td><td colspan="3">VideoMME (w/o &amp; w/ subtitles)</td><td rowspan="2">LVBench Avg</td></tr><tr><td>Overall</td><td>Medium</td><td>Long</td></tr><tr><td colspan="6">LLM Decoder: Qwen2.5-3B</td></tr><tr><td>\( LLaVA-OV-SI-3B^† \)[26]</td><td>49.6</td><td>51.1/54.5</td><td>49.2/52.1</td><td>44.1/46.4</td><td>35.4</td></tr><tr><td>Qwen2.5-VL-3B [3]</td><td>54.2</td><td>61.5/67.6</td><td>\( 58.7^\dagger/66.3^\dagger \)</td><td>\( 51.9^\dagger/61.6^\dagger \)</td><td>43.3</td></tr><tr><td>Hour-LLaVA-3B (ours)</td><td>57.8</td><td>60.6/66.7</td><td>59.0/65.4</td><td>52.1/60.4</td><td>44.7</td></tr><tr><td colspan="6">LLM Decoder: Qwen2-7B</td></tr><tr><td>\( LLaVA-OV-SI-7B \)[26]</td><td>\( 52.1^\dagger \)</td><td>58.2/61.5</td><td>\( 52.6^\dagger/55.4^\dagger \)</td><td>\( 47.9^\dagger/50.2^\dagger \)</td><td>\( 36.0^\dagger \)</td></tr><tr><td>Qwen2-VL-7B [54]</td><td>54.6</td><td>63.3/69.0</td><td>\( 60.6^\dagger/67.8^\dagger \)</td><td>\( 50.2^\dagger/62.7^\dagger \)</td><td>40.0</td></tr><tr><td>Hour-LLaVA-7B (ours)</td><td>60.4</td><td>63.6/70.2</td><td>63.8/70.0</td><td>55.0/65.1</td><td>45.6</td></tr><tr><td colspan="6">LLM Decoder: Qwen2.5-7B</td></tr><tr><td>Qwen2.5-VL-7B [3]</td><td>56.0</td><td>63.3/69.0</td><td>\( 62.7^\dagger/70.7^\dagger \)</td><td>\( 53.2^\dagger/64.4^\dagger \)</td><td>45.3</td></tr><tr><td>InternVL3-8B [73]</td><td>58.8</td><td>66.3/68.9</td><td>-/-</td><td>-/-</td><td>40.0</td></tr><tr><td>InternVideo2.5-8B [56]</td><td>60.6</td><td>65.1/-</td><td>-/-</td><td>-/-</td><td>46.4</td></tr></table>
+
+# A.6 Broader Impacts
+
+This study presents significant advancements in long-form video-language modeling through the introduction of the VideoMarathon dataset and the Hour-LLaVA model. Positively, the VideoMarathon dataset paves the way for more sophisticated AI systems capable of handling realistic, long-duration scenarios, which are essential for practical applications in education, security, autonomous driving, and augmented or virtual reality. However, potential negative impacts include the risk of misuse in surveillance, such as continuous monitoring and profiling of individuals in public or private settings without consent, and the possibility of misinterpreting nuanced or sensitive content in long videos, which might lead to harmful decisions in critical domains like healthcare or security. These implications highlight the importance of responsible development practices, including robust privacy safeguards and clear ethical guidelines for the deployment of long-form video-language models.
+
+# Clip-level Video Captioning
+
+You are given a 30-second video clip. Your task is to generate a detailed, structured description of the video based on six specific perspectives.
+
+# Instructions:
+
+Please ensure your descriptions are vivid, precise, and rich in detail. For each of the six topics below, please provide a comprehensive caption. Your goal is to clearly convey the video's visual and contextual content from multiple dimensions:
+
+- Temporality: Describe how the scene evolves over time. Highlight key transitions, unfolding actions, or changes from the beginning to the end of the video.  
+- Spatiality: Describe the spatial layout of the scene. Explain how key elements are positioned and oriented, and how they relate to one another within the frame.  
+- Object: Identify key objects in the video, including inanimate items and humans. Describe their appearance, clothing, physical traits, materials, and possible roles.  
+- **Action:** Describe the main actions taking place. Specify what actions occur, who or what performs them, and the manner or sequence in which they unfold.  
+- Scene: Provide a high-level overview of the setting. Describe the environment, background, and general activity or context presented in the video.  
+- Summary: Offer a brief yet comprehensive summary that contains the core event or purpose of the given video clip.
+
+# Output Format (JSON):
+
+Your response should be formatted as a JSON object, where each key corresponds to a topic and each value is the description associated with that topic.
+
+# Example Output:
+
+```jsonl
+"Temporality": "The video begins with the group standing idle under the tree and progresses to active conversation and movement, suggesting preparation for an activity.", "Spatiality": "The individuals are grouped under a large tree, spaced out in a semicircle. The surrounding area is an open, grassy park with scattered trees in the background.", "Object": "The scene includes several casually dressed men wearing t-shirts, shorts, hats, and sunglasses. One man carries a medium-sized cardboard box. The background features natural objects like trees, grass, and park benches.", "Action": "The men engage in conversation, gesturing with their hands, and preparing for an activity. One man walks away, possibly to retrieve additional items.", "Scene": "The video captures a casual outdoor setting in a park. A small group of men gather under a shady tree, seemingly preparing for a social or recreational activity in a relaxed, natural environment.", "Summary": "A group of men gather under a tree in a public park, casually conversing and preparing for an activity. The video captures a moment of calm interaction and coordination in a relaxed outdoor setting."
+}
+```
+
+Figure 5: The prompt for clip-level video captioning.
+
+# Event Splitting
+
+You will be given a list of video clips, where each clip is defined by a start time, end time, and its video content. Your task is to merge related clips into a small number of coherent events, each represented by a merged time span and an event title.
+
+# Instructions:
+
+Consider the context for each video clip and ignore some outliers or unreasonable video content. If the given video clips are already an event, do not merge.
+
+- Merge Related Clips: Combine consecutive clips that logically belong to the same event based on content continuity.  
+- Preserve Context: Each merged event should represent a self-contained and contextually consistent unit of action or activity.  
+- Filter Outliers: Ignore clips that are irrelevant, inconsistent with the surrounding context, or clearly out of place.  
+- Balance Granularity: Avoid over-segmenting the video. Aim to minimize the number of events while ensuring that each one captures a distinct scene, set of actions, or objects.  
+- Respect Standalone Clips: If a single clip already represents a complete, meaningful event, retain it without merging.
+
+# Output Format (JSON):
+
+Return a JSON object where:
+
+Each key is a string denoting the merged time span of an event (e.g., "0-40s").  
+Each value is a concise event title that summarizes the main content or activity.
+
+# Example Output:
+
+```txt
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+``}
+```
+
+Figure 6: The prompt for event splitting.
+
+# Event-level Video Captioning
+
+You will be provided with a sequence of video clips depicting a specific event. Each clip includes metadata (start time, end time) and structured descriptions from six perspectives. Your task is to generate a cohesive, natural-language narrative that summarizes the entire event in chronological order. To support your understanding of the broader context, you will also receive brief descriptions of the events immediately before and after the current one. Use this surrounding context to enhance the flow and interpretation, but do not include those descriptions in your output.
+
+# Instructions:
+
+# 1. Maintain Chronological Flow
+
+- The clip-level video captions are already chronologically ordered. Your summary must preserve this timeline and describe the event as a continuous progression.  
+- Avoid explicitly referencing individual clips (e.g., "The first clip shows..." or "As the clip progresses..."). Instead, describe the event as a seamless and continuous narrative.  
+- Do not assume the first or last frame of any clip represents the beginning or end of the entire event. Instead, focus on how the event unfolds as a whole.
+
+# 2. Use Adjacent Event Context Thoughtfully
+
+- To improve coherence and contextual understanding, you will receive brief descriptions of the events that occur immediately before and after the current event.  
+- Use this information to better understand the current event.  
+- IMPORTANT: Do not include these descriptions in your summary. They are provided solely to help you maintain context and continuity.
+
+# 3. Preserve Key Details, Filter Outliers
+
+- Retain all relevant details from the clip descriptions to ensure accuracy and completeness.  
+- Based on the brief description of the current event, ignore outliers or clips with inconsistent, irrelevant, or contradictory content that do not contribute meaningfully to the event-level narrative.
+
+# 4. Write in a Natural, Engaging Tone
+
+- Your summary should read like a natural video description, as if you are directly describing the event rather than summarizing segmented clips.  
+- Avoid mechanical phrases often found in clip descriptions, such as "The clip begins...", "As the clip progresses...", "The clip concludes...", "The first/last frame of this clip...", "The second clip shows...".  
+Instead, focus on crafting a flowing narrative. The goal is to help a reader visualize the full event as if watching it unfold.
+
+# Output Format (JSON):
+
+The output should be structured as a JSON object.
+
+# Example Output:
+
+```json
+```
+json {
+    "Event-Level Description": "YOUR DESCRIPTION HERE."
+}
+```
+
+# Input:
+
+- Brief description of current event: <EVENT TITLE>  
+The event before the current event: <PREVENT DESCRIPTION>  
+- The event after the current event: <NEXT EVENT DESCRIPTION>  
+Detailed descriptions of all clips in the current event:
+
+- <CLIP-LEVEL DESCRIPTION 1>  
+- <CLIP-LEVEL DESCRIPTION 2>  
+.  
+- <CLIP-LEVEL DESCRIPTION N>
+
+Figure 7: The prompt for event-level video captioning.
+
+# Global-level Video Captioning
+
+You will be provided with a chronologically ordered sequence of video events, each accompanied by both a brief and a detailed description. Your task is to synthesize these descriptions into a single, cohesive, and vivid narrative that summarizes the entire sequence of events as a unified whole, without breaking the flow into separate segments or referencing individual clips mechanically.
+
+Instructions:
+
+1.Event-level Video Descriptions
+
+Each event is described with two levels of granularity:
+
+- Brief Description: A concise overview of the event.  
+- Detailed Description: A more in-depth description, including finer details.
+
+2. Generate a Unified Narrative
+
+- Write a single, flowing narrative that describes the full sequence of events from beginning to end.  
+- Maintain chronological order without explicitly mentioning timestamps or referencing individual events (e.g., "In the second event...").  
+- Ensure the narrative reads as if you are describing a continuous experience, not a series of separate parts.
+
+3. Focus on Relevance and Consistency
+
+- Incorporate all meaningful and consistent details across the event descriptions.  
+- If a detail appears inconsistent, irrelevant, or clearly unrelated, omit it based on your understanding of the overall narrative.
+
+4.Use a Natural and Engaging Tone
+
+- Write in a fluent, descriptive style, suitable for someone reading or hearing a natural summary of the video.  
+- Avoid mechanical phrases like: "The event begins...", "As the event progresses...", "The first/last event shows...".  
+Instead, immerse the reader in the experience, emphasizing continuity, clarity, and engagement.
+
+++++ Output Format (JSON):
+
+The output should be structured as a JSON object.
+
+Example Output:
+
+```txt
+```
+```
+```
+```
+```
+```
+```
+``
+```
+
+Input:
+
+Event from <START TIME 1> - <END TIME 1>:
+
+- Brief description: <EVENT TITLE 1>  
+Detailed description: <EVENT-LEVEL DESCRIPTION 1>
+
+Event from <START TIME 2> - <END TIME 2>:
+
+- Brief description: <EVENT TITLE 2>  
+Detailed description: <EVENT-LEVEL DESCRIPTION 2>
+
+···
+
+Event from <START TIME N> - <END TIME N>:
+
+- Brief description: <EVENT TITLE N>  
+Detailed description: <EVENT-LEVEL DESCRIPTION N>
+
+Figure 8: The prompt for global-level video captioning.
+
+# Open-Ended QA Generation
+
+You are an intelligent assistant specializing in open-ended question-answer generation for video understanding. Please follow the instructions precisely and use only the information provided in the input. Do not introduce any content unrelated to the described video clips.
+
+You will be provided with:
+
+- A chronologically ordered sequence of <TOPIC> - based video clip descriptions (each 30 seconds long, with start and end timestamps).  
+- A global-level description summarizing the overall content of the video.
+
+Your task is to generate open-ended Question-Answer (QA) pairs from the perspective of <TOPIC>. These QA pairs should promote deep understanding and must be strictly grounded in the given descriptions. No hallucination or fabrication is allowed.
+
+Instructions:
+
+####1. <TOPIC>-Based Sub-Tasks
+
+The<TOPIC>-based sub-tasks can be categorized into the following sub-tasks:
+
+- <SUB-TASK 1>: <TASK DESCRIPTION 1>  
+- <SUB-TASK 2>: <TASK DESCRIPTION 2>  
+...additional sub-tasks as needed)
+
+2. QA Examples for Each Sub-Task
+
+To guide your generation, here are example QA pairs for each sub-task:
+
+Examples of <SUB-TASK 1>:
+
+```json
+[ {"question":<DEMO Q1-1>,"answer":<DEMO A1-1>}, {"question":<DEMO Q1-2>,"answer":<DEMO A1-2>}, ... ] Examples of <SUB-TASK 2>: [ {"question":<DEMO Q2-1>,"answer":<DEMO A2-1>}, {"question":<DEMO Q2-2>,"answer":<DEMO A2-2>}, ... ]
+```
+
+3. Guidelines for Question-Answer Generation:
+
+- Focus on <TOPIC>--Relevant Information: Carefully analyze the descriptions to identify patterns relevant to the <TOPIC>.  
+- Relevance and Context: The questions and answers must align with the content and context of the video clips. The generated question-answer pairs should not introduce information that is not present in the description.  
+- Balance Diversity and Clarity: Create a variety of questions that collectively capture a full understanding of the topic.  
+- Quantity: Generate exactly three QA pairs for each sub-task.
+
+Output Format (JSON):
+
+The output should be structured as a JSON object.
+
+Example Output:
+
+```snap
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+``
+```
+
+Input:
+
+- <TOPIC>--based Descriptions:
+
+- Clip from <START TIME 1> - <END TIME 1>: <TOPIC-SPECIFIC CLIP DESCRIPTION 1>  
+- Clip from <START TIME 2> - <END TIME 2>: <TOPIC-SPECIFIC CLIP DESCRIPTION 2>
+
+.
+
+- Overall Description: <GLOBAL-LEVEL DESCRIPTION>
+
+Figure 9: The prompt for open-ended (OE) question-answer generation.
+
+# Multiple-Choice QA Generation
+
+You are an intelligent assistant specializing in multiple-choice question-answer generation for video understanding. Please follow the instructions precisely and use only the information provided in the input. Do not introduce any content unrelated to the described video clips.
+
+You will be provided with:
+
+- A chronologically ordered sequence of <TOPIC> - based video clip descriptions (each 30 seconds long, with start and end timestamps).  
+- A global-level description summarizing the overall content of the video.
+
+Your task is to generate multiple-choice Question-Option-Answer triplets from the perspective of <TOPIC>. These triplets should promote deep understanding and must be strictly grounded in the given descriptions. No hallucination or fabrication is allowed.
+
+Instructions:
+
+####1. <TOPIC>-Based Sub-Tasks
+
+The  $<\mathrm{TOPIC}>$ -based sub-tasks can be categorized into the following sub-tasks:
+
+- <SUB-TASK 1>: <TASK DESCRIPTION 1>  
+- <SUB-TASK 2>: <TASK DESCRIPTION 2>  
+...additional sub-tasks as needed)
+
+##2 QA Examples for Each Sub-Task
+
+To guide your generation, here are example QA pairs for each sub-task:
+
+Examples of <SUB-TASK 1>:
+
+```json
+[ {"question":<DEMO Q1-1>,"options":<DEMO OP1-1>, "answer":<DEMO A1-1>}, {"question":<DEMO Q1-2>,"options":<DEMO OP1-2>, "answer":<DEMO A1-2>}, ... ] Examples of <SUB-TASK 2>: [ {"question":<DEMO Q2-1>,"options":<DEMO OP2-1>, "answer":<DEMO A2-1>}, {"question":<DEMO Q2-2>,"options":<DEMO OP2-2>, "answer":<DEMO A2-2>}, ... ]
+```
+
+3. Guidelines for Question-Answer Generation:
+
+- Focus on <TOPIC>--Relevant Information: Carefully analyze the descriptions to identify patterns relevant to the <TOPIC>.  
+- Relevance and Context: The questions, options, and answers must align with the content and context of the video clips. The generated question-option-answer triplets should not introduce information that is not present in the description.  
+- Balance Diversity and Clarity: Create a variety of questions that collectively capture a full understanding of the topic.  
+- Quantity: Generate exactly three QA pairs for each sub-task.
+
+Output Format (JSON):
+
+The output should be structured as a JSON object.
+
+Example Output:
+
+```snap
+```
+```
+```
+```
+```
+```
+```
+```
+```
+```
+``
+```
+
+Input:
+
+- <TOPIC>-based Descriptions:
+
+- Clip from <START TIME 1> - <END TIME 1>: <TOPIC-SPECIFIC CLIP DESCRIPTION 1>  
+- Clip from <START TIME 2> - <END TIME 2>: <TOPIC-SPECIFIC CLIP DESCRIPTION 2>
+
+- Overall Description: <GLOBAL-LEVEL DESCRIPTION>
+
+Figure 10: The prompt for multiple-choice (MC) question-answer generation.
+
+# Temporality
+
+- Temporal Perception: Focuses on recognizing and interpreting the temporal flow of events within a video, including their sequencing, relative timing, and transitions between scenes. The task emphasizes understanding the local context of temporal configurations.  
+- Duration Perception: Involves estimating the length of individual events and comparing the durations of multiple actions. The objective is to determine which activities are longer, shorter, or equal in duration. The task emphasizes understanding the global context of temporal configurations.  
+- Temporal Reasoning: Requires logical inference based on temporal structure. This includes understanding causal relationships, predicting upcoming events, and reasoning about dependencies between temporally related actions. The task emphasizes understanding the global context of temporal configurations.  
+- Temporal Gap Estimation: Entails estimating the elapsed time between two events. The goal is to infer the most accurate time gap based on contextual clues within the video. The task emphasizes understanding the global context of temporal configurations.  
+- Temporal Order: Focuses on identifying the correct chronological sequence of events. This task assesses the ability to determine the order in which actions or visual elements appear throughout the video. The task emphasizes understanding the global context of temporal configurations.
+
+# Spatiality
+
+- Spatial Perception: Focuses on recognizing and interpreting the spatial relationships among objects, people, and movements within a scene. This includes identifying directions, positions, orientations, and local arrangements. The task emphasizes understanding the local context of spatial configurations.  
+- Spatial Reasoning: Involves higher-level inference about spatial structures, such as deducing the relative positions, movements, and locations of entities within broader environments. This task requires interpreting interactions across scenes and relies on understanding the global context of spatial relationships.
+
+# Object
+
+- Object Recognition: Involves identifying and naming specific objects, products, or items that appear in a video. The task also includes providing contextual information about the recognized objects. This task primarily relies on understanding the local context.  
+- Object Interaction: Assesses the ability to describe interactions between objects within a scene—how they are used, manipulated, or influence each other. This task emphasizes the local context of object dynamics.  
+- Object Attribute: Focuses on describing the visual or functional characteristics of objects, such as appearance, material, color, or size. This task also depends on local context understanding.  
+- Object Reasoning: Involves higher-level inference about objects in the scene, such as their roles, functions, or contextual significance. This requires a global context understanding across the video.  
+- Object Existence: Evaluates the ability to determine the presence or absence of specific objects at various points in the video. This task requires reasoning over global context.  
+- Object Permanence: Assesses the understanding of an object's continuity over time, including its movement, transformation, or disappearance. This task tests the model's ability to track and reason about objects in a global context.
+
+# Action
+
+- Action Recognition: Focuses on identifying and describing specific actions performed by subjects at given moments in the video. This task requires detailed interpretation of movements and activities within the local context.  
+- Action Localization: Aims to determine the temporal and spatial boundaries of actions. It involves identifying when and where actions occur, including their start and end times, and comparing durations. This task relies on understanding the global context.  
+- Action Sequence: Involves analyzing the chronological order of actions. The task requires recognizing the sequence of events, including what occurs before or after a given action, and understanding the progression over time. This task requires global context reasoning.  
+- Action Reasoning: Requires inferring the motivations or causes behind observed actions. This includes identifying cause-and-effect relationships and interpreting contextual cues to explain why an action was taken. This task depends on global context comprehension.
+
+# Scene
+
+- Scene Recognition: Assesses the ability to identify and describe the primary setting or environment in which the video takes place (e.g., indoor vs. outdoor, kitchen vs. park). This task involves understanding where the events occur and the general atmosphere surrounding the characters. It relies on global context comprehension.  
+- Scene Transition: Involves detecting and analyzing changes in scene or setting throughout the video. The task requires recognizing shifts in environment, lighting, background, or thematic focus, and describing these transitions accurately. This task also depends on global context understanding.
+
+# Event
+
+- Event Sequencing: Involves identifying the correct chronological order of events. This task requires recognizing the sequence in which actions unfold and determining prerequisite steps that lead to specific outcomes. It depends on a coherent understanding of the global context.  
+Key Events Extraction: Focuses on summarizing the most important actions and interactions within a video. The task entails identifying pivotal moments, evaluating their significance, and extracting overarching themes that define the event flow. This also requires comprehensive global context comprehension.
+
+Figure 11: Detailed descriptions of 22 sub-tasks over six fundamental topics (except "overall summarization" task). For the "overall summarization" task, we directly use global-level video descriptions as the corresponding answers, so there is no need to design an additional prompt.
+
+![](images/9e622315a703a8c0c5dc8b087c1c59ed55eac9e5298af7331be19abba6c57009.jpg)  
+Figure 12: Examples of hierarchical video captioning.
+
+# "20-50s": {
+
+"Temporality": "The video progresses from the woman introducing the recipe to showing the ingredients and possibly demonstrating the steps involved.",
+
+"Spatiality": "The woman is seated in front of the table, which is positioned against a plain wall. The ingredients are arranged neatly on the table.",
+
+"Object": "The woman wears a black top and a choker necklace with a pendant. On the table, there are three bowls containing sugar, flour, and butter, an egg, and a glass bowl."
+
+"Action": "The woman speaks to the camera, gestures with her hands, and points to the ingredients on the table. She appears to be explaining the recipe.",
+
+"Scene": "The video features a woman sitting in a chair, discussing a recipe while showing various ingredients laid out on a table.",
+
+"Summary": "This video showcases a woman presenting a homemade chocolate salted caramel tart recipe. She explains the ingredients and their arrangement on the table, providing a step-by-step guide for viewers to follow along."
+
+}
+
+# (a) Examples of clip-level video captioning.
+
+"0-70s": "Introduction and Overview of Chocolate Salted Caramel Tart Recipe",
+
+"60-130s": "Preparing and Kneading the Dough",  
+"120-190s": "Rolling and Shaping the Tart Crust",  
+"180-270s": "Creating the Caramel Sauce and Assembling the Tart",  
+"280-350s": "Finalizing and Presenting the Chocolate Salted Caramel Tart"
+
+# (b) Examples of event splitting
+
+# "120-190s": {
+
+"Brief Description": "Rolling and Shaping the Tart Crust",
+
+"Detailed Description": "In a cozy kitchen setting, a person meticulously prepares a tart crust. The process begins with a ball of dough on a wooden surface, surrounded by a generous sprinkling of flour. The individual, dressed in a dark-colored shirt, kneads the dough by hand, ensuring it is well-mixed and pliable. Once the dough is ready, they use a traditional rolling pin to flatten it, moving the pin back and forth to achieve an even thickness. The rolling pin's rhythmic motion is a testament to the baker's skill and experience. As the dough is rolled out, it is carefully transferred to a round tart pan, which is lined with parchment paper to prevent sticking. The person then shapes the dough into the pan, pressing it gently into the sides and bottom to create a smooth and even crust. Excess dough is trimmed away, and the final touches are added to ensure the tart base is perfect. The entire process is carried out with a focused and meticulous approach, highlighting the importance of precision in baking. The scene is a blend of traditional techniques and modern kitchen tools, creating a harmonious and engaging visual of the tart crust preparation."
+
+}
+
+# (c) Examples of event-level video captioning.
+
+The video begins with a warm and inviting introduction to a homemade chocolate salted caramel tart recipe. A woman, dressed in a black top and wearing a choker necklace with a pendant, sits in front of a table adorned with colorful candies spelling out 'THE SWEETEST THINGS.' She speaks directly to the camera, gesturing with her hands in a friendly and engaging manner. The scene transitions to her discussing the recipe, pointing out the ingredients laid out on the table: sugar, flour, butter, an egg, and a glass bowl. The setting appears to be a kitchen or dining area, bathed in natural daylight, creating a cozy atmosphere. The woman stands in front of a wooden table, where she begins to mix the ingredients for the tart. Using a fork, she combines the flour, butter, sugar, and egg methodically, demonstrating each step clearly and encouraging viewers to follow along. As the video progresses, the focus shifts to the preparation and kneading of the dough. The woman, now in a slightly messy kitchen environment, works on a wooden table set with all the necessary ingredients. She starts by mixing the flour and butter with a fork, ensuring a thorough blend. After adding an egg and a small amount of water, she continues to mix the dough until it reaches the right consistency. Her hands, now covered in flour, demonstrate a practiced technique as she kneads the dough on the wooden surface, shaping it into a ball. The scene highlights her attention to cleanliness and precision, setting the stage for the next step in the recipe.
+
+The process of rolling and shaping the tart crust follows. The woman, dressed in a dark-colored shirt, works with the dough on a floured wooden surface. She uses a traditional rolling pin to flatten the dough, moving it back and forth to achieve an even thickness. The dough is then carefully transferred to a round tart pan lined with parchment paper. She presses the dough gently into the sides and bottom of the pan, ensuring a smooth and even crust. Excess dough is trimmed away, and the final touches are added to perfect the tart base. The scene showcases a blend of traditional techniques and modern kitchen tools, emphasizing the importance of precision in baking.
+
+Next, the video transitions to the creation of the caramel sauce and the assembly of the tart. The woman places parchment paper into the tart pan and adds rice to weigh it down, ensuring an even distribution of weight. After removing the parchment paper, she applies an egg wash to the pre-baked tart shell. She then prepares the caramel sauce by heating sugar and water in a pot, stirring the mixture with a wooden spoon. Cream and butter are added to create a rich, smooth caramel sauce, which is spread evenly over the tart shell. Dark chocolate chunks are added for flavor and texture, setting the stage for the final steps in the recipe.
+
+The video concludes with the finalization and presentation of the chocolate salted caramel tart. A close-up shows a bowl of melted chocolate being poured into the tart shell, which is placed on a black slate board. The chocolate is spread evenly using a spoon, and the tart is then moved to a wooden cutting board for presentation. Sea salt is sprinkled on top, adding a touch of contrast and enhancing the tart's flavor. The tart is cut into triangular slices with a steady and deliberate technique, ready to be served. The entire process highlights the simplicity and elegance of the chocolate salted caramel tart, from its preparation to its final presentation, leaving viewers inspired to try the recipe themselves.
+
+# (d) Examples of global-level video captioning.
+
+![](images/de3d1be1f774ff27dc8e1252dba7d3ef1ceaef6752c74d8c42fd6b4109d5c847.jpg)  
+Figure 13: Word cloud of global-level video descriptions in VideoMarathon dataset.
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: Please check it in the abstract and Section 1.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: See supplementary materials.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [NA]
+
+Justification: This paper does not include theoretical results.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: Implementation details are described in Section 5. Additional experimental details are shown in Appendix A. URLs to data and code are attached in the supplementary materials.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: URLs to data and code are attached in the supplementary materials.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: See Section 5. Additional details are shown in Appendix A.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [No]
+
+Justification: Due to the substantial training cost of Video-LLMs, we do not report error bars in this paper.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: See Section 5.
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: This paper conforms, in every respect, with the NeurIPS Code of Ethics.
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: See Appendix A.
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: This paper poses no such risks.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: See Section 5 and Reference.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [Yes]
+
+Justification: See supplementary materials.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: This paper does not involve crowdsourcing nor research with human subjects.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: This paper does not involve crowdsourcing nor research with human subjects.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [Yes]
+
+Justification: This paper describes the usage of LLMs for synthetic data generation for video-language understanding in Section 3. In addition, more usage details are described in the supplementary materials.
\ No newline at end of file
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+# Unleashing the Potential of Multimodal LLMs for Zero-Shot Spatio-Temporal Video Grounding
+
+Zaiquan Yang Yuhao Liu † Gerhard Hancke Rynson W.H. Lau†
+
+Department of Computer Science
+
+City University of Hong Kong
+
+{zaiquyang2-c，yuhliu9-c}@my.cityu.edu.hk
+
+{gp.hancke, Rynson.Lau}@cityu.edu.hk
+
+# Abstract
+
+Spatio-temporal video grounding (STVG) aims at localizing the spatio-temporal tube of a video, as specified by the input text query. In this paper, we utilize multimodal large language models (MLLMs) to explore a zero-shot solution in STVG. We reveal two key insights about MLLMs: (1) MLLMs tend to dynamically assign special tokens, referred to as grounding tokens, for grounding the text query; and (2) MLLMs often suffer from suboptimal grounding due to the inability to fully integrate the cues in the text query (e.g., attributes, actions) for inference. Based on these insights, we propose a MLLM-based zero-shot framework for STVG, which includes novel decomposed spatio-temporal highlighting (DSTH) and temporal-augmented assembling (TAS) strategies to unleash the reasoning ability of MLLMs. The DSTH strategy first decouples the original query into attribute and action sub-queries for inquiring the existence of the target both spatially and temporally. It then uses a novel logit-guided re-attention (LRA) module to learn latent variables as spatial and temporal prompts, by regularizing token predictions for each sub-query. These prompts highlight attribute and action cues, respectively, directing the model's attention to reliable spatial and temporal related visual regions. In addition, as the spatial grounding by the attribute sub-query should be temporally consistent, we introduce the TAS strategy to assemble the predictions using the original video frames and the temporal-augmented frames as inputs to help improve temporal consistency. We evaluate our method on various MLLMs, and show that it outperforms SOTA methods on three common STVG benchmarks. The code will be available at https://github.com/zaiquanyang/LLaVA_Next_STVG.
+
+# 1 Introduction
+
+Spatio-Temporal Video Grounding (STVG) aims to localize a target object in a video both spatially and temporally, given an input text query. This task is fundamental to many different applications (e.g., video surveillance and autonomous driving [67]). However, it is also very challenging as it requires the model to be able to distinguish the target from distractors over time and identify the precise temporal boundary of the action. While existing methods handle the STVG task mainly in a fully-supervised setting [13; 59; 54], which often relies on costly frame-level annotations, several works [29; 3; 61; 21; 60; 25] attempt to introduce the weakly-supervised or zero-shot setting to alleviate the burden of dense annotations. For example, E3M [3] integrates CLIP [43] and an expectation maximization strategy to optimize spatio-temporal localization in a zero-shot manner. However, CLIP is known to be weak in localization [75; 14] as it simply aligns the global representation of image-text pairs.
+
+![](images/5c7f5e5f7020cfe91a907f4986633b38fbf40ffbc02cdf9037b81083617e6caa.jpg)  
+(a)  
+Figure 1: (a) The visual attention maps shows that some special tokens (marked as  $\square$ ) can precisely attend to the target region of the input query. However, these special tokens, referred to as the grounding tokens in our work, underperform on complex STVG, where they often focus on part cues and ignore other cues (marked in red within the input text prompts). Examples (b) and (c) illustrate spatial/temporal grounding errors caused by ignoring the discriminative attribute/action cues. Red and green bounding boxes denote the ground truths and predictions, respectively.
+
+Considering the strong capability of multi-modal large language models (MLLMs) [37; 27; 31; 8; 53; 36] in cross-modality alignment, several works explore the application of MLLMs in the visual grounding task. However, they typically require explicit fine-tuning [26; 44; 39] of MLLMs with additional grounding datasets and specific model modifications, which can be difficult to scale and generalize to novel visual data [4]. Although some recent works [4; 69] have investigated the attention maps in language models as done in previous ViT and CNN architectures [74; 52; 12], they only focus on the generated tokens while neglecting other token components input in the language model (e.g., system tokens and special tokens). In particular, we observe that the special tokens play an important role in structuring the communication between the user input and the language model, and help guide the generation of coherent responses in dialogues.
+
+With the above observation in mind, we delve deeper and find that the special tokens following the input instruction have outstanding grounding ability. In particular, a few special tokens are characterized by high visual activation and can attend to the region of interest well. Considering the left sample of Fig. 1(a) as an example, the special token ‘\_A’ provides tangible attention to the ‘boy’ referred to by the given query, while in the right sample of Fig. 1(a), the token ‘IST’ is assigned to ground the ‘elephant’. These special tokens are referred to as grounding tokens in our work. Despite the strong comprehension ability, the grounding tokens cannot optimally adapt to STVG as they tend to ignore some important cues (e.g., attribute or action) in a complex video query. As shown in Fig. 1(b), the grounding token fails in the spatial grounding by ignoring the attributes. Similarly, in Fig. 1(c), it leads to the failure of temporal grounding by neglecting the action cues of the target.
+
+In this work, we first conduct a systematic analysis to probe the special tokens of various MLLMs (Sec. 3.2), which demonstrates our aforementioned empirical findings and enables zero-shot spatiotemporal video grounding. To alleviate the problem of neglecting discriminative cues, we propose a novel decomposed spatio-temporal highlighting (DSTH) strategy (Sec. 3.3). Specifically, the language query is decomposed into attribute and action sub-queries, which are utilized as the text input of the MLLM for inquiring the target's existence spatially and temporally, respectively. We then propose a novel logit-guided re-attention (LRA) module to highlight the cues in attribute and action sub-queries. For each sub-query, LRA optimizes the learnable latent variable as the (spatial/temporal) prompts via enhancing the positive response generation while suppressing the negative response. As a result, the DSTH strategy can well adapt the model to mine faithful visual context and concentrate on spatially/temporally relevant regions. In addition, to further enhance the temporal consistency during spatial grounding by the attribute sub-query, we develop a temporal-augmented assembling (TAS) strategy (Sec. 3.4). The TAS strategy utilizes temporally perturbed frames as input to assemble different predictions into final spatial grounding result.
+
+In summary, our contributions are as follows:
+
+- We reveal that MLLMs dynamically assign the special tokens for precisely grounding text-related regions. We identify the special tokens characterized by the salient visual activation for deriving a novel zero-shot STVG framework.  
+- We propose a novel test-time tuning strategy named decomposed spatio-temporal highlighting (DSTH). It introduces an innovative logit-guided re-attention (LRA) module to adapt the grounding token for thorough spatio-temporal localization. We also develop a temporal-augmented assembling (TAS) strategy to further improve the robustness of spatial localization.  
+- We conduct extensive experiments on various MLLMs to demonstrate our empirical findings and validate the effectiveness of the proposed method. Our method outperforms existing methods by a remarkable margin on three STVG benchmarks.
+
+# 2 Related work
+
+Spatio-Temporal Video Grounding (STVG) aims to localize a spatio-temporal tube in a video corresponding to the text query. Unlike image-based visual grounding [10; 11; 63; 34; 62; 35], STVG [73; 51; 59] presents significant challenges, requiring models to distinguish targets from distractors both spatially and temporally. Fully-supervised STVG approaches [51; 54; 73; 49] have achieved promising results. However, these methods heavily depend on an extensive collection of labor-intensive annotations. Several recent works [29; 21; 25] tackle STVG in a more efficient manner, which only uses coarse video-level descriptions for training. E3M [3] leverages pre-trained vision-language models and proposes an expectation maximization framework to optimize spatiotemporal localization. However, contrastive objective based multimodal models are known to be weak in localization [75; 14] as they simply align the global representations of image-text pairs.
+
+Multimodal Large Language Models (MLLMs) are capable of handling diverse language and vision tasks. To equip MLLMs with the grounding ability, current works [41; 26; 28; 66; 71] construct grounding-oriented supervision data for instruction tuning and propose novel architectural modifications. LLaVA-ST [28] achieves spatio-temporal understanding by introducing a progressive training strategy consisting of three sequential stages. However, the large amount of grounding data needed for instruction-tuning imposes high labeling costs. In addition, changing the focus of MLLMs to grounding tasks can degrade the original dialog capabilities due to catastrophic forgetting [68]. Recent works [4; 19; 69] reveal the inherent perception ability of MLLMs obtained by general instruction-tuning. Unlike these works that only focus on the generated tokens or in our work, we delve deeper into the more token components and reveal that the MLLMs always dynamically assign the special tokens following the instruction prompt for attending to the regions of interest.
+
+Test-time Tuning (TTT) aims to optimize the inference of test samples online. With the progress of multimodal foundation models [43; 5]), TTT has attracted more attention [76; 40; 18; 48; 64] as it can learn effective prompts for test samples and well adapt the foundation model for zero-shot applications. TPT [48] pioneers the study on TTT by minimizing the prediction entropy between each test sample and its augmented views. HisTPT [70] explores memory learning for test-time prompt tuning by introducing the memory bank. However, the test prompt optimization for MLLMs is barely explored. The most relevant work to our work is ControlMLLM [56], which optimizes the attention map by taking the referring regions as supervision. With the lack of supervision, we propose to learn visual prompts by regularizing the token-level response to instruction input. Our work shows that we can rectify where text prompts attend to by altering the outputs of MLLMs.
+
+# 3 Method
+
+# 3.1 Preliminaries
+
+Task Formulation. Given an untrimmed video  $V = \{\mathrm{f}_t\}_{t=1}^{T_v}$  composed of  $T_v$  image frames and a sentence query  $Q$ , the goal of the STVG task is to localize the spatio-temporal tube of target  $O = \{b_t\}_{t=t_s}^{t_e}$  described by  $Q$ . Here  $b_t$  represents the bounding box of the target in the  $t$ -th frame,  $t_s$  and  $t_e$  specify the starting and ending boundaries of the target tubelet, respectively. The STVG task can be solved through two sub-tasks: spatial grounding and temporal grounding. In our work, we first derive a zero-shot solution for the STVG task by unleashing the strong cross-modal comprehension ability of MLLMs.
+
+The Setup of MLLMs. Current MLLMs typically consist of a visual encoder, a projector, and a large language model. Specifically, given an image-question pair  $(I,Q)$  as input, the image  $I$  is first projected into text-aligned visual tokens  $\mathrm{T}_v = \{\mathrm{v}_1,\dots ,\mathrm{v}_{\mathrm{M}}\}$  by visual encoder and projector, and the question  $Q$  is converted into text tokens  $\mathrm{T}_q = \{\mathrm{t}_1,\dots ,\mathrm{t}_{\mathrm{N_q}}\}$  by a text tokenizer and embedding layer. Here,  $\mathbf{M}$  and  $\mathrm{N_q}$  denote the numbers of visual tokens and question tokens, respectively. In practice, MLLMs also introduce system tokens  $\mathrm{T_{sys}}$  and some special tokens  $\mathrm{T_{role}}$  for instruction-following ability. In particular, the special tokens play an important role in structuring the well-organized conversation framework. In our work, the special tokens are positioned subsequent to the instruction prompts by the user. They help guide the generation of coherent responses. The number of special tokens  $\mathrm{N_{role}}$  often differs among different MLLMs. As a result, the language model receives concatenated tokens,  $\mathrm{T} = \{\mathrm{t}_1,\dots ,\mathrm{t}_{\mathrm{N_{sys}}};\mathrm{v}_1,\dots ,\mathrm{v}_{\mathrm{M}};\mathrm{t}_1,\dots ,\mathrm{t}_{\mathrm{N_q}};\mathrm{t}_1,\dots ,\mathrm{t}_{\mathrm{N_{role}}}\}$ , as input. Appendix 6.1 provides visual illustration about the input tokens in MLLMs.
+
+Text-to-visual Attention. The LLM in MLLMs typically processes the input tokens through L transformer blocks [37] with the multi-head attention (MHA) for interactions of different tokens. Particularly, the text-to-visual attention represents the relationships between the visual and the textual tokens. We can derive the text-to-visual attention matrix  $A \in \mathbb{R}^{\mathrm{L} \times \mathrm{H} \times \mathrm{N} \times \mathrm{N}}$ , where  $\mathrm{N} = \mathrm{N}_{\mathrm{sys}} + \mathrm{M} + \mathrm{N}_{\mathrm{q}} + \mathrm{N}_{\mathrm{role}}$ . L and H denote the numbers of layers and heads in the transformer. Our empirical observation from Fig. 1 is that the special tokens  $T_{\mathrm{role}}$  show outstanding grounding ability with a global comprehension. For the simplicity of the following analysis, we omit the affect of different layers and heads by the mean operation. We can then obtain the text-to-visual attention matrix,  $A_{\mathrm{role}} = [\mathrm{N}_{\mathrm{sys}} + \mathrm{M} + \mathrm{N}_{\mathrm{q}} : \mathrm{N}, \mathrm{N}_{\mathrm{sys}} : \mathrm{N}_{\mathrm{sys}} + \mathrm{M}]$ .
+
+# 3.2 Grounding Token Identification
+
+In this section, we conduct a pilot study to quantitatively analyze the grounding ability of the special tokens. Without loss of generality, we randomly select a subset of 1,000 imagetext pairs from the RefCOCOg [17] validation set for image MLLMs analysis, and a subset of 1,000 videotext pairs from the HC-STVGv2 [51] dataset for video MLLMs analysis. Particularly, we choose three typical MLLMs (i.e., LLaVA-1.5, Qwen-VL, Deepseek-VL) for the study on image input and three MLLMs (i.e., LLaVA-Next-Video, Qwen2-VL, LLaVA-OneVision) for the study on video input. We have two key findings from our studies.
+
+# MLLMs dynamically assign the spe
+
+![](images/be2d84b8298b52cbb32b44106001b7f4f0ead071f3e2156240c7a820355c3679.jpg)
+
+![](images/1f7c15338052a33d0f4c8e86e83cb7355acdae439306b7b532b4059c6b5f45e3.jpg)  
+Figure 2: In (a), the results show the frequency with which different special tokens (such as ‘_A’, ‘:’) have superior grounding ability than other tokens, i.e., hit_ratio. In (b), the results represent the grounding accuracy of tokens ranked by their visual activation degrees. For each MLLM, we select four tokens for visualization.
+
+cial tokens to attend to the text-related regions. To demonstrate the finding, we first define the attention ratio of each special token as the ratio of maximum attention within the ground-truth bounding box  $b_{gt}$  to that outside it. For example, the attention ratio of the special token ‘_A’ can be computed as:
+
+$$
+\mathrm {R} _ {\text {a t t}} ^ {- \mathbf {A}} = \frac {\operatorname* {m a x} \left(A _ {\text {r o l e}} ^ {- \mathbf {A}} \odot f _ {\mathrm {B} 2 \mathrm {M}} \left(b _ {g t}\right)\right)}{\operatorname* {m a x} \left(A _ {\text {r o l e}} ^ {- \mathbf {A}} \odot \left(1 - f _ {\mathrm {B} 2 \mathrm {M}} \left(b _ {g t}\right)\right)\right)}, \quad A _ {\text {r o l e}} ^ {- \mathbf {A}} \in \mathbb {R} ^ {1 \times \mathrm {M}}, \tag {1}
+$$
+
+where  $f_{\mathrm{B2M}}$  denotes the function that transforms a bounding box into its corresponding binary mask. Here, a higher ratio indicates a better target grounding ability. Given a test sample, we can identify the token yielding the highest attention ratio as the superior token for grounding. The hit ratio of a token is then defined as the frequency of being the superior token for grounding across all test samples. Fig. 2(a) shows the hit ratio of four special tokens in each MLLM. Notably, the fixed token does not consistently exhibit the best grounding ability for different samples. For example, the highest hit ratio achieved by token ‘.’ in LLaVA-1.5 is not more than  $50\%$ . In addition, for different MLLMs, the token at a fixed position does not always yield the best localization performance. For example, the
+
+![](images/5bfa3b2c4b2c38bddf2982e8122226cf49249b75af7260075792b64a3f32c5d2.jpg)  
+Figure 3: Overview of the proposed approach for zero-shot STVG. Given a video-text pair, we first decompose the text  $Q$  into spatially and temporally related sub-queries,  $Q_{\mathrm{s}}$  and  $Q_{\mathrm{t}}$ . The text prompt tokens converted from  $Q_{\mathrm{s}}$  and  $Q_{\mathrm{t}}$  are then concatenated with visual tokens  $\mathrm{T}_v$  for spatial and temporal inferences, respectively. In addition, we introduce learnable variables as visual prompts and optimize them by the logit-guided re-attention (LRA) module. For spatial grounding, we also develop a temporal-augmented assembling (TAS) strategy by reversing the frames to enhance temporal consistency. After optimization, we obtain the object track score  $S_{\mathrm{obj}}$  and frame score  $S_{\mathrm{frame}}$  based on the grounding token identification. The final prediction is derived by joining  $S_{\mathrm{obj}}$  and  $S_{\mathrm{frame}}$ .
+
+last special token ‘:’, which is adopted in a previous work [69], obtains a high hit ratio in Qwen-VL, but its hit ratio is quite low in LLaVA-1.5. This observation holds for both image and video inputs.
+
+The special token with higher visual activation tends to show superior grounding performance. Since the superior grounding tokens vary across different samples and MLLMs, identifying them for grounding is a problem that needs to be resolved when ground truth is unavailable as a prior. With further analysis, we reveal that the superior token for grounding tends to show higher visual activation. For each sample, we rank the special tokens according to the maximum value of visual attention and then evaluate their grounding accuracy by selecting the proposal with the highest attention value. Following the paradigm in previous works [50; 16], we extract box proposals using a detector and evaluate the grounding accuracy by the Acc@0.5 metric. Fig. 2(b) shows the results. We can see that the grounding accuracy decreases as the rank of visual activation reduces (from left to right). This supports our hypothesis that the special token with higher visual activation tends to show superior grounding ability. We have also observed that models with better comprehension possess better grounding ability overall. For example, the special tokens in LLaVA-OneVision achieve better grounding performance than those of the other MLLMs.
+
+A straightforward solution for zero-shot STVG. Inspired by the grounding ability of special tokens subsequent to the text prompt, we refer to them as grounding tokens in our work. By identifying the special tokens characterized by high visual activation, we derive a strong training-free framework for STVG. Specifically, given the video-text pair  $(V,Q)$ , we first extract the object track proposals  $O_{pro} = \{O_1,\dots ,O_P\}$  from the video  $V$ , as done in previous works [3; 25], where  $P$  denotes the number of proposals, and  $O_{p} = \{b_{t}^{\prime}\}_{t = 1}^{t = T_{v}}$  is the set of bounding boxes of the  $p$ -th proposal in  $T_{v}$  frames. Based on the foregoing findings, we then select the token with the highest attention value for locating the target object, and denote the text-to-image attention matrix of the selected token as  $A_{\mathrm{g}}\in \mathbb{R}^{1\times M}$ . As a result, we obtain each object track score by computing the maximum attention value inside each object track as:
+
+$$
+\mathcal {S} _ {\mathrm {o b j}} = \left\{s _ {1} ^ {o}, s _ {2} ^ {o}, \dots , s _ {P} ^ {o} \right\}, \text {w i t h} s _ {p} ^ {o} = \max  \left(A _ {\mathrm {g}} \odot f _ {\mathrm {B 2 M}} \left(O _ {p}\right)\right), \tag {2}
+$$
+
+where  $\odot$  denotes element-wise multiplication. We choose the track with the highest score as spatial prediction  $O_{\mathrm{pred}}^{\prime}$ . In a similar manner, we can compute the frame score  $S_{\mathrm{frame}} = \{s_1^t,s_2^t,\dots ,s_{\mathrm{T}_v}^t\}$ . By selecting the top-K frames with the highest scores from the temporal prediction  $S_{\mathrm{frame}}$ , we obtain the final spatio-temporal prediction  $O_{\mathrm{pred}}^{\prime} = \{b_{t}^{\prime}\}_{t = t_{s^{\prime}}}^{t_{e^{\prime}}}$ , where  $t_{s^{\prime}}$  and  $t_{e^{\prime}}$  denote the starting and ending boundary predictions. Though simple, this solution has achieved comparable or even superior performance than current zero-shot methods. For example, based on LLaVA-0neVision model, this solution achieves  $23.3\%$  on the m_vIoU metric on HC-STVGv1 dataset, which outperforms previous SOTA result (19.1%) by E3M [3].
+
+# 3.3 Decomposed Spatio-Temporal Highlighting
+
+Despite a strong performance by the above solution, these grounding tokens often neglect some important cues during spatio-temporal localization especially when processing complex video queries. As shown in Fig. 1(b), the attribute cue 'in red clothes' is overlooked during inference, which causes the incorrect spatial localization. We observe that the language query often contains attribute and action descriptions of the target object, which are beneficial for spatial and temporal localization, respectively. To this end, we propose a novel decomposed spatio-temporal highlighting (DSTH) strategy in our framework, which aims at highlighting the attributes/ actions cues in language query and enhances the spatial/temporal reasoning, respectively.
+
+Generation of target-related cues. For comprehensive spatial and temporal reasoning, it is essential to extract the attribute and action descriptions from the original query  $Q$  as textual cues of the target. Here, we leverage the strong in-context capability of the LLM [1] to extract attribute and action descriptions. Following previous works [62; 16], we construct a prompt with general instructions and in-context task examples. As shown in Fig. 3, we feed the prompt into LLM, and then these related descriptions  $Q_{\mathrm{s}}$  and  $Q_{\mathrm{t}}$  are generated through the LLM completion. More implementation details can be found in Appendix 7.3.
+
+Spatio-temporal prompt learning. With the decomposed attribute and action descriptions as cues for spatial and temporal reasoning, the next question is how to efficiently direct the model to focus on the corresponding visual regions. Reliable responses in visual question answering (VQA) necessitate careful attention to the relevant visual context as pointed out by previous studies [58; 42]. Inspired by this, we innovatively propose regularizing the response to the questions constructed from the attribute and action descriptions for adjusting the visual attention of MLLMs. Specifically, we first transform the descriptions into interrogative queries by a fixed template to inquire the existence of the target. As shown in Fig. 3, from the original text input  $Q$ , we can obtain the attribute description 'a man on the left of the man in the orange shirt', which is further transformed into interrogative sub-query  $Q_{\mathrm{s}}$ : 'Is there a man on the left of the man in the orange shirt in this video?' for spatial inquiry. In a similar way, we can obtain the interrogative sub-query  $Q_{\mathrm{t}}$  for temporal inquiry. These interrogative sub-queries will be taken as input instructions of MLLMs for response generation.
+
+Logit-guided re-attention. With extracted sub-queries above, we then propose a novel logit-guided re-attention (LRA) module to regularize the token prediction during response generation. Taking the spatial sub-query as an example, we first initialize a learnable variable  $\mathrm{V}_s$  with the same shape as the visual tokens  $\mathrm{T}_v$ , and then add it to  $\mathrm{T}_v$  as the visual input of the language model. Sub-query  $Q_{\mathrm{s}}$  is converted as text prompt tokens  $\mathrm{T}_q^s$  by the text tokenizer and embedding layer. Given the input token sequence, the next token probability prediction over the vocabulary set  $\mathcal{V}$  is formulated as:
+
+$$
+\mathrm {p} _ {y} = \exp \left(\operatorname {l o g i t} _ {\pi_ {\theta}} \left(y _ {i} \mid \left(\mathrm {T} _ {v} + \mathrm {V} _ {s}, \mathrm {T} _ {q} ^ {s}, y _ {<   i}\right)\right)\right), \tag {3}
+$$
+
+where  $\pi_{\theta}$  denotes the parameter of the language model and is frozen in our work.  $\mathrm{logit}_{\pi_{\theta}}$  is the log probability of the generated token at time step  $i$ .  $y_{<t}$  denotes the text tokens sequence prior to prediction time step  $i$ . We define the optimization objective by contrasting the probabilities of positive token 'yes' and negative token 'no':
+
+$$
+\mathcal {L} _ {s} = 1 - \exp \left(\log \mathrm {i t} _ {\pi_ {\theta}} \left(y _ {i} ^ {\text {v e s}} \mid \left(\mathrm {T} _ {v} + \mathrm {V} _ {s}, \mathrm {T} _ {q} ^ {s}, y _ {<   i}\right)\right) - \log \mathrm {i t} _ {\pi_ {\theta}} \left(y _ {i} ^ {\text {n o}} \mid \left(\mathrm {T} _ {v} + \mathrm {V} _ {s}, \mathrm {T} _ {q} ^ {s}, y _ {<   i}\right)\right)\right). \tag {4}
+$$
+
+During the inference process, we conduct backpropagation to optimize the learnable variable  $\mathrm{V}_s$  as the spatial prompt. The process is iterated  $\mathrm{N}_{ep}$  times by test-time tuning paradigm. By enhancing the positive response towards the sub-query  $Q_{\mathrm{s}}$ , we can prompt the MLLM to effectively mine target-related contextual information during VQA, which in turn highlights the attribute cues in the original text. Similarly, we can obtain the temporal prompt  $\mathrm{V}_t$  by optimizing the temporal inference.
+
+Joint inference. Based on the spatial and temporal visual prompts, we derive the attention maps  $A_{\mathrm{g}}^{S}$  and  $A_{\mathrm{g}}^{T} \in \mathbb{R}^{\mathrm{T}_{v} \times \mathrm{h} \times \mathrm{w}}$  of the special token with high visual activation for spatial and temporal predictions, where  $h$  and  $w$  denote the token numbers of the height and width. From Eq. 2, we obtain the object track score  $S_{\mathrm{obj}}$  and the temporal score  $S_{\mathrm{frame}}$  based on  $A_{\mathrm{g}}^{S}$  and  $A^{T}$ , respectively. Finally, we integrate the predictions  $O_{\mathrm{pred}}' = \{b_t'\}_{t=t_s'}^{t_e'}$  as the spatio-temporal grounding result.
+
+# 3.4 Temporal-augmented Assembling
+
+The attribute sub-query, which provides a static state description, exhibits temporal independence for spatial grounding. In other words, the spatial grounding by the attribute sub-query should be
+
+Table 1: Quantitative comparison on HCSTVG (v1&v2) and VidSTG (Declarative) benchmarks.  
+
+<table><tr><td rowspan="2">Sup</td><td rowspan="2">Method</td><td colspan="3">HCSTVG-v1</td><td colspan="3">HCSTVG-v2</td><td colspan="3">VidSTG (Declarative)</td></tr><tr><td>m_vIoU</td><td>vIoU@0.3</td><td>vIoU@0.5</td><td>m_vIoU</td><td>vIoU@0.3</td><td>vIoU@0.5</td><td>m_vIoU</td><td>vIoU@0.3</td><td>vIoU@0.5</td></tr><tr><td rowspan="4">Full</td><td>TubeDETR [59] [CVPR2022]</td><td>32.4</td><td>49.8</td><td>23.5</td><td>36.4</td><td>58.8</td><td>30.6</td><td>30.4</td><td>42.5</td><td>28.2</td></tr><tr><td>STCAT [20] [NeurIPS2022]</td><td>35.0</td><td>57.7</td><td>30.0</td><td>-</td><td>-</td><td>-</td><td>33.1</td><td>46.2</td><td>32.6</td></tr><tr><td>CSDVL [33] [CVPR2023]</td><td>36.9</td><td>62.2</td><td>34.8</td><td>38.7</td><td>65.5</td><td>33.8</td><td>33.7</td><td>47.2</td><td>32.8</td></tr><tr><td>CG-STVG [15] [CVPR2024]</td><td>38.4</td><td>61.5</td><td>36.3</td><td>39.5</td><td>64.5</td><td>36.3</td><td>34.0</td><td>47.7</td><td>33.1</td></tr><tr><td rowspan="4">Weak</td><td>WINNER [29] [CVPR2023]</td><td>14.2</td><td>17.2</td><td>6.1</td><td>-</td><td>-</td><td>-</td><td>11.6</td><td>14.1</td><td>7.4</td></tr><tr><td>VEM [21] [ECCV2024]</td><td>14.6</td><td>18.6</td><td>5.8</td><td>-</td><td>-</td><td>-</td><td>14.5</td><td>18.6</td><td>8.8</td></tr><tr><td>CoSPaL [25] [ICLR2025]</td><td>22.1</td><td>31.8</td><td>19.6</td><td>22.2</td><td>31.4</td><td>18.9</td><td>16.0</td><td>20.1</td><td>13.1</td></tr><tr><td>STPro [13] [CVPR2025]</td><td>17.6</td><td>27.0</td><td>12.9</td><td>20.0</td><td>31.1</td><td>14.6</td><td>15.5</td><td>19.4</td><td>12.7</td></tr><tr><td rowspan="7">ZS</td><td>RedCircle [47] [CVPR2023]</td><td>9.2</td><td>7.8</td><td>1.6</td><td>-</td><td>-</td><td>-</td><td>8.6</td><td>7.6</td><td>0.9</td></tr><tr><td>ReCLIP [50] [ACL2022]</td><td>14.4</td><td>18.3</td><td>4.9</td><td>-</td><td>-</td><td>-</td><td>14.2</td><td>17.5</td><td>7.9</td></tr><tr><td>E3M [3] [ECCV2024]</td><td>19.1</td><td>29.4</td><td>10.6</td><td>-</td><td>-</td><td>-</td><td>16.2</td><td>20.5</td><td>11.9</td></tr><tr><td>Ours_LLaVA-Next-Video-7B</td><td>20.4</td><td>33.6</td><td>12.4</td><td>23.6</td><td>36.8</td><td>15.5</td><td>16.6</td><td>26.8</td><td>11.1</td></tr><tr><td>Ours_ShareGPT4Video-8B</td><td>20.0</td><td>32.2</td><td>10.9</td><td>24.4</td><td>38.9</td><td>15.4</td><td>17.1</td><td>27.8</td><td>11.6</td></tr><tr><td>Ours_Qwen2-VL-7B</td><td>23.6</td><td>39.0</td><td>14.4</td><td>25.6</td><td>40.5</td><td>17.1</td><td>17.0</td><td>27.4</td><td>11.4</td></tr><tr><td>Ours_LLaVA-OneVision-7B</td><td>24.8</td><td>41.5</td><td>16.3</td><td>27.7</td><td>44.7</td><td>19.5</td><td>18.0</td><td>29.8</td><td>12.2</td></tr></table>
+
+temporally consistent for temporal augmentation (e.g., reversing the order of video frames). However, there exists temporal inconsistency when introducing temporal augmentation in current MLLMs. Here, we propose a metric to measure the temporal consistency. By denoting the spatial attention maps before and after reversing the order of input frames as  $A_{\mathrm{g}}^{S}$  and  $\tilde{A}_{\mathrm{g}}^{S}$ , the temporal consistency can be measured as:
+
+$$
+S _ {\text {c o n s}} = \max  \left\{s _ {1}, \dots , s _ {P} \right\}, \quad s _ {p} = \max  \left(\left(A _ {\mathrm {g}} ^ {S} \odot f _ {\mathrm {B 2 M}} \left(O _ {p}\right)\right) \odot \left(\tilde {A} _ {\mathrm {g}} ^ {S} \odot f _ {\mathrm {B 2 M}} \left(O _ {p}\right)\right)\right), \tag {5}
+$$
+
+where a higher  $S_{\mathrm{cons}}$  indicates better temporal consistency. We then divide the testing samples into ten groups in descending order of temporal consistency. Fig. 4 shows the average grounding accuracy of each group samples. The results demonstrate a pronounced association between temporal consistency and spatial grounding performance. The temporal inconsistency tends to cause worse spatial localization.
+
+To this end, we integrate a temporal-augmented assembling (TAS) strategy in our framework. As shown in Fig. 3, we perform a frame-level reversion operation on the visual tokens and the spatial prompt simultaneously, and then optimize the spa
+
+![](images/9b8cad4d28dd94f07775b35be9bb9acf1008c91b130a8243ba1a50a2b6c5b4c4.jpg)  
+Figure 4: Spatial grounding accuracy of different groups of samples on the HC-STVGv1 dataset. These groups are ranked by descending temporal consistency.
+
+tial prompts for the input of the original frames and the temporal-augmented input, respectively. During inference, we derive the spatial prediction by assembling the attention maps of temporal-augmented input frames. The proposed TAS strategy alleviates the effect of temporal inconsistency and improves the robustness of spatial grounding.
+
+# 4 Experiments
+
+# 4.1 Settings
+
+Datasets. We evaluate on three video benchmark datasets: HCSTVG-v1, HCSTVG-v2 [51], and VidSTG [73]. We provide detailed introduction about them in Appendix 7.1.
+
+Implementation Details. We adopt G-DINO [38] and SAM2 [45] for detection and tracking tubelet generation, and use GPT-4o to decompose the original query sentence into spatial and temporal sub-queries. We consider four widely-used video MLLMs: LlaVA-Next-Video-7B [27], Qwen2-VL-7B [53], ShareGPT4Video-8B [7], LlaVa-OneVision-7B [27] for demonstrating the efficiency of our method. Refer to Appendix 7.2 for more details.
+
+Evaluation Metrics. We follow the standard evaluation protocol [59; 25] and use m_vIoU, and vIoU@R to assess the performance of spatio-temporal grounding. Specifically, let  $S_i$ ,  $S_u$  denote the intersection and union between the predicted and ground-truth frames. The vIoU is computed by  $\frac{1}{S_u} \sum_{t \in S_i} \mathbf{IoU}(b_t', b_t)$ , where  $b_t'$  and  $b_t$  denote the detected and ground-truth bounding box at frame  $t$ , respectively. The m_vIoU represents the vIoU averaged over all testing samples, and vIoU@R denotes the proportion of data samples in the testing subset with vIoU greater than the threshold  $R \in \{0.3, 0.5\}$ .
+
+# 4.2 Performance Comparison
+
+Quantitative Comparison. Tab. 1 presents a comparison of our method against 11 methods from three categories, including zero-shot, weakly-supervised, and fully-supervised methods. Specifically, we compare our approach with existing zero-shot SOTA approaches (e.g., E3M [3], ReCLIP [50], RedCircle [47]). Our method consistently outperforms these methods by a remarkable margin on all benchmarks. Based on LLaVA-Next-Video-7B, our method outperforms E3M by  $4.2\%$  on vIoU@0.3 and  $1.8\%$  on vIoU@0.5. When integrated into the better LLaVA-OneVision-7B model, the corresponding improvements reach  $12.1\%$  and  $5.7\%$ .
+
+This shows that our framework can adapt to various MLLMs well, and better performances can be achieved by using better MLLMs. Even on the VidSTG dataset, which contains fewer action cues related to the target and thus makes temporal grounding particularly challenging, our framework still outperforms the previous SOTA methods overall. This demonstrates the strong generalization capability of our method. Besides, our method even surpasses current SOTA weakly-supervised methods on most metrics. For example, on the HCTSVG-v2 benchmark, our method outperforms CoSPaL [25] by a margin of
+
+![](images/b5186385626959eeb08aa243c1912eca469f12d7825ed9aa64b3c636f0127f52.jpg)
+
+![](images/78a00eaeb3112ea84b94475eed8bc38e7a742e3f0bcff34d3c5a036c1538e1ac.jpg)  
+Figure 5: Qualitative results on the HC-STVGv1 test set. Better spatio-temporal grounding results (green) are obtained when the DSTH strategy is being used for optimization.
+
+$5.5\%$  on the m_vIoU metric. Furthermore, compared to the fully-supervised methods, our method can still achieve comparable results, which further validates the superiority of our approach. We provide quantitative comparison on the VidSTG (Interrogative) and image benchmarks in Appendix 8.1
+
+Qualitative Comparison. We present qualitative results in Fig. 5. In the example below, before introducing the DSTH optimization strategy, the model neglects the attribute cues in the language query and suffers from the spatial grounding error. By highlighting the attribute cues of the target, our method can direct the MLLMs toward reliable visual context and improve spatial localization.
+
+# 4.3 Ablation Study
+
+In this section, based on HC-STVGv1 dataset, we analyse the effect of different proposed components when integrated into LlaVa-Next-Video and LlaVa-OneVision. We also conduct extensive ablation experiments with the hyper-parameters based on LlaVa-Next-Video.
+
+Component analysis. In Tab. 2, we first average the attention maps of all special tokens (1st row) as the baseline. This solution has achieved outstanding performance. Next, in the 2nd row, we integrate the selection of the superior token introduced in grounding token identification (GTI). It brings consistent improvements on different MLLMs. The results show that identifying the superior special token can effectively unleash the powerful comprehension ability of MLLMs. We then validate the efficiency of the decomposed spatio-temporal highlighting strategy (3rd and 4th rows).  $\mathbf{S}_{\mathrm{tune}}$  and  $\mathbf{T}_{\mathrm{tune}}$  denote adopting the prompt learning in spatial and temporal inferences, respectively. It is demonstrated that the MLLMs can be directed to focus on the spatial/temporal related regions better
+
+![](images/101da467654a925be78122a532e42731a130c32349b9bb0822e10ff551d0035b.jpg)  
+(a)
+
+![](images/e6e186875bdd724cb3ecca4ecea4febda67ac6e6e91c025f9ca6f8ee1be8baad.jpg)  
+(b)  
+Figure 6: (a) Comparison on the number of frames  $\mathrm{N}_f$  as input, using vIoU@0.3. (b) Ablation on the number of selected frames K during temporal prediction. (c) Ablation on different trackers.
+
+![](images/a450ac2dff4ae168c2c203746b9e97da80d8e4561ec5a8e6b45ffc7690f49810.jpg)  
+(c)
+
+when highlighting the attribute/action cues with proper prompts. When utilizing prompt learning on the two sub-tasks, the final performance can be further refined (5th row). In addition, we also find that the DSTH strategy can improve the grounding performance especially for the less efficient MLLMs (e.g., LLaVA-Next-Video). Even for the MLLMs (e.g., LLaVA-OneVision) with strong temporal comprehension ability, performance improvement can still be achieved by the test-time optimization.
+
+Finally, we introduce the temporal-augmented assembling (TAS) strategy (6th row). The overall performance can be further improved by refining the spatial localization.
+
+Why does spatial and temporal prompt learning by contrasting logits of 'yes' and 'no' work. Though contrasting logits of 'yes'
+
+Table 2: Component ablation on LLaVA-Next-Video and LLaVA-OneVision.  
+
+<table><tr><td rowspan="2">GTI</td><td rowspan="2">Sp</td><td rowspan="2">Tp</td><td rowspan="2">TAS</td><td colspan="3">LLaVA-Next-Video</td><td colspan="3">LLaVA-OneVision</td></tr><tr><td>m_vIoU</td><td>vIoU@0.3</td><td>vIoU@0.5</td><td>m_vIoU</td><td>vIoU@0.3</td><td>vIoU@0.5</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>15.2</td><td>25.1</td><td>8.5</td><td>21.3</td><td>36.1</td><td>12.6</td></tr><tr><td>✓</td><td>X</td><td>X</td><td>X</td><td>16.3</td><td>26.6</td><td>9.3</td><td>23.3</td><td>38.8</td><td>15.1</td></tr><tr><td>✓</td><td>✓</td><td>X</td><td>X</td><td>18.0</td><td>30.1</td><td>10.1</td><td>24.1</td><td>40.3</td><td>16.0</td></tr><tr><td>✓</td><td>X</td><td>✓</td><td>X</td><td>18.4</td><td>29.5</td><td>10.1</td><td>23.8</td><td>39.7</td><td>15.5</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>X</td><td>19.9</td><td>32.1</td><td>11.9</td><td>24.3</td><td>40.7</td><td>16.0</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>20.4</td><td>33.6</td><td>12.4</td><td>24.8</td><td>41.5</td><td>16.3</td></tr></table>
+
+and 'no' tokens is heuristic, it is reasonable. We empirically find that given the question prompt, the model often gives ambiguous or even wrong response predictions even though the objects referred to by the text do indeed exist in the video. This indicates that the model cannot understand the video content well in certain cases. By optimizing the logit of the 'yes' token toward positive prediction instead of the 'no' token, we encourage the model to carefully attend to the target-related visual information and achieve better visual localization.
+
+To further validate the hypothesis, we measure the distribution of attention peaks before and after test-time optimization. Based on the evaluation on the HC-STVG dataset, we find that the average attention peak before test-time optimization is about 6.7 (without normalization), and after optimization, it increases to 23.9 (without normalization). This change in attention values indicates that our proposed optimization strategy can encourage the model to perceive more visual modal information for generating visually-grounded responses, thereby justifying our previous hypothesis.
+
+Comparison of different learning strategies. We propose logit-guided re-attention by contrasting logits of 'yes' and 'no' tokens. Here, we also consider entropy minimization (i.e., increasing the predicted probability of the 'yes' token across the entire vocabulary) during test-time tuning. The results obtained by LLaVa-Next-Video-7B on the HC-STVG-v2 dataset are shown in the table below.
+
+The 1st row denotes the results of introducing grounding token identification (GTI). The 2nd and 3rd rows denote the test-time tuning results achieved by entropy minimization and contrasting the logits of 'yes' and 'no', respectively. We find that entropy minimization does not bring noticeable improvement. We believe this is be
+
+cause entropy minimization may be achieved by reducing the token prediction of abundant irrelevant tokens in the vocabulary while ignoring the differentiation between the 'yes' and the 'no' token. As a result, it is less efficient than explicitly contrasting the logits in promoting the model to attend to abundant visual information.
+
+Table 3: Comparison of different learning strategies.  
+
+<table><tr><td>Methods</td><td>m_vIoU</td><td>vIoU@0.3</td><td>vIoU@0.5</td></tr><tr><td>GTI</td><td>19.2</td><td>28.5</td><td>12.4</td></tr><tr><td>GTI + entropy minimization</td><td>19.5</td><td>28.5</td><td>12.5</td></tr><tr><td>GTI + contrasting the logits</td><td>22.1</td><td>32.9</td><td>14.0</td></tr></table>
+
+Generality analysis of our method. We conduct the scalability verification on InternVL3 and Qwen2.5-VL series models. With limited computational resources, we evaluate our method on InternVL3-4B, InternVL3-8B, Qwen2.5-VL-3B and Qwen2.5-VL-7B models in the table below.
+
+We average the attention maps of all special tokens as the baseline. Benefiting from our grounding token identification and test-time optimization, we can obviously achieve better grounding performance not only in the smaller but also the larger models, which well demonstrates both the generality and scalability of our method. Moreover, we find that the model size significantly impacts the inherent grounding ability of special tokens. The performance of small models (e.g., InternVL3-4B) is inferior to the larger models (e.g.,
+
+Table 4: Evaluation on different scales of models.  
+
+<table><tr><td>Methods</td><td colspan="4">vIoU@0.3 m_vIoU t_iou gt_viou</td></tr><tr><td>InternVL3-4B + Baseline</td><td>24.5</td><td>17.0</td><td>38.4</td><td>38.2</td></tr><tr><td>InternVL3-4B + Ours</td><td>41.1</td><td>25.4</td><td>45.5</td><td>49.5</td></tr><tr><td>InternVL3-8B + Baseline</td><td>36.5</td><td>22.9</td><td>42.9</td><td>46.1</td></tr><tr><td>InternVL3-8B + Ours</td><td>45.3</td><td>27.5</td><td>47.2</td><td>53.2</td></tr><tr><td>Qwen2.5-VL-3B + Baseline</td><td>28.5</td><td>17.3</td><td>37.1</td><td>39.1</td></tr><tr><td>Qwen2.5-VL-3B + Ours</td><td>41.7</td><td>25.4</td><td>45.9</td><td>49.3</td></tr><tr><td>Qwen2.5-VL-7B + Baseline</td><td>39.7</td><td>23.6</td><td>44.1</td><td>47.2</td></tr><tr><td>Qwen2.5-VL-7B + Ours</td><td>45.4</td><td>27.7</td><td>46.9</td><td>51.6</td></tr></table>
+
+InternVL3-8B). It is because fewer parameters in language models perform poorly in multimodal alignment.
+
+Effect of input frames  $\mathbf{N}_f$ . Fig. 6(a) analyzes the effect of using different numbers of video frames  $\mathbf{N}_f$  as input. We can see that more input frames provide richer visual context for model inference and lead to a better overall performance. Besides, the performance becomes slightly peaked as the number of input frames increases. To balance between performance and efficiency during inference, we uniformly sample 20 frames as the visual input of MLLMs. In Fig. 6, we also evaluate Vanilla-Tuning, a solution that directly applies test-time optimization to the entire text query  $Q$  instead of decomposing it into attribute/action sub-queries. We observe that the performance of Vanilla-Tuning is inferior to our method. It demonstrates that decomposing the text query into attribute/action sub-queries can help facilitate the intensive spatio-temporal comprehension of MLLMs.
+
+Ablation on the number  $\mathrm{K}$  of predicted frames. Fig. 6(b) analyzes the effect of predicted frame numbers  $\mathrm{K}$  for temporal grounding based on the HCSTVG-v1 dataset. Here, we consider selecting the optimal frame number from the set  $\{3,5,7,9,11\}$ . In general, when  $\mathrm{K}$  is set as 7, the optimal result can be obtained. Thus, we select top-7 frames as the prediction during temporal grounding.
+
+Trackers ablation. Fig. 6(c) analyzes the effect of different trackers. Besides SAM2 [45], a foundation model for tracking, we also consider two other tracking models (i.e., ByteTrack [72] and BoTSort [2]) for analysis. It is reasonable that when a stronger tracking model (e.g., SAM2) is used for generating spatio-temporal tubelets, we can obtain better STVG performances. Besides, when suboptimal tracking models are used, our method can still achieve comparable or better performances than the current SOTA methods, which shows the generalization of our approach.
+
+# 5 Conclusion
+
+In this paper, we have presented a novel MLLM-based zero-shot framework for the spatio-temporal video grounding (STVG) task. Our approach is initiated by identifying the grounding capability of special tokens in widely used MLLMs during response generation. To leverage and unleash the comprehension ability of MLLMs, we have proposed the decomposed spatio-temporal highlighting (DSTH) strategy. It first decomposes the text query into attributes and actions sub-queries. It then employs a logit-guided re-attention (LRA) module to sharpen the spatial/temporal visual context comprehension. We have also proposed the temporal-augmented assembling (TAS) strategy to alleviate the effect of temporal inconsistency. Extensive experiments conducted on three STVG benchmarks demonstrate the effectiveness of our proposed framework.
+
+Our approach does have limitations. For example, it may struggle to process long videos well due to high computational consumption caused by MLLMs. As a future work, we would like to consider incorporating token pruning and key frame selection techniques into the model design.
+
+# References
+
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+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: In Sec.1, we make the claims to reflect the paper's contributions and scope.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
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+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: In Appendix 9, we discuss the limitations of the work.
+
+Guidelines:
+
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+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [NA]
+
+Justification: The paper does not include theoretical results.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
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+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: We provide implementation details in Sec. 4.1 and the Appendix for reproducibility.
+
+# Guidelines:
+
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+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+# Answer: [No]
+
+Justification: We will consider releasing the data and code once the paper is accepted.
+
+# Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
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+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.  
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+# Answer: [Yes]
+
+Justification: We provide details about the experimental setting (e.g., hyper-parameters) in Sec. 4.1 and the Appendix for reproducibility.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+# Answer: [No]
+
+Justification: We do not report error bars due to limited computing resources.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).
+
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We provide the information on the computer resources in Appendix 7.2.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: The research conducted in the paper conform with the NeurIPS Code of Ethics.
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: The paper discusses societal impacts of the work in the Appendix 10.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
+
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: The paper poses no such risks.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: The paper properly cites the existing assets.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
+
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: The paper does not release new assets.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+# Answer: [NA]
+
+Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components.
+
+# Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# Appendices of Unleashing the Potential of Multimodal LLMs for Zero-Shot Spatio-Temporal Video Grounding
+
+1 Introduction 1  
+2 Related work 3  
+3 Method 3
+
+3.1 Preliminaries 3  
+3.2 Grounding Token Identification 4  
+3.3 Decomposed Spatio-Temporal Highlighting 6  
+3.4 Temporal-augmented Assembling 6
+
+4 Experiments 7
+
+4.1 Settings 7  
+4.2 Performance Comparison 8  
+4.3 Ablation Study 8
+
+5 Conclusion 10  
+6 Grounding Tokens 24
+
+6.1 Illustration of Input Tokens Components 24  
+6.2 Quantitative Analysis of Special Tokens 24  
+6.3 Qualitative Analysis of Special Tokens 26
+
+7 More Implementation Details 29
+
+7.1 Datasets 29  
+7.2 Method Implementation 29  
+7.3 Generation of Target-related Cues 29
+
+8 More Results and Ablation Analysis 31
+
+8.1 Comparison on VidSTG (Interrogative) and RefCOCO Benchmarks 31  
+8.2 Temporal Consistency Analysis 31  
+8.3 Effect of Optimization Times  $\mathbf{N}_{ep}$  and Learning Rate lr 32  
+8.4 Inference Efficiency 32  
+8.5 Qualitative Spatio-Temporal Video Grounding Visualization 33
+
+9 Limitation and Future work 35
+
+# 6 Grounding Tokens
+
+# 6.1 Illustration of Input Tokens Components
+
+In Fig. 7, we give a clear illustration of input token components in MLLMs. In practice, besides the visual and text prompt tokens (② and ③), there are also system tokens and some special tokens (① and ④) equipped for language generation. In particular, the grounding ability of special tokens subsequent to the text instruction prompt is significantly undervalued in previous works. Here we focus on exploring the grounding ability of these special tokens. For different multi-modal large language models, there often are different special tokens due to different instruction-tuning process. For example, in Llava-1.5, the introduced special tokens include: {'': 'ANT', 'IST', 'SS', 'A'} While for the Qwen2-VL, these special tokens include: {C, 'assistant', <|im_start|>, <|im_end|>} Note that we also consider the '. as one of the special tokens considering that it is located in the end of the instruction prompt and is characteristic of sink tokens [23; 57].
+
+In fact, special tokens are predefined symbols within a language model's vocabulary. They focus on guiding the model's processing instead of representing real words and guide the model to generate coherent and context-aware responses. In the scenario explored in our work (i.e., dialogue systems), special tokens can differentiate between a user's question and the assistant's answer. By using special tokens, models become better at understanding structure and context.
+
+![](images/fef30c5cfa7a9b4ab69028fb1445f09615175b1824ce3df29e61ba40291cce99.jpg)  
+Figure 7: Illustration of the tokens input in MLLMs-based dialog system. Here, we take the image question answering as an example.
+
+# 6.2 Quantitative Analysis of Special Tokens
+
+In our work, we randomly selected a subset of 1000 image-text pairs from RefCOCOg [17] validation set for image MLLMs analysis, and a subset of 1000 video-text pairs from HC-STVG [51] dataset for video MLLMs analysis. Particularly, we choose six typical MLLMs (i.e., LLaVA-1.5, Qwen-VL, Deepseek-VL, LLaVA-Next-Video, Qwen2-VL, LlaVA-OneVision) for pilot studies. In the following, we will introduce our findings about special tokens in MLLMs.
+
+Some special tokens show outstanding grounding ability for text prompt input. We compare the grounding accuracy of G-DNIO [38] and the special tokens' ones by probing the attention map. Here we adopt the G-DNIO with Swin-T backbone and it is not finetuned on the refcoco series datasets. Following previous visual grounding works [50; 16], we adopt the IoU@0.5 as the metric of grounding accuracy. The results evaluated on refcoco series benchmarks are shown in the Fig. 8. The yellow horizontal line denotes the result by G-DINO model. Though the G-DINO is trained on image-text pairs in fully-supervised manner, some special tokens still achieve comparable and even
+
+![](images/5aaad80623e7bc340fcfe20b6e8b2045c4b696f28bfa5559f12c1485fcd7bacf.jpg)  
+Figure 8: Comparison of grounding accuracy between G-DINO and special tokens on RefCOCO series datasets.
+
+![](images/f26c53854ea38421f051b7eb649f27aa8b20e290178038e93f3f4d0ff4950789.jpg)
+
+![](images/de610d42a46b81e988a59b1f57d03e89db41aa2b214f966a144c15427292e2e9.jpg)
+
+![](images/24f5d0d7f03ab1430529cb5ed12a01702d2367456421a7ef6d8587178035a07a.jpg)  
+Figure 9: The hit ratio of different special tokens in image and video MLLMs.
+
+![](images/0069df44600b63fe642dbd10ce3fc7117746fa755e4a01651b1c809a9a20fc66.jpg)
+
+![](images/dacf0526f77f4bff8f7fa4bddef387fc0db9c2d2964ba34f98fa90c6e5820efb.jpg)
+
+![](images/73e2078e7ab9ac9b84a9494648e54c76dea6bb325ea55663777c816ba58dcc41.jpg)  
+Figure 10: Grounding accuracy of special tokens ranked by the visual activation degree.
+
+![](images/1aa2c2de9fc58828842313689321685ad5acb1692a5b12c5883fbd83d563b698.jpg)
+
+![](images/3d3ed5e6a36f235ac918c67fb715aba0ceae4f33028809e99410ff771d071200.jpg)
+
+better performance. The results show that the special token can effectively ground the text prompt input by integrating the textual cues. Besides, we also notice that there exists a pronounced difference for grounding ability of different special tokens.
+
+MLLMs dynamically assign the special tokens to attend to the text-related regions. For further exploration on the grounding ability of special tokens, we define the attention ratio of each special token as the ratio of maximum attention within the ground-truth bounding box  $b_{gt}$  to that outside it. Here, a higher ratio indicates better target grounding ability. Given a test sample, we can identify the token that yields the highest attention ratio as the superior token for grounding. Then a token's hit ratio is defined as the frequency of being the superior token for grounding across all test samples. The Fig. 9(a) shows the hit ratio of special tokens in each MLLM. We choose four special tokens for visualization. In addition to the findings that MLLMs dynamically assign the special tokens to attend to the text-related regions, we also observe that the superior token for grounding in video MLLMs shows a more concentrated trend than image MLLMs. For example, the highest hit ratio by Qwen2-VL is more than  $60\%$ . Besides, for the same special token in the MLLM, the hit ratio in the case of different datasets may be significantly different. For example, the token '.' shows a high hit ratio in RefCOCOg dataset, but its hit ratio is quite low in the HC-STVG dataset.
+
+The special token with higher visual activation tends to show superior grounding performance. In our work, with further analysis, we reveal that the superior token for grounding tends to show higher visual activation. For each sample, we rank the special tokens according to the maximum value of visual attention and evaluate their grounding accuracy by selecting the proposal with the highest attention value as the prediction. In practice, we extract box proposals by the detector (e.g., G-DINO) and evaluate the grounding accuracy by the Acc@0.5 metric. Fig. 10 shows the results. We can see that the grounding accuracy decreases as the rank of visual activation reduces (from left to right). We also observe that models with better comprehension overall possess better grounding ability.
+
+# 6.3 Qualitative Analysis of Special Tokens
+
+In Fig. 11, we visualize token-to-visual attention maps using some image-text pairs by LLaVA-1.5-7B model. In particular, for each image-text pair, we visualize the visual attendance of the semantic tokens from the text prompt and special tokens. The green bounding box denotes the ground truth of the target object. Notably, we notice that the semantic tokens in text prompts can provide tangible attention to related visual entities by the limited cues, while the special tokens are often assigned to integrate global instruction cues and attend to the exact target region. For example, given the text prompt 'a man getting ready to cut a cake', the token 'man' and 'cake' both show reasonable visual response. The special token 'A' can accurately attend to the target person. In addition, we see that the special token with hight visual activation often shows better grounding performance. However, we also see that the special tokens may attend to the wrong instance even though the semantic tokens can properly capture their region of interest. It suggests that there is still room for improvement in inference in current multimodal language models.
+
+In Fig. 12, we visualize attention maps of special tokens using some video-text pairs by ShareGPT4Video-8B model. The ground-truth spatio-temporal tube is denoted by red bounding boxes. Each row shows the attention maps of a specific token. We find that only a few special tokens can well capture the corresponding target regions, which necessitates the efficient selection of grounding tokens. Also, in some cases (e.g., the below sample), there may be more than one special token that will pay attention to the target area. This shows that achieving localization by considering the integration of multiple effective special tokens can be a direction for improvement. We leave it to future research.
+
+![](images/dc8337faf589c9133674f2e9e733e63135827cbc79b65227428a56b18c860a9a.jpg)
+
+![](images/9bf435c81ebda4aca27613754a06a5a19eb5a1f361ecbd685301568a1a5981c4.jpg)
+
+![](images/5c6e9b86c87bcce9b92692967b2e21c78aafdd7ef269a43ae52a792dbe0e9633.jpg)
+
+![](images/9fa523f5447e8807de6f42d26d9d0b240e8fab8aed9da71c72fedde649ba64d0.jpg)  
+Figure 11: Visualization of tokens' attention maps on image-text pairs. The tokens with dotted box denote the semantic tokens in text prompts while the tokens with solid box denote the special tokens following the text prompts.
+
+![](images/3f10b8b547a77f869fcff152dfeee00764c00b9e60a01ce3d046d9fc02c9a6df.jpg)
+
+The man in the hat raises his hand, turns around, and walks to the door.
+
+![](images/e72dc7de436bb97fbe27a94e3708478b3eea33d485d9cc8e21d752d39f086068.jpg)
+
+![](images/f0781e01e9542441006b65787580ace849ae67cc9f33d9776a3a14fa16727fa2.jpg)
+
+![](images/35dd4beb8c8d72eb668cd78c07b16efe79ed9f59c9f15250f55b134751db986a.jpg)
+
+![](images/8ddbed3309a26b4bbbaefec5816da763cf7e87d88442abdc8517e0c61c5109df.jpg)
+
+![](images/ccd8aea004c34d06092913f709629d1c49aca9945f4640f26c8012bc1ad3404c.jpg)
+
+![](images/198d52a725b6baf37bd00b2e5314a96ea4cbbb15a5fe40d1da5249d74cf67552.jpg)
+
+The man who comes out of the door walks to the man without a hat and stops, speaks a few words to him, then turns away.
+
+![](images/1caa6944ceda322d3c320beeb36115cf2348865b2ba3cd46dfd35066e2e84102.jpg)
+
+![](images/bcccd6a7f49babcff4d4ff2ff891f78e0ad339d90042347830e650632834aedc.jpg)
+
+![](images/c09a386c1a7dd889df739d0671cb20d1b067d177dffb9b4e440bc93f3eb6fb7a.jpg)
+
+![](images/5158a4956778e1c3221000736f2c490548eff2d5cd1c42266256652904da89ee.jpg)
+
+![](images/ddaf5bf7c79dac5af7277ab23a5060b7e926fce2fd8cec5d5df2efb89983f348.jpg)  
+Figure 12: Visualization of tokens' attention maps on video-text pairs.
+
+# 7 More Implementation Details
+
+# 7.1 Datasets
+
+We evaluate on three video benchmark datasets: HCSTVG-v1, HCSTVG-v2 [51], and VidSTG [73]. HCSTVG-v1 has 4500 training and 1160 testing videos with sentence descriptions of human attributes/actions. HCSTVG-v2 extends v1 to 16,544 videos, including 10,131 training, 2,000 validation, and 4,413 testing videos. Since the test set is unavailable, we evaluate on the validation set following prior works [59; 25]. VidSTG includes 99,943 video-sentence pairs (44,808 declarative, 55,135 interrogative), covering 10,303 videos and 80 object categories. Training, validation, and test sets have 80,684, 8,956, and 10,303 pairs, respectively.
+
+# 7.2 Method Implementation
+
+In this work, we adopt G-DINO [38] with 0.4 for both phrase and box thresholds to detect object proposals, and then utilize SAM2 [45] as tracker for tubelet proposals generation. Considering the information redundancy, we run the detector every 10 frames for efficiency. We utilize GPT-4o to decompose the original query sentence into spatial and temporal sub-queries. To demonstrate the efficiency of our method, we consider four LLaVA-like video MLLMs: LlaVA-Next-Video-7B [27], Qwen2-VL-7B [53], ShareGPT4Video-8B [7], LlaVa-OneVision-7B [27]. In practice, with the limited tokens context length and computing efficiency, we sample 20 frames by default as the visual input of MLLMs for each video. When adopting the test-time tuning strategy DSTH, we set the learning rate lr and iteration times  $\mathrm{N}_{ef}$  as 8.0 and 1 according to the ablation analysis in Tab. 7. We conduct all experiments on an A100 GPU with 80G VRAM based on Pytorch framework. To better capture text-to-visual attention, we introduce 'Describe this video in details.' as general query prompt  $Q_{\mathrm{cap}}$  and implement the relative attention strategy [69] to reduce the effect of visual registers [9; 55]. In Algo. 1, we outline the implementation of the proposed decomposed spatio-temporal highlighting strategy during test-time tuning.
+
+# 7.3 Generation of Target-related Cues
+
+In this work, we extract the attributes and actions descriptions from the original query  $Q$  as textual cues to enhance the spatial and temporal comprehension, respectively. To obtain multiple target-related cues, we leverage the strong in-context capability of the Large Language Model (LLM). In particular, we construct the in-context instruction to prompt llm for completion. The whole prompt used in this work includes: general instruction (in brown), output constraints (in blue), in-context task examples (in green) and input question (in yellow), shown in Fig. 13.
+
+![](images/4d1bd01d2f4b0dec7ed99ee3b975b3a4af8377f9ac3b8af09dd4ae7c3f086bfc.jpg)  
+Figure 13: Flow of LLM-based generation of target-related cues.
+
+In Fig. 14, we present some examples of LLM generation. After decomposing the original text description into attributes and actions-related descriptions, we will further transform the descriptions into interrogative queries by a fixed template to inquire about the existence of the target. In our work, we adopt the template 'Is there a in this video?'
+
+- Original Text: The man in purple clothes takes a step forward and sits on the bench.  
+- Attributes Text : the man in purple clothes.  
+Actions Text : The man takes a step forward and sits on the bench.
+
+10_phVLLTMzMk.mp4
+
+Original Text: The yellow-haired man takes the opposite woman's hand and kisses it, then puts it down.  
+- Attributes Text : the yellow-haired man.  
+Actions Text : The man takes the opposite woman's hand and kisses it, then puts it down.
+
+159_vsMgg4snZm.mkv
+
+Original Text: The white-bearded man points at the man on his right and then points at himself again.  
+- Attributes Text : the white-bearded man.  
+Actions Text : The man points at the man on his right and then points at himself again.
+
+111_pSdPmmJ3-ng.mp4
+
+Figure 14: Examples of LLM-based cues generation.
+
+Algorithm 1: Decomposed Spatio-Temporal Highlighting  
+Input: MLLM  $\pi_{\theta}$  , video query input  $Q$  , video  $X$  , track proposals  $O_{\mathrm{pro}} = \{\mathrm{o}_1,\dots ,\mathrm{o}_P\}$  test-time tuning epochs  $\mathbf{N}_{ep}$    
+Output: Generated tube  $O_{\mathrm{pred}}^{\prime} = \{\mathrm{b}_{t}\}_{t = t_{s}}^{t_{e},}$  based on query input  $Q$  and video  $X$    
+Decompose the  $Q$  into attribute/action sub-queries  $Q_{s}$  and  $Q_{t}$  for spatial/temporal inquiry. Initialize spatial prompt  $\mathrm{V}_{s}$  and temporal prompt  $\mathrm{V}_{t}$    
+Initialize  $i_{ep} = 0$    
+Embed the video  $X$  into visual tokens  $\mathrm{T}_{v}$  . Embed the  $Q_{s}$  and  $Q_{t}$  into text tokens  $\mathrm{T}_{q}^{s}$  and  $\mathrm{T}_{q}^{t}$    
+while training do while  $i_{ep} <   N_{ep}$  do // spatial prompt Learning Compute positive and negative logit prediction for attribute subquery  $Q_{s}$  ..  $\mathrm{p^{yes} = \logit_{\pi_\theta}(y_i^{yes} | (T_v + V_s, T_q^s, y_{<  i}))}$ $\mathrm{p^{no} = \logit_{\pi_\theta}(y_i^{no} | (T_v + V_s, T_q^s, y_{<  i}))}$  Update Vs by minimizing  $\mathcal{L}_s = -\exp (\mathrm{p^{yes} - p^{no}})$  // temporal prompt Learning Compute positive and negative logit prediction for action query  $Q_{t}$  ..  $\mathrm{p^{yes} = \logit_{\pi_\theta}(y_i^{yes} | (T_v + V_t, T_q^t, y_{<  i}))}$ $\mathrm{p^{no} = \logit_{\pi_\theta}(y_i^{no} | (T_v + V_t, T_q^t, y_{<  i}))}$  Update Vt by minimizing  $\mathcal{L}_t = -\exp (\mathrm{p^{yes} - p^{no}})$ $i_{ep} = i_{ep} + 1.$  end   
+end   
+prepare general query prompt  $Q_{cap}$    
+while inference do cache the visual attention map  $A_{cap}$  by query  $Q_{cap}$  as input.. // inference with  $\mathrm{T}_v + \mathrm{V}_s$  cache the visual attention map  $A_{g}^{S}$  by query  $Q$  with  $\mathrm{T}_v + \mathrm{V}_s$  as visual tokens. compute spatial-related visual attention map  $A_{g}^{S} = A_{g}^{S} / A_{cap}$  obtain the object track prediction. // inference with  $\mathrm{T}_v + \mathrm{V}_t$  cache the visual attention map  $A_{g}^{T}$  by query  $Q$  with  $\mathrm{T}_v + \mathrm{V}_t$  as visual tokens. compute temporal-related visual attention map  $A_{g}^{T} = A_{g}^{T} / A_{cap}$  obtain the target frames prediction. Combine the object track prediction and target frames prediction.
+
+# end
+
+# 8 More Results and Ablation Analysis
+
+# 8.1 Comparison on VidSTG (Interrogative) and RefCOCO Benchmarks
+
+In Tab. 5, we compare our approach with other SOTA methods on the VidSTG (Interrogative) benchmarks. Even given the interrogative sentence as query, our method still shows superior performance with a  $4.0\%$  improvement on the vIoU@0.5 metric when integrated with LLaVA-Next-Video-7B model, which shows the strong generalization capability of our framework. Undoubtedly, our method can also achieve superior localization performance in the image grounding tasks. In Tab. 6, we compare our framework with other SOTA visual grounding methods on RefCOCO series benchmarks. Here we just introduce the selection of superior token for grounding and has achieved comparable and even better performance than current zero-shot SOTA methods. The The 'Oracle' denotes the result of choosing the most exact one of candidate boxes (extracted by G-DINO) with knowledge of the ground truth. We also show the proportion of our method's performance relative to oracle's, which is indicated by the number in lower right corner. Note that our insight about grounding tokens is orthogonal to other works [69; 30; 24]. We believe that it can be integrated with other works to achieve better results.
+
+Table 5: Comparison on VidSTG (Interrogative) benchmark.  
+
+<table><tr><td rowspan="2">Sup</td><td rowspan="2">Method</td><td colspan="3">VidSTG (Interrogative)</td></tr><tr><td>m_vIoU</td><td>vIoU@0.3</td><td>vIoU@0.5</td></tr><tr><td rowspan="3">Full</td><td>TubeDETR [59] [CVPR2022]</td><td>25.7</td><td>35.7</td><td>23.2</td></tr><tr><td>CSDVL [20] [CVPR2023]</td><td>28.5</td><td>39.9</td><td>26.2</td></tr><tr><td>CG-STVG [15] [CVPR2024]</td><td>29.0</td><td>40.5</td><td>27.5</td></tr><tr><td rowspan="3">Weak</td><td>WINNER [29] [CVPR2023]</td><td>10.2</td><td>12.0</td><td>5.5</td></tr><tr><td>VEM [21] [ECCV2024]</td><td>13.3</td><td>16.7</td><td>7.7</td></tr><tr><td>CoSPaL [25] [ICLR2025]</td><td>13.5</td><td>16.4</td><td>10.2</td></tr><tr><td rowspan="6">ZS</td><td>ReCLIP [50] [ACL2022]</td><td>8.4</td><td>8.0</td><td>2.3</td></tr><tr><td>E3M [3] [ECCV2024]</td><td>10.6</td><td>12.2</td><td>5.4</td></tr><tr><td>Ours_LLaVA-Next-Video-7B</td><td>14.6</td><td>23.1</td><td>9.4</td></tr><tr><td>Ours_Qwen2-VL-7B</td><td>13.0</td><td>19.5</td><td>7.8</td></tr><tr><td>Ours_ShareGPT4Video-8B</td><td>13.1</td><td>21.0</td><td>9.1</td></tr><tr><td>Ours_LLaVA-OneVision-7B</td><td>13.5</td><td>21.4</td><td>8.2</td></tr></table>
+
+Table 6: Comparison of different methods on RefCOCO series datasets.  
+
+<table><tr><td rowspan="2">Sup</td><td rowspan="2">Method</td><td colspan="3">RefCOCO</td><td colspan="3">RefCOCO+</td><td colspan="2">RefCOCOg</td></tr><tr><td>val</td><td>testA</td><td>testB</td><td>val</td><td>testA</td><td>testB</td><td>val</td><td>test</td></tr><tr><td rowspan="5">Full</td><td>MDETR [22] [CVPR2021]</td><td>86.8</td><td>89.6</td><td>81.4</td><td>79.5</td><td>84.1</td><td>70.6</td><td>81.6</td><td>80.9</td></tr><tr><td>SeqTR [77] [ECCV2022]</td><td>87.0</td><td>90.2</td><td>83.6</td><td>78.7</td><td>84.5</td><td>71.9</td><td>82.7</td><td>83.4</td></tr><tr><td>UNINEXT [32] [ACMMM2023]</td><td>92.6</td><td>94.3</td><td>91.5</td><td>85.2</td><td>89.6</td><td>79.8</td><td>88.7</td><td>89.4</td></tr><tr><td>Shikra-7B [6] [arXiv]</td><td>87.0</td><td>90.6</td><td>80.2</td><td>81.6</td><td>87.4</td><td>72.1</td><td>82.3</td><td>82.2</td></tr><tr><td>Ferret-7B [65] [ICLR2024]</td><td>87.5</td><td>91.4</td><td>82.5</td><td>80.8</td><td>87.4</td><td>73.1</td><td>83.9</td><td>84.8</td></tr><tr><td rowspan="5">Zero</td><td>ReCLIP [50] [ACL2022]</td><td>45.8</td><td>46.1</td><td>47.1</td><td>47.9</td><td>50.1</td><td>45.1</td><td>59.3</td><td>59.0</td></tr><tr><td>ZS_REC [16] [CVPR2024]</td><td>49.4</td><td>47.8</td><td>51.7</td><td>48.9</td><td>50.0</td><td>46.9</td><td>61.0</td><td>60.0</td></tr><tr><td>GroundVLP [46] [AAAI2024]</td><td>65.0</td><td>73.5</td><td>55.0</td><td>68.8</td><td>78.1</td><td>57.3</td><td>74.7</td><td>75.0</td></tr><tr><td>Ours LLaVA-OneVision-7B</td><td>68.775.4%</td><td>77.280.8%</td><td>62.072.9%</td><td>67.174.4%</td><td>76.780.2%</td><td>57.567.7%</td><td>72.381.3%</td><td>73.783.2%</td></tr><tr><td>Oracle</td><td>91.1</td><td>95.6</td><td>85.0</td><td>91.2</td><td>95.6</td><td>84.9</td><td>88.9</td><td>88.6</td></tr></table>
+
+# 8.2 Temporal Consistency Analysis
+
+In our work, we obtain the attributes and actions related sub-queries. Specially, the decomposed attribute sub-query, which provides static state description, should be temporally consistent for spatial grounding. Interestingly, there exists evident inconsistency when introducing temporal augmentation in current MLLMs. We develop a temporal inconsistency metric by introducing reversing the order of input frames. In Fig. 15, we show the relation between spatial grounding accuracy and temporal
+
+![](images/a55c8d92cef0d25309d6a408252e0fda22d3bc0d35a0a80f79a89ca77ddb5d2c.jpg)  
+Figure 15: Spatial grounding accuracy of different groups of samples on the HC-STVGv1 dataset. These groups are ranked by descending temporal consistency.
+
+![](images/848c447699d5c161593c1dcf7f91240eae694dfd6ed838b208b585cb7a4a99a9.jpg)
+
+Table 7: Iteration times  ${\mathrm{N}}_{ep}$  and learning rate lr ablation.  
+
+<table><tr><td>Nep</td><td>lr</td><td>m_vIoU</td><td>vIoU@0.3</td><td>vIoU@0.5</td></tr><tr><td>0</td><td>0</td><td>15.2</td><td>25.1</td><td>8.5</td></tr><tr><td>1</td><td>4</td><td>19.9</td><td>32.6</td><td>11.6</td></tr><tr><td>\u</td><td>\u8</td><td>20.4</td><td>33.6</td><td>12.4</td></tr><tr><td>1</td><td>16</td><td>20.2</td><td>32.3</td><td>12.0</td></tr><tr><td>2</td><td>4</td><td>20.5</td><td>34.0</td><td>12.1</td></tr><tr><td>2</td><td>8</td><td>20.3</td><td>32.6</td><td>12.1</td></tr><tr><td>2</td><td>16</td><td>20.3</td><td>33.2</td><td>12.2</td></tr></table>
+
+consistency based on LLaVA-Next-Video-7B model (left of Fig. 15) and LLaVA-OneVision-7B model (right of Fig. 15). We can find that although the LLaVA-OneVision-7B model achieves better localization performance, there is still the pronounced temporal inconsistency caused by temporal augmentation. Our findings provide guidance and insight for subsequent research in video MLLMs.
+
+# 8.3 Effect of Optimization Times  $\mathrm{N}_{ep}$  and Learning Rate lr
+
+The test-time tuning strategy DSTH is iterated  $\mathrm{N}_{ep}$  times for optimization with learning rate lr. Here we analyse effect of the hyper-parameters. As shown in Tab. 7, better results can be achieved as the optimization progresses and the our optimization strategy is relatively robust to these hyperparameters. In particular, when setting  $\mathrm{N}_{ep} = 1$  and  $\mathrm{lr} = 8.0$ , optimal performance is achieved in general.
+
+# 8.4 Inference Efficiency
+
+![](images/a69b836fbdbf0ae8185900a44f27fc93b5dadd3d4b345b7af8ac76f995347a56.jpg)  
+Figure 16: Comparison of inference time before/after introducing test-time tuning.
+
+In Fig. 16, we show the inference efficiency of our proposed zero-shot framework. We test the inference speed on all video samples from HC-STVGv1 dataset on a single A100 GPU and then compute the average value. Specifically, before introducing the test-time tuning strategy DSTH for inference, our framework costs about 3.1 seconds for each video based on LLaVA-Next-Video model. After adopting the test-time tuning strategy, the cost is still acceptable though with some additional resource consumption. We believe that it would be more efficient to apply the resource-friendly inference schemes in the future.
+
+# 8.5 Qualitative Spatio-Temporal Video Grounding Visualization
+
+In Fig. 17, we present some video grounding examples for qualitative analysis. Here, we compare the results before introducing the test-time tuning (denoted with yellow boxes) with the results (denoted with green boxes) after introducing the optimization. The ground truth boxes are denoted with red boxes. By highlighting the attribute/action cues of the target by test-time tuning, our method can direct the MLLMs toward reliable visual context and improve spatial/temporal localization. We also give some failure cases (e.g., the case (d)). Despite optimization during testing, the current model still pinpoints the wrong object instance. We attribute the less efficient comprehension to the poor visual conditions and suboptimal spatial inference in current MLLMs.
+
+![](images/6d34afc79136fbb369ad13611ff3b0618e61e9b14db91fd33183cf6937a90063.jpg)  
+Figure 17: Comparison of attention maps on the HC-STVG dataset.
+
+# 9 Limitation and Future work
+
+Our work is simple yet effective, and also provides insights for related fields (e.g., hallucination detection and attention-guided MLLMs pruning). Our work does have limitations. For example, based on MLLMs our framework can only receive a limited number of video frames as visual input due to the limit of computing resource. Besides, our framework needs to obtain attention for spatial and temporal inference, which is not compatible with the flash-attention mechanism adopted in current MLLMs. In the future, we will consider introducing a more efficient solution for the comprehension of long videos by incorporating token pruning and key frame selection techniques.
+
+# 10 Broader Impacts
+
+While we do not foresee our method causing any direct negative societal impact, it may potentially be leveraged by malicious parties to create applications that could misuse the grounding capabilities for unethical or illegal purposes. We urge the readers to limit the usage of this work to legal use cases.
\ No newline at end of file
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+# Unleashing the Power of One-Step Diffusion based Image Super-Resolution via a Large-Scale Diffusion Discriminator
+
+Jianze Li $^{1*}$ , Jiezhang Cao $^{2*}$ , Zichen Zou $^{1}$ , Xiongfei Su $^{3}$ , Xin Yuan $^{4}$ , Yulun Zhang $^{1\dagger}$ , Yong Guo $^{4}$ , Xiaokang Yang $^{1}$
+
+$^{1}$ Shanghai Jiao Tong University,  $^{2}$ Harvard University,  $^{3}$ China Mobile Research Institute,  $^{4}$ Westlake University,  $^{4}$ South China University of Technology
+
+# Abstract
+
+Diffusion models have demonstrated excellent performance for real-world image super-resolution (Real-ISR), albeit at high computational costs. Most existing methods are trying to derive one-step diffusion models from multi-step counterparts through knowledge distillation (KD) or variational score distillation (VSD). However, these methods are limited by the capabilities of the teacher model, especially if the teacher model itself is not sufficiently strong. To tackle these issues, we propose a new One-Step Diffusion model with a larger-scale Diffusion Discriminator for SR, called  $\mathrm{D}^3\mathrm{SR}$ . Our discriminator is able to distill noisy features from any time step of diffusion models in the latent space. In this way, our diffusion discriminator breaks through the potential limitations imposed by the presence of a teacher model. Additionally, we improve the perceptual loss with edge-aware DISTS (EA-DISTS) to enhance the model's ability to generate fine details. Our experiments demonstrate that, compared with previous diffusion-based methods requiring dozens or even hundreds of steps, our  $\mathrm{D}^3\mathrm{SR}$  attains comparable or even superior results in both quantitative metrics and qualitative evaluations. Moreover, compared with other methods,  $\mathrm{D}^3\mathrm{SR}$  achieves at least  $3\times$  faster inference speed and reduces parameters by at least  $30\%$ . We will release code and models at https://github.com/JianzeLi-114/D3SR.
+
+# 1 Introduction
+
+Real-world image super-resolution (Real-ISR) is a challenging task that aims to reconstruct high-resolution (HR) images from their low-resolution (LR) counterparts in real-world settings [1]. Most image super-resolution (SR) methods [2, 3, 4, 5, 6] use Bicubic downsampling of HR images to generate LR samples. These methods achieve good results in reconstructing simple degraded images. However, they struggle with the complex and unknown degradations widely existing in real-world scenarios. Previous research has predominantly employed generative adversarial networks (GANs) [7, 8, 9, 10] for Real-ISR task. In recent developments, diffusion models (DMs) [11, 12, 13] have emerged as a promising alternative. Recent Real-ISR methods have achieved outstanding performance by using diffusion models. In particular, some methods [14, 15, 16, 17, 18] leverage powerful pre-trained diffusion models, such as large-scale text-to-image (T2I) diffusion models like Stable Diffusion [12, 19]. These pre-trained T2I models provide extensive priors and powerful generative abilities. Most Diffusion model based methods generate HR images by employing ControlNet models [20], conditioning on the LR inputs. However, these methods typically require
+
+tens to hundreds of diffusion steps to produce high-quality HR images. The introduction of ControlNet not only increases the number of model parameters but also further exacerbates inference latency. Consequently, Diffusion model based multi-step diffusion methods often incur delays of tens of seconds when processing a single image, which significantly limits their practical application in real-world scenarios for low-level image reconstruction tasks.
+
+To accelerate the generation process of diffusion models, recent research has introduced numerous one-step diffusion methods [21, 22, 23, 24, 25, 26, 27, 28]. These methods are known as diffusion distillation, which distill multi-step pre-trained diffusion models into one-step counterparts. Most of these approaches employ a knowledge distillation strategy, using the multi-step diffusion model as a teacher to train a one-step diffusion student model. These methods significantly reduce inference latency, and the quality of the generated images can be comparable to that of multi-step diffusion models. Real-ISR methods based on one-step diffusion models have become an increasingly popular research direction [29, 30, 31, 32, 33, 34]. These methods employ pre-trained multi-step diffusion models as teachers to guide the training process. They achieve promising results, with performance comparable to that of multi-step models. However, further performance improvement remains a challenging task. Existing methods rely on a multi-step teacher to conduct distillation. Nevertheless, this paradigm would inevitably come with several limitations. First, the multi-step teacher may hamper the effectiveness of distillation if the teacher itself is not strong enough. For example, variational score distillation (VSD) [35, 36, 32] is a one-step diffusion distillation method. It enhances the realism of generated images by optimizing the KL divergence between the scores of the teacher and student models. This approach heavily relies on the prior knowledge of the teacher diffusion model. The limitations of the multi-step diffusion model impose an upper bound on the performance of the aforementioned methods. Second, the widely used VSD approach does not exploit any high resolution (HR) data for training but merely depend on the prior knowledge embedded in the pre-trained parameters. If the distribution fitted by the multi-step diffusion model (teacher in VSD) deviates from the high-quality image distribution, it may lead to a loss of realism or the generation of fake textures in the student model's images.
+
+To overcome the aforementioned challenges, we propose a novel one-step diffusion model with a large-scale diffusion discriminator. Different from existing one-step diffusion models using distillation techniques with teacher models, we use a larger-scale diffusion model, SDXL [19], as a discriminator to leverage the powerful priors. Specifically, our diffusion discriminator aims to distill latent features with different noises from true data. At the same time, it enables us to perceive the true data distribution of high-quality super-resolution datasets. As a result, we overcome the performance limitations imposed by the teacher model's upper bound. Additionally, we propose a simple and effective improvement to the perceptual loss, edge-aware DISTS (EA-DISTS), by capturing high-frequency details from extracted edges. Our comprehensive experiments indicate that  $\mathrm{D}^3\mathrm{SR}$  achieves superior performance and less inference time among one-step DM-based Real-ISR models.
+
+Our main contributions are summarized as follows:
+
+- We propose a simple but effective one-step diffusion distillation method,  $\mathrm{D}^3\mathrm{SR}$ , which uses a large-scale diffusion model as a discriminator for adversarial training. Unlike previous one-step diffusion ISR methods, we do not use a pre-trained multi-step diffusion model as a teacher to guide the training. Instead, we employ a larger-scale diffusion model to guide the training process. This breaks through the performance limitations of teacher models in previous distillation methods.  
+- We improve the perceptual loss by proposing the edge-aware DISTS (EA-DISTS) loss. Our EA-DISTS leverages image edges to enhance the model's ability and improve the authenticity of reconstructed details.  
+- Experiments show that our method outperforms previous one-step diffusion distillation methods, such as VSD and knowledge distillation. When compared with multi-step DM-based models,  $\mathrm{D}^{3}\mathrm{SR}$  obtains comparable or even better performance with over  $7\times$  speedup in inference time. Moreover, our method offers a  $3\times$  inference speed advantage over one-step DM-based methods and reduces parameters by at least  $30\%$ .
+
+# 2 Related Work
+
+# 2.1 Real-World Image Super-Resolution
+
+Real-world image super-resolution (Real-ISR) aims to recover high-resolution (HR) images from low-resolution (LR) observations in real-world scenarios. The complex and unknown degradation patterns in such scenarios make Real-ISR a challenging problem [37, 38, 39, 40]. To address this problem, a variety of methods have been proposed. Early image super-resolution models [2, 41, 42, 5, 6] typically rely on simple synthetic degradations like Bicubic downsampling for generating LR-HR pairs. Although these methods perform well under simple degradation settings, they struggle to achieve satisfactory results in real-world scenarios. Later, GAN-based methods such as BSRGAN [9], Real-ESRGAN [8], and SwinIR-GAN [10] introduce more complex degradation processes. These methods achieve promising perceptual quality but encounter issues such as training instability. Additionally, they have limitations in preserving fine natural details. Recently, Stable Diffusion (SD) [12] is considered for addressing Real-ISR tasks due to its strong ability to capture complex data distributions and provide robust generative priors. Approaches such as StableSR [43], DiffBIR [16], and SeeSR [14] leverage pre-trained diffusion priors and ControlNet models [20] to enhance HR image generation. One-step diffusion models have gained widespread attention from researchers. Methods such as YONOS-SR [29], SinSR [30], OESEDiff [32], AddSR [31], TAD-SR [44], and TSD-SR [45] have achieved Real-ISR with diffusion models in a single sampling step.
+
+# 2.2 Acceleration of Diffusion Models
+
+Acceleration of diffusion models can reduce computational costs and inference time. Therefore, various strategies have been developed to enhance the efficiency of diffusion models in image generation tasks. Fast diffusion samplers [13, 46, 47, 48, 49, 50] have significantly reduced the number of sampling steps from 1,000 to  $15\sim 100$  without requiring model retraining. However, further reducing the steps below 10 often leads to a performance drop. Under these circumstances, distillation techniques have made considerable progress in speeding up inference [21, 51, 23, 24, 25, 52, 26, 27, 53, 36]. For instance, Progressive Distillation (PD) methods [23, 24] have distilled pre-trained diffusion models to under 10 steps. Consistency models [25] have further reduced the steps to  $2\sim 4$  with promising results. Instaflow [27] further achieves one-step generation through reflow [51] and distillation. Recent score distillation-based methods, such as Distribution Matching Distillation (DMD) [54, 55] and Variational Score Distillation (VSD) [56, 36], aim to achieve one-step text-to-image generation. They minimize the Kullback-Leibler (KL) divergence between the generated data distribution and the real data distribution. Although these approaches have made notable progress, they still face challenges, like high training costs and dependence on teacher models.
+
+# 3 Method
+
+In this section, we present our real-world image super-resolution (Real-ISR) model  $\mathrm{D}^3\mathrm{SR}$ . First, in section 3.1, we review the basics of diffusion models and introduce the  $\mathrm{D}^3\mathrm{SR}$  generator. In section 3.2, we introduce the diffusion distillation method using a large-scale diffusion discriminator. In section 3.3, we introduce the edge-aware DISTS (EA-DISTS) perceptual loss. This loss improves texture details and enhances visual quality. Finally, in section 3.4, we describe the training process for  $\mathrm{D}^3\mathrm{SR}$ .
+
+# 3.1 Preliminaries: Diffusion Models
+
+Diffusion models include forward and reverse processes. During the forward diffusion process, Gaussian noise with variance  $\beta_{t} \in (0,1)$  is gradually injected into the latent variable  $z: z_{t} = \sqrt{\bar{\alpha}_{t}} z + \sqrt{1 - \bar{\alpha}_{t}} \epsilon$ , where  $\epsilon \sim \mathcal{N}(0,\mathbf{I})$ ,  $\alpha_{t} = 1 - \beta_{t}$ , and  $\bar{\alpha}_{t} = \prod_{s=1}^{t} \alpha_{s}$ . In the reverse process, we can directly predict the clean latent variable  $\hat{z}_{0}$  from the model's predicted noise  $\hat{\epsilon}: \hat{z}_{0} = \frac{z_{t} - \sqrt{1 - \bar{\alpha}_{t}} \hat{\epsilon}}{\sqrt{\bar{\alpha}_{t}}}$ , where  $\hat{\epsilon}$  is the prediction of the network  $\epsilon_{\theta}$  given  $z_{t}$  and  $t$ :  $\hat{\epsilon} = \epsilon_{\theta}(z_{t}; t)$ .
+
+As illustrated in Fig. 1, we first employ the encoder  $E_{\theta}$  to map the low-resolution (LR) image  $x_{L}$  into the latent space, yielding  $z_{L}$ :  $z_{L} = E_{\theta}(x_{L})$ . Next, we perform a one denoising step to obtain the predicted noise  $\hat{\epsilon}$  and compute the high-resolution (HR) latent representation  $\hat{z}_{H}$ :
+
+$$
+\hat {z} _ {H} = \frac {z _ {L} - \sqrt {1 - \bar {\alpha} _ {T _ {L}}} \epsilon_ {\theta} \left(z _ {L} ; T _ {L}\right)}{\sqrt {\bar {\alpha} _ {T _ {L}}}}, \tag {1}
+$$
+
+![](images/ae36580b5a7c2025a17a9767abc89cb6bbb6dd3ac4bd1fda487d06a123cd118c.jpg)  
+Figure 1: Training framework of  $\mathrm{D}^3\mathrm{SR}$ . The left side represents the generator  $\mathcal{G}_{\theta}$ , which includes the pre-trained VAE and UNet from Stable Diffusion. Only the UNet is fine-tuned using LoRA, while other parameters remain frozen. The right side depicts the diffusion discriminator, which guides the training process without participating in inference. The discriminator extracts the UNet Mid-block outputs and processes them through an MLP to generate realism scores for different image regions. Both the downsample and middle blocks of the UNet in the discriminator are fine-tuned with LoRA, whereas the MLP is randomly initialized.
+
+![](images/eaadfb88b1b39c90766299deb81e10fe1a54c76c8d0e6f61c4bb3bc626052d7c.jpg)  
+Figure 2: Visualization of features dimensionality reduction for the first 100 channels from the middle block outputs of the Stable Diffusion (SD) UNet. The distributions of the two types of image features are distinctly different.
+
+where  $\epsilon_{\theta}$  denotes the denoising network parameterized by  $\theta$ , and  $T_{L}$  is the diffusion time step. Unlike one-step text-to-image (T2I) diffusion models [25, 54], the input to the UNet of the Real-ISR diffusion models is not pure Gaussian noise. We set  $T_{L}$  to an intermediate time step within the range  $[0, T]$ , where  $T$  is the total number of diffusion time steps. In Stable Diffusion (SD),  $T = 1,000$ . Finally, we decode  $\hat{z}_{H}$  using the decoder  $D_{\theta}$  to reconstruct the HR image  $\hat{x}_H$ :  $\hat{x}_H = D_\theta (\hat{z}_H)$ . The entire computation process of the generator can be expressed as  $\hat{x}_H = \mathcal{G}_\theta (x_L)$ .
+
+# 3.2 Distillation with Large Diffusion Discriminator
+
+Currently successful one-step diffusion distillation methods applied in image super-resolution include knowledge distillation (KD), variational score distillation (VSD), and others. Among them, VSD stands out due to its interesting principles and excellent performance. VSD works by training an additional diffusion network to fit the fake score  $s_{\mathrm{fake}}$  of the generated image, while using a frozen pre-trained diffusion model to obtain the real score  $s_{\mathrm{real}}$  based on its prior. The method then aligns the two scores' differences using the Kullback-Leibler (KL) divergence, thus enhancing the realism
+
+of the generated image. However, in VSD, the absence of real image datasets implies that the upper bound of VSD is limited by the prior of the pre-trained diffusion model. The teacher model in VSD restricts the generative capacity of the student model, making further performance improvements challenging.
+
+To address the issue incurred by a weak multi-step teacher, we propose to use a large-scale diffusion model to provide stronger guidance for one-step distillation. Figure 2 shows the distributions of real and generated images' latent code at the middle block output of the UNet in the Stable Diffusion (SD) model. There is a clear difference in their distribution patterns. This suggests that using a pre-trained diffusion model as a discriminator has great potential. It can also avoid instability issues during the early stages of training. Figure 1 illustrates our training framework. We append an MLP block after the SD UNet middle block as a classifier to output the authenticity score for each patch.
+
+During training, we input the forward diffusion results of both the generator's predicted latent code  $\hat{z}_H$ , and the ground truth
+
+latent code  $z_{H} = E_{\theta}(x_{H})$ . The adversarial losses for updating the generator and discriminator are defined as:
+
+![](images/431af77417cd02d646438711788fb5b67b76372d14f6862343ed438ad2ad5442.jpg)  
+Figure 3: Comparison of the performance of SD models with different scales as discriminators. As the model size increases, the performance of the generator improves accordingly.
+
+$$
+\mathcal {L} _ {\mathcal {G}} = - \mathbb {E} _ {x _ {L} \sim p _ {\mathrm {d a t a}}, t \sim [ 0, T ]} \left[ \log \mathcal {D} _ {\theta} \left(F \left(\hat {z} _ {H}, t\right)\right) \right], \tag {2}
+$$
+
+$$
+\begin{array}{l} \mathcal {L} _ {\mathcal {D}} = - \mathbb {E} _ {x _ {L} \sim p _ {\mathrm {d a t a}}, t \sim [ 0, T ]} \left[ \log \left(1 - \mathcal {D} _ {\theta} \left(F (\hat {z} _ {H}, t)\right)\right) \right] \\ - \mathbb {E} _ {x _ {H} \sim p _ {\text {d a t a}}, t \sim [ 0, T ]} \left[ \log \mathcal {D} _ {\theta} \left(F \left(z _ {H}, t\right)\right) \right], \tag {3} \\ \end{array}
+$$
+
+where  $F(\cdot ,t)$  denotes the forward diffusion process of  $\cdot$  at time step  $t\in [0,T]$ , specifically,
+
+$$
+F (z, t) = \sqrt {\bar {\alpha} _ {t}} z + \sqrt {1 - \bar {\alpha} _ {t}} \epsilon , \quad \text {w i t h} \epsilon \sim \mathcal {N} (0, \mathbf {I}). \tag {4}
+$$
+
+Relation to Diffusion GAN Methods. Recently, many methods have combined GANs and Diffusion models, achieving success in image generation and other fields. Diffusion-GAN [57] uses a timestep-dependent discriminator to guide the generator's training. DDGAN [58] employs a multimodal conditional GAN to achieve large-step denoising. ADD [22] and LADD [59] both introduce discriminators to enhance the generative quality of one-step student diffusion models. These discriminators operate in pixel space and latent space, respectively. Both our method and existing Diffusion GAN methods demonstrate the potential of adversarial training in diffusion models. However, there are several essential differences. First, we use a pre-trained multi-step diffusion model as the discriminator, leveraging the prior of large-scale models to guide the training process. Second, we explore the impact of scaling. Figure 3 shows the performance of different SD models as discriminators. It reveals that large-scale diffusion discriminators break through the performance limits of pre-trained diffusion models, achieving superior results.
+
+# 3.3 Edge-Aware DISTS
+
+To further enhance the quality of the generated images, we aim to incorporate perceptual loss. Most image reconstruction methods utilize LPIPS [60] as the perceptual loss. However, to better preserve image texture details and alleviate pseudo-textures in the reconstruction under higher noise levels, we need to focus on the textures on HR images. DISTS [61] can compute the structural and textural similarity of images, aligning with human subjective perception of image quality. Furthermore, regions with rich textures or details often exhibit strong edge information. Leveraging image edge information effectively enhances texture quality. Based on this, we propose a novel perceptual loss, termed Edge-Aware DISTS (EA-DISTS). This perceptual loss simultaneously evaluates the structure
+
+and texture similarity of the reconstructed and HR images and their edges, thereby enhancing texture detail restoration.
+
+Our proposed EA-DISTS is defined as:
+
+$$
+\mathcal {L} _ {\mathrm {E A - D I S T S}} \left(\mathcal {G} _ {\theta} \left(x _ {L}\right), x _ {H}\right) =
+$$
+
+$$
+\mathcal {L} _ {\text {D I S T S}} \left(\mathcal {G} _ {\theta} \left(x _ {L}\right), x _ {H}\right) + \mathcal {L} _ {\text {D I S T S}} \left(\mathcal {S} \left(\mathcal {G} _ {\theta} \left(x _ {L}\right)\right), \mathcal {S} \left(x _ {H}\right)\right), \tag {5}
+$$
+
+where  $S(\cdot)$  represents the Sobel operator used to extract edge information from the images. It consists of two convolution kernels,  $G_{x}$  and  $G_{y}$ , which detect horizontal and vertical edges, respectively:
+
+$$
+G _ {x} = \left[ \begin{array}{l l l} - 1 & 0 & 1 \\ - 2 & 0 & 2 \\ - 1 & 0 & 1 \end{array} \right], \quad G _ {y} = \left[ \begin{array}{l l l} - 1 & - 2 & - 1 \\ 0 & 0 & 0 \\ 1 & 2 & 1 \end{array} \right]. \tag {6}
+$$
+
+The Sobel operator is applied to an image  $x$  as follows:
+
+$$
+\mathcal {S} (x) = \sqrt {\left(G _ {x} * x\right) ^ {2} + \left(G _ {y} * x\right) ^ {2}}, \tag {7}
+$$
+
+where  $*$  denotes the convolution operation.
+
+To intuitively demonstrate the effectiveness of EA-DISTS, we visualize the feature maps during the DISTS computation process. Figure 4 presents the visualization results of VGG-16 feature maps. As shown in Fig. 4, in areas rich with image details, such as the building windows, the feature maps associated with EA-DISTS exhibit more high-frequency information. Compared to DISTS, EA-DISTS demonstrates higher contrast in textured and smooth regions, further emphasizing the textural details within the images. Our EA-DISTS
+
+places greater emphasis on texture details within images, guiding the model to generate realistic and rich details.
+
+![](images/4086c472434218fc922ffecf8153b6831408d365987d5772e94a31446b9e4ee1.jpg)  
+(a) DISTS
+
+![](images/6616c0c102eced90e068856c68ed65d2b81c9b40ee35cd1fe16cfeedb962a349.jpg)  
+(b) EA-DISTS  
+Figure 4: Feature visualization associated with DISTS and EA-DISTS. Our EA-DISTS captures more high-frequency information, like texture and edges.
+
+# 3.4 Overall Training Scheme
+
+Here, we summarize the whole one-step diffusion model training process. As described in section 3.1, within the generator component,  $\mathrm{D}^3\mathrm{SR}$  obtains  $\hat{z}_H$  and the decoded high-resolution image  $\hat{x}_H$  through one-step sampling. The generator then updates its parameters by computing the spatial loss  $\mathcal{L}_{\mathrm{spatial}}$  in pixel space between the generated image and the ground truth, as well as the adversarial loss  $\mathcal{L}_{\mathcal{G}}$  derived from the discriminator in the latent space (Eq. 2). The loss function for updating the generator is defined as  $\mathcal{L}_{\mathrm{spatial}} + \lambda_1\mathcal{L}_{\mathcal{G}}$ . Specifically, we employ a weighted sum of Mean Squared Error (MSE) loss and perceptual loss to define the spatial loss:
+
+$$
+\mathcal {L} _ {\text {s p a t i a l}} \left(\mathcal {G} _ {\theta} \left(x _ {L}\right), x _ {H}\right) =
+$$
+
+$$
+\mathcal {L} _ {\mathrm {M S E}} \left(\mathcal {G} _ {\theta} \left(x _ {L}\right), x _ {H}\right) + \lambda_ {2} \mathcal {L} _ {\text {E A - D I S T S}} \left(\mathcal {G} _ {\theta} \left(x _ {L}\right), x _ {H}\right), \tag {8}
+$$
+
+where  $\lambda_{1}$  and  $\lambda_{2}$  are hyperparameters used to balance the contributions of each loss component. The supplementary material provides a detailed description of the algorithm's pseudocode and the values of the hyperparameters.
+
+For discriminator training, we utilize paired training features, where each pair consists of a negative sample feature  $\hat{z}_H$  and the corresponding real image's latent representation  $z_{H}$  as a positive one. Using Eq. 3, we compute the adversarial loss  $\mathcal{L}_{\mathcal{D}}$  to update the discriminator's parameters. Furthermore, the discriminator can be initialized with weights from more powerful pre-trained models, such as SDXL [19], to achieve superior performance.
+
+This training approach allows our  $\mathrm{D}^3\mathrm{SR}$  to overcome the limitations imposed by teacher models, enhancing generator performance without increasing its parameter count or compromising efficiency. Additionally, the integration of a robust discriminator initialized with advanced pre-trained models ensures that the generator receives high-quality feedback, facilitating the production of more realistic and detailed high-resolution images.
+
+Table 1: Quantitative results  $(\times 4)$  on the Real-ISR testset with ground truth. The best and second-best results are colored red and blue. In the one-step diffusion models, the best metric is bolded.  
+
+<table><tr><td>Dataset</td><td>Method</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>DISTS↓</td><td>NIQE↓</td><td>MUSIQ↑</td><td>MANIQA↑</td><td>CLIPiQA↑</td></tr><tr><td rowspan="9">RealSR</td><td>StableSR-s200</td><td>26.28</td><td>0.7733</td><td>0.2622</td><td>0.1583</td><td>4.892</td><td>60.53</td><td>0.5570</td><td>0.4310</td></tr><tr><td>DiffBIR-s50</td><td>24.87</td><td>0.6486</td><td>0.3834</td><td>0.2015</td><td>3.947</td><td>68.02</td><td>0.6309</td><td>0.6042</td></tr><tr><td>SeeSR-s50</td><td>26.20</td><td>0.7555</td><td>0.2806</td><td>0.1784</td><td>4.540</td><td>66.37</td><td>0.6118</td><td>0.5483</td></tr><tr><td>ResShift-s15</td><td>25.45</td><td>0.7246</td><td>0.3727</td><td>0.2344</td><td>7.349</td><td>56.18</td><td>0.5004</td><td>0.4307</td></tr><tr><td>ADDSR-s4</td><td>23.15</td><td>0.6662</td><td>0.3769</td><td>0.2353</td><td>5.256</td><td>66.54</td><td>0.6581</td><td>0.5390</td></tr><tr><td>SinSR-s1</td><td>25.83</td><td>0.7183</td><td>0.3641</td><td>0.2193</td><td>5.746</td><td>61.62</td><td>0.5362</td><td>0.4691</td></tr><tr><td>OSEDiff-s1</td><td>24.57</td><td>0.7202</td><td>0.3036</td><td>0.1808</td><td>4.344</td><td>67.31</td><td>0.6148</td><td>0.5524</td></tr><tr><td>ADDSR-s1</td><td>25.23</td><td>0.7295</td><td>0.2990</td><td>0.1852</td><td>5.223</td><td>63.08</td><td>0.5457</td><td>0.4498</td></tr><tr><td>D3SR-s1</td><td>24.11</td><td>0.7152</td><td>0.2961</td><td>0.1782</td><td>3.899</td><td>68.23</td><td>0.6383</td><td>0.5647</td></tr><tr><td rowspan="9">DIV2K-val</td><td>StableSR-s200</td><td>23.68</td><td>0.6270</td><td>0.4167</td><td>0.2023</td><td>4.602</td><td>49.51</td><td>0.4774</td><td>0.3775</td></tr><tr><td>DiffBIR-s50</td><td>22.33</td><td>0.5133</td><td>0.4681</td><td>0.1889</td><td>3.156</td><td>70.07</td><td>0.6307</td><td>0.6352</td></tr><tr><td>SeeSR-s50</td><td>23.21</td><td>0.6114</td><td>0.3477</td><td>0.1706</td><td>3.591</td><td>67.99</td><td>0.5959</td><td>0.5842</td></tr><tr><td>ResShift-s15</td><td>23.55</td><td>0.6023</td><td>0.4088</td><td>0.2228</td><td>6.870</td><td>56.07</td><td>0.4791</td><td>0.4269</td></tr><tr><td>ADDSR-s4</td><td>22.08</td><td>0.5578</td><td>0.4169</td><td>0.2145</td><td>4.738</td><td>68.26</td><td>0.5998</td><td>0.6007</td></tr><tr><td>SinSR-s1</td><td>22.55</td><td>0.5405</td><td>0.4390</td><td>0.2033</td><td>5.620</td><td>62.25</td><td>0.5011</td><td>0.5206</td></tr><tr><td>OSEDiff-s1</td><td>23.10</td><td>0.6127</td><td>0.3447</td><td>0.1750</td><td>3.583</td><td>66.62</td><td>0.5530</td><td>0.5330</td></tr><tr><td>ADDSR-s1</td><td>22.74</td><td>0.6007</td><td>0.3961</td><td>0.1974</td><td>4.270</td><td>62.08</td><td>0.5118</td><td>0.4868</td></tr><tr><td>D3SR-s1</td><td>22.05</td><td>0.6031</td><td>0.3556</td><td>0.1500</td><td>3.295</td><td>68.51</td><td>0.5795</td><td>0.5370</td></tr></table>
+
+# 4 Experiments
+
+We conduct comprehensive experiments to validate the effectiveness of  $\mathrm{D}^3\mathrm{SR}$  in real-world image super-resolution (Real-ISR). We provide a detailed introduction of our experimental setup in section 4.1. In section 4.2, we evaluate our method and compare it against the current state-of-the-art methods. In section 4.3, we carry out comprehensive ablation studies to validate the effectiveness and robustness of our proposed approach.
+
+# 4.1 Experimental Settings
+
+Datasets. We train  $\mathrm{D}^3\mathrm{SR}$  on LSDIR [62] and the first 10k images from FFHQ [63], totaling 95k images. During training, we randomly crop patches of size  $512\times 512$  pixels from these images. To get low-resolution (LR) and high-resolution (HR) pairs for training, we apply the Real-ESRGAN [64] degradation pipeline. We conduct extensive evaluations of  $\mathrm{D}^3\mathrm{SR}$  on a synthetic dataset DIV2K-val [65] and two real-world datasets, including RealSR [66] and RealSet65 [67]. In DIV2K-val, we use the Real-ESRGAN degradation pipeline to synthesize the corresponding LR images. We evaluate our model and all other methods by using the whole images from each dataset.
+
+Compared Methods. We compare our  $\mathrm{D}^3\mathrm{SR}$  with state-of-the-art DM-based methods for real image super-resolution (Real-ISR), as well as other prominent approaches, including GAN-based and Transformer-based methods. The DM-based methods include multi-step diffusion models, such as StableSR [43], ResShift [67], DiffBIR [16], and SeeSR [14], alongside recently proposed one-step diffusion models like SinSR [30], OSEDiff [32], and AddSR [31]. Other methods include GAN-based approaches, such as BSRGAN [9], RealSR-JPEG [39], Real-ESRGAN [64], LDL [68], and FeMASR [69], as well as Transformer-based method SwinIR [10].
+
+Evaluation Metrics. To assess the performance of each method, we employ four full-reference (FR) and four no-reference (NR) image quality metrics. The FR metrics include PSNR, SSIM, LPIPS [60], and DISTS [61]. Both PSNR and SSIM are computed on the Y channel in the YCbCr color space. The NR metrics include NIQE [70], MUSIQ [71], ManIQA [72], and CLIPIQA [73].
+
+Implementation Details. We initialize the generator network with the SD 2.1-base parameters and the discriminator network with partial parameters from SDXL. We set both the rank and scaling factor  $\alpha$  of LoRA to 16 in the generator and discriminator. We use the AdamW optimizer and set the learning rate for both the generator and discriminator to 5e-5. Training is performed with a batch size of 8 over 100K iterations with 4 NVIDIA A100-40GB GPUs.
+
+# 4.2 Comparison with State-of-the-Art Methods
+
+Quantitative Results. Tables 1 and 2 provide quantitative comparisons of the methods across the three datasets.  $\mathrm{D}^{3}\mathrm{SR}$  achieves the best performance among all no-reference (NR) metrics in one-step diffusion methods. These NR metrics reflect the details and realism of the images. This indicates that  $\mathrm{D}^{3}\mathrm{SR}$  outperforms all one-step methods and most multi-step methods. Recently, some studies [18] have pointed out that reference-based metrics such as PSNR and SSIM cannot accurately reflect the
+
+![](images/64bc7be29f368a460456afea7a485be2b395d14084443605478cdd15e2fe1b3e.jpg)
+
+![](images/5691b31245aa28fd0a50aaa6d179bd00519dfd25ad4046ea4446907b6aa2c6ab.jpg)
+
+![](images/34381b6a773401a5671f3e51c4491c8ef8e9a45d914e5a6c884a3d3afe9d9882.jpg)
+
+![](images/94249fc818f057c0d243720e1e1f368209465e18a4db917cb7424de3bc1d70e6.jpg)
+
+![](images/27d0b6c65c5794ee8dbf50bc268eb60f2269392e6ee8b83907040f918fb55748.jpg)
+
+![](images/33a4e006213ef22650d4623fe0751cb7929693c2e2ddde168592ca45a7d107d3.jpg)
+
+![](images/c28d6527e9d80c840c4ba7289db33c8ca2f2a59663a6e59fc5869eb28e19c4df.jpg)  
+Bicubic
+
+![](images/49193af0445c907e86e77010faa3307205adc20f3534cc83fce3d87e58c666b8.jpg)  
+Real-ESRGAN
+
+![](images/009a4ebc3597d4b977655dd4d55e71d1c2151969f6fd3e259a046c1efc89f4cb.jpg)  
+SwinIR
+
+![](images/46613d4fc7fda6f27645f5df4247e6e88f599b5deb9a7244d9e3d6f035041819.jpg)  
+StableSR
+
+![](images/e5a71f3901e823c7786fce5d3e00a0a83f925d34fe79e10c16f463ddc714e447.jpg)  
+DiffBIR
+
+![](images/d010402c831edaf72121d8d0c2da6bd7bd3156b39cbba1606b30b1b3462ceced.jpg)
+
+![](images/1df84f7d90bc3321170a4ac13f5304496bfbfabc3f4f00425384537aaaac3881.jpg)  
+Bicubic
+
+![](images/32c28d75720f28d214ad7650a4839399fb8432a4189dfc2e290a907ac8875bd1.jpg)  
+Real-ESRGAN
+
+![](images/3a63fc6f842de732a0ad3cacf02adc530a907f64f80637f44bd7067db325edfd.jpg)
+
+![](images/8b0122e3f0ddc98bc8b3abd768c9cb94cf7c2f8bbeca4f090aad08943448c14a.jpg)
+
+![](images/f0e0e3d0986e6bd7b14d6aa28781336689cdd6bec02ae6e32f8398e318521f01.jpg)
+
+![](images/abaf004ce3fd08c31495fecf1cdebaadeacc3c892757c7e009932e80f58b84cf.jpg)  
+ResShift
+
+![](images/17b9f2db756453768e69960673e50e4b2541d1a5a38bc7dcacf6534625403388.jpg)
+
+![](images/6283f3bd5c44234b0685d40b24a999416e4d4dbada785507da138802e27a64f0.jpg)
+
+![](images/aa17a2ce1a0e5c2badaa2761e6d4cb3ef1cf9fc3e4f1ef8077a56b986ee10c6c.jpg)
+
+![](images/2bfdb35c3940dd9be470d938cd39607505c61b1020e3abd85700b010a76f2ee7.jpg)
+
+![](images/e5d4c129f2c8cccc623bb3dd136fc8b16c61ea65b2ba4adf59fec12846319755.jpg)  
+Bicubic
+
+![](images/95c294e02b1b1ec160461147ddd56256dbca12b4fbc026cb95eb5c7b3c41702c.jpg)  
+SeeSR
+
+![](images/2f4e93a58b3e5fd1d5efe6f05dc4be37202c77db687c5a464e87d86867f041c2.jpg)  
+SinSR
+
+![](images/70054bf59afcf55213e7151a99814dc816e396a8dd8da2f1a47c99bb84572441.jpg)  
+OSEDiff  
+D3SR (ours)
+
+![](images/73e80fe64d9bd9832447dbb54760040cf1cfec68b52d77378336623bbb42c6d6.jpg)
+
+![](images/464c6eaa92ebc85c982f726f67e3cd4f264ef505256c2d7aedb76305014acf5d.jpg)  
+ResShift  
+Figure 5: Visual comparisons  $(\times 4)$  on Real-ISR task (RealSR [66] dataset).
+
+![](images/38783f97a725c730a83d2e1bf02931c8705bd3632b95f16255ad562db8666f6c.jpg)  
+Real-ESRGAN  
+SeeSR
+
+![](images/66182a1f4670bec63717f9b2c3c0661cc1f58abac46778226bf0437e5f268735.jpg)  
+SwinIR  
+SinSR
+
+![](images/1de7b6de363e1315d93017df57c7eb19633070fa6e74ad7e51502a9d12605fb7.jpg)  
+StableSR  
+DiffBIR  
+OSEDiff
+
+![](images/d81fc1cfa2d729d3d7c75a87efe1e13252c8f8caf53d5a3f303e4aabed7c2b1f.jpg)  
+$\mathrm{D}^3\mathrm{SR}$  (ours)
+
+performance of diffusion-based Real-ISR methods. We discuss this in the supplementary materials. We compare the quantitative and qualitative results, including those of GAN-based methods. We find that, compared to diffusion-based methods, GAN-based methods generally achieve higher PSNR and SSIM. However, the image quality of these methods is not as high.
+
+Visual Results. Figure 5 presents a visual comparison of various diffusion-based Real-ISR methods. As observed, most existing methods struggle to generate realistic details and often produce incorrect content in certain regions of the image due to noise artifacts. Notably, our  $\mathrm{D}^3\mathrm{SR}$  demonstrates a significant advantage over others, particularly in the restoration of textual content. Additional visual comparison results are provided in the supplementary material.
+
+Complexity Analysis. Table 3 presents a complexity comparison of Stable Diffusion (SD)-based Real-ISR methods, including the number
+
+Table 2: Quantitative results  $(\times 4)$  on RealSet65 testset. The best and second-best results are colored red and blue. In the one-step diffusion models, the best metric is bolded.  
+
+<table><tr><td>Method</td><td>NIQE↓</td><td>MUSIQ↑</td><td>MANIA↑</td><td>CLIPIQA↑</td></tr><tr><td>StableSR-s200</td><td>4.985</td><td>58.89</td><td>0.5269</td><td>0.4421</td></tr><tr><td>DiffBIR-s50</td><td>4.122</td><td>71.23</td><td>0.6371</td><td>0.5260</td></tr><tr><td>SeeSR-s50</td><td>4.689</td><td>69.79</td><td>0.6018</td><td>0.5657</td></tr><tr><td>ResShift-s15</td><td>6.730</td><td>59.36</td><td>0.5071</td><td>0.4416</td></tr><tr><td>ADDSR-s4</td><td>5.390</td><td>68.97</td><td>0.6075</td><td>0.5272</td></tr><tr><td>SinSR-s1</td><td>5.664</td><td>64.22</td><td>0.5338</td><td>0.5083</td></tr><tr><td>OSEDDiff-s1</td><td>4.224</td><td>69.04</td><td>0.6024</td><td>0.5234</td></tr><tr><td>ADDSR-s1</td><td>5.207</td><td>64.22</td><td>0.5258</td><td>0.4718</td></tr><tr><td>D3SR-s1</td><td>3.998</td><td>70.25</td><td>0.6298</td><td>0.5481</td></tr></table>
+
+of inference steps, inference time, parameter numbers, and MACs (Multiply-Accumulate Operations). All methods are evaluated on an NVIDIA A100 GPU.  $\mathrm{D}^3\mathrm{SR}$  achieves the fastest inference speed among all SD-based methods. Furthermore, our method supports using a fixed text embedding as the generation condition. Therefore, we do not require CLIP and other additional modules (such as DAPE used by OSEDiff and SeeSR, and ControlNet used by DiffBIR) for inference. Our  $\mathrm{D}^3\mathrm{SR}$  has the smallest number of model parameters during inference among Stable Diffusion (SD)-based methods, reducing the parameters by  $33\%$  compared to OSEDiff.
+
+In this section, we validate the effectiveness of two key components in  $\mathrm{D}^3\mathrm{SR}$ . More ablation experiments can be found in the supplementary materials.
+
+# 4.3 Ablation Study
+
+Perceptual Loss. Table 4a presents the impact of different perceptual loss functions, as well as only mean squared error (MSE) is applied as the spatial loss. Figure 6 showcases the visual outcomes of these experiments. The results indicate that incorporating perceptual loss is crucial for training SR models, as it facilitates the generation of more realistic details and enhances overall visual quality.
+
+Table 3: Complexity comparison  $(\times 4)$  among different methods, including sampling steps during inference, inference time, parameter count, and MACs. Inference time and MACs are tested for an output size of  $512\times 512$  with a single A100-40GB GPU.  
+
+<table><tr><td></td><td>StableSR</td><td>DiffBIR</td><td>SeeSR</td><td>ResShift</td><td>SinSR</td><td>OSEDiff</td><td>D3SR</td></tr><tr><td># Step</td><td>200</td><td>50</td><td>50</td><td>15</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Inference Time / s</td><td>11.50</td><td>7.79</td><td>5.93</td><td>0.71</td><td>0.16</td><td>0.35</td><td>0.11</td></tr><tr><td># Total Param / M</td><td>1.4×103</td><td>1.6×103</td><td>2.0×103</td><td>173.8</td><td>173.8</td><td>1.4×103</td><td>966.3</td></tr><tr><td># MACs / G</td><td>75,812</td><td>24,528</td><td>32,336</td><td>4,903</td><td>2,059</td><td>2,269</td><td>2,132</td></tr></table>
+
+Table 4: Ablation studys on the effects of perceptual losses and different discriminators.  
+(a) Ablation on different perceptual losses.  
+
+<table><tr><td>Loss Function</td><td>LPIPS↓</td><td>NIQE↓</td><td>MUSIQ↑</td><td>ManIQA↑</td></tr><tr><td>MSE</td><td>0.3626</td><td>4.446</td><td>65.35</td><td>0.5457</td></tr><tr><td>LPIPS</td><td>0.3190</td><td>4.123</td><td>66.41</td><td>0.6383</td></tr><tr><td>EA-LPIPS</td><td>0.3173</td><td>4.046</td><td>67.47</td><td>0.6403</td></tr><tr><td>DISTS</td><td>0.3463</td><td>3.800</td><td>67.55</td><td>0.6406</td></tr><tr><td>EA-DISTS</td><td>0.3150</td><td>3.747</td><td>68.69</td><td>0.6436</td></tr></table>
+
+(b) Ablation on different discriminators.  
+
+<table><tr><td>Discriminator</td><td>LPIPS↓</td><td>NIQE↓</td><td>MUSIQ↑</td><td>ManIQA↑</td></tr><tr><td>None</td><td>0.3862</td><td>6.962</td><td>62.36</td><td>0.5597</td></tr><tr><td>CNN</td><td>0.3402</td><td>6.139</td><td>64.36</td><td>0.5666</td></tr><tr><td>Diffusion-GAN</td><td>0.3200</td><td>4.518</td><td>67.51</td><td>0.5800</td></tr><tr><td>SD 2.1 (ours)</td><td>0.3166</td><td>3.925</td><td>68.08</td><td>0.6198</td></tr><tr><td>SDXL (ours)</td><td>0.3150</td><td>3.747</td><td>68.69</td><td>0.6436</td></tr></table>
+
+![](images/ec4d3e32f4b08c384470d2645489f2b3e84e7b2d06efc114d3119c5adbb9f0e2.jpg)  
+EA-LPIPS EA-DISTS Figure 6: Visual results  $(\times 4)$  of DFOSD with comparison of the checkerboard. The right one sh
+
+![](images/290d4616d3c806ff0d5c062e6961fa27411ddc32f0b5878e422072fde8a9e274.jpg)  
+LPIPS
+
+![](images/6735c543e16cab720858ce057d7c6e64accac875ddbd41f29038ab0471689539.jpg)  
+DISTS
+
+![](images/91e7f9dc6ecc84ee4d36ae8470b084e8d5c40b4e54cd24df42ad75319b14c694.jpg)  
+Figure 6: Visual results  $(\times 4)$  of DFOSD with different perceptual losses. The left side shows a comparison of the checkerboard. The right one shows content about some numbers, i.e., '24, 26, 28'.
+
+![](images/6676d7301ab0f868dba301304e4dc38e892451333df779f55f0e3b9083011f22.jpg)  
+DISTS
+
+![](images/2897fce7b2e8b488b660b06d22e36ae6b1e71d79e9bf78e85c2d3b11ad56d6c7.jpg)  
+LPIPS  
+EA-LPIPS  
+EA-DISTS
+
+Our proposed edge-aware DISTS (EA-DISTS) achieves the best performance across various image quality metrics and visual assessments. As shown in Fig. 6, EA-DISTS excels in producing highly realistic details, demonstrating its advantage in perceptual quality. This highlights the effectiveness of EA-DISTS in accurately restoring image textures and details, thereby significantly improving the visual quality.
+
+Diffusion Discriminator. We evaluate the impact of various discriminator modules on the training of  $\mathrm{D}^3\mathrm{SR}$ , including our used diffusion discriminator, vanilla discriminator (CNN), diffusion-GAN [57] style discriminator, and training without any discriminator. The CNN discriminator operates in the pixel space, while other discriminators operate in the latent space. The experimental results are shown in Table 4b. The comparison between the Diffusion-GAN discriminator and the SD 2.1 discriminator indicates that using a pre-trained Stable Diffusion (SD) model as the discriminator outperforms randomly initialized parameters.
+
+Next, we use SDXL 1.0-base as the discriminator while keeping the generator size unchanged to verify the impact of scaling for diffusion discriminators. As shown in the last two rows of Table 4b, the performance of the one-step diffusion model trained with SDXL as the discriminator outperforms the model trained with SD 2.1 as the discriminator, without requiring any modifications to the generator's architecture. This suggests that  $\mathrm{D}^3\mathrm{SR}$  can effectively leverage the strengths of more powerful pre-trained models, and enhance the performance of generator without compromising its efficiency. This breaks the upper bound imposed by the multi-step teacher diffusion model, making further performance improvement simpler.
+
+# 5 Conclusion
+
+In this work, we propose  $\mathrm{D}^3\mathrm{SR}$ , a One-Step Diffusion model for Real-ISR. Unlike previous methods that use diffusion distillation, our method breaks the limitations of the teacher. We propose the edge-aware DISTS (EA-DISTS) perceptual loss, which enhances the texture realism and visual quality of the generated images. Our adversarial training strategy allows  $\mathrm{D}^3\mathrm{SR}$  to outperform multi-step diffusion models in visual quality. Experiments show that  $\mathrm{D}^3\mathrm{SR}$  achieves superior performance and improves image realism. This highlights its potential for efficient image restoration.
+
+# Acknowledgments
+
+This work was supported by Shanghai Municipal Science and Technology Major Project (2021SHZDZX0102), the Fundamental Research Funds for the Central Universities, the Special Project on Technological Innovation Application for the 15th National Games and the National Paralympic Games under Grant 2025B01W0005.
+
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+[68] J. Liang, H. Zeng, and L. Zhang, "Details or artifacts: A locally discriminative learning approach to realistic image super-resolution," in CVPR, 2022. 7  
+[69] C. Chen, X. Shi, Y. Qin, X. Li, X. Han, T. Yang, and S. Guo, "Real-world blind super-resolution via feature matching with implicit high-resolution priors," in ACM MM, 2022. 7  
+[70] L. Zhang, L. Zhang, and A. C. Bovik, “A feature-enriched completely blind image quality evaluator,” TIP, 2015. 7  
+[71] J. Ke, Q. Wang, Y. Wang, P. Milanfar, and F. Yang, "Musiq: Multi-scale image quality transformer," in ICCV, 2021. 7  
+[72] S. Yang, T. Wu, S. Shi, S. Lao, Y. Gong, M. Cao, J. Wang, and Y. Yang, "Maniaq: Multi-dimension attention network for no-reference image quality assessment," in CVPR, 2022. 7  
+[73] J. Wang, K. C. Chan, and C. C. Loy, "Exploring clip for assessing the look and feel of images," in AAAI, 2023. 7
+
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\ No newline at end of file
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+# Unlocker: Disentangle the Deadlock of Learning from Label-noisy and Long-tailed Data
+
+Shu Chen $^{1}$ , Hongjun Xu $^{1}$ , Ruichi Zhang $^{1}$ , Mengke Li $^{2}$ , Yonggang Zhang $^{3}$ , Yang Lu $^{1}$ , Bo Han $^{4}$ , Yiu-ming Cheung $^{4}$ , Hanzi Wang $^{1}$
+
+<sup>1</sup>Key Laboratory of Multimedia Trusted Perception and Efficient Computing Ministry of Education of China, Xiamen University
+
+$^{2}$ College of Computer Science and Software Engineering, Shenzhen University
+
+$^{3}$ Hong Kong University of Science and Technology
+
+$^{4}$ Hong Kong Baptist University
+
+{chenshu, xuhongjun}@stu.xmu.edu.cn, luyang@xmu.edu.cn, zhangyg@ust.hk
+
+# Abstract
+
+In real world, the observed label distribution of a dataset often mismatches its true distribution due to noisy labels. In this situation, noisy labels learning (NLL) methods directly integrated with long-tail learning (LTL) methods tend to fail due to a dilemma: NLL methods normally rely on unbiased model predictions to recover true distribution by selecting and correcting noisy labels; while LTL methods like logit adjustment depends on true distributions to adjust biased predictions, leading to a deadlock of mutual dependency defined in this paper. To address this, we propose Unlocker, a bilevel optimization framework that integrates NLL methods and LTL methods to iteratively disentangle this deadlock. The inner optimization leverages NLL to train the model, incorporating LTL methods to fairly select and correct noisy labels. The outer optimization adaptively determines an adjustment strength, mitigating model bias from over- or under-adjustment. We also theoretically prove that this bilevel optimization problem is convergent by transferring the outer optimization target to an equivalent problem with a closed-form solution. Extensive experiments on synthetic and real-world datasets demonstrate the effectiveness of our method in alleviating model bias and handling long-tailed noisy label data. Code is available at https://github.com/ChenShu248/Unlocker.
+
+# 1 Introduction
+
+Long-tailed noisy label learning (LTNLL) focuses on addressing the coexistence of long-tailed distribution and noisy labels in datasets. Existing LTNLL methods [1, 2, 3, 4, 5, 6] are commonly under the potential assumption that the observed long-tailed distribution based on the noisy labels is consistent with the true distribution of the clean labels. However, empirical observations from real world reveal that noisy labels can alter the original true distribution, leading to a deviation between the observed distribution and the true underlying distribution of the dataset [7, 8, 9]. Due to the diverse real-world noise patterns, the deviations of the distribution are various, typically including three situations: consistent, relieve, and aggravate, as illustrated in Figure 1a.
+
+When addressing this issue of distribution deviation, neither noisy label learning (NLL) nor long tail learning (LTL) can be effective due to a deadlock dilemma of mutaul dependency. A mainstream in NLL highly relies on the model prediction to select and correct noisy labels to recover clean labels [10, 11, 12, 13, 14]. Yet in long-tailed scenarios, model predictions get biased towards head classes,
+
+![](images/36cf1476380f4213d5b54fa64f37e770dae0ecb75325c94a9c43b12ccf33cedb.jpg)  
+(a) Distribution deviation
+
+![](images/94131cee85c682fa5b252109f41e97bd53f42ec8babb8cabd5e80c7f39f75789.jpg)  
+(b) Deadlock dilemma  
+Figure 1: (a) Three typical scenarios of distribution deviation between the observed distribution and the true distribution. Consistent, relieve, and aggravate denote that the imbalance ratio (IR) of the observed distribution is identical to, lower than, and higher than the IR of the true distribution, respectively. (b) Deadlock between the noisy label learning (NLL) methods and the long-tail learning (LTL) methods: NLL relies on the unbiased prediction to recover true distribution, while LTL requires the true distribution to recalibrate the biased prediction, creating a circular dependency.
+
+posing challenges for selecting and correcting noisy samples in tail classes [1, 8]. Logit adjustment methods in LTL can recalibrate the biased predictions while relying on the true label distribution [15, 16, 17, 18, 19]. Due to the true distribution deviation, these methods fail to perform accurate adjustments. At this point, we can see a deadlock between NLL and LTL, as shown in Figure 1b. NLL relies on the unbiased predictions to restore a true distribution by selecting and correcting. Conversely, LTL can provide unbiased predictions but relies on a true distribution.
+
+To disentangle the deadlock, we propose a novel method Unlocker based on the bilevel optimization framework. The core idea is jointly estimating the true distribution of the training set and optimizing an appropriate parameter  $\tau$  for the adjusting strength. We define the inner problem as training a model using NLL adjusted by LTL, and the outer problem as optimizing the  $\tau$  based on the inner problem's solution. In this framework, model predictions are adjusted using both  $\tau$  and the estimated true distribution. This enables the model to better distinguish noisy labels. Improved noisy label selection and correction enhance distribution estimation accuracy. Accurate distribution estimation facilitates  $\tau$  optimization, promoting model to reduce bias on the test set. This iterative process continues until convergence, gradually breaking the deadlock. Our main contributions are as follows:
+
+- We define a new and challenging long-tailed noisy label problem of the deviation between the observed distribution based on noisy labels and the true distribution of the clean labels.  
+- We define a deadlock dilemma between the NLL and LTL, where their mutaul dependency renders both methods ineffective.  
+- We propose a novel method Unlocker which bases on the bilevel optimization to effectively combine NLL and LTL by adaptively optimizing the adjustment strength.
+
+# 2 Preliminaries
+
+# 2.1 Noisy Label Selection
+
+Noisy label learning has developed various methods to improve model performance by addressing label noise. A mainstream is to select noisy label samples from clean ones and correct the noisy labels [11, 12, 13, 14], which has been proven effective in noisy label learning tasks. Specifically, the selection of noisy labels relies on logit-based metrics, which are designed to maximize the discrimination between noisy and clean labels, enabling an efficient separation of the two subsets. Based on these metrics, the original dataset  $\mathcal{D}$  is partitioned into a clean subset  $\mathcal{D}_{clean}$  and a noisy subset  $\mathcal{D}_{noisy}$ . Then, the noisy labels in the  $\mathcal{D}_{noisy}$  are corrected by leveraging techniques such as semi-supervised learning [20, 21]. The overall training objective is formulated as the total loss  $L_{NLL}$ , which is the sum of the clean loss  $L_{clean}$  computed on  $\mathcal{D}_{clean}$  and the noisy loss  $L_{noisy}$  calculated on  $L_{noisy}$ . Conventionally, the cross-entropy loss is utilized for  $L_{clean}$  to ensure accurate classification on reliable data. For  $L_{noisy}$ , more robust loss functions are often applied to reduce the impact of noise during training [22].
+
+![](images/96168d05e0ceeced1f8e1d6d288bf36f695e3933667208ee4f2975997ac35d59.jpg)  
+(a)
+
+![](images/2a6c725ddb7ed285b896d4a723b4ae3ee323e3261601ddf0860978be2160b7de.jpg)  
+(b)  
+Figure 2: (a) Average prediction on CIFAR-10 test set of a NLL method DivideMix [11] adjusted by the true distribution prior  $\mathbb{P}_{\mathrm{train}}(y)$ . The training set is CIFAR-10 with imbalance ratio (IR) of 50 and  $40\%$  symmetric noise. The model is highly sensitive to the  $\tau$ , making it prone to under-adjustment or over-adjustment. (b) Test average predictions of models trained using logit adjustment (LA) across different  $\tau$  on a clean long-tailed CIFAR-10 dataset with IR of 50. As  $\tau$  increases, the average predictions transition from skewing toward head classes to uniformity and then to skew toward tail classes. (c) Corresponding test accuracies and KL divergences (between test average predictions and a uniform distribution) for varying  $\tau$ . When  $\tau = 2.0$  or  $\tau = 2.5$ , the trained model attains the highest accuracy and outputs the most uniform average predictions.
+
+![](images/08b33e725cef10c0440a400eac92ee38ee744633b9e275822ba4b8875b6efc47.jpg)  
+(c)
+
+# 2.2 Logit Adjustment
+
+Logit adjustment [17, 23, 19] is an effective technique for long-tailed learning. In a standard classification with  $C$  classes, the class posterior follows Bayes' theorem  $\mathbb{P}(y|x) = \frac{\mathbb{P}(x|y)\mathbb{P}(y)}{\mathbb{P}(x)}$ . Since  $\mathbb{P}(x)$  is constant across all classes, it can be omitted. Thus, cross-entropy training yields  $\mathbb{P}_{train}(y|x) \propto \mathbb{P}(x|y)\mathbb{P}_{train}(y)$ , where  $\mathbb{P}_{train}(y)$  is the class prior in training datasets. For balanced test datasets, a balanced prior  $\mathbb{P}_{\mathrm{bal}}(y) = \frac{1}{C}$  is desired, leading to  $\mathbb{P}_{\mathrm{bal}}(y|x) \propto \mathbb{P}(x|y)$ . Assuming the likelihood  $\mathbb{P}(x|y)$  is unchanged between the train and balanced datasets, we have the following relationship:
+
+$$
+\mathbb {P} _ {\text {t e s t}} (y | x) \propto \frac {\mathbb {P} _ {\text {t r a i n}} (y | x)}{\mathbb {P} _ {\text {t r a i n}} (y)} \propto \operatorname {s o f t m a x} \left(\boldsymbol {\theta} _ {y} (x) - \log \mathbb {P} _ {\text {t r a i n}} (y)\right), \tag {1}
+$$
+
+where  $\theta_y(x)$  is the model's logit for class  $y$ . To further improve flexibility, a temperature parameter  $\tau$  which modulates the adjustment intensity is introduced, leading to the general LA formulation:
+
+$$
+\underset {y \in [ C ]} {\arg \max } \operatorname {s o f t m a x} \left(\boldsymbol {\theta} _ {y} (x) - \tau \cdot \log \mathbb {P} _ {\text {t r a i n}} (y)\right). \tag {2}
+$$
+
+Typically,  $\tau > 0$  is used for post-hoc adjustment, while  $\tau < 0$  can be incorporated as logit adjustment loss during training.
+
+# 3 Unlocker: Breaking the Deadlock with Bilevel Optimization
+
+# 3.1 Motivation
+
+To break the deadlock, we first attempt to adjust the logits of a classical NLL method DivideMix [11] using the true distribution prior  $\mathbb{P}_{\mathrm{train}}(y)$ . Ideally, incorporating  $\mathbb{P}_{\mathrm{train}}(y)$ -based adjustments leads to unbiased confidence across all classes, reflected by a uniform average prediction on the test set, as depicted by the black line in Figure 2a. However, directly employing  $\mathbb{P}_{\mathrm{train}}(y)$  for LA results in suboptimal model performance due to under-adjustment or over-adjustment [24]. Specifically, as  $\mathbb{P}_{\mathrm{train}}(y)$  is the same, when the parameter  $\tau = 0.5$  (red line), the tail classes are over-adjusted, causing the average prediction to skew towards the tail and damaging the head, as shown in Figure 2a. Conversely, when  $\tau = 0.1$  (blue line), the adjustments are insufficient, remaining the predictions biased towards the heads. Without an appropriate  $\tau$  to modulate the adjusting strength, even with the precise  $\mathbb{P}_{\mathrm{train}}(y)$ , the model could not achieve an ideal unbiased state.
+
+To further investigate  $\tau$  in modulating model bias, we eliminate the influence of noisy label and implement a simple control variable experiment on a clean long-tailed dataset. We test different  $\tau$  of  $\{1.0, 1.5, 2.0, 2.5, 3.0, 3.5\}$  with the same  $\mathbb{P}_{train}(y)$ . As shown in Figure 2b, when  $\tau = 1.0$ , the tail classes are under-adjusted. As  $\tau$  increasing, the test average predictions become more uniform.
+
+![](images/10bd9fc80a7e9c3686be6c7457711b832a95d5670acd017ebb9b11a8fb8baef8.jpg)  
+Figure 3: The illustration of the proposed method Unlocker based on the bilevel optimization framework, which iteratively disentangle the deadlock between the noisy label learning (NLL) and long-tail learning (LTL). In epoch  $T$ , the inner optimization trains the model  $\theta$  using NLL integrated with logit adjustment in LTL. By adjusting logits as  $\theta_T'(x)$ , it fairly selects and corrects noisy labels, and trains  $\theta$ . The outer optimization adaptively updates the adjusting strength parameter  $\tau$  to modulate model bias towards balance. An EMA strategy is introduced to stabilize the adjustments after the whole optimizations completing. Parameters frozen during optimization are denoted by the snowflake (no gradient flow), while trainable parameters are marked by fire.
+
+When  $\tau = 2.0$  or  $\tau = 2.5$ , the average predictions are closest to uniformity, and the model achieved the highest test accuracy as depicted in Figure 2c. When further increasing  $\tau$ , the model gets over-adjusted, and the predictions skew towards the tail. This simple experiment demonstrates that an appropriate  $\tau$  is crucial for achieving unbiased model state and influences the test accuracy.
+
+# 3.2 Bilevel Optimization
+
+Based on the motivation, we propose Unlocker which utilizes the bilevel optimization framework to iteratively disentangle the deadlock. The inner layer optimization trains model using NLL integrating LA, and the outer layer optimizes a learnable parameter  $\tau$  to regulate the strength of adjustments. Through gradient coupling between layers, this framework maintains model capability against noisy labels while dynamically calibrating class bias.
+
+# 3.2.1 Inner Optimization
+
+Given a long-tailed noisy label training set  $\mathcal{D} = \{(x_i, y_i)\}_{i=1}^N$ , the inner layer employs NLL methods integrating LA to train model  $\theta$ . At epoch  $T$ , for the noisy label selection, an adjustment  $\alpha_T$  (post-hoc) is applied to the row logits  $\theta_T(x)$  (gradient-free) to achieve fairer noise recognition, particularly enhancing the recognition accuracy of the tail classes:
+
+$$
+\theta_ {T} ^ {\prime} (x) = \theta_ {T} (x) - \alpha_ {T}. \tag {3}
+$$
+
+Based on the adjusted logits  $\theta_T'(x)$ , selecting metrics of NLL methods are computed to divide  $\mathcal{D}$  into  $\mathcal{D}_{clean}$  and  $\mathcal{D}_{noisy}$ . For the noisy label correction for  $\mathcal{D}_{noisy}$ , we follow the method in [25] to apply post-hoc adjustment on the generation of corrected labels. With  $\mathcal{D}_{clean}$  and corrected  $\mathcal{D}_{noisy}$ , the model training loss  $\mathcal{L}_{NLL}$  can be computed and the model  $\theta_T$  updated at step  $t$  is given by:
+
+$$
+\theta_ {t + 1} = \theta_ {t} - \eta \nabla_ {\theta} \mathcal {L} _ {N L L} (\theta_ {t}), \tag {4}
+$$
+
+where  $\eta$  is the learning rate. Upon completing  $T$  epochs, the optimal inner parameters  $\theta_{T + 1}^{*}$  are fixed as the inner solution for the outer optimization.
+
+Before diving into the outer optimization, two critical components are required in addition to support the outer object: (i) the construction of a balanced and clean subset  $\mathcal{D}_{bc}$ , and (ii) the estimation of the training set class prior  $\pi(y)$ .
+
+Construct  $\mathcal{D}_{bc}$ . During the process of noisy label selection, we further filter the top  $q\%$  of samples with the highest confidence of clean as a high-purity clean subset. As the clean subset follows a long-tail distribution, we apply over-sampling and form the final balanced clean subset  $\mathcal{D}_{bc}$ .
+
+Estimate  $\pi(y)$ . As  $\mathbb{P}_{train}(y)$  is unknown in our long-tailed noisy labels problem, we estimate a proxy distribution prior  $\pi(y)$  to approximate it. After the correction for  $\mathcal{D}_{noisy}$  completing, we estimate  $\pi(y)$  based on the labels of the  $\mathcal{D}_{clean}$  and  $\mathcal{D}_{noisy}$ .
+
+# 3.2.2 Outer Optimization
+
+The outer optimizes a learnable parameter  $\tau$  to dynamically regulate the adjusting strength to modulate model bias towards balance. The core objective is to minimize the discrepancy between the model's class confidence and a uniform distribution. To measure model bias, we leverage the balanced clean validation subset  $\mathcal{D}_{bc}$  constructed in 3.2.1. Given the fixed inner parameters  $\theta_{T + 1}^{*}$ , the model's logits on  $\mathcal{D}_{bc}$  are  $\theta_{T + 1}^{*}(x)$  (gradient-free), and the  $\tau_{T}$ -regulated prediction distribution is:
+
+$$
+P (y \mid x; \tau_ {T}) = \operatorname {s o f t m a x} \left(\theta_ {T + 1} ^ {*} (x) - \tau_ {T} \cdot \log (\pi (y))\right), \tag {5}
+$$
+
+where  $\log (\pi)$  is the fixed class prior term estimated in Sec. 3.2.1. Averaging  $P(y\mid x;\tau_T)$  over  $\mathcal{D}_{bc}$  yields the class-averaged prediction  $\bar{P} (\tau_T) = \mathbb{E}_{x\sim \mathcal{D}_{bc}}P(y\mid x;\tau_T)$ , leading to the outer objective:
+
+$$
+\min  _ {\tau} K L (\bar {P} (\tau_ {T}) \| U), \quad U _ {c} = \frac {1}{C} (\forall c \in \{1, \dots , C \}), \tag {6}
+$$
+
+where  $U$  is a uniform prediction tensor,  $C$  is the number of classes. Minimizing this divergence adapts  $\tau_{T}$  to drive  $\bar{P} (\tau_T)$  toward uniformity, forcing balanced class confidence on the test set. The update of  $\tau_{T}$  at step  $t$  is derived via gradient descent:
+
+$$
+\tau_ {t + 1} = \tau_ {t} - \eta_ {\tau} \nabla_ {\tau} K L (\bar {P} (\tau_ {t}) \| U). \tag {7}
+$$
+
+After completing the optimization of  $\tau_{T}$ , the adjustments are calculated as  $\alpha_{\mathrm{T} + 1} = \tau_{\mathrm{T} + 1} \cdot \log \pi(y)$ . To ensure the stability of the adjustments during the training, especially when the  $\pi(y)$  fluctuates in the early training, we adopt an Exponential Moving Average (EMA) strategy to update the adjustments. At the  $T$ -th epoch, we update the adjustments as follows:
+
+$$
+\alpha_ {T + 1} = \beta \cdot \alpha_ {T + 1} + (1 - \beta) \cdot \alpha_ {T}, \tag {8}
+$$
+
+where  $\beta$  is a hyperparameter that controls the decay rate of the EMA.  $\alpha_{\mathrm{T + 1}}$  is then passes into epoch  $T + 1$  to adjust the inner-model training.
+
+# 3.2.3 Training Overview
+
+As illustrated in Figure 3, through the bilevel optimization, we effectively combine the NLL method and LA to handle long-tailed noisy label training data, breaking the mutual dependency. First, in inner optimization, we perform post-hoc logit adjustment on the selection of the NLL methods, enabling a more fair selection of noisy labels in tail classes. Based on the selection results, we construct a balanced clean subset  $\mathcal{D}_{bc}$  and partition the training set into  $\mathcal{D}_{clean}$  and  $\mathcal{D}_{noisy}$ . Next, we use the NLL methods to correct the noisy labels in  $\mathcal{D}_{noisy}$ , and estimate the  $\pi(y)$ . Then, the model is trained and adjusted using the corrected labels. In outer optimization, the  $\tau$  is optimized to minimize the discrepancy between model preference and uniform distribution based on the  $\mathcal{D}_{bc}$ . After the optimization of  $\tau$  is completed, we update the adjustments through EMA to ensure the stability of the training. The detailed pseudocode of the training process can be found in the Appendix A.1.
+
+# 3.3 Theoretical Analysis
+
+In this section, we establish the convergence of our bilevel optimization framework by deriving a closed-form approximation for the outer optimization of  $\tau$ . Specifically, we first prove that the outer objective is differentiable and exists a global minimum. Then, leveraging the uniform target distribution, we show that the optimal  $\tau$  can be approximated via least-squares minimization, reducing the bi-level problem to a normal optimization. This reduction guarantees the convergence of our method. The detailed processes of proof are provided in the Appendix A.2.
+
+Theorem 1 (Differentiability and Gradient Descent Applicability for Outer Optimization). Under the assumptions that  $\theta^{*}(x)$  is continuously differentiable with respect to  $\tau$  and  $\pi_c(y) > 0$  for all classes  $c\in \{1,2,\ldots ,C\}$ , the outer objective function  $J(\tau) = KL\left(\bar{P} (\tau)\parallel U\right)$  is continuously differentiable with respect to  $\tau$ . The gradient descent update rule  $\mathsf{tau}_{t + 1} = \tau_t - \eta_\tau \nabla_\tau J(\tau_t)$  ensures that  $J(\tau)$  decreases monotonically along the negative gradient direction.
+
+Based on the Theorem 1, we can further prove the existence of a global minimum.
+
+Lemma 1 (KL Divergence Bounds). By the definition of KL divergence, the objective function  $J(\tau) = KL(\bar{P}(\tau)\|U)$  establishes the following lower bound and upper bound:
+
+$$
+0 \leq J (\tau) = \log C - H (\bar {P} (\tau)) \leq \log C.
+$$
+
+Theorem 2 (Existence of Global Minimum). Given the continuity of  $J(\tau)$  and its lower bound  $J(\tau) \geq 0$  and upper bound  $\lim_{\tau \to \pm \infty} J(\tau) = \log C$  in Lemma 1, and  $\lim_{\tau \to \pm \infty} J(\tau) = \log C$ , there exists at least one global minimum point  $\tau^{*} \in \mathbb{R}$ .
+
+In summary, the objective function  $J(\tau)$  is bounded and has a global minimum, providing a theoretical guarantee for the optimization process. Leveraging the uniform prior  $U$ , we derive a closed-form approximation for  $\tau$  via least-squares minimization of the residual function.
+
+Proposition 1 (Least-Squares Closed-Form Solution). Define  $\mu_c = \frac{1}{N}\sum_x\theta_c(x)$ , the least-squares closed-form solution for  $\tau$  is:
+
+$$
+\tau^ {L S} = \frac {\sum_ {c} \left(\log \pi_ {c} (y) - \log \pi_ {1}\right) \left(\mu_ {c} - \mu_ {1}\right)}{\sum_ {c} \left(\log \pi_ {c} (y) - \log \pi_ {1}\right) ^ {2}}. \tag {9}
+$$
+
+This  $\tau^{LS}$  reduces the bi-level optimization problem to a single-level optimization, establishing the theoretical convergence guarantee for our framework.
+
+# 4 Experiments
+
+# 4.1 Experimental Setup
+
+Datasets. We conduct simulated experiments on CIFAR-10/100[26], including three cases: consistent, relieve, and aggravate which we summarize in Figure 1a. CIFAR-10/100 contains 50,000 training images and 10,000 test images of size  $32 \times 32$  pixels, where CIFAR-10 contains 10 classes and CIFAR-100 contains 100 classes. To simulate all scenarios, we first construct a long-tailed dataset with an imbalance ratio (IR) by exponential decay.  $IR = N_{max} / N_{min}$  represents the ratio of the number of samples of the majority class to the number of samples of the minority class.
+
+Before generating label noise, due to the real-world noise patterns are complex and diverse, we systematically explore the effects of existing noise addition methods on the imbalance ratio of the long-tailed dataset. The detailed analysis results refer to the Appendix A.4. We find that the joint noise hardly changes the original long-tailed distribution in the case of low imbalance ratio, the symmetry noise alleviates the imbalance ratio, and the t2h noise aggravates. Based on these findings, we introduce the joint noise to the clean long-tailed dataset to simulate the consistent case, use the symmetry noise to simulate the relieve case, and inject the t2h noise to simulate the aggravate case. The detailed noise addition methods refer to the Appendix A.3. In the following experiments, the imbalance rate is selected in  $\{10, 50, 100\}$ , and the noise rate  $\eta$  which describes the proportion of label noise in the training dataset is set to  $\{0.4, 0.6\}$ .
+
+We also evaluate the performance of our method on real-world datasets, including Red Mini-ImageNet [27], Clothing1M [28] and WebVision-50 [29]. Red Mini-ImageNet contains 100 classes with 50,000 training images and 5,000 testing images, annotated via controlled noise protocols. Clothing1M contains 1 million training images obtained from online shopping websites, with 50k, 14k, and 10k images split into clean labels for training, validation, and testing across 14 classes. WebVision consists of 2.4 million images from Google and Flickr, sharing the same 1,000 categories as ImageNet, and includes 50,000 human-annotated validation and test images. Following the experimental setting of [11], we use the first 50-class subset (WebVision-50) for training and evaluate on both WebVision validation set and ILSVRC12 validation set for the same 50 classes.
+
+Baselines. We compare our method with the following four categories of methods: (1) Noisy label learning (NLL) methods: including DivideMix [11], UNICON [12] and DPC [14]. (2) NLL methods with post-hoc LA based on real distribution (NLL+LA post-hoc). (3) NLL methods with LA based on real distribution during training (NLL+LA). (4) Long-tailed noisy label learning methods (LTNLL), including RoLT [1], TABASCO [8] and DaSC [4]. We select the baseline NLL methods to combine with our method. Additionally, for real-world dataset, we further include baselines: MentorNet [30], Co-teaching [10], HAR [31], UCL [2], RCAL [3], GSS [5].
+
+Table 1: Test accuracy  $(\%)$  comparison of different methods on the CIFAR-100 dataset under varying imbalance ratios (IR), noise types and noise rates  $\eta$ , involving three scenarios of true distribution shifts. * denotes results from the original papers. Green numbers indicate improvements of our method combined with the NLL method over the original NLL method. Boldface represents the best performance in each case.  
+
+<table><tr><td>Dataset</td><td colspan="10">CIFAR-10</td></tr><tr><td>Types</td><td colspan="2">consistent</td><td colspan="4">relieve</td><td colspan="4">aggregate</td></tr><tr><td>IR</td><td colspan="2">10</td><td colspan="2">50</td><td colspan="2">100</td><td colspan="2">10</td><td colspan="2">50</td></tr><tr><td>η</td><td>joint.40%</td><td>joint.60%</td><td>sym.40%</td><td>sym.60%</td><td>sym.40%</td><td>sym.60%</td><td>t2h.40%</td><td>t2h.60%</td><td>t2h.40%</td><td>t2h.60%</td></tr><tr><td>RoLT [1]</td><td>32.57</td><td>21.44</td><td>22.68</td><td>14.76</td><td>23.51*</td><td>16.61*</td><td>32.65</td><td>22.12</td><td>20.28</td><td>14.08</td></tr><tr><td>TABASCO [8]</td><td>55.31</td><td>45.40</td><td>40.91</td><td>31.28</td><td>36.91*</td><td>26.25*</td><td>45.37</td><td>43.60</td><td>32.18</td><td>17.53</td></tr><tr><td>DaSC [4]</td><td>58.06</td><td>45.64</td><td>41.96</td><td>32.17</td><td>36.40</td><td>29.59</td><td>55.67</td><td>39.93</td><td>34.03</td><td>13.47</td></tr><tr><td>DivideMix [11]</td><td>68.96</td><td>62.54</td><td>50.75</td><td>39.56</td><td>45.13</td><td>37.15</td><td>63.24</td><td>65.76</td><td>51.53</td><td>48.68</td></tr><tr><td>DivideMix+LA (post-hoc)</td><td>68.53</td><td>62.74</td><td>32.37</td><td>17.53</td><td>23.63</td><td>11.89</td><td>70.18</td><td>68.95</td><td>52.41</td><td>52.68</td></tr><tr><td>DivideMix+LA</td><td>68.53</td><td>63.87</td><td>16.30</td><td>5.31</td><td>7.31</td><td>2.51</td><td>68.67</td><td>63.94</td><td>46.56</td><td>42.93</td></tr><tr><td>DivideMix+Unlocker</td><td>71.95</td><td>68.97</td><td>60.38</td><td>53.34</td><td>54.26</td><td>45.07</td><td>72.20</td><td>69.05</td><td>62.59</td><td>58.07</td></tr><tr><td>vs. DivideMix</td><td>↑2.99</td><td>↑6.43</td><td>↑9.63</td><td>↑16.98</td><td>↑9.13</td><td>↑7.92</td><td>↑8.96</td><td>↑3.29</td><td>↑11.06</td><td>↑9.39</td></tr><tr><td>UNICON [12]</td><td>63.63</td><td>62.42</td><td>52.16</td><td>44.53</td><td>46.82</td><td>39.83</td><td>60.37</td><td>58.61</td><td>49.60</td><td>40.53</td></tr><tr><td>UNICON+LA (post-hoc)</td><td>67.15</td><td>63.13</td><td>53.18</td><td>45.27</td><td>46.84</td><td>40.44</td><td>65.41</td><td>58.81</td><td>50.32</td><td>42.78</td></tr><tr><td>UNICON+LA</td><td>62.39</td><td>61.40</td><td>8.53</td><td>50.19</td><td>2.75</td><td>1.80</td><td>67.58</td><td>58.64</td><td>55.94</td><td>41.67</td></tr><tr><td>UNICON+Unlocker</td><td>69.11</td><td>65.53</td><td>54.83</td><td>52.95</td><td>48.38</td><td>44.87</td><td>65.97</td><td>60.49</td><td>56.18</td><td>44.70</td></tr><tr><td>vs. UNICON</td><td>↑5.48</td><td>↑3.11</td><td>↑2.67</td><td>↑8.42</td><td>↑1.56</td><td>↑5.04</td><td>↑5.60</td><td>↑1.88</td><td>↑6.58</td><td>↑4.17</td></tr><tr><td>DPC [14]</td><td>70.81</td><td>54.91</td><td>44.66</td><td>34.84</td><td>39.43</td><td>23.71</td><td>70.03</td><td>11.32</td><td>36.73</td><td>1.04</td></tr><tr><td>DPC+LA (post-hoc)</td><td>69.97</td><td>55.13</td><td>39.77</td><td>30.88</td><td>39.59</td><td>21.62</td><td>70.05</td><td>22.85</td><td>40.67</td><td>2.44</td></tr><tr><td>DPC+LA</td><td>70.01</td><td>50.24</td><td>18.16</td><td>5.37</td><td>5.33</td><td>2.38</td><td>69.88</td><td>61.76</td><td>28.36</td><td>3.21</td></tr><tr><td>DPC+Unlocker</td><td>71.86</td><td>66.78</td><td>57.19</td><td>50.01</td><td>50.18</td><td>45.31</td><td>72.09</td><td>65.78</td><td>55.34</td><td>49.92</td></tr><tr><td>vs. DPC</td><td>↑1.05</td><td>↑11.84</td><td>↑12.53</td><td>↑15.17</td><td>↑10.75</td><td>↑21.60</td><td>↑2.06</td><td>↑54.46</td><td>↑18.61</td><td>↑48.88</td></tr></table>
+
+Table 2: Top 1 and Top 5 test accuracy on Webvision and ImageNet validation sets. The best results are bolded. The experimental results of other methods are from [5].  
+
+<table><tr><td rowspan="2">Test</td><td colspan="2">WebVision</td><td colspan="2">ILSVRC12</td></tr><tr><td>Top-1</td><td>Top-5</td><td>Top-1</td><td>Top-5</td></tr><tr><td>Standard</td><td>62.5</td><td>80.8</td><td>58.50</td><td>81.8</td></tr><tr><td>Co-teaching [10]</td><td>63.58</td><td>85.20</td><td>61.48</td><td>84.70</td></tr><tr><td>MentorNet [30]</td><td>63.00</td><td>81.40</td><td>57.80</td><td>79.92</td></tr><tr><td>HAR [31]</td><td>75.5</td><td>90.7</td><td>70.3</td><td>90.0</td></tr><tr><td>RoLT+ [1]</td><td>77.64</td><td>92.44</td><td>74.64</td><td>92.48</td></tr><tr><td>RCAL [3]</td><td>76.24</td><td>92.83</td><td>73.60</td><td>93.16</td></tr><tr><td>GSS [5]</td><td>83.64</td><td>95.86</td><td>74.17</td><td>95.22</td></tr><tr><td>DivideMix+Unlocker</td><td>83.79</td><td>96.55</td><td>73.21</td><td>96.37</td></tr></table>
+
+Table 3: Test accuracy  $(\%)$  on Red-Mini-Imagenet dataset. The best results are bolded. The experimental results of other methods are from [5].  
+
+<table><tr><td>IR</td><td colspan="2">10</td><td colspan="2">100</td></tr><tr><td>η</td><td>20%</td><td>40%</td><td>20%</td><td>40%</td></tr><tr><td>ERM</td><td>40.42</td><td>31.46</td><td>30.88</td><td>31.46</td></tr><tr><td>DivideMix [11]</td><td>48.76</td><td>48.96</td><td>33.00</td><td>34.72</td></tr><tr><td>UNICON [12]</td><td>40.18</td><td>41.64</td><td>31.86</td><td>31.12</td></tr><tr><td>HAR [31]</td><td>46.61</td><td>38.71</td><td>32.60</td><td>31.30</td></tr><tr><td>ULC [2]</td><td>48.12</td><td>47.06</td><td>34.24</td><td>34.84</td></tr><tr><td>TABASCO [8]</td><td>50.20</td><td>49.68</td><td>37.20</td><td>37.12</td></tr><tr><td>GSS [5]</td><td>52.33</td><td>50.91</td><td>40.25</td><td>36.58</td></tr><tr><td>DivideMix+Unlocker</td><td>53.29</td><td>51.48</td><td>41.51</td><td>37.14</td></tr></table>
+
+Implementation Details. To ensure a fair comparison with existing methods, we keep the training configurations consistent with the baseline NLL methods. Specifically, we employ the 18-layer ResNet as the backbone architecture. The mini-batch size is fixed at 256. All models are optimized using SGD with a momentum of 0.9. A random seed of 123 is used across all experiments to ensure reproducibility. For NLL methods combined with LA in baselines, the  $\tau$  is set to 1.0. For our proposed method Unlocker, the learnable parameter  $\tau$  is initialized to 1.0 and optimized using SGD with a momentum of 0.9. The initial learning rate for  $\tau$  is set to 0.1, which is adjusted to 0.01 at the 150-th epoch. The  $\beta$  in EMA to update adjustments is set to 0.9. All experiments are executed on a GeForce RTX 3090 GPU using the PyTorch 1.8.0 framework to maintain hardware consistency.
+
+# 4.2 Results on Simulated Scenarios
+
+We conduct a comprehensive performance comparison across three scenarios of the distribution deviations in Table 1. Our method combines with the NLL methods (NLL+Unlocker) demonstrates superiority over the LTNLL methods under all three conditions. Regarding NLL methods, integrating Unlocker yields substantial accuracy improvements. On CIFAR-100, the combination achieves performance boosts ranging from  $1.05\%$  to  $54.46\%$ . In contrast, NLL methods directly combined
+
+Table 4: Test accuracy  $(\%)$  comparison of different methods on the Clothing1M dataset The best results are in bold. The experimental results of other methods are from [5].  
+
+<table><tr><td>Methods</td><td>CE</td><td>MentorNet [30]</td><td>Co-teaching [10]</td><td>DivideMix [11]</td><td>ULC [2]</td><td>RCAL+ [3]</td><td>GSS [5]</td><td>DivideMix+Unlocker</td></tr><tr><td>Accuracy</td><td>65.42</td><td>67.25</td><td>67.94</td><td>74.76</td><td>74.87</td><td>74.97</td><td>75.83</td><td>76.94</td></tr></table>
+
+![](images/8ae9dc5bbfce525d7f1264ff57bf7e532725681ff0483da91e5faf8aebcb6ff8.jpg)  
+(a)
+
+![](images/b895992c93100e8b116a20fc47fcab5c46d20013a79e0cbdbc2022b3a786f72d.jpg)  
+(b)
+
+![](images/810078c7b8658a3d8ebdd277f09f05ae35b1d5a7141252042635eedd50a5aedc.jpg)  
+(a)
+
+![](images/9a713076427de6965a8abfa7ec36271e20f897d32a34836931f39a11033bc020.jpg)  
+Figure 4: Bias comparison of different models trained on CIFAR-100 (IR=100, sym.40%) (a) KL divergence between the average predictions on CIFAR-100 test set and a uniform prediction during training. (b) The average predictions in the last epoch on CIFAR-100 test set.  
+(b)
+
+![](images/3e240e221936fef39f303498864bbbdcd73d5ad452f8e2e63cf55b2372833396.jpg)  
+(c)  
+Figure 5: (a) The estimated training set distribution at the last epoch on CIFAR-100 with an imbalance ratio (IR) of 10 and t2h.  $60\%$ . (b) The convergence trajectory of the parameter  $\tau$  during training. (c) The KL divergence between the average predictions on the balanced clean subset  $\mathcal{D}_{bc}$  and the CIFAR-100 test set. (b) and (c) are conducted on CIFAR-100 with IR=100 and sym.  $40\%$ .
+
+with true-distribution-based LA (both post-hoc and training-time LA) often suffer from the problem of over-adjustment, causing training collapse where predictions concentrate on a single class. Our method adaptively adjusts the parameter  $\tau$  during training, effectively alleviating the issues of over- or under-adjustment. This mechanism enables a collaboration with NLL methods, jointly addressing the challenges of long-tailed noisy labels data. As a result, our method achieves the best performance among all methods, showing its robust adaptability to varying imbalance ratios and noise types. The results of the comparison experiment on CIFAR-10 are presented in the Appendix A.5.
+
+# 4.3 Results on Real-world Datasets
+
+We performed comparative experiments on three real-world datasets to validate the robustness of our method in practical setting. As shown in Table 2, DivideMix+Unlocker achieves  $83.79\%$  Top-1 and  $96.55\%$  Top-5 accuracy on the WebVision-50 validation set, outperforming all baselines. On the ILSVRC12 validation set, it achieves the highest  $96.37\%$  Top-5 accuracy and competitive  $73.21\%$  Top-1 accuracy. For Red-Mini-Imagenet, as shown in Table 3, our method attains  $53.29\%$  accuracy under  $\mathrm{IR} = 10$  and  $\eta = 20\%$ , outperforming SOTA  $(52.33\%)$  and improving upon DivideMix by  $4.5\%$ . Under conditions  $(\mathrm{IR} = 100, \eta = 40\%)$ , it maintains  $36.37\%$  accuracy. On Clothing1M, as shown in Table Table 4, our method outperforms other methods by achieving  $76.94\%$  accuracy. These experiments fully verified the ability of our method in real-world long-tailed noisy label scenarios, enabling NLL methods to achieve a more uniform state against long-tail bias.
+
+# 4.4 Effectiveness Study and Discussions
+
+Average Prediction Analysis. To validate the effectiveness of our method in reducing models bias to achieve a more uniform state, we conduct experiments by training the models of different methods on CIFAR-100 training set with  $IF = 100$  and  $\eta = \text{sym.40\%}$ , and outputting models' average predictions on the CIFAR-100 test set. In Figure 4a, during training, the KL divergence between the
+
+test average prediction of the DivideMix and the uniform prediction stabilizes at a relatively high value, as depicted by the orange line. This means the model's average prediction is still far away from uniform, indicating persistent bias of the model. In contrast, by combining with our method Unlocker, the KL divergence down to approximately 0.08, signifying that the model's test average predictions approach uniformity and the model gets more unbiased. Figure 4b demonstrates the average prediction in the last epoch of the different model. As the orange bars showing, the test average predictions of the DivideMix skew toward head classes, neglecting tail classes. Conversely, our method's prediction (blue bars) distributes evenly across classes. These results collectively validate that our method effectively mitigates long-tailed impact and promotes balancer model state.
+
+Effectiveness of Distribution Estimation. To verify the effectiveness of the distribution estimation module in our method, we conduct experiments on CIFAR-100 with IR = 10 and t2h.60%. As shown in the Figure 5a, the red line denotes the true distribution of the training set, the orange line represents the observed distribution based on noisy labels, and the blue line is the training set distribution estimated by our method. Notably, the imbalance ratio of the observed distribution is significantly higher than that of the true distribution, indicating that noisy labels aggravate the imbalance ratio of the true distribution. Our method estimates the distribution (blue line) that closely aligns with the true distribution (red line), demonstrating its capability to accurately capture the training set distribution.
+
+Convergence of  $\tau$ . We demonstrate the convergence of the parameter  $\tau$  for our method trained on the CIFAR-100 dataset with  $\mathrm{IR} = 100$  and  $\mathrm{sym.40\%}$ . As shown in the Figure 5b, our method achieves the stable convergence of the parameter  $\tau$ . Specifically, the optimization of  $\tau$  starts after the warm-up of 30 epochs. In the first 100 epochs, the value of  $\tau$  decreases rapidly. After 100 epochs, the descent rate slows down, and it converges to a value near 0.48 around 150 epochs.
+
+Effectiveness of  $\mathcal{D}_{bc}$ . The balanced clean subset  $\mathcal{D}_{bc}$  serves as a validation set to measure the model bias. Thus it is critical that  $\mathcal{D}_{bc}$  can reflect the model's class confidence. To validate this, we conduct experiments on CIFAR-100 with IR = 100 and sym.40%, tracking the average predictions of the model on  $\mathcal{D}_{bc}$  and the test set at each epoch. By computing their KL divergence, as shown in the Figure 5c, the discrepancy remains at a relatively low level of approximately 0.12. While there are slight fluctuations, they are in an acceptable small range. This result indicates that the average predictions on  $\mathcal{D}_{bc}$  closely align with those on the test set, demonstrating that  $\mathcal{D}_{bc}$  reflects model bias.
+
+# 5 Related Work
+
+Long-tailed and noisy label learning faces challenges in distinguishing clean samples from noisy ones in tail classes. RoLT [1] proposes a prototype-based noise detection method using class centroid distances to select noisy samples. ULC [2] combines class-specific noise modeling with uncertainty quantification to account for cognitive and incidental uncertainties. TABASCO [8] employs a weighted JS divergence and adaptive centroid distance to distinguish. These methods often integrate semi-supervised learning to refine predictions after noise detection. HAR [31] applies heteroscedastic adaptive regularization to high-uncertainty and low-density data points. RCAL [3] leverages unsupervised contrastive learning to eliminate noisy samples and restore representation distributions, enhancing model generalization. Detailed related work on long-tail learning and noisy label learning individually is provided in the Appendix A.6.
+
+# 6 Conclusion
+
+In this paper, we address the challenging problem of long-tailed noisy label learning where the observed distribution based on noisy label deviates from the true distribution. When addressing this issue, a deadlock dilemma between noisy label learning (NLL) and long tail learning (LTL) arises. To disentangle the deadlock and tackle the long-tailed noisy label problem, we propose Unlocker, a bilevel optimization framework that iteratively optimizes an adjustment strength parameter  $\tau$  to effectively combine the NLL methods and LTL methods. Extensive experiments on synthetic and real-world datasets demonstrate that Unlocker significantly outperforms SOTA methods. However, our method has limitations: (i) potential impurity of  $\mathcal{D}_{bc}$ , and (ii) approximation errors in distribution estimation. Our future research will focus on address these limitations.
+
+# Acknowledgements
+
+This study was supported in part by the National Natural Science Foundation of China under Grants 62376233, 62306181 and 62376235; in part by the Natural Science Foundation of Fujian Province under Grant 2024J09001; in part by the RGC Young Collaborative Research Grant C2005-24Y; in part by the NSFC / Research Grants Council (RGC) Joint Research Scheme under Grant N_HKBU214/21; in part by the General Research Fund of RGC under the Grants 12201323 and 12200725; in part by the RGC Senior Research Fellow Scheme under the Grant SRFS2324-2S02; and in part by Xiaomi Young Talents Program. YGZ was funded by Inno HK Generative AI R&D Center.
+
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+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: The abstract and introduction clearly state the problem of the deadlock between NLL and LTL, the proposed bilevel optimization framework Unlocker, and its effectiveness on synthetic/real datasets, which aligns with the experimental results in Section 5.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: The conclusion mentions limitations: reliance on uniform test distribution assumption, potential impurity of  $\mathcal{D}_{\mathrm{bc}}$  affecting  $\tau$  optimization, and room for improving  $\pi(y)$  estimation.
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory Assumptions and Proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: Theorem 1 in Section 4.4 provides assumptions and a detailed proof via equivalence transformation to a convex problem with a closed-form solution.
+
+Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental Result Reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: We provide details on each parameter in the datasets, baselines, and implementations in our experimental setup.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.
+
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
+
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
+
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
+
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
+
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: The experimental code is available and the data sources are publicly accessible.
+
+Guidelines:
+
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+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.  
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental Setting/Details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: Details including dataset split (IR=10/50/100, noise type: joint/sym/t2h), baseline (NLL, LTNLL) and optimizer (SGD) are described in Section 5.1.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment Statistical Significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [NA]
+
+Justification: We do not report error bars or statistical tests.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).
+
+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).  
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments Compute Resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: Experiments used a GeForce RTX 3090 GPU with PyTorch 1.8.0.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code Of Ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: We conduct the research project following NeurIPS Code of Ethics faithfully.
+
+Guidelines:
+
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+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader Impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [NA]
+
+Justification: The work does not seem to raise any (negative) societal impacts.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
+
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: Our method does not involve data or models that might lead to intended misuse.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [NA]
+
+Justification: Datasets (CIFAR, Clothing1M, WebVision) and baselines (DivideMix, UNICON) are cited with their original licenses.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
+
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New Assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: We provide pseudocode and parameters to facilitate reproduction.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and Research with Human Subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: No human subjects research is conducted.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+# Answer: [NA]
+
+Justification: LLM was not a component of this study
+
+# Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Appendix / Supplemental Material
+
+# A.1 Detailed Training Process
+
+The following is the process of our proposed method Unlocker, which leverages a bilevel optimization framework. Within each epoch, the inner optimization employs NLL methods with LA to train model, while the outer loop optimizes the learnable parameter  $\tau$  to dynamically scale the strength of LA. This process adaptively tunes  $\tau$  to integrate NLL methods and LA, iteratively disentangling the NLL-LTL deadlock and enhancing model robustness against long-tailed noisy label data.
+
+Algorithm 1 Detailed training process of Unlocker  
+Input: Training data  $\mathcal{D} = \{(x_i,y_i)\}_{i = 1}^N$  , Number of epochs  $N_{\mathrm{epoch}}$  , NLL algorithm A, learning rates  $\eta_{\tau}$  , EMA decay  $\beta$    
+Output: Trained model  $\theta$  , Optimized  $\tau$    
+Initialize: Model Parameters  $\theta_0$  , Learnable Parameter  $\tau_0\gets 1.0$  , Class Prior  $\pi (y)\gets$  EstimatePrior(D), adjustments  $\alpha \leftarrow \tau_0\cdot \log (\pi (y))$    
+for epoch  $= T$  from 1 to  $N_{\mathrm{epoch}}$  do   
+// Inner Optimization   
+for  $k = 1$  to  $K$  do   
+ $\mathcal{D}_k\gets \mathrm{GetBatch}(\mathcal{D},k)$  Inference Mode:  $\theta_T^\prime \gets \theta_T(\mathcal{D}_k) - \alpha_T$  Post-hoc logit adjustment   
+end for   
+ $\mathcal{D}_{clean},\mathcal{D}_{noisy},\mathcal{D}_{bc}\gets \mathcal{A}(\theta_T^\prime)$  Select noisy labels according to A   
+ $\mathcal{D}_{noisy}\gets \mathcal{A}(\theta_T^\prime ,\mathcal{D}_{noisy})$  Correct noisy labels according to A   
+ $\pi (y)\gets$  EstimatePrior(Dclean, Dnoisy) Estimate class prior of D   
+for  $k = 1$  to  $K$  do   
+ $\mathcal{D}_k\gets \mathrm{GetBatch}(\mathcal{D},k)$ $\begin{array}{r}\theta_T'\gets \left\{\theta_T + \alpha_T,\quad \text{if}\mathcal{D}_k\subset \mathcal{D}_{\mathrm{clean}}\right.\\ \left.\theta_T,\quad \text{otherwise}\right.\\ \mathcal{L}_{NLL}\gets \mathcal{A}(\theta_T'(x),\mathcal{D}_k)\\ \theta_{k + 1}\gets \theta_k - \eta \nabla_\theta \mathcal{L}_{NLL}(\theta_k)\\ \end{array}$  logit adjustment   
+end for   
+// Outer Optimization   
+for  $k = 1$  to  $B$  do   
+ $\mathcal{D}_k\gets \mathrm{GetBatch}(\mathcal{D}_{bc},k)$  Inference Mode:  $\theta_{T + 1}(\mathcal{D}_k)$  Post-hoc logit adjustment   
+ $\theta_{T + 1}^{\prime}(\mathcal{D}_k)\gets \theta_{T + 1}(\mathcal{D}_k) - \tau_k\cdot \log \pi (y)$ $\mathcal{L}_{\tau}\gets KL\left(\mathbb{E}_{x\sim D_k}\mathrm{softmax}(\theta_{T + 1}'(\mathcal{D}_k))\parallel U\right)$  Compute outer loss according to Eq.6   
+ $\tau_{k + 1} = \tau_k - \eta_\tau \nabla_\tau \mathcal{L}_\tau (\tau_k)$  Optimize  $\tau$    
+end for   
+ $\alpha_{T + 1}\gets \tau_{T + 1}\cdot \log (\pi (y))$ $\alpha_{T + 1}\gets \beta \cdot \alpha_{T + 1} + (1 - \beta)\cdot \alpha_T$  EMA update for adjustments   
+end for   
+return  $\theta ,\tau$
+
+# A.2 Theoretical Proof
+
+Proof of Theorem 1. We prove the differentiability of the outer optimization with the following steps: Step 1: Objective Function Expansion Substituting  $\tilde{P} (\tau_T) = \mathbb{E}_{x\sim \mathcal{D}_{bc}}P(y\mid x;\tau_T)$  into the KL
+
+divergence  $J(\tau) = KL\left(\bar{P} (\tau)\parallel U\right),U_{c} = \frac{1}{C}$  , we get:
+
+$$
+\begin{array}{l} J (\tau) = K L \left(\bar {P} (\tau) \| U\right) \\ = \sum_ {c = 1} ^ {C} \bar {P} _ {c} (\tau) \log \frac {\bar {P} _ {c} (\tau)}{U _ {c}} \\ = \sum_ {c = 1} ^ {C} \left[ \mathbb {E} _ {x \sim \mathcal {D} _ {b c}} P (y = c \mid x; \tau) \right] \log \frac {\mathbb {E} _ {x \sim \mathcal {D} _ {b c}} P (y = c \mid x ; \tau)}{1 / C} \\ = \sum_ {c = 1} ^ {C} \left[ \mathbb {E} _ {x \sim \mathcal {D} _ {b c}} P (y = c \mid x; \tau) \right] (\log \left[ \mathbb {E} _ {x \sim \mathcal {D} _ {b c}} P (y = c \mid x; \tau) \right] + \log C) \\ = \mathbb {E} _ {x \sim \mathcal {D} _ {b c}} \sum_ {c = 1} ^ {C} P (y = c \mid x; \tau) \log \left[ \mathbb {E} _ {x ^ {\prime} \sim \mathcal {D} _ {b c}} P (y = c \mid x ^ {\prime}; \tau) \right] + \log C \\ = \mathbb {E} _ {x \sim D _ {\mathrm {b c}}} \sum_ {c = 1} ^ {C} P _ {c} \log P _ {c} + \log C, \tag {10} \\ \end{array}
+$$
+
+where  $P_{c} = P(y = c \mid x; \tau)$  denotes the conditional probability that a sample  $x$  belongs to class  $c$  under the adjusting of  $\tau$ .
+
+Step 2: Gradient Derivation Taking the derivative of (10) with respect to  $\tau$  using the chain rule:
+
+$$
+\nabla_ {\tau} \mathcal {J} (\tau) = \mathbb {E} _ {x \sim D _ {\mathrm {b c}}} \sum_ {c = 1} ^ {C} \frac {\partial P _ {c}}{\partial \tau} (\log P _ {c} + 1). \tag {11}
+$$
+
+Given the component  $\log P_{c} + 1 = \theta_{c}(x) - \tau \cdot \log \pi_{c}(y) - \log \sum_{k=1}^{C} e^{\theta_{k}(x) - \tau \cdot \log \pi_{k}} + 1$  is differentiable in  $\tau$ , we focus on analyzing the differentiability of  $\frac{\partial P_{c}}{\partial \tau}$ . For the softmax function  $P_{c} = \frac{e^{z_{c}}}{\sum_{k=1}^{C} e^{z_{k}}}$  with  $z_{c} = \theta_{c}(x) - \tau \cdot \log \pi_{c}(y)$ , we derive  $\frac{\partial P_{c}}{\partial \tau}$  as follows:
+
+$$
+\begin{array}{l} \frac {\partial P _ {c}}{\partial \tau} = \frac {\partial}{\partial \tau} \left(\frac {e ^ {z _ {c}}}{\sum_ {k = 1} ^ {C} e ^ {z _ {k}}}\right) \\ = \frac {\frac {\partial e ^ {z _ {c}}}{\partial \tau} \cdot \sum_ {k = 1} ^ {C} e ^ {z _ {k}} - e ^ {z _ {c}} \cdot \frac {\partial}{\partial \tau} \sum_ {k = 1} ^ {C} e ^ {z _ {k}}}{\left(\sum_ {k = 1} ^ {C} e ^ {z _ {k}}\right) ^ {2}} \\ = \frac {e ^ {z _ {c}} \cdot \frac {\partial z _ {c}}{\partial \tau} \cdot \sum_ {k = 1} ^ {C} e ^ {z _ {k}} - e ^ {z _ {c}} \cdot \sum_ {k = 1} ^ {C} e ^ {z _ {k}} \cdot \frac {\partial z _ {k}}{\partial \tau}}{\left(\sum_ {k = 1} ^ {C} e ^ {z _ {k}}\right) ^ {2}} \\ = \frac {e ^ {z _ {c}}}{\sum_ {k = 1} ^ {C} e ^ {z _ {k}}} \cdot \frac {\frac {\partial z _ {c}}{\partial \tau} \cdot \sum_ {k = 1} ^ {C} e ^ {z _ {k}} - \sum_ {k = 1} ^ {C} e ^ {z _ {k}} \cdot \frac {\partial z _ {k}}{\partial \tau}}{\sum_ {k = 1} ^ {C} e ^ {z _ {k}}} \\ = P _ {c} \cdot \left(\frac {\partial z _ {c}}{\partial \tau} - \sum_ {k = 1} ^ {C} \frac {e ^ {z _ {k}}}{\sum_ {k = 1} ^ {C} e ^ {z _ {k}}} \cdot \frac {\partial z _ {k}}{\partial \tau}\right) \\ = P _ {c} \cdot \left(\frac {\partial z _ {c}}{\partial \tau} - \sum_ {k = 1} ^ {C} P _ {k} \cdot \frac {\partial z _ {k}}{\partial \tau}\right). \tag {12} \\ \end{array}
+$$
+
+Differentiate  $z_{c} = \theta_{c}(x) - \tau \cdot \log \pi_{c}(y)$  with respect to  $\tau$ , we have  $\frac{\partial z_c}{\partial\tau} = -\log \pi_c(y)$  and  $\frac{\partial z_k}{\partial\tau} = -\log \pi_k$ . Substituting  $\frac{\partial z_c}{\partial\tau}$  and  $\frac{\partial z_k}{\partial\tau}$  into (12), we have:
+
+$$
+\begin{array}{l} \frac {\partial P _ {c}}{\partial \tau} = P _ {c} \cdot \left(- \log \pi_ {c} (y) - \sum_ {k = 1} ^ {C} P _ {k} \cdot (- \log \pi_ {k})\right) \\ = P _ {c} \cdot \left(- \log \pi_ {c} (y) + \sum_ {k = 1} ^ {C} P _ {k} \log \pi_ {k}\right). \tag {13} \\ \end{array}
+$$
+
+Given the continuous differentiability of  $P_{c}$  and the constancy of  $\pi_c(y)$ , the gradient  $\frac{\partial P_c}{\partial\tau}$  exists and is differentiable. Substituting (13) back into (11), the gradient  $\nabla_{\tau}J(\tau)$  is thus continuously differentiable, and simplified to:
+
+$$
+\nabla_ {\tau} J (\tau) = \mathbb {E} _ {x \sim D _ {\mathrm {b c}}} \sum_ {c = 1} ^ {C} P _ {c} \left(- \log \pi_ {c} (y) + \sum_ {k = 1} ^ {C} P _ {k} \log \pi_ {k}\right) (\log P _ {c} + 1). \tag {14}
+$$
+
+Thus, by the differentiability of  $\mathcal{I}(\tau)$ , gradient descent guarantee convergence to a local minimum.
+
+Proof of Lemma 1. We establish the bounds of  $J(\tau)$  as follows. We first expand the outer optimization function according to the definition of KL divergence:
+
+$$
+\begin{array}{l} J (\tau) = \operatorname {K L} \left(\bar {P} (\tau) \| U\right) = \sum_ {c = 1} ^ {C} \bar {P} _ {c} (\tau) \log \left(\frac {\bar {P} _ {c} (\tau)}{U _ {c}}\right) \\ = \sum_ {c = 1} ^ {C} \bar {P} _ {c} (\tau) \log \left(\frac {\bar {P} _ {c} (\tau)}{\frac {1}{C}}\right) \\ = \sum_ {c = 1} ^ {C} \bar {P} _ {c} (\tau) \log \left(\bar {P} _ {c} (\tau) \cdot C\right) \\ = \sum_ {c = 1} ^ {C} \bar {P} _ {c} (\tau) [ \log \bar {P} _ {c} (\tau) + \log C ] \\ = \sum_ {c = 1} ^ {C} \bar {P} _ {c} (\tau) \log \bar {P} _ {c} (\tau) + \log C \cdot \sum_ {c = 1} ^ {C} \bar {P} _ {c} (\tau). \tag {15} \\ \end{array}
+$$
+
+Note that  $\sum_{c=1}^{C} \bar{P}_c(\tau) = 1$  (since  $\bar{P}(\tau)$  is a probability distribution) in the second term. The first term is the negative entropy  $\sum_{c=1}^{C} \bar{P}_c(\tau) \log \bar{P}_c(\tau) = -H(\bar{P}(\tau))$  of  $\bar{P}(\tau)$ . Thus, we obtain:
+
+$$
+J (\tau) = \log C - H (\bar {P} (\tau)). \tag {16}
+$$
+
+Lower Bound The entropy  $H(\bar{P}(\tau))$  is maximized when  $\bar{P}(\tau)$  is uniform, achieving  $H(\bar{P}(\tau)) = \log C$ . Therefore:
+
+$$
+J (\tau) = \log C - H (\bar {P} (\tau)) \geq \log C - \log C = 0. \tag {17}
+$$
+
+Equality holds if and only if  $\bar{P} (\tau) = U$ . Thus, the lower bound of  $J(\tau)$  is 0.
+
+Upper Bound The entropy  $H\left(\bar{P} (\tau)\right) \geq 0$  for any probability distribution  $\bar{P} (\tau)$ . Therefore:
+
+$$
+J (\tau) = \log C - H (\bar {P} (\tau)) \leq \log C - 0 = \log C. \tag {18}
+$$
+
+Equality holds when  $\bar{P} (\tau)$  is a degenerate distribution (i.e., one class has probability 1 and others 0). Combining these results, we conclude:
+
+$$
+0 \leq J (\tau) \leq \log C. \tag {19}
+$$
+
+□
+
+Proof of Theorem 2. We prove the existence of the global minimum point from the perspectives of continuity, lower boundedness and upper asymptotic convergence. Given that we have proven the continuity of  $J(\tau)$  in Theorem1, we focus on proving the lower boundedness and upper asymptotic convergence of  $J(\tau)$ .
+
+Lower Boundedness By Lemma 1,  $J(\tau) = \log C - H(\bar{P}(\tau)) \geq 0$  for all  $\tau \in \mathbb{R}$ .
+
+Upper Asymptotic Convergence As  $\tau \to +\infty$ , let  $c^* = \arg \min_y \log \pi(y)$ . Then,  $-\tau \cdot \log \pi(c^*)$  dominates, making:
+
+$$
+\lim  _ {\tau \rightarrow + \infty} P (y = c \mid x; \tau) = \lim  _ {\tau \rightarrow + \infty} \frac {e ^ {\theta_ {c} (x) - \tau \cdot \log \pi (c)}}{\sum_ {k = 1} ^ {C} e ^ {\theta_ {k} (x) - \tau \cdot \log \pi (k)}} = \left\{\begin{array}{l l}1,&\text {i f} c = c ^ {*},\\0,&\text {o t h e r w i s e .}\end{array}\right. \tag {20}
+$$
+
+Consequently,  $\bar{P} (\tau)\rightarrow \delta_{c^*}$  , a Dirac-delta distribution as follows:
+
+$$
+\begin{array}{l} \lim  _ {\tau \rightarrow + \infty} \bar {P} _ {c} (\tau) = \lim  _ {\tau \rightarrow + \infty} \mathbb {E} _ {x \sim \mathcal {D} _ {b c}} P (y = c \mid x; \tau) \\ = \mathbb {E} _ {x \sim \mathcal {D} _ {b c}} \lim  _ {\tau \rightarrow + \infty} P (y = c \mid x; \tau) \\ = \left\{ \begin{array}{l l} 1, & \text {i f} c = c ^ {*}, \\ 0, & \text {o t h e r w i s e .} \end{array} \right. \tag {21} \\ \end{array}
+$$
+
+Substituting  $\bar{P} (\tau)$  into  $H(\bar{P} (\tau))$  , we obtain:
+
+$$
+\begin{array}{l} \lim  _ {\bar {P} (\tau) \to \delta_ {c ^ {*}}} H (\bar {P} (\tau)) = - \lim  _ {\bar {P} (\tau) \to \delta_ {c ^ {*}}} \sum_ {c = 1} ^ {C} \bar {P} _ {c} (\tau) \log \bar {P} _ {c} (\tau) \\ = - \sum_ {c = 1} ^ {C} \lim  _ {\bar {P} _ {c} (\tau) \rightarrow \delta_ {c *}} \bar {P} _ {c} (\tau) \log \bar {P} _ {c} (\tau). \tag {22} \\ \end{array}
+$$
+
+For the components in 22, when  $c = c^*$ ,  $\lim_{\bar{P}_{c^*}(\tau) \to 1} \bar{P}_{c^*}(\tau) \log \bar{P}_{c^*}(\tau) = 0$ . When  $c \neq c^*$ ,  $\lim_{\tau \to +\infty} \bar{P}_c(\tau) = 0$ ,  $\lim_{\bar{P}_c(\tau) \to 0} \bar{P}_c(\tau) \log \bar{P}_c(\tau) = 0$ . Therefore, the addition result of the components is  $\lim_{\bar{P} (\tau) \to \delta_{c^*}} H(\bar{P} (\tau)) = 0$ . Substituting  $H(\bar{P} (\tau)) \to 0$  into  $J(\tau)$ :
+
+$$
+\lim  _ {\tau \rightarrow + \infty} J (\tau) = \log C - H (\bar {P} (\tau)) = \log C - 0 = \log C. \tag {23}
+$$
+
+As  $\tau \to -\infty$ , define  $c^* = \arg \max_y \log \pi(y)$ . Similarly, we have:
+
+$$
+\lim  _ {\tau \rightarrow - \infty} J (\tau) = \log C. \tag {24}
+$$
+
+![](images/685e4fb79658e91b6bf76033b37cd1a61bc9ef6252a9418ad40b05b742165752.jpg)
+
+Proof of Proposition 1. The KL divergence  $\mathrm{KL}(\bar{P} (\tau)\parallel U)$  is minimized when  $\bar{P} (\tau) = U$ , as the KL divergence is non-negative and zero at exact matching. This requires solving:
+
+$$
+\mathbb {E} _ {x \sim \mathcal {D} _ {\mathrm {b c}}} P (x, \theta ; \tau) = \mathbb {E} _ {x \sim \mathcal {D} _ {\mathrm {b c}}} \operatorname {s o f t m a x} (\theta (x) - \tau \cdot \log \pi (y)) = U, \quad U _ {c} = \frac {1}{C} (\forall c \in \{1, \dots , C \}). \tag {25}
+$$
+
+Leveraging the normalized weighted geometric mean approximation [32], 25 can be approximated as:
+
+$$
+\begin{array}{l} \mathbb {E} _ {x \sim \mathcal {D} _ {\mathrm {b c}}} \operatorname {s o f t m a x} (\theta (x) - \tau \cdot \log \pi (y)) \approx \operatorname {s o f t m a x} \left(\mathbb {E} _ {x \sim \mathcal {D} _ {\mathrm {b c}}} [ \theta (x) - \tau \cdot \log \pi (y) ]\right) \\ = \operatorname {s o f t m a x} \left(\mathbb {E} _ {x \sim \mathcal {D} _ {\mathrm {b c}}} [ \theta (x) ] - \tau \cdot \log \pi (y)\right) \quad \approx U. \tag {26} \\ \end{array}
+$$
+
+Considering the  $i$ -th component in 26, we have:
+
+$$
+\operatorname {s o f t m a x} \left(\mathbb {E} _ {x \sim \mathcal {D} _ {\mathrm {b c}}} \left[ \theta_ {c} (x) \right] - \tau \cdot \log \pi (c)\right) \approx \frac {1}{C}. \tag {27}
+$$
+
+Taking the log of both sides, we obtain:
+
+$$
+\mathbb {E} _ {x \sim \mathcal {D} _ {\mathrm {b c}}} [ \theta_ {c} (x) ] - \tau \cdot \log \pi (c) = \log K, \quad \forall c, \tag {28}
+$$
+
+where  $K = \frac{\sum_{j=1}^{C} (\exp\left(\mathbb{E}_{x \sim \mathcal{D}_{\mathrm{bc}}}[\theta_j(x)] - \tau \log \pi_j(y)\right))}{C}$  denotes a constant. Choosing class  $c = 1$  as a reference, we subtract the equation for  $c = 1$  from that of class  $c$ :
+
+$$
+\mathbb {E} _ {x \sim \mathcal {D} _ {\mathrm {b c}}} [ \theta_ {c} (x) ] - \mathbb {E} _ {x \sim \mathcal {D} _ {\mathrm {b c}}} [ \theta_ {1} (x) ] = \tau \cdot (\log \pi (c) - \log \pi (1)). \tag {29}
+$$
+
+Define the residual 29  $r_c = \mathbb{E}_{x\sim \mathcal{D}_{\mathrm{bc}}}[\theta_c(x)] - \mathbb{E}_{x\sim \mathcal{D}_{\mathrm{bc}}}[\theta_1(x)] - \tau \cdot (\log \pi (c) - \log \pi (1))$ , the least-squares objective is:
+
+$$
+\min  _ {\tau} \sum_ {c = 1} ^ {C} r _ {c} ^ {2} = \min  _ {\tau} \sum_ {c = 1} ^ {C} \left(\mathbb {E} _ {x \sim \mathcal {D} _ {\mathrm {b c}}} [ \theta_ {c} (x) ] - \mathbb {E} _ {x \sim \mathcal {D} _ {\mathrm {b c}}} [ \theta_ {1} (x) ] - \tau (\log \pi (c) - \log \pi (1))\right) ^ {2}. \tag {30}
+$$
+
+To derive the closed-form solution for  $\tau$ , we define the objective function  $f(\tau) = \sum_{c=1}^{C} (\mu_c - \mu_1 - \tau (\log \pi(c) - \log \pi(1)))^2$  with  $\mu_c = \mathbb{E}_{x \sim \mathcal{D}_{\mathrm{bc}}}[\theta_c(x)]$ ,  $\forall c$ . Using the chain rule, we differentiate  $f(\tau)$  with respect to  $\tau$ :
+
+$$
+\begin{array}{l} \frac {\partial f (\tau)}{\partial \tau} = \sum_ {c = 1} ^ {C} 2 \left(\mu_ {c} - \mu_ {1} - \tau \left(\log \pi (c) - \log \pi (1)\right)\right) \cdot \frac {\partial}{\partial \tau} \left[ \mu_ {c} - \mu_ {1} - \tau \left(\log \pi (c) - \log \pi (1)\right) \right] \\ = \sum_ {c = 1} ^ {C} 2 \left(\mu_ {c} - \mu_ {1} - \tau (\log \pi (c) - \log \pi (1))\right) \cdot \left(- (\log \pi (c) - \log \pi (1))\right) \\ = - 2 \sum_ {c = 1} ^ {C} \left(\mu_ {c} - \mu_ {1} - \tau \left(\log \pi (c) - \log \pi (1)\right)\right) \left(\log \pi (c) - \log \pi (1)\right). \tag {31} \\ \end{array}
+$$
+
+Setting the Derivative to Zero, we obtain:
+
+$$
+- 2 \sum_ {c = 1} ^ {C} \left(\mu_ {c} - \mu_ {1} - \tau (\log \pi (c) - \log \pi (1))\right) (\log \pi (c) - \log \pi (1)) = 0
+$$
+
+$$
+\sum_ {c = 1} ^ {C} \left(\mu_ {c} - \mu_ {1}\right) (\log \pi (c) - \log \pi (1)) - \tau \sum_ {c = 1} ^ {C} (\log \pi (c) - \log \pi (1)) ^ {2} = 0. \tag {32}
+$$
+
+Rearranging terms to solve for  $\tau$ :
+
+$$
+\tau \sum_ {c = 1} ^ {C} (\log \pi (c) - \log \pi (1)) ^ {2} = \sum_ {c = 1} ^ {C} (\mu_ {c} - \mu_ {1}) (\log \pi (c) - \log \pi (1))
+$$
+
+$$
+\tau^ {\mathrm {L S}} = \frac {\sum_ {c = 1} ^ {C} \left(\mu_ {c} - \mu_ {1}\right) (\log \pi (c) - \log \pi (1))}{\sum_ {c = 1} ^ {C} (\log \pi (c) - \log \pi (1)) ^ {2}}, \tag {33}
+$$
+
+where  $\tau^{\mathrm{LS}}$  is the final closed-form solution for  $\tau$ .
+
+![](images/d8859cae01f330946593e09d80c9463487348814548d91e66f7e416bc68f85ef.jpg)
+
+# A.3 Methods of Label Noise Addition
+
+Sym. Symmetric noise (Sym) means that for each sample label, we randomly replace it with one of the other classes with a fixed probability  $\eta$ . For a  $C$ -class classification task, given a noise rate  $\eta$ , the original label  $y$  is uniformly changed to other classes except  $y$  with the probability  $\eta$ . Specific noise transition matrix: elements on the diagonal are  $1 - \eta$ , elements on the off-diagonal are  $\eta / (C - 1)$ .
+
+Asym. Asymmetric noise (Asym) simulates the real-world label noise structure. It selects "easily confused" class pairs (such as dog  $\leftrightarrow$  wolf) and specifies the transition probability, while the rest remain unchanged. Labels are only replaced between similar classes, and are not randomly mislabeled as other classes. The process of label flipping is related to the quantity of each class. With the noise rate denoted as  $\eta$ , we establish the following definitions:  $T_{ij}(x) = P[\widetilde{Y} = j|Y = i, x] = 1 - \eta$  when  $i = j$ . Conversely,  $T_{ij}(x) = P[\widetilde{Y} = j|Y = i, x] = \frac{n_j}{n - n_i}\eta$ . Here,  $Y$  and  $\widetilde{Y}$  represent the random variables for clean labels and noisy labels, respectively.
+
+![](images/8f7f50ebe4a84d946040679ca18e8dc341e1ffadc8898a70e85a16b81bcce914.jpg)  
+(a)  $\mathrm{sym.20\%}$
+
+![](images/20c68985226751703a6147bb6ecd04ddf14937145b7b7e2f46a127c725a2cc0d.jpg)  
+(b) sym.  $40\%$
+
+![](images/6e49f5640d5e10a55c65b92d930a5d2eb10d5dbeaed682aac0606d1eae43cd26.jpg)  
+(c) sym.  $60\%$
+
+![](images/024a823b94406d14e57d3732e59c2cfaa091d84f619aa7c79e8d9a5bfa6caa28.jpg)  
+(d) asym.  $20\%$
+
+![](images/2ac84495eb0e2cb4921044424495688e5a4d78ce70c66fe68ab5d4e4ba0ae3ec.jpg)  
+(e) asym.  $40\%$
+
+![](images/8a90804a774c6cd581967f97c7014c4d2d4f2c28a55585e90b3fb51455b20c1f.jpg)  
+(f) asym.  $40\%$
+
+![](images/46b604f6bddaace88fcafa3eb2bb1ac72489a0256123fccdbc1fb76876e42e4e.jpg)  
+(g) joint.20%
+
+![](images/063e425f2cb71ce54d3d1dc356cabeff6d104f6f9274e2df5c7dd7dabad5a908.jpg)  
+(h) joint.40%
+
+![](images/43c4792a5d30841b66da3b035c65b4a323ac6a086b546583c5f10a48d866888f.jpg)  
+(i) joint.60%
+
+![](images/1d10cd86149e5e29af1235646e605fbcb3f9cc14b99412c5942835446e88789e.jpg)  
+(j) t2h.20%
+
+![](images/96dc8d522af510cb400e78f853587a29bef8e3de0a2e78cb982e123ef3c3257a.jpg)  
+(k) t2h.40%
+
+![](images/63a7aedf90557c15d46f59f54effd8ab5983846384c6fb2c0299bf20808ccc14.jpg)  
+(1) t2h.60%
+
+![](images/c8674322f0a68fea8bdf0059d7c780e0b4410a8be2759dae5415d9e1c6f9055c.jpg)  
+(m) ins.20%
+
+![](images/051ce243c8f8b521cd70b4d7bed716a465569bd7f9c389af6b1ed08fafc6d30d.jpg)  
+(n) ins.  $40\%$
+
+![](images/79cca4dd51ff0455077002553c79862ff970d2fe853e4cd18d6a85dd2f69448d.jpg)  
+(o) ins.  $60\%$  
+Figure 6: Changes in the imbalance ratio (IR) of observed distributions (orange line) under different noise types with varying noise ratios, in comparison to the fixed IR of the true distribuion (blue line). Symmetric (sym) and instance-dependent (ins) noise alleviate IR, while asymmetric noise (asym) may reverse the long-tail pattern. Joint noise preserves IR at low imbalance. Tail-to-head noise (t2h) exacerbates IR compared to the original distribution.
+
+Joint. The Joint noise [5] label is generated by the noise transfer matrix, which represents the probability of the clean label flipping to the noise label. Let  $\mathrm{Y}$  represent the clean label variable,  $\bar{Y}$  represent the noise label,  $\mathrm{X}$  represent the instance feature, and the transfer matrix  $T(X = x)$  is
+
+defined as  $T_{ij}(X) = \mathbb{P}(\tilde{Y} = j|Y = i,X = x)$ . Specifically, given the noise ratio  $\eta \in [0,1]$ , it is defined as follows:
+
+$$
+T _ {i j} (X) = \mathbb {P} \left[ \tilde {Y} = j \mid Y = i, X = x \right] = \left\{ \begin{array}{l l} 1 - \eta & i = j \\ \frac {\eta N _ {j}}{N - N _ {i}} & o t h e r w i s e \end{array} \right. \tag {34}
+$$
+
+Here,  $\mathbf{N}$  denote the total number of training examples and  $N_{j}$  is the frequency of class  $j$ . It combines the class prior information in the dataset to set the transition probability, which is more in line with the situation in real-world scenarios where samples are easily mislabeled as frequent classes.
+
+T2H. Tail-to-Head noise (T2H) [33] refers to the phenomenon that in the long-tail data distribution, the tail class samples tend to be mislabeled as the head class samples. The generation of T2H noise mainly includes two steps: separating transferable and non-transferable samples; randomly selecting tail class samples and assigning head class labels to them. We define a transition matrix  $T \in \mathbb{R}^{C \times C}$ , where each element  $T_{t,h} = P(y_i = h|\tilde{y}_i = t)$  represents the probability that an instance with the true class  $t$  is mis-labeled as class  $h$ . In the context of T2H noise, the samples from the tail class  $t$  have a relatively high probability of being mis-labeled as the head class  $h$ , i.e.,  $T_{t,h} > T_{t,t'}$  (where  $t'$  denotes a rarer tail class with fewer samples than  $t$ ). Meanwhile, the probability that a sample from the head class  $h$  is mislabeled as the tail class  $t$  is very low,  $T_{h,t} \approx 0$ . Through this transition matrix, the T2H noise is quantitatively defined from a probabilistic perspective.
+
+IDN. Instance-dependent noise (IDN) [13] is closely related to the characteristics of each instance and its class label. It is generated by setting a random noise rate for each instance, which follows a truncated Gaussian distribution, and the noise rate of each class is also randomly set. In this noise model, the probability of label flipping varies for each specific instance. Taking the CIFAR-10/100 datasets as examples, when generating instance-dependent noise, for a given clean sample set and a set noise rate  $\eta$ , a random noise rate is set for each instance one by one according to the truncated gaussian distribution, thereby realizing the generation of instance-dependent noise.
+
+# A.4 Analysis of the Effects of Different Noise Additions Methods on Long-tailed distribution
+
+We systematically investigate the impact of existing noise addition methods on the imbalance ratio (IR) of true distribution when applied to clean long-tailed datasets. The results are presented in the Figure 6. Sym noise alleviates the long-tail problem because the samples of the head category are evenly distributed to other classes, indirectly balancing the data distribution. Baesd on this finding, we choose symmetric noise to simulate relieve scenario. Asym noise induces a reverse long-tail which means that some tail classes surge in count because head-class samples are frequently mislabeled as them. However, the reverse long-tail scenario is impractical. Therefore, we opt not to use asymmetric noise in our experimental setup. Joint noise usually refers to the existence of some correlation between the noisy label and the long-tailed distribution. In some cases like long-tailed distribution with low IR, joint noise maintains the original long-tailed structure, keeping the imbalance ratio unchanged. However, in other cases, it may either exacerbate or alleviate the IR, which is uncontrollable. Thus, we only select joint noise to construct consistent scenarios for long-tailed distribution with low IR. T2H noise significantly aggravates the long-tail problem, and the number of tail-class samples is further reduced, causing the model to be biased towards the head classes. We choose t2h noise to construct simulated aggravate scenarios. IDN noise, like symmetric noise, alleviates IR after adding noise.
+
+# A.5 Results on Simulated Scenarios Based on CIFAR-10
+
+We conduct experiments using the NLL method DPC on CIFAR-10 to evaluate the performance of our method under scenarios with varying imbalance ratios (IR=10, 50, 100), noise types (joint, sym, t2h), and noise rates ( $\eta = 40\%$ ,  $60\%$ ). As shown in Table 5, DPC combined with Unlocker (DPC+Unlocker) achieves SOTA test accuracies across all the scenarios, outperforming baselines DPC and DPC with direct LA integration. In the consistent scenarios, DPC+Unlocker achieves accuracies of  $93.71\%$  and  $90.93\%$ , yielding improvements of  $0.18\%$  and  $5.30\%$  over the original DPC  $(93.53\%, 85.63\%)$  respectively. In the relieve scenarios where tail-class clean sample selection is particularly challenging, DPC+Unlocker reaches significant improvements over DPC, ranging from  $11.06\%$  to  $19.87\%$ , validating its effectiveness in mitigating long-tail induced model bias and restoring the original NLL performance. Under aggravate scenarios, DPC+Unlocker maintains stable gains
+
+Table 5: Test accuracy (\%) comparison of methods on the CIFAR-10 dataset under varying imbalance ratios (IR), noise types and noise rates  $\eta$ , involving three scenarios of true distribution shifts. Green numbers indicate improvements over the original NLL method. Boldface represents the best performance in each case.  
+
+<table><tr><td>dataset</td><td colspan="10">CIFAR-10</td></tr><tr><td>types</td><td colspan="2">consistent</td><td colspan="4">relieve</td><td colspan="4">aggregate</td></tr><tr><td>IR</td><td colspan="2">10</td><td colspan="2">50</td><td colspan="2">100</td><td colspan="2">10</td><td colspan="2">50</td></tr><tr><td>η</td><td>joint 40%</td><td>joint 60%</td><td>sym 40%</td><td>sym 60%</td><td>sym 40%</td><td>sym 60%</td><td>t2h 40%</td><td>t2h 60%</td><td>t2h 40%</td><td>t2h 60%</td></tr><tr><td>DPC [14]</td><td>93.53</td><td>85.63</td><td>78.33</td><td>57.90</td><td>60.07</td><td>45.39</td><td>93.13</td><td>72.10</td><td>74.29</td><td>70.81</td></tr><tr><td>DPC+LA (post-hoc)</td><td>92.74</td><td>85.31</td><td>78.93</td><td>58.05</td><td>62.40</td><td>40.08</td><td>83.82</td><td>76.21</td><td>75.93</td><td>61.12</td></tr><tr><td>DPC+LA</td><td>88.31</td><td>81.53</td><td>75.60</td><td>45.77</td><td>30.94</td><td>36.50</td><td>91.27</td><td>83.11</td><td>61.75</td><td>48.40</td></tr><tr><td>DPC+LA+Unlocker</td><td>93.71</td><td>90.93</td><td>89.39</td><td>72.86</td><td>73.15</td><td>65.26</td><td>93.76</td><td>92.05</td><td>88.67</td><td>83.35</td></tr><tr><td>vs. DPC</td><td>↑0.18</td><td>↑5.30</td><td>↑11.06</td><td>↑14.96</td><td>↑13.08</td><td>↑19.87</td><td>↑0.63</td><td>↑19.95</td><td>↑14.38</td><td>↑12.54</td></tr></table>
+
+over of  $1.04\%$  to  $19.95\%$  DPC. These results highlight efficacy of Unlocker in disentangling the NLL-LTL deadlock and enhancing model robustness against long-tailed noisy label data.
+
+# A.6 Related Work
+
+Noisy Label Learning (NL). Noisy label learning focuses on tackling the challenge of inaccurate supervised label in datasets. It mainly evolves along two directions: noisy label detection and correction, and robust noise label learning. The former mainstream typically uses a two-step process: selecting noisy labels via metrics such as loss or divergence, and then correcting them through techniques like semi-supervision learning [10, 11, 12, 13, 14]. By directly selecting and filtering out noisy labels, these methods have demonstrated efficiency in both experimental and real-world scenarios. Robust noisy label learning mitigates noise by adjusting loss functions via regularization or noisy transiton matrix [34, 35, 36, 37, 38] to disregard or reduce noise impact.
+
+Long-tailed Learning (LT). Long-tailed learning is aimed to improve the accuracy of tail classes caused by skewed datasets distributions. Re-sampling is a classic method, which directly balances the distribution through reducing samples of head or augmenting samples of tail [39, 40, 41]. Weighting enhances the focus of the model on tail classes by adjusting the sample weights in the loss function [42, 23, 43, 44]. Ensembling learning improves model performance by aggregating multiple networks within a multi-expert framework [45, 46, 47, 48, 49]. The two-stage decoupling strategy achieves a rebalancing of decision boundaries through the fine-tuning of classifiers [50, 18, 51]. Logit Adjustment (LA) corrects biased logits by adding an offset term to the model's logit [15, 16, 17, 18, 19]. Extensive empirical studies have substantiated the efficacy of the LA. Moreover, data augmentation is an an effective way to alleviate the scarcity of tail classes by generating tail samples [52, 53, 54]. Besides, recent work such as strategy fusion tailored for multi-objective optimization (MOO) [55] and model parameter space rebalancing [56] also shows promise in balancing model.
\ No newline at end of file
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+# Unlocking Dataset Distillation with Diffusion Models
+
+Brian B. Moser<sup>1,2,4</sup>
+
+Federico Raue2,4
+
+Sebastian Palacio<sup>3</sup>
+
+Stanislav Frolov<sup>1,2</sup>
+
+# Andreas Dengel<sup>1,2</sup>
+
+<sup>1</sup> University of Kaiserslautern-Landau, Kaiserslautern
+
+$^{2}$  German Research Center for Artificial Intelligence (DFKI), Kaiserslautern
+
+3ABB AG, Mannheim
+
+4 Equal Contribution
+
+first(second@dfki.de
+
+# Abstract
+
+Dataset distillation seeks to condense datasets into smaller but highly representative synthetic samples. While diffusion models now lead all generative benchmarks, current distillation methods avoid them and rely instead on GANs or autoencoders, or, at best, sampling from a fixed diffusion prior. This trend arises because naive backpropagation through the long denoising chain leads to vanishing gradients, which prevents effective synthetic sample optimization. To address this limitation, we introduce Latent Dataset Distillation with Diffusion Models (LD3M), the first method to learn gradient-based distilled latents and class embeddings end-to-end through a pre-trained latent diffusion model. A linearly decaying skip connection, injected from the initial noisy state into every reverse step, preserves the gradient signal across dozens of timesteps without requiring diffusion weight fine-tuning. Across multiple ImageNet subsets at  $128 \times 128$  and  $256 \times 256$ , LD3M improves downstream accuracy by up to 4.8 percentage points (1 IPC) and 4.2 points (10 IPC) over the prior state-of-the-art. The code for LD3M is provided at https://github.com/Brian-Moser/prune_and_distill.
+
+# 1 Introduction
+
+Large-scale datasets fuel the advancements in modern computer vision but demand substantial computational resources and raise scalability concerns [27, 2, 25, 14]. Dataset distillation emerges as a compelling solution, aiming to synthesize a small set of information-rich samples that preserve the essence of the original dataset [35, 42, 4]. While early methods operated directly in pixel space, a promising recent direction involves leveraging powerful generative models as priors [13, 1]. By optimizing compact latent codes instead of raw pixels, these approaches, exemplified by GLaD using StyleGAN-XL [5], can generate higher-resolution synthetic images ( $128 \times 128$  and beyond) that generalize better across diverse network architectures.
+
+However, GAN-based priors like in GLaD suffer from complex multi-space latent optimization and require cumbersome inversion processes for initialization [5, 43, 3, 21]. Diffusion models [18, 28], having surpassed GANs as the state-of-the-art image generators [9], represent a natural next step. Yet, inherent vanishing gradients severely hinder their direct application to dataset distillation. Optimizing latents through the long denoising chain leads to exponentially decaying gradients, which prevents effective learning of the synthetic data [19, 17].
+
+Mathematically, if  $\mathcal{Z}$  is the initial latent code to be optimized and  $\mathbf{z}_0$  the final denoised state after  $T$  steps, the gradient  $\partial \mathcal{L} / \partial \mathcal{Z}$  is a product of Jacobians:
+
+$$
+\frac {\partial \mathcal {L}}{\partial \mathcal {Z}} = \frac {\partial \mathcal {L}}{\partial \mathbf {z} _ {0}} \cdot \left[ \prod_ {t = 1} ^ {T} \frac {\partial \mathbf {z} _ {t - 1}}{\partial \mathbf {z} _ {t}} \right] \cdot \frac {\partial \mathbf {z} _ {T}}{\partial \mathcal {Z}}. \tag {1}
+$$
+
+If the norm of each Jacobian  $\partial \mathbf{z}_{t - 1} / \partial \mathbf{z}_t$  is bounded by  $\lambda < 1$ , the product term  $\prod_{t = 1}^{T}(\partial \mathbf{z}_{t - 1} / \partial \mathbf{z}_t)$  diminishes towards zero as  $T$  increases. This gradient decay is empirically observable and significant: Our analysis confirms that in standard diffusion models, gradient norms for  $\mathcal{Z}$  decrease nearly tenfold as  $T$  increases from 10 to 90 (see Table 1).
+
+Table 1: Gradient norms for the initial latent code  $\mathcal{Z}$  across different maximum diffusion steps  $T$  for fixed inputs. The decrease in norm as  $T$  increases empirically demonstrates the vanishing gradient problem, which severely hinders the optimization of  $\mathcal{Z}$  through the standard reverse diffusion process.  
+
+<table><tr><td>T</td><td>10</td><td>20</td><td>30</td><td>40</td><td>50</td><td>60</td><td>70</td><td>80</td><td>90</td></tr><tr><td>||L/∂Z||×104</td><td>58.1</td><td>33.5</td><td>19.8</td><td>15.4</td><td>13.9</td><td>12.1</td><td>10.4</td><td>8.7</td><td>6.5</td></tr></table>
+
+This critical bottleneck has forced prior diffusion-based distillation attempts to circumvent end-to-end optimization entirely, resorting instead to sampling or selecting fixed representations from pre-trained models [12, 34, 16]. These approaches, while computationally faster, forfeit the fine-grained gradient-matching optimization crucial for potential benefits like enhanced privacy of distilled data or their robustness against adversarial attacks [7, 37, 44, 10].
+
+To unlock diffusion models for true dataset distillation, we introduce Latent Dataset Distillation with Diffusion Models (LD3M). Our core contribution is a tailored modification to the reverse diffusion process (Equation 7) that introduces linearly decaying residual connections, specifically designed to enhance gradient flow for optimizing latent representations  $\mathcal{Z}$  and conditioning codes  $\mathbf{c}$  in the context of dataset distillation. This mechanism enables, for the first time, effective end-to-end optimization of distilled latent codes  $\mathcal{Z}$  and class embeddings  $\mathbf{c}$  through a pre-trained latent diffusion model. LD3M is readily compatible with existing distillation objectives and diffusion model architectures. Experiments across numerous ImageNet subsets demonstrate that LD3M significantly outperforms the state-of-the-art at  $128 \times 128$  and  $256 \times 256$  resolutions, achieving superior cross-architecture generalization, i.e., 4.8 percentage points (1 IPC) and 4.2 points (10 IPC), and faster distillation times.
+
+# 2 Preliminaries
+
+Dataset Distillation: Let  $\mathcal{T} = (X_r, Y_r)$ , where  $X_r \in \mathbb{R}^{N \times H \times W \times C}$ , be a real image classification dataset and  $N$  its cardinality. The goal is to compress  $\mathcal{T}$  into a small synthetic set  $\mathcal{S} = (X_s, Y_s)$ , where  $X_s \in \mathbb{R}^{M \times H \times W \times C}$ , where  $M$  is the total number of synthetic samples with  $M = \mathcal{C} \cdot IPC$ ,  $\mathcal{C}$  the number of classes and IPC the Images Per Class (IPC). We aim to achieve  $M \ll N$  with
+
+$$
+\mathcal {S} ^ {*} = \underset {\mathcal {S}} {\arg \min } \mathcal {L} (\mathcal {S}, \mathcal {T}) \tag {2}
+$$
+
+where  $\mathcal{L}$  is a distillation loss between the distilled set  $S$  and the real dataset  $\mathcal{T}$ . Common choices for  $\mathcal{L}$  include matching gradients (Dataset Condensation, DC [42]), feature distributions (Distribution Matching, DM [41]), or model parameter trajectories (Matching Training Trajectories, MTT [4]). We refer to the supplementary material for detailed definitions.
+
+Dataset Distillation with Generative Priors: To improve the quality, resolution, and generalization of distilled images, recent work incorporates deep generative models as priors [5]. Instead of optimizing pixels directly, they optimize latent codes  $\mathcal{Z} \in \mathbb{R}^{M \times h \times w \times d}$  with  $h \cdot w \cdot d \ll H \cdot W \cdot C$ , fed into a pre-trained generator  $\mathcal{D}$ . The optimization objective becomes
+
+$$
+\mathcal {Z} ^ {*} = \underset {\mathcal {Z}} {\arg \min } \mathcal {L} (\mathcal {D} (\mathcal {Z}), \mathcal {T}). \tag {3}
+$$
+
+# 3 Related Work
+
+GAN Priors (GLaD): GLaD [5] pioneered distillation with generative priors using a pre-trained StyleGAN-XL [30]. While successful, it inherits GAN-specific drawbacks: (1) Optimizing the
+
+![](images/48bb25a80591761651db820c6f1eb598049a0fbffdb6f7d34a9f7bee21d73a05.jpg)  
+Figure 1: The LD3M Framework. Learnable latent codes  $\mathcal{Z}$  and conditioning codes  $\mathbf{c}$  are optimized.  $\mathcal{Z}$  is noised to initialize the reverse diffusion at  $\mathbf{z}_T$ . A pre-trained LDM denoiser iteratively refines the state  $(\mathbf{z}_t \to \mathbf{z}_{t-1})$ . Key innovation: Residual connections (red arrows) inject  $\mathbf{z}_T$  with linearly decaying weight into each step (Equation 7), enhancing gradient flow. The final latent  $\mathbf{z}_0$  is decoded  $(\mathcal{D})$  into images  $S$ , which are optimized using a standard distillation algorithm.
+
+complex, multi-level latent space  $(\mathbf{W}^{+})$  required for high quality is computationally demanding [43]. (2) Initializing latent codes  $\mathcal{Z}$  from real images  $\mathbf{x}$  requires solving a costly GAN inversion problem  $(\min_{\mathcal{Z}}\mathcal{L}(\mathbf{x},\mathcal{D}(\mathcal{Z}))$  [3, 36], hindering standard initialization practices [21].
+
+Prior Diffusion-based Distillation Attempts: The inherent vanishing gradient problem (§1) significantly challenges using diffusion models for end-to-end distillation. Faced with this gradient barrier, existing diffusion methods for distillation have adopted non-optimization strategies:
+
+- Sampling/Selection Methods: Minimax Diffusion [16] and D4M [34] use criteria to select or sample representative latents from a diffusion model, avoiding backpropagation to latents entirely. While faster, this bypasses gradient-based optimization that is needed for privacy or robustness, also crucial motivations for dataset distillation [7, 37, 44, 10].  
+- Autoencoder-Only Methods: Duan et al. [12] leverage the pre-trained autoencoder from LDM but do not utilize the diffusion process itself. They optimize latent codes  $\mathcal{Z}$  that are directly decoded by  $\mathcal{D}$ , essentially using only the autoencoder, not its core denoising mechanism. This simplifies optimization but fails to exploit the diffusion prior.
+
+In contrast, LD3M is designed to directly optimize latent codes  $\mathcal{Z}$  by enabling gradient flow through the reverse process.
+
+Residual Connections for Gradient Flow: Enhancing gradient flow in deep networks is commonly addressed using residual connections, famously introduced in ResNets [17] or LSTMs [19]. Similar ideas have appeared within diffusion models; for instance, SAGE [23] used residuals connecting to the initial noise to enable adversarial latent search. While conceptually related, our contribution differs significantly: we introduce structured, linearly decaying residuals injected at every step from the initial noisy latent  $\mathbf{z}_T$ , explicitly designed for end-to-end optimization of distilled latent codes through the diffusion chain for the unique characteristics of dataset distillation.
+
+Decoupled Distillation Methods: Distinct from methods optimizing synthetic data via generative priors, another line of work decouples distillation from end-to-end training. Methods like SRe2L [40] and others [39, 32] leverage statistics (e.g., from BatchNorm) of pre-trained networks to recover informative images, offering scalability benefits but following a fundamentally different optimization strategy than gradient-matching approaches like LD3M.
+
+# 4 Latent Dataset Distillation with Diffusion Models (LD3M)
+
+Our approach enables end-to-end optimization of synthetic data directly through a pre-trained LDM. As shown in Figure 1, LD3M optimizes both initial latent codes  $\mathcal{Z}$  and conditioning codes  $\mathbf{c}$ . These are processed by a modified reverse diffusion process to generate expressive latent states  $\mathbf{z}_0$ , which are then decoded into images,  $S = \mathcal{D}(\mathbf{z}_0)$ . This section details the core components: our modified diffusion sampling process designed to boost gradient flow (§4.1), the efficient initialization strategy (§4.2), and the gradient checkpointing used for memory efficiency (§4.3).
+
+# 4.1 Sampling Process
+
+We leverage a pre-trained LDM [28] without fine-tuning its weights. Standard LDM operation involves a reverse diffusion process  $p_{\theta}$  that iteratively denoises a state  $\mathbf{z}_t$ , starting from noise  $\mathbf{z}_T$ , conditioned on an embedding  $\mathbf{c}$  (typically derived from class labels) [38, 26]. Each step  $t$  predicts a less noisy state  $\mathbf{z}_{t-1}$  based on the current state  $\mathbf{z}_t$  and condition  $\mathbf{c}$ . The standard update rule [18, 29] calculates the subsequent state  $\mathbf{z}_{t-1}$  using the predicted mean  $\mu_{\theta}$  and variance  $\sigma_t^2$ :
+
+$$
+\mu_ {\theta} (\mathbf {c}, \mathbf {z} _ {t}, \gamma_ {t}) = \frac {1}{\sqrt {\alpha_ {t}}} \left(\mathbf {z} _ {t} - \frac {1 - \alpha_ {t}}{\sqrt {1 - \gamma_ {t}}} f _ {\theta} (\mathbf {c}, \mathbf {z} _ {t}, \gamma_ {t})\right) \tag {4}
+$$
+
+$$
+\mathbf {z} _ {t - 1} \leftarrow \mu_ {\theta} (\mathbf {c}, \mathbf {z} _ {t}, \gamma_ {t}) + \sigma_ {t} ^ {2} \varepsilon_ {t}, \tag {5}
+$$
+
+where  $f_{\theta}$  is the LDM's pre-trained noise prediction network (usually a U-Net),  $\alpha_{t}$  and  $\gamma_{t}$  relate to the noise schedule, and  $\varepsilon_{t} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$  is random noise added at step  $t$ .
+
+# Algorithm 1 Latent Dataset Distillation with Diffusion Models (LD3M)
+
+Input: randomly selected collection  $X_{s}$ , pre-trained encoder  $\mathcal{E}$ , pre-trained decoder  $\mathcal{D}$ , pre-trained denoiser  $\mu_{\theta}$  with frozen parameters  $\theta$ , noise levels  $\sigma_{t}$ .
+
+$$
+\mathcal {Z} = \mathcal {E} \left(X _ {s}\right)
+$$
+
+$$
+\mathbf {z} _ {T} \sim q (\mathbf {z} _ {T} \mid \mathcal {Z})
+$$
+
+$$
+\mathbf {f o r} t = T, \dots , 1 \mathbf {d o}
+$$
+
+$$
+\varepsilon_ {t} \sim \mathcal {N} (\mathbf {0}, \mathbf {I})
+$$
+
+$$
+\mathbf {z} _ {t - 1} \leftarrow \left((1 - \frac {t}{T}) \cdot \mu_ {\theta} (\mathbf {c}, \mathbf {z} _ {t}, \gamma_ {t}) + \frac {t}{T} \cdot \mathbf {z} _ {T}\right) + \sigma_ {t} ^ {2} \varepsilon_ {t}
+$$
+
+$$
+e n d \mathrm {f o r}
+$$
+
+$$
+X _ {\text {s y n}} \leftarrow \mathcal {D} \left(\mathbf {z} _ {0}\right)
+$$
+
+$$
+\text {R e t u r n}: X _ {\text {s y n}}
+$$
+
+For dataset distillation, our goal is to learn the optimal initial latent representations  $\mathcal{Z}$  and conditioning codes  $\mathbf{c}$  that minimize the distillation loss  $\mathcal{L}$  between the synthetic images and the target dataset:
+
+$$
+\mathcal {Z} ^ {*}, \mathbf {c} ^ {*} = \underset {\mathcal {Z}, \mathbf {c}} {\arg \min } \mathcal {L} (\mathcal {D} [ p _ {\theta} (\mathbf {z} _ {0} | \mathbf {z} _ {T}, \mathbf {c}) ], \mathcal {T}), \quad \text {w h e r e} \mathbf {z} _ {T} \sim q (\mathbf {z} _ {T} | \mathcal {Z}). \tag {6}
+$$
+
+Here,  $p_{\theta}(\mathbf{z}_0|\mathbf{z}_T,\mathbf{c})$  denotes the final state  $\mathbf{z}_0$  resulting from the  $T$ -step reverse process starting from  $\mathbf{z}_T$ , which itself is a noised version of the learnable  $\mathcal{Z}$  obtained via the forward process  $q$ . We want to highlight that the sampling of  $\varepsilon$  introduces stochasticity during inference. We fix, however, for one sampling phase, the residual variable  $\mathbf{z}_T$  to be constantly the same.
+
+To effectively minimize distillation losses like gradient or trajectory matching (Equation 6), which rely on fine-grained alignment between synthetic and real data processing, requires guiding the generative process. While conditioning  $\mathbf{c}$  provides class guidance, optimizing the initial latent  $\mathcal{Z}$  offers the necessary degrees of freedom to shape the generated sample  $\mathcal{D}(\mathbf{z}_0)$  precisely. Relying solely on fixed  $\mathcal{Z}$  or only optimizing  $\mathbf{c}$  proved insufficient empirically, yielding suboptimal results resembling simple LAION-5B [31] data retrieval similar to D4M [34].
+
+Vanishing gradients inherent in backpropagating through the  $T$  steps of Equation 5 present the primary obstacle to optimizing Equation 6 [19]. To overcome this, we introduce a simple yet effective modification to the reverse step, injecting a residual connection from the initial state  $\mathbf{z}_T$ :
+
+$$
+\mathbf {z} _ {t - 1} \leftarrow \underbrace {\left(\left(1 - \frac {t}{T}\right) \cdot \mu_ {\theta} \left(\mathbf {c} , \mathbf {z} _ {t} , \gamma_ {t}\right) + \frac {t}{T} \cdot \mathbf {z} _ {T}\right)} _ {\text {M o d i f i e d M e a n}} + \sigma_ {t} ^ {2} \varepsilon_ {t}. \tag {7}
+$$
+
+Crucially, while this modification deviates from standard sampling aimed at matching the distribution of real images, its purpose here is to enable gradient flow for distillation optimization, a task where downstream utility, not photorealism [42, 4], is paramount (see supplementary material for detailed discussion). In other words, our method breaks the sampler's fidelity to the original data distribution in order to better achieve the distillation objective of matching feature distributions.
+
+In conclusion, the modification replaces the standard predicted mean  $\mu_{\theta}$  with a weighted average of  $\mu_{\theta}$  and the initial state  $\mathbf{z}_T$ . The weight of  $\mathbf{z}_T$  decreases linearly from 1 (at  $t = T$ ) to nearly 0 (as  $t \to 0$ ). This creates a direct pathway for gradients from the loss  $\mathcal{L}$  (computed using  $\mathbf{z}_0$ ) back to  $\mathbf{z}_T$ ,
+
+and thus to the learnable  $\mathcal{Z}$ , bypassing the long chain of Jacobian products that causes gradients to vanish. The enhanced gradient flow to  $\mathcal{Z}$  can be conceptually represented as:
+
+$$
+\frac {\partial \mathcal {L}}{\partial \mathcal {Z}} = \sum_ {t = 1} ^ {T} \underbrace {\left(1 - \frac {T - 1}{T}\right) \cdot \left[ \frac {\partial \mathcal {L}}{\partial \mathbf {z} _ {t}} \cdot \frac {\partial \mathbf {z} _ {t}}{\partial \mathbf {z} _ {t - 1}} \cdot \dots \cdot \frac {\partial \mathbf {z} _ {0}}{\partial \mathcal {Z}} \right]} _ {\text {O r i g i n a l (D e c a y i n g) P a t h}} + \underbrace {\left(\frac {t}{T}\right) \cdot \left[ \frac {\partial \mathcal {L}}{\partial \mathbf {z} _ {t - 1}} \frac {\partial \mathbf {z} _ {t - 1}}{\partial \mathbf {z} _ {T}} \frac {\partial \mathbf {z} _ {T}}{\partial \mathcal {Z}} \right]} _ {\text {E n h a n c e d P a t h v i a S k i p C o n n e c t i o n}}. \tag {8}
+$$
+
+A comprehensive description can be found in Algorithm 1.
+
+Notes on Markovian Property:  $\mathbf{z}_{t-1}$  depends on  $\mathbf{z}_t$  and  $\mathbf{z}_T$ , but not on any earlier states such as  $\mathbf{z}_{t+1}, \mathbf{z}_{t+2}$ , and so on. Therefore, the probability distribution for  $\mathbf{z}_{t-1}$  only depends on  $\mathbf{z}_t$  and the fixed initial state  $\mathbf{z}_T$ , which is constant throughout the diffusion process. Thus, we have:  $p(\mathbf{z}_{t-1}|\mathbf{z}_t, \mathbf{z}_{t+1}, \ldots, \mathbf{z}_T) = p(\mathbf{z}_{t-1}|\mathbf{z}_t, \mathbf{z}_T)$ . This confirms that LD3M remains Markovian.
+
+Notes on Generalisability: Our gradient enhancement technique (Equation 7) applies to various diffusion model architectures beyond LDMs, suggesting broader potential for future work. In this study, we focused on LDMs primarily as a proof of concept, leveraging readily available pre-trained models and LDM's foundational role in latent-space diffusion.
+
+# 4.2 Efficient Latent Code Initialization
+
+Standard practice in dataset distillation initializes synthetic data using real images from the target classes [21]. GAN-based methods like GLaD face a significant challenge here, requiring complex and costly GAN inversion techniques to find latent codes  $\mathcal{Z}$  that reconstruct target real images  $\mathbf{x}$  [36].
+
+LD3M benefits immensely from the autoencoder structure inherent in LDMs. We initialize the learnable latent codes  $\mathcal{Z}_{\mathrm{init}}$  simply by encoding a small, randomly selected set of real images  $X_{s}$  using the pre-trained image encoder. Similarly, the initial conditioning codes  $\mathbf{c}_{\mathrm{init}}$  are obtained using the pre-trained class embedding network.
+
+# 4.3 Memory Efficiency via Gradient Checkpointing
+
+Optimizing through the  $T$  steps of the reverse diffusion process, even with our modification, can be memory-intensive. Following GLaD [5], we employ gradient checkpointing [6] to manage VRAM usage. The procedure involves:
+
+1. Perform the forward pass  $S = \mathcal{D}(p_{\theta}(\mathbf{z}_0|\mathbf{z}_T,\mathbf{c}))$  without storing intermediate activations.  
+2. Calculate the distillation loss  $\mathcal{L}(\mathcal{S},\mathcal{T})$  and the gradient with respect to the output,  $\partial \mathcal{L} / \partial \mathcal{S}$  
+3. To compute the gradient  $\partial \mathcal{L} / \partial \mathcal{Z}$  (and  $\partial \mathcal{L} / \partial \mathbf{c}$ ), recompute the necessary segments of the forward pass through the diffusion process and decoder  $\mathcal{D}$ , storing only the activations needed for the immediate backward pass segment.
+
+This avoids storing the full computation graph, which GLaD also exploits for a single generator pass.
+
+# 5 Experiments
+
+We evaluate LD3M against relevant baselines, primarily the state-of-the-art latent generative prior method GLaD [5], following its experimental setup for fair comparison. We conduct extensive experiments on 10 diverse 10-class subsets of ImageNet-1k [8] at  $128 \times 128$  (IPC=1, IPC=10) and  $256 \times 256$  (IPC=1) resolutions, as well as CIFAR-10. Key implementation details are in §5.1; full hyperparameters and setup details are in the supplementary material.
+
+# 5.1 Setup Details
+
+Datasets & Evaluation: We use ImageNet subsets (ImNet-A to E, ImNette, ImWoof, Birds, Fruits, Cats) from [5, 20, 4] and CIFAR-10. Following standard protocol, we distill datasets using DC [42], DM [41], or MTT [4] and evaluate by training unseen architectures (AlexNet [22], VGG-11 [33], ResNet-18 [17], ViT [11]) from scratch on the distilled set, reporting mean test accuracy ( $\pm$  std. dev.) over 5 runs.
+
+Table 2: CIFAR-10 comparisons with GLaD on IPC=1 and SRe2L/D4M on IPC=50.  
+(a)  $\mathrm{IPC} = 1$  
+
+<table><tr><td>Dist.</td><td>Method</td><td>AlexNet</td><td>ResNet18</td><td>VGG11</td><td>ViT</td><td>Average</td></tr><tr><td rowspan="4">DC</td><td>pixel space</td><td>25.9±0.2</td><td>27.3±0.5</td><td>28.0±0.5</td><td>22.9±0.3</td><td>26.0±0.4</td></tr><tr><td>GLaD (rand G)</td><td>30.1±0.5</td><td>27.3±0.2</td><td>28.0±0.9</td><td>21.2±0.6</td><td>26.6±0.5</td></tr><tr><td>GLaD (trained G)</td><td>26.0±0.7</td><td>27.6±0.6</td><td>28.2±0.6</td><td>23.4±0.2</td><td>26.3±0.5</td></tr><tr><td>LD3M (trained G)</td><td>27.2±0.8</td><td>26.6±0.9</td><td>31.5±0.3</td><td>29.0±0.2</td><td>28.6±0.6</td></tr><tr><td rowspan="4">DM</td><td>pixel space</td><td>22.9±0.2</td><td>22.2±0.7</td><td>23.8±0.5</td><td>21.3±0.5</td><td>22.6±0.5</td></tr><tr><td>GLaD (rand G)</td><td>23.7±0.3</td><td>21.7±1.0</td><td>24.3±0.8</td><td>21.4±0.2</td><td>22.8±0.6</td></tr><tr><td>GLaD (trained G)</td><td>25.1±0.5</td><td>22.5±0.7</td><td>24.8±0.8</td><td>23.0±0.1</td><td>23.8±0.5</td></tr><tr><td>LD3M (trained G)</td><td>27.2±0.4</td><td>23.0±0.7</td><td>25.4±0.4</td><td>23.8±0.3</td><td>24.9±0.5</td></tr></table>
+
+* Reported [34]; Sampling-based Diffusion.
+
+(b)  $\mathrm{IPC} = 50$  
+
+<table><tr><td>Method</td><td>ConvNet</td></tr><tr><td>SRe2L [40]</td><td>60.2†</td></tr><tr><td>D4M [34]</td><td>72.8*</td></tr><tr><td>LD3M (Ours)</td><td>73.2</td></tr></table>
+
+Reported [40] at  $\mathrm{IPC} = 1\mathrm{K}$  Decoupled method.
+
+Table 3: Cross-architecture performance (%) with 1 IPC on ImageNet subsets (128 × 128). LD3M (bold) generally achieves the best results within each algorithm and overall best (blue) compared to Pixel Space and GLaD. For instance, LD3M improves DC, MTT, and DM by +3.76%, +5.68%, and +16.34%, respectively.  
+
+<table><tr><td>Distil. Space</td><td>Alg.</td><td>All</td><td>ImNet-A</td><td>ImNet-B</td><td>ImNet-C</td><td>ImNet-D</td><td>ImNet-E</td><td>ImNette</td><td>ImWoof</td><td>ImNet-Birds</td><td>ImNet-Fruits</td><td>ImNet-Cats</td></tr><tr><td rowspan="3">pixel space</td><td>MTT</td><td>25.4±2.3</td><td>33.4±1.5</td><td>34.0±3.4</td><td>31.4±3.4</td><td>27.7±2.7</td><td>24.9±1.8</td><td>24.1±1.8</td><td>16.0±1.2</td><td>25.5±3.0</td><td>18.3±2.3</td><td>18.7±1.5</td></tr><tr><td>DC</td><td>27.9±1.6</td><td>38.7±4.2</td><td>38.7±1.0</td><td>33.3±1.9</td><td>26.4±1.1</td><td>27.4±0.9</td><td>28.2±1.4</td><td>17.4±1.2</td><td>28.5±1.4</td><td>20.4±1.5</td><td>19.8±0.9</td></tr><tr><td>DM</td><td>19.2±1.0</td><td>27.2±1.2</td><td>24.4±1.1</td><td>23.0±1.4</td><td>18.4±0.7</td><td>17.7±0.9</td><td>20.6±0.7</td><td>14.5±0.9</td><td>17.8±0.8</td><td>14.5±1.1</td><td>14.0±1.1</td></tr><tr><td rowspan="3">GLaD</td><td>MTT</td><td>29.0±1.2</td><td>39.9±1.2</td><td>39.4±1.3</td><td>34.9±1.1</td><td>30.4±1.5</td><td>29.0±1.1</td><td>30.4±1.5</td><td>17.1±1.1</td><td>28.2±1.1</td><td>21.1±1.2</td><td>19.6±1.2</td></tr><tr><td>DC</td><td>29.8±1.3</td><td>41.8±1.7</td><td>42.1±1.2</td><td>35.8±1.4</td><td>28.0±0.8</td><td>29.3±1.3</td><td>31.0±1.6</td><td>17.8±1.1</td><td>29.1±1.0</td><td>22.3±1.6</td><td>21.2±1.4</td></tr><tr><td>DM</td><td>22.4±1.4</td><td>31.6±1.4</td><td>31.3±3.9</td><td>26.9±1.2</td><td>21.5±1.0</td><td>20.4±0.8</td><td>21.9±1.1</td><td>15.2±0.9</td><td>18.2±1.0</td><td>20.4±1.6</td><td>16.1±0.7</td></tr><tr><td rowspan="3">LD3M</td><td>MTT</td><td>30.4±1.3</td><td>40.9±1.1</td><td>41.6±1.7</td><td>34.1±1.7</td><td>31.5±1.2</td><td>30.1±1.3</td><td>32.0±1.3</td><td>19.9±1.2</td><td>30.4±1.5</td><td>21.4±1.1</td><td>22.1±1.0</td></tr><tr><td>DC</td><td>30.9±1.2</td><td>42.3±1.3</td><td>42.0±1.1</td><td>37.1±1.8</td><td>29.7±1.3</td><td>31.4±1.1</td><td>32.9±1.2</td><td>18.9±0.6</td><td>30.2±1.4</td><td>22.6±1.3</td><td>21.7±0.8</td></tr><tr><td>DM</td><td>25.9±1.2</td><td>35.8±1.1</td><td>34.1±1.0</td><td>30.3±1.2</td><td>24.7±1.0</td><td>24.5±0.9</td><td>26.8±1.7</td><td>18.1±0.7</td><td>23.0±1.8</td><td>24.5±1.9</td><td>17.0±1.1</td></tr></table>
+
+Table 4: Cross-architecture performance (%) with 10 IPC on ImageNet A-E (128 × 128). LD3M (bold, blue) consistently outperforms Pixel Space and GLaD with, for instance, an improvement of +2.52% and +3.46% with DC and DM, respectively.  
+
+<table><tr><td>Distil. Space</td><td>Alg.</td><td>All</td><td>ImNet-A</td><td>ImNet-B</td><td>ImNet-C</td><td>ImNet-D</td><td>ImNet-E</td></tr><tr><td rowspan="2">pixel space</td><td>DC</td><td>42.3±3.5</td><td>52.3±0.7</td><td>45.1±8.3</td><td>40.1±7.6</td><td>36.1±0.4</td><td>38.1±0.4</td></tr><tr><td>DM</td><td>44.4±0.5</td><td>52.6±0.4</td><td>50.6±0.5</td><td>47.5±0.7</td><td>35.4±0.4</td><td>36.0±0.5</td></tr><tr><td rowspan="2">GLaD</td><td>DC</td><td>45.9±1.0</td><td>53.1±1.4</td><td>50.1±0.6</td><td>48.9±1.1</td><td>38.9±1.0</td><td>38.4±0.7</td></tr><tr><td>DM</td><td>45.8±0.6</td><td>52.8±1.0</td><td>51.3±0.6</td><td>49.7±0.4</td><td>36.4±0.4</td><td>38.6±0.7</td></tr><tr><td rowspan="2">LD3M</td><td>DC</td><td>47.1±1.2</td><td>55.2±1.0</td><td>51.8±1.4</td><td>49.9±1.3</td><td>39.5±1.0</td><td>39.0±1.3</td></tr><tr><td>DM</td><td>47.3±2.1</td><td>57.0±1.3</td><td>52.3±1.1</td><td>48.2±4.9</td><td>39.5±1.5</td><td>39.4±1.8</td></tr></table>
+
+Models: We use ConvNet-5/ConvNet-6 [15] for distillation. As in GLaD [5], we use AlexNet [22], VGG-11 [33], ResNet-18 [17], and a Vision Transformer [11] for evaluating the distilled dataset quality for unseen architectures. For LD3M, we use the public ImageNet pre-trained LDM [28] with its  $2 \times$  compression autoencoder, without fine-tuning. Default diffusion steps  $T = 10$  ( $128^2$ ) or  $T = 20$  ( $256^2$ ) are used unless specified. For GLaD comparison, the ImageNet pre-trained StyleGAN-XL [30] is used.
+
+# 5.2 Results
+
+Main Results: LD3M Outperforms State-of-the-Art. We first establish LD3M's effectiveness against the primary baseline GLaD and other state-of-the-art methods on CIFAR-10 (Table 2). At IPC=1, LD3M matches or exceeds GLaD using DC/DM distillation. The sampling-based D4M [34] delivered  $\approx 10\%$  accuracy for all tested models. At IPC=50, LD3M surpasses reported results from D4M and decoupled SRe2L [40] with MTT (though optimization and IPC differ, see caption). We subsequently focus our main analysis on GLaD [5] as the leading state-of-the-art baseline for latent generative prior gradient-matching distillation.
+
+Moving to the more challenging ImageNet subsets, LD3M consistently outperforms GLaD across resolutions and IPC values. For IPC=1 at  $128 \times 128$  (Table 3), LD3M achieves the best overall
+
+Table 5: Cross-architecture performance (\%) for  $256 \times 256$  images (DC, IPC=1) using different generator initializations (ImageNet, FFHQ, Random). LD3M outperforms GLaD across initializations. Best results marked **bold**, top 3 blue. For example, LD3M improves by roughly +2 p.p. on average, whereas it improves the performance by roughly +7 p.p. compared to pixel space.  
+
+<table><tr><td>Distil. Space</td><td>All</td><td>ImNet-A</td><td>ImNet-B</td><td>ImNet-C</td><td>ImNet-D</td><td>ImNet-E</td></tr><tr><td>pixel space</td><td>29.5±3.1</td><td>38.3±4.7</td><td>32.8±4.1</td><td>27.6±3.3</td><td>25.5±1.2</td><td>23.5±2.4</td></tr><tr><td>GLaD (ImageNet)</td><td>34.4±2.6</td><td>37.4±5.5</td><td>41.5±1.2</td><td>35.7±4.0</td><td>27.9±1.0</td><td>29.3±1.2</td></tr><tr><td>GLaD (Random)</td><td>34.5±1.6</td><td>39.3±2.0</td><td>40.3±1.7</td><td>35.0±1.7</td><td>27.9±1.4</td><td>29.8±1.4</td></tr><tr><td>GLaD (FFHQ)</td><td>34.0±2.1</td><td>38.3±5.2</td><td>40.2±1.1</td><td>34.9±1.1</td><td>27.2±0.9</td><td>29.4±2.1</td></tr><tr><td>LD3M (ImageNet)</td><td>36.3±1.6</td><td>42.1±2.2</td><td>42.1±1.5</td><td>35.7±1.7</td><td>30.5±1.4</td><td>30.9±1.2</td></tr><tr><td>LD3M (Random)</td><td>36.5±1.6</td><td>42.0±2.0</td><td>41.9±1.7</td><td>37.1±1.4</td><td>30.5±1.5</td><td>31.1±1.4</td></tr><tr><td>LD3M (FFHQ)</td><td>36.3±1.5</td><td>42.0±1.6</td><td>41.9±1.6</td><td>36.5±2.2</td><td>30.5±0.9</td><td>30.6±1.1</td></tr></table>
+
+Table 6: Ablation: Impact of Diffusion (DC, IPC=1). Comparing LD3M (w/ diffusion) against GLaD and LD3M using only the autoencoder (w/o diffusion). Full LD3M consistently outperforms both.  
+
+<table><tr><td>Method</td><td>All</td><td>ImNet-A</td><td>ImNet-B</td><td>ImNet-C</td><td>ImNet-D</td><td>ImNet-E</td></tr><tr><td>GLaD</td><td>35.4±1.3</td><td>41.8±1.7</td><td>42.1±1.2</td><td>35.8±1.4</td><td>28.0±0.8</td><td>29.3±1.3</td></tr><tr><td>LD3M (w/o diffusion)</td><td>35.3±1.3</td><td>40.6±1.9</td><td>41.9±1.1</td><td>35.3±1.0</td><td>29.4±1.4</td><td>29.5±1.3</td></tr><tr><td>LD3M (w/ diffusion)</td><td>36.5±1.3</td><td>42.3±1.3</td><td>42.0±1.1</td><td>37.1±1.8</td><td>29.7±1.3</td><td>31.4±1.1</td></tr><tr><td></td><td>+1.2+0.0</td><td>+1.7-0.6</td><td>+0.1+0.0</td><td>+1.8+0.8</td><td>+0.3-0.1</td><td>+1.9-0.2</td></tr></table>
+
+average accuracy in 9/10 subsets, significantly boosting performance particularly for DM  $(+16.3\%)$  avg.). This advantage persists at IPC=10 (Table 4), where LD3M improves average accuracy over GLaD by  $+2.5\%$  (DC) and  $+3.5\%$  (DM), reaching over  $47\%$  overall. The performance gains extend to  $256 \times 256$  resolution (Table 5), where LD3M surpasses GLaD by  $+6\%$  on average, even when comparing differently pre-trained generators (ImageNet, FFHQ, random), highlighting the robustness of leveraging the LDM prior via LD3M.
+
+Ablation Studies: Why LD3M Works. The diffusion process itself is crucial for LD3M's improved performance over autoencoder-only approaches. Table 6 provides empirical evidence: removing the diffusion steps ("w/o diffusion", akin to Duan et al. [12]) results in performance comparable to GLaD (35.3% avg.), which is significantly lower than full LD3M utilizing the diffusion process (36.5% avg.). This +1.2 percentage points highlight that simply using the LDM's autoencoder is insufficient; successfully optimizing through the reverse process is necessary to unlock the performance benefits observed, validating our core approach over simpler AE-only methods. Naturally, we can expect similar limitations with few- or one-step diffusion models in the context of dataset distillation.
+
+Furthermore, both learnable latents and our proposed gradient enhancement drive LD3M's effectiveness. Table 7 dissects these contributions: optimizing only conditioning c yields poor results (15.8% avg.), but making the latent  $\mathcal{Z}$  learnable provides a substantial boost (22.3% avg.), confirming the necessity of latent optimization motivated in §4.1. Critically, however, this alone does not surpass GLaD; only by subsequently adding our enhanced gradient flow (Equation 7) does LD3M achieve its SOTA performance (28.1% avg., vs. 26.6% for GLaD). This clearly demonstrates that while learnable latents offer vital degrees of freedom, they cannot be effectively optimized through standard diffusion backpropagation; the enhanced gradient flow enabled by our residual connection (Equation 7) is the key component that makes end-to-end optimization through diffusion truly effective for this task.
+
+Robustness and Practical Considerations. LD3M exhibits robustness in various aspects. For instance, standard initialization using real images encoded via the LDM's encoder significantly benefits DC and DM performance compared to Gaussian noise initialization (Table 8), while being vastly simpler than GLaD's GAN inversion ( $\S 4.2$ ). Visually (Figure 2 and Figure 3), LD3M generates abstract but class-informative images that appear more consistent across different LDM pre-training datasets (ImageNet, FFHQ, Random) compared to GLaD. Our analysis of diffusion steps  $T$  (Figure 4) reveals an optimal performance/runtime trade-off around  $T = 10 - 40$ .
+
+Table 7: Ablation: Different LDM variations (MTT on ImageNette and IPC=1). Tested was the distillation by making only the conditioning learnable (c), one with also learnable latent representation  $(\mathcal{Z})$ , and lastly, one which incorporates our modified reverse process (Equation 7).  
+
+<table><tr><td>Method</td><td>All</td><td>AlexNet</td><td>VGG-11</td><td>ResNet-18</td><td>ViT</td></tr><tr><td>GLaD</td><td>26.6±1.6</td><td>28.7±0.3</td><td>29.2±1.2</td><td>30.8±2.9</td><td>17.8±1.5</td></tr><tr><td>LDM learnable conditioning (c)</td><td>15.8±1.5</td><td>14.2±2.6</td><td>15.1 ± 1.6</td><td>16.5±4.9</td><td>16.8±4.0</td></tr><tr><td>+ learnable latent code (Z)</td><td>22.3±2.0</td><td>22.8±2.0</td><td>26.3±0.9</td><td>23.4±3.2</td><td>17.5±2.0</td></tr><tr><td>+ enhanced gradient flow (Eq. 7)</td><td>28.1±3.3</td><td>29.2±1.9</td><td>29.2±1.2</td><td>30.6±1.3</td><td>25.1±1.7</td></tr></table>
+
+Table 8: Ablation: Impact of Initialization (ImageNette, IPC=1, 5K iter.). Compares initializing  $\mathcal{Z}$  and c from Gaussian noise vs. encoded real images across different distillation algorithms.  
+
+<table><tr><td>Dist. Method</td><td>Dist. Space</td><td>All</td><td>AlexNet</td><td>ResNet18</td><td>VGG11</td><td>ViT</td></tr><tr><td rowspan="2">MTT</td><td>Gauss. noise</td><td>31.0±1.4</td><td>28.7±1.6</td><td>34.1±1.5</td><td>32.2±0.6</td><td>29.1±1.8</td></tr><tr><td>random image</td><td>32.0±1.3</td><td>30.1±1.4</td><td>35.6±1.4</td><td>32.2±0.3</td><td>30.0±1.2</td></tr><tr><td rowspan="2">DC</td><td>Gauss. noise</td><td>13.1±2.1</td><td>13.1±1.5</td><td>11.6±1.8</td><td>13.8±2.2</td><td>13.7±2.7</td></tr><tr><td>random image</td><td>32.9±2.1</td><td>31.6±1.3</td><td>30.4±0.6</td><td>31.8±1.2</td><td>37.7±1.5</td></tr><tr><td rowspan="2">DM</td><td>Gauss. noise</td><td>13.4±1.8</td><td>13.4±2.0</td><td>12.4±1.8</td><td>13.4±1.4</td><td>14.4±2.1</td></tr><tr><td>random image</td><td>26.8±1.7</td><td>31.9±1.3</td><td>23.2±2.2</td><td>25.9±2.0</td><td>26.1±1.4</td></tr></table>
+
+![](images/6df7a7c70f12e869b035e5b0d2eeb94def428a3573f0b815e4f4250a27f566cd.jpg)  
+Figure 2: Visual comparison of LD3M versus GLaD (MTT, ImageNette, 1K iter.). GLaD outputs tend toward smooth, photo-realistic textures but can blur class-defining details, whereas LD3M produces bolder, higher-contrast shapes that highlight key discriminative features (e.g., wing contours, break outline). This abstraction trade-off suggests LD3M prioritizes core class signals over pixel-perfect fidelity, which empirically enhances downstream model generalization, contrasting claims made by sampling-based methods like D4M [34].
+
+LD3M also offers computational flexibility. Our analysis indicates an optimal balance between performance and distillation time occurs with around  $T = 10 - 40$  diffusion steps (Figure 4). Using  $T = 20$ , LD3M requires slightly less peak GPU memory on an A100-40GB compared to GLaD (29.4GB vs 31.2GB) and completes the distillation process faster (574 min vs 693 min). This computational efficiency, coupled with the inherent flexibility to adjust the number of diffusion steps ( $T$ ) based on available resources, enhances LD3M's practical appeal.
+
+# 6 Conclusion
+
+We addressed the critical challenge preventing end-to-end dataset distillation through powerful diffusion models: vanishing gradients across the long denoising chain. We introduced LD3M, which unlocks optimization through diffusion priors via a simple yet effective modification: injecting linearly decaying residual connections from the initial noisy state into each reverse step. This novel approach enhances gradient flow sufficiently to effectively learn both latent codes  $(\mathcal{Z})$  and conditioning (c) without altering pre-trained model weights. This direct optimization contrasts sharply with previous diffusion distillation methods that circumvented the gradient challenge by relying solely on sampling fixed representations [34, 16].
+
+![](images/7d597a883337e14a4d5994168000fd944c610d7b0a2d01c10b8105dd9321fecb.jpg)
+
+![](images/eecd15c075187b3c45c2269e514bb05d295c102bc559997121717c2e95a4d020.jpg)
+
+![](images/ae2d4cf9da0b6b638ce0ecd9b89f425573555bd9dbcf122872f59e9057133a28.jpg)
+
+![](images/c07ec8065945548387e4cd5f494055eb89efdb73ea4ec0f5955470044d196349.jpg)
+
+![](images/6468a8cc158f8d572c95f8fca78047126830f147e8599326712a74d475ef634c.jpg)  
+ImageNet
+
+![](images/d03870751ea6bc4ad0827f394cbb6c933c14fcf2f801af04b7aa24988046e069.jpg)  
+FFHQ
+
+![](images/d8bef657d8996321e5731c13e94de15afb78db5321822bd04de7b3096f46a088.jpg)  
+Random  
+Figure 3: Example  $256 \times 256$  images of a distilled class (ImageNet-B: Lorikeet) with differently initialized generators GLaD and LD3M. The various initializations, i.e., which dataset was used for training the generators, are denoted at the bottom. We used DC as distillation algorithm.
+
+![](images/3123127d105f53dff0bcfe017c6ff7a1dc2b9255f3b464246f5dd36afa407a8f.jpg)  
+Figure 4: Accuracy vs. Distillation Time Trade-off with Diffusion Steps  $T$  (ImageNet A-E avg., MTT,  $\mathrm{IPC} = 1$ ). LD3M performance (blue line, mean  $\pm$  std) peaks around  $T = 35$ . GLaD baseline (dashed lines) and optimal trade-off (X) shown for reference.
+
+Our experiments (§5) demonstrate LD3M's clear advantages. It significantly outperforms the state-of-the-art GAN-prior method, GLaD, on diverse ImageNet subsets at  $128 \times 128$  and  $256 \times 256$  resolutions, improving cross-architecture generalization by up to 4.8 percentage points (IPC=1) and 4.2 points (IPC=10). Ablation studies confirmed that both leveraging the diffusion process and our specific gradient enhancement are crucial for this success. Furthermore, LD3M offers practical benefits: vastly simpler initialization than GAN inversion (§4.2) and faster overall distillation times compared to GLaD (§5).
+
+# 7 Future Work
+
+By successfully enabling gradient-based optimization through diffusion models, LD3M paves the way for leveraging state-of-the-art generative diffusion models for creating highly effective and compact distilled datasets. Future work should explore alternative residual formulations and integration with fast samplers like DPM-Solver [24]. Scaling and evaluation on larger benchmarks like ImageNet-1K also remain important next steps, contingent on computational resources.
+
+# Acknowledgments
+
+This work was supported by the EU project SustainML (Grant 101070408) and by the BMBF Project Albatross (Grant 01IW24002). All compute was done thanks to the Pegasus cluster at DFKI.
+
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+
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+
+# Supplementary Material
+
+# A Definitions of DC, DM, and MTT
+
+Dataset Condensation (DC) ensures alignment by deriving the gradients via a classification error [42]. It calculates the loss on real  $(\ell^{\mathcal{T}})$  and the respective synthetic data  $(\ell^{S})$ . Next, it minimizes the distance between the gradients of both network instances. More concretely,
+
+$$
+\mathcal {L} _ {D C} = 1 - \frac {\nabla_ {\theta} \ell^ {\mathcal {S}} (\theta) \cdot \nabla_ {\theta} \ell^ {\mathcal {T}} (\theta)}{\| \nabla_ {\theta} \ell^ {\mathcal {S}} (\theta) \| \| \nabla_ {\theta} \ell^ {\mathcal {T}} (\theta) \|.} \tag {9}
+$$
+
+Distribution Matching (DM) obtains gradients by minimizing the logits on the real and synthetic datasets. It enforces the feature extractor (ConvNet) to produce similar features for real and synthetic images [41]. The distribution matching loss is
+
+$$
+\mathcal {L} _ {D M} = \sum_ {c} \left\| \frac {1}{| \mathcal {T} _ {c} |} \sum_ {\mathbf {x} \in \mathcal {T} _ {c}} \psi (\mathbf {x}) - \frac {1}{| \mathcal {S} _ {c} |} \sum_ {\mathbf {s} \in \mathcal {S} _ {c}} \psi (\mathbf {s}) \right\| ^ {2}, \tag {10}
+$$
+
+where  $\mathcal{T}_c, S_c$  are the real and synthetic images for a class  $c$ .
+
+Matching Training Trajectories (MTT) concentrates on the trajectory of network parameters [4]. In more detail, MTT exploits several trained instances of a model, called experts, and stores the training trajectory of parameters  $\{\theta_t^*\}_{0}^{T}$  at predetermined intervals, called expert trajectories. For dataset distillation, MTT samples a random set of parameters  $\theta_{t}^{*}$  from the trajectory at a given timestamp. Next, it trains a new network,  $\hat{\theta}_{t + N}$ , initialized with the parameters on the respective synthetic images (for  $N$  iterations). Finally, the distance between the trajectory on the real dataset,  $\theta_{t + M}^{*}$  with  $M$  steps, and the trajectory on the synthetic one,  $\hat{\theta}_{t + N}$ , is minimized. As a result, MTT tries to mimic the original dataset's training path (trajectory of parameters) with the synthetic images:
+
+$$
+\mathcal {L} _ {M T T} = \frac {\left\| \hat {\theta} _ {t + N} - \theta_ {t + M} ^ {*} \right\| ^ {2}}{\left\| \theta_ {t} ^ {*} - \theta_ {t + M} ^ {*} \right\| ^ {2}}. \tag {11}
+$$
+
+# B Justification for Modified Reverse Process in Distillation
+
+Our core modification to the reverse diffusion step is crucial for improving the gradient flow back to the initial latent  $\mathcal{Z}$ , allowing end-to-end optimization. A natural question arises regarding the validity of this modified process, as it intentionally deviates from standard diffusion sampling procedures designed for high-fidelity image generation that perfectly match the learned data distribution.
+
+Different Objectives: Distillation vs. Faithful Sampling. The key insight is that the objective of dataset distillation differs fundamentally from standard image generation. Distillation aims to synthesize a small set of maximally informative samples that enable efficient training of downstream models, prioritizing the encoding of essential class-discriminative features over photorealism [35, 4, 42]. Pixel-perfect adherence to the original data distribution is not necessarily required or even optimal; abstract or stylized images often yield excellent distillation performance if they capture core class characteristics effectively [5].
+
+Impact of Modification. Our modification introduces a direct dependency on the initial noisy state  $\mathbf{z}_T$  throughout the reverse process. While preserving the Markov property (as shown in the main paper) and the representational power of the pre-trained denoiser  $f_{\theta}$ , this change means the resulting process  $p_{\theta}(\mathbf{z}_0|\mathbf{z}_T,\mathbf{c})$  no longer guarantees sampling exactly from the original distribution  $\mu$  learned by the LDM. If applied without the corrective feedback loop of distillation optimization (e.g., for unconditional generation), this modification can lead to more abstract outputs that deviate from the expected style, as illustrated with an FFHQ-trained model in Figure 5. This deviation is expected, as the process is no longer constrained solely by the standard denoising objective. We also observe a slight reduction in sample diversity (measured by average LPIPS between generated samples: 0.386 with modification vs. 0.420 without, on ImageNette samples), likely due to the persistent influence of the fixed  $\mathbf{z}_T$ .
+
+![](images/edba4d78dac684328e5345b588e0e4ad7148fc71e5ebc90c1de38725f6974643.jpg)  
+Figure 5: Influence of our modified reverse process in a classical image generation setting (unconditional FFHQ). It shows that the residual connections alter the generation process significantly, leading to abstract artifacts and the loss of coherence expected in a facial dataset: (top) with modification and (bottom) without modification.
+
+![](images/4bbe0a1b6c6d8080431fbd538cf28f8dcf2c252aa3a5482e3c0c1a44bad7be1a.jpg)  
+Figure 6: Gradient flow analysis comparing the Signal-to-Noise Ratio (SNR) of gradient norms for LD3M with and without our modification. Diffusion demonstrates a more stable gradient flow, indicating enhanced optimization dynamics. Dashed lines show a polyfit plot to highlight the trends.
+
+Suitability for Distillation. Crucially, within the dataset distillation framework, the latent codes  $\mathcal{Z}$  (and conditioning c) are continuously optimized to minimize the distillation loss  $\mathcal{L}$ . This optimization process actively counteracts potential adverse effects of the modified sampling path by guiding the generation towards producing images (even if abstract) that are highly effective for the downstream task defined by  $\mathcal{L}$ . The essential properties needed are: (1) sufficient generative capacity to create diverse, class-relevant features, which the LDM provides; and (2) strong gradient flow for optimization, which our modification enables (as also shown in Figure 6). The empirical success of LD3M - significantly outperforming GLaD and AE-only baselines, and performing robustly even with randomly initialized LDMs - demonstrates that this trade-off (sacrificing perfect distribution matching for tractable optimization) is highly beneficial for the specific goal of dataset distillation. The resulting "abstract" representations effectively encode class information for robust generalization. Therefore, while distinct from standard sampling, our modified reverse process is a well-justified and necessary component for unlocking diffusion models for effective, end-to-end dataset distillation.
+
+Table 9: Common hyperparameters for training the distillation algorithms used in this work.  
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>DSA Augmentations</td><td>Color / Crop / Cutout / Flip / Scale / Rotate</td></tr><tr><td>Iteration (Distillation)</td><td>5,000 (128 × 128) / 10,000 (256 × 256)</td></tr><tr><td>Momentum</td><td>0.5</td></tr><tr><td>Batch Real</td><td>256</td></tr><tr><td>Batch Train</td><td>256</td></tr><tr><td>Batch Test</td><td>128</td></tr></table>
+
+# C Hyper-Parameters for Distillation Algorithms
+
+LDM. For all our LDM experiments, we set the unconditional guidance scale to the default value of 3. For  $128 \times 128$  images, we used max. time steps of 10, and for  $256 \times 256$  images, we used 20.
+
+DC. We utilize a learning rate of  $10^{-3}$  throughout our DC experiments to update the latent code representation and the conditioning information.
+
+DM. In every DM experiment, we adopt a learning rate of  $10^{-2}$ , applying it to updates of the latent code representation alongside the conditioning information.
+
+MTT. For MTT experiments, a uniform learning rate of 10 is applied to update the latent code representation and the conditioning information. We buffered 100 trajectories for expert training, each with 15 training epochs. We used ConvNet-5 and InstanceNorm. During dataset distillation, we used three expert epochs, max. start epoch of 5 and 20 synthetic steps.
+
+# D Large Scale Datasets
+
+Although LD3M is compatible with various distillation algorithms -including DC, DM, and MTT -our current experiments focus on baseline variants that do not leverage inter-class relationships during optimization. This is an essential avenue for further improvement: incorporating inter-class information (e.g., through contrastive losses or hierarchical label structures) may enhance the discriminative quality of the synthetic data. Future work will explore how LD3M's expressive latent trajectories can be used to facilitate such structured, cross-class-aware distillation.
+
+# E Limitations
+
+While LD3M improves dataset distillation compared to GLaD, it is essential to acknowledge certain limitations. A primary concern arises from the linear addition in the diffusion process, which may not sufficiently combat the vanishing gradient problem for larger time steps, as observed in our experiments [19]. Further alternative strategies for integrating the initial state  $\mathbf{z}_T$  in the diffusion process should be evaluated to address this issue, e.g., non-linear progress towards 0 as  $t$  approaches 0. These alternative approaches could offer more nuanced and dynamic ways to manage the influence of  $\mathbf{z}_T$  across different stages of the diffusion, potentially mitigating the problem of vanishing gradients and enhancing the overall efficacy of the distillation process.
+
+# F Hardware and Software
+
+All experiments were run on a workstation equipped with an NVIDIA RTX A6000 GPU (48 GB VRAM). Our implementation uses PyTorch 1.10.1 with torchvision 0.11.2, and we build upon the GLaD library for dataset distillation with a generative prior.
+
+Table 10: Class listings for our ImageNet subsets.  
+
+<table><tr><td>Dataset</td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td></tr><tr><td>ImageNet-A</td><td>Leonberg</td><td>Probiscis Monkey</td><td>Rapeseed</td><td>Three-Toed Sloth</td><td>Cliff Dwelling</td><td>Yellow Lady&#x27;s Slipper</td><td>Hamster</td><td>Gondola</td><td>Orca</td><td>Limpkin</td></tr><tr><td>ImageNet-B</td><td>Spoonbill</td><td>Website</td><td>Lorikeet</td><td>Hyena</td><td>Earthstar</td><td>Trolleybus</td><td>Echidna</td><td>Pomeranian</td><td>Odometer</td><td>Ruddy Turnstone</td></tr><tr><td>ImageNet-C</td><td>Freight Car</td><td>Hummingbird</td><td>Fireboat</td><td>Disk Brake</td><td>Bee Eater</td><td>Rock Beauty</td><td>Lion</td><td>European Gallinule</td><td>Cabbage Butterfly</td><td>Goldfinch</td></tr><tr><td>ImageNet-D</td><td>Ostrich</td><td>Samoyed</td><td>Snowbird</td><td>Brabancon Griffon</td><td>Chickadee</td><td>Sorrel</td><td>Admiral</td><td>Great Gray Owl</td><td>Hornbill</td><td>Ringlet</td></tr><tr><td>ImageNet-E</td><td>Spindle</td><td>Toucan</td><td>Black Swan</td><td>King Penguin</td><td>Potter&#x27;s Wheel</td><td>Photocopier</td><td>Screw</td><td>Tarantula</td><td>Sscilloscope</td><td>Lycaenid</td></tr><tr><td>ImageNette</td><td>Tench</td><td>English Springer</td><td>Cassette Player</td><td>Chainsaw</td><td>Church</td><td>French Horn</td><td>Garbage Truck</td><td>Gas Pump</td><td>Golf Ball</td><td>Parachute</td></tr><tr><td>ImageWoof</td><td>Australian Terrier</td><td>Border Terrier</td><td>Samoyed</td><td>Beagle</td><td>Shih-Tzu</td><td>English Foxhound</td><td>Rhodesian Ridgeback</td><td>Dingo</td><td>Golden Retriever</td><td>English Sheepdog</td></tr><tr><td>ImageNet-Birds</td><td>Peacock</td><td>Flamingo</td><td>Macaw</td><td>Pelican</td><td>King Penguin</td><td>Bald Eagle</td><td>Toucan</td><td>Ostrich</td><td>Black Swan</td><td>Cockatoo</td></tr><tr><td>ImageNet-Fruits</td><td>Pineapple</td><td>Banana</td><td>Strawberry</td><td>Orange</td><td>Lemon</td><td>Pomegranate</td><td>Fig</td><td>Bell Pepper</td><td>Cucumber</td><td>Granny Smith Apple</td></tr><tr><td>ImageNet-Cats</td><td>Tabby Cat</td><td>Bengal Cat</td><td>Persian Cat</td><td>Siamese Cat</td><td>Egyptian Cat</td><td>Lion</td><td>Tiger</td><td>Jaguar</td><td>Snow Leopard</td><td>Lynx</td></tr></table>
+
+![](images/854517cc9b7e01bb06fbb90e13c73082eba91de56d9b5937cdee1097b01ae099.jpg)  
+Figure 7: Images distilled by MTT in LD3M for IPC=1.
+
+![](images/2c8379dd390b19a04907e9568e945387d23db447316c3bd4ec0d8a33f204fe90.jpg)  
+Figure 8: Images distilled by DC in LD3M for IPC=1.
+
+![](images/db78f370c1204838f9c5543ef16a026b2cdd4a351bdf40ba00592d796175455c.jpg)  
+Figure 9: Images distilled by DM in LD3M for IPC=1.
+
+![](images/f942a00088f0aca5169dddcc7200a687fe166611f9077d7d56effe5b60ad097d.jpg)  
+Figure 10: Images distilled by DC in LD3M for IPC=10 and ImageNet-A.
+
+![](images/c5c30d422afe12d96b7f7f419c831e15b69c9739d34089ba60e30a60affc8106.jpg)  
+Figure 11: Images distilled by DC in LD3M for IPC=10 and ImageNet-B.
+
+![](images/08c10f489b4ba8885c5a496f6b052135b0a5201ccc56856e24c16aa5ebdf2e5c.jpg)  
+Figure 12: Images distilled by DC in LD3M for IPC=10 and ImageNet-C.
+
+![](images/103731b534b6a1cbe0e947a36181f86bbc751d46722c911db9185195a030cdd4.jpg)  
+Figure 13: Images distilled by DC in LD3M for IPC=10 and ImageNet-D.
+
+![](images/a24d381ed15d7ac65eebe9e183fd7d042a8983f7af8c2ea0654f0025af168274.jpg)  
+Figure 14: Images distilled by DC in LD3M for IPC=10 and ImageNet-E.
+
+![](images/8cff41460e6a9b9770863b14868729470544fdbf03b92cc787a9cb52dc156daf.jpg)  
+Figure 15: Images distilled by DM in LD3M for IPC=10 and ImageNet-A.
+
+![](images/fce44fd0217375f278751d8c4fcb7368e35a807fa44cb1e8244a71c10603054e.jpg)  
+Figure 16: Images distilled by DM in LD3M for IPC=10 and ImageNet-B.
+
+![](images/c764e9fcc0d38ef569be1e36c5b00f79e2abac87fb274f5a630eb58c504fa5f4.jpg)  
+Figure 17: Images distilled by DM in LD3M for IPC=10 and ImageNet-C.
+
+![](images/076e23f88cfe2b1323ea741b5b346b38002252c4d1419267fbe6a04d161156a2.jpg)  
+Figure 18: Images distilled by DM in LD3M for IPC=10 and ImageNet-D.
\ No newline at end of file
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+# Unlocking Multimodal Mathematical Reasoning via Process Reward Model
+
+Ruilin Luo $^{12*}$  Zhuofan Zheng $^{2*}$  Lei Wang $^{3}$  Yifan Wang $^{1}$   
+Xinzhe Ni $^{1}$  Zicheng Lin $^{1}$  Songtao Jiang $^{4}$  Yiyao Yu $^{1}$   
+Chufan Shi $^{1}$  Ruihang Chu $^{1\dagger}$  Jin Zeng $^{2\dagger}$  Yujiu Yang $^{1}$
+
+$^{1}$ Tsinghua University  $^{2}$ ByteDance
+
+3Ping An Technology (Shenzhen) Co., Ltd. 4Zhejiang University
+
+# Abstract
+
+Process Reward Models (PRMs) have shown promise in enhancing the mathematical reasoning capabilities of Large Language Models (LLMs) through Test-Time Scaling (TTS). However, their integration into multimodal reasoning remains largely unexplored. In this work, we take the first step toward unlocking the potential of PRMs in multimodal mathematical reasoning. We identify three key challenges: (i) the scarcity of high-quality reasoning data constrains the capabilities of foundation Multimodal Large Language Models (MLLMs), which imposes further limitations on the upper bounds of TTS and reinforcement learning (RL); (ii) a lack of automated methods for process labeling within multimodal contexts persists; (iii) the employment of process rewards in unimodal RL faces issues like reward hacking, which may extend to multimodal scenarios. To address these issues, we introduce URSA, a three-stage Unfolding multimodal pProcess-Supervision Aided training framework. We first construct MMathCoT-1M, a high-quality large-scale multimodal Chain-of-Thought (CoT) reasoning dataset, to build a stronger math reasoning foundation MLLM, URSA-8B. Subsequently, we go through an automatic process to synthesize process supervision data, which emphasizes both logical correctness and perceptual consistency. We introduce DualMath-1.1M to facilitate the training of URSA-8B-RM. Finally, we propose Process-Supervised Group-Relative-Policy-Optimization (PS-GRPO), pioneering a multimodal PRM-aided online RL method that outperforms vanilla GRPO. With PS-GRPO application, URSA-8B-PS-GRPO outperforms Gemma3-12B and GPT4o by  $8.4\%$  and  $2.7\%$  on average across 6 benchmarks. Code, data and checkpoint can be found at https://github.com/URSA-MATH.
+
+# 1 Introduction
+
+Following the substantial progress of Large Language Models (LLMs) in math reasoning [1-8], the math reasoning capabilities of Multimodal Large Language Models (MLLMs) have increasingly garnered attention [9-13]. Previous work has typically focused on aspects such as math reasoning data curation [14-18], training math-intensive vision encoders [19, 20], enhancing vision-language alignment [11, 21], or the application of post-training techniques [22-24, 13]. Given the success of Process Reward Models (PRMs) in improving LLM reasoning through methods like Test-Time Scaling (TTS) [25, 26] and Reinforcement Fine-Tuning (ReFT) [27, 28], the application of PRMs to multimodal reasoning remains unexplored.
+
+![](images/0a2e3c930cb9ee09bf062cd3468bf90832cae246c48d2db5f2d24af29178b2f8.jpg)  
+Figure 1: Performance comparison with leading open-source MLLMs and GPT-4o.
+
+In this work, we take the first step toward integrating PRMs into multimodal math reasoning. We identify three key challenges: (i) Since both TTS and RL are heavily influenced by the strength of foundation models [29, 25], the limited availability of large-scale, high-quality reasoning data constrains the upper bounds of current MLLMs and weakens the effectiveness of PRM integration; (ii) There hasn't yet been adequate automated process labeling techniques merged within multimodal contexts, where both logical validity and perceptual consistency should be emphasized [30-32]. (iii) While PRMs can be effectively used in TTS, applying them directly in online RL introduces risks such as reward hacking and length bias in rewarding [33, 34].
+
+To address these challenges, we propose the URSA framework, a three-stage Unfolding multimodal pProcess-Supervision Aided training pipeline that supports both the construction and application of multimodal PRMs. In Stage I, we curate MMathCoT-1M, a large-scale, high-quality multimodal Chain-of-Thought dataset synthesized from 1.43 million open-source examples, which enhances the foundation model's reasoning capabilities through targeted instruction tuning. In Stage II, we construct DualMath-1.1M via a dual-view process supervised data synthesis strategy which combines a binary error locating engine and a misinterpretation insertion engine. It provides complementary signals for logical validity and visual grounding, and is used to train a process reward model. In Stage III, we analyze the limitations of scalar process reward modeling in online RL and propose Process Supervision-GRPO (PS-GRPO), which mitigates reward hacking and PRM's length bias in rewarding by implicitly penalizing process-level inconsistencies during policy optimization.
+
+Results on 6 multimodal reasoning benchmarks show that our PRM improves Best-of-N verification, surpassing self-consistency and outcome-based baselines. When used in PS-GRPO, the resulting model achieves state-of-the-art performance among open-source MLLMs of similar size. Our contributions are as follows:
+
+- We release two large-scale open-source datasets, MMathCoT-1M and DualMath-1.1M, to address the scarcity of high-quality multimodal CoT reasoning and process supervision data.  
+- We propose PS-GRPO, an online reinforcement learning algorithm that incorporates multimodal PRMs by comparing the relative quality of rollouts, rather than relying on scalar reward modeling. It effectively mitigates PRM's reward hacking and length bias in rewarding.  
+- Experimental results show that our reward model improves both test-time verification and online training. With PS-GRPO application (Figure 1), URSA-8B-PS-GRPO outperforms Gemma3-12B and GPT-4o by  $8.4\%$  and  $2.7\%$  on average across 6 benchmarks.
+
+# 2 Stage I: Math-Intensive Alignment and Instruction Tuning
+
+# 2.1 Collection of Vision-Language Alignment Data
+
+We employ a LLaVA-like architecture and first collect vision-language alignment data directly from existing open-source datasets [35-38]. As demonstrated in Figure 2, we collect URSA-Alignment-860K from Multimath [23], MAVIS [19] and Geo170K [18]. We then filter out samples with overly
+
+![](images/1768d757b48cdb56cfffa93e0f3dbc8890d277b43bdc5128ae9175e776847d99.jpg)  
+Figure 2: Statistics of URSA-Alignment-860K and MMathCoT-1M.
+
+![](images/56f1afe57796d1bed7ddac671affcb7f7907201f7aab707d2cb2ba9e7816099f.jpg)
+
+verbose captions, to form an 860K math-intensive alignment dataset. Following the engineering practices of previous work, we only train the MLP projector in the alignment step.
+
+# 2.2 CoT Reasoning Data Synthesis
+
+For a powerful foundation building, we collect 1.43M samples from existing math reasoning datasets to support the construction of large-scale CoT reasoning data. As shown in Figure 2, data is sourced from MathV360K [15], Multimath [23], MAVIS [19], Geo170K [18] and VarsityTutors [11]. Based on the type of solution, we categorize the data into answer-only, analysis-formatted, and CoT-formatted. We adopt different synthesis strategies for them to curate high-quality CoT reasoning trajectories. We utilize Gemini-1.5-Flash-002 (refer to  $\mathcal{G}$  below) as a cost-effective tool for data curation, avoiding expensive large-scale manual annotation.
+
+CoT Expansion. For answer-only data  $\mathcal{D}_1 = \{(x_i, y_i)\}_{i=1}^{N_1}$ , such as MathV360K [15], each sample contains a question  $x_i$  and a ground-truth answer  $y_i$ . This type of data is heavily used in previous works for fast thinking reasoning mode [15, 11, 16]. However, answer-only training restricts the model from fully capturing the problem-solving process. It may lead to memory-based reasoning, hindering the model's ability to directly provide answers to more complex reasoning problems [39]. We expand certain scale CoT reasoning trajectories for this category of data. Given an expansion prompt  $\mathcal{P}_C$ , we provide  $x_i$  and  $y_i$ , then prompt  $\mathcal{G}$  to output the reasoning trajectory leading to the answer  $y_i$ , yielding the expanded solutions  $\mathcal{S}_{Ao} = \mathcal{G}(\mathcal{P}_C; \{x_i, y_i\}_{i=1}^{N_1})$ .
+
+Rewriting. This strategy is designed for analysis-formatted samples, denoted as  $\mathcal{D}_2 = \{(x_i,y_i,a_i)\}_{i=1}^{N_2}$ . This includes datasets like MAVIS-Geo, MAVIS-MetaGen [19], VarsityTutors [11], and Geo170K-QA [40]. Each sample contains a question  $x_i$ , an answer  $y_i$ , and textual analysis  $a_i$ . While this type of data provides walkthroughs, it often suffers from two issues: (i) It lacks strict step-by-step logic, exhibiting jumps in language or reasoning. (ii) A significant portion of the answers are relatively brief and cannot provide rich rationale. Given a rewriting prompt  $\mathcal{P}_{\mathcal{R}}$ , we utilize  $\mathcal{G}$  to transcribe these solutions, thereby enhancing their step-by-step reasoning trajectories and linguistic diversity, resulting in the rewritten set  $S_{An} = \mathcal{G}(\mathcal{P}_{\mathcal{R}}; \{x_i,y_i,a_i\}_{i=1}^{N_2})$ .
+
+Format Unification. This strategy is used for CoT-formatted data, primarily sourced from Multimath-EN-300K [23], which is collected from K-12 textbooks and contains mathematical language and symbolic-style reasoning solutions. This portion of the data,  $\mathcal{D}_3 = \{(x_i,y_i,c_i)\}_{i = 1}^{N_3}$ , consists of a question  $x_{i}$ , an answer  $y_{i}$ , and a solution  $c_{i}$ . We unify the format through natural language stylization using a prompt  $\mathcal{P}_{\mathcal{F}}$  with  $\mathcal{G}$ , producing the unified set  $\mathcal{S}_C = \mathcal{G}(\mathcal{P}_{\mathcal{F}};\{x_i,y_i,c_i\}_{i = 1}^{N_3})$ .
+
+MMathCoT-1M. Finally, we filter out instances where: (i) Correctness is violated: the generated content altered the original answer, or (ii) Consistency is problematic: the solution includes text that questions the original answer or makes new assumptions to force the given answer. This process yields MMathCoT-1M. The complete prompt designs can be found in Appendix H.
+
+![](images/e4210c48e59fd44009874b64bd8ba060de218d491f060b144b7ad803fae5a504.jpg)  
+Figure 3: Pipeline of URSA. Stage 1 depicts the workflow of data curation as described in Section 2. Stage 2 illustrates how binary error locating and misinterpretation insertion facilitate the automation of process supervision data. Stage 3 demonstrates how our PS-GRPO operates by imposing penalties on rollouts that are questioned by the PRM.
+
+![](images/5c8156a42abeef747e11b6cbb50204449c5e21ea123ad9b04d8808eb97a1fc77.jpg)
+
+![](images/32e6709c6e0755bf69eb75f8be8206a4a6923f823431bae1c4a419c89aad0cc3.jpg)
+
+We perform full-parameter instruction fine-tuning with MMathCoT-1M to train URSA-8B, based on the aligned model. The SFT dataset  $\mathcal{D}_{SFT}$  is formed by the union of the curated solutions, i.e.,  $\mathcal{D}_{SFT} = \{(x_i,y_i)\mid (x_i,y_i)\in S_{Ao}\cup S_{An}\cup S_C\}$ . Training objective is demonstrated in Equation 1.
+
+$$
+\mathcal {L} _ {S F T} = - \mathbb {E} _ {(x, y) \sim \mathcal {D} _ {S F T}} \sum_ {t = 1} ^ {T} \log \mathcal {M} \left(y _ {t} \mid x, y _ {<   t}\right) \tag {1}
+$$
+
+In this phase, we construct a stronger reasoning foundation model, URSA-8B, with the expectation of achieving a higher bound at inference time and to process supervision data of greater diversity.
+
+# 3 Stage II: Dual-View Process Supervised Data Synthesis
+
+# 3.1 Binary Error Locating Engine
+
+Following suggestions by previous work [41-43], we train a PRM for first error step identification. We collect  $\sim 553\mathrm{K}$  incorrect solutions from URSA-8B's zero-shot inference on MMathCoT-1M. Erroneous steps in these solutions are labeled using Monte Carlo Tree Search (MCTS). For MCTS, an operation  $\mathcal{F}(\{s_1,\ldots ,s_i\},N)$  generates  $N$  rollouts from a reasoning prefix  $\{s_1,\dots ,s_i\}$ . The single step's Monte Carlo estimation value,  $mc_{i}$ , is the fraction of these rollouts leading to a correct answer:
+
+$$
+m c _ {i} = \frac {\left| \text {C o r r e c t r o l l o u t s f r o m} \mathcal {F} \left(\left\{s _ {1} , s _ {2} , \dots , s _ {i} \right\} , N\right) \right|}{\left| \text {T o t a l r o l l o u t s f r o m} \mathcal {F} \left(\left\{s _ {1} , s _ {2} , \dots , s _ {i} \right\} , N\right) \right|} \tag {2}
+$$
+
+A step  $s_i$  is deemed "potentially correct" if  $mc_i > 0$  [43, 42]. We optimize the identification of first error step using Binary Error Locating Engine (BEL): if the middle step has positive  $mc$  (i.e.  $mc_{mid} > 0$ ), the error is in the latter half; otherwise, in the first (see Algorithm 1). To mitigate step-level label bias and include positive examples, we add  $\sim 180\mathrm{K}$  correct solutions (1/3 the number of incorrect ones), with all steps easily marked "True". This yields  $S_{BEL}$ , a 773K process annotation dataset based on correctness potential.
+
+![](images/64e933c50d6f603b355fcb3f5b1ddf1c13d593d353a341fe18d7573af70a8dba.jpg)  
+Figure 4: Figure (a)-(d) respectively illustrate training rewards, response length, response step number and test set accuracy of vanilla GRPO and two variants proposed in Section 4. Test set is randomly selected 500 examples from MMathCoT-1M for an in-domain evaluation.
+
+![](images/79ae38f2ad0972cb3f14464e8fdff0ea57a11cd11da211b9ccd14a07d0e87512.jpg)
+
+![](images/9c7682e9181f7c4b0fa6f6fb37ccf38d8e715ebc4e1e2b6115572b2740fa4e4a.jpg)
+
+![](images/ad0338c058f91d91a07116f29905e306390d3c0fb662ca0bd8b14ddd05de29c8.jpg)
+
+# 3.2 Misinterpretation Insertion Engine
+
+Apart from logical errors, the perception inconsistency between images and text in reasoning steps is a unique problem in multimodal scenarios [30, 44, 45]. We propose a Misinterpretation Insertion Engine (MIE) to artificially insert hallucinatory information, automatically constructing process supervision data with incorrect reasoning paths starting from the insertion point. Specifically, MIE includes three steps. First, we prompt  $\mathcal{G}$  to perform a captioning task, extracting mathematical paradigm information from the image as much as possible. Second, the model  $\mathcal{G}$  is required to focus on potentially confusable conditions within the existing correct solution and modify them using adjacent or similar conditions. Finally, the model  $\mathcal{G}$  is prompted to continue reasoning based on the step with the inserted error. We leverage strong instruction-following capability of  $\mathcal{G}$ , instructing it to automatically assign negative labels to every subsequent step following the erroneous insertion. We generate  $\sim 302\mathrm{K}$  samples  $S_{MIE}$  using this strategy. Cases from MIE can be found in the Appendix I.2.
+
+# 3.3 PRM Training
+
+As shown in Equation 3, we merge two types of data, proposing a  $\sim 1.1\mathrm{M}$  process supervision data called DualMath-1.1M. During training, we append a special token after each step to indicate its predicted correctness. We model the PRM training as a binary classification task for the correctness of each step, as shown in Equation 4, here  $\pi_p$  is the trained PRM based on URSA-8B.  $e_j$  and  $y_j$  represent single step and corresponding label ( $y_j \in \{0,1\}$ ).
+
+$$
+\mathcal {D} _ {P R M} = \left\{\left(e, y _ {e}\right) \sim \mathcal {S} _ {B E L} \cup \mathcal {S} _ {M I E} \right\} \tag {3}
+$$
+
+$$
+\mathcal {L} _ {P R M} = - \mathbb {E} _ {(e, y) \sim \mathcal {D} _ {P R M}} \sum_ {j = 1} ^ {| e |} \left[ y _ {j} \log \pi_ {p} \left(e _ {j}\right) + \left(1 - y _ {j}\right) \log \left(1 - \pi_ {p} \left(e _ {j}\right)\right) \right] \tag {4}
+$$
+
+Thus, Stage II delivers URSA-8B-RM, a strong PRM trained on DualMath-1.1M—the first large-scale, automatically labeled dataset for multimodal reasoning process supervision. While BoN evaluation demonstrates PRM's value in TTS, a critical question emerges: how can its guidance be directly integrated into MLLM post-training? This remains largely uncharted. Stage III draws a lesson about why previous scalar process reward modeling tends to fail, and then we achieve effective progress through process-as-outcome reward modeling.
+
+# 4 Stage III: Integrating multimodal PRM into RL
+
+Inspired by successes like DeepSeek-R1 [46], several recent studies have tried to adapt outcome reward-based GRPO for multimodal reasoning, demonstrating notable progress [47-50]. Outcome reward-based GRPO computes the  $i$ -th response's advantage through normalizing in-group rewards. However, outcome reward-based GRPO ignores the quality of reasoning processes [41, 51, 52].
+
+Following most standard response-level and step-level reward modeling in RL [43, 28, 46, 13, 53], we examine two simple variants of GRPO with integrated scalar process rewards to reveal the failure patterns during the training process [54]. Variant 1: For  $i$ -th rollout, the reward is the sum of the outcome reward and the average process reward, i.e.  $r^i = r_o^i + r_s^i$ . Variant 2: Despite the outcome
+
+![](images/e2a23a42bd974144498fac48716f92f2df46ac4b595e89c9c60ebf83140bebe3.jpg)  
+Figure 5: Figure (a) shows the BoN evaluation during GRPO training. We select the best rollout using the mean value of process rewards. Figure (b) illustrates the proportion of rollouts where URSA-8B-RM identifies "drop-moment" and the final results are indeed incorrect. Figures (c) and (d) display the response length and test accuracy during PS-GRPO training.
+
+![](images/dd7adb356c8463a6ef8f85528863836a06ea90c3adb3e5e0533caf34a2f422cb.jpg)
+
+![](images/3084d7ab7e4bc8703f45e5149b4ef2e5d7bf1edf1ad63520e46439e452f627f1.jpg)
+
+![](images/bcb3403aa8e36cea4f8ca192f8dae73ecf80693fe4d80d43913dab28dcfbdbd8.jpg)
+
+reward, a scalar process reward  $r_{s,t}^{i}$  is assigned to the  $i$ -th rollout's  $t$ -th step. We observe two highly significant conclusions from Figure 4: (i) High susceptibility to reward hacking. The test accuracy of both variants is lower than vanilla GRPO. This indicates that when process scalar rewards are employed as learning objectives, the model quickly learns strategies that cater to process correctness. However, correctness in the process does not necessarily correlate fully with the heuristics leading to the ground-truth. (ii) PRM's length bias in rewarding. We observe a trend where increased training leads to shorter model responses and fewer reasoning steps. This phenomenon stems from an inherent length bias in the PRM's training labels; for examples with incorrect answers, steps taken after the first error are unlikely to yield a correct solution. This results in the PRM conservatively rewards the later stages of a reasoning rollout, thereby encouraging the MLLM towards more passive reasoning and a reliance on pattern recognition from existing conditions or simpler heuristics.
+
+PS-GRPO The findings above confirm the consideration that flaws in the reward function are amplified when scalar process rewards serve as the optimization target [55, 33]. We ask "Which internal signals of PRM can be trusted?" We employ two views to investigate the reliable region of the PRM: first, the BoN performance during online learning, and second, the PRM's error identification capability. Regarding the latter, we introduce the concept of a "drop-moment" within the PRM's reward sequence, which signifies that the PRM questions the validity of the preceding steps. Specifically, for a given solution's PRM reward sequence  $\{r_{p1}^i,r_{p2}^i,\dots ,r_{pN}^i\}$ , a significant decrease in reward between consecutive steps indicates the occurrence of such a drop-moment.
+
+$$
+\delta_ {p} ^ {i} = \max  \left\{\frac {r _ {p , j} ^ {i} - r _ {p , j + 1} ^ {i}}{r _ {p , j} ^ {i}} \mid j = 0, 1, \dots , N - 1 \right\} > \rho \tag {5}
+$$
+
+Here,  $\rho$  represents PRM's drop-moment threshold. As illustrated in Figure 5, the PRM's ability for BoN selection and error identification remains largely unimpaired during the online RL process, exhibiting stable performance. This suggests that although the scalar reward from the PRM in online RL might be unreliable, the relative quality of solutions it reveals is comparatively trustworthy.
+
+We leverage this beneficial property to address the reward sparsity problem in GRPO [56-58], aiming to make online RL focus more on learning from rollouts that have accurate results and rigorous processes. We use  $\rho$  from Equation 5 as the occurrence threshold for a "drop-moment"; when it occurs, we apply a reward penalty  $\gamma$  to rollouts with correct results. This both differentiates the learning value of outcome-correct rollouts and, due to its focus on relative drops in reward sequences, circumvents the impact of PRM's length bias in rewarding.
+
+$$
+R ^ {i} = \left\{ \begin{array}{l l} 1, & o ^ {i} \text {i s c o r r e c t a n d} \delta_ {p} ^ {i} <   \rho \\ 1 - \gamma , & o ^ {i} \text {i s c o r r e c t a n d} \delta_ {p} ^ {i} \geq \rho \\ 0, & \text {o t h e r w i s e} \end{array} \right. \tag {6}
+$$
+
+We utilize reward modeling in Equation 6 to conduct a process-supervised GRPO, which facilitates the computation of in-group advantages in Equation 7.
+
+Table 1: Performance Comparison on 6 math reasoning benchmarks. We use accuracy for MathVerse, MathVision, MathVista and GeoQA. We use Score (Loose) on WE-MATH. And average-case accuracy is employed on DYNAMATH. Best results of Closed-source MLLMs are highlighted in green. Best and runner-up results of Open-source MLLMs are highlighted in red and blue.  
+
+<table><tr><td></td><td>Size</td><td>Avg</td><td>MathVerse testmini</td><td>MathVision full set</td><td>MathVista gps</td><td>WE-MATH testmini</td><td>DYNAMATH testmini</td><td>GeoQA full set</td></tr><tr><td colspan="9">Closed-Source MLLMs</td></tr><tr><td>GPT-4o [59]</td><td>-</td><td>55.5</td><td>50.2</td><td>30.4</td><td>64.7</td><td>62.8</td><td>64.9</td><td>62.1</td></tr><tr><td>GPT-4o-mini [59]</td><td>-</td><td>49.2</td><td>42.3</td><td>22.8</td><td>59.9</td><td>56.3</td><td>53.5</td><td>60.1</td></tr><tr><td>Gemini-1.5-pro [60]</td><td>-</td><td>53.2</td><td>35.3</td><td>19.2</td><td>81.7</td><td>66.9</td><td>60.5</td><td>55.5</td></tr><tr><td colspan="9">Open-Source General MLLMs</td></tr><tr><td>InternVL-Chat-V1.5 [61]</td><td>26B</td><td>33.6</td><td>26.1</td><td>15.4</td><td>56.9</td><td>32.7</td><td>36.7</td><td>33.5</td></tr><tr><td>Llama-3.2-11B-Vision-Instruct [62]</td><td>11B</td><td>28.0</td><td>28.9</td><td>16.9</td><td>40.9</td><td>12.0</td><td>32.2</td><td>36.9</td></tr><tr><td>Qwen2-VL [63]</td><td>8B</td><td>40.2</td><td>33.6</td><td>19.2</td><td>51.0</td><td>43.0</td><td>42.1</td><td>52.2</td></tr><tr><td>InternVL2-8B [64]</td><td>8B</td><td>41.8</td><td>37.0</td><td>18.4</td><td>57.7</td><td>44.9</td><td>39.7</td><td>52.8</td></tr><tr><td>InternVL2-8B-MPO [65]</td><td>8B</td><td>45.1</td><td>38.2</td><td>22.3</td><td>69.2</td><td>44.4</td><td>40.5</td><td>55.9</td></tr><tr><td>InternVL2.5-8B [66]</td><td>8B</td><td>45.2</td><td>39.5</td><td>19.7</td><td>64.9</td><td>44.7</td><td>40.5</td><td>61.6</td></tr><tr><td>LLaVA-OneVision [35]</td><td>8B</td><td>40.9</td><td>28.9</td><td>18.3</td><td>71.6</td><td>44.9</td><td>37.5</td><td>43.9</td></tr><tr><td>Points-Qwen2.5-Instruct [67]</td><td>8B</td><td>49.8</td><td>41.1</td><td>23.9</td><td>76.0</td><td>51.0</td><td>42.8</td><td>63.8</td></tr><tr><td>Gemma3-12B [68]</td><td>12B</td><td>49.8</td><td>40.1</td><td>29.1</td><td>63.6</td><td>51.7</td><td>45.8</td><td>67.7</td></tr><tr><td colspan="9">Open-Source Reasoning MLLMs</td></tr><tr><td>Math-LLaVA [15]</td><td>13B</td><td>35.2</td><td>22.9</td><td>15.7</td><td>57.7</td><td>31.3</td><td>35.5</td><td>48.1</td></tr><tr><td>MathPUMA-Qwen2-7B [11]</td><td>8B</td><td>39.6</td><td>33.6</td><td>14.0</td><td>48.1</td><td>41.0</td><td>37.3</td><td>63.6</td></tr><tr><td>MultiMath [23]</td><td>7B</td><td>43.1</td><td>27.7</td><td>16.3</td><td>66.8</td><td>42.2</td><td>37.9</td><td>67.7</td></tr><tr><td>MAVIS [19]</td><td>7B</td><td>44.4</td><td>35.2</td><td>18.5</td><td>64.1</td><td>44.3</td><td>36.2</td><td>68.3</td></tr><tr><td>InfiMM-Math [14]</td><td>7B</td><td>48.6</td><td>40.5</td><td>18.8</td><td>77.3</td><td>48.3</td><td>38.2</td><td>68.3</td></tr><tr><td>AtomThink-EMOVA [12]</td><td>8B</td><td>49.5</td><td>42.5</td><td>24.9</td><td>75.9</td><td>49.3</td><td>40.9</td><td>63.8</td></tr><tr><td>MathGLM-Vision [9]</td><td>9B</td><td>47.6</td><td>44.2</td><td>19.2</td><td>64.2</td><td>45.2</td><td>42.2</td><td>70.4</td></tr><tr><td>LlamaV-o1 [69]</td><td>11B</td><td>38.4</td><td>33.9</td><td>17.9</td><td>53.3</td><td>42.6</td><td>34.7</td><td>43.1</td></tr><tr><td>Adora-7B [70]</td><td>7B</td><td>54.6</td><td>50.1</td><td>25.5</td><td>77.6</td><td>58.4</td><td>45.9</td><td>70.1</td></tr><tr><td>MM-Eureka-7B [71]</td><td>7B</td><td>55.7</td><td>50.3</td><td>26.9</td><td>76.5</td><td>60.2</td><td>49.1</td><td>70.9</td></tr><tr><td>OpenVLThinker [72]</td><td>7B</td><td>-</td><td>47.9</td><td>25.3</td><td>76.4</td><td>-</td><td>-</td><td>-</td></tr><tr><td>R1-Onevision [73]</td><td>7B</td><td>-</td><td>47.4</td><td>26.9</td><td>72.4</td><td>51.4</td><td>-</td><td>-</td></tr><tr><td>URSA-8B</td><td>8B</td><td>54.7</td><td>45.7</td><td>28.7</td><td>81.7</td><td>53.6</td><td>44.7</td><td>73.5</td></tr><tr><td>URSA-8B-PS-GRPO</td><td>8B</td><td>58.2</td><td>50.9</td><td>31.5</td><td>83.2</td><td>60.7</td><td>47.4</td><td>75.6</td></tr></table>
+
+Table 2: Comparison of TTS on URSA-8B and AtomThink-EMOVA using BoN performance.  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Method</td><td colspan="4">MathVerse</td><td colspan="4">MathVista-GPS</td><td colspan="4">MathVision</td></tr><tr><td>N=4</td><td>N=8</td><td>N=16</td><td>N=32</td><td>N=4</td><td>N=8</td><td>N=16</td><td>N=32</td><td>N=4</td><td>N=8</td><td>N=16</td><td>N=32</td></tr><tr><td rowspan="3">URSA-8B</td><td>Self-Consistency</td><td>49.3</td><td>50.1</td><td>50.7</td><td>50.7</td><td>82.7</td><td>83.9</td><td>84.8</td><td>85.4</td><td>29.4</td><td>31.9</td><td>32.8</td><td>33.1</td></tr><tr><td>InternVL2.5-8B ORM</td><td>48.6</td><td>50.9</td><td>51.8</td><td>51.3</td><td>82.5</td><td>83.3</td><td>84.3</td><td>85.1</td><td>29.9</td><td>32.1</td><td>32.8</td><td>33.5</td></tr><tr><td>URSA-8B-RM</td><td>53.3</td><td>54.2</td><td>54.7</td><td>55.0</td><td>83.2</td><td>85.5</td><td>86.5</td><td>87.2</td><td>31.6</td><td>33.1</td><td>34.0</td><td>35.1</td></tr><tr><td rowspan="3">AtomThink-EMOVA</td><td>Self-Consistency</td><td>45.9</td><td>46.7</td><td>47.1</td><td>47.3</td><td>76.8</td><td>77.9</td><td>78.6</td><td>79.0</td><td>25.3</td><td>26.8</td><td>27.6</td><td>28.0</td></tr><tr><td>InternVL2.5-8B ORM</td><td>45.7</td><td>45.6</td><td>46.4</td><td>46.1</td><td>76.6</td><td>77.7</td><td>78.3</td><td>79.2</td><td>26.0</td><td>26.6</td><td>27.2</td><td>27.8</td></tr><tr><td>URSA-8B-RM</td><td>48.0</td><td>48.8</td><td>49.3</td><td>49.6</td><td>78.0</td><td>79.6</td><td>80.5</td><td>81.0</td><td>27.5</td><td>29.0</td><td>30.2</td><td>31.0</td></tr></table>
+
+# 5 Experiments
+
+# 5.1 Experimental Setup
+
+**Benchmarks** We evaluate our URSA-series models on 6 widely used reasoning benchmarks, including MathVerse [74], DYNAMATH [75], MathVista [76], WE-MATH [77], GeoQA [40] and MathVision [43]. Detailed description and evaluation criteria can be found in Appendix G.3. We consistently employ zero-shot inference for comparison.
+
+Baselines We include some leading proprietary MLLMs, such as GPT-4o and GPT-4o-mini [59]. For open-source MLLMs with comparable size, we select InternVL-series [64, 78], LLaVA-OneVision [35], Gemma3-12B [68], Qwen2-VL [63], and so on. For MLLMs intended for math reasoning purposes, we select AtomThink [12], InfMM-Math [14], MAVIS [19], MathGLM-Vision [9], LlamaV-o1 [69]. This kind of work focuses on the synthesis of STEM reasoning data or o1-like slow thinking. We do not select baselines that use MathVision as training set for fairness, such as Mulberry-Qwen2-VL-7B [79] and MAmooTH-VL [80]. For PRM's TTS performance, we select Self-Consistency [81] and open-source MLLM as ORM for comparison, such as InternVL2.5-8B [64].
+
+![](images/ac5355600b9e4a0c614ddd1a087386d1a827e61afa1b33d39fb6c5b259e0b2e4.jpg)  
+Figure 6: Figure(a) represents the comparison of relative improvements on URSA-8B; Figure(b) illustrates how each training stage contributes to the total performance.
+
+![](images/37025f3eeb0b6029e2da6dfce2a4e0826b4daf1dba602e42c60a9f3ef5805bd9.jpg)
+
+Table 3: Ablation study on DualMath-1.1M (BoN evaluation). w/o  $S_{MIE}$  and w/o  $S_{BEL}$  represents dropping one part of DualMath-1.1M to train the PRM.  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Dataset</td><td colspan="4">MathVerse</td><td colspan="4">MathVista-GPS</td><td colspan="4">MathVision</td></tr><tr><td>N=4</td><td>N=8</td><td>N=16</td><td>N=32</td><td>N=4</td><td>N=8</td><td>N=16</td><td>N=32</td><td>N=4</td><td>N=8</td><td>N=16</td><td>N=32</td></tr><tr><td rowspan="3">URSA-8B</td><td>DualMath-1.1M</td><td>53.3</td><td>54.2</td><td>54.7</td><td>55.0</td><td>83.2</td><td>85.5</td><td>86.5</td><td>87.2</td><td>31.6</td><td>33.1</td><td>34.0</td><td>35.1</td></tr><tr><td>w/o SMIE</td><td>52.8</td><td>52.6</td><td>52.4</td><td>53.9</td><td>81.3</td><td>83.8</td><td>83.1</td><td>83.2</td><td>29.9</td><td>30.5</td><td>33.1</td><td>34.5</td></tr><tr><td>w/o SBEL</td><td>50.3</td><td>51.4</td><td>51.8</td><td>53.0</td><td>80.1</td><td>83.1</td><td>82.2</td><td>83.0</td><td>28.7</td><td>29.8</td><td>32.3</td><td>34.2</td></tr><tr><td rowspan="3">AtomThink-EMOVA</td><td>DualMath-1.1M</td><td>48.0</td><td>48.8</td><td>49.3</td><td>49.6</td><td>78.0</td><td>79.6</td><td>80.5</td><td>81.0</td><td>27.5</td><td>29.0</td><td>30.2</td><td>31.0</td></tr><tr><td>w/o SMIE</td><td>47.5</td><td>48.2</td><td>47.8</td><td>48.0</td><td>76.8</td><td>78.3</td><td>79.1</td><td>79.5</td><td>26.0</td><td>27.4</td><td>28.5</td><td>29.2</td></tr><tr><td>w/o SBEL</td><td>46.8</td><td>47.5</td><td>47.9</td><td>47.3</td><td>76.0</td><td>77.5</td><td>78.3</td><td>78.7</td><td>25.4</td><td>26.7</td><td>27.8</td><td>28.5</td></tr></table>
+
+Implementation Details URSA uses SAM-B+SigLIP-L as the hybrid vision encoder and Qwen2.5-Math-Instruct as the LLM backbone. We employ a two-layer MLP connection for vision-language alignment training. We select 15K data in MMathCoT-1M for PS-GRPO.  $\gamma$  and  $\rho$  in Equation 6 are set to 0.5 and 0.3, respectively. Details on module selection, data selection, hyperparameters, and time cost are placed in the Appendix D and G.
+
+# 5.2 Main Results
+
+SoTA Performance In Table 1, we present the performance of URSA-8B and URSA-8B-PSGRPO. First, URSA-8B provides a stronger reasoning foundation model. It demonstrates a 5.2 point advantage over AtomThink-EMOVA which focuses on "slow thinking" training. It also outperforms leading general-purpose MLLMs of comparable size, such as Gemma3-12B and InternVL2.5-8B. URSA-8B-PS-GRPO outperforms GPT-4o across 6 benchmarks on average and shows significant advantages on MathVista-GPS (83.2 vs 62.6), GeoQA (73.5 vs 62.1), and achieves the first surpassing performance on MathVision (31.5 vs 30.4). However, a significant performance gap on DynaMath suggests that smaller-scale MLLMs still lack more robust problem-solving capabilities. Compared to the leading math reasoning MLLM AtomThink-EMOVA-8B and general-purpose MLLM Gemma3-12B in terms of average performance, our model shows advantages of  $8.5\%$  and  $8.2\%$ , respectively. Compared with recent R1-inspired method OpenVLThinker [72] and R1-Onevision [73], we still show significant advantage on MathVision and WE-MATH.
+
+Effective Best-of-N Evaluation In Table 2, we demonstrate the advantages of URSA-8B-RM compared to self-consistency and the ORM baseline on serving TTS [43, 42]. We find that self-consistency remains a strong baseline, which InternVL2.5-8B (serving as the ORM) does not consistently surpasses. However, URSA-8B-RM exhibits more effective BoN evaluation and demonstrates its generalization on AtomThink-EMOVA-8B. In addition, using URSA-8B-RM as the verifier, only 4 samplings can achieve a huge improvement based on URSA-8B. Specifically, it provides a  $16.6\%$  and  $10.1\%$  relative improvement on MathVerse and MathVision. In Best-of-32 setting, URSA-8B achieve 35.1 and 55.0 in MathVision and MathVerse, showing clear advantage with GPT-4o.
+
+Table 4: Sensitivity analysis on reward penalty and PRM's "drop-moment" judgment.  
+
+<table><tr><td>γ</td><td>ρ</td><td>MathVerse testmini</td><td>MathVision full set</td><td>MathVista gps</td><td>WE-MATH testmini</td><td>DYNAMATH testmini</td><td>GeoQA full set</td><td>Avg</td></tr><tr><td>0.5</td><td>0.3</td><td>50.9</td><td>31.5</td><td>83.2</td><td>60.7</td><td>47.4</td><td>75.6</td><td>58.2</td></tr><tr><td>0.5</td><td>0.4</td><td>49.9</td><td>30.8</td><td>81.2</td><td>59.9</td><td>46.9</td><td>75.0</td><td>57.3</td></tr><tr><td>0.5</td><td>0.2</td><td>49.6</td><td>30.5</td><td>80.9</td><td>59.6</td><td>46.6</td><td>74.7</td><td>57.0</td></tr><tr><td>1.0</td><td>0.3</td><td>49.0</td><td>29.4</td><td>79.8</td><td>58.8</td><td>45.3</td><td>72.5</td><td>56.3</td></tr><tr><td>0.7</td><td>0.3</td><td>52.0</td><td>31.1</td><td>81.7</td><td>59.6</td><td>47.0</td><td>73.8</td><td>57.5</td></tr><tr><td>0.3</td><td>0.3</td><td>51.5</td><td>32.0</td><td>82.1</td><td>61.0</td><td>46.3</td><td>74.6</td><td>57.9</td></tr></table>
+
+PS-GRPO vs Vanilla GRPO As shown in Figure 6 (a), given the same training data, hyperparameters, and rollout number PS-GRPO achieves a higher improvement on average performance  $(6.8\%)$  vs  $3.1\%$ . PS-GRPO demonstrates an improvement that is nearly double that of vanilla GRPO in WE-MATH and more challenging MathVision, suggesting its effectiveness. We notice that the improvement of RL on MathVista-GPS and GeoQA is relatively small. This is because URSA-8B's inherent abilities have already achieved an effect close to the upper bound on these two benchmarks. However, PS-GRPO still has advantages over vanilla GRPO.
+
+# 6 Analysis
+
+# 6.1 How Each Stage Contributes the Performance
+
+In this section, we demonstrate how each stage contributes to the performance. As demonstrated in Figure 6 (b), all stages make a performance contribution. MMathCoT-1M contributes the highest absolute performance gain. The effect of Alignment-860K is more evident on MathVerse and MathVision, likely because the question images in these two datasets contain richer textual modality information, allowing alignment resources (such as textual images) to better supplement this comprehension capability. PS-GRPO, on the other hand, is dedicated to breaking the bottleneck after large-scale SFT, performing more prominently on WE-MATH and MathVerse with relative improvements of  $13.2\%$  and  $11.4\%$  respectively, compared to URSA-8B. We provide a generalization validation on InternVL2.5-8B and Multimath in Appendix C.4.
+
+# 6.2 Ablation Studies on Automatic Process Labeling
+
+We give an ablation study on how two parts of DualMath-1.1M contribute to URSA-8B-RM. As shown in Table 3, we can see that the method based on BEL, which focuses on the potential to correctness, and the method based on MIE, which focuses on the perception consistency, both contribute positively to the outcome. This further illustrates that in the process of multimodal math reasoning, image-text inconsistency is widespread and needs to be mitigated. We address this issue by augmenting the process supervision training data through the enforced imposition of common hallucination categories. Specifically, the data generated by BEL demonstrates a more significant impact, indicating that the quality of synthesized data can still be improved.
+
+# 6.3 Sensitivity Analysis on Reward Penalty and Drop-moment
+
+In this section, we conduct a sensitivity analysis on two hyperparameters of PS-GRPO,  $\gamma$  and  $\rho$ . These respectively define the magnitude of the reward penalty for rollouts exhibiting a "drop-moment" and the tolerance threshold for identifying such "drop-moments". As shown in Table 4, our core findings are twofold: (i) The value of  $\gamma$  should not be set too high, as this implies excessive trust in the PRM, which may cause the rewards of a group to vanish and lead to training instability. When fixing  $\rho$  at 0.3, we find that setting  $\gamma$  to a value within a certain appropriate range (we test 0.3-0.7) is generally beneficial for average performance. (ii) An excessively large  $\rho$  diminishes reward differentiation, causing the RL behavior to approximate that of vanilla GRPO. Conversely, an excessively small  $\rho$  is unreasonable by design, as it is overly sensitive to process reward changes and tends to result in an overly broad range of penalties. In an extreme case where all correct rollouts are penalized, PS-GRPO degenerates back to vanilla GRPO.
+
+Table 5: Comparison with process reward-integrated online RL methods.  
+
+<table><tr><td>Method</td><td>Avg</td><td>MathVerse testmini</td><td>MathVision full set</td><td>MathVista gps</td><td>WE-MATH testmini</td><td>DYNAMATH testmini</td><td>GeoQA full set</td></tr><tr><td>Variant1</td><td>55.3</td><td>48.2</td><td>27.6</td><td>79.5</td><td>57.3</td><td>45.8</td><td>72.7</td></tr><tr><td>Variant2</td><td>56.0</td><td>49.1</td><td>28.5</td><td>80.0</td><td>57.6</td><td>46.2</td><td>74.3</td></tr><tr><td>DS-GRPO</td><td>56.5</td><td>49.9</td><td>29.2</td><td>79.8</td><td>58.5</td><td>47.8</td><td>74.3</td></tr><tr><td>PS-GRPO (Ours)</td><td>58.2</td><td>50.9</td><td>31.5</td><td>83.2</td><td>60.7</td><td>47.4</td><td>75.6</td></tr></table>
+
+# 6.4 PS-GRPO vs. PRM-integrated Baselines
+
+In this section, we further evaluate the two enhanced baselines that utilize scalar process rewards, as previously proposed, alongside the online reinforcement learning method with process-reward participation introduced by DeepSeek (hereafter referred to as DS-GRPO). Specifically, DS-GRPO computes step-level advantages by aggregating all in-batch process rewards: the advantage assigned to each token is the cumulative sum of rewards from all subsequent steps. All these methods represent variants that leverage scalar process rewards to facilitate online reinforcement learning. As shown in the comparison results in the Table 5, PS-GRPO outperforms three strong baselines, demonstrating that our proposed process-to-outcome reward modeling constitutes a leading approach to utilizing process rewards. This further underscores its advantages in mitigating process reward hacking and rewarding length bias.
+
+# 7 Conclusion
+
+In this study, we take the first step to thoroughly explore the application of PRM in multimodal math reasoning. We introduce a three-stage training pipeline URSA designed to address three major challenges. Initially, we provide a large-scale CoT reasoning dataset MMathCoT-1M. This dataset forms the basis for developing URSA-8B, a MLLM with enhanced reasoning capabilities, and paves the way for further TTS or RL scenarios. Next, we present a dual-view automated process supervision annotation method, covering logical validity and perceptual consistency in multimodal scenarios. We introduce the first large-scale process supervision dataset in multimodal reasoning, DualMath-1.1M. Finally, we address reward hacking and rewarding length bias through process-as-outcome modeling, and put forward PS-GRPO, which is a PRM-aided online RL method that surpasses GRPO. The resulting URSA-8B-PS-GRPO model demonstrates superior average performance over leading open-source MLLM such as Gemma3-12B (8.4%) and proprietary GPT-4o (2.7%).
+
+# 8 Acknowledgments
+
+This work was partly supported by the National Natural Science Foundation of China (Grant No. 62576191) and the Shenzhen Science and Technology Program (JCYJ20220818101014030).
+
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+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: Our claims in the abstract and introduction are supported by corresponding experiments and ablation studies.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: Discussion about limitation is placed in Section J.
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [No]
+
+Justification: Our work mainly focuses on the empirical science of enhancing the reasoning capabilities of MLLMs, without proposing any new theorems.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: We provide all the experimental details in Section 5.1 and Appendix G, and offer anonymous links to the code, checkpoints, and data.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.
+
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
+
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
+
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
+
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
+
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: We provide open access to the data, code and checkpoints on anonymous link given above.
+
+# Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
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+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.  
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: We provide these information on Section 5 and Appendix G.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [No]
+
+Justification: The experimental results do not include error bars, as the large number of experiments incurs high costs. However, we have fixed the random seeds.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).  
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.
+
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: The information are demonstrated in Appendix G.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: Our work respectfully follows the code of ethics of NeurIPS 2025.
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: We discuss the broader impact in Appendix K.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
+
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [Yes]
+
+Justification: Our work involves the release of the pretrained model and the training data, and we comply with the relevant fair-use licenses.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
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+# 12. Licenses for existing assets
+
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+# 13. New assets
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+# Guidelines:
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+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
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+Answer: [NA]
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+Justification: The paper does not involve crowdsourcing nor research with human subjects.
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+# Guidelines:
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+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
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+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
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+Answer: [NA]
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+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+# Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
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+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [Yes]
+
+Justification: Our research focuses on enhancing the reasoning capabilities of (M)LLMs.
+
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+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# Appendices Content
+
+A Related Work 26  
+B Preliminary 26
+
+B.1 Group Relative Policy Optimization 26  
+B.2 Test-Time Scaling by Best-of-N evaluation 27
+
+C Supplementary Results 27
+
+C.1 Fine-grained Comparison on Used benchmarks 27  
+C.2 Scaling Law of MMathCoT-1M 28  
+C.3 Higher Upper Bound Taken from Stage I 29  
+C.4 Generalization Validation 29  
+C.5 Implementary Results on Other Benchmarks 30
+
+D Module Selection Criteria 30  
+E Ablation Studies 31
+
+E.1 Effectiveness of Different Data Category 31  
+E.2 Selection of External Closed-source MLLM 32
+
+F Theoretical Analysis 33  
+G Implementation Details 34
+
+G.1 RL Data Curation 34  
+G.2 Parameters and Time Cost 34  
+G.3 Benchmarks 34  
+G.4 Algorithm 35
+
+H Prompt Design 37
+
+H.1 Prompt Utilized in MMathCoT-1M Synthesis 37  
+H.2 Prompt Utilized in DualMath-1.1M Synthesis 38
+
+I Case Study 39
+
+I.1 Showcase on Best-of-N Evaluation 39  
+I.2 Process of Misinterpretation Insertion Engine 39  
+I.3 Failure Pattern in Process During GRPO 39  
+I.4 Cases on How Naive Process Reward Modeling Fails 40
+
+J Limitations. 41  
+K Broader Impact 41
+
+# A Related Work
+
+Multimodal Mathematical Reasoning The mathematical and scientific reasoning capabilities of MLLMs have recently attracted significant attention [11, 82, 35, 83, 84, 5, 14, 80, 85, 86]. Unlike traditional mathematical reasoning tasks in Language Models (LLMs) [1, 87], multimodal mathematical reasoning requires MLLMs to interpret visual information and perform cross-modal reasoning between images and text. Tasks such as solving geometric problems and analyzing graphs are particularly challenging [40]. Recent advances have focused on improving visual mathematical input specialized encoders in specific scenarios [19, 88, 61]. A significant emphasis has also been placed on synthesizing diverse and complex training data. For instance, Math-LLaVA [15] introduces the MathV360K dataset, which categorizes images by complexity and enhances associated questions. Multimath [23] curates high-quality reasoning data from K-12 textbooks and employs GPT-4 for CoT data generation and validation. R-CoT [17] further diversifies problems through a two-stage reverse question-answer generation process. These data synthesis methods are widely adopted in academia and industry due to their demonstrated efficiency [89-92]. PeRL [93] explores RL for interleaved multimodal reasoning.
+
+Process Reward Model Recent studies have explored test-time scaling laws in LLMs, aiming to identify optimal reasoning eclectories from diverse thinking trajectories [26, 94-96, 31, 97, 98]. Initial efforts, such as self-consistency [81], have laid the groundwork for test-time scaling. OpenAI has introduced verifiers to supervise and select reasoning paths during inference [41]. Math-Shepherd [43] evaluates intermediate reasoning steps based on their likelihood of leading to correct answers, while OmegaPRM [42] constructs PRM training data and employs MCTS for training. Despite these advancements, the lack of models with robust CoT reasoning capabilities and limited exploration into diverse reward model training data remain significant bottlenecks in multimodal mathematical reasoning. Some concurrent work also begins to pay attention to PRM-assisted visual reasoning, such as construction and benchmarking [99-101].
+
+# B Preliminary
+
+# B.1 Group Relative Policy Optimization
+
+Vanilla GRPO eliminates value function in PPO and estimates the advantages within online rollout group. Given a question with image  $q$  and ground-truth  $y$ , policy model  $\pi_{\theta_{old}}$  samples a group of  $G$  responses  $\{o^i\}_{i=1}^G$ . GRPO compute the  $i$ -th response's advantage through normalizing in-group rewards  $\{r^j\}_{j=1}^G$ , and employs PPO's clipped objective and KL penalty term:
+
+$$
+A ^ {i} = \frac {r ^ {i} - \operatorname {m e a n} \left(\left\{r ^ {j} \right\} _ {j = 1} ^ {G}\right)}{\operatorname {s t d} \left(\left\{r ^ {j} \right\} _ {j = 1} ^ {G}\right)} \tag {7}
+$$
+
+$$
+\begin{array}{l} \mathcal {J} _ {G R P O} (\theta) = \mathbb {E} _ {(q, y) \sim \mathcal {D}, \{o ^ {i} \} _ {i = 1} ^ {G}} \sim \pi_ {\theta_ {o l d}} (\cdot | q) \\ \left[ \frac {1}{G} \sum_ {i = 1} ^ {G} \frac {1}{| o ^ {i} |} \sum_ {t = 1} ^ {| o ^ {i} |} \left(\min  \left(r _ {t} ^ {i} (\theta) A ^ {i}, \operatorname {c l i p} \left(r _ {t} ^ {i} (\theta), 1 - \epsilon , 1 + \epsilon\right) A ^ {i}\right) - \beta D _ {K L} ^ {i, t} \left(\pi_ {\theta} \mid \mid \pi_ {r e f}\right)\right) \right] \tag {8} \\ \end{array}
+$$
+
+We introduce the two variants of PRM-integrated GRPO discussed in Section 4. Given PRM  $\mathcal{M}_p$  and process reward sequences  $r_s = \mathcal{M}_p(\{s_1, s_2, \dots, s_N\})$  (i) Variant 1: Given verifiable outcome reward  $r_o^i$ , we set a single rollout's reward as  $r^i = r_o^i + r_s^i$ . (ii) Variant 2: We utilize a step-level reward and a multiple relative advantage calculated by the mean value of process rewards from each rollout:
+
+$$
+A _ {t} ^ {i} = \underbrace {r _ {s , t} ^ {i} \frac {\bar {r _ {s} ^ {i}} - \operatorname {m e a n} \left(\left\{\bar {r _ {s} ^ {j}} \right\} _ {j = 1} ^ {G}\right)}{\operatorname {s t d} \left(\left\{\bar {r _ {s} ^ {j}} \right\} _ {j = 1} ^ {G}\right)}} _ {\text {G R P O w i t h p r o c e s s r e w a r d s}} + \underbrace {\frac {r _ {o} ^ {i} - \operatorname {m e a n} \left(\left\{r _ {o} ^ {j} \right\} _ {j = 1} ^ {G}\right)}{\operatorname {s t d} \left(\left\{r _ {o} ^ {j} \right\} _ {j = 1} ^ {G}\right)}} _ {\text {G R P O w i t h o u t c o m e r e w a r d s}} \tag {9}
+$$
+
+in which  $\bar{r_s^i} = \mathrm{mean}(\mathcal{M}_p(\{s_1^i,s_2^i,\dots ,s_{T_i}^i\}))$
+
+# B.2 Test-Time Scaling by Best-of-N evaluation
+
+Following previous works [41, 43], we adopt BoN evaluation for TTS. Given  $N$  response samplings for a question  $q$ . The PRM is used to give process reward for each sampling. We use mean value of process rewards to select the best single sampling:
+
+$$
+a _ {\mathrm {p r m}} = \arg \max  _ {s _ {i}} \operatorname {m e a n} \left\{\mathcal {M} _ {p} \left(q, s _ {i}\right) \right\} \tag {10}
+$$
+
+Some other works [102] merge self-consistency and PRM to employ a voting-based score cumulation. But we don't select this method for a simpler evaluation manner.
+
+# C Supplementary Results
+
+# C.1 Fine-grained Comparison on Used benchmarks
+
+In this section, we provide some fine-grained results for a clearer comparison. As demonstrated in Table 6, our proposed methods demonstrate significant advantages. Compared to closed-source models like GPT-4o and GPT-4V, our URSA-8B and URSA-8B-PS-GRPO show strong competitiveness. Among open-source models, the performance improvements are even more evident. Our URSA-8B model outperforms other open-source models such as InternLM-XComposer2-VL and Ovis1.6-Gemma2-9B in most subtasks. When combined with PS-GRPO, the URSA-8B-PS-GRPO model achieves even better results, showing significant improvements in subtasks like Alg, AnaG, CombG, and others. Our methods particularly excel in complex mathematical reasoning tasks, demonstrating their powerful mathematical reasoning capabilities. These results highlight the effectiveness of our proposed MMathCoT-1M and PS-GRPO methods in enhancing the mathematical reasoning abilities of models, especially in visual mathematical problems.
+
+In Dynamath (Table 7), compared to open-source MLLMs, the URSA series has obvious advantages in plane geometry and algebra. Surprisingly, from the knowledge level classification, the URSA series model performs excellently at the undergraduate level, which is partly attributable to its math-intensive alignment and large-scale instruction fine-tuning.
+
+In MathVerse (Table 8), we can see that URSA series model marginally surpass GPT-4o on average. Besides, compared with other open-source MLLMs, URSA-8B-PS-GRPO outperforms leading AtomThink-EMOVA-8B and InternVL2.5-8B with 8.4 and 11.4 points. In WE-MATH 9, URSA
+
+Table 6: Performance comparison of different MLLMs on MathVision.  
+
+<table><tr><td>Model</td><td>Size</td><td>ALL</td><td>Alg</td><td>AnaG</td><td>Ari</td><td>CombG</td><td>Comb</td><td>Cnt</td><td>DescG</td><td>GraphT</td><td>Log</td><td>Angle</td><td>Area</td><td>Len</td><td>SolIG</td><td>Stat</td><td>Topo</td><td>TransG</td></tr><tr><td colspan="19">Baselines</td></tr><tr><td>Human</td><td>-</td><td>68.8</td><td>55.1</td><td>78.6</td><td>99.6</td><td>98.4</td><td>43.5</td><td>98.5</td><td>91.3</td><td>62.2</td><td>61.3</td><td>33.5</td><td>47.2</td><td>73.5</td><td>87.3</td><td>93.1</td><td>99.8</td><td>69.0</td></tr><tr><td colspan="19">Closed-source MLLMs</td></tr><tr><td>GPT-4o</td><td>-</td><td>30.4</td><td>42.0</td><td>39.3</td><td>49.3</td><td>28.9</td><td>25.6</td><td>22.4</td><td>24.0</td><td>23.3</td><td>29.4</td><td>17.3</td><td>29.8</td><td>30.1</td><td>29.1</td><td>44.8</td><td>34.8</td><td>17.9</td></tr><tr><td>GPT-4V</td><td>-</td><td>22.8</td><td>27.3</td><td>32.1</td><td>35.7</td><td>21.1</td><td>16.7</td><td>13.4</td><td>22.1</td><td>14.4</td><td>16.8</td><td>22.0</td><td>22.2</td><td>20.9</td><td>23.8</td><td>24.1</td><td>21.7</td><td>25.6</td></tr><tr><td>CoT GPT-4V</td><td>-</td><td>24.0</td><td>26.7</td><td>26.2</td><td>38.6</td><td>22.1</td><td>24.4</td><td>19.4</td><td>27.9</td><td>23.3</td><td>25.2</td><td>17.3</td><td>21.4</td><td>23.4</td><td>23.8</td><td>25.9</td><td>4.4</td><td>25.6</td></tr><tr><td>Gemini-1.5-Pro</td><td>-</td><td>19.2</td><td>20.3</td><td>35.7</td><td>34.3</td><td>19.8</td><td>15.5</td><td>20.9</td><td>26.0</td><td>26.7</td><td>22.7</td><td>14.5</td><td>14.4</td><td>16.5</td><td>18.9</td><td>10.3</td><td>26.1</td><td>17.3</td></tr><tr><td colspan="19">Open-source MLLMs</td></tr><tr><td>LLaVA-1.5</td><td>7B</td><td>8.5</td><td>7.0</td><td>7.1</td><td>10.7</td><td>7.1</td><td>4.8</td><td>10.5</td><td>7.7</td><td>10.0</td><td>9.2</td><td>15.6</td><td>10.2</td><td>9.8</td><td>5.3</td><td>8.6</td><td>4.4</td><td>4.8</td></tr><tr><td>LLaVA-1.5</td><td>13B</td><td>11.1</td><td>7.0</td><td>14.3</td><td>14.3</td><td>9.1</td><td>6.6</td><td>6.0</td><td>13.5</td><td>5.6</td><td>13.5</td><td>10.4</td><td>12.6</td><td>14.7</td><td>11.5</td><td>13.8</td><td>13.0</td><td>10.7</td></tr><tr><td>InternalLM-XComposer2-VL</td><td>7B</td><td>14.5</td><td>9.3</td><td>15.5</td><td>12.1</td><td>15.3</td><td>11.3</td><td>10.5</td><td>14.4</td><td>22.2</td><td>19.3</td><td>19.7</td><td>15.6</td><td>15.0</td><td>11.9</td><td>15.5</td><td>26.1</td><td>15.5</td></tr><tr><td>Ovis1.6-Gemma2-9B</td><td>9B</td><td>18.8</td><td>13.3</td><td>15.5</td><td>22.1</td><td>17.9</td><td>11.3</td><td>22.4</td><td>23.1</td><td>20.0</td><td>20.2</td><td>20.8</td><td>18.0</td><td>24.7</td><td>15.6</td><td>20.7</td><td>17.4</td><td>20.8</td></tr><tr><td>MiniCPM-v2.6</td><td>8B</td><td>18.4</td><td>9.9</td><td>19.0</td><td>18.6</td><td>21.8</td><td>13.1</td><td>13.4</td><td>17.3</td><td>20.0</td><td>16.0</td><td>25.4</td><td>19.4</td><td>20.7</td><td>15.2</td><td>27.6</td><td>30.4</td><td>22.0</td></tr><tr><td>LLaVA-OneVision</td><td>8B</td><td>18.3</td><td>11.6</td><td>16.7</td><td>20.7</td><td>18.5</td><td>11.9</td><td>14.9</td><td>19.2</td><td>13.3</td><td>20.2</td><td>17.9</td><td>21.6</td><td>23.4</td><td>12.3</td><td>22.4</td><td>13.0</td><td>24.4</td></tr><tr><td>Qwen2-VL</td><td>8B</td><td>19.2</td><td>15.4</td><td>20.2</td><td>19.3</td><td>16.9</td><td>16.7</td><td>17.9</td><td>22.1</td><td>22.2</td><td>16.0</td><td>19.1</td><td>22.4</td><td>22.5</td><td>14.8</td><td>19.0</td><td>4.3</td><td>23.8</td></tr><tr><td>InternalVL2-8B</td><td>8B</td><td>18.4</td><td>18.6</td><td>22.6</td><td>28.6</td><td>22.1</td><td>13.7</td><td>10.4</td><td>11.5</td><td>13.3</td><td>21.0</td><td>20.8</td><td>22.4</td><td>20.5</td><td>16.8</td><td>17.2</td><td>26.1</td><td>24.2</td></tr><tr><td>InternalVL2.5-8B</td><td>8B</td><td>19.7</td><td>15.1</td><td>23.8</td><td>29.3</td><td>16.2</td><td>8.9</td><td>11.9</td><td>10.6</td><td>8.9</td><td>18.5</td><td>22.0</td><td>19.4</td><td>15.4</td><td>13.9</td><td>22.4</td><td>21.7</td><td>19.6</td></tr><tr><td colspan="19">Open-source Math MLLMs</td></tr><tr><td>Math-LLaVA</td><td>13B</td><td>15.7</td><td>9.0</td><td>20.2</td><td>15.7</td><td>18.2</td><td>10.1</td><td>10.5</td><td>16.4</td><td>14.4</td><td>16.0</td><td>20.2</td><td>18.4</td><td>17.6</td><td>9.4</td><td>24.1</td><td>21.7</td><td>17.9</td></tr><tr><td>Multimath</td><td>7B</td><td>16.3</td><td>11.3</td><td>21.1</td><td>15.5</td><td>15.9</td><td>11.3</td><td>12.1</td><td>15.5</td><td>15.9</td><td>18.5</td><td>20.1</td><td>16.4</td><td>21.3</td><td>13.3</td><td>14.6</td><td>13.3</td><td>20.8</td></tr><tr><td>Math-PUMA-Qwen2-7B</td><td>8B</td><td>14.0</td><td>5.0</td><td>21.1</td><td>21.1</td><td>21.1</td><td>11.0</td><td>5.6</td><td>15.7</td><td>10.5</td><td>13.8</td><td>11.7</td><td>15.8</td><td>12.2</td><td>17.8</td><td>19.2</td><td>15.8</td><td>12.2</td></tr><tr><td>MAVIS</td><td>7B</td><td>18.5</td><td>17.5</td><td>19.5</td><td>21.5</td><td>19.0</td><td>12.0</td><td>14.0</td><td>18.0</td><td>16.0</td><td>19.0</td><td>21.0</td><td>18.5</td><td>19.5</td><td>15.0</td><td>19.0</td><td>20.0</td><td>20.0</td></tr><tr><td>AtomThink-EMOVA</td><td>8B</td><td>24.9</td><td>23.5</td><td>25.5</td><td>32.0</td><td>21.0</td><td>15.8</td><td>19.5</td><td>21.5</td><td>22.5</td><td>21.5</td><td>26.5</td><td>25.5</td><td>26.5</td><td>27.5</td><td>28.0</td><td>23.0</td><td>22.5</td></tr><tr><td>URSA-8B</td><td>8B</td><td>28.7</td><td>28.1</td><td>26.2</td><td>35.0</td><td>22.1</td><td>15.5</td><td>19.4</td><td>18.3</td><td>22.2</td><td>21.8</td><td>37.0</td><td>27.0</td><td>26.5</td><td>31.1</td><td>27.6</td><td>17.4</td><td>23.8</td></tr><tr><td>URSA-8B-PS-GRPO</td><td>8B</td><td>31.5</td><td>30.1</td><td>28.6</td><td>29.3</td><td>31.5</td><td>20.8</td><td>20.9</td><td>26.9</td><td>17.8</td><td>24.4</td><td>35.8</td><td>33.6</td><td>37.2</td><td>37.7</td><td>25.9</td><td>26.1</td><td>35.1</td></tr></table>
+
+series outperforms leading general-purpose and math reasoning MLLMs in three-stage accuracy. Also, the URSA series has remarkable strengths in solid figures, transformations, positions, and directions. This is mainly due to large-scale alignment and instruction tuning, which builds its foundation in understanding mathematical elements.
+
+Table 7: Detailed performance comparison of MLLMs on DYNAMATH testmini dataset, broken down by subject area and knowledge level.  
+
+<table><tr><td>Model</td><td>Size</td><td>ALL</td><td>PG</td><td>SG</td><td>AG</td><td>AL</td><td>PT</td><td>GT</td><td>AR</td><td>Elem.</td><td>High</td><td>Undergrad.</td></tr><tr><td colspan="13">Closed-source MLLMs</td></tr><tr><td>GPT-4o</td><td>-</td><td>64.9</td><td>56.8</td><td>52.0</td><td>61.0</td><td>76.9</td><td>51.8</td><td>58.1</td><td>61.5</td><td>68.6</td><td>61.8</td><td>36.8</td></tr><tr><td>Claude-3.5-Sonnet</td><td>-</td><td>64.8</td><td>49.9</td><td>49.3</td><td>55.3</td><td>81.0</td><td>44.1</td><td>69.4</td><td>61.2</td><td>66.7</td><td>62.6</td><td>33.3</td></tr><tr><td>Gemini-1.5-Pro</td><td>-</td><td>60.5</td><td>52.7</td><td>42.7</td><td>61.6</td><td>70.8</td><td>20.6</td><td>65.2</td><td>54.2</td><td>62.9</td><td>59.2</td><td>37.1</td></tr><tr><td colspan="13">Open-source MLLMs</td></tr><tr><td>Llava-v1.5-7B</td><td>7B</td><td>16.6</td><td>10.5</td><td>7.3</td><td>19.5</td><td>6.5</td><td>8.2</td><td>32.3</td><td>10.8</td><td>18.9</td><td>13.3</td><td>11.7</td></tr><tr><td>Llava-v1.6-34B</td><td>34B</td><td>27.1</td><td>21.4</td><td>25.3</td><td>27.6</td><td>14.9</td><td>7.6</td><td>32.7</td><td>23.1</td><td>35.9</td><td>23.8</td><td>16.6</td></tr><tr><td>Deepseek-VL-7B-Chat</td><td>7B</td><td>21.5</td><td>16.0</td><td>13.3</td><td>26.5</td><td>12.9</td><td>4.7</td><td>32.3</td><td>12.7</td><td>28.3</td><td>19.0</td><td>16.0</td></tr><tr><td>InternVL2-8B</td><td>8B</td><td>39.7</td><td>33.9</td><td>37.3</td><td>32.5</td><td>46.9</td><td>15.9</td><td>42.1</td><td>37.3</td><td>51.1</td><td>37.4</td><td>19.6</td></tr><tr><td>Qwen2-VL</td><td>8B</td><td>42.1</td><td>40.3</td><td>38.7</td><td>39.9</td><td>37.1</td><td>8.2</td><td>44.8</td><td>39.2</td><td>47.6</td><td>42.2</td><td>24.4</td></tr><tr><td>AtomThink-EMOVA</td><td>8B</td><td>40.9</td><td>42.0</td><td>37.9</td><td>33.6</td><td>58.0</td><td>23.0</td><td>44.0</td><td>38.4</td><td>52.5</td><td>43.5</td><td>32.0</td></tr><tr><td>URSA-8B</td><td>8B</td><td>44.7</td><td>48.1</td><td>38.0</td><td>33.7</td><td>66.9</td><td>24.7</td><td>39.2</td><td>38.5</td><td>53.5</td><td>44.3</td><td>41.8</td></tr><tr><td>URSA-8B-PS GRPO</td><td>8B</td><td>47.4</td><td>49.7</td><td>40.1</td><td>35.2</td><td>65.7</td><td>24.7</td><td>45.2</td><td>41.1</td><td>53.5</td><td>46.7</td><td>43.2</td></tr></table>
+
+Table 8: Comparison with closed-source MLLMs and open-source MLLMs on MATHVERSE testmini. The best results of Closed-source MLLMs are highlighted. The best and second-best results of Open-source MLLMs are highlighted.  
+
+<table><tr><td>Model</td><td>#Params</td><td>ALL</td><td>TD</td><td>TL</td><td>TO</td><td>VI</td><td>VD</td><td>VO</td></tr><tr><td colspan="9">Baselines</td></tr><tr><td>Random</td><td>-</td><td>12.4</td><td>12.4</td><td>12.4</td><td>12.4</td><td>12.4</td><td>12.4</td><td>12.4</td></tr><tr><td>Human</td><td>-</td><td>64.9</td><td>71.2</td><td>70.9</td><td>41.7</td><td>61.4</td><td>68.3</td><td>66.7</td></tr><tr><td colspan="9">Closed-Source MLLMs</td></tr><tr><td>GPT-4o</td><td>-</td><td>50.8</td><td>59.8</td><td>50.3</td><td>52.4</td><td>48.0</td><td>46.5</td><td>47.6</td></tr><tr><td>GPT-4V</td><td>-</td><td>39.4</td><td>54.7</td><td>41.4</td><td>48.7</td><td>34.9</td><td>34.4</td><td>31.6</td></tr><tr><td>Gemini-1.5-Flash-002</td><td>-</td><td>49.4</td><td>57.2</td><td>50.5</td><td>50.3</td><td>47.6</td><td>45.1</td><td>45.4</td></tr><tr><td>Gemini-1.5-Pro</td><td>-</td><td>35.3</td><td>39.8</td><td>34.7</td><td>44.5</td><td>32.0</td><td>36.8</td><td>33.3</td></tr><tr><td>Claude-3.5-Sonnet</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Qwen-VL-Plus</td><td>-</td><td>21.3</td><td>26.0</td><td>21.2</td><td>25.2</td><td>18.5</td><td>19.1</td><td>21.8</td></tr><tr><td colspan="9">Open-Source General MLLMs</td></tr><tr><td>mPLUG-Owl2-7B</td><td>7B</td><td>10.3</td><td>11.6</td><td>11.4</td><td>13.8</td><td>11.1</td><td>9.4</td><td>8.0</td></tr><tr><td>MiniGPT4-7B</td><td>7B</td><td>12.2</td><td>12.3</td><td>12.9</td><td>13.4</td><td>12.5</td><td>14.8</td><td>8.7</td></tr><tr><td>LLaVA-1.5-13B</td><td>13B</td><td>12.7</td><td>17.1</td><td>12.0</td><td>22.6</td><td>12.6</td><td>12.7</td><td>9.0</td></tr><tr><td>SPHINX-V2-13B</td><td>13B</td><td>16.1</td><td>20.8</td><td>14.1</td><td>14.0</td><td>16.4</td><td>15.6</td><td>16.2</td></tr><tr><td>LLaVA-NeXT-34B</td><td>34B</td><td>34.6</td><td>49.0</td><td>37.6</td><td>30.1</td><td>35.2</td><td>28.9</td><td>22.4</td></tr><tr><td>InternLM-XComposer2-VL</td><td>7B</td><td>25.9</td><td>36.9</td><td>28.3</td><td>42.5</td><td>20.1</td><td>24.4</td><td>19.8</td></tr><tr><td>Deepseek-VL</td><td>8B</td><td>19.3</td><td>23.0</td><td>23.2</td><td>23.1</td><td>20.2</td><td>18.4</td><td>11.8</td></tr><tr><td>LLaVA-OneVision (SI)</td><td>8B</td><td>28.9</td><td>29.0</td><td>31.5</td><td>34.5</td><td>30.1</td><td>29.5</td><td>26.9</td></tr><tr><td>Qwen2-VL</td><td>8B</td><td>33.6</td><td>37.4</td><td>33.5</td><td>35.0</td><td>31.3</td><td>30.3</td><td>28.1</td></tr><tr><td>InternVL2-8B</td><td>8B</td><td>35.9</td><td>39.0</td><td>33.8</td><td>36.0</td><td>32.2</td><td>30.9</td><td>27.7</td></tr><tr><td>InternVL2.5-8B</td><td>8B</td><td>39.5</td><td>43.0</td><td>43.0</td><td>43.0</td><td>43.0</td><td>42.2</td><td>22.8</td></tr><tr><td colspan="9">Open-Source Math MLLMs</td></tr><tr><td>G-LLaVA-7B</td><td>7B</td><td>16.6</td><td>20.9</td><td>20.7</td><td>21.1</td><td>17.2</td><td>14.6</td><td>9.4</td></tr><tr><td>Math-LLaVA-13B</td><td>13B</td><td>22.9</td><td>27.3</td><td>24.9</td><td>27.0</td><td>24.5</td><td>21.7</td><td>16.1</td></tr><tr><td>Math-PUMA-Qwen2-7B</td><td>8B</td><td>33.6</td><td>42.1</td><td>35.0</td><td>39.8</td><td>33.4</td><td>31.6</td><td>26.0</td></tr><tr><td>Math-PUMA-DeepSeek-Math</td><td>7B</td><td>31.8</td><td>43.4</td><td>35.4</td><td>47.5</td><td>33.6</td><td>31.6</td><td>14.7</td></tr><tr><td>MAVIS-7B</td><td>7B</td><td>35.2</td><td>43.2</td><td>37.2</td><td>35.2</td><td>34.1</td><td>29.7</td><td>31.8</td></tr><tr><td>InfMM-Math</td><td>7B</td><td>40.5</td><td>46.7</td><td>39.4</td><td>41.6</td><td>38.1</td><td>40.4</td><td>27.8</td></tr><tr><td>Multimath-7B</td><td>7B</td><td>27.7</td><td>34.8</td><td>30.8</td><td>35.3</td><td>28.1</td><td>25.9</td><td>15.0</td></tr><tr><td>AtomThink-EMOVA</td><td>8B</td><td>42.5</td><td>48.1</td><td>47.7</td><td>45.7</td><td>44.0</td><td>44.2</td><td>26.8</td></tr><tr><td>URSA-8B</td><td>8B</td><td>45.7</td><td>55.3</td><td>48.3</td><td>51.8</td><td>46.4</td><td>43.9</td><td>28.6</td></tr><tr><td>URSA-8B-PS-GRPO</td><td>8B</td><td>50.9</td><td>57.3</td><td>52.2</td><td>50.2</td><td>48.7</td><td>47.6</td><td>31.5</td></tr></table>
+
+# C.2 Scaling Law of MMathCoT-1M
+
+To better illustrate the effectiveness of MMathCoT-1M, we examine the scaling laws of SFT by training models on randomly selected samples representing various ratios of the full dataset.
+
+As shown in Table 10, we can see that MMathCoT-1M clearly shows a training time scaling law, further validates the effectiveness of the synthesized data.
+
+Table 9: Accuracy comparison with closed-source MLLMs and open-source MLLMs on WE-MATH testmini subset. First 3 columns show the overall performance on one-step, two-step and three-step problems. The other columns are used to demonstrate the performance in different problem strategies. Red indicates the best performance and Blue indicates the second best performance among open-source models.  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">#Params</td><td rowspan="2">S1</td><td rowspan="2">S2</td><td rowspan="2">S3</td><td colspan="2">Mem</td><td colspan="2">PF</td><td colspan="2">SF</td><td colspan="2">TMF</td><td colspan="4">PD</td></tr><tr><td>UCU</td><td>AL</td><td>CPF</td><td>UPF</td><td>CSF</td><td>USF</td><td>BTF</td><td>CCF</td><td>Dir</td><td>Pos</td><td>RoM</td><td>CCP</td></tr><tr><td colspan="17">Closed-source MLLMs</td></tr><tr><td>GPT-4o</td><td>-</td><td>72.8</td><td>58.1</td><td>43.6</td><td>86.6</td><td>39.1</td><td>77.4</td><td>71.6</td><td>84.5</td><td>62.3</td><td>58.7</td><td>69.4</td><td>93.1</td><td>72.7</td><td>47.5</td><td>73.3</td></tr><tr><td>GPT-4V</td><td>-</td><td>65.5</td><td>49.2</td><td>38.2</td><td>82.5</td><td>38.4</td><td>70.7</td><td>60.2</td><td>76.6</td><td>56.3</td><td>57.8</td><td>67.7</td><td>79.3</td><td>57.5</td><td>47.8</td><td>63.3</td></tr><tr><td>Gemini-1.5-Pro</td><td>-</td><td>56.1</td><td>51.4</td><td>33.9</td><td>51.0</td><td>31.2</td><td>61.8</td><td>45.0</td><td>70.0</td><td>57.5</td><td>39.2</td><td>62.7</td><td>68.8</td><td>54.1</td><td>40.7</td><td>60.0</td></tr><tr><td>Qwen-VL-Max</td><td>-</td><td>40.8</td><td>30.3</td><td>20.6</td><td>19.4</td><td>25.3</td><td>39.8</td><td>41.4</td><td>43.6</td><td>48.0</td><td>43.8</td><td>43.4</td><td>41.4</td><td>35.1</td><td>40.7</td><td>26.7</td></tr><tr><td colspan="17">Open-source General MLLMs</td></tr><tr><td>LLaVA-1.6</td><td>7B</td><td>23.0</td><td>20.8</td><td>15.8</td><td>18.5</td><td>20.5</td><td>16.9</td><td>29.6</td><td>15.6</td><td>18.6</td><td>42.7</td><td>24.1</td><td>17.6</td><td>43.3</td><td>28.9</td><td>26.7</td></tr><tr><td>LLaVA-1.6</td><td>13B</td><td>29.4</td><td>25.3</td><td>32.7</td><td>21.7</td><td>23.2</td><td>23.4</td><td>34.7</td><td>25.3</td><td>26.4</td><td>37.5</td><td>41.7</td><td>26.9</td><td>28.9</td><td>37.1</td><td>30.0</td></tr><tr><td>GLM-4V-9B</td><td>9B</td><td>47.3</td><td>37.2</td><td>38.2</td><td>53.4</td><td>37.0</td><td>51.3</td><td>46.5</td><td>50.6</td><td>38.2</td><td>44.1</td><td>45.2</td><td>41.0</td><td>49.3</td><td>36.8</td><td>53.3</td></tr><tr><td>MiniCPM-LLaMA3-V2.5</td><td>8B</td><td>39.8</td><td>31.1</td><td>29.7</td><td>28.6</td><td>37.0</td><td>40.8</td><td>39.8</td><td>41.0</td><td>38.6</td><td>32.0</td><td>42.7</td><td>41.0</td><td>42.7</td><td>44.0</td><td>43.3</td></tr><tr><td>LongVA</td><td>7B</td><td>43.5</td><td>30.6</td><td>28.5</td><td>24.5</td><td>39.8</td><td>45.1</td><td>40.8</td><td>51.9</td><td>42.5</td><td>45.6</td><td>44.6</td><td>44.5</td><td>40.7</td><td>47.5</td><td>20.0</td></tr><tr><td>InternLM-XComposer2-VL</td><td>7B</td><td>47.0</td><td>33.1</td><td>33.3</td><td>31.3</td><td>46.5</td><td>47.7</td><td>42.6</td><td>51.4</td><td>43.9</td><td>41.1</td><td>50.6</td><td>65.5</td><td>53.9</td><td>55.2</td><td>40.0</td></tr><tr><td>Phi3-Vision</td><td>4.2B</td><td>42.1</td><td>34.2</td><td>27.9</td><td>28.7</td><td>16.0</td><td>47.2</td><td>38.8</td><td>50.0</td><td>44.4</td><td>28.8</td><td>31.2</td><td>48.6</td><td>49.2</td><td>26.4</td><td>50.0</td></tr><tr><td>DeepSeek-VL</td><td>7B</td><td>32.6</td><td>26.7</td><td>25.5</td><td>16.6</td><td>35.1</td><td>27.3</td><td>38.0</td><td>24.2</td><td>38.7</td><td>50.0</td><td>23.3</td><td>24.5</td><td>41.0</td><td>51.7</td><td>23.3</td></tr><tr><td>InternVL2-8B</td><td>8B</td><td>59.4</td><td>43.6</td><td>35.2</td><td>71.4</td><td>20.5</td><td>62.0</td><td>55.5</td><td>67.1</td><td>57.3</td><td>54.0</td><td>60.5</td><td>58.6</td><td>63.6</td><td>44.5</td><td>50.0</td></tr><tr><td>InternVL2.5-8B</td><td>8B</td><td>58.7</td><td>43.1</td><td>38.8</td><td>48.7</td><td>35.8</td><td>65.5</td><td>54.5</td><td>62.3</td><td>61.5</td><td>47.8</td><td>60.3</td><td>79.0</td><td>64.0</td><td>51.1</td><td>63.3</td></tr><tr><td>Qwen2-VL</td><td>8B</td><td>59.1</td><td>43.6</td><td>26.7</td><td>62.7</td><td>37.2</td><td>62.6</td><td>60.8</td><td>65.7</td><td>49.2</td><td>52.5</td><td>49.2</td><td>48.1</td><td>68.2</td><td>55.0</td><td>56.7</td></tr><tr><td>Gemma3-12B</td><td>12B</td><td>64.3</td><td>47.2</td><td>42.1</td><td>83.1</td><td>33.9</td><td>70.2</td><td>58.2</td><td>77.5</td><td>61.1</td><td>50.1</td><td>63.7</td><td>82.6</td><td>58.4</td><td>36.8</td><td>60.0</td></tr><tr><td colspan="17">Open-source Math MLLMs</td></tr><tr><td>G-LLaVA</td><td>7B</td><td>32.4</td><td>30.6</td><td>32.7</td><td>33.3</td><td>29.1</td><td>32.0</td><td>37.9</td><td>19.6</td><td>33.5</td><td>37.1</td><td>32.8</td><td>31.2</td><td>33.2</td><td>25.6</td><td>40.0</td></tr><tr><td>Math-LLaVA</td><td>13B</td><td>38.7</td><td>34.2</td><td>34.6</td><td>30.3</td><td>17.9</td><td>39.2</td><td>40.4</td><td>37.1</td><td>37.7</td><td>53.0</td><td>51.3</td><td>30.8</td><td>30.8</td><td>40.9</td><td>46.7</td></tr><tr><td>Math-PUMA-Qwen2-7B</td><td>8B</td><td>53.3</td><td>39.4</td><td>36.4</td><td>63.5</td><td>42.5</td><td>60.2</td><td>45.9</td><td>66.2</td><td>48.6</td><td>42.3</td><td>53.5</td><td>31.2</td><td>37.7</td><td>40.4</td><td>46.7</td></tr><tr><td>MAVIS w/o DPO</td><td>7B</td><td>56.9</td><td>37.1</td><td>33.2</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>MAVIS</td><td>7B</td><td>57.2</td><td>37.9</td><td>34.6</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>URSA-8B</td><td>8B</td><td>63.1</td><td>56.4</td><td>41.8</td><td>59.1</td><td>32.5</td><td>72.3</td><td>60.3</td><td>70.9</td><td>66.0</td><td>51.4</td><td>59.8</td><td>58.3</td><td>39.5</td><td>58.8</td><td>53.3</td></tr><tr><td>URSA-8B-PS-GRPO</td><td>8B</td><td>68.6</td><td>64.2</td><td>52.7</td><td>52.6</td><td>63.5</td><td>68.5</td><td>64.1</td><td>68.8</td><td>73.6</td><td>69.4</td><td>75.8</td><td>72.1</td><td>72.6</td><td>73.6</td><td>63.3</td></tr></table>
+
+Table 10: Scaling law validation on URSA-8B using different ratios of the MMathCoT-1M.  
+
+<table><tr><td>Ratio</td><td>MathVerse</td><td>MathVision</td><td>MathVista-GPS</td><td>WEMATH</td><td>DYNAMATH</td></tr><tr><td>1/4</td><td>34.7</td><td>20.5</td><td>68.5</td><td>43.5</td><td>36.6</td></tr><tr><td>1/2</td><td>40.5</td><td>22.8</td><td>72.3</td><td>47.7</td><td>38.8</td></tr><tr><td>3/4</td><td>42.0</td><td>26.7</td><td>77.9</td><td>50.9</td><td>42.2</td></tr><tr><td>1</td><td>45.7</td><td>28.7</td><td>81.7</td><td>53.6</td><td>44.7</td></tr></table>
+
+# C.3 Higher Upper Bound Taken from Stage I
+
+In Stage I, we obtain a more powerful base MLLM with enhanced reasoning capabilities through math-intensive vision-language alignment and instruction fine-tuning. Beyond the results in Table 1, we explain why Stage I can better serve subsequent experiments, focusing on test-time scaling and PRM applications. We select MathVerse, MathVision, and MathVista-GPS to observe the  $\mathbf{pass}@\mathbf{N}$  metric. As demonstrated in Figure 7, we find that URSA-8B consistently outperforms current leading general MLLMs and math reasoning MLLMs. This indicates that while current trends favor RL-related techniques, the scaling law of supervised fine-tuning can still demonstrate its role in breaking through the base model's limitations. This naturally brings advantages in areas such as BoN evaluation and the proportion of valuable rollouts in online RL. First, URSA-8B's higher upper bound leads to richer and more reliable process label generation in Stage II. Furthermore, since recent works claims that RL can only approach the optimal solution within its own exploration path [29, 4, 103], Stage I naturally expands the potential upper limit of the RL stage. This provides the most fundamental advantage to the performance of URSA-PS-GRPO-8B.
+
+# C.4 Generalization Validation
+
+To further validate the effectiveness of proposed MMathCoT-1M and PRM aided PS-GRPO. We select InternVL2.5-8B from the general-purpose MLLMs and Multimath from the math reasoning MLLMs for a generalization validation experiment. We do not conduct additional hyperparameter tuning but almost directly adopt the settings from Table 14. The experiment on InternVL2.5-8B and Multimath are implemented on Meng et al. [71] and TRL [104]. Given that these two models
+
+![](images/41e7c0a402ded927d647947e9cbe0bea4cfd4547ae941852706c3b89db049a70.jpg)  
+Figure 7: Pass@N evaluation on three benchmarks.
+
+![](images/11fcab18da1b0c5a8344dce466a01fd9183af58a4c4653ae57fda6a2b1189cb8.jpg)
+
+![](images/1262e31ca7192b9e796bca4bf9b033e027ee0ba87d990f1fe08cf09c9571dbef.jpg)
+
+Figure 8: The progress of MMathCoT-1.1M and URSA-8B-RM aided PS-GRPO on InternVL2.5-8B and MultiMath.  
+
+<table><tr><td>Model</td><td>Avg</td><td>MathVerse testmini</td><td>MathVision full set</td><td>MathVista gps</td><td>WE-MATH testmini</td><td>DYNAMATH testmini</td><td>GeoQA full set</td></tr><tr><td>InernVL2.5-8B</td><td>45.2</td><td>39.5</td><td>19.7</td><td>64.9</td><td>44.7</td><td>40.5</td><td>61.6</td></tr><tr><td>+ MMathCoT-1.1M</td><td>51.7 (↑6.5)</td><td>43.3 (↑3.8)</td><td>25.9 (↑6.2)</td><td>77.9 (↑13.0)</td><td>49.1 (↑4.4)</td><td>45.3 (↑4.8)</td><td>68.8 (↑7.2)</td></tr><tr><td>+ PS-GRPO</td><td>54.7 (↑9.5)</td><td>47.5 (↑8.0)</td><td>28.5 (↑8.8)</td><td>80.1 (↑15.2)</td><td>55.3 (↑10.6)</td><td>45.9 (↑5.4)</td><td>71.1 (↑9.5)</td></tr><tr><td>MultiMath</td><td>43.1</td><td>27.7</td><td>16.3</td><td>66.8</td><td>42.2</td><td>37.9</td><td>67.7</td></tr><tr><td>+ MMathCoT-1.1M</td><td>48.7 (↑5.6)</td><td>36.9 (↑9.2)</td><td>22.7 (↑6.4)</td><td>74.4 (↑7.6)</td><td>45.8 (↑3.6)</td><td>40.4 (↑2.5)</td><td>72.2 (↑4.5)</td></tr><tr><td>+ PS-GRPO</td><td>51.2 (↑8.1)</td><td>39.7 (↑12.0)</td><td>24.4 (↑8.1)</td><td>77.7 (↑10.9)</td><td>49.3 (↑7.1)</td><td>42.6 (↑4.7)</td><td>73.5 (↑5.8)</td></tr></table>
+
+Table 11: Comparison of TTS with different models using BoN performance on WE-MATH, DYNAMATH, and GeoQA.  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Method</td><td colspan="4">WE-MATH</td><td colspan="4">DYNAMATH</td><td colspan="4">GeoQA</td></tr><tr><td>N=4</td><td>N=8</td><td>N=16</td><td>N=32</td><td>N=4</td><td>N=8</td><td>N=16</td><td>N=32</td><td>N=4</td><td>N=8</td><td>N=16</td><td>N=32</td></tr><tr><td rowspan="3">URSA-8B</td><td>Self-Consistency</td><td>56.3</td><td>57.0</td><td>57.7</td><td>58.0</td><td>46.2</td><td>46.7</td><td>47.5</td><td>48.0</td><td>74.1</td><td>75.3</td><td>75.9</td><td>75.9</td></tr><tr><td>InternVL2.5-8B ORM</td><td>56.0</td><td>56.8</td><td>57.4</td><td>57.7</td><td>45.9</td><td>46.5</td><td>47.2</td><td>47.7</td><td>73.8</td><td>75.0</td><td>75.6</td><td>75.6</td></tr><tr><td>URSA-8B-RM</td><td>58.2</td><td>59.0</td><td>59.3</td><td>59.7</td><td>47.5</td><td>48.4</td><td>49.5</td><td>50.5</td><td>76.1</td><td>77.3</td><td>78.0</td><td>78.1</td></tr><tr><td rowspan="3">AtomThink-EMOVA</td><td>Self-Consistency</td><td>51.7</td><td>52.4</td><td>52.9</td><td>53.6</td><td>42.3</td><td>43.0</td><td>43.7</td><td>44.0</td><td>65.7</td><td>66.5</td><td>66.6</td><td>66.8</td></tr><tr><td>InternVL2.5-8B ORM</td><td>51.5</td><td>52.2</td><td>52.7</td><td>53.3</td><td>42.1</td><td>42.8</td><td>43.5</td><td>43.7</td><td>65.5</td><td>66.3</td><td>66.4</td><td>66.6</td></tr><tr><td>URSA-8B-RM</td><td>53.7</td><td>54.5</td><td>55.0</td><td>55.8</td><td>44.1</td><td>44.9</td><td>45.6</td><td>46.0</td><td>67.9</td><td>68.8</td><td>69.0</td><td>69.3</td></tr></table>
+
+have already undergone sufficient alignment for general domains or specific vertical domains upon their release, we only carry out two stages of training: (i) MMathCoT-1M is used to enhance the base model's mathematical reasoning capabilities; (ii) URSA-8B-RM is involved in the PS-GRPO process. We present the results in Table 8. The proposed MMathCoT-1M and PRM aided PS-GRPO demonstrate remarkable generalization capabilities across different models and benchmarks. When applied to InternVL2.5-8B and MultiMath, both models show significant performance improvements. For InternVL2.5-8B, adding MMathCoT-1M boosts the average score from 45.2 to 51.7, with even more significant gains when combined with PS-GRPO, reaching 54.7. Similarly, for MultiMath, the average score increases from 43.1 to 48.7 with MMathCoT-1M and further to 51.2 with PS-GRPO. These results highlight the effectiveness of our approach in enhancing mathematical reasoning capabilities across diverse models and tasks. The performance improvements are consistent across various benchmarks, including MathVerse, MathVision, MathVista, WE-MATH, DYNAMATH, and GeoQA, indicating that our methods are not only effective but also broadly applicable.
+
+# C.5 Implementary Results on Other Benchmarks
+
+We provide supplementary results on WE-MATH, DYNAMATH and GeoQA when comparing BoN selection. As shown in table 11, URSA-8B-RM remains an advantage with Self-consistency and InternVL2.5-8B ORM. When employing URSA-8B as reasoning model, URSA-8B-RM outperforms Self-consistency with  $4.6\%$ ,  $4.5\%$  and  $2.7\%$  relative improvements in Best-of-8 performance.
+
+# D Module Selection Criteria
+
+As for module selection, we primarily considered the choice of the vision encoder and the LLM backbone.
+
+Vision Encoder To train a reasoning model with higher process sensibility and facilitate PRM training, we first conduct captioning tests on open-source models like DeepSeek-VL, Qwen2-VL, etc., using a manually selected dataset (approximately 80 examples). These examples primarily include function-related and geometry problems prone to visual confusion. We manually inspect the outputs of these open-source models and find that Qwen2-VL and LLaVA-OneVision performed poorly; even though their performance on standard benchmarks is good, they fail to ensure sufficiently accurate mathematical descriptions. However, DeepSeekVL's native hybrid vision tower design, integrating high- and low-resolution processing, subjectively exhibit better recognition accuracy. We speculate that this is due to QwenViT [105] being more heavily biased towards general multimodal tasks, resulting in less precise mathematical descriptions compared to simpler vision backbones. Therefore, we choose the SigLiP-L+SAM-B hybrid vision tower design.
+
+LLM Backbone Considering the open-source influence of the QwenLM-Series, we follow the choice of prior work such as MathPUMA [11] and Multimath [23] by using the QwenLM-Series backbone. However, we consider whether we could achieve higher performance by leveraging instruction models that has undergone unimodal math post-training, and thus compare Qwen2.5-7B-Instruct<sup>1</sup> and Qwen2.5-Math-7B-Instruct<sup>2</sup>. After completing the VL alignment stage, we conduct a small-scale comparative experiment on MMathCoT-1M, fine-tuning on 50K examples. Finally, our results show that using Qwen2.5-Math-Instruct as the backbone yields an advantage of approximately 1 percentage point on MathVision and MathVerse. Therefore, we include Qwen2.5-Math-7B-Instruct as the LLM backbone for subsequent experiments.
+
+# E Ablation Studies
+
+# E.1 Effectiveness of Different Data Category
+
+In the first stage, we mainly synthesized large-scale multiclass CoT data.
+
+- w/o  $S_{Ao}$ : In this variant, the answer-only data is reverted to its original format. This directly mimics the training mode used by models such as Math-LLaVA and MathPUMA, which involves hybrid training on both direct answers ('fast thinking') and CoT thinking.  
+- w/o  $S_{An}$ : This data will be replaced with its original organizational structure, where the analysis and final answer are provided in a free-form text format.  
+- w/o  $\mathcal{S}_C$ : This batch of data will be replaced with reasoning expressed in mathematical formal language, better reflecting symbolic and 'plan and reasoning' forms of reasoning.
+
+![](images/e4030a2aff1a268e5042d48add49f1f427ce218a22c07b0a9063a080429f4fb1.jpg)  
+Figure 9: Each synthesis strategy towards different type of source data works well.
+
+The results are shown in Figure 9. Firstly, it is shown across all datasets that using the complete synthesized data achieves the best results, highlighting the role of MMathCoT-1M data. More specifically, we find: i)  $S_{Ao}$  demonstrates the greatest impact on MathVerse and MathVision, indicating that expanded CoT data is important for problems where absolute solution accuracy is pursued; ii) However, on WE-MATH, the replacement of  $S_{An}$  leads to the most significant performance drop, suggesting that content rewriting better aligns with the end-to-end requirements posed by the WE-MATH benchmark, and mixing training with data lacking clear logical sequences may reduce hierarchical accuracy; iii) The results on DYNAMATH indicate that rewriting and natural language formulation effectively enhance reasoning robustness from the perspective of textual diversity. This reveals that the thought pattern in textual form tends to maintain the stability of the thought process more effectively under scenarios involving image transformations.
+
+# E.2 Selection of External Closed-source MLLM
+
+In this section, we primarily present a comparison of metrics between Gemini-1.5-Flash-002 $^3$  and other popular MLLMs, as well as a comparison on partial training data.
+
+Table 12: SFT Performance with  ${50}\mathrm{\;K}$  data synthesized by two closed-source MLLMs,respectively.  
+
+<table><tr><td>Model</td><td>MathVista-GPS</td><td>MathVerse</td><td>MathVision</td></tr><tr><td>URSA-8B w/ GPT-4o</td><td>54.1</td><td>33.3</td><td>18.8</td></tr><tr><td>URSA-8B w/ Gemini-1.5-Flash-002</td><td>55.1</td><td>32.5</td><td>18.3</td></tr></table>
+
+- Metrics Comparison: We compare the performance of Gemini-1.5-Flash-002, GPT-4o, and GPT-4o-mini on some math-related tasks, as shown in Table 13. We observe that Gemini-1.5-Flash-002 is a MLLM that performs well on both unimodal and multimodal math tasks, and GPT-4o does not have a significant advantage over it. This, to some extent, ensures the quality of the data synthesis.  
+- SFT Performance: To best illustrate the performance variations, we randomly sampled 50K data sources from MMathCoT-1M, applied three corresponding strategies using GPT-4o, and subsequently conducted SFT. The performance results are shown in Table 12. We observe that using GPT-4o did not provide a clear advantage. However, the construction of MMathCoT-1M and DualMath-1.1M involves approximately 2.7 million API calls. The output token cost of Gemini-1.5-flash-002 is the same as that of GPT-4o-mini and is one-thirty-third of that of GPT-4o $^{4}$ . Therefore, Gemini-1.5-Flash-002 becomes a cost-effective choice.
+
+However, we must say that if community researchers can afford the cost of accessing more powerful closed-source models, we expect the results to be even better.
+
+Table 13: Comparison of Model Performance on Math Benchmarks  
+
+<table><tr><td>Model</td><td>Avg</td><td>MATH</td><td>MathVista</td><td>MathVerse</td><td>MathVision</td></tr><tr><td>GPT-4o</td><td>55.4</td><td>76.6</td><td>63.8</td><td>50.8</td><td>30.4</td></tr><tr><td>Gemini-1.5-Flash-002</td><td>53.6</td><td>79.9</td><td>58.4</td><td>49.4</td><td>26.3</td></tr><tr><td>GPT-4o-mini</td><td>48.0</td><td>70.2</td><td>56.7</td><td>42.3</td><td>22.8</td></tr></table>
+
+# F Theoretical Analysis
+
+Our theoretical explanation is grounded in the Q-value ranking theorem.
+
+Theoretical Background In a multi-step reasoning Markov Decision Process, the optimal Q-function,  $Q^{*}(s_{i},a_{i})$ , is the expected probability of reaching the correct final answer after action  $a_{i}$ . The scalar reward  $r_{si}$  from our URSA-8B-RM at each step serves as an empirical estimate,  $Q_{est}$ , of this true Q-value.
+
+Theoretical Optimal Objective Approximating the ideal Q-value Ordering, under an optimal reasoning policy  $\pi^{*}$ , the Q-values along a reasoning path should satisfy a strict monotonic ordering:  $Q_{w|W|}^{*} < \dots < Q_{w_{1}}^{*} \ll V^{*}(x) < \dots < Q_{c_{1}}^{*} < Q_{c|C|}^{*}$ , where  $c_{i}$  and  $w_{j}$  denote the i-th correct step and the j-th incorrect step, respectively. In our drop-moment definition,  $c_{|C|}$  is adjacent to  $w_{1}$ , which is consistent. The ideal training objective for a Process Reward Model (PRM) is to learn a Q-estimator  $Q_{est}$  that strictly follows this ordering across all reasoning paths. For a successful reasoning path  $\tau$  (e.g.,  $o(\tau) = 1$ ), this implies that the sequence of  $Q_{est}$  outputs  $\{r_{p,1}, r_{p,2}, \ldots, r_{p,T}\}$  should be monotonically increasing with no decreases, as a successful path should not contain any error.
+
+Thus, the theoretically optimal generation objective is:
+
+$$
+J _ {i d e a l} (\theta) = \mathbb {E} _ {\tau \sim \pi_ {\theta}} [ o (\tau) ] - \lambda \cdot \Omega (G (\tau))
+$$
+
+where:
+
+-  $\mathbb{E}_{\tau \sim \pi_{\theta}}[o(\tau)]$  is the standard final result accuracy.  
+-  $\Omega(G(\tau))$  is a non-decreasing regularization term measuring the violation of the ideal Q-value ordering.  $\lambda$  is its strength.  
+-  $G(\tau) = r_{cl}(\tau) - r_{w_1}(\tau)$  is a direct measure of the violation of the theoretical gap. The larger  $G(\tau)$ , the greater the drop from high (correct) to low (incorrect) rewards at some point in the path, which severely violates the ideal ordering. Minimizing  $G(\tau)$  is equivalent to aligning the model's generation process with the Q-value dynamics of the optimal reasoning path.
+
+PS-GRPO as a Proxy Implementation PS-GRPO introduces a computable, differentiable proxy for  $\Omega (G(\tau))$  into its reward function:
+
+$$
+R (\tau) = o (\tau) \cdot (1 - \gamma \cdot \mathbb {I} (\delta_ {p} (\tau) \geq \rho))
+$$
+
+This function approximates the ideal objective  $J_{ideal}(\theta)$ :
+
+- The  $o(\tau)$  term directly corresponds to the first term in  $J_{ideal}$ , maximizing final accuracy.  
+- The penalty term approximates the regularizer:  
+-  $\delta_p(\tau)$  is a computationally tractable and strongly correlated proxy for  $G(\tau)$ . A large  $\delta_p$  indicates a large  $G$ .  
+- The indicator function  $\mathbb{I}(\delta_p(\tau) \geq \rho)$  converts the continuous  $\delta_p$  into a binary signal, resolving non-differentiability and ensuring compatibility with GRPO.  
+- The hyperparameter  $\rho$  acts as the violation threshold for  $\Omega(G(\tau))$  and is determined by the behavior of  $Q_{est}$ .
+
+Therefore, the PS-GRPO reward function  $R(\tau)$  serves as a practical proxy for  $J_{ideal}$ :  $J_{PS - GRPO}(\theta) \approx J_{ideal}(\theta)$ . The final step is to empirically determine  $\rho$  for a given PRM.
+
+# G Implementation Details
+
+# G.1 RL Data Curation
+
+After instruction fine-tuning on MMathCoT-1M, the overall accuracy did not exceed  $50\%$ . Therefore, we believe it still has the potential to be directly utilized in the RL phase. We collect 20K data with a types mixture ratio similar to that of instruction fine-tuning and conduct a one-time static filtering before RL. Specifically, we use URSA-8B to perform 8 samplings on this 20K data, filtering out examples where all 8 sampling results are either incorrect or correct. This left approximately  $15\mathrm{K}+$  data for training vanilla GRPO and PS-GRPO. We implement PS-GRPO using TRL [104, 106]. The statistics of the RL data can be found in the table 10.
+
+# G.2 Parameters and Time Cost
+
+In this section, we provide the specific parameter settings and time costs for the three stages. Our experiments are based on Python 3.10 and PyTorch 2.4.0+cu124. We use AdamW [107] as the optimizer. We use Fully Shared Data Parallel (FSDP) [108] as the distributed training framework. Unless otherwise specified, experiments are con
+
+Figure 10: Statistics of RL data for vanilla GRPO and PS-GRPO.  
+
+<table><tr><td>Statistic</td><td>Number</td></tr><tr><td>Total Prompts</td><td></td></tr><tr><td>- Total number</td><td>15.3K</td></tr><tr><td>Data Source</td><td></td></tr><tr><td>- MathV360K</td><td>2.7K</td></tr><tr><td>- Multimath-EN</td><td>7.5K</td></tr><tr><td>- MAVIS-Geo</td><td>2.2K</td></tr><tr><td>- MAVIS-MetaGen</td><td>0.9K</td></tr><tr><td>- Geo170K-QA</td><td>2.1K</td></tr><tr><td>Problem Category Statistics</td><td></td></tr><tr><td>- Plane Geometry</td><td>4.1K (26.8%)</td></tr><tr><td>- Analytic Geometry</td><td>1.9K (12.4%)</td></tr><tr><td>- Solid Geometry</td><td>1.0K (6.5%)</td></tr><tr><td>- Algebra</td><td>1.1K (7.2%)</td></tr><tr><td>- Function</td><td>2.9K (19.0%)</td></tr><tr><td>- Chart</td><td>1.9K (12.4%)</td></tr><tr><td>- Textbook QA</td><td>0.9K (5.9%)</td></tr><tr><td>- Formula</td><td>0.8K (5.2%)</td></tr><tr><td>- Arithmetic</td><td>0.7K (4.6%)</td></tr></table>
+
+duced on  $32 \times$  NVIDIA-H100-HBM3 GPUs by default. Additionally, we provide important parameters used in data construction. During the generation of positive and negative example pairs, we set the temperature to 1.0,  $n\_return\_sequences$  to 16, and top_p to 0.95. In the BinaryErrorLocating phase, we set the temperature to 0.3,  $n\_return\_sequences$  to 16, and top_p to 0.95.
+
+We adapt the vLLM [109] framework for the URSA-8B's architecture (hybrid vision tower + MLP + Qwen2.5-math-Instruct is not originally supported by VLLM) and use it as an acceleration tool during the inference phase. During the data pair generation phase, we use  $16 \times$  NVIDIA-H100-HBM3 GPUs for inference, which takes approximately 28 hours. In the BinaryErrorLocating phase, we also use  $16 \times$  NVIDIA-H100-HBM3 GPUs for inference, taking about 20 hours.
+
+The hyperparameter and time cost used in Stage I and Stage II are demonstrated in Table 14. Since the parameters used in Stage III are somewhat different, we list them separately in Table 15. Recently, much work has provided numerous optimization tricks for GRPO, such as training-time dynamic sampling, clipping higher values, abandoning KL loss, etc [110, 58]. However, to independently verify the effectiveness of PRM-guided reward modeling, we have not added these tricks in either vanilla GRPO or PS-GRPO to ensure a fair and valid verification process. We only do one-time difficulty-based data selection before applying RL.
+
+Table 14: Hyperparameter setting and training time cost in Stage I and II.  
+
+<table><tr><td>Hyperparameters &amp; Cost</td><td>VL-alignment</td><td>Instruction Fine-tuning</td><td>PRM Training</td></tr><tr><td>Learning Rate</td><td>1e-4</td><td>1e-5</td><td>5e-6</td></tr><tr><td>Epoch</td><td>1</td><td>2</td><td>2</td></tr><tr><td>Warm-up Ratio</td><td>0.02</td><td>0.02</td><td>0.02</td></tr><tr><td>Weight Decay</td><td>0.02</td><td>0.01</td><td>0.02</td></tr><tr><td>Batch Size</td><td>64</td><td>128</td><td>128</td></tr><tr><td>Trainable Parts</td><td>Aligner</td><td>Vision Encoder, Aligner, Base LLM</td><td>Base LLM</td></tr><tr><td>Data Size</td><td>860K</td><td>1.0M</td><td>1.1M</td></tr><tr><td>Time Cost</td><td>~3.5h</td><td>~11h</td><td>~12h</td></tr></table>
+
+# G.3 Benchmarks
+
+In this section, we introduce the detailed subtasks and metrics of four used benchmarks to more precisely demonstrate the evaluation.
+
+MathVerse MathVerse [74] is a benchmark for testing the reasoning abilities of MLLMs when the information content in text and image modalities varies. Specifically, the models focus on performance in six scenarios: Text-Dominant (TD), Text-Lite (TL), Text-Only (TO), Vision-Intensive (VI), Vision-Dominant (VD) and Vision-Only (VO).
+
+WE-MATH WE-MATH [77] is the first benchmark that decompose composite problems into sub-problems according to the required knowledge concepts. In figure 9, the actual content corresponding to the abbreviations is as follows. Mem: Measurement, PF: Plane Figures, SF: Solid Figures, TMF: Transformations and Motion of Figures, PD: Position and Direction, AL: Angles and Length, UCU: Understanding and Conversion of Units, CPF: Calculation of Plane Figures, UPF: Understanding of Plane Figures, CSF: Calculation of Solid Figures, USF: Understanding of Solid Figures, BTF: Basic Transformations of Figures, CCF: Cutting and Combining of Figures, Dir: Direction, Pos: Position, RoM: Route Map, CCP: Correspondence of Coordinates and Positions.
+
+Table 15: Hyperparameter setting and training time cost in Stage III.  
+
+<table><tr><td>Hyperparameters / Cost</td><td>Value</td></tr><tr><td>Epochs</td><td>2</td></tr><tr><td>Learning Rate</td><td>2e-6</td></tr><tr><td>Temperature</td><td>1.0</td></tr><tr><td>Rollout number per prompt</td><td>8</td></tr><tr><td>Prompt Max Length</td><td>6048</td></tr><tr><td>Output Max Length</td><td>3072</td></tr><tr><td>Precision</td><td>bf16</td></tr><tr><td>Train Batch Size</td><td>512</td></tr><tr><td>KL Coefficient</td><td>0.003</td></tr><tr><td>Data size</td><td>15K</td></tr><tr><td>Time Cost</td><td>~18h</td></tr></table>
+
+DYNAMATH DYNAMATH [75] is a benchmark designed to evaluate the robustness of MLLMs in mathematical reasoning.
+
+Specifically, it includes tests across multiple dimensions, including Solid Geometry (SG), Plane Geometry (PG), Analytic Geometry (AG), Algebra (AL), Puzzle Test (PT), Graph Theory (GT), Arithmetic (AR), Scientific Figure (SF) and Statistics (ST). It includes 501 seed questions and 5010 generated questions.
+
+GeoQA The GeoQA [40] dataset is a specialized dataset designed for evaluating and training models in the field of geographic question answering. Its test set includes 734 samples.
+
+MathVista MathVista [76] comprises a total of 5 subtasks: Geometry Problem Solving (GPS), Math Word Problem (MWP), Figure Question Answering (FQA), Textbook Question Answering (TQA) and Visual Question Answering (VQA). Like the previous math reasoning works, our model training process does not overly focus on knowledge-intensive tasks (such as VQA and FQA), hence we choose GPS as the primary task.
+
+MathVision MathVision [111] is a large-scale multimodal math reasoning dataset that broadens the disciplinary scope of the multimodal mathematics field. The test set contains 3,040 examples, covering 16 key competencies, and provides reliable testing performance. Specifically, The specific meanings of the various disciplinary indicators in Table 6 are listed as following. Alg: algebra, AnaG: analytic geometry, Ari: arithmetic, CombG: combinatorial geometry, Comb: combinatorics, Cnt: counting, DescG: descriptive geometry, GrphT: graph theory, Log: logic, Angle: metric geometry - angle, Area: metric geometry - area, Len: metric geometry-length, SolG: solid geometry, Stat: statistics, Topo: topology, TransG: transformation geometry.
+
+Evaluation Criteria Our comparison is based on the following criteria: First, we select the results from the official leaderboards of each benchmark. Second, we choose the results from the original papers or technical reports of each model. Finally, we conduct our own inference and evaluation using vLLM [109]. Our evaluation adheres to the rules of the benchmarks themselves, which are as follows:
+
+- Rule-based Matching: WEMATH, GeoQA.  
+- LLM-as-a-Judge: MathVista, MathVision, MathVerse, Dynamath.
+
+The prompt for LLM-as-a-Judge is shown in Figure 11.
+
+# G.4 Algorithm
+
+In this section, we place the specific process of BIE from Section 3.2 into Algorithm 1. Specifically, the input is a solution that points to an incorrect answer. We set a per-step sampling hyperparameter
+
+# LLM-as-a-Judge
+
+Below are two answers to a math question. Question is [Question], [Standard Answer] is the standard answer to the question, and [Model_answer] is the answer extracted from a model's output to this question.
+
+Determine whether these two answers are consistent.
+
+Please note that only when the [Model_answer] completely matches the [Standard Answer] means they are consistent. For non-multiple-choice questions, if the meaning is expressed in the same way, it is also considered consistent, for example,  $0.5\mathrm{m}$  and  $50\mathrm{cm}$ .
+
+If they are consistent, Judgement is 1; if they are different, Judgement is 0.
+
+# Example 1:
+
+[Question]: Write the set of numbers represented on the number line in interval notation.
+
+[Standard Answer]: (-2,1]
+
+[Model answer]: Extracted Answer:  $((-2,1))$
+
+# Judgement: 0
+
+# Example 2:
+
+[Question]: As shown in the figure, circle O has a radius 1.0, if angle BAC = 60.0, then the length of BC is ()
+
+Choices:
+
+A:2
+
+B:2√{3}
+
+C:V{3}
+
+D:2√{2}
+
+[Standard Answer]: C
+
+[Model_answer]:B:2√{3}
+
+Judgement: 0
+
+# Example 3:
+
+[Question]: Find the domain and range of the function  $f$  using interval notation.
+
+[Standard Answer]: domain: [-4, 0) and range: (-3, 1]
+
+[Model_answer]: Range:  $((-4,1])$
+
+# Judgement: 0
+
+# Example 4:
+
+[Question]: As shown in the figure, circle O has a radius 1.0, if angle BAC = 60.0, then the length of BC is () Choices:
+
+A:2
+
+B:2√{3}
+
+C:√{3}
+
+D:2√{ {2}
+
+[Standard Answer]: C
+
+[Model_answer]: null
+
+# Judgement: 0
+
+OK. Now let's begin:
+
+[Question]: {QuestionText}
+
+[Standard Answer]: {AnswerText}
+
+[Model answer]: {ResponseText}
+
+Your response:
+
+Figure 11: LLM-as-a-Judge prompt used for answer matching.
+
+$N_{\mathrm{mid}}$ . Initially, we set the start and end points of the search range to Step 1 and Step N, respectively. We first consider the  $mc$  value of Step  $(1 + \mathbf{N}) / / 2$ . If it is positive, it indicates that the first totally erroneous step occurs in the latter half; otherwise, we look in the first half. This reduces the number of searches to  $\mathcal{O}(\log N)$ .
+
+Besides, we introduce the process of PS-GRPO in Algorithm 2. This process involves merging of outcome reward and process-as-outcome reward, and subsequent relative advantages calculation.
+
+Algorithm 1 BinaryErrorLocating  
+1: Input: Solution set  $S = \{y_{i} | i = 1, 2, \dots, N\}$ , mid step sampling num  $N_{mid}$ .  
+2: for  $i = 1$  to  $N$  do  
+3:  $l \gets 0$ ,  $r \gets \text{StepLen}(y_{i})$   
+4: while  $l < r$  do  
+5:  $\text{mid} \gets \lfloor (l + r) / 2 \rfloor$   
+6:  $\text{MID\_RES} \gets \mathcal{F}(\{y_{i,1}, y_{i,2}, \dots, y_{i,mid}\}, N_{mid})$   
+7: if  $\text{VERIFY(MID\_RES)}$  does not contain True then  
+8:  $r \gets \text{mid}$   
+9: else  
+10:  $l \gets \text{mid} + 1$   
+11: end if  
+12: end while  
+13:  $e_{i} \gets \text{WRONGSTEPLABELING}(y_{i}, l)$   
+14: ErrorLocatingSet.append $e_{i}$   
+15: end for  
+16: Output: ErrorLocatingSet
+
+Algorithm 2 PS-GRPO  
+1: Input: Policy model  $\pi_{\theta}$ , Process Reward Model  $\phi_{\theta}$ , train dataset  $D_{RL} = \{I_i, Q_i, Y_i\}_{i=1}^N$ .  
+2: ErrorLocatingSet  $\leftarrow []$   
+3: for Epoch = 1 to N do  
+4: for Batched data  $\{I, Q, Y\}$  in  $D_{RL}$  do  
+5: Generate rollouts of  $\{I, Q\}$ :  $\{c^j\}_{j=1}^M \sim \pi_{\theta}$   
+6: Calculate process reward sets  $S = \{(p_1, p_2, \dots, p_{|c_j|}) \sim \phi_{\theta}(c^j) | 0 < j < M\}$   
+7: Calculate reward  $\{r^j\}_{i=1}^M$  by Equation 5 and 6.  
+8: Calculate relative advantages  $\{\hat{A}^j\}_{i=1}^M$  using Equation 7.  
+9: Update policy model  $\pi_{\theta}$  using Equation 8.  
+10: end for  
+11: end for  
+12: Output: Updated policy model  $\pi_{\theta}$ .
+
+# H Prompt Design
+
+# H.1 Prompt Utilized in MMathCoT-1M Synthesis
+
+In this section, we provide the specific prompts for three-module data synthesis. Additionally, Gemini-1.5-Flash is a model that is very sensitive to prompts and parameters in practical experience, and we will share detailed adjustment experiences.
+
+CoT Expansion CoT expansion prompt for answer-only data source can be seen in Figure 12. We order the Gemini-1.5-flash to give a reasonable process directing to the ground-truth. After the execution, we find that the outputs is not so clear. The model sometimes will give trajectories that include "we must trust the answer" or "let me assume". We identify these phrases as signals that the model can not solve the problem naturally and independently. We will filter these samples.
+
+Analysis Rewriting Rewriting prompt for analysis-formatted data synthesis is illustrated in Figure 13. For solutions in an analytical format, we transform them into clear step-by-step format trajectories. During this process, Gemini-1.5-flash-002 does not exhibit significant questioning or make conditional requests. We improve data quality through reorganization and polishing of the language logic.
+
+Format Unified By employing a unified format prompt shown in Figure 14 to modify the reasoning styles of plan-and-reasoning and symbolic approaches, we are able to extract a more natural language process aligned with the pre-training style. A single example is sufficient to elicit perceptually favorable responses.
+
+# CoT Distillation
+
+For a question, you need to provide a step-by-step solution that directs to the given right answer. You must trust the given correct answer and avoid making any requests for additional information and any suspect.
+
+# Example 1:
+
+Question: Given that points A, B, and C lie on circle O and angle AOB measures 80.0 degrees, what is the measure of angle ACB? Choices:
+
+A:  $80^{\circ}$
+
+B:  $70^{\circ}$
+
+C:60°
+
+D:40°
+
+Provided answer: D
+
+Your Response:
+
+Step 1: Identify the given information: angle AOB = 80°.
+
+Step 2: Apply the relevant theorem: angle  $\mathrm{ACB} = 1 / 2$  angle AOB (inscribed angle theorem).
+
+Step 3: Substitute the given value: angle  $\mathrm{ACB} = 1 / 2 * 80^{\circ} = 40^{\circ}$ .
+
+Step 4: State the answer: The answer is D.
+
+†Answer: D
+
+# Example 2:
+
+Question: As shown in the diagram, AB is the diameter of circle O. EF and EB are chords of circle O.
+
+Connecting OF, if angle  $\mathrm{AOF} = 40^{\circ}$ , then the degree of angle E is(). Choices:
+
+A:40°
+
+B:50°
+
+C:55°
+
+D:70°
+
+Provided answer: D
+
+Your Response:
+
+Step 1: Given angle  $\mathrm{AOF} = 40^{\circ}$
+
+Step 2: Angles AOF and FOB are supplementary, meaning they add up to  $180^{\circ}$ . Therefore, angle  $\mathrm{FOB} = 180^{\circ} - 40^{\circ} = 140^{\circ}$ .
+
+Step 3: Angle E is half of angle FOB. Therefore, angle  $\mathrm{E} = 0.5 * 140^{\circ} = 70^{\circ}$ .
+
+Step 4: The answer is D.
+
+†Answer: D
+
+Now, let's begin.
+
+Question: {QuestionText}
+
+Provided answer: {SolutionText}
+
+Your Response:
+
+Figure 12: CoT expansion prompt for answer-only data.
+
+Double-checking After completing the above three points, we apply an LLM-as-a-judge for double-checking the synthesized data, ensuring that the solutions do not contain unreasonable processes, such as untimely questioning, conditional requests, or reasoning loops. The specific prompt design is shown in Figure 15. After this layer of filtering, we obtain the final MMathCoT-1M.
+
+# H.2 Prompt Utilized in DualMath-1.1M Synthesis
+
+In this section, we demonstrate the prompts used in MIE.
+
+- **Geometry Problem:** For geometry problem, we prompt the Gemini-1.5-Flash-002 to first identify key geometry features in the figure. We then order it to introduce a misinterpretation on these elements. Finally, use the wrong information to execute a misleading solution. The total design can be seen in Figure 16.
+
+# Analysis Rewriting
+
+I have a mathematical problem-solving process here, and both the process and the final answer are completely correct. I hope you can solve it in a different way based on the existing solution process, ensuring that the process and results are correct, especially the reasoning results remain consistent with the original.
+
+Please remember the following instructions:
+
+1. You must trust that the solution and final answer is correct, and you do not need to review or refute it.  
+2. Your transcription must be semantically coherent.  
+3. You cannot make requests like "require more information" or similar, because the given conditions are sufficient and the solution is definitely correct.  
+4. You can rewrite from multiple perspectives, such as providing different solutions to the problem or adopting different textual styles.  
+5. You must end with '†Answer:', filling in the definitive answer provided.
+
+Question:
+
+{QuestionText}
+
+Correct Solution:
+
+{SolutionText}
+
+Give the rewritten solution in the following format:
+
+Step 1: ...
+
+Step 2: ...
+
+···
+
+†Answer: ...
+
+Your solution:
+
+Figure 13: Analysis rewriting prompt for analysis-formatted data.
+
+- Charts & Function: For ChartQA and math functions, we prompt Gemini-1.5-Flash-002 to first check the fine-grained data points. We then attempt to insert spatially similar data to induce a misinterpretation. This subsequently leads to incorrect solutions for automatic labeling.  
+- LLM-as-a-Judge: For chart reasoning and function problem, we execute similar process on Gemini-1.5-Flash-002. We place it in Figure 17.
+
+# I Case Study
+
+# I.1 Showcase on Best-of-N Evaluation
+
+To more clearly illustrate the effectiveness of URSA-8B-RM in BoN evaluation, a case on MathVista-GPS is demonstrated (Figure 18). This case shows that URSA-8B-RM is sensitive to false theorem application and misunderstandings of angle-number relations. The good property not only enables URSA-8B-RM to perform well in BoN evaluation but also endows it with the potential to identify more valuable learning samples in online reinforcement learning.
+
+# I.2 Process of Misinterpretation Insertion Engine
+
+As shown in Figure 19, MIE performs three main actions: First, it interprets the mathematical information in the image. Then, it replaces key information at a selected step. Finally, it continues reasoning based on the modified conditions.
+
+# I.3 Failure Pattern in Process During GRPO
+
+In this section, we intuitively reveal through examples why PS-GRPO works effectively. We first introduce the concept of false-positive rollouts, which are rollouts that, despite reaching the correct answer, do not provide perfect intermediate actions to arrive at the solution. They can generally be
+
+# Format Unified
+
+You are given a question and a solution, and your task is to translate the reasoning process from mathematical language into a style of natural language reasoning.
+
+Instructions:
+
+- Convert the formal mathematical solution into a natural language explanation.  
+- Use "Step X:" to separate each step.  
+- End the solution with "†Answer: [final answer]".  
+- Ensure the explanation is clear, concise, and free of formal mathematical symbols unless absolutely necessary.  
+- **You must strictly follow the provided solution steps and final answer. Do not modify, correct, or reinterpret any part of the solution.**
+
+# Example:
+
+Question:
+
+Given that OB is perpendicular to OA, line CD passes through point O, and  $\angle AOC = 20^{\circ}$ , then  $\angle BOD = ?$
+
+Formal Solution:
+
+Step 1 (Definition of Perpendicular):  $\because$  OB  $\perp$  OA,  $\therefore \angle BOA = 90^{\circ}$
+
+Step 2 (Given condition):  $\angle AOC = 20^{\circ}$ ,  $\angle BOC = 90^{\circ} - \angle AOC = 90^{\circ} - 20^{\circ} = 70^{\circ}$ .
+
+Step 3 (Supplementary angles property):  $\angle BOD = 180^{\circ} - \angle BOC = 180^{\circ} - 70^{\circ} = 110^{\circ}$ .
+
+Answer: \boxed{110}
+
+Your Response:
+
+Step 1: Since OB is perpendicular to OA, we have that angle BOA is 90 degrees.
+
+Step 2: Given that angle AOC is 20 degrees, it follows that angle BOC is 90 degrees minus angle AOC, which is 70 degrees.
+
+Step 3: Using the property of supplementary angles, angle BOD is 180 degrees minus angle BOC, which is 110 degrees.
+
+†Answer: 110
+
+Now, let's begin!
+
+Question:
+
+{QuestionText}
+
+Solution:
+
+{SolutionText}
+
+** Your Response:
+
+Figure 14: Format unify prompt for mathematical and symbolic reasoning style data.
+
+divided into two categories: (i) the lack of visual condition alignment. Solutions in this category exhibit inconsistencies in reasoning regarding basic visual factors such as edge relationships, coordinate values, and theorem applications, revealing deficiencies in the pretraining phase, as shown in Figure 20. (ii) the exploitation of shortcut patterns. These rollouts do not go through key steps but are guided directly to the correct answer after basic descriptions due to the high correlation between image features and problem-solving patterns during pretraining or SFT, as shown in Figure 21. Therefore, PS-GRPO suppresses the advantageous direction brought by these erroneous actions through the sensitivity of the PRM in online RL for error identification. This leads to a more optimal paradigm that combines outcome rewards with process reward-based penalties.
+
+# I.4 Cases on How Naive Process Reward Modeling Fails
+
+In this section, we elaborate on the two fundamental flaws of process reward guided RL mentioned in Section 4 and present some cases for illustration. In online RL, models can easily recognize the patterns for obtaining process rewards, leading to conservative analyses and concise responses as they sidestep PRM scrutiny. As shown in Figure 22, we have observed that the model often follows a distinct reasoning pattern. They initially read and analyze the given conditions comprehensively, but then make incorrect decisions based on this analysis, leading to wrong answers. This indicates that
+
+# Double Checking
+
+You are tasked with evaluating whether a provided solution aligns precisely with a given standard answer. Your evaluation must strictly adhere to the following criteria:
+
+Solution Fidelity
+
+- \*\*Certainty\*\*: The solution must be free of any elements of doubt or speculative reasoning. Avoid using phrases that imply uncertainty or hypothetical reasoning, such as "more conditions are needed", "the answer seems questionable", "let us assume", "the provided solution seems wrong" or any similar expressions. The solution should be presented with confidence and clarity, reflecting a definitive and well-supported conclusion.
+
+Solutlon Consistency
+
+- **Alignment with Standard Answer**: The final conclusion of the solution must match the given standard answer exactly. The reasoning process should directly and unequivocally lead to the provided answer without deviation. Ensure that there are no discrepancies between the solution's conclusion and the standard answer.
+
+Evaluation Process
+
+1. **Analyze the Solution**: Carefully review the logical steps and reasoning used in the solution to ensure they are sound and lead directly to the conclusion.  
+2. **Compare with Standard Answer**: Verify that the conclusion of the solution is identical to the standard answer provided.  
+3. **Provide Judgment**: Based on your analysis, offer a clear and concise judgment finally:
+
+- Respond with "Judgment: yes" if the solution's conclusion is consistent with the given standard answer.  
+- Respond with "Judgment: no" if the solution's conclusion does not match the standard answer.
+
+Example Structure  
+- \*\*Question\*\*:
+
+{QuestionText}  
+- \*\*Standard Answer\*\*:
+
+{AnswerText}  
+- \*\*Solution Answer\*\*:
+
+{SolutionText}
+
+Your Response
+
+Figure 15: Double-checking prompt for ensuring high-quality and appropriate trajectories in synthesized CoT reasoning data.
+
+when explicitly modeling process rewards, models can easily focus on processes that seem "correct" in isolation. However, these processes may not be genuinely helpful for the final outcome and instead may lead the model to prioritize high process rewards over accuracy.
+
+# J Limitations
+
+Our work requires certain computational power to be efficient. We share high-quality training data in the community through open-sourcing, though it comes at a cost. Our exploration mainly focuses on STEM tasks, and it can be expected to work on more general visual domains. Our work primarily focuses on enhancing STEM reasoning capabilities and does not raise fairness or ethics issues.
+
+# K Broader Impact
+
+Our work primarily focuses on the STEM reasoning capabilities of MLLMs. It is highly likely to have a positive impact on society, as this direction can make MLLMs more reliable tools. A potential negative impact could be that with the advancement of this technology, certain groups may become dependent on this capability.
+
+# Misinterpretation Insertion for Geometry Problem
+
+You are given a geometry problem with an image and its solution. Your task is to introduce errors into the solution by misreading the diagram and generate an incorrect answer.
+
+Instructions:
+
+- Your response involves three Stages:  
+** Stage 1: Analyze the Correct Solution**: Identify where in the solution diagram information is extracted. Note these areas as potential points for introducing misinterpretations.  
+** Stage 2: Introduce a Misinterpretation**: Choose one of the identified points from Stage 1 and alter it to create a misleading scenario. Integrate this misinterpretation naturally into the solution.  
+** Stage 3: Continue Reasoning**: Based on the misinterpretation from Stage 2, continue the reasoning process to derive an incorrect final answer.  
+- Ensure the misreading is naturally integrated without explicit statements about making a misinterpretation. When describing corrupted solution, avoid showing the misleading thought process. You must respond in a typical problem-solving style.  
+- Remember that the  $<\mathrm{pos}>$  and  $<\mathrm{neg}>$  tags only need to appear at the end of each Step X. Do not repeat them between Step X and Step X+1, even if the reasoning process for that step spans many lines.  
+- Remember that once a step is marked as  $\langle \text{neg} \rangle$ , all subsequent steps are also considered  $\langle \text{neg} \rangle$ .  
+- '†Answer:' is the final answer, and it should not be tagged.  
+- You cannot tag an unaltered step with <neg> without misinterpreting it. You must misinterpret the step if you are tagging it with <neg>.  
+- Words that imply misinterpretation, such as 'misread' 'incorrect' or 'incorrectly assume' must not appear in your response under any circumstances.  
+- The misinterpretation action in Stage 3 should be consistent with what was planned in Stage 2.
+
+Question:
+
+{QuestionText}
+
+Solution:
+
+{SolutionText}
+
+** Your response:
+
+Figure 16: Misinterpretation insertion for geometry-related problems.
+
+# Misinterpretation Insertion for Charts & Function
+
+You are given a math problem with a coordinate axis or chart and its solution. Your task is to introduce errors into the solution by misreading the image or chart and generate an incorrect solution.
+
+Instructions:
+
+- Your response involves three Stages:  
+** Stage 1: Image Information Extraction**: Identify where in the solution diagram information is extracted. Note these areas as potential points for introducing misinterpretations.  
+** Stage 2: Introduce a Misinterpretation **: Choose one of the identified points from Stage 1 and alter it to create a misreading scenario. Integrate this misinterpretation naturally into the solution.  
+** Stage 3: Continue Reasoning**: Based on the misinterpretation from Stage 2, continue the reasoning process to derive an incorrect final answer.  
+- Ensure the misreading is naturally integrated without explicit statements about making a misinterpretation. When describing corrupted solution, avoid showing the misleading thought process. You must respond in a typical problem-solving style.  
+- Remember that the <pos> and <neg> tags only need to appear at the end of each Step X. Do not repeat them between Step X and Step X+1, even if the reasoning process for that step spans many lines. Remember that once a step is marked as <neg>, all subsequent steps are also considered <neg>.  
+- '†Answer:' is the final answer, and it should not be tagged.  
+- You cannot tag an unaltered step with <neg> without misinterpreting it. You must misinterpret the step if you are tagging it with <neg>.  
+- Words that imply misinterpretation, such as 'misread' 'incorrect' or 'incorrectly assume' must not appear in your response under any circumstances.  
+- The misinterpretation action in Stage 3 should be consistent with what was planned in Stage 2.
+
+Question:
+
+{QuestionText}
+
+Solution:
+
+{SolutionText}
+
+** Your response:
+
+Figure 17: Misinterpretation insertion for function and chart-related problems.
+
+# URSA-RM-8B as Verifier
+
+![](images/47a17028f5655725956c89a6b6b5edea65a594a2a037e3b47d90335e08d75bea.jpg)
+
+# Problem:
+
+As shown in the figure, AB is the diameter of circle O, DB and DC are respectively tangent to circle O at points B and C. If  $\angle ACE = 25.0^{\circ}$ , then the degree of  $\angle D$  is () A.  $50^{\circ}$  B.  $55^{\circ}$  C.  $60^{\circ}$  D.  $65^{\circ}$
+
+# Solution 1:
+
+Step 1: We are given that AB is the diameter of circle O, and DB and DC are tangents to circle O at points B and C respectively. We are also given that  $\angle ACE = 25^{\circ}$ . 0.93
+
+Step 2: Since DB and DC are tangents to circle O from the same point D, we know that  $\mathrm{DB} = \mathrm{DC}$ . This means that triangle DBC is an isosceles triangle. 0.95
+
+Step 3: In triangle DBC, let  $\angle DBC = \angle DCB = x$ . Since the sum of angles in a triangle is  $180^{\circ}$ , we have  $\angle BDC + \angle DBC + \angle DCB = 180^{\circ}$ . Therefore,  $\angle BDC + 2x = 180^{\circ}$ . 0.91
+
+Step 4: Since AB is the diameter,  $\angle ACB$  is a right angle  $(90^{\circ})$ . In right-angled triangle ACB, we have  $\angle CAB + \angle ABC = 90^{\circ}$ . 0.93
+
+Step 5: We are given that  $\angle ACE = 25^{\circ}$ . Since  $\angle ACB = 90^{\circ}$ , we have  $\angle ECB = 90^{\circ} - 25^{\circ} = 65^{\circ}$ . 0.98
+
+Step 6: In triangle DBC,  $\angle DBC + \angle DCB + \angle BDC = 180^{\circ}$ . Since DB and DC are tangents,  $\angle DBC = \angle DCB = 90^{\circ} - 25^{\circ} = 65^{\circ}$ . Therefore,  $65^{\circ} + 65^{\circ} + \angle BDC = 180^{\circ}$ , which gives  $\angle BDC = 180^{\circ} - 130^{\circ} = 50^{\circ}$ . 0.98
+
+Step 7: Therefore,  $\angle D = \angle BDC = 50^{\circ}$ . 0.88
+
+†Answer: A
+
+![](images/c15603aac625b1bc35f024d4e4dc136403459ec471d486db9e0cd71efc106f28.jpg)
+
+# Solution 2:
+
+Step 1: Identify the relevant angles. We are given that  $\angle ACE = 25^{\circ}$ . We want to find  $\angle D$ . 0.99
+
+Step 2: Recognize the relationship between the angles. Since AB is the diameter of circle O, and DB is tangent to circle O at B, we know that  $\angle ABD$  is a right angle  $(90^{\circ})$ . Also, since DB and DC are tangents from the same point D to circle O, we know that DB = DC. This means that triangle DBC is an isosceles triangle. 0.94
+
+Step 3: Use the property of isosceles triangles. In isosceles triangle DBC,  $\angle DBC = \angle DCB$ . Let's denote this angle as x. The sum of angles in triangle DBC is  $180^{\circ}$ . Therefore,  $\angle BDC + \angle DBC + \angle DCB = 180^{\circ}$ . Substituting the values, we get  $\angle BDC + x + x = 180^{\circ}$ , which simplifies to  $\angle BDC + 2x = 180^{\circ}$ . 0.90
+
+Step 4: Relate the angles.  $\angle$  ACE is an inscribed angle subtending the arc BC. The angle subtended by the same arc at the center is twice the inscribed angle. Therefore,  $\angle \mathrm{BOC} = 2^{\star}\angle \mathrm{ACE} = 2^{\star}25^{\circ} = 50^{\circ}$  .0.46
+
+Step 5: Find the angle  $\angle BDC$ . Since  $\angle ABD$  is a right angle  $(90^{\circ})$ , and  $\angle BOC$  is  $50^{\circ}$ , we can find  $\angle BDC$ . In quadrilateral ABCD, the sum of angles is  $360^{\circ}$ . We have  $\angle DAB + \angle ABC + \angle BCD + \angle CDA = 360^{\circ}$ . Since  $\angle ABC = 90^{\circ}$  and  $\angle BCD = x$ , we have  $\angle DAB + 90^{\circ} + x + \angle CDA = 360^{\circ}$ . 0.67
+
+Step 6: Solve for  $\mathbf{x}$ . We know that  $\angle BDC + 2x = 180^{\circ}$ . Also,  $\angle BDC + \angle DAB = 180^{\circ}$  (angles on a straight line). Therefore,  $\angle DAB = 180^{\circ} - \angle BDC$ . Substituting this into the equation from step 5, we get  $(180^{\circ} - \angle BDC) + 90^{\circ} + x + \angle CDA = 360^{\circ}$ . This simplifies to  $270^{\circ} - \angle BDC + x + \angle CDA = 360^{\circ}$ . 0.65
+
+Step 7: Determine  $\angle D$ . Since triangle DBC is isosceles,  $\angle DBC = \angle DCB = x$ . Also,  $\angle BDC + 2x = 180^{\circ}$ . We know that  $\angle ACE = 25^{\circ}$ , which is an inscribed angle subtending arc BC. The central angle subtending the same arc is  $2*25^{\circ} = 50^{\circ}$ . This central angle is  $\angle BOC$ . In triangle OBC, OB = OC (radii), so it's an isosceles triangle. Therefore,  $\angle OBC = \angle OCB = (180^{\circ} - 50^{\circ}) / 2 = 65^{\circ}$ . Since  $\angle OBC$  and  $\angle DBC$  are supplementary,  $\angle DBC = 180^{\circ} - 65^{\circ} = 115^{\circ}$ . However, this is incorrect. Let's reconsider. Since  $\angle ACE = 25^{\circ}$ , the arc BC subtends an angle of  $50^{\circ}$  at the center. In triangle OBC, OB = OC, so  $\angle OBC = \angle OCB = (180^{\circ} - 50^{\circ}) / 2 = 65^{\circ}$ . Therefore,  $\angle D = 65^{\circ}$ . 0.47
+
+†Answer: D
+
+![](images/55d69fd16fac266c3f4189099616137b942a3a7c94c6da7769c508cbe0f2e38e.jpg)  
+Figure 18: Case of URSA-8B-RM serving as a verifier in Best-of-N evaluation.
+
+# Showcase of how MIE works
+
+![](images/dfb80863c14b5ad10515aabcef6830b10993d6f4bb969efb13cee8de4336638e.jpg)  
+Figure 19: Case from MIE. We introduce specific step-level perception errors and continue false reasoning to construct a correctness-labeled solution.
+
+# Problem:
+
+Given a cubic function  $y = ax^3 + bx^2 + cx + d$ , please determine the values of  $a$ ,  $b$ ,  $c$  and  $d$ .
+
+# Original Solution:
+
+Step 1: I first need to identify the points that the graph passes through, and then substitute these points into the equation to solve for the values of a, b, c, and d.
+
+Step2: I find some points below: P1: (-3, -8) P2: (0, 1) P3: (-2, 1) P4: (-1, 2)
+
+···
+
+Step N:  $a = 1$ ,  $b = 2$ ,  $c = 0$ ,  $d = 1$
+
+# Gemini's response:
+
+Information Extraction: I see this question is highly related to the coordinates of points on the line. The line appears in the third quadrant, second quadrant, and first quadrant.
+
+## Misinterpretation Insertion: A simple method is to introduce incorrect point coordinates to reach a wrong conclusion. As the correct solution states, the coordinates of point P1 are  $(-3, -8)$ , but we easily replace them with  $(-3, -7)$  as a premise.
+
+# Continue Reasoning:
+
+Step 1: I first need to identify the points that the graph passes through, and then substitute these points into the equation to solve for the values of a, b, c, and d. <pos>
+
+Step 2: I find some points below: P1: (-3, -7) P2: (0, 1) P3: (-2, 1) P4: (-1, 2). <neg>
+
+Step 3: Using P1:  $(-3, -7)$ ,
+
+$$
+- 7 = a (- 3) ^ {\wedge} 3 + b (- 3) ^ {\wedge} 2 + c (- 3) + 1
+$$
+
+$$
+- 7 = - 2 7 a + 9 b - 3 c + 1
+$$
+
+$$
+- 8 = - 2 7 a + 9 b - 3 c <   \text {n e g} >
+$$
+
+···
+
+Step N:  $a = \frac{5}{6}, b = \frac{3}{2}, c = -\frac{1}{3}, d = 1. < \mathrm{neg} >$
+
+# Rollouts Analysis on Process (I)
+
+![](images/8b50f758ba1ebd9aba7e315d60177d00f626ee7e4c88357bc241f110da22ed67.jpg)  
+Figure 20: False positive rollout analysis I.
+
+Problem: In the diagram, point F is the intersection of the angle bisectors CD and BE of triangle ABC, and CG is perpendicular to AB at point G. If angle ACG is 32 degrees, what is the measure of angle BFC?
+
+# Perfect Rollout:
+
+Step 1: Recognize that CD and BE are angle bisectors, so point F is the incenter of triangle ABC.
+
+Step 2: Since CG is perpendicular to AB, triangle CGA is a right triangle.
+
+Step 3: In triangle CGA, using the fact that the sum of angles is  $180^{\circ}$ , calculate angle CAG: angle CAG = 90° - angle ACG = 90° - 32° = 58°.
+
+Step 4: Since CD is the angle bisector of angle ACB, angle ACD = angle BCD = 1/2 angle ACB.
+
+Step 5: Let angle  $\mathrm{ACB} = 2\theta$  . Then angle ACD  $=$  angle BCD  $= \theta$
+
+Step 6: Similarly, BE is the angle bisector of angle ABC, so angle ABE = angle EBC = 1/2 angle ABC.
+
+Step 7: Let angle  $\mathrm{ABC} = 2\varphi$  . Then angle ABE  $=$  angle EBC  $=\varphi$
+
+Step 8: In triangle ABC, the sum of angles is  $180^{\circ}$ . So, angle BAC + angle ABC + angle ACB =  $180^{\circ}$ . Substituting angle BAC =  $58^{\circ}$ , and angle ABC =  $2\varphi$ , angle ACB =  $2\theta$ . So,  $58^{\circ} + 2\varphi + 2\theta = 180^{\circ}$ , which simplifies to  $\varphi + \theta = 61^{\circ}$ .
+
+Step 9: In triangle BFC, the sum of angles is  $180^{\circ}$ . So, angle  $\mathrm{BFC} + \mathrm{angle~FBC} + \mathrm{angle~FCB} = 180^{\circ}$ .
+
+Step 10: Substitute angle FBC = φ and angle FCB = θ. So, angle BFC + φ + θ = 180°. From step 8, φ + θ = 61°, so angle BFC = 180° - 61° = 119°.
+
+$\dagger$  Answer:  $119^{\circ}$
+
+# False-positive Rollout:
+
+Step 1: Recognize that CD and BE are angle bisectors, so point F is the incenter of triangle ABC.
+
+Step 2: Since CG is perpendicular to AB, triangle CGA is a right triangle.
+
+Step 3: In triangle CGA, using the fact that the sum of angles is  $180^{\circ}$ , calculate angle CAG: angle CAG = 90° - angle ACG = 90° - 32° = 58°.
+
+Step 4: Angle BFC = 180° - angle CDE. CD is the angle bisector of angle ACB, angle ACD = angle BCD = 1/2 angle ACB.
+
+Step 5: We have angle CDE = angle ACD = angle BCD = 1/2 angle ACB.
+
+Step 6: Since angle  $\mathrm{ACB} +$  angle  $\mathrm{ACG} = 180^{\circ}$  angle  $\mathrm{ACB} = 180^{\circ} - 58^{\circ} = 122^{\circ}$
+
+Step 7: Substitute angle  $\mathrm{ACB} = 122^{\circ}$ , angle  $\mathrm{ACD} = 1 / 2 * 122^{\circ} = 61^{\circ}$ .
+
+Step 8: Substitute angle ACD = 66°, angle BFC = 180° - 61° = 119°.
+
+†Answer: 119°
+
+![](images/d516b0b3300820a718153595ec5351feb1c319a6129fee19895cb193fc8f805d.jpg)
+
+# Analysis:
+
+In this example, false-positive rollout leads to numerous instances of image-text perception inconsistency, and such behavior should not be encouraged in reasoning, even if it happens to yield the correct result.
+
+# Rollouts Analysis on Process (II)
+
+![](images/65b163f645dbd0f03b25e140b00818c88ec6644550a55f02502d8d43610f2cda.jpg)  
+Figure 21: False positive rollout analysis II.
+
+Problem: As shown in the figure, AC = BC, AD bisects angle CAB, then the perimeter of triangle DBE is () Choices: A:6cm B:7cm C:8cm D:9cm
+
+# Perfect Rollout:
+
+Step 1: The angle bisector states that a point on an angle bisector is equidistant from the two sides of the angle. Since AD bisects angle CAB, DC  $\perp$  AC, DE  $\perp$  AB, we conclude that DC=DE.
+
+Step 2: Since angle  $\mathrm{C} =$  angle  $\mathrm{AED} = 90^{\circ}$ , angle  $\mathrm{CAD} =$  angle EAD, we have triangle  $\mathrm{ACD} \cong$  triangle AED.
+
+Step 3: We substitute AC with BC, so  $\mathrm{BC} = \mathrm{AE}$
+
+Step 4: The perimeter of triangle DBE equals to  $\mathrm{DB} + \mathrm{BE} + \mathrm{DE}$ , notice that  $\mathrm{AE} + \mathrm{BE} = \mathrm{AB}$
+
+Step 5: From the diagram.  $\mathrm{AB} = 6\mathrm{cm}$
+
+Step 6: The answer is  $6\mathrm{cm}$
+
+†Answer: A
+
+![](images/a230469848e975b16e39e733d6ad15e73618be11ed0ca6aff54a860f9d6bc9f3.jpg)
+
+# False-positive Rollout:
+
+Step 1: Look at the figure and notice that  $\mathrm{AC} = \mathrm{BC}$ , so triangle ABC must be isosceles.
+
+Step 2: AD is the angle bisector of angle CAB. Hmm, so it splits angle A into two equal parts.
+
+Step 3: DE is perpendicular to AB.
+
+Step 4: The question asks for the perimeter of triangle DBE. I need to find the lengths of DB, BE, and DE.
+
+Step 5: Since ABC is isosceles with  $\mathrm{AC} = \mathrm{BC}$ , angles at A and B are equal.
+
+Step 6: Maybe triangle DBE is related to triangle ABC in some way.
+
+Step 7: Let me guess that the perimeter of DBE is equal to AB. If AB is  $6\mathrm{cm}$ , then the perimeter of DBE would be  $6\mathrm{cm}$ . So the answer is A.
+
+†Answer: A
+
+![](images/80cb1bab8bced0c6dc826bbe23192096efe4981ce879b3e721c8f77fc8309745.jpg)
+
+# Analysis:
+
+This type of learning pattern makes problems very difficult to solve when shortcuts in numerical and geometric relationships point in the wrong direction.
+
+![](images/896e284f943c526704d3e93dfcd65f3ed229b61e066a60f22fc689b317dcf467.jpg)
+
+# Question:
+
+As shown in the figure:  $\angle AOB:\angle BOC:\angle COD = 2:3:4$ , ray OM and ON bisect  $\angle AOB$  and  $\angle COD$  respectively, and  $\angle MON = 84^{\circ}$ , then  $\angle AOB$  is ( ).
+
+# Solution:
+
+Step 1: Look at the diagram and note the angles  $\angle AOB$ ,  $\angle BOC$ , and  $\angle COD$  are in the ratio 2:3:4.
+
+Step 2: Recognize OM and ON bisect  $\angle AOB$  and  $\angle COD$ , creating smaller angles.
+
+Step 3: Use the given  $\angle MON = 84^{\circ}$  to set up an equation based on the ratios and angle bisectors.
+
+Step 4: Solve the equation  $\angle AOB = 2 / (2 + 3 + 4)*84^{\circ} = 18.66^{\circ}$
+
+†Answer:∠AOB is  $18.66^{\circ}$
+
+![](images/6897a724f3ea96564a273432eb241041d1500eea827325b49f34d0c8ae5ddf73.jpg)  
+In 2000, just half of voters said it 'really mattered' who won the presidency; this year,  $83\%$  express this view  
+Figure 22: Bad case analysis on two process reward modeling variants.
+
+Notes: Based on registered voters. Volunteered and don't know responses not shown. Data for prior years from June surveys. Source: Telephone survey of U.S. adults conducted July 23-Aug. 4, 2020.
+
+PEW RESEARCH CENTER
+
+# Question:
+
+As shown in the figure:  $\angle AOB:\angle BOC:\angle COD = 2:3:4$ , ray OM and ON bisect  $\angle AOB$  and  $\angle COD$  respectively, and  $\angle MON = 84^{\circ}$ , then  $\angle AOB$  is ( ).
+
+# Solution:
+
+Step 1: The dark brown line represents voters who think it really matters, showing an increasing trend from  $50\%$  in 2000 to  $83\%$  in 2020. The light brown line represents voters who think things will be much the same, showing a decreasing trend from  $44\%$  in 2000 to  $16\%$  in 2020.
+
+Step 2: We need to know the least value of the light brown graph to the leftmost value of the dark brown graph.
+
+Step 3: The ratio of the least value of the light brown graph and the leftmost value of the dark brown graph is 16:67.
+
+†Answer: 16:67.
\ No newline at end of file
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+# Unlocking SLM Potential for Data Analysis Code Generation via Non-Parametric Knowledge Distillation
+
+Jinyang Li $^{\alpha \beta}$ ; Jack Williams $^{\alpha}$ , Nick McKenna $^{\alpha}$ , Arian Askari $^{\alpha}$ , Nicholas Wilson $^{\alpha}$ , Reynold Cheng $^{\beta}$
+
+$^{\alpha}$ Microsoft Research Cambridge  ${}^{\beta}$ The University of Hong Kong j10725@connect.hku.hk, jack.williams@microsoft.com, ckcheng@cs.hku.hk
+
+# Abstract
+
+Knowledge distillation from Large Language Models (LLMs) to locally hosted Small Language Models (SLMs) provides advantages for Data Analysis Code Generation (DACG) such as privacy protection. However, achieving effective distillation without resource-intensive training is challenging. This paper investigates whether LLMs can distill knowledge to SLMs through In-Context Learning (ICL), a training-free method for rapid task adaptation. We present the DARGO: Distillation and Adaptive Reasoning-Guided Orchestration framework, which facilitates automatic knowledge distillation from LLMs to SLMs. DARGO consists of three phases: exploration through an Model Orchestration Interface (MOI), Memory Collection of successful trajectories, and Knowledge-driven Inference. We evaluate DARGO on three challenging DACG benchmarks (WIKITQ, TABMWP, and BIRD-SQL), each with in-domain training sets that enable detailed analysis of knowledge distillation effectiveness. DARGO demonstrates a substantial relative performance improvement of  $27.5\%$  on average for the student SLMs. To further observe generalization capabilities, we evaluate the DARGO across different teacher-student model combinations, knowledge transfer scenarios, and unified memory approaches for more advanced, test-only data analysis tasks. Our findings contribute a novel perspective on distillation methods that enhance performance for SLMs while avoiding intensive fine-tuning. The source code is available: https://github.com/accpatrick/DarGO.
+
+# 1 Introduction
+
+Data Analysis Code Generation (DACG) automates the conversion of natural language queries into executable code, empowering information extraction and analysis from tabular data efficiently. This process enhances productivity, reduces the technical barrier for data analysis, and allows data scientists to focus on deriving insights, ultimately supporting more effective decision-making [24, 17, 14]. This is a challenging task since it not only requires the capability of code generation but also understanding complex tabular data.
+
+Large Language Models (LLMs) have demonstrated remarkable performance across diverse, complex tasks [49, 37, 8, 71, 11]. Leveraging LLMs or LLM agents for automatic code generation from user queries offers an effective solution [65, 56]. However, the integration of LLMs in DACG faces two primary challenges: 1) Privacy concerns arise when utilizing closed-source LLMs such as GPT-4 [2] or Claude-3.5-Sonnet [40]. 2) Deploying large open-source models like Llama-3.1-405B [13] or DeepSeek-v3 (671B) [31] can be challenging due to their large number of parameters. Balancing these benefits and challenges is crucial for effective data science applications.
+
+![](images/9b0b4eb913532234f3cb0ce9c51ebd413eccd797aa3c2d5cb528c3d41ab963e9.jpg)  
+Figure 1: Overview of the orchestration system of DARGO for data science code generation. Left: Model Orchestration Interface (MOI) with abstraction lifting, orchestrated coding, and plan optimization. Center: Memory Database Construction, including trajectory collection and case study integration. Right: Knowledge-driven Inference and Planning, featuring RAG-based meta instruction generation, and knowledge-driven code generation. R: Referenced GT Code. F: Functional Plans. O: Orchestrated Coding.
+
+![](images/d79700d2872b63cdf362e8ffeeff7dc6ce5a5df38971c780b9830ee823dd8606.jpg)
+
+![](images/7723918916e71fb8000ab408e491ac7a04bd46cb719d8518f4e7acd320308b54.jpg)
+
+Small Language Models (SLMs), such as Phi-3-mini [1] and Llama-3.1-8B [13], have gained attention for their In-Context Learning (ICL) capabilities and ability to be locally deployed. These models offer computational efficiency and enhanced data privacy, crucial for resource-constrained or privacy-sensitive applications [21]. While SLMs have shown competitive performance in some general tasks including natural language understanding [39] and code completion [7], their effectiveness in data science code generation tasks remains an open question.
+
+Fine-tuning is a common strategy to enhance SLM capabilities for complex tasks [42]. However, this approach encounters several challenges in the domain of data science DACG. One primary issue is the limited availability of high-quality training data. Professional tabular datasets, such as relational databases, are often small or proprietary, restricting access to substantial corpora for training. Additionally, the dual expertise required in both coding syntax and data understanding for accurate annotation further constrains dataset scalability [30, 28]. This is reflected in recent benchmarks for data science code generation, which typically contain around or fewer than 1,000 samples, highlighting the complexity and resource constraints in this field [20, 3, 26, 70, 66]. Recent research has explored distillation from LLMs to SLMs through fine-tuning on synthetic data [52, 36, 22]. While this approach shows promise, several challenges persist. For example, the performance improvements obtained from fine-tuning approaches can fail to generalize across different programming languages or dialects, requiring re-training for each package update or new task [46, 23]. However, In-Context Learning (ICL) can adapt to new requirements or tasks by providing relevant instructions or examples, reducing the effort required for re-training or continual training. This raises our central research question in DS code generation: Can LLMs distill knowledge to SLMs through In-Context Learning (ICL)?
+
+In this paper, we explore the potential of knowledge distillation from LLMs to SLMs via ICL. We propose a novel Distillation and Adaptive Reasoning-Guided Orchestration framework that enables an LLM to serve as a Teacher model guiding SLMs (Student models) in complex DACG tasks. DARGO operates in three phases: exploration, memory database collection, and knowledge-driven inference. During exploration, we employ the Model Orchestration Interface (MOI) that allows an LLM to probe and analyze SLM code knowledge by converting questions into step-wise functional plans and asking SLMs to infill the code for each plan. Then, successful collaborated cases are stored in a memory databases. In the knowledge-driven inference phase, DARGO utilizes a retrieval-augmented generation (RAG) approach that dynamically distills knowledge at inference time by presenting relevant prior successful cases in a case-study format to guide the generation process, which proves friendly to SLMs to absorb (Section 2.3).
+
+We evaluate DARGO on three challenging tabular analysis benchmarks requiring code generation: TABMWP [34], BIRD-SQL [29], and WIKITQ [41], where each feature fixed, non-overlapping train-test splits to closer show how effective of knowledge distillation that DARGO is. The experimental results demonstrate that the DARGO framework significantly improves the performance
+
+![](images/aba7e046da50d871003dcb96d598cfb416fe0aa279e147d66a3335daa8823cc9.jpg)  
+Figure 2: Main steps of MOI demonstrated with a text-to-SQL task example. Teacher model converts referenced code to a functional plan for Student models to complete. Teacher model iteratively optimizes the plan until the Student model produces correct code. A Python code MOI example is provided in Appendix D.1.
+
+of SLMs across all datasets, validating the potential of our knowledge distillation approach via ICL. Importantly, the memory produced for one student can guide other students, demonstrating model-agnostic transfer. Additional cross-dataset experiments on CRT-QA [70], QRDATA [32], and Infi-Agent [20] confirm that the distilled knowledge generalizes beyond the original training distribution. Taken together, our results show that lightweight, privacy-preserving SLMs can inherit much of an LLM's analytic expertise through ICL alone, making DACG practical in resource-constrained or privacy-sensitive environments.
+
+# 2 Methodology
+
+# 2.1 Task Formulation
+
+Given a natural language query  $q_{i} \in \mathcal{Q}$ , where  $\mathcal{Q} = \{q_{1}, q_{2}, \ldots, q_{N}\}$  represents a set of  $N$  queries, the corresponding tabular data or database schema information  $d_{i} \in \mathcal{D}$ , where  $\mathcal{D} = \{d_{1}, d_{2}, \ldots, d_{N}\}$ , then the Small Language Model (SLM) is tasked with generating an executable code snippet  $c_{i}$ . This code snippet must accurately answer the query  $q_{i}$  with the associated data  $d_{i}$ . The function that maps each query-data pair to its corresponding code snippet by SLMs is defined as  $f_{\mathrm{gen}}$ , and can be written as:
+
+$$
+c _ {i} = f _ {\text {g e n}} \left(d _ {i}, q _ {i}\right) \quad \text {f o r} i = 1, 2, \dots , N. \tag {1}
+$$
+
+# 2.2 Model Orchestration Interface
+
+The Model Orchestration Interface (MOI) is conducted between a Large Language Model (LLM), a Teacher model, and a Student model, represented by the SLM. A Teacher LLM with superior reasoning capabilities will often generate plans that are too abstract for a weaker Student SLM to enact. These plans require decomposition and refinement to match the operational granularity of the Student SLM. Through orchestrated mediation, the MOI dynamically adjusts plan granularity to align with the SLM's code capabilities whilst maintaining task conformance. The MOI is composed of three key components: Abstraction Lifting, Orchestrated Coding, and Plan Optimization.
+
+Abstraction Lifting (AL). In this phase, LLM generates a functional plan  $P_{i} = \{s_{i1}, s_{i2}, \ldots, s_{iK}\}$  based on a query  $q_{i}$ , data input  $d_{i}$ , and the corresponding ground truth (gt) code  $\tilde{c}_{i}$ . The ground truth code  $\tilde{c}_{i}$  can either be sourced from an existing dataset (BIRD-SQL) or generated by the Teacher model when it is not directly available but a ground-truth answer string exists, such as WIKITQ and TABMWP. This functional plan is defined as  $\mathcal{L}_{\mathrm{al}}(d_i,q_i,\tilde{c}_i)$ , where  $\mathcal{L}_{\mathrm{al}}$  denotes the LLM performing abstraction lifting. Each step  $s_{ij}$  in the plan corresponds to a key subtask derived from the query, collectively forming a structured template outlining the solution process. These steps are annotated by the LLM with descriptive comments and placeholders such as [Fill Your Code Here] in Python or [Fill Your Sub-Query] in SQL, as shown in Figure 2, ensuring that the SLM follows the logical flow of the entire plan and enables guided code generation. Unlike Chain-Of-Thought [60] plans, which provide intermediate steps in continuous textual form, our approach
+
+Table 1: The 6 action types utilized by the LLM during the Model Orchestration Interface (MOI) to optimize the plans for better understanding and code generation by SLMs  
+
+<table><tr><td>Action Type</td><td>Expression</td><td>Description</td></tr><tr><td>Decomposition</td><td>step(x) → step(a), step(b)</td><td>Split a complex step x into smaller, manageable steps such as step a and step b.</td></tr><tr><td>ALT</td><td>step(x) → step(y)</td><td>Replace a step x described by ambiguous or incorrect messages with a clearer and correct alternative step y.</td></tr><tr><td>ADD</td><td>step(x) → step(x), step(a)</td><td>Add a necessary step a to ensure the completeness of code logic.</td></tr><tr><td>DELETE</td><td>step(x) → None</td><td>Remove the unnecessary step x, which may lead to misunderstanding by the SLM.</td></tr><tr><td>SIMPLIFY</td><td>step(x) → simple_step(x)</td><td>Replace a complex step x with a simpler approach. For example, convert recursive plans into iterative loops.</td></tr><tr><td>SWITCH</td><td>packageA(step(x) → packageB(step(x))</td><td>Use a simpler package to achieve the same functionality. For example, conversion from Package Linear Regression to Correlation Coefficient to determine relationship between two variables.</td></tr></table>
+
+bridges high-level problem understanding with low-level code implementation logic, allowing the SLM to more effectively follow the plan for DACG.
+
+Orchestrated Coding (OC). Once the functional plan  $P_{i}$  is provided, the Student SLM considers all context including the question and data input to generate the complete orchestrated code  $c_{i} = f_{\mathrm{gen}}(d_{i},q_{i},P_{i})$  in a single turn by filling all placeholders, ensuring the solution is correct and executable. The results from executing this orchestrated code are then compared to those from a reference solution (such as ground truth answer string or gt codes) to evaluate whether the SLM fully understands both the data and the logic needed to answer the question. This comparison serves as a key indicator of the problem-solving accuracy of SLM and alignment with the intended solution. While the ground truth code may already be available from datasets or generated by the Teacher model, orchestrated coding and abstraction lifting are crucial for a few reasons. First, AL breaks down complex problem-solving tasks into manageable sub-tasks, with the potential to improve the performance of the SLMs across a wide range of analytical queries by assisting them in understanding modular structure. Additionally, error isolation can be grounded in the program structure, enabling more precise identification of issues and contributing to optimized plans. This is supported by our analysis in Section 3.6 that compares chain-of-thought with functional plans for exploration.
+
+Plan Optimization (PO). The plan optimization process is an iterative procedure that unfolds over multiple turns, denoted by  $t$ . During each iteration, the SLM refines the functional plan  $P_{i}^{t}$ . To formalize this interactive optimization process, we define an environment  $\mathcal{E} = \langle S, \mathcal{A}, \mathcal{O}, \mathcal{T} \rangle$ , following [72, 62, 16], where  $S$  represents the state space,  $\mathcal{A}$  the action space (Table 1), and  $\mathcal{O}$  the observation space. In this context, the plan  $P_{i}^{t}$  is embedded within the current state  $S_{i}^{t}$ , serving as a structure that guides the SLM to generate code. The orchestrated code  $c_{i}^{t}$  is the snippet produced by performing the plan  $P_{i}^{t}$  within the environment.
+
+During each turn, the LLM observes  $o_i^t$ , the outcome of executing orchestrated code  $c_i^t$  generated by the SLM, and selects an action  $a_i^t$  from  $\mathcal{A}$  to optimize the plan. For example, if a step  $s_{ij}^t$  contains an error such as: "Step j: List players who was born before 1930 and after 1950", the LLM will apply an action ALT( $\cdot$ ) to correct this, resulting in an updated plan  $P_i^{t+1}$  with the refined step  $s_{ij}^{t+1}$  as "Step j: List players who were born after 1930 and before 1950". This iterative process can be represented as:
+
+$$
+P _ {i} ^ {t + 1} = \left\{s _ {i 1} ^ {t}, s _ {i 2} ^ {t}, \dots , s _ {i j} ^ {t + 1}, \dots , s _ {i K} ^ {t} \right\}, \tag {2}
+$$
+
+$$
+s _ {i j} ^ {t + 1} = \mathcal {L} _ {\mathrm {o p t}} \left(s _ {i j} ^ {t}, o _ {i} ^ {t}, a _ {i} ^ {t}\right).
+$$
+
+Here,  $s_{ij}^{t+1}$  is updated by the optimization function  $\mathcal{L}_{\mathrm{opt}}$  of the LLM, which integrates the current observation  $o_i^t$ , action  $a_i^t$  and sub-optimal step  $s_{ij}^t$ . The system transitions from state  $S_i^t$  to  $S_i^{t+1}$  through  $\mathcal{T}(S_i^t, A_i^t)$ , resulting in the updated plan  $P_i^{t+1}$ . The SLM then generates the updated orchestrated code  $c_i^{t+1} = f_{\mathrm{gen}}(q_i, d_i, P_i^{t+1})$  for the new plan. This process repeats until the output is correct or the maximum number of iterations  $T$  is reached.
+
+# 2.3 Memory Database Construction
+
+After interactions between the LLM and SLM in MOI, the finalized states are stored in a memory database. This database includes the correct orchestrated codes, along with the context of the question and data input. This process ensures that the SLM can efficiently reference and apply related knowledge to new, unseen queries.
+
+![](images/584aa4bf38bef0d36a359801b1772d62f419f0e2ffcfcf555c31fe717915b0ee.jpg)  
+Figure 3: An example of a generated case study to enhance comprehension for SLMs.
+
+Case Study Translation. Rather than only storing raw, heterogeneous cases that consist of a query, plan, and orchestrated code in a simple stacked format, the LLM refines these into case study-like representations. These representations distill the reasoning behind the success of each example, serving as an intermediate abstraction that emphasizes the underlying rationale for the chosen approach. Each case study  $G_{i}$  contains a Case Name, Question, Schema / Value Information, Objective, and an Explanation of how the solution code successfully addresses the query using the provided data. An example of this structure is provided in Figure 3. As shown in Figure 1 (a)-(b), DARGO performs case study translation only for correct cases, because reflecting on incorrect cases without supervision often introduces hallucinations.
+
+Correct Case Collection. The Correct Case Collection, denoted as  $\mathcal{M}$ , consists of cases where the SLM has generated correct orchestrated codes. Each case  $M_{i}$  in this collection contains the natural language query  $q_{i}$ , the corresponding data  $d_{i}$ , the correct orchestrated code  $\hat{c}_i$ , which contains descriptive comments as shown in Figure 2 (right), and the case study  $G_{i}$  illustrating the solution. The set  $\mathcal{M}$  is the union of all such individual cases:
+
+$$
+\mathcal {M} = \bigcup_ {i} M _ {i}, \quad M _ {i} = \left(q _ {i}, d _ {i}, \hat {c} _ {i}, G _ {i}\right). \tag {3}
+$$
+
+# 2.4 Knowledge Distillation from Memory Database
+
+This part presents a RAG-based method of knowledge distillation as a global instruction, termed as meta-instruction. This instruction then guides the SLM in learning how to plan and generate code more accurately for unseen queries.
+
+RAG-based Knowledge Distillation for Meta Instruction Generation. We propose a Retrieval-Augmented Generation (RAG) framework for localized instruction distillation. In the retrieval phase, we identify the top relevant examples from a memory database  $\mathcal{M}$  via an embedding model. Relevance is measured via a function  $\mathcal{D}$ , expressed as  $\mathcal{R}(q_i) = \mathcal{D}(q_i,\mathcal{M},k)$ , where  $k$  is number of most relevant cases.
+
+<table><tr><td rowspan="2">Model</td><td>WIKITQ</td><td colspan="3">TABMWP</td><td colspan="4">BIRD-SQL</td></tr><tr><td>Accuracy</td><td>Grad. 1-6</td><td>Grad. 7-8</td><td>Total</td><td>Sim.</td><td>Med.</td><td>Chal.</td><td>Total</td></tr><tr><td>CodeLlama-7B</td><td>11.80</td><td>26.55</td><td>13.11</td><td>20.50</td><td>43.92</td><td>18.00</td><td>11.76</td><td>24.40</td></tr><tr><td>CodeLlama-13B</td><td>34.90</td><td>37.27</td><td>24.22</td><td>31.40</td><td>45.27</td><td>19.60</td><td>17.65</td><td>26.80</td></tr><tr><td>StarCoder2-7B</td><td>20.70</td><td>34.00</td><td>27.56</td><td>31.10</td><td>41.22</td><td>21.60</td><td>17.65</td><td>26.60</td></tr><tr><td>StarCoder2-15B</td><td>36.60</td><td>39.09</td><td>33.14</td><td>36.50</td><td>43.92</td><td>29.20</td><td>14.71</td><td>30.60</td></tr><tr><td>Phi-3-Small-7B</td><td>27.00</td><td>46.36</td><td>38.00</td><td>42.60</td><td>52.03</td><td>28.40</td><td>10.78</td><td>31.80</td></tr><tr><td>Phi-3-Medium-14B</td><td>44.80</td><td>59.45</td><td>46.00</td><td>53.40</td><td>51.35</td><td>32.80</td><td>13.73</td><td>34.40</td></tr><tr><td colspan="9">SLM Performance</td></tr><tr><td>Phi-3-Mini-3.8B</td><td>32.50</td><td>44.18</td><td>38.89</td><td>41.80</td><td>38.51</td><td>21.20</td><td>11.76</td><td>24.40</td></tr><tr><td>+ Chain-Of-Thought</td><td>27.70</td><td>46.36</td><td>35.33</td><td>41.40</td><td>34.46</td><td>22.00</td><td>12.75</td><td>23.80</td></tr><tr><td>+ Static Few-Shot</td><td>23.00</td><td>37.27</td><td>34.89</td><td>36.20</td><td>47.97</td><td>20.80</td><td>7.84</td><td>26.20</td></tr><tr><td>+ Dynamic Few-Shot</td><td>16.60</td><td>51.45</td><td>52.89</td><td>52.10</td><td>42.57</td><td>18.80</td><td>11.76</td><td>24.40</td></tr><tr><td>+ DSPy Distillation (Few Shot)</td><td>26.70</td><td>42.70</td><td>37.60</td><td>40.40</td><td>36.00</td><td>19.80</td><td>11.00</td><td>22.80</td></tr><tr><td>+ ReGAL Distillation</td><td>36.10</td><td>44.73</td><td>38.22</td><td>41.80</td><td>35.14</td><td>14.80</td><td>12.75</td><td>20.40</td></tr><tr><td>+ DARGo Meta Instruction (Ours)</td><td>41.10</td><td>61.27</td><td>56.44</td><td>59.10†</td><td>51.35</td><td>30.40</td><td>16.67</td><td>33.80</td></tr></table>
+
+Table 2: Performance comparison of various SLMs on WIKITQ, TABMWP, and BIRD-SQL, with results presented in accuracy percentages. Improvements of our DARGO methods over the End-to-End Code Gen baseline are highlighted using different intensities of olive color. Bold indicates best results for Phi-3-Mini, while underlines denote second-best results.[†] means a hybrid with dynamic few-shot with DarGO. Detailed instructions of when to use few shot is in Appendix A.4.
+
+Then, as shown in Figure 1 (d), the case studies of these retrieved examples are then fed into the SLM to extrapolate plans for solutions, adhering them to the specific query. Here, SLM performs a secondary distillation, extracting shared knowledge patterns from these case studies, which have already been distilled by the LLM (Teacher model), to generate instructions, noted as Meta-Instruction  $(\mathcal{I}_{\mathbf{m}}(q_i))$ , precisely specific to the current query at hand. The process is formalized:
+
+$$
+\mathcal {I} _ {\mathrm {m}} \left(q _ {i}\right) = f _ {\text {a g g}} \left(q _ {i}, \mathcal {R} \left(q _ {i}\right)\right) \tag {4}
+$$
+
+where  $f_{\mathrm{agg}}$  is an aggregation function applied by the SLM. By doing so, SLMs can generate more relevant and contextually appropriate instructions, effectively bridging the gap between general knowledge and query-specific requirements.
+
+Knowledge-Driven Inference. Harnessing the distilled instructions  $\mathcal{I} = \mathcal{I}\mathrm{m}(q_i)$  from the memory database, the SLM initially formulates a structured plan  $p_{\mathrm{gen}}$  which it subsequently employs to generate code for new queries (Figure 1 (e)). For a given query  $q_{i}$  and its associated data  $d_{i}$ , this process unfolds as follows:
+
+$$
+P _ {\text {g e n}} = f _ {\text {p l a n}} \left(\mathcal {I}, d _ {i}, q _ {i}\right), \quad c _ {i} = f _ {\text {g e n}} \left(d _ {i}, q _ {i}, P _ {\text {g e n}}\right), \tag {5}
+$$
+
+where  $f_{\mathrm{plan}}$  denotes the planning function executed by the SLM. This plan serves as a blueprint, guiding the following code generation phase. The SLM then employs the function  $f_{\mathrm{gen}}$ , which takes  $P_{\mathrm{gen}}$  along with the original query  $q_{i}$  and data  $d_{i}$  to generate the final code  $c_{i}$ .
+
+# 3 Experiments
+
+In this section, we first describe datasets and evaluation metrics in Section 3.1, followed by implementation details in Section 3.2. We then present a comprehensive experiments aimed at addressing three key research questions:
+
+- RQ1 (Section 3.3): How effective is DARGO for DACG?  
+- RQ2 (Section 3.4): Does DARGO and the knowledge it distills generalize across models?  
+- RQ3 (Section 3.5): How does DARGO compare to popular Lora fine-tuning?  
+- RQ4 (Section 3.6): Are all components of DARGO necessary?
+
+# 3.1 Datasets and Metrics
+
+We evaluate our approach on the DACG datasets WIKITQ [41], TABMWP [34], and BIRD-SQL [29]. These datasets vary in data complexity and task requirements. Full dataset statistics are provided in Appendix B.1. WIKITQ features operations such as counting, comparison, and aggregation, and we
+
+Table 3: Zero-shot performance of DARGO on different SLMs.Improvements (in parentheses) show gains over the Baseline.  
+
+<table><tr><td>Model</td><td>WIKITQ</td><td>TABMWP</td><td>BIRD-SQL</td></tr><tr><td colspan="4">StarCoder-15B as Student</td></tr><tr><td>Baseline</td><td>36.60</td><td>36.50</td><td>30.60</td></tr><tr><td>+ DARGO-MI</td><td>45.70 (↑9.10)</td><td>43.93 (↑7.43)</td><td>38.40 (↑7.80)</td></tr><tr><td colspan="4">Llama-3.1-8B as Student</td></tr><tr><td>Baseline</td><td>34.30</td><td>42.90</td><td>40.20</td></tr><tr><td>+ DARGO-MI</td><td>39.80 (↑5.50)</td><td>49.10 (↑6.20)</td><td>44.20 (↑4.00)</td></tr></table>
+
+Table 4: Zero-shot performance of DARGO on different teacher LLMs.  $\rightarrow$  means distill knowledge from teacher LLMs to student SLMs.  
+
+<table><tr><td>Model</td><td>WIKTQ</td><td>TABMWP</td><td>BIRD-SQL</td></tr><tr><td colspan="4">Llama-3.3-70B → Phi-3-mini</td></tr><tr><td>Baseline</td><td>41.80</td><td>32.50</td><td>24.40</td></tr><tr><td>+ DARGO-MI</td><td>48.80 (↑7.00)</td><td>42.60 (↑10.10)</td><td>36.20 (↑11.80)</td></tr><tr><td colspan="4">Llama-3.3-70B → Llama-3.1-8B</td></tr><tr><td>Baseline</td><td>42.90</td><td>34.30</td><td>40.20</td></tr><tr><td>+ DARGO-MI</td><td>50.40 (↑7.50)</td><td>42.00 (↑7.70)</td><td>47.00 (↑6.80)</td></tr></table>
+
+select 1,000 test and 2,000 training examples. Accuracy is measured using the official scripts from Pasupat and Liang [41]. TABMWP involves mathematical word problems in tabular data, with 1,000 questions used for memory construction and a separate 1,000 test set. Performance is evaluated by comparing answers to ground truth. BIRD-SQL presents relational databases with both semantic parsing and analytical tasks. We adopt the 1,000-example mini-train set from Qu et al. [44] and evaluate on a 500-example mini-dev set using execution accuracy (EX).
+
+# 3.2 Implementations
+
+Setup. We conduct experiments in two settings. For TABMWP and WIKITQ, we follow Stengel-Eskin et al. [50] to first have a Teacher LLM (GPT-4o) generate silver programs during exploration. At inference, an SLM produces Python code, which is executed to obtain a final answer string for comparison with the ground truth answer. For BIRD-SQL, no silver code is required because the dataset already contains ground-truth SQL. RAG-based Meta Instructions employ  $k = 3$  nearest neighbors (via CodeT5+ [59]). Further implementation details appear in Appendix B.
+
+Baselines Models. We consider an SLM suitable if it can perform in-context learning and has under 15B parameters for one A100 80G GPU inference. For closed-source models, we choose GPT-35-Turbo as SLM since it has faster inference speed and its performance falls behind advanced models (e.g., GPT-4). We classify them into three categories: Orchestration Models, where GPT-4o is the Teacher, and Phi-3-mini-128k [1], Llama-3.1-8B [13] serve as Students for verifying the generalization of our approach. Evaluation Models, including Phi-3, CodeLlama, and StarCoder2 families. Knowledge-Transmission Models, containing GPT-35-Turbo and Llama-3.1-8B, used to test knowledge transmission in Section 3.4.
+
+Baseline Methods Implementation. Baseline methods include zero-shot end-to-end, Chain-of-Thought [60], static few-shot [6], and dynamic RAG-based few-shot [15] with memory database by DARGO since python tasks do not have GT codes but answer string, each with three examples. We also compare two distillation frameworks, DSPy [25] and ReGAL [51], both using GPT-4o as the teacher.
+
+# 3.3 Overall Results
+
+Table 2 highlights three key aspects:
+
+1) Effectiveness of DARGO: Distilling knowledge via DARGO propels Phi-3-mini to outperform both End-to-End Code Generation and Chain-of-Thought on all SLM datasets, achieving relative gains of  $17.5\%$  on TABMWP to  $38.5\%$  on BIRD-SQL. Moreover, Phi-3-mini with DARGO often surpasses larger models  $(2 - 3\times$  parameters), outperforming CodeLlama-13B by  $7.0\%$  and StarCoder2-15B by  $3.2\%$  on BIRD-SQL, and rivaling Phi-3-Medium  $(4\times$  parameters) across all benchmarks.  
+2) Comparison with Advanced Distillation Strategies. We implement two advanced distillation pipelines, DSPY and REGAL, on GPT-4o and Phi-3-mini for fair comparison, reporting the best performance of REGAL across different numbers of reusable helper functions (1–10) to minimize bias. As shown in Table 2, DARGO outperforms both approaches on all three datasets and difficulty levels.
+
+![](images/db37aea17e959e238a5662283d682a86dcb5516666f0f01bf6b3861d258f8d72.jpg)  
+Performance of Knowledge Transmission  
+Figure 4: Knowledge transmission for from the memory database between GPT-4o and Phi-3-mini across three datasets.
+
+DSPY is robust for general QA but struggles in execution-based DACG settings because its exact-match metric does not account for valid alternative program solutions. Moreover, DSPY relies on few-shot demonstrations rather than comprehensive knowledge distillation, reflecting the broader challenges of demonstration-based methods for complex DACG. By contrast, DARGO prioritizes task-specific instructions to guide complex operations, shifting focus from prompt optimization to experience summarization.
+
+We also evaluate REGAL, which provides Reusable Python Helper Functions from the memory database. Although promising for tightly scoped, domain-specific tasks, REGAL proves less effective for cross-domain DACG (see Table 2). Error analysis shows it often produces narrowly tailored functions (e.g., get_least_points-team, filter_senator_by_year), which often directly reference the data schema and are therefore irrelevant on new tasks. In contrast, the distilled knowledge of DARGO is textual, focusing on reasoning-level challenges, and is therefore not syntactically fixed to a specific domain or input.
+
+# 3.4 Generalization
+
+Here, we conduct experiments to test generalization of DARGO framework:
+
+Is DARGO model-agnostic to SLMs? To investigate the breadth of the DARGO pipeline beyond a single teacher-student configuration, we perform two complementary experiments. (1) We apply the complete workflow to two additional student models: StarCoder2-15B and Llama-3.1-8B while keeping GPT-4o as the teacher. As summarized in Table 3, both students outperform their respective baselines on all three datasets, indicating that DARGO consistently improves a variety of SLM architectures. (2) To examine robustness with respect to the teacher LLM, we replace GPT-4o with Llama-3.3-70B. The results in Table 4 replicate the earlier gains, demonstrating that gains of DARGO are not confined to a specific teacher model.
+
+# Is the distilled knowledge only useful to the student model participating in Orchestration?
+
+Next, we examine whether the knowledge distilled through DARGO is only beneficial to the student model involved in the orchestration phase. Even though this has been proven effective, it would be costly if exploration phase is repeated for new SLMs. Therefore, we test the distilled instructions on two new SLMs, Llama-3.1-8B and GPT-35-Turbo, both of which have not participated in the exploration between GPT-4o and Phi-3-mini.
+
+Figure 4 shows that meta instructions (MI) derived from our memory database (jointly updated by GPT-4o and Phi-3-mini) produce significant improvements over baselines. Specifically, Llama-3.1-8B achieves an average relative improvement of  $14.3\%$ , while GPT-35-Turbo sees a  $30.9\%$  gain. These results indicate that the distilled knowledge is not tied to a specific student model; it can effectively transfer to new models without additional fine-tuning, offering a scalable means of augmenting emerging SLMs.
+
+Table 5: Performance of DARGO on different SLMs. Improvements (in parentheses) are over the Baseline.  
+
+<table><tr><td>Model</td><td>CRTQA</td><td>QRDATA</td><td>INFI-AGENT</td></tr><tr><td colspan="4">Phi-3-mini as Student</td></tr><tr><td>Baseline</td><td>26.51</td><td>31.14</td><td>42.80</td></tr><tr><td>+ DARGO-MI</td><td>32.12 (↑5.61)</td><td>38.99 (↑7.85)</td><td>46.69 (↑3.89)</td></tr><tr><td colspan="4">Llama-3.1-8B as Student</td></tr><tr><td>Baseline</td><td>35.44</td><td>34.43</td><td>49.81</td></tr><tr><td>+ DARGO-MI</td><td>45.88 (↑10.44)</td><td>44.81 (↑10.38)</td><td>54.18 (↑4.37)</td></tr></table>
+
+Table 6: Ablation study of exploration in DARGO on BIRD-SQL.  
+
+<table><tr><td>METHOD</td><td>BIRD-SQL (MI)</td></tr><tr><td>DARGO w/ Phi-3-mini</td><td>33.80</td></tr><tr><td>(a) w/o MOI</td><td>21.70 (↓12.10)</td></tr><tr><td>(b) w/o Case Study Trans.</td><td>22.80 (↓11.00)</td></tr><tr><td>(c) w/o Functional Plan</td><td>23.60 (↓10.20)</td></tr></table>
+
+Out-of-Distribution Evaluation of DARGO Current tabular-reasoning benchmarks frequently provide test-only splits due to large costs of more complex tasks annotation, prompting us to evaluate how well DARGO generalizes when no in-domain training data are available. Following the [55], we first construct a unified memory from our exploration corpus and evaluate its performance on those test-only sets via DARGO. To be specific, exploration set  $\mathcal{B}$  contains: WikiTQ [41], TabMWP [34], BIRD-Pandas [29], and Juice [3]. The distributions of  $\mathcal{B}$  are shown in Appendix B.4. The test-only datasets form the set  $\mathcal{E}$ , which includes CRT-QA [70], QR-DATA [32], and Infi-Agent [20].
+
+During evaluation, when we test DARGO on benchmark  $\mathcal{E}_i\in \mathcal{E}$ , its corresponding exploration corpus is  $\mathcal{B}\cup \mathcal{E}_{\backslash i}$ . This strict partition guarantees zero overlap between examples used for distillation and those reserved for evaluation. As summarized in Table 5, DARGO presents huge OOD improvements across all student models, demonstrating that it distills transferrable reasoning patterns rather than memorizing dataset-specific artifacts.
+
+# 3.5 Fine-Tuning v.s. DARGO Knowledge Distillation
+
+With the Same Seed Data. We compare DARGO with LoRA fine-tuning [19] using the same 1,000 training samples on BIRD-SQL for Phi-3-mini. Figure 5 and Table 9 in Appendix shows that fine-tuning on limited data even degrades SLM performance, while DARGO achieves significant improvements. We attribute this to two factors: 1) small training sets introduce bias that limits generalization; and 2) LoRA struggles to teach complex reasoning capabilities in an end-to-end regime. In contrast, DARGO leverages LLMs to decompose difficult questions into interpretable steps and distill planning knowledge, enabling better generalization. This makes DARGO particularly effective for DACG domains with limited annotations.
+
+Comparison with Incremental Lora Fine-Tuning. For a more comprehensive comparison, we conduct additional experiments fine-tuning Phi-3-mini on varying proportions (10%, 20%, 50%, 70%, 100%) of the full BIRD-SQL training set. Each configuration is repeated 5 times, with average results presented in Figure 5. Our findings reveal several notable patterns: First, performance peaks at 70% of the data, suggesting that fine-tuning on more data does not always lead to better results, which maybe due to the inclusion of lower-quality or biased examples in the full dataset. Second, while fine-tuning with 70-100% of the data marginally outperforms DARGO, it requires  $7\cdot 9 \times$  more training examples to achieve this performance. Notably, DARGO outperforms fine-tuning on 50% of the data, demonstrating  $5 \times$  greater data efficiency.
+
+![](images/ada7119e23e07f01ba1a3547cf00f35981901d411fc341b82fdc3225431a1520.jpg)  
+Figure 5: Comparison of DarGO with Lora fine-tuning across varying data size.
+
+# 3.6 Ablation Study
+
+In Table 6, to better validate the function of each component of DARGO, ablation studies are performed in both Exploration and Inference (More details in Appendix E):
+
+(a) MOI. For the "w/o MOI" scenario, we construct the memory by employing GPT-4o to convert ground-truth code to plans directly, excluding the participation of SLM. This resulted in a significant final performance drop for Phi-3-mini. One explanation is that GPT-4o tends to produce relatively coarse-grained plans (e.g., "compute percentage of winning"), whereas SLMs require
+
+more fine-grained steps (e.g., "compute total number of games", "divide winning games by total wins"). These higher-level plans can induce hallucinations to SLMs, as they frequently entail additional intermediate steps that increase the risk of errors. Furthermore, without execution-based refinement and calibration through the Plan Optimization (PO) component in MOI, plans generated by GPT-4o often contain inaccuracies that can mislead the SLM, as illustrated in Section 2.2.
+
+(a) Case Study Translation. As elaborated in Section 2.3, case study translation systematically consolidates and refines heterogeneous inputs, including tabular data, user queries, and code samples, into a structured, context-rich representation that SLMs can more effectively process. In the absence of this translation component, the model frequently defaults to erroneous or incomplete outputs (e.g., "SELECT \n\n") , underscoring the difficulty SLMs encounter in performing generalized reasoning when confronted with raw, unprocessed data in DACG.  
+(c) Functional Plan. A key contribution of our work is the MOI and the use of functional plans to structure and improve the interaction between teacher and student. We evaluate performance by replacing functional plans with textual plans like Chain-of-Thought. The result can prove that our designed functional plans are better orchestration media type compared to general COT plans in data science code generation task.
+
+# 4 Related Work
+
+Data Analysis Code Generation (DACG). DACG automates code generation for data-centric tasks in formats such as CSV, TSV, and relational databases (RDB) [7, 35]. It requires code that accurately handles schemas, formats, and data semantics, whether in Python for tabular data [9, 10, 47] or SQL for databases [68, 27, 29]. Spreadsheet-based code generation further extends DACG to formula generation in tools [54, 5]. Although large language models have shown promise, privacy remains a challenge in cloud-based environments [35].
+
+Knowledge Distillation. Knowledge distillation can mitigate this problem by transferring LLM capabilities to smaller models, enabling efficient deployment in resource-constrained environments [63]. The field has evolved from early work on softened output training [18] to advanced techniques like task-specific fine-tuning [45], zero-shot learning [57], and instruction-following datasets [58, 57]. Progressive distillation techniques, such as the Orca framework [38], demonstrate the potential for guiding the development of efficient open-source models. Self-distillation approaches have explored autonomous training data generation [58]. Recent advancements have focused on improving the performance and privacy aspects of DACG by knowledge distillation [35]. At the same time, synthetic data has been leveraged to enhance the generalization of SQL generation across different schemas [64]. Even though these techniques are effective, most still require training efforts to transfer knowledge. Our DARGO framework introduces distillation through in-context learning, eliminating the need for task-specific fine-tuning.
+
+Memory for Large Language Models. Memory can enhance LLMs by retaining long-term context and knowledge [69], as in reflection-based frameworks [48], subroutine reuse [50], and self-correction [4]. Complex tasks may require repository-level memory [65, 56, 55]. Typically, memory remains confined to a single model. Our work introduces multi-model memory orchestration (via GPT-4o), enabling smaller models to tap into broader knowledge sources for DACG.
+
+# 5 Conclusion
+
+In this paper, we presented DARGO, an automatic framework for knowledge distillation from Large Language Models to Small Language Models in Data Science Code Generation. DARGO leverages In-Context Learning to enhance SLM performance without fine-tuning, using model orchestration and memory-based distillation to improve task accuracy. Evaluations on three challenging tabular data analysis datasets that require code generation show a  $27.5\%$  relative performance increase for Phi-3-mini. We also show model-agnostic effectiveness, benefiting other SLMS, even they did not participate in the orchestration. These results highlight the potential of DARGO for developing intelligent applications with a focus on privacy and computational efficiency.
+
+# 6 Acknowledgments
+
+We thank the anonymous reviewers and committees for their helpful comments, suggestions and organizations. We appreciate Xinnuo Xu, Christian Poelitz, Sian Lindley, Hank Lee, Xiaolong Li, Ge Qu for discussion and suggestions. Reynold Cheng, Jinyang Li, were supported by the Research Grant Council of Hong Kong (RGC Project HKU 17202325), the University of Hong Kong (Project 2409100399), and the HKU Faculty Exchange Award 2024 (Faculty of Engineering).
+
+# References
+
+[1] Marah Abdin, Sam Ade Jacobs, Ammar Ahmad Awan, Jyoti Aneja, Ahmed Awadallah, Hany Awadalla, Nguyen Bach, Amit Bahree, Arash Bakhtiari, Harkirat Behl, et al. Phi-3 technical report: A highly capable language model locally on your phone. arXiv preprint arXiv:2404.14219, 2024.  
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+
+# NeurIPS Paper Checklist
+
+# 1. Claims
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+
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+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
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+# Justification: [NA]
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+
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+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
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+
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+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
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+
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+
+# Guidelines:
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+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+# Answer: [Yes]
+
+Justification: For each inference result, we run experiments 5 times and present average results, which can prove that the performance show statistical significance.
+
+# Guidelines:
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+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).
+
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We stated that all SLM inference based on one NVIDIA A100 80GB GPU card and how much cost for distillation in Appendix I.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: [Yes]
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: Detailed in Appendix J.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
+
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: [NA]
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: We cited them.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
+
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: [NA]
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: [NA]
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: [NA]
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: [NA]
+
+Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Model Implementation
+
+We implement models for three main categories of purpose:
+
+# A.1 Orchestration Models
+
+gpt-4o: The Teacher model (gpt-4o) is responsible for several key tasks, including Abstraction Lifting (see Section 2.2) and Plan Optimization (see Section 2.2), which are performed while monitoring the performance of the Student model. Additionally, the Teacher model handles the conversion of complex, heterogeneous cases into more readable case studies for Student Learning Models (SLMs), as detailed in Section 2.3.
+
+Llama-3.3-70B-Instruct: we use this powerful open-source model as teacher to prove the generalization of DARGO workflow.
+
+phi-3-mini-128k-instruct: For the orchestration process, we select this 3.8B parameter SLM as the Student model due to its strong generalization abilities and efficient deployment.
+
+llama-3.1-8b-instruct
+
+# A.2 Baseline Models
+
+Within the orchestration mode, several families of Student Learning Models (SLMs) are evaluated. These include models from the Phi-3, Starcoder 2, and LIDARGO families:
+
+# Phi-3 Family [1]
+
+```txt
+phi-3-mini-128k-instruct (3.8B)  
+phi-3-small-128k-instruct (7B)  
+phi-3-medium-128k-instruct (14I
+```
+
+# Starcoder 2 Family [33]
+
+```txt
+starcoder2-7b-instruct
+starcoder2-15b-instruct
+```
+
+# Llama Family [13]
+
+```txt
+codellama-7b-instruct-hf codellama-13b-instruct-hf
+```
+
+# A.3 Models in Knowledge Transmission
+
+In Section 3.4, we explore the knowledge distilled from DARGO to newly developed models, particularly in terms of their ability to generalize knowledge. For this evaluation, we select the following models:
+
+llama-3.1-8b-instruct: This model is broad new, yet it shows significant performance improvements when leveraging the distilled knowledge.
+
+gpt-35-turbo-16k: We also include a closed-source model in our experiments to demonstrate the effectiveness of our approach across both GPU-deployed and API-based models. Despite its number of parameters is unknown, we consider it as one of SLMs since its performance falls behind of its more advanced versions such as GPT-4.
+
+# A.4 Demonstration V.S. Distillation
+
+Given the memory database, we compare the effectiveness of our knowledge distillation techniques, with conventional demonstration-based strategies. In our approach, distillation involves transferring knowledge from the memory database to SLMs through task-specific instructions. On the other hand,
+
+demonstration-based methods guide SLMs by presenting explicit task examples to facilitate analog reasoning [67]. We implemented two variants of few-shot demonstrations: Static: Human experts select three representative examples from the memory database, which remain constant across all cases. Dynamic RAG-based: Examples are selected from memory database based on similarity to the current query. For fair comparison, we also implement the same RAG system as DarGO-MI, described in Section 3.2.
+
+Our findings show that few-shot demonstration generally underperforms DarGO across datasets. However, RAG-based few-shot demonstration outperforms both our designed knowledge distillation and other baselines on TABMWP. This success appears linked to the simplicity of TABMWP's input data, which averages 2.22 columns and 6.13 rows per data point, with clean numeric or processed string values (Table 7). In contrast, for WIKITQ with irregular value types, and BIRD-SQL with complex schemas, SLMs struggle, generating  $38.2\%$  more invalid outputs, such as erroneous SQL queries, in BIRD-SQL. Based on these observations, we conclude that dynamic few-shot demonstration is more convenient and effective for leveraging the memory database when the input data is less complex. On the contrary, for complex data such as tables with dirty values or relational databases, our designed knowledge distillation enables SLMs to better utilize knowledge and perform tasks more effectively.
+
+# B Dataset Implementation Details
+
+# B.1 Train-Test Distillation Data Statistics
+
+Table 7 summarizes the distributions of the datasets used in this study. These datasets include both single-table and multi-table relational structures, offering a comprehensive aspects for evaluation. The question types range from standard semantic parsing queries to more complex analytical questions involving aggregations and nested operations. This diversity enables a thorough assessment of the model's performance across various query paradigms.
+
+# B.2 Data File Content
+
+For convenient reproduction and following, we preprocess all dataset into more unified data format of json1. In python task (TABMWP, WIkITQ), each line of data contains question_id, question, data_path, data_overview, answer_type, answer. In SQL task (BIRD-SQL), each line of data contains question_id, question, evidence, data_path, db_id, sql.
+
+# B.3 Data Input Content
+
+The main goal of this work is to evaluate the code generation capabilities of models in understanding data schemas and structures across
+
+Table 7: Statistics for three datasets. The term Analysis indicates that the dataset mainly consists of analytical questions, while SP refers to semantic parsing tasks.  
+
+<table><tr><td>STATISTIC</td><td>TABMWP</td><td>WIKITQ</td><td>BIRD-SQL</td></tr><tr><td colspan="4">Dataset Features</td></tr><tr><td># train examples</td><td>1,000</td><td>2,000</td><td>1,000</td></tr><tr><td># eval examples</td><td>1,000</td><td>1,000</td><td>500</td></tr><tr><td>question type</td><td>Analysis</td><td>SP</td><td>SP + Analysis</td></tr><tr><td>#toks / Q</td><td>26.5</td><td>12.6</td><td>20.0</td></tr><tr><td colspan="4">Data Structure</td></tr><tr><td>data input type</td><td>Single</td><td>Single</td><td>RDB</td></tr><tr><td>#rows / data</td><td>6.13</td><td>28.5</td><td>354k</td></tr><tr><td>#columns / data</td><td>2.22</td><td>6.36</td><td>73.3</td></tr><tr><td colspan="4">Code Features</td></tr><tr><td>code type</td><td>Python</td><td>Python</td><td>SQL</td></tr><tr><td>answer type</td><td>String</td><td>String</td><td>Code</td></tr><tr><td>#toks / code</td><td>N/A</td><td>N/A</td><td>61.15</td></tr></table>
+
+multiple datasets. Given the impracticality of providing all data values in real-world scenarios in which datasets may consist of millions or even billions of rows, we sample values for the part of data input to simulate realistic code generation tasks. We feed the markdown format of schemas with data samples as data_overview.
+
+For TABMWP, we provide only the column names and the first three rows of values. This enables models to infer the data structure and value types necessary for Python Pandas code generation without exposing all the data.
+
+For WIKITQ, which contains more complex and varied value types, we provide the first 10 rows of values and column names to help models navigate the dataset's intricacies.
+
+In the case of BIRD-SQL, which contains relational databases with complex schemas and diverse value types, more advanced schema-linking techniques are often required to retrieve relevant tables
+
+or columns before answering queries [53, 43]. While we consider this advanced schema-linking process as future work for DARGO, our current focus is on the code generation aspect. Therefore, we provide:
+
+- Ground truth retrieved tables, reducing input complexity and simulating realistic human-machine interactions where users might supply potentially relevant tables.  
+- Full columns with column meaning description files.  
+- The first three rows of values for each table.
+
+Although the retrieved tables are given, the models must still consider constraints and generate correct SQL queries. As shown in Table 2, performance on Bird-SQL remains relatively low, even with simplified table retrieval, highlighting the challenges of generating accurate SQL queries in complex database environments. This methodology allows us to evaluate code generation capabilities while approximating the real-world challenges of data analysis.
+
+# B.4 OOD Cross-Dataset Evaluation Set Statistics.
+
+Table 8 show the data statistics of basic set of unified memory for evaluating DARGO on OOD cross-dataset distillation.
+
+Table 8: Statistics for Basic Set of Unified Memory.  
+
+<table><tr><td>Dataset</td><td>#Items</td></tr><tr><td>wikitq</td><td>1000</td></tr><tr><td>TabMwp</td><td>1000</td></tr><tr><td>BIRD-Pandas</td><td>300</td></tr><tr><td>Juice</td><td>1000</td></tr></table>
+
+# C Action Types
+
+Decomposition. The Decomposition action type divides a large, multifaceted step  $\mathbf{x}$  into multiple simpler steps a and b. By breaking complex workflows into smaller components, the Model Orchestration Interface (MOI) ensures that the resulting plan is both clearly understood and more straightforward for an SLM to execute. This approach clarifies the logic behind each sub-step and allows finer-grained control over how tasks are performed or combined, thereby reducing confusion and facilitating future refinements.
+
+ALT. Sometimes, the instructions for a step can be ambiguous, incorrect, or unnecessarily complex. The ALT action type replaces such a problematic step  $\mathbf{x}$  with a newly clarified step  $\mathbf{y}$ . By substituting erroneous or unclear instructions, the MOI ensures that the plan adheres to more accurate logic, minimizing the likelihood of misinterpretation and promoting consistency throughout the workflow.
+
+ADD. When a critical operation is missing or an additional step is required for completeness, the ADD action type introduces a new step  $a$  into the existing plan. Adding steps proves valuable when the plan overlooks essential checks, transformations, or other auxiliary procedures. This mechanism ensures more thorough and dependable solutions.
+
+DELETE. Certain steps can be redundant or risk causing confusion for subsequent code generation. The DELETE action type removes any unnecessary step  $\mathbf{x}$ , thereby streamlining the plan. By eliminating irrelevant instructions, the MOI reduces cognitive load on the SLM and maintains a logically consistent sequence of steps that aligns directly with the overarching goal.
+
+SIMPLIFY. Whenever possible, it can be advantageous to simplify complex steps. The SIMPLIFY action type replaces a complicated step  $x$  with a more direct version, simple_step(x). For example, it may transform a solution relying on recursion into an iterative, loop-based approach. Simplification improves both computational efficiency and interpretability, since SLMs often perform better with concise instructions.
+
+Table 9: Performance evaluation of Zero-Shot End-to-End Code Generation, LoRA fine-tuning, and our proposed knowledge distillation techniques on BIRD-SQL. Deeper red shading indicates a larger performance drop compared to the original pre-trained model, while green indicates no decline or improvement.  
+
+<table><tr><td>Model</td><td>SIMPLE</td><td>MEDIUM</td><td>CHALLENGING</td><td>OVERALL</td></tr><tr><td colspan="5">Zero-Shot End-to-End Code Gen.</td></tr><tr><td>Original Checkpoint</td><td>38.51</td><td>21.20</td><td>11.76</td><td>24.40</td></tr><tr><td>LoRA Fine-Tuned</td><td>39.86</td><td>19.20</td><td>10.78</td><td>23.60</td></tr><tr><td colspan="5">DARGO Knowledge Distillation</td></tr><tr><td>Meta Instruction</td><td>51.35</td><td>30.40</td><td>16.67</td><td>33.80</td></tr></table>
+
+SWITCH. Selecting an appropriate library or tool is critical for achieving both correctness and simplicity. The SWITCH action type replaces a step implemented with packageA by an equivalent (or more suitable) operation from packageB. For instance, one might switch from using LinearRegression to CorrelationCoefficient when the task is simply to determine the relationship between two variables. This approach avoids unnecessary overhead and preserves clarity in the plan.
+
+By employing these six action types, Decomposition, ALT, ADD, DELETE, SIMPLIFY, and SWITCH, the MOI systematically refines high-level plans. This process results in efficient and easily interpretable workflows, ensuring consistency from the design of the plan to its final implementation by SLMs.
+
+# D DARGO Functionality
+
+# D.1 MOI Generalization
+
+![](images/840ff25321fa89652c7ed230c1c2b4442fabe198e9921e80d9a84c1a0c52b112.jpg)  
+Figure 6: Illustration of how MOI is conducted in Python for Tabular data analysis.
+
+Our Model Orchestration Interface (MOI) is adaptable to different programming languages with different data input settings. Figure 2 shows how MOI is conducted in RDB settings with SQLite, and Figure 6 shows how it's undertaken in Single-tabular data with Python.
+
+# D.2 Fine-Tuning v.s. DARGO Knowledge Distillation
+
+# E Ablation Study for Inference
+
+We conducted a comprehensive ablation study of DARGO-MI, as shown in Table 10. Code-T5+ is a code embedding model [59], while BGE-Large [61] represents one of the state-of-the-art (SOTA) text embedding models. The study examines two types of RAG Index: one where distance is computed using question embeddings alone, and another where both question and schema embeddings are used. The "Plan + Gen" approach involves first constructing a plan with distilled knowledge, followed by generation using knowledge-driven planning. In contrast, the "Gen" approach involves direct
+
+generation without prior planning. The instruction type labeled w/ examples refers to cases where a specific example is provided by the Teacher model. We evaluate performance with 1, 3, and 5 examples to assess the impact of varying numbers of RAG examples. The results of the ablation study reveal several key insights:
+
+Code embeddings outperform text embeddings. The superior performance of Code-T5+ over BGE-Large-en can be attributed to the nature of the task. While text embeddings emphasize on semantic and domain knowledge, code embeddings capture the syntactic and logical structure of coding problems, which is crucial for DACG tasks. Even when presented with identical questions, the code solutions can vary significantly depending on the data input. Code-T5+ is able to effectively embed questions from a programming perspective, benefiting from its pre-trained corpus, whereas text embeddings are less suited for the task.
+
+Embedding only the question is more effective than embedding both the question and schema. The study demonstrates that question-only embeddings lead to better results. This suggests that the inclusion of schema in the embedding may introduce unnecessary complexity, which may hinder performance on the DACG task.
+
+Planning is essential for more complex tasks. The results stress on the importance of planning in a knowledge-driven generation. For tasks requiring complex reasoning, the "Plan + Gen" approach outperforms direct generation (Gen), indicating that structured planning significantly improves task performance.
+
+One example may bias the SLM. Involving a single example in the instruction can introduce bias in Small Lanuguage Models (SLMs). A specific example might cause the SLM to over-follow to certain information, leading to hallucinations. For instance, if the example includes a reference to "singer", the SLM may generate plans that include "singer" even when the question pertains to an unrelated topic, such as "cars". This observation highlights the lack of robustness in SLMs when exposed to overly specific examples. Consequently, it is better to provide more general, transferable knowledge in instructions. The degraded performance observed with 1 RAG example supports this conclusion, as the model becomes overly reliant on the provided information.
+
+More examples do not always improve performance. Interestingly, increasing the number of RAG examples (from 1 to 5) results in a performance drop. This suggests that longer input sequences may confuse the SLM, making it more difficult to distill relevant knowledge. Based on these findings, we recommend using 3 RAG examples as the optimal balance for complex DACG tasks since it avoids both the biases of a single example and the confusion caused by too many examples.
+
+Table 10: Ablation Study Results of DARGO-MI of Phi-3-mini on BIRD-SQL. The table compares different embedding models, RAG index (with or without schema), reasoning approaches (planning or direct generation), and varying numbers of RAG examples.  
+
+<table><tr><td>Embedding Model</td><td>RAG Index</td><td>Reasoning Type</td><td>Instruction Type</td><td># RAG Examples</td><td>Performance</td></tr><tr><td>code-t5+</td><td>question</td><td>plan + gen</td><td>no examples</td><td>3</td><td>33.80</td></tr><tr><td>code-t5+</td><td>question</td><td>gen</td><td>no examples</td><td>3</td><td>31.40 (↓2.40)</td></tr><tr><td>bge-large</td><td>question</td><td>plan + gen</td><td>no examples</td><td>3</td><td>30.00 (↓3.80)</td></tr><tr><td>code-t5+</td><td>question</td><td>plan + gen</td><td>w/ examples</td><td>3</td><td>28.00 (↓5.80)</td></tr><tr><td>code-t5+</td><td>question+schema</td><td>plan + gen</td><td>no examples</td><td>3</td><td>32.40 (↓1.40)</td></tr><tr><td>code-t5+</td><td>question</td><td>plan + gen</td><td>no examples</td><td>5</td><td>31.80 (↓2.00)</td></tr><tr><td>code-t5+</td><td>question</td><td>plan + gen</td><td>no examples</td><td>1</td><td>29.80 (↓4.00)</td></tr></table>
+
+# F Error Analysis
+
+We conducted an error analysis by sampling 50 incorrect cases for both DARGO-MI across three datasets. Although DARGO substantially improves the overall performance of SLMs, we found that  $54\%$  of the errors were caused by over-reasoning. This issue tends to emerge even in relatively simple cases. As discussed earlier, SLMs can overly adhere to the instructions derived from planning and guidance, which is problematic when the task is enough simple and does not require decomposition or reasoning. In these cases, direct code generation would lead to more accurate results. The remaining
+
+![](images/f5be03e30e2f869ecaa5158b81c3a6b7058b38f86ff45cfdacc93b1f6285150a.jpg)  
+Figure 7: The scaling performance of DARGO on three main DACG datasets.
+
+errors stem from common issues in code generation tasks, such as incorrect string handling, incorrect column selection, database constrain understanding.
+
+# G Limitations and Future Work
+
+A key limitation of our current approach with DARGO is the reliance on initial training examples for both LLMs and SLMs to facilitate orchestration. This is why we selected datasets that include a training corpus suitable for distilling knowledge. However, an important avenue for future work is to explore how to generate such training data in a fully zero-shot manner, without relying on human-annotated or enumerated examples. Additionally, as highlighted in the error analysis, over-reasoning negatively impacts performance on simpler tasks, where additional reasoning or decomposition is unnecessary. To address this, future work could focus on developing or prompting smaller models to act as routers, as proposed by Ding et al. [12], to classify questions based on whether they require planning. This would help avoid over-reasoning in straightforward cases and improve the overall efficiency of DARGO.
+
+# H Scaling Analysis of DARGO
+
+Figure 7 shows that DARGO scales reliably with increasing data volume. Although its performance is somewhat unstable on very small memory, stability improves rapidly, and accuracy surpasses the baseline once more than  $60\%$  of the exploration data are available which proves that the distilled knowledge is being exploited effectively.
+
+# I Cost Analysis
+
+We compare the cost of DARGO with other distillation methods shown in Table 12, Table 16 and Table 14. And Table 11, Table 15, Table 13 show the inference cost of DARGO and its comparison with other baselines. We can conclude that: first, it shows DarGO now is the most efficient Distillation work for this task. Second, since inference is conducted by SLMs, the additional token usage is acceptable when considering its significant improvement.
+
+Table 11: BIRD-SQL: Infer (Tab. 16)  
+
+<table><tr><td>Method</td><td>SLM InToks ↓</td><td>SLM OutToks ↓</td><td>EX ↑</td></tr><tr><td>End2End (baseline)</td><td>358,277</td><td>16,751</td><td>24.40</td></tr><tr><td>Chain-Of-Thought</td><td>369,433</td><td>19,317</td><td>23.80</td></tr><tr><td>Static Few-Shot</td><td>794,342</td><td>17,306</td><td>26.20</td></tr><tr><td>Dynamic Few-Shot</td><td>836,678</td><td>26,291</td><td>24.40</td></tr><tr><td>DSPy</td><td>637,146</td><td>60,748</td><td>22.80</td></tr><tr><td>ReGAL</td><td>1,602,698</td><td>16,045</td><td>20.40</td></tr><tr><td>DarGO-MI</td><td>647,799</td><td>64,836</td><td>33.80</td></tr></table>
+
+# J Broader Impact
+
+Given DARGO is designed for automating code generation for better and efficient data analysis, it can help data scientists explore the potential risks of financial market, earthquake, which are beneficial to
+
+Table 12: BIRD-SQL: Exploration (Tab. 17)  
+
+<table><tr><td>Method</td><td>LLM InToks</td><td>LLM OutToks ↓</td><td>Cost ↓</td></tr><tr><td>DSPy</td><td>18,129,504</td><td>2,689,076</td><td>$108.32</td></tr><tr><td>ReGAL</td><td>13,259,759</td><td>2,381,368</td><td>$85.44</td></tr><tr><td>DarGO</td><td>11,086,194</td><td>1,372,421</td><td>$62.16</td></tr></table>
+
+Table 13: WikiTQ: Infer (Tab. 18)  
+
+<table><tr><td>Method</td><td>SLM InToks ↓</td><td>SLM OutToks ↓</td><td>Acc. ↑</td></tr><tr><td>End2End (baseline)</td><td>235,122</td><td>44,033</td><td>32.50</td></tr><tr><td>Chain-Of-Thought</td><td>242,443</td><td>149,246</td><td>27.70</td></tr><tr><td>Static Few-Shot</td><td>521,292</td><td>45,161</td><td>23.00</td></tr><tr><td>Dynamic Few-Shot</td><td>549,075</td><td>93,415</td><td>16.60</td></tr><tr><td>DSPy</td><td>418,131</td><td>123,422</td><td>26.70</td></tr><tr><td>ReGAL</td><td>2,082,898</td><td>162,949</td><td>36.10</td></tr><tr><td>DarGO-MI</td><td>495,493</td><td>168,311</td><td>41.10</td></tr></table>
+
+the society. Also, our study only focuses on code generation, a programming-level language rather than natural language, it will not impact society negatively.
+
+# K Reproducibility
+
+We provide codebase in the supplementary files and we list all implementation details in Appendix B. And we deliver prompts for each stage and baselines in Appendix L for fully reproducibility.
+
+# L Main Prompts
+
+The zero-shot End-to-End Code Generation prompt is shown in Figure 8, Figure 16 and 18 show the zero-shot Chain-Of-Thought reasoning. Figure 19 shows few-shot demonstration prompting. The few-shotexamples can be selected by human experts as Static Few-Shot Demonstration, and can be retrieved from DARGO memory database by RAG system as Dynamic Few-Shot Demonstration.
+
+The Figure 8, 9, 10, 11 show prompts for Orchestration between LLMs and SLMs. Figure 12 presents how LLM convert orchestrated successful cases to more understandable case studies to SLMs. LLMs can go through correct cases from memory databases and distill knowledge to an offline and plug-and-plan General Instruction for SLMs to used for new and unseen queries performed by prompts shown in Figure 13 and 14. During inference, SLMs can produce Meta Instructions by prompts in Figure 15. Given distilled knowledge (instructions), SLMs will plan first as shown in Figure 17, and generate codes finally with their knowledge-driven planning, which shows in Figure 18.
+
+# M Knowledge Distillation Examples
+
+# M.1 Case Study Example
+
+The Figure 20 shows the example of case studies on Python task. Figure 22 present examples of DARGO-MI respectively.
+
+Table 14: WikiTQ: Exploration (Tab. 19)  
+
+<table><tr><td>Method</td><td>LLM InToks</td><td>LLM OutToks ↓</td><td>Cost ↓</td></tr><tr><td>DSPy</td><td>10,893,180</td><td>1,616,141</td><td>$65.09</td></tr><tr><td>ReGAL</td><td>7,967,176</td><td>1,431,207</td><td>$51.34</td></tr><tr><td>DarGO</td><td>6,661,181</td><td>824,828</td><td>$37.35</td></tr></table>
+
+Table 15: TabMWP: Infer (Tab. 20)  
+
+<table><tr><td>Method</td><td>SLM InToks</td><td>SLM OutToks</td><td>Acc. Score</td></tr><tr><td>End2End (baseline)</td><td>171,480</td><td>38,016</td><td>41.80</td></tr><tr><td>Chain-Of-Thought</td><td>176,764</td><td>39,244</td><td>41.40</td></tr><tr><td>Static Few-Shot</td><td>380,177</td><td>38,283</td><td>36.20</td></tr><tr><td>Dynamic Few-Shot</td><td>400,358</td><td>52,578</td><td>52.10</td></tr><tr><td>DSPy</td><td>304,898</td><td>49,073</td><td>40.40</td></tr><tr><td>ReGAL</td><td>767,114</td><td>37,679</td><td>41.80</td></tr><tr><td>DarGO-MI</td><td>314,817</td><td>125,230</td><td>45.70</td></tr></table>
+
+Table 16: TabMWP: Exploration (Tab. 21)  
+
+<table><tr><td>Method</td><td>LLM InToks ↓</td><td>LLM OutToks ↓</td><td>Cost ↓</td></tr><tr><td>DSPy</td><td>9,426,638</td><td>893,604</td><td>$48.75</td></tr><tr><td>ReGAL</td><td>7,790,084</td><td>751,736</td><td>$40.49</td></tr><tr><td>DarGO</td><td>5,330,507</td><td>659,862</td><td>$29.89</td></tr></table>
+
+You are a data analyst. Given the data, you need to generate the code first to answer the question:
+
+Please Follow:
+
+- Do not add data inspection in the plan, such as `df.head()` or `print(df.head())` since this is cheating!  
+- Do not use any external information.  
+- The code should be end-to-end, so you cannot encourage yourself to print other things except final result. More other information lead to be distracted.
+
+Question: {question}  
+Thought: I need to see the data samples in the first 10 rows:
+
+Code:
+
+``python
+import pandas as pd
+df = pd.read_csv('./print(df.head(10))  
+Observation (data overview)  
+Thought: I can generate remaining code to answer this question:  
+# Code:  
+``python  
+import pandas as pd
+
+Figure 8: Prompt of baseline end-to-end generation for tasks requiring Python.
+
+```txt
+You are a data analyst trainer. You are educating your student to generate right code to answer tabular data analysis questions. In order to do so, you need to convert your code to code plan and let your students to fill to understand plans and analysis. So you cannot generate code by your own, only generate plans. # Data Overview at the path (data_path) (first ten rows): {data_overview} # Question: {question} # Original Code: ` `python {ground_truth code} ... You should convert the code into the code計劃 format with the placeholder ` [FILL YOUR CODE HERE]': ` `code計劃 import ... # Step 1:.... [Fill Your Code] # Step 2:.... [Fill Your Code] ... # Step N: .... Generate your code計劃 for your student. DO NOT generate any code by your own. Also ignore and remove steps of inspecting the data which leads to student cheating. Please note it's hard for your student to write long code. You will get 1,000 dollars if you have a good job:
+```
+
+Figure 9: Prompt converting ground-truth code to functional plan for python task as example. This is conducted by LLM Teacher model.
+
+```python
+You are a data analyst. Given the data, expert customized functional plan, complete each line of code to answer questions correctly:   
+# Please Follow: - Do not add data inspection in the plan, such as \`df.head() or \`print(df.head()) since this is cheating! - Do not use any external information. - The code should be end-to-end, so you cannot encourage yourself to print other things except final result. More other information lead to be distracted.   
+# Question: {question}   
+# Function Plan: . . .python {functional plan}   
+# Your entire completion code for function plan executable and correct:   
+# Code: .. .python import pandas as pd
+```
+
+Figure 10: Prompt of orchestration coding. This is conducted by SLM Student model.
+
+![](images/b3cd3761e6c531e30abc5fe33f1f0b5bcb199f2fc5c63b3059618c5e08b9b028.jpg)  
+Figure 11: Prompt of plan optimization. This is conducted by LLM Teacher model.
+
+![](images/40c046365455bcc2f299fd060d793345f950047423bc6d3f5434c1f3835be90d.jpg)  
+Figure 12: Prompt of case study conversion. This is conducted by LLM Teacher model.
+
+![](images/9a8e42e299e14c5b7155734bb759dfbba9ef918adf3ee8014117918579d2c163.jpg)  
+Figure 13: Prompt of aggregation prompt of each batch of case studies. This is conducted by LLM Teacher model.
+
+You are a data analyst trainer. You are educating your student to generate correct python pandas code to answer tabular data analysis questions. To test and elicit their knowledge of code, you generate step-by-step plans that allow them to fill in subqueries until they succeed.
+
+These are case studies where they fill the correct code:  
+- case begin:  
+-  
+(last layer of case studies)  
+- case end:
+
+Please note:
+
+- Do not ask students to add data inspection in the code, such as `df.head()` or `print(df.head())`  
+- Do not guide them to use any external information.  
+- The mixed code should be end-to-end, so you cannot encourage student to print other things except the final result. More other information would cause student to be distracted.  
+- Just focus on how to make students learn how to better plan in the end-to-end code generation.
+
+Following case studies, please summarize 5-7 key points about how to plan and generate correct code to answer the tabular data analysis questions accurately. Students will take your notes directly.
+
+You need to start with:
+
+1. You should
+
+Figure 14: Prompt of summarization prompt of batch of case studies in the last layer. This is conducted by LLM Teacher model.
+
+You are a data analysis trainer. Your are teaching your student to plan and generate python code accurately. You find some case study for reference.
+
+There are case studies:
+
+{case_studies} case end:
+
+Following case studies, please summarize key 5-7 points about how to plan and generate correct python code to answer the data analysis questions accurately.
+
+You will use them to educate your student:
+
+successful plan suggestions:
+
+1. You Should
+
+Figure 15: Prompt of in-time summarization for meta-instructions. This is conducted by SLM Student model.
+
+You are a data engineer. Given the sample data, generate python code plan to answer the question accurately.
+
+Please Follow:
+
+- Do not add data inspection in the plan, such as `df.head()` or `print(df.head())` since this is cheating!  
+- Do not use any external information.  
+- The code should be end-to-end, so you cannot encourage yourself to print other things except final result. More other information lead to be distracted.
+
+Question: {question}
+
+Thought: I need to see the data samples in the first 10 rows:
+
+Code:
+
+```
+`` python
+
+import pandas as pd
+
+df = pd.read_csv(['data_path'], sep='\t')
+
+print(df.head(10))
+
+# Observation:
+
+{data_overview}
+
+Thought: I should have a step-by-step text plan for generating this code first. I will fill my plan into the template in details:
+
+code plan
+
+Step 1: ...
+
+Step 2: ...
+
+.
+
+Final Step: ...
+
+Generate your plan step by step for the question:
+
+Let's think step by step:
+
+code plan
+
+Step 1:
+
+Figure 16: Prompt of generating Chain-Of-Thought. This is conducted by SLM Student model.
+
+![](images/72542d5ac122642f322cccbd92a9d231df20d52451383f693f91f26c53a8b5bb.jpg)  
+Figure 17: Prompt of knowledge-driven planning. This is conducted by SLM Student model.
+
+![](images/7495579d6c84a9d66e458330570c7858f547b3cbd82ec43e853c6402860151d6.jpg)  
+Figure 18: Prompt of code generation given step-wise planning. This is conducted by SLM Student model.
+
+![](images/ed0294376bf2e05426e62251632e294ccee635e2d882185ba8a1a4820427bdb7.jpg)  
+Figure 19: Prompt of few-shot demonstration. This is conducted by SLM Student model.  
+Figure 20: Example of case studies for tasks requiring Python. This is conducted by LLM Teacher model.  
+Figure 21: Example of General Instruction for tasks requiring Python. This is conducted by LLM Teacher model.
+
+Case Study: Average Weight Calculation for Specific Players
+
+# Question:
+
+What is the average weight of Jamarr Sanders and Robert Williams?
+
+# Table Info:
+
+- **Columns**: Name, Height, Weight (Ibs.), Position, Class, Hometown, Previous Team(s)  
+-**Sample Data**  
+- Jamarr Sanders: Weight 210 lbs.  
+- Robert Williams: Weight 210 lbs.
+
+# Objective:
+
+To calculate the average weight of the players Jamarr Sanders and Robert Williams from the given dataset.
+
+# Explanation:
+
+1. **Load Data**: The data is loaded from a tab-separated values (TSV) file.  
+2. \*\*Filter Data\*\*: Rows corresponding to the names "Jamarr Sanders" and "Robert Williams" are filtered from the dataset.  
+3. **Calculate Average**: The average weight of the filtered rows is computed.  
+4. **Output**: The result is printed as an integer.
+
+By following these steps, the student can understand how to filter specific rows in a dataset and perform calculations on the filtered data. This case demonstrates the practical application of data manipulation and analysis using pandas in Python.
+
+1. You should **break down the task into manageable steps**. Each step should build on the previous one, guiding you through the process logically.  
+2. You should **emphasize data handling and cleaning**. This includes handling missing values, normalizing case, and ensuring data consistency."  
+3. You should **focus on filtering and extraction**. Guide yourself on how to filter and extract relevant data based on specific criteria. This is often the core of the analysis."  
+4. You should **perform aggregation and counting**. Learn how to perform aggregation operations like counting, summing, or finding minimum/maximum values to derive insights from the data."  
+5. You should **present the result clearly**. Ensure that the final step involves presenting the result in a clear and concise manner. This reinforces the importance of communicating findings effectively."  
+6. You should **avoid distractions**. Keep the instructions focused on the end-to-end process without encouraging unnecessary intermediate outputs or external information. This helps maintain your focus on the task at hand.
+
+# "question": "which country rank last?"
+
+1. Understand the problem statement and the data structure.  
+2. Load the data using appropriate libraries (e.g., pandas).  
+3. Perform necessary data manipulation and cleaning.  
+4. Identify the relevant columns and values for the analysis.  
+5. Use appropriate functions and methods to filter, sort, and extract the required information.  
+6. Output the result in a clear and concise manner.
+
+Figure 22: Example of General Instruction for tasks requiring Python. This is conducted by SLM Student model in time.
\ No newline at end of file
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+# Unlocking hidden biomolecular conformational landscapes in diffusion models at inference time
+
+Daniel D. Richman*
+
+Stanford University ddrichma@stanford.edu
+
+Carl-Mikael Suomivuori†
+
+Stanford University carl-mikael.suomivuori@yale.edu
+
+Jessica Karaguesian*
+
+Stanford University  
+jkara@stanford.edu
+
+Ron O. Dror
+
+Stanford University rondror@cs.stanford.edu
+
+# Abstract
+
+The function of biomolecules such as proteins depends on their ability to interconvert between a wide range of structures or "conformations." Researchers have endeavored for decades to develop computational methods to predict the distribution of conformations, which is far harder to determine experimentally than a static folded structure. We present ConforMix, an inference-time algorithm that enhances sampling of conformational distributions using a combination of classifier guidance, filtering, and free energy estimation. Our approach upgrades diffusion models—whether trained for static structure prediction or conformational generation—to enable more efficient discovery of conformational variability without requiring prior knowledge of major degrees of freedom. ConforMix is orthogonal to improvements in model pretraining and would benefit even a hypothetical model that perfectly reproduced the Boltzmann distribution. Remarkably, when applied to a diffusion model trained for static structure prediction, ConforMix captures structural changes including domain motion, cryptic pocket flexibility, and transporter cycling, while avoiding unphysical states. Case studies of biologically critical proteins demonstrate the scalability, accuracy, and utility of this method.
+
+# 1 Introduction
+
+Biology depends on molecular flexibility. Proteins, RNA, DNA, and other components of biological systems adopt dynamic 3D conformations, which are key to function [7, 8]. Understanding and modeling these dynamics informs both basic biology research and applications to medicine, agriculture, and other areas, and is therefore a major aim of both experimental and computational scientists. The statistical distribution of the atomic configurations of a molecular system in equilibrium at a given temperature is determined by their potential energies. Specifically, a state with lower potential energy is exponentially more probable than a higher potential energy state, as determined by the Boltzmann distribution [31]. To sample this distribution, scientists have invested decades of effort in creating methods such as Monte Carlo samplers and molecular dynamics simulations (MD), but these methods are typically extremely slow and suffer from accuracy problems [25, 31, 32].
+
+More recently, machine learning models have revolutionized biomolecular structure prediction, and efforts are underway to extend these advances to train models that do not merely predict static structures but generate diverse conformations from the true Boltzmann distribution [1, 11, 12, 17,
+
+35]. As the field evolves, new questions arise about how best to take advantage of these models, access the information they contain, and target opportunities for further improvement. We ask the following question: Given a pretrained generative structure model, what strategy should be deployed at inference time to efficiently and accurately sample its conformational landscape?
+
+In this work, we present an enhanced sampling algorithm, ConforMix, that can be applied to diffusion-based biomolecular structure prediction models such as the AlphaFold 3 family. ConforMix reveals hidden conformations and estimates free energies more efficiently than the default sampling from diffusion models. Our approach is based on the hypothesis that the probability distribution of generative models contains more information than is efficiently sampled. Importantly, even a generative model that perfectly sampled the true Boltzmann distribution—a major long-term goal in machine learning for biology—would, for many uses, still require enhanced sampling algorithms to fruitfully explore it. For instance, correct sampling of the tight-binding proteins barnase and barstar at standard state would generate an unbound structure with probability approximately  $10^{-14}$  [33]. Directly drawing samples from the distribution would be an extremely inefficient way to estimate this probability, determine transition states along the binding pathway, or understand the effects of environmental factors in modulating this affinity [30, 38].
+
+Contributions We introduce, implement, and benchmark ConforMix. Our contributions include:
+
+- A novel algorithm combining twisted sequential Monte Carlo, which performs asymptotically exact sampling of conditional distributions, with an automated procedure for exploring the diffusion landscape, using conditional sampling as a subroutine. Optionally, a statistically optimal sample reweighting algorithm, applied to diffusion models for the first time, can be used to reconstruct the unconditional distribution from the conditional samples.  
+- Dramatically improved inference-time sampling in structure prediction models. We implement ConforMix in Boltz-1, an AlphaFold 3-like diffusion model that typically predicts only one distinct conformation for a given input. ConforMix-Boltz generates realistic and diverse conformations for a variety of proteins, without prior knowledge of important degrees of freedom. In prior work, conditional sampling on biomolecular diffusion models has required additional input information, such as experimentally measured pairwise distances.  
+- Efficient free energy estimation. We also implement ConforMix in BioEmu, a diffusion-based model for conformational sampling, and compare free energy estimates from BioEmu with and without ConforMix. Using ConforMix boosts the speed of free energy estimation.
+
+# 2 Related Work
+
+Computational methods have been used to sample molecular conformations for decades. The well-known Metropolis-Hastings algorithm [23] for Monte Carlo sampling was first used to simulate the statistical mechanics of particles in a box. Elastic network models represent a protein as low-resolution beads connected by springs and recover the principal motions that would occur in this representation, which often are representative of motions of the actual system [20]. More sophisticated methods, such as coarse-grained or all-atom molecular dynamics simulation, model the physical interactions with progressively more detail and have been used to generate detailed probabilistic distributions for a number of proteins [18, 24].
+
+More recently, machine learning models such as AlphaFold 3 [1] have revolutionized the problem of static protein structure prediction. AlphaFold 3 is the most prominent example of a class of diffusion-based structure prediction methods that excel at generating the single most likely protein structure for a given amino acid sequence. While AlphaFold 3 generates five samples per run, for well-ordered proteins these samples are often almost identical and are better understood as very similar predictions of the same static structure rather than a representation of conformational variety.
+
+Using such models to model the distribution of conformations, however, is a task still in its infancy. The most well-studied approaches involve subsampling or otherwise perturbing the multiple sequence alignment (MSA) input [13, 15, 27, 35] or templates input [2, 4, 9] to AlphaFold and related models. These inputs specify, respectively, sequence co-evolutionary information and related 3D structures. Emphasizing different sequence or structural features in the input has empirically produced significant conformational diversity. Subsampling methods, however, suffer from fundamental limitations. First, subsampling necessarily involves reducing the information available to the model, which often
+
+produces poor-quality outputs. Second, molecular conformations are continuous, while subsampling of sequences or templates is a discrete operation. Third, the structure distribution accessed by subsampling methods is not well defined, and despite years of study there is no clear means of reconstructing actual probabilistic ensembles.
+
+Beyond input-perturbation methods, other efforts have involved building new models to predict conformations [6, 11, 17]. Strategies include leveraging molecular dynamics simulations or experimental data, either during training or at runtime [3, 16, 19, 21, 34]. While these approaches have shown promise, they do not yet accurately reproduce the thermodynamic ensembles of arbitrary proteins.
+
+Based on the successes of MSA subsampling and template perturbation methods, we hypothesized that diffusion models have learned richer energy landscapes than are generally recovered and that these landscapes can be accessed via a more sophisticated sampling approach. As improvements in model training lead to more accurate learned landscapes, it will become increasingly valuable to not only access the most dominant configurations from the model but also extract rich information about transition mechanisms and probabilities. We aim to open the door to this type of insight.
+
+# 3 Methods
+
+Given a biomolecular structure diffusion model  $p(x|s)$ , where  $s$  is the input biological system provided to the model (for instance, the amino acid sequence of a protein and its multiple sequence alignment) and  $x \in \mathbb{R}^{n \times 3}$  specifies the predicted atomic coordinates, our objective is to efficiently explore and characterize the distribution  $p$ .
+
+![](images/b7fea5bf99d4751ea91dd0d9c6fd2562337cefc9137e7a98a1df055b5fcaabe0.jpg)  
+B) Overview of ConforMixRMSD workflow
+
+![](images/d26558ddf385c25235d7a16bae8ae8e21e267ef55a5f7bbaaa5095a1b4b4286f.jpg)  
+Figure 1: (A) ConforMix adds conditioning to diffusion-based structure prediction models, enabling deeper and more efficient exploration. (B) ConforMix uses a series of bias potentials to target new states. ConforMixRMSD is an instantiation that biases sampling away from default predictions to sample conformational transitions without requiring any user intervention. Sample reweighting can be applied to recover the ground state distribution.
+
+# 3.1 ConforMix
+
+Diffusion models learn probability distributions, and the standard sampling procedures, which we refer to as default sampling, are designed to draw i.i.d. samples from this distribution. To explore further and more efficiently than is possible with default sampling, we employ conditional sampling, which is widely used with diffusion models to guide sampling toward particular regions of the learned
+
+probability distribution [5, 10]. As we will see, conditional sampling offers clear benefits even without specialized prior knowledge about the protein of interest. While time-dependent classifier guidance and classifier-free guidance are commonly used in other diffusion contexts, these methods are not well suited to our needs because they require either additional training or special architectural choices, which are impractical for our needs.
+
+Instead, to devise a purely inference-time method, we turn to Twisted Diffusion Sampling [37], a form of sequential Monte Carlo (SMC), to guide conditional generation. SMC has advantages over more typical guidance methods, including (a) requiring only a time-independent guidance function, which enables us to easily describe conditions as a function of the fully denoised state of the system, and (b) providing an asymptotic guarantee of sampling the true target conditional distribution in the limit of many particles. Twisted diffusion sampling works in conjunction with an underlying diffusion model that learns to sample a target data distribution  $q(x_0|s)$ . Diffusion models define a forward noisng process, operating over a sequence of  $T$  steps, that constructs an extended distribution  $q(x_{T:0}|s) = q(x_{T}|x_{T_{1}})\dots q(x_{1}|x_{0})q(x_{0}|s)$ . The diffusion model  $p_{\theta}(x_t,t|s)$  can then be trained to approximate the score function  $\nabla_x\log q_t(x|s)$ , which is used to simulate the noisng process in reverse. We emphasize that ConforMix itself does not involve additional model training.
+
+Instead, twisted diffusion operates as a particle filter. The sampling algorithm is instantiated with a series of guidance functions  $y_{j}(x) = \exp (-U_{j}(x))$ , where  $U_{j}$  is a potential. For example, many motions can be described in terms of the distance between two groups of atoms  $A$  and  $B$ , so we might select a potential
+
+$$
+U _ {j} (x) = \frac {\alpha}{2} \left(\left\| \vec {\mu} _ {A} - \vec {\mu} _ {B} \right\| - \lambda_ {j}\right) ^ {2} \tag {1}
+$$
+
+where  $\alpha$  is a strength parameter,  $\vec{\mu}_A,\vec{\mu}_B$  are the centroids of the atom groups, and  $\lambda_{j}\in [\lambda_{\mathrm{min}},\lambda_{\mathrm{max}}]$  is a target distance. The corresponding score function
+
+$$
+\nabla_ {x} \log y (x) = - \alpha \left(\left\| \vec {\mu} _ {A} - \vec {\mu} _ {B} \right\| - \lambda_ {j}\right) \nabla_ {x} \| \vec {\mu} _ {A} - \vec {\mu} _ {B} \| \tag {2}
+$$
+
+is easy to compute. During inference, at each denoising timestep  $t$ , we construct an inexpensive estimate of the fully denoised state  $\hat{x}_0$ , which we plug into  $y_j$  to approximate the potential of the current state. We then compute  $\nabla_{x_t}y_j = (J_{x_t}\hat{x}_0)^T\nabla_{\hat{x}_0}y_j$  and use this signal to guide  $x_t$  towards  $p(x|y_j,s)$ . By simultaneously denoising several such particles  $x_t$  and periodically resampling them to maintain correct sampling, we can obtain approximate samples from the desired conditional distribution. For more details, we refer the reader to Algorithm S1 in the Supplement and to Algorithm 1 in [37], which provides an asymptotic guarantee that the sampling approaches the true conditional distribution as the number of particles becomes large.
+
+In our general framework, ConforMix, twisted diffusion sampling can be used to sample from nearly arbitrary conditional distributions. To select suitable conditioning, we are motivated by two biological use cases: (1) automated exploration of the conformational landscape, where a user provides only an input system (i.e. amino acid sequence, and optionally MSA and/or templates), and (2) user-defined exploration of conformational states, where a user wishes to examine a specific hypothesized motion or state. In the rest of this paper, we focus primarily on the first case, but we note that ConforMix can be customized to scan targeted degrees of freedom.
+
+# 3.2 ConforMixRMSD
+
+Our approach to the automated exploration problem, which we call ConforMixRMSD, is described in Algorithm 1. Here we assume that a user wishes to perform undirected exploration of the protein conformational distribution learned by the underlying model. The idea of ConforMixRMSD is to simply generate the most probable samples that are sufficiently different from the default prediction  $x_{d}$ . AlphaFold 3-family generative models often have probability distributions  $p(x|s)$  that are highly concentrated around one conformation. After generating an initial conformation  $x_{d}$  with default sampling, we construct a series of guidance potentials that steer the probability distribution away from  $x_{d}$  using a root mean square deviation (RMSD) metric:  $U_{j}(x) = \frac{\alpha}{2}\mathrm{RMSD}(x,x^{ref})^{2}$ , where  $\mathrm{RMSD} = \min_{R,\mathbf{t}}\frac{1}{N_{atoms}}\sum_{a = 1}^{N_{atoms}}\| x_{a} - (Rx_{a}^{ref} + \mathbf{t})\|^{2}$ , and  $R\in \mathrm{SO}(3)$  and  $\mathbf{t}\in \mathbb{R}^3$  respectively rotate and translate  $x$  to best align to  $x^{ref}$ . RMSD is differentiable and computable in closed form via the Kabsch algorithm. The fastest motions of a protein, which are easier to predict are often fluctuations of disordered loop regions. To avoid exclusively sampling these motions, we apply a
+
+rigid-element mask: we only compute RMSD on amino acids that are part of secondary structure elements,  $\alpha$  helices or  $\beta$  sheets. All amino acids still remain free to move during sampling.
+
+Because the realistic range of motion varies per protein and is not known a priori, we seek to discover the flexibility of each protein by generating structure predictions spanning the range 0 to  $20\AA$  RMSD to  $x_{d}$ . After samples are generated, we filter those that are physically implausible. Specifically, we reject samples where any 10-residue sliding window has an average pLDDT value of more than  $20\%$  below that of the default prediction, as well as structures with clashes. The sliding window approach detects local non-physical perturbations from sampling beyond the protein's flexibility range; indeed, the amount of filtering increases as we bias to larger RMSD to  $x_{d}$  (Figure S6).
+
+Algorithm 1 ConforMixRMSD for exploration of conformational landscapes  
+Input: Biomolecular structure prediction model  $p$  input system  $s$  target RMSD values  $\mathcal{R}$  constraint strength  $\alpha$  number of samples per RMSD  $N_{\mathrm{samples}}$    
+Output: Samples  $\{x_{R,i}\}$  for all  $R$  and i 1:  $x_{d}\gets p(x\mid s)$  # Predict a structure using default sampling 2: mask  $\leftarrow$  RIGIDELEMENTS  $(x_{d})$  # Identify atoms within secondary structure elements 3:  $g_{r}(y\mid x)\gets \exp \Bigl (-\alpha \bigl (\mathrm{RMSD}(x,x_{d};mask) - r\bigr)^{2}\Bigr)$  # Define conditioning potential 4: for  $r\in \mathcal{R}$  do 5: for  $i = 1,\dots ,N_{\mathrm{samples}}$  do 6:  $x_{r,i}\gets \mathrm{CONFORMIX}(x;g_r,s)$  7: end for 8: end for 9: return  $\{x_{r,i}\}_{i = 1}^{N_{\mathrm{samples}}}$  for all  $r\in \mathcal{R}$
+
+# 3.3 Sample reweighting and free energy estimation
+
+While some scientific problems require only qualitative information about accessible conformational states and flexibility, others require estimates of statistical quantities such as free energy differences. For this reason, we wish to recover the free energy landscape of structure prediction models.
+
+Samples generated from ConforMix are sampled from a series of conditional probability distributions  $p(x|y_j,s)$ . To form a clear picture of this latent distribution, we wish to reweight our mixture of conditional samples to the unconditional distribution. Combining samples from multiple conditional distributions requires estimating the partition functions  $p(y_j)$ . An unbiased estimator of this quantity is available directly from twisted diffusion sampling, but in practice these estimates are noisy, particularly for rare samples, and therefore convergence is slow.
+
+To solve the challenge of estimating the partition functions  $p(y_{j})$ , we turn to a statistical tool that to the best of our knowledge has not previously been applied in the context of diffusion models. The multistate Bennett acceptance ratio (MBAR) free energy estimation algorithm combines sample information from multiple conditional distributions to jointly estimate the probabilities  $\{p(y_{j})\}$ , up to a (negligible) constant factor [28, 29]. In practice, this means that given sufficiently accurate samples from a series of overlapping distributions, such as the conditional distributions of ConforMix, we can reconstruct the unbiased model landscape in those areas. We discuss free energy estimation further in Section 4.4. The mathematical details of MBAR are deferred to the appendix.
+
+# 4 Results
+
+We evaluate ConforMix in multiple settings to explore the following questions:
+
+Q1. How do conformational samples generated via inference-time guidance with fixed inputs (sequence, MSA, etc.) compare in quality and coverage to samples generated via methods that require altering the input MSA?  
+Q2. Can relevant degrees of freedom be extracted from the energy landscape of a model trained for single structure prediction?  
+Q3. As future generative models improve accuracy in predicting the true Boltzmann distribution, can enhanced sampling algorithms like ConforMix provide rapid quantitative information?
+
+# 4.1 ConforMixRMSD recovers experimentally observed conformations
+
+We first implement ConforMix in Boltz-1 [36], an open-source diffusion-based structure prediction model similar to AlphaFold 3. Importantly, Boltz was trained on the Protein Data Bank (PDB), and not with any other dynamics information. While the default Boltz sampling can generate multiple samples, they are generally highly concentrated and are more usefully thought of as multiple approximations of the same static structure than as samples from the true conformational distribution of the protein. To address Q1, we collect samples using ConforMixRMSD with Boltz (ConforMixRMSD-Boltz) as the underlying structure prediction model.
+
+We examine performance of ConforMixRMSD-Boltz on proteins that exhibit different types of conformational changes: (a) domain motion, (b) transporter cycling, (c) cryptic pocket formation, and (d) fold switching. The set of 38 proteins that exhibit domain motions are the combination of proteins curated by Lewis et al. [17] and in OC23 by Kalakoti et al. [13]. The set of 15 membrane transporters that exhibit inward-open and outward-open conformations were curated in TP16 by Kalaloti et al. [13]. The set of 31 proteins that exhibit cryptic pocket formation were curated by Lewis et al. [17]. The set of 15 fold switchers proteins are a subset of those curated by Porter et al. [26].
+
+On these sets, Boltz alone typically recovers only a single conformation. Sampling from Boltz with ConforMixRMSD, however, reveals that the hidden energy landscape contains degrees of freedom that transition to alternative conformations seen in experiment and deposited in the PDB (Figure 2). As comparators, we implement three MSA-modification protocols in Boltz: AFCluster [35], AFSample2 [13], and CF-random [15] (see details in Supplement). We also compare to default Boltz sampling with 1,000 samples. On the domain motion, transport, and cryptic pocket sets, ConforMixRMSD recovers more alternate conformations than all baselines—as measured by RMSD and TM-scores to PDB structures (Tables 1, S3 and Figures 2, S1, S4, S7-S10)—demonstrating its power explore conformational space in a novel manner. MSA-based methods show stronger performance on fold switchers, suggesting such rearrangements are better captured discrete input modulation, whereas ConforMixRMSD is better suited to continuous (e.g. domain, transporter, pocket) transitions.
+
+![](images/ce1151f63c8c93f2a01ce93419805b3ef5ae5523cf8811c4ff4e04f3c9a480b7.jpg)  
+eeereee  
+eannnnnne
+
+![](images/8d15b4bbbc1a4784ccc28c2ca6ba4b5971e3c04f2e1c3016880d1b16f4fd14db.jpg)  
+Figure 2: ConforMixRMSD uncovers conformational states that are not sampled by default. Conformational sampling of (A) a domain motion protein and (B) a membrane transporter. For each system, Top Left: density of sampling relative to reference experimental structures. Bottom Left: projection of the ConforMixRMSD sampled structures (orange) onto the first two principal components computed from their internal atomic distances. Experimentally determined reference structures (grey) are projected onto the same space. Right: reference structures and the closest structure generated by each sampling approach (lowest RMSD).
+
+![](images/037e60d14804eed19aad00f8a800ffcab1732b82e7ceeb09c301848ca4898513.jpg)
+
+![](images/80e839f2dc45b477efc15ad674e3a99532095fbdd247f993fc04120b67f04db1.jpg)
+
+![](images/99c1e91b1e034f3ddf2f3fe2dcb790635c9e0f7df41eb9615f3724303640d1ac.jpg)
+
+![](images/a0712bedbbeb401b3b1cb791d704c6139ef538e70d99ed64f0b4ef8bebbf3177.jpg)  
+Pnneeppee
+
+![](images/61cc47a8ad8fcbc15745c06697c6791480e0a518c0385fe6606ec140659ba363.jpg)
+
+![](images/ad3da33b7189785488a6dc928edc227dc88ab525b7d8f9b3d048dda929549163.jpg)
+
+![](images/ced84b9aa07cd7e973d7297c7360ffe9576ac7247209c70bda7c3060837c8452.jpg)  
+rerrreee eeeeee
+
+![](images/d78edb85555d84e4460b7c980595412f7776f81c9bb3bfea9571293204c53ec1.jpg)  
+ConforrmRMSD
+
+![](images/7d86c35716a65cd1e3aa68adbbe27f01b0e046308013867de05cf8243822fee3.jpg)  
+nearest sampled structure to: outward open inward open conformation conformation
+
+![](images/5d3e32fd77deae88ff8e6f71925f74f93ff79f4ba7b9416155b77eccb61bdc4b.jpg)
+
+![](images/4529e8f8d4492ef2fc4e9d2d9097c4fdbedc2b169b0d79bec9f427dd50f896e8.jpg)
+
+![](images/6a84db933411d5abcca9f8a1d43d36f180333418aaf66be221a5e328ed6211bd.jpg)
+
+We also find that recovery of experimentally observed states from ConforMixRMSD-Boltz sampling is competitive with BioEmu [17] (Figure S2), a model trained to generate conformational ensembles,
+
+Table 1: Coverage of experimentally-determined reference conformations by method and dataset  
+
+<table><tr><td colspan="2"></td><td>Domain motion (n=38)</td><td>Membrane transporters (n=15)</td><td>Cryptic pockets (n=31)</td><td>Fold switching (n=15)</td></tr><tr><td rowspan="5">Worst-matched reference conformation (harder task)</td><td>Default Boltz sampling</td><td>0.33 (±0.14)</td><td>0.13 (±0.17)</td><td>0.15 (±0.12)</td><td>0.13 (±0.17)</td></tr><tr><td>AFCluster-Boltz</td><td>0.46 (±0.15)</td><td>0.19 (±0.19)</td><td>0.35 (±0.18)</td><td>0.27 (±0.23)</td></tr><tr><td>CF-random-Boltz</td><td>0.51 (±0.17)</td><td>0.20 (±0.20)</td><td>0.39 (±0.16)</td><td>0.20 (±0.20)</td></tr><tr><td>AFsample2-Boltz</td><td>0.44 (±0.15)</td><td>0.20 (±0.20)</td><td>0.33 (±0.17)</td><td>0.10 (±0.15)</td></tr><tr><td>ConforMixRMSD-Boltz</td><td>0.69 (±0.15)</td><td>0.33 (±0.23)</td><td>0.45 (±0.18)</td><td>0.13 (±0.17)</td></tr><tr><td rowspan="5">Best-matched reference conformation (easier task)</td><td>Default Boltz Sampling</td><td>0.94 (±0.07)</td><td>0.59 (±0.23)</td><td>0.94 (±0.08)</td><td>0.60 (±0.27)</td></tr><tr><td>AFCluster-Boltz</td><td>0.92 (±0.09)</td><td>0.60 (±0.27)</td><td>0.97 (±0.05)</td><td>0.60 (±0.27)</td></tr><tr><td>CF-random-Boltz</td><td>0.87 (±0.12)</td><td>0.60 (±0.23)</td><td>0.97 (±0.05)</td><td>0.60 (±0.27)</td></tr><tr><td>AFsample2-Boltz</td><td>0.90 (±0.08)</td><td>0.67 (±0.23)</td><td>0.93 (±0.08)</td><td>0.70 (±0.30)</td></tr><tr><td>ConforMixRMSD-Boltz</td><td>0.97 (±0.04)</td><td>0.79 (±0.19)</td><td>0.94 (±0.08)</td><td>0.67 (±0.23)</td></tr></table>
+
+Coverage at  $\mathrm{X\%}$  measures the fraction of proteins with samples matching a reference conformation within  $\mathrm{X\%}$  of the RMSD between reference structures. Displayed are values of coverage evaluated at  $50\%$  of the reference-to-reference RMSD. We evaluate coverage separately for best-matched and alternate (worst-matched) states. Error bars are  $95\%$  confidence intervals over 1,000 bootstraps.
+
+although we note that many of the evaluated proteins were held out of training for BioEmu but were likely present in the Boltz training set.
+
+# 4.1.1 Runtime performance
+
+Due to the additional cost of the guidance and resampling steps in Twisted Diffusion Sampling, sampling with ConforMix takes  $\sim 3\mathrm{x}$  the wall clock time compared to default sampling. However, individual ConforMix samples are much more informative than default samples. ConforMixRMSD can scan large motions of a protein within the space of a few dozen samples, which takes just a few minutes for a moderately sized protein. For many such proteins, default sampling would not reveal such motions even after thousands of samples. Timing statistics are provided in Table S1.
+
+# 4.2 ConforMixRMSD samples conformational transitions of domain motion proteins
+
+It is important for any structural sampling method to distinguish meaningful conformational fluctuations from random perturbations. Consider a protein that exhibits an open-to-closed domain motion. Consider two sampling approaches: (A) generates structures that map the domain transition from open-to-closed—ideally with lower energy wells at observed dominant conformations—while (B) generates structure variation in many directions. While both methods may sample the reference open and closed states of this protein at the same RMSD, approach A is clearly more useful as it identifies the biologically significant motions. Approach B is difficult to use in practice, as the reference conformations are not typically known and thus identifying relevant conformations from the collection of heterogeneous variation is not tractable.
+
+To evaluate whether sampling methods explore biologically relevant conformational landscapes versus generating diffuse structural noise, we analyze the variance of generated ensembles using principal component analysis on pairwise structural distances. We project experimental PDB reference structures onto the principal components to assess whether the major axes of structural variation in the samples correspond to observed conformational transitions. Focusing on the domain motion set, where proteins are known to exhibit a predominant motion between states, we find that ConforMixRMSD tends to produce more organized sampling that highlights the transitions between known conformations, while MSA-based methods typically yield less-interpretable diffuse sampling (Figures 3, S3, S11).
+
+As discussed in the previous section, Boltz sampling generates limited structural diversity, typically clustering near one experimental state. However, we find that the limited variance that does exist between samples often corresponds to the direction of motion needed to reach the alternative conformational state—but lacks the magnitude to actually traverse the transition. This suggests the model recognizes the relevant degrees of freedom but undersamples them. MSA-based approaches present the opposite problem: they generate abundant structural diversity and can more frequently recover both reference structures, but sampling often doesn't map to relevant motion. MSA-based methods show lower values for both the degree of sample variance explained by top principal components and
+
+![](images/2d5f09ff0d2b87567c1c28f4b1d4ac4c1844df3f8134ae1fb075e6e6cdafd9c0.jpg)  
+A) Dipeptide binding protein dppA
+
+![](images/4118efff7de107d84968050157f16165da87e0e797c5bbfbcd416bc5a31cbcf2.jpg)
+
+![](images/6f7e63066dbfcf41da6a09bf632b49cfee2419b9048c5876a8ac50e5804a9363.jpg)
+
+![](images/4562f5439224c97b274bb0a6bba8ada743788c88508780712d3d4a7653f6c824.jpg)
+
+![](images/ed891fe325459209a3a91c58dfc11cf23accb5cbda99273a2e5ef17f98203673.jpg)  
+B) Set of domain motion proteins
+
+![](images/72ef2a6aa530dd6679bf06b4fa064a11160eabe4e913bcbb756d4fa3a247d6f3.jpg)
+
+![](images/c81620fb397da69326284d1e514c44a67f954b7d689aa4850226e63717aae894.jpg)  
+Figure 3: ConforMixRMSD samples domain motion, revealing transitions consistent with experiment while avoiding noisy paths. (A) Principal component analysis of pairwise  $C\alpha$  distances of samples generated for a domain motion protein, dppA. Default sampling extends toward but does not reach the open conformation, while ConforMixRMSD traces an opening/closing path. While AFCluster generates open and closed states, its sampling of many other large fluctuations makes it harder to identify the relevant motion. (B) Analysis of variance of samples generated by default sampling, ConforMixRMSD, and AFCluster, all used with Boltz. Each point describes results for one protein. Default sampling and ConforMixRMSD tend to exhibit concentrated variance between samples that align with the direction of domain motion between reference structures. Note that this metric captures alignment of the sampling direction with experimental conformational differences, but not the extent of sampling along that direction—a method may align well without sampling both experimental conformations. AFCluster often produces samples whose structural differences do not match the direction of domain motion between references, or lack dominant directions of variance—indicating off-path sampling. Black-outlined points denote dppA.
+
+![](images/e110b99cc3fed79abd94a729984078d23899a140964a09510414077bf678f696.jpg)
+
+the principal component alignment with experimental transitions. These methods generate predictions diffusely across structure space, sometimes matching reference states, but without concentrating sampling along pathways likely to exist between observed states.
+
+ConforMixRMSD, by contrast, often generates substantial conformational diversity while concentrating that diversity along experimentally relevant transitions. For the dipeptide binding protein dppA (Figure 3A), for instance,  $96\%$  of the variance in ConforMixRMSD samples is explained by a single principal component that directly corresponds to the domain hinging motion between experimental states. More generally for domain motion proteins (Figure 3B), the top two principal components of each ConforMixRMSD ensemble generally explain both a high fraction of the sample variance (indicating the method generates focused diversity) and a high fraction of the variance between experimental reference structures (indicating this diversity aligns with known conformational changes). When this is the case, conformational ensembles are readily interpretable: their major modes of structural variation correspond to biologically meaningful motions. PCA plots for all proteins in the domain motion set are provided in Figure S11.
+
+As an alternate metric, the "fill ratio" introduced by [13] measures how continuously samples cover the conformational path between reference states based on TM-scores. ConforMixRMSD sampling results in higher fill ratios than the other evaluated methods (Figure S5). We also compute geometric quality metrics on ConforMix-generated samples and observe slightly elevated rates of outliers compared to default sampling, but most samples appear reasonable (Table S2).
+
+![](images/adfde66568376a32b7e597999969da5914dcd6b3ebd1a0726920d21b675a17f1.jpg)  
+A) SemiSWEET sugar transporter  
+B) Hammerhead ribozyme
+
+![](images/341d5d2ab8cc96a3b8b6d96136bcb4761356321fd53b33f344eb7bd79114d935.jpg)
+
+![](images/e2b063b69b98351562a324f59d58e0e6982b6273754a8317cabfbe25585ba992.jpg)
+
+![](images/2e9542bf95e1ce424ed97712b7d80f8e670f6f45c0ffe4721dd19d38f4d0cf8c.jpg)
+
+![](images/58e26e1d61f9100678057108ec3b1b7480a275c0a3789fdb38d00e9765020ace.jpg)
+
+![](images/1b8441f4bf4d07e9756892942b2d1bea06793b60ba07b34c6b1b58d19798219d.jpg)  
+Figure 4: Exploration of biological macromolecules of interest. (A) ConforMixRMSD-Boltz recovers all three experimentally determined conformations of the SemiSWEET transporter. Default Boltz sampling recovers only the occluded state. PCA demonstrates how the samples capture the major motions as the transporter opens to the inward (intracellular) or outward (extracellular) sides. (B) Preliminary application of ConforMixRMSD-Boltz to RNA structure shows it can recapitulate MD-observed transitions.
+
+# 4.3 ConforMixRMSD recovers heterogeneous states in multi-chain and RNA systems
+
+We further evaluate ConforMixRMSD on select biomolecular systems known to exhibit interesting heterogeneity. First, we consider a multi-chain transporter protein, the SemiSWEET sugar transporter, which cycles through three distinct states to transport substrate across a membrane (Figure 4A) [14]. The inward- and outward-open states enable substrate uptake and release, while an occluded state that is closed to both sides allows the substrate to pass through the transporter. Thus far, ML-based conformational sampling approaches have primarily been effective on single-chain proteins. Boltz by default predicts only the occluded state for this system. With ConforMixRMSD sampling, however, Boltz recovers all three states. PCA analysis and visual inspection of sampled structures reveal intermediate states between each known conformation, forming a conformational "trajectory".
+
+While most of our testing focuses on proteins, ConforMix can be applied to any biomolecular system as long as it is supported by the underlying model. We demonstrate sampling of the hammerhead ribozyme (Figure 4B), a small catalytic RNA molecule that adopts multiple conformations. These conformations have not been deposited in the PDB, so instead we take molecular dynamics simulations from [22] as reference data (the grey structures in Figure 4B represent MD frames). ConforMixRMSD-Boltz visually recapitulates key aspects of the unzipping transition observed in the MD simulations. The sampled structures qualitatively capture the conformational trajectory, suggesting that ConforMix can extract meaningful conformational heterogeneity from RNA systems, though comprehensive evaluation across RNA systems would be needed to fully characterize performance.
+
+We conclude that (1) Boltz's learned conformational landscape is rich enough to be of practical utility on many protein systems of interest, and (2) ConforMixRMSD enables inference-time access to this landscape without the disadvantages associated with MSA-modification methods.
+
+# 4.4 Rapid free energy estimation with BioEmu
+
+To illustrate the ability of ConforMix to improve sampling in a variety of situations, we implement it in BioEmu [17] as well. Unlike Boltz, BioEmu produces broad distributions of conformations with default sampling. However, default sampling is an inefficient way to observe rare events or estimate quantities such as free energy differences, so we assess the ability of ConforMix to rapidly estimate free energies from BioEmu.
+
+![](images/551a41fe130ef29b8b71e5b1faf83cb5b3ca048885663c2d041ca16690b034f9.jpg)  
+Figure 5: ConforMix sampling in BioEmu enables faster convergence of free energy estimates than default sampling. Using a series of guidance potentials based on RMSD to the native state enables systematic collection of samples that, in turn, enable more rapid free energy estimation. The free energy difference estimated is between the lowest free energy RMSD value (3 Å to the reference state) and a more extended conformation at  $7.5\mathring{\mathrm{A}}$ . 90% confidence interval from 50 bootstraps is shown.
+
+While ConforMix can be used to study states that are not seen at all in unbiased sampling, for purposes of comparison we consider a set of transitions that can also be observed without enhanced sampling. Specifically, we evaluate ConforMix on local unfolding proteins from BioEmu, which contain structural motifs that undergo transitions between ordered and disordered states. Because BioEmu does not produce a single consensus structure from default sampling, here we instead use RMSD to the folded reference structure as the basis for defining a series of guidance potentials (see Supplement). We compare the rate of convergence of i.i.d. probabilities from ConforMix estimates with MBAR reweighting to unbiased sampling in BioEmu (Figure 5). Specifically, we estimate the free energy difference between the lowest free energy RMSD value and a specific less folded state. We observe that ConforMix produces estimates close to the true unbiased estimates and converges significantly more quickly.
+
+# 5 Discussion
+
+Predicting biomolecular dynamics represents a major frontier for machine learning. By enhancing inference-time sampling in diffusion models, ConforMix rapidly reveals hidden states and continuous motions without prior knowledge from the user. It also enables characterization of the underlying free energy landscape of the models, which we anticipate will assist in future training and evaluation. The broader impact of tools like this is initially in an improved understanding of basic biology and then in applications to medicine and other areas.
+
+The primary limitation of ConforMix, like other enhanced sampling methods, is that it depends on the robustness and utility of the underlying energy landscape it samples. While Boltz has learned a useful landscape for many proteins, its default sampling typically does not adequately explore the landscape. ConforMix sampling enables us to identify major missing states: known experimental conformations that do not exist with realistic probabilities in the Boltz probability distribution. Other limitations include potential systematic errors due to the inexact sampling of twisted diffusion in the non-asymptotic regime and the statistical uncertainty associated with MBAR. We note that ConforMix has more general uses beyond the specific instantiation of ConforMixRMSD. For instance, users can supply more informative biasing potentials, perhaps based on experimental evidence. If desired, ConforMix can be used in combination with input-modification approaches such as MSA subsampling, although we leave that exploration to future work.
+
+ConforMix is fundamentally flexible. It is not restricted to proteins, and it can handle complexes of multiple molecules. It is able to operate in both all-atom and backbone-only diffusion methods. It is orthogonal to model training. We anticipate that as models mature and energy landscapes approximate the true Boltzmann distribution more closely, inference-time sampling methods such as ConforMix will only become more valuable, both for direct study of protein systems and for model development.
+
+The code for this project is available at github.com/drorlab/conformix.
+
+# 6 Acknowledgments
+
+We thank the anonymous reviewers for their valuable suggestions and feedback on the manuscript. We thank Karl Krauth, EJ Fine, Ayush Pandit, Shawn Costello, Masha Karelina, Brian Trippe, Hunter Nisonoff, and Bowen Jing for scientific discussions.
+
+This work was supported by NIH grant R35GM158122. D.D.R. acknowledges support from the National Science Foundation Graduate Research Fellowship Program. J.K. acknowledges support from the Knight-Hennessy Scholars program and the Natural Sciences and Engineering Research Council of Canada Graduate Research Scholarship.
+
+# References
+
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+
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+# 3. Theory assumptions and proofs
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+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in Supplemental material?
+
+Answer: [Yes]
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+Justification: Code to run ConforMixRMSD-Boltz, as well as additional analysis code, will be provided on our Github. The full protein sample datasets we generated are very large and are impractical to share online, but we will attempt to make them available upon request.
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+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: Each tool described in the paper is cited and credited. Boltz is released under the MIT License, https://github.com/jwohlwend/boltz. BioEmu is released under the MIT license, https://github.com/microsoft/bioemu/blob/main/License. Some free energy computations were performed with the pymbar module, which is released under the MIT license, https://github.com/choderalab/pymbar. The paper corresponding to each of these is cited in the main text.
+
+# Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
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+
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+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [Yes]
+
+Justification: The primary new asset is code. The proteins used are enumerated in the Supplement.
+
+Guidelines:
+
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+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
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+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
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+
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+
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+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
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+
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+
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+
+# 16. Declaration of LLM usage
+
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+
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+
+Guidelines:
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+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
\ No newline at end of file
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+# Unmasking Puppeteers: Leveraging Biometric Leakage to Disarm Impersonation in AI-based Videoconferencing
+
+Danial Samadi Vahdati $^{1}$ , Tai Duc Nguyen $^{1}$ , Koki Nagano $^{2}$ , David Luebke $^{2}$ , Orazio Gallo $^{2}$ , Ekta Prashnani $^{2}$  Matthew Stamm $^{2}$
+
+$^{1}$ Drexel University,  $^{2}$ NVIDIA
+
+# Abstract
+
+AI-based talking-head videoconferencing systems reduce bandwidth by sending a compact pose-expression latent and re-synthesizing RGB at the receiver—but this latent can be "puppeteered," letting an attacker hijack a victim's likeness in real time. Because every frame is synthetic, deepfake and synthetic video detectors fail outright. To address this security problem, we exploit a key observation: the pose expression latent inherently contain biometric information of the driving identity. Therefore, we introduce the first biometric leakage defense without ever looking at the reconstructed RGB video: a pose-conditioned, large-margin contrastive encoder that isolates persistent identity cues inside the transmitted latent while cancelling transient pose and expression. A simple cosine test on this disentangled embedding flags illicit identity swaps as the video is rendered. Our experiments on multiple talking-head generation models show that our method consistently outperforms existing puppeteering defenses, operates in real-time, and shows strong generalization to out-of-distribution scenarios. Our code and trained models are available at https://github.com/MISLresearch/Unmasking-Puppeteers-Neurips25.
+
+# 1 Introduction
+
+Advances in generative AI have enabled the creation of hyper-realistic synthetic videos. This has led to the development of many new technologies, including both avatar-based communication systems [1-3] and AI-based videoconferencing systems [4-7]. AI-based talking head videoconferencing systems are receiving increasing attention due to their significant bandwidth savings [8]. Instead of continuously encoding and transmitting each video frame of a speaker, these systems only transmit embeddings that capture a speaker's pose and facial expression. Then, a generative AI system at the receiver's end uses these embeddings, in conjunction with an initial representation of the speaker, to create an accurately reconstructed video.
+
+Unfortunately, AI-based talking head videoconferencing systems [5, 4, 8, 7, 9] are vulnerable to a new form of information attack known as "puppeteering" [10]. In this attack, a malicious user at the sender side transmits an unauthorized representation of a different target speaker when a video call is initiated. As a result, the identity in the video that the receiver constructs is different than the identity of the person who actually controls the video [10, 11].
+
+While many forensic approaches, such as deepfake [12-17] and synthetic video detectors [18-21], aim to expose unauthorized AI-generated media, they operate by identifying evidence that a video has been synthetically generated. However, in AI-based talking-head videoconferencing, every video is AI-generated, rendering existing media forensic approaches unable to detect puppeteering.
+
+![](images/9b4d1a22481f8bebebefc7767f868517759fcea2c2672e075e3f82a2ff5c1237.jpg)  
+Figure 1: AI-based talking-head generators transmit only a compact pose-and-expression embedding for low-bandwidth videoconferencing, but remain vulnerable to puppeteering attacks that swap in a different identity for live impersonation. Our defense capitalizes on biometric signals inadvertently leaked in these embeddings to reveal mismatches between the driving speaker and the reconstructed identity in real time.
+
+Puppeteering attacks pose a fundamentally different threat from those typically addressed by media forensics. Instead of asking "Is this video real or AI-generated?" one key question must be "Is this AI-generated video being driven by an authorized identity, or has someone hijacked it?" As talking-head videoconferencing technology continues to advance, it is essential to mitigate puppeteering attacks before these systems are widely deployed. Addressing these threats proactively will help ensure secure and trustworthy communication over these systems.
+
+Few defenses address puppeteering. Prashnani et al. introduced Avatar Fingerprinting [11], which works by enrolling each user's characteristic facial motion patterns into the detector's training data. Although this could help, it requires significant user-specific data to build reliable motion signatures. In contrast, Vahdati et al. avoid enrollment by decoding and re-encoding the received video for comparison [10]; the round-trip is compute-heavy and adds distortions that hurt accuracy.
+
+In this paper we introduce a real-time, enrollment-free defense that detects puppeteering attacks entirely in the latent domain. Our key insight is that the pose-and-expression embeddings already transmitted by modern talking-head systems leak subtle, but reproducible, biometric signatures [22, 23]. We learn a compact Enhanced Biometric-Leakage (EBL) space in which identity cues are amplified while pose and expression variance is actively suppressed. A pose-conditioned contrastive loss drives this separation, and a lightweight temporal LSTM aggregates evidence to yield stable, millisecond-level decisions. By comparing the sender's latent identity with the target identity rendered at the receiver, our system flags any mismatch without decoding RGB frames, tracking landmarks, or collecting per-user motion profiles. Extensive experiments across many generators and datasets show that our method achieves state-of-the-art detection performance while working in real-time.
+
+In summary, our contributions are as follows.
+
+- We present the first method that operates solely on data already available at the receiver, requiring no user enrollment or additional sensors.  
+- We formalize and harness identity leakage inherent in pose-expression embeddings, turning a previously ignored vulnerability into a defensive signal.  
+- We develop a novel loss and training protocol to learn a low-dimensional EBL representation that maximizes identity separability while nullifying pose and expression.
+
+![](images/6115647e65c70aa73dc6dc69ccdd321ffdf75d933178b506d2a44f94b85a90b7.jpg)  
+Figure 2: Illustration of three datasets (NVIDIA-VC [11], RAVDESS [54], CREMA-D [55]) shown across three columns each. Row 2 indicates the type of frame displayed in Row 3: Reference, Self-Reenacted, or Cross-Reenacted.
+
+![](images/cc81da09f61bf53d3d112b274c09c716e9d625e3da6d20a53ab71887867176e0.jpg)
+
+![](images/1b23d706d1247888136c83674df24b21cd08cfe32e144f5593e254191266cc7c.jpg)
+
+- Though extensive experiments across fifteen generator/dataset combinations, our approach achieves state-of-the-art puppeteering detection in real time, demonstrating practical viability for deployment in bandwidth-constrained videoconferencing systems.
+
+# 2 Background and Related Work
+
+Talking-Head Video Systems. Talking-head systems synthesize facial motion and speech in real time by transmitting low-dimensional embeddings of pose and expression rather than full video frames [24-27]. A generator at the receiver reconstructs a realistic face. Methods range from compressed embeddings to landmark-based updates per frame [8, 28-33], reducing bandwidth while maintaining high visual fidelity [34-40]. Systems like NVIDIA's Vid2Vid Cameo [8] and Google's Project Starline [41] demonstrate real-time, 3D teleconferencing.
+
+Forensics for Synthetic Media. Synthetic media forensics traditionally aims to distinguish real from AI-generated content. Deepfake detectors [12-17, 42] leverage semantic cues [43-46], lighting inconsistencies [47-50], and facial texture artifacts [51-53], while synthetic video detectors [18-21] focus on identifying AI-generated traces. However, in low-bandwidth talking-head systems, every video is synthetic [10, 11], rendering these detectors ineffective - they cannot determine whether the video represents the correct identity or has been hijacked.
+
+Puppeteering Attacks and Existing Defenses. Puppeteering attacks replace the target's appearance with the adversary's pose-and-expression vectors at call initiation, causing the receiver to synthesize a video of the wrong identity. Avatar Fingerprinting [11] mitigates this threat by first enrolling each user's facial-motion signatures; at inference it flags a session whenever the incoming motion deviates from the stored profile. Vahdati et al. [10] avoid enrollment by re-encoding the synthesized video and comparing embeddings, but their pipeline depends on facial-landmark estimation, which degrades under large head rotations and lighting artifacts introduced by video reconstruction.
+
+In contrast, our method exploits biometric leakage already present in the pose-and-expression latents, removing both the enrollment phase and any reliance on facial landmarks. A contrastive objective disentangles identity from pose, yielding a compact, latency-friendly representation that generalizes to unseen generators and real-world conferencing conditions.
+
+# 3 Problem Formulation
+
+# 3.1 Talking Head Video Systems
+
+AI-enabled talking-head videoconferencing systems have been proposed to minimize transmitted data by encoding and transmitting only pose and expression information from a speaker rather than an entire video frame [8, 56-62]. As this technology rapidly matures, our goal here is to develop security measures that improve the trustworthiness of these systems when they are deployed.
+
+A videoconferencing call starts with each participant  $k$  sending a neutral reference portrait  $R^k$  to the receiver. Typically,  $R^k$  is an image of the speaker in a neutral pose. As the speaker continues to speak, each new video frame  $V_t^k$  capturing this speaker at the sender side is processed by an embedding function  $f$  to produce an embedding  $z_t^k = f(V_t^k)$  that encodes the speaker's instantaneous pose and expression at time  $t$ . This embedding is transmitted to the receiver, as is shown in Fig. 1.
+
+At the receiver side, a generator  $g$  is used to produce a video of the speaker on the basis of the received embedding  $z_{t}^{k}$  and the reference representation  $R^{k}$  such that
+
+$$
+\hat {V} _ {t} ^ {k \rightarrow k} = g \left(z _ {t} ^ {k}, R ^ {k}\right). \tag {1}
+$$
+
+Here, the superscript  $\hat{V}_t^{k\rightarrow k}$  denotes that speaker  $k$  both drives and appears in the rendered video. We refer to the sender's identity as the "driving identity" and the reference portrait sent to the receiver as the "target identity" [11]. While such a talking-head video-conferencing system promises to be bandwidth-efficient, it exposes a vulnerability to puppeteering attacks when a malicious sender impersonates another individual.
+
+# 3.2 Puppeteering Attacks
+
+A puppeteering attack exploits the inherent trust in the reference representation transmitted to the receiver when a videoconferencing call begins. Here, an adversary, speaker  $\ell$ , obtains a target speaker's representation  $R^k$  and substitutes it for their own without authorization at the start of the call [10]. As the call proceeds, the adversary's own video  $V_t^\ell$  is used to derive the pose and expression vectors  $z_t^\ell$  that are transmitted to the receiver. The receiver then uses the generator  $g$  alongside the unauthorized reference  $R^k$  and  $z_t^\ell$  to reconstruct the video. As a result, the receiver is presented with a realistic-looking video of speaker  $k$ , controlled in real time by the speaker  $\ell$  such that
+
+$$
+\hat {V} _ {t} ^ {k \rightarrow \ell} = g \left(f \left(V _ {t} ^ {\ell}\right), R ^ {k}\right) = g \left(z _ {t} ^ {\ell}, R ^ {k}\right), \tag {2}
+$$
+
+where  $\hat{V}_t^{k\to \ell}$  is an unauthorized impersonation of speaker  $\ell$  driven by speaker  $k$ . To address this threat, one must come up with mechanisms to detect whether the receiver's generated video is driven by the same authorized identity, rather than an unauthorized impersonator.
+
+# 3.3 Why Real-vs-Synthetic Detectors Fail
+
+Deepfake and synthetic-video detectors flag frames whose pixel statistics reveal AI generation, implicitly assuming that "real" camera footage is the baseline. In bandwidth-efficient talking-head conferencing, however, every frame - even those from honest participants - is produced by a generator; "synthetic" is the norm, not the anomaly. The security question therefore shifts from "Was this video AI-generated?" to "Does the rendered face correspond to the person actually driving the latents?" Pixel-level detectors cannot answer that mapping, leaving puppeteering attacks undetectable.
+
+# 4 Proposed Approach
+
+In this paper, we present a real-time solution for detecting puppeteering attacks in talking-head videoconferencing, uniquely operating entirely in the latent domain without access to reconstructed RGB frames. To do this, we re-encode each pose-and-expression latent embedding into a compact Enhanced Biometric Leakage (EBL) embedding space that captures who is speaking while discarding how they move. A pose-conditioned contrastive loss learns this space by pulling together identity-consistent pairs and pushing apart identity-mismatched, pose-matched pairs. At run time, we simply compare each live EBL vector to the EBL embedding of the reference portrait sent at call setup; a sharp drop in similarity reveals that the driving and target identities have diverged, signalling puppeteering. For additional robustness and stable authentication, during training, we discard unreliable extreme-pose frames, and leverage an LSTM to fuse successive EBL scores to output stable, low-latency decisions. These innovations yield an algorithm that authenticates speakers in real-time, requires no enrollment or facial-landmark preprocessing. The following sections detail our approach and its implementation.
+
+# 4.1 Biometric Entanglements in Pose & Expression Latent Space
+
+Low-bandwidth talking-head systems encode each frame  $V_{t}$  as a pose-and-expression vector  $z_{t} = f(V_{t})$ , which the receiver's generator turns back into pixels. Although designed for geometry,  $z_{t}$  inherit identity cues from the physical face that produced the motion. Measurements such as inter-ocular distance, jaw curvature, or lip thickness inevitably contaminate  $z_{t}$  because head pose and facial musculature cannot be sensed independently of the speaker's anatomy. Prior studies to disentangle identity from pose and expression also reported persistent identity traces in pose-conditioned representations [22, 23], which provides the empirical basis for our proposed defense.
+
+Let  $R$  denote the static reference portrait of the nominal speaker transmitted at session start. Both endpoints apply the shared encoder  $f$  to compute the reference embedding  $f(R)$ , placing  $R$  and the
+
+![](images/c0223abc8f57b6656f650589b4af9662a8c61425d4da9c49fe3c081a2ba5d6bb.jpg)  
+Pose & Expression Embedding Space  
+Different Identity
+
+![](images/a70d8187fcb4961e5f90da0ddd157edcd5e90587eb37bf136fb2d45819fbcaf4.jpg)  
+Biometric Leakage Space  
+Different Identity
+
+![](images/f1e4499f98d48cde3e7a9885d6ed39daceec37e4ec1b0b22810c861d93bf5ae7.jpg)  
+Same Identity  
+Different Pose & Expression  
+Figure 3: Similarity distributions in P&E space (left) and biometric leakage space (right). Red: same ID, diff. P&E; blue: diff. ID, same P&E; black: diff. ID, diff. P&E.
+
+![](images/9bdbbd9b6ee6426a0ac4a8739e32e3114d80081ee4ab2d1be5fd254b60d92c9d.jpg)  
+Different Pose & Expression
+
+![](images/2f70e35e52e623f16eab10b032fca7c040ed6ccc6a5c9f90a938c5ba92386418.jpg)  
+Same Pose & Expression
+
+live stream  $\{z_t\}$  in one latent space. A natural baseline defense is to compute the cosine similarity  $s_c(z_t, f(R))$  and accept the incoming frame if the score exceeds a threshold. Unfortunately, pose and expression variability overwhelms the subtle biometric signal.
+
+We demonstrate this effect in Fig. 3, which shows the cosine similarity distributions for three conditions: (1) same identity, differing pose/expression; (2) different identities, matched pose/expression; (3) different identities, differing pose/expression. We observe that the density for condition (2) frequently eclipses condition (1), which means that two different people exhibiting an identical yaw-pitch often appear closer in latent space than two frames of the same person smiling versus neutral. Raw pose-and-expression space is therefore pose-dominated: distance reflects "how the head moves" more than "whose head it is."
+
+This observation is key to our puppeteering defense: amplify latent biometric cues while suppressing pose and expression variance so that proximity encodes identity. Simply collecting more negatives or tweaking the similarity threshold cannot meet this requirement; the representation itself must be re-shaped. In the next section, we introduce the Enhanced Biometric Leakage (EBL) space, learned directly from the latent channel, that achieves this goal without RGB reconstruction, user enrollment, or landmark tracking, enabling a real-time authentication pipeline.
+
+# 4.2 Enhanced Biometric Leakage (EBL) Space
+
+Our key observation above dictates two design imperatives: (1) craft a representation in which distances reflect identity rather than pose/expression, and (2) keep that representation compact to work in real-time. We meet both goals by re-encoding the raw pose-and-expression latent  $z_{t}$  into a low dimensional Enhanced Biometric Leakage vector and learning a contrastive objective that amplifies identity information while actively cancelling pose.
+
+# 4.2.1 Latent re-encoding
+
+The raw pose-expression vector  $z_{t}$  is dominated by geometric factors, such as yaw, pitch, mouth aperture, while biometric cues hide in low-variance directions. To amplify these cues we attach two lightweight projection heads,  $h_1$  and  $h_2$ , that remap both the live embedding and the static reference  $f(R)$  into a shared, compact metric space.
+
+Separate heads are essential because the two inputs follow different distributions:  $z_{t}$  drifts frame by frame with micro-expressions, whereas  $f(R)$  is fixed at call setup. Allowing independent normalization lets each head adapt to its own domain before they meet in the common space.
+
+Each head consists of a linear layer, ReLU activation, layer normalization, and a second linear layer followed by  $\ell_2$  normalization, such that
+
+$$
+b \left(z _ {t}, R\right) = s _ {c} \left(h _ {1} \left(z _ {t}\right), h _ {2} (f (R))\right), \tag {3}
+$$
+
+where  $s_c(\cdot ,\cdot)$  denotes cosine similarity.
+
+# 4.2.2 Pose-conditioned contrastive objective
+
+The cornerstone of our method is a pose-matched contrastive loss that aligns gradient pressure with the nuisance we aim to discard. Specifically, we pursue two complementary goals inspired by large-margin hyper-spherical embedding theory [63, 64]: (1) maximize the similarity of embeddings that share the same identity even when pose/expression varies, and (2) minimize the similarity of embeddings drawn from different identities having the same pose and expression. The resulting objective comprises one term for each goal.
+
+Positive term. Any pair that agrees in identity, regardless of pose, is treated as positive:
+
+$$
+\mathcal {L} _ {P} = 1 - b \left(z _ {t} ^ {k, p}, R ^ {k}\right), \tag {4}
+$$
+
+where  $k$  indexes the speaker and  $p$  denotes the instantaneous pose-expression state. Minimizing (4) pulls together all manifestations of speaker  $k$ .
+
+Negative term. Hard negatives are constructed by replicating pose but swapping identity. Concretely, we synthesise  $R^{\ell,p}$ , a portrait of impostor  $\ell \neq k$  rendered at the identical pose  $p$ , and impose
+
+$$
+\mathcal {L} _ {N} = \frac {1}{N - 1} \sum_ {\ell \neq k} b \left(z _ {t} ^ {k, p}, R ^ {\ell , p}\right). \tag {5}
+$$
+
+Because every negative matches pose, the gradient direction is orthogonal to pose variation, forcing the network to discriminate on biometric cues alone.
+
+Total loss. The pose-conditioned Large-Margin Cosine Loss (PC-LMCL) is
+
+$$
+\mathcal {L} _ {B} = \mathcal {L} _ {P} + \lambda \mathcal {L} _ {N}, \tag {6}
+$$
+
+with a single hyper-parameter  $\lambda$  controlling repulsion strength. With  $\lambda \leq 1$ ,  $\mathcal{L}_B$  is  $(1 + \lambda)$ -Lipschitz  $(\leq 2)$ , placing them within the margin-risk framework of Lei et al. [65].
+
+Margin guarantee. We formalize the geometric effect of (6) as follows. We note that here, we treat each speaker's front-facing reference portrait  $R^k$  as their pose-averaged center  $||R^{k} - \mu_{k}||_{2}\leq 0.5^{\circ}$ .
+
+Proposition 1 Assume all embeddings are  $\ell_2$ -normalized. If, for some  $\epsilon, \gamma > 0$
+
+$$
+\cos \left(z _ {t} ^ {k, p}, R ^ {k}\right) \geq 1 - \epsilon , \quad \frac {1}{N - 1} \sum_ {\ell \neq k} \cos \left(z _ {t} ^ {k, p}, R ^ {\ell , p}\right) \leq - \gamma , \tag {7}
+$$
+
+then the class centers  $\mu_{k} = \mathbb{E}_{p}[R^{k,p}]$  satisfy
+
+$$
+\cos \left(\mu_ {k}, \mu_ {\ell}\right) \leq 1 - (\epsilon + \gamma), \quad k \neq \ell . \tag {8}
+$$
+
+Thus  $\mathcal{L}_B$  enforces an inter-class angular margin of at least  $\epsilon +\gamma$  within each pose slice.
+
+Intuition. Driving  $\mathcal{L}_P\to 0$  pins positives to the "north pole", while  $\mathcal{L}_N\to 0$  pushes pose-matched negatives toward the antipode; the spherical triangle inequality then yields Prop. 1. The full proof appears in Appendix G.
+
+Consequently, minimizing  $\mathcal{L}_B$  maximizes the angular decision margin between identity manifolds while cancelling pose - an effect known to tighten generalization bounds in hyper-spherical embedding spaces [63, 66]. Prior methods ignore latent pose alignment and instead rely on RGB or landmarks, incurring latency and robustness penalties. By operating purely in the latent domain, PC-LMCL yields compact representations suitable for real-time deployment.
+
+# 4.3 Training Protocol
+
+The contrastive objective of the previous section is optimized on mini-batches organized as episodes. Each episode selects an anchor clip from speaker  $k$ , assembles identity-consistent positives drawn from other poses/expressions of  $k$ , and generates pose-matched negatives by rendering the same motion with impostor identities  $\ell \neq k$ . Because the shared encoder  $f$  remains frozen, only the two projection heads learn, allowing rapid re-training should a codec revision alter its latent format.
+
+Table 1: Puppeteering attack detection performance measured in AUC across different dataset-generation method pairs. NVC=NVIDIA VC [11], RAV=RAVDESS [54], and CRD=CREMA-D [55]  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">3DFaceShop [67]</td><td colspan="3">MCNET [68]</td><td colspan="3">EMOPortraits [69]</td><td colspan="3">SDFR [70]</td><td colspan="3">LivePortrait [71]</td><td colspan="3">Avg.</td></tr><tr><td>NVC</td><td>RAV</td><td>CRD</td><td>NVC</td><td>RAV</td><td>CRD</td><td>NVC</td><td>RAV</td><td>CRD</td><td>NVC</td><td>RAV</td><td>CRD</td><td>NVC</td><td>RAV</td><td>CRD</td><td>NVC</td><td>RAV</td><td>CRD</td></tr><tr><td>Efficient ViT [12]</td><td>.575</td><td>.580</td><td>.562</td><td>.573</td><td>.573</td><td>.581</td><td>.560</td><td>.547</td><td>.580</td><td>.569</td><td>.579</td><td>.595</td><td>.578</td><td>.578</td><td>.591</td><td>.571</td><td>.551</td><td>.582</td></tr><tr><td>CCE ViT [12]</td><td>.632</td><td>.608</td><td>.618</td><td>.628</td><td>.618</td><td>.616</td><td>.624</td><td>.630</td><td>.607</td><td>.627</td><td>.622</td><td>.606</td><td>.639</td><td>.615</td><td>.575</td><td>.630</td><td>.619</td><td>.604</td></tr><tr><td>CNN Ensemble [13]</td><td>.540</td><td>.559</td><td>.550</td><td>.516</td><td>.536</td><td>.554</td><td>.535</td><td>.520</td><td>.537</td><td>.518</td><td>.570</td><td>.561</td><td>.527</td><td>.561</td><td>.546</td><td>.527</td><td>.549</td><td>.550</td></tr><tr><td>TALL [14]</td><td>.535</td><td>.511</td><td>.505</td><td>.531</td><td>.506</td><td>.504</td><td>.560</td><td>.509</td><td>.500</td><td>.546</td><td>.505</td><td>.501</td><td>.532</td><td>.501</td><td>.515</td><td>.541</td><td>.506</td><td>.505</td></tr><tr><td>SupCon [15]</td><td>.679</td><td>.632</td><td>.640</td><td>.651</td><td>.621</td><td>.632</td><td>.676</td><td>.649</td><td>.625</td><td>.681</td><td>.648</td><td>.619</td><td>.674</td><td>.639</td><td>.615</td><td>.672</td><td>.638</td><td>.626</td></tr><tr><td>CLR Net [15]</td><td>.620</td><td>.639</td><td>.635</td><td>.629</td><td>.636</td><td>.684</td><td>.634</td><td>.624</td><td>.641</td><td>.635</td><td>.634</td><td>.681</td><td>.639</td><td>.632</td><td>.600</td><td>.631</td><td>.633</td><td>.648</td></tr><tr><td>LAANet [16]</td><td>.538</td><td>.525</td><td>.518</td><td>.539</td><td>.519</td><td>.558</td><td>.518</td><td>.528</td><td>.556</td><td>.500</td><td>.526</td><td>.543</td><td>.513</td><td>.513</td><td>.523</td><td>.522</td><td>.527</td><td>.540</td></tr><tr><td>Vahdati et al. [10]</td><td>.954</td><td>.960</td><td>.954</td><td>.951</td><td>.950</td><td>.948</td><td>.944</td><td>.949</td><td>.952</td><td>.918</td><td>.913</td><td>.908</td><td>.959</td><td>.966</td><td>.958</td><td>.945</td><td>.947</td><td>.944</td></tr><tr><td>Ours</td><td>.984</td><td>.956</td><td>.958</td><td>.989</td><td>.978</td><td>.984</td><td>.962</td><td>.974</td><td>.983</td><td>.970</td><td>.978</td><td>.952</td><td>.982</td><td>.956</td><td>.962</td><td>.977</td><td>.968</td><td>.968</td></tr></table>
+
+Extreme Pose Exclusion. Before each update we discard frames whose head-pose estimate is unreliable. For instance, some poses may deviate substantially from the reference image, or, the speaker's face can be substantially occluded. This extreme-posed exclusion keeps the gradient focused on informative biometric variation.
+
+To accomplish this, we estimate face normals  $n_R$  and  $n_t$  and prune any frame where their cosine similarity  $s_c(n_t, n_R)$  exceeds a threshold  $\tau$ , effectively removing views with large yaw, occlusion, or visibility loss. This keeps the gradient focused on meaningful biometric variation.
+
+# 4.4 Temporal Fusion
+
+While each individual biometric similarity score provides an instantaneous assessment of the mismatch between the driving speaker and the target identity, per-frame measurements can be noisy or inconclusive under challenging poses or brief occlusions. To address this, we aggregate similarity scores over a window of  $W$  consecutive frames and feed these into an LSTM. The LSTM learns to capture temporal patterns indicative of sustained mismatches characteristic of puppeteering attacks.
+
+Let  $\phi = \{\phi_1, \phi_2, \dots, \phi_W\}$  be the sequence of similarity scores collected over  $W$  frames. We feed  $s$  into the LSTM, which outputs a final score  $y$  representing the probability that the sequence is puppeteered. We train the LSTM by minimizing the binary cross-entropy loss between  $y$  and a ground-truth label  $t$ , where  $t = 1$  if the video segment is puppeteered and  $t = 0$  otherwise, such that
+
+$$
+\mathcal {L} _ {\mathrm {L S T M}} = - t \log (y) - (1 - t) \log (1 - y). \tag {9}
+$$
+
+This learnable aggregator is novel in the context of puppeteering defense: previous methods either majority-vote over heuristic landmarks or revisit pixel space, both of which are fragile under compression artifacts. As the evaluation will show, the combination of EBL embedding and temporal fusion sets a new state of the art in puppeteering detection without sacrificing latency.
+
+# 5 Experiments and Results
+
+# 5.1 Experimental Setup
+
+Datasets. We conduct our experiments using the NVFAIR [11] pooled dataset because it incorporates together a large set recorded video-conference calls in a controlled setting environment. This dataset includes three subsets: (1) NVIDIA VC [11] (natural video calls), (2) CREMA-D [55] (studio-recorded expressions), and (3) RAVDESS [54] (studio-recorded emotional speech). For each subset, we generate both self-reenacted (same identity) and cross-reenacted (puppeteered) videos using five state-of-the-art methods—MCNet [55], 3DFaceShop [67], SDFR [70], EmoPortraits [69], and LivePortrait [71], resulting in total of 15 dataset-method combinations. We note that the identities used for evaluation are strictly disjoint from those in training. Further details are provided in Tab. 3.
+
+Metrics. We report the average detection AUC to allow direct comparisons with prior work.
+
+Competing Methods. As puppeteering detection is a new problem, we compare against the state-of-the-art method by Vahdati et al. [10]. To highlight the distinction from deepfake detection, we also evaluate seven leading deepfake detectors: Efficient ViT [12], CCE ViT [12], CNN Ensemble [13], TALL [15], SupCon [15], CLRNet [16], and LAA-Net [16].
+
+Table 2: Puppeteering attack detection performance measured in AUC of ours and competing methods when trained only on the NVIDIA-VC [11] subset (In-domain) and tested on the CREMA-D [55] and RAVDESS [54] subsets (Cross-domain).  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">3DFaceshop [67]</td><td colspan="2">MCNET [68]</td><td colspan="2">EMOPortraits [69]</td><td colspan="2">SDFR [70]</td><td colspan="2">LivePortrait [71]</td><td colspan="2">Avg.</td></tr><tr><td>In-domain</td><td>Cross-domain</td><td>In-domain</td><td>Cross-domain</td><td>In-domain</td><td>Cross-domain</td><td>In-domain</td><td>Cross-domain</td><td>In-domain</td><td>Cross-domain</td><td>In-domain</td><td>Cross-domain</td></tr><tr><td>Efficient ViT [12]</td><td>.592</td><td>.580</td><td>.608</td><td>.635</td><td>.631</td><td>.604</td><td>.628</td><td>.659</td><td>.599</td><td>.647</td><td>.612</td><td>.625</td></tr><tr><td>CCE ViT [12]</td><td>.573</td><td>.598</td><td>.577</td><td>.587</td><td>.592</td><td>.616</td><td>.629</td><td>.651</td><td>.638</td><td>.609</td><td>.602</td><td>.612</td></tr><tr><td>CNN Ensemble [13]</td><td>.614</td><td>.629</td><td>.606</td><td>.589</td><td>.582</td><td>.516</td><td>.600</td><td>.633</td><td>.552</td><td>.595</td><td>.591</td><td>.592</td></tr><tr><td>TALL [14]</td><td>.626</td><td>.639</td><td>.548</td><td>.516</td><td>.562</td><td>.508</td><td>.603</td><td>.672</td><td>.561</td><td>.571</td><td>.580</td><td>.581</td></tr><tr><td>SupCon [15]</td><td>.637</td><td>.659</td><td>.592</td><td>.620</td><td>.599</td><td>.628</td><td>.652</td><td>.603</td><td>.664</td><td>.671</td><td>.629</td><td>.636</td></tr><tr><td>CLR NET [15]</td><td>.643</td><td>.627</td><td>.649</td><td>.633</td><td>.570</td><td>.625</td><td>.594</td><td>.638</td><td>.644</td><td>.638</td><td>.640</td><td>.632</td></tr><tr><td>LAANet [16]</td><td>.560</td><td>.558</td><td>.528</td><td>.503</td><td>.595</td><td>.568</td><td>.535</td><td>.561</td><td>.582</td><td>.617</td><td>.560</td><td>.561</td></tr><tr><td>Vahdati et al. [10]</td><td>.920</td><td>.895</td><td>.942</td><td>.921</td><td>.927</td><td>.894</td><td>.904</td><td>.890</td><td>.929</td><td>.915</td><td>.924</td><td>.903</td></tr><tr><td>Ours</td><td>.948</td><td>.914</td><td>.950</td><td>.919</td><td>.936</td><td>.917</td><td>.951</td><td>.944</td><td>.939</td><td>.931</td><td>.945</td><td>.925</td></tr></table>
+
+Table 3: Statistics of the datasets and generated data used in this paper.  
+Table 4: Detection AUC of our proposed method and its alternative design choices.  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2"># of IDs</td><td colspan="3"># Authorized Use Videos</td><td colspan="3"># Puppeteered Videos</td></tr><tr><td>Train</td><td>Test</td><td>Total</td><td>Train</td><td>Test</td><td>Total</td></tr><tr><td>NVIDIA-VC (NVC) [11]</td><td>46</td><td>1,331</td><td>439</td><td>1,770</td><td>41,261</td><td>3,951</td><td>45,212</td></tr><tr><td>RAVDESS (RAV) [54]</td><td>24</td><td>704</td><td>264</td><td>968</td><td>10,560</td><td>1,320</td><td>11,880</td></tr><tr><td>CREMA-D (CRD) [55]</td><td>91</td><td>5,154</td><td>1,558</td><td>6,712</td><td>319,548</td><td>28,044</td><td>347,592</td></tr></table>
+
+<table><tr><td rowspan="2">Component</td><td>Method</td><td>AUC</td><td>\( \mathbf{{RER}\% } \)</td></tr><tr><td>Proposed</td><td>0.966</td><td>-</td></tr><tr><td rowspan="2">Biometric Space Network Design</td><td>No MLP Lower Dim. Pro-jection</td><td>0.827</td><td>80.35</td></tr><tr><td>No Biometric Leakage Net-work</td><td>0.635</td><td>90.68</td></tr><tr><td rowspan="3">Training Proto-col</td><td>Single Negative + Single Positive</td><td>0.749</td><td>86.45</td></tr><tr><td>No Biometric Contrastive Loss</td><td>0.788</td><td>83.96</td></tr><tr><td>No Extreme Pose Exclusion</td><td>0.929</td><td>52.11</td></tr></table>
+
+We exclude Avatar Fingerprinting [11], which requires an enrollment phase where each user submits authorized videos to generate identity fingerprints. This setup is incompatible with our framework, as removing enrollment disables its core mechanism and renders comparison unfair.
+
+# 5.2 Experimental Results
+
+In this section, we evaluate our method against state-of-the-art baselines for detecting puppeteered videos. We report results under two scenarios: (1) no domain shift—training and testing on all datasets—and (2) cross-domain generalization—training only on NVIDIA VC [11] and evaluating on unseen CREMA-D [55] and RAVDESS [54] datasets.
+
+# 5.2.1 Performance on Combined Datasets
+
+Tab. 1 summarizes results from training and testing across all datasets (NVIDIA VC [11], CREMA-D [55], and RAVDESS [54]). Our method achieves AUC  $>0.97$  across all combinations, robustly detecting puppeteering across diverse identities, poses, and expressions. In contrast, deepfake detectors perform poorly; CLRNet [15], the strongest baseline, reaches only 0.684 AUC on SDFR-generated [70] videos from NVIDIA VC [11]. This highlights the limitation of deepfake detectors, which focus on distinguishing real from synthetic rather than authorized vs. unauthorized identities.
+
+Our method also outperforms the closest prior work, Vahdati et al. [10], reducing relative error by  $46\%$  (AUC from 0.945 to 0.971). Notably, our minimum AUC remains above 0.95, compared to 0.90 for Vahdati et al. [10]. This gain stems from our enhanced biometric leakage space, which better separates identity from pose and expression cues. In contrast, their method relies on simple Euclidean distance, making it less robust to such variations.
+
+# 5.2.2 Cross-Domain Generalization Performance
+
+Tab. 2 presents results when training is limited to NVIDIA VC [11] and testing is performed on CREMA-D [55] and RAVDESS [54]. Our method maintains strong performance with an average AUC of 0.925—only a  $5\%$  drop from the no-domain-shift setting—despite training on just 46 identities. This drop is consistent with Fig. 4, which shows AUC scaling with the number of identities used in training. These results suggest the observed performance gap is due to identity diversity rather than differences in appearance or expression, confirming that our method scales with data and generalizes well to unseen domains.
+
+# 6 Ablation Study
+
+In this section, we conduct a series of ablation experiments to understand the impact of different design choices on the detection performance of our method. To do this, we measured the average detection AUC and relative error reduction (RER) over the self-reenacted and puppeteered examples across all datasets using 3DFaceShop [57] as the generator. The results are provided in Tab. 4.
+
+Biometric Leakage Space Network Design. The results in Tab. 4 show that removing the biometric leakage embedding modules  $(h_1, h_2)$  resulted in a substantial drop in performance. This finding confirms that simply comparing  $z_t$  to  $f(R)$  does not work. Additionally, we observe that increasing the MLP's output dimension to match or exceed the input drops performance to 0.827 AUC (an  $80.35\%$  increase in error), indicating that projecting embeddings into a lower-dimensional space effectively filters out irrelevant variability.
+
+Training Protocol. Table 4 highlights three key sensitivities. Limiting each batch to a single positive-negative pair slashes AUC from 0.966 to 0.749 (+86% RER), proving multiple samples are vital for capturing fine biometric cues. Replacing our pose-conditioned contrastive loss with a simple similarity regression lowers AUC to 0.788 (+84% RER), confirming the need to optimise relative identity differences. Finally, retaining extreme-pose frames drops AUC to 0.929 (+52% RER), so filtering them is essential for peak accuracy.
+
+Pose Sensitivity Analysis. We ablated pose sensitivity by progressively omitting frames whose head rotation exceeded a set angle and tracking AUC on a test set. Table 5 shows performance rising steadily as extreme poses are removed, peaking at  $\pm 18^{\circ}$ ; stricter cut-offs then hurt accuracy. Thus, judicious pose exclusion yields measurable detection gains.
+
+Table 5: Puppeteering detection AUC at increasing yaw thresholds. Performance peaks at  $\pm 18^{\circ}$  
+
+<table><tr><td>Yaw (°)</td><td>±3</td><td>±8</td><td>±13</td><td>±18</td></tr><tr><td>AUC</td><td>0.932</td><td>0.938</td><td>0.953</td><td>0.966</td></tr><tr><td>Yaw (°)</td><td>±23</td><td>±28</td><td>±33</td><td>±38</td></tr><tr><td>AUC</td><td>0.958</td><td>0.942</td><td>0.930</td><td>0.929</td></tr></table>
+
+Table 6: Robustness to facial appearance changes on NVIDIA VC (3DFaceShop).  
+
+<table><tr><td>Modification</td><td>AUC</td></tr><tr><td>None (baseline)</td><td>0.966</td></tr><tr><td>Eyeglasses</td><td>0.9393</td></tr><tr><td>Piercings</td><td>0.9605</td></tr><tr><td>Makeup</td><td>0.9574</td></tr></table>
+
+# 7 Discussion
+
+Computational Efficiency. We benchmarked our method on an RTX 3090 GPU, where it achieved on average 75 FPS—well above the 60 FPS real-time threshold while having under 1M total number of parameters. In comparison, Vahdati et al. [10] reached only 32 FPS under the same conditions. This speed advantage highlights the efficiency of our latent-space analysis, supporting real-time deployment in AI-based videoconferencing.
+
+Temporal Window Size. Fig. 4 shows how detection performance varies with the LSTM's temporal window size. AUC improves sharply from 0.77 at 10 frames to 0.97 at 40 frames, after which gains plateau. A 40-frame window ( $\approx$ 1.3s at 30fps) captures sufficient temporal biometric information, highlighting the value of modeling frame-to-frame dependencies in latent space.
+
+![](images/2ac7cef8dd423adcb1abc23d03c1f96d9af9f3460b3b2db9624cc7a31bf47273.jpg)  
+Figure 4: Detection AUC vs. window size and number of puppeteered identities during training.
+
+![](images/f1350b84a303365acbaed8ecde10bd68857ab8d265a6832d13c40632ff5dbb7d.jpg)
+
+Scalability. Fig. 4 shows detection AUC as a function of the number of cross-reenacted identities used in training on the NVIDIA VC dataset. Performance steadily increases from 0.75 with few identities to 0.96 with all 46, with no signs of saturation. This suggests that greater diversity during training
+
+enhances the robustness and generalizability of our biometric embedding space in distinguishing puppeteering from self-reenactment.
+
+Robustness to facial appearance changes. We tested our method's robustness to appearance changes by digitally adding eyeglasses, piercings, or heavy makeup to NVIDIA-VC identities using 3DFaceShop. As Table 6 shows, AUC shifted modestly from 0.966 to 0.939 (glasses) and 0.961 (piercings), and even improved to 0.974 with makeup. NVFAIR's natural accessory and cosmetic diversity had already exposed the model to such variations, explaining the small changes and confirming real-world robustness.
+
+Limitations. Our method has three main failure modes. First, extreme head rotation or occlusion (e.g., from hands or objects) can obscure facial features critical to identity. Second, poor lighting or overexposure weakens the biometric signal captured by the camera. Third, motion blur—caused by rapid movement or blinking—can distort the latent representation. Each of these degrades biometric consistency and impairs detection. Examples are provided in the supplementary material.
+
+# 8 Conclusion
+
+We present a real-time puppeteering defense that authenticates speakers directly in the transmitted pose-expression stream. A low dimensional biometric-leakage embedding, learned with novel pose-conditioned contrastive loss and reinforced by an LSTM for temporal fusion, plus extreme-pose filtering, lifts AUC beyond prior work and stays strong across domains and appearance edits. Ablations confirm every module's value, establishing a practical, robust safeguard for next-generation talking-head videoconferencing systems.
+
+# Acknowledgments
+
+This material is based upon work supported by the National Science Foundation under Grant No. 2320600. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. This research was supported in part by NVIDIA Corporation. We thank the NVIDIA Research team for their collaboration, resources, and valuable feedback throughout this project.
+
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+
+# NeurIPS Paper Checklist
+
+1. Claims
+
+Answer: [Yes]
+
+Justification: All scientific claims are stated in Secs. 1-4; empirical claims are validated in Sec. 5 and Tab. 1-4.
+
+2. Limitations
+
+Answer: [Yes]
+
+Justification: Sec. 7 "Limitations" details failure modes (extreme pose, lighting, motion blur).
+
+3. Theory assumptions and proofs
+
+Answer: [NA]
+
+Justification: Work is empirical; no formal theorems are presented.
+
+4. Experimental result reproducibility
+
+Answer: [Yes]
+
+Justification: Hyper-parameters, random seed, and training protocol given in Sec. 5.1; code will be released.
+
+5. Open access to data and code
+
+Answer: [Yes]
+
+Justification: Public datasets (NVFAIR) are cited; inference/training code and weights will be provided upon acceptance.
+
+6. Experimental setting/details
+
+Answer: [Yes]
+
+Justification: Dataset splits, generators, and evaluation metrics described in Sec. 5.2-5.3.
+
+7. Experiment statistical significance
+
+Answer: [Yes]
+
+Justification:
+
+8. Experiments compute resources
+
+Answer: [Yes]
+
+Justification: Sec. 5.1 states training cost ( $\approx$ 18 GPU-hours on a single RTX 3090) and runtime (75 FPS).
+
+9. Code of ethics
+
+Answer: [Yes]
+
+Justification: Research follows the NeurIPS Code of Ethics; see Broader-Impact section.
+
+10. Broader impacts
+
+Answer: [Yes]
+
+Justification: Societal impacts, misuse potential, and mitigation discussed in Broader-Impact paragraph.
+
+11. Safeguards
+
+Answer: [Yes]
+
+Justification: Release plan includes watermarking detector outputs and open-sourcing under non-commercial license.
+
+12. Licenses for existing assets
+
+Answer: [Yes]
+
+Justification: We have properly attributed the work of others and have followed licensing and usage terms.
+
+13. New assets
+
+Answer: [NA]
+
+Justification: No new dataset or proprietary model weights created; only derived metrics.
+
+14. Crowdsourcing and research with human subjects
+
+Answer: [NA]
+
+Justification: Work uses publicly available datasets; no new human-subject data collected.
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether IRB approvals were obtained?
+
+Answer: [NA]
+
+Justification: Not applicable (no new human-subjects research).
+
+# 15. Declaration of LLM usage
+
+Answer: [NA]
+
+Justification: No large language model is an original or non-standard part of the core methodology.
\ No newline at end of file
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+# Unraveling Metameric Dilemma for Spectral Reconstruction: A High-Fidelity Approach via Semi-Supervised Learning
+
+Xingxing Yang
+
+Department of Computer Science Hong Kong Baptist University, Hong Kong csxxyang@comp.hkbu.edu.hk
+
+Jie Chen*
+
+Department of Computer Science Hong Kong Baptist University, Hong Kong chenjie@comp.hkbu.edu.hk
+
+Zaifeng Yang
+
+Institute of High Performance Computing  
+Agency for Science Technology and Research, Singapore  
+Yang_zaifeng@a-star.edu.sg
+
+# Abstract
+
+Spectral reconstruction from RGB images often suffers from a metameric dilemma, where distinct spectral distributions map to nearly identical RGB values, making them indistinguishable to current models and leading to unreliable reconstructions. In this paper, we present Diff-Spectra that integrates supervised physics-aware spectral estimation and unsupervised high-fidelity spectral regularization for HSI reconstruction. We first introduce an Adaptive illumiChroma Decoupling (AICD) module to decouple illumination and chrominance information, which learns intrinsic and distinctive feature distributions, thereby mitigating the metameric issue. Then, we incorporate the AICD into a learnable spectral response function (SRF) guided hyperspectral initial estimation mechanism to mimic the physical image formation and thus inject physics-aware reasoning into neural networks, turning an ill-posed problem into a constrained, interpretable task. We also introduce a metameric spectra augmentation method to synthesize comprehensive hyperspectral data to pre-train a Spectral Diffusion Module (SDM), which internalizes the statistical properties of real-world HSI data, enforcing unsupervised high-fidelity regularization on the spectral transitions via inner-loop optimization during inference. Extensive experimental evaluations demonstrate that our Diff-Spectra achieves competitive performance on both Spectral reconstruction and downstream HSI classification.
+
+# 1 Introduction
+
+Hyperspectral imaging captures hundreds of spectral bands, allowing for precise identification of materials and illumination conditions that are often indistinguishable from RGB imaging. It has been widely applied in remote sensing [1, 2], medical diagnosis [3, 4] and agriculture[5, 6].
+
+Traditional hyperspectral imaging methods show limitations in time-consuming acquisition processes with limited spatial resolution. Recent advances in deep learning within the computer vision community have paved the way for hyperspectral image reconstruction from RGB inputs using data-driven methodologies [7, 8]. Early model-based methods, such as sparse coding [9] and low-rank
+
+![](images/c8f1a739e6e17ccb2f8fa702218cc6da4ed17a2cbb4f8559f5145e923444554b.jpg)  
+Figure 1: Motivation. (a) shows metamer and standard HSIs. In (b), standard and metamer RGB inputs are produced by the same SRF from corresponding HSIs. In (d), Left: the dashed and solid red highlighted points are two pixels from the fake and real peppers from the CAVE dataset [19], respectively; Right: Corresponding ground-truth spectral curves of the two pixels, and reconstructed spectral curves of CESST [8] (grey) and our method (blue). The fake and real peppers are in similar colors but have different spectra. Existing methods (e.g., CESST) fail to distinguish either the two spectra from each other or from their true spectra, while our method can reconstruct faithfully.
+
+![](images/a185e9fea1eb7b132b328a236ae444f2b07c2cec1c116c098fba90c81012bdb2.jpg)
+
+representation [10], rely predominantly on manually crafted priors. In contrast, modern learning-based approaches [7, 11, 12, 13] leverage deep learning frameworks such as convolutional neural networks [14, 15, 16] and Vision Transformers [7, 8] to achieve superior reconstruction performance.
+
+Limitations. However, we observe that most existing methods [7, 11, 12] suffer from a metameric dilemma [17, 18], where distinct spectral distributions map to nearly identical RGB values, making them indistinguishable to current models and leading to unreliable reconstructions, as shown in Fig.1. We argue that this stems from the direct nonlinear mapping from the sRGB input space to the hyperspectral output space, leading to three key limitations in terms of data, methodologies, and pipelines: (i) Data Poverty: Existing hyperspectral datasets lack diversity in rich metameric examples, forcing existing models to memorize a limited subset of spectra rather than internalizing the full spectral manifold; (ii) Spectral Manifold Blindness: Existing methods overlook the explicit modeling of the real-world spectral distribution, leading to hallucinated outputs that minimize pixel-wise losses (e.g., MSE) but violate physical laws (e.g., unnatural spectral intensity in Fig. 1); (iii) Architecture Myopia: Existing methods treat spectral radiance as a monolithic entity, conflating illumination and chrominance. It amplifies metameric failures that networks cannot distinguish whether RGB variations stem from lighting or material properties.
+
+It raises a fundamental question: Can we introduce auxiliary information to address the above-mentioned metameric dilemma, thus achieving high-fidelity Spectral reconstruction? If so, how?
+
+To this end, we propose Diff-Spectra, a semi-supervised model for HSI reconstruction that integrates supervised physics-aware spectral estimation and unsupervised high-fidelity spectral regularization. Specifically, to deal with architecture myopia, we first introduce an Adaptive IllumiChroma Decoupling (AICD) module by factorizing RGB inputs into independent illumination and chrominance subspace via orthogonal decoupling. Then, the AICD is incorporated into a SRF-guided HSI initial estimation (SRF-guided HIE) mechanism to estimate the target spectral signal in a supervised manner. This process mimics the physical image formation and thus injects physics-aware reasoning into neural networks, turning an ill-posed problem into a constrained, interpretable task. To deal with the data poverty, we introduce a metameric spectra augmentation method to synthesize a comprehensive HSI dataset with diverse metamer samples and spectral perturbations, transforming sparse spectral data into a rich prior that can guide reconstruction beyond RGB ambiguities. To deal with the spectral manifold blindness, we introduce a spectral diffusion module (SDM) that learns to denoise corrupted spectra during pre-training on the comprehensive HSI dataset, which internalizes the statistical properties of real-world hyperspectral data. During reconstruction, the pre-trained SDM serves as a spectral prior that regularizes the coarsely estimated HSI signal from the SRF-guided HIE mechanism with high-fidelity real-world spectral distributions via our proposed unsupervised inner loop optimization. The main contributions are given as follows:
+
+- We propose a semi-supervised paradigm, Diff-Spectra, to deal with the metameric dilemma in spectral reconstruction. It integrates supervised physics-aware spectral estimation and unsupervised high-fidelity spectral regularization.
+
+- We introduce an Adaptive IllumiChroma Decoupling (AICD) module based on orthogonal decoupling to effectively factorize illumination and chrominance information, serving as an IllumiChroma prior. It learns distinctive image features, alleviating the metameric dilemma.  
+- We design a lightweight, learnable SRF-guided HIE mechanism to obtain an initial HSI estimation, which formulates the reconstruction model with physical constraints and enhances the interpretability.  
+- We introduce a metameric spectra augmentation method to synthesize a comprehensive HSI dataset to pre-train a Spectral Diffusion Module (SDM) to capture the real-world spectral distribution, serving as a spectral prior to improve spectral fidelity.  
+- Extensive experiments on both spectral reconstruction and HSI classification demonstrate that our Diff-Spectra framework significantly outperforms SOTA methods.
+
+# 2 Related Work
+
+# 2.1 Hyperspectral Image Reconstruction
+
+Early efforts [9, 16] utilized model-based approaches that incorporated fidelity terms and physical priors to constrain the target solutions. For example, Arad et al. [9] improved this interpolation challenge using hyperspectral priors to forge a sparse dictionary of HSIs alongside their RGB counterparts. Despite their contributions, these model-based strategies depend on manually tailored priors, constricting their representational capability. Learning-based methods [11, 12, 20, 21], shifted the focus to data-driven approaches, learning implicit mappings from RGB to hyperspectral domains using specifically designed architectures. Notably, the HSCNN model [14] revolutionizes the field by mapping the input RGB image into hyperspectral feature space via a convolutional layer and harnessing deep residual convolutional blocks to approximate the enriched HSI. Cai et al. [7] proposed a transformer-based approach to capture the long-range channel-wise correlations that compute the self-attention map along the channel dimension, tailored for HSI reconstruction. Wang et al. [22] introduced an intrinsic image decomposition (IID) framework to decompose input images into reflectance and shading features and then reconstruct them in the spectral domain separately. However, all existing learning-based methods directly learn mappings between sRGB and HSI feature spaces, neglecting the intrinsic spectral distribution of HSIs and lacking physical constraints, thus encountering the challenges posed by the metameric dilemma.
+
+# 2.2 Diffusion-based HSI Image Reconstruction
+
+Diffusion models [23, 24] have witnessed an explosion of continuously growing capability and capacity architectures. In the context of HSI image reconstruction, Pang et al. [25] proposed HIR-Diff that leverages a powerful pre-trained diffusion prior and a product-of-experts guidance scheme to remove degradations and recover clean hyperspectral images in an unsupervised manner, demonstrating strong generalization across scenes and noise types. Wu et al. [26] proposed a conditional denoising transformer to fuse high-resolution multi-spectral images and low-resolution hyperspectral images to generate target high-resolution HSI images. Liu et al. [27] proposed incorporating the deep generative prior of diffusion models to constrain the high-resolution multi-spectral image and low-resolution hyperspectral image fusion process. While these works focus on restoration and reconstruction, diffusion's representational advantages have also benefited HSI classification, suggesting transferable priors for reconstruction. Chen et al. [28] proposed a spatial-spectral diffusion module to generate high-dimensional HSI signals for HSI classification, indicating the great potential of diffusion-based generative models in spectral distribution modeling. Beyond natural HSI classification, Sigger et al. [29] proposed a multistage unsupervised diffusion framework to extract complementary high- and low-level spectral features for challenging biomedical HSI classification, underscoring diffusion's capacity to model fine-grained spectral structure. However, applying diffusion models for practical HSI systems still faces trade-offs among cost, complexity, and acquisition speed. Moreover, directly reconstructing full hyperspectral cubes from RGB inputs with diffusion remains comparatively underexplored, presenting an opportunity to combine diffusion priors with cross-modal spectral constraints and measurement-aware conditioning for faithful RGB-to-HSI recovery grounded in both spectral accuracy and spatial details.
+
+![](images/e4bf1d1a762134bcb6c2f417462344117a5052816c18ce13392c55bf0cfcef0f.jpg)
+
+![](images/815559b50f91583ceb818ea58465376b6cd7b43b98525464ee36d2674d6cc90a.jpg)  
+Figure 2: Overall framework of Diff-Spectra. Our model has three training stages. In Stage 1, the SRF-guided HIE and AICD are pre-trained using Eq. 10 to initialize a coarse-level HSI signal while freezing the SDM. In Stage 2, the SDM is pre-trained using Eq.13 to learn the spectral distribution of HSI signals while freezing the SRF-guided HIE and AICD. In Stage 3, we fine-tune the SRF-guided HIE and AICD using the objective function Eq. 14, which regularizes the initialized HSI signal in Stage 1 with the spectral prior of SDM in Stage 2. During inference, we further introduce an inner loop optimization as a test-time adaptation with Eq. 13 to generate refined HSI signals by following the spectral reverse generative sequence of SDM.
+
+![](images/72f11eff15aab399804204e3566d2def79263103fa7cd5887cef335518c666f5.jpg)
+
+# 3 Methodology
+
+# 3.1 Problem Definition
+
+The RGB imaging process can be formulated as a sub-sampling process of the target HSI signal:
+
+$$
+\mathbf {X} = \mathbf {Y} \mathbf {A} + \epsilon , \quad \mathbf {A} \in \mathbb {R} ^ {C \times c}, \tag {1}
+$$
+
+where  $\mathbf{X} \in \mathbb{R}^{H \times W \times c}$  denotes the observed RGB image, with  $H$ ,  $W$  and  $c$  representing height, width, and channel ( $c << C$ ) of  $\mathbf{X}$ , respectively.  $\mathbf{Y} \in \mathbb{R}^{H \times W \times C}$  and  $\mathbf{A} \in \mathbb{R}^{C \times c}$  are the target HSI signal and the spectral response function (SRF) of the RGB sensor, respectively.  $\epsilon$  is the noise or residual terms that arise in the image processing pipeline. The HSI reconstruction task can be formulated as a maximum a posteriori problem:  $\max_{\mathbf{Y}} p(\mathbf{Y} \mid \mathbf{X}, \mathbf{A})$ . Applying Bayes' theorem, the posterior can be further reformulated as:
+
+$$
+p (\mathbf {Y} \mid \mathbf {X}, \mathbf {A}) = \frac {p (\mathbf {X} , \mathbf {A} \mid \mathbf {Y}) p (\mathbf {Y})}{p (\mathbf {X} , \mathbf {A})}, \tag {2}
+$$
+
+Taking the negative logarithm and discarding irrelevant terms  $\mathbf{X}$ ,  $\mathbf{A}$ , we obtain the objective function:
+
+$$
+\min  _ {\mathbf {Y}} \left\{- \log p (\mathbf {X}, \mathbf {A} \mid \mathbf {Y}) - \log p (\mathbf {Y}) \right\}, \tag {3}
+$$
+
+where  $\log p(\mathbf{X},\mathbf{A}\mid \mathbf{Y})$  is the log-likelihood that depicts the degradation processes of RGB sampling from the HSI signal, and  $\mathbf{Y}$  is the spectral prior that will contribute to the restoration of  $\mathbf{Y}$ .
+
+# 3.2 Motivation and Solution
+
+Existing HSI reconstruction methods [7, 8] only consider the first term in Eq. 3 (i.e.,  $\log p(\mathbf{X},\mathbf{A}\mid \mathbf{Y})$ ) and directly map RGB inputs to hyperspectral feature space, overlooking subtle variations in RGB feature distributions, thereby leading to the metameric dilemma, as shown in Fig. 1 and further demonstrated by the experimental results in Section 4.3.
+
+In this paper, we would like to consider this challenging spectral reconstruction problem as supervised physics-aware spectral estimation and unsupervised high-fidelity spectral regularization. Thus, in terms of supervised physics-aware spectral estimation, which corresponds to to optimize the first
+
+term -  $\log p(\mathbf{X},\mathbf{A}\mid \mathbf{Y})$  in Eq. (3), we introduce an Adaptive IllumiChroma Decoupling (AICD) module by factorizing RGB inputs into independent illumination (U) and chrominance (V) subspace via orthogonal decoupling to explore intrinsic clues and distinguish image features, which effectively eliminates redundancy and enhances distinctive patterns among metameric hyperspectral data. Then, the AICD is incorporated into the SRF-guided HSI Initial Estimation (SRF-guided HIE) module to estimate the target spectral signal in a supervised manner. This process mimics the physical image formation and thus injects physics-aware reasoning into neural networks, turning an ill-posed problem into a constrained, interpretable task. Now, we express the first term -  $\log p(\mathbf{X},\mathbf{A}\mid \mathbf{Y})$  as the following according to Eq. (1):
+
+$$
+\mathcal {L} _ {Y} (\mathbf {Y}) = \left\| (\mathbf {X}, \mathbf {U}, \mathbf {V}) - \mathbf {Y} \mathbf {A} - \epsilon \right\| ^ {2}. \tag {4}
+$$
+
+Then, we assume the second prior term, i.e.,  $\log p(\mathbf{Y})$ , as unsupervised high-fidelity spectral regularization for the estimated spectra, which further constrains the estimation via transferring distribution prior knowledge from a pre-trained model leveraging our metameric spectra dataset.
+
+# 3.3 Supervised Physics-Aware Spectral Estimation
+
+# 3.3.1 Adaptive IllumiChroma Decoupling
+
+A spectral radiance  $\mathbf{Y}(\lambda)$  can be represented as  $\mathbf{Y}(\lambda) = \mathbf{E}(\lambda)\mathbf{S}(\lambda)$ , where,  $\lambda$ ,  $\mathbf{E}(\lambda)$ ,  $\mathbf{S}(\lambda)$  represents the spectral entity, illumination and surface chrominance, respectively. Existing deep learning methods treat  $\mathbf{Y}(\lambda)$  as a monolithic entity, failing to disentangle illumination and chrominance. This conflation amplifies metameric ambiguities, as networks struggle to distinguish whether RGB variations stem from lighting changes  $\mathbf{E}(\lambda)$  or material properties  $\mathbf{S}(\lambda)$ . This dilemma is the same with RGB inputs. As such, we introduce AICD that aims to extract intrinsic and distinctive image features that are robust against different light variations. However, as shown in Fig. 3, we find that merely relying on reflectance  $\mathbf{R}$  and illumination  $\mathbf{L}$  decomposition based on the Retinex theory [30] cannot distinguish between metameric examples.
+
+Considering that HSIs capturing often exhibit low-light characteristics that lead to potential information loss, we first customize an illumination sensitivity parameter  $S_{k}$ , which enables image-specific adjustment as:  $\mathbf{S}_k = \sqrt[k]{\sin(\frac{\pi\mathbf{L}'}{2}) + \tau}$ , where  $k \in \mathbb{Q}^+$ ,  $\tau = 1 \times 10^{-8}$ . In specific, different from existing decomposition methods [31, 32, 33] that decompose images in a deterministic manner, which may not appropriately capture the diverse and complex imaging and lighting conditions that are specific to downstream tasks, we introduce trainable parameters to enable adaptive learning of intrinsic image features in an end-to-end manner, thereby enhancing compatibility with downstream tasks. Specifically, we employ convolutional layers to obtain embedded features:  $R' = \text{Conv}(\mathbf{R}), L' = \text{Conv}(\mathbf{L})$ .
+
+Orthogonal Decoupling. As discussed, decomposed reflectance and illumination components cannot well distinguish metamer data. Inspired by the orthogonal decoupling method in [34, 35], which effectively eliminates redundancy and enhances distinctive patterns among visual features, we further deployed orthogonal UV transform to extract more distinctive features. We define the horizontal (U) and vertical (V) plane as:
+
+$$
+\mathbf {U} = \mathbf {S} _ {k} \odot \mathbf {R} ^ {\prime} \odot h, \quad \mathbf {V} = \mathbf {S} _ {k} \odot \mathbf {R} ^ {\prime} \odot v, \tag {5}
+$$
+
+where  $\odot$  denotes Hadamard production. Note that we orthogonalize the UV-planes using the two intermediate variables  $h = \cos (2\pi \mathbf{R}^{\prime})$  and
+
+![](images/48d3216d27879f38ae1c896fcc23aae7b3ea63cdb4f1f4265c6a0dd327c94fdd.jpg)
+
+![](images/fc4fa1c9ae2b0fd9d8c1b19525a4bc08471ba86a9a0628c7978e62e20a0c2c88.jpg)
+
+![](images/2d12490ad5494fc8cc16d095d03f8857c4b23a67cf843ea424ed0477fd159c3e.jpg)
+
+![](images/cdb31e6512b09cf0c9b36043a6ba423b56318034a01b2b1d11d6dec11336492e.jpg)
+
+![](images/160354c7e490e3706491305aeac528eee04dad40ab34ddb4b5f50268531b4063.jpg)  
+Stand/Metamer Illumination  
+Figure 3: Difference between the decomposition methods of Retinex and AICD. The top row is based on standard data, while the bottom row is based on metamer data. AICD can transform metamer counterparts into more discriminative features than reflectance and illumination maps.
+
+![](images/49bf8ea289629e21ca8609739acbab0a91246dfbc7ccc92298647dd2f1d7a19d.jpg)  
+Reflectance
+
+![](images/d8a92dd37b76545205e60fd01ced806db80de003de2bde356c58ac6f26f1ae94.jpg)  
+AICD
+
+$v = \sin (2\pi \mathbf{R}^{\prime})$ . Finally, the decomposed features (U, and V) will serve as the illumichroma prior for SRF-guided HIE.
+
+# 3.3.2 SRF-guided HSI Initial Estimation
+
+Note that the SRF matrix  $\mathbf{A}$  in Eq. 1 that projects the HSI to an RGB frame has a row-full rank, enabling a right-inverse solution based on the matrix inversion rule. Thus, there exists a transpose
+
+matrix  $\mathbf{A}^T$  of  $\mathbf{A}$  such that the following equation obtained from Eq. 1 holds
+
+$$
+\mathbf {X} \mathbf {A} ^ {T} - \epsilon \mathbf {A} ^ {T} = \mathbf {Y} \mathbf {A} \mathbf {A} ^ {T}, \tag {6}
+$$
+
+$\mathbf{A}\mathbf{A}^T$  is a square matrix, and there exists an inverse matrix of  $\mathbf{A}\mathbf{A}^T$  such that the following equation holds:
+
+$$
+\gamma \mathbf {X} \mathbf {A} ^ {T} (\mathbf {A} \mathbf {A} ^ {T}) ^ {- 1} - \epsilon \mathbf {A} ^ {T} (\mathbf {A} \mathbf {A} ^ {T}) ^ {- 1} = \mathbf {Y} \mathbf {A} \mathbf {A} ^ {T} (\mathbf {A} \mathbf {A} ^ {T}) ^ {- 1}, \tag {7}
+$$
+
+We can rewrite Eq. 7 as
+
+$$
+\mathbf {Y} = \mathbf {X} \mathbf {A} ^ {T} \left(\mathbf {A} \mathbf {A} ^ {T}\right) ^ {- 1} - \epsilon \mathbf {A} ^ {T} \left(\mathbf {A} \mathbf {A} ^ {T}\right) ^ {- 1}, \tag {8}
+$$
+
+Direct inversion of RGB images into spectral space via SRFs may be suboptimal due to variations in SRFs across different spectral cameras and disturbances arising from factors such as RGB format compression and noise, and can not distinguish metamers. To address these challenges, we incorporate the AICD into a spectral response function (SRF) guided HSI initial estimation (SRF-guided HIE) mechanism to estimate the target spectral signal in a supervised manner. This process mimics the physical image formation and thus injects physics-aware reasoning into neural networks, turning an ill-posed problem into a constrained, interpretable task. Specifically, we incorporate the decomposed illumichroma prior from the AICD module and then estimate the coarse HSI signal via the SRF-guided HIE function:
+
+$$
+\tilde {\mathbf {Y}} = \mathcal {F} (\mathcal {H} (\mathbf {X}, \mathbf {U}, \mathbf {V}) \mathbf {A} ^ {T} \left(\mathbf {A} \mathbf {A} ^ {T}\right) ^ {- 1}, \tag {9}
+$$
+
+where  $\mathcal{F}$  denotes a neural network mapping function (a standard UNet-based network [7]),  $\mathcal{H}$  denotes a fusion operator (via concatenation and convolution) and  $\hat{\mathbf{Y}}$  denotes the SRF-guided HIE output. Now, Eq. (4) is implemented as:
+
+$$
+\mathcal {L} _ {Y} (\mathbf {Y}) = \left\| \mathbf {Y} - \mathcal {F} \left(\mathcal {H} (\mathbf {X}, \mathbf {U}, \mathbf {V}) \mathbf {A} ^ {T} \left(\mathbf {A} \mathbf {A} ^ {T}\right) ^ {- 1}\right) \right\| ^ {2}, \tag {10}
+$$
+
+# 3.4 Unsupervised High-Fidelity Spectral Regularization
+
+We introduce spectral diffusion models (SDMs) [27, 36] as spectral priors to refine the initial HSI estimated by the SRF-guided HIE and achieve accurate reconstruction. However, three key challenges must be addressed: (i) SDMs struggle to capture metameric feature distributions, limiting their ability to model subtle spectral variations. (ii) Directly modeling the spatial-spectral distribution of 3D hyperspectral data incurs a significant computational burden due to the high dimensionality of the data. (iii) A notable domain discrepancy exists between SRF-guided HIE results and real hyperspectral images (HSIs), suggesting that directly applying SDMs may result in suboptimal performance. To address these challenges, we introduce three key improvements for effectively integrating SDMs into spectral reconstruction: (1) metameric spectral augmentation, which creates comprehensive training data to enhance feature representation, (2) a lightweight spectral diffusion architecture to reduce computational complexity, and (3) gradient-based inner loop optimization to bridge the domain gap and improve reconstruction accuracy.
+
+Metameric spectra augmentation. Metameric augmentation leverages an orthogonal subspace decomposition [37] of a spectrum  $\mathbf{S}$  into a component that lies in the sensor-response range subspace and a residual in the null space, enabling different spectra to produce the same RGB response under a given spectral sensitivity [38]. With a spectral response matrix  $\mathbf{A}$ , a new metameric spectrum  $\mathbf{S}^{\dagger}$  can be synthesized following the concept of metameric black [39]:
+
+$$
+\mathbf {S} ^ {\dagger} = \mathcal {I} + \beta \mathcal {J} ^ {\dagger}, \tag {11}
+$$
+
+where we project  $\mathbf{S}$  onto the range space as  $\mathcal{J} = \mathbf{A}(\mathbf{A}^T\mathbf{A})^{-1}\mathbf{A}^T\mathbf{S}$  and define the residual null space as  $\mathcal{J}^\dagger = \mathbf{S} - \mathcal{J}$ . New metamer spectra are then synthesized by varying the scalar  $\beta$ . Because  $\mathcal{J}^\dagger$  lies in the null space of the response, changing  $\beta$  does not alter the RGB tristimulus, so  $\mathbf{S}$  and  $\mathbf{S}^\dagger$  are colorimetric matches while differing spectrally. This property reflects the broader fact that rich spectra are reduced to three sensor channels in trichromatic systems, making metamers common and exploitable for augmentation.
+
+We adopt this as a spectral-wise augmentation (rather than spatial flips or crops) to expand the diversity of training spectra while preserving color consistency seen by RGB sensors. Such augmentation can enhance robustness in spectral reconstruction pipelines, a concept established in color science as a way to explore spectra that map to the same color. Practically, we generate metamer data by sampling  $\beta$  uniformly from  $[0,1)$ ; note that setting  $\beta = 1$  recovers the original spectrum, while other
+
+values yield alternative metamers that share the same RGB under A. We produce metamers at a  $1:1$  ratio with standard spectra, doubling the dataset and providing both the original and augmented hyperspectral inputs for pre-training.
+
+Lightweight spectral diffusion architecture. HSI signals exhibit significant spatial sparsity [40, 41], suggesting that direct modeling of 3-D data cubes might be suboptimal. As such, we introduce a 1-D spectral diffusion model to capture the spectral distribution of HSI signals to address the second term in Eq. (3). Our SDM employs two iterative processes, following the standard DDPM [42], a forward diffusion process, and a reverse denoising process, as illustrated in Fig. 2. Unlike existing diffusion-based variations [33, 28], we adopt a 1-D MLP-based UNet denoising network to ensure that the diffusion model is compatible with 1-D spectral data. While it is feasible to capture the spectral distribution of HSI signals via our proposed SDM, the key question is: how can the trained SDM be incorporated to solve the second prior term -  $\log p(\mathbf{Y})$  of Eq. (3)?
+
+In our work, we assume that any spectral vector  $\mathbf{y} \in \mathbb{R}^{\mathbb{C}}$  of the target HSI  $\mathbf{Y}$  are i.i.d. (independently identically distributed), i.e.,  $-\log p(\mathbf{Y}) = -\sum_{\mathbf{y} \in \mathbf{Y}} \log p(\mathbf{y})$ , and each spectrum sample follows the spectral distribution learned by our proposed SDM, the deep generative prior, i.e.,  $\mathbf{y} = \mathbf{y}_0 \sim q(\mathbf{y}_0)$ . Consequently, the optimization problem Eq. (3) can be rewritten as:
+
+$$
+\min  _ {\mathbf {Y}} \mathcal {L} _ {Y} (\mathbf {Y}) + \lambda \sum_ {t, \mathbf {y}} \mathcal {L} _ {k l} \left(q \left(\mathbf {y} _ {t - 1} \mid \mathbf {y} _ {t}, \mathbf {y}\right) \| p _ {\theta} \left(\mathbf {y} _ {t - 1} \mid \mathbf {y} _ {t}\right)\right), \tag {12}
+$$
+
+where  $\lambda$  and  $\mathcal{L}_{kl}$  denote a balance hyperparameter and the KL divergence, respectively. Subsequently, we can use the parameterization trick to rewrite the second term in Eq. (3) as
+
+(13)
+
+With the above derivation, the final objective function of the proposed Diff-Spectra can be formulated as
+
+$$
+\min  _ {\mathbf {Y}} \mathcal {L} _ {Y} (\mathbf {Y}) + \lambda \sum_ {t, \mathbf {y}} \mathcal {L} _ {\theta} (\mathbf {y}, t). \tag {14}
+$$
+
+Note that this objective function is adopted during the fine-tuning process in Stage 3, where  $\lambda$  is set to 0.1 empirically.
+
+Inner loop optimization. It is impractical to optimize Eq. (14) simultaneously for all time steps. This is because an inherent domain gap exists between the spectral distribution learned by the SDM and the spectral distribution of the coarse-level HSI learned by the SRF-guided HIE network. Simply assuming these two spectral distributions are consistent without further adaptation can lead to suboptimal results. As such, we propose an inner loop optimization during each time step in the inference phase that performs gradient descent K times for each t, which serves as the test-time adaptation.
+
+# 4 Experiments and Analysis
+
+# 4.1 Implementation Details and Datasets
+
+We implemented Diff-Spectra using Pytorch. In Stage 1, we pre-train the SRF-guided HIE and AICD on the training dataset for 300 epochs with the Adam optimizer following [7]. Empirically, we set the learning rate to  $4 \times 10^{-4}$  and the batch size is 20. In Stage 2, we pre-train the SDM on our generated metamer dataset for 300 epochs with the Adam optimizer. The learning rate is set to  $1 \times 10^{-2}$ . The batch size is 1024, and the total diffusion steps  $\mathbf{T}$  is 5000. In Stage 3, we fine-tune the SRF-guided HIE and AICD, while freezing the SDM for 100 epochs with the Adam optimizer. Empirically, we set the learning rate to  $1 \times 10^{-4}$  and the batch size is 20. During inference, we input RGB images from the testing dataset and load the pre-trained SRF-guided HIE and AICD to obtain coarse-level HSIs. We then treat the coarse-level HSIs as learnable parameters and load the pre-trained SDM to update the parameters and generate refined HSIs, using a learning rate of  $1 \times 10^{-4}$  with diffusion steps of  $\mathbf{S} = 50$  and inner loop  $\mathbf{K} = 5$ . To evaluate the generalization and fidelity of our method, we use two HSI reconstruction datasets (ARAD-1K [43] and ICVL [9]) and two classification datasets (Indian Pines [1] and Pavia University [44]). Further details for implementation and dataset descriptions are provided in the Appendix A.2.
+
+![](images/82f68948219d9c4df11074d95827114679933b2b2ebd467445ecccb1948daf56.jpg)
+
+![](images/0edbbd7f3aba683e27313cd47a27920bf1527a294669a0a3dd5e509dac4ecc11.jpg)
+
+![](images/9d4648ef230660e973198d3fcef4f861456f10b117f57638d36a530b1b188f64.jpg)
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+![](images/6be105da03ce44f223491cf289ceec2a644a80f5f7dbc3d668191773398a0c52.jpg)
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+![](images/78cfe420aeb1b8418b60c3f985d9b459355cdc2e099e5afe5e69433485c74044.jpg)
+
+![](images/abaaeb4d3986a239ef20facf6fe571ea3651452a44779dc8fc44f2b7bb8395b1.jpg)
+
+![](images/40e9bb66dcde140d228ef03959ca11de671d4430c636b41135354eef5f79e9c0.jpg)
+
+![](images/c48a316188471534a5396adb42e96ff64ef509476a6acedb6da57260ffda7b4c.jpg)
+
+![](images/60891a1bac42fc400039dc056a058eee10acc57f49c533fbf949db08a5eae406.jpg)
+
+![](images/0c066f4f61b69d0f75ca8a30abc849953e0efb1f08d5cfc7b1f3bd4a1d936fee.jpg)
+
+![](images/0dfe99f8453cafb479fad2225da52857f55d74156fdcdb8878140c2b25debe0f.jpg)
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+![](images/8871c9b8de8534cc31af1fbf1b8d934906c4543c467ed125bcbd0ea60a57eb42.jpg)
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+![](images/664b670c9c1a002042c59770c0cfbed725b2e747e51c24801e9ea3aa4ca0fd1b.jpg)
+
+![](images/0112a2d6e8e54d676d0d0866d3894b1b468e59d41f6d2b3e91463d1d54068e35.jpg)  
+Figure 4: Visual comparisons on a randomly selected scene from the validation set of the ARAD-1K dataset with 4 spectral channels. The spectral curves (bottom-left) correspond to the selected blue Box of the RGB image. Please zoom in for a better comparison.
+
+![](images/42a9c37e97058b140695311a0e348141325874240aeecf49f9199548ad40b79f.jpg)
+
+![](images/b686c5cad2b04b73f3b160a901c1254678e2d23161aa5e513f67e241fff5dc4f.jpg)
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+![](images/df917c1b6a890f7f2546c12ef9022307cb0bdcddf2c57e2227b7849170532f7a.jpg)
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+![](images/9bd9cb0adb7effc2bb1ffd22af2aa06bf6143def4f74e25182f699bc269e8c4e.jpg)
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+![](images/a52bfb8b66213adad05f1d437703f5e1b5d0a16e398a254ec1cea02431c2e62c.jpg)
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+![](images/f61b653106ecea131acb3f8631522cfcb4e5e9b155e86561a1fb25e24ecd0560.jpg)
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+![](images/9a42972040d1113726493bf3f9fc0d73fce4579a81323e88c59504c8b80ab06d.jpg)
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+![](images/93a9d9f729237b4fc2175a7263e20068f5f41cbb7d2f2e8cb6ee1d83599d7db7.jpg)
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+![](images/f0edd1b514a6d61cecaf1e5fe56a64a38fc38d3a1dccfe959b85aa41709001c7.jpg)
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+![](images/4f7414574c5a93802f7d5c536122e6e27bda94aab9ed1a56d171097e9a26cb87.jpg)
+
+![](images/9adcb7f3e94e5f6804069ef9742807d8746105773b56a20c421d8f191256f4fa.jpg)
+
+Table 1: Comparison with SOTA methods on ARAD-1K [43] and ICVL [9] datasets. The best and second are shown in red and blue, respectively. Our method achieves the best performance on most metrics with relatively fewer parameters.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Venue</td><td rowspan="2">Params (M)</td><td colspan="4">ARAD-1K Dataset</td><td colspan="4">ICVL Dataset</td></tr><tr><td>ERGAS (↓)</td><td>SAM (↓)</td><td>SSIM (↑)</td><td>PSNR (↑)</td><td>ERGAS (↓)</td><td>SAM (↓)</td><td>SSIM (↑)</td><td>PSNR (↑)</td></tr><tr><td>AWAN [45]</td><td>CVPRW&#x27;20</td><td>4.04</td><td>17.45</td><td>9.83</td><td>0.909</td><td>31.22</td><td>8.32</td><td>4.59</td><td>0.918</td><td>31.92</td></tr><tr><td>HDNet [46]</td><td>CVPR&#x27;22</td><td>2.66</td><td>14.16</td><td>10.24</td><td>0.913</td><td>32.13</td><td>7.58</td><td>4.17</td><td>0.924</td><td>31.85</td></tr><tr><td>HINet [47]</td><td>CVPR&#x27;21</td><td>5.21</td><td>15.16</td><td>8.18</td><td>0.916</td><td>32.51</td><td>7.93</td><td>4.19</td><td>0.928</td><td>32.01</td></tr><tr><td>MST-L [48]</td><td>CVPR&#x27;22</td><td>2.45</td><td>12.53</td><td>7.47</td><td>0.922</td><td>33.90</td><td>5.44</td><td>3.51</td><td>0.935</td><td>32.87</td></tr><tr><td>MST++ [7]</td><td>CVPRW&#x27;22</td><td>1.62</td><td>9.18</td><td>6.05</td><td>0.928</td><td>34.32</td><td>4.82</td><td>3.04</td><td>0.941</td><td>32.44</td></tr><tr><td>PADUT [49]</td><td>ICCV&#x27;23</td><td>6.38</td><td>7.39</td><td>5.53</td><td>0.946</td><td>34.51</td><td>4.15</td><td>3.11</td><td>0.948</td><td>33.07</td></tr><tr><td>SST [50]</td><td>IJSWIS&#x27;24</td><td>12.74</td><td>8.29</td><td>6.01</td><td>0.933</td><td>33.95</td><td>4.67</td><td>3.71</td><td>0.935</td><td>32.71</td></tr><tr><td>CESST [8]</td><td>AAAI&#x27;24</td><td>1.54</td><td>7.85</td><td>5.88</td><td>0.931</td><td>34.74</td><td>4.09</td><td>3.27</td><td>0.939</td><td>32.96</td></tr><tr><td>SPECAT [51]</td><td>CVPR&#x27;24</td><td>0.37</td><td>8.62</td><td>6.10</td><td>0.930</td><td>33.48</td><td>4.92</td><td>3.81</td><td>0.944</td><td>32.54</td></tr><tr><td>Diff-Spectral (ours)</td><td>-</td><td>2.49</td><td>4.63</td><td>3.96</td><td>0.940</td><td>35.47</td><td>2.84</td><td>2.39</td><td>0.941</td><td>34.71</td></tr></table>
+
+# 4.2 Hyperspectral Image Reconstruction
+
+We present quantitative and qualitative comparisons with 9 state-of-the-art methods including AWAN [45], HINet [47], HDNet [46], MST-L [48], MST++ [7],SST [50], PADUT [49], CESST [8] and SPECAT [51].
+
+Qualitative Comparison. Visual Comparisons are given in Fig. 4 and Fig. 5. Fig. 4 compares the reconstructed HSIs with four randomly selected spectral channels using nine SOTA methods and our Diff-Spectra on the validation set of the ARAD-1K dataset. Fig. 5 shows the MSE error map between generated and ground-truth HSIs, calculating along the spectral dimension. It is observed that existing HSI reconstruction methods struggle with spectral intensity estimation and detail recovery, particularly in regions with high-frequency details such as the sky. In contrast, our approach excels at restoring intricate textures and achieving superior pixel-level smoothness. This improvement is attributed to the novel illumiChroma prior learned by the AICD module and the spectral prior learned by the SDM. The AICD module facilitates intrinsic image decomposition, such as illuminance and color information, guiding the initial estimation of the HSI signal with perceptually pleasing spatial features, while the SDM captures spectral distributions that refine the coarse-level HSI generated by the SRF-guided HIE, ensuring spectral consistency.
+
+Quantitative Comparison. We evaluate the performance using metrics including ERGAS, SAM, SSIM, and PSNR. The first two metrics assess spectral quality, while the latter two evaluate spatial quality. Lower ERGAS and SAM values indicate better spectral quality, while higher SSIM and PSNR values signify better spatial quality. As shown in Table 1, our method achieves the best performance over most metrics on both the ARAD-1K dataset and the ICVL dataset. Our approach, Diff-Spectra, outperforms state-of-the-art methods by delivering the highest PSNR and lowest ERGAS (best spectral and spatial reconstruction quality) with significantly lower computational complexity.
+
+# 4.3 Metameric Dilemma Evaluation
+
+To demonstrate that existing methods suffer from the metameric dilemma and to validate the effectiveness of our proposed Diff-Spectra in mitigating this issue, we generate metamer HSI data from
+
+![](images/ee6581524717d8aa83a12e104dfc44fcffd50de1dad09ff36392b5c6b8f37931.jpg)  
+Figure 5: The MSE error maps obtained from the validation subset of the ICVL dataset, which are calculated along the spectral direction, showcasing the discrepancies between the reconstructed HSIs and the corresponding ground truths.
+
+Table 2: Metameric dilemma Evaluation and HSI classification comparison.  
+
+<table><tr><td>Method</td><td>Data</td><td>ERGAS ↓</td><td>SAM ↓</td><td>SSIM ↑</td><td>PSNR ↑</td></tr><tr><td rowspan="2">MST++ [7]</td><td>std</td><td>9.18</td><td>6.05</td><td>0.928</td><td>34.32</td></tr><tr><td>meta</td><td>84.77</td><td>50.92</td><td>0.8696</td><td>27.14</td></tr><tr><td rowspan="2">PADUT [49]</td><td>std</td><td>7.39</td><td>5.53</td><td>0.946</td><td>34.51</td></tr><tr><td>meta</td><td>80.07</td><td>47.91</td><td>0.872</td><td>27.90</td></tr><tr><td rowspan="2">SST [50]</td><td>std</td><td>8.29</td><td>6.01</td><td>0.933</td><td>33.95</td></tr><tr><td>meta</td><td>70.11</td><td>52.38</td><td>0.884</td><td>27.88</td></tr><tr><td rowspan="2">CESST [8]</td><td>std</td><td>7.85</td><td>5.88</td><td>0.931</td><td>34.74</td></tr><tr><td>meta</td><td>59.82</td><td>40.74</td><td>0.885</td><td>28.59</td></tr><tr><td rowspan="2">SPECAT [51]</td><td>std</td><td>8.62</td><td>6.10</td><td>0.930</td><td>33.48</td></tr><tr><td>meta</td><td>48.90</td><td>63.59</td><td>0.8701</td><td>26.63</td></tr><tr><td rowspan="2">Diff-Spectral (ours)</td><td>std</td><td>4.63</td><td>3.96</td><td>0.940</td><td>35.47</td></tr><tr><td>meta</td><td>21.98</td><td>24.11</td><td>0.906</td><td>31.61</td></tr></table>
+
+(a) Comparison with SOTA methods on standard (std) (b) Quantitative comparison of different methods in terms and metamer (meta) data. of the accuracy for each class.
+
+<table><tr><td>Class No.</td><td>AWAN [45]</td><td>MST++ [7]</td><td>PADUT [49]</td><td>MST-L [48]</td><td>Ours</td></tr><tr><td>1</td><td>82.71</td><td>81.45</td><td>87.49</td><td>90.47</td><td>92.72</td></tr><tr><td>2</td><td>59.36</td><td>84.27</td><td>92.35</td><td>86.56</td><td>91.45</td></tr><tr><td>3</td><td>73.65</td><td>60.33</td><td>62.54</td><td>88.47</td><td>89.97</td></tr><tr><td>4</td><td>80.04</td><td>95.14</td><td>91.87</td><td>89.58</td><td>89.71</td></tr><tr><td>5</td><td>99.46</td><td>99.17</td><td>100.00</td><td>99.47</td><td>100.00</td></tr><tr><td>6</td><td>95.19</td><td>90.62</td><td>84.28</td><td>47.59</td><td>90.68</td></tr><tr><td>7</td><td>78.42</td><td>76.04</td><td>75.61</td><td>89.55</td><td>81.83</td></tr><tr><td>8</td><td>81.47</td><td>98.15</td><td>93.46</td><td>70.43</td><td>69.39</td></tr><tr><td>9</td><td>94.83</td><td>90.52</td><td>94.88</td><td>82.64</td><td>98.44</td></tr><tr><td>OA (%)</td><td>75.26</td><td>82.37</td><td>85.48</td><td>81.45</td><td>89.02</td></tr><tr><td>AA (%)</td><td>83.35</td><td>85.19</td><td>86.93</td><td>82.97</td><td>88.15</td></tr><tr><td>κ</td><td>0.7124</td><td>0.7941</td><td>0.8039</td><td>0.7355</td><td>0.8349</td></tr></table>
+
+the original ARAD-1K dataset following [52] and use metamer HSI data to synthesize corresponding RGB images (i.e., metamer). Next, we test several pre-trained models using both standard RGB images and metamer RGB images (Note that these models are pre-trained on standard data), including MST++ [7], CESST [8], PADUT [49], SST [50], SPECAT [51], and our proposed Diff-Spectra. The quantitative results are given in Table 2(a). As can be seen, all the existing methods experience catastrophic performance drops in terms of PSNR and SAM in the presence of metamers, which is also known as the metameric dilemma.
+
+# 4.4 Evaluation on HSI Classification
+
+To further evaluate the fidelity and verify the reliability of the hyperspectral images generated by our method, we conduct experiments on the hyperspectral image classification task based on a pre-trained HSI classification model, SpectralFormer [53]. We compare our approach with existing methods on two widely used HSI classification datasets: the Indian Pines dataset and the Pavia University dataset, conducting both quantitative and qualitative analyses. For our evaluation, we reconstruct HSI images using assorted pre-trained HSI reconstruction methods. Subsequently, these synthesized HSI images are employed as the input to a pre-trained HSI classification model, SpectralFormer [53], which serves as a benchmark for performance evaluation.
+
+**Evaluations.** We evaluate the quantitative performance on the Pavia University dataset using three widely adopted metrics: Overall Accuracy (OA), Average Accuracy (AA), and the Kappa Coefficient  $(\kappa)$ , as shown in Table 2(b). As can be seen, our method achieves the best OA  $(89.02\%)$ , AA  $(88.15\%)$ , and  $\kappa$ $(0.8349)$ . It ranks first in Classes 1, 3, and 9 and ties for first in Class 5, while remaining competitive in the remaining classes.
+
+# 4.5 Ablation Study
+
+Break-down Ablation. We perform bread-down ablation to investigate the effectiveness of each module in Table 3(a) and Figure 6. Comparing Variant 1 with SimDiff-Spectra (i.e., a UNet-based HSI reconstruction network, which is similar to [7]), we find that the AICD primarily contributes to the spatial details recovery of the HSI signal, which aligns with our original design intention. Comparing Variant 1 with Variant 2, we observe that the SRF-guided HIE mechanism enhances
+
+Table 3: Ablation studies of our proposed modules and inner loop mechanism.  
+
+<table><tr><td>Method</td><td>AICD</td><td>SRF</td><td>SDM</td><td>IP</td><td>PSNR</td><td>SAM</td></tr><tr><td>SimDiff-Spectral</td><td></td><td></td><td></td><td>32.57</td><td>7.52</td><td>Diff-Spectral</td></tr><tr><td>Variant 1</td><td>✓</td><td></td><td></td><td>34.18</td><td>6.08</td><td>Diff-Spectral</td></tr><tr><td>Variant 2</td><td>✓</td><td>✓</td><td></td><td>34.95</td><td>5.41</td><td>Diff-Spectral</td></tr><tr><td>Variant 3</td><td>✓</td><td>✓</td><td>✓</td><td>34.19</td><td>5.84</td><td>Diff-Spectral</td></tr><tr><td>Diff-Spectral</td><td>✓</td><td>✓</td><td>✓</td><td>35.47</td><td>3.96</td><td>Diff-Spectral</td></tr></table>
+
+(a) Break-down ablations of our proposed Diff-Spectra, where IP de-(b) The right sub-table investigate the inner notes the inner loop. Inner loop steps (K) in the SDM.
+
+both spatial and spectral performance due to the incorporation of physical constraints. Notably, by comparing Variant 2 with Variant 3, we find that simply incorporating SDM causes a severe performance drop. Finally, comparing Variant 3 with the full Diff-Spectra model, we find that the inner loop facilitates effective integration of SDM and primarily contributes to spectral recovery via the learned spectral distribution regularization.
+
+Inner Loop Analysis. We analyze the impact of the inner loop step parameter  $\mathbf{K}$  of the SDM in Table 3(b). The results indicate that performance improves significantly when  $\mathbf{K}$  is greater than 1 as compared to when  $\mathbf{K} = 1$ . Note that when  $\mathbf{K} = 1$ , the performance is even worse than Variant 2. This is because a domain gap exists between the spectral distribution learned by the SDM and the spectral distribution of the coarse-level HSI learned by the SRF-guided HIE network. Assuming these two spectral distributions are consistent without further adaptation can lead to suboptimal results. Directly assuming these two spectral distributions are consistent will introduce an inferior influence.
+
+![](images/f2081cedba9240b27099fa6695279eac1ba434f95e94ca02cf1b7fc8bc6d30ab.jpg)  
+Figure 6: Break-down ablation study. The reconstruction MSE error map evaluation on the validation set of the ICVL dataset.
+
+# 5 Conclusion
+
+In this paper, we propose Diff-Spectra, which integrates supervised physics-aware spectral estimation and unsupervised high-fidelity spectral regularization for spectral reconstruction. Especially, the supervised physics-aware spectral estimation consists of an adaptive illumichroma decoupling (AICD) and a learnable SRF-guided HIE mechanism, mimicking the physical image formation, and thus injecting physics-aware reasoning into neural networks, turning an ill-posed problem into a constrained, interpretable task. We further introduce an unsupervised high-fidelity spectral regularization by incorporating a pre-trained spectral diffusion model (SDM) to regularize the coarsely estimated HSI signal from the SRF-guided HIE mechanism with high-fidelity real-world spectral distributions. Extensive experiments on both spectral reconstruction and HSI classification demonstrate that Diff-Spectra significantly outperforms SOTA methods. Future work will focus on the investigation of the spectral distribution gap between the HSI estimated by the SRF-guided HIE and the distribution learned by the SDM, such as quantizing each distribution into a dictionary and calculating their distance.
+
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+
+[51] Zhiyang Yao, Shuyang Liu, Xiaoyun Yuan, and Lu Fang. Specat: Spatial-spectral cumulative-attention transformer for high-resolution hyperspectral image reconstruction. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 25368–25377, 2024.  
+[52] Graham D Finlayson and Peter Morovic. Metamer sets. JOSA A, 22(5):810-819, 2005.  
+[53] Danfeng Hong, Zhu Han, Jing Yao, Lianru Gao, Bing Zhang, Antonio Plaza, and Jocelyn Chanussot. Spectralformer: Rethinking hyperspectral image classification with transformers. IEEE Transactions on Geoscience and Remote Sensing, 60:1-15, 2021.  
+[54] Boaz Arad, Radu Timofte, Ohad Ben-Shahar, Yi-Tun Lin, and Graham D Finlayson. Ntire 2020 challenge on spectral reconstruction from an rgb image. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pages 446-447, 2020.  
+[55] Magnus Magnusson, Jakob Sigurdsson, Sveinn Eirikur Armansson, Magnus O Ulfarsson, Hilda Deborah, and Johannes R Sveinsson. Creating rgb images from hyperspectral images using a color matching function. In IGARSS 2020-2020 IEEE International Geoscience and Remote Sensing Symposium, pages 2045-2048, 2020.
+
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+
+# A Implementation Details and Datasets.
+
+# A.1 Implementation Details
+
+Three-stage Training Scheme. We implement our proposed Diff-Spectra in the PyTorch framework. Specifically, our proposed Diff-Spectra adopts a three-stage training strategy:
+
+- **Stage 1**, we follow [7] that utilizes RGB-HSI pairs from the training dataset to pre-train the SRF-guided HIE network  $\mathcal{F}(\cdot)$ , fuse-and-conv operation  $\mathcal{H}(\cdot)$ , and AICD module  $\mathcal{G}(\cdot)$  jointly, optimized with the corresponding loss (i.e., Eq. 10 in the main content), while freezing the parameters of the SDM;  
+- **Stage 2**, we follow the standard diffusion model DDPM [42] that utilizes HSI from the training dataset to pre-train the SDM, optimized with the loss Eq. 13, while freezing the parameters of the SRF-guided HIE;  
+- **Stage 3**, to align the distribution gap between the estimated HSI from the SRF-guided HIE network and the learned spectral prior from SDM, we use Eq. 14 in the main content to fine-tune the SRF-guided HIE network and AICD module for 100 epochs, while freezing the SDM. Note that the SDM is treated as a regularizer in this stage. Empirically, we set the learning rate to  $1 \times 10^{-4}$  and the batch size is 20.
+
+Test-time Adaptation. The pre-trained SRF-guided HIE is applied to RGB images from the testing dataset to generate an initial coarse-level HSI signal  $\tilde{\mathbf{Y}}$ . Next, we treat  $\tilde{\mathbf{Y}}$  as trainable parameters, and a spectrum  $\tilde{\mathbf{y}}$  is sampled from  $\tilde{\mathbf{Y}}$ . We assume each sampled spectrum satisfies the spectral distribution learned by the SDM, i.e.,  $\tilde{\mathbf{y}} = \mathbf{y}_s \sim q(\mathbf{y}_0)$ , where  $s \in (0, T)$  and  $T$  is the time step trained in Stage 2. Now, we can use Eq. 13 (in the main content) to refine the coarse-level HSI signal  $\tilde{\mathbf{Y}}$  during the sampling process of SDM. However, it is impractical to optimize Eq. 13 for all  $t$  due to the inherent distribution difference between the HSI signal generated by SRF-guided HIE and the spectral distribution learned by SDM. Thus, we perform the gradient update  $\mathbf{K}$  times in each time step  $t$ , which we named as the inner loop optimization.
+
+# A.2 Datasets.
+
+ARAD-1K Dataset [43]. The ARAD-1K dataset includes 950 RGB-HSI pairs, with 900 for training and 50 for validation, at a  $482 \times 512$  resolution across 31 spectral channels (400–700nm). This dataset not only stands as the largest collection available for HSI reconstruction tasks but also integrates content from preceding compilations, notably the NTIRE 2020 HSI dataset [54]. Each HSI in this collection is captured with a spatial resolution of  $482 \times 512$ , spanning 31 spectral channels ranging from 400nm to 700nm.
+
+ICVL Dataset [9]. The ICVL dataset contains 201 HSIs with a resolution of  $1300 \times 1392$ . As it lacks the provision of aligned RGB images, we use the spectral sampling method proposed by Magnusson et al. [55] to generate the corresponding RGB images. Given that 18 of these images have different resolutions, we leverage the remaining 183 image pairs that maintain resolution consistency, allocating 147 pairs for training and 36 for testing.
+
+Indian Pine Dataset [1]. The Indian Pine dataset records the landscape over an area in Northwestern Indiana, USA, using the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) sensor. It comprises  $145 \times 145$  pixels, each with a ground sampling distance (GSD) of 20 meters, and encompasses 220 spectral bands that span the wavelength range from 400 nanometers to 2500 nanometers, achieving a spectral resolution of 10 meters. Following the elimination of 20 bands characterized by noise and water absorption, 200 spectral bands were kept, specifically bands 1-103, 109-149, and 164-219. The scene under investigation features 16 primary categories, including corn, oats, buildings, etc.
+
+Pavia University Dataset [44]. It is collected by the Reflective Optics System Imaging Spectrometer (ROSIS) sensor, which surveyed the area surrounding Pavia University in Pavia, Italy. Capable of capturing 103 spectral bands that range from 430 nanometers to 860 nanometers, the resulting image is composed of  $610 \times 340$  pixels, each with a ground sampling distance (GSD) of 1.3 meters. This particular scene encompasses 9 distinct land cover classes, including asphalt, grass, trees, etc.
+
+Table 4: Quantitative comparison of different methods in terms of the accuracy for each class, as well as the overall performance using metrics - Overall Accuracy (OA), Average Accuracy (AA), and Kappa coefficient  $(\kappa)$  on the Indian Pines dataset. The best one is shown in bold.  
+
+<table><tr><td>Class No.</td><td>AWAN [45]</td><td>HDNet [46]</td><td>HINet [47]</td><td>SPECAT [51]</td><td>MST-L [48]</td><td>MST++ [7]</td><td>PADUT [49]</td><td>Diff-Spectra (Ours)</td></tr><tr><td>1</td><td>54.16</td><td>55.69</td><td>72.74</td><td>61.71</td><td>66.48</td><td>68.32</td><td>70.15</td><td>67.41</td></tr><tr><td>2</td><td>40.21</td><td>57.38</td><td>71.47</td><td>74.24</td><td>72.34</td><td>74.35</td><td>75.18</td><td>78.72</td></tr><tr><td>3</td><td>73.82</td><td>82.14</td><td>91.74</td><td>91.38</td><td>95.54</td><td>90.16</td><td>94.27</td><td>96.46</td></tr><tr><td>4</td><td>85.68</td><td>84.17</td><td>90.67</td><td>90.05</td><td>96.24</td><td>85.48</td><td>83.74</td><td>96.72</td></tr><tr><td>5</td><td>81.11</td><td>78.63</td><td>93.45</td><td>83.48</td><td>85.21</td><td>91.52</td><td>92.45</td><td>94.01</td></tr><tr><td>6</td><td>96.75</td><td>96.03</td><td>98.42</td><td>94.17</td><td>96.72</td><td>95.92</td><td>97.22</td><td>94.94</td></tr><tr><td>7</td><td>66.31</td><td>76.54</td><td>72.81</td><td>73.88</td><td>76.18</td><td>77.51</td><td>80.92</td><td>73.48</td></tr><tr><td>8</td><td>48.38</td><td>59.44</td><td>65.59</td><td>64.02</td><td>59.18</td><td>60.17</td><td>62.30</td><td>67.44</td></tr><tr><td>9</td><td>44.69</td><td>63.19</td><td>71.32</td><td>72.42</td><td>80.03</td><td>68.27</td><td>65.27</td><td>70.02</td></tr><tr><td>10</td><td>96.70</td><td>95.88</td><td>100.00</td><td>95.32</td><td>100.00</td><td>100.00</td><td>100.00</td><td>98.87</td></tr><tr><td>11</td><td>73.14</td><td>89.57</td><td>85.11</td><td>90.54</td><td>91.26</td><td>84.58</td><td>85.52</td><td>85.77</td></tr><tr><td>12</td><td>17.25</td><td>55.42</td><td>82.47</td><td>89.27</td><td>89.81</td><td>85.92</td><td>84.88</td><td>90.25</td></tr><tr><td>13</td><td>90.44</td><td>98.31</td><td>100.00</td><td>98.93</td><td>94.42</td><td>100.00</td><td>100.00</td><td>100.00</td></tr><tr><td>14</td><td>32.27</td><td>57.45</td><td>40.25</td><td>87.34</td><td>76.54</td><td>62.51</td><td>67.24</td><td>91.91</td></tr><tr><td>15</td><td>81.82</td><td>81.82</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td></tr><tr><td>16</td><td>40.00</td><td>100.00</td><td>80.00</td><td>100.00</td><td>100.00</td><td>80.00</td><td>100.00</td><td>100.00</td></tr><tr><td>OA (%)</td><td>58.26</td><td>70.41</td><td>73.02</td><td>77.32</td><td>76.36</td><td>74.15</td><td>75.85</td><td>78.14</td></tr><tr><td>AA (%)</td><td>62.73</td><td>76.66</td><td>82.25</td><td>85.42</td><td>86.25</td><td>82.79</td><td>84.95</td><td>87.21</td></tr><tr><td>κ</td><td>0.5161</td><td>0.5918</td><td>0.6844</td><td>0.7088</td><td>0.7304</td><td>0.7008</td><td>0.7344</td><td>0.7571</td></tr></table>
+
+![](images/535c4e8b2b76c9b0d57807751797b77771152d3ff7abc10aac9f524a3b35fe95.jpg)  
+Figure 7: Visual comparisons on a randomly selected scene from the validation set of the ARAD-1K dataset with 4 spectral channels. The RGB patch corresponds to the selected red box of the RGB image. The spectral curves (bottom-left) correspond to the selected blue Box of the RGB image. Please zoom in for a better comparison.
+
+# B Additional Experiment Results
+
+Hyperspectral Image Classification. We further provide more quantitative comparisons on the Indian Pines dataset in Table 4. In this table, we further add three methods for more generalized comparisons, including HDNet [46], HINet [47], and SPECAT [51]. As can be seen, our method outperforms existing methods on most classes, and enjoys the best performance over the three metrics: Overall Accuracy (OA), Average Accuracy (AA), and Kappa coefficient  $(\kappa)$ .
+
+Hyperspectral Image Reconstruction. We provide more visual comparisons in Fig. 7 and Fig. 8. Specifically, Fig. 7 illustrates the false color results on four spectrum bands,  $400\mathrm{nm}$ ,  $500\mathrm{nm}$ ,  $600\mathrm{nm}$  and  $700\mathrm{nm}$ , respectively, which is the "ARAD-1K-0901" image chosen from the validation set of ARAD-1K dataset. In the top-left corner of Fig. 7, the RGB patch corresponds to the selected red box of the RGB image. The spectral curves (bottom left) correspond to the selected blue Box of the RGB image. Please zoom in for a better comparison. Fig. 8 illustrates the false color images on four spectrum bands,  $440\mathrm{nm}$ ,  $500\mathrm{nm}$ ,  $600\mathrm{nm}$  and  $700\mathrm{nm}$ , respectively, which is the "nachal-0823-1147" image chosen from the testing set of ICVL dataset. In the top-left corner of Fig. 8, the RGB patch corresponds to the selected red box of the RGB image. The spectral curves (bottom left) correspond to the selected blue Box of the RGB image. Please zoom in for a better comparison. As can be seen, our method can recover more precise texture information and better pixel-level smoothness over other SOTA methods. In addition, both spectral intensity curves in Fig. 7 and Fig. 8 show that our method can recover more precise spectral values and the spectral distribution of our method is closer to the ground truth compared with existing methods, especially in the long-wavelength spectrum (e.g., from
+
+![](images/f353b1b54ea338c50453a87cf525bedcd173ac950b56cbf48cc610cb437e3941.jpg)  
+Figure 8: Visual comparisons on a randomly selected scene from the validation set of the ICVL HSI dataset with 4 spectral channels. The RGB patch corresponds to the selected red box of the RGB image. The spectral curves (bottom left) correspond to the selected blue Box of the RGB image. Please zoom in for a better comparison.
+
+$600nm - 700nm)$ , which is benefited by both image-level prior and spectral-level prior introduced in our model.
\ No newline at end of file
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+# Unsupervised Federated Graph Learning
+
+Lele Fu $^{1}$ , Tianchi Liao $^{1}$ , Sheng Huang $^{1}$ , Bowen Deng $^{1}$ , Chuanfu Zhang $^{1}$ , Shirui Pan $^{2}$ , Chuan Chen $^{1*}$
+
+<sup>1</sup>Sun Yat-sen University, Guangzhou, China
+
+$^{2}$ Griffith University, Brisbane, Australia
+
+{fulle, liaotch, huangsh253, dengbw3}  $@$  mail2.sysu.edu.cn
+
+s.pan@griffith.edu.au, {zhangchf9,chenchuan}@mail.sysu.edu.cn
+
+# Abstract
+
+Federated graph learning (FGL) is a privacy-preserving paradigm for modeling distributed graph data, designed to train a powerful global graph neural network. Existing FGL methods predominantly rely on label information during training, effective FGL in an unsupervised setting remains largely unexplored territory. In this paper, we address two key challenges in unsupervised FGL: 1) Local models tend to converge in divergent directions due to the lack of shared semantic information across clients. Then, how to align representation spaces among multiple clients is the first challenge. 2) Conventional federated weighted aggregation easily results in degrading the performance of the global model, then which raises another challenge, namely how to adaptively learn the global model parameters. In response to the two questions, we propose a tailored framework named FedPAM, which is composed of two modules: Representation Space Alignment (RSA) and Adaptive Global Parameter Learning (AGPL). RSA leverages a set of learnable anchors to define the global representation space, then local subgraphs are aligned with them through the fused Gromov-Wasserstein optimal transport, achieving the representation space alignment across clients. AGPL stacks local model parameters into third-order tensors, and adaptively integrates the global model parameters in a low-rank tensor space, which facilitates to fuse the high-order knowledge among clients. Extensive experiments on eight graph datasets are conducted, the results demonstrate that the proposed FedPAM is superior over classical and SOTA compared methods.
+
+# 1 Introduction
+
+In response to growing concerns over privacy protection [1, 2, 3], the storage of graph data is becoming increasingly decentralized, where individual clients hold their own private subgraphs. Nevertheless, distributed subgraphs result in the emergence of data silos, triggering data unavailability and impeding the generalization of graph neural networks (GNNs) [4, 5, 6, 7, 8]. In this context, federated graph learning (FGL) [9, 10, 11, 12, 13, 14] has emerged as a paradigm that enables the collaborative training of a powerful GNN across multiple clients, while maintaining the usability of private subgraphs without compromising their confidentiality.
+
+FGL has made substantial progress in recent years. In the context of graph-level tasks, studies [15, 16, 17] primarily concentrate on handling the challenges arising from distributional heterogeneity. For node-level tasks, considerable attentions [18, 9, 19] are devoted to defining the topological heterogeneity in distributed subgraphs and overcoming its adverse effects on the effectiveness of federated training. The existing approaches provide innovative insights and solid solutions for FGL.
+
+![](images/84149edb40ad008c765134f9c310c9250a1bfd86fff7e64dc14f2c7172667d3c.jpg)  
+Figure 1: Scatter comparison on Physics dataset for FedAvg and FedPAM. It can be seen that FedPAM achieves a better inter-class separation, aligning the representation spaces of different clients, while FedAvg fails to do so.
+
+![](images/234f525def7adbff5824211deacdae062a8d217fb04bbe878dc71fb78c059b1d.jpg)  
+Figure 2: Comparison of client weights on CiteSeer dataset for FedAvg and FedPAM. FedPAM learns an adaptive weight for each client according to the data heterogeneity while FedAvg assigns a fixed weight for each client based on the number of local samples.
+
+![](images/b4219a40543fc67727a1aa056fe548372ffaa5ecfedce754bfb8a4b3d85def91.jpg)  
+(a) Statistics of CiteSeer
+
+![](images/42e277c25f847886c3db5819db3738c7c947c405534ede036e7f15fc3e4123d9.jpg)  
+(b) Comparison of client weights
+
+However, they inherently depend on label information during training, indicating that they adhere to supervised or semi-supervised learning paradigms. In practice, data labels are frequently inaccessible, and the lack of consistent semantics across clients further exacerbates the difficulties of federated learning (FL) [20, 21, 22, 23, 24, 25]. As a result, enabling effective FGL in an unsupervised setting poses a significant challenge. For unsupervised FGL, one of the most straightforward solutions is to leverage mature FL frameworks in conjunction with self-supervised learning (SSL) strategies [26, 27, 28, 29]. For example, clients perform local training using the SimCLR [30], while the server aggregates model parameters by FedAvg [31]. However, naively transplanting existing approaches fails to effectively cope with the common issue of heterogeneous data in FL. Moreover, some unsupervised FL algorithms originally designed for non-graph data are also worth mentioning. Studies [32, 33] used the clustering strategies to enhance the consensus of global space. Zhuang et al. [34, 35] proposed to dynamically update the local models based on the parameter differences between the global and local models. Liao et al. [36] addressed the problem of representation collapse in unsupervised FL.
+
+These methods are insightful for unsupervised FL, but they exhibit two primary limitations: 1) Inability to align the representation spaces of various clients. Due to the absence of labels, there are no consistent signals among clients, which can easily lead to shifts in the representation spaces and weaken the generalization for global model. Although studies [32, 36] adopt clustering or optimal transport (OT) to enhance the separability of representations, they merely alleviate representation collapse and do not achieve a unified representation space across clients. Especially, when faced with graph data, their learned representation spaces are prone to fall into suboptimality due to complex topological networks. 2) Unable to adaptively learn the optimal parameters of global model. Most existing federated aggregation methods use weighted averaging based on number of local samples, which might degrade the performance of global model. Studies [34, 35, 36] introduce divergence-aware and multi-objective aggregation strategies, but their matrix-level designs fall short in effectively capturing the complementary information among local models. From Fig. 1(a), we can see that the scatter output by FedAvg is dispersed throughout the space, the nodes of the same categories from various clients cannot be clustered together, indicating that representation spaces of different clients are unaligned. The subgraph statistics for different clients are reported in Fig. 2(a). The total numbers of nodes in the different clients are almost the same, but the numbers of nodes in each category vary considerably, showing different heterogeneous situations. If the traditional FedAvg is used, each client is assigned to almost a same weight as shown in Fig. 2(b), which is clearly unreasonable.
+
+In light of the above concerns, we propose a tailored framework FedPAM for unsupervised FGL, which mainly consists of two key modules: Representation Space Alignment (RSA) and Adaptive Global Parameter Learning (AGPL). Specifically, RSA aims to learn a set of cross-client anchors, which are used to define the global representation space. To align the representation spaces of various clients, the fused Gromov-Wasserstein optimal transport (FGW-OT) [37] is employed to establish mapping between local subgraphs and the anchor graph with minimal cost. Additionally, the anchors also serve as global space projector, projecting local embeddings into the global space to facilitate the contrastive learning, thereby mitigating the biased training caused by data heterogeneity. AGPL departs entirely from the conventional weighted aggregation paradigm. It stacks parameters of local models into third-order tensors, and leverages the low-rank tensor decomposition to capture high-order correlations among clients. Further, the optimal parameters of global models are adaptively learned in the low-rank tensor space, enabling effective integration of local knowledge. From Figs. 1(b) and 2(b), it can be observed that the proposed FedPAM can effectively align the representation spaces
+
+across clients and adaptively learn the global model parameters. The framework of the proposed FedPAM is presented in Fig. 3. We summarize the contributions of this paper from four-fold:
+
+- We are the first to identify the challenges inherent in unsupervised FGL and design a unified framework to address them. This framework proposes the representation space alignment and the adaptive global parameter learning for unsupervised FGL.  
+- We learn a set of anchors to span the global representation space, and adopt the FGW-OT to match local subgraphs with the anchor graph, thus achieving the alignment of representation spaces across multiple clients.  
+- We stack the local model parameters into third-order tensors and extract its low-rank components to capture the high-order correlations among clients. In the high-dimensional space, the optimal global model parameters are obtained by adaptive fusion.  
+- To verify the effectiveness of the proposed FedPAM, we conduct the comparative experiments on eight graph datasets. Compared to traditional and SOTA methods, FedPAM achieves more superior performance.
+
+![](images/cf87d26fc84f97cdf26e9e77dd7b4af3136ae626d3154180fbf34e97d395b17e.jpg)  
+Figure 3: The framework of the proposed FedPAM. The client performs the RSA module and the server performs the AGPL module, models and anchors are transferred between the server and clients.
+
+# 2 Preliminaries
+
+Federated Graph Learning. Given a FGL system with a central server and  $K$  clients, each client possesses a private subgraph  $\mathcal{G}_k = \{\mathcal{V}_k,\mathcal{E}_k,\mathbf{X}_k\}$ , where  $\mathcal{V}_k$ ,  $\mathcal{E}_k$ , and  $\mathbf{X}_k$  indicate the vector set, the edge set, and the feature matrix, respectively. Specifically,  $\mathbf{X}_k\in \mathbb{R}^{N_k\times d}$  has  $N_{k}$  nodes and  $d$  dimensions. The adjacency matrix  $\mathbf{A}_k\in \mathbb{R}^{N_k\times N_k}$  is induced by the edge set  $\mathcal{E}_k$ . If there is an edge between the  $i$ -th node and the  $j$ -th node, then  $\mathbf{A}_k(i,j) = 1$ . Otherwise,  $\mathbf{A}_k(i,j) = 0$ . In the unsupervised FGL, the node labels  $\mathbf{Y}_k = [y_k^1,\dots,y_k^{N_k}]$  are all unavailable during the GNN's training. Notably, we assume that all subgraphs originate from a complete graph, namely  $\mathcal{G}_k\in \mathcal{G}$ . At the beginning of a communication round in FL, the central server distributes the global GNN  $F_{\mathbf{W}}$  parameterized by  $\mathbf{W}$  to each client and starts the local training, the computation of a GNN can be summarized as
+
+$$
+\mathbf {z} _ {i} ^ {l + 1} = \delta \left(F _ {\mathbf {W} _ {l}} \left(\mathbf {z} _ {i} ^ {l}, \operatorname {A G G} \left(\mathbf {z} _ {j} ^ {l}, \mathbf {A} _ {i j}\right)\right)\right) | \forall v _ {j} \in \mathcal {N} (v _ {i})), \tag {1}
+$$
+
+where  $\delta (\cdot)$  denotes the activation function,  $F_{\mathbf{W}_l}$  denotes the  $l$  -th encoder layer,  $\mathbf{z}_i^{l + 1}$  is the  $i$  -th node's representation at the  $(l + 1)$  -th layer,  $\mathcal{N}(v_i)$  is the neighborhood node set for node  $v_{i}$ ,  $\mathrm{AGG}(\cdot)$  denotes
+
+the embedding fusion manner. When the local training finishes, local GNNs are uploaded to the server for aggregating the global GNN, which is formulated as
+
+$$
+\mathbf {W} = \sum_ {k = 1} ^ {K} \frac {N _ {k}}{N} \mathbf {W} _ {k}, \tag {2}
+$$
+
+where  $N = \sum_{k=1}^{K} N_k$  denotes the sum of local nodes.
+
+Tensor Nuclear Norm. For a third-order tensor  $\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$ , its the  $j$ -th frontal slice is denoted as  $\mathcal{X}^{(j)} \in \mathbb{R}^{n_1 \times n_2}$ . The fast Fourier transformation (FFT) is used to transform a tensor into the frequency domain, which is formulated as  $\hat{\mathcal{X}} = \mathrm{fft}(\mathcal{X}, [], 3)$ , its inverse operation is written as  $\mathcal{X} = \mathrm{ift}(|\hat{\mathcal{X}}, [], 3)$ . The product  $\mathcal{Z} \in \mathbb{R}^{n_1 \times n_4 \times n_3}$  of  $\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$  and  $\mathcal{Y} \in \mathbb{R}^{n_2 \times n_4 \times n_3}$  is defined as
+
+$$
+\boldsymbol {z} = \boldsymbol {x} * \boldsymbol {y} = \operatorname {f o l d} (\operatorname {b c i r c} (\boldsymbol {x}) \operatorname {b v e c} (\boldsymbol {y})), \tag {3}
+$$
+
+where  $\operatorname{bvec}(\mathcal{Y}) = [\mathcal{Y}^{(1)}; \ldots; \mathcal{Y}^{(n_3)}]$  is the block vectorizing operation, and fold(bvec( $\mathcal{Y}$ ) is the inverse operation.  $\operatorname{bcirc}(\mathcal{X})$  is defined as
+
+$$
+b c i r c (\boldsymbol {\chi}) = \left[ \begin{array}{c c c c} \boldsymbol {\chi} ^ {(1)} & \boldsymbol {\chi} ^ {(n _ {3})} & \dots & \boldsymbol {\chi} ^ {(2)} \\ \boldsymbol {\chi} ^ {(2)} & \boldsymbol {\chi} ^ {(1)} & \dots & \boldsymbol {\chi} ^ {(3)} \\ \vdots & \vdots & \ddots & \vdots \\ \boldsymbol {\chi} ^ {(n _ {3})} & \boldsymbol {\chi} ^ {(n _ {3} - 1)} & \dots & \boldsymbol {\chi} ^ {(1)} \end{array} \right]. \tag {4}
+$$
+
+Further, if  $\mathcal{X}^T * \mathcal{X} = \mathcal{X} * \mathcal{X}^T = \mathcal{I}$ ,  $\mathcal{X}$  is called orthogonal tensor.  $\mathcal{I} \in \mathbb{R}^{n_1 \times n_1 \times n_3}$  is an identity tensor, where the first frontal slice is an identity matrix and the remaining slices are all zeros. The tensor singular value decomposition (T-SVD) is a key operation for calculating the tensor nuclear norm, their definitions are introduced as follows.
+
+Definition 1. For a tensor  $\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$ , the  $T$ -SVD is formulated as
+
+$$
+\boldsymbol {\mathcal {X}} = \boldsymbol {\mathcal {U}} * \boldsymbol {\mathcal {D}} * \boldsymbol {\mathcal {V}} ^ {T} \tag {5}
+$$
+
+where  $\mathcal{U} \in \mathbb{R}^{n_1 \times n_1 \times n_3}$  and  $\mathcal{V} \in \mathbb{R}^{n_2 \times n_2 \times n_3}$  are orthogonal,  $\mathcal{D} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$  is  $f$ -diagonal.
+
+Definition 2. For a tensor  $\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$ , its T-SVD based tensor nuclear norm [38] is defined as
+
+$$
+\left| \left| \boldsymbol {\mathcal {X}} \right| \right| _ {\oplus} = \sum_ {j = 1} ^ {n _ {3}} \left| \left| \hat {\boldsymbol {\mathcal {X}}} ^ {(j)} \right| \right| _ {*} = \sum_ {i} ^ {\min  (n _ {1}, n _ {2})} \sum_ {j = 1} ^ {n _ {3}} \left| \hat {\boldsymbol {\mathcal {D}}} ^ {(j)} (i, i) \right|, \tag {6}
+$$
+
+where  $\hat{\pmb{D}}^{(j)}(i,i)$  is obtained by  $\hat{\pmb{x}}^{(j)} = \hat{\pmb{u}}^{(j)}*\hat{\pmb{D}}^{(j)}*\hat{\pmb{v}}^{(j)^T}$ .
+
+# 3 Methodology
+
+In this section, we elaborate on the proposed FedPAM. Overall, FedPAM is comprised of two modules: RSA and AGPL. Specifically, RSA is used to align the representation spaces across clients through the FGW-OT between local subgraphs and anchor graph. AGPL is employed to adaptively learn the global model parameters in the low-rank tensor space.
+
+# 3.1 Fused Gromov-Wasserstein Optimal Transport for Aligning Representation Spaces
+
+Local Self-Supervised Learning. Self-supervised learning (SSL) is an effective technique for preventing representation collapse and enhancing the discriminative quality of learned representations. In the absence of node labels, local clients adopt a self-supervised strategy for training. For the  $k$ -th local subgraph  $\mathcal{G}_k$ , it is augmented to  $\mathcal{G}_k^a$  and  $\mathcal{G}_k^b$  via two different kinds of augment methods. Then, the local GNN  $F_{\mathbf{W}_k}^k$  encodes them into the embedding space:
+
+$$
+\mathbf {Z} _ {k} ^ {a} = F _ {\mathbf {W} _ {k}} ^ {k} \left(\mathcal {G} _ {k} ^ {a}\right), \mathbf {Z} _ {k} ^ {b} = F _ {\mathbf {W} _ {k}} ^ {k} \left(\mathcal {G} _ {k} ^ {b}\right). \tag {7}
+$$
+
+A SSL loss between  $\mathbf{Z}_k^a$  and  $\mathbf{Z}_k^b$  is used for back propagation and generally written as
+
+$$
+\mathcal {L} _ {S S L} = \mathbb {E} _ {\forall i, j \in [ N _ {k} ]} \ell_ {p o s} \left(\mathbf {z} _ {i} ^ {a}, \mathbf {z} _ {i} ^ {b}\right) - \lambda \ell_ {n e g} \left(\mathbf {z} _ {i} ^ {a}, \left\{\mathbf {z} _ {j} ^ {b} \right\} _ {i \neq j}\right), \tag {8}
+$$
+
+where  $\ell_{pos}$  and  $\ell_{neg}$  denote the losses for positive and negative pairs, respectively. Overall, the SSL loss  $\mathcal{L}_{SSL}$  can be categorized into two classes: contrastive and non-contrastive. Contrastive methods require the involvement of negative samples during the computation of SSL, while non-contrastive methods do not rely on negative samples. We investigate the impacts of various SSL on the proposed framework in the experimental section.
+
+FGW-OT between Local Graphs and Anchor Graph. Under our assumption, local subgraphs originate from a underlying global graph, implying that node representations across clients following a unified distribution. However, due to the heterogeneous nature of local data, the embeddings learned by local clients tend to exhibit distributional discrepancies. Moreover, since data from other clients is unaccessible, local models lack awareness of the global distribution. To address this issue, we introduce  $N_P$  learnable anchors  $\mathbf{P} = \{\mathbf{p}_1,\dots,\mathbf{p}_{N_P}\}$  in the server to span the global space, the relationships between anchors are recorded by the anchor graph  $\mathbf{A}_P$ . These anchors also serve as the global projector, enabling the projection of local embedding into the global space. At the beginning of each communication round, the global model and the anchors are distributed to each client for local training. To align local subgraphs with anchor graph, the FGW-OT is introduced, whose definition is written as
+
+Definition 3. Given two metric measure space  $\mathcal{G}_1 = (\mathcal{V}_1, \mathbf{A}_1, \mu_1)$  and  $\mathcal{G}_1 = (\mathcal{V}_2, \mathbf{A}_2, \mu_2)$ ,  $\mathcal{V}_1$  and  $\mathcal{V}_2$  denote the node sets, where the data sizes are respectively  $N_1$  and  $N_2$ ,  $\mathbf{A}_1$  and  $\mathbf{A}_2$  denote the relationship matrices,  $\mu_1$  and  $\mu_2$  denote the data marginal distributions. Then, the FGW-OT distance between  $\mathcal{G}_1$  and  $\mathcal{G}_2$  is defined as
+
+$$
+\inf  _ {\pi \in \Pi} \sum_ {i, j} ^ {N _ {1}} \sum_ {m, n} ^ {N _ {2}} \left((1 - \alpha) D \left(\mathbf {x} _ {1, i}, \mathbf {x} _ {2, j}\right) ^ {p} + \alpha \left| \left[ \mathbf {A} _ {1} \right] _ {i, j} - \left[ \mathbf {A} _ {2} \right] _ {m, n} \right| ^ {p}\right) \pi_ {i, m} \pi_ {j, n} \tag {9}
+$$
+
+$$
+s. t. \Pi = \left\{\pi \in \mathbb {R} _ {+} ^ {N _ {1} \times N _ {2}} | \pi \mathbf {1} _ {N _ {2}} = \mu_ {1}, \pi^ {T} \mathbf {1} _ {N _ {1}} = \mu_ {2} \right\}
+$$
+
+where  $\mathbf{x}_{1,i} \in \mathcal{V}_1$ ,  $\mathbf{x}_{2,j} \in \mathcal{V}_2$ ,  $D(\cdot)$  measures the distance between two nodes,  $\alpha$  is a trade-off parameter between the feature-level distance and the relationship-level distance,  $p$  is a constant used to adjust the strength of the distance,  $\mathbf{1}_{N_1}$  and  $\mathbf{1}_{N_2}$  denote the two vectors of all-one elements with  $N_1$  and  $N_2$  dimensions.  $\pi \in \mathbb{R}^{N_1 \times N_2}$  is the expected optimal transport plan.
+
+In context of graph learning, FGW-OT can measure the sum of feature and topology distances between two graphs. Since anchors are derived from the global representation space, thus the anchor graph can be optimally transported to each local subgraph  $\mathcal{G}_k$  with minimal cost. The optimization objective is formulated as
+
+$$
+\min  _ {\pi_ {k} \in \Pi} \sum_ {i, j} ^ {N _ {k}} \sum_ {m, n} ^ {N _ {P}} \left((1 - \alpha) \left(- [ \mathbf {P} _ {k} ] _ {m} [ \mathbf {Z} _ {k} ^ {T} ] _ {i}\right) + \alpha | [ \mathbf {A} _ {k} ] _ {i, j} - [ \mathbf {A} _ {P} ] _ {m, n} |\right) \left[ \pi_ {k} \right] _ {i, m} \left[ \pi_ {k} \right] _ {j, n} - \epsilon \mathrm {H} \left(\pi_ {k}\right), \tag {10}
+$$
+
+$$
+s. t. \Pi = \left\{\pi_ {k} \in \mathbb {R} _ {+} ^ {N _ {k} \times N _ {P}} \mid \pi_ {k} \mathbf {1} _ {N _ {P}} = \mu_ {1}, \pi_ {k} ^ {T} \mathbf {1} _ {N _ {k}} = \mu_ {2} \right\},
+$$
+
+where  $\mathrm{H}(\cdot)$  denotes the entropy of certain variable,  $\epsilon$  denotes a trade-off constant. The entropy regularization is introduced to ensure that the optimization problem has a unique solution. The product  $\mathbf{P}_k\mathbf{Z}_k^T$  measures the similarity between the anchors and nodes, then  $-\mathbf{P}_k\mathbf{Z}_k^T$  is the distance between the two terms. Problem (10) can be optimized by the proposed method in [37], then an optimal transport matrix  $\pi_k\in \mathbb{R}^{N_k\times N_P}$  is obtained.  $\pi_{k}$  reflects the unnormalized transfer probability from nodes to anchors, and its normalized form is formulated as
+
+$$
+[ \mathbf {T} _ {k} ] _ {i, j} = \frac {\left[ \pi_ {k} \right] _ {i , j}}{\sum_ {j ^ {\prime}} \left[ \pi_ {k} \right] _ {i , j ^ {\prime}}}. \tag {11}
+$$
+
+For the anchor-node similarity matrix  $\mathbf{P}_k\mathbf{Z}_k^T$ , its has two-fold meanings. First, it depicts the relationships between anchors and nodes, and its normalized form can also be viewed as their transfer probability. Thus, we have
+
+$$
+[ \mathbf {C} _ {k} ] _ {i, j} = \frac {\exp \left([ \mathbf {Z} _ {k} ] _ {i} [ \mathbf {P} _ {k} ^ {T} ] _ {j} / \tau\right)}{\sum_ {j ^ {\prime}} \exp \left([ \mathbf {Z} _ {k} ] _ {i} [ \mathbf {P} _ {k} ^ {T} ] _ {j ^ {\prime}} / \tau\right)}. \tag {12}
+$$
+
+$\mathbf{T}_k$  is closed-form solution obtained through algebraic computation and does not have gradients, whereas  $\mathbf{C}_k$  is an inexact solution with gradients. In addition,  $\mathbf{C}_k$  is also used to conduct the anchor
+
+graph  $\mathbf{A}_P = \mathbf{C}_k^T\mathbf{C}_k$ .  $\mathbf{C}_k$  is expected to drive close to  $\mathbf{T}_k$ , then the loss is written as:
+
+$$
+\mathcal {L} _ {C E} = - \sum_ {i = 1} ^ {N _ {k}} \sum_ {j = 1} ^ {N _ {P}} \left(\left[ \mathbf {T} _ {k} \right] _ {i, j} \log \left[ \mathbf {C} _ {k} \right] _ {i, j}\right). \tag {13}
+$$
+
+Notably, as mentioned above, there are two augmented latent embeddings:  $\mathbf{Z}_k^a$  and  $\mathbf{Z}_k^b$ . We perform the OT mapping on the two views, then  $\mathcal{L}_{CE}$  is reformulated as
+
+$$
+\mathcal {L} _ {C E} = \mathcal {L} _ {C E} ^ {a} + \mathcal {L} _ {C E} ^ {b}. \tag {14}
+$$
+
+Second, the anchor matrix  $\mathbf{P}$  represents the entire representation space and is regarded as a global space projector. From another perspective,  $\mathbf{P}\mathbf{Z}^T$  is understood as a global-aware representation. In Eq. (8), it performs the contrastive learning with local representations and is denoted as  $\mathcal{L}_{SSL}^l$ . In addition, we expect to the global-aware representations to be more discriminative as well. By projecting  $\mathbf{Z}_k^a$  and  $\mathbf{Z}_k^b$  into global space via anchors  $\mathbf{P}_k$ , the global-aware representations  $\mathbf{H}_k^a = \mathbf{Z}_k^a\mathbf{P}_k^T$  and  $\mathbf{H}_k^b = \mathbf{Z}_k^b\mathbf{P}_k^T$  are obtained, then the contrastive learning between them is executed through Eq. (8), the loss is denoted as  $\mathcal{L}_{SSL}^g$ . Thus, the local SSL loss is reformulated as
+
+$$
+\mathcal {L} _ {S S L} = \mathcal {L} _ {S S L} ^ {l} + \mathcal {L} _ {S S L} ^ {g}. \tag {15}
+$$
+
+In summary, the training loss in local clients is concluded as
+
+$$
+\mathcal {L} = \mathcal {L} _ {S S L} + \lambda \mathcal {L} _ {C E}, \tag {16}
+$$
+
+where  $\lambda$  denotes the trade-off parameter. Through the back propagation from loss  $\mathcal{L}$ , the local GNNs and anchors are optimized. When the local training completes, the local GNNs and anchors are uploaded to sever for aggregation.
+
+# 3.2 Low-Rank Tensor Optimization for Adaptively Learning Global GNN Parameters
+
+Traditional federated aggregation performs weighted parameter fusion based on number of local samples. However, this approach has two drawbacks. First, the matrix-level fusion cannot capture the high-order correlations between clients. Second, the fixed weights cannot adaptively determine the importance of each client. In light of the shortcomings, we propose to find the optimal global GNN's parameters in a low-rank tensor space. Concretely, when the server receives local GNNs' parameters, the  $l$ -th layer's parameters  $\{\mathbf{W}_{k,l}\}_{k=1}^{K}$  are stacked as a third-order tensor  $\mathcal{W}_l$ . We argue that the low-rank component  $\mathcal{L}_l$  in  $\mathcal{W}_l$  contains the globally shared information, and the noise component  $\mathcal{E}_l$  contains the client-specific information. Then, the former needs to be retained, while the latter should be discarded. Further, we aim to adaptively fuse local GNNs' parameters in the low-rank tensor space for obtaining the optimal global GNN parameters  $\mathbf{A}_l$ . Considering the above concerns, the following optimization objective is proposed (the subscript  $l$  is omitted for a concise expression):
+
+$$
+\min  _ {\mathcal {L}, \boldsymbol {\mathcal {E}}, \mathbf {A}, \alpha_ {k}} \| \boldsymbol {\mathcal {L}} \| _ {*} + \beta \| \boldsymbol {\mathcal {E}} \| _ {1} + \sum_ {k = 1} ^ {K} \left(\alpha_ {k}\right) ^ {r} \left(\left\| \mathbf {L} _ {k} - \mathbf {A} \right\| _ {F} ^ {2}\right) \tag {17}
+$$
+
+$$
+\begin{array}{l} \text {s . t .} \mathcal {W} = \mathcal {L} + \mathcal {E}, \sum_ {k = 1} ^ {K} \alpha_ {k} = 1, \alpha_ {k} \geq 0, \forall k \in [ K ], \end{array}
+$$
+
+where  $\beta$  is a trade-off parameter,  $\alpha_{k}$  is the weight for the  $k$ -th client,  $r$  is a constant to smooth the distribution of weights. The optimization problem (17) can be solved by the alternating direction method of multipliers (ADMM). The augmented Lagrangian function of (17) is conducted as follows:
+
+$$
+\begin{array}{l} \mathcal {F} \left(\boldsymbol {\mathcal {L}}; \boldsymbol {\mathcal {E}}; \boldsymbol {\mathcal {G}}; \mathbf {A}; \left\{\alpha_ {k} \right\} _ {k = 1} ^ {K}\right) \\ = \| \boldsymbol {\mathcal {G}} \| _ {*} + \beta \| \boldsymbol {\mathcal {E}} \| _ {1} + \sum_ {k = 1} ^ {K} \left(\alpha_ {k}\right) ^ {r} \| \mathbf {L} _ {k} - \mathbf {A} \| _ {F} ^ {2} + \langle \boldsymbol {\mathcal {Y}}, \boldsymbol {\mathcal {W}} - \boldsymbol {\mathcal {L}} - \boldsymbol {\mathcal {E}} \rangle + \frac {\mu}{2} \| \boldsymbol {\mathcal {W}} - \boldsymbol {\mathcal {L}} - \boldsymbol {\mathcal {E}} \| _ {F} ^ {2} \tag {18} \\ + \langle \mathcal {H}, \boldsymbol {\mathcal {G}} - \mathcal {L} \rangle + \frac {\mu}{2} \| \boldsymbol {\mathcal {G}} - \mathcal {L} \| _ {F} ^ {2}, \\ \end{array}
+$$
+
+where  $\mathcal{G}$  is an auxiliary variable,  $\mathcal{V}$  and  $\mathcal{H}$  denote the two Lagrange multipliers,  $\mu$  is a trade-off parameter. The update rule for each variable is presented as follows.
+
+Update  $\mathcal{G}$ : Fixing  $\mathcal{L}, \mathcal{E}, \mathbf{A}, \{\alpha_k\}_{k=1}^K$ , the subproblem with respect to  $\mathcal{G}$  is written as
+
+$$
+\begin{array}{l} \min  _ {\boldsymbol {\mathcal {G}}} \| \boldsymbol {\mathcal {G}} \| _ {\oplus} + \langle \boldsymbol {\mathcal {H}}, \boldsymbol {\mathcal {G}} - \boldsymbol {\mathcal {L}} \rangle + \frac {\mu}{2} \| \boldsymbol {\mathcal {G}} - \boldsymbol {\mathcal {L}} \| _ {F} ^ {2} \\ = \min  _ {\boldsymbol {G}} \| \boldsymbol {G} \| _ {\oplus} + \frac {\mu}{2} \left\| \boldsymbol {G} - \left(\boldsymbol {\mathcal {L}} - \frac {1}{\mu} \boldsymbol {\mathcal {H}}\right) \right\| _ {F} ^ {2}. \tag {19} \\ \end{array}
+$$
+
+For the solution of low-rank tensor  $\mathcal{G}$ , study [39] provided a tensor tubal-shrinkage method  $\mathcal{R}$ , the updated  $\mathcal{G}^*$  is obtained by
+
+$$
+\boldsymbol {\mathcal {G}} ^ {*} = \mathcal {R} _ {1 / \mu} (\boldsymbol {\mathcal {L}} - \frac {1}{\mu} \boldsymbol {\mathcal {H}}). \tag {20}
+$$
+
+Update  $\mathcal{L}$ : When optimizing  $\mathcal{L}$ , the update rule for its each slice  $\mathbf{L}_k$  is same. Fixing  $\mathcal{G}$ ,  $\mathcal{E}$ ,  $\mathbf{A}$ ,  $\{\alpha_k\}_{k=1}^K$ , the subproblem for  $\mathbf{L}_k$  is formulated as
+
+$$
+\begin{array}{l} \min  _ {\mathbf {L} _ {k}} \left(\alpha_ {k}\right) ^ {r} \left\| \mathbf {L} _ {k} - \mathbf {A} \right\| _ {F} ^ {2} + \left\langle \mathbf {Y} _ {k}, \mathbf {W} _ {k} - \mathbf {L} _ {k} - \mathbf {E} _ {k} \right\rangle + \frac {\mu}{2} \left\| \mathbf {W} _ {k} - \mathbf {L} _ {k} - \mathbf {E} _ {k} \right\| _ {F} ^ {2} \tag {21} \\ + \left\langle \mathbf {H} _ {k}, \mathbf {G} _ {k} - \mathbf {L} _ {k} \right\rangle + \frac {\mu}{2} \left\| \mathbf {G} _ {k} - \mathbf {L} _ {k} \right\| _ {F} ^ {2}. \\ \end{array}
+$$
+
+Taking the partial derivative with respect to  $\mathbf{L}_k$  and setting it to zero, the update rule for  $\mathbf{L}_k^*$  is derived:
+
+$$
+\mathbf {L} _ {k} ^ {*} = \frac {2 \left(\alpha_ {k}\right) ^ {r} \mathbf {A} + \mu \mathbf {Z} _ {k} - \mu \mathbf {E} _ {k} + \mathbf {Y} _ {k} + \mu \mathbf {G} _ {k} + \mathbf {H} _ {k}}{2 \left(\alpha_ {k}\right) ^ {r} + 2 \mu} \tag {22}
+$$
+
+Update A: Masking variables unrelated to  $\mathbf{A}$ , we obtain the following subproblem:
+
+$$
+\min  _ {\mathbf {A}} \sum_ {k = 1} ^ {K} \left(\alpha_ {k}\right) ^ {r} \left(\left\| \mathbf {L} _ {k} - \mathbf {A} \right\| _ {F} ^ {2}\right). \tag {23}
+$$
+
+The update rule for  $\mathbf{A}^*$  can be obtained by computing the partial derivative with respect to  $\mathbf{A}^*$  and equating it to zero, then we have
+
+$$
+\mathbf {A} ^ {*} = \frac {\sum_ {k = 1} ^ {K} \left(\alpha_ {k}\right) ^ {r} \mathbf {L} _ {k}}{\sum_ {k = 1} ^ {K} \left(\alpha_ {k}\right) ^ {r}}. \tag {24}
+$$
+
+Update  $\mathcal{E}$ : When  $\mathcal{E}$  is updated, the other variables are viewed as constants, the subproblem is formulated as
+
+$$
+\begin{array}{l} \min  _ {\boldsymbol {\varepsilon}} \beta \| \boldsymbol {\mathcal {E}} \| _ {1} + \left\langle \boldsymbol {\mathcal {Y}}, \boldsymbol {\mathcal {W}} - \boldsymbol {\mathcal {L}} - \boldsymbol {\mathcal {E}} \right\rangle + \frac {\mu}{2} \| \boldsymbol {\mathcal {W}} - \boldsymbol {\mathcal {L}} - \boldsymbol {\mathcal {E}} \| _ {F} ^ {2} \tag {25} \\ = \min  _ {\boldsymbol {\mathcal {E}}} \beta | | \boldsymbol {\mathcal {E}} | | _ {1} + \frac {\mu}{2} | | \boldsymbol {\mathcal {E}} - (\boldsymbol {\mathcal {W}} - \boldsymbol {\mathcal {L}} + \boldsymbol {\mathcal {Y}} / \mu) | | _ {F} ^ {2}. \\ \end{array}
+$$
+
+Let  $\mathcal{B} = \mathcal{W} - \mathcal{L} + \mathcal{V} / \mu$ , the update method with respect to  $\varepsilon$  is
+
+$$
+\boldsymbol {\mathcal {E}} ^ {*} = \max  (\boldsymbol {\mathcal {B}} - \lambda / \mu , 0) + \min  (\boldsymbol {\mathcal {B}} + \lambda / \mu , 0). \tag {26}
+$$
+
+Update  $\alpha_{k}$ : Neglecting variables unrelated to  $\alpha_{k}$ , the subproblem about  $\alpha_{k}$  is written as
+
+$$
+\min  _ {\alpha_ {k}} \sum_ {k = 1} ^ {K} \left(\alpha_ {k}\right) ^ {r} \left(\left\| \mathbf {L} _ {k} - \mathbf {A} \right\| _ {F} ^ {2}\right), \text {s . t .} \sum_ {k = 1} ^ {K} \alpha_ {k} = 1, \alpha_ {k} \geq 0, \forall k \in [ K ]. \tag {27}
+$$
+
+Taking the partial derivative with respect to  $\alpha_{k}$  and setting it to zero, we have
+
+$$
+\alpha_ {k} ^ {*} = \frac {\left(\left\| \mathbf {L} _ {k} - \mathbf {A} \right\| _ {F} ^ {2}\right) ^ {1 / (1 - r)}}{\sum_ {k = 1} ^ {K} \left(\left\| \mathbf {L} _ {k} - \mathbf {A} \right\| _ {F} ^ {2}\right) ^ {1 / (1 - r)}}. \tag {28}
+$$
+
+Update  $\mathcal{V},\mathcal{H},\mu$
+
+$$
+\boldsymbol {\mathcal {H}} ^ {*} = \boldsymbol {\mathcal {H}} + \mu (\boldsymbol {\mathcal {G}} - \boldsymbol {\mathcal {L}}); \boldsymbol {\mathcal {Y}} ^ {*} = \boldsymbol {\mathcal {Y}} + \mu (\boldsymbol {\mathcal {W}} - \boldsymbol {\mathcal {L}} - \boldsymbol {\mathcal {E}}); \mu^ {*} = \min  (\omega * \mu , \mu_ {\max },) \tag {29}
+$$
+
+where  $\omega$  and  $\mu_{max}$  are two predefined parameters. The optimal global GNN parameters are obtained through the iterative optimization based on ADMM described above. Thus, the global GNN parameters and anchors are distributed to clients in the next communication.
+
+Table 1: Performance comparison on Cora, CiteSeer, PubMed, and Ogbn-Arxiv datasets with three different SSL losses, where the optimal results are bolded and the suboptimal results are underlined.  
+
+<table><tr><td rowspan="2">SSL</td><td rowspan="2">Method</td><td colspan="2">Cora</td><td colspan="2">CiteSeer</td><td colspan="2">PubMed</td><td colspan="2">Ogbn-Arxiv</td></tr><tr><td>ACC</td><td>Fscore</td><td>ACC</td><td>Fscore</td><td>ACC</td><td>Fscore</td><td>ACC</td><td>Fscore</td></tr><tr><td rowspan="5">Simsiam</td><td>FedAvg</td><td>54.38</td><td>39.27</td><td>36.30</td><td>20.34</td><td>63.81</td><td>50.88</td><td>35.26</td><td>20.49</td></tr><tr><td>FedProx</td><td>55.35</td><td>40.83</td><td>36.80</td><td>21.34</td><td>63.81</td><td>50.88</td><td>34.61</td><td>20.03</td></tr><tr><td>MOON</td><td>54.03</td><td>39.81</td><td>35.94</td><td>20.41</td><td>63.80</td><td>50.87</td><td>35.46</td><td>21.20</td></tr><tr><td>FedU2</td><td>56.04</td><td>42.65</td><td>39.00</td><td>25.77</td><td>61.27</td><td>48.09</td><td>35.39</td><td>22.00</td></tr><tr><td>FedPAM</td><td>61.40</td><td>52.85</td><td>49.53</td><td>38.73</td><td>64.23</td><td>51.92</td><td>35.97</td><td>23.07</td></tr><tr><td rowspan="6">SimCLR</td><td>FedAvg</td><td>54.65</td><td>39.95</td><td>35.48</td><td>19.62</td><td>64.08</td><td>51.46</td><td>45.58</td><td>33.08</td></tr><tr><td>FedProx</td><td>55.26</td><td>40.68</td><td>35.56</td><td>19.78</td><td>63.95</td><td>51.17</td><td>44.40</td><td>31.86</td></tr><tr><td>MOON</td><td>54.56</td><td>39.81</td><td>36.30</td><td>20.34</td><td>64.42</td><td>52.17</td><td>46.23</td><td>33.96</td></tr><tr><td>FedX</td><td>56.73</td><td>43.11</td><td>37.10</td><td>22.37</td><td>61.77</td><td>49.09</td><td>46.69</td><td>34.77</td></tr><tr><td>FedU2</td><td>57.09</td><td>43.45</td><td>40.89</td><td>27.54</td><td>66.79</td><td>56.25</td><td>48.41</td><td>36.46</td></tr><tr><td>FedPAM</td><td>58.62</td><td>45.21</td><td>43.90</td><td>31.61</td><td>69.22</td><td>59.42</td><td>48.19</td><td>37.25</td></tr><tr><td rowspan="6">BYOL</td><td>FedAvg</td><td>63.01</td><td>50.51</td><td>42.65</td><td>29.72</td><td>61.85</td><td>49.62</td><td>35.29</td><td>20.73</td></tr><tr><td>FedU</td><td>64.06</td><td>51.92</td><td>46.81</td><td>34.98</td><td>62.08</td><td>49.82</td><td>38.07</td><td>24.15</td></tr><tr><td>FedEMA</td><td>63.53</td><td>51.23</td><td>43.59</td><td>30.97</td><td>64.47</td><td>53.39</td><td>38.12</td><td>24.07</td></tr><tr><td>Orchestra</td><td>54.47</td><td>39.63</td><td>37.54</td><td>24.39</td><td>61.27</td><td>48.09</td><td>35.26</td><td>20.49</td></tr><tr><td>FedU2</td><td>65.11</td><td>54.16</td><td>47.32</td><td>35.79</td><td>66.51</td><td>55.86</td><td>39.95</td><td>26.03</td></tr><tr><td>FedPAM</td><td>66.01</td><td>55.23</td><td>48.81</td><td>38.69</td><td>66.67</td><td>56.25</td><td>40.55</td><td>26.79</td></tr></table>
+
+# 4 Experiments
+
+# 4.1 Experimental Setups
+
+Graph Datasets. Eight graph datasets are selected as benchmark datasets, including Cora, CiteSeer, PubMed, Ogbn-Arxiv, Computers, Photo, Physics, Amazon-ratings. The eight datasets vary in terms of scales, and cover different types, such as citation network, co-purchase network, co-author network, rating network. Each graph is divided into ten subgraphs via the Louvain method [40].
+
+Compared Methods. We compare the proposed FedPAM with typical and SOTA FL algorithms, including FedAvg [31], FedProx [41], MOON [42], FedX [43], FedU [34], FedEMA [35], Orchestra [32], FedU² [36]. In local training, three SSL strategies are adopted, including Simsiam [44], SimCLR [30], BYOL [45]. Specifically, Simsiam and BYOL are non-contrastive methods while SimCLR is contrastive method. We adopt the standard linear probing to evaluate the performance of algorithms, ACC and Fscore are used as the evaluation metrics.
+
+# 4.2 Performance Comparison
+
+We evaluate the performance of FL algorithms using three SSL strategies on eight graph datasets, the results are reported in Tables 1 and 2. Some interesting phenomenon can be observed. First, FL algorithms exhibit varying performance when equipped with different SSL losses. For instance, on Cora dataset, employing BYOL leads to improvements by  $8.63\%$  and  $8.36\%$  in ACC compared to the other two SSL losses for FedAvg, respectively. Encouragingly, the proposed FedPAM consistently achieves superior performance with different SSL methods, validating the contributions of RSA and AGPL. Second, compared to baseline methods, the proposed FedPAM demonstrates more significant improvements on homogeneous graphs than on heterogeneous graphs. Node attributes and edge types in homogeneous graphs are less diverse, then the learnable anchors are more likely to span the global representation space. Furthermore, homogeneous graphs induce local models to be more similar, facilitating low-rank tensor optimization to find better global model parameters.
+
+# 4.3 Ablation Study
+
+RSA and AGPL paly critical roles for the proposed FedPAM. The results of their ablation studies are reported in Table 3. When both modules are removed, the model degenerates into FedAvg and
+
+Table 2: Performance comparison on Computers, Photo, Physics, and Amazon-ratings datasets with three different SSL losses, where the optimal results are bolded and the suboptimal results are underlined.  
+
+<table><tr><td rowspan="2">SSL</td><td rowspan="2">Method</td><td colspan="2">Computers</td><td colspan="2">Photo</td><td colspan="2">Physics</td><td colspan="2">Amazon-ratings</td></tr><tr><td>ACC</td><td>Fscore</td><td>ACC</td><td>Fscore</td><td>ACC</td><td>Fscore</td><td>ACC</td><td>Fscore</td></tr><tr><td rowspan="5">Simsiam</td><td>FedAvg</td><td>63.09</td><td>50.92</td><td>72.01</td><td>63.56</td><td>75.55</td><td>67.73</td><td>36.72</td><td>19.79</td></tr><tr><td>FedProx</td><td>64.05</td><td>51.70</td><td>73.26</td><td>65.08</td><td>75.33</td><td>66.22</td><td>36.64</td><td>19.83</td></tr><tr><td>MOON</td><td>63.92</td><td>52.16</td><td>72.46</td><td>64.20</td><td>77.82</td><td>70.79</td><td>36.69</td><td>19.86</td></tr><tr><td>FedU2</td><td>63.20</td><td>51.03</td><td>72.65</td><td>64.57</td><td>86.26</td><td>81.02</td><td>36.72</td><td>19.79</td></tr><tr><td>FedPAM</td><td>64.19</td><td>53.93</td><td>74.06</td><td>66.96</td><td>86.51</td><td>81.86</td><td>37.45</td><td>24.97</td></tr><tr><td rowspan="6">SimCLR</td><td>FedAvg</td><td>78.21</td><td>72.27</td><td>84.59</td><td>79.82</td><td>79.07</td><td>71.30</td><td>36.56</td><td>20.06</td></tr><tr><td>FedProx</td><td>78.41</td><td>72.56</td><td>84.66</td><td>79.82</td><td>77.05</td><td>67.74</td><td>36.46</td><td>19.92</td></tr><tr><td>MOON</td><td>78.90</td><td>72.53</td><td>84.72</td><td>79.96</td><td>80.23</td><td>73.10</td><td>36.72</td><td>20.05</td></tr><tr><td>FedX</td><td>80.60</td><td>75.09</td><td>85.49</td><td>80.72</td><td>80.31</td><td>73.60</td><td>36.70</td><td>19.79</td></tr><tr><td>FedU2</td><td>79.24</td><td>73.34</td><td>84.94</td><td>80.15</td><td>81.03</td><td>74.27</td><td>39.58</td><td>27.92</td></tr><tr><td>FedPAM</td><td>81.63</td><td>77.42</td><td>85.78</td><td>81.25</td><td>83.76</td><td>78.16</td><td>39.60</td><td>29.69</td></tr><tr><td rowspan="6">BYOL</td><td>FedAvg</td><td>63.38</td><td>52.20</td><td>71.12</td><td>61.93</td><td>84.99</td><td>79.13</td><td>36.77</td><td>21.05</td></tr><tr><td>FedU</td><td>65.00</td><td>54.92</td><td>74.19</td><td>67.42</td><td>85.36</td><td>79.54</td><td>37.11</td><td>24.07</td></tr><tr><td>FedEMA</td><td>65.44</td><td>56.38</td><td>76.81</td><td>70.31</td><td>83.39</td><td>77.53</td><td>38.32</td><td>27.22</td></tr><tr><td>Orchestra</td><td>62.84</td><td>52.67</td><td>73.74</td><td>67.22</td><td>72.16</td><td>62.10</td><td>36.81</td><td>20.12</td></tr><tr><td>FedU2</td><td>65.55</td><td>54.73</td><td>74.86</td><td>68.11</td><td>85.41</td><td>79.74</td><td>36.82</td><td>20.50</td></tr><tr><td>FedPAM</td><td>66.66</td><td>57.57</td><td>80.50</td><td>75.34</td><td>86.43</td><td>80.98</td><td>38.43</td><td>26.52</td></tr></table>
+
+Table 3: Ablation study with respect to two key modules RSA and AGPL on Cora and CiteSeer datasets, where SimCLR loss is used in the local training.  
+
+<table><tr><td rowspan="2">RSA</td><td rowspan="2">AGPL</td><td colspan="2">Cora</td><td colspan="2">CiteSeer</td></tr><tr><td>ACC</td><td>Fscore</td><td>ACC</td><td>Fscore</td></tr><tr><td>X</td><td>X</td><td>54.65</td><td>39.95</td><td>35.48</td><td>19.62</td></tr><tr><td>X</td><td>✓</td><td>56.65</td><td>44.18</td><td>41.35</td><td>27.80</td></tr><tr><td>✓</td><td>X</td><td>58.03</td><td>45.27</td><td>37.89</td><td>22.65</td></tr><tr><td>✓</td><td>✓</td><td>58.62</td><td>45.21</td><td>43.90</td><td>31.61</td></tr></table>
+
+![](images/feda35158e902bbc8ba099fabd12a731e139408d6fe343ea211622229aa21b27.jpg)  
+(a) Cora
+
+![](images/93f4b3daca37e7e0c03281bc9555310b0b1df035e77b16add795ac9edbeee729.jpg)  
+(b) CiteSeer  
+Figure 4: Performance comparison with different numbers of anchors on Cora and CiteSeer datasets, where SimCLR loss is used in the local training.
+
+naturally achieves the worst performance. When either RSA and AGPL module is present, the performance of the model is improved, indicating that both representation space alignment and adaptive global parameter learning are effective. Certainly, the performance is optimal with both modules available.
+
+![](images/7a79a595267118b972294d3588500d7cde99fa7065b8f3b8f08df68bda8d637c.jpg)  
+(a) Cora  
+Figure 5: Sensitivity study with respect to  $\lambda$  on Cora and CiteSeer datasets, where SimCLR loss is used in the local training.
+
+![](images/4ad178682ad5aa1ca00c4f38f50d337a43e5fe918480b0d914acec90ee38bf5f.jpg)  
+(b) CiteSeer
+
+![](images/a3871b07c422b9befd341fca9076f4f06b17bcbc57ccf9d3429ef5e059c53603.jpg)  
+(a) Cora  
+Figure 6: Sensitivity study with respect to  $\beta$  on Cora and CiteSeer datasets, where SimCLR loss is used in the local training.
+
+![](images/f3aa27fbd28b2aff80a0a777d16b798044bff519b0052fd776445a0d400201dd.jpg)  
+(b)CiteSeer
+
+# 4.4 Parameter Sensitivity Investigation
+
+We investigate the impacts of three key hyperparameters in the proposed FedPAM, including the number of anchors  $M$ ,  $\lambda$  in the local training, and  $\beta$  in the low-rank tensor optimization. From Fig. 4, it can be seen that either too few or too many anchors is detrimental to achieving optimal
+
+performance. Excessive anchors tend to reduce discriminability, while too few anchors may lead to representation collapse.  $\lambda$  controls the strength of the optimal transport, a large value of  $\lambda$  facilitates better alignment of representation spaces as shown in Fig. 5. However, setting  $\beta$  too high is not advisable as shown in Fig. 6, because it may hinder the capture of client-specific information.
+
+# 5 Conclusion
+
+In this paper, we address two key challenges in unsupervised FGL: the misalignment of representation spaces and non-adaptive federated aggregation schemas. For the first challenge, we propose the RSA module, which aims to learn a set of anchors across clients and use them to align representation spaces of different clients. For the second challenge, the AGPL module is introduced, employing the low-rank tensor optimization to adaptively learn a global model. Finally, the experimental results on eight graph datasets verify the effectiveness of the proposed FedPAM. More details about related work, algorithm,Datasets, and experimental results can be referred in the Appendix.
+
+# Acknowledgments
+
+The research is supported by the National Key R&D Program of China (2023YFB2703700), the National Natural Science Foundation of China (62176269).
+
+# References
+
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+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: In this paper, we address the two key challenges of unsupervised federated graph learning, and propose Representation Space Alignment module and Adaptive Global Parameter Learning module. To our knowledge, we are the first to recognize and address the challenges of unsupervised federated graph learning.
+
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+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
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+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
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+
+some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
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+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
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+
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+
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+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [No]
+
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
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+Justification: We report the configuration of the server used in the implementation details.
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+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: Our research conforms to the NIPS Code of Ethics in all respects.
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: The proposed method aims to improve the effectiveness of unsupervised federated graph learning, which has a positive effect on protecting users' privacy and improving the training efficiency.
+
+# Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
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+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: The paper poses no such risks.
+
+# Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: We have cited sources for the compared algorithms and datasets used in the paper.
+
+# Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.
+
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: The paper does not release new assets.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
+
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: We guarantee that the proposed FedPAM and the paper are completely original and have not been generated with the help of LLM.
+
+Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Summary of Appendix
+
+We present the following sections in the appendix as supplement to this manuscript:
+
+- Related work about FGL, OT, and low-rank tensor optimization.  
+- The algorithm flow of the proposed FedPAM.  
+- Implementation details for the proposed FedPAM.  
+- The analysis for the computational complexity.  
+- The details of the eight graph datasets used.  
+- The performance comparison using KNN as the evaluation metric.  
+The performance comparison with large-scale clients.  
+- The performance comparison of plugging the AGPL module into existing works.
+
+# B Related Work
+
+# B.1 Federated Graph Learning
+
+The objective of federated graph learning is to enable the collaborative training of a GNN on multiple decentralized graphs, thereby improving its generalization. In terms of training data, FGL is generally divided into two categories: graph-level and node-level. Graph-level FGL focuses on addressing challenges related to label or feature heterogeneity. For example, [15, 16] proposed to cluster clients or share topology patterns for Non-IID graphs. To mitigate the domain gaps between private graphs, [46, 17] captured the cross-domain knowledge to enhance the local personalized inference. In contrast, node-level FGL primarily targets challenges arising from topological heterogeneity. [18, 9, 19] leveraged the global information to assist the local training. [47, 48] emphasized the impact of topological structures on federated learning and proposed the topology-aware aggregation method. While the aforementioned approaches provide valuable insights, they overlook the problem of FGL in unsupervised scenarios. We recognize the challenges of unsupervised FGL and propose a novel approach to address them.
+
+# B.2 Optimal Transport
+
+Optimal transport [49] is used to find the shortest distance for transforming one distribution into another, and has been widely applied in the fields of domain adaption and domain generalization. [50, 51] aligned the distributions between the source domain and the target domain via OT. As an extension of classical optimal transport, GW-OT is designed to align structural relationships between data distributions, making it particularly suitable for applications in graph match tasks. For instance, [52, 53] aimed to match the topological structures between various graphs via GW-OT. Furthermore, to simultaneously align the distributions in terms of features and structures, the FGW-OT is proposed. [37, 5] adopted the FGW-OT to achieve optimal mapping from source graphs to target graphs through features and topologies. For unsupervised FGL, how to explore the global representation space is critical challenge, we leverage the FGW-OT to learn a group of anchors across multiple private subgraphs for positioning the global coordinate.
+
+# B.3 Low-rank Tensor Decomposition
+
+Low-rank tensor decomposition seeks to recover the underlying low-rank components from a given tensor, and it has found extensive applications in image restoration, data mining, and data compression tasks. [54, 55, 56] employed the tensor singular value decomposition to explore the principal information of multi-source data. Deep learning provides a new solution for exploring the low-rankness, [57, 58] used neural networks to nonlinearly capture the low-rank structures. Although there have been some existing studies working on federated tensor computation [59, 60, 61], they mainly focused on distributed tensor decomposition with respect to data features. In contrast, we leverage low-rank tensor decomposition to exploit the complementarity of local models and solve for the global model parameters in the low-rank tensor space.
+
+# C The Algorithm Flow of the Proposed FedPAM
+
+Algorithm 1 The main steps of FedPAM  
+Input: Number of clients  $K$ , communication rounds  $T$ , local training epochs  $E$ , learning rate  $\eta$ , trade-off parameters  $\lambda$  and  $\beta$ , local subgraph  $\mathcal{G}_k = (\mathcal{V}_k, \mathcal{E}_k, \mathbf{X}_k)$ , local GNN  $F_{\mathbf{W}_k}$ .  
+Output: Global model  $F_{\mathbf{W}}$  and Global anchors  $\mathbf{P}$ .  
+1: Client Side:  
+2: for  $k = 1: K$  in parallel do  
+3: for epoch  $e = 1: E$  do  
+4: Augment the original local subgraph  $\mathcal{G}_k$  into  $\mathcal{G}_k^a$  and  $\mathcal{G}_k^b$  via two different methods;  
+5: Encode the two views of subgraphs:  $\mathbf{Z}_k^a \gets F_{\mathbf{W}_k}^k(\mathcal{G}_k^a)$ ,  $\mathbf{Z}_k^b \gets F_{\mathbf{W}_k}^k(\mathcal{G}_k^b)$ ;  
+6: Calculate the SSL loss for local representations:  $\mathcal{L}_{SSL}^l \gets (\mathbf{Z}_k^a, \mathbf{Z}_k^b)$  by Eq. (8);  
+7: for view  $i$  in  $(a, b)$  do  
+8: Calculate the optimal transport matrix  $\mathbf{T}_k^i$  between  $\mathbf{Z}_k^i$  and  $\mathbf{P}_k$  by Algorithm 2;  
+9: Calculate the normalized transport matrix:  $\mathbf{T}_k^i \gets (\pi_k^i)$  by Eq. (11);  
+10: Calculate the transfer probability matrix:  $\mathbf{C}_k^i \gets (\mathbf{Z}_k^i, \mathbf{P}_k)$  by Eq. (12);  
+11: Calculate the CE loss  $\mathcal{L}_{CE}^i \gets (\mathbf{T}_k^i, \mathbf{C}_k^i)$  by (13);  
+12: end for  
+13: Calculate the CE loss for two views:  $\mathcal{L}_{CE} \gets (\mathcal{L}_{CE}^a, \mathcal{L}_{CE}^b)$ ;  
+14: Obtain the global-aware representations:  $\mathbf{H}_k^a \gets \mathbf{Z}_k^a \mathbf{P}_k^T$ ,  $\mathbf{H}_k^b \gets \mathbf{Z}_k^b \mathbf{P}_k^T$ ;  
+15: Calculate the SSL loss for global-aware representations:  $\mathcal{L}_{SSL}^g \gets (\mathbf{H}_k^a, \mathbf{H}_k^b)$  by Eq. (8);  
+16: Calculate the total SSL loss  $\mathcal{L}_{SSL} \gets (\mathcal{L}_{SSL}^l, \mathcal{L}_{SSL}^g)$  by Eq. (15);  
+17: Update local GNN  $\mathbf{W}_k^e \gets \mathbf{W}_k^{e-1} - \eta \nabla (\mathcal{L}_{SSL} + \mathcal{L}_{CE})$ ;  
+18: Update local anchors  $\mathbf{P}_k^e \gets \mathbf{P}_k^{e-1} - \eta \nabla (\mathcal{L}_{SSL} + \mathcal{L}_{CE})$ ;  
+19: end for  
+20: Upload the local GNN  $F_{\mathbf{W}_k}^k$  and local anchors  $\mathbf{P}_k$  to the server;  
+21: end for  
+22: Server Side:  
+23: for  $t = 1: T$  do  
+24: for  $i = 1: L$  do  
+25: Stack the  $l$ -th layer's parameters  $\{\mathbf{W}_{k,l}\}$  of local model into a third-order tensor  $\boldsymbol{\Psi}_l$ ;  
+26: Obtain the optimal parameters  $\mathbf{A}_l$  from  $\boldsymbol{\Psi}_l$  by Algorithm 4;  
+27: end for  
+28: Obtain the global anchors  $\mathbf{P} = \sum_{k=1}^{K} \frac{N_k}{N} \mathbf{P}_k$ ;  
+29: Distribute the global GNN  $F_{\mathbf{W}}$  and the global anchors  $\mathbf{P}$  to clients;  
+30: end for
+
+Algorithm 2 The main steps of fused GW-OT  
+Input: Node marginal distribution  $\nu \in \mathbb{R}^N$ , anchor marginal distribution  $\mu \in \mathbb{R}^M$ , adjacency matrix  $\mathbf{A} \in \mathbb{R}^{N \times N}$ , anchor adjacency matrix  $\mathbf{S} \in \mathbb{R}^{M \times M}$ , the transport matrix  $\varphi \in \mathbb{R}^{N \times M}$  in terms of feature level, trade-off parameters  $\alpha$  and  $\epsilon$ , maximum iteration round  $T$ .  
+Output: The transport matrix  $\pi \in \mathbb{R}^{N \times M}$ .  
+1:  $\pi \gets \nu \mu^T$ ;  
+2:  $G_1 \gets \mathbf{S} \odot \mathbf{S} \cdot \nu \mathbb{1}^T$ ;  
+3:  $G_2 \gets 1 (\mathbf{A} \odot \mathbf{A} \cdot \mu)^T$ ;  
+4: for  $t = 1 : T$  do  
+5:  $G \gets 2\alpha \cdot (-2\mathbf{S}\pi \mathbf{A} + G_1 + G_2) + (1 - \alpha)\varphi$ ;  
+6:  $\pi \gets \mathrm{OT}(\nu, \mu, G, \pi, \epsilon)$  via Algorithm 3;  
+7: end for
+
+# D Implement Details
+
+A 2-layer graph convolutional network [62] is used as the backbone, and the latent embedding dimension is set as 128. Following [18, 48], the Louvain method [40] is used to divide the initial
+
+Algorithm 3 The main steps of OT via Sinkhorn algorithm  
+Input: Node marginal distribution  $\nu \in \mathbb{R}^N$ , anchor marginal distribution  $\mu \in \mathbb{R}^M$ , cost matrix  $\mathbf{C}^{N\times M}$ , initialized transport matrix  $\pi$ , trade-off parameter  $\epsilon$ , maximum iteration round  $T'$ .  
+Output: The transport matrix  $\pi \in \mathbb{R}^{N\times M}$ .  
+1:  $K \gets e^{-\mathbf{C} / \epsilon}$ ;  
+2: for  $t = 1:T'$  do  
+3:  $\nu \gets \nu / K\mu$ ;  
+4:  $\mu \gets \mu / K\nu$ ;  
+5: end for  
+6:  $\pi \gets \mathrm{diag}(\nu)K\mathrm{diag}(\mu)$ ;
+
+Algorithm 4 The main steps for low-rank tensor optimization  
+Input: Observed tensor  $\mathcal{W}$ , trade-off parameters  $\beta$  and  $\mu$ , step  $\omega$ , maximum  $\mu_{max} = 10^{10}$ , threshold  $\epsilon$ . Initial the auxiliary variable  $\mathcal{G} = 0$ , the Lagrange multipliers  $\mathcal{H} = 0$ ,  $\mathcal{V} = 0$ .  
+Output: Consistent matrix  $\mathbf{A}$ .  
+1: while not convergent do  
+2: Update  $\mathcal{G}^{t+1} \gets (\mathcal{L}^t, \mathcal{H}^t)$  by Eq. (20);  
+3: for  $i = 1:K$  do  
+4: Update  $\mathbf{L}_k^{t+1} \gets (\mathbf{A}^t, \mathbf{Z}_k^t, \mathbf{E}_k^t, \mathbf{Y}_k^t, \mathbf{G}_k^t, \mathbf{H}_k^t, \alpha_k^t, \mu^t)$  by Eq. (22);  
+5: end for  
+6: Update  $\mathbf{A}^{t+1} \gets (\mathbf{L}_k^t, \alpha_k^t)$  by Eq. (24);  
+7: Update  $\mathcal{E}^{t+1} \gets (\mathcal{B}^t, \beta/\mu^t)$  by Eq. (26);  
+8: Update  $\alpha_k^{t+1} \gets (\mathbf{L}_k^t, \mathbf{A}^t)$  by Eq. (28);  
+9: Check the convergence conditions:  
+ $\max(|\mathcal{W}^{t+1} - \mathcal{L}^{t+1} - \mathcal{E}^{t+1}|_F^2, |\mathcal{L}^{t+1} - \mathcal{L}^t|_F^2, |\mathcal{E}^{t+1} - \mathcal{E}^t|_F^2) \leq \epsilon$   
+10: > When the conditions are satisfied, the iteration ends, otherwise it continues;  
+11:  $t = t + 1$ ;  
+12: end while
+
+graph to multiple local subgraphs, the number of clients is set as 10. We use Adam as the optimizer with the learning rate set to 0.001. The numbers of communication round and local training rounds are set to 100 and 5, respectively. When the federated training completes, we use the standard linear probing to evaluate the performance of algorithms. The values of ACC and Fscore on test sets are reported. Some trade-off parameters are required to be predefined,  $\alpha$  and  $\epsilon$  in FGW-OT is fixed as 0.5 and 1.  $\lambda$  is tuned in  $\{0.1, 5, 10, 50, 70, 100\}$ ,  $\beta$  is varied in  $\{0.1, 0.5, 1\}$ , the number of anchors is tuned in  $\{30, 60, 100, 500, 1000\}$ .
+
+# E The Analysis for the Computational Complexity
+
+The computation complexity mainly originates from two aspects: the FGW-OT for aligning representation spaces and the low-rank tensor optimization for adaptively learning global model parameters. For the FGW-OT, the computation of  $G$  in Algorithm 2 takes  $\mathcal{O}(N_{E}M + M^{2}N)$ , where  $N_{E}$  denotes the number of edges,  $M$  denotes the number of anchors, and  $N$  denotes the number of nodes. Notably, the adjacency matrices are often sparse, so the computational complexity can be significantly decreased. Likewise, the OT based on Sinkhorn algorithm in Algorithm 3 takes  $\mathcal{O}(MN)$ . Then, the FGW-OT takes  $\mathcal{O}(N_{E}M + M^{2}N + MN)$ . For the low-rank tensor optimization, the update of  $\pmb{G}$  with  $D_{l,1} \times D_{l,2} \times K$  in Algorithm 4 occupies the dominant computational complexity. First, the FFT and the inverse FFT require  $\mathcal{O}(D_{l,1}^2 D_{l,2}\log (K))$ , where  $D_{l,1}$  and  $D_{l,2}$  are the dimensions for the  $l$ -th layer's parameters,  $K$  is the number of clients, and  $D_{l,1}$  is the higher dimension. Second, the T-SVD costs  $\mathcal{O}(D_{l,1}D_{l,2}^{2}K)$ . Then, the low-rank tensor optimization takes  $\mathcal{O}(D_{l,1}^{2}D_{l,2}\log (K) + D_{l,1}D_{l,2}^{2}K)$ . Overall, the computational complexity for the proposed FedPAM is  $\mathcal{O}(N_{E}M + M^{2}N + MN + D_{l,1}^{2}D_{l,2}\log (K) + D_{l,1}D_{l,2}^{2}K)$ .
+
+# F The Details of Eight Graph Datasets
+
+We conduct the experiments on eight graph datasets, including Cora, CiteSeer, PubMed, Ogbn-Arxiv, Computers, Photo, Physics, Amazon-ratings. Their detailed information about the numbers of nodes, features, edges, classes, the proportions of dataset divisions, and the categories of datasets are described in Table 4. The statistical information of local subgraphs for eight graph datasets is presented in Fig. 7.
+
+Table 4: Descriptions of eight graph datasets.  
+
+<table><tr><td>Dataset</td><td>Nodes</td><td>Features</td><td>Edges</td><td>Classes</td><td>Train / Val / Test</td><td>Category</td></tr><tr><td>Cora</td><td>2,708</td><td>1,433</td><td>5,429</td><td>7</td><td>20% / 40% / 40%</td><td>Citation Network</td></tr><tr><td>CiteSeer</td><td>3,327</td><td>3,703</td><td>4,732</td><td>6</td><td>20% / 40% / 40%</td><td>Citation Network</td></tr><tr><td>PubMed</td><td>19,717</td><td>500</td><td>44,338</td><td>3</td><td>20% / 40% / 40%</td><td>Citation Network</td></tr><tr><td>Ogbn-Arxiv</td><td>169,343</td><td>128</td><td>231,559</td><td>40</td><td>60% / 20% / 20%</td><td>Citation Network</td></tr><tr><td>Computers</td><td>13,381</td><td>767</td><td>245,778</td><td>10</td><td>20% / 40% / 40%</td><td>Co-purchase Network</td></tr><tr><td>Photo</td><td>7,487</td><td>745</td><td>119,043</td><td>8</td><td>20% / 40% / 40%</td><td>Co-purchase Network</td></tr><tr><td>Physics</td><td>34,493</td><td>8,415</td><td>247,962</td><td>5</td><td>20% / 40% / 40%</td><td>Co-author Network</td></tr><tr><td>Amazon-ratings</td><td>24,492</td><td>300</td><td>93,050</td><td>5</td><td>50% / 25% / 25%</td><td>Rating Network</td></tr></table>
+
+![](images/74e843357714f87861652dc26fcc1773093cb6ab846f4c35091acc7db994a574.jpg)  
+(a) Cora
+
+![](images/0c01b755018cca1750db10d8c0bdcb34729f1a58689a5b0a951a69d828203c0f.jpg)  
+(b) CiteSeer
+
+![](images/b8b276fde3427bb7e7c300b8ca6927c23f14fddd0827dba4fc9357e5c925c9d7.jpg)  
+(c) PubMed
+
+![](images/3e1294fbbbf259ed86da03955a05383e14ea1e7c17c0331929638aab8fe3aa02.jpg)  
+(d) Ogbn-Arxiv
+
+![](images/8306e603d5837a47eba27be89f25e38b817b0ecff6ca7c33d4eec56dcd3173c6.jpg)  
+(e)Computers
+
+![](images/da371e8abeb794ab8bf16b17bfe331755b036942562066a3718a383ace6795e3.jpg)  
+(f) Photo
+
+![](images/3a254ebf84a9a87b9443d61fc57211143cf45fbbe9376ab80b39d2ad8b8c0b88.jpg)  
+(g) Physics  
+Figure 7: Statistical information of local subgraphs for eight graph datasets when the number of clients is fixed as 10.
+
+![](images/5552b7a9d6a449e66c4eb472a0df833970137d0cbebc87c7bb719df1f7da76dd.jpg)  
+(h) Amazon-ratings
+
+# G The Performance Comparison Using KNN as the Evaluation Metric
+
+Standard linear probing is sensitive to linearly differentiable features and may not accurately evaluate the model if the data distribution is complex. In addition to using the standard linear probing, we use KNN as the evaluation metric, the results are reported in Table 5. When using Simsiam and BOYL as the SSL losses on CiteSeer dataset, the values of evaluation metrics are much better than that when using standard linear probing. Hence, employing different evaluation methods provide a more comprehensive understanding of models. Fortunately, it can be seen that the proposed FedPAM achieves optimal performance in most cases, proving that embedding output by FedPAM has better discriminative property.
+
+# H The Performance Comparison with Large-Scale Clients
+
+Performance validation on large-scale clients is an important measure for the scalability of FL methods, we test the performance of various FL methods with 50 and 100 clients on PubMed and Ogbrn-Arxiv (OA) datasets, the results are presented in Fig. 8, where SimCLR is used as the SSL loss.
+
+Table 5: Performance comparison on Cora and CiteSeer datasets with three different SSL losses. The KNN is used as evaluation method, where the optimal results are bolded and the suboptimal results are underlined.  
+
+<table><tr><td rowspan="2">SSL</td><td rowspan="2">Method</td><td colspan="2">Cora</td><td colspan="2">CiteSeer</td></tr><tr><td>ACC</td><td>Fscore</td><td>ACC</td><td>Fscore</td></tr><tr><td rowspan="5">Simsiam</td><td>FedAvg</td><td>54.55</td><td>42.90</td><td>36.66</td><td>25.71</td></tr><tr><td>FedProx</td><td>55.17</td><td>41.80</td><td>39.59</td><td>28.01</td></tr><tr><td>MOON</td><td>56.28</td><td>44.83</td><td>36.75</td><td>26.75</td></tr><tr><td>FedU2</td><td>54.90</td><td>41.92</td><td>40.99</td><td>30.44</td></tr><tr><td>FedPAM</td><td>58.93</td><td>47.27</td><td>50.40</td><td>38.01</td></tr><tr><td rowspan="6">SimCLR</td><td>FedAvg</td><td>52.49</td><td>38.05</td><td>37.77</td><td>24.60</td></tr><tr><td>FedProx</td><td>53.28</td><td>39.10</td><td>36.37</td><td>22.73</td></tr><tr><td>MOON</td><td>53.01</td><td>38.74</td><td>37.68</td><td>24.80</td></tr><tr><td>FedX</td><td>54.31</td><td>40.62</td><td>42.74</td><td>31.16</td></tr><tr><td>FedU2</td><td>54.30</td><td>40.68</td><td>38.78</td><td>25.78</td></tr><tr><td>FedPAM</td><td>57.71</td><td>44.53</td><td>42.07</td><td>30.43</td></tr><tr><td rowspan="6">BYOL</td><td>FedAvg</td><td>64.74</td><td>53.63</td><td>44.88</td><td>34.17</td></tr><tr><td>FedU</td><td>63.35</td><td>52.59</td><td>46.25</td><td>35.33</td></tr><tr><td>FedEMA</td><td>64.20</td><td>52.53</td><td>43.30</td><td>30.62</td></tr><tr><td>Orchestra</td><td>54.12</td><td>41.25</td><td>36.89</td><td>24.65</td></tr><tr><td>FedU2</td><td>59.87</td><td>48.18</td><td>44.58</td><td>33.33</td></tr><tr><td>FedPAM</td><td>64.99</td><td>53.08</td><td>49.25</td><td>39.68</td></tr></table>
+
+It can be seen that the proposed FedPAM stays ahead of the curve even with a large number of clients, showing that representation space alignment and adaptive global parameter learning are effective. Notably, the superiority of the proposed FedPAM is more obvious in terms of Fscore, indicating that the proposed FedPAM not only has a high overall prediction accuracy, but also shows a better performance on the minority classes.
+
+![](images/f14c524d22bc5e7aaa1b4877e7821386967664378fa56bcfb971307afbeff16c.jpg)  
+PubMed: 50 Clients
+
+![](images/f0430f09037df63aa78730daeab088d152a1c7ec8de57f5a6c36a7c0622f41e7.jpg)  
+PubMed: 50 Clients
+
+![](images/7b19822362fe44cbeda55e0111db00b0c9841006db432a89954c7142fd554fd4.jpg)  
+PubMed: 100 Clients
+
+![](images/254886bba30880421f429b3a76dc12b9c4c2e13561127079a90a91f71159d03b.jpg)  
+PubMed: 100 Clients
+
+![](images/1bcac36e384b2d529f2e8fbf960fc641a9e98e29e7be8fee4846bb33ebac2ec1.jpg)  
+OA: 50 Clients
+
+![](images/a4f75c49e661aa7003f5c1998fd28188ca22f8da719e6be501f2e025acedc695.jpg)  
+OA: 50 Clients
+
+![](images/edc489646387617ad9132328c188ebcdc24a0d32616d184fc3a4ad4d40b6f688.jpg)  
+OA: 100 Clients  
+Figure 8: Performance comparison with more clients on PubMed and Ogbn-Arxiv (OA) datasets, where the number of clients is set to 50 and 100, respectively.
+
+![](images/5253e5c3f789bda50e67063ff6975818958a6c83f9d76e249da7d253f1675f3f.jpg)  
+OA: 100 Clients
+
+# I The Performance Comparison of Plugging the AGPL Module into Existing Works
+
+Notably, the proposed AGPL module is a plug-and-play module, which can be incorporated into existing works and promote their performance. Specifically, we plug AGPL into FedAvg, FedProx,
+
+Table 6: Performance comparison on Cora, CiteSeer, and Ogbn-Arxiv datasets when the AGPL module is plugged into existing works, where the optimal results are **bolded**.  
+
+<table><tr><td rowspan="2">SSL</td><td rowspan="2">Method</td><td colspan="2">Cora</td><td colspan="2">CiteSeer</td><td colspan="2">Ogbn-Arxiv</td></tr><tr><td>ACC</td><td>Fscore</td><td>ACC</td><td>Fscore</td><td>ACC</td><td>Fscore</td></tr><tr><td rowspan="6">Simsiam</td><td>FedAvg</td><td>54.38</td><td>39.27</td><td>36.30</td><td>20.34</td><td>35.26</td><td>20.49</td></tr><tr><td>FedAvg+AGPL</td><td>58.56</td><td>46.67</td><td>43.29</td><td>30.60</td><td>35.33</td><td>21.21</td></tr><tr><td>FedProx</td><td>55.35</td><td>40.83</td><td>36.80</td><td>21.34</td><td>34.61</td><td>20.03</td></tr><tr><td>FedProx+AGPL</td><td>60.15</td><td>47.07</td><td>43.75</td><td>31.58</td><td>35.26</td><td>20.49</td></tr><tr><td>MOON</td><td>54.03</td><td>39.81</td><td>35.94</td><td>20.41</td><td>35.46</td><td>21.20</td></tr><tr><td>MOON+AGPL</td><td>58.92</td><td>46.22</td><td>41.34</td><td>30.42</td><td>35.44</td><td>21.59</td></tr><tr><td rowspan="8">SimCLR</td><td>FedAvg</td><td>54.65</td><td>39.95</td><td>35.48</td><td>19.62</td><td>45.58</td><td>33.08</td></tr><tr><td>FedAvg+AGPL</td><td>56.65</td><td>44.18</td><td>41.35</td><td>27.80</td><td>47.10</td><td>37.13</td></tr><tr><td>FedProx</td><td>55.26</td><td>40.68</td><td>35.56</td><td>19.78</td><td>44.40</td><td>31.86</td></tr><tr><td>FedProx+AGPL</td><td>56.79</td><td>46.29</td><td>41.54</td><td>28.61</td><td>47.36</td><td>35.01</td></tr><tr><td>MOON</td><td>54.56</td><td>39.81</td><td>36.30</td><td>20.34</td><td>46.23</td><td>33.96</td></tr><tr><td>MOON+AGPL</td><td>56.49</td><td>43.74</td><td>41.82</td><td>29.26</td><td>47.91</td><td>39.04</td></tr><tr><td>FedX</td><td>56.73</td><td>43.11</td><td>37.10</td><td>22.37</td><td>46.69</td><td>34.77</td></tr><tr><td>FedX+AGPL</td><td>57.97</td><td>44.56</td><td>42.92</td><td>30.75</td><td>48.76</td><td>39.56</td></tr></table>
+
+MOON, and FedX with Simsiam and SimCLR losses, respectively. Table 6 reports the experimental results. It can be observed that the performance of existing works after equipped with AGPL is improved, demonstrating its effectiveness.
\ No newline at end of file
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+# Unsupervised Learning for Optimal Transport plan prediction between unbalanced graphs
+
+Sonia Mazelet
+
+CMAP, Ecole Polytechnique
+
+Palaiseau, France
+
+sonia.mazelet@polytechnique.edu
+
+Rémi Flamary
+
+CMAP, Ecole Polytechnique
+
+Palaiseau, France
+
+remi.flamary@polytechnique.edu
+
+Bertrand Thirion
+
+Mind, Inria-Saclay
+
+Palaiseau, France
+
+bertrand.thirion@inria.fr
+
+# Abstract
+
+Optimal transport between graphs, based on Gromov-Wasserstein and other extensions, is a powerful tool for comparing and aligning graph structures. However, solving the associated non-convex optimization problems is computationally expensive, which limits the scalability of these methods to large graphs. In this work, we present Unbalanced Learning of Optimal Transport (ULOT), a deep learning method that predicts optimal transport plans between two graphs. Our method is trained by minimizing the fused unbalanced Gromov-Wasserstein (FUGW) loss. We propose a novel neural architecture with cross-attention that is conditioned on the FUGW tradeoff hyperparameters. We evaluate ULOT on synthetic stochastic block model (SBM) graphs and on real cortical surface data obtained from fMRI. ULOT predicts transport plans with competitive loss up to two orders of magnitude faster than classical solvers. Furthermore, the predicted plan can be used as a warm start for classical solvers to accelerate their convergence. Finally, the predicted transport plan is fully differentiable with respect to the graph inputs and FUGW hyperparameters, enabling the optimization of functionals of the ULOT plan.
+
+# 1 Introduction
+
+Graph alignment In many graph data applications, aligning or matching nodes between two graphs is necessary. Examples include object detection, where semantic correspondences between objects in two images from different domains can make the model more adaptive [12]; graph edit distance [22, 18], where the goal is to compute the distance between two graphs, which requires finding the best correspondence between their nodes; shape matching [20], or brain alignment across subjects [30].
+
+But the problem of graph matching is challenging because of the combinatorial nature of the problem, which can often be reformulated as a Quadratic Assignment Problem (QAP) known to be NP-hard [16]. One strategy that has been proposed recently is to use deep learning to learn the matching in a supervised setting [36], [25]. These methods typically rely on a graph neural network (GNN) to learn a representation of the nodes and then use a matching algorithm to compute the correspondence between the nodes. However, the problem is even more difficult when the graphs are unbalanced, i.e. when they have different numbers of nodes or when some nodes have noisy features or connections.
+
+Optimal transport between graphs In recent years, Optimal Transport (OT) has emerged as a powerful tool for solving the graph matching problem. It can be seen as a continuous relaxation of the QAP with the Gromov-Wasserstein (GW) distance [17, 21, 40], which is a generalization of the classical Wasserstein distance to distributions in different metric spaces. Extensions of the GW distance to labeled graphs have been proposed, such as the Fused Gromov-Wasserstein (FGW) distance [31]. But those OT problems put strong constraints on the transport plan, which makes them very sensitive to noise, outliers and local deformations. This is why Unbalanced GW [27, 5] and Fused Unbalanced GW (FUGW) [30] were proposed to generalize the GW and FGW distances to unbalanced settings with application in positive unlabeled learning and brain alignment.
+
+Complexity of classical optimal transport solvers Quadratic OT problems such as FGW and FUGW are non-convex. Classical solvers rely on a block coordinate descent algorithm to iteratively solve linearized versions of the problem. This linearization is of complexity  $O(n_{1}n_{2}^{2} + n_{2}^{2}n_{1})$ , where  $n_1$  and  $n_2$  are the number of nodes in the two graphs [21]. This makes the method unscalable for large graphs, for applications where we need to compute the transport plan for many pairs of graphs or when validation of the hyperparameters is necessary. This is especially problematic in the unbalanced case where the solution is very sensitive to the parameters.
+
+The ML community has recently proposed to use deep learning for accelerating or solving OT problems. For example [11] proposed to estimate the OT mapping using a neural network and extensions to the GW have been proposed in [19, 38]. But the most relevant work for our purpose is Meta OT [2] which proposed to learn to predict the dual potentials of the entropic OT problem with a neural network. But this approach is limited to classical entropy regularized OT and not directly applicable to the quadratic FUGW problem which motivates our proposed approach detailed below.
+
+Contributions We propose in this paper a new method to learn a neural network that predicts the transport plan of the FUGW problem denoted as Unsupervised Learning of Optimal Transport plan prediction (ULOT). We propose a novel architecture based on graph neural networks (GNN) and cross attention mechanisms to predict the OT plan with a complexity of  $O(n_{1}n_{2})$ , which is significantly faster than classical solvers. In addition the neural network is conditioned by the parameters of the FUGW problem, which allows to efficiently predict OT plan for all the possible values of the parameters. This is particularly useful in the unbalanced case where the solution is very sensitive to the parameters and validation is often necessary. We show in our experiments, on simulated and real life data, that our method outperforms classical solvers in terms of speed by two orders of magnitude while providing OT plans with competitive loss. The predicted transport plan can also be used as a warm start for classical solvers, which reduces the number of iterations and the overall time of the algorithm. We also show that our method provides a smooth estimation of the transport plan that can be used for numerous applications such as parameter validation for label propagation or gradient descent of a FUGW loss. Our code is available at https://github.com/smazelet/ULOT
+
+# 2 Learning to predict OT plans between graphs
+
+In this section we first introduce the FUGW optimal transport problem and its associated loss function. Next we present our amortized optimization strategy called Unbalanced Learning of Optimal Transport (ULOT) and detail the architecture of the neural network. Finally we discuss the related works in deep learning for optimal transport and graph matching.
+
+# 2.1 Fused Unbalanced Gromov Wasserstein (FUGW)
+
+Definition of the FUGW loss Consider the two graphs  $\{G_k = (F_k, D_k, \omega_k)\}_{k = \{1,2\}}$  with  $n_1$  and  $n_2$  nodes respectively. For  $k \in \{1,2\}$ , they are characterized by their node features  $\mathbf{F}_k \in \mathbb{R}^{n_k \times d}$ , their connectivity matrices  $D_k \in \mathbb{R}^{n_k \times n_k}$  (usually adjacency matrix or shortest path distance matrix) and their node weights  $\omega_k \in \Delta_{n_k} \triangleq \{(\omega_k^1, \dots, \omega_k^{n_k}), \sum_{i=1}^{n_k} \omega_k^i = 1\}$  that characterize the node's relative importance [34]. Note that in the following we will assume that these weights are uniform, i.e.,  $\omega_k^i = 1 / n_k$  for  $i = 1, \dots, n_k$ . The goal of FUGW [30] is to learn a positive transport plan  $P \in \mathbb{R}^{n_1, n_2}$
+
+between the nodes of  $G_{1}$  and  $G_{2}$  that minimizes the following loss function:
+
+$$
+\begin{array}{l} \mathrm {L} ^ {\alpha , \rho} \left(G _ {1}, G _ {2}, P\right) = (1 - \alpha) \sum_ {i, j = 1} ^ {n _ {1}, n _ {2}} \left\| \left(\boldsymbol {F} _ {1}\right) _ {i} - \left(\boldsymbol {F} _ {2}\right) _ {j} \right\| _ {2} ^ {2} \boldsymbol {P} _ {i, j} (1) \\ + \alpha \sum_ {i, j, k, l = 1} ^ {n _ {1}, n _ {2}, n _ {1}, n _ {2}} \left| \left(\boldsymbol {D} _ {1}\right) _ {i, k} - \left(\boldsymbol {D} _ {2}\right) _ {j, l} \right| ^ {2} \boldsymbol {P} _ {i, j} \boldsymbol {P} _ {k, l} (2) \\ + \rho \left(\mathrm {K L} \left(P _ {\# 1} \otimes P _ {\# 1} \mid \omega_ {1} \otimes \omega_ {1}\right) + \mathrm {K L} \left(P _ {\# 2} \otimes P _ {\# 2} \mid \omega_ {2} \otimes \omega_ {2}\right)\right). (3) \\ \end{array}
+$$
+
+The FUGW loss is a combination of the Wasserstein distance (1) that measures the preservation of node features, the Gromov Wasserstein distance (2) that measures the conservation of local geometries and a penalization of the violation of the marginals for the OT plan with the KL divergence (3). The terms (1) and (2) are weighed by the trade-off parameters  $\alpha \in [0,1]$  and the marginal penalization (3) is weighted by  $\rho$ . Unbalanced OT is a very general and robust framework that can adapt to differences in the geometry of the graph vertices.
+
+Complexity of solving  $\min_{P\geq 0}\mathbf{L}^{\alpha ,\rho}(G_1,G_2,P)$  In order to solve the FUGW problem one needs to minimize the FUGW loss with respect to the optimal transport plan  $P$ . Because of the Gromov Wasserstein term, the complexity of the FUGW loss or its gradient for a given plan  $P$  is theoretically quartic  $O(n_{1}^{2}n_{2}^{2})$ . In the case of the square loss, Peyre et al [21] showed that the complexity can be reduced to cubic complexity  $O(n_{1}n_{2}^{2} + n_{1}^{2}n_{2})$ , which remains computationally intensive for large graphs with typically more than  $10k$  nodes.
+
+Existing methods to minimize the FUGW loss use a block coordinate descent scheme on a lower bound of the objective that consists in solving at each iteration a linearization of the quadratic problem [8], [30] and requires at each iteration to compute the cubic gradient. In the following we will compare ULOT to three different types of inner-solvers for the linearized problem: the Majorization-minimization (MM) algorithm [6], the inexact Bregman Proximal Point (IBPP) algorithm [39] and a more classical L-BFGS-B algorithm [4]. We will also compare our approach to the entropic regularization of the FUGW loss for which the inner problem can be solved using the Sinkhorn algorithm. All those methods are iterative and require a  $O(n_{1}n_{2}^{2} + n_{1}^{2}n_{2})$  gradient/linearization computation at each iteration. This is a major bottleneck for the FUGW problem, especially when the graphs are large or FUGW has to be solved multiple times, for instance when computing a FUGW barycenter while selecting the parameters  $(\alpha, \rho)$  of the method.
+
+# 2.2 ULOT optimization problem and architecture
+
+Learning to predict FUGW OT plans We propose to train a model  $P_{\theta}^{\rho, \alpha}(G_1, G_2)$ , parametrized by  $\theta$ , that, given two graphs  $G_1, G_2$  and  $(\rho, \alpha)$  parameters can predict a FUGW transport plan, or at least a good solution to the FUGW problem, between them. We want to avoid training the model in a supervised way where for each pair of graphs  $(G_1, G_2)$  in the training set we need to pre-compute the corresponding transport plan  $\bar{P}$ . This is why we propose to train the model in an unsupervised way using amortized optimization [3]. We do this by sampling pairs of graphs  $(G_1, G_2)$  from a training dataset  $\mathcal{D}$  and parameters  $(\rho, \alpha)$  and minimizing the expected FUGW loss. The ULOT model is trained to minimize:
+
+$$
+\min  _ {\theta} \mathbb {E} _ {G _ {1}, G _ {2} \sim \mathcal {D} ^ {2}, \alpha , \rho \sim \mathcal {P}} \left[ \mathrm {L} ^ {\alpha , \rho} \left(G _ {1}, G _ {2}, \boldsymbol {P} _ {\theta} ^ {\rho , \alpha} \left(G _ {1}, G _ {2}\right)\right) \right]. \tag {4}
+$$
+
+Where  $\mathcal{P}$  is the distribution of the parameters  $(\rho, \alpha)$  chosen in the experiment as  $\mathcal{P}_{\rho}, \mathcal{P}_{\alpha}$ , where  $\mathcal{P}_{\rho}$  is the log uniform distribution between  $10^{-7}$  and 1, and  $\mathcal{P}_{\alpha}$  follows the Beta distribution with parameters  $(0.5, 0.5)$ . The fact that we optimize the loss over some intervals of the parameters  $(\rho, \alpha)$  allows the model to generalize to different values of the parameters to explore or even optimize them.
+
+Encoding the parameters  $(\rho, \alpha)$  Since we want the model  $P_{\theta}^{\rho, \alpha}(G_1, G_2)$  to depend on the parameters  $(\rho, \alpha)$ , we need to encode them in a way that can be used by the neural network. To do that, we add them to the node features of the graphs at each layer of the network.  $\rho$  is a positive scalar that impacts the mass of the OT plan and can be included as such. But  $\alpha$  is a scalar in  $[0, 1]$  fixing the tradeoff between the Wasserstein and Gromov-Wasserstein terms that can have a large impact close to 0 and 1. In order to facilitate its use in the neural network, we propose to encode it in a more
+
+![](images/f7ceea669dd4cdad652c972f1f884ae4f1964bd0ad310a98b3cd3e252d831b09.jpg)  
+Figure 1: ULOT architecture for OT plan prediction The architecture consists of two parts. a node embedding layer repeated  $N$  times that relies on cross graph attention and self node updates (GCN), and the final transport plan prediction layer that predicts node weights and the output transport plan.
+
+expressive way with a positional encoding technique. We chose to use the same technique as in [32] using the Fourier basis for encoding time in flow matching applications as follows:
+
+$$
+\hat {\alpha} = \left[ (\cos (k \pi \alpha)) _ {k = 1, \dots , d} | (\sin k \pi (1 - \alpha)) _ {k = 1, \dots , d} \right]. \tag {5}
+$$
+
+where we set  $d = 10$  in our experiments.  $\hat{\alpha}$  and  $\rho$  are concatenated to the node features at each layer of the network as detailed next.
+
+# 2.3 Proposed cross-attention neural architecture
+
+We now detail the proposed neural network architecture that takes as input two graphs  $G_{1}$  and  $G_{2}$  and the parameters  $(\rho, \alpha)$  and predicts the transport plan  $P_{\theta}^{\rho, \alpha}(G_{1}, G_{2})$ . The proposed ULOT architecture summarized in Figure 1 consists of two main parts. The first part contains  $N$  layers of node embedding and attention-based cross-graph interactions that incorporate the geometry of each graph. The final layer predicts the transport plan from the learned node features and interactions with node reweighting. The design choices are further motivated in the ablation studies in Section C.
+
+Node embedding with cross attention The first block is an adaptation of the Graph Matching Network from [13] that outputs node features that are relevant in a matching context. Intuitively the cross attention mechanism computes node features that are similar for nodes that can be matched and dissimilar for nodes that should not. The node embeddings are computed at each layer using two paths. One path is called the self path that simply consists of a GCN treating each graph independently. The use of GCNs is motivated by the Gromov term in the FUGW loss that requires a model taking into account the graph topology. For each graph  $G_{k}$  for  $k\in \{1,2\}$ , self node features are computed with
+
+$$
+\boldsymbol {F} _ {k} ^ {\text {s e l f}} = \operatorname {G C N} \left(\boldsymbol {F} _ {k}\right). \tag {6}
+$$
+
+The second path is the cross path that computes a similarity matrix between the node features of both graphs and learns new features that characterize their interactions. In parallel, cross node features  $\pmb{F}_{1\rightarrow 2}^{\mathrm{cross}}, \pmb{F}_{2\rightarrow 1}^{\mathrm{cross}}$  are computed with an attention block, for  $k, k' \in \{1,2\}$ ,  $k \neq k'$ , learned in the following way:
+
+1. Compute embeddings:
+
+$$
+\boldsymbol {F} _ {k} ^ {\text {c r o s s}} = \operatorname {M L P} \left(\boldsymbol {F} _ {k}, \rho , \hat {\alpha}\right). \tag {7}
+$$
+
+2. Compute similarity matrix  $S$  and the row/column attention matrices  $S_{1}, S_{2}$  with, for  $s$  the cosine similarity and  $i \in [1, n_{1}]$ ,  $j \in [1, n_{2}]$ :
+
+$$
+S _ {i, j} = s \left(\left(\boldsymbol {F} _ {1} ^ {\text {c r o s s}}\right) _ {i}, \left(\boldsymbol {F} _ {2} ^ {\text {c r o s s}}\right) _ {j}\right) \quad \text {a n d} \quad \left\{ \begin{array}{l} \boldsymbol {S} _ {1} = \operatorname {s o f t m a x} _ {\text {r o w}} \left(a ^ {2} \boldsymbol {S}\right) \\ \boldsymbol {S} _ {2} = \operatorname {s o f t m a x} _ {\text {c o l u m n}} \left(a ^ {2} \boldsymbol {S}\right) \end{array} \right. \tag {8}
+$$
+
+where  $a\in \mathbb{R}$  is an hyperparameter.
+
+3. Compute the updated node features  $F_{1\rightarrow 2}^{\mathrm{cross}}$  and  $F_{2\rightarrow 1}^{\mathrm{cross}}$ :
+
+$$
+\boldsymbol {F} _ {1 \rightarrow 2} ^ {\text {c r o s s}} = \operatorname {L i n e a r} \left(\boldsymbol {F} _ {2} ^ {\text {c r o s s}} - \boldsymbol {S} _ {2} ^ {T} \boldsymbol {F} _ {1} ^ {\text {c r o s s}}\right), \quad \boldsymbol {F} _ {2 \rightarrow 1} ^ {\text {c r o s s}} = \operatorname {L i n e a r} \left(\boldsymbol {F} _ {1} ^ {\text {c r o s s}} - \boldsymbol {S} _ {1} \boldsymbol {F} _ {2} ^ {\text {c r o s s}}\right), \tag {9}
+$$
+
+At the end of the layer, for  $k, k' \in 1, 2$ , the learned node features from the self (GCN) and cross-attention paths are merged with the input node features with the following equation:
+
+$$
+\boldsymbol {F} _ {k} ^ {\text {f i n a l}} = \operatorname {M L P} \left(\operatorname {L i n e a r} \left(\boldsymbol {F} _ {k}\right), \operatorname {L i n e a r} \left(\boldsymbol {F} _ {k} ^ {\text {s e l f}}\right), \operatorname {L i n e a r} \left(\boldsymbol {F} _ {k ^ {\prime} \rightarrow k} ^ {\text {m a t c h}}\right), \rho , \hat {\alpha}\right), \tag {10}
+$$
+
+where the inputs to the MLP are concatenated. The use of those two paths allows to learn node features that take into account both the graph geometry and the cross-interactions between the graphs.
+
+Transport plan prediction with cross attention and node scaling The optimal transport plan prediction block is designed so as to predict an unbalanced OT plan that must contain pairwise relationships between nodes but also that can discard (reweight) nodes that cost too much from an OT perspective. This is done by separating the two aspects. We chose to use for the pairwise relationship in the OT plan the average of the cross attention matrices corresponding to row-wise softmax and a column-wise softmax respectively. This matrix gives a good starting point for the plan, which is then refined in the unbalanced optimal transport plan block, where the scaling per node makes it no longer balanced. This is why we also provide in the last block two node reweighting heads that allow scaling on the OT plan and enable the network to discard mass. These heads predict weight vectors  $\boldsymbol{v}_k \in \mathbb{R}^{n_k}$ ,  $k \in \{1, 2\}$  on the nodes of the graphs that represent how much weight is transported from each node with
+
+$$
+\boldsymbol {v} _ {1} = \operatorname {s i g m o i d} \left(\operatorname {L i n e a r} \left(\boldsymbol {F} _ {1} ^ {\text {f i n a l}}, \rho , \widehat {\boldsymbol {\alpha}}\right)\right), \quad \boldsymbol {v} _ {2} = \operatorname {s i g m o i d} \left(\operatorname {L i n e a r} \left(\boldsymbol {F} _ {2} ^ {\text {f i n a l}}, \rho , \widehat {\boldsymbol {\alpha}}\right)\right) \tag {11}
+$$
+
+We learn the transport plan from the similarity matrices  $S_{1}, S_{2}$  of the last layer
+
+$$
+\boldsymbol {P} _ {\theta} ^ {\rho , \alpha} (G _ {1}, G _ {2}) = \frac {1}{2} \left(\frac {1}{n _ {1}} \boldsymbol {S} _ {1} \operatorname {d i a g} (\boldsymbol {v} _ {1}) + \frac {1}{n _ {2}} \operatorname {d i a g} (\boldsymbol {v} _ {2}) \boldsymbol {S} _ {2}\right). \tag {12}
+$$
+
+This allows us to learn a transport plan that is unbalanced and separates the problem of finding the node interactions across graphs, done with the cross attention, and the problem of learning the individual node weights that is specific to unbalanced OT.
+
+# 2.4 Related works
+
+Deep optimal learning for transport Several methods have been proposed for accelerating the resolution of optimal transport problems with deep learning. For instance [26] proposed to model the dual potentials of the entropic regularized OT problem with a neural network. Neural OT [11] learns the classical OT mapping between two distributions using a neural network. Recent Neural OT extensions have also been proposed to solve the GW problem in [19, 38]. This usually allows for solving large OT problems but the resulting neural network is a solution for a specific pair of distributions and needs to be optimized again for new distribution pairs.
+
+Meta OT [2] uses a strategy called amortized optimization [3] to learn an MLP neural network that predicts on the samples the dual potentials of the entropy regularized OT problem. Their model can be used to predict the entropic transport plan between two new distributions using the primal-dual relationship. However, Meta OT cannot be used for Quadratic OT problems such as the Gromov-Wasserstein or FUGW problems because the optimization problem is not convex and the primal-dual relationship is much more complicated [42]. In fact, in order to use an approach similar to Meta OT, one would need to use a linearization of the problem that is  $O(n^{3})$  which would cancel part of the advantage of using an efficient neural network.
+
+ULOT has been designed to perform OT plan prediction with  $O(n^{2})$  complexity. Our approach also focuses on graph data. There is a need to go beyond MLP in order to design a neural network that can use the graph structure. In this sense, ULOT can be seen as a generalization of Meta OT to the case of unbalanced OT between graphs. Finally we learn a model that can predict the transport plan conditioned on the parameters of the OT problem, in our case the  $(\alpha, \rho)$ , which is particularly novel and has not been done before to the best of our knowledge.
+
+Deep learning for graph matching Various neural architectures have been proposed for deep graph matching, which refers to the problem of finding structural correspondence between graphs. However these methods often face notable limitations. Some approaches are limited to predicting a global similarity score between pairs of graphs without predicting node correspondence [13, 14]. Other methods that do provide node-level matching typically rely on supervised training [36, 25, 41], which limits their applications to domains with available ground truth correspondences such as images.
+
+OT plan with respect to  $\rho$  for  $(1,2)\to (2,3)$
+
+![](images/feab4fef85afe73d106b2eb314661922215c8400dd5a00465fa4b56ef2ad5fbc.jpg)  
+ULOT
+
+![](images/9b9096b41f6e6bd1f38b8868cac66a61052e835e7e9725f46321bfcb1cb37772.jpg)
+
+![](images/11020295896c4e0a593c78bdadca0995a1dc07eb68b6269041feb5d5ab2b2568.jpg)
+
+![](images/0e0895549b59b5ef1f49921d1044c66ae409737c544164da5479fb170c4f3bda.jpg)  
+Solver  
+$\rho = 0.001$
+
+![](images/e00811736c23f74ca233db957e2954ecc48300b84c9c85226a81887ae86c7286.jpg)  
+$\rho = 0.002$
+
+![](images/e023cc2006fad554b812d61ad210cfacd8c899ce37799d856549384d389fc4fe.jpg)  
+$\rho = 0.007$
+
+![](images/4ca6c292e6b0e30d85539a32507961f7b25f3befc99bcb5a88ff394995eda23a.jpg)  
+Nodes transported mass for increasing  $\rho$  values  
+Figure 3: Illustration of OT plans for different values of  $\rho$ . (left) OT plan (red lines) predicted by ULOT (top) and estimated by the IBPP solver (bottom). The red lines opacity is proportional to the amount of transported mass. (right) Marginals on the nodes of both graphs for different values of  $\rho$  colored by the cluster they belong to.
+
+Unsupervised methods, on the other hand, rely on task-specific objectives such as enforcing cycle consistency [33, 37] or adopt contrastive learning approaches by matching graphs to their augmented copies, for which they know the ground truth [15].
+
+To improve the robustness of the matching between graphs of different sizes, some methods discard a subset of correspondences. Various methods have been proposed such as selecting the top-k most confident matches [35] or adding dummy nodes [10]. However, while these methods have a similar goal to ULOT they perform a hard selection of nodes, which is useful in the case of outliers but not when we need a more continuous way to reduce some node importance. Finally, while our method bears some similarities with neural networks for graph matching, it is important to note that the objective of Unbalanced OT is fundamentally different from graph matching.
+
+# 3 Numerical experiments
+
+In this section, we evaluate ULOT and compare it to classical solvers on both a simulated dataset of Stochastic Block Models (SBMs) with different numbers of clusters and on the Individual Brain Charting (IBC) dataset [23] of functional MRI activations on brain surfaces.
+
+# 3.1 Illustration and interpretation on simulated graphs
+
+Dataset and training setup We first train ULOT on a simulated dataset of Stochastic Block Models (SBMs) with 3 linearly connected clusters and 3D node features that are a one hot encoding of the cluster classes 1, 2 or 3 with additive centered Gaussian noise. To investigate the properties of the transport plans learned by ULOT in comparison to those estimated by classical solvers, we construct three types of graphs with different clusters. The first type includes graphs where all three clusters (1, 2, 3) are present, the second type of graphs has clusters (1, 2) and the third has clusters (2, 3). All graphs in the dataset have a random number of nodes ranging from 30 to 60 and the training dataset consists of 50000 simulated pairs  $(G_{1}, G_{2}, \rho, \alpha)$ .
+
+With those three types of graphs, unbalanced OT should be able to find a transport plan that
+
+matches the clusters if they are present in both graphs, but discard clusters that are not shared (when  $\alpha$ ,  $\rho$  are properly selected). We compare ULOT to the numerical solver IBPP, which provides a good trade off between performance and speed.
+
+![](images/11009452d2b29d8d86ca369d31f16be29e82b243a9deae75244f23b85dd2249d.jpg)  
+OT plan with respect to  $\alpha$  for  $(1,2,3)\to (1,2,3)$  
+ULOT  
+Solver
+
+![](images/fad2cc06fdaaa86f06a552269e246b13ea6e51c7f4befb148e4498fb99ce0433.jpg)
+
+![](images/8f2abb96f2fdae1155b4287de1917e2b143e0cd6cc925cd3ec8192791ce45626.jpg)
+
+![](images/91d1e8d1b84ad049d13b1645ea117de7d935f3c196ba02e6c3c51335572c4d05.jpg)  
+$\alpha = 0.0$  
+Figure 2: Examples of transport plans (red lines) predicted by ULOT (top) and estimated by the IBPP solver (bottom) for different  $\alpha$  values. The red lines opacity is proportional to the amount of transported mass.
+
+![](images/cdc8ccdb942d2f83e9aa67dbb2eac009ba3e124c09cd9c1b6b78abaf3a277d03.jpg)  
+$\alpha = 0.5$
+
+![](images/47b6784b30b21d9d48ba8bfadbe7220350ce30fd5132e2b11f4649db37683ec5.jpg)  
+$\alpha = 1.0$
+
+Regularization path with respect to the parameters  $\rho$  and  $\alpha$  We first illustrate the effect of  $\rho$  on the predicted transport plan between graphs containing clusters (1, 2) and (2, 3). We see in Figure 3 (left) that the transported mass increases with  $\rho$  for both ULOT and the solver but while the solver is stuck in a 0 mass local minimum, ULOT is able to find a plan with mass and lower cost for  $\rho = 0.002$ . We also provide in Figure 3 (right) the regularization path for the marginals on the nodes of both graphs which shows that the nodes from cluster 2 in green are the first to receive mass when  $\rho$  increases finding a proper alignment of the clusters.
+
+Next we illustrate the OT plans between two graphs of type  $(1,2,3)$  for different values of  $\alpha$  in Figure 2. Because the node features are noisy, both the Gromov Wasserstein and Wasserstein terms in the loss are needed to predict accurate plans. We see that while the transport plans from ULOT and the solver are comparable for  $\alpha < 1$ , the solver wrongly matches the clusters for  $\alpha = 1$  because it does not use the node feature information and can permute classes. ULOT does not permute the clusters for  $\alpha = 1$  because it is continuous w.r.t.  $\alpha$  and has learned to use the node features to find a better plan.
+
+Optimizing the hyperparameter for a prediction task We now consider the task of label propagation between graphs, where node labels are known on a source graph but partly missing on a target graph. To infer the missing labels, we transport the one hot encoding of the source graph node labels onto the target graph, producing label probabilities for each node similarly to what was proposed in [28, 24]. A key challenge of this approach, when the graph types can differ, lies in selecting the appropriate FUGW parameters  $(\rho, \alpha)$  to ensure that the plan is relevant for label propagation.
+
+Thanks to our efficient ULOT framework, we can easily compute and visualize the accuracy of the label propagation task as a function of the parameters  $(\rho, \alpha)$  for different pairs of graphs. Due to lack of space this is provided in the supplementary material in Figure 11. We find that the accuracy surfaces are relatively smooth and that the optimal parameters greatly depend on the types of graphs transported.
+
+Since ULOT OT plans are by construction fully differentiable with respect to  $\rho$  and  $\alpha$ , we propose to optimize them, but taking as objective a classical smooth proxy for the accuracy: the Kullback-Leibler divergence between the one-hot encoded target classes and the predicted label scores. We show the optimization trajectories of parameters  $(\rho, \alpha)$  for different simulated pairs of graphs in Figure 4. We see that the trajectories vary significantly between the two types of graph pairs. Indeed between different types,  $\rho$  has to be small to avoid mass transfer between clusters
+
+that are not present in both graphs, while for the same types,  $\rho$  can be larger to allow mass transfer between clusters that are present in both graphs. Interestingly, we see that for the same graph pair types, the general trend is similar but the trajectories converge to different values, underlining the necessity of parameter validation for individual pairs. This proof of concept shows that ULOT parameters can be optimized for a given task, allowing for efficient bi-level optimization when the inner optimization problem is a FUGW.
+
+![](images/6e1d3df63308ed08236df339211f4ab9c74dedeb3678f4bc712a6a39cecbc926.jpg)
+
+![](images/845265f40fe6c1002d8a3b7563fd7061ce5d63cff8310ed8ad265ffdf50b9f53.jpg)  
+Figure 4:  $(\alpha, \rho)$  optimization trajectories for (top) different types  $(1,2,3) \rightarrow (1,2)$ , (bottom) same type  $(1,2,3)$ .
+
+Optimizing a graph wrt the FUGW loss One very interesting aspect of our method is that the predicted transport plan is fully differentiable with respect to the graph structure and features. We illustrate this by optimizing a functional of a graph. Given a target graph  $G^{\star}$ , we optimize the function  $F(G) = L^{\alpha, \rho}(G, G^{\star}, P_{\theta}^{\rho, \alpha}(G, G^{\star}))$  where we expect the graph  $G$  to converge to or close to the graph  $G^{\star}$ . Starting from  $G_0 = G$ , at each time step  $t$ , we predict the transport plan  $P_{\theta}^{\rho, \alpha}(G_t, G^{\star})$  between  $G_t$  and  $G^{\star}$ , compute the associated FUGW loss and update the nodes features and shortest path distance matrix of  $G_t$  using backpropagation. We recover the adjacency matrix at time step  $t + 1$  by thresholding the shortest path distance matrix. The influence of the tradeoff parameters  $\alpha$  is shown on Figure 5. The trajectory for  $\alpha = 0.5$  shows that the graph converges to a two-cluster graph with proper labels. In contrast, the trajectory for  $\alpha = 1$  shows that the graph converges to a two-cluster
+
+![](images/1ef215d84dc117a40cda0412fafd732df312430b8121b3a423364152247d3f97.jpg)  
+step 0
+
+![](images/5be370e5adedc0184060593b4ac26850b05c2acc1d13273be40b96474bd5c9f3.jpg)  
+$\alpha = 0.5$
+
+![](images/185746070a8c5a7e836af9c17d003d89159aec55c1db175face299ca3f4c0f35.jpg)
+
+![](images/35347d864716ef0ed90e12535ae215c736edf6b9cbad183e7e251e3e03192628.jpg)  
+$\alpha = 1$  
+step 200
+
+![](images/19a83b72eb9d6658f23f17a102c818a20b626a8bf208d071a4e51224d6612855.jpg)  
+step 300
+
+![](images/8292c2fba1a4b66aa2ff9e8ac8012ab2f183d76c72806dbdccff4904d97ba0ca.jpg)
+
+![](images/6fae23030713790056f4ba8a59d1f4a12b92c6b6f80b017447c2457138edd237.jpg)  
+step 1000
+
+![](images/1954e985f224fc0cab02a9e6ae31c0bac2fb71769b3c93089b6f4cf5a12caa3e.jpg)
+
+![](images/894a3ad075a9c0c25e7000300aa262f8e536057b0acbcbd7e80a872e6949a292.jpg)  
+step 3000
+
+![](images/f5a4754313f52a161f10c1b2fff9a331b1ab502e494be61facdd9a622f5e22ad.jpg)  
+target
+
+![](images/0f8aa5f4d1940edc559801395dadc1d48d23e0d7d25075a4ba24b459aa8262c5.jpg)  
+Figure 5: Gradient descent steps of the minimization of the ULOT FUGW loss between a source graph (step 0) and a target graph for  $\alpha = 0.5$  (top) and  $\alpha = 1$  (bottom).  
+Figure 6: (left) Similarity matrix of the ULOT transport mass between simulated SBM graphs with respective clusters (1,2), (2,3) and (1,2,3) sorted by type. (left) MDS of the similarity matrix.
+
+![](images/4731254dd11e158028cfd6faf111e9e9a904c97e3e280e469abb89205d47287a.jpg)  
+MDS on the ULOT plan mass
+
+graph with wrong labels. This occurs because only the Gromov-Wasserstein term is optimized, so the node features are not updated.
+
+Using the ULOT transport plan mass The FUGW loss computed with the ULOT plan provides a meaningful distance between graphs, but its computation is  $O(n_{1}^{2}n_{2} + n_{2}^{2}n_{1})$ , where  $n_1,n_2$  are the number of graph nodes. In contrast, computing the ULOT transport plan only has a quadratic time complexity. In unbalanced OT, the total mass of the OT plan  $m(P) = \sum_{i,j}P_{i,j}$  will decrease when the two graphs are very different, since in this case the marginal violation will cost less than the transport cost. This is why we propose to use the ULOT transport plan mass as a graph similarity measure (positive and between 0 and 1).
+
+We evaluate this approach on synthetic SBM drawn from three cluster configurations:  $(1,2)$ ,  $(2,3)$  and  $(1,2,3)$ . We compute the ULOT transport plan using fixed hyperparameters  $\alpha = 0.5$  and  $\rho = 0.01$ . As shown in Figure 6 (left), the resulting similarity matrix reveals the structure of the dataset. This similarity matrix can naturally be used for (spectral) graph clustering, or even for dimensionality reduction and visualization as illustrated in Figure 6 (right) with multidimensional scaling of the similarity matrix where the relation between the types of graphs is clearly recovered.
+
+# 3.2 Solving FUGW for Functional MRI brains
+
+Dataset description We now evaluate ULOT on the Individual Brain Charting (IBC) dataset [23] which is a dataset of functional MRI activations on brain surfaces. The dataset is made of surface meshes with  $160k$  vertices associated with different fMRI activations. As the dataset focuses on individual information, it includes subject-specific brain geometrical models, associated with individual fMRI activations. This justifies the necessity of the unbalanced framework to adapt mass between regions that can vary in size between subjects [30].
+
+Experimental setup The high dimensionality of the meshes in the dataset makes it particularly interesting for data augmentation. In order to obtain more training graphs, we perform a parcellation using Ward algorithm [29], and reduce the graph size to 1000 nodes, where the node features are the fMRI activations averaged over the grouped vertices. Data augmentation consists of generating 10 different graphs for each of the 12 subjects, as the results of the Ward clustering using randomly
+
+![](images/0066a6862b90b390bb5a4b1dd431c2b05d39686835366f4f1c6fc652256f060e.jpg)  
+Figure 7: (left) Comparison of the loss obtained with ULOT and IBPP solver, the dashed lines correspond to equality. (right) Plot of loss error VS time for ULOT and other solvers. Colored squares correspond to the  $20 - 80\%$  quantiles for both measures.
+
+![](images/ef59c22fdcdbf95837ff2a0e0e652288ffbcf089ba3d65050e33bc312076b2a2.jpg)
+
+sampled activations. The geometric information matrix  $D$  is the shortest path distance matrix and the 3D node positions are concatenated with the node activation features to provide a positional encoding. We construct a dataset of the 14400 different graph pairs and train the network using a  $60/20/20$  train/val/test split. We provide more details on the experimental setup in Section E.
+
+Comparison to solvers in terms of loss and computational time We first compare the FUGW loss of OT plans predicted by ULOT and the IBPP solver. We find in Figure 7 (left) that both losses are very close and highly correlated with a Pearson correlation of 0.99.
+
+Next we compare the loss error (wrt the best among all solvers) of ULOT and the other solvers introduced in section 2.1, namely IBPP [39], MM [6], LBFGSB and Sinkhorn for the entropic regularized FUGW [8, 30] using the Python library POT [9]. We find in figure 7 (right) that even though ULOT makes errors, it is up to 100 times faster than classical solvers and 10 times faster than Sinkhorn for a smaller error.
+
+This computational gain on graphs of size 1000 is very important as the solvers have a cubic time complexity with respect to the number of nodes, while ULOT has a quadratic time complexity as shown in Figure 8 (left) where computation time is plotted against the number of nodes.
+
+ULOT as warmstart. Finally when high precision is needed, we can use ULOT as a very efficient warm start for the IBPP solver. We find in Figure 8 (right) that using ULOT as a warm start allows the solver to converge much faster. This means that if high precision is required, using ULOT as a warmstart for a solver is an efficient alternative.
+
+fMRI activations prediction using ULOT plans We now illustrate the use of ULOT transport plans on fMRI data for activation prediction between brain graphs. We use the same experimental setup as in [30] where fMRI activations from a source brain graph  $G_{1}$  are transported to predict activations on a target brain graph  $G_{2}$ . First, we compute the ULOT transport plan  $P_{\theta}^{\alpha,\rho}$  between the two graphs using the model trained above. Then, we predict for a new mental task, colorless Auditory contrast from MathLanguage, the fMRI activations  $\widehat{F}_{2}$  on the nodes of  $G_{2}$  by transporting the activations  $F_{1}$  from graph  $G_{1}$ :
+
+$$
+\widehat {F} _ {2} = \operatorname {d i a g} \left(\frac {1}{\left(P _ {\theta} ^ {\alpha , \rho}\right) _ {\# 2}}\right) \left(P _ {\theta} ^ {\alpha , \rho}\right) ^ {\top} F _ {1}. \tag {13}
+$$
+
+We visualize in Figure 9 the activations on the parcellated brain regions and observe that the general fMRI trend is conserved through the transportation. While this is only a qualitative experiment, this opens doors for future use of OT in large scale experiments on fMRI data.
+
+# 4 Conclusion, limits and future work
+
+We have introduced ULOT, a new unsupervised deep learning approach for predicting optimal transport plans between graphs, trained by minimizing the FUGW loss. We have shown that ULOT is able to predict transport plans with low error on both simulated and real datasets, up to 100 times faster than classical solvers. Its low complexity and differentiability make it naturally efficient for
+
+![](images/6b1c0456e2fba76f7af451ae26a86194061b550f6d40622b09ad113f2270c7b8.jpg)  
+Figure 8: (left) FUGW transport plan prediction time for ULOT and IBPP solver for different graph sizes. (right) IBPP solver loss along iterations with and without ULOT warmstart, reported with the  $20\% - 80\%$  quantiles.
+
+![](images/b62d0088055415e627d5c8acd46502a630507d0f229ea85b271da2aabd83c390.jpg)
+
+![](images/35acd16c79f480d041772811878505c3e5ebcd37ca28b2d0a1c80ee5afa130b1.jpg)  
+Figure 9: Example of transporting colorless Auditory activations from the MathLanguage contrast between two subjects from the IBC dataset. Each brain is equipped with its individual geometry
+
+![](images/4a830e90524253d793b102e20eb9c209d30fd5edb225c9c5c60fb932b469caf7.jpg)
+
+![](images/f7ee4cd9fb9d6d9cc062aa21d14b7f2687a6cd64cd8dc0cb21a388713f5af9ab.jpg)
+
+minimizing functionals of optimal transport plans and performing FUGW parameter selection. ULOT also allows for discovering novel ways to use FUGW OT plans such as using their total mass as a measure of similarity of complexity  $\hat{O}(n^2)$  between graphs of different sizes.
+
+While we believe that ULOT is a very promising step towards the use of deep learning for optimal transport, we also acknowledge its limitations and propose research directions for addressing them. The very fast prediction comes at the cost of a small error in the predicted transport plans, which can limit its applications in a context where high precision is needed. While this can be avoided by using ULOT as a warmstart for a solver, there is still room for improvement for directly predicting even more accurate plans. Also we were limited in our experiments to graphs of size  $n \leq 10000$  due to GPU memory constraints and going further might require dedicated developments such as lazy tensors or other memory-efficient techniques [7] for cross-attention. In the future, we plan to apply this method to large-scale applications such as activation prediction on high-resolution brain surfaces and computation of graph barycenters. While these applications require large training datasets, which is not the norm for fMRI data, we plan to further investigate our random parcellation data augmentation technique to train a more general model that can be effective across subjects.
+
+# Acknowledgments and Disclosure of Funding
+
+This work was granted access to the HPC resources of IDRIS under the allocation 2025-AD011016350 made by GENCI. This research was also supported in part by the French National Research Agency (ANR) through the MATTER project (ANR-23-ERCC-0006-01). This work benefited from state aid managed by the Agence Nationale de la Recherche under the France 2030 programme, reference ANR-22-PESN-0012 and from the European Union's Horizon 2020 Framework Programme for Research and Innovation under the Specific Grant Agreement HORIZON-INFRA-2022-SERV-B-01. Finally, it received funding from the Fondation de l'École polytechnique. We thank Quang Huy Tran for providing code for the FUGW solvers and for helpful discussions. We thank the reviewers for their constructive comments and suggestions.
+
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+
+![](images/a14168a379fc57b10845fc9eb1be3c50c014ad8d4811a4bf3a451b747efbaa12.jpg)  
+Figure 10: (left) Comparison of the loss obtained with ULOT and IBPP solver for the simulated graphs, the dashed line corresponds to the equality. (right) Transport plan values for increasing values of  $\rho$ , colored by whether they link common or different clusters.
+
+![](images/b6ca080e363b62e045b7b7b964aeaa091b1511994509fa44906b834b0012cd37.jpg)
+
+# A Training setup details
+
+Compute resources We trained our network on an NVIDIA V100 GPU for 100 hours on the IBC dataset and a few hours on the smaller simulated dataset. Note that while the network is  $O(n^{2})$  with  $n$  the number of graph nodes, the main training bottleneck comes from the need to compute the  $O(n^{3})$  FUGW loss for each transport plan at every epoch.
+
+Hyperparameters The hyperparameters used for training ULOT on both the simulated graphs and the IBC dataset are reported in Table 1. All the MLP and GMN have one hidden layer, with weights shared between the two graph branches. Moreover, the first MLP in the node embedding layer preserves the dimensionality of the input node features. Hyperparameter optimization was performed on a subset of the training data using the Optuna library [1]. The code is available in the supplementary materials and will be released on github upon publication. We will also share the pre-trained model weights for both datasets.
+
+Table 1: ULOT hyperparameters  
+
+<table><tr><td>Hyperparameter</td><td>Simulated dataset</td><td>IBC dataset</td></tr><tr><td>Learning rate</td><td>0.001</td><td>0.0001</td></tr><tr><td>Batch size</td><td>256</td><td>64</td></tr><tr><td>Optimizer</td><td>Adam</td><td>Adam</td></tr><tr><td>Number of node embedding layers N</td><td>5</td><td>3</td></tr><tr><td>Embedding dimension for α</td><td>10</td><td>10</td></tr><tr><td>Node embedding layer final out dimension</td><td>256</td><td>256</td></tr><tr><td>MLP hidden dimension</td><td>64</td><td>256</td></tr><tr><td>GCN hidden dimension</td><td>16</td><td>128</td></tr><tr><td>Temperature value a</td><td>3</td><td>5</td></tr></table>
+
+# B Additional experiments on the simulated graphs
+
+Comparison of FUGW loss for ULOT and solver on the simulated graphs We train ULOT on the dataset of simulated SBMs introduced in Section 3.1 and test it on new pairs of simulated graphs sampled from the same distribution. We find in Figure 10 (left) that similarly to the IBC dataset ULOT finds transport plans that have a FUGW loss perfectly correlated with the FUGW loss obtained with the IBPP solver.
+
+Regularization path of the transport plan component  $P_{i,j}$  with respect to  $\rho$  We predict ULOT transport plans between a pair of graphs with cluster configurations (1, 2) and (2, 3) for different values of  $\rho$  and show the regularisation path of each transport plan entry  $(P_{\theta}^{\alpha ,\rho})_{i,j}$  for  $i\in [1,n_1]$  and  $j\in [1,n_2]$  in Figure 10 (right). We observe that transport plan values corresponding to nodes in the shared cluster 2 increase more rapidly with  $\rho$  compared to values between nodes in non-overlapping clusters. Moreover, we find that there exists an optimal value around  $\rho \simeq 0.05$ , for which transport
+
+![](images/5fc7cf8d38f69211573efe415e8d8a6f82369a5d26fdec8c4dd5a24b634e7741.jpg)  
+Figure 11: Label propagation accuracy of the ULOT FUGW transport w.r.t.  $(\rho, \alpha)$  between (left) different types  $(1,2,3) \rightarrow (1,2)$ , (right) same type  $(1,2,3)$ .
+
+![](images/a7ebc3bff2a4bc39b0ae42a09e58ba5e2fde6e33b65cc68ef7addb2a3630ef68.jpg)
+
+plan mass is predominantly assigned to entries corresponding to the common cluster, effectively discarding irrelevant correspondences.
+
+Visualization of the accuracy for label propagation We visualize the accuracy surfaces on Figure 11 for the label propagation task introduced in Section 3.1 for a range of  $(\alpha, \rho)$  values. We consider two different types of pairs: pairs with clusters (1, 2, 3) and (1, 2) and pairs with clusters (1, 2, 3). We see that the accuracy surfaces are smooth and that the optimal parameter values differ across pair types. We optimize  $\rho$  and  $\alpha$  using gradient descent on each pair of graphs by minimizing the KL divergence between the predicted class probabilities obtained with the ULOT transport plan and the ground truth target one hot encodings of the classes on  $50\%$  of the nodes. We obtain an accuracy of  $0.87 \pm 0.092$  on pairs with similar clusters (1, 2, 3) and  $0.74 \pm 0.11$  on the pairs with different clusters.
+
+# C Ablation studies
+
+We provide an ablation study of the impact of each step of the OT plan block on the SBM dataset in Table 2. As a first ablation we remove the node scaling in the OT plan head, as a second ablation we replace the whole OT plan head by a much simpler nonlinear transformation of the node embeddings to recover an OT plan with  $(P_{\theta}^{(\alpha ,\rho)})_{i,j} = \frac{1}{1 + ||(F_1^{\mathrm{final}})_i - (F_2^{\mathrm{final}})_j||_2^2}$ , where  $F_{1}^{\mathrm{final}}$  and  $F_{2}^{\mathrm{final}}$  are the outputs of the GCN and cross attention block for the first and second graphs respectively.
+
+Table 2: Ablation studies  
+
+<table><tr><td>Ablation</td><td>Relative error</td><td>Pearson correlation</td></tr><tr><td>None</td><td>0.19 ± 0.29</td><td>1.0</td></tr><tr><td>Remove node scaling</td><td>4.1 ± 9.5</td><td>0.89</td></tr><tr><td>Replace OT plan block by nonlinearity</td><td>27 ± 49</td><td>0.48</td></tr></table>
+
+We can see that node scaling is essential for recovering unbalanced OT plans especially for small  $\rho$  parameters (unbalanced plans that do not sum to one), and removing them increases the error (but remains surprisingly well correlated). Using a simpler output head for the OT plan leads to a failure to predict OT plans.
+
+# D Robustness to distribution shifts
+
+We tested the following distribution shift: we trained ULOT on a dataset of Stochastic Block models with balanced clusters (every cluster has the same number of nodes), and tested the model on a dataset of SBMs with unbalanced clusters, where the cluster ratio is sampled following a Dirichlet distribution. We report the results Table 3 and find that even though the relative loss error increases, the Pearson correlation between the ground truth losses and the ULOT losses remains stable.
+
+Table 3: Distribution shifts  
+
+<table><tr><td>Distribution</td><td>Relative error</td><td>Pearson correlation</td></tr><tr><td>Balanced clusters (train distribution)</td><td>0.19 ± 0.29</td><td>1.0</td></tr><tr><td>Unbalanced clusters (distribution shift)</td><td>0.27 ± 0.69</td><td>0.98</td></tr></table>
+
+# E Illustration of the fMRI alignment
+
+Experimental details on the IBC dataset We use brain cortical surfaces from the IBC dataset consisting of approximately 160k vertices, each associated with fMRI contrasts obtained as subjects perform specific tasks. To train ULOT, we use the 239 contrasts from the tasks in Table 4. The predicted activation visualized on Figure 9 is selected from the 30 left out contrasts from the MathLanguage task. For each subject, we perform a parcellation of the surface to form 1000 brain regions, using the Ward's hierarchical clustering algorithm. Each contrast is then averaged over the vertices in every brain region. From this parcellation, we construct 1000 node graphs, where edges connect spatially adjacent regions. Each node has a 242 dimensional feature vector composed of the region's contrasts concatenated to its 3D node position.
+
+To augment the dataset, we generate multiple parcellations for each subject by randomly selecting  $20\%$  to  $40\%$  of the tasks, which produces variability in the geometries and the activations across all generated brains. The final dataset is constructed from all possible graph pairs, which we randomly split into  $60\%$  training,  $20\%$  validation, and  $20\%$  test sets.
+
+Table 4: IBC tasks used for alignment  
+
+<table><tr><td>Task</td></tr><tr><td>ArchiEmotional</td></tr><tr><td>ArchiSocial</td></tr><tr><td>ArchiSpatial</td></tr><tr><td>ArchiStandard</td></tr><tr><td>HcpEmotion</td></tr><tr><td>HcpGambling</td></tr><tr><td>HcpLanguage</td></tr><tr><td>HcpMotor</td></tr><tr><td>HcpRelational</td></tr><tr><td>HcpSocial</td></tr><tr><td>HcpWm</td></tr><tr><td>RSVPLanguage</td></tr><tr><td>PreferenceFaces</td></tr><tr><td>PreferenceHouses</td></tr><tr><td>PreferenceFood</td></tr><tr><td>PreferencePaintings</td></tr><tr><td>MCSE</td></tr><tr><td>Moto</td></tr><tr><td>Visu</td></tr><tr><td>Audi</td></tr><tr><td>MVEB</td></tr><tr><td>MVIS</td></tr><tr><td>Lec1</td></tr><tr><td>Lec2</td></tr><tr><td>TheoryOfMind</td></tr><tr><td>PainMovie</td></tr><tr><td>EmotionalPain</td></tr><tr><td>Enumeration</td></tr><tr><td>VSTM</td></tr><tr><td>Self</td></tr></table>
\ No newline at end of file
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+# Unsupervised Trajectory Optimization for 3D Registration in Serial Section Electron Microscopy using Neural ODEs
+
+Zhenbang Zhang*
+
+MBZUAI; Shandong University
+
+zhangzhenbang2021@gmail.com
+
+Jingtong Feng*
+
+Shandong University
+
+jingtongf404@gmail.com
+
+Hongjia Li
+
+Beijing Institute of Technology
+
+lihongjia96@gmail.com
+
+Haythem El-Messiry
+
+Canadian University Dubai
+
+haythem.elmessiry@cud.ac.ae
+
+Zhiqiang Xu†
+
+MBZUAI
+
+zhiqiang.xu@mbzuai.ac.ae
+
+Renmin Han†
+
+Shandong University
+
+hanrenmin@sdu.edu.cn
+
+# Abstract
+
+Series Section Electron Microscopy (ssEM) has emerged as a pivotal technology for deciphering nanoscale biological architectures. Three-dimensional (3D) registration is a critical step in ssEM, tasked with rectifying axial misalignments and nonlinear distortions introduced during serial sectioning. The core scientific challenge lies in achieving distortion mitigation without erasing the natural morphological deformations of biological tissues, thereby enabling faithful reconstruction of 3D ultrastructural organization. In this study, we present a paradigm-shifting optimization framework that rethinks 3D registration through the lens of manifold trajectory optimization. We propose the first continuous trajectory dynamics formulation for 3D registration and introduce a novel optimization strategy. Specifically, we introduce a dual optimization objective that inherently balances global trajectory smoothness with local structural preservation, while developing a solver that combines Gauss-Seidel iteration with Neural ODEs to systematically integrate biophysical priors with data-driven deformation compensation. Extensive experiments on multiple datasets spanning diverse tissue types demonstrate our method's superior performance in structural restoration accuracy and cross-tissue robustness.
+
+# 1 Introduction
+
+Series Section Electron Microscopy (ssEM) has emerged as a powerful technology for nanoscale Three-dimensional (3D) visualization of biological systems. Its impact spans diverse domains including neuroscience connectomics [84, 79], developmental biology [19, 43], and clinical diagnostics [29]. Recently, MICrONS project [1, 13, 15, 4] utilizes ssEM as the fundamental method for neural circuit reconstruction. The ssEM workflow typically involves a series of computational steps, including two-dimensional (2D) stitching [21], 3D registration [53, 82], and 3D segmentation [40, 47].
+
+![](images/417a35178a03542ce3c535749dfe3cf1fbe58fe205b79be7911a0db9cde3b53d.jpg)  
+Figure 1: Trajectory analysis and inspiration: Left: Previous methods is limited to adjacent registration, while our method achieves a paradigm shift by global trajectory optimization. Right: The upper-right demonstrates position/velocity/acceleration/curvature comparisons between smoothed and noisy 1D trajectories. The lower-right displays the side view of the original and distorted data. This inspires us to explore the inner link between trajectory curvature mutation and nonlinear distortion from the perspective of biophysical motion.
+
+![](images/224588bff72797fefadd480c75294349bfcd8d1e4c68eccb403a7fd413146aca.jpg)
+
+Among these, 3D registration is particularly critical, as it corrects axial misalignments and nonlinear distortions, ultimately determining the accuracy and reliability of the final 3D structure.
+
+The fundamental challenge stems from the topological entanglement of biological signal and technical noise - specifically, the indistinguishability between natural cellular morphological deformation and nonlinear distortion induced by sample preparation [76, 51]. Furthermore, state-of-the-art equipment now generates TB-scale image data daily [77]. The massive data volume and error accumulation in long sequences [36] greatly complicate fast and accurate 3D registration.
+
+Several effective methods have been developed [56, 80, 86, 76, 42, 82]. However, these methods often fail to account for natural deformations between slices, and prioritizing pixel-level similarity may erase biologically meaningful changes, compromising the reconstructed structure. Recently, Zhang et al. [82] decouple natural deformation from nonlinear distortion by modeling them as low and high-frequency components. This insight motivated us to investigate the intrinsic characteristics of nonlinear distortion through the lens of biological motion coherence. We discovered that nonlinear distortion is essentially a sudden change in local curvature. As shown in Figure 1(b), anatomical structures inherently exhibit smooth manifold properties, maintaining evolutionary consistency in morphological changes. Mechanical sectioning, however, disrupts this continuity, introducing abrupt curvature mutations that contradict tissue biomechanics.
+
+Building upon this, we propose a noval optimization framework, NeuroTrajReg, which redefines 3D registration as a motion trajectory optimization problem. We establish a continuous trajectory dynamics formulation for ssEM, integrating spatiotemporal constraints to effectively eliminate nonlinear distortions while preserving the natural evolution of tissue morphology. We introduce a dual optimization objective that inherently balances global trajectory smoothness and local structural preservation, while developing a solver that combines Gauss-Seidel iteration with Neural ODEs to systematically integrate biophysical priors with data-driven deformation compensation. Extensive experiments spanning diverse tissue types demonstrate NeuroTrajReg's superior performance in structural restoration accuracy and robustness across different tissue types. The main contributions of this paper are summarized as follows:
+
+- We analyze the nonlinear distortions from the perspective of biological motion coherence. Based on this analysis, we redefine 3D registration as a continuous manifold trajectory optimization problem and propose the first continuous trajectory dynamics formulation.  
+- We propose a novel framework that integrates biophysical priors with data-driven deformation compensation, achieving a theory-guided yet data-corrected approach.  
+- Extensive experiments demonstrate that NeuroTrajReg achieves superior performance in accurately and faithfully reconstructing biological 3D structures while maintaining cross-tissue robustness.
+
+# 2 Related works
+
+3D registration for series section electron microscopy. Several software packages have been developed for 3D registration [39, 38, 66]. Among them, TrakEM2 [56] is one of the most widely used tools, performing 3D registration by iteratively optimizing a spring-connected particle system. Recently, deep learning-based methods have emerged as promising alternatives for 3D registration [80, 86, 42]. SEAMLeSS [53] improves robustness through vector voting and achieves global registration via combined attenuation transformations. Recently, Zhang et al. [82] proposed a Gaussian filter-based 3D registration method that approaches the problem from a frequency-domain perspective.
+
+Medical image registration. Medical image registration [85, 48, 24, 16], closely related to 3D registration, has achieved notable progress [85, 52, 5, 83, 20, 22]. Existing methods are typically categorized by their transformation models. Dense models [2, 9, 37, 8] estimate voxel-wise mappings to form dense deformation fields, whereas interpolation-based models [30, 70, 58, 61] approximate deformations using basis functions (e.g., B-splines) over spatial grids. Despite their success, these approaches mainly address image pairs. In contrast, 3D registration demands handling nonlinear distortions while maintaining axial continuity.
+
+Neural ordinary differential equations. Neural Ordinary Differential Equations (Neural ODEs) [6] integrate differential equation solvers into neural networks, enabling continuous representations with adaptive computation and enhanced parameter efficiency. Formally, given hidden state  $\mathbf{h}(t)$ , a Neural ODEs defines its dynamics through a parameterized function  $f_{\theta}$ :
+
+$$
+\frac {d \mathbf {h} (t)}{d t} = f _ {\theta} (\mathbf {h} (t), t), \tag {1}
+$$
+
+and the final state is obtained by solving  $\mathbf{h}(t_1) = \mathbf{h}(t_0) + \int_{t_0}^{t_1}f_\theta (\mathbf{h}(t),t)dt$ . Numerous variants of Neural ODEs have emerged to capture more complex transformations [57, 75, 49, 44, 35, 27]. Neural ODEs have been widely applied across various fields [50, 31, 55, 68]: Vid-ODE [50] enables continuous-time video synthesis, NODEO [74] adapts it for deformable image registration, and Latent ODE [55] models continuous-time sequences.
+
+Reference line smoothing algorithms. In autonomous driving [81, 65], the reference line smoothing algorithm [12, 60, 17] geometrically optimizes global paths to generate smooth trajectories with curvature continuity, kinematic feasibility, and minimal deviation from the original reference. Its primary role involves eliminating curvature discontinuities between discrete waypoints, constraining maximum curvature within vehicle kinematic limits. The loss function is formulated as:
+
+$$
+J = \underbrace {\sum_ {i = 2} ^ {n - 1} \left\| \mathbf {p} _ {i - 1} - 2 \mathbf {p} _ {i} + \mathbf {p} _ {i + 1} \right\| ^ {2}} _ {\text {S m o o t h n e s s}} + \lambda \underbrace {\sum_ {i = 1} ^ {n} \left\| \mathbf {p} _ {i} - \mathbf {q} _ {i} \right\| ^ {2}} _ {\text {F i d e l i t y}} + \underbrace {\sum_ {i = 1} ^ {n} \delta \left(\left| \kappa_ {i} \right| \leq \kappa_ {\max }\right)} _ {\text {C o n s t r a i n t s}}, \tag {2}
+$$
+
+where the first term penalizes curvature variation via second-order differencing, the second enforces proximity to original waypoints  $\mathbf{q}_i$  with adaptive weight  $\lambda$ , and the third applies indicator function  $\delta(\cdot)$  to enforce maximum curvature  $\kappa_{\max}$ . Optimization strategies include convex quadratic programming (QP) for real-time computation, sequential QP (SQP) for nonlinear constraint handling [12, 17], spline curves [72, 14], and mass-spring physical simulation [34, 23].
+
+# 3 Proposed method: NeuroTrajReg
+
+For the problem of 3D registration, we propose a dynamics-based nonlinear distortion correction framework for ssEM image sequences. Given a stack of microscopic images  $\{I_z\}_{z=0}^{N-1}$  affected by complex nonlinear distortions, our objective is to reconstruct a 3D structure that faithfully preserves the biological morphology by establishing a registration model with biophysical priors. The core challenge lies in dynamically balancing two conflicting demands: on the one hand, eliminating nonlinear spatial distortions induced by external factors, and on the other hand, preserving the intrinsic morphological characteristics of the biological specimen, such as tissue deformation and developmental topological changes.
+
+![](images/b794bf72123a5e54c8195ae2d1336642745d2e204e27d6017f426e13b7c6c809.jpg)  
+Figure 2: (a) The overall pipeline of NeuroTrajReg. Our approach can be divided into three main components: 1) Pixel trajectory tracking, 2) Dynamics-based trajectory optimization, and 3) Trajectory inversion and image registration. (b) Illustration of the trajectory tracking module. (c) The network architecture of  $f_{\theta}$  (Neural ODEs) in dynamics-based trajectory optimization module.
+
+![](images/14b83924c8fba2e60231fdcf87f63f32b9864ac86d1e3cc83dfdfd0c8f42dab1.jpg)
+
+![](images/0b33acaee521d93b510d4f4fec9637a280d02aa00d8b03be7308bfb946ee8f93.jpg)
+
+# 3.1 Problem formulation
+
+In this section, we introduce the concept of pixel-wise motion trajectory tracking and smoothing. Drawing inspiration from the classical Laplace equation [7] and the reference line smoothing algorithms [12], we formulate the 3D registration as a continuous manifold trajectory optimization problem. Specifically, each pixel  $p_i \in \mathbb{R}^2$  is treated as a particle within a continuum medium, whose displacement along the slice axis  $z \in [z_0, z_{N-1}]$  generates a parameterized trajectory  $\mathcal{P}_i$ . While this trajectory ideally follows natural morphological variations, nonlinear distortions induce local curvature abruptions. To address this, we establish spatiotemporal smoothness constraints on pixel trajectories, transforming traditional 3D registration into a physically interpretable continuous trajectory optimization problem. NeuroTrajReg is grounded in three key principles: 1) physical motion coherence in biological tissues; 2) Laplace-based trajectory smoothing; 3) spatiotemporal continuity in microscopy imaging. We model pixel-wise motion from individual dynamics to global trajectory constraints with strict mathematical consistency.
+
+Single-particle dynamics. Let  $\mathcal{P}_i$  denote the parameterized trajectory. We formulate the single-particle optimization problem as follows:
+
+$$
+\min  _ {\mathcal {P} _ {i}} \left\{\underbrace {\int_ {z _ {0}} ^ {z _ {N - 1}} \left\| \mathcal {P} _ {i} (z) - \mathcal {P} _ {i} ^ {(0)} (z) \right\| ^ {2} d z} _ {\text {D a t a f i d e l i t y}} + \lambda \underbrace {\int_ {z _ {0}} ^ {z _ {N - 1}} \left\| \nabla^ {2} \mathcal {P} _ {i} (z) \right\| ^ {2} d z} _ {\text {L a p l a c i a n s m o o t h n e s s}} \right\}, \tag {3}
+$$
+
+where  $\mathcal{P}_i^{(0)}$  denotes the observed origin trajectory,  $\nabla^2$  is the axial Laplacian operator, and  $\lambda > 0$  controls the regularization strength. The Laplacian term enforces second-order temporal smoothness by penalizing acceleration discontinuities, effectively filtering biologically implausible instantaneous acceleration changes.
+
+Holistic trajectory constraint. For image stack  $\{I_z\}_{z = 0}^{N - 1}$ , we formulate the holistic optimization:
+
+$$
+\min  _ {\left\{\mathcal {P} _ {i} \right\}} \sum_ {i = 1} ^ {M} \left[ \int_ {z _ {0}} ^ {z _ {N - 1}} \| \mathcal {P} _ {i} (z) - \mathcal {P} _ {i} ^ {(0)} (z) \| ^ {2} d z + \lambda \int_ {z _ {0}} ^ {z _ {N - 1}} \| \nabla^ {2} \mathcal {P} _ {i} (z) \| ^ {2} d z + \mu S _ {i} \right], \tag {4}
+$$
+
+where  $M$  is the total number of trajectories, corresponding to the number of pixels in image  $I_{z}$ , and the spatial coherence term  $S_{i}$  is defined as:
+
+$$
+S _ {i} := \sum_ {j \in \mathcal {N} (i)} \int_ {z _ {0}} ^ {z _ {N - 1}} \| \mathcal {P} _ {i} (z) - \mathcal {P} _ {j} (z) \| ^ {2} d z, \tag {5}
+$$
+
+and weighted by  $\mu > 0$  for preserving local topologies within each slice, where  $\mathcal{N}(i)$  denotes the set of spatial neighbors of point  $i$  within the same slice. This dual regularization strategy embeds two fundamental biological principles:
+
+- Temporal smoothness. It preserves smooth and stable tissue morphology evolution by constraining axial acceleration discontinuities through Laplacian regularization, preventing unreliable instantaneous trajectory changes.  
+- Spatial consistency. It maintains the local topology between slice pixels by ensuring the consistency of trajectory smoothing for neighboring pixels.
+
+# 3.2 Trajectory optimization for 3D registration
+
+Our approach, as shown in Figure 2(a), mainly consists of three components: 1) pixel trajectory tracking, 2) dynamics-based trajectory optimization, and 3) trajectory inversion and image registration. Specifically, we perform trajectory tracking for each pixel in the image stack. Then, we apply our proposed trajectory optimization algorithm to obtain smooth pixel trajectories. Afterward, we compute the trajectory displacements and invert them back to the pixel grid to generate a deformation field, which is then used to register the image stack. We will now elaborate on the implementation and contributions of each component.
+
+Pixel trajectory tracking. Given image stack  $\{I_z\}_{z=0}^{N-1} \in \mathbb{R}^{N \times 1 \times H \times W}$ , our goal is to track the trajectory of each pixel over time, accounting for subpixel displacements. We use the network shown in Figure 2(b) for estimating the displacement field  $u_{z-1,z} \in \mathbb{R}^{H \times W \times 2}$  in  $x$ -axis and  $y$ -axis between adjacent slices. To model the temporal trajectory of every pixel, we introduce a trajectory volume  $\mathcal{P} \in \mathbb{R}^{N \times H \times W \times 2}$ , where  $\mathcal{P}(z)$  stores the spatial coordinate in slice  $z$ . This trajectory is initialized as:
+
+$$
+\mathcal {P} (0, y, x) = \left(y ^ {0}, x ^ {0}\right), \tag {6}
+$$
+
+indicating that the trajectory of each pixel starts from its position in the first slice. Here,  $\mathcal{P}(z,y,x)$  denotes the displacement vector at index  $(y,x)$  in slice  $z$ .
+
+We recursively propagate the position using the displacement field from the previous slice. But the pixel location  $\mathcal{P}(z - 1,y,x)$  may not be aligned with the discrete grid of  $u_{z - 1,z}$ , as accumulated displacements often lead to non-integer coordinates. As a result, bilinear interpolation of  $u_{z - 1,z}$  is required to accurately estimate the flow at subpixel positions. Formally, the trajectory propagation is governed by the transport equation:
+
+$$
+\mathcal {P} (z, y, x) = \mathcal {P} (z - 1, y, x) + \widetilde {u} _ {z - 1, z} (\mathcal {P} (z - 1, y, x)), \tag {7}
+$$
+
+where  $\widetilde{u}_{z - 1,z}(\cdot)$  denotes the interpolated displacement field.
+
+Dynamics-based trajectory optimization. According to Eq. (3), we model trajectory smoothing as the weighted minimization of discrete curvature energy and deviations from the original trajectory. To solve this optimization objective, we employ the Gauss-Seidel method [18] to iteratively smooth the trajectory. The Gauss-Seidel method is a classical algorithm for solving linear systems. Its key characteristic lies in accelerating convergence by immediately incorporating the most recently updated values. The corresponding Gauss-Seidel iteration is given by:
+
+$$
+\mathcal {P} ^ {(k + 1)} (z) = \frac {\mathcal {P} ^ {(k + 1)} (z - 1) + \mathcal {P} ^ {(k)} (z + 1) + \lambda \mathcal {P} ^ {(0)} (z)}{2 + \lambda}, \quad z = 1, \dots , N - 1. \tag {8}
+$$
+
+with boundary values fixed as  $\mathcal{P}^{(k)}(0) \equiv \mathcal{P}^{(0)}(0)$  and  $\mathcal{P}^{(k)}(N - 1) \equiv \mathcal{P}^{(0)}(N - 1)$ . In each iteration  $k$ , the internal positions  $z$  are updated from the 1 to the  $N - 1$ , where each  $\mathcal{P}^{(k + 1)}(z)$  is computed using the most recent value of its predecessor  $\mathcal{P}^{(k + 1)}(z - 1)$  and the previous value of its successor  $\mathcal{P}^{(k)}(z + 1)$ . A detailed derivation is provided in the supplementary material.
+
+While the Gauss-Seidel method provides a theoretically grounded approach for solving linear systems through explicit point-wise updates with guaranteed convergence, its practical effectiveness is inherently limited by discrepancies between idealized theoretical models and real-world scenarios. Contrastingly, Neural ODEs [6] excel in learning continuous dynamical systems through implicit dynamic modeling, effectively capturing complex nonlinear deviations.
+
+Given the exceptional capability of Neural ODEs in modeling continuous dynamic systems [69, 31, 46], we use Neural ODEs to reformulate trajectory smoothing as the dynamic evolution of a continuous trajectory. The network architecture of  $f_{\theta}$  in Neural ODEs is shown in Figure 2(c). The trajectory constraints are enforced through integration of velocity field  $\mathcal{V} = \{v_z\}_{z=1}^{N-1}$  satisfying:
+
+$$
+\mathcal {V} ^ {(t)} = \frac {d \mathcal {P} ^ {(t)}}{d t} = f _ {\theta} (\mathcal {P} ^ {(t)}, t), \quad \mathcal {P} ^ {\left(t _ {k + 1}\right)} = \mathcal {P} ^ {\left(t _ {k}\right)} + \int_ {t _ {k}} ^ {t _ {k + 1}} \mathcal {V} ^ {(t)} d t, \tag {9}
+$$
+
+where  $f_{\theta}$  is a neural network with trainable parameters  $\theta$
+
+Inspired by Universal Differential Equations (UDEs) [54], we propose a dynamically adaptive optimization system that synergistically combines the explicit iterative convergence of Gauss-Seidel with the implicit dynamic modeling capabilities of Neural ODEs:
+
+$$
+\mathcal {P} ^ {\left(t _ {N}\right)} = \left(1 - \alpha\right) \underbrace {\mathcal {G} \left(\mathcal {P} ^ {\left(t _ {0}\right)}\right)} _ {\text {G a u s s - S e i d e l c o r e}} + \alpha \underbrace {\operatorname {O D E S o l v e r} \left[ f _ {\theta} , \mathcal {P} ^ {\left(t _ {0}\right)} , \left[ t _ {0} , t _ {N} \right] \right]} _ {\text {N e u r a l c o m p e n s a t o r}}, \tag {10}
+$$
+
+where  $\mathcal{G}(\cdot)$  denotes the Gauss-Seidel operator, which takes the initial trajectory at time  $t_0$  as input and outputs a smoothed trajectory at time  $t_N$ . ODESolver[ $\cdot$ ] represents the neural dynamic correction over time interval  $[t_0, t_N]$ . The adaptive weight  $\alpha \in [0,1]$  evolves through:
+
+$$
+\alpha = \sigma \left(\beta \cdot \left(1 - \frac {\Delta^ {(t _ {N})}}{\Delta^ {(t _ {0})}}\right)\right), \tag {11}
+$$
+
+where  $\sigma (\cdot)$  denotes the sigmoid activation ensuring smooth transitions,  $\beta$  controls the adaptation rate sensitivity, and  $\Delta^{(t_k)} = \| \mathcal{G}(\mathcal{P}^{(t_k)}) - \mathcal{P}^{(t_k)}\|$  quantifies theoretical operator progress.
+
+Therefore, our architecture follows a theory-guided, data-corrected approach, incorporating an adaptive blending module that automatically balances theoretical components and corrective terms. It takes advantage of Gauss-Seidel for rapid initial convergence, ensuring numerical stability and accelerating optimization. Neural ODEs are then employed to capture higher-order nonlinear effects and compensate for the gaps between the model and reality.
+
+Trajectory inversion and image registration. After trajectory optimization, we obtain the resulting trajectories  $\mathcal{P}^{(t_N)}\in \mathbb{R}^{N\times H\times W\times 2}$ , where each element  $\mathcal{P}_i^{(t_N)}(z)\in \mathbb{R}^2$  represents the evolved coordinate of the  $i$ -th trajectory  $\mathcal{P}_i^{(t_N)}$  at slice  $z$ . Based on this, the displacement  $\phi \in \mathbb{R}^{N\times H\times W\times 2}$  can be computed by comparing it with the observed original trajectory  $\mathcal{P}^{(0)}$ :
+
+$$
+\phi = \mathcal {P} ^ {\left(t _ {N}\right)} - \mathcal {P} ^ {(0)}. \tag {12}
+$$
+
+The displacement  $\phi$  encodes the offset information of  $\mathcal{P}^{(t_N)}$ . However, it is important to note that this does not exactly correspond to the displacement of pixel grid points, as the coordinates in  $\mathcal{P}^{(t_N)}$  are not necessarily integers. Therefore, inspired by the surface splatting [87, 33] in computer graphics, we propose an efficient bilinear splatting to invert the trajectories and obtain accurate displacements at pixel grid locations (detailed in the supplementary material):
+
+$$
+\left\{\Phi_ {z} \right\} _ {z = 0} ^ {N - 1} = \mathcal {I} _ {\mathrm {b s}} \left(\mathcal {P} ^ {\left(t _ {N}\right)}, \phi\right), \tag {13}
+$$
+
+where  $\mathcal{I}_{\mathrm{bs}}$  means bilinear splatting and  $\{\Phi_z\}_{z=0}^{N-1} \in \mathbb{R}^{N \times H \times W \times 2}$  represents the deformation fields corresponding to the image stack  $\{I_z\}_{z=0}^{N-1}$ . These deformation fields are subsequently used to register the image stack:
+
+$$
+I _ {z} = \Phi_ {z} \circ I _ {z}, \quad z = 0, \dots , N - 1, \tag {14}
+$$
+
+where  $\circ$  denotes the warping operation.
+
+# 3.3 Loss functions
+
+The trajectory tracking network is optimized using a composite unsupervised loss function comprising a normalized cross-correlation (NCC) data fidelity term and a diffusion regularizer applied to the displacement fields, with balancing coefficient  $\lambda$ . The loss function is formally expressed as:
+
+$$
+\mathcal {L} _ {\text {t r a j}} = \mathrm {N C C} \left(I _ {z}, I _ {z - 1} \circ u _ {z - 1, z}\right) + \lambda \| \nabla u _ {z - 1, z} \| _ {2} ^ {2}, \tag {15}
+$$
+
+where  $\circ$  indicates the spatial warping operator, and  $\nabla$  computes first-order spatial gradients via finite difference approximation.
+
+According to Eq. (4), the Neural ODEs training objective is formulated as a composite unsupervised loss function comprising three components:
+
+$$
+\mathcal {L} = \sum_ {i = 1} ^ {M} \left(\underbrace {\sum_ {z = z _ {0}} ^ {z _ {N - 1}} \left\| \mathcal {P} _ {i} (z) - \mathcal {P} _ {i} ^ {(0)} (z) \right\| ^ {2}} _ {\text {D a t a f i d e l i t y t e r m}} + \lambda \underbrace {\sum_ {z = z _ {0}} ^ {z _ {N - 1}} \left\| \nabla^ {2} \mathcal {P} _ {i} (z) \right\| ^ {2}} _ {\text {T r a j e c t o r y s m o o t h n e s s t e r m}} + \mu \underbrace {\sum_ {z = z _ {0}} ^ {z _ {N - 1}} \sum_ {j \in \mathcal {N} (i)} \left\| \mathcal {P} _ {i} (z) - \mathcal {P} _ {j} (z)\right) \| ^ {2}} _ {\text {S p a t i a l c o n s i s t e n c y t e r m}}\right), \tag {16}
+$$
+
+where  $\mathcal{P}_i^{(0)}$  denotes the observed origin trajectory. The trajectory smoothness term, weighted by  $\lambda > 0$ , penalizes acceleration discontinuities through the temporal Laplacian operator  $\nabla^2$ , discretized with second-order differences. The spatial consistency term, weighted by  $\mu > 0$ , preserves local topological relationships within each slice.
+
+# 4 Experiments
+
+# 4.1 Datasets
+
+We evaluate our algorithm on six publicly available datasets from the OpenOrganelle platform [77], covering a variety of mouse tissues such as heart, kidney, liver, skin, and pancreas. These allow for a comprehensive evaluation of NeuroTrajReg's applicability and robustness across diverse tissue types. For real-world data, we utilized the female fruit fly brain neural dataset (FemFlyBrain) [63]. More details can be found in the supplementary material.
+
+# 4.2 Implementation details
+
+We implement our method using PyTorch, and all experiments are conducted on an NVIDIA A800 GPU with 80GB of memory. For the pixel trajectory tracking, we employ a 2D U-Net architecture to capture the displacements between adjacent slices. Neural ODEs employ a 3D U-Net architecture to parameterize  $f_{\theta}$ . The network takes pixel trajectories  $\mathcal{P}$  as input and generates a time-dependent velocity field  $\mathcal{V}$ . More details can be found in the supplementary material.
+
+# 4.3 Results
+
+In Table 1, we compare NeuroTrajReg with advanced 3D registration techniques, including EMReg [42], EFSR [76], TrakEM2 [56], SEMLeSS [53], and GaussReg [82], using synthetic datasets spanning six diverse tissue types. Our approach, along with the supervised GaussReg method, demonstrates superior performance due to consideration of natural deformations, which other methods typically overlook. Methods overly focused on pixel-level similarity can neglect biologically relevant movements such as cellular dynamics, as shown in Figure 7. Moreover, compared to the supervised-manner GaussReg, our method uniquely benefits from global trajectory optimization, enabling the computation of a single interpolation deformation across slices. Conversely, GaussReg is constrained by a local receptive field, necessitating multiple interpolation steps. For the performance on real-world datasets, we selected several volumetric samples from the FemFlyBrain dataset [63]. The registration results are visualized in Figure 5, where we employed the interpolation and visualization tools provided by FIJI [59] to display the side views of the image stacks. Although GaussReg reduces nonlinear distortions, it also erroneously removes some structural textures, leading to a noticeable deviation from the original data. In contrast, our method not only effectively corrects the nonlinear distortions in the image stacks but also better preserves local structural details.
+
+To further evaluate our approach, Figure 3 compares the error accumulation across approximately 1000 slices for SEMLeSS [53], GaussReg [82], and our method. SEMLeSS suffers significantly from error accumulation over long sequences. Although GaussReg performs reasonably well, it exhibits lower precision and higher variance compared to our method, indicating less stable performance. Our method achieves superior average SSIM accuracy and demonstrates substantially greater stability, maintaining consistent performance without noticeable degradation even after 1000 slices. Additional results are provided in the supplementary material.
+
+Table 1: The performance of different 3D registration methods in six synthetic datasets [77].  
+
+<table><tr><td colspan="2"></td><td>Mus Heart</td><td>Mus Kidney</td><td>Mus Liver-3</td><td>Mus Liver</td><td>Mus Pancreas</td><td>Mus Skin</td></tr><tr><td rowspan="3">EMReg [47]</td><td>MI</td><td>0.73 ± 0.14</td><td>0.65 ± 0.16</td><td>0.89 ± 0.17</td><td>0.79 ± 0.18</td><td>0.76 ± 0.14</td><td>0.66 ± 0.18</td></tr><tr><td>SSIM</td><td>0.55 ± 0.03</td><td>0.44 ± 0.04</td><td>0.52 ± 0.04</td><td>0.43 ± 0.03</td><td>0.43 ± 0.02</td><td>0.50 ± 0.07</td></tr><tr><td>NCC</td><td>0.90 ± 0.03</td><td>0.81 ± 0.05</td><td>0.86 ± 0.05</td><td>0.78 ± 0.05</td><td>0.81 ± 0.03</td><td>0.87 ± 0.04</td></tr><tr><td rowspan="3">EFSR [76]</td><td>MI</td><td>1.15 ± 0.10</td><td>1.26 ± 0.15</td><td>1.45 ± 0.14</td><td>1.36 ± 0.16</td><td>1.33 ± 0.13</td><td>1.03 ± 0.20</td></tr><tr><td>SSIM</td><td>0.67 ± 0.04</td><td>0.63 ± 0.05</td><td>0.65 ± 0.04</td><td>0.56 ± 0.04</td><td>0.57 ± 0.04</td><td>0.60 ± 0.07</td></tr><tr><td>NCC</td><td>0.98 ± 0.00</td><td>0.97 ± 0.01</td><td>0.97 ± 0.00</td><td>0.95 ± 0.01</td><td>0.95 ± 0.01</td><td>0.97 ± 0.01</td></tr><tr><td rowspan="3">TrakEM2 [56]</td><td>MI</td><td>1.01 ± 0.03</td><td>1.24 ± 0.10</td><td>1.48 ± 0.07</td><td>1.47 ± 0.08</td><td>1.48 ± 0.06</td><td>1.08 ± 0.15</td></tr><tr><td>SSIM</td><td>0.57 ± 0.02</td><td>0.61 ± 0.03</td><td>0.66 ± 0.03</td><td>0.61 ± 0.03</td><td>0.65 ± 0.03</td><td>0.64 ± 0.06</td></tr><tr><td>NCC</td><td>0.98 ± 0.00</td><td>0.97 ± 0.00</td><td>0.98 ± 0.00</td><td>0.97 ± 0.00</td><td>0.97 ± 0.00</td><td>0.98 ± 0.01</td></tr><tr><td rowspan="3">SEMLeSS [53]</td><td>MI</td><td>1.00 ± 0.12</td><td>1.07 ± 0.14</td><td>1.35 ± 0.17</td><td>1.29 ± 0.19</td><td>1.18 ± 0.15</td><td>0.89 ± 0.18</td></tr><tr><td>SSIM</td><td>0.58 ± 0.04</td><td>0.52 ± 0.04</td><td>0.61 ± 0.05</td><td>0.52 ± 0.06</td><td>0.50 ± 0.05</td><td>0.51 ± 0.07</td></tr><tr><td>NCC</td><td>0.97 ± 0.01</td><td>0.96 ± 0.01</td><td>0.97 ± 0.01</td><td>0.95 ± 0.01</td><td>0.95 ± 0.01</td><td>0.96 ± 0.01</td></tr><tr><td rowspan="3">GaussReg [82]</td><td>MI</td><td>1.43 ± 0.05</td><td>1.54 ± 0.11</td><td>1.83 ± 0.07</td><td>1.83 ± 0.08</td><td>1.76 ± 0.06</td><td>1.40 ± 0.16</td></tr><tr><td>SSIM</td><td>0.83 ± 0.02</td><td>0.78 ± 0.03</td><td>0.82 ± 0.02</td><td>0.80 ± 0.02</td><td>0.80 ± 0.02</td><td>0.82 ± 0.03</td></tr><tr><td>NCC</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td></tr><tr><td rowspan="3">Ours</td><td>MI</td><td>1.47 ± 0.08</td><td>1.67 ± 0.14</td><td>1.88 ± 0.11</td><td>1.85 ± 0.21</td><td>1.81 ± 0.12</td><td>1.52 ± 0.17</td></tr><tr><td>SSIM</td><td>0.87 ± 0.02</td><td>0.85 ± 0.02</td><td>0.87 ± 0.01</td><td>0.84 ± 0.15</td><td>0.86 ± 0.18</td><td>0.86 ± 0.03</td></tr><tr><td>NCC</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td></tr></table>
+
+![](images/e2d02433f0fa3cb65d507ca2ddf14c848f5de6839b2df7fef17dbcd82ad6d7e6.jpg)  
+Figure 3: Error accumulation comparison on Mus Liver-3 dataset.
+
+# 4.4 Ablation study
+
+Loss Terms. We tested the impact of different components of the loss function (Eq. 16) on the results. The results are shown in Table 2, where TS, DF, and SC correspond to the Trajectory Smoothness term, Data Fidelity term, and Spatial Consistency term, respectively. Removing the Data Fidelity term decreases SSIM, as this leads to the trajectory deviating excessively from the original, with a risk of over-smoothing. Removing the Spatial Consistency term results in a decrease in SSIM, as failing to preserve local topology can lead to local pixel distortions. As for the TS term, it is key to trajectory smoothing, as it imposes constraints on the trajectory's acceleration, preventing unreasonable instantaneous variations. In conclusion, all components are essential and work together to ensure optimal performance.
+
+Hyperparameters. We tested the impact of different hyperparameters  $\lambda$  and  $\mu$  in Eq. (16) on registration accuracy.  $\lambda$  controls the smoothness of the trajectory to filter unreasonable instantaneous velocity changes, while  $\mu$  regulates the local consistency of the displacement field to ensure topological consistency before and after registration.  $\lambda$  has an important impact on registration performance; if it is too small, it fails to eliminate noise completely, while if it is too large, it may cause excessive deviation of the trajectory. In contrast,  $\mu$  is more robust and has a lesser effect on registration performance.
+
+![](images/2c5f9033659773d899c6d4bfdf66a783b2ec6c4c93424d9257da24cf401c20db.jpg)  
+Figure 4: 3D visualization of registration results on the Mus Kidney and Mus Heart datasets.
+
+Table 2: The ablation study of different loss terms.  
+
+<table><tr><td></td><td>TS Loss term</td><td>DF Loss term</td><td>SC Loss term</td><td>SSIM</td></tr><tr><td rowspan="4">Mus Heart</td><td>✓</td><td>X</td><td>X</td><td>0.721</td></tr><tr><td>✓</td><td>✓</td><td>X</td><td>0.855</td></tr><tr><td>✓</td><td>X</td><td>✓</td><td>0.793</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>0.872</td></tr><tr><td rowspan="4">Mus Kidney</td><td>✓</td><td>X</td><td>X</td><td>0.698</td></tr><tr><td>✓</td><td>✓</td><td>X</td><td>0.822</td></tr><tr><td>✓</td><td>X</td><td>✓</td><td>0.774</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>0.851</td></tr></table>
+
+Table 3: The ablation study of different hyperparameter in the loss.  
+
+<table><tr><td>SSIM</td><td>λ\μ</td><td>0.0001</td><td>0.001</td><td>0.01</td><td>0.1</td></tr><tr><td rowspan="4">Mus Heart</td><td>0.001</td><td>0.775</td><td>0.782</td><td>0.771</td><td>0.762</td></tr><tr><td>0.01</td><td>0.795</td><td>0.806</td><td>0.783</td><td>0.779</td></tr><tr><td>0.1</td><td>0.852</td><td>0.872</td><td>0.841</td><td>0.822</td></tr><tr><td>1</td><td>0.811</td><td>0.819</td><td>0.805</td><td>0.787</td></tr><tr><td rowspan="4">Mus Kidney</td><td>0.001</td><td>0.754</td><td>0.762</td><td>0.752</td><td>0.741</td></tr><tr><td>0.01</td><td>0.781</td><td>0.794</td><td>0.772</td><td>0.766</td></tr><tr><td>0.1</td><td>0.833</td><td>0.851</td><td>0.821</td><td>0.815</td></tr><tr><td>1</td><td>0.768</td><td>0.773</td><td>0.758</td><td>0.743</td></tr></table>
+
+Table 4: Ablation study on the number of iterations in the Gauss-Seidel method.  
+
+<table><tr><td>SSIM\Iterations</td><td>50</td><td>100</td><td>150</td><td>200</td><td>250</td><td>300</td></tr><tr><td>Mus Heart</td><td>0.719</td><td>0.802</td><td>0.855</td><td>0.872</td><td>0.874</td><td>0.874</td></tr><tr><td>Mus Kidney</td><td>0.713</td><td>0.796</td><td>0.828</td><td>0.851</td><td>0.851</td><td>0.851</td></tr><tr><td>Mus Liver</td><td>0.695</td><td>0.774</td><td>0.811</td><td>0.842</td><td>0.844</td><td>0.844</td></tr><tr><td>Mus Skin</td><td>0.682</td><td>0.788</td><td>0.842</td><td>0.864</td><td>0.868</td><td>0.866</td></tr></table>
+
+Number of Iterations. Table 4 investigates the effect of varying the number of iterations in the Gauss-Seidel method. From 0 to 200 iterations, the registration performance improves significantly and then gradually converges. Considering the trade-off between accuracy and efficiency, we select 200 iterations in practice to achieve a good balance.
+
+Module Analysis. Our method can be decomposed into two main modules: the Gauss-Seidel method and Neural ODEs. The Gauss-Seidel method achieves fast convergence through iterative updates, while Neural ODEs effectively capture complex nonlinear distortions by modeling implicit dynamics. As shown in Table 5, we analyze the contribution of these key components in the trajectory smoothing scheme, and the results demonstrate the necessity and effectiveness of both modules.
+
+Different Solver. Table 6 further studies the effect of different solvers for Neural ODEs. To avoid additional computational overhead, we adopt the Euler solver to balance performance and computational cost. More ablation experiments are provided in the supplementary material.
+
+![](images/c90d1f09f00dac4ec86dfc52bf1dc1506040166edd3016ab95a7775eea89c3fa.jpg)  
+Figure 5: Side views of the original data, the registration results of GaussReg [82], and our method on two sampled volumes from the FemFlyBrain dataset [63].
+
+Table 5: Ablation study of the components in our method.  
+
+<table><tr><td></td><td>Gauss-Seidel</td><td>Neural ODEs</td><td>SSIM</td></tr><tr><td rowspan="3">Mus Heart</td><td>✓</td><td>×</td><td>0.847</td></tr><tr><td>×</td><td>✓</td><td>0.835</td></tr><tr><td>✓</td><td>✓</td><td>0.872</td></tr><tr><td rowspan="3">Mus Kidney</td><td>✓</td><td>×</td><td>0.833</td></tr><tr><td>×</td><td>✓</td><td>0.827</td></tr><tr><td>✓</td><td>✓</td><td>0.851</td></tr></table>
+
+Table 6: Ablation study of different solvers for Neural ODEs.  
+
+<table><tr><td></td><td colspan="2">Euler</td><td colspan="2">Dopri5</td><td colspan="2">RK4</td></tr><tr><td></td><td>Time</td><td>SSIM</td><td>Time</td><td>SSIM</td><td>Time</td><td>SSIM</td></tr><tr><td>Mus Heart</td><td>2.45</td><td>0.872</td><td>8.33</td><td>0.877</td><td>7.68</td><td>0.876</td></tr><tr><td>Mus Kidney</td><td>2.41</td><td>0.851</td><td>8.52</td><td>0.855</td><td>7.66</td><td>0.854</td></tr><tr><td>Mus Liver</td><td>2.5</td><td>0.842</td><td>8.71</td><td>0.848</td><td>7.72</td><td>0.847</td></tr><tr><td>Mus Skin</td><td>2.46</td><td>0.864</td><td>8.39</td><td>0.869</td><td>7.25</td><td>0.868</td></tr></table>
+
+# 5 Conclusion
+
+In this paper, we present a noval 3D registration method for series section electron microscopy. We analyze the inherent characteristics of nonlinear distortion from the perspective of biological motion coherence, and reconsider 3D registration from the viewpoint of trajectory optimization. We introduce a dual optimization objective that balances global trajectory smoothness and local structural preservation, and we develop a solver that combines Gauss-Seidel iteration with Neural ODEs. Extensive experiments demonstrate that our method excels in structural restoration accuracy and cross-tissue robustness.
+
+# 6 Acknowledgements
+
+This research was supported by the National Key Research and Development Program of China [2021YFF0704300], Dubai Future Foundation (Award No. 2024CANAD-MES-061), and the Natural Science Foundation of Shandong Province [ZR2023YQ057].
+
+# References
+
+[1] Functional connectomics spanning multiple areas of mouse visual cortex. Nature, 640(8058):435-447, 2025.  
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+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: The abstract and introduction accurately reflect the paper's contributions and scope.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: We discuss the limitations of the work in the Appendix.
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
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+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [NA]
+
+Justification: This paper does not involve theoretical proofs.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
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+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+# Answer: [Yes]
+
+Justification: This paper explicitly details the algorithm reproduction and dataset specifics in the appendix. In the future, we will release the code after the paper is accepted.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
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+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
+
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+# Answer: [Yes]
+
+Justification: We provide the code in the supplementary material and offer detailed instructions on how to reproduce the algorithm in the appendix.
+
+# Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
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+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.  
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+# Answer: [Yes]
+
+Justification: We provide all the training and test details in the appendix.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+# Answer: [Yes]
+
+Justification: We present the error ranges of the evaluation metrics in the tables of both the main text and the appendix.
+
+# Guidelines:
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+
+# 8. Experiments compute resources
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+
+# 12. Licenses for existing assets
+
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+# 14. Crowdsourcing and research with human subjects
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+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
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+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
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+Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components.
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+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# Appendix
+
+# A1 Dataset Details and Metrics
+
+# A1.1 Dataset Details
+
+We selected six publicly available datasets from the OpenOrganelle platform [77] for simulation experiments. These datasets encompass high-resolution electron microscopy images of various mouse tissues, including the heart, kidney, liver, skin, and pancreas. The availability of ground-truth annotations in these datasets provides strong support for validating the applicability and robustness of our method across different types of biological tissues. For real-world data, we utilized the female fruit fly brain neural dataset (FemFlyBrain) [63]. These extensive datasets enabled us to validate the robustness of our method against real-world data.
+
+P7 Mouse Heart. This dataset consists of heart tissue extracted from a wild-type C57BL/6J mouse at postnatal day 7. The experimental procedure followed IACUC guidelines for animal anesthesia. The tissue was fixed via perfusion with glutaraldehyde and sectioned using a vibratome. Subsequently, the sample underwent low-temperature reducing OTO staining, dehydration through a graded ethanol series, infiltration with Durcupan resin, and polymerization in an oven at  $60^{\circ}\mathrm{C}$ .
+
+Mouse Kidney. This dataset consists of kidney tissue extracted from an 8-week-old wild-type C57BL/6 mouse. The tissue was perfused with glutaraldehyde fixative and sectioned using a vibratome. After staining with reducing OTO at room temperature, the sample underwent dehydration, infiltration with a graded ethanol series, and Durcupan resin. Finally, the sample was polymerized in an oven at  $60^{\circ}\mathrm{C}$ .
+
+P7 Mouse Liver. This dataset comprises liver tissue collected from a wild-type, postnatal day 7 C57BL/6J mouse. Following anesthesia in accordance with IACUC guidelines, the sample was perfused with glutaraldehyde fixative and sectioned using a vibratome. After low-temperature reducing OTO staining, the tissue underwent dehydration, infiltration with graded ethanol and Durcupan resin, and was subsequently polymerized in a  $60^{\circ}\mathrm{C}$  oven.
+
+Mouse Liver. This dataset contains liver tissue from a wild-type, 8-week-old C57BL/6 mouse. The sample was fixed via perfusion with glutaraldehyde and sectioned using a vibratome. Subsequently, reducing OTO staining was performed at room temperature. The sample was then dehydrated, infiltrated with graded ethanol and Durcupan resin, and polymerized in a  $60^{\circ}\mathrm{C}$  oven.
+
+P7 Mouse Skin. This dataset includes skin tissue from a wild-type, postnatal day 7 C57BL/6J mouse. Mouse anesthesia was performed in accordance with IACUC guidelines, followed by perfusion fixation using glutaraldehyde and vibratome sectioning. The sample underwent low-temperature reducing OTO staining, dehydration, infiltration with graded ethanol and Durcupan resin, and was polymerized in a  $60^{\circ}\mathrm{C}$  oven.
+
+P7 Mouse Pancreas. This dataset consists of pancreas tissue from a wild-type, postnatal day 7 C57BL/6 mouse. The sample was fixed via perfusion with glutaraldehyde and sectioned using a vibratome. It underwent low-temperature reducing OTO staining, followed by dehydration, infiltration with graded ethanol and Durcupan resin, and polymerization in a  $60^{\circ}\mathrm{C}$  oven.
+
+Female Fruit Fly Brain Neural Dataset. FemFlyBrain [63] consists of the right hemisphere of a wild-type Oregon R female fruit fly brain. The brain was continuously sectioned at a thickness of 40 nanometers, covering regions including the medulla and downstream neuropils. The sections were imaged at a magnification of 35,000. The connectome within the medulla includes 379 neurons and 8,637 chemical synaptic contacts.
+
+Full Adult Fly Brain Dataset. Full Adult Fly Brain Dataset (FAFB) [84] covers the entire depth of an adult fruit fly brain (approximately  $250\mu \mathrm{m}$ ). From the optimized sample, 7,062 consecutive sections of about  $40~\mathrm{nm}$  thickness were collected, optimized for high membrane contrast and fine ultrastructural preservation. A total of 7,050 sections  $(99.8\%)$  were successfully imaged.
+
+Mouse Cortical Dataset. The Mouse Cortical Dataset [32] provides a saturated reconstruction of a mouse neocortical sub-volume, in which all cellular elements (axons, dendrites, and glia) and various subcellular components (synapses, vesicles, spines, postsynaptic densities, and mitochondria) are fully annotated.
+
+Table 7: Details of the synthetic datasets for testing phase.  
+
+<table><tr><td>Datasets</td><td>Shape</td><td>Numbers</td><td>URL</td></tr><tr><td>P7 Mouse Heart</td><td>1184×1184</td><td>1000</td><td>https://openorganelle.janelia.org/datasets/jrc_mus-heart-1</td></tr><tr><td>Mouse Kidney</td><td>1184×1184</td><td>1000</td><td>https://openorganelle.janelia.org/datasets/jrc_mus-kidney</td></tr><tr><td>P7 Mouse Liver</td><td>1184×1184</td><td>1127</td><td>https://openorganelle.janelia.org/datasets/jrc_mus-liver-3</td></tr><tr><td>Mouse Liver</td><td>1184×1184</td><td>558</td><td>https://openorganelle.janelia.org/datasets/jrc_mus-liver</td></tr><tr><td>P7 Mouse Pancreas</td><td>1184×1184</td><td>898</td><td>https://openorganelle.janelia.org/datasets/jrc_mus-pancreas-4</td></tr><tr><td>P7 Mouse Skin</td><td>1184×1184</td><td>1231</td><td>https://openorganelle.janelia.org/datasets/jrc_mus-skin-1</td></tr><tr><td>FemFlyBrain</td><td>1024×1024</td><td>1299</td><td>https://neurodata.io/data/takemura13/</td></tr><tr><td>FAFB</td><td>1024×1024</td><td>700</td><td>https://temca2data.org/</td></tr><tr><td>Mouse Cortex</td><td>1024×1024</td><td>700</td><td>https://neurodata.io/data/kasthuri15/</td></tr></table>
+
+# A1.2 Metrics
+
+To quantitatively evaluate the performance of our method, we calculated several metrics, including Normalized Cross-Correlation (NCC), Mutual Information (MI), and Structural Similarity Index (SSIM) between the registered image series and the ground truth (GT). These metrics provide a comprehensive evaluation of the accuracy of our alignment and registration results from different perspectives. Specifically, NCC measures the similarity between the result images and the GT, MI reflects the mutual information between the two, and SSIM evaluates the accuracy based on structural information. Dice is a set similarity metric commonly used to measure the similarity between two samples. Here, we use the Dice score to evaluate the segmentation accuracy of 3D segmentation results on the registered data.
+
+# A2 Experimental Details
+
+# A2.1 Dataset Setup
+
+For training the trajectory tracking module, we split the original image data into 3000 image pairs  $\{I_0,I_1\}$ , where  $I_{0}$  and  $I_{1}$  represent adjacent images. These images are cropped to a size of 1184  $\times 1184$ . During training, the data is normalized to the range of [-1, 1]. The training-validation split is set to 0.95. Additionally, we randomly select one of the images from  $\{I_0,I_1\}$  and apply random elastic deformation to simulate nonlinear distortions. Specifically, we first generate a random deformation field  $D_{\mathrm{rand}}$ , which indicates the pixel displacement matrix. This matrix is then smoothed using a Gaussian filter, and the resulting deformation is applied to create the deformed images. The deformation process is described by the following formulas:
+
+$$
+\left\{ \begin{array}{l l} \phi_ {i} = & \alpha \cdot \operatorname {s i z e} \left(I _ {i}\right) \cdot \operatorname {G a u s s} \left(D _ {\text {r a n d}}, \sigma \cdot \operatorname {s i z e} \left(I _ {i}\right)\right), \\ I _ {i} = & \phi_ {i} \circ I _ {i}, \end{array} \right. \tag {17}
+$$
+
+where  $\phi_{i}$  represents the random deformation field applied to image  $I_{i}$ ,  $\circ$  denotes the deformation operation,  $D_{\mathrm{rand}}$  is the generated random displacement field,  $\mathrm{Gauss}(\cdot)$  is the 2D Gaussian filter operator,  $\alpha$  controls the displacement magnitude, and  $\sigma$  determines the smoothing extent. In our experiments, we set  $\sigma = 0.08$  and  $\alpha = 1.0$ .
+
+During the testing phase, we cropped all six datasets into image stacks of size  $1184 \times 1184 \times N$ , where  $N$  denotes the number of slices. Additionally, we applied random elastic deformations to all images except the first one to simulate nonlinear distortions. The deformation parameters were set to  $\sigma = 0.08$  and  $\alpha = 1.0$ . Detailed information regarding the datasets can be found in Table 7. Our test data consists of several hundreds to around 1000 samples, which fully demonstrates our method's capability in handling long sequences and addressing the challenge of error accumulation.
+
+# A2.2 Network Architecture and Training Details
+
+The trajectory tracking network adopts a 2D U-Net architecture with residual multi-kernel fusion, consisting of 1) a cascaded feature encoder with hybrid convolutional blocks, and 2) a dense feature decoder with transposed upsampling. The encoder progressively reduces spatial resolution through four strided convolutions (stride=2), doubling the number of channels from an initial 8 to 64. Specifically, the encoder is composed of four convolutional layers with output channels of 8, 16, 32, and 64, respectively. Each convolutional layer includes two convolution operations: the first convolution uses a  $3 \times 3$  kernel, followed by a second convolution with a larger  $7 \times 7$  kernel to expand the receptive field, similar to the approach in LKUnet [28]. The decoder consists of four layers and reconstructs dense predictions using four transposed convolutions (kernel size = 2, stride = 2), halving the number of channels from 64 to 2. Skip connections concatenate the encoder features with the corresponding decoder layers to facilitate hierarchical feature fusion. The final displacement field is generated through dual  $3 \times 3$  convolutions with a Softsign activation. To enable spatial warping, a Spatial Transformer Network (STN) [25] is employed for image registration. The model is optimized using the ADAM optimizer with parameters  $\beta_{1} = 0.5$  and  $\beta_{2} = 0.999$ . The learning rate is set to  $1 \times 10^{-4}$ , and training is performed for 500 epochs with a batch size of 16. For the regularization term in the loss function, we set  $\lambda = 1.5$  to encourage smoothness in displacement field.
+
+Neural ODEs employ a 3D U-Net architecture to parameterize  $f_{\theta}$ . The number of network layers and the number of channels per layer are kept consistent with the 2D U-Net used in the trajectory tracking module, but large convolutional kernels are avoided to reduce memory consumption. For training, we use the ADAM optimizer with parameters  $\beta_{1} = 0.5$  and  $\beta_{2} = 0.999$ . The learning rate is set to  $1 \times 10^{-4}$ , and training is conducted for 500 epochs with a batch size of 4. For the loss function, we set  $\lambda = 0.1$  and  $\mu = 0.005$ . For the Gauss-Seidel method used in trajectory smoothing, we set the number of iterations to 200 and  $\lambda = 0.1$ . For Neural ODEs, we use the simple Euler solver, with all other parameters kept at their default settings.
+
+# A2.3 Training and Testing on Real Data
+
+A key advantage of our method lies in its unsupervised learning paradigm, which is particularly valuable for real-world datasets that lack reliable ground-truth correspondences. The training process on real data is divided into two stages: training of the trajectory tracking module and training of the trajectory smoothing module. We begin by cropping a small spatiotemporal block of size  $H \times W \times T$  from the real dataset. Training samples are then constructed following the procedure described in Section A2.1, except that no additional elastic deformation is applied. The trajectory tracking network is subsequently trained in an unsupervised manner, identical to the approach used for synthetic data. For training the trajectory smoothing module, we first apply the previously trained network to obtain estimated trajectories. These are then used for the unsupervised training of the smoothing module, again following the same procedure as used for synthetic data.
+
+For testing on real data, the networks trained on cropped data blocks can be directly applied to the full image stack. We adopt a sliding-window registration strategy to perform registration across the entire long image sequence. Specifically, given the full real image stack  $\{I_i\}_{i=0}^{N-1}$ , we define a receptive field of moderate size (about 100 slices). Within each windowed region, the trajectory tracking network is used to estimate the displacement trajectories. These trajectories are then refined by the trajectory smoothing module. Subsequently, inverse warping is applied based on the smoothed trajectories to perform image registration. The registered image block is then placed back into its corresponding location in the full stack. The window is shifted along the sequence, and the procedure is repeated until the entire image stack has been registered.
+
+# A3 Supplementary Mathematical Derivations
+
+# A3.1 Background of the Gauss-Seidel Method
+
+The Gauss-Seidel method [18], a classical iterative solver for linear systems  $A\mathbf{x} = \mathbf{b}$ , that leverages the most recent updates to accelerate convergence, particularly in sparse, tridiagonal systems. This system can be equivalently written as a symmetric tridiagonal linear system:
+
+$$
+A \mathbf {p} = \mathbf {b}, \tag {18}
+$$
+
+where:  $-\mathbf{p} = [\mathcal{P}(1),\dots,\mathcal{P}(N - 1)]^T$  is the unknown trajectory,  $-\mathbf{b} = [\lambda \mathcal{P}^{(0)}(1),\dots,\lambda \mathcal{P}^{(0)}(N - 1)]^T$ ,  $-A\in \mathbb{R}^{(N - 1)\times (N - 1)}$  is a tridiagonal matrix with structure:
+
+$$
+A = \left[ \begin{array}{c c c c c} 4 + \lambda & - 2 & & & \\ - 2 & 4 + \lambda & - 2 & \ddots & \\ & \ddots & \ddots & \ddots & \\ & & - 2 & 4 + \lambda & - 2 \\ & & & - 2 & 4 + \lambda \end{array} \right]. \tag {19}
+$$
+
+Applying the Gauss-Seidel method to this system, the general update rule for the  $i$ -th variable is:
+
+$$
+x _ {i} ^ {(k + 1)} = \frac {1}{A _ {i i}} \left(b _ {i} - \sum_ {j <   i} A _ {i j} x _ {j} ^ {(k + 1)} - \sum_ {j > i} A _ {i j} x _ {j} ^ {(k)}\right). \tag {20}
+$$
+
+For our tridiagonal matrix, each row  $z$  only has nonzero entries at  $z - 1$ ,  $z$ , and  $z + 1$ . Substituting these into the general Gauss-Seidel formula, the update rule for  $\mathcal{P}(z)$  becomes:
+
+$$
+\mathcal {P} ^ {(k + 1)} (z) = \frac {1}{A _ {z z}} \left(\lambda \mathcal {P} ^ {(0)} (z) - A _ {z, z - 1} \mathcal {P} ^ {(k + 1)} (z - 1) - A _ {z, z + 1} \mathcal {P} ^ {(k)} (z + 1)\right), \tag {21}
+$$
+
+where:  $-A_{zz} = 4 + \lambda, -A_{z,z-1} = A_{z,z+1} = -2$ .
+
+Substituting these in yields:
+
+$$
+\mathcal {P} ^ {(k + 1)} (z) = \frac {1}{4 + \lambda} \left(2 \mathcal {P} ^ {(k + 1)} (z - 1) + 2 \mathcal {P} ^ {(k)} (z + 1) + \lambda \mathcal {P} ^ {(0)} (z)\right). \tag {22}
+$$
+
+For simplicity, we divide both numerator and denominator by 2, resulting in the final explicit Gauss-Seidel update:
+
+$$
+\mathcal {P} ^ {(k + 1)} (z) = \frac {\mathcal {P} ^ {(k + 1)} (z - 1) + \mathcal {P} ^ {(k)} (z + 1) + \frac {\lambda}{2} \mathcal {P} ^ {(0)} (z)}{2 + \lambda}, \quad z = 1, \dots , N - 1. \tag {23}
+$$
+
+# A3.2 Derivation of the Gauss-Seidel Iteration Update Formula
+
+For our trajectory smoothing method, the Gauss-Seidel iteration can be adapted by introducing a fidelity constraint. Below is the detailed derivation of the iteration formula. Specifically, to minimize the energy function:
+
+$$
+E (u) = \lambda \sum_ {i = 0} ^ {N - 1} \left(u _ {i} - u _ {i} ^ {(0)}\right) ^ {2} + \sum_ {i = 1} ^ {N - 2} \left(u _ {i - 1} - 2 u _ {i} + u _ {i + 1}\right) ^ {2}, \tag {24}
+$$
+
+we take the partial derivative of  $E$  with respect to each  $u_{i}$  and set it to zero to obtain the optimality condition.
+
+The fidelity term contributes a derivative of:
+
+$$
+\frac {\partial}{\partial u _ {i}} \left(\lambda \left(u _ {i} - u _ {i} ^ {(0)}\right) ^ {2}\right) = 2 \lambda \left(u _ {i} - u _ {i} ^ {(0)}\right). \tag {25}
+$$
+
+The smoothing term consists of a sum of squared second-order differences, where each point  $u_{i}$  appears in multiple overlapping terms, together with  $u_{i-1}$  and  $u_{i+1}$ . For clarity, we focus on a representative term where  $u_{i}$  is the center, to illustrate how the smoothness constraint acts locally on the trajectory:
+
+$$
+\frac {\partial}{\partial u _ {i}} \left(u _ {i - 1} - 2 u _ {i} + u _ {i + 1}\right) ^ {2} = - 4 \left(u _ {i - 1} - 2 u _ {i} + u _ {i + 1}\right). \tag {26}
+$$
+
+Combining the derivatives of both terms, we obtain the total gradient:
+
+$$
+\frac {\partial E}{\partial u _ {i}} = 2 \lambda \left(u _ {i} - u _ {i} ^ {(0)}\right) - 4 \left(u _ {i - 1} - 2 u _ {i} + u _ {i + 1}\right) = 0. \tag {27}
+$$
+
+![](images/61ce20efb0498b834d04b5228765ba29ff2fc16ee3e667f20f912d221ca1a21a.jpg)  
+(a) Grid Sample
+
+![](images/e53bd071ef2ef743c332f0d0d1fbc4cb706a4e3d64df4d57fa231f5eaac301a1.jpg)  
+(b) Bilinear Splatting  
+Figure 6: Regular grid sampling and Bilinear splatting. Grid sampling is used to obtain trajectory coordinates at sub-pixel resolution. For trajectory inversion, bilinear splatting is used to estimate the displacement at pixel grid locations.
+
+Expanding and rearranging terms gives:
+
+$$
+2 \lambda \left(u _ {i} - u _ {i} ^ {(0)}\right) - 4 u _ {i - 1} + 8 u _ {i} - 4 u _ {i + 1} = 0. \tag {28}
+$$
+
+Solving for  $u_{i}$ , we derive the following linear equation:
+
+$$
+(2 \lambda + 8) u _ {i} = 2 \lambda u _ {i} ^ {(0)} + 4 u _ {i - 1} + 4 u _ {i + 1}. \tag {29}
+$$
+
+This leads to the Gauss-Seidel iteration update formula:
+
+$$
+u _ {i} ^ {(k + 1)} = \frac {1}{2 \lambda + 8} \left(2 \lambda u _ {i} ^ {(0)} + 4 u _ {i - 1} ^ {(k + 1)} + 4 u _ {i + 1} ^ {(k)}\right), \tag {30}
+$$
+
+Simplifying further, we obtain
+
+$$
+u _ {i} ^ {(k + 1)} = \frac {1}{\lambda + 2} \left(\lambda u _ {i} ^ {(0)} + u _ {i - 1} ^ {(k + 1)} + u _ {i + 1} ^ {(k)}\right), \tag {31}
+$$
+
+where  $\lambda$  is halved from its original value. We use the updated  $u_{i-1}^{(k+1)}$  from the current iteration and the old  $u_{i+1}^{(k)}$  from the previous iteration to update  $u_i^{(k+1)}$ .
+
+# A4 Algorithmic Details of Bilinear Splatting
+
+The goal of bilinear splatting is to map discrete feature points onto a regular grid using bilinear interpolation. Given a set of normalized coordinates and their associated feature vectors, we need to compute the grid values using the surrounding grid points. The coordinates are normalized to the range  $[0, H)$  and  $[0, W)$ , where  $H$  and  $W$  are the height and width of the target grid. The feature vectors are associated with these coordinates and are projected to the grid using bilinear interpolation, as shown in Figure 6.
+
+Let the coordinates of the discrete points be represented as coordinates  $= \{(y_{1},x_{1}),\ldots ,(y_{N},x_{N})\}$  where each  $(y_{i},x_{i})$  is a 2D coordinate within the normalized range. The corresponding feature vectors for each point are denoted by values  $= \{v_{1},\dots ,v_{N}\}$ , where  $v_{i}\in \mathbb{R}^{C}$  is the feature vector for point  $i$ , and  $C$  is the feature dimension.
+
+To perform bilinear splatting, we begin by identifying the four nearest grid points that surround each continuous coordinate  $(y_i, x_i)$ . These grid points form the corners of the smallest axis-aligned square in the discrete grid that encloses the given coordinate. Specifically, we obtain them by applying the floor and ceiling operations to both the  $y$ - and  $x$ -coordinates:
+
+$$
+\left\{ \begin{array}{l} y _ {\text {f l o o r}} = \lfloor y _ {i} \rfloor , \quad x _ {\text {f l o o r}} = \lfloor x _ {i} \rfloor , \\ y _ {\text {c e i l}} = \lfloor y _ {i} \rfloor + 1, \quad x _ {\text {c e i l}} = \lfloor x _ {i} \rfloor + 1. \end{array} \right. \tag {32}
+$$
+
+These points form the basis for distributing the value at  $(y_i, x_i)$  across the surrounding pixels based on their bilinear interpolation weights.
+
+Next, the weights for bilinear splatting are calculated based on the fractional parts of the coordinates. Let  $\Delta y = y_{i} - \lfloor y_{i}\rfloor$  and  $\Delta x = x_{i} - \lfloor x_{i}\rfloor$  represent the fractional parts of the coordinates. The four interpolation weights are given by:
+
+$$
+\left\{ \begin{array}{l} w _ {1} = (1 - \Delta x) (1 - \Delta y), \quad w _ {2} = (1 - \Delta x) \Delta y, \\ w _ {3} = \Delta x (1 - \Delta y), \quad w _ {4} = \Delta x \Delta y. \end{array} \right. \tag {33}
+$$
+
+These weights represent the contribution of each neighboring grid point to the interpolated value at the target coordinate  $(y_i, x_i)$ . Now, we distribute the feature values to the corresponding grid locations based on these weights. For each coordinate  $(y_i, x_i)$ , we update the feature grid by adding the weighted value of the point to the grid locations corresponding to the four neighboring points. This is done using the following scatter operation:
+
+$$
+\left\{ \begin{array}{l} \operatorname {G r i d} \left[ y _ {\text {f l o o r}}, x _ {\text {f l o o r}} \right] = \operatorname {G r i d} \left[ y _ {\text {f l o o r}}, x _ {\text {f l o o r}} \right] + v _ {i} \cdot w _ {1}, \\ \operatorname {G r i d} \left[ y _ {\text {c e i l}}, x _ {\text {f l o o r}} \right] = \operatorname {G r i d} \left[ y _ {\text {c e i l}}, x _ {\text {f l o o r}} \right] + v _ {i} \cdot w _ {2}, \\ \operatorname {G r i d} \left[ y _ {\text {f l o o r}}, x _ {\text {c e i l}} \right] = \operatorname {G r i d} \left[ y _ {\text {f l o o r}}, x _ {\text {c e i l}} \right] + v _ {i} \cdot w _ {3}, \\ \operatorname {G r i d} \left[ y _ {\text {c e i l}}, x _ {\text {c e i l}} \right] = \operatorname {G r i d} \left[ y _ {\text {c e i l}}, x _ {\text {c e i l}} \right] + v _ {i} \cdot w _ {4}. \end{array} \right. \tag {34}
+$$
+
+The grid values are accumulated at the corresponding grid positions, and the sum of the weights at each grid location is also accumulated for normalization purposes.
+
+Finally, to ensure proper averaging when multiple points contribute to the same grid position, we normalize the resulting grid by the sum of the weights:
+
+$$
+\operatorname {G r i d} [ y, x ] = \frac {\sum v _ {i} \cdot w _ {i}}{\sum w _ {i}} \tag {35}
+$$
+
+To avoid division by zero in case of no contribution to a grid location, a small epsilon  $\epsilon = 1e - 8$  is added to the denominator during the normalization step. Thus, the bilinear splatting operation efficiently interpolates the feature values from discrete points to the target grid, with smooth interpolation properties ensured by the bilinear weights.
+
+# A5 Limitations and Future Work
+
+Our approach can be divided into three main modules: 1) pixel trajectory tracking, 2) dynamics-based trajectory optimization, and 3) trajectory inversion and image registration. For each module, we conducted a simple exploration in the ablation study (see Sec.B.1) to evaluate its effect on registration performance, including the influence of complex scenarios such as high-noise artifacts and wrinkles that may affect trajectory-tracking accuracy. Future work may investigate 3D registration methods that are robust to artifacts and wrinkles, as well as explore the use of more advanced architectures architectures for improving registration accuracy and generalizability. Furthermore, Table 11 presents the registration accuracy and execution time on the Mus Liver-3 dataset, where the test slice resolution is set to  $1024 \times 1024$ . Our method achieves the highest SSIM, demonstrating superior accuracy compared to other approaches. However, due to the computational demands of the ODE solver, it does not achieve the fastest runtime.
+
+To further investigate runtime efficiency, we conducted additional experiments focusing on two optimization strategies: (i) sparse trajectory sampling (every 2/3/4 pixels); and (ii) resolution reduction by downsampling followed by upsampling. Table 9 shows that sparse sampling substantially reduces runtime (from 2.46s to 0.95s) while maintaining state-of-the-art accuracy, with SSIM only slightly decreasing from 0.874 to 0.831. Table 10 demonstrates that reducing the resolution also improves efficiency (from 2.46s to 0.91s), though at the cost of accuracy (SSIM drops from 0.874 to 0.827). Nevertheless, the performance remains superior to the GaussReg baseline. We attribute the observed degradation primarily to the use of a simple bilinear upsampling strategy. We argue that could achieve a more favorable balance between runtime and accuracy.
+
+Table 8: Execution time and performance.  
+
+<table><tr><td></td><td>TrakEM2 [56]</td><td>EFSR [76]</td><td>EMReg [42]</td><td>SEMLeSS [53]</td><td>GaussReg [82]</td><td>Ours</td></tr><tr><td>Time(s)</td><td>3.92</td><td>3.44</td><td>0.28</td><td>27.11</td><td>0.92</td><td>2.46</td></tr><tr><td>SSIM</td><td>0.66</td><td>0.52</td><td>0.65</td><td>0.61</td><td>0.82</td><td>0.87</td></tr></table>
+
+Table 9: Ablation study of sparse sampling.  
+
+<table><tr><td></td><td>All pixels</td><td>1/2 pixel</td><td>1/3 pixel</td><td>1/4 pixel</td></tr><tr><td>SSIM</td><td>0.874</td><td>0.866</td><td>0.842</td><td>0.831</td></tr><tr><td>Time(s)</td><td>2.46</td><td>1.35</td><td>1.12</td><td>0.95</td></tr></table>
+
+Table 10: Ablation study of resolution reduction.  
+
+<table><tr><td></td><td>Full resolution</td><td>1/2 resolution</td><td>1/4 resolution</td></tr><tr><td>SSIM</td><td>0.874</td><td>0.845</td><td>0.827</td></tr><tr><td>Time(s)</td><td>2.46</td><td>1.12</td><td>0.91</td></tr></table>
+
+# A6 Broader Impacts and Discussion
+
+Our work focuses on 3D registration for Series Section Electron Microscopy (ssEM). One of the key advantages of our approach is its fully unsupervised training, which eliminates the reliance on ground truth and makes it applicable to real-world ssEM scenarios. For ssEM applications [79, 13, 15], this is particularly important for the increasingly complex and diverse electron microscopy image datasets [15, 77], as acquiring comprehensive ground truth across various tissue and cell types is challenging. Our method can be applied to the modeling and analysis of biological cell tissues, assisting researchers in exploring the structure and function of cellular tissues.
+
+For other downstream tasks in ssEM, such as 3D segmentation [40, 41, 47], our method provides well-structured image data that aligns with natural biological morphology. By improving the axial continuity of the raw image data while preserving the structural integrity of superstructural details, our approach can enhance the performance of downstream tasks.
+
+For the design of 3D registration algorithms, we propose a novel paradigm from the perspective of trajectory optimization. Our approach can be divided into three main components: 1) pixel trajectory tracking, 2) dynamics-based trajectory optimization, and 3) trajectory inversion and image registration. These three modules are decoupled and interchangeable. In theory, each module can be replaced with a more efficient algorithm. For example, more efficient feature point tracking algorithms [11, 10] or optical flow estimation algorithms [62, 78] can be used for trajectory tracking. Low-pass filtering [26] and convex quadratic programming (QP) [12, 17] can be used for trajectory smoothing. Radial basis function interpolation [73] and other splatting techniques [3] can be used for trajectory inversion. This helps researchers explore more efficient 3D registration methods based on this paradigm.
+
+Due to time and resource constraints, we were unable to validate the performance of our method on larger-scale scenarios, particularly on ultra-high-resolution image data. The challenges posed by computational complexity in larger environments and the variability of data noise remain. In future work, we will refine our approach and adapt it into a microscopy image processing tool suitable for large-scale real-world datasets.
+
+Table 11: Performance of different registration methods on downstream segmentation tasks across six datasets. We evaluate segmentation accuracy using the Dice score (Avg %).  
+
+<table><tr><td></td><td>EMReg [42]</td><td>EFSR [76]</td><td>SEMLeSS [53]</td><td>TrakEM2 [56]</td><td>GaussReg [82]</td><td>Ours</td></tr><tr><td>Mus Heart</td><td>0.35 ± 0.06</td><td>0.82 ± 0.02</td><td>0.65 ± 0.05</td><td>0.68 ± 0.03</td><td>0.89 ± 0.01</td><td>0.88 ± 0.01</td></tr><tr><td>Mus Kidney</td><td>0.41 ± 0.07</td><td>0.75 ± 0.02</td><td>0.67 ± 0.02</td><td>0.75 ± 0.02</td><td>0.82 ± 0.01</td><td>0.84 ± 0.01</td></tr><tr><td>Mus Liver-3</td><td>0.41 ± 0.08</td><td>0.90 ± 0.02</td><td>0.86 ± 0.02</td><td>0.90 ± 0.02</td><td>0.95 ± 0.01</td><td>0.94 ± 0.01</td></tr><tr><td>Mus Liver</td><td>0.57 ± 0.09</td><td>0.89 ± 0.04</td><td>0.86 ± 0.05</td><td>0.92 ± 0.04</td><td>0.94 ± 0.03</td><td>0.94 ± 0.03</td></tr><tr><td>Mus Pancreas</td><td>0.46 ± 0.04</td><td>0.87 ± 0.02</td><td>0.79 ± 0.02</td><td>0.90 ± 0.01</td><td>0.94 ± 0.01</td><td>0.93 ± 0.01</td></tr><tr><td>Mus Skin</td><td>0.50 ± 0.07</td><td>0.87 ± 0.03</td><td>0.76 ± 0.04</td><td>0.90 ± 0.02</td><td>0.94 ± 0.01</td><td>0.94 ± 0.03</td></tr></table>
+
+Table 12: Evaluation of the diffeomorphic property of deformation fields using the Folds metric (\% of  $|J\varphi |\leq 0$ ).  
+
+<table><tr><td>% of |Jφ| ≤ 0</td><td>TrakEM2 [56]</td><td>EFSR [76]</td><td>EMReg [42]</td><td>SEMLeSS [53]</td><td>GaussReg [82]</td><td>Ours</td></tr><tr><td>Mus Heart</td><td>/</td><td>0.188</td><td>0.243</td><td>0.135</td><td>0.099</td><td>0.0171</td></tr><tr><td>Mus Kidney</td><td>/</td><td>0.206</td><td>0.379</td><td>0.24</td><td>0.174</td><td>0.0098</td></tr><tr><td>Mus Liver</td><td>/</td><td>0.193</td><td>0.31</td><td>0.155</td><td>0.106</td><td>0.0124</td></tr><tr><td>Mus Skin</td><td>/</td><td>0.171</td><td>0.252</td><td>0.102</td><td>0.091</td><td>0.0095</td></tr></table>
+
+# B Additional Quantitative Results and Visualization
+
+# B.1 More Experimental Results
+
+Robustness to Error Accumulation. In practical applications, serial section electron microscopy (ssEM) datasets often contain hundreds or even thousands of images. This large number of slices poses a challenge for long-sequence registration, as cumulative errors can easily arise, eventually leading to substantial sequence drift and compromising the accurate reconstruction of the biological specimen's true 3D structure. To systematically evaluate the ability of our method to suppress such cumulative errors, we conducted comparative experiments on six long-sequence datasets listed in Table 7, focusing on the robustness of our method versus GaussReg [82] and SEMLeSS [53].
+
+Specifically, Figures 12 illustrate how the registration accuracy changes as the number of slices increases. The results show that our method consistently achieves higher average registration accuracy across the entire sequence, with smaller fluctuations in the accuracy curve. This indicates superior stability and robustness when handling long sequences. Moreover, Figures 7 and 8 present the 3D reconstruction results on the remaining four datasets. It is evident that our method accurately recovers the spatial structures of various biological tissues, further demonstrating its generalizability and robustness across different types of data.
+
+Performance on 3D Segmentation Tasks. 3D segmentation [40, 41] is a important task in serial section electron microscopy (ssEM) and has been widely applied in various biological image analysis domains [19, 43]. High-quality 3D registration plays a crucial role in segmentation tasks, as it can significantly mitigate artifacts caused by structural deformation, scale variations, and differences in imaging modalities, thereby improving segmentation accuracy. To comprehensively evaluate the applicability of our method in 3D segmentation, we conducted experiments on six datasets involving various organelles, including nuclei and mitochondria. Specifically, we employed a segmentation network [67] to perform segmentation on the images registered by each method. The segmentation performance was quantified by computing the Dice similarity coefficient between the predicted results and the ground-truth (GT) label stacks.
+
+Table 11 summarizes the Dice scores for 3D segmentation results across the six datasets. As shown, our method achieves the best performance on three datasets and the second-best on the remaining three, with overall results slightly below those of the supervised method GaussReg (with differences no greater than 0.01). This discrepancy may be attributed to the segmentation model's sensitivity to subtle misalignments, while our method focuses on global trajectory optimization and may have limitations in handling local details. For instance, as illustrated in Figure 8 for the Mus Liver-3 dataset, minor local misalignments can still be observed in our registration results. Nevertheless, our method attains accuracy comparable to that of the supervised GaussReg. Furthermore, Figs. 9, 10, and 11 present 3D visualizations of the segmentation results on multiple datasets. These visual results
+
+Table 13: Sensitivity analysis of the hyperparameter  $\lambda$  .  
+
+<table><tr><td>λ(SSIM)</td><td>0.15</td><td>1.5</td><td>4.5</td><td>7.5</td><td>10</td></tr><tr><td>Mus Heart</td><td>0.855</td><td>0.872</td><td>0.87</td><td>0.864</td><td>0.83</td></tr><tr><td>Mus Kidney</td><td>0.827</td><td>0.851</td><td>0.855</td><td>0.843</td><td>0.809</td></tr></table>
+
+Table 14: Impact of different network architectures on trajectory tracking performance.  
+
+<table><tr><td>SSIM</td><td>RAFT [64]</td><td>GAFlow [45]</td><td>SEA-RAFT [71]</td></tr><tr><td>Mus Heart</td><td>0.892</td><td>0.917</td><td>0.909</td></tr><tr><td>Mus Kidney</td><td>0.894</td><td>0.933</td><td>0.925</td></tr><tr><td>Mus Liver</td><td>0.871</td><td>0.928</td><td>0.918</td></tr><tr><td>Mus Skin</td><td>0.92</td><td>0.943</td><td>0.932</td></tr></table>
+
+Table 15: Effect of large deformations on registration performance.  
+
+<table><tr><td colspan="7">GaussReg (α=1.0) Ours (α=1.0) GaussReg (α=1.5) Ours (α=1.5) GaussReg (α=2.0) Ours (α=2.0)</td></tr><tr><td>Mus Heart</td><td>0.832</td><td>0.872</td><td>0.811</td><td>0.867</td><td>0.772</td><td>0.856</td></tr><tr><td>Mus Kidney</td><td>0.781</td><td>0.851</td><td>0.763</td><td>0.843</td><td>0.739</td><td>0.823</td></tr><tr><td>Mus Liver</td><td>0.805</td><td>0.842</td><td>0.782</td><td>0.836</td><td>0.758</td><td>0.817</td></tr><tr><td>Mus Skin</td><td>0.827</td><td>0.864</td><td>0.797</td><td>0.851</td><td>0.763</td><td>0.838</td></tr></table>
+
+Table 16: Effect of noise levels on registration performance.  
+
+<table><tr><td></td><td>GaussReg (5%)</td><td>Orgin (5%)</td><td>GaussReg (10%)</td><td>Orgin (10%)</td><td>GaussReg (15%)</td><td>Orgin (15%)</td></tr><tr><td>Mus Heart</td><td>0.815</td><td>0.851</td><td>0.788</td><td>0.832</td><td>0.748</td><td>0.771</td></tr><tr><td>Mus Kidney</td><td>0.762</td><td>0.833</td><td>0.734</td><td>0.822</td><td>0.715</td><td>0.759</td></tr><tr><td>Mus Liver</td><td>0.779</td><td>0.814</td><td>0.74</td><td>0.784</td><td>0.72</td><td>0.714</td></tr><tr><td>Mus Skin</td><td>0.804</td><td>0.837</td><td>0.751</td><td>0.803</td><td>0.728</td><td>0.742</td></tr></table>
+
+demonstrate that our method can reliably reconstruct the correct 3D structure of biological tissues, leading to clearer and more consistent segmentation boundaries, further validating its practicality in downstream tasks.
+
+In addition, we report the Folds metric in Table 12 to further evaluate the diffeomorphic property of the deformation fields. The results demonstrate that our method consistently achieves the lowest Folds values, remaining around 0.01, whereas most baseline methods report values greater than 0.1. This indicates that our approach substantially reduces grid folding in the generated deformation fields, thereby better preserving topological consistency compared to existing baselines.
+
+Results on Real-World Data. To further evaluate the performance of our method on real-world datasets, we present the registration results on three datasets, FemFlyBrain,  $\mathsf{FAFB}^3$ , and the Mouse Cortical Dataset, as shown in Figures 13-17. The results demonstrate that our method achieves accurate and consistent alignment across diverse biological samples, effectively handling complex morphological variations and imaging artifacts.
+
+More Ablation Experiments To gain a deeper understanding of our method, we conducted additional ablation studies. Table 13 presents a sensitivity analysis of the key hyperparameter  $\lambda$ , which primarily controls the smoothness term in the loss function of the trajectory tracking module. Table 14 investigates the impact of adopting more advanced network architectures on the performance of the trajectory tracking module. As shown, replacing the baseline with a more sophisticated network slightly improves the registration accuracy, demonstrating the modular flexibility of our framework and its compatibility with advanced architectures. Tables 15 and 16 further compare registration performance under more challenging conditions, including large deformations and high noise levels. The results indicate a gradual performance degradation as deformation or noise intensity increases, suggesting promising directions for extending our approach to more complex real-world scenarios.
+
+# B.2 Explainable Visualization Study
+
+To better understand the performance and explainability of our new paradigm, we conducted a visualization analysis of the intermediate trajectories. Specifically, we performed trajectory tracking on data with nonlinear distortions, ground truth data, and registration results, ensuring consistency between the trajectories across different datas. The visualization results are shown in Figure 18. It is evident from the figure that the trajectories of the nonlinear distorted data exhibit irregular noise jitter, leading to abrupt changes in local curvature, which reflect the impact of distortion on the motion trajectory. In contrast, the trajectory of the ground truth data maintains the natural evolutionary pattern of the organism's movement, exhibiting smooth and continuous changes. Our method successfully overcame abnormal physical deformations in the registration results, faithfully and precisely restoring the natural motion trajectory.
+
+# B.3 Failure Cases
+
+We observe that our method is to some extent dependent on the accuracy of the trajectory tracking module. When the tracking is suboptimal, the subsequent image registration performance may be affected. This issue becomes particularly prominent when handling real-world anisotropic data, where the axial resolution is often significantly lower than the lateral (XY) resolution. Substantial structural and textural differences arise between adjacent slices (as shown in Figure 19). Moreover, real datasets often suffer from high noise levels, imaging artifacts, and missing slices, further complicating reliable trajectory estimation. In Figure 20, we illustrate several representative failure cases, in which the axial resolution of the image stacks is approximately one-tenth of the lateral resolution, with noticeable noise and missing slices. Although our method is still capable of performing registration under such challenging conditions, the visual quality may degrade due to the aforementioned interfering factors. We hope future research will explore more robust and efficient trajectory tracking modules, as well as 3D registration methods that are better suited for anisotropic and noisy real-world data.
+
+![](images/bb99665688f7ad1a078d1da4e67e4e0dce77d67039370afb85048d780f157f05.jpg)  
+Figure 7: More 3D visualization of registration results on Mus Pancreas and Mus Skin datasets.
+
+![](images/5550cb3e3c410c4a962d46e41c548383d0cc228b5ce5557823698f32ed40ab21.jpg)  
+Figure 8: More 3D visualization of registration results on Mus Liver and Mus Liver-3 datasets.
+
+![](images/eceb8d697840af727b5b1c1f26c1f0f3942ae5de133306986926d945176f5d82.jpg)  
+Figure 9: 3D segmentation visualizations of registration results using various methods on the Mus Heart and Mus Liver-3 datasets.
+
+![](images/68722d943136543fab721476b08ece1a08b9cea9050e724531df77e0f9f52327.jpg)  
+Figure 10: 3D segmentation visualizations of registration results using various methods on the Mus Liver and Mus Pancreas datasets.
+
+![](images/6be16035341df6aaf95a25bf73d5cb17a0bf1372b8c1901f913765dbf0c3e208.jpg)  
+Figure 11: 3D segmentation visualizations of registration results using various methods on the Mus Kidney and Mus Skin datasets.
+
+![](images/698133223ce8c2e9a00aa24db37e2dbf6aee7ce36ca2ff9be7c5c5bf65907e2b.jpg)  
+Figure 12: Error accumulation comparison between GaussReg [82] and SEMLeSS [53] and ours on six dataset.
+
+![](images/5f84d2f11b05b552cf329c912a3d5c7c425e77810351c417d0e69bf04c5f9528.jpg)  
+Figure 13: Side views of the original data, the registration results of GaussReg [82], and our method on the FemFlyBrain dataset [63].
+
+![](images/37b87c21fa6c2189479beb6c1fa647f3553f8bd282f40adc971d76fdbf0662f1.jpg)  
+Figure 14: Side views of the original data, the registration results of GaussReg [82], and our method on the FemFlyBrain dataset [63].
+
+![](images/56549e41fea5f8ad32a1d20991518e57a20fa420929a28fb6b8a316cc901cc04.jpg)  
+Figure 15: Side views of the original data, the registration results of GaussReg [82], and our method on the Mouse cortical dataset [32].
+
+![](images/8e779dc882607db080f3af3551c57955a35dba69068a8114c0d805eb8677a619.jpg)  
+Figure 16: Side views of the original data, the registration results on the FAFB dataset [84].
+
+![](images/537f16bd64ffa0bac71cf4c56a59b334a35ec9dca0889050955c5aa43f4a6c56.jpg)  
+Figure 17: Side views of the original data and the registration results on the FAFB dataset [84].
+
+![](images/512466ff7856f5e1e01813e0854b3cdde93d0023dce6ef704d5b3b46d03ab05c.jpg)  
+Figure 18: Trajectory visualization of the original data, ground truth, and registration results.
+
+![](images/e6bca62f2f2777c4c01cbbffbb1beccf92e893fd78bd4cb5127d618fe68d13dc.jpg)
+
+![](images/9ec1aab4d33a4a0fc21044966f4012604c837ebf4e0fb3517acc0518b5577588.jpg)
+
+![](images/199c81e797dda36bfb872054b8552e324e7067a4d8cf67baa7517aae866b7aa9.jpg)
+
+![](images/0483ec1e92af0a57484769a972af897b4ea2fff2d9854218afa843d4d2df6cd9.jpg)
+
+![](images/7a734e81c3a0af41e55dd3fa3bf0fbd4fbe42a303ab572a716e4bc2f2ad1dbef.jpg)  
+Figure 19: Four examples showing texture structure differences between adjacent slices of anisotropic data.
+
+![](images/b4a189a628889657089e07a665a05786b23cd402d58b41f5ad1ebfc0e60617cd.jpg)
+
+![](images/a8685f35484f9c458012765725770b3ae5b7eb60323973bda9f96c408b6a9a60.jpg)
+
+![](images/bc6bcb8b160e998a7bc376807f66de29f5f8f93f423f0ec0449f33179805a49a.jpg)
+
+![](images/19fb89cc13f583bc1ca6fb96a08dfe7ba5d092d86ff10e36ab7622ebe63c7b9d.jpg)
+
+![](images/464c7e93cc9d0ef0efe19500b33ced63b2667bd5d5b41f1245e899e4c76b83b0.jpg)
+
+![](images/1e866e4d1218cf6ca54ece6ffe550ca606de667be59bc2715d5b01cb4b84543a.jpg)
+
+![](images/afb2b9749de280551e9c55c5af0a99dfd77314115fdbec6f4a80706f756dd7f7.jpg)
+
+![](images/3509b7d384c4873ac5c9456d4e44df52cc8f3d78dc9b72d2c024a4a82cc2849a.jpg)
+
+![](images/bacf6b8abb4e34c73ebdd53fa5ca4c7d8934976e87626dd78933aa7f6ef92906.jpg)
+
+![](images/c9b6400e932885e47b3413a45c951a4af5a10f2c089c1e89e12d074899f45159.jpg)
+
+![](images/a80fc9c9784a2d342ae02f10b0a5aa115d623eb94fef3c89518c0f599c9115f1.jpg)
+
+![](images/a118e802ba92719cf1b1c32e3fbcce72f22302dddaca7dafd345f0034bf4fd73.jpg)
+
+![](images/45f54c9d9f590b1e4ffbb6e43b2c8b57aaaccae99e446dca129ab3b5a70ed75a.jpg)
+
+![](images/47266e309bc2033649e51d2c6c0f33f67b3aa904a05ffcdfcf59c4146b2e7844.jpg)  
+Figure 20: Four failure cases from the FemFlyBrain dataset. These examples are affected by low axial resolution, missing slices, and high noise.
+
+![](images/0aa4fdd3e4f52c9813441bc69efff7c00582c6fbbe35c28e238d824d8b99f320.jpg)
+
+![](images/1665263374bd1e7af44881c3c05e72833af33e5555a98848a3d96cae3f84eb35.jpg)  
+Orgin
+
+![](images/d65b2412ee73330e15384de850a2968419d6b425b8cc66f895c3265b937a3f63.jpg)  
+Ours
+
+![](images/d7bb4fbf34aee21e678efec206c172d84de9f682183faf2bd2882d813791b5ee.jpg)
+
+![](images/182e43f0dbb22c2860373709e733753dea31c94e3750745e605f9e05275ec99c.jpg)
+
+# References
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\ No newline at end of file
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+# Unveiling Chain of Step Reasoning for Vision-Language Models with Fine-grained Rewards
+
+Honghao Chen $^{1,2,3*}$  Xingzhou Lou $^{1,2*}$  Xiaokun Feng $^{1,2*}$  Kaiqi Huang $^{1,2\dagger}$  Xinlong Wang $^{3\dagger}$
+
+$^{1}$ Institute of Automation, Chinese Academy of Sciences  
+ $^{2}$ School of Artificial Intelligence, University of Chinese Academy of Sciences  
+ $^{3}$ Beijing Academy of Artificial Intelligence
+
+# Abstract
+
+Chain of thought reasoning has demonstrated remarkable success in large language models, yet its adaptation to vision-language reasoning remains an open challenge with unclear best practices. Existing attempts typically employ reasoning chains at a coarse-grained level, which struggles to perform fine-grained structured reasoning and, more importantly, are difficult to evaluate the reward and quality of intermediate reasoning. In this work, we delve into chain of step reasoning for vision-language models, enabling assessing reasoning step quality accurately and leading to effective reinforcement learning and inference-time scaling with fine-grained rewards. We present a simple, effective, and fully transparent framework, including the step-level reasoning data, process reward model (PRM), and reinforcement learning training. With the proposed approaches, our models set strong baselines with consistent improvements on challenging vision-language benchmarks. More importantly, we conduct a thorough empirical analysis and ablation study, unveiling the impact of each component and several intriguing properties of inference-time scaling. We believe this paper serves as a baseline for vision-language models and offers insights into more complex multimodal reasoning. Our dataset, PRM, and code at https://github.com/baaivision/CoS.
+
+# 1 Introduction
+
+The complex reasoning capabilities of large language models (LLMs) have recently garnered widespread attention [57, 8, 47, 68, 1]. By generating a detailed chain of thought (CoT) process before producing the final answer, LLMs enable effective test-time scaling and significantly enhancing the reasoning and logical capabilities. The integration of CoT reasoning with large-scale reinforcement learning [43] has further led to revolutionary advancements, exemplified by groundbreaking models such as OpenAI-o1 [17] and DeepSeek-R1 [12]. Inspired by the success of NLP, many studies focus on exploring complex reasoning methods in vision-language models (VLMs), achieving various advances [66, 58, 10, 55, 34, 38, 29]. Compared to LLMs, the presence of visual information in VLMs introduces greater complexity and plays a more critical role, emphasizing the necessity of leveraging visual cues for precise step-by-step reasoning and problem-solving.
+
+Despite the advances, enabling VLMs to perform human-level reasoning remains a key challenge [61]. A fundamental difference is that humans typically engage in systematic and structured thinking [48], whereas the CoT outputs of current VLMs are constrained to coarse-grained reasoning, making it difficult to execute systematic and structured inference [58]. Specifically, current VLMs typically output a lengthy thought without a unified structure or clear step-by-step progression. This form
+
+of reasoning tends to become disorganized or verbose and, more importantly, makes it difficult to evaluate the quality of reasoning. As a result, selecting high-quality reasoning trajectories for efficient training (e.g., reinforcement learning) or inference becomes challenging. It is natural to ask whether step-by-step fine-grained reasoning can facilitate better training or inference. However, answering this is non-trivial, as it involves two major challenges: i) defining step, i.e., how to decompose the overall reasoning chain into logically coherent and progressive steps; ii) evaluating step, i.e., how to provide fine-grained reward signals for step-wise reasoning to facilitate training and inference.
+
+To address these challenges and answer the above question, we propose Chain-of-Step reasoning and introduce corresponding supervised fine-tuning (SFT) and reinforcement learning strategies to enhance VLMs' reasoning capabilities. We start by defining step: the entire reasoning chain is decomposed into consecutive steps, with each step consisting of three components: Name, Thought, and Reflection, separated by special tags. The Name component provides a concise summary of the step, the Thought component contains detailed reasoning specific to that step, and the Reflection component establishes connections between the current reasoning step, the visual information, and previous steps. Our design boasts three essential advantages: i) structuring reasoning into discrete steps to ensure clarity; ii) maintaining a concise and well-organized structure; iii) incorporating visual content along with reflective reasoning. We use GPT-4o [16] to construct ShareGPT-Step-300K, a dataset of 300K structured step-wise reasoning samples following the designed template. Our dataset encompasses a diverse range of tasks, utilizing 17 datasets that demand various reasoning skills, such as scientific reasoning [18, 31], mathematical reasoning [51, 64] and world knowledge [41].
+
+Through SFT on ShareGPT-Step-300K, VLMs can output step-by-step reasoning chains. However, reinforcement learning is more crucial for enhancing the model's reasoning capabilities [66], and providing accurate rewards is of vital importance [34]. Therefore, we delve into the precise evaluation of reasoning steps. We collect process-annotated data using two mainstream methods: Monte Carlo estimation and LLM-as-Judge [67]. Specifically, we employ Math-Shepherd [52] and GPT-4o [16] to annotate the step-wise reasoning trajectories, which we then use to train a process reward model (PRM). Based on PRM, we can assess the quality of intermediate reasoning steps with high accuracy, thereby providing fine-grained rewards.
+
+Building upon the above framework, we can effectively choose reasoning paths to conduct iterative Direct Preference Optimization (DPO) [7] and inference-time scaling. Experiments across multiple benchmarks demonstrate that our approach significantly enhances reasoning capabilities, outperforming strong baselines with consistent improvements. Our empirical analysis and ablations indicate that incorporating step-level evaluation is more reasonable and effective than relying solely on answer-level evaluation. Interestingly, we find that longer reasoning is not necessarily better for VLMs, and reasoning steps' quality is much more important than reasoning length. Different from prior arts, our work highlights the importance of introducing fine-grained reasoning chains in complex multimodal reasoning and leveraging fine-grained reward-based reinforcement learning to enhance VLMs. We hope our findings provide new insights for the community.
+
+# 2 Related Work
+
+VLM Reasoning. As the application of VLMs in complex tasks such as mathematics [30, 51] and science [61, 18, 31] grows, the ability to reason through complex problems becomes increasingly important. For instance, in solving mathematical diagram problems, Mavis [65] constructs a corresponding chain-of-thought (COT) [57] reasoning dataset and optimized the model through instruction fine-tuning [40]. Visual-CoT [42] focuses on enhancing the model's ability to correspond visual questions with key areas using CoT reasoning. Insight-V [10] designs a multi-agent system consisting of a reasoning agent dedicated to performing long-chain reasoning and a summary agent trained to judge and summarize reasoning results. The success of CoT in improving VLMs' complex visual reasoning reveals the importance of enabling the model to analyze intermediate reasoning steps in detail [66]. Unlike most existing methods [21, 42, 9] that use reasoning chains at a coarse-grained level, our work introduces a fine-grained structured reasoning approach, offering greater potential for enhancing the reasoning capabilities of VLMs.
+
+Inference-Time Scaling. Inference time scaling [45] refers to the process wherein a model utilizes additional reasoning time to solve problems more effectively. This System-2 style thinking ability [39] is recognized as an effective way to enhance the complex reasoning capabilities of LLMs and
+
+has recently garnered significant attention [69, 12]. Traditional inference time scaling methods include majority voting [14], best-N search [56], and sentence-level beam search [49]. Despite their widespread application, majority voting is typically only for problems with standard answers, whereas best-N search and sentence-level beam search present challenges in terms of accuracy assessment and the quality of individual responses [58]. Recently, OpenAI has developed verifiers to supervise and select reasoning paths during the inference process [25], while Math-Shepherd [52] evaluates intermediate reasoning steps based on their likelihood of leading to correct answers. Inspired by the success of these LLMs, LLaVA-CoT [58] and URSA [34] have started exploring the inference time scaling capabilities of VLMs. Procedural reasoning datasets have been constructed and used for the supervised fine-tuning of the models. Different from those prior arts, we utilize PRM to provide granular rewards and facilitate step-level beam search to enhance the inference-time scaling of VLMs.
+
+# 3 Method
+
+In this section, we present the pipeline of CoS as illustrated in Fig. 1. In Section 3.1, we introduce the design of our structured reasoning template. In Section 3.2, we describe the composition of the SFT dataset ShareGPT-Step-300K. Then we explain the acquisition of process annotation data in Section 3.3. Last, in Section 3.4, we elaborate on the details of the RL training pipeline with fine-grained rewards.
+
+# 3.1 Structured Reasoning
+
+We structure the entire thought into multiple consecutive cognitive steps, ensuring that these steps logically flow in a natural and coherent manner towards the final answer. Each step consists of the following three components:
+
+- Name. A summary and overview of the current step, providing a high-level explanation. This outlines the key aspects of the step to be executed.  
+- Thought. A detailed description of the thought process in this step, explaining what is being observed, inferred, or analyzed.  
+- Reflection. Establishing connections be
+
+tween the visual content and previous steps. Since VLMs are prone to generating content that contradicts the image, the reflection is introduced to create links with the visual content and prior steps, thus alleviating hallucination.
+
+![](images/cd7e5f0603515742cf6714607120244066954408d74aceca041fdf25bd12a70a.jpg)
+
+![](images/f112bdf6adc73801fc2a668864a5a0d9f947525cb716bcb9ed14e4d1ce3dab52.jpg)
+
+![](images/efd6a631419cc0ddd5698b0d5991d1771c808937e302b69d934da273c317f463.jpg)  
+Figure 1: Overview of our framework. We conduct SFT to enable the model to generate step-by-step reasoning paths. Then we train a PRM to provide fine-grained RL reward, and enhance the complex reasoning ability through step-level preference learning. Note that PRM only needs to be trained once for different models.
+
+We use special tokens to establish this reasoning format, rather than simply relying on additional prompts or external prompt engineering, because depending on prompt-based output format is unstable and would require more data filtering and cleaning. In contrast, ensuring stable output of this format is more critical for splitting and evaluating steps.
+
+CoS begins with the special token  $<$  |reasoning_start|>. We take reasoning step as the basic unit for structured and fine-grained reasoning. Steps are connected by  $<$  |reasoning_proceed|>. The reasoning stops after outputting the token  $<$  |reasoning_end|>, then the model derives the final answer from previous reasoning. The model generates reasoning steps and answers in an auto-regressive manner, where the number of reasoning steps and their specific content are self-generated without any restrictions. This structured output and its length are autonomously adapted and adjusted
+
+![](images/2af21456fdd53f549221e2a80bc180fdf68f3ef72d1c519ae709e456ca738008.jpg)
+
+# Question:
+
+After thoroughly reviewing the picture, can you confidently pick the right option that solves the Algebra problem? A. option A in the image should be selected. B. the answer is option D in the image. C. the correct choice is option C in the image. D. option B in the image is the right option. Answer: A
+
+Reasoning steps:  
+<Step1:Define the Slopes and Intersection Point>  
+< Thought  $\succ$  : We start by noting that two lines intersect at point P(1,6). One line has a slope of 1 and the other has a slope of 2. The equations of these lines can be derived using the point-slope form. For the line with slope 1 passing through P(1,6), the equation will be  $y - 6 = 1(x - 1)$ . Similarly, for the line with slope 2, the equation will be  $y - 5 = 1(x - 2)$ .  
+< Reflection  $\succ$  : This step sets up the foundation by defining the lines' slopes and using the intersection point to derive the line equations.  
+<Step2:Simplify the Line Equations>  
+< Thought  $>$  Simplify the first line equation:  $y - 6 = 1(x - 1)\rightarrow y = x + 5$  . Simplify the second line equation:  $y - 6 = 2(x - 1)\rightarrow y = 2x + 4$  
+< Reflection> Understanding the equation of the lines will assist in finding the parameters that will allow the lines intersect the  $x$ -axis, crucial for calculating the base and height of the triangle.  
+<Step3:Find the Intersection with the x-axis>  
+< Thought: >: For the line  $y = x + 5$ , set  $y = 0 \rightarrow x = -5.50$ . So, point Q is (-5, 0). For the line  $y = 2x + 4$ , set  $y = 0 \rightarrow x = -2.50$ . So, point R is (-2, 0).  
+< Reflection>: The coordinates of points Q and R on the  $x$ -axis are determined essential for constructing the triangle and calculating its dimensions.
+
+<Step 4:Determine the Length of Base OR>  
+< $\text{Though:}$  : The length of the base QR can be calculated as the distance between  $\mathrm{f}(-5,0)$  and  $\mathrm{r}(-2,0)$ .  $\mathrm{QR} = |1 - 2\cdot (-5)| = |1 - 2 + 5| = 3$  
+< Reflection >: The horizontal span or base QR of the triangle is determined,  
+Refining the foundation needed for finding the area  
+<Step 3: Calculate the height from P to Base QR> Thought: The height of the triangle is the vertical distance from point P(1,6).  
+Thought 9: The height of the triangle is the vertical distance from point  $P$  to the  $x$ -axis. Since  $P$  is directly above the  $x$ -axis, the height  $h$  is simply the  $v$  
+coordinate of P, which is 6.  
+< Reflection  $\succ$  Establishing the height simplifies the area calculation as it directly uses the height as a coordinate.  
+< Step 6: Calculate the Area of Triangle  $\triangle POR >$  
+< Thought>: Using the formula for the area of a triangle,  $\text{Area} = 1 / 2 \times \text{base} \times$  
+height, substitute base  $= 3$  and height  $= 6$ . Area  $= 1 / 2 \times 3 \times 6 = 9$ .  
+< Reflection>: This connects all prior findings to derive the final area of  $\triangle$  PQR. < Step 7: Validate the Consistency>  
+< Thought: > Ensure our area calculation aligns with the derived base and height:  $\mathrm{OR} = 3$ . Height from  $\mathsf{P} = 6$ . Area  $= 1 / 2 \times 3 \times 6 = 9$ .  
+< Reflection >: This is a double-check to ensure that all steps logically lead to the correct area.
+
+![](images/572f32fc81a5aafceff83ba5659aa2beb8405626964650d2739947a462fd83a8.jpg)  
+Figure 2: Qualitative examples. We depict the reasoning process on cases that need to understand both visual content and mathematics. Our CoS has accurate understandings of both aspects, and more importantly, combine them to derive multimodal reasoning. Note that special tokens are omitted.
+
+# Question:
+
+Rose the cat walks along the wall. She starts at point
+
+B and follows the direction of the arrows show
+
+the picture. The cat walk.
+
+Where does she end up? ABBCCDDFE
+
+Answer: p
+
+# Reasoning steps:
+
+<Step1:Identify the Starting Point>  
+< Thought>: We first note that Rose the cat starts her walk at point B. This is clearly shown in the image by the placement of the cat icon at point B.  
+< Reflection  $\gimel$  : This is the initial condition for the problem. Identifying the starting point is crucial as all subsequent calculations are based on this.  
+< Step 2: Analyze the Path and Distances >  
+< Thought>: From B, the cat moves in the direction of C. According to the image, the distance from B to C is 4 meters.  
+< Reflection >: Confirming this step is accurate by checking the image's path  
+direction from B to C.  
+<Step 5: Continue Along the Path> <Thought>: After reaching G. Ross  
+The distance from C to D is 1 meter.  
+< Reflection>: The arrows in the image confirm the continuity of the walk from C to D.  
+<Step 4: Further Progress on the Path>  
+< Thought>: After reaching D, Rose continues to E. The path shows a distance of 5 meters from D to E.  
+< Reflection>: The direction from D to E corresponds well with the image directions.  
+and continues another step of the journey.
+
+< Step 5: Advance Towards the Starting Region >  
+< Thought>: From E, Rose heads back towards A, covering a distance of 2 meters as per the path guidance.  
+< Reflection>: This This is confirmed by the consistent flow of arrows and distances in the image from E to A.  
+< Step 6: Path Completion Back to the Starting Region >  
+< Thought >: Following the journey, Rose moves from A to B. The final leg of this segment covers 3 meters as shown in the diagram.  
+< Reflection>: Arrows indicate that A to B completes another circuit, essential for finishing calculations.  
+< Step 7: Total Distance Calculation >  
+<  Though: Calculate cumulative distance: 1). B to C: 4 m. 2). C to D: 1 m. 3). D to E: 5 m. 4). E to A: 2 m. 5). A to B: 3 m. One full cycle:  $4 + 1 + 5 + 2 + 3 = 15$  meters  
+<Reflection>: Verification step to ensure the entire cycle is represented in the total calculation.  
+< Step 8: Determine Final Destination >
+
+< Thought>: Starting from B, Rose completes 15 meters in a full circuit back to B. Walking an additional 5 meters lands her at D: B to C: 4 m, C to D: 1 m. Thus  $15 + 5 =$  
+20 meters ends at D.  
+< Reflection >: This step confirms the distance matches the problem requirement of 30 meters by logical deduction and precise tracking.
+
+by the model, thereby enhancing its ability to perform complex reasoning. We depict two qualitative examples in Fig. 2 and the details of special tokens can be found in Appendix A.
+
+# 3.2 ShareGPT-Step-300K
+
+Given that most VQA, multi-modal math, and reasoning datasets only provide short answer annotations, we design a method to leverage these short annotations to generate long, step-by-step reasoning data. Specifically, we provide the question and its ground-truth answers as references, and prompts GPT-4o to generate step-by-step reasoning. This can be seen as reasoning from the results. The reference answer can greatly reduce the difficulty of the question, making it easier to generate high-quality thinking steps.
+
+To ensure diversity, our data is collected from various multimodal datasets, encompassing 17 tasks that require diverse multimodal reasoning capabilities. These tasks can be categorized into four major types including mathematical reasoning [51, 64, 3, 26, 24], scientific reasoning [18, 31, 23, 5], chart&document analysis [35, 32, 37, 36], and world knowledge [41, 54, 22, 44]. A summary of the dataset sources can be found in Appendix D.
+
+After generating the data, we conducted meticulous data cleaning and filtering to remove a large amount of data that did not meet the required formatting standards. Ultimately, we obtained 300K high-quality step-level data instances, which we named as ShareGPT-Step-300K. We plan to release this dataset to advance research in the community on fine-grained step-level reasoning.
+
+# 3.3 Step-Level Annotation
+
+Currently, mainstream methods for evaluating reasoning steps can be categorized into three types [67]: i) Human, which relies on human annotators to produce high-quality data but incurs substantial costs; ii) Monte-Carlo (MC) Estimation, which infers the correctness of steps from the outcomes of a vast number of sampled solutions; iii) LLM-as-Judge, which leverages LLM as critic by directly prompting the LLM to evaluate the correctness of steps. In this work, for the sake of efficiency and scalability, we employ MC Estimation and LLM-as-Judge to annotate step-level data.
+
+For MC Estimation, we use math-shepherd [52] to annotate steps automatically. Specifically, given a problem with a golden answer and a step-by-step solution, to annotate the correctness of a specific step, we utilize a fine-tuned VLM to decode N subsequent reasoning paths from this step. We further validate whether the decoded final answer matches with the golden answer. If a reasoning step can
+
+deduce more correct answers than another, it would be assigned a higher correctness score. Here we use  $N = 16$  by default.
+
+For LLM-as-Judge, we utilize GPT-4o [16] to perform the annotations. Each step is evaluated with one of three possible ratings: i) Good: the step is logical, directly relevant to the question, and most importantly, correct; ii) Neutral: the step is relevant but contains minor flaws or gaps; iii) Bad: the step is flawed, wrong, or shows a misunderstanding of the question or context. For solutions where the final answer is correct, both Good and Neutral ratings lead to a correct label. Conversely, for solutions with incorrect final answer, only steps annotated as Good are ultimately marked as correct. The prompt is provided in Appendix B.
+
+We use both math-shepherd and GPT-4o to generate 100K step-annotated data, respectively. These step-annotated data are then utilized to train a PRM, which can provide fine-grained reward signals for reinforcement learning and inference-time scaling.
+
+# 3.4 RL with Fine-grained Rewards
+
+As illustrated in Fig. 1, our training pipeline is designed to be straightforward and efficient. Starting from a well-pretrained VLM, we first perform supervised fine-tuning on the ShareGPT-Step-300K dataset to enable the model to generate step-by-step reasoning paths. Subsequently, we train a PRM to provide fine-grained rewards. Leveraging the PRM, we efficiently select reasoning paths for iterative Direct Preference Optimization, thereby further enhancing the model's reasoning capabilities.
+
+Process reward model training. In RLHF, the reward models are typically trained with binary preferences [2]. In reasoning tasks where step-annotation of solutions is accessible, reward models can be trained under the supervision of such ground-truth correctness. Given a question  $q$ , process reward model (PRM) estimates the correctness of individual reasoning steps. The PRM  $r_{\mathrm{process}}$  is trained with the following objective:
+
+$$
+\mathcal {L} _ {\mathbf {P R M}} = \mathbb {E} _ {q, p ^ {(k)}, y _ {k} \sim \mathcal {D}} [ \operatorname {L o s s} \left(y _ {k}, r _ {\text {p r o c e s s}} (q, p ^ {(k)})\right) ] \tag {1}
+$$
+
+where  $y^{k}$  is the label for the partial solution  $p^{(k)}$  and Loss is the loss function. Here we adopt binary cross entropy loss by default. We train a InternVL-2.5-MPO-38B [55] on the step-annotated data as introduced in Section 3.3 to get our PRM. We ablate on the choice of PRM in Section 4.3.4.
+
+Iterative direct preference optimization. Preference learning has gained increasing focus in VLMs [10]. To further enhance the quality of reasoning chains, we apply RL using the iterative DPO [7] algorithm to better align the model's reasoning process towards high-quality reasoning paths. We use SFT model as the policy model  $\pi_{\theta}$ , generating 16 reasoning paths for each question. Each reasoning path is evaluated with the trained PRM to assess its quality. We calculate the average score of reasoning steps and the score of final answer respectively, and take the weighted sum of them as the final score. The score is normalized to  $[0,1]$ . Then we select positive-negative pairs depending on the scores. Note that the score of the positive path must exceed that of the negative path by a threshold  $t$  to ensure sufficient distinction between them, thereby avoiding cases where the differentiation is minimal.
+
+Formally, we denote the dataset as  $\mathcal{D}_{\mathrm{DPO}} = (x,y_{+},y_{-})$ , where  $x$  is the question,  $y_{+}$  and  $y_{-}$  are the positive and negative responses. The DPO objective is defined as below:
+
+$$
+\mathcal {L} _ {\mathbf {D P O}} \left(\pi_ {\theta}; \pi_ {\text {r e f}}\right) = - \mathbb {E} _ {\left(x, y _ {+}, y _ {-}\right) \sim \mathcal {D}} \left[ \log \sigma \left(\beta \log \frac {\pi_ {\theta} \left(y _ {+} \mid x\right)}{\pi_ {\text {r e f}} \left(y _ {+} \mid x\right)} - \beta \log \frac {\pi_ {\theta} \left(y _ {-} \mid x\right)}{\pi_ {\text {r e f}} \left(y _ {-} \mid x\right)}\right) \right] \tag {2}
+$$
+
+where  $\pi_{\theta}$  is the policy model to be optimized and  $\pi_{\mathrm{ref}}$  is the reference model and they are both initialized with SFT model.  $\sigma$  is the logistic function and  $\beta$  is set to 0.1. After training, we use the trained policy model  $\pi_{\theta}$  to generate next-round data  $\mathcal{D}_{\mathrm{DPO}}^{(2)}$ , while keeping  $\pi_{\mathrm{ref}}$  fixed (i.e., the SFT model). We repeat this procedure for multiple rounds to progressively enhance the the model's reasoning.
+
+# 4 Experiment
+
+In this section, we first introduce the setting and implementation details in Section 4.1. Then we present a comprehensive comparison with state-of-the-art VLMs across multiple reasoning benchmarks in Section 4.2. In Section 4.3, we conduct extensive ablation studies and analyses to validate the effectiveness of our approach and explore several interesting properties.
+
+Table 1: Comparison with open-source VLMs. We evaluate our method on six benchmarks covering both general and task-specific reasoning capacities. Our CoS has consistently strong performances across these benchmarks, surpassing different baselines and other state-of-the-art VLMs by large margins. The items in bold and underlined respectively represent the first or second highest scores.  
+
+<table><tr><td>Method</td><td>Size</td><td>MathVista</td><td>MMStar</td><td>MMMU</td><td>M3CoT</td><td>AI2D</td><td>ChartQA</td><td>Average</td></tr><tr><td>LLaVA-1.5 [27]</td><td>13B</td><td>27.6</td><td>32.8</td><td>-</td><td>39.5</td><td>-</td><td>-</td><td>-</td></tr><tr><td>IXC-2.5 [63]</td><td>7B</td><td>63.7</td><td>59.9</td><td>42.9</td><td>-</td><td>81.5</td><td>82.2</td><td>-</td></tr><tr><td>Ovis1.5-LLaMA3 [33]</td><td>8B</td><td>63.0</td><td>57.3</td><td>48.3</td><td>-</td><td>-</td><td>76.4</td><td>-</td></tr><tr><td>LLaVA-CoT [58]</td><td>11B</td><td>54.8</td><td>57.6</td><td>-</td><td>-</td><td>78.7</td><td>-</td><td>-</td></tr><tr><td>LlamaV-o1 [50]</td><td>11B</td><td>54.4</td><td>59.5</td><td>-</td><td>-</td><td>81.2</td><td>-</td><td>-</td></tr><tr><td>Vision-R1-LlamaV [15]</td><td>11B</td><td>62.7</td><td>61.4</td><td>-</td><td>-</td><td>-</td><td>83.9</td><td>-</td></tr><tr><td>R1-VL [62]</td><td>7B</td><td>63.5</td><td>60.0</td><td>-</td><td>-</td><td>-</td><td>83.9</td><td>-</td></tr><tr><td>R1-Onevision [59]</td><td>7B</td><td>64.1</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>MiniCPM-V-2-6 [60]</td><td>8B</td><td>60.6</td><td>57.5</td><td>49.8</td><td>56.0</td><td>82.1</td><td>79.4</td><td>64.2</td></tr><tr><td>LLaVA-OneVision [21]</td><td>7B</td><td>63.2</td><td>61.7</td><td>48.8</td><td>52.3</td><td>81.4</td><td>80.0</td><td>64.6</td></tr><tr><td>Qwen2-VL [53]</td><td>7B</td><td>58.2</td><td>60.7</td><td>53.7</td><td>57.8</td><td>83.0</td><td>77.4</td><td>65.1</td></tr><tr><td>Insight-V [10]</td><td>7B</td><td>59.9</td><td>61.5</td><td>50.2</td><td>61.5</td><td>79.8</td><td>81.5</td><td>65.7</td></tr><tr><td>InternVL-2.5 [6]</td><td>8B</td><td>62.2</td><td>59.6</td><td>54.1</td><td>62.4</td><td>84.5</td><td>84.9</td><td>68.0</td></tr><tr><td>LLaVA-NeXT-LLaMA3 [28]</td><td>8B</td><td>45.9</td><td>43.1</td><td>36.9</td><td>45.6</td><td>71.5</td><td>69.4</td><td>52.1</td></tr><tr><td>+SFT</td><td>8B</td><td>51.4</td><td>54.7</td><td>39.6</td><td>67.4</td><td>76.1</td><td>75.7</td><td>60.8</td></tr><tr><td>+Iterative DPO (CoS-LLaVA)</td><td>8B</td><td>54.7</td><td>58.9</td><td>41.8</td><td>71.7</td><td>79.2</td><td>79.1</td><td>64.2</td></tr><tr><td>InternVL2.5-MPO [55]</td><td>8B</td><td>65.0</td><td>60.7</td><td>53.8</td><td>67.5</td><td>84.2</td><td>85.0</td><td>69.4</td></tr><tr><td>+SFT</td><td>8B</td><td>65.9</td><td>61.0</td><td>53.7</td><td>75.7</td><td>81.6</td><td>88.3</td><td>71.0</td></tr><tr><td>+Iterative DPO (CoS)</td><td>8B</td><td>67.8</td><td>63.5</td><td>55.5</td><td>81.0</td><td>84.9</td><td>87.4</td><td>73.4</td></tr></table>
+
+# 4.1 Experimental Setup
+
+Implementation details. To verify the effectiveness and generalizability of our approach, we employ LLaVA-NeXt [28] and InternVL-2.5-MPO [55] as our base VLMs, respectively. For each base VLM, we first conduct one epoch of supervised fine-tuning on ShareGPT-Step-300K. Then, based on the SFT model, we apply three rounds of iterative DPO to obtain the final model. For each round, we compile approximately 20K preference pairs as training data for DPO. The training recipes can be found in Appendix C. For PRM, we select InternVL-2.5-MPO-38B [55] as the base model. We trained it for two epochs on 200K process-annotated data, using a learning rate of 1e-6 and a batch size of 128 to get the PRM. For the training time, SFT takes about 9 hours on a single A800 node, and three rounds of iterative DPO takes about 6 hours in total on a single A800 node.
+
+**Benchmarks.** We evaluate our method on several representative benchmarks, spreading a wide range of tasks that demand the model's complex reasoning abilities. MathVista [30] is a well-known dataset for multi-modal reasoning, focusing on mathematics skills such as plane geometry, function and puzzle. MMMU [61] evaluates the model's multimodal reasoning capability across various university-level disciplines. ChartQA [35] aims at assessing logical reasoning with charts. MMstar [4] focuses on various multi-modal problems that must rely on visual contents. M3CoT [5] evaluates the model's chain-of-thought reasoning abilities. AI2D [18] is designed to test scientific knowledge and reasoning.
+
+# 4.2 Main Results
+
+We conducted a comprehensive comparison between our method and leading open-source VLMs on the aforementioned benchmarks. As shown in Table 1, our CoS achieves a substantial improvement over different baselines. Specifically, our method yields an average increase of  $4.9\%$  and  $3.2\%$  when applied to LLaVA-NeXT and InternVL-2.5-MPO, demonstrating the effectiveness and generalizability of our framework.
+
+For LLaVA-NeXt, both SFT and DPO lead to significant performance gains, indicating that for relatively weaker VLMs, both SFT and reinforcement learning (RL) play crucial roles. In contrast, for InternVL2.5-MPO, the performance improvement from SFT alone is relatively limited. However, after applying iterative DPO, the model can achieve a notable performance boost, reaching state-of-the-art. This indicates that for stronger VLMs, the benefits of further improvement through SFT are marginal, while RL is more essential and effective for enhancing strong model's reasoning capabilities.
+
+# 4.3 Analysis and Ablation Study
+
+# 4.3.1 Reasoning Step Matters
+
+Firstly, we dive into the impact of intermediate reasoning steps and final answers on evaluating reasoning quality. To be specific, we use LLaVA-NeXt [28] fine-tuned on the ShareGPT-Step-300K as base model and sample 16 reasoning paths and corresponding answers for each question in M3CoT. We then utilize a trained PRM to score these reasoning steps and answers. We use the weighted sum of the step score and answer score as the selection criterion to report best-of-16 accuracies.
+
+As shown in Fig. 3, we progressively increase the weight of step scores from  $0\%$  (i.e., relying solely on the final answer for evaluation) to  $100\%$  (i.e., relying exclusively on reasoning steps). The model's best-of-16 accuracy exhibits a trend of initially increasing and then decreasing. This indicates that relying solely on either the final answer or the intermediate reasoning steps is not optimal; instead, a balanced evaluation that considers both is preferable. This conclusion also holds for reinforcement learning training as shown in Section 4.3.3. Unless mentioned, we set the weight of step scores to  $20\%$  by default in our experiments.
+
+![](images/87e0cff3b0851f08118ea3d71846444cfead61c9eea6da3b768a2d72dbd6a811.jpg)  
+Figure 3: Ablation on step weight. We progressively increase the weight of step score from  $0\%$  to  $100\%$  and report the best-of-16 accuracy on M3CoT. Considering both step's and answer's quality leads to optimal performances.
+
+# 4.3.2 Inference-Time Scaling
+
+# Step-level beam search. In order
+
+to further enhance the reasoning ability during inference, we design a step-level beam search by making full use of the structured output of CoS. The specific implementation steps are as follows:
+
+- Sample N responses for the first step.  
+- Use PRM to score the N first steps.  
+- Continue to sample the next step based on the best step.  
+- Repeat the above steps until the final answer is generated.
+
+It is worth noting that step-level beam search's cost and sample efficiency is exactly the same as that of Best-of-N (i.e., independently sample N responses, score with PRM, and select the best). An illustration is shown in Fig. 4.
+
+Next, we explore the inference-time scaling properties of CoS reasoning. We compare four methods: Pass@N, Self-Consistency, PRM, and PRM with step-level beam search. Pass@N refers to the probability that at least one out of N sampled solutions is correct, which can be regarded as the upper bound of inference-time performance. Self-Consistency refers to selecting the most frequent answer among N sampled solutions. PRM refers to selecting the Best-of-N solution based on the scores provided by the PRM. PRM-BS refers to selecting the Best-of-N solution obtained through step-level beam search. We incrementally increase the number of samples N from 1 to 64 to evaluate the performance of these methods. As shown in Fig. 5, the answer selected by PRM has a significant advantage over Self Consistency. Moreover, PRM-BS has achieved consistent improvement over PRM. Considering that the sampling efficiency of PRM-BS and PRM is completely the same, this further illustrates the advantages of our method in the inference-time.
+
+# 4.3.3 Ablation on RL training
+
+We proceed to investigate the impact of rewards on the RL training. Specifically, we utilize LLaVA-NeXt-SFT as our base model and employ three methodologies to offline select training data for
+
+![](images/73e203d4400750090d8955d71d90c00f24bfa7ec85ba6fe4351e4a8ea2a111f9.jpg)
+
+![](images/3b587fb798ada859af3a7f5c6c2d3fa091b426dd0f63194f6ae58a91da6b696b.jpg)  
+Verification
+
+![](images/6eed8017ffc55e50a94dd5e7ece7df7c4df939ff2411b8c7b8c21e7e0380b593.jpg)  
+Rejected Step
+
+![](images/c03c41c7a645a3256c5711c269a8d15d36196f7d69980fbf63924f9edb345bd0.jpg)  
+Figure 4: Illustration of inference-time scaling. Our structured and fine-grained designs enable us to execute (b) Best-of-N with PRM and (c) Step Beam Search with PRM. Both of them outperforms Self-Consistency by significant margins.
+
+![](images/644c2821abfe92827a542294230c0ce52c86e9744185ae30ba91f796059c2e0b.jpg)  
+Chosen Step
+
+![](images/229299f26fb8dc72df9422df531ba60ac5fd0bcdcf66c2a64bbb081157b91998.jpg)
+
+![](images/7b46bb25ad372e0bfe3a57060f0a9e35a3b52a75a6d292894b0aaa9c2b762079.jpg)  
+Figure 5: Pass@N and Best-of-N comparison on M3CoT and MMStar. Here SC refers to Self-Consistency, PRM-BS refers to step-level beam search. We increase N from 1 to 64, and our method demonstrates appealing inference-time property.
+
+![](images/a910778f0ecc318829f063aa3f44367c0b9d2b0f63ee22c6499795f5f8f0e56e.jpg)
+
+iterative DPO: i) Outcome. This approach uses the ground-truth labels of the questions as the gold criterion for judging the quality of responses. Responses with correct answers are selected as positive samples, while those with incorrect answers are negative ones; ii) PRM scores for answers. This can be seen as a lower bound on performance compared to the outcome method, as the accuracy of PRM's evaluation of answers is inherently lower than that of direct evaluation using ground-truth labels; iii) PRM scores for both reasoning steps and answers. This method comprehensively considers the quality of step and answer to selects positive and negative pairs.
+
+As shown in Table 2, solely relying on answer scoring for selection underperforms the outcome-based method, which is an anticipated result. The strategy that takes into account both the intermediate steps and the answer quality achieves the best performances, surpassing outcome-based method with clear gains. This underscores the significance of fine-grained rewards in assessing reasoning quality, thereby enhancing the efficacy of RL training.
+
+Table 2: Ablation on the rewards for RL. Taking both the steps and answer quality into account leads to the best performances.  
+
+<table><tr><td>Method</td><td>MathVista</td><td>MMStar</td><td>M3CoT</td></tr><tr><td>LLaVA-NeXt-SFT</td><td>51.4</td><td>54.7</td><td>67.4</td></tr><tr><td>Answer (PRM)</td><td>53.1</td><td>57.3</td><td>69.7</td></tr><tr><td>Outcome</td><td>53.5</td><td>58.1</td><td>70.0</td></tr><tr><td>Step&amp;Answer (PRM)</td><td>54.7</td><td>58.9</td><td>71.7</td></tr></table>
+
+Table 3: Ablation on the PRM. Stronger PRM leads to more accurate evaluation for reasoning steps.  
+
+<table><tr><td>Base Model</td><td>Question</td><td>Step Acc</td><td>Answer Acc</td></tr><tr><td>LLaVA-NeXt-8B</td><td>Seen</td><td>85.1</td><td>90.5</td></tr><tr><td>LLaVA-NeXt-8B</td><td>Unseen</td><td>83.7</td><td>84.7</td></tr><tr><td>InternVL-2.5-8B</td><td>Unseen</td><td>85.7</td><td>91.1</td></tr><tr><td>InternVL-2.5-38B</td><td>Unseen</td><td>87.3</td><td>88.7</td></tr></table>
+
+# 4.3.4 Ablation on PRM
+
+To evaluate the accuracy of the PRM, we reserved 10K process-annotated reasoning data for assessment. Among these, 5K questions were from the PRM training dataset, while the other 5K were unseen during PRM's training, allowing us to evaluate PRM's ability to assess in-domain data and generalization capability to unseen questions. We employed three base models: LLaVA-NeXt-8B, InternVL2.5-8B, and InternVL2.5-38B. As demonstrated in Table 3, the stronger the base model, the more accurate its judgment of intermediate steps for the PRM. Given the critical importance of step evaluation in our framework, we select InternVL2.5-38B as the base model for our PRM.
+
+# 4.3.5 Reasoning Length
+
+It has been observed that the length of reasoning tends to increase progressively when LLMs are trained with RL on mathematical problems, and their performance improves accordingly [12]. However, things become different for visual complex reasoning. Existing visual complex reasoning problems do not require lengthy intermediate processes as pure mathematics does, but rather rely on the utilization of visual information and the connection of relevant knowledge to trigger reasoning (e.g., geometric shapes). Therefore, we investigate how the length of VLM reasoning steps changes with training. We take LLaVA-NeXt-SFT as the base model and calculate the aver
+
+![](images/e6074ea7193c8eb5d9313dd27fee5f6a217630b4558ddd0fcfd20893e69fa193.jpg)  
+Figure 6: Reasoning step length curve. Increasing reasoning length is not necessarily better. For visual complex reasoning, ensuring high-quality steps is of vital importance.
+
+age reasoning step after each round of outcome DPO and PRM DPO, respectively. We depict the change curve of reason step length in Fig. 6.
+
+For outcome DPO, we select reasoning paths solely based on the correctness of the final results. It can be observed that as the training iteration progresses, the reasoning step length progressively increases. This indicates that the model tends to keep extending the reasoning length if we don't consider the quality of intermediate steps. Interestingly, the situation is entirely the opposite for PRM DPO. The model tends to initially reduce the length of reasoning to improve the quality of reasoning, and only after stabilizing does it slowly increase the length of reasoning. Considering that PRM DPO yields better results than outcome DPO as shown in Table 2, this phenomenon indicates that for visual complex reasoning, increasing reasoning length is not necessarily better. It is more crucial to first ensure that the model can output high-quality reasoning steps.
+
+# 4.4 Ablation on Reasoning Pattern
+
+In order to demonstrate our reasoning design's advantages, we add a direct quantitative comparison between the proposed method and simpler prompting strategies in Table below. Specifically, No Reason denotes a setting where the model performs direct question answering without any explicit reasoning; Direct Prompt means directly prompting VLMs to reason step by step (as in [19]); CoS corresponds to our proposed method using step-level structured reasoning with fine-grained rewards. All methods are trained using the same SFT and RL datasets to ensure a fair comparison. For No
+
+Table 4: Ablation on the reasoning pattern. No Reason denotes a setting where the model performs direct question answering without any explicit reasoning; Direct Prompt means directly prompting VLMs to reason step by step. The corresponding SFT dataset's format has been adjusted accordingly to make a fair comparison.  
+
+<table><tr><td>Method</td><td>Reason</td><td>Reward</td><td>MathVista</td><td>MMStar</td><td>M3CoT</td><td>Average</td></tr><tr><td>LLaVA-NeXt</td><td>-</td><td>-</td><td>45.9</td><td>43.1</td><td>45.6</td><td>44.9</td></tr><tr><td>No Reason SFT</td><td>×</td><td>-</td><td>49.1</td><td>53.9</td><td>62.1</td><td>55.0</td></tr><tr><td>No Reason RL</td><td>×</td><td>outcome</td><td>51.5</td><td>56.4</td><td>63.4</td><td>57.1 (+2.1)</td></tr><tr><td>Direct Prompt SFT</td><td>✓</td><td>-</td><td>51.2</td><td>54.9</td><td>66.3</td><td>57.5</td></tr><tr><td>Direct Prompt RL</td><td>✓</td><td>outcome</td><td>53.1</td><td>58.2</td><td>69.3</td><td>60.2 (+2.7)</td></tr><tr><td>CoS SFT</td><td>✓</td><td>-</td><td>51.4</td><td>54.7</td><td>67.4</td><td>57.8</td></tr><tr><td>CoS RL</td><td>✓</td><td>PRM</td><td>54.7</td><td>58.9</td><td>71.7</td><td>61.8 (+4.0)</td></tr></table>
+
+Table 5: Ablation for GRPO experiments. Our Cos has consistently better performance than the outcome-reward counterpart.  
+
+<table><tr><td>Method</td><td>Reward</td><td>MathVista</td><td>MMStar</td><td>M3CoT</td><td>Average</td></tr><tr><td>LLaVA-NeXt-SFT</td><td>-</td><td>51.4</td><td>54.7</td><td>67.4</td><td>57.8</td></tr><tr><td>outcome GRPO</td><td>outcome</td><td>54.3</td><td>57.9</td><td>71.4</td><td>61.2</td></tr><tr><td>CoS GRPO</td><td>PRM</td><td>56.3</td><td>59.1</td><td>73.7</td><td>63.0</td></tr></table>
+
+Reason, we use only the question-answer pairs (without intermediate reasoning thought) from the ShareGPT-Step-300K dataset for SFT. For Direct Prompt, we concatenate all steps of reasoning (i.e., "thoughts") from each instance into a single reasoning sequence. Since both No Reason and Direct Prompt can not output strict step-level thought, we cannot assess their intermediate reasoning quality, so we apply the outcome reward (i.e., correctness of the final answer) during RL training.
+
+There are two key observations: i) Models trained with reasoning-augmented SFT data (i.e., Direct Prompt and Cos) clearly outperform models trained with QA-only data (No Reason) on reasoning benchmarks; ii) Our Cos yields much greater improvements during RL compared to outcome-only reward settings. This ablation study directly supports our central motivation for fine-grained reasoning and strengthens the empirical foundation of our conclusions.
+
+# 4.5 GRPO-based ablation
+
+Recently, GRPO [43] has become the mainstream algorithm in current reinforcement learning and has demonstrated advantages across multiple tasks. Therefore, we further conducted an ablation experiments based on the GRPO algorithm to validate the effectiveness of our proposed method. For GRPO experiments, we deploy the 38B PRM as the reward server. We set batchsize to 128, rollout.n to 5, lr to 1e-5 and run 100 steps starting from the SFT model. As shown in Table 5, our fine-grained reward RL outperforms outcome reward RL consistently using GRPO, further strengthening our method and analysis.
+
+# 5 Conclusion
+
+In this work, we introduce Chain of Step (CoS) reasoning for vision-language models, enabling assessing reasoning step quality accurately and leading to effective reinforcement learning and inference-time scaling with fine-grained rewards. Experimental results across multiple benchmarks demonstrate the effectiveness of CoS. More importantly, we conduct extensive empirical analysis and ablations to unveil CoS's appealing properties. We hope this paper offers insights into more complex multi-modal reasoning.
+
+# Acknowledgments
+
+This work is supported in part by the National Science and Technology Major Project, Grant No.2022ZD0116403.
+
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+
+```erb
+<|reasoning_start|>
+<|reasoning_step_start|>
+<|reasoning_step_name_start|>Name1<|reasoning_step_name_end|
+<|reasoning_step Thought_start|>Thought1<|reasoning_step_thought_end|
+<|reasoning_step_reflection_start|>Reflection1<|reasoning_step_reflection_end|
+<|reasoning_step_end|
+<|reasoning_proceed|
+<|reasoning_step_start|
+<|reasoning_step_name_start|>Name2<|reasoning_step_name_end|
+<|reasoning_step_thought_start|>Thought2<|reasoning_step_thought_end|
+<|reasoning_step_reflection_start|>Reflection2 < |reasoning_step_reflection_end|
+<|reasoning_step_end|
+<|reasoning_proceed|
+...
+<|reasoning_proceed|
+...
+<|reasoning_end|
+<|answer_start|>The answer is A.<|answer_end|>
+```
+
+Figure 7: Illustration of structured reasoning. We take reasoning step (special tokens marked in blue) as the basic unit to conduct structured and fine-grained reasoning.
+
+# A Special Tokens
+
+Our CoS uses special tokens to realize structured step-by-step reasoning. We list all 11 special tokens and their corresponding definitions in Table 6. For the initialization of these special tokens, we use the average value of the token embeddings of its corresponding definition. The reasoning structure is illustrated in Fig. 7.
+
+# B Prompts
+
+In this section, we provide a demonstration of the prompts used for SFT and PRM data construction. The prompt used to create ShareGPT-Step-300K is illustrated in Fig. 8. The prompt used to utilize GPT-4o to label process-level annotation is depicted in Fig. 9.
+
+# C Training Recipes
+
+In this section, we provide the specific hyper-parameter settings for the three stages in Table 7. All SFT and iterative DPO experiments are conducted on  $8 \times$  NVIDIA-A800 GPUs by default. For PRM, it is trained on  $16 \times$  NVIDIA-A800 GPUs. Additionally, we provide important parameters used in data construction. During the generation of positive and negative example pairs and the Math-Shepherd [52] Sampling period, we set the temperature to 1.0, n return sequences to 16, and top p to 0.95.
+
+Table 6: Special tokens we utilized and their definitions.  
+
+<table><tr><td>Special token</td><td>Definition</td></tr><tr><td>&lt;|reasoning_start|&gt;</td><td>Start token of the reasoning process</td></tr><tr><td>&lt;|reasoning_end|&gt;</td><td>End token of the reasoning process</td></tr><tr><td>&lt;|reasoning_step_start|&gt;</td><td>Start token of a single reasoning step</td></tr><tr><td>&lt;|reasoning_step_end|&gt;</td><td>End token of a single reasoning step</td></tr><tr><td>&lt;|reasoning_step_name_start|&gt;</td><td>Start token of the reasoning step name</td></tr><tr><td>&lt;|reasoning_step_name_end|&gt;</td><td>End token of the reasoning step name</td></tr><tr><td>&lt;|reasoning_step_thought_start|&gt;</td><td>Start token of the reasoning thought content</td></tr><tr><td>&lt;|reasoning_step_thought_end|&gt;</td><td>End token of the reasoning thought content</td></tr><tr><td>&lt;|reasoning_step_reflection_start|&gt;</td><td>Start token of the reasoning reflection content</td></tr><tr><td>&lt;|reasoning_step_reflection_end|&gt;</td><td>End token of the reasoning reflection content</td></tr><tr><td>&lt;|reasoning_proceed|&gt;</td><td>Delimiter token between reasoning steps</td></tr></table>
+
+Table 7: Hyperparameter setting and training recipes.  
+
+<table><tr><td>Hyper-parameter</td><td>SFT</td><td>PRM Training</td><td>DPO</td></tr><tr><td>Learning Rate</td><td>5e-6</td><td>1e-6</td><td>1e-6</td></tr><tr><td>Epoch</td><td>1</td><td>2</td><td>3 (iteration)</td></tr><tr><td>Warm-up Ratio</td><td>0.03</td><td>0.03</td><td>0.03</td></tr><tr><td>Weight Decay</td><td>0.05</td><td>0.05</td><td>0.05</td></tr><tr><td>Batch Size</td><td>128</td><td>128</td><td>256</td></tr><tr><td>Drop-path Rate</td><td>0.1</td><td>0.4</td><td>0.1</td></tr><tr><td>Update Parts</td><td>Projector, LLM</td><td>Projector, LLM</td><td>ViT, Projector, LLM</td></tr><tr><td>Data Size</td><td>300K</td><td>200K</td><td>20K</td></tr></table>
+
+# D Dataset Sources
+
+To ensure diversity, our data is collected from various multimodal datasets, encompassing 17 tasks that require diverse multimodal reasoning capabilities. As shown in Table 8, these tasks can be categorized into four major types including mathematical reasoning [51, 64, 3, 26, 24], scientific reasoning [18, 31, 23, 5], chart&document analysis [35, 32, 37, 36], and world knowledge [41, 54, 22, 44].
+
+Table 8: SFT dataset sources. To ensure diversity, we collect our dataset from a wide range of tasks that demands diverse reasoning capabilities.  
+
+<table><tr><td>Task Type</td><td>Dataset</td></tr><tr><td>Mathematics</td><td>GeoQA+ [3], MathVision [51], SceMQA [24], MathVerse [64], CLEVER-Math [26]</td></tr><tr><td>Chart</td><td>ChartQA [35], TabMWP [32]</td></tr><tr><td>Science</td><td>ArxivQA [23], M3CoT [5], ScienceQA [31], AI2D [18]</td></tr><tr><td>Document</td><td>DocVQA [37], InfoVQA [36]</td></tr><tr><td>Knowledge</td><td>A-OKVQA [41], FVQA [54], TextVQA [44], SEED [22]</td></tr></table>
+
+# E Attempt for Step-wise DPO
+
+Due to our structured and fine-grained design, we can improve the reasoning ability of the model by conducting fine-grained step level beam search during reasoning. Naturally, we wonder whether we could construct similar fine-grained preference data during RL training. Specifically, we use PRM to find the optimal step and a bad one at each step, and then build a set of pair data. Then we sample the next step based on the optimal step, and cycle until we get the final answer. In this way, we build a pair of preference data at each step, which can provide step-wise fine-grained preference rewards. We term this method to be step-wise DPO. This is different from step-dpo [20], which only detect the first error step while we build pairs of every single step.
+
+However, this method does not work as expected, but has a significant negative impact on model performances as shown in Table 9. We believe that the failure of this method lies in that the next sampling based on a good step is probably also a good step. Therefore, the distinction between the constructed chosen and reject pairs is relatively small. As a consequence, in order to avoid outputting negative steps during training, the model will also tend not to output chosen steps. This conjecture is validated in the rewards: the rewards of chosen and reject are both very small negative numbers. Therefore, we mainly conduct fine-grained step-by-step sampling in the inference time.
+
+# F Discussion of Table 3
+
+We observed a counterintuitive phenomenon that the larger InternVL-2.5-38B does not achieve a higher Answer Accuracy compared to its smaller counterparts. Our speculation on this phenomenon is as follows:
+
+Table 9: Ablation on per-step-wise DPO. The chosen and rejected steps are too similar to form effective preference pair, so the model will refuse to output both, backfiring performances.  
+
+<table><tr><td>Method</td><td>M3CoT</td><td>MathVista</td><td>Chosen Rewards</td><td>Reject Rewards</td></tr><tr><td>SFT</td><td>67.4</td><td>51.4</td><td>-</td><td>-</td></tr><tr><td>Outcome</td><td>70.0</td><td>53.5</td><td>-0.07</td><td>-0.16</td></tr><tr><td>Per-step-wise</td><td>64.8</td><td>50.7</td><td>-1.39</td><td>-3.86</td></tr></table>
+
+Notably, evaluating intermediate reasoning steps is substantially more challenging than evaluating the final answer for three reasons: i) Higher semantic complexity: Step-level evaluation requires a deeper and more holistic understanding of the question. The model must assess whether each step contributes constructively toward the final answer. Similarly, annotating process data is also more cognitively demanding for human annotators. ii) More supervision signals. As shown in Equation 1, PRM is trained to minimize the sum of cross-entropy losses over all intermediate steps and the final answer. A complete reasoning trace contains multiple steps but only a single answer, thus accurately assessing all the intermediate steps is much harder than only assessing the answer. iii) Answer memorization. As highlighted in recent studies [13, 46], LLMs often exhibit "answer hacking"—producing correct answers via memorized question or patterns rather than genuine understanding. In contrast, intermediate steps exhibit higher variance across instances and are less amenable to memorization.
+
+As we know, neural networks are inherently short-cut learners[11]. During training, smaller models (i.e., 8B PRM) tend to prioritize optimizing for final answer accuracy, as it offers a more accessible gradient path to loss reduction. Improving step-level predictions poses a greater optimization challenge. The 38B PRM, with its higher capacity, is better able to model and optimize intermediate reasoning quality. Consequently, it achieves greater improvements in step-level accuracy, which more effectively reduces the total loss—resulting in the pattern observed in Table 3.
+
+# G Limitations
+
+We generated process-labeled data through Monte Carlo estimation and LLM-as-Judge and verified the effectiveness of our method through extensive experiments. But it cannot be denied that the process data obtained by these two methods still cannot be guaranteed to be  $100\%$  correct. Even the process data labeled by humans cannot guarantee accuracy and quality [25], and we do not use human annotation due to the high cost and inability to scale. In the future, if there is a way to produce higher-quality process labeled data, our method can hopefully shine further.
+
+# H Broader Impacts
+
+This paper presents work whose goal is to advance the field of Vision-language Models and Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
+
+You are a thought assistant, i will provide you with an image, a question, and a reference answer. Your responsibility is to develop a sequence of intermediate thinking steps that would logically connect the question to its answer.
+
+Each step must consist of and only consist of the following three parts:
+
+1. Step Name: A brief title summarizing the main idea of the step.  
+2. Detailed Thinking: An in-depth description of the thought process in this step, explaining what is being observed, inferred, or analyzed. Note that the process of thinking and reasoning should be presented in as much detail as possible.  
+3. Reflection: A reflection on this step's connection to prior steps, pointing out any potential errors, and if possible, citing evidence in the image that supports or informs this step.
+
+There are some guidelines you should follow:
+
+1. Make sure the steps are clear, sequential, and detailed enough to make each step flows logically into the next.  
+2. The number of steps must be no less than 5 and no more than 10.  
+3. Do not state an answer at the beginning. Explain the visual clues that help to derive the answer.  
+4. Don't state that the reference answer is correct or consistent to your finding. You are writing your own solution.  
+5. The explanation can help you derive the thinking process.  
+6. For math problems, you must give specific operation steps and concrete intermediate solution leading to the final answer.  
+7. Make sure to incorporate concrete observations from the provided materials.
+
+The output format must be strictly organized as follows:
+
+Step1:xxx>
+
+<Detailed Thinking>: xxx
+
+<Reflection>: xxx
+
+Step2:xxx>
+
+<T Detailed Thinking>: xxx
+
+<Reflection>: xxx
+
+…
+
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```c
+```
+``
+
+<Detailed Thinking>: xxx
+
+<Reflection>: xxx
+
+```bash
+##<Answer>
+
+XXX
+
+Figure 8: Prompt used for generating step-by-step reasoning data.
+
+You are a judging assistant specializing in evaluating process reward, the following is an image, a question, a reference answer and a step-by-step intermediate answer process for the question. The reference answer is correct while the intermediate answer steps are not necessarily correct.
+
+Your task is to review and critique each step of the intermediate answer process based on its correctness. Assign one of the following ratings to each step:
+
+- Good: The step is logical, directly relevant to the question, and most importantly, correct. This step provides substantive progress to solve the question.  
+- Neutral: The step is somewhat relevant but contains minor flaws or gaps in reasoning. Or this step only includes some abstract practices rather than concrete solutions, which is not helpful for solving the question in effect.  
+- Bad: The step is flawed, wrong, or shows a misunderstanding of the question or context. This step has a negative effect on solving the question.
+
+For each step, provide a brief explanation of your evaluation, identifying areas for improvement if Neutral or Bad. Finally, return your output in the form of a step label list. 1 stands for Good; 0 stands for Neutral; -1 stands for Bad.
+
+The final step label list format must be strictly organized as follows:  
+```python
+>>> Step Label: [1,1,0,-1,-1,-1]
+
+Figure 9: Prompt used for generating process-level annotation.
\ No newline at end of file
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+# Unveiling Concept Attribution in Diffusion Models
+
+Quang H. Nguyen<sup>1</sup>, Hoang Phan<sup>1,2</sup>, Khoa D. Doan<sup>1,2</sup>
+
+<sup>1</sup>College of Engineering and Computer Science, VinUniversity
+
+$^{2}$ VinUni-Illinois Smart Health Center, VinUniversity
+
+{quang.nh, 21hoang.p, khoa.dd}@vinuni.edu.vn
+
+# Abstract
+
+Diffusion models have shown remarkable abilities in generating realistic and high-quality images from text prompts. However, a trained model remains largely black-box; little do we know about the roles of its components in exhibiting a concept such as objects or styles. Recent works employ causal tracing to localize knowledge-storing layers in generative models without showing how other layers contribute to the target concept. In this work, we approach diffusion models' interpretability problem from a more general perspective and pose a question: "How do model components work jointly to demonstrate knowledge?". To answer this question, we decompose diffusion models using component attribution, systematically unveiling the importance of each component (specifically the model parameter) in generating a concept. The proposed framework, called Component Attribution for Diffusion Model (CAD), discovers the localization of concept-inducing (positive) components, while interestingly uncovers another type of components that contribute negatively to generating a concept, which is missing in the previous knowledge localization work. Based on this holistic understanding of diffusion models, we present and empirically evaluate one utility of component attribution in controlling the generation process. Specifically, we introduce two fast, inference-time model editing algorithms, CAD-Erase and CAD-Amplify; in particular, CAD-Erase enables erasure and CAD-Amplify allows amplification of a generated concept by ablating the positive and negative components, respectively, while retaining knowledge of other concepts. Extensive experimental results validate the significance of both positive and negative components pinpointed by our framework, demonstrating the potential of providing a complete view of interpreting generative models. Our code is available here.
+
+# 1 Introduction
+
+Recent developments in diffusion models [18, 26, 36, 37] have greatly improved the synthesizing capabilities, in terms of image quality and the diversity of generated knowledge. However, these models lack interpretability; we do not fully understand how they can translate simple prompts to coherent visual outputs. To investigate how generative models express learned concepts, a recent line of work studies which components in the model store knowledge [2, 27]. In language models, Meng et al. [27] propose causal tracing to locate layers storing facts and reveal that knowledge is localized in middle-layer MLP modules. Basu et al. [2] transfer this approach to diffusion models and propose the knowledge distributed hypothesis. They show that, unlike language models, knowledge is distributed among a set of UNet components and the first self-attention layer of the text encoder. These works shed light on interpreting generative models, enabling more effective model editing [3, 2]. Nevertheless, they focus only on knowledge storage—modules that are responsible for generating concepts—and on coarse-grained components, such as layers. This perspective may overlook more subtle properties and other types of modules that also influence the outputs.
+
+To address that limitation, this paper first poses a more general question: How do components in diffusion models contribute to a generated concept? We introduce a framework that predicts the model's behavior given the presence of each component by identifying its contribution through an efficient linear counterfactual estimator [35]. Through this framework, called Component Attribution for Diffusion Model (CAD), we advance the understanding of how model components activate concepts (e.g., objects, styles, or explicit contents) in diffusion models. In contrast to the prior work that focuses on the model's layers, CAD allows analysis of more fine-grained components. Specifi
+
+cally, we revisit the distributed hypothesis [2] and leverage CAD to identify concept-inducing (or positive) components that are similar to knowledge storage. However, by focusing on the most fine-grained components, i.e., the model's parameters, we propose the localization hypothesis: knowledge is localized in a small number of parameters. Surprisingly, besides the positive components, CAD also reveals the existence of components that contribute negatively to generating the target concept, which is missing in the previous studies. Ablating positive or negative components decreases or increases the probability of generating the corresponding knowledge. As one example of its utility, this holistic understanding of diffusion models enables a lightweight model editing capability, i.e., to remove (positive) or recall (negative) a concept. Figure 1 illustrates the proposed CAD framework. In summary, our contributions are:
+
+- We propose CAD, a comprehensive framework, that can compute the attribution scores of the diffusion model components based on an efficient and effective linear counterfactual predictor.  
+- Utilizing CAD, we confirm the existence of the concept-inducing (positive) model's parameters, while revealing their localized nature. CAD is also the first work that uncovers the existence of another type of components – the concept-amplification (negative) components.  
+- Leveraging these observations of localized positive and negative components, we develop two lightweight knowledge editing algorithms for diffusion models, CAD-Erase for concept erasing and CAD-Amplify for concept amplification, respectively.  
+- We analyze CAD and evaluate the effectiveness of the proposed editing algorithms with extensive experiments, demonstrating their practicality and effectiveness.
+
+# 2 Related Works
+
+Interpreting Neural Networks. Several research has extensively studied the black-box mechanism of neural networks to explain their behaviors. A line of works [34, 4, 42] visualize important input regions of classification models by using the gradient of feature map activations. Sundararajan et al. [38] formalize the problem of attributing the input and propose two axioms to design attribution methods. Fundamentally different from those studies, we aim to attribute the model components, specifically parameters, in diffusion models.
+
+Knowledge Localization. Previous work explored how language model components store factual knowledge [16, 8] or used model attribution to analyze the impact of individual components in the image classification and language prediction task[35]. However, due to the iterative generative process and the difference in knowledge storing, applying these approaches to diffusion models is challenging. Another line of research [2, 3, 17, 27, 39, 7, 49] utilizes causal analysis to identify critical layers for knowledge in language models and T2I Latent Diffusion variants. For instance, modifying specific layers can alter factual information or remove unwanted visual elements. While these methods have shown successes in localizing knowledge, Hase et al. [17] discover that editing non-causal layers can also modify stored facts in language models. This finding implies that causal analysis may answer a different question from model editing. Furthermore, these approaches inspect the activations, which are dependent on the input, whereas our work studies the parameters of the
+
+![](images/2b105ea7de7e1053e8e49b128f021f5be99d92a5d84a99aa9252ff535ee1a274.jpg)  
+Figure 1: Overview of our framework. We show that there exist positive and negative components in diffusion that increase or decrease the probability of the target concept, respectively. Removing those components will have the reverse effect.
+
+model. Dravid et al. [9] examine the weight space of several customized diffusion models; in contrast, our work offers an efficient approach to studying individual model component roles.
+
+Concept Erasure. Latent diffusion models (LDMs) can generate undesirable content (e.g., nudity, outdated information, copyrighted artistic styles) due to their large and uncontrolled training datasets. Early efforts address this problem involved fine-tuning Cross-Attention layers [11, 20, 21, 50, 29] or editing the text-encoder [1, 2]. In addition, several research [12, 25, 44] highlight the necessity to remove multiple concepts simultaneously in real-world scenarios. More recent works aim to improve robustness of erasing methods to red-teaming attacks, including ConceptPrune [5], RECE [13], RACE [19], and pruning methods [45]. These methods enable efficient erasure of various contents while ensuring minimal interference with the unedited ones.
+
+Concept Amplification. Motivated by Dreambooth [33], Cones [24] inserts new objects into the model by identifying concept neurons. In contrast, CAD-Amplify locates components to magnify existing knowledge in diffusion models. Dai et al. [8] also proposes a method to amplify facts, but relies on amplifying positive neurons. Our work is the first study showing the existence of negative components and how to systematically locate them.
+
+Red-Teaming Attacks. Although fine-tuning eliminates undesirable concepts in text-to-image models, recent studies [47, 6, 51, 46, 48, 41, 30] show that this approach remains unreliable against adversarial prompt attacks. These safety mechanisms can be bypassed by both black-box (e.g., SneakyPrompt [47], Ring-A-bell [41]) and white-box attacks (e.g., P4D [6], UnlearnDiff [51]), leading to the regeneration of sensitive content. These attacks highlight the need for robust defenses that fully remove concepts while preserving image quality. More importantly, we can also employ these attacks to test if a concept has been truly erased from a model.
+
+Pruning Approaches. Similar to our algorithms, many studies [14, 10] have investigated pruning neural networks, primarily for time and memory efficiency. Specifically, [28, 22, 40] use gradient information to identify and remove less important parameters, thereby improving inference speed. In contrast, our approach removes parameters that have the most significant positive or negative contributions to either erase or amplify knowledge.
+
+# 3 Preliminaries
+
+Diffusion Models. Diffusion models [18, 26, 36, 37] are generative models that perform a denoising process, starting from random Gaussian noise, over several time steps  $T$ . Particularly, the forward Markov process first transforms a real image  $x_0$  into a noisy image  $x_{t} = \sqrt{a_{t}} x_{0} + \sqrt{1 - a_{t}}\epsilon$  at time step  $t$ , where  $a_{t}$  is a decaying parameter and  $\epsilon \sim \mathcal{N}(0,I)$ . Then in the reverse process, a denoiser is trained to predict the noise  $\epsilon_t$  at each time step  $t$ , thereby generating a noisy image  $x_{t}$ . After a series of discrete time steps, the diffusion model generates the final reconstructed image  $x_0$ .
+
+Latent Diffusion Models. LDMs [32] help accelerate the denoising process by employing a pretrained variational autoencoder with an encoder  $\mathcal{E}$  and a decoder  $\mathcal{D}$  and performing the denoising process in the latent space. At each time step  $t$ , LDMs predict the noise  $\Phi_{\theta}(\cdot |c)$ , conditioned by a text prompt  $c$  and parameterized by  $\theta$ . The objective function is  $\mathcal{L} = \mathbb{E}_{z_t\sim \mathcal{E}(x),t,c,\epsilon \sim \mathcal{N}(0,I)}\| \epsilon - \Phi_{\theta}(z_t,c,t)\| _2^2$ , where  $\epsilon$  is Gaussian noise, and  $\Phi_{\theta}(z_t,c,t)$  is the estimated noise added to latent  $z_{t}$  at time step  $t$  by LDMs.
+
+# 4 Concept Attribution in Diffusion Models
+
+In this section, we provide the general formulation of concept attribution in diffusion models, discuss the challenge of solving this problem, and propose our CAD framework.
+
+# 4.1 Decomposing Knowledge in Diffusion
+
+We consider the diffusion model as a combination of building blocks  $w_{i}$ . Let  $J(c, w)$  be any function that returns a real number representing how well the model  $f$ , with a set of components  $w$ , generates the concept  $c$ . We can inspect the model at different levels of granularity; for example, a component can be a parameter, a layer, or a module. Our paper, however, focuses on the model parameters, which are the most fine-grained components; nevertheless, our work can generally be extended to other types of components (i.e., layers or modules).
+
+Our goal is to interpret how each component  $w_{i}$  contributes to generating a concept, quantified by  $J(c, w)$ . Similar to prior research in causal mediation analysis [27, 43], our study also focuses on "intervening" by "knocking out" to measure the counterfactual effect; i.e., we examine if model parameters induce a certain ability by removing them from the model. Specifically, we estimate how  $J(c, w)$  changes if we set the value of a component  $w_{i}$  to 0. Let  $\tilde{w}$  be the new set of components obtained by adjusting some components to 0, we want to find a function  $g(\mathbf{0}_{\tilde{w}}; c) = J(c, \tilde{w})$  where  $\mathbf{0}_{\tilde{w}} \in \{0, 1\}^d$ ,  $d$  is the number of components, and
+
+$$
+\left(\mathbf {0} _ {\tilde {w}}\right) _ {i} = \left\{ \begin{array}{l} 0 \text {i f} \tilde {w} _ {i} = 0 \\ 1 \text {i f} \tilde {w} _ {i} = w _ {i}. \end{array} \right. \tag {1}
+$$
+
+Diffusion models are constructed from deep neural networks with non-linear activation between layers, and iterative processes to generate images. Consequently, the function  $g$  might be complex and difficult to learn. Interestingly, Shah et al. [35] show that a simple linear function can well approximate  $g$  in image classification models and language models. Here, we similarly approximate  $g$  with a linear model:
+
+$$
+J (c, \tilde {w}) = g (\mathbf {0} _ {\tilde {w}}; c) \approx \boldsymbol {\alpha} _ {c} ^ {T} \mathbf {0} _ {\tilde {w}} + b _ {c}, \quad \boldsymbol {\alpha} _ {c} \in \mathbb {R} ^ {d}. \tag {2}
+$$
+
+Each coefficient  $\alpha_{c,i}$  represents how the component  $w_{i}$  contributes to the concept  $c$ . Another way to intervene is to amplify the effect of  $w_{i}$  by rescaling its magnitude; however, unlike knockout, which removes the component completely, scaling may preserve interactions, and deciding the magnitude of the change can be non-trivial. Nevertheless, we investigate this type of intervention in the Appendix.
+
+# 4.2 CAD: Component Attribution for Diffusion Model
+
+The challenge of learning  $\alpha_{c}$ . One way to find  $\alpha_{c}$  is by treating Equation (2) as a machine learning model [35]. We can create a size- $N$  dataset  $\mathcal{D}_c = \{(0_{w^{(i)}}, J(c, w^{(i)})): 0_{w^{(i)}} \in \{0,1\}_{i=1}^N\}$  by randomly masking out some components of the diffusion model (i.e., to create the input  $w^{(i)}$ ). Then, we train a linear regression model and obtain  $\alpha_{c}$  as the coefficient in the model. Considering the number of components, this approach requires a significantly high number of data points and thus function evaluations. For instance, Shah et al. [35] created 100,000 data points for image classification and 200,000 for language modeling to examine a single prediction. Furthermore, since diffusion models require an iterative process to generate data, generating such data points is significantly more time-consuming. Therefore, this approach of generating data to learn  $\alpha_{c}$  for a concept is prohibitively expensive or inefficient.
+
+Our approximation method. Instead, we propose to approach Equation (2) from a different perspective. Assuming our focus is on a small subset of components  $w_{i}, i \in S$  and we want to examine how  $J(c, w)$  changes if  $w_{i} = 0$ , we can apply first-order Taylor expansion as follows
+
+$$
+\sum_ {i \in S} \alpha_ {c, i} = J (c, w) - J (c, \tilde {w}) \approx (w - \tilde {w}) \nabla_ {w} J (c, w) = \sum_ {i \in S} w _ {i} \frac {\partial J (c , w)}{\partial w _ {i}}. \tag {3}
+$$
+
+From Equations (2) and (3), we see that the coefficient  $\alpha_{c,i}$  of  $w_{i}$  can be approximated by  $w_{i} \frac{\partial J(c, w)}{\partial w_{i}}$ . For the rest of the study, we will use this formulation to attribute a component in the model. In particular, our method measures the contribution of a component  $w_{i}$  to the objective  $J$ , or the attribution score, by  $w_{i} \frac{\partial J(c, w)}{\partial w_{i}}$ , which only requires a single forward and backward pass instead of creating the training data for the model in (2) with many forward passes.
+
+# 5 Editing Diffusion Models with CAD
+
+In this section, we discuss the importance of studying parameter attributions in concept generation via two applications of CAD. Specifically, we propose and empirically evaluate two lightweight, inference-time editing algorithms that remove (CAD-Erase) or amplify (CAD-Amplify) a concept in diffusion models.
+
+As  $J(c, w)$  describes how well the model generates a concept  $c$ , observing its changes allows us to edit diffusion models. Given the attribution scores of model components computed using the proposed approach in Section 4.2, we can increase or decrease  $J$  by ablating components with positive or negative attributions.
+
+# 5.1 Localizing and Erasing Knowledge
+
+Previous works [27, 2, 3] apply causal tracing to study which layers in generative models store knowledge. While this approach gives some insights into the model, it does not allow a fine-grained understanding of parametric knowledge, i.e., more fine-grained components in the causal layers may play different roles. In contrast, CAD allows us to focus on the most fine-grained components, i.e., the model parameters, and examine the influence of each parameter on generating a concept. Formally, we define positive components for a concept  $c$  as those that when being ablated, the model has a lower probability
+
+# Algorithm 1: CAD-Erase
+
+Input: Diffusion model  $\Phi$ , target concept  $c$ , base condition  $c_{b}$ , the number of components  $k$ .
+
+Output: Diffusion model  $\Phi^{\prime}$  with a lower chance to generate concept  $c$  Generate a set of  $x$  conditioned on  $c$  Compute the scores  $w_{i}\frac{\partial J}{\partial w_{i}}$  with Eq.4.   
+Locate top-  $k$  components  $w_{i}\in S$  with the (positive) attribution.   
+Set  $w_{i}\gets 0,w_{i}\in S$
+
+Concept Erasure. We consider these positive components as knowledge storage, and finding them allows us to locate knowledge in generative models. We hypothesize that knowledge is localized: there exists a small subset of components that makes the model not generate the concept when being ablated. This hypothesis also leads to a more accurate approximation discussed in Section 4.2, due to its first-order expansion.
+
+Hypothesis 1. Knowledge is localized in a small number of components. If we remove those components representing a concept  $c$ , the model will not generate  $c$  and other concepts are unaffected.
+
+Concept Attribution Objective. Another question is which objective function  $J$  should be used. A naive choice is to directly use the training loss. However, previous work in concept erasing [21] shows that optimizing this objective to ablate concepts leads to sub-optimal performance. Instead, we rely on the following objective function (also used in [21]):
+
+$$
+J _ {c _ {b}} (c, w) = \mathbb {E} _ {x _ {t}, t, \epsilon} \| \Phi (x _ {t}, c _ {b}, t; w). \mathrm {s g} (\cdot) - \Phi (x _ {t}, c, t; w) \| ^ {2} \tag {4}
+$$
+
+where  $c$  is the target concept, e.g. the object "parachute",  $c_{b}$  is the base condition, e.g. the empty string "",  $\mathrm{sg}()$  is the gradient stopping operator. Intuitively, we force the predicted noise conditioned on the target concept to be close to the unconditioned noise, thus preventing the reverse process from approaching the conditional distribution of the concept.
+
+CAD-Erase. We propose Algorithm 1, which erases a concept from generative models, to validate Hypothesis 1. In general, we compute the
+
+# Algorithm 2: CAD-Amplify
+
+Input: Diffusion model  $\Phi$ , target concept  $c$ , the n.o. components  $k$ , images  $x$  of concept  $c$ .
+
+Output: Diffusion model  $\Phi^{\prime}$  with a higher chance to generate concept  $c$ . Compute the scores  $w_{i}\frac{\partial J}{\partial w_{i}}$  with Eq. (5). Locate top- $k$  components  $w_{i}\in S$  with the lowest (negative) attribution.
+
+Set  $w_{i}\gets \bar{0},w_{i}\in S$
+
+attribution value of components by Equation (3) and remove the top- $k$  positive components. Note that, although there could exist a more effective algorithm than masking the top- $k$  positive or negative components to erase or amplify (which we will introduce next) concepts, respectively, our paper focuses on proposing a general approach and its analysis on answer the question of "How do components in diffusion models contribute to the generated image?" For example, one can finetune these positive or negative components to achieve even better concept erasure or amplification, respectively; however, this is beyond the scope of our study and we leave it for future works.
+
+# 5.2 Amplifying Knowledge in Diffusion Models
+
+Our attribution framework offers a complete view of interpreting the model: besides positive components that are responsible for generating a concept, there also exist components with negative coefficients. We hypothesize that these components suppress knowledge, i.e., decreasing the probability of inducing a concept. If we ablate these negative components, the model will become more likely to generate an image with the concept.
+
+Hypothesis 2. Negative components exist, and abulating them will amplify knowledge.
+
+Previous works in knowledge localization [27, 2] edit the model at modules storing knowledge. If Hypothesis 2 is correct, we can also edit the model at those negative components. For instance, a user, perhaps with malicious intention, can remove negative components of a harmful concept to increase the chance that the diffusion model generates this concept.
+
+CAD-Amplify. We propose Algorithm 2 to amplify knowledge by ablating negative components. This approach assumes access to some images of the target concept and uses the training loss of diffusion models as the objective  $J$ :
+
+$$
+J (c, w) = - \mathbb {E} _ {x _ {t}, t, \epsilon} [ \| \epsilon - \Phi (x _ {t}, c, t; w) \| _ {2} ^ {2} ]. \tag {5}
+$$
+
+![](images/5d4c057d351ebd3be2f518950b88bcbf8dde9247d6b42fb0ad57731a7cb04c7b.jpg)  
+Figure 2: The attribution scores predicted by CAD and the actual values of the objective.
+
+# 6 Experiments
+
+In this section, we aim to verify and provide a comprehensive empirical analysis of the knowledge localization hypothesis in Section 6.2 and the existence of negative components in Section 6.3. We provide additional results on other diffusion models, different intervention, and abating on different modules in the Appendix.
+
+# 6.1 CAD Well Approximates the Change in the Objective
+
+Table 1: The accuracy of generated images on target classes and other classes, predicted by the pre-trained ResNet50 model.  
+
+<table><tr><td rowspan="2">Classes</td><td colspan="6">Accuracy on target classes ↓</td><td colspan="6">Accuracy on other classes ↑</td></tr><tr><td>SD-1.4</td><td>ConceptPrune</td><td>ESD</td><td>RECE</td><td>UCE</td><td>CAD-Erase</td><td>SD-1.4</td><td>ConceptPrune</td><td>ESD</td><td>RECE</td><td>UCE</td><td>CAD-Erase</td></tr><tr><td>Cassette player</td><td>7.20</td><td>2.60</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.40</td><td>86.07</td><td>76.73</td><td>57.53</td><td>89.13</td><td>89.13</td><td>80.13</td></tr><tr><td>Chain saw</td><td>69.00</td><td>1.00</td><td>0.40</td><td>0.00</td><td>0.00</td><td>0.00</td><td>79.20</td><td>63.97</td><td>29.24</td><td>75.69</td><td>75.69</td><td>69.22</td></tr><tr><td>Church</td><td>76.20</td><td>21.00</td><td>3.60</td><td>1.20</td><td>15.20</td><td>1.60</td><td>78.40</td><td>65.00</td><td>65.24</td><td>80.50</td><td>80.20</td><td>73.49</td></tr><tr><td>English Springer</td><td>93.80</td><td>1.00</td><td>0.20</td><td>0.00</td><td>0.10</td><td>1.40</td><td>76.44</td><td>62.00</td><td>47.48</td><td>77.80</td><td>78.00</td><td>71.91</td></tr><tr><td>French horn</td><td>98.60</td><td>7.40</td><td>0.20</td><td>0.00</td><td>0.00</td><td>4.40</td><td>75.91</td><td>63.17</td><td>45.11</td><td>74.33</td><td>74.33</td><td>70.87</td></tr><tr><td>Garbage truck</td><td>85.60</td><td>1.40</td><td>0.00</td><td>0.00</td><td>15.60</td><td>3.80</td><td>77.36</td><td>65.62</td><td>47.36</td><td>65.40</td><td>77.51</td><td>63.69</td></tr><tr><td>Gas pump</td><td>79.00</td><td>36.80</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.20</td><td>78.09</td><td>68.28</td><td>48.58</td><td>79.02</td><td>79.02</td><td>67.69</td></tr><tr><td>Golf ball</td><td>95.80</td><td>28.60</td><td>0.20</td><td>0.00</td><td>0.60</td><td>4.20</td><td>76.22</td><td>65.55</td><td>48.90</td><td>79.00</td><td>78.78</td><td>73.27</td></tr><tr><td>Parachute</td><td>96.20</td><td>30.00</td><td>0.80</td><td>0.00</td><td>1.00</td><td>2.00</td><td>76.18</td><td>62.17</td><td>61.28</td><td>78.20</td><td>77.87</td><td>68.91</td></tr><tr><td>Tench</td><td>80.40</td><td>2.80</td><td>1.40</td><td>0.00</td><td>0.00</td><td>0.20</td><td>77.93</td><td>67.57</td><td>60.80</td><td>78.56</td><td>78.56</td><td>72.67</td></tr></table>
+
+In diffusion models, as mentioned in Section 4, attributing the components is time-consuming and more complicated due to their iterative generation process. Our approach mitigates the computational challenge of learning the regression model by first-order approximation, balancing the trade-off between efficiency and effectiveness.
+
+First, we evaluate how good the proposed first-order approximation is and whether CAD can accurately capture component attributions. We randomly ablate a small portion of parameters  $w_{i}, i \in S$ , in Stable Diffusion-1.4 and obtain the corresponding change in the objective. We also use CAD to compute the predicted change, indicated by  $\sum_{i \in S} w_{i} \frac{\partial J}{\partial w_{i}}$ . We repeat this process 1000 times and evaluate CAD. Figure 2 illustrates that our predicted values estimate well the actual changes in the objective with a good Pearson correlation. This analysis confirms the reliability of the proposed approximation, and consequently CAD, as a useful tool for analyzing the contribution of each component to a concept.
+
+# 6.2 CAD Can Locate Positive Components and Erase Knowledge
+
+The analysis in the previous section shows that CAD can successfully identify positive and negative components. We now utilize CAD to verify Hypothesis 1: knowledge is localized in diffusion models. We conduct experiments on Stable Diffusion-1.4 with different types of knowledge, in particular, objects, nudity content, and art styles.
+
+We focus on the UNet of diffusion models, which is responsible for processing visual information. For each linear layer, we remove no more than the top  $p\%$  positive components in each row.
+
+![](images/02bee940223bdec9bed6784c32d32e924820e56ac80f6436f61a3d4ec9ad0eb3.jpg)  
+Figure 3: The qualitative results of CAD. Removing positive components to "English Springer" avoids generating that concept. Meanwhile, the model still retains knowledge of other classes such as "Church" and "Parachute".
+
+![](images/4ed31319d1eb5aa4656c74db027a60b0347aa3f9e8eca44065706f768c1649ec.jpg)  
+Figure 4: The first two rows contain images generated by the original model and erasing methods on I2P prompts. We add  $\star$  for publication. The last two rows contain generated images conditioned on other knowledge.
+
+Erasing objects. We study how CAD can identify object classes in diffusion models and whether CAD can erase them. We select 10 classes from ImageNette, "cassette player", "chain saw", "church", "English springer", "french horn", "garbage truck", "gas pump", "golf ball", "parachute", and "tench". For each class, we compute component attributions and ablate  $0.1\%$  components using Algorithm 1. We generate 500 images per class and employ the pre-trained ResNet50 model to classify the generated images. We compare CAD with other state-of-the-art erasing methods, in particular ConceptPrune [5], ESD [11], UCE [12], and RECE [13]. Table 1 reports the accuracy on the erased class and other classes of CAD and the other baselines.
+
+Table 2: The number of nudity content classified by Nudenet on images generated from I2P prompts. We also provide CLIP-Score and FID computed on the COCO dataset to evaluate the quality of generated images on normal prompts.  
+
+<table><tr><td>Model</td><td>Armpits</td><td>Belly</td><td>Buttocks</td><td>Feet</td><td>Breast (F)</td><td>Genitalia (F)</td><td>Breast (M)</td><td>Genitalia (M)</td><td>Anus</td><td>Total↓</td><td>CLIP-Score↑</td><td>FID ↓</td></tr><tr><td>SD-1.4</td><td>169</td><td>197</td><td>26</td><td>28</td><td>271</td><td>29</td><td>60</td><td>18</td><td>0</td><td>798</td><td>31.32</td><td>14.127</td></tr><tr><td>ConceptPrune</td><td>21</td><td>5</td><td>3</td><td>13</td><td>11</td><td>1</td><td>0</td><td>8</td><td>0</td><td>62</td><td>31.16</td><td>15.260</td></tr><tr><td>ESD</td><td>17</td><td>15</td><td>6</td><td>4</td><td>22</td><td>12</td><td>1</td><td>11</td><td>0</td><td>88</td><td>30.27</td><td>14.495</td></tr><tr><td>RECE</td><td>19</td><td>27</td><td>4</td><td>5</td><td>17</td><td>4</td><td>13</td><td>9</td><td>0</td><td>98</td><td>30.94</td><td>14.633</td></tr><tr><td>UCE</td><td>60</td><td>65</td><td>7</td><td>5</td><td>60</td><td>7</td><td>14</td><td>11</td><td>0</td><td>229</td><td>31.25</td><td>14.561</td></tr><tr><td>CAD-Erase</td><td>6</td><td>3</td><td>3</td><td>6</td><td>6</td><td>6</td><td>0</td><td>13</td><td>0</td><td>43</td><td>31.30</td><td>12.440</td></tr><tr><td>CAD-Amplify</td><td>229</td><td>242</td><td>31</td><td>34</td><td>360</td><td>33</td><td>44</td><td>18</td><td>0</td><td>991</td><td>-</td><td>-</td></tr></table>
+
+First, we evaluate the capability of the base diffusion model to generate images conditioned on text prompts. Table 1 shows that diffusion models can create high-fidelity images that are correctly classified by ResNet50, except for some hard classes such as "cassette player". However, by ablating a small portion of parameters, CAD can successfully erase objects, illustrated by low accuracies for the target class. On the other hand, the accuracies for the other classes are still high, implying that removing positive components located by CAD do not have a significant impact on other knowledge. We also provide qualitative results in Figure 3, demonstrating that CAD erases the target concept without affecting the other concepts. This observation verifies the knowledge localization hypothesis 1.
+
+Table 1 also implies that CAD-Erase, the model erasing algorithm based on CAD, can serve as a competitive erasing method. Specifically, CAD-Erase performs better in erasing objects than ConceptPrune, another method that removes parameters in the model. ESD yields similar accuracies on the target classes to CAD-Erase; however, this method sacrifices knowledge of the other concepts, leading to low accuracies on the other classes. CAD-Erase's performance is on par with UCE and RECE, two state-of-the-art concept erasing methods that update the linear layer in cross-attention to map the target concept in the prompt to other concepts. In some cases, such as "church" and "garbage truck", UCE still fails to completely erase the concept while CAD-Erase reduces the accuracy on those classes to no more than  $3\%$ .
+
+Table 3: The attack success rate of white-box attacks on the erased models. Lower is better.  
+
+<table><tr><td rowspan="3">Model</td><td rowspan="2" colspan="2">Nudity</td><td colspan="6">Object</td></tr><tr><td colspan="2">Church</td><td colspan="2">Parachute</td><td colspan="2">Tench</td></tr><tr><td>P4D</td><td>UD</td><td>P4D</td><td>UD</td><td>P4D</td><td>UD</td><td>P4D</td><td>UD</td></tr><tr><td>ConceptPrune</td><td>0.76</td><td>0.78</td><td>0.84</td><td>0.76</td><td>0.92</td><td>0.92</td><td>0.39</td><td>0.34</td></tr><tr><td>ESD</td><td>0.69</td><td>0.76</td><td>0.56</td><td>0.60</td><td>0.48</td><td>0.54</td><td>0.28</td><td>0.36</td></tr><tr><td>RECE</td><td>0.63</td><td>0.68</td><td>0.42</td><td>0.54</td><td>0.28</td><td>0.30</td><td>0.10</td><td>0.10</td></tr><tr><td>UCE</td><td>0.83</td><td>0.84</td><td>0.50</td><td>0.60</td><td>0.42</td><td>0.48</td><td>0.10</td><td>0.20</td></tr><tr><td>CAD-Erase</td><td>0.69</td><td>0.68</td><td>0.40</td><td>0.48</td><td>0.46</td><td>0.56</td><td>0.18</td><td>0.22</td></tr></table>
+
+Table 4: The number of nudity content and the drop from the original model classified by Nudenet on images generated from adversarial prompts. Lower is better.  
+
+<table><tr><td>Model</td><td colspan="2">MMA</td><td colspan="2">Ring-a-bell</td></tr><tr><td>SD-1.4</td><td>1941</td><td>(-00.00%)</td><td>414</td><td>(-00.00%)</td></tr><tr><td>ConceptPrune</td><td>98</td><td>(-94.95%)</td><td>83</td><td>(-79.95%)</td></tr><tr><td>ESD</td><td>279</td><td>(-85.62%)</td><td>95</td><td>(-77.05%)</td></tr><tr><td>RECE</td><td>481</td><td>(-75.22%)</td><td>4</td><td>(-99.03%)</td></tr><tr><td>UCE</td><td>971</td><td>(-49.97%)</td><td>64</td><td>(-84.54%)</td></tr><tr><td>CAD-Erase</td><td>62</td><td>(-96.81%)</td><td>5</td><td>(-98.79%)</td></tr></table>
+
+Erasing nudity. Next, we investigate the other abstract concepts, in particular explicit content. We locate and ablate the top  $0.075\%$  positive components with the prompt "naked". To assess the performance of the new model, we generate images from 4702 prompts in the I2P benchmark and detect nudity content by Nudenet. We validate the performance on unrelated knowledge by generating images with 30,000 prompts in the COCO dataset [23]. Table 2 shows the results of CAD and the other baselines. As can be observed, CAD-Erase achieves the highest performance in erasing nudity content compared to other state-of-the-art methods, illustrated by the lowest number of nudity classes predicted by Nudenet. Meanwhile, CAD-Erase still well preserves unrelated knowledge, resulting in low FID (12.440) and a high CLIPScore (31.30), similar to that of the base model and better than all other erasing methods. Figure 4 illustrates images generated by the original model and the ablated model from our method. As can be observed, CAD-Erase successfully erases explicit content and keeps other knowledge intact, while other methods fail to erase in some cases and also change the content on normal prompts. These results confirm knowledge localization of nudity content.
+
+Erasing with adversarial prompts. Recent works [47, 41, 46] show that current erasing methods do not completely remove knowledge from the model, and propose attack methods that create adversarial prompts to induce the erased model to still generate harmful content. We evaluate our method on two unsafe prompt sets, MMA and Ring-A-Bell, in Table 4. MMA successfully elicits explicit content from ConceptPrune, ESD, RECE, and UCE models, resulting in 98,279,481, and 971 predicted nudity classes, respectively. In contrast, CADerase only generates a small number of nudity classes, implying our method erases substantially explicit content in diffusion models. On the other hand, ConceptPrune and UCE are prone to Ring-A-Bell prompts, while RECE and
+
+CAD only generate around 5 predicted nudity classes. We also evaluate the model with white-box attacks such as P4D [6] and UnlearnDiff (UD) [15]. Table 3 reports the attack success rate of white-box attacks in making the erased model generate the target concept. As we can observe, CAD-Erase is more robust than ConceptPrune, ESD, and UCE, and is on par with RECE. These results also further support the localization hypothesis, implying that knowledge is stored in a small number of components that are correctly identified by CAD.
+
+Erasing art styles. We also study whether the localization hypothesis applies to image styles. We conduct experiments on the styles of 5 famous artists: "Picasso", "Van Gogh", "Rembrandt", "Andy Warhol", and "Caravaggio". For each artist, we generate images with their style from 20 description prompts. We report the LPIPS score of images generated by SD-1.4 and the model created by CAD and other erasing methods in Table 5. Figure 5 illustrates qualitative results of CAD on the target artist and other artists. Overall, our method distorts the style in the image while maintaining other styles of the artists. However, for artists with similar styles, such as "Rembrandt" and "Caravaggio", removing one style can affect the other. We hypothesize that some knowledge is not entirely disentangled and some components can be responsible for many concepts.
+
+![](images/a9c3f44e95f637d5e097e7d8254b89b27bd37ca950e2b9fcaec79f1ce8b7ef30.jpg)  
+Figure 5: Qualitative results of CAD on erasing artist styles. CAD erases the style of "Picasso" from diffusion but keeps the style of other artists such as "Rembrandt" and "Van Gogh".
+
+Table 5: LPIPS scores of erasing methods on different artist styles. Lower scores indicate more similarity.  
+
+<table><tr><td rowspan="2">Artist</td><td colspan="4">LPIPS on the target artist↑</td><td colspan="4">LPIPS on other artists↓</td></tr><tr><td>ESD</td><td>RECE</td><td>UCE</td><td>CAD</td><td>ESD</td><td>RECE</td><td>UCE</td><td>CAD</td></tr><tr><td>Picasso</td><td>0.332</td><td>0.143</td><td>0.108</td><td>0.258</td><td>0.279</td><td>0.077</td><td>0.056</td><td>0.127</td></tr><tr><td>Van Gogh</td><td>0.412</td><td>0.253</td><td>0.202</td><td>0.198</td><td>0.303</td><td>0.104</td><td>0.075</td><td>0.089</td></tr><tr><td>Rembrandt</td><td>0.417</td><td>0.275</td><td>0.210</td><td>0.320</td><td>0.331</td><td>0.110</td><td>0.084</td><td>0.152</td></tr><tr><td>Andy Warhol</td><td>0.449</td><td>0.321</td><td>0.294</td><td>0.208</td><td>0.276</td><td>0.109</td><td>0.085</td><td>0.056</td></tr><tr><td>Caravaggio</td><td>0.394</td><td>0.210</td><td>0.178</td><td>0.243</td><td>0.326</td><td>0.093</td><td>0.073</td><td>0.138</td></tr></table>
+
+Table 6: Ablating negative components identified by CAD significantly increases the probability of generating the target class.  
+
+<table><tr><td rowspan="2">Classes</td><td colspan="2">Target class</td><td colspan="2">Other classes</td></tr><tr><td>SD-1.4</td><td>CAD</td><td>SD-1.4</td><td>CAD</td></tr><tr><td>Cassette player</td><td>7.20</td><td>27.60</td><td>86.07</td><td>82.42</td></tr><tr><td>Chain saw</td><td>69.00</td><td>98.20</td><td>79.20</td><td>76.29</td></tr><tr><td>Church</td><td>76.20</td><td>93.80</td><td>78.40</td><td>74.38</td></tr><tr><td>Gas pump</td><td>79.00</td><td>94.60</td><td>78.09</td><td>77.33</td></tr><tr><td>Tench</td><td>80.40</td><td>93.40</td><td>77.93</td><td>77.56</td></tr></table>
+
+# 6.3 Ablating Negative Components Strengthens Knowledge
+
+This section investigates the ability of CAD-Amplify, which is based on CAD's attribution framework, to amplify knowledge, when removing the negative components.
+
+Amplify objects. Table 1 shows that Stable Diffusion still struggles to generate some classes, such as "cassette player", "chain saw", "church", "gas pump". To compute the objective in Equation (5), we select 5 images, for each class, from the ImageNette dataset that are correctly classified by the pre-trained ResNet50. We compute the attribution scores and remove the negative components with CAD-Amplify(Algorithm 2). Table 6 shows that CAD-Amplify improves the accuracy of the target classes significantly. More particularly, the accuracy of "cassette player" is increased from  $7.2\%$  to  $27.6\%$ , and those of the other classes
+
+![](images/66c91f00045226b5b5982daa2ee60bd431e053ab006b198df09c2d288822fccd.jpg)  
+Figure 6: The first row contains generated images conditioned on "church" and "chain saw" but are incorrectly classified by ResNet50. The second row contains images generated from the model in which negative components are ablated, with the same seed as the first row.
+
+are more than  $90\%$ . These results indicate the existence of the negative components, verifying Hypothesis 2.
+
+We additionally provide qualitative results in Figure 6 to further demonstrate that CAD-Amplify can amplify knowledge. This figure illustrates pairs of images generated by the original model and the ablated model, using the same seeds. As can be observed, CAD-Amplify adds details of the concept to the images, unleashing the target knowledge.
+
+Amplify nudity content. We also investigate how CAD-Amplify (Algorithm 2) increases the probability of generating images with explicit content. Similar to previous experiments, we remove the top  $0.1\%$  negative components of the concept "naked" and evaluate on I2P prompts with Nudenet. We also study to what extent other erasing methods remove knowledge, and whether we can restore knowledge by ablating negative components with CAD-Amplify. Table 2 illustrates Nudenet's detections on images generated by the base SD-1.4 and CAD-Amplify. As can be observed, CAD-Amplify increases the chance of eliciting nudity images, compared to the base model SD-1.4, by removing only a small number of parameters.
+
+# 7 Conclusion
+
+In this work, we study the contribution of each component, i.e., the model parameter, in generating images in diffusion models. We propose a framework based on the first-order approximation to efficiently compute the attribution scores and two editing algorithms to erase or amplify knowledge in the diffusion model. Our empirical analysis confirms the localization hypothesis, showing that knowledge is localized in a small number of components. We also show the existence of negative components that suppress knowledge, and ablating them increases the probability of generating the target concept. Our study provides a complete view of interpreting diffusion models by analyzing both positive and negative components. This understanding allows us to build more trustworthy and reliable generative models.
+
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+
+This Appendix provides additional details, analysis, and quantitative and qualitative results to support the main paper. Section A and B discuss the limitations and societal impacts of our work. We report experimental setups and hyperparameters in Section C. Section D shows the performance of CAD on different modules. Section E discusses the results of CAD with different ablation ratios. We report the computational cost of our method in Section F. Section G studies the other type of intervention on model components. We present experimental results for Stable Diffusion v2.1, Stable Diffusion XL, and Stable Diffusion v3.5 in Section H and additional qualitative results in Section I.
+
+# A Limitations
+
+In this work, we only focus on the most fine-grained model components, i.e., the model parameters, and study their contributions to concept generation. We do not examine other types of components, such as layers or modules, which can potentially influence multiple concepts at once. Furthermore, we study the contribution of model components to a concept represented in the generated image, which is the final result of the reverse process in diffusion models. Extending our work to analyze model attribution to a specific stage in the reverse process or a spatial location in the image is an interesting direction for future work.
+
+In addition, as our work only focuses on identifying and analyzing positive and negative components in diffusion models, the proposed lightweight erasing and amplification algorithms may not be the most performant. Nevertheless, one can develop more sophisticated approaches, e.g., fine-tuning the highly influential components, that may achieve better concept-editing performance than ours. Again, we leave this for future work.
+
+When removing objects, we observe that CAD-Erase slightly compromises some other knowledge, i.e., decreases the accuracies on other classes. This means that although knowledge is generally localized, there could still exist some components of those being removed that are responsible for multiple pieces of knowledge. Studying the entanglement of parametric knowledge would be an interesting future direction.
+
+# B Societal Impacts
+
+Our work proposes a framework that facilitates the analysis of diffusion models and allows us to understand how model components work. On the one hand, this framework could be potentially misused to induce harmful behaviors in generative models, such as amplifying explicit content or misinformation in generated images. On the other hand, future research could employ our approach to safeguard the model by identifying harmful components.
+
+# C Experimental Setup
+
+In our study, we compare our method with other concept erasure techniques and test its robustness against red-teaming attacks. We conduct the experiments on RTX A5000 GPUs. To evaluate erasing methods and prompt attacks, we use their official implementations. We provide details on the hyperparameters and setups used from these methods as follows:
+
+- For Stable Diffusion v1.4:
+
+- ESD: We follow the setting in the original paper and fine-tune the UNet with a learning rate of  $1e - 5$ . To compute the objective, we generate images of the target class with a guidance scale of 3. The scale of negative guidance in the objective is set to 1.  
+- UCE. We apply UCE across ten objects within the Imagenette class and for the artistic styles of Picasso, Van Gogh, Rembrandt, Andy Warhol, and Caravaggio, including the nudity concept. The method includes a "preserve" parameter in artist styles, which retains styles not targeted for erasure. We follow that setting, by erasing only one artist style at each checkpoint while keeping the rest.  
+- RECE. This method continues to fine-tune models using checkpoints previously erased by UCE. We utilize public checkpoints, which are available at https://huggingface.co/ChaoGong/RECE. These checkpoints include models fine-tuned to erase concepts
+
+such as nudity and Van Gogh style, besides 5 objects such as church, garbage truck, English springer, golf ball, and parachute.
+
+- ConceptPrune. We follow the setting provided by the author. Note that the original paper only evaluates on SD-v1.5. For the nudity concept, we apply a mask at the initial denoising step with  $\hat{t} = 9$  and a sparsity level of  $k = 1\%$ . For object removal in the Imagenette classes, we use  $\hat{t} = 10$  and  $k = 2\%$ . The same parameters are applied to the erasure of artist styles. Additionally, the "select ratio" parameter  $m$  determines the threshold for applying the binary mask to the model weights. The method prunes only those neurons that exceed  $m\%$  throughout the initial time steps  $\hat{t}$ . As this parameter is not detailed in their work, we set  $m = 0.5$  to balance the removal and retaining ability.
+
+- For Stable Diffusion v2.1:
+
+- UCE. We conduct the same experiments with Stable Diffusion v1.4 for all the concepts: object, artistic style, and nudity.  
+- RECE. For nudity content, we set  $\lambda$  at  $1e - 1$ . In object removal scenarios where UCE has successfully erased four objects with an accuracy of  $0.00\%$ , RECE focuses on the remaining objects. For the difficult object "church", we use  $\lambda = 1e - 3$ , and for easy objects like "golf ball", "parachute", "cassette player", "gas pump", and "garbage truck", we use  $\lambda = 1e - 1$ . We fine-tune for 10 epochs for nudity and 5 epochs for object removal, consistent with the hyperparameters used in the paper.
+
+- For nudity and object evaluation:
+
+- We follow the settings in prior studies.  
+- To accelerate the benchmark process, we use a batch size of 16 for Stable Diffusion v1.4 and 8 for Stable Diffusion v2.1. This allows us to evaluate using a single A5000 GPU. We maintain a consistent seed of 0 for all benchmark experiments.
+
+Table 7: The effect of ablating parameters in different modules.  
+
+<table><tr><td rowspan="2">Classes</td><td colspan="4">Accuracy on the target class↓</td><td colspan="4">Accuracy on other classes↑</td></tr><tr><td>FF</td><td>Attn1</td><td>Attn2</td><td>Residual</td><td>FF</td><td>Attn1</td><td>Attn2</td><td>Residual</td></tr><tr><td>Cassette player</td><td>0.40</td><td>0.00</td><td>2.00</td><td>11.60</td><td>80.13</td><td>59.38</td><td>37.44</td><td>34.44</td></tr><tr><td>Chain saw</td><td>0.00</td><td>0.40</td><td>13.60</td><td>16.00</td><td>69.22</td><td>44.80</td><td>50.13</td><td>20.38</td></tr><tr><td>Church</td><td>1.60</td><td>0.80</td><td>43.80</td><td>3.80</td><td>73.49</td><td>60.27</td><td>39.82</td><td>10.20</td></tr><tr><td>English Springer</td><td>1.40</td><td>1.00</td><td>21.60</td><td>16.20</td><td>71.91</td><td>61.96</td><td>34.49</td><td>15.38</td></tr><tr><td>French horn</td><td>4.40</td><td>3.00</td><td>30.60</td><td>46.40</td><td>70.87</td><td>66.93</td><td>51.47</td><td>18.93</td></tr><tr><td>Garbage truck</td><td>3.80</td><td>6.40</td><td>1.40</td><td>2.20</td><td>63.69</td><td>50.71</td><td>39.64</td><td>35.91</td></tr><tr><td>Gas pump</td><td>0.20</td><td>8.20</td><td>15.60</td><td>16.60</td><td>67.69</td><td>58.51</td><td>31.16</td><td>40.49</td></tr><tr><td>Golf ball</td><td>4.20</td><td>29.20</td><td>61.60</td><td>35.20</td><td>73.27</td><td>69.40</td><td>44.80</td><td>5.89</td></tr><tr><td>Parachute</td><td>2.00</td><td>3.80</td><td>54.20</td><td>28.00</td><td>68.91</td><td>55.96</td><td>36.58</td><td>14.33</td></tr><tr><td>Tench</td><td>0.20</td><td>0.00</td><td>9.60</td><td>13.60</td><td>72.67</td><td>52.27</td><td>57.73</td><td>12.73</td></tr><tr><td>Average</td><td>1.82</td><td>5.28</td><td>25.40</td><td>18.96</td><td>71.19</td><td>58.02</td><td>42.33</td><td>20.87</td></tr></table>
+
+# D Ablation Study
+
+In this section, we study our framework in different modules of diffusion models. Specifically, we prune positive parameters in different modules, such as feed-forward layers (FF), self-attention (Attn1), cross-attention(Attn2), and residual connections. Table 7 reports the accuracy of images generated by CAD-Eraseon different modules on the erased class and other classes. As can be observed, parameters in modules other than feed-forward layers are highly entangled, removing positive parameters of a concept affects other concepts.
+
+# E The Effect of The Ratio of Ablated Components
+
+As mentioned in Section 6, some components may be responsible for many concepts. Thus, ablating too many positive components can lead to degradation in the generation quality of other concepts. To investigate this behavior, we evaluate CAD in erasing objects with different numbers of ablated
+
+Table 8: The accuracy of generated images by SD v2.1 on target classes and other classes, predicted by the pretrained ResNet50 model.  
+
+<table><tr><td rowspan="2">Classes</td><td colspan="4">Accuracy on target classes↓</td><td colspan="4">Accuracy on other classes↑</td></tr><tr><td>SD-2.1</td><td>UCE</td><td>RECE</td><td>CAD-Erase</td><td>SD-2.1</td><td>UCE</td><td>RECE</td><td>CAD-Erase</td></tr><tr><td>Cassette player</td><td>15.60</td><td>0.20</td><td>0.00</td><td>0.20</td><td>88.22</td><td>79.17</td><td>69.95</td><td>87.38</td></tr><tr><td>Chain saw</td><td>98.40</td><td>0.00</td><td>0.00</td><td>1.40</td><td>71.95</td><td>71.95</td><td>71.95</td><td>74.40</td></tr><tr><td>Church</td><td>90.60</td><td>23.20</td><td>6.80</td><td>38.00</td><td>79.88</td><td>69.97</td><td>65.57</td><td>81.60</td></tr><tr><td>English Springer</td><td>98.60</td><td>0.00</td><td>0.00</td><td>4.00</td><td>70.73</td><td>70.73</td><td>70.73</td><td>77.13</td></tr><tr><td>French horn</td><td>98.80</td><td>0.00</td><td>0.00</td><td>2.40</td><td>78.97</td><td>74.28</td><td>74.28</td><td>76.82</td></tr><tr><td>Garbage truck</td><td>84.00</td><td>0.60</td><td>0.20</td><td>4.20</td><td>80.62</td><td>74.33</td><td>64.17</td><td>78.60</td></tr><tr><td>Gas pump</td><td>90.00</td><td>0.20</td><td>0.00</td><td>6.40</td><td>79.95</td><td>69.88</td><td>57.57</td><td>76.98</td></tr><tr><td>Golf ball</td><td>93.80</td><td>0.20</td><td>0.00</td><td>1.80</td><td>79.53</td><td>75.68</td><td>64.15</td><td>79.22</td></tr><tr><td>Parachute</td><td>63.20</td><td>0.80</td><td>0.00</td><td>0.20</td><td>82.93</td><td>73.00</td><td>69.64</td><td>78.87</td></tr><tr><td>Tench</td><td>76.60</td><td>0.00</td><td>0.00</td><td>1.00</td><td>81.44</td><td>71.42</td><td>71.42</td><td>78.29</td></tr></table>
+
+Table 9: The number of nudity content classified by Nudenet on images generated from I2P prompts. We also provide CLIP-Score and FID computed on the COCO dataset to evaluate the quality of generated images on normal prompts.  
+
+<table><tr><td>Model</td><td>Armpits</td><td>Belly</td><td>Buttocks</td><td>Feet</td><td>Breast (F)</td><td>Genitalia (F)</td><td>Breast (M)</td><td>Genitalia (M)</td><td>Anus</td><td>Total↓</td><td>CLIP-Score↑</td><td>FID ↓</td></tr><tr><td>SD-2.1</td><td>232</td><td>106</td><td>35</td><td>116</td><td>225</td><td>13</td><td>15</td><td>19</td><td>0</td><td>761</td><td>31.58</td><td>12.860</td></tr><tr><td>RECE</td><td>4</td><td>0</td><td>1</td><td>7</td><td>4</td><td>0</td><td>0</td><td>2</td><td>0</td><td>18</td><td>29.32</td><td>15.760</td></tr><tr><td>UCE</td><td>93</td><td>42</td><td>2</td><td>48</td><td>79</td><td>1</td><td>18</td><td>21</td><td>0</td><td>304</td><td>31.33</td><td>12.785</td></tr><tr><td>CAD-Erase</td><td>79</td><td>19</td><td>13</td><td>74</td><td>73</td><td>1</td><td>0</td><td>18</td><td>0</td><td>277</td><td>31.57</td><td>12.872</td></tr><tr><td>CAD-Amplify</td><td>230</td><td>106</td><td>36</td><td>124</td><td>240</td><td>13</td><td>19</td><td>18</td><td>0</td><td>786</td><td>-</td><td>-</td></tr></table>
+
+![](images/8e0eb099e25835f5ebfdfb86901e08060bdb476a267f1109667a4c73188eb87b.jpg)  
+Figure 7: The accuracy on CAD with different ablation ratios on the erased class and other classes. components. Figure 7 illustrates the accuracy with different ablation ratios, showing that high ratios decrease the accuracy of other classes. However, this drop occurs after the accuracy on the erased class reaches almost  $0\%$ , thus, we can expect a high disentanglement of knowledge in the model.
+
+# F Computational Cost of Our Methods
+
+In this section, we investigate the computation overhead of our editing methods. Since CAD edits diffusion models by only updating their weights once, it does not incur any computational overhead. In addition, as discussed in Section 4, unlike other methods such as ESD, CAD does not require training but instead leverages first-order approximation to efficiently compute model attribution. Therefore, the time and memory for editing are insignificant. We quantify the computational cost of CAD and different editing methods in Table 10. We observe that ESD takes a significant amount of
+
+time since it requires training for 1000 epochs. ConceptPrune requires generating several images to collect activations for a single concept, leading to high editing time. UCE and RECE offer fast editing due to the closed-form update; however, for a certain type of knowledge, such as artistic styles, they require additional computation to preserve many unrelated concepts. Meanwhile, CAD consumes a small amount of computation, enabling scalability to larger models while maintaining zero impact on final inference speed.
+
+Table 10: Time (second) and memory (MB) comparison across editing methods.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Object</td><td colspan="2">Nudity</td><td colspan="2">Artist</td></tr><tr><td>Time</td><td>Memory</td><td>Time</td><td>Memory</td><td>Time</td><td>Memory</td></tr><tr><td>ESD</td><td>5,691</td><td>10,225</td><td>5,624</td><td>10,233</td><td>5,617</td><td>10,238</td></tr><tr><td>ConceptPrune</td><td>230</td><td>4,047</td><td>294</td><td>4,075</td><td>321</td><td>4,035</td></tr><tr><td>RECE</td><td>26</td><td>15,561</td><td>66</td><td>15,562</td><td>470</td><td>15,665</td></tr><tr><td>UCE</td><td>9</td><td>7,420</td><td>19</td><td>7,658</td><td>386</td><td>8,610</td></tr><tr><td>CAD</td><td>66</td><td>8,152</td><td>65</td><td>8,157</td><td>66</td><td>8,223</td></tr></table>
+
+# G Intervention by Amplifying Components
+
+Table 11: Intervening diffusion by knocking out or amplifying components.  
+
+<table><tr><td rowspan="3">Classes</td><td colspan="4">Accuracy on target classes↓</td><td colspan="4">Accuracy on other classes↑</td></tr><tr><td colspan="3">Amplifying</td><td rowspan="2">Knocking out</td><td colspan="3">Amplifying</td><td rowspan="2">Knocking out</td></tr><tr><td>scale=1.5</td><td>scale=2</td><td>scale=3</td><td>scale=1.5</td><td>scale=2</td><td>scale=3</td></tr><tr><td>Cassette player</td><td>7.80</td><td>0.20</td><td>0.00</td><td>0.40</td><td>86.09</td><td>80.11</td><td>41.42</td><td>81.33</td></tr><tr><td>Chain saw</td><td>69.40</td><td>0.20</td><td>0.00</td><td>0.20</td><td>79.24</td><td>65.80</td><td>6.71</td><td>71.87</td></tr><tr><td>Church</td><td>76.60</td><td>1.40</td><td>0.00</td><td>3.00</td><td>78.44</td><td>74.47</td><td>33.16</td><td>74.24</td></tr><tr><td>English Springer</td><td>93.60</td><td>1.20</td><td>0.00</td><td>0.60</td><td>76.56</td><td>72.22</td><td>42.20</td><td>69.36</td></tr><tr><td>French horn</td><td>98.80</td><td>11.40</td><td>0.20</td><td>0.60</td><td>75.98</td><td>71.60</td><td>51.18</td><td>68.09</td></tr><tr><td>Garbage truck</td><td>85.60</td><td>9.00</td><td>0.00</td><td>2.20</td><td>77.44</td><td>62.78</td><td>27.96</td><td>64.73</td></tr><tr><td>Gas pump</td><td>78.00</td><td>0.20</td><td>0.00</td><td>1.60</td><td>78.29</td><td>66.71</td><td>28.40</td><td>66.04</td></tr><tr><td>Golf ball</td><td>95.80</td><td>8.20</td><td>1.40</td><td>5.40</td><td>76.31</td><td>73.84</td><td>65.13</td><td>73.20</td></tr><tr><td>Parachute</td><td>96.20</td><td>2.80</td><td>0.00</td><td>1.60</td><td>76.27</td><td>67.56</td><td>32.49</td><td>67.44</td></tr><tr><td>Tench</td><td>80.80</td><td>0.00</td><td>0.00</td><td>0.20</td><td>77.98</td><td>71.33</td><td>29.29</td><td>67.93</td></tr><tr><td>Average</td><td>78.26</td><td>3.46</td><td>0.16</td><td>1.58</td><td>78.26</td><td>70.64</td><td>35.79</td><td>70.42</td></tr></table>
+
+In Section 4, we study the causal effect of model components by removing them from the model. We also perform another intervention that amplifies the effect of model components by rescaling the magnitude of model components. Intuitively, increasing the magnitude of negative components could also suppress the target concept, although knowledge may still exist in positive components. The main problem of this approach is that it's hard to determine the scale for a meaningful intervention; choosing a low value may not be enough to erase the target concept, while a high value may affect other knowledge. We evaluate the performance of the model when model components are scaled up by different values. Table 11 reports the performance when amplifying negative components or knocking out positive components, showing that not all scales are suitable to verify the role of model components. With an appropriate value, i.e., 2, intervening negative components also remove the target knowledge while retaining other knowledge, confirming the effect of those components.
+
+# H Additional Results on Other Diffusion Models
+
+In this section, we report the performance of our two algorithms on other diffusion models, including Stable Diffusion v2.1, Stable Diffusion XL [31], and Stable Diffusion v3.5 to further support our analysis.
+
+# H.1 Stable Diffusion v2.1
+
+Erasing objects. Table 8 shows the accuracy of SD-2.1 erased by Algorithm 1 on the target class and other classes. As can be observed, CAD erases the target knowledge significantly while maintaining unrelated knowledge.
+
+Erasing nudity. Table 9 evaluates CAD in erasing nudity, showing that removing positive components in SD-2.1 also significantly decreases the probability of generating explicit contents and keeps the quality of generated images on normal prompts.
+
+Amplifying objects. We also apply Algorithm 2 to amplify knowledge in SD-2.1. Table 12 demonstrates that CAD increases objects in SD-2.1. CAD can also amplify knowledge of explicit contents, as shown in Table 12.
+
+Table 12: Ablating negative components on SD-2.1.  
+
+<table><tr><td>Classes</td><td>SD-2.1</td><td>CAD</td></tr><tr><td>Cassette player</td><td>15.60</td><td>18.60</td></tr><tr><td>Parachute</td><td>63.20</td><td>96.40</td></tr></table>
+
+Table 13: The CLIP-Score of SD-XL and SD-3.5 with CAD-Erase.  
+
+<table><tr><td>Model</td><td>Method</td><td>CLIP-Score</td></tr><tr><td rowspan="2">SD-3.5</td><td>Base model</td><td>32.08</td></tr><tr><td>CAD-Erase</td><td>30.63</td></tr><tr><td rowspan="2">SD-XL</td><td>Base model</td><td>31.93</td></tr><tr><td>CAD-Erase</td><td>31.67</td></tr></table>
+
+# H.2 Stable Diffusion XL and Stable Diffusion v3.5
+
+We also evaluate the editing performance on Stable Diffusion XL and Stable Diffusion v3.5. We ablate no more than  $0.03\%$  of the model's parameters and measure the number of nudity content in images generated from prompts in I2P, Ring-A-Bell, and MMA datasets in Table 14, as well as CLIP-Score on COCO30k in Table 13. The results show that in different diffusion models, CAD can locate positive components, and knowledge is still localized. In particular, CAD reduces the number of nudity labels by  $97.39\%$  and  $98.63\%$  on SD-XL and jailbreaking prompts such as Ring-A-Bell and MMA datasets, while inducing minimal impact on CLIP-Score. This experiment further indicates the generalization and practicality of our framework.
+
+Table 14: The number of nudity content classified by Nudenet on images generated by SD-XL and SD-3.5 from I2P, Ring-a-bell, and MMA prompts.  
+
+<table><tr><td>Dataset</td><td>Method</td><td>Armpits</td><td>Belly</td><td>Buttocks</td><td>Feet</td><td>Breast (F)</td><td>Genitalia (F)</td><td>Breast (M)</td><td>Genitalia (M)</td><td>Anus</td><td>Total</td></tr><tr><td rowspan="4">12P</td><td>SD-3.5 Base</td><td>158</td><td>130</td><td>19</td><td>41</td><td>137</td><td>3</td><td>16</td><td>10</td><td>0</td><td>514</td></tr><tr><td>SD-3.5 CAD Erase</td><td>30</td><td>32</td><td>3</td><td>2</td><td>64</td><td>5</td><td>10</td><td>11</td><td>0</td><td>157 (-69.46%)</td></tr><tr><td>SD-XL Base</td><td>222</td><td>169</td><td>32</td><td>44</td><td>211</td><td>14</td><td>42</td><td>14</td><td>0</td><td>748</td></tr><tr><td>SD-XL CAD Erase</td><td>46</td><td>9</td><td>2</td><td>7</td><td>19</td><td>2</td><td>1</td><td>9</td><td>0</td><td>95 (-87.30%)</td></tr><tr><td rowspan="4">Ring-A-Bell</td><td>SD-3.5 Base</td><td>66</td><td>73</td><td>6</td><td>30</td><td>110</td><td>4</td><td>25</td><td>0</td><td>0</td><td>314</td></tr><tr><td>SD-3.5 CAD Erase</td><td>6</td><td>7</td><td>0</td><td>0</td><td>14</td><td>0</td><td>3</td><td>0</td><td>0</td><td>30 (-90.45%)</td></tr><tr><td>SD-XL Base</td><td>106</td><td>110</td><td>19</td><td>33</td><td>203</td><td>50</td><td>16</td><td>0</td><td>0</td><td>537</td></tr><tr><td>SD-XL CAD Erase</td><td>3</td><td>1</td><td>0</td><td>0</td><td>10</td><td>0</td><td>0</td><td>0</td><td>0</td><td>14 (-97.39%)</td></tr><tr><td rowspan="4">MMA</td><td>SD-3.5 Base</td><td>216</td><td>96</td><td>51</td><td>21</td><td>68</td><td>1</td><td>49</td><td>9</td><td>0</td><td>511</td></tr><tr><td>SD-3.5 CAD Erase</td><td>8</td><td>13</td><td>6</td><td>3</td><td>7</td><td>2</td><td>5</td><td>4</td><td>0</td><td>48 (-90.61%)</td></tr><tr><td>SD-XL Base</td><td>209</td><td>185</td><td>47</td><td>75</td><td>212</td><td>11</td><td>52</td><td>12</td><td>0</td><td>803</td></tr><tr><td>SD-XL CAD Erase</td><td>0</td><td>2</td><td>0</td><td>6</td><td>0</td><td>0</td><td>0</td><td>3</td><td>0</td><td>11 (-98.63%)</td></tr></table>
+
+# I Additional Qualitative Results
+
+In this section, we provide additional qualitative results to demonstrate how CAD augments knowledge in diffusion models compared to other methods.
+
+Figure 8 illustrates generated images conditioned on sensitive prompts of the original SD-1.4 and different erasing methods. CAD removes explicit content in the model and maintains the quality on normal prompts.
+
+Figure 9 shows images generated from a SD-1.4 that has been erased knowledge of "Van Gogh" style by different methods. CAD successfully erases the target art style and maintains the quality of other styles. RECE and UCE also keep knowledge of other styles but change the original content.
+
+![](images/6b130657fc8ba76694b83fb365b918e3d9649ddd3c69d45eda56d7631daae77d.jpg)  
+Figure 8: Qualitative results of erasing sensitive content in SD-1.4 with CAD. We add  $\star$  for publication. Prompt (1): "High-detail image of a city skyline at dusk, lights on"; Prompt (2): "Hyperrealistic image of a car on a winding road, motion blur"; Prompt (3): "Photo of a cozy living room with a fireplace, warm lighting"; Prompt (4): "Realistic depiction of a serene beach at sunset, calm waves".  
+Figure 10 provides generated images after erasing knowledge of objects in SD-2.1 We also show qualitative results of erasing explicit content in SD-2.1 in Figure 11.  
+Figure 12 demonstrates how CAD amplifies knowledge in SD-2.1.
+
+![](images/dd4efc04b849b974dfaab523f415c8c56e435825bbc8fd75e1d926a1e5b66708.jpg)  
+Figure 9: Erasing "Van Gogh" style with different methods.
+
+![](images/4f8110b7658c536e9cd9a3a3e684b7200606180f68d4747769507b1e61ab2373.jpg)  
+Figure 10: Qualitative results of erasing objects in SD-2.1 with CAD.
+
+![](images/cadfeb61a247acc5a43b7550dbfe3445f35083e85503997632c023dd44594a5e.jpg)  
+Figure 11: Qualitative results of erasing sensitive content in SD-2.1 with CAD. We add  $\clubsuit$  for publication. Prompt (1): "Impressionist landscape of a Japanese garden in autumn, with a bridge over a koi pond"; Prompt (2): "Art Nouveau painting of a female botanist surrounded by exotic plants in a greenhouse"; Prompt (3): "Gothic painting of an ancient castle at night, with a full moon, gargoyles, and shadows"; Prompt (4): "Black and white street photography of a rainy night in New York, reflections on wet pavement"; Prompt (5): "Underwater photography of a coral reef, with diverse marine life and a scuba diver for scale"; Prompt (6): "Documentary-style photography of a bustling marketplace in Marrakech, with spices and textiles"; Prompt (7): "Cyberpunk cityscape with towering skyscrapers, neon signs, and flying cars"; Prompt (8): "Concept art for a post-apocalyptic world with ruins, overgrown vegetation, and a lone survivor".
+
+![](images/e17b8e3a5c0458a8df34fb9fe71189ae8e61273a1322cde76a06308fd52da9a4.jpg)  
+Figure 12: Qualitative results of amplifying knowledge in SD-2.1 with CAD.
+
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+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
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+
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+
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+
+- The answer NA means that the paper does not use existing assets.  
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+
+# 13. New assets
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+
+# 14. Crowdsourcing and research with human subjects
+
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+
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+
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+
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+
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+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
\ No newline at end of file
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+# Unveiling Environmental Sensitivity of Individual Gains in Influence Maximization
+
+Xinyan Su $^{1,2}$ , Zhiheng Zhang $^{3,4*}$ , Jiyan Qiu $^{1,2}$ , Zhaojuan Yue $^{1}$ , Jun Li $^{1*}$
+
+<sup>1</sup>Computer Network Information Center, Chinese Academy of Sciences, Beijing, China
+
+$^{2}$ University of Chinese Academy of Sciences, Beijing, China
+
+$^{3}$ School of Statistics and Data Science, Shanghai University of Finance and Economics, Shanghai 200433, P.R. China
+
+$^{4}$ Institute of Data Science and Statistics, Shanghai University of Finance and Economics, Shanghai 200433, P.R. China
+
+{suxinyan, qiujiyan, yuezhaojuan, lijun}@cnic.cn
+
+zhangzhiheng@mail.shufe.edu.cn
+
+# Abstract
+
+Influence Maximization (IM) seeks a seed set to maximize information dissemination in a network. Elegant IM algorithms could naturally extend to cases where each node is equipped with a specific weight, reflecting individual gains to measure its importance. In prevailing literature, these gains are typically assumed to remain constant throughout diffusion and are solvable through explicit formulas based on node characteristics and network topology. However, this assumption is not always feasible due to two key challenges: 1) Unobservability: The individual gains of each node are primarily evaluated by the difference between the outputs in the activated and non-activated states. In practice, we can only observe one of these states, with the other remaining unobservable post-propagation. 2) Environmental sensitivity: Beyond nodes' inherent properties, individual gains are also sensitive to the activation status of surrounding nodes, which change dynamically during propagation even when the network topology is fixed. To address these uncertainties, we introduce a Causal Influence Maximization (CauIM) framework, leveraging causal inference techniques to model dynamic individual gains. We propose two algorithms, G-CauIM and A-CauIM, where the latter incorporates a novel acceleration technique. Theoretically, we establish the generalized lower bound of influence spread and provide robustness analysis. Empirically, experiments on synthetic and real-world datasets validate the effectiveness and reliability of our approach.
+
+# 1 Introduction
+
+Information propagation over networks has been booming in recent years. Due to the power of the "word-of-mouth" phenomenon, influence spread has proven to be essential in various applications, such as viral marketing [7], HIV prevention [55], and recommendations [10]. The problem of selecting the seed set to maximize information spread is known as the Influence Maximization (IM) [28].
+
+Beyond optimizing the total number of infected nodes, current research has focused on investigating the individual gains of each node in real-world scenarios, referred to as weighted-IM [52, 53, 19]. Researchers endeavor to address the question: how can limited resources be utilized to maximize total gains? This challenge manifests in various network scenarios, such as student networks and email networks, involving activities like awareness dissemination and product promotion. For example,
+
+![](images/69621065832467ed566e06601982de4aca646839f24a1d3e03e4271ff6cb132c.jpg)  
+Figure 1: Illustration of individual gains during a certain propagation iteration in a product promotion scenario, focusing on the starred (*) node. The leftmost part represents an iteration state of the network represented as hyperedges (blue ovals). Each node is either activated (blue), indicating a purchase behavior, or inactive (gray). The orange star indicates the target individual whose gain we aim to evaluate, and “?” represents an unknown status. The middle panel illustrates the key question at the current propagation step: “if we promote now, what gain would the starred node contribute?” The individual gain is defined as the counterfactual difference  $Y_{i}(\text{buy}) - Y_{i}(\text{not buy})$ , representing the difference between the profit of the node in its activated and its non-activated state. Two core challenges are highlighted: unobservability, i.e., only one status of each node can be observed, with the counterfactual scenario unknown, and environmental sensitivity, which indicates the individual gains of a node are influenced by the activation status of surrounding nodes.
+
+when targeting users with varying purchasing power in product promotion, these users exhibit diverse purchasing behaviors, resulting in varying profits for the seller. Here, by regarding purchasing power as individual gains, the goal is to identify specific users for product advertising and optimize the overall difference in profit gains pre- and post-product promotion dissemination.
+
+Researchers usually assume that the purchasing power of each node remains observable and stable [29] throughout the entire process. Such a weighted IM setting appears to be a natural extension of traditional IM and hence leads to relatively limited exploration [28]. However, in practice, this setting would be violated, and we summarize it as two fundamental properties, as illustrated in Fig. 1: (i) Unobservability. Accurately quantifying the actual purchasing power of each user is hindered by the limited observations of purchase occurrences (where only two outcomes are observable: activated or not, corresponding to purchase or non-purchase for each node; one represents the "factual", and the other "counterfactual"), thereby complicating the determination of the actual increase in benefit gains that each user can deliver to sellers; (ii) Environmental sensitivity. The expected purchasing power of each person is not only associated with the individuals themselves but also influenced by the attitudes of their social contacts. For example, as more people in a friend circle make purchases, individuals become increasingly susceptible to influence and are likely to make additional purchases. These properties indicate that individual gains in IM are environmentally dynamic and challenging to ascertain, posing challenges in their computation.
+
+To tackle these challenges, we employ causality techniques to model the advertising problem, inspired by the concept of Individual Treatment Effect (ITE) [45]. It measures the disparity between the factual world and the counterfactual one, which goes beyond traditional observational studies, such as those based on network structure or direct assignment of other feature weights [28, 14]. By incorporating the process of inference, we transition the problem from observational study to the direct utilization of ground truth for measuring individual gains at the node level. Drawing from this, we propose the Causal Influence Maximization (CauIM) framework. Specifically, in the hypergraph modeling $^2$ , we redefine the objective function of traditional IM by incorporating ITE as weights assigned to each node. This formulation integrates both internal covariate information and external environmental information for each node (Fig. 1).
+
+Noteworthy, it is not merely a simple causality-plugin-in interdisciplinary attempt since we should proactively challenge the Stable Unit Treatment Value Assumption (SUTVA) in causality [1], which permits the presence of interference between different nodes as illustrated above. Taking a step forward, even if there is pioneering exploration upon interference-based causality [37, 36, 32], distinguishing CauIM from such literature, we delve deeper into practical constraints: the original treatment policy can solely influence a "limited seed set", and then the causality estimator should be considered under a propagation process that involves temporal instability and computational burden. This process is highly non-trivial due to the dynamic environmental information associated with each node. In this context, we further provide an efficient and stable computational enhancement, which replaces the conventional greedy selection with vectorized derivative-based operations to improve scalability in practice.
+
+In sum, our contributions are summarized as follows: i) We propose CauIM, a novel framework for influence maximization in networks that incorporates community structures and environment-sensitive individual node effects, providing a new perspective on modeling dynamic individual gains in networks. We also justify its practical applicability in real-world scenarios.ii) We demonstrate the effectiveness (approximate optimal guarantee), robustness, and acceleration feasibility of CauIM, particularly when ITE values are not strictly positive or suffer from estimation bias. In correspondence, we provide the greedy-based implementation (G-CauIM) and further design the Gradient-based Accelerated CauIM (A-CauIM). iii) We conduct experiments on three real-world datasets and one synthetic dataset. It not only supports our theoretical claim upon effectiveness and robustness but also validates the efficiency improvement of A-CauIM in practice.
+
+# 2 Related work
+
+Influence Maximization (IM) IM is first identified as an algorithm problem by Kempe in [28] and has given rise to several notable variants, including simulation-based (CELF [31], CELF++ [16]), sketch-based (RIS [5], TIM [48], IMM [49]), and heuristic algorithms (HADP [56]), etc. Three key elements of the problem are 1) graph structure, 2) diffusion process and 3) seed selection. For instance, CELF, RIS, and TIM emphasize iterative algorithmic enhancements while ensuring theoretical assurances within triggering models. In contrast, the latest heuristic algorithm, HADP, prioritizes computational efficiency at the cost of sacrificing theoretical guarantees. Additionally, recent learning-based IM methods [6, 30, 35] focus on understanding the inherent nature of individual node representations concerning the marginal influence gain. Nevertheless, these methods exhibit limitations in model generalization and the reliability of final results. The original optimization objective requires reassessment in diverse scenarios. Dynamic IM studies, such as Peng [44], focus on evolving edge probabilities while maintaining submodularity guarantees. Their objective differs from ours: the edge set changes over time, whereas we consider static networks where node rewards vary with unobserved environmental factors and treatment assignments. Hence, these methods are orthogonal to our framework. Researchers have increasingly focused on exploring the heterogeneity of importance between nodes based on these methods [52, 23, 15]. Nevertheless, most existing studies rely on fixed topological or attribute-based priors and treat individual importance as constant, lacking a unified framework that accounts for environmental sensitivity when estimating dynamic individual effects. We refer readers to the Appendix I for more details.
+
+Treatment effect estimation How to recover the ITE directly from the observational data instead of randomization test [46] is currently receiving a lot of attention. There are two main strategies for estimation: 1) weighting-based methods [33, 34, 12], and 2) representation-based methods [47, 26]. In this paper, we follow the second strategy. Ma et al. [37] estimated the causal effect via representation learning on the more general hypergraph. However, these methods do not consider the IM question.
+
+# 3 Preliminary
+
+Notations and Basic Concepts We develop our model on an undirected hypergraph  $\mathcal{G}(\mathcal{V},\mathcal{H},\mathbb{H})$  where  $\mathcal{V}\coloneqq \{v_{1},v_{2},\ldots v_{n}\}$ ,  $\mathcal{H}\coloneqq \{h_1,h_2,\dots h_m\}$ , representing the node set and the hyperedge set respectively, and  $\mathbb{H}\in \{0,1\}^{m\times n}$  denotes the incidence matrix between hyperedges and nodes. Each undirected hyperedge represents a social connection among the nodes it contains. For each
+
+node  $v_{i}$ , covariate  $X_{v_{i}}$  denotes its node feature and  $\mathcal{N}_{v_i}$  indicates the set of its neighborhood<sup>3</sup>. To maintain clarity, we abuse the notation that  $X_{v_i} = X_i$  and  $N_{v_i} = N_i$ , applying the same convention for symbols with the subscript  $v_{i}$ .
+
+Stepping forward, we would like to introduce several broad concepts in causal inference: the potential outcome (individual profit under one status) of each node  $v_{i}$  is denoted as  $Y_{i}(T_{i} = t;X_{i},\pmb{T}_{-i},\pmb{X}_{-i})$ . Here  $t = 0,1$  refers to the case where node  $v_{i}$  is activated in the diffusion process or not. Moreover,  $T_{-i},X_{-i}$  represent the environmental information, namely,
+
+$$
+\boldsymbol {T} _ {- i} := \left\{T _ {1}, T _ {2}, \dots T _ {i - 1}, T _ {i + 1}, \dots T _ {n} \right\}, \tag {1}
+$$
+
+$$
+\boldsymbol {X} _ {- i} := \left\{X _ {1}, X _ {2}, \dots X _ {i - 1}, X _ {i + 1}, \dots X _ {n} \right\}.
+$$
+
+The individual treatment effect (ITE) is defined as
+
+$$
+\tau_ {i} := Y _ {i} \left(T _ {i} = 1; X _ {i}, \boldsymbol {T} _ {- i}, \boldsymbol {X} _ {- i}\right) \tag {2}
+$$
+
+$$
+- Y _ {i} \left(T _ {i} = 0; X _ {i}, \boldsymbol {T} _ {- i}, \boldsymbol {X} _ {- i}\right).
+$$
+
+Here "treatment"  $\{T_i, T_{-i}\}$  indicate the activation status of  $v_i$  and its surrounding nodes. ITE represents the difference in node outcomes between the activated and inactivated cases. As illustrated in the introduction, it cannot be directly extracted from observations (property  $(i)$ ), and it also depends on the activation state of surrounding nodes (property  $(ii)$ ).
+
+Problem Formulation We adopt the widely-used Susceptible-Infected Contact Process (SICP) diffusion model [56]. Starting from an initial seed set  $S_0$ , diffusion unfolds in discrete steps. In each iteration, every activated node  $v$  randomly selects one of its affiliated hyperedges and attempts to activate its inactive neighbors  $u \in \mathcal{N}_v$  within that hyperedge, each with a certain probability. This process continues until no new activations occur, as illustrated in Appendix Fig.4. Our framework also supports alternative diffusion models, such as Linear Threshold (LT) model [18, 17], which can be seamlessly integrated with the algorithms presented in Section 4.
+
+We define  $ap(v_i; S_0)$  as the probability that node  $v_i$  gets infected in the entire propagation process initiated by seed set  $S_0$ . According to Wang et al. [52],  $ap(v_i; S_0) = \sum_{u \in S_0} p_r(u, v_i)$ , where  $p_r(u, v_i)$  denotes the probability of reachability from node  $u$  to  $v_i$  inclusive of all reachable paths. Finally, we identify the objective function as the expected total causal influence during the diffusion:
+
+$$
+\sigma (S) = \mathbb {E} \left[ \sum_ {v _ {i} \in \mathcal {V}} a p \left(v _ {i}; S\right) \tau_ {i} \right], S \subseteq \mathcal {V}. \tag {3}
+$$
+
+Here  $\tau_{i}$  is identified in Eq. 2 and the expectation takes upon activation status  $\{T_i, T_{-i}\}$  in all possible propagation process. The goal of CauIM is to find
+
+$$
+S ^ {*} = \operatorname {a r g m a x} _ {S} \left\{\sigma (S) \right\}, s. t. | S | \leq K. \tag {4}
+$$
+
+where  $K$  is a fixed budget. Compared with the traditional definition, this formulation provides a concise yet general expression at the cost of introducing the relatively difficult-to-calculate term  $a p(\cdot)$ . Due to its computational difficulty, we will elaborate on an efficient approximation and acceleration process in the next section.
+
+Assumption 3.1 (Bounded ITE and Consistency). 1) Bounded ITE:  $\max_{v_i\in \mathcal{V}}|\tau_i|\leq M$  , where  $M$  is a constant. 2) Consistency [9]: Potential outcomes  $Y_{i}(T_{i} = t;X_{i},T_{-i},X_{-i})$  are deterministic in Eq. 2 and equal to the observational values of  $Y$  for  $t = 0,1$  given fixed  $\{X_i,T_{-i},X_{-i}\}$
+
+These are standard assumptions for ITE in causal inference [43]. Notably, Assumption 3.1 accounts for interference effects that violate SUTVA [24], i.e., "the potential outcomes for any unit do not vary with the treatment assigned to other units". We incorporate such effects into an environment function:  $O_{i} = \mathrm{ENV}(\mathbb{H}, T_{-i}, X_{-i})$ . This function follows [37], where environmental covariates and treatment assignment are summarized as a low-dimensional representation. We formalize this concept in Assumption 3.2.
+
+Assumption 3.2 (Environment Assumption under Interference [37]). For each node  $v_{i}$ , the two potential outcomes in Eq. 2 are conditionally independent given  $\{T_i,O_i\}$ .
+
+Assumption 3.2 extends the standard ignorability assumption [25] to the graphical case, where the authors claimed that the pair of potential outcomes is independent of the treatment assignment, given the covariates of each node. It guarantees there are no unmeasured covariates in the graph, which is a fairly broad assumption and has been adopted by Ma et al. [37]. Assumption 3.2 essentially ensures that the two effects  $(\{X_{i},T_{i}\} ,\{X_{-i},T_{-i}\})$  together constitute a sufficient statistic for ITE. In other words, ITE could be legitimately estimated via observations under these assumptions and hence provides the potential to design the estimator  $^5$
+
+# 4 Methodology
+
+In this section, we first identify three primary challenges when designing our algorithms. Subsequently, we provide a detailed introduction to two algorithms within our CauIM framework. The first is an offline greedy-based implementation for causal influence maximization named G-CauIM (Alg. 1). We then improve the efficiency of G-CauIM by speeding up the diffusion and greedy selection process, and propose a Gradient-based Accelerated CauIM, referred to as A-CauIM (Fig. 2). Finally, we theoretically demonstrate the algorithm's effectiveness. Key notations for this section can be found in Appendix 3. Notably, classic IM maximizes a monotone submodular count of activations under an implicit assumption of fixed, context-independent, nonnegative marginal gains. In CauIM, node-level gains are weighted and context dependent (they may vary with neighbors' states and be negative), so submodularity may break, and sketch-based pipelines [49, 48] offer no guarantees. We thus begin by summarizing three main challenges as follows.
+
+Algorithm 1: G-CauIM  
+15 return  $S^{*}$  
+Input:  $\mathcal{G}(\mathcal{V},\mathcal{H},\mathbb{H})$  ; seed number  $K$  .  $X_{i}$  , initial treatment  $T_{i}$  and  $T_{-i}$  of each node  $v_{i}$  ; observational data  $D = \{Y_{i}(t;\cdot)\}_{v_{i}\in \mathcal{V}}$  , where  $t = T_{i}$  ; the ITE bound  $M$    
+Output: Deterministic seed set  $S^{*}$  with  $|S^{*}| = K$    
+1 Function  $\widehat{\mathrm{ITE}} (X_i,T_i,T_{-i},\mathcal{G};\theta)$  ..   
+2 Compute the representation  $Z_{i}$  of  $X_{i}$  via representation learning;   
+3 Compute the high-order interference representation  $O_{i}\coloneqq \mathrm{ENV}(\mathbb{H},T_{-i},Z_{-i};\theta)$  (Assumption 3.1 and Assumption 3.2);   
+4 Concatenate  $Z_{i}$ $O_{i}$  and feed them into a Multi-Layer Perceptron (MLP):  $\{\hat{Y}_i(1;\cdot),\hat{Y}_i(0;\cdot)\} \sim \mathrm{MLP}([Z_i||O_i])$    
+5 Compute the ITE  $\hat{\tau}_i = \hat{Y}_i(1;\cdot) - \hat{Y}_i(0;\cdot)$  for  $v_{i}$  .   
+6 return  $\hat{\tau}_i$  ..   
+7 Function Main:   
+8 (Initialization)  $S^{*} = \emptyset$  ; Loss  $= 0$  .   
+9 (Training) Using the above  $\widehat{\mathrm{ITE}} (\cdot ;\theta)$  function, compute the cumulative loss by  $D$  .. Loss  $= \sum_{v_{i}\in \mathcal{V},t = 0,1}\left|\left(\hat{Y}_i(t;\cdot) - Y_i(t;\cdot)\right)\mathbb{I}(T_i = t)\right|$  (only the factual term is active via  $\mathbb{I}(T_i = t)$  );   
+10 (Projection to bounded-ITE set) Define the feasible set  $\Theta_M\coloneqq \{\theta :\max_{v_i\in \mathcal{V}}|\hat{\tau}_i(\theta)|\leq M\}$  . Set  $\theta^{\prime}\coloneqq \Pi_{\Theta_{M}}(\theta^{opt}):=\arg\min _{\theta\in\Theta_{M}}\|\theta-\theta^{opt}\|_{2}.$    
+11 for  $|S^{*}| <   K$  do   
+12 Conduct propagation under current seed set  $S^{*}$  , generate  $\hat{\tau}_{i} = \widehat{\mathrm{ITE}} (X_{i},T_{i},T_{-i},\mathcal{G};\theta^{'})$  for  $v_{i}\notin S^{*}$  where  $T_{i}$  is changed to its current activated state( 0 or 1), and  $T_{-i}$  is changed based on other nodes' activated states,  $\theta^{\prime}\coloneqq \theta^{opt} + \triangle_{\theta},\triangle_{\theta}\coloneqq \min \{\| \theta_q\| :\exists \delta \leq \| \theta_q\| ,\widehat{\mathrm{ITE}} (\cdot ;\theta +\delta)\leq M\}$  repeat the process and get the mean;   
+13  $v_{0} = \arg \max_{v\notin S^{*}}\{\sigma (S^{*}\cup \{v\}) - \sigma (S^{*})\}$  .   
+14  $S^{*} = S^{*}\cup \{v_{0}\}$
+
+# 4.1 Three Challenges
+
+(i) Unmeasured individual effect(ITE) (Eq. 2): the inherent limitations in causal inference necessitate the recovery of the counterfactual to address the "missing data problem" in our objective function Eq. 4. Furthermore, it may vary across iterations due to its dependence on environmental information  $\mathbf{X}_{-i}, \mathbf{T}_{-i}$ . (ii) Approximate optimal guarantee: The traditional greedy-based IM might not guarantee sub-optimal properties due to the unknown individual effect as mentioned above and, therefore, requires re-analysis. (iii) Estimation bias: CauIM exhibits robustness against biases stemming from individual effect estimation and the sampling strategy.
+
+Address challenge (i): ITE estimation. Motivated by Ma et al. [37], Ma and Tresp [38], we recover the individual ITE from observational data using a neural network model represented by  $\widehat{\mathrm{ITE}} (\cdot)$  function in both G-CauIM and A-CauIM. To handle dynamic characteristics of ITE, we incorporate a model parameter adjustment strategy into Alg. 1 of G-CauIM. Additionally, we employ an approximation strategy in A-CauIM, as depicted in the ITE Estimator section of Fig. 2. Further details of this procedure are provided in the subsequent Algorithms section.
+
+Address challenge (ii): To achieve approximate optimum, we inductively select the seed candidate via greedy search. We show this greedy strategy will also hold a weaker but analogous order of  $\left(1 - \frac{1}{e}\right)$  approximate optimum level of the traditional IM in further theoretical parts:
+
+$$
+v _ {0} = \operatorname {a r g m a x} _ {v \notin S ^ {*}} \left\{\sigma \left(S ^ {*} \cup \{v \}\right) - \sigma \left(S ^ {*}\right) \right\}. \tag {5}
+$$
+
+It can be seen as a more general result since the traditional IM serves as the special case with  $\tau = 1$ . However, the analysis of the above settings is harder since the submodularity might not always exist due to the potential negative ITE. Additionally, for practical issues, We refer readers to I for the Monte Carlo-based greedy CauIM.
+
+Address challenge (iii): Our CauIM model demonstrates robustness against bias in estimating Individual Treatment Effects (ITE), which will be outlined in the theoretical discussion. This resilience is ensured under broad assumptions regarding the controlled probability  $p_r(v, u)$  of reaching all nodes  $u \in \mathcal{V}$  in the complete graph from the current propagation node  $v$ , which is readily feasible in real-world scenarios, as evidenced in the experimental findings. In a word, this property provides us with more flexibility and possibilities in adjusting the estimator parameters.
+
+# 4.2 Proposed Algorithms
+
+**Environmental Function.** We define the environment of node  $i$  as  $O_{i} = \mathrm{ENV}(\mathbb{H}, T_{-i}, Z_{-i})$ , which summarizes the influence from its surroundings. Here,  $\mathbb{H}$  denotes the network structure (e.g., hyperedges involving  $i$ ), and  $(T_{-i}, Z_{-i})$  represents the activation states and representation vector of all other nodes. The function  $\mathrm{ENV}(\cdot)$  encodes this context into a low-dimensional vector. Since  $\mathbb{H}$  is static in our setting, ENV mainly reflects the dynamic states of  $i$ 's neighbors, e.g., the number or embedding of active peers at a given time. Here ENV is a transmission function using Hypergraph Convolution module [3]; more details are in Appendix I.
+
+G-CauIM. Our primary procedures are executed in function Main (line 7). We initially train the ITE estimation model offline (line 9), represented by function  $\widehat{\mathrm{ITE}}(\cdot)$  (line 1). In this model, for node  $v_{i}$ , we construct  $Z_{i}$  via each covariate  $X_{i}$  using representation techniques (line 2). Additionally, we construct  $O_{i}$  to denote the higher-order interference representation of node  $v_{i}$  with its environments:  $O_{i} = \mathrm{ENV}(\mathbb{H},\pmb{T}_{-i},\pmb{Z}_{-i})$  (line 3), and  $Z_{-i} \coloneqq \{Z_1,\dots Z_{i - 1},Z_{i + 1},\dots Z_n\}$ . Finally, with the combination of representation  $Z_{i}$  and  $O_{i}$ , we obtain the estimation value  $\hat{Y}_i(1),\hat{Y}_i(0)$  via MLP model (line 4). During training, we employ a balancing mechanism to ensure covariate balance between the treatment group  $\{v_{i}:T_{i} = 1\}$  and control group  $\{v_{j}:T_{j} = 0\}$ , achieved by incorporating an additional penalty term to the representation vector. Such technique is not unique, referring to Yao et al. [57], Harshaw et al. [20]. In addition, line 10 is to ensure the estimated ITE is bounded by  $M$  (identified in assumption 3.1) via controlling the estimator parameter  $\theta$ . Diffusion process and greedy selection take place in line 11-14, where we employ the traditional greedy algorithm strategy and re-analyze the marginal benefit function as the incremental form of ITE. It is noteworthy that as the seed set expands, the activated states of each node change in the propagation, leading to diverse values of  $\hat{\tau}_i$ . To bound the varying  $\hat{\tau}_i$ , we adjust the parameters  $\theta^{opt}$  of the trained  $\widehat{\mathrm{ITE}}(\cdot)$ , as depicted in line 12. The detailed procedure is also depicted in Fig. 5.
+
+![](images/19c94cf66284fe307a60200f34e7e801170f4fb7a18f2d663b06cc41d29dcb31.jpg)  
+Figure 2: A-CauIM. Compared with G-CauIM (Alg. 1), we add a storage table for activation probabilities  $ap(\cdot; \cdot)$  and then simplify the complex greedy selection (Eq. 5) into more efficient derivative operations (Eq. 7). In addition, we transform  $ap(\cdot; \cdot)$  into continuous values close to 0, 1 to signify the activated states  $T_i$  of each node on average. And by this procedure, we obtain  $\widehat{\mathbb{E}\tau_i}$  which is the approximation of the expectation on unobserved  $\tau_i$ .
+
+A-CauIM. Four components are presented in Fig. 2. Initialization is to obtain a trained  $\widehat{ITE} (\cdot ;\theta^{opt})$  the same as that of G-CauIM. In the diffusion process, our objective is to calculate  $ap(\cdot)$ . To address the inefficiency loops of G-CauIM, we enhance the computation by utilizing PYTORCH for fast graph computation. In sum, during each round of seed selection, we initially utilize a bipartite graph, where two sides are constructed from hyperedges and nodes, to approximately compute the activated probability  $ap(v_{i};S)$  of each node  $v_{i}$  under  $S$  iteratively. Specifically, during  $j$ -th iteration of propagation (given the current seed set  $S$ ), according to the SICP model mentioned in Section 3, suppose that hyperedge  $h_p$  is chosen, and its internal node  $v_q$  is activated. The activation probability of  $h_p$  and  $v_q$  in  $j$ -th iteration are computed as  $Prh(h_p,j)$  and  $Prv(v_q,j)$ , respectively, as defined in Eq. 6. Here  $\mathcal{H}_q$  denotes the set of hyperedges containing  $v_q$ ,  $PSICP^6$  represents the basic activation probability of a node. In this sense, we derive  $ap(v_q;S) = \lim_{n\to +\infty}Prv(v_q,n)$ . This process provides a rapid approximation of the original multiple randomizations of propagation.
+
+$$
+P r h (h _ {p}, j) = \sum_ {v _ {k} \in h p} P r v (v _ {k}, j - 1) / | \mathcal {H} _ {k} |,
+$$
+
+$$
+P r v \left(v _ {q}, j\right) = 1 - \prod_ {h _ {p} \in \mathcal {H} _ {q}} \left(1 - P _ {S I C P}\right) ^ {P r h \left(h _ {p}, j\right)} \tag {6}
+$$
+
+The next step is ITE estimation. We convert  $ap(v_i; S)$  into values  $1 - e^{C[ap(.S) - C_d] + }$  to represent the activated states  $T_i$  of each node on average according to the obtained probabilities. Here  $C, C_d$  are a priori constants. It aims to be close to binary treatment 0, 1 to fit the  $\widehat{ITE}$  model and maintain the differentiability. Using  $ITE(; \theta^{opt})$ , we determine  $\widehat{\mathbb{E}\tau_i}$ . For the Greedy selection process, we calculate the marginal gain using  $\sigma(S) = \sum_{v_i \in \mathcal{V}} ap(v_i, S) \widehat{\mathbb{E}\tau_i}$ . To approximate  $\mathbb{E}\left[\sum_{v_i \in \mathcal{V}} ap(v_i, S) \tau_i\right]$ . aforementioned in Section 3. Ideally, we hope A-CauIM would utilize PYTORCH to differentiate the objective function (Eq. 5) to identify the node that maximizes marginal gains. However, it is usually unreliable since the indices of nodes are discrete and not amenable to differentiation. To address this issue, we use the asymptotic approximate version of  $ap(v_q; S)$  (which is "parameter-based continuous") to replace the indicator of 0, 1 on each node. On this basis, Eq. 5 has been transformed into an operation that is continuously differentiable by PYTORCH, despite the cost of losing certain information from the unused diffusion process.
+
+$$
+v := \arg \max  _ {v \notin S} \left\{\frac {\partial (\sigma (S))}{\partial (a p (v ; S))} * a p (v; S) \right\}. \tag {7}
+$$
+
+Finally, in each round, we select the node with the highest derivative value multiplied by its activated probability as the seed node in Eq. 7, which exerts the greatest impact on  $\sigma(S)$  under a small perturbation to the connection probability between  $v$  and  $S$  in the whole propagation.
+
+Time Complexity. We have reduced the complexity of this problem in our settings from  $O(KRnm)$  to  $O(Km\mathbb{E}_{h\in \mathcal{H}}|h|)$ , where  $m, n$  are the numbers of hyperedges and nodes,  $K$  is seed set number identified in our preliminaries, and  $R$  is a simulation number of propagation process (The complexity of the ITE estimation module, utilized in both algorithms, is excluded here). Such improvement is especially significant on relatively sparse graphs. Since  $O(Km\mathbb{E}_{h\in \mathcal{H}}|h|)\leq O(Kmn)$  when  $\mathbb{E}_{h\in \mathcal{H}}|h|\ll n$ .
+
+# 4.3 Theoretical analysis
+
+In this section, we first prove that the traditional greedy algorithm can be naturally extended to hypergraphs and maintains the  $(1 - 1 / e)$  approximate guarantee. Then we demonstrate that this approximate guarantee still holds for our CauIM algorithm (Theorem 4.4, challenge 2). In addition, we show that CauIM's performance is robust to the estimation error of ITE (Theorem 4.5, challenge 1 and challenge 3).<sup>7</sup>
+
+Proposition 4.1. Our CauIM problem is NP-hard.
+
+Lemma 4.2 (Approximately optimal guarantee of greedy IM on hypergraph). The greedy method on the hypergraph can achieve the  $\left(1 - \frac{1}{e}\right)$  approximate optimal guarantee.
+
+Condition 4.3 (Bounded increase of reachable probability). We define the reachable probability from a set (or node)  $v_{1} \in \mathcal{V}$  to a set (or node)  $v_{2} \in \mathcal{V}$  as  $p_{v_1v_2}$ .  $\forall v_{1} \subseteq v_{2} \subseteq \mathcal{V}, |v_{2}| = |v_{1}| + 1$ , we have the bounded condition of the increase of the reachable probability:  $\forall v' \in \mathcal{V}, |p_{v'v_1} - p_{v'v_2}| \leq \varepsilon_1$ . Moreover,  $\max_{v \in \mathcal{V}} \sum_{v_i \in R(v)} |\tau_{v_i}| p_{vv_i} \leq \varepsilon_2$ , where  $R(v)$  denotes the successors that  $v$  can arrive during the diffusion process. Here  $\varepsilon_1, \varepsilon_2$  are both a priori constants.
+
+Notice that this condition is fairly broad and model-free.
+
+Theorem 4.4 (Approximate optimal guarantee of CauIM). 1) If  $\tau_{i} > 0$ ,  $i \in \mathcal{V}$ , the CauIM algorithm can achieve the  $(1 - \frac{1}{e})$  optimal approximate guarantee. 2) If we do not have  $\tau > 0$ , then a more generalized guarantee is  $\sigma(S_K^g) \geq (1 - \frac{1}{e})(\sigma(S^*) - K\varepsilon_1\varepsilon_2) - \varepsilon_2e^{\frac{1}{K} - 1}$ .
+
+The estimation error of  $\sigma(\cdot)$  can be traced back to both the Monte-Carlo strategy and the estimation error of  $\tau_{i}$  during representation learning. We summarize it as the result on robustness analysis as follows. The proofs of the theorems are deferred in Appendix D, E, and F.
+
+Theorem 4.5 (Robustness). We denote the MC estimation of  $\sigma(S)$  as  $\sigma(\hat{S})$ . If  $\forall S \subseteq \mathcal{V}, |\frac{\hat{\sigma}(S)}{\sigma(S)} \in [1 - \gamma, 1 + \gamma]$  and  $\gamma \leq \frac{\varepsilon / k}{2 + \varepsilon / k}$ ,  $\gamma > 0$ , then our CauIM problem can achieve the optimal guarantee, which can be transferred to  $\sigma(S_K^g) \geq (1 - \frac{1}{e} - \varepsilon) (\sigma(S^*) - K\varepsilon_1\varepsilon_2) - \varepsilon_2 e^{\frac{1}{K} - 1}$ .
+
+We refer readers to  $\mathbf{G}$  for details. In addition, notice that the estimation error  $\hat{\tau}_i - \tau_i$  also causes the error  $\hat{\sigma}(S) - \sigma(S)$ . Specifically, if we have  $|\hat{\tau}_i - \tau_i| \leq \delta$ , then  $\hat{\sigma}(S) - \sigma(S)$  can also be bounded.
+
+Corollary 4.6 (Robustness of noise). We consider the ideal CauIM case without MC strategy. The traditional IM objective function (i.e.,  $\tau_{i} = 1, \forall v_{i} \in \mathcal{V}$ ) is denoted as  $\sigma_{\text{naive}}$ . If  $|\hat{\tau}_i - \tau_i| \leq \delta$  and  $|\frac{\sigma_{\text{naive}}(S)}{\sigma(S)}| \leq \frac{\gamma}{\delta}$ , then Theorem 4.5 holds. Detailed proof can be found in Appendix H.
+
+# 5 Experiments
+
+We perform experiments on four datasets and validate the findings presented in Section 4.3. We aim to answer the following three questions.
+
+RQ1: Effectiveness (Theorem 4.4) When maximizing the sum of node ITE (overall individual gains), can our G-CauIM and A-CauIM outperform the traditional IM methods and maintain efficiency?
+
+RQ2: Robustness (Theorem 4.5, Corollary 4.6) If ITE estimation is not accurate enough, can
+
+Table 1: RQ1: Performance comparison of four different methods on four datasets (seed number= 15). Our methods gain general improvements compared with baselines: Traditional Greedy (denoted "Baseline") and Random Selection.  
+
+<table><tr><td>Methods</td><td>GoodReads</td><td>Contact</td><td>Email-Eu</td><td>SD-100</td></tr><tr><td>Baseline</td><td>297.56</td><td>68.12</td><td>735.28</td><td>138.91</td></tr><tr><td>Random</td><td>45.86</td><td>66.51</td><td>590.67</td><td>145.97</td></tr><tr><td>G-CauIM</td><td>330.25</td><td>69.53</td><td>804.28</td><td>151.59</td></tr><tr><td>A-CauIM</td><td>302.17</td><td>66.78</td><td>802.41</td><td>160.49</td></tr></table>
+
+Table 2: A-CauIM presents a significant efficiency improvement compared with other competitive baselines (CPU, Torch 1.11.0). From G-CauIM to A-CauIM, the complexity reduces to  $O\left(Km\mathbb{E}_{h\in \mathcal{H}}|h|\right)$  (Section 4).  
+
+<table><tr><td>Methods</td><td>GoodReads</td><td>Contact</td><td>Email-Eu</td><td>SD-100</td></tr><tr><td>Baseline</td><td>1day</td><td>6h40min</td><td>22h</td><td>3h</td></tr><tr><td>Random</td><td>2s</td><td>2s</td><td>2s</td><td>2s</td></tr><tr><td>G-CauIM</td><td>1day02h</td><td>7h</td><td>23h</td><td>3h</td></tr><tr><td>A-CauIM</td><td>28s</td><td>115s</td><td>550s</td><td>53s</td></tr></table>
+
+![](images/09df220c0b9c8318ee175f28b6355be84df392ef46abcd2b3644b2f2bb7ccc10.jpg)  
+(a) Result on GoodReads
+
+![](images/cf6eb823dfa08225e04581848d406b9a47dc03abb9e05415b78313efb1a41be3.jpg)  
+Figure 3: a,b) Performance of CauIM on the GoodReads and Contact dataset. "Iter" refers to the time step in each seed selection round. c) Variance curve trend with different noise in individual ITE, where  $\epsilon_{y_i}$  represents the standard deviation of the injected noise. d) Total sum of ITE under various propagation probabilities  $p$ .
+
+![](images/43f929ccfe07189242ec341c84cb5e21897d8fa9d29d06632ad35af9ce70589b.jpg)  
+(b) Result on Email-Eu
+
+![](images/a5d3a4a53c9e52b9a5b763e2b004640010db95ae2ccf44ac3638f508b171aa4a.jpg)  
+(c) Analysis on noise  $\epsilon$  
+(d) Analysis on various  $p$
+
+CauIM perform robustly? The robustness means that in perturbations, our approach will achieve an approximate result close to the normal state. RQ3: Sensitivity Which components and parameters of the model are essential for the performance of CauIM? Detailed Settings are outlined in Appendix C.
+
+# 5.1 General CauIM Performance
+
+RQ1 Results are summarized in Table 1, with a detailed analysis of two datasets shown in Fig. 3(a) and Fig. 3(b). We can summarize the observations into four phenomena: 1) G-CauIM shows slight improvement over traditional Greedy and improves significantly compared with random selection, while the gap of their curves widens as the seed number increases. 2) A-CauIM achieves comparable performance with G-CauIM while significantly enhancing efficiency. 3) The fluctuation amplitude of the G-CauIM curve is relatively small, as demonstrated in Fig. 1, consistent with its alignment to the dynamic changes in individual effects and support from Theorem 4.4 and Theorem 4.5 in Section 4.3. 4) Traditional Greedy loses its advantage in most situations and approaches Random.
+
+# 5.2 Robustness and Sensitivity Analysis
+
+We address RQ2 and RQ3 using A-CauIM. It is sufficient since A-CauIM is a learning-based method with slightly more uncertainty but outstanding efficiency. We conduct experiments on GoodReads dataset for ease of analysis.
+
+RQ2 & RQ3 To examine robustness of our model, we add Gaussian noise  $\epsilon_y$  to the simulated individual ITE results, where  $\epsilon_y \sim N(0, \epsilon^2)$ . The randomness of experiments thus comes from three sources: 1)  $\epsilon_y$ , 2) the propagation probability  $p$ , and 3) the dynamic ITE of each node. We modify the scale of the noise and plot corresponding curves. Fig. 3(c) can provide the following illustrations: 1) when the noise increases (not exceeding 9), it is not enough to counter the random variance of the propagation process and dynamic ITE itself. This "powerlessness" effect disappears until noise reaches around 10. 2) During the early stage of seed growth, when the number of seeds is smaller than 10, the effect of total noise of the infected nodes is not large. More discussions of the parameter  $\epsilon$  are in Appendix C.5.
+
+We further detect how the components of the algorithm interact and how the diffusion process affects A-CauIM with various parameter  $p$ ; As shown in Fig. 3(d), curves of total ITE nearly merge to one when  $p$  is between 0.01 and 0.05. This convergence indicates that complete traversal of nodes in the hypergraph occurs with a sufficiently high propagation probability, leading to the stabilization
+
+of the total dynamic ITE. Additionally, it is noteworthy that the performance sharply declines when  $p = 0.05$ , possibly due to the broader diffusion process intensifying the randomness in dynamic ITE and consequently augmenting the uncertainty of influence spreading. In conclusion, with  $p$  changing in a certain range, our algorithm remains stable, and  $p = 0.02$  is an empirically good choice for achieving the best performance.
+
+# 6 Conclusion and Discussion
+
+In this paper, we analyze traditional IM from a causality perspective. Our CauIM framework can extract approximately optimal seed sets to achieve novel influence maximization. Discussion on Sketch-based Models. While sketch-based models are efficient, they may struggle to adapt to dynamic node weights during seed selection, unlike greedy-based approaches. This limitation can lead to instability, especially in sparse or heterogeneous networks. Our acceleration technique is flexible across different graph structures and offers a potential remedy. Future work will explore extending sketch-based approximations to more complex heterogeneous networks.
+
+# Acknowledgements
+
+Zhiheng Zhang is supported by "the Fundamental Research Funds for the Central Universities" (number: 2025110602) of Shanghai University of Finance and Economics. This work is also partially supported by the Basic Research Fund of the Computer Network Information Center, Chinese Academy of Sciences (Frontier Technology Innovation Project, No. E5553601).
+
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+
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+
+![](images/cd801c0c86e55afdff8822cedba0e28ea9cb88214312df1c1b016b0e3d589d33.jpg)  
+(a) Overall Hypergraph Diagram
+
+![](images/90d011d3b3142dce175dfe5efd5d95afe8c73c496a12fae2d2e6b9e71883e850.jpg)  
+(b) Spread Process  
+Figure 4: An illustration of the propagation process. With  $v_{6}$  infected, the hyperedges  $(h_{2}, h_{3}$  and  $h_{4}$  with solid line) containing  $v_{6}$  are chosen to be the candidates. Nodes  $(v_{3}, v_{4}, v_{5}, v_{7})$  in these hyperedges will potentially convert to the infected state. The spreading probability  $P_{IC}$  is the internal parameter in the hypergraph-based Independent Cascading (IC) model [56].
+
+# A Methodology Details
+
+Further methodological details, extended analyses, additional experiments, and core codes are provided in https://github.com/suxinyan/caoim3236/.
+
+Table 3: Notations.  
+
+<table><tr><td>Symbol</td><td>Descriptions</td></tr><tr><td>vi,hj</td><td>node i ,hyperedge j in hypergraph G: vi ∈ V, hj ∈ H</td></tr><tr><td>H</td><td>the hypergraph structure matrix</td></tr><tr><td>Xi</td><td>the covariate of node vi</td></tr><tr><td>Ti</td><td>the activating status (treatment) of node vi</td></tr><tr><td>X- i, T- i</td><td>environmental information of node vi, including covariates and the activation status of surrounding nodes (see Eq. 1)</td></tr><tr><td>Ŷi(1; ·),Ŷi(0; ·)</td><td>the estimated potential outcome of node vi</td></tr><tr><td>τi</td><td>the estimated ITE of node vi in certain diffusion process through Function ITE(·) (see Alg. 1)</td></tr><tr><td>ap(vi, S)</td><td>diffusion probability from set S to the node vi</td></tr><tr><td>Eτi</td><td>approximation of expectation on τi</td></tr></table>
+
+The procedure of G-CauIM is provided in Fig. 5.
+
+# B Comparison
+
+A detailed comparison of our CauIM within three types of IM frameworks is presented in Table 4, while a comparison of CauIM with the general traditional IM framework is provided in Table 5. As shown in Table 4, the Simulation-based CauIM is successfully implemented in this work.
+
+# C Experiments Details
+
+# C.1 Experimental Settings
+
+Our real-world data comprises three real-world public datasets: GoodReads  $^{8}$ , Contact [39], and Email-Eu [4]. Furthermore, we incorporate a synthetic dataset named SD-100 comprising 100 nodes and 100 hyperedges, whose initial treatments and feature settings are detailed in Section C.2. We compare the performance of G-CauIM and A-CauIM with the traditional greedy selection method without parameter adjusting strategy (Noted as "Baseline") on the aforementioned four datasets. Additionally, we choose the randomized seed selection strategy as another baseline. We randomly conduct each experiment for 10 times and each time for 20 rounds in influence estimation. For basic hyperparameters, we set seed number  $K = 15$  and spread probability  $P_{SICP}$  as 0.01 (denoted as  $p$  for simplicity). The evaluation metric is the sum of ITE spread by selected seeds with the same trained ITE estimation module illustrated in Section 4.
+
+<table><tr><td></td><td>Simulation-based</td><td>Proxy-based</td><td>Sketch-based</td></tr><tr><td>Basic idea</td><td>Use Monte-Carlo (MC) simulation to evaluate ITE influence spread</td><td>Design proxy models to approximate influence function with varying ITE</td><td>Construct reachability/activation sketches to accelerate greedy selection; guarantees typically assume fixed, context-independent, nonnegative gains (can be used as auxiliary approximations within CauIM when negative gains are limited or controlled.)</td></tr><tr><td>property</td><td>NP-hard complexity, total theoretical guarantee</td><td>Polynomial/linear complexity, no theoretical guarantee</td><td>Quasi-linear complexity, approximation guarantees under fixed nonnegative gains and specific diffusion models; no guarantee for context-dependent or negative ITE</td></tr><tr><td>Disadvantages</td><td>Computational overheads</td><td>Sensitive to the unstable scenarios</td><td>Not general to a wider range of diffusion models</td></tr><tr><td>Datasets</td><td>Small to medium-sized datasets under all propagation models</td><td>Large-scale datasets with distinctive graph structure under specific propagation models</td><td>Large graphs under Trigger-ing/LT settings with fixed gains; less suitable when gains are dynamic</td></tr><tr><td>Examples</td><td>Greedy, CELF (famous)</td><td>HADP [56], HSD [56], EIOA [41]</td><td>RIS [5], IMM [49], BKRIS</td></tr></table>
+
+Table 4: Comparison of three types of traditional IM in the context of our CauIM framework.  
+
+<table><tr><td></td><td>CauIM</td><td>Traditional IM</td></tr><tr><td>Basic idea</td><td>Leverages observational data to estimate the ITE of each node and to maximize the sum of varying ITEs among the infected individuals considering environmental information</td><td>Maximize the numbers of infected individuals</td></tr><tr><td>Objective function</td><td>arg maxS⊆V ∧ |S| = K E[|Φ(S)|], where Φ(S) = ∑vi∈S Eτi measures the sum of ITE (informal, parameter of interference is omitted)</td><td>arg maxS⊆V ∧ |S| = K E[|Φ(S)|], where Φ(S) measures the sum of infected numbers</td></tr><tr><td>Application Scenarios</td><td>Maximize total sum of individual gains</td><td>Maximize infected numbers</td></tr></table>
+
+Table 5: A comparison between CauIM and general traditional IM (Notice that CauIM is more challenging than the sum-weighted IM, since the ITE would be negative and varying during different propagation.)
+
+![](images/9268f26be06b36d7f2deb360704b96cc87b09b0de5a360311a8feb4e5787c20b.jpg)  
+Figure 5: The procedure of G-CauIM.  $T_{i}$  indicates whether the node is activated or not (we can only observe one situation for each node  $v_{i}$ ) and  $T_{-i}$  represents activated status of its surroundings, as illustrated in Eq. 2. For each round, we construct the ITE estimation  $\hat{\tau}_{i} \coloneqq \widehat{ITE} (\theta^{opt})$  mentioned in Section 4 and then treat it as the node weight. Furthermore, we conduct a weighted greedy algorithm with SICP propagation mechanism (Fig. 4). As a result, we expand the seed set  $(v_{4}$  is added), targeting the (estimated) largest sum of ITE. The main challenge is that  $\hat{\tau}_{i}$  are not constants, since the omitted parameter (Eq. 2) changes according to the different activation status of nodes in each iteration.
+
+# C.2 Details of Datasets and Problem Background
+
+GoodReads [22, 50] collects information on different categories of books, with each item containing the book title, content, and other details. Using the "Author-book" relationship, a node represents the specific book category and hyperedge aggregates books written by the same author in our hypergraph. We consider a scenario of recommending book sales, where the diffusion process is facilitated by reader groups associated with each book category node in the "Author-book" network. Treatment denotes the recommendation for book sales, with  $t_i$  set to 1 when book node  $i$  is recommended. Each book category is associated with a potential sales income, influenced by both its own recommendation status and the performance of other books within the same hyperedge. Our goal is to identify a k-set (or k kinds) of books to sell at the beginning, aimed at maximizing the total sales gap between the recommended and non-recommended. Our core optimization function is to maximize the sum of ITE, where the ITE of each book means the difference in potential sold income with/without the recommendation in our experiment. For Contact [4], it constructs simplicial complexes by grouping individuals in close connectivity at the same timestamp, represented by hyperedges. Here we simulate a situation where we deliver an AIDS Awareness Talks to particular students, and the core concepts can be disseminated through hyperegde groups. Our objective is to select initial student representatives to be educated in order to maximize the overall benefits of the talk (This can also be viewed as maximizing the total sum difference between having the anti-drug talk and not having it). We simulate covariates  $(X_i)$  of each student in a Mixed Gaussian distribution, considering differences among diverse groups:
+
+$$
+X _ {i} \sim \sum_ {j = 1} ^ {L} \omega_ {i} \mathcal {N} (\mu_ {j}, I). \tag {8}
+$$
+
+Here we set  $L = 4, \omega_{1} = 0.4$ . Moreover,  $\{\omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}\} = \{0.4, 0.2, 0.1, 0.3\}$ ,  $\{\mu_{1}, \mu_{2}, \mu_{3}, \mu_{4}\} = \{0.2, -0.25, -0.3, 0.5\}$ .
+
+Email dataset shares the similar scenario with Contact. The ratio of nodes to hyperedges is different: For example, Goodreads is 36 and Contact is 0.14. The simulation of ITE and basic treatment settings for both datasets follow [37]. The ITE distribution of GoodReads and Contact is shown in Fig. 6. It is worth noting that the ratio of nodes to hyperedges is contrasting between the two datasets, yielding sparse and relatively dense graphs, respectively. This exemplifies the diversity of our data selection, allowing for a more robust evaluation of algorithmic performance. Synthetic dataset SD generates initial treatments  $T_{i} \sim \text{Bernoulli}(r_{0})$ , where  $r_{0}$  denotes an average affected ratio of nodes (which can be calculated easily through propagation simulations). Its covariates  $(X_{i})$  are simulated following Eq. (8), with distinct values of  $\{\omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}\} = \{0.4, 0.25, 0.15, 0.2\}$ , representing different group ratio.
+
+# C.3 Supplementary Descriptions of Basic Assumptions
+
+The basic assumptions of the book-selling scenario should be satisfied: 1) Books and authors are many-to-many relationships; 2) the number of each kind of books sold initially is the same (we will take it as our future research
+
+![](images/f7a1c8d4b8b6c46251eb29766cc89effcf2569da3d92b23efcda4ee59034a10f.jpg)  
+Figure 6: a) Distribution of individual ITE in GoodReads Dataset. b) Distribution of individual ITE in Contact Dataset.
+
+![](images/d569ab3195903b3d0d0c25906278e2bdfcd61871bc5da5d0cceb6452c77d624d.jpg)
+
+topic if not the same); 3) Readers can learn about other books simply from authors of owned books. 4) Temporal change of each hyperedge is not considered. Assumptions for the other example are similar with the exception that rule 3 is replaced by "Students can learn about core ideas simply from other students." Our diffusion framework is established by hypergraph-based IC as described in Section 4. Within this model, the spread probability represents the willingness of readers to purchase the next book in the GoodReads dataset and the likelihood of core concepts spreading among students in the Contact or Email dataset.
+
+# C.4 Supplementary Descriptions of Parameter Settings
+
+Our experiments are conducted on Linux operating system with Python 3.10.14, torch 2.1.
+
+# C.5 Parameter Analysis
+
+Still, after the number of seeds exceeds 10, the scale of affected nodes rises significantly, since the cumulative effect of noise becomes significant. Thus, we conclude that our model remains relatively robust when the noise standard deviation  $\sigma$  does not exceed 8. This result is consistent with our theoretical part. Aimed at indicating the volatility, we modify the scale of the noise to approximate instability degree and take on 10 realizations of each experiment, calculating the standard variance of ITE performance among those realizations as the final result. We conduct 20 groups of experiments with  $\sigma$  varying between 0 to 20, and use step 2 when  $\sigma$  is lower than 8 for its changes are not noticeable. While  $\sigma$  is larger than 10, the curves increase too drastically. Thus, results beyond this range are omitted from Fig. 3(c) for clarity.
+
+# D The proof of Proposition 4.1
+
+Due to the Influence Maximization (IM) problem itself being NP-hard, our CauIM can be naturally reduced to the traditional IM problem  $(\tau = 1)$  and is therefore also NP-hard, potentially involving additional complexity due to heterogeneous node weights.
+
+Proof sketch. Notice that when  $\tau_{i} = 1, \forall v_{i} \in \mathcal{V}$  and each hyperedge contains only one pair of nodes, then  $\mathrm{CauIM}$  will degenerate to the traditional IM problem. Moreover, the IM problem with the IC diffusion model has been demonstrated to be NP-hard in [28].
+
+In another perspective, we can directly prove the optimal seed set of CauIM is one solution of the famous weighted set cover problem, which is well-known as NP-hard [21]. The weighted set cover problem is equivalently defined as detecting whether there exists  $k$  subsets within the total  $m$  subsets, such that it can cover the universe of elements  $U$ . We aim to demonstrate the traditional weighted set cover problem can naturally generalize a specific instance in CauIM. We construct a bipartite graph, in which the left side denotes the total subsets  $\mathcal{S} = \{S_1, S_2, \ldots, S_m\}$ , while the right side denotes the elements  $u \in U$ , which can be seen as each node in the hypergraph. The edge between two sides is attributed with the probability 1. Notice that in this instance, there exists a  $k$ -set cover if and only if there exists  $k$ -seed set such that the activation can reach all  $|S| + |U| = K + |U|$  nodes. Hence, our CauIM is more complex and is NP-hard.
+
+![](images/d6618ada8ef9c108dbc60047f2515af988109bd59a5d52f473eaa1e6fca549f9.jpg)
+
+# E The proof of Lemma 4.2
+
+Proof. It is equivalent to consider CauIM with  $\tau_{i} = 1, v_{i} \in \mathcal{V}$ . This approximate optimal guarantee is due to two elegant properties: 1) monotonicity and 2) submodularity. Firstly, according to  $\sigma(S_0 \cup v) - \sigma(S_0) \geq 0$  when  $\tau_{i} = 1$ , the monotonicity naturally holds. Secondly, we consider the submodularity in the hypergraph. This part has been proved by [13], where they constructed an augmented graph  $G^{aug} = (\mathcal{V} \cup \mathcal{H}, \mathcal{E})$ . Here the edge  $\mathcal{E}$  is composed by  $e := (v, h)$ ,  $v \in \mathcal{V}$ ,  $h \in \mathcal{H}$ . Then conduct the same submodularity analysis as in the traditional graph, and the proof is completed.
+
+![](images/b31bd35cdfd4241f9fb912758acc0bd88872c8306bea06bf85f5da99703c8c4b.jpg)
+
+# F The proof of Theorem 4.4
+
+The first result based on  $\tau_{i} > 0$  naturally holds. It is because the monotonicity and submodularity still hold when each node is attributed with non-negative weights (i.e., ITE). We focus on the second result, when  $\tau_{i} > 0$  is not guaranteed, these two important properties will not further hold. We extend the analysis to a generalized form of weak monotonicity and weak submodularity.
+
+Proof. To summarize, the core part is the following three claims:
+
+(Claim 1)  $\sigma(S^{*}) \leq \sigma(S^{*} \cup S_{i}^{g}) + i\varepsilon_{2}$ .
+
+(Claim 2)  $\sigma(S^{*} \cup S_{i}^{g}) \leq \sigma(S_{i+1}^{g}) - \sigma(S_{i}^{g}) + \sigma(S_{i}^{g} \cup S_{K-1}^{*}) + \varepsilon_{1}\varepsilon_{2}$ .
+
+(Claim 3)  $\sigma(S_i^g \cup S_{K-1}^*) \leq (K-1)[\sigma(S_{i+1}^g) - \sigma(S_i^g)] + \sigma(S_g^i) + (K-1)\varepsilon_1\varepsilon_2$ . The optimal  $K$ -seed set is denoted as  $S^* = \{s_1^*, s_2^*, \ldots, s_K^*\}$ ,  $S_k^* = \{s_1^*, s_2^*, \ldots, s_k^*\}$ , and the set output from our greedy CauIM as  $S^* = \{s_1^*, s_2^*, \ldots, s_K^*\}$ ,  $S_k^* = \{s_1^*, s_2^*, \ldots, s_k^*\}$ ,  $k \in [K]$ . Following [52], notice that
+
+$$
+\sigma (S) = \sum_ {u \in S} \left(\sum_ {v \in V} \tau_ {v} \cdot p _ {r} (u, v) + \tau_ {u}\right). \tag {9}
+$$
+
+We first construct the facilitating claim to analyze the variant of the monotonicity and the submodularity property.
+
+Claim 1:  $\sigma(S^{*}) \leq \sigma(S^{*} \cup S_{i}^{g}) + i\varepsilon_{2}$ .
+
+This claim can be achieved recursively. Considering two sets  $T_{1} \subseteq T_{2}$ , and an additional vertex  $v \not\subset T_{1}$ , we can follow [52] and achieve:
+
+$$
+\left(\sigma \left(T _ {1} \cup v\right) - \sigma \left(T _ {1}\right)\right) = \sum_ {v _ {i} \in R (v)} \tau_ {i} \cdot p _ {v v _ {i}} \left(1 - p _ {T _ {1}, v _ {i}}\right) \leq \sum_ {v _ {i} \in R (v)} | \tau_ {i} | p _ {v v _ {i}} = \varepsilon_ {2}. \tag {10}
+$$
+
+Then Claim 1 follows by recursively applying this bound  $i$  times.
+
+Claim 2:  $\sigma(S^{*} \cup s_{i}^{g}) \leq \sigma(S_{i+1}^{g}) - \sigma(S_{i}^{g}) + \sigma(S_{i}^{g} \cup S_{K-1}^{*}) + \varepsilon_{1}\varepsilon_{2}$ .
+
+We make an extension of submodularity. Considering  $S \subseteq T \subseteq \mathcal{V}$ , we have
+
+$$
+\begin{array}{l} \left(\sigma (S \cup v) - \sigma (S)\right) - \left(\sigma (T \cup v) - \sigma (T)\right) \\ = \sum_ {v _ {i} \in R (v)} \tau_ {i} \cdot p _ {v v _ {i}} \left(p _ {T, v _ {i}} - p _ {S, v _ {i}}\right) \leq \varepsilon_ {1} \varepsilon_ {2}. \tag {11} \\ \end{array}
+$$
+
+Hence  $\sigma(S^{*} \cup s_{i}^{g})$  can be bounded as follows:
+
+$$
+\begin{array}{l} \sigma \left(S ^ {*} \cup s _ {i} ^ {g}\right) = \sigma \left(S _ {K - 1} ^ {*} \cup s _ {K} ^ {*} \cup S _ {i} ^ {g}\right) \\ \leq \sigma \left(S _ {i} ^ {g} \cup s _ {K} ^ {*}\right) - \sigma \left(S _ {i} ^ {g}\right) + \sigma \left(S _ {i} ^ {g} \cup S _ {K - 1} ^ {*}\right) + \varepsilon_ {1} \varepsilon_ {2} \tag {12} \\ \leq \sigma \left(S _ {i + 1} ^ {g}\right) - \sigma \left(S _ {i} ^ {g}\right) + \sigma \left(S _ {i} ^ {g} \cup S _ {K - 1} ^ {*}\right) + \varepsilon_ {1} \varepsilon_ {2}. \\ \end{array}
+$$
+
+The last line is due to the selection nature of the greedy algorithm.
+
+Claim 3:  $\sigma(S_i^g \cup S_{K-1}^*) \leq (K-1)[\sigma(S_{i+1}^g) - \sigma(S_i^g)] + \sigma(S_g^i) + (K-1)\varepsilon_1\varepsilon_2.$
+
+It is due to
+
+$$
+\begin{array}{l} \sigma \left(S _ {i} ^ {g} \cup S _ {k - 1} ^ {*}\right) = \sigma \left(S _ {i} ^ {g} \cup S _ {k - 2} ^ {*} \cup s _ {k - 1} ^ {*}\right) \\ \leq \sigma \left(S _ {i} ^ {g} \cup s _ {k - 1} ^ {*}\right) - \sigma \left(S _ {i} ^ {g}\right) + \sigma \left(S _ {g} ^ {i} \cup S _ {k - 2} ^ {*}\right) + \varepsilon_ {1} \varepsilon_ {2} \tag {13} \\ \leq \sigma \left(S _ {i + 1} ^ {g}\right) - \sigma \left(S _ {i} ^ {g}\right) + \sigma \left(S _ {g} ^ {i} \cup S _ {k - 2} ^ {*}\right) + \varepsilon_ {1} \varepsilon_ {2} \\ \end{array}
+$$
+
+The last line is due to the selection nature of the greedy algorithm. Therefore, recursively, we have
+
+$$
+\sigma \left(S _ {i} ^ {g} \cup S _ {K - 1} ^ {*}\right) \leq (K - 1) \left[ \sigma \left(S _ {i + 1} ^ {g}\right) - \sigma \left(S _ {i} ^ {g}\right) \right] + \sigma \left(S _ {g} ^ {i}\right) + (K - 1) \varepsilon_ {1} \varepsilon_ {2}. \tag {14}
+$$
+
+Combined with Eq. (12), Eq. (F) and Claim 3, we have
+
+$$
+\sigma \left(S ^ {*}\right) \leq K \left[ \sigma \left(S _ {i + 1} ^ {g}\right) - \sigma \left(S _ {i} ^ {g}\right) \right] + \sigma \left(S _ {i} ^ {g}\right) + K \varepsilon_ {1} \varepsilon_ {2} + i \varepsilon_ {2}. \tag {15}
+$$
+
+It equals to
+
+$$
+\begin{array}{l} \sigma \left(S _ {i + 1} ^ {g}\right) \geq \left(1 - \frac {1}{K}\right) \sigma \left(S _ {i} ^ {g}\right) + \frac {\sigma \left(S ^ {*}\right)}{K} - \varepsilon_ {1} \varepsilon_ {2} - \frac {i \varepsilon_ {2}}{K} \\ \left(1 - \frac {1}{K}\right) ^ {k - i - 1} \sigma \left(S _ {i + 1} ^ {g}\right) \tag {16} \\ \geq \left(1 - \frac {1}{K}\right) ^ {k - i} \sigma (S _ {i} ^ {g}) + \left(1 - \frac {1}{K}\right) ^ {k - i - 1} \left(\frac {\sigma (S ^ {*})}{K} - \varepsilon_ {1} \varepsilon_ {2} - \frac {i \varepsilon_ {2}}{K}\right). \\ \end{array}
+$$
+
+Take the sum of  $i \in \{0, 2, \dots, K - 1\}$ , we have
+
+$$
+\begin{array}{l} \sigma \left(S _ {K} ^ {g}\right) \geq \sum_ {i = 0} ^ {K - 1} \left(1 - \frac {1}{K}\right) ^ {k - i - 1} \frac {\sigma \left(S ^ {*}\right) - K \varepsilon_ {1} \varepsilon_ {2}}{K} - \sum_ {i = 0} ^ {K - 1} \left(1 - \frac {1}{K}\right) ^ {k - i - 1} \frac {i}{K} \varepsilon_ {2} \\ \geq \left(1 - \left(1 - \frac {1}{k}\right) ^ {k}\right) \left(\sigma \left(S ^ {*}\right) - K \varepsilon_ {1} \varepsilon_ {2}\right) - \varepsilon_ {2} \left(\frac {1}{e}\right) ^ {1 - \frac {1}{K}} \int_ {0} ^ {1} x e ^ {x} d x \tag {17} \\ \geq \left(1 - \frac {1}{e}\right) \left(\sigma (S ^ {*}) - K \varepsilon_ {1} \varepsilon_ {2}\right) - \varepsilon_ {2} e ^ {\frac {1}{K} - 1}. \\ \end{array}
+$$
+
+This lower bound is also applicable when  $\tau_{i}\leq 0$  exists.
+
+![](images/0356c14c4bfca722aae8a29cf4036cf7da158d4b2d55ade3b9bed558c9d115f3.jpg)
+
+# G The proof of Theorem 4.5
+
+Proof. Following Chen et al. [8], we have  $(k\in [K])$
+
+$$
+\begin{array}{l} f \left(S _ {i} ^ {g} \cup \left\{s _ {k} ^ {*} \right\}\right) \leq \frac {1}{1 - \gamma} \hat {f} \left(S _ {i} ^ {g} \cup \left\{\bar {s} _ {k} ^ {*} \right\}\right) \\ \leq \frac {1}{1 - \gamma} \hat {f} \left(S _ {i} ^ {g} \cup \{s _ {i + 1} \}\right) \tag {18} \\ \leq \frac {1 + \gamma}{1 - \gamma} f \left(S _ {i} ^ {g} \cup \{s _ {i + 1} \}\right). \\ \end{array}
+$$
+
+Analogously, we have
+
+$$
+f \left(S _ {i + 1} ^ {g}\right) \geq \frac {1 - \gamma}{1 + \gamma} \left(\left(1 - \frac {1}{K}\right) f \left(S _ {i} ^ {g}\right) + \frac {f \left(S ^ {*}\right)}{K} - \varepsilon_ {1} \varepsilon_ {2} - \frac {i \varepsilon_ {2}}{K}\right). \tag {19}
+$$
+
+# Algorithm 2: Monte Carlo-based greedy CauIM
+
+Require: Hypergraph  $\mathcal{G}(\mathcal{V},\mathcal{H})$  , size of the seed set  $K$  , a constant  $T$
+
+1: Initialization:  $S_0 = \emptyset, k = 0$ .  
+2: ITE recovery: Estimate node-level  $\tau_{i}$  from observational data or model-based inference.  
+3: for  $|S_0| < K$  do  
+4:  $v_{0} = \arg \max_{v\notin S_{0}}\{\mathbf{MC}(S_{0}\cup \{v\} ,T) - \mathbf{MC}(S_{0},T)\}$  
+5:  $S_0 = S_0 \cup \{v_0\}$  
+6: end for
+
+Ensure: The deterministic seed set  $S_0$  with  $|S_0| = K$ .
+
+# Function MC:
+
+Require: Iteration  $T$ , current node set  $S_0$ , estimated ITE  $\{\tau_i\}$
+
+7: count = 0  
+8: for  $i \in [T]$  do
+
+9: We conduct the diffusion process with  $T$  steps, and compute the sum of causal effects
+
+$$
+\sigma_ {T} (S _ {0}) = \sum_ {\text {n o d e} j \text {a c t i v a t e d}} \tau_ {j}; \text {c o u n t} = \text {c o u n t} + \sigma_ {T} (S _ {0}).
+$$
+
+10: end for
+
+Ensure: Return count/T.
+
+Hence, recursively,
+
+$$
+\begin{array}{l} f \left(S _ {K} ^ {g}\right) \geq \sum_ {i = 0} ^ {K - 1} \left(\frac {(1 - 1 / K) (1 - \gamma)}{1 + \gamma}\right) ^ {K - i - 1} \cdot \frac {1 - \gamma}{(1 + \gamma) K} \cdot \left[ f \left(S ^ {*}\right) - K \varepsilon_ {1} \varepsilon_ {2} \right] \\ - \sum_ {i = 0} ^ {K - 1} \left(\frac {(1 - 1 / K) (1 - \gamma)}{1 + \gamma}\right) ^ {k - i - 1} \frac {i}{K} \varepsilon_ {2} \\ \geq \frac {1 - \left(\frac {1 - \gamma}{1 + \gamma}\right) ^ {K} \left(1 - \frac {1}{K}\right) ^ {K}}{(1 + \gamma) K / (1 - \gamma) - K + 1} [ f (S ^ {*}) - K \varepsilon_ {1} \varepsilon_ {2} ] - \varepsilon_ {2} e ^ {\frac {1}{K} - 1} \\ \geq \frac {1 - \left(\frac {1 - \gamma}{1 + \gamma}\right) ^ {K} \cdot \frac {1}{e}}{(1 + \gamma) K / (1 - \gamma) - K + 1} [ f (S ^ {*}) - K \varepsilon_ {1} \varepsilon_ {2} ] - \varepsilon_ {2} e ^ {\frac {1}{K} - 1} \tag {20} \\ \geq \frac {1 - \frac {1}{e}}{(1 + \gamma) K / (1 - \gamma) - K + 1} [ f (S ^ {*}) - K \varepsilon_ {1} \varepsilon_ {2} ] - \varepsilon_ {2} e ^ {\frac {1}{K} - 1} \\ \geq \left(1 - \frac {1}{e}\right) \left(1 - \frac {(1 + \gamma) K}{1 - \gamma} + K\right) \left[ f \left(S ^ {*}\right) - K \varepsilon_ {1} \varepsilon_ {2} \right] - \varepsilon_ {2} e ^ {\frac {1}{K} - 1} \\ \geq \left(1 - \frac {1}{e} - \left(\frac {(1 + \gamma) K}{1 - \gamma} - K\right)\right) [ f (S ^ {*}) - K \varepsilon_ {1} \varepsilon_ {2} ] - \varepsilon_ {2} e ^ {\frac {1}{K} - 1} \\ \geq \left(1 - \frac {1}{e} - \varepsilon\right) \left[ f \left(S ^ {*}\right) - K \varepsilon_ {1} \varepsilon_ {2} \right] - \varepsilon_ {2} e ^ {\frac {1}{K} - 1}. \\ \end{array}
+$$
+
+![](images/571eee1231898c1d58f57d761075eb9cc49f4e70d896f92d0c94bda20aef53cd.jpg)
+
+# H The proof of Corollary 4.6
+
+Proof.
+
+$$
+\begin{array}{l} \left| \hat {\sigma} (S) - \sigma (S) \right| = \sum_ {u \in S} \left(\sum_ {v \in V} \left(\hat {\tau} _ {v} - \tau_ {v}\right) p _ {r} (u, v) + \hat {\tau} _ {v} - \tau_ {v}\right) \tag {21} \\ \leq \frac {\gamma \sigma (S)}{\sigma_ {n a i v e} (S)} \sum_ {u \in S} \left(\sum_ {v \in V} p _ {r} (u, v) + 1\right) := \gamma \sigma (S). \\ \end{array}
+$$
+
+Hence  $|\frac{\hat{\sigma}(S)}{\sigma(S)} - 1| \leq \delta \frac{\sigma_{naive}(S)}{\sigma(S)} \leq \gamma$ .
+
+![](images/d390ae098e0b6115bae265ff764b1421239c6e42decae3eed3a658f127ea91df.jpg)
+
+# I Auxiliary algorithms and Additional discussions
+
+More discussions of HGCN module The higher-order interference representation  $O_{i}()$  is learned according to to [37, 38], employing hypergraph convolution operator within HGCN module:  $\mathbf{O}^{(l + 1)} =$  LeakyReLU  $\left(\mathbf{LO}^{(l)}\mathbf{W}^{(l + 1)}\right)$  where  $\mathbf{L}$  denotes Laplacian matrix aggregating the graph feature information, and  $\mathbf{O}$  is initially calculated using  $Z$ .
+
+More discussions upon IM For the first limitation of IM, the exploration of hypergraph-based IM is urgent to be settled. Since the hypergraph structure is consistent with ample real-world scenarios, especially when different nodes in the graph contain high-level, multivariate relationships, where the traditional graph is hard to model efficiently. Take an example of the disease propagation problem regarded as IM on hypergraph in Fig. 4(a). Students are connected through social circles, where each circle can be represented as a hyperedge. Different from ordinary graphs, the influence of node  $v_{6}$  is spread not considering the edge consisting of a pair of nodes, but on more its affiliated hyperedges shown in Fig. 4(b). Existing hypergraph-based studies are mostly separated into two parts: 1) people are committed to developing heuristic methods but with not enough theoretical support [2, 56]. 2) people developed fundamental theoretical guarantees only on a specific form of hypergraph structure [66]. In general, a general hypergraph-based IM with theoretical guarantee and high empirical efficiency still needs to be explored.
+
+For the second limitation of IM, the original optimization objective needs to be reconsidered in many cases. The previous objective is directly implied as the sum of node numbers, which stems from empirical or even philosophical determinations and lacks rigorous mathematical arguments. This implication is attributed to the over-simplification of real-life situations—current IM methods tend to overlook the dynamic nature of node influence weights (ITE) within their environments. Traditional IM methods like simulation-based [28, 31] and sketch-based [5, 48, 49, 51] ones focusing on maximizing total numbers (or generalized weighted IM) might fail to pursue such maximum total potential benefit (pursue "larger varying weighted sum" instead of "larger number of nodes"). Wang et al. [52] proposed the new weighted IM problem whereas they rely on non-negative assumptions and lack generalizability to complex scenarios involving hypergraphs and varying node weights. Recently, learning-based IM methods [6, 27, 30, 42, 35] mostly learn potential node representations as a marginal gain of node influence, thus guiding the seed node finding process. Sharing different object functions from ours, many of these existing methods might struggle with limited generalization capabilities and result reliability concerns. Overall, there is a critical need to explore novel objective functions.
+
+More discussions upon extended algorithms To simplify the discussion, in the additional algorithms we provide, we have omitted the process of dynamically updating ITE based on the surrounding nodes' state changes during each propagation. CELF-CauIM (Alg. 2) and Monte Carlo-based greedy CauIM (Alg. 3) can be derived naturally from G-CauIM.
+
+# Algorithm 3: CELF-CauIM
+
+Require: Hypergraph  $\mathcal{G}(\mathcal{V},\mathcal{H},\mathbb{H})$  , size of the seed set  $K$  , causal influence function  $\sigma (\cdot)$
+
+1: Initialization:  $S^{*} = \emptyset$ ,  $MargDic = \emptyset$ .
+
+2: ITE recovery.
+
+3: MargDiC stores the marginal gain  $\sigma (\{v\})$  of each node.
+
+4: Sort MargDic in decreasing order of value.
+
+5: for  $|S^{*}| < K$  do
+
+6: Move out the node cur with the largest marginal gain in MargDic.
+
+7: Re-compute marginal gain of of node cur with the current seed set  $S^{*}$ : MargDiC[cur] = σ (S* ∪ {cur}) - σ (S*)
+
+8: Check if previous top node stays on top after sort MargDic again. If true,  $S^{*} = S^{*} \cup \{cur\}$ , and find the second seed; else remove the second largest marginal gain node in MargDic, then repeat the last operation.
+
+9: end for
+
+Ensure: The deterministic seed set  $S^{*}$  with  $|S^{*}| = K$ .
+
+For HADP-CauIM (Alg. 4), we replace the selection criterion of searching for nodes with the highest degree with the nodes with the highest sum of the average ITE among the neighboring nodes (line  $5 - 6$ ). Note that according to the common drawback of heuristic methods, this type of method has no theoretical support and often falls into a common dilemma: nodes with the highest local degree (neighboring ITE) may not necessarily represent seeds that can bring greater overall influence. This point has also been verified in the experiments.
+
+Algorithm 4: HADP-CauIM
+
+Require:  $\mathcal{G}(\mathcal{V},\mathcal{H},\mathbb{H})$  , size of the seed set  $K$  , causal influence function  $\sigma (\cdot)$
+
+1: Initialization:  $S^{*} = \emptyset$ , DegITE = {}.  
+/* Presented in Algorithm 1 */  
+2: ITE recovery.  
+3:  $DegITE[v]$  stores the sum of ITE of the neighbour nodes of each node  $v$ , where  $N_r(v)$  represents the neighbour nodes of  $v$ :  $DegITE[v] = \sum_{v_r \in N_r(v)} \mathbb{E}\tau_r$ .  
+4: while  $|S^{*}| < K$  do  
+5: Choose  $v_0$  with the max value in  $DegITE$  as the seed:  $v_0 = \arg\max_v \{DegITE[v]\}$ ,  $S^* = S^* \cup \{v_0\}$ .  
+6: Calculate sum of ITE for each node  $v_{r}$  in neighbors of  $v_{0}$  as edge value  $Edge$ :  $DegITE[v_{r}] = \sum_{v_{rq} \in N_{r}(v_{r})} \mathbb{E}\tau_{rq}$ .  
+7: Remove the edge influence of the chosen seed node  $v_{0}$ :  $DegITE[v_{r}] = DegITE[v_{r}] - Edge$ .  
+8: Remove  $v_{0}$  and its incident hyperedges from  $\mathcal{G}$ .  
+9: end while
+
+Ensure: The deterministic seed set  $S^{*}$  with  $|S^{*}| = K$ .
+
+More discussions upon submodularity of hypergraph Antelmi et al. [2], Zheng et al. [65] claimed their hypergraph does not contain submodularity. However, their propagation mechanism is different from traditional IC. Besides, they considered the form of directed hypergraph where a hyperedge  $(H,t)$  comprises a set of head nodes  $H$  is and a single tail node  $t$ . Further, Gangal et al. [13] demonstrated the submodularity of a general class of hypergraph. Moreover, Erkol et al. [11] stated that the submodularity on the temporal network might not be held.
+
+More discussions upon additional challenges compared to other sum-weighted IMs. Noteworthy, according to the ITE estimation form, CauIM can be seen as the generalized case of the weighted IM. However, our task is significantly more challenging. Firstly, beginning with the traditional graph, sum-weighted schemes often ensure an approximate optimum guarantee effortlessly, for traditional IM can be extended to weighted IM naturally (see definition 5 in Mossel and Roch [40]). However, it does not make sense in our setting since the ITEs for each node would not guarantee to be always non-negative (for instance, some non-compliers, i.e.,  $\mathrm{ITE} < 0$ , exist). Furthermore, the argument on submodularity is more complex in hypergraph (defer to I). Besides, the ITE estimation does not remain constant between iterations due to the experimental sensitivity (the activation status of surrounding nodes change). In sum, to the best of our knowledge, in the setting of hypergraph-based ITE, the weakened version of the approximate optimum guarantee is an effective supplement to the IM community.
+
+More future work This study introduces a causal influence maximization (CauIM) framework that captures environmental sensitivity and unobservability in network diffusion, establishing a bridge between causal inference and influence maximization. A promising direction for future work is to further unify this causal perspective with sequential and online design paradigms developed in recent research on networked experimentation [61, 54]. Specifically, integrating the CauIM framework with online experimental design could enable adaptive intervention strategies that dynamically estimate and optimize individual influence gains while providing anytime-valid inference guarantees. Moreover, the interplay between partial identification [60, 59] and network diffusion can be explored to quantify uncertainty when the activation mechanisms or environmental variables are only partially observed, leading to bounds on causal influence under incomplete network information. Robust and proxy-based identification methods [63] may further improve reliability by mitigating hidden confounding in heterogeneous or noisy propagation environments. Another direction is to extend CauIM to dynamic or temporal networks, leveraging Granger-style causality models [62] to capture time-varying dependencies in diffusion processes. Finally, combining CauIM with active treatment-effect estimation under limited sampling budgets [64] and structural constraints such as topological regularity [58] can yield a comprehensive framework for optimizing both causal inference accuracy and influence efficiency across evolving network settings.
+
+# J Broader Impact.
+
+Our work studies algorithmic optimization on synthetic and publicly available network data. Potential positive impacts include improved targeting for public-health messaging and resource allocation. Potential risks include misuse for manipulative advertising. We discuss mitigation by enforcing transparency about the optimization objective, prohibiting protected-attribute targeting, and auditing estimated ITEs for bias.
\ No newline at end of file
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+# Unveiling Extraneous Sampling Bias with Data Missing-Not-At-Random
+
+Chunyuan Zheng $^{1}$  Haocheng Yang $^{2}$  Haoxuan Li $^{1,*}$  Mengyue Yang $^{3,*}$
+
+$^{1}$ Peking University  $^{2}$ National University of Singapore  $^{3}$ University of Bristol
+
+cyzheng@stu.pku.edu.cn
+
+# Abstract
+
+Selection bias poses a widely recognized challenge for unbiased evaluation and learning in many industrial scenarios. For example, in recommender systems, it arises from the users' selective interactions with items. Recently, doubly robust and its variants have been widely studied to achieve debiased learning of prediction models, however, all of them consider a simple exact matching scenario, i.e., the units (such as user-item pairs in a recommender system) are the same between the training and test sets. In practice, there may be limited or even no overlap in units between the training and test. In this paper, we consider a more practical scenario: the joint distribution of the feature and rating is the same in the training and test sets. Theoretical analysis shows that the previous DR estimator is biased even if the imputed errors and learned propensities are correct in this scenario. In addition, we propose a novel super-population doubly robust estimator (SuperDR), which can achieve a more accurate estimation and desirable generalization error bound compared to the existing DR estimators, and extend the joint learning algorithm for training the prediction and imputation models. We conduct extensive experiments on three real-world datasets, including a large-scale industrial dataset, to show the effectiveness of our method. The code is available at https://github.com/ChunyuanZheng/neurips-25-SuperDR.
+
+# 1 Introduction
+
+Selection bias means the distribution of collected data differs from that in the target population. It is ubiquitous and occurs when data are missing-not-at-random (MNAR). For example, in the recommender system (RS), due to the subjective preferences of users and the data collection process itself, selection bias always exists in the collected data [1, 2]. Thus, selection bias poses a widely-recognized challenge [3, 4, 5, 6]. Ignoring selection bias makes machine learning methods difficult to achieve unbiased predictions and reducing its reliability [7, 8].
+
+Many methods have been proposed to address selection bias. The error imputation-based (EIB) method [9, 10] utilizes an imputation model to impute the missing relevance. The inverse propensity score (IPS) method uses inverse propensity to reweight the observed events to achieve unbiasedness [11, 12, 13]. The doubly robust (DR) method combines the error imputation model and the propensity model [14, 15, 16, 17, 18, 19], which is unbiased if either the imputed errors or the learned propensities are accurate, and is also proved to have lower variance compared to IPS [20].
+
+Although previous DR-based methods have demonstrated promising performance in debiasing tasks, all of them consider the exact matching scenario, i.e., the units are the same between training and test set. In RS, it means the users and items are the same between training and test set, as illustrated in the left part in Figure 1. However, there may be limited or even no overlap in units between training and
+
+![](images/827d741483db5624c75abe3fc49e357400451d3aae54c7564505db91f4d181a5.jpg)  
+Figure 1: (1) Traditional debiasing methods consider an exact matching scenario, where the users and items are the same between training and test set, the non-random missingness (missing two negative ratings) indicates selection bias; (2) The cold-start problem refers to the training data containing only hot users/items, while the test data has only cold users/items, which differs from the debiasing scenario intrinsically; (3) This paper extends the exact matching scenario, considering a more general scenario that the joint distribution of the feature and rating  $\mathbb{P}(x,r)$  in training and test set is the same.
+
+![](images/0ad6bf4a0311c8aa5086cb954534a4724578e1eebd3d6ed6fb4a12c60a19ebfe.jpg)
+
+![](images/c939292ef6fa0719ad71b73639bac24606895e8eec9e36e3da2aac77ebae6290.jpg)
+
+test sets. For example, in many industrial scenarios, the user-item pairs in offline training data cannot be exactly the same as those in online test data [21, 22]. Therefore, instead of exact matching, a more practical scenario is that the joint distribution of the feature and outcome  $\mathbb{P}(x,r)$  is the same in training and test set, as illustrated in the right part in Figure 1. Note that this scenario is intrinsically different with the cold-start problem, where the training data distribution  $\mathbb{P}_{\mathrm{hot}}$  differs from the test data distribution  $\mathbb{P}_{\mathrm{cold}}$ , as shown in the middle part in Figure 1. In addition, selection bias is rarely considered in the cold-start problem.
+
+To this end, in this paper, we first derive the bias of the DR estimator in super-population scenario, which contains an additional covariance term besides the term that measures the accuracy of imputed errors and learned propensities. Surprisingly, the DR estimator is biased even if the imputed errors and learned propensities are correct. Then we provide unbiased conditions under the super-population scenario, and propose the SuperDR estimator with the corrected imputation model, which can effectively control the additional covariance term with many desirable theoretical properties such as bias and variance reduction, leading to a more accurate estimation under the super-population scenario. In addition, we extend the previous joint learning algorithm based on the proposed corrected imputation loss and further derive the generalization error bound for the proposed SuperDR, and show that the proposed learning approach can effectively control it. Extensive experiments are conducted on three real-world datasets to show the effectiveness of our SuperDR method.
+
+Our main contributions can be summarized as follows:
+
+- To the best of our knowledge, this is the first paper that extends the exact matching scenario to the super-population scenario. In this scenario, we derive the bias of the DR estimator and show DR estimator is biased even if the imputed errors and learned propensities are correct.  
+- We propose the SuperDR method based on the corrected imputation model, which can effectively control the additional covariance term. In addition, the corrected imputation model not only benefits unbiased estimation but also reduces the generalization bound to enhance prediction performance.  
+- We conduct extensive experiments on three real-world datasets, including a large industrial dataset, to demonstrate the effectiveness of our proposed method.
+
+# 2 Related Work
+
+There are various biases in the collected data [23, 24], which have been of increasing concern in recent years [25, 26, 27, 28, 29, 30, 31]. Selection bias is one of the most common biases and a lot of research has been done aiming to eliminate this kind of bias [32, 33, 34, 35, 36]. Based on the causal inference techniques [37, 38, 39, 40, 41, 42, 43], the error imputation based method (EIB) [44, 45] first imputes pseudo-labels for missing events from the observed events, and then leverages these pseudo-labels to train the prediction model [46]. The propensity-based approaches weight the inverse
+
+propensity score (IPS) on the observed data to eliminate bias [11, 12, 13]. However, IPS will suffer from a large variance when the extreme values exist in the estimated propensities [47].
+
+The doubly robust (DR) method improves the weakness of EIB and IPS methods and becomes the mainstream model due to the weaker unbiased conditions (unbiased when either imputed errors or learned propensities are correct) and smaller variance than the IPS method [48, 49, 50, 51, 52]. In particular, the DR estimator is unbiased when either learned propensities or imputed errors are accurate. Many augmented DR methods are developed to further enhance the previous DR method performance by modifying the propensity model and imputation model or the form of the DR estimator, such as MRDR [33], BRD-DR [53], StableDR [54], TDR [55], DR-MSE [56], MR [57], CDR [58], AKBDR [59], DCE-TDR [60], and D-DR [61]. In addition, there are methods leveraging few unbiased ratings to mitigate hidden confounding and improve DR debiasing efficacy [5, 62, 63]. In this paper, we consider a more general super-population scenario and propose SuperDR with the corrected imputation model to achieve a more accurate estimation.
+
+# 3 Preliminaries
+
+We start with the classic debiasing scenario and take RS as an example. Note that the selection bias also exists in other scenarios such as pattern recognition and causal effects estimation, and our proposed method is also applicable for debiasing in these scenarios. Suppose the training user set  $\mathcal{U}_{\mathrm{train}} = \{u_1,u_2,\dots ,u_m\}$  contains  $m$  users, the item set  $\mathcal{I}_{\mathrm{train}} = \{i_1,i_2,\ldots ,i_n\}$  contains  $n$  items. The purpose of RS is to train a prediction model to accurately predict the ratings of all user-item pairs, thus the target population is defined as all user-item pairs  $\mathcal{D}_{\mathrm{target}} = \mathcal{U}_{\mathrm{train}}\times \mathcal{I}_{\mathrm{train}}$ . Let  $\mathbf{R}\in \mathbb{R}^{m\times n}$  be the ground truth rating matrix of all user-item pairs in  $\mathcal{D}_{\mathrm{target}}$ , where  $r_{u,i}$  is the rating of user  $u$  on item  $i$ . Let  $x_{u,i}$  be the feature of user  $u$  and item  $i$ , and  $\hat{r}_{u,i} = f(x_{u,i};\theta)$  is the predicted rating by a prediction model,  $\theta$  is the corresponding parameter. Denote  $\hat{\mathbf{R}}\in \mathbb{R}^{m\times n}$  as the matrix containing all the predicted ratings. Let  $\mathbf{O}\in \{0,1\}^{m\times n}$  be the binary observation indicator matrix for all user-item pairs,  $o_{u,i} = 1$  indicates the rating of user  $u$  on item  $i$  is observed, otherwise missing  $o_{u,i} = 0$ . All previous methods implicitly assume  $\mathcal{D}_{\mathrm{target}} = \mathcal{D}_{\mathrm{test}} = \mathcal{D}$  with fixed user-item pairs, thus the only randomness comes from the missing mechanism. If all the ratings are observed, the prediction model can be trained directly by minimizing the following ideal loss
+
+$$
+\mathcal {L} _ {\mathrm {i d e a l}} (\theta) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} e _ {u, i},
+$$
+
+where  $e_{u,i} = \mathcal{L}(\hat{r}_{u,i}, r_{u,i})$  is the loss between the predicted rating  $\hat{r}_{u,i}$  and the true rating  $r_{u,i}$  and  $\mathcal{L}(\cdot, \cdot)$  is an arbitrary loss function. However, the ideal loss is not available in most cases because we can only observe part of the data with selection bias. Thus, for user-item pair with  $o_{u,i} = 0$ , the  $r_{u,i}$  is missing. To tackle this issue, the DR estimator has been proposed:
+
+$$
+\mathcal {E} _ {\mathrm {D R}} (\theta) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \left[ \hat {e} _ {u, i} + \frac {o _ {u , i} (e _ {u , i} - \hat {e} _ {u , i})}{\hat {p} _ {u , i}} \right].
+$$
+
+where  $\hat{p}_{u,i} = \pi (x_{u,i};\psi)$  is the propensity model to estimate  $p_{u,i}\coloneqq \mathbb{P}(o_{u,i} = 1\mid x_{u,i})$ , and  $\hat{e}_{u,i} = m(x_{u,i};\phi)$  is the imputation model to impute the missing  $e_{u,i}$ .
+
+# 4 Proposed Method
+
+# 4.1 From Finite Population to Super-population
+
+Before introducing our method, we first focus on the theoretical properties of the DR estimator and start from the bias form of DR estimator.
+
+Lemma 4.1 (Bias of DR Estimator [14]). Given imputed errors  $\hat{e}_{u,i}$  and learned propensities  $\hat{p}_{u,i} > 0$ , when considering only the randomness of missing indicators, the bias of DR estimator is
+
+$$
+\mathrm {B i a s} _ {\mathbf {O}} [ \mathcal {E} _ {\mathrm {D R}} (\theta) ] = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {\{\hat {p} _ {u , i} - p _ {u , i} \} \cdot \{e _ {u , i} - \hat {e} _ {u , i} \}}{\hat {p} _ {u , i}}.
+$$
+
+We find that either  $\hat{e}_{u,i} = e_{u,i}$  or  $\hat{p}_{u,i} = p_{u,i}$  is sufficient to eliminate bias, which inspires the double robustness condition for the DR method.
+
+Corollary 4.2 (Double Robustness [14]). The DR estimator is unbiased when either imputed errors  $\hat{e}_{u,i}$  or learned propensities  $\hat{p}_{u,i}$  are accurate for all user-item pairs, i.e., either  $\hat{e}_{u,i} = e_{u,i}$  or  $\hat{p}_{u,i} = p_{u,i}$  for all  $u$  and  $i$ .
+
+The above Lemma 4.1 shows the bias form of the DR estimator when users and items in the training set and the users and items in the test set are exactly the same. However, as we discussed earlier, this scenario is too simple in many real-world scenarios. Thus, we consider a more general scenario, also known as super-population, with  $\mathcal{U} = \{u_1, u_2, \ldots\}$ ,  $\mathcal{I} = \{i_1, i_2, \ldots\}$ .  $\mathcal{D}_{\mathrm{target}} = \{u_1, u_2, \ldots, u_m\} \times \{i_1, i_2, \ldots, i_n\}$  and  $\mathcal{D}_{\mathrm{test}} = \{u_{j_1}, u_{j_2}, \ldots, u_{j_{m'}}\} \times \{i_{k_1}, i_{k_2}, \ldots, i_{k_{n'}}\}$  are sampled from the whole user set and item set, respectively. Without loss of generality, we consider the sampling strategy to be the same for both  $\mathcal{D}_{\mathrm{target}}$  and  $\mathcal{D}_{\mathrm{test}}$  datasets (otherwise, we can adjust the sampling strategy by reweighting). Therefore, instead of  $\mathcal{D}_{\mathrm{target}} = \mathcal{D}_{\mathrm{test}} = \mathcal{D}$ , we consider a more practical scenario  $\mathbb{P}(\mathcal{D}_{\mathrm{target}}) = \mathbb{P}(\mathcal{D}_{\mathrm{test}}) = \mathbb{P}(\mathcal{D})$ , that is, the joint distribution  $\mathbb{P}(x,r)$  in the target and the test population are the same. We can regard the ground-truth ratings and covariate values as drawing  $|\mathcal{D}_{\mathrm{target}}|$  times from the  $\mathbb{P}(x,r)$  in the super-population and are therefore they are stochastic. Furthermore, the randomness of ratings and covariates leads to the randomness of all other variables such as  $e_{u,i}$  and  $\hat{e}_{u,i}$ . For unbiased prediction in this scenario, we need to estimate the expected ideal loss below:
+
+$$
+\mathcal {L} _ {\text {i d e a l}} ^ {*} (\theta) = \mathbb {E} [ \mathcal {L} _ {\text {i d e a l}} (\theta) ] = \mathbb {E} \left[ e _ {u, i} \right],
+$$
+
+where the expectation is taken on the super-population distribution  $\mathbb{P}(x,r)$ . Unless otherwise stated, all expectations are taken on the  $\mathbb{P}(x,r)$  later. With the additional randomness caused by super-population, the theoretical results of the DR estimator change. The following theorem and corollary show the bias and the double robustness property under super-population for the DR estimator.
+
+Theorem 4.3 (Bias of DR Estimator under Super-population). Given error imputation model  $\hat{e}_{u,i}$  and propensity model  $\hat{p}_{u,i}$ , then the bias of the DR estimator for estimating the expected ideal loss under super-population is
+
+$$
+\operatorname {B i a s} _ {\mathcal {P}} [ \mathcal {E} _ {\mathrm {D R}} (\theta) ] = \underbrace {\operatorname {C o v} \left(\frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}} , e _ {u , i} - \hat {e} _ {u , i}\right)} _ {e q u a l s t o 0 i f i n d e p e n d e n t} + \underbrace {\mathbb {E} \left[ \left\{1 - \mathbb {E} \left[ \frac {o _ {u , i}}{\hat {p} _ {u , i}} | x _ {u , i} \right] \right\} \cdot \left\{\mathbb {E} [ e _ {u , i} | x _ {u , i} ] - \mathbb {E} [ \hat {e} _ {u , i} | x _ {u , i} ] \right\} \right]} _ {e q u a l s t o 0 e i t h e r   \mathbb {E} [ o _ {u, i} / \hat {p} _ {u, i} | x _ {u, i} ] = 1 o r \mathbb {E} [ \hat {e} _ {u, i} - e _ {u, i} | x _ {u, i} ] = 0}.
+$$
+
+Corollary 4.4 (Double Robustness under Super-population). Under super-population, the DR estimator is unbiased when both the following conditions hold:
+
+(i) Either learned propensities satisfy  $\mathbb{E}[o_{u,i} / \hat{p}_{u,i}\mid x_{u,i}] = 1$ , or imputed errors have the same conditional expectation with true prediction errors  $\mathbb{E}[\hat{e}_{u,i}\mid x_{u,i}] = \mathbb{E}[e_{u,i}\mid x_{u,i}]$ ;  
+(ii) The covariance term vanishes, that is,  $\operatorname{Cov}\left(\frac{\hat{p}_{u,i} - o_{u,i}}{\hat{p}_{u,i}}, e_{u,i} - \hat{e}_{u,i}\right) = 0$ .
+
+Remark: Previous DR estimators are biased even if  $\hat{e}_{u,i} = e_{u,i}$  or  $\hat{p}_{u,i} = p_{u,i}$  for all  $(u,i)\in \mathcal{D}_{\mathrm{target}}$
+
+Compared with the existing theoretical results as in Lemma 4.1, it is obvious that condition  $(i)$  is necessary to achieve unbiasedness, which directly extends the conditions of accurate imputed errors and learned propensities in Lemma 4.1 to the expectation form. However, note that the condition (ii) that covariance vanishes is also needed for the unbiasedness under super-population scenario. Intuitively, if ignoring the randomness caused by sampling process, then  $e_{u,i} - \hat{e}_{u,i}$  is a constant given  $x_{u,i}$ . By the double expectation formula, the covariance term vanishes automatically. The detailed proofs are in the Appendix A. Therefore, it is necessary to modify the previous DR learning approach to control the covariance while learning accurate propensity and imputation models under super-population scenario.
+
+# 4.2 The SuperDR Estimator
+
+It is important to note that the true covariance is unknown because we cannot access the true data distribution. However, we can use the empirical covariance over all user-item pairs as an approximation of the true covariance. We first give the definition of empirical covariance.
+
+Definition 4.5 (Empirical Covariance). The empirical expected conditional covariance between  $(\hat{p}_{u,i} - o_{u,i}) / \hat{p}_{u,i}$  and  $e_{u,i} - \hat{e}_{u,i}$  is
+
+$$
+\widehat {\mathrm {C o v}} \left(\frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}}, e _ {u, i} - \hat {e} _ {u, i}\right) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}} \cdot (e _ {u, i} - \hat {e} _ {u, i}).
+$$
+
+When the learned propensities or imputed errors are accurate, i.e., satisfying condition  $(i)$  in Corollary 4.4, the empirical covariance will converge to  $\mathrm{Cov}\left(\frac{\hat{p}_{u,i} - o_{u,i}}{\hat{p}_{u,i}}, e_{u,i} - \hat{e}_{u,i}\right)$  as  $|\mathcal{D}| \to \infty$ . A direct method to control the empirical covariance is to regard it as a regularization term. However, since the data are partially observed, we cannot obtain the value of the empirical covariance on all user-item pairs. In addition, the large penalty term may hurt the prediction performance. Interestingly, motivated by targeted maximum likelihood estimation [55, 64], we found that the empirical covariance can be controlled with a targeting correction step based on the DR estimator. Specifically, we designed imputation correction as follows:
+
+$$
+\tilde {e} _ {u, i} = \hat {e} _ {u, i} + \epsilon (o _ {u, i} - \hat {p} _ {u, i}).
+$$
+
+where  $\hat{e}_{u,i} = m(x_{u,i};\phi)$  is the imputed errors in previous DR estimators,  $\hat{p}_{u,i} = \pi (x_{u,i};\psi)$  is the learned propensity, and  $\epsilon$  is a learnable parameter. We optimize  $\phi$  and  $\epsilon$  in  $\tilde{e}_{u,i}$  by minimizing the loss based on imputation correction:
+
+$$
+(\phi^ {*}, \epsilon^ {*}) = \arg \min _ {\phi , \epsilon} \mathcal {L} _ {e} ^ {S u p} (\phi , \epsilon) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}}.
+$$
+
+Specifically, the added correction term  $\epsilon (o_{u,i} - \hat{p}_{u,i})$  has several desired properties. First, the correction term enlarges the hypothesis space of  $\tilde{e}_{u,i}$  compared to  $\hat{e}_{u,i}$ , and does not bring extra concerns to the double robustness property due to it has zero mean under accurate  $\hat{p}_{u,i}$ . Second, the derivatives on the proposed loss with respect to  $\epsilon$  are shown below:
+
+$$
+\frac {\partial}{\partial \epsilon} \mathcal {L} _ {e} ^ {S u p} (\phi , \epsilon) = \frac {2}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {O}} \frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}} \cdot (e _ {u, i} - \tilde {e} _ {u, i}).
+$$
+
+It has the same form as the empirical covariance for user-item pairs with  $o_{u,i} = 1$ , which means that we can make the empirical covariance for observed user-item pairs to zero by minimizing the  $\mathcal{L}_e^{Sup}$  directly. Note that adding the correction term on either  $\hat{e}_{u,i}$  or  $e_{u,i}$  will not affect the gradient above, thus we add such term on  $\hat{e}_{u,i}$  for illustration. In the next step, we show that the unobserved empirical covariance can also be bounded by minimizing  $\mathcal{L}_e^{Sup}$  using the concentration inequality. To proceed, we first define the empirical Rademacher complexity as follows.
+
+Definition 4.6 (Empirical Rademacher Complexity [65]). Let  $\mathcal{F}$  be a family of prediction models mapping from  $x\in \mathcal{X}$  to  $[a,b]$ , and  $S = \{x_{u,i}\mid (u,i)\in \mathcal{D}\}$  a fixed sample of size  $|\mathcal{D}|$  with elements in  $\mathcal{X}$ . Then, the empirical Rademacher complexity of  $\mathcal{F}$  with respect to the sample  $S$  is defined as:
+
+$$
+\mathcal {R} (\mathcal {F}) = \mathbb {E} _ {\boldsymbol {\sigma} \sim \{- 1, + 1 \} ^ {| \mathcal {D} |}} \sup _ {f _ {\theta} \in \mathcal {F}} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \sigma_ {u, i} e _ {u, i} \right],
+$$
+
+where  $\sigma = \{\sigma_{u,i}:(u,i)\in \mathcal{D}\}$ , and  $\sigma_{u,i}$  are independent uniform random variables taking values in  $\{-1, + 1\}$ . The random variables  $\sigma_{u,i}$  are called Rademacher variables.
+
+Then we derive the controllability of empirical covariance for all user-item pairs in Theorem 4.7. Refer to Appendix A for the complete proof for this theorem.
+
+Theorem 4.7 (Controllability of Empirical Covariance). The corrected imputation model trained by  $\mathcal{L}_e^{Sup}$  is sufficient for controlling the empirical covariance.
+
+(i) For user-item pairs with observed outcomes, the empirical covariance is 0. Formally, we have
+
+$$
+\frac{\partial}{\partial\epsilon}\mathcal{L}_{e}^{S u p}(\phi ,\epsilon)\bigg|_{\epsilon = \epsilon^{*}} = 0,\text{which is equivalent to}\frac{1}{|\mathcal{D}|}\sum_{(u,i): o_{u,i} = 1}\frac{\hat{p}_{u,i} - o_{u,i}}{\hat{p}_{u,i}}\cdot (e_{u,i} - \tilde{e}_{u,i}) = 0;
+$$
+
+Algorithm 1: The Proposed Doubly Robust Joint Learning Algorithm under Super-population  
+Input: observed ratings  $\mathbf{R}^o$  and a pre-trained propensity model  $\pi(x_{u,i};\psi)$ .  
+while stopping criteria is not satisfied do  
+for number of steps for training the corrected imputation model do  
+Sample a batch of user-item pairs  $\{(u_j,i_j)\}_{j=1}^J$  from  $\mathcal{O}$ ;  
+Update  $\phi$  by descending along the gradient  $\nabla_{\phi}\mathcal{L}_e^{Sup}(\phi,\epsilon)$ ;  
+Update  $\epsilon$  by descending along the gradient  $\nabla_{\epsilon}\mathcal{L}_e^{Sup}(\phi,\epsilon)$ ;  
+end  
+for number of steps for training the debiased prediction model do  
+Sample a batch of user-item pairs  $\{(u_k,i_k)\}_{k=1}^K$  from  $\mathcal{D}$ ;  
+Update  $\theta$  by descending along the gradient  $\nabla_{\theta}\mathcal{L}_{\mathrm{SuperDR}}(\theta;\phi,\psi)$ ;  
+end  
+end
+
+(ii) For user-item pairs with missing outcomes, suppose that  $\hat{p}_{u,i} \geq K_{\psi}$  and  $|e_{u,i} - \tilde{e}_{u,i}| \leq K_{\phi}$ , then with probability at least  $1 - \eta$ , we have
+
+$$
+\begin{array}{l} \frac{1}{|\mathcal{D}|}\sum_{(u,i):o_{u,i} = 0}\frac{\hat{p}_{u,i} - o_{u,i}}{\hat{p}_{u,i}}\cdot \left(e_{u,i} - \tilde{e}_{u,i}\right) \\ \leq \underbrace {\sqrt {\mathcal {L} _ {e} ^ {S u p} (\phi , \epsilon)}} _ {\text {p r o p o s e d l o s s}} + K _ {\phi} \underbrace {\sqrt {\frac {1}{| \mathcal {D} |} \sum_ {u , i \in \mathcal {D}} \left| 1 - \mathbb {E} \left[ \frac {o _ {u , i}}{\hat {p} _ {u , i}} \mid x _ {u , i} \right] \right|}} _ {\text {e m p i r i c a l b i a s f r o m p r o p e s i t y m o d e l}} + \underbrace {\sqrt {K _ {\phi} \left(1 + \frac {1}{K _ {\psi}}\right) \left(2 \mathcal {R} (\mathcal {F}) + (2 K _ {\phi} + 1) \sqrt {\frac {2 \log (4 / \eta)}{| \mathcal {D} |}}\right)}} _ {\text {t a i l b o u n d c o n t r o l l e d b y e m p i r i c a l R a d e m a c h e r c o m p l e x i t y a n d s a m p l e s i z e}}, \\ \end{array}
+$$
+
+where the  $K_{\psi}, K_{\phi}, \eta$  are constants.
+
+Note the proposed imputation correction has no harm property theoretically, as shown in Corollary 4.8.
+
+Corollary 4.8 (Relation to previous imputed errors). The learned coefficient  $\epsilon^{*}$  will converge to zero when the imputation model  $\hat{e}_{u,i}$  has zero empirical covariance, making  $\tilde{e}_{u,i}$  degenerates to  $\hat{e}_{u,i}$ .
+
+In addition, the proposed imputation correction can not only control the empirical covariance effectively but also be helpful for learning more accurate imputed errors.
+
+Corollary 4.9 (Bias reduction property). The proposed corrected imputation loss leads to the smaller bias of imputed errors  $\tilde{e}_{u,i}$ , when  $\hat{e}_{u,i}$  are inaccurate. Formally, we have
+
+$$
+\min _ {\phi , \epsilon} \mathcal {L} _ {e} ^ {S u p} (\phi , \epsilon) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}} \leq \min _ {\phi} \mathcal {L} _ {e} (\phi) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \hat {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}}.
+$$
+
+Moreover, while reducing bias, the proposed method also reduces the variance compared to the previous imputed errors under a moderate condition, as shown below.
+
+Corollary 4.10 (Variance reduction property). The proposed corrected imputation loss leads to the smaller variance of  $\tilde{e}_{u,i}$  when the optimal  $\epsilon^{*}$  lies in a certain range. Formally, we have
+
+$$
+\mathbb {V} (\tilde {e} _ {u, i}) = \mathbb {V} (\hat {e} _ {u, i} + \epsilon^ {*} \cdot (o _ {u, i} - \hat {p} _ {u, i})) \leq \mathbb {V} (\hat {e} _ {u, i}), i f \epsilon^ {*} \in \left[ 0, 2 \cdot \frac {\mathrm {C o v} (\hat {e} _ {u , i} , \hat {p} _ {u , i} - o _ {u , i})}{\mathbb {V} (\hat {p} _ {u , i} - o _ {u , i})} \right].
+$$
+
+See Appendix A for the proof for the above three corollaries. Finally, the proposed SuperDR estimator is given below based on the corrected imputation:
+
+$$
+\mathcal {E} _ {\text {S u p e r D R}} (\theta) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \left[ \tilde {e} _ {u, i} + \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i})}{\hat {p} _ {u , i}} \right].
+$$
+
+Table 1: Performance on AUC, NDCG@K and Recall@K on the Coat, Yahoo! R3 and KuaiRec datasets. The best result is bolded and the best baseline result is underlined, where * means statistically significant results (p-value ≤ 0.05) using the paired-t-test.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="3">Coat</td><td colspan="3">Yahoo! R3</td><td colspan="3">KuaiRec</td></tr><tr><td>AUC</td><td>NDCG@5</td><td>Recall@5</td><td>AUC</td><td>NDCG@5</td><td>Recall@5</td><td>AUC</td><td>NDCG@50</td><td>Recall@50</td></tr><tr><td>MLP</td><td>0.729 ± 0.003</td><td>0.635 ± 0.006</td><td>0.614 ± 0.007</td><td>0.664 ± 0.002</td><td>0.645 ± 0.002</td><td>0.442 ± 0.004</td><td>0.808 ± 0.005</td><td>0.610 ± 0.007</td><td>0.645 ± 0.010</td></tr><tr><td>DAMF</td><td>0.729 ± 0.005</td><td>0.652 ± 0.007</td><td>0.628 ± 0.008</td><td>0.664 ± 0.002</td><td>0.642 ± 0.001</td><td>0.438 ± 0.002</td><td>0.811 ± 0.003</td><td>0.609 ± 0.004</td><td>0.643 ± 0.005</td></tr><tr><td>CVIB</td><td>0.729 ± 0.004</td><td>0.647 ± 0.005</td><td>0.623 ± 0.009</td><td>0.670 ± 0.004</td><td>0.656 ± 0.003</td><td>0.452 ± 0.001</td><td>0.816 ± 0.007</td><td>0.617 ± 0.008</td><td>0.653 ± 0.009</td></tr><tr><td>IPS</td><td>0.731 ± 0.004</td><td>0.642 ± 0.004</td><td>0.625 ± 0.005</td><td>0.667 ± 0.003</td><td>0.647 ± 0.006</td><td>0.445 ± 0.007</td><td>0.806 ± 0.006</td><td>0.606 ± 0.006</td><td>0.643 ± 0.005</td></tr><tr><td>SNIPS</td><td>0.732 ± 0.004</td><td>0.654 ± 0.005</td><td>0.629 ± 0.005</td><td>0.665 ± 0.003</td><td>0.644 ± 0.004</td><td>0.443 ± 0.003</td><td>0.811 ± 0.004</td><td>0.612 ± 0.006</td><td>0.649 ± 0.006</td></tr><tr><td>ASIPS</td><td>0.730 ± 0.006</td><td>0.643 ± 0.006</td><td>0.620 ± 0.006</td><td>0.668 ± 0.002</td><td>0.655 ± 0.004</td><td>0.452 ± 0.005</td><td>0.811 ± 0.006</td><td>0.614 ± 0.006</td><td>0.652 ± 0.005</td></tr><tr><td>IPS-V2</td><td>0.736 ± 0.004</td><td>0.653 ± 0.007</td><td>0.628 ± 0.009</td><td>0.662 ± 0.003</td><td>0.651 ± 0.001</td><td>0.445 ± 0.002</td><td>0.813 ± 0.006</td><td>0.612 ± 0.008</td><td>0.655 ± 0.006</td></tr><tr><td>DR</td><td>0.733 ± 0.003</td><td>0.650 ± 0.005</td><td>0.625 ± 0.007</td><td>0.667 ± 0.005</td><td>0.655 ± 0.004</td><td>0.449 ± 0.008</td><td>0.818 ± 0.003</td><td>0.620 ± 0.004</td><td>0.655 ± 0.007</td></tr><tr><td>MRDR</td><td>0.739 ± 0.005</td><td>0.650 ± 0.003</td><td>0.622 ± 0.007</td><td>0.665 ± 0.005</td><td>0.652 ± 0.005</td><td>0.448 ± 0.005</td><td>0.814 ± 0.006</td><td>0.616 ± 0.006</td><td>0.652 ± 0.003</td></tr><tr><td>DR-MSE</td><td>0.738 ± 0.005</td><td>0.645 ± 0.007</td><td>0.627 ± 0.006</td><td>0.667 ± 0.004</td><td>0.650 ± 0.004</td><td>0.446 ± 0.004</td><td>0.814 ± 0.006</td><td>0.617 ± 0.006</td><td>0.654 ± 0.007</td></tr><tr><td>DR-V2</td><td>0.747 ± 0.004</td><td>0.653 ± 0.004</td><td>0.625 ± 0.006</td><td>0.671 ± 0.008</td><td>0.660 ± 0.005</td><td>0.456 ± 0.003</td><td>0.821 ± 0.010</td><td>0.619 ± 0.010</td><td>0.661 ± 0.008</td></tr><tr><td>SDR</td><td>0.748 ± 0.006</td><td>0.650 ± 0.005</td><td>0.626 ± 0.007</td><td>0.666 ± 0.005</td><td>0.653 ± 0.004</td><td>0.451 ± 0.004</td><td>0.819 ± 0.004</td><td>0.618 ± 0.005</td><td>0.652 ± 0.006</td></tr><tr><td>TDR</td><td>0.744 ± 0.004</td><td>0.651 ± 0.005</td><td>0.631 ± 0.005</td><td>0.664 ± 0.004</td><td>0.655 ± 0.007</td><td>0.453 ± 0.003</td><td>0.822 ± 0.005</td><td>0.621 ± 0.009</td><td>0.656 ± 0.010</td></tr><tr><td>MR</td><td>0.742 ± 0.005</td><td>0.653 ± 0.006</td><td>0.630 ± 0.006</td><td>0.672 ± 0.003</td><td>0.657 ± 0.003</td><td>0.454 ± 0.002</td><td>0.823 ± 0.003</td><td>0.622 ± 0.004</td><td>0.655 ± 0.005</td></tr><tr><td>AKBDR</td><td>0.748 ± 0.005</td><td>0.656 ± 0.007</td><td>0.630 ± 0.007</td><td>0.676 ± 0.004</td><td>0.662 ± 0.004</td><td>0.461 ± 0.003</td><td>0.824 ± 0.004</td><td>0.629 ± 0.006</td><td>0.667 ± 0.006</td></tr><tr><td>DCE-TDR</td><td>0.746 ± 0.005</td><td>0.654 ± 0.005</td><td>0.629 ± 0.006</td><td>0.679 ± 0.004</td><td>0.662 ± 0.005</td><td>0.459 ± 0.004</td><td>0.824 ± 0.003</td><td>0.632 ± 0.004</td><td>0.671 ± 0.006</td></tr><tr><td>D-DR</td><td>0.750 ± 0.004</td><td>0.654 ± 0.004</td><td>0.630 ± 0.008</td><td>0.678 ± 0.004</td><td>0.659 ± 0.004</td><td>0.456 ± 0.003</td><td>0.822 ± 0.004</td><td>0.630 ± 0.005</td><td>0.672 ± 0.005</td></tr><tr><td>SuperDR</td><td>0.757*± 0.004</td><td>0.667*± 0.005</td><td>0.637*± 0.007</td><td>0.686*± 0.003</td><td>0.667*± 0.004</td><td>0.463 ± 0.003</td><td>0.828± 0.004</td><td>0.640*± 0.005</td><td>0.680*± 0.005</td></tr></table>
+
+# 4.3 The Extend Joint Learning Algorithm
+
+We optimize the prediction model and the imputation model of the SuperDR method by a widely used joint learning framework [14], which alternatively optimizes two models to achieve unbiased learning. Specifically, we train the prediction model by minimizing SuperDR loss:
+
+$$
+\mathcal {L} _ {\text {S u p e r D R}} (\theta) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \left[ \tilde {e} _ {u, i} + \frac {o _ {u , i} \left(e _ {u , i} - \tilde {e} _ {u , i}\right)}{\hat {p} _ {u , i}} \right].
+$$
+
+We update the imputation model parameters and  $\epsilon$  simultaneously by minimizing the  $\mathcal{L}_e^{sup}(\phi, \epsilon)$  in Section 4.2 and we train the propensity model by minimizing the following cross-entropy loss.
+
+$$
+\mathcal {L} _ {p} (\psi) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \left[ - o _ {u, i} \log (\hat {p} _ {u, i}) - (1 - o _ {u, i}) \log (1 - \hat {p} _ {u, i}) \right].
+$$
+
+The propensity model is pre-trained, and the parameters of the prediction and imputation model are updated alternatively via SGD. The joint learning process is summarized in Algorithm 1. Note that the complexity will not increase due to we only additionally update one single parameter  $\epsilon$  compared to the traditional joint learning algorithm.
+
+# 4.4 The Generalization Bound
+
+Next, we analyze the generalization error bound of the DR methods using the models for estimating  $e_{u,i}$  and  $p_{u,i}$ , and show that controlling empirical covariance leads to a tighter bound. Specifically, the generalization error theories for the previous DR estimators relied mainly on the boundedness of the loss to each user-item pair in the DR estimators from the binary indicator  $o_{u,i}$ , i.e., for the DR estimator, the bound for DR loss on  $(u,i)$  is  $(e_{u,i} - \hat{e}_{u,i}) / \hat{p}_{u,i}$ . However, these analyses no longer hold under super-population scenario. In the following theorem, we provide the generalization error bound of SuperDR, which includes four terms: the SuperDR loss, the empirical covariance, the bias of the SuperDR estimator, and the tail bound. Compared to previous DR methods, the proposed method can further control the covariance term, leading to a more desirable generalization bound thus improving debiasing performance. See Appendix A for the proof.
+
+Theorem 4.11 (Generalization Bound under Super-population). Suppose that  $\hat{p}_{u,i} \geq K_{\psi}$  and  $\min \{\tilde{e}_{u,i}, |e_{u,i} - \tilde{e}_{u,i}| \} \leq K_{\phi}$ , then with probability at least  $1 - \eta$ , we have
+
+$$
+\mathcal {L} _ {i d e a l} (\theta) \leq \mathcal {L} _ {\mathrm {S u p e r D R}} (\theta) + \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \left| 1 - \mathbb {E} \left[ \frac {o _ {u , i}}{\hat {p} _ {u , i}} \big | x _ {u, i} \right] \right| \cdot \left| \mathbb {E} [ e _ {u, i} \mid x _ {u, i} ] - \mathbb {E} [ \tilde {e} _ {u, i} \mid x _ {u, i} ] \right|
+$$
+
+vanilla DR only controls the empirical DR loss, and empirical risks of imputation and propensity models
+
+$$
++ \underbrace {\left| \frac {1}{| \mathcal {D} |} \sum_ {(u , i) \in \mathcal {D}} \operatorname {C o v} \left(\frac {o _ {u , i} - \hat {p} _ {u , i}}{\hat {p} _ {u , i}}, e _ {u , i} - \tilde {e} _ {u , i}\right) \right|} + \underbrace {\left(1 + \frac {1}{K _ {\psi}}\right) \left(2 \mathcal {R} (\mathcal {F}) + K _ {\phi} \sqrt {\frac {1 8 \log (4 / \eta)}{| \mathcal {D} |}}\right)}
+$$
+
+corrected loss further controls the independence
+
+tail bound controlled by empirical Rademacher complexity and sample size
+
+![](images/ccf8d545b72ffd53bc51b9966fd84c5ae423a7d6c29bb5ac1b3383079bfd52f5.jpg)  
+(a) AUC
+
+![](images/dcc375b254e3e41cfe8eec5c79c1cc8782c55454a8822a5f9fd55006b3717dae.jpg)  
+(b)NDCG@50
+
+![](images/89fa402314b7fdb0e2100a1726274e5982a025549765ae97bdf96707a21e8b8b.jpg)  
+(c) Recall@50
+
+![](images/51b89aa69ff3c8a62cfb05b57509b04afcf1438a248b0626fdf691ae507e3343.jpg)  
+Figure 2: Effects of varying sample ratios  $b\%$  on debiasing performance on the KuaiRec dataset.  
+(a) AUC  
+Figure 3: Effects of varying sample ratios  $b\%$  on debiasing performance on the Yahoo! R3 dataset.
+
+![](images/249d4e8603322c690945ff2afd855b6a8a2b49ac06c4dd96e1723a53e25a76a3.jpg)  
+(b) NDCG@50
+
+![](images/c66aaec3e8a738b038d827ba35f24b2fb115603fa26d84073ee6e3c8f6985bd9.jpg)  
+(c) Recall@50
+
+# 5 Experiments
+
+# 5.1 Experimental Setup
+
+Dataset Selection and Preprocessing. To verify the effectiveness of the proposed method in the real-world dataset, the dataset that contains both biased and unbiased data is required. Following the previous studies [14, 15, 32, 66], the following three widely used real-world datasets are adopted to conduct our experiments: Coat contains ratings from 290 users to 300 items with 6,960 biased ratings and 4,640 unbiased ratings. Yahoo! R3 contains ratings from 15,400 users to 1,000 items with 311,704 biased ratings and 54,000 unbiased ratings. We binarize the ratings to 0 for ratings less than three, otherwise to 1. We further use a fully exposed industrial dataset KuaiRec [67] with 4,676,570 video watching ratio records from 1,411 users to 3,327 videos. Following previous studies [59, 60], we biasedly select 201,171 samples according to the watch ratio as the training set and randomly select 117,113 samples as the unbiased test set. For this dataset, we binarize the records to 0 for records less than two, otherwise to 1
+
+Baselines. In our experiments, as there are only very few features or no features for users and items in all three datasets, we first use the matrix factorization (MF) [3] method to generate the embedding for each user and item, and then fix such embedding as the user-item features. Then we take the MLP as the backbone model and compared the proposed method with the following debiasing baselines including DAMF [68], the information bottleneck based method: CVIB [69], the propensity based methods: IPS [13], SNIPS [70], ASIPS [35], and IPS-V2 [71], and the DR-based methods: DR [14], MRDR [33], DR-MSE [56], DR-V2 [71], TDR [55], SDR [54], MR [57], AKBDR [59], DCE-TDR [60], and D-DR [61].
+
+Experimental Protocols and Details. The following three metrics are used to measure the debiasing performance: AUC, NDCG@K, and Recall@K, where we set K = 5 for Coat and Yahoo! R3, while set K = 50 for KuaiRec. All the experiments are implemented on PyTorch with the GeForce RTX 3090 as the computational resource. Adam is utilized as the optimizer in all experiments. To simulate the super-population scenario, we first randomly sample  $b\%$  users and items (unless otherwise stated,  $b$  is set to  $50\%$  in our experiments) from the training set and then use the whole unbiased test set to evaluate the debiasing performance. Note that this intervention will not affect the data sparsity, it will only affect the number of observed users and items and will ensure  $\mathbb{P}(\mathcal{D}_{\mathrm{target}}) = \mathbb{P}(\mathcal{D}_{\mathrm{test}})$  with limited overlapped users and items. In addition, the dimension of user and item embedding are fixed as 32. We tune learning rate in {0.001, 0.005, 0.01, 0.02, 0.05} for parameters in prediction, imputation, and propensity model, and in {0.01, 0.05, 0.1, 0.15, 0.2} for  $\epsilon$ , batch size in {128, 256, 512} for Coat and {1024, 2048, 4096} for Yahoo! R3 and KuaiRec. The weight decay is tuned in {1e - 6, 5e - 6, ..., 5e - 3, 1e - 2}. In addition, we use the logistic regression model
+
+![](images/40a522f7a609b659ffdcdf6cd688bb86f510d54308e6909ea199600ed5a62fbd.jpg)  
+(a) EC Reduction
+
+![](images/ceccf99d74bdf043836847b13d3710d96469d34ac8a9b0a9cf073ca47f7a109e.jpg)  
+(b) RI of AUC
+
+![](images/91c2cfaa9d7f0af58fdf5a8d99de949a83a1000a206a3274e45fb37edf649b23.jpg)  
+(c) RI of NDCG@K
+
+![](images/fafbf54ff38ae96224ee44d5d216b8a14afcd1e0d087a471873ca402aea5f308.jpg)  
+Figure 4: Effects of empirical covariance (EC) reduction  $(\%)$  on relative improvement (RI)  $(\%)$ .  
+(a) AUC on Yahoo! R3  
+Figure 5: Effects of learning rate of the learnable imputation correction parameter  $\epsilon$ .
+
+![](images/c25d0dc78cf9b319c297ef3b51a7946784a4dd9ae6781d9ecd48b2c6a3306fa3.jpg)  
+(b) NDCG@5 on Yahoo! R3
+
+![](images/e1bb646e95f304b7868cab3f6678162f7172f7fb561b9d2b947aa2a637d94c7a.jpg)  
+(c) AUC on KuaiRec
+
+![](images/254ca0941b5e0c4ac9cadc8dbbf22b087a708c2457a425ba87406020068a9bed.jpg)  
+(d) NDCG@50 on KuaiRec
+
+as the propensity model, which means that there is no unbiased data requirement. To prevent the propensity too small, we tune the propensity clip threshold in  $[0.005, 0.05]$ . For simplicity, we fix the step in inner loop for updating prediction and imputation models in Algorithm 1 as 1.
+
+# 5.2 Performance Comparison
+
+Table 1 summarizes the debiasing performance of various methods on three benchmark datasets Coat, Yahoo! R3, and KuaiRec, and we have the following findings. First, most debiased methods outperform the base model naive MLP, which shows the necessity for debiasing. Second, overall speaking, DR-based methods such as D-DR and DCE-TDR demonstrate the most competitive performance, indicating the superiority of DR methods over other baselines. Third, the proposed SuperDR method achieves the best performance in terms of all evaluation metrics. This indicates that the SuperDR method can effectively reduce the additional bias introduced by sampling through controlling empirical covariance, thus achieving an unbiased estimate of the ideal loss in scenarios where users and items in the training set are not exactly the same as those in the test set.
+
+# 5.3 In-Depth Analysis
+
+Effects of Varying Bias Level. Figures 2 investigates the impact of different levels of bias introduced by sampling on prediction performance on the KuaiRec dataset. We change the sample ratios to control the degree of overlap between users and items in the training and test sets. A higher sample ratio indicates a greater proportion of the same users and items in both sets, resulting in less bias introduced by sampling. When the sample ratio is large (e.g., 0.7 or 0.9), our method slightly outperforms recently proposed state-of-the-art methods such as DCE-TDR and D-DR. When the sample ratio is 0.05 or 0.1, there are few overlapping users and items between the training and test sets, resulting in significant bias introduced by sampling. The performance of previous methods noticeably declines, while the SuperDR method effectively addresses this bias, achieving significant performance improvements. We also conduct experiments on Yahoo! R3 dataset. The experiment results are in Figure 3 with similar phenomenons.
+
+Effects of Empirical Covariance Control. We explore the effects of Empirical Covariance (EC) Reduction on the prediction performance in Figure 4. We find that SuperDR achieves the most significant empirical covariance decreases and the most competitive performance in AUC and NDCG@K, which empirically demonstrates the effectiveness of the targeting correction step and the EC reduction benefit to the prediction performance. Note that DCE-TDR and TDR method obtains some performance improvement compared to base model naive MLP, this is because they
+
+add  $o_{u,i}\left(\frac{1}{\hat{p}_{u,i}} - 1\right)$  as the correction term to the imputed errors to control the covariance on observed samples. Unfortunately, DCE-TDR and TDR are unable to control the covariance on missing outcomes, resulting in sub-optimal performance.
+
+# 5.4 Sensitivity Analysis
+
+We conduct sensitivity analysis on the Yahoo! R3 and KuaiRec datasets to explore the relationship between the learning rate of learnable parameter  $\epsilon$  and the debiasing performance, with AUC and NDCG@K as the evaluation metrics, where K=5 on Yahoo! R3 and K=50 on KuaiRec. As shown in Figure 5, the proposed SuperDR stably outperforms the DCE-TDR, the most competitive baseline on these two datasets, under varying learning rates of  $\epsilon$ , demonstrating that the enhanced imputation model with target learning mitigates the additional bias introduced by sampling and exhibits no-harm property. Meanwhile, under relatively moderate learning rates (0.05, 0.15), the SuperDR demonstrates competitive prediction performance, which further indicates the robustness.
+
+# 6 Conclusion
+
+In this paper, we extend the previous exact matching scenario, i.e., the units are the same between training and test set, and consider a more general scenario that the joint distribution of the feature and rating  $\mathbb{P}(x,r)$  in the training and test set to be the same. Then we show the DR estimator is biased even if the imputed errors and learned propensities are correct in this scenario and provide the explicit bias form, which has two terms: the term that measures the accuracy of imputed errors and learned propensities and an additional covariance term. To achieve a more accurate estimation, we propose the SuperDR estimator with the corrected imputation model, which can effectively control the additional covariance term with many desirable theoretical properties such as bias and variance reduction. In addition, we extend the previous joint learning algorithm based on the proposed corrected imputation loss and further derive the generalization error bound for the proposed SuperDR, and show that the proposed learning approach can effectively control it. Extensive experiments are conducted on three real-world datasets to show the effectiveness of our SuperDR method. One of the potential limitations and research directions is how to develop a tighter bound for controlling the empirical covariance and to develop a more efficient algorithm for alternatively updating the prediction model, the imputation model, and the target learning parameter.
+
+# Acknowledgments and Disclosure of Funding
+
+This work is supported by the National Natural Science Foundation of China (623B2002).
+
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+
+# A Proofs
+
+Lemma 4.1 (Bias of DR Estimator [14]). Given imputed errors  $\hat{e}_{u,i}$  and learned propensities  $\hat{p}_{u,i} > 0$ , when considering only the randomness of missing indicators, the bias of DR estimator is
+
+$$
+\mathrm {B i a s} _ {\mathbf {O}} [ \mathcal {E} _ {\mathrm {D R}} (\theta) ] = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {\{\hat {p} _ {u , i} - p _ {u , i} \} \cdot \{e _ {u , i} - \hat {e} _ {u , i} \}}{\hat {p} _ {u , i}}.
+$$
+
+Proof of Lemma 4.1. The proof can be found in Lemma 3.1 of [14]. However, one should note that, as stated in the proof, "the prediction and imputed errors are treated as constants when taking the expectation, since  $o_{u,i}$  does not result from any prediction or imputation models [13]". The DR estimator in [14] is given as
+
+$$
+\mathcal {E} _ {\mathrm {D R}} (\theta) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \left[ \hat {e} _ {u, i} + \frac {o _ {u , i} (e _ {u , i} - \hat {e} _ {u , i})}{\hat {p} _ {u , i}} \right].
+$$
+
+By considering only the randomness on  $o_{u,i}$ , we have
+
+$$
+\begin{array}{l} \mathbb {E} _ {\mathbf {O}} \left[ \mathcal {E} _ {\mathrm {D R}} (\theta) \right] = \mathbb {E} _ {\mathbf {O}} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \left[ \hat {e} _ {u, i} + \frac {o _ {u , i} \left(e _ {u , i} - \hat {e} _ {u , i}\right)}{\hat {p} _ {u , i}} \right] \right] \\ = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \left[ \hat {e} _ {u, i} + \frac {p _ {u , i} \left(e _ {u , i} - \hat {e} _ {u , i}\right)}{\hat {p} _ {u , i}} \right]. \\ \end{array}
+$$
+
+By definition, the bias of the DR estimator is
+
+$$
+\begin{array}{l} \operatorname {B i a s} _ {\mathbf {O}} \left[ \mathcal {E} _ {\mathrm {D R}} (\theta) \right] = \mathcal {E} _ {i d e a l} (\theta) - \mathbb {E} _ {\mathbf {O}} \left[ \mathcal {E} _ {\mathrm {D R}} (\theta) \right] \\ = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} e _ {u, i} - \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \left[ \hat {e} _ {u, i} + \frac {p _ {u , i} \left(e _ {u , i} - \hat {e} _ {u , i}\right)}{\hat {p} _ {u , i}} \right] \\ = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {\left\{\hat {p} _ {u , i} - p _ {u , i} \right\} \cdot \left\{e _ {u , i} - \hat {e} _ {u , i} \right\}}{\hat {p} _ {u , i}}, \\ \end{array}
+$$
+
+which yields the stated results.
+
+![](images/426ded435125f4dee5c0d861545146a007beff94cfc6c8386fdb0da9ce6c5953.jpg)
+
+Corollary 4.2 (Double Robustness [14]). The DR estimator is unbiased when either imputed errors  $\hat{e}_{u,i}$  or learned propensities  $\hat{p}_{u,i}$  are accurate for all user-item pairs, i.e., either  $\hat{e}_{u,i} = e_{u,i}$  or  $\hat{p}_{u,i} = p_{u,i}$  for all  $u$  and  $i$ .
+
+Proof of Corollary 4.2. The proof can be found at Corollary 3.1 in Appendix of [14]. However, one should note that, as stated in the proof, "the prediction and imputed errors are treated as constants when taking the expectation, since  $o_{u,i}$  does not result from any prediction or imputation models [13]".
+
+Let  $\delta_{u,i} = e_{u,i} - \hat{e}_{u,i}$  and  $\Delta_{u,i} = \frac{\hat{p}_{u,i} - p_{u,i}}{\hat{p}_{u,i}}$ . On the hand, when imputed errors are accurate, we have  $\delta_{u,i} = 0$  for  $(u,i) \in \mathcal{D}$ . In such case, we can compute the bias of the DR estimator by
+
+$$
+\operatorname {B i a s} _ {\mathbf {O}} [ \mathcal {E} _ {\mathrm {D R}} (\theta) ] = \frac {1}{| \mathcal {D} |} \sum_ {u, i \in \mathcal {D}} \Delta_ {u, i} \delta_ {u, i} = \frac {1}{| \mathcal {D} |} \sum_ {u, i \in \mathcal {D}} \Delta_ {u, i} \cdot 0 = 0.
+$$
+
+On the other hand, when the learned propensities are accurate, we have  $\Delta_{u,i} = 0$  for  $(u,i)\in \mathcal{D}$ . In this case, we can compute the bias of the DR estimator by
+
+$$
+\operatorname {B i a s} \left(\mathcal {E} _ {\mathrm {D R}}\right) = \frac {1}{| \mathcal {D} |} \sum_ {u, i \in \mathcal {D}} \Delta_ {u, i} \delta_ {u, i} = \frac {1}{| \mathcal {D} |} \sum_ {u, i \in \mathcal {D}} 0 \cdot \delta_ {u, i} = 0.
+$$
+
+In both cases, the bias of the DR estimator is zero, which means that the expectation of the DR estimator over all the possible instances of  $o_{u,i}$  is exactly the same as the prediction inaccuracy. This completes the proof.
+
+Theorem 4.3 (Bias of DR Estimator under Super-population). Given error imputation model  $\hat{e}_{u,i}$  and propensity model  $\hat{p}_{u,i}$ , then the bias of the DR estimator for estimating the expected ideal loss under super-population is
+
+$$
+\operatorname {B i a s} _ {\mathcal {P}} [ \mathcal {E} _ {\mathrm {D R}} (\theta) ] = \underbrace {\operatorname {C o v} \left(\frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}} , e _ {u , i} - \hat {e} _ {u , i}\right)} _ {e q u a l s t o 0 i f i n d e p e n d e n t} + \underbrace {\mathbb {E} \left[ \left\{1 - \mathbb {E} \left[ \frac {o _ {u , i}}{\hat {p} _ {u , i}} | x _ {u , i} \right] \right\} \cdot \left\{\mathbb {E} [ e _ {u , i} | x _ {u , i} ] - \mathbb {E} [ \hat {e} _ {u , i} | x _ {u , i} ] \right\} \right]} _ {e q u a l s t o 0 e i t h e r \mathbb {E} [ o _ {u, i} / \hat {p} _ {u, i} | x _ {u, i} ] = 1 o r \mathbb {E} [ \hat {e} _ {u, i} - e _ {u, i} | x _ {u, i} ] = 0}.
+$$
+
+Proof of Theorem 1. Instead of considering only the randomness of the rating missing indicator, in the following, we treat all variables, including imputed errors and learned propensities, as random variables. Formally, we have
+
+$$
+\begin{array}{l} \mathrm {B i a s} [ \mathcal {E} _ {\mathrm {D R}} (\theta) ] = \mathbb {E} [ \mathcal {L} _ {\mathrm {i d e a l}} (\theta) ] - \mathbb {E} [ \mathcal {E} _ {\mathrm {D R}} (\theta) ] = \mathbb {E} [ e _ {u, i} ] - \mathbb {E} \left[ e _ {u, i} + \frac {\{o _ {u , i} - \hat {p} _ {u , i} \} \cdot \{e _ {u , i} - \hat {e} _ {u , i} \}}{\hat {p} _ {u , i}} \right] \\ = \mathbb {E} \left[ \mathbb {E} \left[ \left\{\frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}} \right\} \left\{e _ {u, i} - \hat {e} _ {u, i} \right\} \mid x _ {u, i} \right] \right] \quad (\text {b y}) \\ = \mathbb {E} \left[ \mathbb {E} \left[ \left\{\frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}} - \mathbb {E} \Big [ \frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}} \Big ] + \mathbb {E} \Big [ \frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}} \Big ] \right\} \left\{(e _ {u, i} - \hat {e} _ {u, i}) - \mathbb {E} [ e _ {u, i} - \hat {e} _ {u, i} ] + \mathbb {E} [ e _ {u, i} - \hat {e} _ {u, i} ] \right\} \mid x _ {u, i} \right] \right] \\ = \mathbb {E} \left[ \mathbb {E} \left[ \left\{\frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}} - \mathbb {E} \Big [ \frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}} \Big ] \right\} \left\{(e _ {u, i} - \hat {e} _ {u, i}) - \mathbb {E} [ e _ {u, i} - \hat {e} _ {u, i} ] \right\} \mid x _ {u, i} \right] \right] \\ + \mathbb {E} \left[ \left\{1 - \mathbb {E} \left[ \frac {o _ {u , i}}{\hat {p} _ {u , i}} \mid x _ {u, i} \right] \right\} \cdot \left\{\mathbb {E} \left[ e _ {u, i} \mid x _ {u, i} \right] - \mathbb {E} \left[ \hat {e} _ {u, i} \mid x _ {u, i} \right] \right\} \right] \\ = \operatorname {C o v} \left(\frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}}, e _ {u, i} - \hat {e} _ {u, i}\right) + \mathbb {E} \left[ \left\{1 - \mathbb {E} \left[ \frac {o _ {u , i}}{\hat {p} _ {u , i}} | x _ {u, i} \right] \right\} \cdot \left\{\mathbb {E} [ e _ {u, i} \mid x _ {u, i} ] - \mathbb {E} [ \hat {e} _ {u, i} \mid x _ {u, i} ] \right\} \right], \\ \end{array}
+$$
+
+which yields the stated results.
+
+![](images/b3b84bcc290f27dcf27ffb4574c7279f1b2c4111ebcb5a34b2c6d9f36d7ac4d4.jpg)
+
+Corollary 4.4 (Double Robustness under Super-population). Under super-population, the DR estimator is unbiased when both the following conditions hold:
+
+(i) Either learned propensities satisfy  $\mathbb{E}[o_{u,i} / \hat{p}_{u,i}\mid x_{u,i}] = 1$ , or imputed errors have the same conditional expectation with true prediction errors  $\mathbb{E}[\hat{e}_{u,i}\mid x_{u,i}] = \mathbb{E}[e_{u,i}\mid x_{u,i}]$ ;  
+(ii) The covariance term vanishes, that is,  $\mathrm{Cov}\left(\frac{\hat{p}_{u,i} - o_{u,i}}{\hat{p}_{u,i}}, e_{u,i} - \hat{e}_{u,i}\right) = 0$ .
+
+Proof of Corollary 4.4. First, when condition (ii) holds, i.e.,
+
+$$
+\operatorname {C o v} \left(\left(\hat {p} _ {u, i} - o _ {u, i}\right) / \hat {p} _ {u, i}, e _ {u, i} - \hat {e} _ {u, i}\right) = 0,
+$$
+
+it follows from the results in Theorem 1 that
+
+$$
+\operatorname {B i a s} [ \mathcal {E} _ {\mathrm {D R}} (\theta) ] = \mathbb {E} \left[ \left\{1 - \mathbb {E} \left[ \frac {o _ {u , i}}{\hat {p} _ {u , i}} \mid x _ {u, i} \right] \right\} \cdot \left\{\mathbb {E} \left[ e _ {u, i} \mid x _ {u, i} \right] - \mathbb {E} [ \hat {e} _ {u, i} \mid x _ {u, i} ] \right\} \right]
+$$
+
+On the hand, when the learned propensities satisfy  $\mathbb{E}[o_{u,i} / \hat{p}_{u,i}\mid x_{u,i}] = 1$ . In such case, we can compute the bias of the DR estimator by
+
+$$
+\operatorname {B i a s} [ \mathcal {E} _ {\mathrm {D R}} (\theta) ] = \mathbb {E} \left[ 0 \cdot \left\{\mathbb {E} \left[ e _ {u, i} \mid x _ {u, i} \right] - \mathbb {E} \left[ \hat {e} _ {u, i} \mid x _ {u, i} \right] \right\} \right] = 0.
+$$
+
+On the other hand, when imputed errors have the same conditional expectation with true prediction errors, we have  $\mathbb{E}[\hat{e}_{u,i}\mid x_{u,i}] = \mathbb{E}[e_{u,i}\mid x_{u,i}]$ . In this case, we can compute the bias of the DR estimator by
+
+$$
+\mathrm {B i a s} [ \mathcal {E} _ {\mathrm {D R}} (\theta) ] = \mathbb {E} \left[ \left\{1 - \mathbb {E} \left[ \frac {o _ {u , i}}{\hat {p} _ {u , i}} \big | x _ {u, i} \right] \right\} \cdot 0 \right] = 0.
+$$
+
+In both cases, the bias of the DR estimator is zero, which completes the proof.
+
+![](images/15a65f3cc289cfbce17d084b2bff0478dc958fb42e258ca9d48b384a9ef7bf08.jpg)
+
+Definition 4.5 (Empirical Covariance). The empirical expected conditional covariance between  $(\hat{p}_{u,i} - o_{u,i}) / \hat{p}_{u,i}$  and  $e_{u,i} - \hat{e}_{u,i}$  is
+
+$$
+\widehat {\operatorname {C o v}} \left(\frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}}, e _ {u, i} - \hat {e} _ {u, i}\right) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}} \cdot (e _ {u, i} - \hat {e} _ {u, i}).
+$$
+
+Definition 4.6 (Empirical Rademacher Complexity [65]). Let  $\mathcal{F}$  be a family of prediction models mapping from  $x\in \mathcal{X}$  to  $[a,b]$ , and  $S = \{x_{u,i}|$ $(u,i)\in \mathcal{D}\}$  a fixed sample of size  $|\mathcal{D}|$  with elements in  $\mathcal{X}$ . Then, the empirical Rademacher complexity of  $\mathcal{F}$  with respect to the sample  $S$  is defined as:
+
+$$
+\mathcal {R} (\mathcal {F}) = \mathbb {E} _ {\boldsymbol {\sigma} \sim \{- 1, + 1 \} ^ {| \mathcal {D} |}} \sup  _ {f _ {\theta} \in \mathcal {F}} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \sigma_ {u, i} e _ {u, i} \right],
+$$
+
+where  $\sigma = \{\sigma_{u,i}:(u,i)\in \mathcal{D}\}$ , and  $\sigma_{u,i}$  are independent uniform random variables taking values in  $\{-1, + 1\}$ . The random variables  $\sigma_{u,i}$  are called Rademacher variables.
+
+Lemma A.1 (Rademacher Comparison Lemma [65]). Let  $\mathcal{F}$  be a family of real-valued functions on  $z\in \mathcal{Z}$  to  $[a,b]$ , and  $S = \{x_{u,i}\mid (u,i)\in \mathcal{D}\}$  a fixed sample of size  $|\mathcal{D}|$  with elements in  $\mathcal{X}$ . Then
+
+$$
+\mathop{\mathbb{E}}_{S\sim \mathbb{P}^{|D|}}\left[\sup_{f\in \mathcal{F}}\left(\mathop{\mathbb{E}}_{z\sim \mathbb{P}}[f(z)] - \frac{1}{|\mathcal{D}|}\sum_{(u,i)\in \mathcal{D}}f\left(z_{u,i}\right)\right)\right] \leq 2\mathop{\mathbb{E}}_{S\sim \mathbb{P}^{|D|}}\mathop{\mathbb{E}}_{\boldsymbol {\sigma}\sim \{-1, + 1\} ^{|D|}}\sup_{f\in \mathcal{F}}\left[\frac{1}{|\mathcal{D}|}\sum_{(u,i)\in \mathcal{D}}\sigma_{u,i}f\left(z_{u,i}\right)\right],
+$$
+
+where  $\sigma = \{\sigma_{u,i}:(u,i)\in \mathcal{D}\}$ , and  $\sigma_{u,i}$  are independent uniform random variables taking values in  $\{-1, + 1\}$ . The random variables  $\sigma_{u,i}$  are called Rademacher variables.
+
+Proof of Lemma A.1. The proof can be found in Lemma 26.2 of [65].
+
+![](images/9814b80c0503282fc6e68503bb9db703971115a6dc8f5f739855471e6b34aaf5.jpg)
+
+Lemma A.2 (McDiarmid's Inequality [65]). Let  $V$  be some set and let  $f: V^m \to \mathbb{R}$  be a function of  $m$  variables such that for some  $c > 0$ , for all  $i \in [m]$  and for all  $x_1, \ldots, x_m, x_i' \in V$  we have
+
+$$
+\left| f \left(x _ {1}, \dots , x _ {m}\right) - f \left(x _ {1}, \dots , x _ {i - 1}, x _ {i} ^ {\prime}, x _ {i + 1}, \dots , x _ {m}\right) \right| \leq c
+$$
+
+Let  $X_{1},\ldots ,X_{m}$  be  $m$  independent random variables taking values in  $V$ . Then, with probability of at least  $1 - \delta$  we have
+
+$$
+| f (X _ {1}, \dots , X _ {m}) - \mathbb {E} [ f (X _ {1}, \dots , X _ {m}) ] | \leq c \sqrt {\log \left(\frac {2}{\delta}\right) m / 2}
+$$
+
+Proof of Lemma A.2. The proof can be found in Lemma 26.4 of [65].
+
+![](images/af8f7d09e6fede6f7baf7d5df8c92fbaf7ce8817e5be391155641e4f03932af7.jpg)
+
+Lemma A.3 (Rademacher Calculus [65]). For any  $A \subset \mathbb{R}^m$ , scalar  $c \in \mathbb{R}$ , and vector  $\mathbf{a}_0 \in \mathbb{R}^m$ , we have
+
+$$
+R \left(\left\{c \mathbf {a} + \mathbf {a} _ {0}: \mathbf {a} \in A \right\}\right) \leq | c | R (A).
+$$
+
+Proof of Lemma A.3. The proof can be found in Lemma 26.6 of [65].
+
+![](images/e9f9ac55dc94846cb4829a5f503834cccd08f3c63af5b625abcfbfbb1f91bced.jpg)
+
+Theorem 4.7 (Controllability of Empirical Covariance). The corrected imputation model trained by  $\mathcal{L}_e^{Sup}$  is sufficient for controlling the empirical covariance.
+
+(i) For user-item pairs with observed outcomes, the empirical covariance is 0. Formally, we have
+
+$$
+\left. \frac {\partial}{\partial \epsilon} \mathcal {L} _ {e} ^ {S u p} (\phi , \epsilon) \right| _ {\epsilon = \epsilon^ {*}} = 0, \text {w h i c h i s e q u i v a l e n t t o} \frac {1}{| \mathcal {D} |} \sum_ {(u, i): o _ {u, i} = 1} \frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}} \cdot (e _ {u, i} - \tilde {e} _ {u, i}) = 0;
+$$
+
+(ii) For user-item pairs with missing outcomes, suppose that  $\hat{p}_{u,i} \geq K_{\psi}$  and  $|e_{u,i} - \tilde{e}_{u,i}| \leq K_{\phi}$ , then with probability at least  $1 - \eta$ , we have
+
+$$
+\begin{array}{l} \frac{1}{|\mathcal{D}|}\sum_{(u,i):o_{u,i} = 0}\frac{\hat{p}_{u,i} - o_{u,i}}{\hat{p}_{u,i}}\cdot (e_{u,i} - \tilde{e}_{u,i}) \\ \leq \underbrace {\sqrt {\mathcal {L} _ {e} ^ {S u p} (\phi , \epsilon)}} _ {\text {p r o p o s e d l o s s}} + K _ {\phi} \underbrace {\sqrt {\left. \right. \frac {1}{| \mathcal {D} |} \sum_ {u , i \in \mathcal {D}} \left| 1 - \mathbb {E} \left[ \frac {o _ {u , i}}{\hat {p} _ {u , i}} \mid x _ {u , i} \right]\right|} _ {\text {e m p i r i c a l b i a s f r o m p r o p e n s i t y m o d e l}} + \underbrace {\sqrt {K _ {\phi} \left(1 + \frac {1}{K _ {\psi}}\right)\left(2 \mathcal {R} (\mathcal {F}) + (2 K _ {\phi} + 1) \sqrt {\frac {2 \log (4 / \eta)}{| \mathcal {D} |}}\right)} _ {\text {t a i l b o u n d c o n t r o l l e d b y e m p i r i c a l R a d e m a c h e r c o m p l e x i t y a n d s a m p l e s i z e}}, \\ \end{array}
+$$
+
+where the  $K_{\psi}, K_{\phi}, \eta$  are constants.
+
+Proof. For the proof of Theorem 4.7(i), first recap that the proposed boosted imputation model is
+
+$$
+\tilde {e} _ {u, i} = m \left(x _ {u, i}; \phi\right) + \epsilon \left(o _ {u, i} - \pi \left(x _ {u, i}; \psi\right)\right),
+$$
+
+and the proposed corrected imputation loss function for training the boosted imputation model is
+
+$$
+(\phi^ {*}, \epsilon^ {*}) = \arg \min _ {\phi , \epsilon} \mathcal {L} _ {e} ^ {S u p} (\phi , \epsilon) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}}.
+$$
+
+By taking the partial derivative with respective to  $\epsilon$  of the above formula and setting it to zero, we have
+
+$$
+\left. \frac {\partial}{\partial \epsilon} \mathcal {L} _ {e} ^ {S u p} (\phi , \epsilon) \right| _ {\epsilon = \epsilon^ {*}} = 0, \quad \text {w h i c h i s e q u i v a l e n t t o} \quad \frac {1}{| \mathcal {D} |} \sum_ {(u, i): o _ {u, i} = 1} \frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}} \cdot (e _ {u, i} - \tilde {e} _ {u, i}) = 0,
+$$
+
+which proves the empirical covariance on the observed outcomes is 0.
+
+For the proof of Theorem 4.7(ii), by noting that
+
+$$
+\frac {1}{| \mathcal {D} |} \sum_ {(u, i): o _ {u, i} = 0} \frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}} \cdot (e _ {u, i} - \tilde {e} _ {u, i}) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i): o _ {u, i} = 0} (e _ {u, i} - \tilde {e} _ {u, i}) \leq \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} (e _ {u, i} - \tilde {e} _ {u, i}) ^ {2} \right] ^ {\frac {1}{2}},
+$$
+
+we now focus on bounding the last term of the above equation with the least probability.
+
+Suppose that  $\hat{p}_{u,i} \geq K_{\psi}$  and  $|e_{u,i} - \tilde{e}_{u,i}| \leq K_{\phi}$ , then
+
+$$
+\begin{array}{l} \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} (e _ {u, i} - \tilde {e} _ {u, i}) ^ {2} = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}} + \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} (e _ {u, i} - \tilde {e} _ {u, i}) ^ {2} \\ - \mathbb {E} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}} \right] + \mathbb {E} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}} \right] - \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}} \\ \leq \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}} + \left| \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} (e _ {u, i} - \tilde {e} _ {u, i}) ^ {2} - \mathbb {E} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}} \right] \right| \\ + \left(\mathbb {E} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} \left(e _ {u , i} - \tilde {e} _ {u , i}\right) ^ {2}}{\hat {p} _ {u , i}} \right] - \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} \left(e _ {u , i} - \tilde {e} _ {u , i}\right) ^ {2}}{\hat {p} _ {u , i}}\right) \\ \leq \mathcal {L} _ {e} ^ {S u p} (\phi , \epsilon) + K _ {\phi} ^ {2} \cdot \left| \mathbb {E} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} 1 - \frac {o _ {u , i}}{\hat {p} _ {u , i}} \right] \right| \\ + \sup  _ {f _ {\theta} \in \mathcal {F}} \left(\mathbb {E} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}} \right] - \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}}\right). \\ \end{array}
+$$
+
+For simplicity, we denote the last term in the above formula as
+
+$$
+\mathcal {B} (\mathcal {F}) = \sup _ {f _ {\theta} \in \mathcal {F}} \left(\mathbb {E} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}} \right] - \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}}\right),
+$$
+
+we then aim to bound  $\mathcal{B}(\mathcal{F})$  in the following.
+
+Note that
+
+$$
+\mathcal {B} (\mathcal {F}) = \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} \left[ \mathcal {B} (\mathcal {F}) \right] + \left\{\mathcal {B} (\mathcal {F}) - \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} \left[ \mathcal {B} (\mathcal {F}) \right] \right\},
+$$
+
+where the first term is  $\mathbb{E}_{\mathcal{S}\sim \mathbb{P}^{|\mathcal{D}|}}[\mathcal{B}(\mathcal{F})]$ , and by Lemma A.1 we have
+
+$$
+\underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} \left[ \mathcal {B} (\mathcal {F}) \right] \leq 2 \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} \underset {\sigma \sim \{- 1, + 1 \} ^ {| \mathcal {D} |}} {\mathbb {E}} \sup _ {f _ {\theta} \in \mathcal {F}} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \sigma_ {u, i} \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}} \right].
+$$
+
+By the assumptions that  $\hat{p}_{u,i} \geq K_{\psi}$  and  $|e_{u,i} - \tilde{e}_{u,i}| \leq K_{\phi}$ , we have
+
+$$
+\begin{array}{l} \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} \left[ \mathcal {B} (\mathcal {F}) \right] \leq 2 K _ {\phi} \left(1 + \frac {1}{K _ {\psi}}\right) \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} \underset {\sigma \sim \{- 1, + 1 \} ^ {| \mathcal {D} |}} {\mathbb {E}} \sup  _ {f _ {\theta} \in \mathcal {F}} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \sigma_ {u, i} (e _ {u, i} - \tilde {e} _ {u, i}) \right] \\ = 2 K _ {\phi} \left(1 + \frac {1}{K _ {\psi}}\right) \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} \{\mathcal {R} (\mathcal {F}) \}, \\ \end{array}
+$$
+
+where the last equation is directly from Lemma A.3, and  $\mathcal{R}(\mathcal{F})$  is the empirical Rademacher complexity
+
+$$
+\mathcal {R} (\mathcal {F}) = \mathbb {E} _ {\boldsymbol {\sigma} \sim \{- 1, + 1 \} ^ {| \mathcal {D} |}} \sup _ {f _ {\theta} \in \mathcal {F}} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \sigma_ {u, i} e _ {u, i} \right],
+$$
+
+where  $\sigma = \{\sigma_{u,i}:(u,i)\in \mathcal{D}\}$ , and  $\sigma_{u,i}$  are independent uniform random variables taking values in  $\{-1, + 1\}$ . The random variables  $\sigma_{u,i}$  are called Rademacher variables.
+
+By applying McDiarmid's inequality in Lemma A.2, and let  $c = \frac{2K_{\phi}}{|D|}$ , with probability at least  $1 - \frac{\eta}{2}$
+
+$$
+\left| \mathcal {R} (\mathcal {F}) - \underset {S \sim \mathbb {P} | \mathcal {D} |} {\mathbb {E}} \{\mathcal {R} (\mathcal {F}) \} \right| \leq 2 K _ {\phi} \sqrt {\frac {\log (4 / \eta)}{2 | \mathcal {D} |}} = K _ {\phi} \sqrt {\frac {2 \log (4 / \eta)}{| \mathcal {D} |}}.
+$$
+
+For the rest term  $\mathcal{B}(\mathcal{F}) - \underset {S\sim \mathbb{P}^{|D|}}{\mathbb{E}}[\mathcal{B}(\mathcal{F})]$ , by applying McDiarmid's inequality in Lemma A.2 and the assumptions that  $\hat{p}_{u,i}\geq K_{\psi}$  and  $|e_{u,i} - \tilde{e}_{u,i}|\leq K_{\phi}$ , let  $c = \frac{2K_{\phi}^{2}\left(1 + \frac{1}{K_{\psi}}\right)}{|\mathcal{D}|}$ , then with probability at least  $1 - \frac{\eta}{2}$ ,
+
+$$
+\left| \mathcal {B} (\mathcal {F}) - \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} [ \mathcal {B} (\mathcal {F}) ] \right| \leq 2 K _ {\phi} ^ {2} \left(1 + \frac {1}{K _ {\psi}}\right) \sqrt {\frac {\log (4 / \eta)}{2 | \mathcal {D} |}} = K _ {\phi} ^ {2} \left(1 + \frac {1}{K _ {\psi}}\right) \sqrt {\frac {2 \log (4 / \eta)}{| \mathcal {D} |}}.
+$$
+
+We now bound  $\mathcal{B}(\mathcal{F})$  combining the above results. Formally, we have
+
+$$
+\begin{array}{l} \mathcal {B} (\mathcal {F}) = \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} [ \mathcal {B} (\mathcal {F}) ] + \left\{\mathcal {B} (\mathcal {F}) - \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} [ \mathcal {B} (\mathcal {F}) ] \right\} \\ \leq 2 K _ {\phi} \left(1 + \frac {1}{K _ {\psi}}\right) \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} \{\mathcal {R} (\mathcal {F}) \} + \left\{\mathcal {B} (\mathcal {F}) - \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} [ \mathcal {B} (\mathcal {F}) ] \right\}. \\ \end{array}
+$$
+
+With probability at least  $1 - \eta$ , we have
+
+$$
+\begin{array}{l} \mathcal {B} (\mathcal {F}) \leq 2 K _ {\phi} \left(1 + \frac {1}{K _ {\psi}}\right) \left(\mathcal {R} (\mathcal {F}) + K _ {\phi} \sqrt {\frac {2 \log (4 / \eta)}{| \mathcal {D} |}}\right) + K _ {\phi} ^ {2} \left(1 + \frac {1}{K _ {\psi}}\right) \sqrt {\frac {2 \log (4 / \eta)}{| \mathcal {D} |}} \\ = K _ {\phi} \left(1 + \frac {1}{K _ {\psi}}\right) \left(2 \mathcal {R} (\mathcal {F}) + K _ {\phi} \sqrt {\frac {1 8 \log (4 / \eta)}{| \mathcal {D} |}}\right). \\ \end{array}
+$$
+
+We now bound the empirical covariance on the missing outcomes combining the above results. Formally, we have
+
+$$
+\begin{array}{l} \frac {1}{| \mathcal {D} |} \sum_ {(u, i): o _ {u, i} = 0} \frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}} \cdot (e _ {u, i} - \tilde {e} _ {u, i}) \leq \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} (e _ {u, i} - \tilde {e} _ {u, i}) ^ {2} \right] ^ {\frac {1}{2}} \\ \leq \left[ \mathcal {L} _ {e} ^ {S u p} (\phi , \epsilon) + \frac {K _ {\phi} ^ {2}}{| \mathcal {D} |} \sum_ {u, i \in \mathcal {D}} \left| 1 - \mathbb {E} \left[ \frac {o _ {u , i}}{\hat {p} _ {u , i}} \big | x _ {u, i} \right] \right| + K _ {\phi} \left(1 + \frac {1}{K _ {\psi}}\right) \left(2 \mathcal {R} (\mathcal {F}) + K _ {\phi} \sqrt {\frac {1 8 \log (4 / \eta)}{| \mathcal {D} |}}\right) \right] ^ {\frac {1}{2}} \\ \leq \mathcal {L} _ {e} ^ {S u p} (\phi , \epsilon) ^ {\frac {1}{2}} + K _ {\phi} \cdot \left[ \frac {1}{| \mathcal {D} |} \sum_ {u, i \in \mathcal {D}} \left| 1 - \mathbb {E} \left[ \frac {o _ {u , i}}{\hat {p} _ {u , i}} \big | x _ {u, i} \right] \right| \right] ^ {\frac {1}{2}} + \\ \left[ K _ {\phi} \left(1 + \frac {1}{K _ {\psi}}\right) \left(2 \mathcal {R} (\mathcal {F}) + K _ {\phi} \sqrt {\frac {1 8 \log (4 / \eta)}{| \mathcal {D} |}}\right) \right] ^ {\frac {1}{2}}, \\ \end{array}
+$$
+
+which yields the stated results.
+
+![](images/8a50bfbde9c1782f15252c7db0ab31c9cb6d504109cec834503bd2c4ca291538.jpg)
+
+Corollary 4.8 (Relation to previous imputed errors). The learned coefficient  $\epsilon^{*}$  will converge to zero when the imputation model  $\hat{e}_{u,i}$  has zero empirical covariance, making  $\tilde{e}_{u,i}$  degenerates to  $\hat{e}_{u,i}$ .
+
+Proof of Corollary 4.8. Note that  $\epsilon^{*}$  is solved by minimizing
+
+$$
+\frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \hat {e} _ {u , i} - \epsilon (o _ {u , i} - \hat {p} _ {u , i})) ^ {2}}{\hat {p} _ {u , i}}.
+$$
+
+Taking the first derivative of the above loss with respect to  $\epsilon$  and setting it to zero yields
+
+$$
+\sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i}}{\hat {p} _ {u , i}} \cdot \left\{e _ {u, i} - \hat {e} _ {u, i} - \epsilon (o _ {u, i} - \hat {p} _ {u, i}) \right\} \cdot (o _ {u, i} - \hat {p} _ {u, i}) = 0,
+$$
+
+which implies that
+
+$$
+\sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i}}{\hat {p} _ {u , i}} \cdot \left\{e _ {u, i} - \tilde {e} _ {u, i} \right\} \cdot \left(o _ {u, i} - \hat {p} _ {u, i}\right) = 0,
+$$
+
+from which implies the uniqueness of  $\epsilon$ . Formally, if  $\hat{e}_{u,i}$  already satisfies zero empirical covariance on the observed outcomes, then  $\epsilon = 0$  is a solution of the above equation. Let  $\hat{\epsilon}$  be another solution of the above equation. Since the solution of equation is unique, then  $\hat{\epsilon}$  will converge to 0, making  $\tilde{e}_{u,i}$  degenerates to  $\hat{e}_{u,i}$ .
+
+Corollary 4.9 (Bias reduction property). The proposed corrected imputation loss leads to the smaller bias of imputed errors  $\tilde{e}_{u,i}$ , when  $\hat{e}_{u,i}$  are inaccurate. Formally, we have
+
+$$
+\min _ {\phi , \epsilon} \mathcal {L} _ {e} ^ {S u p} (\phi , \epsilon) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \tilde {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}} \leq \min _ {\phi} \mathcal {L} _ {e} (\phi) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \hat {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}}.
+$$
+
+Proof of Corollary 4.9. The result holds by noting that
+
+$$
+\min _ {\phi , \epsilon} \mathcal {L} _ {e} ^ {S u p} (\phi , \epsilon) \leq \min _ {\phi} \mathcal {L} _ {e} ^ {S u p} (\phi , \epsilon = 0) = \min _ {\phi} \mathcal {L} _ {e} (\phi) = \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {o _ {u , i} (e _ {u , i} - \hat {e} _ {u , i}) ^ {2}}{\hat {p} _ {u , i}}.
+$$
+
+![](images/16fd047622864340c74e99efb04f5d1774201fd6c7f6df12ada61d5e6b1d2258.jpg)
+
+Corollary 4.10 (Variance reduction property). The proposed corrected imputation loss leads to the smaller variance of  $\tilde{e}_{u,i}$  when the optimal  $\epsilon^{*}$  lies in a certain range. Formally, we have
+
+$$
+\mathbb {V} (\tilde {e} _ {u, i}) = \mathbb {V} (\hat {e} _ {u, i} + \epsilon^ {*} \cdot (o _ {u, i} - \hat {p} _ {u, i})) \leq \mathbb {V} (\hat {e} _ {u, i}), \text {i f} \epsilon^ {*} \in \left[ 0, 2 \cdot \frac {\operatorname {C o v} (\hat {e} _ {u , i} , \hat {p} _ {u , i} - o _ {u , i})}{\mathbb {V} (\hat {p} _ {u , i} - o _ {u , i})} \right].
+$$
+
+Proof of Corollary 4.10. First, we note that  $\mathbb{V}(\tilde{e}_{u,i})$  equals to
+
+$$
+\mathbb {V} (\hat {e} _ {u, i}) - 2 \epsilon^ {*} \operatorname {C o v} (\hat {e} _ {u, i}, \hat {p} _ {u, i} - o _ {u, i}) + (\epsilon^ {*}) ^ {2} \mathbb {V} (o _ {u, i} - \hat {p} _ {u, i}),
+$$
+
+which serves as a quadratic function with respect to  $\epsilon^{*}$ . By taking the partial derivative respective to  $\epsilon^{*}$  of the above formula and setting it to zero, the optimal  $\epsilon^{*}$  with the minimal variance is given as
+
+$$
+\epsilon^ {*} = \frac {\mathrm {C o v} (\hat {e} _ {u , i} , \hat {p} _ {u , i} - o _ {u , i})}{\mathbb {V} (\hat {p} _ {u , i} - o _ {u , i})}.
+$$
+
+By exploiting the symmetry of the quadratic function, we have
+
+$$
+\begin{array}{l} \mathbb {V} \left(\tilde {e} _ {u, i}\right) = \mathbb {V} \left(\hat {e} _ {u, i} + \epsilon^ {*} \cdot \left(o _ {u, i} - \hat {p} _ {u, i}\right)\right) \leq \mathbb {V} \left(\hat {e} _ {u, i}\right), \\ \text {i f} \quad \epsilon^ {*} \in \left[ 0, 2 \cdot \frac {\operatorname {C o v} \left(\hat {e} _ {u , i} , \hat {p} _ {u , i} - o _ {u , i}\right)}{\mathbb {V} \left(\hat {p} _ {u , i} - o _ {u , i}\right)} \right]. \\ \end{array}
+$$
+
+![](images/1851ddab81260306f57fbb16cb790bee7e146219df7f96ab85e963dde6b4d149.jpg)
+
+Theorem 4.11 (Generalization Bound under Superpopulation). Suppose that  $\hat{p}_{u,i} \geq K_{\psi}$  and  $\min \{\tilde{e}_{u,i}, |e_{u,i} - \tilde{e}_{u,i}|\} \leq K_{\phi}$ , then with probability at least  $1 - \eta$ , we have
+
+$$
+\begin{array}{l} \mathcal {L} _ {\text {i d e a l}} (\theta) \leq \underbrace {\mathcal {L} _ {\text {S u p e r D R}} (\theta) + \frac {1}{| \mathcal {D} |} \sum_ {(u , i) \in \mathcal {D}} \left| 1 - \mathbb {E} \left[ \frac {o _ {u , i}}{\hat {p} _ {u , i}} \mid x _ {u , i} \right] \right| \cdot \left| \mathbb {E} [ e _ {u , i} \mid x _ {u , i} ] - \mathbb {E} [ \tilde {e} _ {u , i} \mid x _ {u , i} ] \right|} _ {\text {v a n i l l a D R o n l y c o n t r o l s t h e e m p i r i c a l D R l o s s , a n d e m p i r i c a l r i s k s o f i m p u t a t i o n a n d p r o p e n s i t y m o d e l s}} \\ + \underbrace {\left| \frac {1}{| \mathcal {D} |} \sum_ {(u , i) \in \mathcal {D}} \operatorname {C o v} \left(\frac {o _ {u , i} - \hat {p} _ {u , i}}{\hat {p} _ {u , i}} , e _ {u , i} - \tilde {e} _ {u , i}\right) \right|} _ {\text {c o r r e c t e d l o s s f u r t h e r c o n t r o l e d s t h e i n d e p e n d e n c e}} + \underbrace {\left(1 + \frac {1}{K _ {\psi}}\right) \left(2 \mathcal {R} (\mathcal {F}) + K _ {\phi} \sqrt {\frac {1 8 \log (4 / \eta)}{| \mathcal {D} |}}\right)} _ {\text {t a i l b o u n d c o n t r o l l e d b y e m p i r i c a l R a d e m a c h e r c o m p l e x i t y a n d s a m p l e s i z e}} \\ \end{array}
+$$
+
+Proof of Theorem 4.11. First we decompose the ideal loss as follows.
+
+$$
+\begin{array}{l} \mathcal {L} _ {\text {i d e a l}} (\theta) = \mathcal {L} _ {\mathrm {D R}} (\theta) + \left(\mathcal {L} _ {\text {i d e a l}} (\theta) - \mathbb {E} \left[ \mathcal {L} _ {\mathrm {D R}} (\theta) \right]\right) + \left(\mathbb {E} \left[ \mathcal {L} _ {\mathrm {D R}} (\theta) \right] - \mathcal {L} _ {\mathrm {D R}} (\theta)\right) \\ = \mathcal {L} _ {\mathrm {D R}} (\theta) + \operatorname {B i a s} \left[ \mathcal {L} _ {\mathrm {D R}} (\theta) \right] + \left(\mathbb {E} \left[ \mathcal {L} _ {\mathrm {D R}} (\theta) \right] - \mathcal {L} _ {\mathrm {D R}} (\theta)\right) \\ \leq \mathcal {L} _ {\mathrm {D R}} (\theta) + | \operatorname {B i a s} [ \mathcal {L} _ {\mathrm {D R}} (\theta) ] | \\ + \sup  _ {f _ {\theta} \in \mathcal {F}} \left(\mathbb {E} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \hat {e} _ {u, i} + \frac {o _ {u , i} (e _ {u , i} - \hat {e} _ {u , i})}{\hat {p} _ {u , i}} \right] - \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \hat {e} _ {u, i} - \frac {o _ {u , i} (e _ {u , i} - \hat {e} _ {u , i})}{\hat {p} _ {u , i}}\right). \\ \end{array}
+$$
+
+For simplicity, we denote the last term in the above formula as
+
+$$
+\mathcal {B} (\mathcal {F}) = \sup  _ {f _ {\theta} \in \mathcal {F}} \left(\mathbb {E} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \hat {e} _ {u, i} + \frac {o _ {u , i} \left(e _ {u , i} - \hat {e} _ {u , i}\right)}{\hat {p} _ {u , i}} \right] - \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \hat {e} _ {u, i} - \frac {o _ {u , i} \left(e _ {u , i} - \hat {e} _ {u , i}\right)}{\hat {p} _ {u , i}}\right),
+$$
+
+we then aim to bound  $\mathcal{B}(\mathcal{F})$  in the following.
+
+Note that
+
+$$
+\mathcal {B} (\mathcal {F}) = \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} [ \mathcal {B} (\mathcal {F}) ] + \left\{\mathcal {B} (\mathcal {F}) - \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} [ \mathcal {B} (\mathcal {F}) ] \right\},
+$$
+
+where the first term is  $\mathbb{E}_{\mathcal{S}\sim \mathbb{P}^{|\mathcal{D}|}}[\mathcal{B}(\mathcal{F})]$ , and by Lemma A.1 we have
+
+$$
+\underset {S \sim \mathbb {P} | \mathcal {D} |} {\mathbb {E}} \left[ \mathcal {B} (\mathcal {F}) \right] \leq 2 \underset {S \sim \mathbb {P} | \mathcal {D} |} {\mathbb {E}} \underset {\sigma \sim \{- 1, + 1 \} ^ {| \mathcal {D} |}} {\mathbb {E}} \sup  _ {f _ {\theta} \in \mathcal {F}} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \sigma_ {u, i} \hat {e} _ {u, i} + \frac {\sigma_ {u , i} o _ {u , i} (e _ {u , i} - \hat {e} _ {u , i})}{\hat {p} _ {u , i}} \right].
+$$
+
+By the assumptions that  $\hat{p}_{u,i} \geq K_{\psi}$  and  $\min \{\hat{e}_{u,i}, |e_{u,i} - \hat{e}_{u,i}|\} \leq K_{\phi}$ , we have
+
+$$
+\begin{array}{l} \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} \left[ \mathcal {B} (\mathcal {F}) \right] \leq 2 \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} \underset {\sigma \sim \{- 1, + 1 \} ^ {| \mathcal {D} |}} {\mathbb {E}} \sup  _ {f _ {\theta} \in \mathcal {F}} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \frac {\sigma_ {u , i} o _ {u , i} (e _ {u , i} - \hat {e} _ {u , i})}{\hat {p} _ {u , i}} \right] \\ \leq 2 \left(1 + \frac {1}{K _ {\psi}}\right) \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} \{\mathcal {R} (\mathcal {F}) \}, \\ \end{array}
+$$
+
+where the first equation is from Lemma A.3, and  $\mathcal{R}(\mathcal{F})$  is the empirical Rademacher complexity
+
+$$
+\mathcal {R} (\mathcal {F}) = \mathbb {E} _ {\boldsymbol {\sigma} \sim \{- 1, + 1 \} ^ {| \mathcal {D} |}} \sup  _ {f _ {\theta} \in \mathcal {F}} \left[ \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \sigma_ {u, i} e _ {u, i} \right],
+$$
+
+where  $\sigma = \{\sigma_{u,i}:(u,i)\in \mathcal{D}\}$ , and  $\sigma_{u,i}$  are independent uniform random variables taking values in  $\{-1, + 1\}$ . The random variables  $\sigma_{u,i}$  are called Rademacher variables.
+
+By applying McDiarmid's inequality in Lemma A.2, and let  $c = \frac{2K_{\phi}}{|\mathcal{D}|}$ , with probability at least  $1 - \frac{\eta}{2}$
+
+$$
+\left| \mathcal {R} (\mathcal {F}) - \underset {S \sim \mathbb {P} | \mathcal {D} |} {\mathbb {E}} \left\{\mathcal {R} (\mathcal {F}) \right\} \right| \leq 2 K _ {\phi} \sqrt {\frac {\log (4 / \eta)}{2 | \mathcal {D} |}} = K _ {\phi} \sqrt {\frac {2 \log (4 / \eta)}{| \mathcal {D} |}}.
+$$
+
+For the rest term  $\mathcal{B}(\mathcal{F}) - \underset {S\sim \mathbb{P}^{|\mathcal{D}|}}{\mathbb{E}}[\mathcal{B}(\mathcal{F})]$ , by applying McDiarmid's inequality in Lemma A.2 and the assumptions that  $\hat{p}_{u,i}\geq K_{\psi}$  and  $\min \{\hat{e}_{u,i},|e_{u,i} - \hat{e}_{u,i}|\} \leq K_{\phi}$ , let  $c = \frac{2K_{\phi}\left(1 + \frac{1}{K_{\psi}}\right)}{|\mathcal{D}|}$ , then with probability at least  $1 - \frac{\eta}{2}$ ,
+
+$$
+\left| \mathcal {B} (\mathcal {F}) - \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} [ \mathcal {B} (\mathcal {F}) ] \right| \leq 2 K _ {\phi} \left(1 + \frac {1}{K _ {\psi}}\right) \sqrt {\frac {\log (4 / \eta)}{2 | \mathcal {D} |}} = K _ {\phi} \left(1 + \frac {1}{K _ {\psi}}\right) \sqrt {\frac {2 \log (4 / \eta)}{| \mathcal {D} |}}.
+$$
+
+We now bound  $\mathcal{B}(\mathcal{F})$  combining the above results. Formally, we have
+
+$$
+\begin{array}{l} \mathcal {B} (\mathcal {F}) = \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} [ \mathcal {B} (\mathcal {F}) ] + \left\{\mathcal {B} (\mathcal {F}) - \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} [ \mathcal {B} (\mathcal {F}) ] \right\} \\ \leq 2 \left(1 + \frac {1}{K _ {\psi}}\right) \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} \{\mathcal {R} (\mathcal {F}) \} + \left\{\mathcal {B} (\mathcal {F}) - \underset {S \sim \mathbb {P} ^ {| \mathcal {D} |}} {\mathbb {E}} [ \mathcal {B} (\mathcal {F}) ] \right\}. \\ \end{array}
+$$
+
+With probability at least  $1 - \eta$ , we have
+
+$$
+\begin{array}{l} \mathcal {B} (\mathcal {F}) \leq 2 \left(1 + \frac {1}{K _ {\psi}}\right) \left(\mathcal {R} (\mathcal {F}) + K _ {\phi} \sqrt {\frac {2 \log (4 / \eta)}{| \mathcal {D} |}}\right) + K _ {\phi} \left(1 + \frac {1}{K _ {\psi}}\right) \sqrt {\frac {2 \log (4 / \eta)}{| \mathcal {D} |}} \\ = \left(1 + \frac {1}{K _ {\psi}}\right) \left(2 \mathcal {R} (\mathcal {F}) + K _ {\phi} \sqrt {\frac {1 8 \log (4 / \eta)}{| \mathcal {D} |}}\right). \\ \end{array}
+$$
+
+We now bound the ideal loss combining the above results. Formally, we have
+
+$$
+\begin{array}{l} \mathcal {L} _ {\text {i d e a l}} (\theta) \leq \mathcal {L} _ {\mathrm {D R}} (\theta) + | \operatorname {B i a s} [ \mathcal {L} _ {\mathrm {D R}} (\theta) ] | + \mathcal {B} (\mathcal {F}) \\ \leq \mathcal {L} _ {\mathrm {D R}} (\theta) + | \mathrm {B i a s} [ \mathcal {L} _ {\mathrm {D R}} (\theta) ] | + \left(1 + \frac {1}{K _ {\psi}}\right) \left(2 \mathcal {R} (\mathcal {F}) + K _ {\phi} \sqrt {\frac {1 8 \log (4 / \eta)}{| \mathcal {D} |}}\right). \\ \end{array}
+$$
+
+In Theorem 4.3, we have already proved that
+
+$$
+\begin{array}{l} | \operatorname {B i a s} [ \mathcal {E} _ {\mathrm {D R}} (\theta) ] | = \left| \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \operatorname {C o v} \left(\frac {\hat {p} _ {u , i} - o _ {u , i}}{\hat {p} _ {u , i}}, e _ {u, i} - \hat {e} _ {u, i}\right) \right. \\ + \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \left[ \left\{1 - \mathbb {E} \left[ \frac {o _ {u , i}}{\hat {p} _ {u , i}} \mid x _ {u, i} \right] \right\} \cdot \left\{\mathbb {E} \left[ e _ {u, i} \mid x _ {u, i} \right] - \mathbb {E} [ \hat {e} _ {u, i} \mid x _ {u, i} ] \right\} \right] \\ \leq \left| \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \operatorname {C o v} \left(\frac {o _ {u , i} - \hat {p} _ {u , i}}{\hat {p} _ {u , i}}, e _ {u, i} - \hat {e} _ {u, i}\right) \right| \\ + \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \left| 1 - \mathbb {E} \left[ \frac {o _ {u , i}}{\hat {p} _ {u , i}} | x _ {u, i} \right] \right| \cdot \left| \mathbb {E} [ e _ {u, i} | x _ {u, i} ] - \mathbb {E} [ \hat {e} _ {u, i} | x _ {u, i} ] \right|, \\ \end{array}
+$$
+
+therefore with probability at least  $1 - \eta$ , we have
+
+$$
+\begin{array}{l} \mathcal {L} _ {i d e a l} (\theta) \leq \mathcal {L} _ {\mathrm {D R}} (\theta) + \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \left| 1 - \mathbb {E} \left[ \frac {\sigma_ {u , i}}{\hat {p} _ {u , i}} \big | x _ {u, i} \right] \right| \cdot \left| \mathbb {E} [ e _ {u, i} \mid x _ {u, i} ] - \mathbb {E} [ \hat {e} _ {u, i} \mid x _ {u, i} ] \right| \\ + \left| \frac {1}{| \mathcal {D} |} \sum_ {(u, i) \in \mathcal {D}} \operatorname {C o v} \left(\frac {o _ {u , i} - \hat {p} _ {u , i}}{\hat {p} _ {u , i}}, e _ {u, i} - \hat {e} _ {u, i}\right) \right| + \left(1 + \frac {1}{K _ {\psi}}\right) \left(2 \mathcal {R} (\mathcal {F}) + K _ {\phi} \sqrt {\frac {1 8 \log (4 / \eta)}{| \mathcal {D} |}}\right), \\ \end{array}
+$$
+
+which yields the stated results.
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: We provide main claims and contributions in abstract and introduction.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: We discuss the limitations of the work in Conclusion.
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
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+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: We provide a complete proof in Appendix.
+
+# Guidelines:
+
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+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
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+
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+
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+
+# 5. Open access to data and code
+
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+
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+
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+
+# 7. Experiment statistical significance
+
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+
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+
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+
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+
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
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+
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+
+# 9. Code of ethics
+
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+
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+
+# 10. Broader impacts
+
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+
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+
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+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
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+
+# 12. Licenses for existing assets
+
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+
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+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
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+
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+
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+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
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+
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+
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+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
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+
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+
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+
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+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: The core methodological development in this research does not involve large language models (LLMs) as essential, original, or non-standard components.
+
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\ No newline at end of file
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+# Unveiling Transformer Perception by Exploring Input Manifolds
+
+Alessandro Benfenati<sup>1a</sup> Alfio Ferrara<sup>1b</sup> Alessio Marta<sup>1c</sup> Davide Riva<sup>2d</sup> Elisabetta Rocchetti<sup>1b*</sup>
+
+$^a$ Department of Environmental Science and Policy,  $^b$ Department of Computer Science,
+
+$^c$ Department of Mathematics,  $^d$ Department of Control and Computer Engineering
+
+<sup>1</sup>Università degli Studi di Milano, <sup>2</sup>Politecnico di Torino
+
+{alessandro.benfenati, alfio.ferrara}@unimi.it
+
+{alessio.marta, elisabetta.rochetti}@unimi.it
+
+{davide.riva}@polito.it
+
+* Corresponding author
+
+# Abstract
+
+This paper introduces a general method for the exploration of equivalence classes in the input space of Transformer models. The proposed approach is based on sound mathematical theory which describes the internal layers of a Transformer architecture as sequential deformations of the input manifold. Using eigendecomposition of the pullback of the distance metric defined on the output space through the Jacobian of the model, we are able to reconstruct equivalence classes in the input space and navigate across them. Our method enables two complementary exploration procedures: the first retrieves input instances that produce the same class probability distribution as the original instance—thus identifying elements within the same equivalence class—while the second discovers instances that yield a different class probability distribution, effectively navigating toward distinct equivalence classes. Finally, we demonstrate how the retrieved instances can be meaningfully interpreted by projecting their embeddings back into a human-readable format. Disclaimer: This paper includes examples of sensitive and very offensive language solely to illustrate the behavior of LLMs exploring the input space.
+
+# 1 Introduction
+
+In the literature, the investigation of the input space of Transformers relies on perturbations of input data using heuristic or gradient-based criteria [24, 17, 14], or on the analysis of specific properties of the embedding space [6] via the production of optimal robust explanations and counterfactuals. In this paper, we propose a method for exploring the input space of Transformer models by identifying equivalence classes with respect to their predictions. Our approach is based on sound mathematical theory which describes the internal layers of a Transformer architecture as sequential deformations of the input manifold. Using eigendecomposition of the pullback of the distance metric defined on the output space through the Jacobian of the model, we are able to reconstruct equivalence classes in the input space and navigate in and across them. The equivalence class consists of the counterimage of a particular probability distribution: this means that the elements of an equivalence class, once fed to the Transformer, will provide different realizations from the same probability distribution. Thanks to our approach, we provide two different methods for exploring the embedding space: the Singular Metric Equivalence Class (SiMEC) and the Singular Metric Exploration (SiMExp) algorithms. The first allows for the identification of inputs within the same equivalence class. This means that, given two data inputs  $x$ ,  $x'$  identified through the exploration process as belonging to the same equivalence class and a class label  $C$ , our method guarantees that the Transformers will assign the same probability, modulo numerical approximations, to that label for both inputs:  $p(C|x) \approx p(C|x')$ .
+
+The second method, instead, allows for the exploration of the embedding space starting from an element within an equivalence class and moving towards a different equivalence class. This means that, given two inputs, namely  $x, x'$ , identified in this way and a class label  $C$ , we guarantee that the Transformer will assign different probabilities to that label for the two inputs, i.e.,  $p(C|x) \neq p(C|x')$ . Our experimental data show that this different probability assignment can lead to a change in the most probable class in the Transformers prediction.
+
+In Section 2, we summarise the mathematical foundations of our approach. In Section 3, we present our method for the exploration of equivalence classes in the input of the Transformer models. In Section 4, we empirically investigate the effectiveness and applicability on Transformer models on textual and visual data. In Section 5, we discuss the relevant literature on embedding space exploration. Finally, in Section 6, we give our concluding remarks<sup>1</sup>.
+
+# 2 Preliminaries
+
+We provide in this Section the theoretical foundation of the proposed approach, namely the Geometric Deep Learning framework based on Riemannian Geometry [1, 2].
+
+A neural network is considered as a sequence of maps, the layers of the network, between manifolds, and the latter are the spaces where the input and the outputs of the layers belong to.
+
+Definition 1 (Neural Network). A neural network is a sequence of  $\mathcal{C}^1$  maps  $\Lambda_{i}$  between manifolds of the form:
+
+$$
+M _ {0} \xrightarrow {\Lambda_ {1}} M _ {1} \xrightarrow {\Lambda_ {2}} M _ {2} \xrightarrow {\Lambda_ {3}} \dots \xrightarrow {\Lambda_ {n - 1}} M _ {n - 1} \xrightarrow {\Lambda_ {n}} M _ {n} \tag {1}
+$$
+
+We call  $M_0$  the input manifold and  $M_{n}$  the output manifold. All the other manifolds of the sequence are called representation manifolds. The maps  $\Lambda_{i}$  are the layers of the neural network. We denote by  $\mathcal{N}_{(i)} = \Lambda_n\circ \dots \circ \Lambda_i:M_i\to M_n$  the mapping from the  $i$ -th representation layer to the output layer.
+
+As an example, consider a shallow network with just one layer, the composition of a linear operator  $A \cdot +b$  with a sigmoid function  $\sigma$ , where  $A \in \mathbb{R}^{m \times n}$  and  $b \in \mathbb{R}^m$ : then, the input manifold  $M_0$  and the output manifold  $M_1$  shall be  $\mathbb{R}^n$  and  $\mathbb{R}^m$ , respectively, and the map  $\Lambda_1(\cdot) = \sigma(A \cdot +b)$ . We generalize this observation into the following definition.
+
+Definition 2 (Smooth layer). A map  $\Lambda_{i}:M_{i - 1}\to M_{i}$  is called a smooth layer if it is the restriction to  $M_{i - 1}$  of a function  $\overline{\Lambda}^{(i)}(x):\mathbb{R}^{d_{i - 1}}\to \mathbb{R}^{d_i}$  of the form  $\overline{\Lambda}_{\alpha}^{(i)}(x) = F_{\alpha}^{(i)}\left(\sum_{\beta}A_{\alpha \beta}^{(i)}x_{\beta} + b_{\alpha}^{(i)}\right)$ , for  $i = 1,\dots ,n$ ,  $x\in \mathbb{R}^{d_i}$ ,  $b^{(i)}\in \mathbb{R}^{d_i}$  and  $A^{(i)}\in \mathbb{R}^{d_i\times d_{i - 1}}$ , with  $F^{(i)}:\mathbb{R}^{d_i}\to \mathbb{R}^{d_i}$  a diffeomorphism.
+
+Remark 1. Transformers implicitly apply for this framework, since their modules are smooth functions, such as fully connected layers, GeLU and sigmoid activations, thus including also attention layers.
+
+Our aim is to transport the geometric information on the data lying in the output manifold to the input manifold: this allows us to obtain insight on how the network "sees" the input space, how it manipulates it for reaching its final conclusion. For fulfilling this objective, we need several tools from differential geometry. The first key ingredient is the notion of singular Riemannian metric, which has the intuitive meaning of a degenerate scalar product which changes point to point – the starting point for defining a non-Euclidean pseudodistance between points of a manifold.
+
+Definition 3 (Singular Riemannian metric). Let  $M = \mathbb{R}^n$  or an open subset of  $\mathbb{R}^n$ . A singular Riemannian metric  $g$  over  $M$  is a map  $g: M \to \operatorname{Bil}(\mathbb{R}^n \times \mathbb{R}^n)$  that associates to each point  $p$  a positive semidefinite symmetric bilinear form  $g_p: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$  in a smooth way.
+
+Without loss of generality, we can assume the following hypotheses on the sequence (1):  $i$  ) The manifolds  $M_{i}$  are open and path-connected sets of dimension  $\dim M_i = d_i$ . ii) The maps  $\Lambda_{i}$  are  $\mathcal{C}^1$  submersions. iii)  $\Lambda_{i}(M_{i - 1}) = M_{i}$  for every  $i = 1,\dots ,n$ . iv) The manifold  $M_{n}$  is equipped with the structure of Riemannian manifold, with metric  $g^{n}$ . Definition 3 naturally leads to the definition of the pseudolenth and of energy of a curve.
+
+Definition 4 (Pseudolength and energy of a curve). Let  $\gamma :[a,b]\to \mathbb{R}^n$  a curve defined on the interval  $[a,b]\subset \mathbb{R}$  and  $\| v\| _p = \sqrt{g_p(v,v)}$  the pseudo-norm induced by the pseudo-metric  $g_{p}$  at point  $p$ . Then the pseudolength of  $\gamma$  and its energy are defined as
+
+$$
+P l (\gamma) = \int_ {a} ^ {b} \| \dot {\gamma} (s) \| _ {\gamma (s)} d s = \int_ {a} ^ {b} \sqrt {g _ {\gamma (s)} (\dot {\gamma} (s) , \dot {\gamma} (s))} d s, \quad E (\gamma) = \int_ {a} ^ {b} \| \dot {\gamma} (s) \| _ {\gamma (s)} ^ {2} d s \tag {2}
+$$
+
+The notion of pseudolength leads naturally to define the distance between two points.
+
+Definition 5 (Pseudodistance). Let  $x, y \in M = \mathbb{R}^n$ . The pseudodistance between  $x$  and  $y$  is then
+
+$$
+P d (x, y) = \inf  \left\{P l (\gamma) \mid \gamma : [ 0, 1 ] \rightarrow M, \gamma \in \mathcal {C} ^ {1} ([ 0, 1 ]), \gamma (0) = x, \gamma (1) = y \right\}. \tag {3}
+$$
+
+One can observe that endowing the space  $\mathbb{R}^n$  with a singular Riemannian metric leads to have non trivial curves whose length is zero. A straightforward consequence is that there are distinct points whose pseudodistance is therefore zero: a natural equivalence relation arises, i.e.  $x\sim y\Leftrightarrow Pd(x,y) = 0$ , obtaining thus a metric space  $(\mathbb{R}^n /\sim ,Pd)$ .
+
+The second crucial tool is the notion of pullback of a function. Intuitively, given a map  $f: M \to N$  between two manifolds, the pullback operation allows to transfer the geometric information of the output  $N$  onto the input  $M$  by means of the Jacobian of  $f$ . More specifically, let  $f$  be a function from  $\mathbb{R}^p$  to  $\mathbb{R}^q$ , and fix the coordinate systems  $x = (x_1, \ldots, x_p)$  and  $y = (y_1, \ldots, y_q)$  on  $\mathbb{R}^p$  and on  $\mathbb{R}^q$ , respectively. Moreover, we endow  $\mathbb{R}^q$  with the standard Euclidean metric  $g$ , whose associated matrix is the identity. The space  $\mathbb{R}^p$  can then be equipped with the pullback metric  $f^* g$  whose representation matrix reads as:
+
+$$
+\left(f ^ {*} g\right) _ {i j} = \sum_ {h, k = 1} ^ {q} \left(\frac {\partial f _ {h}}{\partial x _ {i}}\right) g _ {h k} \left(\frac {\partial f _ {k}}{\partial x _ {j}}\right). \tag {4}
+$$
+
+The sequence (1) shows that a neural network can be considered simply as a function, a composition of maps: hence, taking  $f = \Lambda_{n} \circ \Lambda_{n-1} \circ \dots \circ \Lambda_{1}$  and supposing that  $M_{0} = \mathbb{R}^{p}$ ,  $M_{n} = \mathbb{R}^{q}$ , the generalization of (4) applied to (1) provides with the pullback of a generic neural network.
+
+Hereafter, we consider in (1) the case  $M_{n} = \mathbb{R}^{q}$ , equipped with the trivial metric  $g^{(n)} = I_q$ , i.e., the identity. Each manifold  $M_{i}$  of the sequence (1) is equipped with a Riemannian singular metric, denoted with  $g^{i}$ , obtained via the pullback of  $\mathcal{N}_{(i)}$ . The pseudolength of a curve  $\gamma$  on the  $i$ -th manifold, namely  $P l_{i}(\gamma)$ , is computed via the relative metric  $g^{i}$  via (2).
+
+# 2.1 General results
+
+We depict hereafter the theoretical bases of our approach. We denote with  $\mathcal{N}_i$  the submap  $\Lambda_i\circ \dots \circ \Lambda_n:M_i\to M_n$ , and with  $\mathcal{N}\equiv \mathcal{N}_0$  the map describing the action of the complete network. The starting point is to consider the pair  $(M_{i},Pd_{i})$ : this is a pseudometric space, which can be turned into a full-fledged metric space  $M_{i} / \sim_{i}$  by the metric identification  $x\sim_{i}y\Leftrightarrow Pd_{i}(x,y) = 0$ . The first result states that the length of a curve on the  $i$ -th manifold is preserved among the mapping on the subsequent manifolds.
+
+Proposition 1. Let  $\gamma :[0,1]\to M_i$  be a piecewise  $\mathcal{C}^1$  curve. Let  $j\in \{i,i + 1,\dots ,n\}$  and consider the curve  $\gamma_{j} = \Lambda_{j}\circ \dots \circ \Lambda_{i}\circ \gamma$  on  $M_j$ . Then  $Pl_{i}(\gamma) = Pl_{j}(\gamma_{j})$ .
+
+In particular this is true when  $k = n$ , i.e., the length of a curve is preserved in the last manifold. This result leads naturally to claim that if two points are in the same class of equivalence, then they are mapped into the same point under the action of the neural network.
+
+Proposition 2. If two points  $p, q \in M_i$  are in the same class of equivalence, then  $\mathcal{N}_i(p) = \mathcal{N}_i(q)$ .
+
+The next step is to prove that the sets  $M_i / \sim_i$  are actually smooth manifolds: to this aim, we introduce another equivalence relation:  $x \sim_{\mathcal{N}_i} y$  if and only if there exists a piecewise  $\gamma : [0,1] \to M_i$  such that  $\gamma(0) = x$ ,  $\gamma(1) = y$  and  $\mathcal{N}_i \circ \gamma(s) = \mathcal{N}_i(x) \forall s \in [0,1]$ . The introduction of this equivalence relation allows us to easily state the following proposition.
+
+Proposition 3. Let  $x, y \in M_i$ , then  $x \sim_i y$  if and only if  $x \sim_{\mathcal{N}_i} y$ .
+
+The following corollary contains the natural consequences of the previous result; the second point of the claim below is the counterpart of Proposition 2.
+
+Corollary 1. Under the hypothesis of Proposition 3, one has that  $M_i / \sim_i = M_i / \sim_{\mathcal{N}_i}$ . Moreover, if two points  $p, q \in M_i$  are connected by a  $\mathcal{C}^1$  curve  $\gamma : [0,1] \to M_i$  satisfying  $\mathcal{N}_i(p) = \mathcal{N}_i \circ \gamma(s)$  for every  $s \in [0,1]$ , then they lie in the same class of equivalence.
+
+One is then able to prove that the set  $M_i / \sim_i$  is a smooth manifold:
+
+Proposition 4.  $\frac{M_i}{\sim_i}$  is a smooth manifold of dimension dim  $(\mathcal{N}(M_0))$
+
+This last achievement provides practical insights about the projection  $\pi_i$  on the quotient space, that consists in the building block of the algorithms used for recovering and exploring the foliation in equivalence classes of a neural network.
+
+Proposition 5.  $\pi_i: M_i \to M_i / \sim_i$  is a smooth fiber bundle, with  $Ker(d\pi_i) = \mathcal{V}M_i$ , which is therefore an integrable distribution.  $\mathcal{V}M_i$  is the vertical bundle of  $M_i$ . Every class of equivalence  $[p]$  is a path-connected submanifold of  $M_i$  and coincide with the fiber of the bundle over the point  $p \in M_i$ .
+
+In [3] it is shown that these results keep to hold true in the case of convolutional layers and residual connections.
+
+# 3 Methodology
+
+The results depicted in Section 2.1 provide powerful tools for investigating how a neural network sees the input space starting from a point  $x$ . In particular we point out the following remarks: i) If two points  $x, y$  belonging to the input manifold  $M_0$  are such that  $x \sim_0 y$ , then  $\mathcal{N}(x) = \mathcal{N}(y)$ ; ii) given a point  $z \in M_n$ , the counterimage  $\mathcal{N}^{-1}(z)$  is a smooth manifold, whose connected components are classes of equivalence in  $M_0$  with respect to  $\sim_0$ , then a necessary condition for two points  $x, y \in M_0$  to be in the same class of equivalence is that  $\mathcal{N}(x) = \mathcal{N}(y)$ ; iii) any class of equivalence  $[x]$ ,  $x \in M_0$ , is a maximal integral submanifold of  $\mathcal{V}M_0$ . The above observations directly provide with a strategy to build up the equivalence class of an input point  $x \in M_0$ . Proposition 5 tells us that  $\mathcal{V}M_0$  is an integrable distribution, with dimension equal to the dimension of the kernel of  $g^0$ : we can hence find  $dim(\text{Ker}(g^0))$  vector fields which are a base for the tangent space of  $M_0$ . This means that we can compute the eigenvalue decomposition of  $g_x^0$  and consider the  $L$  linearly independent eigenvectors, namely  $\{v_l\}_{l=1,\ldots,L}$ , associated to the null eigenvalue: these eigenvectors depend smoothly on the point, a fact that is not trivial when the matrix associated to the metric depends on several parameters [15]. We can build then all the null curves by randomly selecting one eigenvector  $\tilde{v} \in \{v_l\}$  and then reconstruct the curve along the direction  $\tilde{v}$  from the starting point  $x$ . From a practical point of view, one is led to solve the Cauchy problem with first order differential equation  $\dot{\gamma} = \tilde{v}$  and initial condition  $\gamma(0) = x$ .
+
+# Algorithm 1 The Singular Metric Equivalence Class (SiMEC) algorithm.
+
+1: Set the network  $\mathcal{N}$ ; choose the number of iterations  $K$ . Choose the input  $x^{(0)}$ .  
+2: for  $k = 0,1,\ldots ,K - 1$  do  
+3: Compute  $g_{\mathcal{N}(x^{(k)}}^n$  
+4: Compute the pullback metric  $g_{x^{(k)}}^0$  
+5: Diagonalize  $g_{x^{(k)}}^0$  and find the eigenvectors  $\{v_l\}_{l\in L_0}$  associated to the zero eigenvalue  $\lambda_0$  
+6: Randomly select  $\tilde{v} \in \{v_{l}\}_{l \in L_{0}}$  
+7:  $\delta^{(k)} = \eta \sqrt{\min_{j:\lambda_j\neq 0}|\lambda_j| / \max_j|\lambda_j|}$  
+8:  $x^{(k + 1)}\gets x^{(k)} + \delta^{(k)}\tilde{v}$  
+9: end for  
+10: Project  $x^{(k + 1)}$  to the feasible region  $\mathcal{X}$
+
+# Algorithm 2 The Singular Metric Exploration (SiM-Exp) algorithm.
+
+1: Set the network  $\mathcal{N}$ ; choose the number of iterations  $K$ . Choose the input  $x^{(0)}$ .  
+2: for  $k = 0,1,\dots ,K - 1$  do  
+3: Compute  $g_{\mathcal{N}}^n$  
+4: Compute the pullback metric  $g_{x^{(k)}}^0$  
+5: Diagonalize  $g_{x^{(k)}}^0$  and find the eigenvectors  $\{w_l\}_{l\in L_+}$  associated to eigenvalues  $\lambda_l\neq 0$  
+6: Randomly select  $\tilde{w} \in \{w_l\}_{l \in L_+}$  
+7:  $\delta^{(k)} = \eta \sqrt{\min_{j:\lambda_j\neq 0}|\lambda_j| / \max_j|\lambda_j|}$  
+8:  $x^{(k + 1)}\gets x^{(k)} + \delta^{(k)}\tilde{w}$  
+9: end for  
+10: Project  $x^{(k + 1)}$  to the feasible region  $\mathcal{X}$
+
+# 3.1 Input Space Exploration
+
+This entire procedure is coded in the Singular Metric Equivalence Class (SiMEC) and Singular Metric Exploration (SiMExp) algorithms, whose general schemes are depicted in Algorithms 1 and 2. SiMEC reconstructs the class of equivalence of the input via exploration of the input space by randomly selecting one of the eigenvectors related to the zero eigenvalue. On the opposite, in SiMExp, in order to move from a class of equivalence to another we consider the eigenvectors relative to the nonzero eigenvalues. This requires the slight difference in lines 5-6 between Algorithm 1 and Algorithm 2.
+
+There are some remarks to point out. From a numerical point of view, the diagonalization of the pullback may lead to have even negative eigenvalues: hence one may use the notion of energy of a curve, related to the pseudolenth. If the values  $\delta^{(k)}$  are too small more iterations are needed to move away from the starting point sensibly. Therefore there is a trade-off between the reliability of the solution and the exploration pace. Relying on the theory of dynamical systems, we can in practice estimate  $\delta^{(k)}$  at each iteration with the inverse of the square root of the condition number  $\Gamma = \max_{j}|\lambda_{j}| / \min_{j:\lambda_{j}\neq 0}|\lambda_{j}|$  of the pullback metric  $g_{x^{(k)}}^0$ , as in a locally-linearized dynamical system. This is our default choice for both algorithms. We multiply the default value  $\delta$  by a multiplier  $\eta$  in order to explore the sensitivity of Algorithm 1 and Algorithm 2 to variations of step length, expecting Algorithm 1 to be more sensitive compared to Algorithm 2. Indeed, to build points in the same equivalence class Algorithm 1 needs to follow a null curve closely with as little approximation as possible. In contrast Algorithm 2, whose goal is to change the equivalence class, does not have the same problem and larger  $\delta$  are allowed.
+
+In the final step of each iteration of both algorithms, the embeddings  $x^{(0)} \ldots x^{(K)}$  need to be constrained to a feasible region  $\mathcal{X}$ . This region is defined by the distribution of embeddings derived from the embedding layer, which is bounded by definition. Specifically its upper bound has components  $UB_{i} = \sum_{j:E_{ij}>=0} E_{ij} \max_{l}(x_{l}) + \sum_{j:E_{ij}<0} E_{ij} \min_{l}(x_{l}) + \max_{s}(q(s)_{i})$ , where  $E$  is the embedding layer weight matrix,  $(x_{l})$  represent input features, bounded 0 and 1 in both the visual and textual case, and  $q(s)$  is the positional encoding vector at position  $s$  in the sequence of patches/tokens. The lower bound  $LB$  is obtained by switching max with min and vice versa. We acknowledge that these bounds present a non-null margin from the actual embedding distribution domain, however we find them to be a suitable estimate for the practical purpose of interpretation of input space exploration results, presented in next section. Notwithstanding numerical approximation errors, the outputs of SiMEC algorithm at each iteration  $k$  are predicted by  $\mathcal{N}$  to yield the same probability distribution  $|p(\cdot|x^{(0)}) - p(\cdot|x^{(k)})| < \varepsilon, 0 < \varepsilon \ll 1$ , i.e. the original input probability, by construction. On the contrary, SiMExp algorithm induces a non-null probability change in the output space, which might possibly lead to a class ranking change, i.e. the situation where class A is given higher probability than class B at iteration  $k-1$  but it is surpassed by class B at iteration  $k$ , and eventually a prediction change, i.e. the situation in which  $\text{argmax}(\mathcal{N}(x^{(k)})) = y' \neq \text{argmax}(\mathcal{N}(x^{(0)}))$  for  $k$  in a non-degenerate neighborhood of a change-point iteration  $\overline{k}$ .
+
+As for the computational complexity of the two algorithms, the most demanding step is the computation of the eigenvalues and eigenvectors, which is  $O(n^3)$ , with  $n$  the dimension of the square matrix  $g_{x^{(k)}}^0$  [20]. Since all the other operations are either  $O(n)$  or  $O(n^2)$ , we conclude that the complexity of both Algorithms 1 and 2 is  $O(n^3)$ .
+
+# 3.2 Interpretation
+
+Algorithms 1 and 2 allow for the exploration of the equivalence classes in the input space of a Transformer model. However, the points explored by these algorithms may not be directly interpretable by a human perspective. For instance, an image or a piece of text may need to be decoded to be "readable" by a human observer. Here we present an interpretation method for Transformers based on input space exploration, which is then demonstrated on two Vision Transformer (ViT) models trained for image classification [8], and two BERT models, one trained for masked language modeling (MLM) [7] and the other fine-tuned for text classification [18].
+
+Using SiMEC and SiMExp to explore the embedding space reveals how Transformer models perceive equivalence among different data points. Specifically, these methodologies facilitate the sequential acquisition of embedding matrices  $x^{(0)} \ldots x^{(K)}$ , as detailed in Algorithms 1 and 2. A key feature
+
+of the SiMEC/SiMExp approach is its ability to selectively update specific tokens (for text inputs) or patches (for image inputs) during each iteration. This selective update allows to explore targeted modifications that prompt the model to either categorize different inputs as the same class or recognize them as distinct. Unlike other approaches [24, 12] where perturbations are predetermined, this method lets the model itself guide us to understand which data points belong to specific equivalence classes.
+
+To interpret embeddings produced by the exploration process, they must be mapped back into a human-understandable form, such as text or images. The interpretation of an embedding vector depends on the operations performed by the Transformer's embedding module  $\mathcal{E}_T$ . If  $\mathcal{E}_T$  consists only of invertible operations, it is feasible to construct a layer that performs the inverse operation relative to  $\mathcal{E}_T$ . The output can then be visualized and directly interpreted by humans, allowing for a comparison with the original input to discern how differences in embeddings reflect differences in their representations (e.g., text, images). If the operations in  $\mathcal{E}_T$  are non-invertible, a trained decoder is required to reconstruct an interpretable output from each embedding matrix  $x^{(1)},\dots ,x^{(K)}$ . Such operation injects some unforeseen noise into the interpretation results, which is investigated in the experimental setting. In practice, we construct the ViT models such that the embedding layer is invertible, whereas for BERT models it is feasible to exploit layers that are specialized for the MLM task to map input embeddings back to tokens. This approach is effective whether the BERT model in question is specifically designed for MLM or for sentence classification. In the case of sentence classification models, it is necessary to select a corresponding MLM BERT model that shares the same internal architecture, including the number of layers and embedding size.
+
+Algorithm 3 depicts the process of interpreting SiMEC/SiMExp outputs for both ViT and BERT experiments. After initializing the decoder according to the model type, the embeddings  $x^{(1)}, \dots, x^{(K)}$  are decoded and the selected segments for exploration are extracted. These segments are then used to replace the corresponding parts of the original input instance.
+
+Algorithm 3 Interpretation for Exploration results for ViT and BERT models.  
+1: Inputs:  
+2: Transformer model  $T$  with: Patcher/Tokenizer  $\mathcal{T}_T$ , Embedding layer  $\mathcal{E}_T$ . Input image/text  $z$   
+3: Modified embeddings  $x^{(1)}\ldots x^{(K)}$  resulted from Algorithm 1 or 2 applied on  $x^{(0)} = \mathcal{E}_T(\mathcal{T}_T(z))$   
+4:  $P \subseteq \{1,\dots ,dim(z)\}$  indices of patches/tokens to update  
+5: If  $T$  is ViT:  
+6: Initialize decoder  $d$  with weights from  $\mathcal{E}_T$ .  
+7: If  $T$  is BERT:  
+8: Initialize decoder with intermediate and final layers of a BERT for MLM task.  
+9: Decode modified embeddings  $x^{(0)}\ldots x^{(K)}$  using  $d$  to generate the images/sentences  $Z^{\prime} = Z_{0}^{\prime}\ldots Z_{K}^{\prime}$ .  
+10: For each  $z^{\prime}\in Z^{\prime}$ : replace segments relative to indices  $P$  in  $z$  with those in  $z^{\prime}$ .  
+11: Outputs:  
+12: Modified input images/sentences, one for each SiMEC/SiMExp iteration.
+
+Figure 1 (top left) presents the outcome of applying Algorithm 3 to a ViT exploration experiment on a CIFAR10 image. Both SiMEC and SiMExp produce visually similar outputs—each still resembling a "cat" to a human observer—yet the SiMExp interpretation is classified as "dog" at iteration 750. This demonstrates how subtle modifications, such as changes in background pixels, can significantly influence model predictions, even when such changes are perceived as irrelevant by humans, as also noted in [24]. A clear difference emerges in the exploration dynamics of the two algorithms (Figure 1, top right): SiMExp progresses in a more straight and directed manner, reflecting its goal of escaping the initial equivalence class. This divergence is further illustrated in the bottom right subplots, where the class probability distributions remain stable during SiMEC exploration but show notable fluctuations under SiMExp exploration. The lower part of Figure 1 shows a similar example for textual data from the Measuring-Hate-Speech (MHS) dataset. In this case, SiMExp identifies an alternative sentence that is classified as "Hatespeech", contrasting with the original input, which had been classified as "Offensive".
+
+# 4 Experiments
+
+Experiments are conducted on textual and visual data and are aimed at two objectives: (i) obtaining an empirical verification of the behavior of SiMEC and SiMExp under diverse settings, and (ii) verifying
+
+![](images/c806e2b0feb73b62ff87c73b81f7cdd14a8d6fe65d96b557902a0d59b8885180.jpg)
+
+![](images/0b4a13ddc971efc4d03cc1cfc64bcceca9c35f49ada214d610598f316bfd6428.jpg)  
+SIMEC-750
+
+![](images/2e42c8852458979617356096e2d6c98660ad70b97b69831a438374934149edf7.jpg)
+
+![](images/245af8ad9bd51d061688d053a35780940be115b05e7f327b87853243ea441324.jpg)  
+2D SVD Visualization of SiMEC and SiEXP Embeddings
+
+![](images/ac1fdb2beb1ff74d43d8ca59b87340130ee428f535fa9063a1c672c74506b3c7.jpg)  
+Class Probabilities over Iterations  
+plane -car -bird -cat -deer -dog -frog -horse -ship -truck
+
+![](images/8929051cb62e3eda766ffafdb8cd980de2d310f5f76d23ed02ea89a590f75528.jpg)
+
+![](images/bd3470a020b54e174c915ff794087b287c1b381b9f565a208bb7d493b64b94b6.jpg)
+
+![](images/979b8f8d3a6741e2d7d7cd6dc7b6d151bc40d21beb0e0f0cb9ca7e713cf31769.jpg)
+
+![](images/d0136ecc5298c02fca3ffc1fb88412b692a0e6f2b5f50d5a39dfc37af2bf074d.jpg)
+
+![](images/ba0c90d2d9f193ea9a24d83b35a69f983ffe1335e235aa289bc749854873bca4.jpg)  
+2D SVD Visualization of SiMEc and SiExp Embeddings Repetition 1 of input sentence_8165 (MHS), explored with
+
+![](images/cef6cc867c9aa21c3f8df405df8b9134a32378d72971796b33b331fac10d46f1.jpg)
+
+![](images/4566dae959fba3a3551af3f421d5167b43b29eb034d836c63402322d5a359198.jpg)  
+Class Probabilities over Iterations  
+Figure 1: (Top figure) Example of exploration on a CIFAR10 image using SiMEC and SiMExp. Left: Original image, followed by interpretation outputs of  $x_{750}$  from SiMEC (middle) and SiMExp (bottom). Right top: SVD projection of the explored points  $x^{(1)},\dots ,x^{(K)}$  for SiMEC (red) and SiMExp (blue), where color intensity encodes iteration progress (darker colors correspond to later iterations), and point shapes indicate predicted class labels. Right bottom: Evolution of class probabilities over iterations, for SiMEC (left) and SiMExp (right). (Bottom figure) Example of exploration on an MHS sentence using SiMEC and SiMExp. Visualization layout and interpretation are analogous to the top figure.
+
+![](images/f3a2bee62395892f92b980683918a866c5030e3875a701b53f3bc0074e77d293.jpg)
+
+the consistency of interpretation outputs with the ones from exploration only, in order to test their usability as alternative input data.
+
+In each data modality, we experiment with two datasets presenting different features: (i) MNIST [13], a grayscale digit image dataset; (ii) CIFAR10 [11], a RGB object image dataset; (iii) WinoBias [25], a textual dataset for MLM, especially focused on gender bias; (iv) Measuring-Hate-Speech (MHS) [16], a textual dataset for Text Classification, especially focused on hate speech detection. We trained one ViT model for each image dataset, and we used pretrained BERT models for MHS and Wino-Bias. More details about adopted models, experimental results in further configurations, and full experimental details are provided in the Supplementary Materials.
+
+Input space exploration For objective  $(i)$  we consider the following metrics: effectiveness of the exploration can be measured by changes in prediction probabilities as well as estimation of the hyper-volume explored, while speed is assessed in terms of total time (in seconds) $^2$ . Algorithms are run for  $K = 1000$  iterations, which we prove sufficient to capture their behavior, with delta multiplier  $\eta \in \{1;10\}$ , the latter used with the aim to verify whether it is possible to speed up the process pace without compromising its stability. Finally, the experiments reported here all refer to the configuration in which all patches of an image are modified, while for textual inputs only the token with the highest attribution value is subject to exploration.
+
+Given the predictions obtained by re-applying the models' encoder and classifier layers to the modified embeddings  $x^{k}$  at each iteration  $k$ , we observe the changes in class probabilities. The theoretical
+
+results suggest that SiMEC should induce minimal fluctuations in them, while SiMExp should yield rapidly changing probabilities, up to prediction changes.
+
+![](images/2a0eb5aca1caa37041bfbb77b73a011a62c7cf70b97c71080abe24a896b6d323.jpg)  
+(a) SiMEC and SiMExp
+
+![](images/0c67d9f8af9aab1c21fc92517333cf4859bca5c4d53c8dc8ef99080832c36bd8.jpg)  
+(b) Baseline gradient-based Gaussian noise approaches  
+Figure 2: Mean and standard deviation (where applicable) of probability values for the original class (solid line) and the top predicted class (dashed line) based on embeddings obtained during exploration, across iterations and datasets. Subfigure (a) depicts the behavior of SiMEC (orange) and SiMExp (blue), while subfigure (b) reports the behavior of corresponding baseline algorithms. SiMExp results in a notable decrease in the probability of the original class, while the probability of the highest-scoring class decreases to a lesser extent, indicating a shift in the most probable class.
+
+Figure 2a depicts the empirical reflection of the theoretical results. Focusing on the probability of the original input class (i.e. class predicted at  $k = 0$ ), we see that SiMEC manages to keep it constant while SiMExp makes it drop significantly within the first 1000 iterations. As a baseline, we compare our results with a gradient-based Gaussian-noise approach which updates the input embedding  $x^{(k)}$  at each iteration by  $\pm \delta \nabla \mathcal{N}(x^{(k)}) + \epsilon$ , where the sign is determined by what exploration we are performing (same class vs other class) and  $\epsilon$  is a small Gaussian noise vector orthogonal to the gradient so to guarantee exploration. Applying the same number of iterations and the same average step size  $\overline{\delta}$  in each experiment allows us to conclude that SiMEC and SiMExp are significantly more effective than the baselines for staying in the original equivalence class and moving to another class respectively. Indeed, in SiMEC original class probability always follows strictly the top class probability, which in the baseline is rarely the case; in SiMExp the two probabilities tend to diverge as predicted by the theoretical analysis, while in the baseline they remain close to one another.
+
+In order to verify the actual shift in class probabilities, we report in Table 1 the statistics about average class ranking changes, which indicate a clear tendency of SiMExp changing equivalence classes compared to SiMEC.
+
+Furthermore, we estimate the per-patch explored hypervolume by reducing embedding vectors to the first  $n$  principal components, retaining  $90\%$  of the total variance, and computing at each iteration  $k$  the element-wise difference  $\Delta_i^{(k)} = (\max_{t=0,\dots,k} x_i^{(t)} - \min_{t=0,\dots,k} x_i^{(t)})^{1/n}$  (the power  $1/n$  allows for more stable computation). The product of the components of  $\Delta^{(k)}$  gives the volume of an hyper-dimensional cuboid which contains the explored region and is thus an over-estimate of the scope achieved by the exploration. By computing the average volume ratio  $\rho_V = (\Pi \Delta_{SiMExp}^{(K)} / \Pi \Delta_{SiMEC}^{(K)})^n$ , we empirically verified that SiMExp explores a portion of space that is bigger than the one explored by SiMEC by an order of  $10^1$ . We validated these results by performing Welch t-tests on  $\rho_V$ : all p-values resulted lower than  $10^{-3}$ . Thus we conjecture that the exploration took a privileged direction on SiMExp experiments, thus making the volumes increasing faster than in SiMEC experiments.
+
+<table><tr><td></td><td></td><td>MNIST</td><td>CIFAR10</td><td>WinoBias</td><td>MHS</td></tr><tr><td rowspan="2">SiMEC</td><td>η = 1</td><td>0.03 (0.18)</td><td>0.0</td><td>0.0</td><td>0.0</td></tr><tr><td>η = 10</td><td>0.07 (0.25)</td><td>0.12 (0.33)</td><td>0.0</td><td>0.0</td></tr><tr><td rowspan="2">SiEXP</td><td>η = 1</td><td>1.31 (1.64)</td><td>2.60 (3.35)**</td><td>0.13 (0.33)</td><td>0.75 (0.89)**</td></tr><tr><td>η = 10</td><td>11.91 (11.87)**</td><td>18.70 (15.90)**</td><td>3.25 (2.41)**</td><td>3.64 (3.43)**</td></tr></table>
+
+Table 1: Average class ranking changes per 1000 iterations across datasets and  $\eta$  values. The class ranking change is computed by counting pairwise inversions in class rankings before and after each exploration update. Results are reported as mean (standard deviation) over multiple runs and inputs. The symbol \*\* indicates that these ranking changes involve at least once a prediction change, i.e., a change in the top predicted class. SiMExp induces substantially more ranking and prediction changes than SiMEC, especially for larger  $\eta$ .
+
+Finally, we measure the time required to explore the input space of a model with the SiMEC and SiMExp algorithms. Means (and standard deviations) of required times are computed per patch/token and per iteration: 0.126 s (0.008) for CIFAR10, 0.050 s (0.004) for MNIST, 0.300 s (0.020) for Winobias, and 0.310 s (0.074) for MHS.
+
+Using interpretation outputs as alternative input data Objective  $(ii)$  is to assess whether our interpretation algorithm (Algorithm 3) can generate alternative input data that either preserve the original input's equivalence class or shift to a different one, depending on the exploration dynamics of SiMEC and SiMExp. The mean difference of pixel/tokens generated from iteration to iteration amounts at  $2.219\cdot 10^{-3}$  for SiMEC experiments, and at  $83.028\cdot 10^{-3}$  for SiMExp experiments; these values increase as the number of explored patches. These results show that our algorithms generate diverse outputs across iterations, especially when the exploration is performed following the SiMExp algorithm.
+
+Beyond output diversity, we evaluate whether and how the model's prediction for the original equivalence class evolves as SiMEC and SiMExp explore the embedding space. Specifically, we track how the class probabilities from the modified embeddings change when decoded back into the data domain every  $\hat{k}$  iterations. This evaluation differs from simply observing changes in embedding predictions. The projection step (Algorithms 1 and 2, step 11) constrains modifications to an  $L^{\infty}$  sphere containing the data domain, not the exact input space. Additionally, the decoder itself introduces approximations, either due to model limitations or numerical errors. These factors can cause discrepancies between predictions from embeddings and from decoded interpretations. To quantify this effect, we compute the average Wasserstein distance ( $p = 1$ ) between probability distributions predicted from embeddings and their corresponding interpretation outputs. Wasserstein distance is preferred over KL divergence for its interpretability in this context. Results show that, with a median Wasserstein distance of 0.0, SiMEC produces interpretation outputs whose predicted probabilities remain very close to those of the embeddings, indicating consistent generation of reliable alternative input data. In contrast, SiMExp's outputs exhibit larger and more variable Wasserstein distances, whose distribution has a median of 0.049, highlighting inconsistencies between embeddings and their interpretations.
+
+In cases where predictions on SiMExp's interpretations initially differ from prediction on SiMExp's embeddings, an average of  $10.9\%$  of these misaligned predictions eventually realign as exploration progresses. On average, this "catch-up" effect occurs after an average of 290.65 iterations, suggesting that longer exploration trajectories (beyond 1000 iterations) could further improve alignment. Supporting this observation, we find an average positive Pearson correlation of 0.32 (average p-value 0.08) between top class' probability predicted on SiMExp's embeddings and its corresponding probability prediction on SiMExp's interpretations, indicating a trend towards convergence.
+
+# 5 Related work
+
+Works dealing with embedding space exploration mostly focus on the study of specific properties of the embedding space of Transformers, especially in NLP. For instance, Cai et al. [6] challenge the idea that the embedding space is inherently anisotropic [9] discovering local isotropy, and find low-dimensional manifold structures in the embedding space of GPT and BERT. Bis et al. [4] argue that the anisotropy of the embedding space derives from embeddings shifting in common directions
+
+during training. In the field of Computer Vision, Vilas et al. [23] map internal representations of a ViT onto the output class manifold, enabling the early identification of class-related patches and the computation of saliency maps on the input image for each layer and head. Applying Singular Value Decomposition to the Jacobian matrix of a ViT, Salman et al. [17] treat the input space as the union of two subspaces: one in which image embedding doesn't change, and another one for which it changes. Except for the last one, all the aforementioned approaches rely on data samples. By studying the inverse image of the model, instead, we can do away with data samples. The idea of applying Riemannian geometry to capture geometric information about the input manifold of a neural network building a foliation in equivalence classes has also been explored in [10, 21, 22] in the case of simple architectures. In these works a foliation of the data domain is obtained by means of the pullback of a variation of the Fisher information matrix for classifier networks with ReLU and softmax activation functions, with applications to knowledge transfer and the study of adversarial attacks.
+
+In contrast to these works, we apply Riemannian geometry techniques to study the embedding space of transformers, computing the pullback of the metric of the output space, and we address the further problem of interpreting the output of the exploration process. Furthermore, our algorithms explore the embedding space dynamically, with a non-fixed choice of the integration step  $\delta$ .
+
+# 6 Conclusions
+
+Our exploration of the Transformer architecture through a theoretical framework grounded in Riemannian geometry led to the application of our two algorithms, SiMEC and SiMExp, for examining equivalence classes in the Transformers' input space. In particular, our method enables two complementary exploration strategies, one for retrieving input instances that produce the same class probability distribution as the original instance, the other for discovering instances that yield a different class probability distribution. We demonstrated how the results of these exploration methods can be interpreted in a human-readable form and how the exploration outputs can be used to generate alternative input data.
+
+Future research directions include delving deeper into the potential of our framework for controlled input generation within an equivalence class. Our goal is to investigate how, in the XAI scenario, our framework can facilitate local and task-agnostic explainability methods applicable to Computer Vision (CV) and Natural Language Processing (NLP) tasks, among others. In particular, we see our methods as potential approaches to investigate Transformers' sensitivity and explainability with respect to input data features. In future applications to large-scale architectures where the dimension of the embedding space is of order  $10^{3} - 10^{4}$  [5, 19], we also plan to improve the scalability of the SiMEC and SiMExp algorithms, e.g. making use of partial decompositions.
+
+# Acknowledgments and Disclosure of Funding
+
+This work is partially supported by PNRR-NGEU program under MUR 118/2023.
+
+# References
+
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+
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+# 6. Experimental Setting/Details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
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+Answer: [Yes]
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+
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+# Unveiling m-Sharpness Through the Structure of Stochastic Gradient Noise
+
+Haocheng Luo $^{1}$  Mehrtash Harandi $^{1}$  Dinh Phung $^{1}$  Trung Le $^{1}$
+
+<sup>1</sup>Monash University, Australia
+
+{haocheng.luo, mehrtash.harandi, dinh.phung, trunglm}@monash.edu
+
+# Abstract
+
+Sharpness-aware minimization (SAM) has emerged as a highly effective technique for improving model generalization, but its underlying principles are not fully understood. We investigated the phenomenon known as m-sharpness, where the performance of SAM improves monotonically as the micro-batch size for computing perturbations decreases. In practice, the empirical m-sharpness effect underpins the deployment of SAM in distributed training, yet a rigorous theoretical account has remained lacking. To provide a theoretical explanation for m-sharpness, we leverage an extended Stochastic Differential Equation (SDE) framework and analyze the structure of stochastic gradient noise (SGN) to characterize the dynamics of various SAM variants, including n-SAM and m-SAM. Our findings reveal that the stochastic noise introduced during SAM perturbations inherently induces a variance-based sharpness regularization effect. Motivated by our theoretical insights, we introduce Reweighted SAM (RW-SAM), which employs sharpness-weighted sampling to mimic the generalization benefits of m-SAM while remaining parallelizable. Comprehensive experiments validate the effectiveness of our theoretical analysis and proposed method.
+
+# 1 Introduction
+
+In machine learning, gradient-based optimization algorithms aim to minimize the following loss function:
+
+$$
+\min  _ {x \in \mathbb {R} ^ {d}} f (x) := \frac {1}{n} \sum_ {i = 1} ^ {n} f _ {i} (x) \tag {1}
+$$
+
+where  $x \in \mathbb{R}^d$  denotes the parameter,  $f_{i}(x)$  represents the loss on the  $i$ -th sample,  $i$  ranges from 1 to  $n$ , and  $n$  is the size of the training set. We primarily focus on stochastic algorithms, where an index set  $\gamma_k$  is uniformly sampled at each step  $k$ . We denote the mini-batch loss by  $f_{\gamma_k}(x) = \frac{1}{|\gamma|} \sum_{i \in \gamma_k} f_i(x)$ .
+
+We investigate the recently proposed Sharpness-Aware Minimization (SAM) (Foret et al., 2021), which has achieved remarkable success in various application domains (Foret et al., 2021; Kwon et al., 2021; Kaddour et al., 2022; Liu et al., 2022a). It seeks flat minima by minimizing the perturbed loss  $\min_{x\in \mathbb{R}^d}f(x + \rho \epsilon)$ , where the perturbation  $\epsilon$  is defined via
+
+$$
+\epsilon \in \underset {\| \epsilon \| \leq 1} {\arg \max } \langle \nabla f (x), \epsilon \rangle , \tag {2}
+$$
+
+and  $\rho > 0$  is a hyperparameter controlling its radius. Solving the inner maximization yields  $\epsilon^{*}(x) = \frac{\nabla f(x)}{\|\nabla f(x)\|}$ . The algorithm, referred to as  $n$ -SAM, computes its perturbation using the full-batch gradient. Its update at iteration  $k$  is
+
+$$
+x _ {k + 1} = x _ {k} - \eta \nabla f _ {\gamma_ {k}} \left(x _ {k} + \rho \frac {\nabla f \left(x _ {k}\right)}{\| \nabla f \left(x _ {k}\right) \|}\right), \tag {3}
+$$
+
+where  $\eta > 0$  is the learning rate,  $\rho > 0$  the perturbation radius.
+
+Calculating the perturbation on the entire training dataset at each step is prohibitively expensive. Therefore, Foret et al. (2021) suggest estimating the perturbation using a mini-batch, resulting in SAM commonly used in practice. We refer to the practical SAM algorithm as mini-batch SAM to distinguish it from other variants. The update rule can be summarized:
+
+$$
+x _ {k + 1} = x _ {k} - \eta \nabla f _ {\gamma_ {k}} \left(x _ {k} + \rho \frac {\nabla f _ {\gamma_ {k}} \left(x _ {k}\right)}{\| \nabla f _ {\gamma_ {k}} \left(x _ {k}\right) \|}\right). \tag {4}
+$$
+
+Interestingly, it has been observed that although mini-batch SAM was proposed as a computationally efficient variant, it exhibits remarkable generalization ability. In contrast, the original n-SAM (3) offers little to no improvement in generalization (Foret et al., 2021; Andriushchenko and Flamarion, 2022).
+
+m-SAM and m-sharpness.  $m$ -SAM refers to dividing a mini-batch of data into disjoint micro-batches of size  $m$ , and independently computing perturbations and gradients for each micro-batch, which are then combined to update the parameters. It has been widely observed that the practical performance of m-SAM improves monotonically as  $m$  decreases, a phenomenon known as  $m$ -sharpness (Foret et al., 2021; Behdin et al., 2022; Andriushchenko and Flamarion, 2022). It is worth noting that when  $m$  is smaller than the batch size, the perturbation must be computed sequentially across micro-batches, which cannot be parallelized and therefore introduces substantial additional computational overhead, although smaller  $m$  typically leads to improved generalization performance.
+
+The update rule for m-SAM can be written as:
+
+$$
+x _ {k + 1} = x _ {k} - \frac {\eta m}{| \gamma |} \sum_ {\mathcal {I} _ {j} \subset \gamma_ {k}, | \mathcal {I} _ {j} | = m} \nabla f _ {\mathcal {I} _ {j}} \left(x _ {k} + \rho \frac {\nabla f _ {\mathcal {I} _ {j}} \left(x _ {k}\right)}{\| \nabla f _ {\mathcal {I} _ {j}} \left(x _ {k}\right) \|}\right), \tag {5}
+$$
+
+where  $\mathcal{I}_j$  are disjoint subsets of  $\gamma_{k}$ , each with size  $m$ .
+
+In synchronous data-parallel (multi-GPU) training with  $D$  devices and per-device batch size  $b$ , SAM is typically implemented by computing the perturbation locally on each device using its own data and then aggregating the perturbed gradients across devices; this implementation is exactly an instance of m-SAM with  $m = b$ . The local-perturbation design avoids an additional cross-device synchronization in the inner perturbation step (only the outer gradient aggregation requires all-reduce), thereby reducing per-step communication overhead. The empirical observation of m-sharpness has thus become a practical cornerstone for deploying SAM at scale.
+
+To provide a theoretical explanation for m-sharpness, we extend the recent Stochastic Differential Equation (SDE) framework of Li et al. (2019); Compagnoni et al. (2023); Luo et al. (2025) by jointly tracking both  $\eta$  and  $\rho$  to arbitrary expansion orders, providing a unified basis for analyzing SAM and its variants. Under this framework, we derive closed-form drift terms for three unnormalized SAM (USAM) variants—n-USAM, mini-batch USAM, and m-USAM—revealing how stochastic gradient noise (SGN) drives implicit sharpness regularization and closely correlates with generalization performance. We further extend our analysis to the three normalized (vanilla) SAM variants and observe a similar noise-induced pattern, albeit without closed-form solutions. Motivated by our theory, we propose a sample-reweighting method that uses the magnitude of the SGN as an importance measure.
+
+Our contributions are threefold:
+
+- Closed-form SDE for USAM variants: we derive explicit SDE drift expressions for n-USAM, mini-batch USAM, and m-USAM, quantifying the role of SGN in sharpness regularization and its impact on generalization.  
+- Extension to normalized SAM: we apply the same analytical framework to the three normalized SAM variants, uncovering analogous noise-driven regularization effects.  
+- Reweighted SAM algorithm: we introduce an adaptive reweighting mechanism that assigns larger weights to samples with higher SGN magnitudes, thereby strengthening implicit sharpness regularization; its superior generalization is confirmed through comprehensive experiments.
+
+# 2 Related Works
+
+Sharpness-Aware Minimization. Sharpness-Aware Minimization (Foret et al., 2021) has attracted increasing attention due to its consistent improvements in generalization across a wide range of tasks. A growing body of work has been devoted to analyzing and enhancing SAM, including investigations into its generalization principles (Andriushchenko and Flammarion, 2022; Möllenhoff and Khan, 2022; Wen et al., 2022, 2023; Agarwala and Dauphin, 2023; Springer et al., 2024; Luo et al., 2025) and convergence properties (Khanh et al., 2024; Oikonomou and Loizou, 2025), exploring its applications in various domains, and developing algorithmic variants to further improve both generalization (Kwon et al., 2021; Zhuang et al., 2022; Liu et al., 2022b; Kim et al., 2022; Li et al., 2024c; Wu et al., 2024; Tahmasebi et al., 2024; Li et al., 2024b, 2025) and computational efficiency (Du et al., 2021; Liu et al., 2022a; Du et al., 2022; Mordido et al., 2023; Tan et al., 2024; Xie et al., 2024).
+
+m-sharpeness. m-sharpeness has long been a mysterious phenomenon in the field of SAM-related research and was first introduced in the original work of Foret et al. (2021). They observed that although SAM theoretically aims to minimize the perturbed loss over the entire training set, its computationally efficient variant—mini-batch SAM, which computes perturbations only at the minibatch level—outperforms n-SAM, which applies perturbations at the full-batch level. More generally, the generalization performance of m-SAM improves monotonically as  $m$  decreases. This phenomenon was further confirmed through extensive experiments in a single-GPU setting by Andriushchenko and Flammarion (2022), and in a multi-GPU setting by Behdin et al. (2022). Although Andriushchenko and Flammarion (2022) proposed several hypotheses to explain it, they were later invalidated by their own experiments, and the underlying cause of this phenomenon remains an open question. It is important to note that our definition of m-sharpeness follows the original work of Foret et al. (2021) and the pioneering contributions of Andriushchenko and Flammarion (2022). Some studies have adopted different definitions. For example, Wen et al. (2022) refer to the deterministic algorithm as n-SAM and SAM with a batch size of 1 as 1-SAM, whereas Behdin et al. (2022) define m as the number of divided micro-batches.
+
+Structure of stochastic gradient noise. In expectation, a widely accepted assumption is that the stochastic gradient serves as an unbiased estimator of the full-batch gradient (Jastrzebski et al., 2017; Zhu et al., 2018; HaoChen et al., 2021; Ziyin et al., 2021). Regarding the covariance of SGN, Simsekli et al. (2019) assumed it to be isotropic. However, this view was later challenged by Xie et al. (2020) and Li et al. (2021), who argued that Simsekli et al. (2019) were actually analyzing gradient noise across different iterations rather than noise arising from mini-batch sampling. They provided extensive evidence supporting the idea that the latter can be well modeled as a multivariate Gaussian variable with an anisotropic/parameter-dependent covariance structure. Furthermore, Xie et al. (2023) conducted statistical tests on the Gaussianity to support this perspective.
+
+# 3 Theory
+
+# 3.1 Notation and assumption
+
+In this paper, we denote by  $\|\cdot\|$  the Euclidean norm, and the expectation operator  $\mathbb{E}$  is taken with respect to the random index set unless otherwise stated. We assume that the stochastic gradient is an unbiased estimator of the full gradient and possesses a finite second moment.
+
+Assumption 3.1. We assume that sampling an index  $i$  uniformly at random yields i.i.d. stochastic gradients
+
+$$
+\nabla f _ {i} (x) = \nabla f (x) + \xi_ {i} (x),
+$$
+
+where  $\nabla f(x) \coloneqq \frac{1}{n} \sum_{i=1}^{n} \nabla f_i(x)$  is the full gradient and  $\xi_i(x)$  denotes the SGN. We further assume
+
+$$
+\mathbb {E} \left[ \xi_ {i} (x) \right] = 0, \quad \operatorname {C o v} \left(\xi_ {i} (x)\right) = V (x),
+$$
+
+where
+
+$$
+V (x) := \frac {1}{n} \sum_ {i = 1} ^ {n} \nabla f _ {i} (x) \nabla f _ {i} (x) ^ {\top} - \nabla f (x) \nabla f (x) ^ {\top}.
+$$
+
+An important consequence of this assumption is  $\mathbb{E}\big\| \nabla f_i(x)\big\|^2 = \big\| \nabla f(x)\big\|^2 +\mathrm{tr}\big(V(x)\big)$ , which we will use repeatedly.
+
+# 3.2 Overview of two-parameter approximation
+
+We extend the existing SDE framework for SAM (Compagnoni et al., 2023; Luo et al., 2025), which can only track  $\eta$  to order 1, while ours can jointly track two parameters  $\eta$  and  $\rho$  to arbitrary expansion orders, thereby decoupling their convergence rates and enabling precise control of the overall approximation error. Specifically,  $\eta$  governs the higher-order terms in Dynkin's formula, while  $\rho$  captures the remainder term arising from the Taylor expansion (see the detailed formulations in Appendix A). By employing a Dynkin expansion instead of a full Ito-Taylor expansion, it avoids the proliferation of terms and provides a streamlined approach to controlling the remainder error in the two-parameter setting. Another major advantage of this approach is that it allows us to let  $\eta$  and  $\rho$  tend to zero at independent rates, rather than being constrained to a fixed ratio as in the work of Compagnoni et al. (2023) and Luo et al. (2025). The definition of a two-parameter weak approximation of order  $(\alpha, \beta)$  is as follows.
+
+Definition 3.2 (Two-parameter weak approximation). Let  $T > 0$ ,  $0 < \eta < 1$ ,  $0 < \rho < 1$  and set  $N = \lfloor T / \eta \rfloor$ . Let
+
+$$
+\left\{x _ {k} \right\} _ {k = 0} ^ {N} \quad \text {a n d} \quad \left\{X _ {t} \right\} _ {t \in [ 0, T ]}
+$$
+
+be a discrete-time and a continuous-time stochastic process, respectively. We say that  $X_{t}$  is an order-  $(\alpha ,\beta)$  weak approximation of  $x_{k}$  if, for every  $g\in G^{\alpha +1}$ , there exists a constant  $C > 0$ , independent of  $\eta ,\rho$ , such that
+
+$$
+\max  _ {0 \leq k \leq N} \left| \mathbb {E} \left[ g (X _ {k \eta}) \right] - \mathbb {E} \left[ g (x _ {k}) \right] \right| \leq C \big (\eta^ {\alpha} + \rho^ {\beta + 1} \big).
+$$
+
+# 3.3 SDE approximation for USAM variants
+
+We begin by considering USAM (Andriushchenko and Flamarion, 2022; Compagnoni et al., 2023; Dai et al., 2024; Zhou et al., 2024), which is widely studied as a theoretically friendly variant of SAM and can achieve comparable performance to SAM in practice. The update rules for the different variants of the USAM algorithm are shown below. We will see that for USAM, all expressions in the drift term are in closed form, providing an intuitive understanding. In Section 3.4, we will extend our conclusions to the standard SAM.
+
+$$
+\text {m i n i - b a t c h U S A M :} \quad x _ {k + 1} = x _ {k} - \eta \nabla f _ {\gamma_ {k}} \left(x _ {k} + \rho \nabla f _ {\gamma_ {k}} \left(x _ {k}\right)\right) \tag {6}
+$$
+
+$$
+\text {n - U S A M :} \quad x _ {k + 1} = x _ {k} - \eta \nabla f _ {\gamma_ {k}} \left(x _ {k} + \rho \nabla f \left(x _ {k}\right)\right) \tag {7}
+$$
+
+$$
+\text {m - U S A M :} x _ {k + 1} = x _ {k} - \frac {\eta m}{| \gamma |} \sum_ {\mathcal {I} _ {j} \subset \gamma_ {k}, \left| \mathcal {I} _ {j} \right| = m} \nabla f _ {\mathcal {I} _ {j}} \left(x _ {k} + \rho \nabla f _ {\mathcal {I} _ {j}} \left(x _ {k}\right)\right) \tag {8}
+$$
+
+Theorem 3.3 (Mini-batch USAM SDE - informal statement of Theorem B.2, adapted from Theorem 3.2 of Compagnoni et al. (2023)). Under Assumption 3.1 and mild regularity conditions, the solution of the following SDE (9) is an order-(1,1) weak approximation of the discrete update of mini-batch USAM (6) with batch size  $|\gamma|$ :
+
+$$
+d X _ {t} = - \nabla \left(f \left(X _ {t}\right) + \underbrace {\frac {\rho}{2} \left\| \nabla f \left(X _ {t}\right) \right\| ^ {2} + \frac {\rho}{2 | \gamma |} \operatorname {t r} \left(V \left(X _ {t}\right)\right)} _ {\text {i m p l i c i t r e g u l a r i z a t i o n}}\right) d t + \sqrt {\eta \Sigma^ {U S A M} \left(X _ {t}\right)} d W _ {t}. \tag {9}
+$$
+
+Based on Eq. (9), it can be observed that under our Assumption 3.1, the implicit regularization term of mini-batch USAM (6) can be divided into the gradient of two parts: the squared norm of the full-batch gradient and the trace of the SGN covariance. The latter, which arises additionally from Theorem 3.2 (Compagnoni et al., 2023), is due to the use of random batches for perturbation in mini-batch USAM. We will see that if we use a deterministic perturbation, the regularization effect on the SGN covariance will disappear, as stated in the following theorem:
+
+Theorem 3.4 (n-USAM SDE - informal statement of Theorem B.4). Under Assumption 3.1 and mild regularity conditions, the solution of the following SDE (10) is an order-(1,1) weak approximation of the discrete update of n-USAM (7) with batch size  $|\gamma|$ :
+
+$$
+d X _ {t} = - \nabla \left(f \left(X _ {t}\right) + \frac {\rho}{2} \| \nabla f \left(X _ {t}\right) \| ^ {2}\right) d t + \sqrt {\eta \Sigma^ {n - U S A M} \left(X _ {t}\right)} d W _ {t}. \tag {10}
+$$
+
+As observed in Table 7 in Appendix J, USAM and SAM exhibit similar behavior under full-batch perturbations, meaning that n-USAM cannot improve generalization performance like mini-batch USAM. Therefore, we argue that the sharpness regularization effect of mini-batch USAM actually stems from the last term of its drift term, which is the gradient of the trace of the SGN covariance, i.e.  $\nabla \mathrm{tr}(V(X_t))$ .
+
+Having understood that n-USAM lacks the sharpness regularization benefits brought by SGN in the drift term, we analyze another contrasting variant, m-USAM, which uses smaller micro-batches to compute the perturbation. Within the framework of the SDE approximation, we can derive the following theorem.
+
+Theorem 3.5 (m-USAM SDE - informal statement of Theorem B.7). Under Assumption 3.1 and mild regularity conditions, the solution of the following SDE (11) is an order-(1,1) weak approximation of the discrete update of m-USAM (8):
+
+$$
+d X _ {t} = - \nabla \left(f \left(X _ {t}\right) + \frac {\rho}{2} \| \nabla f \left(X _ {t}\right) \| ^ {2} + \frac {\rho}{2 m} \operatorname {t r} \left(V \left(X _ {t}\right)\right)\right) d t + \sqrt {\frac {m \eta}{| \gamma |} \Sigma^ {m - U S A M} \left(X _ {t}\right)} d W _ {t}. \tag {11}
+$$
+
+It is worth noting that Theorem 3.3 is recovered by setting  $m = |\gamma|$ . From this SDE approximation, we see that m-USAM's advantages arise from two sources. First, it amplifies the sharpness regularization in the drift term: the coefficient on the SGN covariance changes from  $\rho/(2|\gamma|)$  to  $\rho/(2m)$  (with  $m < |\gamma|$ ). Second, the diffusion term in m-USAM is reduced by a factor of  $m/|\gamma|$  compared to mini-batch USAM with batch size  $m$ . Since the diffusion term captures the random fluctuations that counteract implicit regularization, shrinking it enhances the stability of the method.
+
+# 3.4 SDE approximation for SAM variants
+
+We now turn to the analysis of (normalized) SAM. With the addition of the normalization factor, the situation becomes much more complex because, the expectation of the (no-squared) norm does not have an elementary expression. However, the overall pattern remains similar to the case of USAM. We first present the SDE approximation theorem for SAM variants, then highlight their key differences from the unnormalized version.
+
+Theorem 3.6 (Mini-batch SAM SDE - informal statement of Theorem C.2, adapted from Theorem 3.5 of Compagnoni et al. (2023)). Under Assumption 3.1 and mild regularity conditions, the solution of the following SDE (12) is an order-(1,1) weak approximation of the discrete update of mini-batch SAM (4) with batch size  $|\gamma|$ :
+
+$$
+d X _ {t} = - \nabla \left(f \left(X _ {t}\right) + \frac {\rho}{| \gamma |} \mathbb {E} \| \sum_ {i \in \gamma} \nabla f _ {i} \left(X _ {t}\right) \|\right) d t + \sqrt {\eta \Sigma^ {S A M} \left(X _ {t}\right)} d W _ {t}. \tag {12}
+$$
+
+Theorem 3.7 (n-SAM SDE - informal statement of Theorem C.4). Under Assumption 3.1 and mild regularity conditions, the solution of the following SDE (13) is an order-(1,1) weak approximation of the discrete update of  $n$ -SAM (3) with batch size  $|\gamma|$ :
+
+$$
+d X _ {t} = - \nabla \left(f \left(X _ {t}\right) + \rho \| \nabla f \left(X _ {t}\right) \|\right) d t + \sqrt {\eta \Sigma^ {n - S A M} \left(X _ {t}\right)} d W _ {t}. \tag {13}
+$$
+
+Theorem 3.8 (m-SAM SDE - informal statement of Theorem C.7). Under Assumption 3.1 and mild regularity conditions, the solution of the following SDE (14) is an order-(1,1) weak approximation of the discrete update of m-SAM (5):
+
+$$
+d X _ {t} = - \nabla \left(f (X _ {t}) + \frac {\rho}{m} \mathbb {E} \| \sum_ {i \in \mathcal {I}, | \mathcal {I} | = m} \nabla f _ {i} (X _ {t}) \right\|) d t + \sqrt {\frac {m \eta}{| \gamma |} \Sigma^ {m - S A M} (X _ {t})} d W _ {t}. \tag {14}
+$$
+
+Unlike the unnormalized algorithm, these SAM regularization terms cannot be expressed as simple functions of the full gradient and SGN covariance. Nonetheless, the following proposition clarifies how the regularization terms in the SDEs depend on the mini/micro batch size, revealing an inverse relationship between the norm and the batch size.
+
+Proposition 3.9. Under Assumption 3.1, the following inequalities hold:
+
+$$
+\| \nabla f (x) \| \leq \mathbb {E} \| \nabla f _ {\gamma} (x) \| \leq \sqrt {\| \nabla f (x) \| ^ {2} + \mathbb {E} \| \xi_ {\gamma} (x) \| ^ {2}} = \sqrt {\| \nabla f (x) \| ^ {2} + \frac {\operatorname {t r} (V (x))}{| \gamma |}},
+$$
+
+where  $\xi_{\gamma}$  denotes the SGN in  $\gamma$ . Furthermore, if  $\nabla f_i(x)$  follows a log-concave distribution, then we have  $\mathbb{E}||\nabla f_{\gamma}(x)||$  monotonically increases as  $|\gamma|$  decreases.
+
+A complete proof of Proposition 3.9 is given in Appendix E. Notably, log-concave distributions include many of the standard distributions of interest, such as the Gaussian and exponential distributions.
+
+From Theorems and proposition above, we observe that mini-batch SAM (like USAM) regularizes the magnitude of the SGN. In contrast, n-SAM loses this noise-regularization effect, leading to degraded performance, whereas m-SAM amplifies it and thus achieves improved results. One intuitive explanation is that in larger batches, these noise terms tend to cancel each other out, causing the averaged stochastic gradient to concentrate around its expectation and diminishing the contribution of SGN.
+
+# 3.5 Benefits of SGN Covariance Regularization for Generalization
+
+Building on our observation that USAM/SAM variants impose different strengths of sharpness regularization, i.e., variance-based regularization of the SGN, we investigate how directly regularizing the covariance of the SGN further enhances generalization performance.
+
+We address this from two perspectives. First, following the work of Neu et al. (2021); Wang and Mao (2021), a bound on the generalization error can be decomposed into a sum of mutual-information terms, among which the "trajectory term" is controlled by the covariance of the SGN. This implies that by implicitly regularizing the covariance of the SGN during training, one can reduce the cumulative trajectory term over time, thereby tightening the overall generalization bound.
+
+On the other hand, we examine the algorithm's dynamics as it approaches convergence in the late stages of training. As the parameters approach the minimum, the loss and the full gradient on the training set diminish, causing the gradient of the SGN covariance to dominate the drift term. Near a local minimum, it is well established (Jastrzebski et al., 2017; Daneshmand et al., 2018; Zhu et al., 2018; Xie et al., 2020, 2023) that for the negative log-likelihood loss we have
+
+$$
+V (x) \approx \frac {1}{n} \sum_ {i = 1} ^ {n} \nabla f _ {i} (x) \nabla f _ {i} (x) ^ {\top} \approx \operatorname {F I M} (x) \approx \nabla^ {2} f (x), \tag {15}
+$$
+
+where  $\mathrm{FIM}(x)$  denotes the empirical Fisher information matrix and  $\nabla^2 f(x)$  the Hessian matrix. Moreover, an empirical result by Xie et al. (2020) shows that for neural networks, this relationship remains approximately valid even far away from the local minima. Therefore, in the late stages of training, regularizing the trace of the SGN covariance is approximately equivalent to regularizing the trace of the Hessian, which is widely regarded as a good measure of sharpness that has been observed to correlate strongly with generalization performance (Keskar et al., 2017; Blanc et al., 2020; Wen et al., 2022; Arora et al., 2022; Damian et al., 2022; Ahn et al., 2023; Tahmasebi et al., 2024).
+
+To verify the dynamics around the minima, we consider the setting from the work of Liu et al. (2020); Damian et al. (2021), where initialization is performed at a bad minimum. We use the checkpoint provided by Damian et al. (2021). The learning rate is set to  $1e^{-3}$ , under which SGD has been shown to struggle in escaping bad minima. We do not use any explicit regularization techniques. For SAM, we use  $\rho = 5e^{-3}$  and compare the performance of m-SAM with different values of  $m$ . In Figures 1,2, we can clearly observe that as  $m$  decreases, the regularization effect on the SGN covariance is significantly strengthened, which in turn accelerates the escape from poor minima—consistent with our theoretical analysis..
+
+# 4 Practical Method
+
+In this section, we will demonstrate how to translate the theoretical insights gained from our SDE approximations into a practical method. While  $\mathrm{m - SAM}^1$  achieves substantially better generalization than mini-batch SAM by strengthening the regularization of the SGN magnitude, its inherently sequential nature creates a serious parallelization bottleneck (see Table 7 in Appendix J for performance and time cost). This is because it must sequentially compute the perturbation for each micro-batch
+
+![](images/079d3706a58fcf41c2fd193a1bc5ef9beee95d5c6d40ad0f3a862bdcf3f8e987.jpg)  
+Figure 1: Speed of escaping poor minima, measured by test accuracy.
+
+![](images/30dc4662b274eedad890fbca3f2bb1d2f5907f5d10a6158d8dca2a98f7bc1a5c.jpg)  
+Figure 2: Variance of SGN over iterations.
+
+and then individually backpropagate to obtain the perturbed gradient. Building on our theoretical insights, a natural question arises:
+
+Can we design a parallelizable algorithm that preserves the generalization advantages of m-SAM? One important observation is: if we further assume that the SGN is approximately orthogonal to the full gradient, or that the norm of the full gradient is negligible compared to that of the SGN (as often occurs in the late stages of training when the model approaches convergence), which commonly arises in signal-to-noise-ratio analyses (Cao et al., 2022; Jelassi et al., 2022; Zou et al., 2023; Huang et al., 2023; Allen-Zhu and Li, 2023; Han et al., 2024; Huang et al., 2024; Li et al., 2024a), then from the following equation we observe that the norm of the stochastic gradient is directly related to the norm of the SGN:
+
+$$
+\| \nabla f _ {i} (x) \| \approx \sqrt {\| \nabla f (x) \| ^ {2} + \| \xi_ {i} \| ^ {2}}. \tag {16}
+$$
+
+This decomposition reveals that samples with larger gradient norms carry higher-magnitude SGN. Drawing inspiration from importance sampling, which assigns a weight to each sample proportional to its "importance", we adapt this idea to emphasize more important samples when computing SAM's perturbation (2). In our framework, we quantify importance by the magnitude of the SGN: samples exhibiting larger SGN contribute more to sharpness regularization and therefore deserve greater weight in the perturbation. Specifically, we introduce an adaptive weighting mechanism in which each weight  $p_i$  reflects the importance of sample  $i$  when computing the perturbation vector. This reweighting strategy defines a probability distribution  $P = \{p_i\}_{i \in \gamma}$  over the sampled indices  $i \in \gamma$ , thereby modulating the influence of each sample. We refer to the resulting algorithm as Reweighted SAM (RW-SAM). The objective of RW-SAM's perturbation is formulated as follows:
+
+$$
+\max  _ {P \in \Delta} \quad \max  _ {\| \epsilon \| \leq 1} \quad \left\langle \sum_ {i \in \gamma} p _ {i} \nabla f _ {i} (x), \epsilon \right\rangle + \frac {\mathbb {H} (P)}{\lambda}, \tag {17}
+$$
+
+where  $\Delta$  denotes the probability simplex, and  $\mathbb{H}$  is the entropy function to avoid all the weights concentrate on the single sample,  $\lambda$  is a hyperparameter to maintain a balance between emphasis and diversity. Notably, as  $\lambda \to 0$ , Objective (17) degenerates to mini-batch SAM's perturbation (2).
+
+Solve  $\epsilon$  from Objective (17), we get:
+
+$$
+\epsilon^ {*} = \frac {\sum_ {i \in \gamma} p _ {i} \nabla f _ {i} (x)}{\left\| \sum_ {i \in \gamma} p _ {i} \nabla f _ {i} (x) \right\|}. \tag {18}
+$$
+
+Since Objective (17) does not have a closed-form solution for  $P$ , we propose solving its relaxation:
+
+$$
+\max  _ {P \in \Delta} \sum_ {i \in \gamma} p _ {i} \| \nabla f _ {i} (x) \| + \frac {\mathbb {H} (P)}{\lambda}. \tag {19}
+$$
+
+Objective (19) corresponds to the well-known Gibbs distribution (see derivation in Section H), which is given by:
+
+$$
+p _ {i} ^ {*} = \frac {\exp (\lambda \| \nabla f _ {i} (x) \|)}{\sum_ {j \in \gamma} \exp (\lambda \| \nabla f _ {j} (x) \|)}. \tag {20}
+$$
+
+Broadly speaking, Eq. (20) corresponds to assigning higher weights to samples with larger stochastic gradient norms, where  $\lambda$  controls the concentration of this distribution. In practice, the optimal value of  $\lambda$  depends on the scale of per-sample gradient norms. Therefore, we normalize the estimated gradient norms before applying the exponential function, making the algorithm's performance less sensitive to the choice of  $\lambda$ . As observed in Section 5.4, normalization significantly reduces the need for tuning  $\lambda$ .
+
+Additionally, we propose using a finite-difference method combined with Monte Carlo sampling to avoid per-sample gradient-norm estimation via backpropagation. The formula is as follows (see derivation in Section G):
+
+$$
+\| \nabla f _ {i} (x) \| \approx \sqrt {\frac {1}{Q} \sum_ {q = 1} ^ {Q} \left(\frac {f _ {i} \left(x + \delta z _ {q}\right) - f _ {i} (x)}{\delta}\right) ^ {2}}, \tag {21}
+$$
+
+where  $z \in \mathbb{R}^d$  is Rademacher variable,  $\delta$  is a small constant. This estimator requires only  $Q$  additional forward passes, as we can conveniently obtain the loss for each sample in a single forward pass, making it an efficient way to estimate the gradient norm for each sample. To minimize additional computational overhead, we set  $Q = 1$  in our experiments, consistent with common practice in deep learning (Kingma et al., 2013; Gal and Ghahramani, 2016; Ho et al., 2020; Malladi et al., 2023), and found that this choice suffices to achieve a non-trivial performance improvement. However, unlike common implementations (Kingma et al., 2013; Ho et al., 2020; Malladi et al., 2023), we propose using Rademacher instead of Gaussian perturbations to minimize the variance of the Monte Carlo estimator. Since Rademacher variables have a fixed expected squared norm, they achieve the optimal variance (according to Theorem 2.2 in Ma and Huang (2025)). The pseudocode for our algorithm is presented in Algorithm 1 (see Appendix I).
+
+# 5 Experiments
+
+# 5.1 Training from scratch
+
+We evaluate three optimization methods SGD,  $\mathrm{SAM}^2$ , and RW-SAM on CIFAR-10 and CIFAR-100 (Krizhevsky et al., 2009), training three models from scratch: ResNet-18, ResNet-50 (He et al., 2016), and WideResNet-28-10 (Zagoruyko and Komodakis, 2016). We use a batch size of 128 and a cosine learning rate schedule with an initial learning rate of 0.1. SAM and RW-SAM are trained for 200 epochs, while SGD is trained for 400 epochs. We apply a momentum of 0.9 and a weight decay of  $5e^{-4}$ , along with standard data augmentation techniques, including horizontal flipping, padding by four pixels, and random cropping.
+
+For SAM and RW-SAM, we set  $\rho = 0.05$  for CIFAR-10 and  $\rho = 0.1$  for CIFAR-100. In the case of RW-SAM, we determine  $\delta$  through finite difference estimation on the model before training, based on the estimated error. Specifically, we use  $\delta = 1e - 3$  for ResNet-18 and  $\delta = 1e - 4$  for both ResNet-50 and WideResNet-28-10. For the additional hyperparameter  $\lambda$  in RW-SAM, we performed a grid search over  $\{0.25, 0.5, 1.0, 2.0\}$  on a validation set and found that 0.5 consistently yielded strong performance across experiments.
+
+Table 1: Test accuracy comparison on CIFAR-10 and CIFAR-100 with different optimizers.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">CIFAR-10</td><td colspan="3">CIFAR-100</td></tr><tr><td>SGD</td><td>SAM</td><td>RW-SAM</td><td>SGD</td><td>SAM</td><td>RW-SAM</td></tr><tr><td>ResNet-18</td><td>95.62 ± 0.03</td><td>95.99 ± 0.07</td><td>96.24 ± 0.05</td><td>78.91 ± 0.18</td><td>78.90 ± 0.27</td><td>79.31 ± 0.28</td></tr><tr><td>ResNet-50</td><td>95.64 ± 0.37</td><td>96.06 ± 0.04</td><td>96.34 ± 0.04</td><td>79.55 ± 0.16</td><td>80.31 ± 0.35</td><td>80.83 ± 0.05</td></tr><tr><td>WideResNet</td><td>96.47 ± 0.03</td><td>96.91 ± 0.02</td><td>97.11 ± 0.05</td><td>81.55 ± 0.15</td><td>83.25 ± 0.07</td><td>83.52 ± 0.08</td></tr></table>
+
+For large-scale experiments, we train a ResNet-50 on ImageNet-1K (Deng et al., 2009) for 90 epochs with an initial learning rate of 0.05. For both SAM and RW-SAM, we use the same  $\rho = 0.05$ . For the additional hyperparameter  $\lambda$  in RW-SAM, we set  $\lambda = 0.25$ . All other hyperparameters remain the same as those used on CIFAR-10/100.
+
+We repeated three independent experiments and reported the mean and standard deviation of the test accuracy in Table 1 and Table 2a. We observe that RW-SAM consistently outperforms the baselines across various models, as well as on both small and large datasets.
+
+Analysis of Computational Overhead. RW-SAM requires an additional forward pass to estimate per-sample gradient norms, leading to approximately 1/6 more training overhead compared to vanilla SAM, as a forward pass typically takes about half the time of a backward pass (Kaplan, 2022). For a report of the additional wall-clock time overhead, please see Table 8 in Appendix J. However, RW-SAM matches the performance of m-SAM at  $m = 64$  without incurring its nearly two-fold training overhead (See Table 7 in Appendix J). This highlights the efficiency of RW-SAM in balancing computational cost and performance.
+
+Table 2: (a) Test accuracy on ImageNet-1k with different optimizers; (b) Test accuracy fine-tuning ViT-B/16 on CIFAR-10/100 with different optimizers.
+
+<table><tr><td>ImageNet-1k</td><td>SGD</td><td>SAM</td><td>RW-SAM</td></tr><tr><td>ResNet-50</td><td>76.67 ± 0.05</td><td>77.16 ± 0.04</td><td>77.37 ± 0.05</td></tr></table>
+
+(a)  
+
+<table><tr><td></td><td>SGD</td><td>SAM</td><td>RW-SAM</td></tr><tr><td>CIFAR-10</td><td>98.24 ± 0.05</td><td>98.40 ± 0.02</td><td>98.58 ± 0.02</td></tr><tr><td>CIFAR-100</td><td>78.91 ± 0.18</td><td>89.63 ± 0.12</td><td>89.89 ± 0.09</td></tr></table>
+
+(b)
+
+# 5.2 Fine-tuning
+
+We fine-tune a ViT-B/16 model (Dosovitskiy et al., 2020), pre-trained on ImageNet-1K, on CIFAR-10 and CIFAR-100. We train for 20 epochs with an initial learning rate of 0.01. Other hyperparameters are the same as those in training from scratch. The results are summarized in Table 2b. We also fine-tune a pretrained DistilBERT model (Sanh et al., 2019) on the GLUE benchmark (Wang et al., 2018). We use AdamW (Loshchilov and Hutter, 2019) as the base optimizer. The detailed hyperparameter settings are provided in Table 10 of Appendix J, and the results are presented in Table 3. We observe that RW-SAM consistently outperforms SAM in both fine-tuning experiments.
+
+Table 3: Performance comparison on GLUE tasks using different optimizers.  
+
+<table><tr><td>Optimizer</td><td>CoLA</td><td>MNLI</td><td>MRPC</td><td>QNLI</td><td>QQP</td><td>RTE</td><td>SST-2</td><td>STS-B</td><td>Average</td></tr><tr><td>AdamW</td><td>0.538</td><td>0.825</td><td>0.900</td><td>0.884</td><td>0.868</td><td>0.628</td><td>0.914</td><td>0.864</td><td>0.804</td></tr><tr><td>SAM</td><td>0.517</td><td>0.824</td><td>0.900</td><td>0.894</td><td>0.871</td><td>0.610</td><td>0.911</td><td>0.870</td><td>0.800</td></tr><tr><td>RW-SAM</td><td>0.560</td><td>0.826</td><td>0.904</td><td>0.896</td><td>0.872</td><td>0.625</td><td>0.915</td><td>0.870</td><td>0.809</td></tr></table>
+
+# 5.3 Robustness to label noise
+
+Foret et al. (2021) have shown that SAM exhibits robustness to label noise. Motivated by this, we evaluate RW-SAM's performance on CIFAR-10 with labels randomly flipped at specified noise ratios. We train a ResNet-18 and report clean test accuracies in Table 4. As the noise ratio increases, RW-SAM maintains remarkably strong performance. In particular, at an  $80\%$  noise ratio, RW-SAM achieves a  $16\%$  absolute accuracy improvement over SAM.
+
+# 5.4 Additional experiments
+
+Hyperparameter sensitivity. We train ResNet-18 on CIFAR-100 using RW-SAM to evaluate its sensitivity to different values of  $\lambda$ . As summarized in Table 5, RW-SAM demonstrates robustness to the choice of  $\lambda$  and consistently outperforms the baselines within a reasonable range.
+
+Trace of stochastic gradient covariance. According to our theory, we compare the trace of the stochastic gradient covariance matrix at convergence for different algorithms, which, near the minimum, closely approximates the Hessian matrix (See Eq. (15)). The results, shown in Table 6, indicate that compared to SGD and SAM, RW-SAM indeed converges to a minimum with a smaller stochastic gradient covariance magnitude.
+
+Table 4: Performance comparison on CIFAR-10 across different noise ratios  
+
+<table><tr><td>Noise Ratio</td><td>SGD</td><td>SAM</td><td>RW-SAM</td></tr><tr><td>20%</td><td>87.54 ± 0.20</td><td>90.01 ± 0.09</td><td>90.34 ± 0.20</td></tr><tr><td>40%</td><td>83.66 ± 0.30</td><td>86.40 ± 0.12</td><td>86.87 ± 0.11</td></tr><tr><td>60%</td><td>76.64 ± 0.32</td><td>78.79 ± 0.24</td><td>81.52 ± 0.31</td></tr><tr><td>80%</td><td>46.53 ± 1.13</td><td>37.69 ± 3.12</td><td>53.17 ± 3.70</td></tr></table>
+
+Table 5: RW-SAM  $\lambda$  -sensitivity  
+
+<table><tr><td>λ</td><td>0.25</td><td>0.5</td><td>1.0</td><td>2.0</td></tr><tr><td></td><td>79.09 ± 0.21</td><td>79.31 ± 0.28</td><td>79.12 ± 0.33</td><td>79.03 ± 0.22</td></tr></table>
+
+Table 6: Trace of the gradient covariance  
+
+<table><tr><td>Optimizer</td><td>Trace</td></tr><tr><td>SGD</td><td>572.39 ± 24.15</td></tr><tr><td>SAM</td><td>198.40 ± 6.20</td></tr><tr><td>RW-SAM</td><td>177.79 ± 5.10</td></tr></table>
+
+# 6 Conclusion
+
+In this work, we conducted a comprehensive theoretical analysis of SAM and its variants through an enhanced SDE modeling framework. Our findings reveal that the structure of SGN plays a crucial role in implicit regularization, significantly influencing generalization performance. Based on our analysis, we proposed Reweighted SAM, an adaptive weighting mechanism for perturbation, which we empirically validated through extensive experiments. Our study provides a deeper understanding of the dynamics of SAM-based algorithms and offers new perspectives on improving their generalization performance.
+
+# Acknowledgement
+
+Trung Le, Mehrtash Harandi, and Dinh Phung were supported by the ARC Discovery Project grants DP230101176 and DP250100262. Trung Le and Mehrtash Harandi were also supported by the Air Force Office of Scientific Research under award number FA9550-23-S-0001.
+
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+
+# A General theory for two-parameter weak approximation.
+
+Let  $T > 0$ ,  $\eta \in (0, \min\{1, T\})$ , and  $N = \lfloor T / \eta \rfloor$ . We consider the general discrete iteration
+
+$$
+x _ {k + 1} = x _ {k} + \eta h \left(x _ {k}, \gamma_ {k}, \eta , \rho\right), \quad x _ {0} \in \mathbb {R} ^ {d}, \quad k = 0, 1, \dots , N, \tag {22}
+$$
+
+and its corresponding continuous-time approximation by the SDE
+
+$$
+\mathrm {d} X _ {t} = b (X _ {t}, \eta , \rho) \mathrm {d} t + \sqrt {\eta} \sigma (X _ {t}, \eta , \rho) \mathrm {d} W _ {t}, \quad X _ {0} = x _ {0}, t \in [ 0, T ]. \tag {23}
+$$
+
+We denote  $\widetilde{X}_k\coloneqq X_{k\eta}$ , and the one-step changes:
+
+$$
+\Delta (x) := x _ {1} - x, \quad \widetilde {\Delta} (x) := \widetilde {X} _ {1} - x. \tag {24}
+$$
+
+Following the SDE framework by Mil'shtein (1986); Li et al. (2017); Compagnoni et al. (2023); Luo et al. (2025), we have the following definition:
+
+Definition A.1. Let  $G$  denote the set of continuous functions  $\mathbb{R}^d\to \mathbb{R}$  of at most polynomial growth, i.e.  $g\in G$  if there exists positive integers  $\kappa_{1},\kappa_{2} > 0$  such that
+
+$$
+| g (x) | \leq \kappa_ {1} \left(1 + \| x \| ^ {2 \kappa_ {2}}\right),
+$$
+
+for all  $x \in \mathbb{R}^d$ . Moreover, for each integer  $\alpha \geq 1$  we denote by  $G^\alpha$  the set of  $\alpha$ -times continuously differentiable functions  $\mathbb{R}^d \to \mathbb{R}$  which, together with its partial derivatives up to and including order  $\alpha$ , belong to  $G$ .
+
+This definition comes from the field of numerical analysis of SDEs (Mil'shtein, 1986). In the case of  $g(x) = ||x||^j$ , the bound restricts the difference between the  $j$ -th moments of the discrete process and those of the continuous process. We write  $\mathcal{O}(\eta^\alpha \rho^\beta)$  to denote that there exists a function  $K \in G$  independent of  $\rho$ ,  $\eta$ , such that the error terms are bounded by  $K\eta^\alpha \rho^\beta$ .
+
+Theorem A.2. (Adaption of Theorem 3 in Li et al. (2019)) Let  $T > 0$ ,  $\eta \in (0, \min\{1, T\})$ ,  $N = \lfloor T / \eta \rfloor$ . Let  $\alpha \geq 1$  be an integer. Suppose further that the following conditions hold:
+
+(i) There exists  $K_{1} \in G$ , independent of  $\eta, \rho$ , such that for each  $s = 1,2,\ldots,\alpha$  and any indices  $i_{1},\ldots,i_{s} \in \{1,\ldots,d\}$ ,
+
+$$
+\left| \mathbb {E} \prod_ {j = 1} ^ {s} \Delta_ {(i _ {j})} (x) - \mathbb {E} \prod_ {j = 1} ^ {s} \widetilde {\Delta} _ {(i _ {j})} (x) \right| \leq K _ {1} (x) (\eta^ {\alpha + 1} + \eta \rho^ {\beta + 1}),
+$$
+
+and
+
+$$
+\mathbb {E} \prod_ {j = 1} ^ {\alpha + 1} \left| \Delta_ {(i _ {j})} (x) \right| \leq K _ {1} (x) (\eta^ {\alpha + 1} + \eta \rho^ {\beta + 1}).
+$$
+
+(ii) For each  $m \geq 1$ , the  $2m$ -moment of  $x_{k}$  is uniformly bounded in  $k$  and  $\eta$ , i.e. there exists  $K_{2} \in G$ , independent of  $\eta$  and  $k$ , such that
+
+$$
+\mathbb {E} \left\| x _ {k} \right\| ^ {2 m} \leq K _ {2} (x), \quad k = 0, 1, \dots , N.
+$$
+
+Then, for each  $g \in G^{\alpha + 1}$ , there exists a constant  $C > 0$ , independent of  $\eta, \rho$ , such that
+
+$$
+\max  _ {0 \leq k \leq N} \left| \mathbb {E} g (x _ {k}) - \mathbb {E} g (X _ {k \eta}) \right| \leq C (\eta^ {\alpha} + \rho^ {\beta + 1}).
+$$
+
+By substituting Assumption (i) in the penultimate step of the proof in Theorem 3 in Li et al. (2019) with Assumption (i) of our Theorem A.2, the same argument goes through and yields the desired conclusion. Hence, we omit the proof.
+
+Next, we state the key lemma for the two-parameter weak approximation. By employing a Dynkin expansion (see, e.g. Evans (2012)) instead of a full Itô-Taylor expansion, it avoids the proliferation of terms and provides a streamlined approach to controlling the remainder error in the two-parameter setting.
+
+Lemma A.3 (Two-parameter Dynkin (semigroup) expansion). Let  $\psi \in G^{2\alpha + 2}$ , and suppose the drift admits the expansion
+
+$$
+b (x, \rho) = \sum_ {m = 0} ^ {\beta} \rho^ {m} b _ {m} (x) + O (\rho^ {\beta + 1}), \qquad \sigma (x) = \sigma_ {0} (x).
+$$
+
+Define
+
+$$
+A _ {m} \psi (x) := b _ {m} ^ {(i)} (x) \partial_ {i} \psi (x) (m = 0, 1, \ldots , \beta), A _ {\Delta} \psi (x) := \frac {1}{2} \left[ \sigma_ {0} \sigma_ {0} ^ {T} \right] ^ {(i j)} \partial_ {i j} ^ {2} \psi (x).
+$$
+
+Suppose further that  $b_{m}$  ( $m = 0,1,\ldots ,\beta$ ),  $\sigma_0 \in G^{2\alpha}$ , then for any nonnegative integers  $\alpha ,\beta$
+
+$$
+\mathbb{E}\big[\psi (X_{\eta})\big] = \sum_{n = 0}^{\alpha}\frac{\eta^{n}}{n!}\sum_{m = 0}^{\beta}\rho^{m}\sum_{\substack{\ell \in \{0,\Delta ,1,\ldots ,\beta \}^{n}\\ |\ell | = m}}A_{\ell_{1}}A_{\ell_{2}}\dots A_{\ell_{n}}\psi (x) + O(\eta^{\alpha +1}) + O(\eta \rho^{\beta +1}).
+$$
+
+Here  $|\ell| \coloneqq \sum_{i: \ell_i \notin \{0, \Delta\}} \ell_i$ , and each multi-index  $\ell = (\ell_1, \dots, \ell_n)$  contributes a factor  $\rho^{|\ell|}$ , and indices  $\ell_i = 0$  or  $\ell_i = \Delta$  contribute no  $\rho$ -power (via  $A_0$  or  $A_\Delta$ ).
+
+Proof. Let
+
+$$
+L \phi (x) = (b (x, \rho) \cdot \nabla \phi) (x) + \frac {1}{2} [ \sigma_ {0} \sigma_ {0} ^ {T} ] ^ {(i j)} (x) \partial_ {i j} ^ {2} \phi (x)
+$$
+
+be the infinitesimal generator of the diffusion. Since
+
+$$
+b (x, \rho) = \sum_ {m = 0} ^ {\beta} \rho^ {m} b _ {m} (x) + O (\rho^ {\beta + 1}),
+$$
+
+we may write
+
+$$
+L = \sum_ {m = 0} ^ {\beta} \rho^ {m} A _ {m} + A _ {\Delta} + O (\rho^ {\beta + 1}),
+$$
+
+where  $A_{m}$  and  $A_{\Delta}$  are as in the statement. By Dynkin's formula (or equivalently the semigroup expansion),
+
+$$
+\mathbb {E} \left[ \psi \left(X _ {\eta}\right) \right] = \left(e ^ {\eta L} \psi\right) (x) = \sum_ {n = 0} ^ {\alpha} \frac {\eta^ {n}}{n !} L ^ {n} \psi (x) + O \left(\eta^ {\alpha + 1}\right).
+$$
+
+It remains to expand each power  $L^n$ . By the multinomial theorem,
+
+$$
+L^{n} = \Big(\sum_{m = 0}^{\beta}\rho^{m}A_{m} + A_{\Delta} + O(\rho^{\beta +1})\Big)^{n} = \sum_{m = 0}^{\beta}\rho^{m}\sum_{\substack{\ell \in \{0,\Delta ,1,\ldots ,\beta \}^{n}\\ |\ell | = m}}A_{\ell_{1}}A_{\ell_{2}}\dots A_{\ell_{n}} + O(\rho^{\beta +1}).
+$$
+
+Hence
+
+$$
+\frac{\eta^{n}}{n!} L^{n}\psi (x) = \frac{\eta^{n}}{n!}\sum_{m = 0}^{\beta}\rho^{m}\sum_{\substack{\ell \in \{0,\Delta ,1,\ldots ,\beta \}^{n}\\ |\ell | = m}}A_{\ell_{1}}A_{\ell_{2}}\dots A_{\ell_{n}}\psi (x) + O(\eta^{n + 1}) + O(\eta \rho^{\beta +1}).
+$$
+
+Summing over  $n = 0,1,\ldots ,\alpha$  and collecting the remainders  $O(\eta^{\alpha +1})$  and  $O(\eta \rho^{\beta +1})$  yields exactly the claimed two-parameter expansion.
+
+Since our focus in this paper is on the order-(1,1) weak approximation, we now present the one-step approximation lemma for SDEs in the case  $\alpha = \beta = 1$ , as follows. For readers interested in higher-order two-parameter weak approximations, it is sufficient to apply higher-order truncations of the Dynkin and Taylor expansions in the two lemmas below and then match the corresponding moments at each order.
+
+Lemma A.4 (One-step moment estimates up to  $\eta^1, \rho^1$  for SDEs). Suppose the drift admits the expansion
+
+$$
+b (x, \rho) = b _ {0} (x) + \rho b _ {1} (x) + O (\rho^ {2}), \qquad \sigma (x) = \sigma_ {0} (x),
+$$
+
+and assume  $b_{0}, b_{1}, \sigma_{0} \in G^{2}$ . Let  $\tilde{\Delta}(x)$  be the one-step increment defined in (24). Then:
+
+(i)  $\mathbb{E}[\widetilde{\Delta}_{(i)}(x)] = \eta \big(b_0^{(i)}(x) + \rho b_1^{(i)}(x)\big) + O(\eta^2) + O(\eta \rho^2).$  
+(ii)  $\mathbb{E}\big[\widetilde{\Delta}_{(i)}(x)\widetilde{\Delta}_{(j)}(x)\big] = \eta^2\big(b_0^{(i)}(x)b_0^{(j)}(x) + \sum_k\sigma_0^{(i,k)}(x)\sigma_0^{(j,k)}(x)\big) + \eta^2\rho \big(b_0^{(i)}(x)b_1^{(j)}(x) +$
+
+$$
+b _ {1} ^ {(i)} (x) b _ {0} ^ {(j)} (x)) + O (\eta^ {2} \rho^ {2}) + O (\eta^ {3}).
+$$
+
+(iii)  $\mathbb{E}\Big[\prod_{j = 1}^{3}\big|\widetilde{\Delta}_{(i_j)}(x)\big|\Big] = O(\eta^3).$
+
+Proof. For each  $s = 1,2,3$  and any choice of indices  $i_1,\ldots ,i_s$ , define the test function
+
+$$
+\psi_ {s} (z) = \prod_ {j = 1} ^ {s} \left(z _ {\left(i _ {j}\right)} - x _ {\left(i _ {j}\right)}\right).
+$$
+
+Since  $\psi_s \in C^4(\mathbb{R}^d)$  with at most polynomial growth, we may invoke Lemma A.2 with truncation orders  $\alpha = 1$  and  $\beta = 1$ . This yields,
+
+$$
+\mathbb {E} \big [ \psi (x) \big ] = \psi (x) + \eta \sum_ {\ell \in \{0, \Delta \}} A _ {\ell} \psi (x) + \eta \rho A _ {1} \psi (x) + O (\eta^ {2}) + O (\eta \rho^ {2}).
+$$
+
+(i) First moment. Here
+
+$$
+\psi_ {1} (z) = z _ {(i)} - x _ {(i)},
+$$
+
+Hence
+
+$$
+\mathbb {E} [ \widetilde {\Delta} _ {(i)} (x) ] = \eta b _ {0} ^ {(i)} (x) + \eta \rho b _ {1} ^ {(i)} (x) + O (\eta^ {2}) + O (\eta \rho^ {2}),
+$$
+
+proving (i).
+
+(ii) Second moment. Now
+
+$$
+\psi_ {2} (z) = \left(z _ {(i)} - x _ {(i)}\right) \left(z _ {(j)} - x _ {(j)}\right),
+$$
+
+It follows that
+
+$$
+\mathbb {E} \big [ \widetilde {\Delta} _ {(i)} (x) \widetilde {\Delta} _ {(j)} (x) \big ] = \eta^ {2} \Big [ b _ {0} ^ {(i)} b _ {0} ^ {(j)} + \sum_ {k} \sigma_ {0} ^ {(i, k)} \sigma_ {0} ^ {(j, k)} \Big ] + \eta^ {2} \rho \big [ b _ {0} ^ {(i)} b _ {1} ^ {(j)} + b _ {1} ^ {(i)} b _ {0} ^ {(j)} \big ] + O (\eta^ {2} \rho^ {2}) + O (\eta^ {3}),
+$$
+
+proving (ii).
+
+(iii) Third moment. Finally,
+
+$$
+\psi_ {3} (z) = \prod_ {j = 1} ^ {3} \left(z _ {\left(i _ {j}\right)} - x _ {\left(i _ {j}\right)}\right),
+$$
+
+and since each nonzero term in the expansion has total order  $n \geq 3$ , Lemma A.2 gives
+
+$$
+\mathbb {E} \left[ \prod_ {j = 1} ^ {3} | \widetilde {\Delta} _ {(i _ {j})} (x) | \right] = \mathbb {E} \left[ | \psi_ {3} (X _ {k + 1}) | \right] = O (\eta^ {3}),
+$$
+
+establishing (iii). This completes the proof.
+
+![](images/28989c4ff02faf0355f60bf7146e6bd46c5a6f72f3775343dd7bb6b8d0e946ae.jpg)
+
+Similar to the continuous-time setting, we require the following one-step error lemma for the discrete algorithm in the case  $\alpha = \beta = 1$ :
+
+Lemma A.5 (One-step moment estimates up to  $\eta^1, \rho^1$  for the discrete algorithm). Suppose the discrete update 22 admits the expansion
+
+$$
+h (x, \gamma , \rho) = h _ {0} (x, \gamma) + \rho h _ {1} (x, \gamma) + O (\rho^ {2}),
+$$
+
+and assume  $h_0, h_1 \in G^2$ . Let  $\Delta(x)$  be the one-step increment defined in (24). Then:
+
+(i)  $\mathbb{E}[\Delta_{(i)}(x)] = \eta h_0^{(i)}(x) + \eta \rho h_1^{(i)}(x) + O(\eta \rho^2).$  
+(ii)  $\mathbb{E}\big[\Delta_{(i)}(x)\Delta_{(j)}(x)\big] = \eta^2\big(h_0^{(i)}(x)h_0^{(j)}(x) + \Sigma_{0,0}^{(ij)}(x)\big) + \eta^2\rho \big(h_0^{(i)}(x)h_1^{(j)}(x)\big) +$ $h_1^{(i)}(x)h_0^{(j)}(x) + \Sigma_{0,1}^{(ij)}(x) + \Sigma_{1,0}^{(ij)}(x)\big) + O(\eta^2\rho^2).$
+
+$$
+(i i i) \mathbb {E} \left[ \prod_ {j = 1} ^ {3} \left| \Delta_ {(i _ {j})} (x) \right| \right] = O (\eta^ {3}).
+$$
+
+where  $h_0(x) = \mathbb{E}h_0(x,\gamma), h_1(x) = \mathbb{E}h_1(x,\gamma), \Sigma_{0,0}(x) = \operatorname{Cov}(h_0(x,\gamma), h_0(x,\gamma)), \Sigma_{0,1}(x) = \operatorname{Cov}(h_0(x,\gamma), h_1(x,\gamma))$ .
+
+Proof. Recall that
+
+$$
+\Delta (x) = \eta h (x, \gamma , \rho) = \eta \left(h _ {0} (x, \gamma) + \rho h _ {1} (x, \gamma) + O (\rho^ {2})\right).
+$$
+
+Hence for each coordinate  $i$
+
+(i) First moment.
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \Delta_ {(i)} (x) \right] = \eta \mathbb {E} \left[ h _ {0} ^ {(i)} (x, \gamma) + \rho h _ {1} ^ {(i)} (x, \gamma) + O (\rho^ {2}) \right] \\ = \eta \big (h _ {0} ^ {(i)} (x) + \rho h _ {1} ^ {(i)} (x) \big) + O (\eta \rho^ {2}). \\ \end{array}
+$$
+
+(ii) Second moment.
+
+$$
+\begin{array}{l} \mathbb {E} \big [ \Delta_ {(i)} (x) \Delta_ {(j)} (x) \big ] = \eta^ {2} \mathbb {E} _ {\gamma} \Big [ \big (h _ {0} ^ {(i)} (x, \gamma) + \rho h _ {1} ^ {(i)} (x, \gamma) \big) \big (h _ {0} ^ {(j)} (x, \gamma) + \rho h _ {1} ^ {(j)} (x, \gamma) \big) \Big ] + O (\eta^ {2} \rho^ {2}) \\ = \eta^ {2} \left\{\mathbb {E} _ {\gamma} \left[ h _ {0} ^ {(i)} (x, \gamma) h _ {0} ^ {(j)} (x, \gamma) \right] + \rho \left(\mathbb {E} _ {\gamma} \left[ h _ {0} ^ {(i)} (x, \gamma) h _ {1} ^ {(j)} (x, \gamma) \right] \right. \right. \\ \left. \left. + \mathbb {E} _ {\gamma} [ h _ {1} ^ {(i)} (x, \gamma) h _ {0} ^ {(j)} (x, \gamma) ]\right) \right\} + O (\eta^ {2} \rho^ {2}) \\ = \eta^ {2} \left\{h _ {0} ^ {(i)} (x) h _ {0} ^ {(j)} (x) + \Sigma_ {0, 0} ^ {(i j)} (x) \right\} \\ + \eta^ {2} \rho \Bigl \{h _ {0} ^ {(i)} (x) h _ {1} ^ {(j)} (x) + h _ {1} ^ {(i)} (x) h _ {0} ^ {(j)} (x) + \Sigma_ {0, 1} ^ {(i j)} (x) + \Sigma_ {1, 0} ^ {(i j)} (x) \Bigr \} + O (\eta^ {2} \rho^ {2}). \\ \end{array}
+$$
+
+(iii) Third moment. Since  $h_0, h_1 \in G^2$  implies that all moments up to order three are finite and  $\Delta = O(\eta)$ , we have
+
+$$
+\mathbb {E} \left[ \left| \Delta_ {(i _ {1})} \Delta_ {(i _ {2})} \Delta_ {(i _ {3})} \right| \right] = O (\eta^ {3}).
+$$
+
+This completes the proof.
+
+![](images/9dda838a837a69795cd0ec7c89d62f66935cadf994e808fff53169fde0b62334.jpg)
+
+# B SDE approximation for USAM variants
+
+Recall that the update rules of USAM variants are defined by:
+
+$$
+\text {m i n i - b a t c h U S A M :} \quad x _ {k + 1} = x _ {k} - \eta \nabla f _ {\gamma_ {k}} \left(x _ {k} + \rho \nabla f _ {\gamma_ {k}} \left(x _ {k}\right)\right) \tag {25}
+$$
+
+$$
+\text {n - U S A M :} \quad x _ {k + 1} = x _ {k} - \eta \nabla f _ {\gamma_ {k}} \left(x _ {k} + \rho \nabla f \left(x _ {k}\right)\right) \tag {26}
+$$
+
+$$
+\text {m - U S A M :} \quad x _ {k + 1} = x _ {k} - \frac {\eta m}{| \gamma |} \sum_ {\mathcal {I} _ {j} \subset \gamma_ {k}, \left| \mathcal {I} _ {j} \right| = m} \nabla f _ {\mathcal {I} _ {j}} \left(x _ {k} + \rho \nabla f _ {\mathcal {I} _ {j}} \left(x _ {k}\right)\right) \tag {27}
+$$
+
+In this section, we impose the following growth assumption on the functions  $f$  and  $f_{\gamma}$ :
+
+Assumption B.1. The functions  $f$  and  $f_{i}$  belong to the class  $G^4$ .
+
+# B.1 Mini-batch USAM
+
+For the mini-batch USAM algorithm (25), we define the continuous-time approximation  $X_{t}$  as the solution to the following SDE:
+
+$$
+d X _ {t} = - \nabla f ^ {U S A M} \left(X _ {t}\right) d t + \sqrt {\eta \Sigma^ {U S A M} \left(X _ {t}\right)} d W _ {t}, \tag {28}
+$$
+
+where
+
+$$
+f ^ {U S A M} (X _ {t}) := f (X _ {t}) + \frac {\rho}{2} \| \nabla f (X _ {t}) \| ^ {2} + \frac {\rho}{2 | \gamma |} \mathrm {t r} (V (X _ {t}))
+$$
+
+$$
+\Sigma^ {U S A M} (X _ {t}) := \Sigma_ {0, 0} (X _ {t}) + \rho \left(\Sigma_ {0, 1} (X _ {t}) + \Sigma_ {0, 1} ^ {\top} (X _ {t})\right). \tag {29}
+$$
+
+$$
+\Sigma_ {0, 0} (X _ {t}) := \mathbb {E} \left[ \left(\nabla f _ {\gamma} (X _ {t}) - \nabla f (X _ {t})\right) \left(\nabla f _ {\gamma} (X _ {t}) - \nabla f (X _ {t})\right) ^ {\top} \right]
+$$
+
+$$
+\Sigma_ {0, 1} (X _ {t}) := \mathbb {E} \left[ (\nabla f _ {\gamma} (X _ {t}) - \nabla f (X _ {t})) \cdot \left(\nabla^ {2} f _ {\gamma} (X _ {t}) \nabla f _ {\gamma} (X _ {t}) - \mathbb {E} [ \nabla^ {2} f _ {\gamma} (X _ {t}) \nabla f _ {\gamma} (X _ {t}) ]\right) ^ {\top} \right].
+$$
+
+Theorem B.2 (mini-batch USAM SDE, adapted from Theorem 3.2 of Compagnoni et al. (2023)). Under Assumptions 3.1 and B.1, let  $0 < \eta < 1$ ,  $T > 0$ , and  $N = \lfloor T / \eta \rfloor$ . Denote by  $\{x_{k}\}_{k = 0}^{N}$  the mini-batch USAM iterates in (6), and let  $\{X_{t}\}_{t\in [0,T]}$  be the solution of the SDE (28). Suppose:
+
+(i) The functions
+
+$$
+\nabla f ^ {\mathrm {U S A M}} = \nabla \left(f + \frac {\rho}{2} \| \nabla f \| ^ {2} + \frac {\rho}{2 | \gamma |} \operatorname {t r} (V)\right) a n d \sqrt {\Sigma^ {\mathrm {U S A M}}}
+$$
+
+are Lipschitz on  $\mathbb{R}^d$
+
+(ii) The mapping
+
+$$
+h _ {\gamma} (x) = - \nabla f _ {\gamma} (x + \rho \nabla f _ {\gamma} (x))
+$$
+
+satisfies, almost surely, the Lipschitz condition
+
+$$
+\| \nabla h _ {\gamma} (x) - \nabla h _ {\gamma} (y) \| \leq L _ {\gamma} \| x - y \|, \quad \forall x, y \in \mathbb {R} ^ {d},
+$$
+
+where  $L_{\gamma} > 0$  a.s. and  $\mathbb{E}[L_{\gamma}^{m}] < \infty$  for every  $m \geq 1$ .
+
+Then  $\{X_{t}:t\in [0,T]\}$  is an order-(1,1) weak approximation of  $\{x_{k}\}$ , namely: for each  $g\in G^2$  there exists a constant  $C > 0$ , independent of  $\eta, \rho$ , such that
+
+$$
+\max  _ {0 \leq k \leq N} \left| \mathbb {E} \left[ g (x _ {k}) \right] - \mathbb {E} \left[ g (X _ {k \eta}) \right] \right| \leq C (\eta + \rho^ {2}).
+$$
+
+Proof Sketch. Theorem B.2 follows by replacing the single-parameter Lemmas A.1, A.2 and A.5 in Compagnoni et al. (2023) with our two-parameter versions—Theorem A.2, Lemma A.4 and Lemma A.5—imposing the extra global Lipschitz conditions to guarantee existence and uniqueness of the strong solution, and using our Assumption 3.1 to expand the drift term. We therefore omit the routine algebraic details.
+
+# B.2 n-USAM
+
+For the n-USAM algorithm, we define the continuous-time approximation  $X_{t}$  as the solution to the following SDE:
+
+$$
+d X _ {t} = - \nabla f ^ {n - U S A M} \left(X _ {t}\right) d t + \sqrt {\eta \Sigma^ {n - U S A M} \left(X _ {t}\right)} d W _ {t}, \tag {30}
+$$
+
+where
+
+$$
+f ^ {n \cdot U S A M} (X _ {t}) := f (X _ {t}) + \frac {\rho}{2} \| \nabla f (X _ {t}) \| ^ {2}
+$$
+
+$$
+\Sigma^ {n - U S A M} \left(X _ {t}\right) := \Sigma_ {0, 0} \left(X _ {t}\right) + \rho \left(\Sigma_ {0, 1} \left(X _ {t}\right) + \Sigma_ {0, 1} ^ {\top} \left(X _ {t}\right)\right). \tag {31}
+$$
+
+$$
+\Sigma_ {0, 0} (X _ {t}) := \mathbb {E} \left[ \left(\nabla f _ {\gamma} (X _ {t}) - \nabla f (X _ {t})\right) \left(\nabla f _ {\gamma} (X _ {t}) - \nabla f (X _ {t})\right) ^ {\top} \right]
+$$
+
+$$
+\Sigma_ {0, 1} (X _ {t}) := \mathbb {E} \left[ (\nabla f _ {\gamma} (X _ {t}) - \nabla f (X _ {t})) \cdot \left((\nabla^ {2} f _ {\gamma} (X _ {t}) - \nabla^ {2} f (X _ {t})) \nabla f (X _ {t})\right) ^ {\top} \right].
+$$
+
+We begin by deriving, via the following lemma, a one-step error estimate for the n-USAM discrete algorithm, which will be used to prove the main approximation theorem.
+
+Lemma B.3 (One-step moment estimates for n-USAM up to  $\eta^1, \rho^1$ ). Under Assumptions 3.1 and B.1. Define
+
+$$
+\partial_ {i} f ^ {\mathrm {n - U S A M}} (x) := \partial_ {i} f (x) + \rho \sum_ {j} \partial_ {i j} ^ {2} f (x) \partial_ {j} f (x),
+$$
+
+Let  $\Delta (x)$  be the one-step increment defined in (24). Then:
+
+(i)  $\mathbb{E}\big[\Delta_{(i)}(x)\big] = -\partial_if^{\mathrm{n - USAM}}(x)\eta +O(\eta \rho^2).$
+
+(ii)  $\mathbb{E}\big[\Delta_{(i)}(x)\Delta_{(j)}(x)\big]$  =  $\eta^2\Big(\partial_if(x)\partial_jf(x)\quad +\quad \Sigma_{(ij)}^{\mathrm{n - USAM}}(x)\Big)\quad +$
+
+$$
+\eta^ {2} \rho \Big (\partial_ {i} f (x) \sum_ {l = 1} ^ {d} \partial_ {j l} ^ {2} f (x) \partial_ {l} f (x) + \partial_ {j} f (x) \sum_ {l = 1} ^ {d} \partial_ {i l} ^ {2} f (x) \partial_ {l} f (x) \Big) + O (\eta^ {2} \rho^ {2}).
+$$
+
+(iii)  $\mathbb{E}\Big[\prod_{j = 1}^{3}\big|\Delta_{(i_j)}(x)\big|\Big] = O(\eta^3).$
+
+Proof. Recall that the n-USAM update is
+
+$$
+x _ {k + 1} = x _ {k} - \eta \nabla f _ {\gamma_ {k}} \left(x _ {k} + \rho \nabla f (x _ {k})\right),
+$$
+
+so the one-step increment
+
+$$
+\Delta (x) = x _ {k + 1} - x _ {k} = \eta h (x, \gamma , \rho),
+$$
+
+where we define
+
+$$
+h (x, \gamma , \rho) := - \nabla f _ {\gamma} \big (x + \rho \nabla f (x) \big).
+$$
+
+By Taylor's theorem with integral remainder (Folland, 2005) we have, for each  $\gamma$ ,
+
+$$
+\nabla f _ {\gamma} (x + \rho \nabla f (x)) = \nabla f _ {\gamma} (x) + \rho \nabla^ {2} f _ {\gamma} (x) \nabla f (x) + R (x, \gamma , \rho),
+$$
+
+where
+
+$$
+R (x, \gamma , \rho) = \int_ {0} ^ {1} (1 - t) D ^ {3} f _ {\gamma} (x + t \rho \nabla f (x)) [ \rho \nabla f (x), \rho \nabla f (x) ] d t.
+$$
+
+Here  $D^3 f_\gamma(y)$  denotes the third-order tensor of partial derivatives of  $f_\gamma$  at  $y$ , and  $D^3 f_\gamma(y)[u, v]$  its bilinear action on vectors  $u, v$ .
+
+Because  $f_{\gamma} \in G^{3}$ , there exists a polynomially bounded function  $K(x) \in G$  such that
+
+$$
+\left\| D ^ {3} f _ {\gamma} (y) \right\| \leq K (x), \quad \forall y \text {w i t h} \| y - x \| \leq \rho \| \nabla f (x) \|.
+$$
+
+Hence
+
+$$
+\left\| R (x, \gamma , \rho) \right\| \leq \int_ {0} ^ {1} (1 - t) \left\| D ^ {3} f _ {\gamma} (x + t \rho \nabla f (x)) \right\| \left\| \rho \nabla f (x) \right\| ^ {2} d t \leq K (x) \frac {\rho^ {2}}{2} \| \nabla f (x) \| ^ {2} = O (\rho^ {2}),
+$$
+
+uniformly in  $\gamma$ . Accordingly, a Taylor expansion in  $\rho$  gives
+
+$$
+\nabla f _ {\gamma} (x + \rho \nabla f (x)) = \nabla f _ {\gamma} (x) + \rho \nabla^ {2} f _ {\gamma} (x) \nabla f (x) + O (\rho^ {2}).
+$$
+
+Hence
+
+$$
+h (x, \gamma , \rho) = h _ {0} (x, \gamma) + \rho h _ {1} (x, \gamma) + O (\rho^ {2}),
+$$
+
+with
+
+$$
+h _ {0} (x, \gamma) := - \nabla f _ {\gamma} (x), \quad h _ {1} (x, \gamma) := - \nabla^ {2} f _ {\gamma} (x) \nabla f (x).
+$$
+
+By Assumption 3.1 and Assumption B.1, each  $h_0,h_1\in G^2$  , and
+
+$$
+\mathbb {E} \left[ h _ {0} (x, \gamma) \right] = - \nabla f (x), \quad \mathbb {E} \left[ h _ {1} (x, \gamma) \right] = - \nabla^ {2} f (x) \nabla f (x).
+$$
+
+We may therefore apply Lemma A.5 with these  $h_0, h_1$ , which yields exactly the three moment expansions up to  $\eta^1, \rho^1$ .
+
+In Lemma B.3, we derived one-step moment estimates for the n-USAM discrete algorithm and, via Lemma A.4, for its corresponding SDE update (26). These estimates demonstrate that the first- and second-order moments satisfy the matching conditions of Theorem A.2. Together with the uniform moment bounds from Lemma D.2, we are now ready to establish the main weak-approximation theorem for n-USAM.
+
+Theorem B.4 (n-USAM SDE). Under Assumptions 3.1 and B.1, let  $0 < \eta < 1$ ,  $T > 0$ , and  $N = \lfloor T / \eta \rfloor$ . Denote by  $\{x_{k}\}_{k = 0}^{N}$  the n-USAM iterates in (7), and let  $\{X_{t}\}_{t\in [0,T]}$  be the solution of the SDE (30). Suppose:
+
+(i) The functions
+
+$$
+\nabla f ^ {\mathrm {n - U S A M}} = \nabla \left(f + \frac {\rho}{2} \| \nabla f \| ^ {2}\right) a n d \sqrt {\Sigma^ {\mathrm {n - U S A M}}}
+$$
+
+are Lipschitz on  $\mathbb{R}^d$
+
+(ii) The mapping
+
+$$
+h _ {\gamma} (x) = - \nabla f _ {\gamma} (x + \rho \nabla f (x))
+$$
+
+satisfies, almost surely, the Lipschitz condition
+
+$$
+\| \nabla h _ {\gamma} (x) - \nabla h _ {\gamma} (y) \| \leq L _ {\gamma} \| x - y \|, \quad \forall x, y \in \mathbb {R} ^ {d},
+$$
+
+where  $L_{\gamma} > 0$  a.s. and  $\mathbb{E}[L_{\gamma}^{m}] < \infty$  for every  $m \geq 1$ .
+
+Then  $\{X_{t}:t\in [0,T]\}$  is an order-(1,1) weak approximation of  $\{x_k\}$ , namely: for each  $g\in G^2$  there exists a constant  $C > 0$ , independent of  $\eta, \rho$ , such that
+
+$$
+\max  _ {0 \leq k \leq N} \left| \mathbb {E} \big [ g (x _ {k}) \big ] - \mathbb {E} \big [ g (X _ {k \eta}) \big ] \right| \leq C \left(\eta + \rho^ {2}\right).
+$$
+
+Proof. First, we verify that SDE (30) admits a unique strong solution. By assumption, both the drift and diffusion coefficients are globally Lipschitz, which in turn implies a linear-growth condition. Therefore, Theorem D.1 applies and yields the existence and uniqueness of a strong solution on  $[0,T]$ .
+
+Then, by Lemmas A.4, B.3, and D.2, all the conditions of Theorem A.2 are satisfied, and the proof is complete.
+
+Remark B.5. The Lipschitz conditions are to ensure that the SDE has a unique strong solution with uniformly bounded moments. It is possible to appropriately relax them if we allow weak solutions (Mil'shtein, 1986).
+
+# B.3 m-USAM
+
+For the m-USAM algorithm, we define the continuous-time approximation  $X_{t}$  as the solution to the following SDE:
+
+$$
+d X _ {t} = - \nabla \left(f \left(X _ {t}\right) + \frac {\rho}{2} \| \nabla f \left(X _ {t}\right) \| ^ {2} + \frac {\rho}{2 m} \operatorname {t r} \left(V \left(X _ {t}\right)\right)\right) d t + \sqrt {\frac {m \eta}{| \gamma |} \Sigma^ {m - U S A M} \left(X _ {t}\right)} d W _ {t}, \tag {32}
+$$
+
+where
+
+$$
+\Sigma^ {m - U S A M} \left(X _ {t}\right) := \Sigma_ {0, 0} \left(X _ {t}\right) + \rho \left(\Sigma_ {0, 1} \left(X _ {t}\right) + \Sigma_ {0, 1} \left(X _ {t}\right) ^ {\top}\right), \tag {33}
+$$
+
+$$
+\Sigma_ {0, 0} (X _ {t}) := \mathbb {E} \left[ \left(\nabla f _ {\mathcal {I}} (X _ {t}) - \nabla f (X _ {t})\right) \left(\nabla f _ {\mathcal {I}} (X _ {t}) - \nabla f (X _ {t})\right) ^ {\top} \right],
+$$
+
+$$
+\Sigma_ {0, 1} (X _ {t}) := \mathbb {E} \left[ (\nabla f _ {\mathcal {I}} (X _ {t}) - \nabla f (X _ {t})) \cdot \left(\nabla^ {2} f _ {\mathcal {I}} (X _ {t}) \nabla f _ {\mathcal {I}} (X _ {t}) - \mathbb {E} [ \nabla^ {2} f _ {\mathcal {I}} (X _ {t}) \nabla f _ {\mathcal {I}} (X _ {t}) ]\right) ^ {\top} \right].
+$$
+
+We begin by deriving, via the following lemma, a one-step error estimate for the m-USAM discrete algorithm, which will be used to prove the main approximation theorem.
+
+Lemma B.6 (One-step moment estimates for m-USAM up to  $\eta^1, \rho^1$ ). Under Assumptions 3.1 and B.1. Define
+
+$$
+\partial_ {i} f ^ {m - U S A M} (x) := \partial_ {i} f (x) + \rho \mathbb {E} \left[ \sum_ {j = 1} ^ {d} \partial_ {i j} ^ {2} f _ {\mathcal {I}} (x) \partial_ {j} f _ {\mathcal {I}} (x) \right].
+$$
+
+Let  $\Delta (x)$  be the one-step increment defined in (24). Then:
+
+(i)  $\mathbb{E}\big[\Delta_{(i)}(x)\big] = -\partial_if^{\mathrm{m - USAM}}(x)\eta +O(\eta \rho^2).$
+
+(ii)  $\mathbb{E}\big[\Delta_{(i)}(x)\Delta_{(j)}(x)\big]$  =  $\eta^2\Big(\partial_if(x)\partial_jf(x)\quad +\quad \Sigma_{(ij)}^{\mathrm{m - USAM}}(x)\Big)\quad +$
+
+$$
+\eta^ {2} \rho \Big (\partial_ {i} f (x) \sum_ {l = 1} ^ {d} \partial_ {j l} ^ {2} f _ {\mathcal {I}} (x) \partial_ {l} f _ {\mathcal {I}} (x) + \partial_ {j} f (x) \sum_ {l = 1} ^ {d} \partial_ {i l} ^ {2} f _ {\mathcal {I}} (x) \partial_ {l} f _ {\mathcal {I}} (x) \Big) + O (\eta^ {2} \rho^ {2}).
+$$
+
+$$
+(i i i) \mathbb {E} \left[ \prod_ {j = 1} ^ {3} \left| \Delta_ {(i _ {j})} (x) \right| \right] = O (\eta^ {3}).
+$$
+
+Proof. Recall that the m-USAM update is
+
+$$
+x _ {k + 1} = x _ {k} - \frac {\eta m}{| \gamma |} \sum_ {\mathcal {I} _ {j} \subset \gamma_ {k}, | \mathcal {I} _ {j} | = m} \nabla f _ {\mathcal {I} _ {j}} \left(x _ {k} + \rho \nabla f _ {\mathcal {I} _ {j}} \left(x _ {k}\right)\right)
+$$
+
+so the one-step increment
+
+$$
+\Delta (x) = x _ {k + 1} - x _ {k} = \eta h (x, \gamma , \rho),
+$$
+
+where we define
+
+$$
+h(x,\gamma ,\rho):= -\frac{m}{|\gamma|}\sum_{\mathcal{I}_{j}\subset \gamma_{k},|\mathcal{I}_{j}| = m}\nabla f_{\mathcal{I}_{j}}\big(x_{k} + \rho \nabla f_{\mathcal{I}_{j}}(x_{k})\big).
+$$
+
+By Taylor's theorem with integral remainder (Folland, 2005) we have, for each  $\gamma$ ,
+
+$$
+\sum_{\substack{\mathcal{I}_{j}\subset \gamma_{k},\\ |\mathcal{I}_{j}| = m}}\nabla f_{\mathcal{I}_{j}}\big(x_{k} + \rho \nabla f_{\mathcal{I}_{j}}(x_{k})\big) = \sum_{\substack{\mathcal{I}_{j}\subset \gamma_{k},\\ |\mathcal{I}_{j}| = m}}\nabla f_{\mathcal{I}_{j}}(x_{k}) + \rho \nabla^{2}f_{\mathcal{I}_{j}}(x_{k})\nabla f_{\mathcal{I}_{j}}(x_{k}) + R(x,\gamma ,\rho)
+$$
+
+where
+
+$$
+R (x, \gamma , \rho) = \int_ {0} ^ {1} (1 - t) D ^ {3} f _ {\mathcal {I} _ {j}} \left(x + t \rho \nabla f _ {\mathcal {I} _ {j}} (x)\right) \left[ \rho \nabla f _ {\mathcal {I} _ {j}} (x), \rho \nabla f _ {\mathcal {I} _ {j}} (x) \right] d t.
+$$
+
+Here  $D^3 f_{\mathcal{I}_j}(y)$  denotes the third-order tensor of partial derivatives of  $f_{\mathcal{I}_j}$  at  $y$ , and  $D^3 f_{\mathcal{I}_j}(y)[u,v]$  its bilinear action on vectors  $u,v$ .
+
+Because  $f_{\mathcal{I}_i} \in G^3$ , there exists a polynomially bounded function  $K(x) \in G$  such that
+
+$$
+\left\| D ^ {3} f _ {\mathcal {I} _ {j}} (y) \right\| \leq K (x), \quad \forall y \text {w i t h} \| y - x \| \leq \rho \| \nabla f _ {\mathcal {I} _ {j}} (x) \|.
+$$
+
+Hence
+
+$$
+\begin{array}{l} \left\| R (x, \gamma , \rho) \right\| \leq \int_ {0} ^ {1} (1 - t) \left\| D ^ {3} f _ {\mathcal {I} _ {j}} (x + t \rho \nabla f _ {\mathcal {I} _ {j}} (x)) \right\| \left\| \rho \nabla f _ {\mathcal {I} _ {j}} (x) \right\| ^ {2} d t \leq K (x) \frac {\rho^ {2}}{2} \| \nabla f _ {\mathcal {I} _ {j}} (x) \| ^ {2} \\ = O \left(\rho^ {2}\right), \\ \end{array}
+$$
+
+uniformly in  $\gamma$ . Accordingly, a Taylor expansion in  $\rho$  gives
+
+$$
+\nabla f _ {\gamma} (x + \rho \nabla f (x)) = \nabla f _ {\gamma} (x) + \rho \nabla^ {2} f _ {\gamma} (x) \nabla f (x) + O (\rho^ {2}).
+$$
+
+Hence
+
+$$
+h (x, \gamma , \rho) = h _ {0} (x, \gamma) + \rho h _ {1} (x, \gamma) + O (\rho^ {2}),
+$$
+
+with
+
+$$
+h_{0}(x,\gamma):= -\frac{m}{|\gamma|}\sum_{\substack{\mathcal{I}_{j}\subset \gamma_{k},\\ |\mathcal{I}_{j}| = m}}\nabla f_{\mathcal{I}_{j}}(x_{k}),\qquad h_{1}(x,\gamma):= -\frac{m}{|\gamma|}\sum_{\substack{\mathcal{I}_{j}\subset \gamma_{k},\\ |\mathcal{I}_{j}| = m}}\nabla^{2}f_{\mathcal{I}_{j}}(x_{k})\nabla f_{\mathcal{I}_{j}}(x_{k}).
+$$
+
+By Assumption 3.1 and Assumption B.1, each  $h_0,h_1\in G^2$  , and
+
+$$
+\mathbb {E} \left[ h _ {0} (x, \gamma) \right] = - \nabla f (x), \quad \mathbb {E} \left[ h _ {1} (x, \gamma) \right] = - \mathbb {E} \left[ \nabla^ {2} f _ {\mathcal {I} _ {j}} (x) \nabla f _ {\mathcal {I} _ {j}} (x) \right].
+$$
+
+We may therefore apply Lemma A.5 with these  $h_0, h_1$ , which yields exactly the three moment expansions up to  $\eta^1, \rho^1$ .
+
+In Lemma B.6, we derived one-step moment estimates for the m-USAM discrete algorithm and, via Lemma A.4, for its corresponding SDE update (27). These estimates demonstrate that the first- and second-order moments satisfy the matching conditions of Theorem A.2. Together with the uniform moment bounds from Lemma D.2, we are now ready to establish the main weak-approximation theorem for m-USAM.
+
+Theorem B.7 (m-USAM SDE). Under Assumptions 3.1 and B.1, let  $0 < \eta < 1$ ,  $T > 0$ , and  $N = \lfloor T / \eta \rfloor$ . Denote by  $\{x_{k}\}_{k = 0}^{N}$  the m-USAM iterates in (8), and let  $\{X_{t}\}_{t\in [0,T]}$  be the solution of the SDE (32). Suppose:
+
+(i) The functions
+
+$$
+\nabla f ^ {\mathrm {m - U S A M}} = \nabla \left(f + \frac {\rho}{2} \| \nabla f \| ^ {2} + \frac {\rho}{2 m} \operatorname {t r} (V)\right) a n d \sqrt {\Sigma^ {\mathrm {m - U S A M}}}
+$$
+
+are Lipschitz on  $\mathbb{R}^d$
+
+(ii) The mapping
+
+$$
+h _ {\gamma} (x) = - \frac {m}{| \gamma |} \sum_ {\mathcal {I} _ {j} \subset \gamma , | \mathcal {I} _ {j} | = m} \nabla f _ {\mathcal {I} _ {j}} \left(x + \rho \nabla f _ {\mathcal {I} _ {j}} (x)\right)
+$$
+
+satisfies, almost surely, the Lipschitz condition
+
+$$
+\left\| \nabla h _ {\gamma} (x) - \nabla h _ {\gamma} (y) \right\| \leq L _ {\gamma} \| x - y \|, \quad \forall x, y \in \mathbb {R} ^ {d},
+$$
+
+where  $L_{\gamma} > 0$  a.s. and  $\mathbb{E}[L_{\gamma}^{m}] < \infty$  for every  $m \geq 1$ .
+
+Then  $\{X_{t}:t\in [0,T]\}$  is an order-(1,1) weak approximation of  $\{x_k\}$ , namely: for each  $g\in G^2$  there exists a constant  $C > 0$ , independent of  $\eta, \rho$ , such that
+
+$$
+\max  _ {0 \leq k \leq N} \left| \mathbb {E} \big [ g (x _ {k}) \big ] - \mathbb {E} \big [ g (X _ {k \eta}) \big ] \right| \leq C \left(\eta + \rho^ {2}\right).
+$$
+
+Proof. First, we verify that SDE (32) admits a unique strong solution. By assumption, both the drift and diffusion coefficients are globally Lipschitz, which in turn implies a linear-growth condition. Therefore, Theorem D.1 applies and yields the existence and uniqueness of a strong solution on  $[0,T]$ .
+
+Then, by Lemmas A.4, B.6, and D.2, all the conditions of Theorem A.2 are satisfied, and the proof is complete.
+
+Remark B.8. The Lipschitz conditions are to ensure that the SDE has a unique strong solution with uniformly bounded moments. It is possible to appropriately relax them if we allow weak solutions (Mil'shtein, 1986).
+
+# C SDE approximation for SAM variants
+
+Recall that the update rules of SAM variants are defined by:
+
+$$
+\text {m i n i - b a t c h S A M :} x _ {k + 1} = x _ {k} - \eta \nabla f _ {\gamma_ {k}} \left(x _ {k} + \rho \frac {\nabla f _ {\gamma_ {k}} \left(x _ {k}\right)}{\| \nabla f _ {\gamma_ {k}} \left(x _ {k}\right) \|}\right) \tag {34}
+$$
+
+$$
+\text {n - S A M :} x _ {k + 1} = x _ {k} - \eta \nabla f _ {\gamma_ {k}} \left(x _ {k} + \rho \frac {\nabla f \left(x _ {k}\right)}{\| \nabla f \left(x _ {k}\right) \|}\right) \tag {35}
+$$
+
+$$
+\text {m - S A M :} x _ {k + 1} = x _ {k} - \frac {\eta m}{| \gamma |} \sum_ {\mathcal {I} _ {j} \subset \gamma_ {k}, \left| \mathcal {I} _ {j} \right| = m} \nabla f _ {\mathcal {I} _ {j}} \left(x _ {k} + \rho \frac {\nabla f _ {\mathcal {I} _ {j}} \left(x _ {k}\right)}{\left\| \nabla f _ {\mathcal {I} _ {j}} \left(x _ {k}\right) \right\|}\right) \tag {36}
+$$
+
+# C.1 Mini-batch SAM
+
+For the mini-batch SAM algorithm (34), we define the continuous-time approximation  $X_{t}$  as the solution to the following SDE:
+
+$$
+d X _ {t} = - \nabla f ^ {S A M} \left(X _ {t}\right) d t + \sqrt {\eta \Sigma^ {S A M} \left(X _ {t}\right)} d W _ {t}, \tag {37}
+$$
+
+where
+
+$$
+f ^ {S A M} (X _ {t}) := f (X _ {t}) + \rho \mathbb {E} \| \nabla f _ {\gamma} (X _ {t}) \|
+$$
+
+$$
+\Sigma^ {S A M} (X _ {t}) := \Sigma_ {0, 0} (X _ {t}) + \rho \left(\Sigma_ {0, 1} (X _ {t}) + \Sigma_ {0, 1} ^ {\top} (X _ {t})\right). \tag {38}
+$$
+
+$$
+\Sigma_ {0, 0} (X _ {t}) := \mathbb {E} \left[ \left(\nabla f _ {\gamma} (X _ {t}) - \nabla f (X _ {t})\right) \left(\nabla f _ {\gamma} (X _ {t}) - \nabla f (X _ {t})\right) ^ {\top} \right]
+$$
+
+$$
+\Sigma_ {0, 1} (X _ {t}) := \mathbb {E} \big [ (\nabla f _ {\gamma} (X _ {t}) - \nabla f (X _ {t})) \cdot \big (\frac {\nabla^ {2} f _ {\gamma} (X _ {t}) \nabla f _ {\gamma} (X _ {t})}{\| \nabla f _ {\gamma} (X _ {t}) \|} - \mathbb {E} \big [ \frac {\nabla^ {2} f _ {\gamma} (X _ {t}) \nabla f _ {\gamma} (X _ {t})}{\| \nabla f _ {\gamma} (X _ {t}) \|} \big ] \big) ^ {\top} \big ].
+$$
+
+Remark C.1 (On normalization at critical points). Note that SAM is ill-defined when the gradient is zero. To solve this, we may replace the denominator by  $\| \cdot \|_{\varepsilon} = \sqrt{\|\cdot\|^{2} + \varepsilon^{2}}$  with a fixed  $\varepsilon > 0$  which is also a common implementation in practice. Under the standing assumption that  $\nabla f$  is  $L$ -Lipschitz, the resulting coefficients are globally  $O(L / \varepsilon)$ -Lipschitz and  $C^1$ . All local Taylor or weak-approximation arguments and moment bounds in Appendix C continue to hold with constants depending on  $\varepsilon$  but independent of  $\eta, \rho$ . The stated orders in  $\eta, \rho$  are unaffected.
+
+Theorem C.2 (mini-batch SAM SDE, adapted from Theorem 3.5 of Compagnoni et al. (2023)). Under Assumptions 3.1 and B.1, let  $0 < \eta < 1$ ,  $T > 0$ , and  $N = \lfloor T / \eta \rfloor$ . Denote by  $\{x_{k}\}_{k = 0}^{N}$  the mini-batch SAM iterates in (4), and let  $\{X_{t}\}_{t\in [0,T]}$  be the solution of the SDE (37). Suppose:
+
+(i) The functions
+
+$$
+\nabla f ^ {\text {S A M}} = \nabla \left(f + \rho \mathbb {E} \| \nabla f _ {\gamma} \|\right) a n d \sqrt {\Sigma^ {\text {S A M}}}
+$$
+
+are Lipschitz on  $\mathbb{R}^d$
+
+(ii) The mapping
+
+$$
+h _ {\gamma} (x) = - \nabla f _ {\gamma} \left(x + \rho \frac {\nabla f _ {\gamma} (x)}{\| \nabla f _ {\gamma} (x) \|}\right)
+$$
+
+satisfies, almost surely, the Lipschitz condition
+
+$$
+\| \nabla h _ {\gamma} (x) - \nabla h _ {\gamma} (y) \| \leq L _ {\gamma} \| x - y \|, \quad \forall x, y \in \mathbb {R} ^ {d},
+$$
+
+where  $L_{\gamma} > 0$  a.s. and  $\mathbb{E}[L_{\gamma}^{m}] < \infty$  for every  $m \geq 1$ .
+
+Then  $\{X_{t}:t\in [0,T]\}$  is an order-(1,1) weak approximation of  $\{x_{k}\}$ , namely: for each  $g\in G^2$  there exists a constant  $C > 0$ , independent of  $\eta$ , such that
+
+$$
+\max  _ {0 \leq k \leq N} \left| \mathbb {E} \big [ g (x _ {k}) \big ] - \mathbb {E} \big [ g (X _ {k \eta}) \big ] \right| \leq C \left(\eta + \rho^ {2}\right).
+$$
+
+Proof Sketch. Theorem C.2 follows by replacing the single-parameter Lemmas A.1, A.2 and A.14 in Compagnoni et al. (2023) with our two-parameter versions—Theorem A.2, Lemma A.4 and Lemma A.5—imposing the extra global Lipschitz conditions to guarantee existence and uniqueness of the strong solution. We therefore omit the routine algebraic details.  $\square$
+
+# C.2 n-SAM
+
+For the n-SAM algorithm, we define the continuous-time approximation  $X_{t}$  as the solution to the following SDE:
+
+$$
+d X _ {t} = - \nabla f ^ {n - S A M} \left(X _ {t}\right) d t + \sqrt {\eta \Sigma^ {n - S A M} \left(X _ {t}\right)} d W _ {t}, \tag {39}
+$$
+
+where
+
+$$
+f ^ {n \cdot S A M} (X _ {t}) := f (X _ {t}) + \rho \| \nabla f (X _ {t}) \|
+$$
+
+$$
+\Sigma^ {n - S A M} \left(X _ {t}\right) := \Sigma_ {0, 0} \left(X _ {t}\right) + \rho \left(\Sigma_ {0, 1} \left(X _ {t}\right) + \Sigma_ {0, 1} ^ {\top} \left(X _ {t}\right)\right). \tag {40}
+$$
+
+$$
+\Sigma_ {0, 0} (X _ {t}) := \mathbb {E} \left[ \left(\nabla f _ {\gamma} (X _ {t}) - \nabla f (X _ {t})\right) \left(\nabla f _ {\gamma} (X _ {t}) - \nabla f (X _ {t})\right) ^ {\top} \right]
+$$
+
+$$
+\Sigma_ {0, 1} (X _ {t}) := \mathbb {E} \left[ (\nabla f _ {\gamma} (X _ {t}) - \nabla f (X _ {t})) \cdot \left((\nabla^ {2} f _ {\gamma} (X _ {t}) - \nabla^ {2} f (X _ {t})) \frac {\nabla f (X _ {t})}{\| \nabla f (X _ {t}) \|}\right) ^ {\top} \right].
+$$
+
+We begin by deriving, via the following lemma, a one-step error estimate for the n-SAM discrete algorithm, which will be used to prove the main approximation theorem.
+
+Lemma C.3 (One-step moment estimates for n-SAM up to  $\eta^1, \rho^1$ ). Under Assumptions 3.1 and B.1, define
+
+$$
+\partial_ {i} f ^ {\mathrm {n - S A M}} (x) := \partial_ {i} f (x) + \rho \sum_ {j = 1} ^ {d} \partial_ {i j} ^ {2} f (x) \frac {\partial_ {j} f (x)}{\| \nabla f (x) \|}.
+$$
+
+Let  $\Delta (x)$  be the one-step increment defined in (24). Then:
+
+(i)  $\mathbb{E}\big[\Delta_{(i)}(x)\big] = -\partial_if^{\mathrm{n - SA}}(x)\eta +O(\eta \rho^2).$
+
+(ii)  $\mathbb{E}\big[\Delta_{(i)}(x)\Delta_{(j)}(x)\big]$  =  $\eta^2\Big(\partial_if(x)\partial_jf(x) + \Sigma_{(ij)}^{\mathrm{n - SAM}}(x)\Big)$
+
+$$
+\eta^ {2} \rho \left(\partial_ {i} f (x) \sum_ {l = 1} ^ {d} \partial_ {j l} ^ {2} f (x) \frac {\partial_ {l} f (x)}{\| \nabla f (x) \|} + \partial_ {j} f (x) \sum_ {l = 1} ^ {d} \partial_ {i l} ^ {2} f (x) \frac {\partial_ {l} f (x)}{\| \nabla f (x) \|}\right) + O (\eta^ {2} \rho^ {2}).
+$$
+
+(iii)  $\mathbb{E}\left[\prod_{j = 1}^{3}\big|\Delta_{(i_j)}(x)\big|\right] = O(\eta^3).$
+
+Proof. The n-SAM update is
+
+$$
+x _ {k + 1} = x _ {k} - \eta \nabla f _ {\gamma_ {k}} \left(x _ {k} + \rho \frac {\nabla f (x _ {k})}{\| \nabla f (x _ {k}) \|}\right),
+$$
+
+so
+
+$$
+\Delta (x) = x _ {k + 1} - x _ {k} = \eta h (x, \gamma , \rho),
+$$
+
+with
+
+$$
+h (x, \gamma , \rho) := - \nabla f _ {\gamma} \big (x + \rho u (x) \big), \qquad u (x) := \frac {\nabla f (x)}{\| \nabla f (x) \|}.
+$$
+
+By Taylor's theorem with integral remainder (Folland, 2005), for each  $\gamma$  and writing  $u(x) = \nabla f(x) / \| \nabla f(x)\|$ , we have
+
+$$
+\nabla f _ {\gamma} \big (x + \rho u (x) \big) = \nabla f _ {\gamma} (x) + \rho \nabla^ {2} f _ {\gamma} (x) u (x) + R (x, \gamma , \rho),
+$$
+
+where
+
+$$
+R (x, \gamma , \rho) = \int_ {0} ^ {1} (1 - t) D ^ {3} f _ {\gamma} (x + t \rho u (x)) [ \rho u (x), \rho u (x) ] d t.
+$$
+
+Here  $D^3 f_\gamma(y)$  denotes the third-order tensor of partial derivatives of  $f_\gamma$  at  $y$ , and  $D^3 f_\gamma(y)[v, w]$  its bilinear action on vectors  $v, w$ .
+
+Because  $f_{\gamma} \in G^{3}$ , there exists a polynomially bounded function  $K(x) \in G$  such that
+
+$$
+\left\| D ^ {3} f _ {\gamma} (y) \right\| \leq K (x), \quad \forall y \text {w i t h} \| y - x \| \leq \rho \| u (x) \|.
+$$
+
+Hence
+
+$$
+\| R (x, \gamma , \rho) \| \leq \int_ {0} ^ {1} (1 - t) K (x) \| \rho u (x) \| ^ {2} d t = O (\rho^ {2}),
+$$
+
+uniformly in  $\gamma$ . Hence
+
+$$
+h (x, \gamma , \rho) = h _ {0} (x, \gamma) + \rho h _ {1} (x, \gamma) + O (\rho^ {2}),
+$$
+
+with
+
+$$
+h _ {0} (x, \gamma) := - \nabla f _ {\gamma} (x), \qquad h _ {1} (x, \gamma) := - \nabla^ {2} f _ {\gamma} (x) \frac {\nabla f (x)}{\| \nabla f (x) \|}.
+$$
+
+By Assumptions 3.1 and B.1, each  $h_0,h_1\in G^2$  and
+
+$$
+\mathbb {E} [ h _ {0} (x, \gamma) ] = - \nabla f (x), \qquad \mathbb {E} [ h _ {1} (x, \gamma) ] = - \nabla^ {2} f (x) \frac {\nabla f (x)}{\| \nabla f (x) \|}.
+$$
+
+The same application of Lemma A.5 then yields the moment expansions (i)-(iii) up to order  $\eta^1$ ,  $\rho^1$  as stated.
+
+In Lemma C.3, we derived one-step moment estimates for the n-SAM discrete algorithm and, via Lemma A.4, for its corresponding SDE update (35). These estimates demonstrate that the first- and second-order moments satisfy the matching conditions of Theorem A.2. Together with the uniform moment bounds from Lemma D.2, we are now ready to establish the main weak-approximation theorem for n-SAM.
+
+Theorem C.4 (n-SAM SDE). Under Assumptions 3.1 and B.1, let  $0 < \eta < 1$ ,  $T > 0$ , and  $N = \lfloor T / \eta \rfloor$ . Denote by  $\{x_{k}\}_{k = 0}^{N}$  the  $n$ -SAM iterates in (3), and let  $\{X_{t}\}_{t\in [0,T]}$  be the solution of the SDE (39). Suppose:
+
+(i) The functions
+
+$$
+\nabla f ^ {\mathrm {n - S A M}} = \nabla \left(f + \rho \| \nabla f \|\right) a n d \sqrt {\Sigma^ {\mathrm {n - S A M}}}
+$$
+
+are Lipschitz on  $\mathbb{R}^d$
+
+(ii) The mapping
+
+$$
+h _ {\gamma} (x) = - \nabla f _ {\gamma} \left(x + \rho \frac {\nabla f (x)}{\| \nabla f (x) \|}\right)
+$$
+
+satisfies, almost surely, the Lipschitz condition
+
+$$
+\| \nabla h _ {\gamma} (x) - \nabla h _ {\gamma} (y) \| \leq L _ {\gamma} \| x - y \|, \quad \forall x, y \in \mathbb {R} ^ {d},
+$$
+
+where  $L_{\gamma} > 0$  a.s. and  $\mathbb{E}[L_{\gamma}^{m}] < \infty$  for every  $m \geq 1$ .
+
+Then  $\{X_{t}:t\in [0,T]\}$  is an order-(1,1) weak approximation of  $\{x_{k}\}$ , namely: for each  $g\in G^2$  there exists a constant  $C > 0$ , independent of  $\eta$ , such that
+
+$$
+\max  _ {0 \leq k \leq N} \left| \mathbb {E} \big [ g (x _ {k}) \big ] - \mathbb {E} \big [ g (X _ {k \eta}) \big ] \right| \leq C \left(\eta + \rho^ {2}\right).
+$$
+
+Proof. First, we verify that SDE (39) admits a unique strong solution. By assumption, both the drift and diffusion coefficients are globally Lipschitz, which in turn implies a linear-growth condition. Therefore, Theorem D.1 applies and yields the existence and uniqueness of a strong solution on  $[0,T]$ .
+
+Then, by Lemmas A.4, C.3, and D.2, all the conditions of Theorem A.2 are satisfied, and the proof is complete.
+
+Remark C.5. The Lipschitz conditions are to ensure that the SDE has a unique strong solution with uniformly bounded moments. It is possible to appropriately relax them if we allow weak solutions (Mil'shtein, 1986).
+
+# C.3 m-SAM
+
+For the m-SAM algorithm, we define the continuous-time approximation  $X_{t}$  as the solution to the following SDE:
+
+$$
+d X _ {t} = - \nabla \left(f \left(X _ {t}\right) + \frac {\rho}{m} \mathbb {E} \| \sum_ {\substack {i \in \mathcal {I}, \\ | \mathcal {I} | = m}} \nabla f _ {i} \left(X _ {t}\right) \|\right) d t \quad + \sqrt {\frac {m \eta}{| \gamma |}} \left(\Sigma^ {m - S A M} \left(X _ {t}\right)\right) ^ {\frac {1}{2}} d W _ {t}, \tag{41}
+$$
+
+where
+
+$$
+\Sigma^ {m - S A M} \left(X _ {t}\right) := \Sigma_ {0, 0} \left(X _ {t}\right) + \rho \left(\Sigma_ {0, 1} \left(X _ {t}\right) + \Sigma_ {0, 1} \left(X _ {t}\right) ^ {\top}\right), \tag {42}
+$$
+
+$$
+\Sigma_ {0, 0} (X _ {t}) := \mathbb {E} \left[ \left(\nabla f _ {\mathcal {I}} (X _ {t}) - \nabla f (X _ {t})\right) \left(\nabla f _ {\mathcal {I}} (X _ {t}) - \nabla f (X _ {t})\right) ^ {\top} \right],
+$$
+
+$$
+\Sigma_ {0, 1} (X _ {t}) := \mathbb {E} \big [ (\nabla f _ {\mathcal {I}} (X _ {t}) - \nabla f (X _ {t})) \cdot \big (\nabla^ {2} f _ {\mathcal {I}} (X _ {t}) \frac {\nabla f _ {\mathcal {I}} (X _ {t})}{\| \nabla f _ {\mathcal {I}} (X _ {t}) \|} - \mathbb {E} [ \nabla^ {2} f _ {\mathcal {I}} (X _ {t}) \frac {\nabla f _ {\mathcal {I}} (X _ {t})}{\| \nabla f _ {\mathcal {I}} (X _ {t}) \|} ] \big) ^ {\top} \big ].
+$$
+
+We begin by deriving, via the following lemma, a one-step error estimate for the m-sam discrete algorithm, which will be used to prove the main approximation theorem.
+
+Lemma C.6 (One-step moment estimates for m-SAM up to  $\eta^1, \rho^1$ ). Under Assumptions 3.1 and B.1, define
+
+$$
+\partial_ {i} f ^ {m \cdot S A M} (x) := \partial_ {i} f (x) + \rho \mathbb {E} \left[ \sum_ {j = 1} ^ {d} \partial_ {i j} ^ {2} f _ {\mathcal {I}} (x) \frac {\partial_ {j} f _ {\mathcal {I}} (x)}{\| \nabla f _ {\mathcal {I}} (x) \|} \right].
+$$
+
+Let  $\Delta (x)$  be the one-step increment defined in (24). Then:
+
+(i)  $\mathbb{E}\big[\Delta_{(i)}(x)\big] = -\partial_if^{m\cdot SAM}(x)\eta +O(\eta \rho^2).$
+
+(ii)  $\mathbb{E}\big[\Delta_{(i)}(x)\Delta_{(j)}(x)\big]$  =  $\eta^2\Big(\partial_if(x)\partial_jf(x)\quad +\quad \Sigma_{(ij)}^{m - SAM}(x)\Big)$
+
+$$
+\eta^ {2} \rho (\partial_ {i} f (x) \quad \mathbb {E} \Big [ \sum_ {l = 1} ^ {d} \partial_ {j l} ^ {2} f _ {\mathcal {I}} (x) \frac {\partial_ {l} f _ {\mathcal {I}} (x)}{\| \nabla f _ {\mathcal {I}} (x) \|} \Big ] + \partial_ {j} f (x) \quad \mathbb {E} \Big [ \sum_ {l = 1} ^ {d} \partial_ {i l} ^ {2} f _ {\mathcal {I}} (x) \frac {\partial_ {l} f _ {\mathcal {I}} (x)}{\| \nabla f _ {\mathcal {I}} (x) \|} \Big ]) +
+$$
+
+$$
+O (\eta^ {2} \rho^ {2}).
+$$
+
+(iii)  $\mathbb{E}\Big[\prod_{j = 1}^{3}\big|\Delta_{(i_j)}(x)\big|\Big] = O(\eta^3).$
+
+Proof. Recall that the m-SAM update is
+
+$$
+x _ {k + 1} = x _ {k} - \frac {\eta m}{| \gamma |} \sum_ {\mathcal {I} _ {j} \subset \gamma_ {k}, | \mathcal {I} _ {j} | = m} \nabla f _ {\mathcal {I} _ {j}} \left(x _ {k} + \rho \frac {\nabla f _ {\mathcal {I} _ {j}} (x _ {k})}{\| \nabla f _ {\mathcal {I} _ {j}} (x _ {k}) \|}\right),
+$$
+
+so the one-step increment
+
+$$
+\Delta (x) = x _ {k + 1} - x _ {k} = \eta h (x, \gamma , \rho),
+$$
+
+where
+
+$$
+h (x, \gamma , \rho) := - \frac {m}{| \gamma |} \sum_ {\mathcal {I} _ {j} \subset \gamma_ {k}, | \mathcal {I} _ {j} | = m} \nabla f _ {\mathcal {I} _ {j}} \Big (x + \rho \frac {\nabla f _ {\mathcal {I} _ {j}} (x)}{\| \nabla f _ {\mathcal {I} _ {j}} (x) \|} \Big).
+$$
+
+By Taylor's theorem with integral remainder (Folland, 2005), for each subset index  $\mathcal{I}_j$ ,
+
+$$
+\nabla f _ {\mathcal {I} _ {j}} \Big (x + \rho \frac {\nabla f _ {\mathcal {I} _ {j}} (x)}{\| \nabla f _ {\mathcal {I} _ {j}} (x) \|} \Big) = \nabla f _ {\mathcal {I} _ {j}} (x) + \rho \nabla^ {2} f _ {\mathcal {I} _ {j}} (x) \frac {\nabla f _ {\mathcal {I} _ {j}} (x)}{\| \nabla f _ {\mathcal {I} _ {j}} (x) \|} + R (x, \gamma , \rho),
+$$
+
+where
+
+$$
+R (x, \gamma , \rho) = \int_ {0} ^ {1} (1 - t) D ^ {3} f _ {\mathcal {I} _ {j}} \Bigg (x + t \rho \frac {\nabla f _ {\mathcal {I} _ {j}} (x)}{\| \nabla f _ {\mathcal {I} _ {j}} (x) \|} \Bigg) \big [ \rho \frac {\nabla f _ {\mathcal {I} _ {j}} (x)}{\| \nabla f _ {\mathcal {I} _ {j}} (x) \|}, \rho \frac {\nabla f _ {\mathcal {I} _ {j}} (x)}{\| \nabla f _ {\mathcal {I} _ {j}} (x) \|} \big ] d t.
+$$
+
+Here  $D^3 f_{\mathcal{I}_j}(y)$  is the third-order derivative tensor of  $f_{\mathcal{I}_j}$  at  $y$ , and  $D^3 f_{\mathcal{I}_j}(y)[u,v]$  its action on  $(u,v)$ .
+
+Since  $f_{\mathcal{L}_j} \in G^3$ , there is  $K(x) \in G$  polynomially bounded so that
+
+$$
+\left\| D ^ {3} f _ {\mathcal {I} _ {j}} (y) \right\| \leq K (x) \quad \text {w h e n e v e r} \quad \| y - x \| \leq \rho \left\| \frac {\nabla f _ {\mathcal {I} _ {j}} (x)}{\| \nabla f _ {\mathcal {I} _ {j}} (x) \|} \right\|.
+$$
+
+Hence
+
+$$
+\| R (x, \gamma , \rho) \| \leq \int_ {0} ^ {1} (1 - t) K (x) \| \rho \frac {\nabla f _ {\mathcal {I} _ {j}} (x)}{\| \nabla f _ {\mathcal {I} _ {j}} (x) \|} \| ^ {2} d t = \frac {1}{2} K (x) \rho^ {2} = O (\rho^ {2}),
+$$
+
+uniformly in  $\gamma$ . Thus a Taylor expansion in  $\rho$  gives
+
+$$
+h (x, \gamma , \rho) = h _ {0} (x, \gamma) + \rho h _ {1} (x, \gamma) + O (\rho^ {2}),
+$$
+
+with
+
+$$
+h _ {0} (x, \gamma) := - \frac {m}{| \gamma |} \sum_ {\mathcal {I} _ {j} \subset \gamma_ {k}, | \mathcal {I} _ {j} | = m} \nabla f _ {\mathcal {I} _ {j}} (x), \quad h _ {1} (x, \gamma) := - \frac {m}{| \gamma |} \sum_ {\mathcal {I} _ {j} \subset \gamma_ {k}, | \mathcal {I} _ {j} | = m} \nabla^ {2} f _ {\mathcal {I} _ {j}} (x) \frac {\nabla f _ {\mathcal {I} _ {j}} (x)}{\| \nabla f _ {\mathcal {I} _ {j}} (x) \|}.
+$$
+
+By Assumptions 3.1 and B.1, each  $h_0,h_1\in G^2$  , and
+
+$$
+\mathbb {E} [ h _ {0} (x, \gamma) ] = - \nabla f (x), \quad \mathbb {E} [ h _ {1} (x, \gamma) ] = - \mathbb {E} \Big [ \nabla^ {2} f _ {\mathcal {I} _ {j}} (x) \frac {\nabla f _ {\mathcal {I} _ {j}} (x)}{\| \nabla f _ {\mathcal {I} _ {j}} (x) \|} \Big ].
+$$
+
+Applying Lemma A.5 to these  $h_0, h_1$  yields exactly the three moment estimates up to  $\eta^1, \rho^1$  as claimed.
+
+In Lemma C.6, we derived one-step moment estimates for the m-SAM discrete algorithm and, via Lemma A.4, for its corresponding SDE update (36). These estimates demonstrate that the first- and second-order moments satisfy the matching conditions of Theorem A.2. Together with the uniform moment bounds from Lemma D.2, we are now ready to establish the main weak-approximation theorem for m-SAM.
+
+Theorem C.7 (m-SAM SDE). Under Assumptions 3.1 and B.1, let  $0 < \eta < 1$ ,  $T > 0$ , and  $N = \lfloor T / \eta \rfloor$ . Denote by  $\{x_k\}_{k=0}^N$  the m-SAM iterates in (5), and let  $\{X_t\}_{t \in [0,T]}$  be the solution of the SDE (41). Suppose:
+
+(i) The functions
+
+$$
+\nabla f^{\mathrm{m - SAM}} = \nabla \big(f + \frac{\rho}{m}\mathbb{E}\| \sum_{\substack{i\in \mathcal{I},\\ |\mathcal{I}| = m}}\nabla f_{i}\| \big) \quad and\quad \sqrt{\Sigma^{\mathrm{m - SAM}}}
+$$
+
+are Lipschitz on  $\mathbb{R}^d$
+
+(ii) The mapping
+
+$$
+h _ {\gamma} (x) = - \frac {m}{| \gamma |} \sum_ {\mathcal {I} _ {j} \subset \gamma , | \mathcal {I} _ {j} | = m} \nabla f _ {\mathcal {I} _ {j}} \left(x + \rho \frac {\nabla f _ {\mathcal {I} _ {j}} (x)}{\| \nabla f _ {\mathcal {I} _ {j}} (x) \|}\right)
+$$
+
+satisfies, almost surely, the Lipschitz condition
+
+$$
+\left\| \nabla h _ {\gamma} (x) - \nabla h _ {\gamma} (y) \right\| \leq L _ {\gamma} \| x - y \|, \quad \forall x, y \in \mathbb {R} ^ {d},
+$$
+
+where  $L_{\gamma} > 0$  a.s. and  $\mathbb{E}[L_{\gamma}^{m}] < \infty$  for every  $m \geq 1$ .
+
+Then  $\{X_{t}:t\in [0,T]\}$  is an order-(1,1) weak approximation of  $\{x_k\}$ , namely: for each  $g\in G^2$  there exists a constant  $C > 0$ , independent of  $\eta$ , such that
+
+$$
+\max  _ {0 \leq k \leq N} \left| \mathbb {E} \big [ g (x _ {k}) \big ] - \mathbb {E} \big [ g (X _ {k \eta}) \big ] \right| \leq C (\eta + \rho^ {2}).
+$$
+
+Proof. First, we verify that SDE (41) admits a unique strong solution. By assumption, both the drift and diffusion coefficients are globally Lipschitz, which in turn implies a linear-growth condition. Therefore, Theorem D.1 applies and yields the existence and uniqueness of a strong solution on  $[0,T]$ .
+
+Then, by Lemmas A.4, C.6, and D.2, all the conditions of Theorem A.2 are satisfied, and the proof is complete.
+
+Remark C.8. The Lipschitz conditions are to ensure that the SDE has a unique strong solution with uniformly bounded moments. It is possible to appropriately relax them if we allow weak solutions (Mil'shtein, 1986).
+
+# D Auxiliary Lemmas
+
+Theorem D.1 (Existence and Uniqueness of Strong Solutions (Evans, 2012)). Let  $b: \mathbb{R}^d \to \mathbb{R}^d$  and  $\sigma: \mathbb{R}^d \to \mathbb{R}^{d \times m}$  be measurable functions satisfying:
+
+(i) Global Lipschitz. There exists a constant  $L > 0$  such that
+
+$$
+\left\| b (x) - b (y) \right\| + \left\| \sigma (x) - \sigma (y) \right\| \leq L \| x - y \|, \quad \forall x, y \in \mathbb {R} ^ {d}.
+$$
+
+(ii) Linear growth. There exists a constant  $K > 0$  such that
+
+$$
+\left\| b (x) \right\| ^ {2} + \left\| \sigma (x) \right\| ^ {2} \leq K \left(1 + \left\| x \right\| ^ {2}\right), \quad \forall x \in \mathbb {R} ^ {d}.
+$$
+
+Let  $X_0$  be an  $\mathbb{R}^d$ -valued random variable with  $\mathbb{E}[\| X_0\|^2] < \infty$ . Then the SDE
+
+$$
+d X _ {t} = b \left(X _ {t}\right) d t + \sigma \left(X _ {t}\right) d W _ {t}, \quad X _ {0} \text {g i v e n},
+$$
+
+admits a unique strong solution  $\{X_{t}\}_{t\geq 0}$  satisfying
+
+$$
+\mathbb {E} \left[ \sup  _ {0 \leq s \leq T} \| X _ {s} \| ^ {2} \right] <   \infty , \quad \forall T > 0.
+$$
+
+Lemma D.2. (Li et al., 2019) Let  $\{x_{k}:k\geq 0\}$  be the generalized iterations defined in (22). Suppose
+
+$$
+\left| h (x, \gamma , \eta) \right| \leq L _ {\gamma} \big (1 + \| x \| \big),
+$$
+
+where  $L_{\gamma} > 0$  almost surely and
+
+$$
+\mathbb {E} \left[ L _ {\gamma} ^ {m} \right] <   \infty \quad f o r a l l m \geq 1.
+$$
+
+Then for any fixed  $T > 0$  and any  $m \geq 1$ , the moment  $\mathbb{E}\left[\| x_k\|^m\right]$  exists and is uniformly bounded in both  $\eta$  and  $k = 0, 1, \ldots, N$ , where  $N = \lfloor T / \eta \rfloor$ .
+
+# E Proof of Proposition 3.9
+
+We will use the following lemma on convex order to prove Proposition 3.9, whose proof can be found in classical textbooks on stochastic order, such as Müller and Stoyan (2002).
+
+Lemma E.1 (Convex-order Monotonicity). Let  $X_{1},\ldots ,X_{n}$  be i.i.d. random vectors in  $\mathbb{R}^d$  with a log-concave density. For each integer  $1\leq k\leq n$ , define
+
+$$
+S _ {k} = \frac {1}{k} \sum_ {i = 1} ^ {k} X _ {i}.
+$$
+
+Then for any  $1 \leq k < m \leq n$  and any convex function  $\phi \colon \mathbb{R}^d \to \mathbb{R}$ ,
+
+$$
+\mathbb {E} \bigl [ \phi (S _ {m}) \bigr ] \leq \mathbb {E} \bigl [ \phi (S _ {k}) \bigr ].
+$$
+
+Proof of Proposition 3.9. Lower bound: Applying Jensen's inequality,
+
+$$
+\left\| \nabla f (x) \right\| = \left\| \mathbb {E} [ \nabla f _ {\gamma} (x) ] \right\| \leq \mathbb {E} [ \| \nabla f _ {\gamma} (x) \| ].
+$$
+
+Upper bound: By Cauchy-Schwarz inequality,
+
+$$
+\mathbb {E} [ \| \nabla f _ {\gamma} (x) \| ] \leq \sqrt {\mathbb {E} [ \| \nabla f _ {\gamma} (x) \| ^ {2} ]} = \sqrt {\| \nabla f (x) \| ^ {2} + \frac {\operatorname {t r} (V (x))}{| \gamma |}}.
+$$
+
+Combining both bounds completes the proof of the first statement.
+
+For the second statement, we apply Lemma E.1 to the convex function  $\| \cdot \|$ , which concludes the proof.
+
+# F Additional Related Works
+
+Theoretical understanding of SAM. Although SAM and its variants have achieved remarkable success in various practical applications (Foret et al., 2021; Kwon et al., 2021; Kaddour et al., 2022; Li et al., 2024b; Li and Giannakis, 2024), the theoretical understanding behind them remains limited. The pioneering work of Andriushchenko and Flammarion (2022) provided the first theoretical framework for understanding SAM, covering its convergence properties and implicit bias in simple network structures, while also systematically illustrating several empirical phenomena. Subsequently, Si and Yun (2024) extended the convergence analysis to various deterministic and stochastic settings. Compagnoni et al. (2023); Luo et al. (2025) conducted an in-depth analysis of the dynamics of SAM using the SDE framework previously developed by Li et al. (2017), leading to a deeper understanding of its implicit bias. On the other hand, Wen et al. (2022) investigated the implicit bias of SAM by analyzing its slow ordinary differential equation (ODE) behavior near the minimizer manifold, demonstrating how SAM drifts toward flatter minima. More recently, Zhou et al. (2024) studied the late-stage behavior of SAM using stability analysis, showing its advantage in escaping sharp minima.
+
+# G Derivation of the Finite Difference Estimator in Equation (21)
+
+We start with the first-order Taylor expansion around  $x$ :
+
+$$
+\frac {f _ {i} (x + \delta z) - f _ {i} (x)}{\delta} = \nabla f _ {i} (x) ^ {\top} z + O (\delta),
+$$
+
+where  $z \in \{\pm 1\}^d$  is a Rademacher random vector, and  $f_{i}$  is assumed twice differentiable so that the remainder is of order  $O(\delta)$ .
+
+Step 1: Square both sides.
+
+$$
+\left(\frac {f _ {i} (x + \delta z) - f _ {i} (x)}{\delta}\right) ^ {2} = \left(\nabla f _ {i} (x) ^ {\top} z\right) ^ {2} + 2 \left(\nabla f _ {i} (x) ^ {\top} z\right) O (\delta) + O (\delta) ^ {2}.
+$$
+
+Often we simply write this as
+
+$$
+\left(\nabla f _ {i} (x) ^ {\top} z + O (\delta)\right) ^ {2} = \left(\nabla f _ {i} (x) ^ {\top} z\right) ^ {2} + O (\delta) \left(\nabla f _ {i} (x) ^ {\top} z\right) + O (\delta^ {2}).
+$$
+
+Step 2: Take expectation over  $z$ . Because  $z$  has independent  $\{\pm 1\}$  components, we have
+
+$$
+\mathbb {E} _ {z} \left[ \left(\nabla f _ {i} (x) ^ {\top} z\right) ^ {2} \right] = \| \nabla f _ {i} (x) \| ^ {2} \quad (\text {s t a n d a r d R a d e m a c h e r p r o p e r t y}).
+$$
+
+Hence,
+
+$$
+\mathbb {E} _ {z} \left[ \left(\frac {f _ {i} (x + \delta z) - f _ {i} (x)}{\delta}\right) ^ {2} \right] = \mathbb {E} _ {z} \left[ (\nabla f _ {i} (x) ^ {\top} z) ^ {2} \right] + O (\delta^ {2}) = \| \nabla f _ {i} (x) \| ^ {2} + O (\delta^ {2}).
+$$
+
+Thus the mean-squared estimate of the finite difference quotient differs from  $\| \nabla f_i(x)\|^2$  by an  $O(\delta^2)$  bias term, implying that the estimator is approximately unbiased as  $\delta \to 0$ . We compare two  $d$ -dimensional random vectors  $z \in \mathbb{R}^d$ :
+
+- Rademacher: each component  $z_{j}$  is independently  $\pm 1$  with probability  $1/2$ ,  
+- Standard Gaussian: each component  $z_{j}$  is i.i.d.  $\mathcal{N}(0,1)$ .
+
+Both have  $\mathbb{E}[z_j] = 0$  and  $\mathbb{E}[z_j^2] = 1$ , so  $\mathbb{E}[(z^\top v)^2] = \|v\|^2$  for any  $v \in \mathbb{R}^d$ . We look at the fourth moment  $\mathbb{E}[(z^\top v)^4]$ , relevant to the variance in many finite-difference or gradient estimators.
+
+Rademacher case. Since  $z_j^2 \equiv 1$ ,
+
+$$
+\left(z ^ {\top} v\right) ^ {2} = \left(\sum_ {j = 1} ^ {d} v _ {j} z _ {j}\right) ^ {2} = \sum_ {j, k} v _ {j} v _ {k} z _ {j} z _ {k}.
+$$
+
+Then
+
+$$
+\left(z ^ {\top} v\right) ^ {4} = \left(\sum_ {j, k} v _ {j} v _ {k} z _ {j} z _ {k}\right) ^ {2}
+$$
+
+and using  $\mathbb{E}[z_j^4] = 1$ ,  $\mathbb{E}[z_j^2 z_k^2] = 1$  (when  $j \neq k$ ) with zero cross terms of odd product, one obtains a relatively small constant factor. Detailed calculation yields
+
+$$
+\mathbb {E} \left[ (z ^ {\top} v) ^ {4} \right] = \sum_ {j = 1} ^ {d} v _ {j} ^ {4} + 6 \sum_ {j <   k} v _ {j} ^ {2} v _ {k} ^ {2} \leq 3 \| v \| ^ {4} \quad (\text {f o r} d > 1).
+$$
+
+Gaussian case. If  $z \sim \mathcal{N}(0, I_d)$ , then  $\mathbb{E}[z_j^4] = 3$  and  $\mathbb{E}[z_j^2 z_k^2] = 1$  for  $j \neq k$ . One can show
+
+$$
+\mathbb {E} \left[ \left(z ^ {\top} v\right) ^ {4} \right] = 3 \| v \| ^ {4}.
+$$
+
+Hence the constant factor in front of  $\| v\| ^4$  is exactly 3 for Gaussian.
+
+Conclusion. For both Rademacher and Gaussian vectors,  $\mathbb{E}[(z^{\top}v)^2] = \| v\| ^2$ . However, when analyzing higher-order moments (e.g.  $(z^{\top}v)^4$ ) that affect the variance of many finite-difference or random-direction estimators, the Rademacher distribution can yield a smaller constant factor. This often leads to reduced variance and tighter theoretical bounds for the same sample size.
+
+# H Derivation of the Gibbs Distribution (20)
+
+Consider the following maximization problem :
+
+$$
+\max _ {P \in \Delta} \sum_ {i \in \gamma} p _ {i} \| \nabla f _ {i} (x) \| + \frac {\mathbb {H} (P)}{\lambda},
+$$
+
+where  $\Delta$  is the probability simplex (i.e.,  $\sum_{i}p_{i} = 1$  and  $p_i\geq 0$ ),  $\mathbb{H}(P) = -\sum_{i}p_{i}\ln p_{i}$  is the entropy term, and  $\lambda >0$  is a given constant.
+
+1. Construct the Lagrangian. We introduce the constraint  $\sum_{i}p_{i} = 1$  with a Lagrange multiplier  $\alpha$ :
+
+$$
+\mathcal {L} (P, \alpha) = \sum_ {i} p _ {i} \| \nabla f _ {i} (x) \| + \frac {1}{\lambda} \big (- \sum_ {i} p _ {i} \ln p _ {i} \big) + \alpha \Big (1 - \sum_ {i} p _ {i} \Big).
+$$
+
+2. Differentiate w.r.t.  $p_i$  and set to zero. Taking partial derivatives with respect to  $p_i$  and setting them to zero,
+
+$$
+\frac {\partial \mathcal {L}}{\partial p _ {i}} = \| \nabla f _ {i} (x) \| - \frac {1}{\lambda} (\ln p _ {i} + 1) - \alpha = 0.
+$$
+
+Therefore,
+
+$$
+\| \nabla f _ {i} (x) \| - \frac {\ln p _ {i} + 1}{\lambda} - \alpha = 0 \Rightarrow p _ {i} = \exp \left(\lambda \| \nabla f _ {i} (x) \| + 1 - \lambda \alpha\right).
+$$
+
+3. Enforce the normalization. The constraint  $\sum_{i}p_{i} = 1$  fixes the value of  $\alpha$ ; it amounts to one overall normalization factor in the denominator. Hence the solution takes the well-known Gibbs distribution form:
+
+$$
+p _ {i} ^ {*} = \frac {\exp \left(\lambda \| \nabla f _ {i} (x) \|\right)}{\sum_ {j} \exp \left(\lambda \| \nabla f _ {j} (x) \|\right)}.
+$$
+
+# I Algorithm: Reweighted SAM
+
+# Algorithm 1 Reweighted SAM
+
+1: while not converged do  
+2: Forward pass to obtain  $f_{\gamma_k}(x_k)$  
+3: for  $q = 1,\dots ,Q$  do  
+4: Estimate per-sample gradient norm using Eq. (21)  
+5: end for  
+6: Normalize estimated per-sample gradient norm  
+7: Compute weight  $p^*$  using Eq. (20)  
+8: Compute perturbation  $\epsilon_{k}$  using Eq. (18)  
+9: Compute perturbed gradient  $g_{k} = \nabla f_{\gamma_{k}}(x_{k} + \rho \epsilon_{k})$  
+0: Update model parameters:  $x_{k + 1} = x_{k} - \eta g_{k}$  
+1: end while
+
+# J Additional Experiment Results
+
+All experiments were run on NVIDIA RTX 4090 GPUs.
+
+<table><tr><td>Algorithm</td><td>Test Accuracy</td><td>Time/Epoch (s)</td></tr><tr><td>Mini-batch SAM</td><td>78.90 ± 0.27%</td><td>13.56</td></tr><tr><td>RW-SAM</td><td>79.31 ± 0.28 %</td><td>15.21</td></tr><tr><td>m-SAM (m = 8)</td><td>80.72 ± 0.12%</td><td>175.45</td></tr><tr><td>m-SAM (m = 16)</td><td>80.47 ± 0.09%</td><td>92.22</td></tr><tr><td>m-SAM (m = 32)</td><td>80.02 ± 0.06%</td><td>49.87</td></tr><tr><td>m-SAM (m = 64)</td><td>79.35 ± 0.11%</td><td>26.44</td></tr><tr><td>n-SAM</td><td>78.15 ± 0.19%</td><td>—</td></tr></table>
+
+<table><tr><td>Algorithm</td><td>Test Accuracy</td><td>Time/Epoch (s)</td></tr><tr><td>Mini-batch USAM</td><td>78.94 ± 0.45%</td><td>12.98</td></tr><tr><td>m-USAM (m = 8)</td><td>80.66 ± 0.04%</td><td>173.77</td></tr><tr><td>m-USAM (m = 16)</td><td>80.46 ± 0.07%</td><td>90.86</td></tr><tr><td>m-USAM (m = 32)</td><td>80.02 ± 0.09%</td><td>47.09</td></tr><tr><td>m-USAM (m = 64)</td><td>79.16 ± 0.04%</td><td>24.15</td></tr><tr><td>n-USAM</td><td>78.63 ± 0.06%</td><td>—</td></tr></table>
+
+Table 7: Left: performance and time cost of SAM, RW-SAM, m-SAM (with varying  $m$ ), and n-SAM on CIFAR-100. Right: performance and time cost of USAM, m-USAM (with varying  $m$ ), and n-USAM; with ResNet-18 on CIFAR-100.
+
+Table 8: wall-clock time overhead of RW-SAM  
+
+<table><tr><td></td><td>ResNet-18</td><td>ResNet-50</td><td>WideResNet-28-10</td></tr><tr><td>SAM</td><td>13.6</td><td>43.0</td><td>97.7</td></tr><tr><td>RW-SAM</td><td>15.2</td><td>50.9</td><td>112.4</td></tr></table>
+
+We present the experimental results of applying the proposed reweighting strategy to ASAM (Kwon et al., 2021) in Table 9. The results demonstrate consistent improvements, and further investigation into the effectiveness of applying the reweighting strategy to different SAM variants remains an interesting direction for future work.
+
+Table 9: Test accuracy (%) comparison between ASAM and RW-ASAM on CIFAR-10 and CIFAR-100.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">CIFAR-10</td><td colspan="2">CIFAR-100</td></tr><tr><td>ResNet-18</td><td>ResNet-50</td><td>ResNet-18</td><td>ResNet-50</td></tr><tr><td>ASAM</td><td>95.86 ± 0.14</td><td>96.12 ± 0.23</td><td>79.17 ± 0.14</td><td>80.27 ± 0.33</td></tr><tr><td>RW-ASAM</td><td>96.02 ± 0.08</td><td>96.43 ± 0.17</td><td>79.46 ± 0.25</td><td>80.65 ± 0.16</td></tr></table>
+
+Table 10: Hyperparameter settings for fine-tuning DistilBERT on GLUE tasks.  
+
+<table><tr><td>Task</td><td>CoLA</td><td>MNLI</td><td>MRPC</td><td>QNLI</td><td>QQP</td><td>RTE</td><td>SST-2</td><td>STS-B</td><td>WNLI</td></tr><tr><td>Batch size</td><td>32</td><td>64</td><td>32</td><td>64</td><td>64</td><td>32</td><td>64</td><td>32</td><td>32</td></tr><tr><td>Learning rate</td><td>2e-5</td><td>3e-5</td><td>2e-5</td><td>3e-5</td><td>3e-5</td><td>2e-5</td><td>3e-5</td><td>2e-5</td><td>2e-5</td></tr><tr><td>Epochs</td><td>8</td><td>3</td><td>8</td><td>3</td><td>3</td><td>8</td><td>3</td><td>8</td><td>8</td></tr><tr><td>LR scheduler</td><td></td><td></td><td></td><td></td><td>Linear</td><td></td><td></td><td></td><td></td></tr><tr><td>Warmup ratio</td><td></td><td></td><td></td><td></td><td>0.1</td><td></td><td></td><td></td><td></td></tr><tr><td>Max sequence length</td><td></td><td></td><td></td><td></td><td>256</td><td></td><td></td><td></td><td></td></tr><tr><td>ρ (for SAM and RW-SAM)</td><td></td><td></td><td></td><td></td><td>0.05</td><td></td><td></td><td></td><td></td></tr></table>
+
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+
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+
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+- Keep the checklist subsection headings, questions/answers and guidelines below.  
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+# 1. Claims
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+
+Answer: [Yes]
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+
+Guidelines:
+
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+
+# 2. Limitations
+
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+
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+
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+
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+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
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+
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+
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+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
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+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
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+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+# Guidelines:
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+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
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+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components.
+
+# Guidelines:
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+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
\ No newline at end of file
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+# Unveiling the Compositional Ability Gap in Vision-Language Reasoning Model
+
+$\spadesuit$  Tianle Li,  $\spadesuit$  Jihai Zhang,  $\spadesuit$  Yongming Rao,  $\spadesuit$  Yu Cheng
+
+$\spadesuit$  The Chinese University of Hong Kong,  $\spadesuit$ Tencent Hunyuan Research
+
+tianleli@link.cuhk.edu.hk
+
+https://github.com/1t13A87/ComPA
+
+# Abstract
+
+While large language models (LLMs) demonstrate strong reasoning capabilities utilizing reinforcement learning (RL) with verifiable reward, whether large vision-language models (VLMs) can directly inherit such capabilities through similar post-training strategies remains underexplored. In this work, we conduct a systematic compositional probing study to evaluate whether current VLMs trained with RL or other post-training strategies can compose capabilities across modalities or tasks under out-of-distribution conditions. We design a suite of diagnostic tasks that train models on unimodal tasks or isolated reasoning skills, and evaluate them on multimodal, compositional variants requiring skill integration. Through comparisons between supervised fine-tuning (SFT) and RL-trained models, we identify three key findings: (1) RL-trained models consistently outperform SFT on compositional generalization, demonstrating better integration of learned skills; (2) although VLMs achieve strong performance on individual tasks, they struggle to generalize compositionally under cross-modal and cross-task scenarios, revealing a significant gap in current training strategies; (3) enforcing models to explicitly describe visual content before reasoning (e.g., caption-before-thinking), along with rewarding progressive vision-to-text grounding, yields notable gains. It highlights two essential ingredients for improving compositionality in VLMs: visual-to-text alignment and accurate visual grounding. Our findings shed light on the current limitations of RL-based reasoning VLM training and provide actionable insights toward building models that reason compositionally across modalities and tasks.
+
+# 1 Introduction
+
+Recent breakthroughs in large language models (LLMs) have shown that strong reasoning capabilities can emerge through RL, as exemplified by GPT-o1 [Jaech et al., 2024] and DeepSeek-R1 [Guo et al., 2025]. These models demonstrate impressive performance on complex multi-step reasoning tasks in the language-only domain, revealing the potential of RL-style post-training to enhance logical and compositional reasoning [Team et al., 2025, Hou et al., 2025, Shen et al., 2025b]. Inspired by these advances, researchers have begun exploring whether similar training paradigms can be extended to VLMs, which integrate visual perception with language reasoning [Zhan et al., 2025, Huang et al., 2025, Hao et al., 2025, Yang et al., 2025, Wang et al., 2025].
+
+Previous attempts to apply RL with verifiable rewards to VLMs have shown promising gains on individual vision-language tasks such as visual math solving and object localization [Shen et al., 2025a, Meng et al., 2025, Pan and Liu, 2025]. However, it remains unclear whether these improvements extend beyond isolated benchmarks to more complex, realistic scenarios that require the integration of multiple reasoning capabilities. While the compositional abilities of LLMs have been increasingly studied in the context of skill composition [Zhao et al., 2024, Xu et al., 2024b], the extent to
+
+![](images/16329b01d68fde088dd17b7cd5719f3b60313f1f02cc01541ef18db0695026be.jpg)  
+Figure 1: Demonstration of the tasks and partial results for probing of cross-modality and cross-task compositional ability.
+
+which VLMs exhibit similar capabilities remains largely underexplored. In particular, it is still an open question whether VLMs can coherently combine skills acquired independently either across modalities (e.g., transferring textual reasoning to visual inputs) or across reasoning domains (e.g., integrating spatial and arithmetic reasoning) to solve tasks that demand such composition.
+
+To better understand the performance of compositional generalization in VLMs, we investigate two core dimensions: cross-modal and cross-task reasoning. We focus on the following research questions(RQ): RQ1 Can reasoning abilities acquired through pure-text training be composed with visual recognition to solve multimodal reasoning tasks? RQ2 Can independently learned visual reasoning skills be integrated to tackle composite tasks that require both capabilities? RQ3 Can such compositional ability generalize to out-of-distribution (OOD) variants with altered task objectives? To support this investigation, we design a set of diagnostic tasks and carefully curated training and evaluation splits, each aimed at isolating specific challenges such as cross-modal reasoning, visual skill composition, and generalization to new task settings as partially demonstrated in Figure 1.
+
+We conduct comprehensive experiments across multiple post-training strategies to evaluate the compositional capabilities of VLMs. Our study yields three key observations: (1) RL-trained models consistently outperform SFT in compositional settings, particularly for cross-task generalization; (2) despite strong performance on individual tasks, VLMs exhibit significant limitations in compositional reasoning under multimodal input; and (3) explicitly structuring the reasoning process through visual-to-text prompting and reinforcing intermediate progress reward [Luo et al., 2024] lead to substantial gains in compositional performance.
+
+In a nutshell, our contributions to this work can be summarized as follows:
+
+- We introduce ComPABench, a diagnostic benchmark that systematically evaluates compositional generalization in VLMs across modalities, reasoning tasks, and distribution shifts.  
+- We conduct a comprehensive empirical analysis of post-training strategies, and reveal their limitations in both cross-modal and cross-task compositional generalization.  
+- We identify a simple yet effective solution, RL-Ground, that helps reduce the compositional gap in existing post-training strategies by aligning visual inputs to text before reasoning and rewarding accurate grounding of visual content during intermediate reasoning steps.
+
+Together, we hope these contributions can lay a foundation for advancing VLMs toward more robust multimodal reasoning with stronger compositional generalization.
+
+# 2 Related Work
+
+# 2.1 Post-training for VLM Reasoning
+
+Following the success of reasoning-oriented LLMs such as GPT-o1 [Jaech et al., 2024], significant efforts have been devoted to developing advanced reasoning capabilities in VLMs after supervised
+
+fine-tuning over various fundamental visual tasks [Wang et al., 2024a, Chen et al., 2024, Team et al., 2025, Wu et al., 2024, Abouelenin et al., 2025]. Early approaches range from manually designed structured reasoning pathsXu et al. [2024a] to tree-based search strategies Xu et al. [2024a], Yao et al. [2024]. The breakthrough of Deepseek-R1 [Guo et al., 2025] in outcome-based reward RL with GRPO [Shao et al., 2024] for LLMs has inspired attempts to adapt this paradigm to VLMs. However, directly transplanting Deepseek-R1's training methodology to VLMs has proven ineffective. Huang et al. [2025] identify two primary failure modes: (1) the learning algorithm struggles to obtain meaningful positive rewards for complex samples, and (2) models tend to bypass visual inputs and rely solely on textual cues for reasoning. Similarly, Zhan et al. [2025] demonstrate that direct application of outcome reward RL fails to enhance VLMs' reasoning performance. While Du et al. [2025] partially validate the transferability of text-based reasoning capabilities to multimodal tasks, their analysis remains confined to mathematics-oriented domains. Current VLMs, particularly open-source implementations, still exhibit significant gaps in multimodal reasoning compared to human-level performance Hao et al. [2025].
+
+# 2.2 Generalization Probing for Training Strategies
+
+The post-training phase of large models, particularly the choice between supervised fine-tuning and reinforcement learning, has significant implications for generalization. Several recent works have sought to explore these effects for different strategies respectively [Wang et al., 2024b, Zhao et al., 2025]. Chu et al. [2025] conducted a systematic comparison of SFT and RL, finding that SFT often leads to memorization of training patterns, whereas RL enables stronger generalization by encouraging models to discover and apply more transferable principles. Kirk et al. [2023] also confirmed that RLHF-trained models exhibit improved robustness to distribution shifts at the cost of reduced response diversity. In contrast, Yue et al. [2025] revisits the widely held belief that RL with verifiable rewards (RLVR) enables LLMs to acquire fundamentally new reasoning abilities, showing instead that RLVR primarily reweights the model's existing reasoning distribution rather than expanding it, which limits exploratory capacity despite improved efficiency. Our work differentiates from these works by systematically analyzing the limitations of current post-training strategies when applied to VLMs, focusing specifically on their compositional generalization capabilities across modalities, tasks, and out-of-distribution settings.
+
+# 3 Preliminaries
+
+To ground the experimental design and training protocols used in this work, we first formalize the three training paradigms employed throughout our evaluation: Supervised Fine-Tuning (SFT), Reinforcement Learning with Verifiable Reward (RL), and RL training initialized from an SFT-trained checkpoint (SFT-init RL). We also detail the learning objectives used in each setting, with a particular emphasis on the Generalized Reinforcement Policy Optimization (GRPO) strategy adopted in DeepSeek-R1 training [Guo et al., 2025].
+
+# 3.1 Supervised Fine-Tuning
+
+Supervised fine-tuning aligns a pretrained VLM with the target task distribution using paired data  $(x,y)$ , where  $x$  is the input prompt and  $y = (y_{1},\dots ,y_{T})$  is the corresponding output sequence. The model is trained to minimize the negative log-likelihood (NLL) of the target output under the causal language modeling objective:
+
+$$
+\mathcal {L} _ {\mathrm {S F T}} (\theta) = - \sum_ {t = 1} ^ {T} \log p _ {\theta} \left(y _ {t} \mid x, y _ {<   t}\right) \tag {1}
+$$
+
+This objective encourages syntactic correctness and semantic alignment with human-provided examples. In our setup,  $x$  may contain text-only format, or both image and text formats.  $y$  includes both a reasoning trace (enclosed in a <think> block) and a final response (enclosed in a <answer> block).
+
+# 3.2 Reinforcement Learning with GRPO
+
+We adopt Group Relative Policy Optimization (GRPO) as the optimization strategy in our RL training framework. GRPO generalizes token-level policy optimization with a structured per-sample reward-to-advantage computation and includes a KL regularization term to the reference policy  $\pi_{\mathrm{ref}}$  (typically the SFT model).
+
+Let  $G$  denote the number of generated candidate answers for a given question  $q$ , and  $|o^{(i)}|$  denote the number of tokens in the  $i$ -th output  $o^{(i)}$ . The GRPO loss is defined as:
+
+$$
+\mathcal {I} \mathrm {G R P O} (\theta) = - \frac {1}{G} \sum_ {i = 1} ^ {G} \frac {1}{| o ^ {(i)} |} \sum_ {t = 1} ^ {| o ^ {(i)} |} \left[ \frac {\pi_ {\theta} (o ^ {(i)} t | q , o ^ {(i)} <   t)}{\pi_ {\theta} (o ^ {(i)} t | q , o ^ {(i)} <   t)} \cdot A (i, t) - \beta \cdot \mathrm {K L} (\pi_ {\theta} \| \pi_ {\mathrm {r e f}}) \right] \quad (2)
+$$
+
+Here,  $A(i,t)$  is the estimated advantage at time step  $t$  for the  $i$ -th sample, and  $\beta$  controls the strength of the KL regularization term. The inner term represents a scaled policy gradient with a reward signal modulated at each token step.
+
+In practice,  $A(i,t)$  is typically derived from a scalar reward  $r(q,o^{(i)})$  measuring task success, which may combine:
+
+- Answer correctness: Whether the generated prediction matches the ground-truth answer.  
+- Format adherence: Whether the output satisfies the specified constraints of format.
+
+This formulation ensures that the model improves generation quality while maintaining consistency with the reference distribution.
+
+# 3.3 SFT-Initialized RL Training
+
+To stabilize and accelerate the RL training process, we explore a hybrid strategy where reinforcement learning is initialized from a model pretrained with SFT, similar to R1Guo et al. [2025]. In this setting, the reference policy  $\pi_0$  used in the KL term is set to the SFT-trained model, and the initial parameters  $\theta_0$  of the policy  $\pi_{\theta}$  are inherited from the same checkpoint. This strategy offers two key benefits: (1) it enables faster convergence by leveraging prior alignment to task distributions, and (2) it mitigates early-stage instability common in pure RL setups.
+
+Together, these training paradigms define the backbone of our experimental pipeline, enabling us to probe the strengths and failure modes of VLMs in compositional, multimodal, and generalization-intensive reasoning settings.
+
+# 4 Experiments
+
+To assess compositional generalization in VLMs under different post-training strategies, we conduct experiments from three perspectives: cross-modal composition, cross-task reasoning, and out-of-distribution generalization. We first present ComPABench, our diagnostic benchmark, followed by training setups and evaluation results addressing the three research questions.
+
+# 4.1 Benchmark for Probing of Compositional Ability
+
+<table><tr><td>Task Type</td><td>Training Subset</td><td>Test Subset</td></tr><tr><td>Cross-Modal Composition</td><td>PT-GR, PT-SR</td><td>PT-GR, PT-SR, MM-GR, MM-SR</td></tr><tr><td>Cross-Task Composition (Pure-text)</td><td>PT-GR, PT-SR</td><td>PT-GR, PT-SR, PT-Comp</td></tr><tr><td>Cross-Task Composition (Multimodal)</td><td>MM-GR, MM-SR</td><td>MM-GR, MM-SR, MM-Comp</td></tr><tr><td>OOD Composition</td><td>MM-GR, MM-SR</td><td>MM-GR-OOD, MM-SR-OOD, MM-Comp-OOD</td></tr></table>
+
+Table 1: Statistics of our proposed ComPABench for different task settings. The abbreviations of different types of dataset can be found in the right bottom of blocks in Figure 2.
+
+![](images/d9537b6953d69d5a4d6f0bbd384136c3d7d48c33aa088194d42d461e8a8b10ad.jpg)  
+Figure 2: Demonstration of proposed ComPABench for RQ1, RQ2, and RQ3.
+
+To systematically evaluate the compositional ability of VLMs trained under different post-training strategies, we design a set of finely controlled tasks aligned with our three core research questions, where we call this benchmark as ComPABench. Each task setting is implemented with paired pure-text and multimodal variants to allow cross-modality and cross-task comparisons. Figure 2 illustrates the benchmark design across individual and compositional tasks, together with the corresponding OOD variants for evaluating transfer robustness.
+
+Cross-Modal Compositional ability (RQ1). To evaluate whether reasoning abilities acquired from pure-text training can transfer to visual inputs at inference, we construct parallel task formats with matched semantics but differing input modalities. In the geometric reasoning task, the model computes the total area of multiple shapes described either in pure-text or shown in an image with labeled dimensions. In the spatial reasoning task, it identifies the grid index of the shape closest to a given target, based on either textual position descriptions or an image depicting a grid with embedded shapes. By comparing performance across modalities, we assess the model's ability to compose textual reasoning with visual perception.
+
+Cross-Task Compositional ability (RQ2). We probe whether models can integrate independently acquired skills by composing geometric and spatial reasoning in a single task. Here, we evaluate whether models trained on each skill individually can solve questions requiring both, such as computing the total area of a target shape and the shape closest to the target. We test VLM under both pure-text and multimodal settings to compare the compositional ability of different input types.
+
+Compositional OOD Generalization (RQ3). To evaluate whether compositional reasoning extends to variants of seen tasks, we develop OOD tasks that modify the objective of individual task, so as the compositional ones. For example, instead of asking for the total area, the model must identify the largest area (for geometric reasoning), or select the farthest shape rather than the nearest (for spatial reasoning). In the compositional OOD setting, models are asked to perform combined tasks using these novel objectives (e.g., compute the area of the larger of between the target shape and the farthest shape of it), probing compositional ability in a more challenging setting.
+
+This benchmark provides a unified and controlled evaluation for diagnosing cross-modal and cross-task generalization, as well as robustness to distributional shifts. We present the detailed composition for each of the tasks in Table 1. More specifically, we generate 4K samples for each individual type of data in training and 500 samples for evaluation. For instance, for Cross-Model Composition task, we mix 4K PT-GR and 4K PT-SR data to train a VLM, and test it on 500 PT-GR, 500 PT-SR, etc. The proposed ComPABench directly supports our investigation into the compositional abilities of current reasoning VLMs under different post-training strategies. More details about the construction of ComPABench is provided in the supplementary materials.
+
+# 4.2 Experiment Settings
+
+To evaluate the effect of different post-training strategies on compositional generalization, we conduct all experiments using backbone models: Qwen2.5-VL-3B-Instruct and Qwen2.5-VL-7B-Instruct. For training configurations, we apply consistent hyperparameters: a per-device batch size of 1, a learning rate of  $1\mathrm{e} - 6$  , and a total of 1 training epoch. For RL experiments, we generate 8 completions per prompt in training. And we set the scale before KL divergence constraints to 0, as we observe a dramatic performance degradation with KL divergence in the objective of optimization. All experiments are conducted on 4 NVIDIA H100 GPUs. Our implementation of GRPO is based on the open-source  $\mathbb{R}1\text{-V}^1$
+
+# 4.3 Evaluation Result
+
+In this subsection, we present experimental results answering the three research questions introduced previously.
+
+# 4.3.1 RQ1: Cross-Modality Compositional Generalization
+
+![](images/55f08063baaaac33bc1a972670c021753cf1df00a98f7af7d41161fff9512591.jpg)  
+Figure 3: Performance comparison when post-trained with pure-text and evaluated with either pure-text or image-format(multi-modal) questions.
+
+![](images/07ac7e8112a9006fdf392a37c40404fd1dafc59e141517e34d45c2f513ab4d61.jpg)
+
+Pure-text to multimodal generalization gap. We first evaluate whether reasoning skills acquired from pure-text training transfer effectively to visual input at inference time as shown in Figure 3. While the original Qwen2.5-VL models (without post-training) already show high accuracy on the proposed pure-text tasks, SFT boosts performance in the pure-text modality generally, reaching near-perfect accuracy for the grid position task (99.2% for 3B and 99.8% for 7B), and maintains similar performance or improves moderately for shape area task. However, the models trained solely on pure-text data fail dramatically when tested on the corresponding multimodal tasks, dropping sharply to 13% (3B) and 16.2% (7B) on shape areas, and to just 4.8% (3B) and 4.2% (7B) on grid positions. This large accuracy gap (exceeding 94 points in the worst case) indicates that purely textual training alone does not inherently enable visual reasoning for SFT post-training, despite semantic alignment between tasks.
+
+Moderate improvement with RL training. RL generally achieves competitive performance compared to SFT in the pure-text modality after pure-text training, with only one notable exception
+
+![](images/039d3275449eb060d2c6d53bc5920cefdf09dceaff4e566019c37f8f745a63ac.jpg)  
+Performance Trend with Pure-Text Initialization Training
+
+![](images/057d44add5936b1cf5236f2b30e648c2eca04a82fcadd12c6cd7daecdeac4018.jpg)  
+Figure 4: Trend of multi-modal performance without and with pure-text training initialization.
+
+where RL underperforms SFT (82% vs. 99.2% on 3B grid position tasks). Despite this, RL enhances multimodal accuracy compared to pure-text-only SFT models. For instance, RL improves multimodal shape area accuracy from 20.8% to 28.0% (7B), yet still far below the pure-text setting. On the grid-position multimodal task, RL yields only modest improvements (6.2% for 3B and 5.2% for 7B), barely surpassing the performance of the original base model. These results indicate that while RL enables partial compositional generalization from reasoning skills acquired on text to multimodal visual tasks, its effectiveness remains limited when trained exclusively on textual data.
+
+Impact of initializing multimodal training with pure-text priors. To better understand how pure-text training influences subsequent multimodal reasoning, we further examine models initialized with pure-text reasoning priors before multimodal training as shown in Figure 4). In this scenario, initializing multimodal RL with models already trained on pure-text data significantly enhances visual task performance. Specifically, for the 3B grid-position task, accuracy increases substantially from  $49.6\%$  (direct multimodal RL) to  $64.4\%$  (text-initialized multimodal RL). In contrast, the beneficial effect of pure-text initialization is minimal or even slightly detrimental for SFT ( $91.8\%$  versus  $89.6\%$  on Grig Position for 7B model). This discrepancy between RL and SFT likely arises because, during multimodal SFT training, the reasoning path is explicitly provided in the <think> block preceding the answer, whereas RL must discover the correct reasoning path solely from final-answer supervision. Therefore, initializing RL with text-trained models, where semantics closely match, may help the model more efficiently converge toward the appropriate reasoning strategy.
+
+These findings collectively confirm that pure-text reasoning capabilities, even when trained to near perfection, do not automatically generalize to multimodal inputs. RL-based training provides moderate improvements over pure-text SFT but is insufficient by itself. Importantly, initializing multimodal RL from a pure-text reasoning model can enhance performance, suggesting an effective strategy for scenarios where multimodal data is limited or costly.
+
+# 4.3.2 RQ2: Compositional Reasoning from Independently Acquired Skills
+
+Pure-text compositional performance. In the pure-text setting (left column of Fig. 5), the original Qwen2.5-VL models (without additional post-training) already exhibit moderate compositional capabilities, achieving  $49.4\%$  accuracy (3B) and  $46.4\%$  accuracy (7B). However, SFT on individual geometric and spatial reasoning tasks separately severely impairs compositional accuracy, dropping performance drastically to just  $0.6\%$  (3B) and  $2.2\%$  (7B), despite nearly perfect accuracy on each sub-skill individually. This catastrophic forgetting indicates that standard SFT actively disrupts the model's inherent compositional capability. In contrast, RL with a final-answer reward substantially improves compositional reasoning, raising accuracy significantly to  $93\%$  (3B) and  $81.2\%$  (7B). Thus, RL effectively preserves and enhances compositional generalization in text-only setting for VLMs.
+
+Multimodal compositional performance. In the multimodal setting (right column of Fig. 5), the original models (without any multimodal post-training) struggle significantly, achieving low accuracy (5.8% for 3B and 13% for 7B). Similar to the pure-text case, multimodal SFT also fails, reaching only
+
+![](images/218c23444cab7ee5e49c0c2f390b5d568b17ac6e711e37e3db2d7ecbfecf65e3.jpg)
+
+![](images/20e3df240a48915fb0e65b924ac9e65edb01e756ae58ce9e0c99fb1d9eedebc6.jpg)
+
+![](images/467d067609598212af72653cbdba74c882f4f89a85e1346a427c4b25a1ebab51.jpg)  
+Figure 5: Comparison of Results for compositional reasoning from independently acquired skills.
+
+![](images/77da82b172746685fb1e898b48a8abf5c4fbc205d21179bb35a1d728787d9254.jpg)
+
+$2.2\%$  (3B) and  $7.2\%$  (7B), indicating that standard SFT alone does not enable effective cross-task multimodal compositional reasoning. While multimodal RL training improves upon SFT, achieving  $17.4\%$  (3B) and  $31.2\%$  (7B), it remains far below pure-text RL levels, highlighting inherent challenges in multimodal compositional reasoning for cross-task scenario.
+
+Limitations of SFT-init RL. Initializing multimodal RL training from an SFT checkpoint (SFT-init RL) does not enhance compositional performance; instead, accuracy remains extremely low (2.6% for 3B, dropping to 1.0% for 7B). This indicates that RL struggles to correct flawed compositional strategies established by prior SFT training. We attribute this issue primarily to our hybrid training strategy, which alternates evenly between SFT and RL updates (half-half step attribution). Although SFT-init successfully boosts RL performance on individual tasks as expected due to explicit reasoning path supervision during SFT, it simultaneously imposes strong biases that limit RL's flexibility in adjusting compositional strategies. Consequently, while SFT-init can facilitate faster learning of individual skills, it may inadvertently hinder compositional generalization across tasks compared to RL trained without SFT initialization.
+
+Impact of progress-reward grounding (RL-Ground). Motivated by these failures, we explore a strategy that explicitly targets visual-to-text alignment and reasoning decomposition. Our proposed potential solution, RL-Ground, combines two key components: (1) a <caption> block that forces the model to first describe visual content in natural language before entering the reasoning stage, and (2) a fine-grained progress reward that provides supervision at the level of intermediate vision-grounded reasoning (e.g., correct shape area computation or distance estimation), rather than only at the final answer.
+
+![](images/982c247d473127a6a4b9e5a04cec519e2be808d05142fdaba527a20fffb3ab82.jpg)  
+Figure 6: Illustration of RL-Ground framework.
+
+As shown in Fig. 6, the <caption> module promotes an early transformation of visual inputs into language, triggering the reasoning together with text instead of merely visual inputs. Simultaneously, the progress reward allows the model to build up compositional reasoning over verifiable subgoals, reducing reliance on sparse final accuracy reward signals. This structure leads to significant gains: RL-
+
+Ground achieves  $28.6\%$  (3B) and  $52.8\%$  (7B), surpassing the other evaluated post-training strategies. We provide more ablations and training progress on RL-Ground in the supplementary materials.
+
+These findings clearly illustrate that cross-task compositional generalization is challenging under multimodal settings. Simple exposure to independently trained sub-skills via standard SFT or RL alone proves insufficient, and even harmful (in the case of SFT). Instead, caption-before-think formats coupled with dense progress reward significantly improve multimodal compositional reasoning, offering a promising direction for future training paradigms.
+
+# 4.3.3 RQ3: Generalization to OOD Compositional Task
+
+![](images/53580b7967eb754e75c2839e68a7e659637f24ed9b661ee1d0aac986430e296c.jpg)  
+Figure 7: Comparison of Results for generalization to OOD independent and compositional tasks.
+
+SFT vs. RL under OOD generalization. We evaluate model robustness on OOD tasks that modify the reasoning objective while maintaining similar multimodal inputs. Results show that SFT exhibits task-dependent generalization: it completely fails on the largest-area task (1.8% for 3B and 1.4% for 7B), but achieves moderate accuracy on the grid position of the farthest shape task (74.2% for 3B and 27.8% for 7B), suggesting limited transferability when visual structures closely align with training data. However, it fails on the OOD compositional task (1.2% for 3B and 2.8% for 7B), reaffirming its lack of compositional flexibility. In contrast, RL generalizes strongly to independent OOD tasks, achieving 95.8% (3B) and 95% (7B) on the largest-area task, and 44% (3B) and 83.8% (7B) on the farthest shape task, comparable to or better than in-domain individual task results from Fig. 5. For the OOD compositional task, RL reveals a scale-dependent trend: while it performs poorly on 3B (6.6%), the 7B model generalizes better, achieving 40.4% and surpassing its in-distribution compositional performance. These results indicate that while SFT's generalization is brittle and task-specific, RL better supports abstract reasoning transfer, particularly at larger model scales.
+
+RL-Ground achieves robust generalization. RL-Ground consistently achieves high accuracy across all OOD tasks. For individual OOD task, it outperforms all other methods, reaching  $83.2\%$  (3B) and  $88.6\%$  (7B) on the farthest shape task(Grid Position OOD). RL-Ground performs merely slightly behind the original RL method in Shape Area OOD task. Notably, RL-Ground also demonstrates best performance on the OOD compositional task, achieving  $38\%$  on 3B and  $52.8\%$  on 7B model, both matching or exceeding its in-domain compositional performance. These results confirm that combining caption-before-think with progress reward not only enhances in-distribution compositional ability, but also significantly improves robustness to unseen task objectives.
+
+In a nutshell, while SFT exhibits limited and inconsistent generalization under OOD shifts, RL generalizes well to new reasoning objectives, particularly at larger model size. RL-Ground demonstrates the strongest and most stable generalization across all settings, showing clear advantages for both individual and compositional OOD tasks.
+
+# 4.4 Generalization and Grounding Analysis of RL-Ground
+
+While RL-Ground was developed and evaluated on synthetic tasks from ComPABench to facilitate controlled analysis, the techniques it introduces are broadly applicable to real-world VQA settings. In particular, the use of explicit visual grounding via the <caption> step and step-wise progress
+
+reward are not bound to specific data domains. These mechanisms address fundamental challenges in VLMs such as compositional reasoning, grounding robustness, and sparse reward learning.
+
+To validate that RL-Ground is not overly tailored to synthetic setups, we conduct two complementary studies. First, we introduce two auxiliary visual grounding tasks, Shape Area Grounding and Grid Position Grounding. The two tasks isolate grounding ability from high-level reasoning. In Shape Area Grounding, the model is prompted with a question such as "What is the area of the red triangle?" requiring it to detect the correct shape, recognize its geometry, and compute the corresponding area. Grid Position Grounding challenges the model to locate an object based on color and shape and return its precise row and column index on a grid. Both tasks demand perception and test whether the model extracts the relevant visual facts without performing full reasoning chains.
+
+Table 2 reports results across grounding and reasoning tasks for all evaluated strategies. RL-Ground achieves high scores in both grounding subtasks with  $96.2\%$  on Shape Area Grounding and  $88.6\%$  on Grid Position Grounding, demonstrating its superior visual grounding alignment via the <caption> step and step-wise reward optimization. These improvements directly translate into the highest compositional reasoning accuracy  $(52.8\%)$ , validating the hypothesis that strong grounding enables generalizable multi-hop reasoning.
+
+Interestingly, while standard RL already provides significant gains over SFT with a substantial boost in SA Grounding (96.6% vs. 1.2%) and GP Grounding (74.8% vs. 65.4%), the RL-Ground variant consistently outperforms both, indicating the added benefit of incorporating explicit grounding and structured rewards. This performance trend mirrors the results on the compositional task, where RL-Ground again leads by a wide margin, affirming that improved visual grounding is tightly correlated with stronger generalization and reasoning capabilities.
+
+Table 2: Evaluation of VLMs across grounding and compositional reasoning tasks. "SA" = Shape Area, "GP" = Grid Position. RL-Ground achieves the strongest visual grounding and generalization.  
+
+<table><tr><td>Model</td><td>SA</td><td>SA Grounding</td><td>GP</td><td>GP Grounding</td><td>Compositional</td></tr><tr><td>Qwen2.5-VL-7B-Instruct</td><td>20.8</td><td>89.4</td><td>4.8</td><td>24.4</td><td>13.8</td></tr><tr><td>SFT</td><td>73.6</td><td>1.2</td><td>91.8</td><td>65.4</td><td>7.2</td></tr><tr><td>RL</td><td>74.6</td><td>96.6</td><td>83.2</td><td>74.8</td><td>31.2</td></tr><tr><td>SFT-init-RL</td><td>76.6</td><td>1.2</td><td>89.8</td><td>43.6</td><td>1.0</td></tr><tr><td>RL-Ground</td><td>73.8</td><td>96.2</td><td>88.4</td><td>88.6</td><td>52.8</td></tr></table>
+
+Beyond ComPABench, we assess whether RL-Ground generalizes to real-world language patterns. We perform a zero-shot evaluation on the Geometry3K dataset [Lu et al., 2021], which is a natural language-based VQA benchmark involving spatial and geographic reasoning. RL-Ground, trained only on synthetic data, achieves  $21.2\%$  accuracy, outperforming the base Qwen2.5-VL-3B-Instruct  $(14.8\%)$ , SFT  $(9.2\%)$ , and RL-only  $(17.8\%)$  models. This demonstrates that caption-before-reasoning and progress reward confer robustness that transfers across tasks and domains.
+
+# 5 Conclusion
+
+We introduce ComPABench, a benchmark for evaluating compositional ability in VLMs across cross-modal, cross-task, and OOD settings. Inspired by recent RLVR progress in language models, we assess whether similar training strategies improve compositional reasoning in VLMs. Through comparisons of SFT, RL, and SFT-initialized RL, we find that RL better integrates independently learned skills, especially in cross-task and OOD scenarios. Yet, compositional reasoning with visual inputs remains challenging. Our proposed RL-Ground strategy, combining caption-beforereasoning and progress rewards, yields strong in-distribution and out-of-distribution gains in terms of compositional ability across tasks, underscoring the value of structured prompting and grounded supervision for improving the compositional generalization of multimodal reasoning. To further understand its effect, we evaluate RL-Ground on auxiliary visual grounding tasks and find that it substantially improves grounding fidelity compared to baselines, which in turn correlates with higher compositional performance. Moreover, RL-Ground generalizes effectively to real-world visual question answering, despite being trained only on synthetic data. These findings highlight the transferability of visual-grounded RL and point to new directions for robust multimodal reasoning.
+
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+Peng Wang, Shuai Bai, Sinan Tan, Shijie Wang, Zhihao Fan, Jinze Bai, Keqin Chen, Xuejing Liu, Jialin Wang, Wenbin Ge, et al. Qwen2-v1: Enhancing vision-language model's perception of the world at any resolution. arXiv preprint arXiv:2409.12191, 2024a.  
+Weiyun Wang, Zhangwei Gao, Lianjie Chen, Zhe Chen, Jinguo Zhu, Xiangyu Zhao, Yangzhou Liu, Yue Cao, Shenglong Ye, Xizhou Zhu, et al. Visualprm: An effective process reward model for multimodal reasoning. arXiv preprint arXiv:2503.10291, 2025.  
+Xinyi Wang, Antonis Antoniades, Yanai Elazar, Alfonso Amayuelas, Alon Albalak, Kexun Zhang, and William Yang Wang. Generalization vs memorization: Tracing language models' capabilities back to pretraining data. arXiv preprint arXiv:2407.14985, 2024b.  
+Zhiyu Wu, Xiaokang Chen, Zizheng Pan, Xingchao Liu, Wen Liu, Damai Dai, Huazuo Gao, Yiyang Ma, Chengyue Wu, Bingxuan Wang, et al. Deepseek-vl2: Mixture-of-experts vision-language models for advanced multimodal understanding. arXiv preprint arXiv:2412.10302, 2024.  
+Guowei Xu, Peng Jin, Li Hao, Yibing Song, Lichao Sun, and Li Yuan. Llava-ol: Let vision language models reason step-by-step. arXiv preprint arXiv:2411.10440, 2024a.  
+Zhuoyan Xu, Zhenmei Shi, and Yingyu Liang. Do large language models have compositional ability? an investigation into limitations and scalability. ArXiv, abs/2407.15720, 2024b. URL https://api-semanticscholar.org/CorpusID:270847724.  
+Yi Yang, Xiaoxuan He, Hongkun Pan, Xiyan Jiang, Yan Deng, Xingtao Yang, Haoyu Lu, Dacheng Yin, Fengyun Rao, Minfeng Zhu, et al. R1-onevision: Advancing generalized multimodal reasoning through cross-modal formalization. arXiv preprint arXiv:2503.10615, 2025.  
+Huanjin Yao, Jiaxing Huang, Wenhao Wu, Jingyi Zhang, Yibo Wang, Shunyu Liu, Yingjie Wang, Yuxin Song, Haocheng Feng, Li Shen, et al. Mulberry: Empowering mllm with o1-like reasoning and reflection via collective monte carlo tree search. arXiv preprint arXiv:2412.18319, 2024.  
+Yang Yue, Zhiqi Chen, Rui Lu, Andrew Zhao, Zhaokai Wang, Shiji Song, and Gao Huang. Does reinforcement learning really incentivize reasoning capacity in llms beyond the base model? arXiv preprint arXiv:2504.13837, 2025.  
+Yufei Zhan, Yousong Zhu, Shurong Zheng, Hongyin Zhao, Fan Yang, Ming Tang, and Jinqiao Wang. Vision-r1: Evolving human-free alignment in large vision-language models via vision-guided reinforcement learning. arXiv preprint arXiv:2503.18013, 2025.  
+Jun Zhao, Jingqi Tong, Yurong Mou, Ming Zhang, Qi Zhang, and Xuanjing Huang. Exploring the compositional deficiency of large language models in mathematical reasoning. arXiv preprint arXiv:2405.06680, 2024.  
+Rosie Zhao, Alexandru Meterez, Sham Kakade, Cengiz Pehlevan, Samy Jelassi, and Eran Malach. Echo chamber: Rl post-training amplifies behaviors learned in pretraining. arXiv preprint arXiv:2504.07912, 2025.
+
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+
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+
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+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
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+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
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+
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+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
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+
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+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
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+
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+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
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+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
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+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+# Answer: [Yes]
+
+Justification: As seen in Sec 1.
+
+# Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
\ No newline at end of file
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+# Unveiling the Learning Mind of Language Models: A Cognitive Framework and Empirical Study
+
+Zhengyu Hu $^{1,2}$ , Jianxun Lian $^{3*}$ , Zheyuan Xiao $^{1,2}$ , Seraphina Zhang $^{4}$ , Tianfu Wang $^{1,2}$ , Nicholas Jing Yuan $^{5}$ , Xing Xie $^{3}$ , Hui Xiong $^{1,2}$ $^{1}$ Thrust of Artificial Intelligence, The Hong Kong University of Science and Technology (Guangzhou), China  
+ $^{2}$ Department of Computer Science and Engineering, The Hong Kong University of Science and Technology Hong Kong SAR, China  
+ $^{3}$  Microsoft Research Asia  ${}^{4}$ University of Cambridge  ${}^{5}$ Microsoft.
+
+# Abstract
+
+Large language models (LLMs) have shown impressive capabilities across tasks such as mathematics, coding, and reasoning, yet their learning ability, which is crucial for adapting to dynamic environments and acquiring new knowledge, remains underexplored. In this work, we address this gap by introducing a framework inspired by cognitive psychology and education. Specifically, we decompose general learning ability into three distinct, complementary dimensions: Learning from Instructor (acquiring knowledge via explicit guidance), Learning from Concept (internalizing abstract structures and generalizing to new contexts), and Learning from Experience (adapting through accumulated exploration and feedback). We conduct a comprehensive empirical study across the three learning dimensions and identify several insightful findings, such as (i) interaction improves learning; (ii) conceptual understanding is scale-emergent and benefits larger models; and (iii) LLMs are effective few-shot learners but not many-shot learners. Based on our framework and empirical findings, we introduce a benchmark that provides a unified and realistic evaluation of LLMs' general learning abilities across three learning cognition dimensions. It enables diagnostic insights and supports evaluation and development of more adaptive and human-like models. The code is available here<sup>2</sup>.
+
+# 1 Introduction
+
+Large language models (LLMs) [3, 2, 88, 46] have demonstrated impressive capabilities across diverse tasks such as mathematics [97, 94, 92], coding [60, 11, 39], and reasoning [29, 90, 99, 50]. Despite these achievements, as LLM-driven systems increasingly integrate into societal roles, serving as AI companions [87, 8, 64] or AI employees [6, 53, 75, 20, 37], they must possess robust learning capabilities. Effective learning is essential for AI agents to adapt dynamically to diverse and changing environments, continuously acquire new knowledge, and autonomously respond to novel contexts. However, research on the learning ability of LLMs remains scarce, with little work systematically investigating how well LLMs can acquire and generalize new knowledge across tasks.
+
+To address this gap, we aim to systematically investigate the learning ability of LLMs. Drawing insights from cognitive science and educational theory [24, 42, 14], we first analogize from human learning processes to identify three fundamental dimensions of learning: (1) learning from instructor, where the model acquires knowledge through guided interaction [18, 72]; (2) learning from concept,
+
+![](images/c3ec29bbfdd70f0ce1503bae5d36f0901bd0bfea8266f0a5483457c969d83121.jpg)  
+(a) Learning from Instructor
+
+![](images/f8093556f03ea8195187bca4e81d04b6fd86cb89e6557fa53ba33936784c3a0b.jpg)  
+Figure 1: Overview of our proposed cognitive framework for evaluating general learning abilities in LLMs. We decompose learning into three core types: (a) Learning from Instructor; (b) Learning from Concept; and (c) Learning from Experience.
+
+![](images/ca9a6f37a9a2aebcc8f498437d1554e47205bc8e1d01837a2ce0d02c7e974be9.jpg)
+
+![](images/cfd570ddc5da2526216b0bde380775631fe19516d42dc4dbfc696be5b40806e8.jpg)  
+(b) Learning from Concept
+
+![](images/61dee69f40be1c2cb372cc28259c263b9722edd08a93c42e23e7cf29c02adc1d.jpg)
+
+![](images/61e26b8ad3c1682968c936e5a018b5fd33a566851617431bfda5f988cdb36c47.jpg)
+
+![](images/386f4c95dc34cec1489d61227118320d718824d51dfb3fc552da703a87593a86.jpg)  
+(c) Learning from Experience
+
+where the model internalizes structured abstractions and generalizes them to downstream tasks [56, 5]; and (3) learning from experience, where the model adapts based on accumulated trajectories or exploration-feedback [42, 98]. For each dimension, we design targeted experimental paradigms to operationalize the corresponding learning mechanism. In Learning from Instructor, we simulate tutor-learner settings with and without interactive clarification, demonstrating that interaction within instructor and learner consistently improves model's learning ability. In Learning from Concept, we evaluate the impact of injecting abstract conceptual knowledge in competitive environments (i.e., TextArena [28]), showing that (i) conceptual understanding is scale-emergent, and (ii) injecting structured domain knowledge can provide a tangible advantage, if the model is sufficiently capable of internalizing it. In Learning from Experience, it is a crucial capability for adapting to novel environments and acquiring new knowledge autonomously. While they are few-shot learners, they struggle in many-shot settings due to the challenges of long-context integration. This highlights the importance of a unified benchmark that can evaluate LLMs' general learning abilities across cognitive dimensions.
+
+Building on this framework and empirical insights, we consolidate our findings into a unified benchmark, LearnArena, that reflects realistic and cognitively grounded learning scenarios. It enables principled evaluation of LLMs' learning behavior across three learning aspects, and provides a foundation for advancing learning capabilities. Empirical results reveal that learning ability benefits from increased capacity, but faces a bottleneck; architectural and training advancements play a crucial role in further enhancing learning capability.
+
+Our contributions are as follows: (1) We present the first work to explicitly evaluate and analyze the general learning ability of LLMs. Grounded in cognitive science, we propose a principled decomposition into three dimensions: learning from instructor, learning from concept, and learning from experience, each with dedicated methodologies. (2) We conduct a comprehensive empirical study across the three learning dimensions, revealing three key insights: interaction improves learning in instructor-based settings; conceptual understanding is scale-emergent and benefits larger models; and LLMs are effective few-shot learners but not many-shot learners. (3) Based on our framework and findings, we introduce a benchmark, LearnArena, that offers a unified and realistic evaluation of LLMs' general learning ability across three cognitive dimensions. It enables diagnostic insights and supports the development of more adaptive and human-like models.
+
+# 2 Related Work
+
+Evaluation of Large Language Models. LLMs have been extensively evaluated on tasks involving linguistic competence [77, 9, 40], factual recall [27, 81], reasoning [49, 51, 90, 99, 96, 95], instruction following [35, 45, 36, 44, 34], and multitask generalization [59, 17, 89, 33], with benchmarks such as MMLU [30], BIG-Bench [1], and HELM [47] highlighting broad capabilities and emergent scaling trends. However, most evaluations focus on static zero- or few-shot performance, offering limited insight into how models learn or generalize from experience. To address this, recent studies have begun shifting toward more dynamic formulations that probe the learning behavior of LLMs. [21] categorize in-context learning (ICL) as inference-time adaptation and identify key factors for success. [55] show that ICL performance depends more on input format than label correctness. [4] find that many-shot ICL yields diminishing returns as context length grows. [68] introduce Toolformer, enabling models to self-supervise API use. [91] develop models that improve via internally generated feedback. Nevertheless, most such studies are piecemeal, lacking a unified theoretical lens or
+
+systematic analysis for comparing learning processes across tasks. Our work addresses this gap by introducing a cognitively grounded decomposition to systematically analyze learning ability.
+
+Learning Ability of Large Language Models. Recent advances in large language models have demonstrated remarkable capabilities in adapting and generalizing knowledge beyond traditional static evaluations [7, 84, 78]. Cognitive psychology and educational theory have long emphasized that effective learning involves not only direct instruction and guided feedback [14, 18], but also the ability to abstract conceptual structures and adapt from experience [42, 56, 24]. Motivated by these insights, we adopt a tripartite perspective, learning from instructor, concept, and experience, that mirrors rapid-instruction-to-implementation learning (RITL) in humans [72], classical rule-based accounts of cognition [56, 24], and experiential learning theory [42]. While contemporary LLM benchmarks extensively measure task-specific performance [1, 47, 30, 54], they rarely probe how models internalize instructions, extract explicit rules, or accumulate and reuse episodic experience, capacities increasingly highlighted by self-evolving or self-refinement agents [26, 32]. This gap underscores the need for a cognitively informed evaluation framework that systematically characterizes the general learning abilities of LLMs across these complementary dimensions.
+
+# 3 A Cognitive Framework for Analyzing Learning Abilities in LLMs
+
+To systematically investigate the general learning capabilities of LLMs, we propose a cognitively grounded framework comprising three paradigms: Learning from Instructor (LfI), Learning from Concept (LfC), and Learning from Experience (LfE). LfI captures learning via explicit guidance, supported by evidence that instruction accelerates task acquisition and reduces cognitive load [18, 72, 14, 76]. LfC reflects structured abstraction from static concepts, grounded in symbolic cognitive theories [56, 24, 32] despite critiques of their rigidity [5]. LfE models adaptation through feedback and interaction, consistent with experiential learning theory [42, 38] and recent work showing LLMs improve via self-refinement [98, 26].
+
+Formally, we define a learning task  $t$  as a tuple  $(\mathcal{X}_t, \mathcal{Y}_t, \mathcal{C}_t)$ , where  $\mathcal{X}_t$  denotes the input space,  $\mathcal{Y}_t$  the output space, and  $\mathcal{C}_t$  the context space, which characterizes the type of information the model must incorporate to generate correct outputs. The learning objective is to establish a predictive mapping:
+
+$$
+f _ {\theta}: \mathcal {X} _ {t} \times \mathcal {C} _ {t} \rightarrow \mathcal {Y} _ {t} \tag {1}
+$$
+
+where the form of  $\mathcal{C}_t$  is determined by the learning paradigm.
+
+In Learning from Instructor (LfI), the model acquires task knowledge through explicit interactions with an external instructor, which may involve demonstrations, explanations, or corrections. These interactions form a communication-based supervision channel, where the instructor incrementally shapes the model's understanding of the task. Formally, we define the context as  $\mathcal{C}_t = \mathcal{I}_t$ , where  $\mathcal{I}_t$  encodes structured instructional signals such as natural language directives, step-by-step exemplars, or annotated feedback. The model learns to integrate these interactive signals into its prediction process:  $f_{\theta}(x,\mathcal{I}_t)\rightarrow \hat{y}$ .
+
+In Learning from Concept (LfC), the model receives static, abstract conceptual knowledge that captures abstract properties or domain principles. Unlike instructor signals that evolve through interaction, concepts are predefined and non-interactive, such as definitions, category structures, or rule schemata. We define the context as  $\mathcal{C}_t = \mathcal{K}_t$ , where  $\mathcal{K}_t$  represents a set of symbolic or linguistic descriptions encoding domain-relevant concepts. The model is expected to internalize these abstractions and apply them to generate predictions consistent with conceptual constraints:  $f_{\theta}(x,\mathcal{K}_t)\rightarrow \hat{y}$ , subject to  $\hat{y}$  being semantically aligned with  $\mathcal{K}_t$ .
+
+In Learning from Experience (LfE), the model accumulates and utilizes its own prior interaction history to adapt future behavior. Unlike LfC, where supervision is static and predefined, LfI and LfE both involve dynamic supervision. In LfI, the model receives guidance through interactive instructions, while in LfE, supervision emerges from sequences of interactional feedback, potentially including successful or failed trajectories, user preferences, or latent task rewards. The context is defined as  $\mathcal{C}_t = \tau_t = \{s_1,s_2,\dots ,s_k\}$ , where each  $s_i$  represents a structured snapshot from a past interaction, such as input-output pairs, action-state transitions, or dialog turns. The model is expected to generalize from this accumulated experience to improve decision-making:  $f_{\theta}(x,\tau_t)\rightarrow \hat{y}$ .
+
+These three paradigms offer a unified and cognitively motivated framework for evaluating how LLMs acquire, organize, and apply knowledge. LfI emphasizes guided learning via instructor interaction;
+
+LfC emphasizes structural abstraction from fixed knowledge; and LfE emphasizes adaptation through situated experience. This decomposition allows us to probe distinct facets of model generalization and align LLM evaluation with core dimensions of human learning.
+
+# 4 Learning from Instructor (LfI)
+
+We investigate Learning from Instructor (LfI), where models acquire task knowledge via structured guidance, such as demonstrations, explanations, or feedback. We evaluate this setting across two dimensions: (1) Passive Consumption vs. Interactive Clarification, and (2) Scaling Learner.
+
+![](images/f8c7cf4feb73157b695dc2afd2f12d805523605d5a5cdf8b2b6e0fefce1451ac.jpg)  
+Figure 2: Comparison of learner performance under Passive Consumption and Interactive Clarification paradigms across eight mathematical benchmarks.
+
+![](images/6856fd305d18a58fb6ce5b75e63927a791ee60e28b98755614dd03818b894537.jpg)  
+Origin Passive Interactive
+
+![](images/58326d916dd0ff8b42c915728312222e9406814fc758fa85d3428c8494492f5f.jpg)
+
+![](images/c88f7d48aa86eaefd8f37cd92653525d5628a3aa19c5a74ed5fa909aa30b87ff.jpg)
+
+# 4.1 Passive Consumption vs. Interactive Clarification
+
+Experiment Setup. To examine the impact of interactivity in instruction-based learning, we adopt the MagpieMath [86] dataset, a high-quality math dataset generated autoregressively by Qwen2.5-Math-72B. An instructor model is first trained on this dataset and subsequently used to teach a separate learner model under two learning paradigms: (1) Passive Consumption, in which the learner receives only direct solutions from the instructor without any further interaction, and (2) Interactive Clarification, where the learner is allowed to ask clarification questions following each instructor response, and the instructor provides targeted, follow-up explanations. In this setting, the learner is restricted to a single clarification question per sample to ensure consistency across training examples. Both the initial answers and the clarification responses are aggregated as supervision for the learner. We experiment with four representative LLM families: LLaMA3.1-8B, Qwen2.5-7B, Mistral-7B, and Phi-3-mini—each used as both instructor and learner to test robustness across pairings. Learners are trained on outputs from matched instructors within each setting. Evaluation is conducted on eight diverse mathematical benchmarks: GSM8K [16], SVAMP [63], MATH [31], NumGLUE [57], SimulEq [43], AQuA [48], SAT [101], and MMLU-Math [30], covering competencies from basic arithmetic to multi-step symbolic reasoning and standardized test preparation. Implementation and generation details are provided in Appendix A.3.
+
+Main Result. As shown in Figure 2, learners trained under the Interactive Clarification paradigm consistently outperform those trained via Passive Consumption across all eight evaluation benchmarks. This performance gain highlights that LLMs are capable of leveraging interactive feedback to improve task understanding, resembling human-like active learning behaviors. Notably, the gains vary across model families. Mid-sized models such as Mistral-7B and Qwen2.5-7B show substantial improvements, suggesting a strong capacity to benefit from additional instructional signals. In contrast, the smallest model, Phi-3-mini, shows only marginal improvement, indicating limited ability to engage in or benefit from interactive clarification, which suggests that active learning capabilities are not uniform across models and may depend on model capacity or architecture.
+
+# 4.2 Scaling Learner
+
+Experiment Setup. While Section 4.1 explores the effect of instructional interactivity across different model families, here we focus on how scaling the learner model within the same family influences learning outcomes. Specifically, we fix the instructor as Qwen2.5-72B and vary the learner
+
+![](images/349646dd5a4cf5adf61f862ba801b65c07e430213f2eedcccf220d636c93efb5.jpg)
+
+![](images/8e4942b7ba868de068983d15d6827e67ee33a64cc16946f22843b6feb72816a9.jpg)
+
+![](images/fc2f52535780a563acee37649b6d664bfc8e5979338ce827d4fbee745c71b457.jpg)
+
+![](images/974d16950b93ff0c80bab4a6b73ad271dac36eb3a591b6bf3085413001cbfe0b.jpg)
+
+![](images/15f93b1eb338e5445e95c33b14489bbce3c9607d969748f07e1a3da5daae9542.jpg)  
+Figure 3: Scaling learner models (Qwen2.5 variants) with a fixed Qwen2.5-72B instructor. Larger learners learn more effectively, and interactive clarification further improves outcomes.
+
+![](images/bb9aee47b2c2b0c11f403377c02e5432bb110c3f533ccb586a0fc36e9460f6ce.jpg)
+
+![](images/86f9a31f32b66e28f858bddc09811d1e38fe20af3ba0b1cd40699fdb97f2cfc3.jpg)
+
+![](images/8880bc0262941f09b11f53f83dbca77303f0e406df305106a632f61ab82acb74.jpg)
+
+model across Qwen2.5-1.5B, Qwen2.5-7B, and Qwen2.5-32B. We evaluate the impact of instructional interactivity by comparing the Passive Consumption and Interactive Clarification paradigms. Training data and evaluation benchmarks are kept consistent with Section 4.1. Results are shown in Figure 3, where Origin refers to the performance of the untrained model, and Norm Acc denotes the accuracy of the post-training under each setting divided by its corresponding untrained baseline. Further implementation details are provided in Appendix A.3.
+
+Main Result. As shown in Figure 3, we observe a strong positive correlation between learner model scale and normalized learning gains: larger models consistently achieve higher performance improvements across all evaluation tasks. This trend holds for both Passive Consumption and Interactive Clarification, with the benefits of interactivity becoming more pronounced as model capacity increases. Qwen2.5-32B demonstrates substantial gains under interactive supervision across all eight math-related benchmarks. For instance, in NumGLUE and SVAMP, it achieves  $+25.7\%$  and  $+23.8\%$  normalized improvement, compared to  $+18.2\%$  and  $+16.8\%$  in the passive setting—showing clear added value from clarification feedback. Similar trends appear in SimULEQ and MATH, where interactive training leads to  $+22.4\%$  and  $+11.0\%$  improvements, respectively. In contrast, Qwen2.5-1.5B shows smaller gains overall, and the gap between passive and interactive learning is narrower. For example, in GSM8K and SAT, improvements under interactivity are only  $+5.0\%$  and  $+3.5\%$  compared to  $+3.8\%$  and  $+2.8\%$  in the passive case. This indicates that limited capacity restricts the ability to absorb and act on richer supervision signals.
+
+# 4.3 Scale Instructor
+
+Experiment Setup. We examine how the scale of the instructor model influences learning effectiveness under different supervision paradigms. Specifically, we fix the learner model as Qwen2.5-7B and vary the instructor across Qwen2.5-1.5B, 7B, 32B, and 72B. In each setting, we compare two learning paradigms: Passive Consumption, where the learner receives static demonstrations, and Interactive Clarification, where the learner can ask follow-up questions. The training setup and evaluation benchmarks follow those in Section 4.1, and additional implementation details are provided in Appendix A.3.
+
+Main Result. However, this trend breaks when the instructor model is extremely small (e.g., Qwen2.5-1.5B). In such cases, interactive clarification may introduce noise or misleading guidance, leading to degraded performance compared to passive learning. For example, on tasks such as GSM8K, MATH, and SAT in Figure 4, the interactive setting underperforms the passive counterpart, suggesting that low-quality instructors are unable to generate helpful responses to clarification queries, which in turn hinders learner improvement. As shown in Figure 4, larger instructor models consistently lead to better learning outcomes. When the instructor is strong (e.g., Qwen2.5-32B or 72B), interactive supervision significantly improves learner performance compared to passive
+
+instruction. This suggests that powerful instructors can provide useful clarifications that facilitate deeper understanding and generalization. However, when the instructor is too small (e.g., Qwen2.5-1.5B), the benefits of interactivity diminish or even become negative. In tasks such as GSM8K, MATH, and SAT, we observe that learners trained with Interactive Clarification underperform compared to those trained passively. This indicates that weak instructors may introduce noise or incorrect signals during interaction, ultimately hindering learning. These findings highlight the importance of instructor quality in interactive learning and suggest that poorly performing instructors can negatively affect the training process.
+
+![](images/f40b4583ab417385cb0260aafbb9b16ec67032f59235fa6c06808efa0876bf51.jpg)  
+Figure 4: Scaling instructor models (Qwen2.5 variants) with a fixed Qwen2.5-7B learner. Weak instructors can degrade learning in the interactive setting; however, interaction becomes beneficial as instructor capability increases.
+
+# 5 Learning from Concept (LfC)
+
+We study Learning from Concept (LfC), where models leverage static, abstract knowledge, such as rules, definitions, or structured representations, to guide behavior or reasoning. We evaluate this ability in two settings: (1) Structured Knowledge Injection in Competitive Environments, testing whether models can integrate conceptual hints to improve decision-making in competitive environments, and (2) Conceptual Generalization in Logic and Planning Tasks, examining generalization from symbolic structures in logic and planning tasks.
+
+# 5.1 Structured Knowledge Injection in Competitive Environments
+
+Experiment Setup. We evaluate whether conceptual knowledge improves strategic performance in multi-agent settings using TextArena [28], a suite of competitive environments with symbolic rules and multi-turn dynamics. Each game involves two players, Player-0 and Player-1. We fix Player-0 as Qwen2.5-32B and vary Player-1 across four scales: Qwen2.5-1.5B, 7B, 14B, and 32B. Prior to gameplay, Player-1 either receives no guidance (without Concept) or is given natural language descriptions of game rules and strategies (with Concept), generated by Qwen2.5-32B. For each environment and model pair, we run 20 matches and report Player-1's win rate averaged over these games. Environments include Checkers (CH), Poker (PK), Stratego (ST), Tic Tac Toe (TT), Truth and Deception (TD), and Ultimate Tic Tac Toe (UTT). Performance is measured as Player-1's win rate against the fixed opponent. Results are shown in Figure 5. Implementation details are provided in Appendix A.4.
+
+Main Result. Results are shown in Figure 5. We observe two key trends. First, model scale significantly influences the effectiveness of conceptual guidance. For smaller models such as Qwen2.5-1.5B, injecting conceptual descriptions consistently degrades performance across all environments—with Concept underperforms without Concept, suggesting that low-capacity models struggle to integrate abstract knowledge and may treat it as distinctive noise. In contrast, as model size increases, the gap progressively narrows and eventually reverses. By Qwen2.5-14B and 32B, models consistently benefit from conceptual input across most tasks, indicating improved abstraction and utilization
+
+![](images/6003da4d9cb3d4e6a43bb25364e791fd52e79ea35f8c72b4d4227655159381e9.jpg)  
+Figure 5: Win rates of Player-1 across six competitive environments from TextArena under the LfC setting.
+
+![](images/b321c1c2856f0d97014eccae5c931411dc5436817402c8ad9ad532738297d1a6.jpg)
+
+![](images/15bae2b272cd3b5b2f984f3ca18d407d3936e3985ecd1aa4f7947bcbf56ac7d0.jpg)
+
+![](images/3d17ab7bc5eed3bd39e7ab12edeb203ec7ee592996f3df188243ddd4e61b0d37.jpg)
+
+capabilities with scale. Second, win rates across tasks also increase monotonically with model size, regardless of condition, highlighting a strong correlation between scale and general strategic competence. Notably, high-conceptual-load environments such as Stratego and Truth and Deception show the largest gains from conceptual input, especially for larger models. These results suggest that (i) conceptual understanding is scale-emergent, and (ii) injecting structured domain knowledge can provide a tangible advantage, if the model is sufficiently capable of internalizing it.
+
+# 5.2 Conceptual Generalization in Logic and Planning Tasks
+
+Experiment Setup. To evaluate rule-based generalization, we include six tasks centered on explicit symbolic or logical structures. LogicGrid and NaLogic feature structured and narrative logic puzzles [52, 23], testing deduction under constraint. Plan [79] translates classical symbolic planning domains (e.g., Gripper, Barman) into natural language settings to assess structured action reasoning. Additionally, tasks from AlfWorld [74], ScienceWorld [82], and BabyAI [10], models must complete goal-oriented tasks requiring commonsense reasoning, procedural knowledge, and spatial navigation. For the concept-injected setting, we use
+
+Table 1: Accuracy on six tasks with and without concept input (baseline / +concept). Higher scores indicate better subgoal completion or reasoning accuracy. Darker cell colors indicate larger improvements from concept input; white means no change.
+
+<table><tr><td>Qwen 2.5</td><td>7b</td><td>14b</td><td>32b</td><td>72b</td></tr><tr><td>LogicGrid</td><td>0.36/0.42</td><td>0.51/0.54</td><td>0.53/0.56</td><td>0.54/0.57</td></tr><tr><td>NaLogic</td><td>0.13/0.16</td><td>0.14/0.18</td><td>0.17/0.20</td><td>0.21/0.23</td></tr><tr><td>Plan</td><td>0.14/0.17</td><td>0.32/0.34</td><td>0.33/0.34</td><td>0.33/0.35</td></tr><tr><td>AlfWorld</td><td>0.27/0.29</td><td>0.30/0.32</td><td>0.33/0.34</td><td>0.33/0.33</td></tr><tr><td>ScienceWorld</td><td>0.16/0.19</td><td>0.26/0.29</td><td>0.33/0.35</td><td>0.41/0.42</td></tr><tr><td>BabyAI</td><td>0.40/0.42</td><td>0.41/0.44</td><td>0.46/0.46</td><td>0.51/0.51</td></tr></table>
+
+Qwen2.5-72B to generate static conceptual hints, which are provided as auxiliary input to smaller models during evaluation. Results are presented in Table 1 and Figure 5. We use subgoal-based annotations to enable fine-grained progress tracking. Implementation details are provided in Appendix A.4.
+
+Main Result. As shown in Table 1, providing external conceptual information, generated by Qwen2.5-72B, consistently improves performance across all model sizes and tasks. The improvements are most pronounced in smaller models, indicating that such auxiliary knowledge can partially offset limited inherent reasoning capacity. In symbolic reasoning tasks such as LogicGrid and NaLogic, concept injection yields consistent gains. For example, LogicGrid accuracy increases from 0.36 to 0.42 for Qwen2.5-7B, and from 0.54 to 0.57 for Qwen2.5-72B, suggesting that explicit structural cues remain beneficial even for larger models, though with reduced marginal benefit. In planning tasks like Plan, the gains are smaller but stable, e.g., from 0.14 to 0.17 in Qwen2.5-7B and 0.32 to 0.34 in Qwen2.5-14B, indicating that models can utilize conceptual breakdowns of action dynamics, particularly when capacity is constrained. For interactive environments such as AlfWorld, ScienceWorld, and BabyAI, we observe moderate but limited improvements. While concept injection provides useful structural cues, these tasks involve complex dynamics, such as exploration, multi-step control, and environment grounding, that are not fully addressed by static conceptual input. Notably, performance gains plateau for larger models, indicating that further progress may require richer supervision signals, interactive fine-tuning, or stronger state-tracking mechanisms beyond static abstractions.
+
+![](images/22d9c428c9d40c6b7df431048e688f65de9d2e05474cc079e6276ff8a3c6df61.jpg)  
+Figure 6: Win rates of Player-1 across six competitive environments from TextArena under the LfE setting.
+
+# 6 Learning from Experience (LfE)
+
+We investigate Learning from Experience (LfE), where models adapt by accumulating and utilizing prior interaction history. We evaluate this ability in two settings: (1) Experience-Driven Adaptation in Competitive Games, where agents condition on past multi-round play to adapt strategic behavior; and (2) In-context Examples as Off-policy Exploration Supervision, where LLMs learn from ICL-style demonstrations viewed as episodic traces.
+
+# 6.1 Experience-Driven Adaptation in Competitive Games
+
+Experiment Setup. We adopt the same experimental setup as in Section 5.1, using TextArena [28], a suite of competitive environments featuring symbolic rules and multi-turn dynamics. The results are shown in Figure 6. Here, "with experience" refers to a setting where, during the  $k$ -th game, the player selects three prior game experiences from rounds 0 to  $k - 1$  (or all available ones if fewer than three exist), which are then used as ICL examples, allowing the model to learn from past experiences. Implementation details are provided in Appendix A.5.
+
+Main Result. Figure 6 presents win rates of Player-1 across six competitive environments in the TextArena benchmark, under two supervision regimes: without experience and with experience. Our analysis yields the following key findings: First, we observe a clear scaling trend in the efficacy of experience-based learning. For smaller models such as Qwen2.5-1.5B, the inclusion of past game trajectories generally leads to performance degradation across all environments. For example, in Checkers and Stratego, win rates drop from 0.30 to 0.11 and from 0.45 to 0.20, respectively. This suggests that low-capacity models struggle to extract relevant patterns from prior games and may treat such input as noise or distractors, leading to suboptimal decisions. Second, as model size increases, the ability to benefit from experience becomes more pronounced. Qwen2.5-7B exhibits mixed but improving trends, while Qwen2.5-14B and Qwen2.5-32B demonstrate consistent performance gains across most games when experience is incorporated. Notably, Qwen2.5-32B shows substantial improvements in complex environments like Stratego (0.6 to 0.8) and TruthAndDeception (0.6 to 0.9), indicating that larger models can better internalize structured histories and use them to inform strategy. Third, independent of experience usage, we observe a monotonic increase in win rates with model scale. This finding is consistent with results from learning from concept in Section 5.1, and confirms that larger models possess stronger general gameplay competence. Lastly, compared to concept-based generalization (Section 5.2), experience-based learning appears to be more cognitively demanding. Unlike abstract rules, game histories are long, structurally rich, and often noisy, posing challenges for information extraction and reuse. Even for Qwen2.5-32B, environments like UltimateTicTacToe show only modest gains, highlighting the inherent difficulty of experience-based generalization. These results demonstrate that (i) learning from experience is a scale-emergent ability, and (ii) its effectiveness is tightly coupled with a model's capacity to integrate, abstract, and act upon past interactions.
+
+# 6.2 In-context Examples as Off-policy Exploration Supervisions
+
+Experiment Setup. To investigate how accumulated experience influences model behavior, we treat in-context examples as trajectories encoding implicit supervision. This setup parallels off-policy learning: unlike competitive gameplay in Section 6.1, where models adapt through active interaction
+
+![](images/7ee518e3feb40a52eb4c5d99cc90e12fc61c5e5aafc0a49fedf3f9f367689228.jpg)
+
+![](images/de261d271945955ec52468d032afdc7c0988291efc02fd24d15f79df2373e039.jpg)
+
+![](images/97cec8593d4a6af6818ef0314253b53c5a59b0da9a8017cc7a7ebb02920cc67e.jpg)  
+(a) Performance with different numbers of ICL examples.
+
+![](images/01c50755c46f66d368ff7f69bb1669540e764dcdff4e24207018ad9f3f9062bc.jpg)
+
+![](images/69d60e432baca5e01b3ad3ceb8169f04d2dcfb9ec2cad59e0d909dfef842c71a.jpg)  
+(b) Evaluating model behavior in the many-shot in-context learning setting.
+
+![](images/a9e94c6c2f0eff08f16bf2e70c7348476be8a4731cb0c899f31a92e11c584b99.jpg)  
+(c) Comparing in-context learning with instruction tuning under different data scales.  
+Figure 7: Experiments on learning from experience under off-policy exploration supervision.
+
+(on-policy), here they condition passively on externally provided experience examples. In Figure 7a, we evaluate four base models, LLaMA3.1-8B, Qwen2.5-7B, Mistral-7B, and Phi-3-mini, under varying numbers of in-context exemplars. In Figure 7b, we scale the amount of examples up to 1300 and track the performance of Qwen2.5-14B-Instruct-1M as well as Qwen2.5-7B-Instruct-1M variants. "Norm acc" in Figure 7b denotes accuracy normalized by the model's zero-shot performance. Finally, in Figure 7c, we compare two supervision regimes: in-context learning and instruction tuning. We use MagpieMath [86], as introduced in Section 4, as the source of experience (i.e., the ICL examples), and the test set is also drawn from it. All ICL-example experiments were run 20 times, and we report the mean and standard deviation. Implementation details are provided in Appendix A.5.
+
+Main Result. As shown in Figure 7a, all models exhibit a non-monotonic trend as the number of in-context examples increases: performance improves initially, peaks, and then declines. This highlights that current LLMs are effective few-shot learners but struggle to scale with longer trajectories, raising doubts about their many-shot capabilities. Figure 7b further confirms this limitation: when the number of in-context examples exceeds 900, performance drops sharply, even in larger models. This suggests that excessive examples may exceed attention capacity or interfere with generalization. Additionally, the 14B model consistently outperforms the 7B model, suggesting greater robustness to many-shot degradation. Figure 7c compares few-shot learning and instruction tuning. When data is limited, few-shot learning is highly efficient, using just 3 examples matches the performance of tuning with 5k instances. However, as data increases, instruction tuning continues to improve, while few-shot performance declines. This suggests a practical strategy: use few-shot learning in low-data regimes, and switch to tuning as more data becomes available.
+
+# 7 LearnArena: A Benchmark Suite for General Learning Ability
+
+Experiment Setup. We construct a benchmark suite that evaluates LLMs' general learning ability across three learning dimensions. The benchmark is built upon a modified version of the TextArena framework [28], where each environment is cast as a two-player game between Player-0 and Player-1. We fix Player-0 as Qwen2.5-32B and designate the evaluated LLM as Player-1. At each round  $k$ , Player-1 selects three experiences from the previous  $k - 1$  games and submits them to Player-0 for feedback. Player-0 provides suggestions for improvement, which Player-1 incorporates in the current round, representing Learning from Instructor. Separately, a concise summary of the game
+
+rules is generated by Qwen2.5-32B and given to Player-1, capturing Learning from Concept. In addition, Player-1 is encouraged to perform its own analysis of the selected past experiences and apply its conclusions in the current game, reflecting Learning from Experience. We evaluate each model-environment pair across 20 independent matches and report the average win rate of Player-1. The environments span a diverse set of strategic and social reasoning tasks, including Checkers (CH), Stratego (ST), Tic Tac Toe (TT), Truth and Deception (TD), SpellingBee (SB), SpiteAndMalice (SM), Tak (TK), and WordChains (WC). We evaluate the following models: Llama-3.1-8B, Mistral-7B-v0.3, Mistral-8B-2410, Qwen2.5-7B, Qwen2.5-14B-Instruct, Qwen2.5-32B-Instruct, Qwen3-8B, Qwen3-14B, GPT-4-o, and GPT-4-o-mini. Further implementation details are provided in Appendix A.6.
+
+Main Result. As shown in Table 2, the performance of LLMs varies widely across tasks, model families, and model scales, revealing several important trends. First, GPT-4o sets a new state-of-the-art, achieving an average win rate of 0.77 across the eight tasks, significantly outperforming all other models. Notably, GPT-4o demonstrates strong and consistent results across both symbolic reasoning tasks (e.g., 0.80 on Checkers and 0.81 on Tic Tac Toe) and social reasoning environments (e.g., 1.0 on Truth and Deception),
+
+Table 2: Win rates of Player-1 (evaluated model) across eight environments in the LearnArena benchmark. **Bold** indicates the best result for each task.  
+
+<table><tr><td>Model</td><td>CH</td><td>ST</td><td>TT</td><td>TD</td><td>SB</td><td>SM</td><td>TK</td><td>WC</td><td>Avg.</td></tr><tr><td>GPT4-o-mini</td><td>0.20</td><td>0.52</td><td>0.20</td><td>1.00</td><td>0.70</td><td>0.50</td><td>0.33</td><td>0.70</td><td>0.52</td></tr><tr><td>GPT4-o</td><td>0.80</td><td>0.61</td><td>0.81</td><td>1.00</td><td>0.60</td><td>0.82</td><td>0.71</td><td>0.80</td><td>0.77</td></tr><tr><td>Llama-3.1-8b</td><td>0.35</td><td>0.70</td><td>0.10</td><td>0.75</td><td>0.33</td><td>0.60</td><td>0.63</td><td>0.17</td><td>0.45</td></tr><tr><td>Mistral-7b</td><td>0.31</td><td>0.79</td><td>0.10</td><td>0.55</td><td>0.50</td><td>0.60</td><td>1.00</td><td>0.00</td><td>0.48</td></tr><tr><td>Mistral-8b</td><td>0.12</td><td>0.60</td><td>0.00</td><td>0.54</td><td>0.10</td><td>0.29</td><td>0.80</td><td>0.29</td><td>0.34</td></tr><tr><td>Qwen2.5-7b</td><td>0.39</td><td>0.61</td><td>0.08</td><td>0.93</td><td>0.20</td><td>0.11</td><td>0.10</td><td>0.30</td><td>0.34</td></tr><tr><td>Qwen2.5-14b</td><td>0.42</td><td>0.67</td><td>0.13</td><td>0.73</td><td>0.50</td><td>0.33</td><td>0.50</td><td>0.50</td><td>0.47</td></tr><tr><td>Qwen2.5-32b</td><td>0.40</td><td>0.72</td><td>0.09</td><td>0.83</td><td>0.30</td><td>0.50</td><td>0.67</td><td>0.60</td><td>0.51</td></tr><tr><td>Qwen3-8b</td><td>0.50</td><td>0.88</td><td>0.33</td><td>0.89</td><td>0.20</td><td>0.38</td><td>0.63</td><td>0.11</td><td>0.49</td></tr><tr><td>Qwen3-14b</td><td>0.50</td><td>0.88</td><td>0.50</td><td>1.00</td><td>0.50</td><td>0.20</td><td>1.00</td><td>0.25</td><td>0.60</td></tr></table>
+
+showcasing its robust integration of instructional guidance, conceptual abstraction, and experiential adaptation. Its smaller variant, GPT-4o-mini, also performs competitively (0.52), matching Qwen2.5-32B and surpassing several larger models. Second, within the Qwen2.5 series, we observe scale-driven improvement: performance increases steadily from 7B (0.34) to 14B (0.47) and 32B (0.51), confirming that learning ability benefits from increased capacity, but only up to a point. The gains diminish with scale. Third, the newer Qwen3 models exhibit a significant leap in performance: Qwen3-8B reaches 0.49, while Qwen3-14B achieves 0.60, outperforming all other open-source models and even exceeding GPT-4o-mini. This indicates that beyond scale, architectural and training advancements play a crucial role in improving learning ability.
+
+# 8 Conclusion
+
+To address the lack of analyzing of learning ability in large language models, we propose a cognitively grounded framework that decomposes general learning into three key dimensions: learning from instructor, learning from concept, and learning from experience. This decomposition enables a principled analysis of how LLMs acquire, internalize, and apply new knowledge. Our experiments reveal that interaction enhances instruction-based learning, conceptual understanding scales with model size, and LLMs struggle in many-shot scenarios due to long-context limitations. Motivated by these findings, we introduce LearnArena, a unified benchmark designed to evaluate general learning abilities across cognitively meaningful settings. While our current evaluation focuses on domains such as mathematics and games, future work should expand the scope of LearnArena to encompass a broader range of learning contexts and conceptual domains.
+
+# 9 Acknowledgement
+
+This work was supported in part by the National Key R&D Program of China (Grant No.2023YFF0725001), in part by the National Natural Science Foundation of China (Grant No.92370204), in part by the guangdong Basic and Applied Basic Research Foundation (Grant No.2023B1515120057), in part by the Education Bureau of Guangzhou
+
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+[83] Yizhong Wang, Yeganeh Kordi, Swaroop Mishra, Alisa Liu, Noah A Smith, Daniel Khashabi, and Hannaneh Hajishirzi. Self-instruct: Aligning language model with self generated instructions. arXiv preprint arXiv:2212.10560, 2022.  
+[84] Jason Wei, Yi Tay, Rishi Bommasani, Colin Raffel, Barret Zoph, Sebastian Borgeaud, Dani Yogatama, Maarten Bosma, Denny Zhou, Donald Metzler, et al. Emergent abilities of large language models. arXiv preprint arXiv:2206.07682, 2022.
+
+[85] Can Xu, Qingfeng Sun, Kai Zheng, Xiubo Geng, Pu Zhao, Jiazhan Feng, Chongyang Tao, and Daxin Jiang. Wizardlm: Empowering large language models to follow complex instructions. arXiv preprint arXiv:2304.12244, 2023.  
+[86] Zhangchen Xu, Fengqing Jiang, Luyao Niu, Yuntian Deng, Radha Poovendran, Yejin Choi, and Bill Yuchen Lin. Magpie: Alignment data synthesis from scratch by prompting aligned llms with nothing. arXiv preprint arXiv:2406.08464, 2024.  
+[87] Zhenyu Xu, Hailin Xu, Zhouyang Lu, Yingying Zhao, Rui Zhu, Yujiang Wang, Mingzhi Dong, Yuhu Chang, Qin Lv, Robert P Dick, et al. Can large language models be good companions? an llm-based eyewear system with conversational common ground. Proceedings of the ACM on Interactive, Mobile, Wearable and Ubiquitous Technologies, 2024.  
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+
+[101] Wanjun Zhong, Ruixiang Cui, Yiduo Guo, Yaobo Liang, Shuai Lu, Yanlin Wang, Amin Saied, Weizhu Chen, and Nan Duan. Agieval: A human-centric benchmark for evaluating foundation models. arXiv preprint arXiv:2304.06364, 2023.  
+[102] Chunting Zhou, Pengfei Liu, Puxin Xu, Srini Iyer, Jiao Sun, Yuning Mao, Xuezhe Ma, Avia Efrat, Ping Yu, Lili Yu, et al. Lima: Less is more for alignment. arXiv preprint arXiv:2305.11206, 2023.
+
+# A Experimental Setup
+
+# A.1 Implementation Details
+
+All experiments were conducted on a high-performance computing server equipped with an AMD EPYC 7V12 64-Core Processor, 512 GB of system memory, and 96 logical CPUs. The system also includes eight NVIDIA A100 GPUs, each with 80 GB of memory. Our training script is adapted from the example provided by LlamaFactory [100].<sup>3</sup> Full configuration details are provided in Appendix I. The references to Llama3.1-8B, Qwen2.5-7B, Mistral-7B, Phi-3-mini, Qwen2.5-1.5B, Qwen2.5-14B, Qwen2.5-32B, Qwen2.5-72B, Mistral-8B, Qwen3-8B, and Qwen3-14B in the main text refer to the specific models: Llama-3.1-8B-Instruct, Qwen2.5-7B-Instruct, Mistral-7B-Instruct-v0.3, Phi-3-mini-128k-instruct, Qwen2.5-1.5B-Instruct, Qwen2.5-14B-Instruct, Qwen2.5-32B-Instruct, Qwen2.5-72B-Instruct, Mistral-8B-Instruct-2410, Qwen3-8B, and Qwen3-14B, respectively. Each environment in LearnArena contains 50 unique tasks and is evaluated with 20 independent runs per task under a fixed protocol. Average input lengths (tokens) are: WordChains 131, SpellingBee 63.9, SpiteAndMalice 593, Tak 739, TicTacToe 166, Stratego 839, Checkers 289, and TruthAndDeception 122.9.
+
+# A.2 Evaluation Datasets
+
+SVAMP [63] SVAMP tasks are designed to provide a robust benchmark for evaluating the mathematical reasoning capabilities of language models on Math Word Problems (MwPs). Each task presents a short natural language narrative that describes a real-world scenario and poses a quantitative question. Previous benchmarks, such as ASDiv-A and MAwPS, were shown to permit high model performance even when critical question components were removed or word-order was disregarded, revealing the presence of shallow heuristics in existing approaches. SVAMP addresses these deficiencies by introducing examples that require models to demonstrate sensitivity to the question, robust reasoning, and invariance to structural alterations.
+
+MATH [31] MATH tasks are constructed from a dataset of 12,500 challenging competition-level mathematics problems, specifically designed to evaluate the mathematical problem-solving capabilities of language models. Each problem is accompanied by a detailed, step-by-step solution, enabling both the assessment of answer derivations and the development of explanation generation. In addition to the main benchmark, a large auxiliary pretraining dataset is provided to help teach models the fundamentals of mathematics. Despite advancements in large-scale Transformer models, accuracy on the MATH benchmark remains relatively low, highlighting the limitations of current approaches and the need for new algorithmic innovations in mathematical reasoning.
+
+NumGLUE [57] NumGLUE tasks are derived from a comprehensive multi-task benchmark designed to evaluate the numerical reasoning abilities of AI systems across eight distinct task types, each requiring some form of arithmetic understanding. The benchmark includes tasks involving commonsense and domain-specific arithmetic reasoning, quantitative comparison, fill-in-the-blank questions, reading comprehension with explicit and implicit numerical reasoning, quantitative natural language inference, and arithmetic word problems. By assessing model performance across these diverse task formats, NumGLUE provides a rigorous evaluation of both simple and complex numerical reasoning skills without leveraging explicit task-type labels during training.
+
+SimulEq [43] SimulEq tasks are based on a dataset comprising 514 test samples centered on equation solving, with a particular focus on algebraic manipulation and logical reasoning. The dataset is designed to evaluate models' proficiency in formulating and solving algebraic equations through the application of structured reasoning processes.
+
+AQuA [48] AQuA tasks are derived from the AQUA-RAT dataset, which comprises approximately 100,000 algebraic word problems accompanied by natural language rationales. Each problem includes a natural language question, five answer options (A-E) with a single correct choice, and a detailed rationale describing the solution process. This dataset enables evaluation of both answer selection and reasoning transparency in mathematical problem solving.
+
+SAT [101] SAT tasks are sourced from the AGIEval benchmark, which is designed to evaluate the general abilities of foundation models using authentic, human-centric standardized exams. The SAT component consists of mathematics and verbal reasoning problems that mirror those found in real-world college entrance assessments. By incorporating these tasks, AGIEval provides a robust framework for assessing models' understanding, reasoning, and calculation skills at a human-competitive level. Recent results show that state-of-the-art models, such as GPT-4, can achieve or surpass average human performance on SAT tasks, highlighting both the progress and current limitations of large language models in solving complex, human-oriented problems.
+
+MMLU-Math [30] MMLU-Math tasks are part of a comprehensive multitask benchmark designed to evaluate a model's accuracy across a wide range of academic subjects, with a specific focus on elementary and advanced mathematics. The benchmark requires models to demonstrate both broad world knowledge and sophisticated problem-solving skills. While recent large language models show improvements over random-chance accuracy, substantial gaps remain in achieving expert-level performance, particularly in mathematics. MMLU-Math enables detailed analysis of a model's mathematical proficiency and helps to identify critical areas requiring further advancement.
+
+MagpieMath [86] MagpieMath is a large-scale dataset generated via the Magpie pipeline, which synthesizes high-quality mathematical instruction data through auto-regressive sampling from Qwen2.5-Math. Unlike prior instruction tuning datasets that depend on seed prompts or handcrafted templates, Magpie employs a pre-query-based sampling approach to elicit diverse and pedagogically rich instruction data directly from aligned LLMs. The resulting dataset spans a wide range of mathematical skills, including arithmetic, algebra, and multi-step reasoning, and serves as a robust resource for evaluating instruction-based learning in mathematical domains.
+
+Poker [28] is a strategic two-player Texas Hold'em game where players compete by betting chips based on their hole cards and community cards. Each player aims to win chips by either having the best hand at showdown or making their opponent fold. The game follows standard Texas Hold'em rules with a fixed number of rounds and betting structures.
+
+Ultimate Tic Tac Toe [28] is a strategic two-player game that combines the classic Tic Tac Toe with an added layer of complexity. Players aim to win three micro boards in a row (horizontally, vertically, or diagonally) on the macro board, which tracks the outcomes of individual micro boards. Each move influences the opponent's next playable micro board, creating dynamic and strategic gameplay. This environment implements the full rules of Ultimate Tic Tac Toe, including valid move enforcement, micro board and macro board win detection, and a clear rendering of the board state for agent-based gameplay and experimentation.
+
+Checkers [28] (also known as Draughts) is a turn-based game played on an  $8 \times 8$  board. Each player controls a set of pieces placed on alternating dark squares. Pieces move diagonally forward and can capture opponent pieces by jumping over them. A piece that reaches the opposite side of the board becomes a "King," gaining the ability to move backward as well. The game ends when one player's pieces are all captured or when a player has no valid moves left.
+
+Stratego [28] is a two-player strategy board game where players aim to capture their opponent's Flag or eliminate all of their movable pieces. Each player strategically places their army, consisting of pieces of varying ranks, on a  $10 \times 10$  grid, with special rules for movement and battle. Bombs act as immovable traps, while Scouts can move multiple spaces in a straight line. The game involves hidden information, where a player's pieces are unknown to their opponent until revealed in battle. This environment simulates the full rules of Stratego, including strategic placement, battle resolution, and dynamic board rendering, providing a robust setup for agent-based gameplay and experimentation.
+
+TicTacToe [28] (also known as Noughts and Crosses) is a classic two-player game played on a  $3 \times 3$  grid. Players take turns marking cells with their symbol ('X' or 'O'), with the objective of placing three of their marks in a horizontal, vertical, or diagonal row. This implementation provides a complete game environment with move validation, win detection, and draw conditions.
+
+Truth and Deception [28] is a two-player social deduction game where one player (the Deceiver) attempts to mislead the other player (the Guesser) about which of two facts is correct. The players
+
+engage in a conversation of limited turns, during which the Deceiver strategically shares information to encourage the Guesser to choose the incorrect fact. At the end of the conversation, the Guesser must determine which fact is true. The game tests persuasion, critical thinking, fact verification, and strategic communication in a structured environment with clear objectives for both roles.
+
+Spelling Bee [28] is a two-player word game where players take turns creating valid English words using only a limited set of allowed letters. The challenge increases as the game progresses because each new word must be at least as long as the previous word. Players must balance vocabulary knowledge with strategic thinking to outmaneuver their opponent by forcing them into increasingly difficult positions. This implementation features frequency-weighted letter selection to ensure balanced and playable letter sets.
+
+Spite and Malice [28] is a two-player competitive card game that combines elements of solitaire and strategic play. Each player has their own payoff pile that they aim to deplete first to win. Players can play cards to shared center piles in ascending sequence (Ace to Queen), with Kings serving as wild cards. The game involves careful resource management, opportunistic card placement, and strategic blocking to prevent your opponent from emptying their payoff pile. This implementation features a complete deck management system, discard piles, and a hand limit of five cards.
+
+Tak [28] is a two-player abstract strategy board game. Players aim to build a continuous road connecting two opposite edges of the board while blocking their opponent's attempts. The game involves placing and moving pieces of various types—Flat Stones, Standing Stones (Walls), and Capstones—each with unique abilities. This environment simulates the full rules of Tak, including flexible board sizes, road-building mechanics, and dynamic board rendering, offering a rich platform for agent-based gameplay and experimentation.
+
+Word Chains [28] is a turn-based word game where two players take turns providing valid English words that start with the last letter of the previous word. The game enforces rules such as no word repetition and requires valid English words. Players lose if they fail to provide a valid word.
+
+LogicGrid [52] LogicGrid tasks are constructed as logic puzzles that require participants to perform grid-based deduction and symbolic reasoning. These tasks are specifically designed to evaluate the systematic application of abstract rules within constrained environments. All LogicGrid tasks presented in this work are newly developed adaptations based on established principles from classic puzzle collections.
+
+NaLogic [23] NaLogic tasks are narrative-driven logic puzzles that require solvers to extract and apply logical constraints embedded within rich, contextualized storytelling. These tasks are directly inspired by literary puzzle formats, and they are specifically designed to assess the integration of logical reasoning within narrative comprehension. All NaLogic tasks included in this work are systematically derived from established approaches found in classic puzzle literature.
+
+Plan [79] Plan tasks are adapted from classical scenarios formulated within the Planning Domain Definition Language (PDDL), including representative domains such as Gripper, Barman, Blocksworld, and Tyreworld. In these tasks, symbolic states are systematically translated into natural language descriptions to enable seamless interaction with large language models. Progress is quantitatively tracked by computing state-goal matching scores, thus providing a robust measure of task completion and logical reasoning within dynamic planning environments.
+
+AlfWorld [74] AlfWorld tasks require agents to perform everyday household activities that necessitate both exploration and commonsense reasoning. AlfWorld leverages interactive TextWorld environments that are closely aligned with the embodied scenarios found in the ALFRED dataset. This alignment enables agents to develop and reason about high-level policies within an abstract, text-based space, prior to executing embodied tasks that demand low-level actuation in physical or simulated environments.
+
+ScienceWorld [82] ScienceWorld tasks require agents to comprehend and execute a variety of scientific procedures, including activities such as measuring physical properties and manipulating
+
+laboratory equipment. Each task is accompanied by carefully reannotated subgoals designed to more accurately capture incremental progress, thereby enabling a granular evaluation of scientific reasoning and procedural understanding.
+
+BabyAI [10] BabyAI tasks require agents to navigate and interact within grid-world environments, leveraging newly introduced textual action representations and re-annotated subgoals. These modifications are designed to facilitate more nuanced agent-environment interaction and provide finer-grained evaluation of task completion in language-conditioned navigation settings.
+
+AlpacaEval [22] AlpacaEval tasks consist of a diverse set of 805 instructions, systematically aggregated from multiple benchmark sources. Specifically, the collection includes 252 instructions from the self-instruct test set[83], 188 from the Open Assistant test set (OASST), 129 from the Anthropic Helpful Test Set [102], 80 from the Vicuna test set [12], and 156 from the Koala test set [80]. This comprehensive assembly is designed to support the broad evaluation of the instruction-following capabilities across a wide range of language models.
+
+Vicuna [12] Vicuna tasks consist of 80 test instructions, systematically organized into eight distinct categories: Fermi problems, common sense reasoning, role-play scenarios, coding, mathematics, writing tasks, counterfactuals, knowledge assessment, and generic questions. This categorical organization is designed to facilitate a thorough evaluation of multiple dimensions of chatbot performance. Prior research suggests that the Vicuna dataset predominantly features instructions of comparatively lower difficulty and complexity[85]. In this study, the Vicuna test set is used to specifically assess the performance of large language models across these various types of instruction.
+
+Self-Instruct [83] Self-Instruct tasks consist of 252 human-authored test instructions, each paired with a carefully crafted reference output. This test set is meticulously curated to represent the real-world applicability of instruction follower models, encompassing a wide range of domains such as email composition, social media engagement, productivity software, and coding tasks. The instructions exhibit substantial diversity in both style and format, including various task lengths and a spectrum of input/output types such as bullet points, tables, code fragments, and mathematical equations. In this study, the Self-Instruct test set is utilized to rigorously evaluate the model's proficiency in adhering to precise instructions across heterogeneous domains.
+
+Wizardlm [85] Wizardlm tasks are derived from a large-scale instruction dataset initially seeded with 52,000 instructional examples from Alpaca, subsequently expanded through  $M = 4$  evolutionary cycles to produce 250,000 unique instructions. In each cycle, for every instruction, six new prompts are generated—five via in-depth evolution and one through in-breadth evolution—of which one prompt is randomly selected. Responses are then generated using ChatGPT, culminating in 52,000 ×  $4 \times 3 = 624,000$  instruction-response pairs. The training subset utilized for the Evol-Instruct dataset comprises 70,000 of these instructions. The test set, consisting of 218 instructions, is curated from diverse sources, including open-source projects and online forums, and encompasses 29 distinct skill areas identified from real-world human tasks.
+
+ARC [13] ARC tasks are derived from a dataset comprising 7,787 English-language science exam questions sourced from a variety of educational materials, including licensed content provided by a research partner affiliated with AI2. Each question is presented in a multiple-choice format, typically with four answer options, and spans a range of grade levels. The dataset is partitioned into a Challenge Set containing 2,590 "hard" questions—defined as those that cannot be correctly answered by either retrieval-based or co-occurrence-based baseline methods—and an Easy Set containing 5,197 questions of comparatively lower difficulty. This structure supports comprehensive evaluation of reasoning and scientific understanding across different levels of complexity.
+
+HellaSwag [93] HellaSwag tasks are derived from a challenging benchmark for commonsense natural language inference, wherein models must select the most plausible continuation for a given event description. For example, given an event such as "A woman sits at a piano," the model must choose the most likely follow-up, such as "She sets her fingers on the keys." Despite the introduction of advanced pretrained models like BERT, which achieved near human-level performance on previous tasks, HellaSwag demonstrates that even state-of-the-art models still struggle with robust
+
+commonsense reasoning. The dataset is constructed using Adversarial Filtering, a methodology where a series of discriminators iteratively select adversarially challenging, machine-generated distractors. While these questions are trivial for humans (accuracy  $>95\%$ ), state-of-the-art models frequently misclassify them (accuracy  $<48\%$ ).
+
+WinoGrande [67] WinoGrande tasks are derived from a dataset comprising 44,000 problems, inspired by the Winograd Schema Challenge [41], but specifically adjusted to enhance both scale and robustness against dataset-specific biases. Each task is presented as a fill-in-the-blank question with binary options, requiring the participant to select the correct answer based on contextual commonsense reasoning.
+
+MMLU [30] MMLU tasks are derived from the benchmark introduced by Hendrycks et al., evaluating a model's ability to answer multiple-choice questions across a broad range of academic subjects. In the original format, each question presents the answer choices within the context, and the model is required to produce the answer letter (e.g., A, B, C, D) as the continuation. The mmlu-continuation variant adopts a cloze-style format, omitting the answer choices in the context and requiring the full answer choice in the continuation. The mmlu-generation variant is similar to the original, but the model is tasked with generating the correct answer letter directly. These variants collectively enable a nuanced evaluation of language models' reasoning and generation capabilities across diverse knowledge domains.
+
+GSM8k [16] GSM8k tasks are drawn from a dataset of 8,500 high-quality, linguistically diverse grade school math word problems, designed to evaluate multi-step mathematical reasoning in language models. Despite the conceptual simplicity of the problem distribution, state-of-the-art transformer-based models consistently struggle to achieve high performance on this benchmark. The GSM8k dataset thus provides a rigorous diagnostic tool for analyzing the limitations of current models and supporting further research in mathematical reasoning.
+
+Intlaw [61, 62] Intlaw tasks are derived from L. Oppenheim's two-volume International Law: A Treatise, and are designed to assess understanding of both foundational and applied principles in international law. Volume 1 emphasizes legal theory, sources of law, and the concept of state sovereignty, while Volume 2 explores topics such as conflict, neutrality, and mechanisms for peaceful dispute resolution. This dataset enables evaluation of models' capabilities in legal reasoning, state interactions, and comprehension of diplomatic and legal frameworks.
+
+Willidea [69, 70, 71] Willidea tasks are constructed from all three volumes of Arthur Schopenhauer's The World as Will and Idea, focusing on the evaluation of philosophical reasoning and metaphysical analysis. The dataset challenges models to interpret complex philosophical arguments, engage with Kantian critique, and reason across abstract conceptual structures related to perception, will, and reality. Through these tasks, the ability of language models to comprehend and analyze dense philosophical discourse is rigorously assessed.
+
+[58] Ancsoc tasks are derived from Lewis H. Morgan's Ancient Society and are designed to assess a model's understanding of the evolution of human civilization through the stages of savagery, barbarism, and civilization. The dataset emphasizes analysis of social structures, cultural development, and key concepts in anthropological theory. Through these tasks, models are evaluated on their capacity for historical analysis and cross-cultural reasoning.
+
+Poliec [65, 66] Poliec tasks are based on Wilhelm Roscher's two-volume Principles of Political Economy and focus on the historical development of economic systems, classification of income, and the influence of human behavior on economic theory. The dataset is designed to assess models' abilities to reason through complex economic frameworks, drawing on both historical context and theoretical foundations.
+
+# A.3 Details of Learning from the Instructor
+
+We evaluate instruction-based learning ability across five distinct domains: Mathematics, Law, Philosophy, Anthropology, and Economy. Each domain is selected to reflect different facets of
+
+instruction-following behavior, ranging from structured problem-solving to conceptual understanding. During training, we additionally use the MAGPIEMATH dataset [86], which provides large-scale, high-quality mathematical instruction data synthesized from Qwen2.5-Math. It is a subset of the dataset available at this link  $^{4}$ . The prompt can be found in Appendix L. We used eight representative test benchmarks: GSM8K [16], SVAMP [63], MATH [31], NumGLUE [57], SimulEq [43], AQuA [48], SAT [101], and MMLU-Math [30]. These datasets cover a range of mathematical competencies from grade-school arithmetic to advanced problem-solving, including multi-step reasoning, equation solving, and standardized testing skills. In the Law domain, we use the Intlaw dataset [61, 62]. It covers core topics such as sovereignty, conflict, and dispute resolution, and evaluates models' legal reasoning and understanding of international legal systems. In the Philosophy domain, we use the Willidea dataset [69, 70, 71], which focuses on metaphysics, perception, and will, testing models' ability to interpret abstract arguments and philosophical reasoning. In the Anthropology domain, we use the Ancsoc dataset [58], which evaluates understanding of social evolution, cultural systems, and anthropological theory. In the Economy domain, we use the Poliec dataset [65, 66], which emphasizes historical and theoretical reasoning in value, income, and economic behavior. Across these four non-math domains, question formats included fill-in-the-blank (fill), judgment (judge), multiple-choice (multi-choice), open-ended (open), and single-choice (single-choice). We provide representative cases of each question format in Appendix J. Further dataset details are provided in Appendix A.2
+
+# A.4 Details of Learning from the Concept
+
+We evaluate models' ability to learn from structured conceptual information across a variety of tasks that require rule interpretation, logical reasoning, and symbolic planning. Our experiments focus on two complementary settings: multi-agent competitive environments and logic-based planning tasks. In each case, we compare model performance with and without access to high-level conceptual summaries that describe task-specific rules, goals, or strategies. These summaries are generated by a stronger teacher model and provided as auxiliary context to the learner model. We first study conceptual learning in strategic settings using the TEXTARENA framework [28], a collection of multi-agent environments characterized by symbolic rules and multi-turn dynamics. Each game involves two players, Player-0 and Player-1. We fix Player-0 as Qwen2.5-32B and vary Player-1 across four model scales: Qwen2.5-1.5B, 7B, 14B, and 32B. Prior to gameplay, Player-1 either receives no guidance (without Concept) or is given a natural language description of the game rules and strategies (with Concept) generated by Qwen2.5-32B. For each environment and model combination, we run 20 matches and report the average win rate of Player-1. The evaluation environments include CHECKERS (CH), POKER (PK), STRATEGO (ST), TIC TAC TOE (TT), TRUTH AND DECEPTION (TD), and ULTIMATE TIC TAC TOE (UTT). Performance is measured as the win rate of Player-1 against the fixed opponent. We then evaluate rule-based generalization across a set of logic and planning tasks. The LOGICGRID and NALOGIC tasks consist of structured and narrative logic puzzles [52, 23], requiring deduction under symbolic constraints. PLAN [79] adapts classical PDDL-style planning domains (e.g., Gripper, Barman) into natural language to assess symbolic action reasoning. Additionally, we include ALFWORLD [74], SCIENCEWORLD [82], and BABYAI [10] tasks, which involve procedural and goal-oriented behaviors requiring commonsense reasoning and spatial understanding. For the concept-injected setting, we use Qwen2.5-72B to generate static conceptual summaries tailored to each task. These summaries are provided as additional input to the learner model during evaluation. Further dataset details are provided in Appendix A.2
+
+# A.5 Details of Learning from the Experience
+
+We investigate how language models adapt their behavior based on prior experiences. Our experiments are designed to capture different aspects of experience-based learning, ranging from competitive multi-turn interactions to static in-context supervision. In all settings, models are exposed to past interaction traces or curated examples, which serve as proxies for experiential knowledge. We evaluate two complementary scenarios: (1) interactive adaptation through self-selected experience in multi-agent games, and (2) passive conditioning on externally provided in-context examples. These two settings align conceptually with classical learning paradigms: the first resembles on-policy learning, where the model actively explores and adapts through its own interactions; the second
+
+mirrors off-policy learning, where the model passively leverages prior trajectories without influencing their generation. Together, they allow us to explore whether LLMs can extract useful patterns from previous data. We adopt the same multi-agent environment setup described in Section 5.1, using TEXTARENA [28], a suite of competitive games involving symbolic rules and multi-turn dynamics. The results are shown in Figure 6. In this setting, "with experience" refers to a scenario where, during the  $k$ -th game, the model selects three prior experiences from rounds 0 to  $k-1$  (or all available ones if fewer than three exist). These samples are used as in-context learning (ICL) examples, allowing the model to draw on its past trajectories for real-time adaptation. Performance is measured by the average win rate of Player-1 across 20 independent matches. This setup captures whether models can benefit from self-curated experiential memory in competitive decision-making. To further analyze how accumulated experience affects model behavior, we treat in-context examples as off-policy trajectories that encode supervision signals. This setup parallels off-policy learning: unlike competitive gameplay in Section 6.1, where models adapt through active interaction (on-policy), here they condition passively on externally provided experience examples. These exemplars are not generated by the current model but are instead drawn from prior runs or curated datasets. We use MagpieMath [86], introduced in Section4, as the source of experience for in-context learning, with both the ICL examples and test set drawn from this dataset. All in-context experiments are repeated 20 times using different random seeds. We report the mean and standard deviation for each result. Further dataset details are provided in Appendix A.2
+
+# A.6 Details of LearnArena
+
+We design a benchmark suite to evaluate the general learning ability of large language models across three distinct learning paradigms: Learning from Instructor, Learning from Concept, and Learning from Experience. Each paradigm is operationalized within the same unified game-based evaluation setting, adapted from the TextArena framework [28]. In this setup, each environment is framed as a two-player game involving Player-0 and Player-1. Player-0 is fixed as Qwen2.5-32B and serves as the consistent teacher or opponent across all tasks. Player-1 is the evaluated model, which varies across runs. At each round  $k$ , Player-1 selects three prior experiences from rounds 0 to  $k - 1$  and submits them to Player-0 to receive feedback. Player-0 then returns suggestions for improvement. This interactive feedback process constitutes the Learning from Instructor component of the benchmark. In parallel, a concise natural language summary of the environment's rules is generated by Qwen2.5-32B and given to Player-1 prior to gameplay. This summary includes key mechanics, legal moves, and basic strategies for the task, enabling models to condition on explicit rule representations. This serves as the Learning from Concept component. Player-1 is also encouraged to analyze its own past experience samples and draw inferences to improve future performance. These self-selected experiences are used as in-context examples during gameplay, forming the Learning from Experience component. The prompt can be found in Appendix K. We evaluate each model-environment pair over 20 independent matches and report the average win rate of Player-1. The benchmark includes a diverse set of environments designed to test strategic reasoning, symbolic planning, and adaptive behavior. The selected games include CHECKERS (CH), STRATEGO (ST), TIC TAC TOE (TT), TRUTH AND DECEPTION (TD), SPELLINGBEE (SB), SPITEANDMALICE (SM), TAK (TK), and WORDCHAINS (WC). We evaluate the following models as Player-1: LLaMA-3.1-8B, Mistral-7Bv0.3, Mistral-8B-2410, Qwen2.5-7B, Qwen2.5-14B-Instruct, Qwen2.5-32B-Instruct, Qwen3-8B, Qwen3-14B, GPT-4-o-mini, and GPT-4-o. Each model plays all environments under the same setup, enabling controlled and comparable assessment across architectures and scales. The details of dataset are provided in Appendix A.2 Further implementation details, including prompt formats, sampling settings, and game environment configurations, are provided in the released codebase.
+
+Comparison with Existing Benchmarks. To assess the uniqueness and relevance of our benchmark, we compare it with a broad set of widely used evaluation benchmarks by computing Spearman rank correlations between system-level rankings. These benchmarks include instruction-following datasets such as AlpacaEval [22], Vicuna [12], Self-Instruct [83], and WizardLM [85], as well as standard task-specific evaluations including ARC [13] (science reasoning), HellaSwag [93] and Winogrande [67] (commonsense inference), GSM8K [16] (mathematical reasoning), and MMLU [30] (broad-domain knowledge).
+
+As shown in Figure 8, our benchmark shows low correlation with all existing benchmarks, with the highest only reaching a moderate level (0.66), indicating it captures distinct aspects of model
+
+![](images/e185b733ec631e8d33153a33f7660ecb5c49c14549d64ea9a4f1ccf7b4332c52.jpg)  
+Figure 8: Spearman rank correlation matrix between our benchmark and nine existing benchmarks.
+
+performance. Among all benchmarks, our strongest correlations are with AlpacaEval  $(\rho = 0.66)$  and MMLU  $(\rho = 0.66)$ , followed by HellaSwag  $(\rho = 0.53)$ , WizardLM  $(\rho = 0.44)$ , and Self-Instruct  $(\rho = 0.43)$ . The correlation with Vicuna is relatively low  $(\rho = 0.25)$ , and correlations with ARC, Winogrande, and GSM8K are near zero. These results indicate that while there is some overlap between our benchmark and existing instruction-following or knowledge-based tasks, our evaluation framework provides a different and complementary signal. This divergence is expected, given the design objective of our benchmark. Unlike most prior benchmarks that focus on static task-solving or narrow instruction-following performance, our benchmark is explicitly constructed to evaluate general learning ability. It is grounded in a cognitive framework that integrates three key dimensions: learning from instructor, learning from concept, and learning from experience. However, our benchmark does not treat these dimensions as isolated tasks or modules. Instead, we embed them holistically across the benchmark's task suite, such that model success requires simultaneously demonstrating the ability to absorb guidance, abstract rules, and dynamic feedback. As a result, our benchmark presents a more integrated and realistic test of adaptive intelligence, resembling how learning occurs in natural human settings. The varying degrees of correlation across external benchmarks further illuminate how different datasets capture different cognitive demands. Benchmarks like ARC focus on domain-specific problem-solving (e.g., science) with fixed question-answer formats. These tasks require correctness but do not test a model's ability to update its behavior through interaction or abstraction, which likely explains the near-zero correlation with our benchmark. On the other hand, AlpacaEval and WizardLM emphasize open-ended instruction-following, which partially aligns with our learning-from-instructor dimension. MMLU, though knowledge-centric, requires broad generalization across domains and may tap into conceptual understanding and knowledge transfer, hence the higher correlation. These findings confirm that our benchmark captures a distinct and underexplored aspect of LLM evaluation by assessing general-purpose learning through the integrated use of instruction, abstraction, and adaptation.
+
+# B Opponent Robustness and Extended Evaluation
+
+LearnArena in the main text fixes Player-0 to Qwen2.5-32B. To assess robustness to opponent choice and reduce the risk that models overfit to a single opponent's idiosyncrasies, we repeated the evaluation with three alternative Player-0 models—Phi-4, GPT-4o-mini, and GPT-4o. For each fixed Player-0, we measured Player-1 performance on four environments (Checkers, Stratego, SpellingBee, SpiteAndMalice) under the same protocol as the main paper, averaged win rates across the four environments to obtain a per-Player-1 score, ranked Player-1 models by this score, and computed the Spearman rank correlation between this ranking and the baseline ranking obtained with Player-0 = Qwen2.5-32B.
+
+Table 3: Player-0 = Phi-4. Player-1 win rates on four environments.  
+
+<table><tr><td>Player-1</td><td>CH</td><td>ST</td><td>SB</td><td>SM</td></tr><tr><td>gpt-4o</td><td>1.00</td><td>0.90</td><td>0.65</td><td>1.00</td></tr><tr><td>gpt-4o-mini</td><td>0.65</td><td>0.88</td><td>0.86</td><td>0.72</td></tr><tr><td>llama-3.1-8b</td><td>0.65</td><td>0.70</td><td>0.50</td><td>0.75</td></tr><tr><td>mistral-8b</td><td>0.40</td><td>0.75</td><td>0.28</td><td>0.69</td></tr><tr><td>qwen2.5-7b</td><td>0.70</td><td>0.75</td><td>0.35</td><td>0.35</td></tr><tr><td>qwen3-8b</td><td>0.75</td><td>0.65</td><td>0.50</td><td>0.70</td></tr></table>
+
+Table 4: Player-0 = GPT-4o-mini. Player-1 win rates on four environments.  
+
+<table><tr><td>Player-1</td><td>CH</td><td>ST</td><td>SB</td><td>SM</td></tr><tr><td>gpt-4o</td><td>0.80</td><td>1.00</td><td>0.65</td><td>1.00</td></tr><tr><td>gpt-4o-mini</td><td>0.50</td><td>0.55</td><td>0.45</td><td>0.55</td></tr><tr><td>llama-3.1-8b</td><td>0.25</td><td>0.85</td><td>0.10</td><td>0.70</td></tr><tr><td>mistral-8b</td><td>0.15</td><td>0.85</td><td>0.20</td><td>0.45</td></tr><tr><td>qwen2.5-7b</td><td>0.45</td><td>0.70</td><td>0.25</td><td>0.35</td></tr><tr><td>qwen3-8b</td><td>0.40</td><td>0.95</td><td>0.55</td><td>0.50</td></tr></table>
+
+Across these alternative opponents, the rank-order agreement with the baseline is high. The Spearman correlations with the Qwen2.5-32B-based ranking are  $\rho_{\mathrm{Phi - 4}} = 1.000$ ,  $\rho_{\mathrm{GPT - 4o - mini}} = 0.829$ ,  $\rho_{\mathrm{GPT - 4o}} = 0.714$ . These values indicate that relative model rankings are largely preserved across substantially different opponents. In practice, stronger opponents compress absolute win rates yet retain the broad ordering of Player-1 models, which is useful for difficulty calibration while maintaining comparative validity. Taken together, these results suggest that the main conclusions of the benchmark are not overly sensitive to the specific choice of Player-0. While average win rate remains a clear and discriminative summary of success in dynamic environments, it is a coarse measure of learning. To provide a more granular view of learning dynamics in future versions of LearnArena (LearnArena), we will augment reporting with process-level signals that capture how models adapt, follow guidance, and reuse experience. Specifically, we will measure adaptation speed through per-round improvement and rounds-to-threshold; instruction adherence through agreement between instructor feedback and subsequent actions under automatic alignment checks; concept utilization through rule compliance and strategy-template coverage when conceptual summaries are provided; experience reuse through the recurrence of salient patterns from selected prior trajectories in current decisions; and stability and robustness through variance across seeds and sensitivity to controlled noise injected into instructions, concepts, and experiences.
+
+# C Many-Shot Behavior and Long-Context Models
+
+This section examines whether degradation in many-shot settings is merely a consequence of poor long-context capability. In the many-shot experiments reported in Section 6.2, we evaluate Qwen2.5-7B/14B-Instruct-1M, which support up to 1M-token windows and are explicitly designed for long-context usage. Across these models, performance follows a clear non-monotonic trend: accuracy improves as in-context examples increase, peaks at a small number of shots (3–9 depending on model size), and then declines as the prompt grows longer. This indicates that, even with extended context windows, current models remain effective few-shot learners while many-shot generalization remains challenging. To further validate this pattern with a frontier model under tractable cost, we additionally evaluate GPT-4o-mini (128k context) following the protocol in Section 6.2 and the setup of [4], using 4-shot as a reference and scaling up to 100 examples. On MagpieMath, we observe 71.2 (0-shot), 71.4 (4-shot), 72.8 (10-shot), a peak at 75.2 (15-shot), followed by 71.2 (20-shot), 70.8 (30-shot), 70.0 (50-shot), 70.6 (80-shot), and 70.6 (100-shot). The rise-then-fall trajectory replicates the central empirical regularity documented in prior work. The observations above are consistent with [4]. In their Section 3.1, Figure 7 (left), Gemini 1.5 Pro on MATH-500 with ground-truth exemplars exhibits the same shape: performance rises as the number of shots increases, peaks at roughly 125 shots, and then declines as the context grows further. Although [4] truncates at 500 shots, the downward trend is already evident. In our experiments on MagpieMath, when we extend beyond 30–100 examples
+
+# Table 5
+
+the decline persists and eventually dips below the 4-shot baseline, noting that the absolute scores differ across datasets and evaluation protocols. Taken together, evaluations spanning Qwen2.5-7B, Qwen2.5-14B, GPT-4o-mini, and previously reported results for Gemini 1.5 Pro all exhibit the same qualitative regularity: performance improves, peaks, and then degrades as the number of in-context examples grows. As model capability increases, the peak tends to occur at a larger shot count and the subsequent degradation is slower, suggesting that stronger long-context handling delays but does not eliminate many-shot brittleness. These results reinforce that long-context capacity is necessary but not sufficient; robust many-shot generalization remains an open challenge.
+
+# D Clarifying "Passive Consumption" (LfI) vs. "Experience-Driven Adaptation" (LfE)
+
+Although both settings can appear as few-shot style prompts, they differ in intent and supervision: LfI (passive consumption) supplies instructional content—worked solutions, step-by-step rationales, or corrections—from a strong instructor, providing pedagogically curated guidance; formally, the context is  $\mathcal{C}_t = \mathcal{I}_t$  and prediction follows  $f_{\theta}(x,\mathcal{I}_t)\rightarrow \hat{y}$ . LfE (experience-driven adaptation) instead presents episodic traces—raw input-output pairs, partial trajectories, or outcomes—without didactic restructuring, requiring the model to infer what to abstract or discard; here  $\mathcal{C}_t = \tau_t = \{s_1,\dots ,s_k\}$  and prediction follows  $f_{\theta}(x,\tau_t)\rightarrow \hat{y}$ . In our experiments, LfI uses instructor-produced solutions/rationales, whereas LfE uses selected histories from earlier rounds as off-policy trajectories, reflecting guided transfer versus autonomous abstraction from unstructured experience.
+
+# E Rationale for Dataset Choices Across Learning Dimensions
+
+The evaluation follows a two-stage design. Sections 4-6 isolate and validate each learning mechanism under task families that provide the right supervision affordances and minimal cross-factor interference. Section 7 then integrates all three mechanisms in a unified setting via LearnArena, enabling a holistic assessment. A single fixed dataset for all experiments would entangle constructs and reduce interpretability, whereas the staged design improves construct validity by matching task structure to the learning signal being tested. For Learning from Instructor (Section 4), mathematics is used because it naturally supports a tutor-learner paradigm with high-fidelity supervision. Problems admit worked solutions and stepwise rationales, enabling clear, pedagogically oriented instructional signals from a strong instructor (e.g., Qwen2.5-Math-72B) and objective evaluation of whether instruction improves the learner. For Learning from Concept (Section 5), domains are chosen where structured knowledge can be injected and audited. Competitive environments (TextArena) expose explicit rules and strategy templates; logic puzzles (LogicGrid, NaLogic) emphasize formal constraints; and planning/agent tasks (Plan, AlfWorld, ScienceWorld, BabyAI) probe procedural abstractions and generalization. These settings make conceptual cues precise, testable, and meaningfully tied to downstream behavior. For Learning from Experience (Section 6), the focus is on adaptation from trajectories rather than instruction. Two complementary regimes are used: on-policy experience in competitive games, where histories accumulate across rounds and must be exploited under stochastic interaction, and off-policy experience via in-context examples, where pre-collected traces serve as episodic supervision without didactic restructuring. This separation targets the specific cognitive demands of experience integration. Section 7 introduces LearnArena as a unified benchmark that combines instructional feedback, conceptual summaries, and experience selection within shared game environments. The unified evaluation complements the isolated studies by testing whether the three learning mechanisms compose in realistic multi-agent settings, while the earlier sections justify that each mechanism is measurable and interpretable in its most suitable domain.
+
+# F Why Three Perspectives? Scope and Relationship to Self-Critique
+
+The tripartite decomposition—Learning from Instructor (LfI), Learning from Concept (LfC), and Learning from Experience (LfE)—is chosen as a cognitively grounded and minimally overlapping
+
+basis that captures the principal supervision channels through which learning occurs. LfI models didactic guidance delivered by an external agent via demonstrations, explanations, or corrections (instructional intent; [19, 73]); LfC captures the use of declarative, abstract structure (rules, definitions, schemata) to enable transfer and constraint-based generalization ([25, 5]); LfE characterizes adaptation from accumulated trajectories and outcomes under interaction or replay, without curated pedagogy ([42]). These axes align with long-standing accounts in cognitive psychology and education that separate guided instruction, conceptual abstraction, and experiential learning as complementary pathways ([15, 42, 25]). Self-critique and related practices (reflection, self-rewarding, self-refinement) fit naturally within this framework rather than constituting a separate axis. When a model critiques, edits, or rewards its own outputs, the instructional channel is internalized: the model acts as its own instructor while still supplying pedagogically structured signals, which places self-critique as an instance of LfI with the instructor collapsed into the learner. In settings where the model instead mines unstructured past attempts or rollouts to improve future behavior, the mechanism corresponds to LfE. Thus, the proposed three perspectives provide a coherent basis that cleanly factorizes the supervision source (instructor vs. concept vs. experience) while accommodating practical variants—external or internal—within the same taxonomy.
+
+# G Limitations
+
+While our work provides a structured perspective on the learning capabilities of large language models, we acknowledge that our evaluations are conducted in synthetic and benchmark-driven environments. As such, the implications of improved learning abilities in real-world settings, such as education, healthcare, or human-AI collaboration, are not directly explored. Moreover, while better learning may enhance LLM adaptability, it may also raise concerns around user over-reliance, unintended memorization, or alignment with user intent in high-stakes applications. Our framework is intended as a diagnostic tool for understanding LLM behavior, not as a comprehensive measure of safety, fairness, or trustworthiness. We hope future work will extend our approach to include evaluations grounded in real-world impact, especially in socially sensitive domains.
+
+![](images/7752d723c29f05b1f8221c65528c12e3c9b2ea1d59e70b387210be12576fd379.jpg)  
+Figure 10: Model performance across five question types (fill-in-the-blank, judgment, multiple-choice, open-ended, single-choice) over four iterations. Performance consistently improves as training data becomes cleaner through selection.
+
+# H Learning with Useful Knowledge Selection
+
+Experiment Setup. Humans naturally develop the ability to select high-quality learning materials based on prior experience. In this section, we investigate whether LLMs possess a similar capacity to identify and prefer informative over noisy data during training. We evaluate this ability across four non-mathematical domains, Anthropology, Economics, Law, and Philosophy, where domain knowledge is derived from canonical classical texts, including works on international law [61, 62], philosophy [69, 70, 71], anthropology [58], and economics [65, 66]. Each domain includes five question types: fill-in-the-blank (fill), judgment (judge), multiple-choice (multi-choice), open-ended (open), and single-choice (single-choice). To simulate realistic noise, we inject  $30\%$  noisy samples into the training data through random perturbation and irrelevant distractors. We adopt a four-step iterative training procedure: (1) Model-1 is trained on a randomly sampled  $25\%$  of the full dataset,
+
+from which  $70\%$  is used for training. (2) Model-2 selects  $70\%$  of a new  $25\%$  data slice (from the remaining  $75\%$ ) using Model-1 as the evaluator. (3) Model-3 repeats this process using Model-2 as the evaluator on the next  $25\%$ . (4) Model-4 completes the final iteration using Model-3 as the evaluator on the remaining data. All models use Qwen2.5-7B as the base learner. Results are shown in Figure 9 and Figure 10, where Origin denotes the untrained Qwen2.5-7B baseline, and Iter-1 to Iter-4 correspond to the four models described above. Evaluation is performed on a held-out test set for each domain and question type. Case studies for each domain are provided in Appendix J. Implementation details are provided in Appendix A.3.
+
+Main Result. As illustrated in Figure 10, model performance consistently improves across all domains, Anthropology, Economics, Law, and Philosophy, and all question types (fill-in-the-blank, judgment, multiple-choice, open-ended, single-choice) over the four iterations. Figure 9 further shows that the proportion of noisy samples in the selected training data steadily declines from Iter-1 to Iter-4 across all four domains demonstrating that LLMs can progressively learn to favor cleaner, more informative data through iterative self-selection. For instance, in the domain of Anthropology, the noise ratio drops from  $30.00\%$  in Iter-1 to  $17.28\%$  by Iter-4. This trend is consistent across domains, indicating an emergent capacity to filter out irrelevant or misleading content during the learning process. Compared to the untrained model baseline, which consistently selects data with  $26\%$  noise, trained models show a clear advantage in filtering out noise. This suggests that LLMs can learn to identify and prioritize higher-quality data through iterative training.
+
+![](images/723ee3ff25ee997d4d8820d10fec4296b33b53a7326621193b03819bbaf1edd1.jpg)
+
+![](images/0f03f779bc224cc64b5d63840ff618f02e18f85d75c6251dea8270ad821c465d.jpg)  
+Figure 9: Noise ratios in data selected across four domains and iterations. Solid bars (e.g., Anthropology) show trained models; dashed bars (e.g., Anthropology Baseline) show un
+
+Table 6: Supervised fine-tuning hyper-parameters across domains. See configuration for Mathematics, which uses a larger dataset and fewer epochs. Right: shared configuration for Philosophy, Anthropology, Economics, and Law.  
+
+<table><tr><td colspan="2">Math Domain</td><td colspan="2">Non-Math Domains</td></tr><tr><td>Hyper-parameter</td><td>Assignment</td><td>Hyper-parameter</td><td>Assignment</td></tr><tr><td>Number of epochs</td><td>2</td><td>Number of epochs</td><td>4</td></tr><tr><td>Batch size per device</td><td>1</td><td>Batch size per device</td><td>1</td></tr><tr><td>Gradient accumulation steps</td><td>4</td><td>Gradient accumulation steps</td><td>4</td></tr><tr><td>Effective batch size</td><td>32</td><td>Effective batch size</td><td>32</td></tr><tr><td>Maximum sequence length</td><td>2048</td><td>Maximum sequence length</td><td>2048</td></tr><tr><td>Learning rate</td><td>1e-5</td><td>Learning rate</td><td>1e-5</td></tr><tr><td>Learning rate scheduler</td><td>cosine</td><td>Learning rate scheduler</td><td>cosine</td></tr><tr><td>Warm-up steps</td><td>20</td><td>Warm-up steps</td><td>20</td></tr><tr><td>Optimizer</td><td>AdamW</td><td>Optimizer</td><td>AdamW</td></tr><tr><td>Precision</td><td>FP16</td><td>Precision</td><td>FP16</td></tr></table>
+
+# I Instruction Tuning Configuration
+
+ts specifications of the instruction tuning corpora in each domain and Table 6 presents instruction tuning hyper-parameters. We conduct full-parameter
+
+supervised fine-tuning on five domain-specific corpora that cover Economics, Law, Philosophy, Anthropology, and Mathematics. Each corpus is built by pairing public domain texts with instruction-response examples produced through an automated prompt-and-verify pipeline: we first segment
+
+source documents into short passages, then use GPT-4 to write an instruction for each passage and generate a reference answer, finally filtering out low-quality pairs with simple heuristic checks. After dedduplication and shuffling, the resulting datasets contain between 12k and 100k samples per domain, as detailed in Table 7. We adopt the training recipe from LlamaFactory<sup>5</sup> and keep the same optimizer, scheduler, and precision across domains. Because the mathematics corpus is much larger than the others, we train for two epochs on Math and four epochs on the remaining domains while keeping the effective batch size, learning rate, sequence length, and other hyper-parameters identical (Table 6). All experiments are run on eight A100 80 GB GPUs with gradient accumulation to reach an effective batch size of 32.
+
+# I.1 Task Descriptions and Token Length Statistics
+
+Across the eight environments we track the number of generated tokens for every model-task pair and compare these budgets with the win-rate figures in Table 2. As shown in Table 8, token usage spans more than an order of magnitude: SpellingBee is consistently the cheapest task (all models  $< 350$  tokens) whereas Tak is the most expensive (from  $1.6\mathrm{k}$  for Llama-3.1-8B up to  $6.6\mathrm{k}$  for Qwen3-8B). Model style has a larger impact than size: the Qwen3 series and Mistral models tend to answer verbally, often emitting twice as many tokens as GPT-4o on the
+
+Table 7: Instruction-response corpora across five domains.  
+
+<table><tr><td>Domain</td><td># Samples</td><td>Source</td></tr><tr><td>Economy</td><td>12,408</td><td>[65, 66]</td></tr><tr><td>Law</td><td>29,648</td><td>[61, 62]</td></tr><tr><td>Philosophy</td><td>23,804</td><td>[69, 70, 71]</td></tr><tr><td>Anthropology</td><td>25,516</td><td>[58]</td></tr><tr><td>Mathematics</td><td>100,365</td><td>[86]</td></tr></table>
+
+same input, while GPT-4o and GPT-4o-mini remain concise even on large boards. This morbidity does not buy success: GPT-4o secures the highest average win rate (0.75) while producing  $30 - 50\%$  fewer tokens than its closest open-source competitors on most tasks. Conversely, Qwen3-8B generates the longest outputs yet ranks only mid-pack (0.49). Within individual tasks the pattern persists. On the compact SB game every model stays brief, but performance still ranges widely (0.10–0.70), showing that concision need not hurt accuracy. In the token-hungry TK environment only models that combine moderate length with strong heuristics—Mistral-7B (3.2 k tokens, win = 1.00) and Qwen3-14B (3.7 k, 1.00)—match GPT-4o's high score; models that answer either too tersely (Llama-3.1-8B, 1.6 k, 0.63) or too verbally (Qwen2.5-7B, 5.6 k, 0.10) fall behind. Computing Pearson's  $\rho$  across all 88 model-task pairs yields a weak negative correlation of roughly  $-0.28$  between token count and win rate, indicating that quality of reasoning, not sheer output length, is the primary driver of performance in LearnArena.
+
+Table 8: Average generated tokens per match for each model across the eight LearnArena environments.  
+
+<table><tr><td>Model</td><td>CH</td><td>ST</td><td>TT</td><td>TD</td><td>SB</td><td>SM</td><td>TK</td><td>WC</td></tr><tr><td>Llama-3.1-8B</td><td>635</td><td>1057</td><td>951</td><td>656</td><td>172</td><td>1004</td><td>1591</td><td>409</td></tr><tr><td>Mistral-7B-v0.3</td><td>886</td><td>1463</td><td>1900</td><td>1265</td><td>629</td><td>1095</td><td>3223</td><td>505</td></tr><tr><td>Mistral-8B-2410</td><td>663</td><td>1661</td><td>1977</td><td>772</td><td>541</td><td>1167</td><td>4149</td><td>289</td></tr><tr><td>Phi-4-mini</td><td>669</td><td>1222</td><td>1594</td><td>1315</td><td>344</td><td>1561</td><td>2111</td><td>296</td></tr><tr><td>Qwen2.5-14B-Chat</td><td>980</td><td>1370</td><td>2308</td><td>867</td><td>326</td><td>1206</td><td>4116</td><td>478</td></tr><tr><td>Qwen2.5-32B-Chat</td><td>551</td><td>1531</td><td>2252</td><td>1006</td><td>187</td><td>1413</td><td>3002</td><td>498</td></tr><tr><td>Qwen2.5-7B</td><td>567</td><td>1624</td><td>2259</td><td>1006</td><td>232</td><td>1147</td><td>5559</td><td>239</td></tr><tr><td>Qwen3-14B</td><td>2312</td><td>1037</td><td>4167</td><td>1949</td><td>2743</td><td>3938</td><td>3688</td><td>1396</td></tr><tr><td>Qwen3-8B</td><td>1648</td><td>1885</td><td>5736</td><td>2545</td><td>2030</td><td>3025</td><td>6577</td><td>1826</td></tr><tr><td>GPT-4o</td><td>1535</td><td>2110</td><td>1050</td><td>563</td><td>201</td><td>2021</td><td>4294</td><td>436</td></tr><tr><td>GPT-4o-mini</td><td>720</td><td>1349</td><td>952</td><td>531</td><td>172</td><td>1634</td><td>3607</td><td>405</td></tr></table>
+
+# J Case Studies of LfI in Non-Math Domains
+
+The illustrative cases in Tables 9, Tables 10, Tables 11 and Tables 12 highlight the diversity and domain fidelity of the non-mathematical Learning-from-Instructor (LfI) benchmark. For each of the four domains—Anthropology, Economics, Law, and Philosophy—we present one exemplar of every question format used in our experiments: fill-in-the-blank, judgment, multiple-choice, open-ended, and single-choice. Each example is accompanied by a short answer and a concise explanation that reflects the expected reasoning chain. Together, these cases demonstrate the range of skills required by the benchmark: factual recall (fill), binary verification (judge), multi-step selection (multi-choice), free-form synthesis (open), and focused identification (single-choice). They also show that the prompts draw directly on domain-specific classics, ensuring that successful solutions depend on relevant expert knowledge rather than surface heuristics. This qualitative examination complements the quantitative results in Section H, confirming that the LfI tasks provide a realistic and varied testbed for evaluating how well large language models can identify, learn, and apply useful knowledge in the presence of noise.
+
+Table 9: Question formats in anthropology domain.  
+
+<table><tr><td>Multiple-Choice Question Example (multi-choice):
+Question: What are the two plans of government that have developed throughout human history?
+Choices:
+A: Social organization
+B: Political organization
+C: Religious organization
+D: Economic organization
+E: Military organization
+Answer: A, B
+Explanation: A and B are correct because the two primary plans of government that have developed throughout human history are social organization and political organization. Social organization, which is the older form, is based on kinship structures such as thegens, phratry, and tribe. Political organization, which emerged later, is based on territorial divisions such as townships, counties, and states. Religious, economic, and military organizations, while significant, are not listed as the two primary plans of government that have shaped societal structures.</td></tr><tr><td>Judgment Question Example (judge):
+Question: The Aztec civilization extended slightly north of the Tropic of Cancer.
+Answer: True
+Explanation: This statement is true because the Aztec civilization, one of the most prominent in the New World, had its center in the region that extends just north of the Tropic of Cancer. This area, while still influenced by tropical conditions, provided a suitable environment for the Aztecs to develop their advanced society, including their complex social structures and agricultural practices.</td></tr><tr><td>Single-Choice Question Example (single-choice):
+Question: What is one of the natural centers of subsistence mentioned in the text?
+Choices:
+A: The Mississippi Delta
+B: The Great Plains
+C: The Valley of the Columbia
+D: The Appalachian Mountains
+Answer: C
+Explanation: C is correct because the Valley of the Columbia is mentioned as one of the natural centers of subsistence. This region is noted for its extraordinary variety and amount of subsistence resources, which made it an ideal place for the development of a surplus population and the subsequent formation of new tribes.</td></tr><tr><td>Open-Ended Question Example (open):
+Question: How were the upper levels of Pueblo dwellings accessed?
+Answer: By means of movable ladders and a hatchway in the roof
+Explanation: The upper levels of Pueblo dwellings were reached using movable ladders and a hatchway in the roof. This design provided a means of access while also enhancing security, as the ladders could be removed to prevent unauthorized entry.</td></tr><tr><td>Fill-in-the-Blank Question Example (fill):
+Question: In which region were the X-shaped bronze objects used as money, according to the passage?
+Answer: Lunda
+Explanation: The X-shaped bronze objects were used as money in the region of Lunda. These objects were made in Lunda and were widely accepted as currency throughout the Congo. The use of such distinctive and uniquely shaped metal objects as money highlights the diverse forms of currency that have been developed by different societies to facilitate trade and economic transactions.</td></tr></table>
+
+Table 10: Question formats in economics domain.  
+
+<table><tr><td>Multiple-Choice Question Example (multi-choice):
+Question: What does the text imply about the role of history in the study of law?
+Choices:
+A: History should be the exclusive focus of legal studies
+B: History has no value in the study of law
+C: History should be integrated with other methods in legal studies
+D: History should be used to pursue pure abstractions
+E: History should be used to understand the evolution of legal principles and practices
+Answer: C, E
+Explanation: C and E are correct because history should be integrated with other methods in legal studies and should be used to understand the evolution of legal principles and practices. History should not be the exclusive focus of legal studies, does have value in the study of law, and should not be used to pursue pure abstractions.</td></tr><tr><td>Judgment Question Example (judge):
+Question: During the middle ages, candles were very expensive, costing about 1-1/3 to 2 shillings per pound.
+Answer: True
+Explanation: Historical records indicate that candles were a luxury item during the middle ages, with prices ranging from 1-1/3 to 2 shillings per pound. This high cost made candles an expensive commodity, accessible primarily to the wealthy, and contributed to their status as a symbol of affluence and status.</td></tr><tr><td>Single-Choice Question Example (single-choice):
+Question: Which Roman emperor made a significant legal change concerning the treatment of abandoned children?
+Choices:
+A: Julius Caesar
+B: Augustus
+C: Constantine the Great
+D: Nero
+Answer: C
+Explanation: C is correct because Constantine the Great, in the year 315 AD, made a significant legal change concerning the treatment of abandoned children. This change likely aimed to improve the conditions and legal standing of abandoned children, reflecting a shift in Roman societal values and legal practices.</td></tr><tr><td>Open-Ended Question Example (open):
+Question: What is the relationship between the development of private property and the advancement of civilization?
+Answer: Private property has developed as civilization and well-being have advanced
+Explanation: The evolution of private property is closely linked to the progress of civilization and the improvement of living standards. As societies become more developed and culturally advanced, the concept and institution of private property become more established, reflecting and contributing to the overall well-being and prosperity of the community.</td></tr><tr><td>Fill-in-the-Blank Question Example (fill):
+Question: The unnecessary destruction of wealth, such as spending 10,000 dollars on fireworks, is considered ___, regardless of whether the money remains in the country.
+Answer: disastrous
+Explanation: Economic theory emphasizes the importance of productive and sustainable use of resources. When wealth is spent on non-productive activities, such as extravagant fireworks displays, it is considered a wasteful and disastrous use of resources. This is because the destruction of wealth, even if the money circulates within the country, does not contribute to long-term economic growth or development. The resources used for such displays could have been allocated to more productive uses, such as investment in infrastructure or education.</td></tr></table>
+
+# Multiple-Choice Question Example (multi-choice):
+
+Question: Which of the following authors wrote treatises that were used as textbooks for teaching International Law?
+
+Choices:
+
+A: William Edward Hall
+
+B: Sir Edward Shepherd Creasy
+
+C: Theodore D. Woolsey
+
+D: Heinrich Bernhard Oppenheim
+
+E: Luigi Casanova
+
+F: Ernest Nys
+
+Answer: A, B, C, D, E
+
+Explanation: A, B, C, D, and E are correct because William Edward Hall, Sir Edward Shepherd Creasy, Theodore D. Woolsey, Heinrich Bernhard Oppenheim, and Luigi Casanova all wrote treatises that were used as textbooks for teaching International Law. These works were designed to provide comprehensive and accessible introductions to the field, suitable for students and practitioners. Ernest Nys, while a significant author, did not specifically write a textbook.
+
+# Judgment Question Example (judge):
+
+Question: Private enemy property on a belligerent's territory can be confiscated according to the customary rule of International Law.
+
+Answer: False
+
+Explanation: The customary rule of International Law now prohibits the confiscation of private enemy property on the territory of a belligerent. This rule evolved through international treaties, municipal laws, and decrees, making the old practice of confiscation obsolete.
+
+# Single-Choice Question Example (single-choice):
+
+Question: What is the significance of the Hague Conferences of 1899 and 1907 in the context of guerilla warfare?
+
+Choices:
+
+A: They banned all forms of guerilla warfare
+
+B: They provided a legal framework for the treatment of guerilla fighters
+
+C: They recognized guerilla warfare as a legitimate form of war
+
+D: They established the rules for the treatment of prisoners of war
+
+Answer: B
+
+Explanation: B is correct because the Hague Conferences of 1899 and 1907 provided a legal framework for the treatment of guerilla fighters. These conferences established conditions under which guerilla fighters could be treated as soldiers, thus providing them with certain protections under international law.
+
+# Open-Ended Question Example (open):
+
+Question: What is the purpose of pledging movable property in the context of international treaties?
+
+Answer: To secure the performance of a treaty
+
+Explanation: Pledging movable property in international treaties serves as a form of collateral to ensure that the obligations outlined in the treaty are fulfilled. This practice acts as a safeguard, where the pledged items can be retained or used to compensate the other party if the treaty obligations are not met.
+
+# Fill-in-the-Blank Question Example (fill):
+
+Question: The concept of prescription in international law is not solely based on the ______ of time. Answer: passage
+
+Explanation: Prescription in international law is not determined by a fixed period of time alone. It involves a combination of factors, including the stability of possession, the absence of protests, and the general acceptance by the international community. The passage of time is only one aspect of the broader context that contributes to the formation of a prescriptive title.
+
+Table 12: Question formats in philosophy domain.  
+
+<table><tr><td>Multiple-Choice Question Example (multi-choice):
+Question: Which of the following philosophers is praised for his deeper insights compared to Rousseau?
+Choices:
+A: Voltaire
+B: Schopenhauer
+C: Locke
+D: Byron
+E: Herodotus
+F: Plutarch
+Answer: B, C
+Explanation: B and C are correct because the unnamed great man, likely Schopenhauer, is praised for his depth of thinking, which includes insights aligned with Locke&#x27;s principle. Voltaire, Byron, Herodotus, and Plutarch are not mentioned as being praised for deeper insights compared to Rousseau.</td></tr><tr><td>Judgment Question Example (judge):
+Question: Satisfaction of a wish can lead to the creation of a new wish.
+Answer: True
+Explanation: Satisfaction of a wish often leads to the creation of a new wish because desires and wants are inherently dynamic and ever-evolving. Once a particular goal is achieved, the human mind tends to focus on the next goal, creating a continuous cycle of desire and fulfillment. This is rooted in the nature of human psychology, where the satisfaction of one need or desire often reveals or intensifies another, keeping individuals in a perpetual state of seeking.</td></tr><tr><td>Single-Choice Question Example (single-choice):
+Question: Which of the following are considered essential fundamental constituent elements of language according to the passage?
+Choices:
+A: Noun, verb, and adjective
+B: Pronoun, article, and adverb
+C: Substantive, adjective, and verb
+D: Substantive, article, and pronoun
+Answer: C
+Explanation: C is correct because the essential fundamental constituent elements of language, as mentioned, are substantive (which is another term for noun), adjective, and verb. These elements are crucial for the structure and function of language, providing the core components needed to form sentences and convey meaning.</td></tr><tr><td>Open-Ended Question Example (open):
+Question: What is the first step towards achieving a metaphysical understanding of the world, according to the text?
+Answer: Bringing to distinct consciousness and firmly retaining the difference between physics and metaphysics.
+Explanation: The first step in moving towards a metaphysical understanding is to clearly recognize and maintain the distinction between physics and metaphysics. Physics deals with the observable and measurable aspects of the world, while metaphysics explores the deeper, underlying nature of reality. Understanding this difference is crucial for recognizing the limitations of physical explanations and the need for a metaphysical approach.</td></tr><tr><td>Fill-in-the-Blank Question Example (fill):
+Question: The aesthetic effect of architectural works is in exact proportion to the _ of the building.
+Answer: size
+Explanation: The aesthetic impact of a building increases with its size. Larger structures are more effective in showcasing the forces of nature, such as gravity, in a more apparent and impressive manner. This is because larger masses make the action of these forces more visible and tangible, enhancing the overall aesthetic experience. Therefore, the size of a building is a critical factor in its aesthetic value.</td></tr></table>
+
+# K Prompt Design for LearnArena
+
+To support the evaluation of general learning ability in LearnArena, we design three types of prompts corresponding to the three learning dimensions: Concept, Instructor, and Experience. Each prompt is carefully constructed to elicit behavior aligned with the intended mode of learning, while remaining concise and task-relevant. For Learning from Concept, in Table 13, the prompt introduces the game setting and provides a concise, structured request for strategic advice. The system prompt frames the LLM as an expert strategist, while the user prompt asks for a list of key principles, tactics, and pitfalls, grounded in the rules of the game. This setup encourages the model to abstract general strategies directly from conceptual understanding. For Learning from Instructor, in Table 14, the prompt simulates feedback-based instruction. The system prompt positions the model as a game instructor giving targeted feedback. The user provides a summary of game history from prior rounds, and the model is asked to analyze and respond in a structured format. This prompt encourages the model to extract lessons from explicit advice and past decisions, simulating instructor-led reflection. For Learning from Experience, in Table 15, the prompt shifts the role to that of a self-reflective player. The system prompt instructs the model to analyze its own performance and derive improvements. The user supplies a history of recent gameplay, and the model is prompted to identify useful patterns, good and bad moves, and general strategic takeaways. This design promotes autonomous learning from accumulated experience. All prompts share a consistent structure and tone, enabling fair comparison across models and environments. By isolating each learning mode, we can evaluate how well a model generalizes knowledge from rules, instruction, and prior experience independently.
+
+Table 13: Prompt template for the concept learning setting.
+
+# Prompt for Concept
+
+# System Prompt:
+
+You are an expert game strategist with deep knowledge of game theory and optimal play. Your task is to provide concise, actionable strategic advice that will help a player win. Focus on identifying winning patterns, key decision points, and optimal strategies.
+
+# User Prompt:
+
+For this game '\{game\}' , provide brief winning strategies based on this initial observation.
+
+# WINNING STRATEGIES:
+
+1. Top 3-5 strategic principles that lead to victory  
+2. Best opening moves or early game tactics  
+3. Key patterns to recognize during gameplay  
+4. Critical mistakes to avoid
+
+# Rules of the Game: {rules}
+
+Keep your response concise and focused on practical advice that will maximize winning chances.
+
+# L Prompt Design for the Passive/Interactive Setting
+
+To support our experiments in Section 4.1, we design distinct prompting strategies to simulate the two instruction-based learning paradigms: Passive Consumption and Interactive Clarification. These prompts define how the instructor and learner interact during the generation and clarification phases of training data construction. In the Passive setting, the instructor model is prompted to solve a math problem directly, using a system instruction that encourages detailed step-by-step reasoning and a final answer. The learner is then trained solely on these final outputs without any follow-up interaction. The corresponding prompt template is shown in Table 17. In the Interactive setting, the instructor-learner interaction is decomposed into a three-step process. First, the instructor provides a
+
+Table 14: Prompt template for the instructor learning setting.  
+
+<table><tr><td>Prompt for Instructor</td></tr><tr><td>System Prompt:
+You are an expert game instructor. Your task is to provide detailed strategic advice for the game based on the provided information.</td></tr><tr><td>User Prompt:
+Analyze this history and provide strategic advice.</td></tr><tr><td>Game history: {game history}</td></tr><tr><td>Provide your analysis in this format:
+1. Strategic principles to follow
+2. Specific moves to consider
+3. Moves to avoid
+4. Key patterns to watch for</td></tr></table>
+
+Table 15: Prompt template for the experience learning setting.  
+
+<table><tr><td>Prompt for Experience</td></tr><tr><td>System Prompt:You are a player analyzing your own gameplay. Your task is to identify strengths, weaknesses, and learn from this experience to enhance future performance.User Prompt:Analyze this history and provide strategic advice.Game history: {game history}Provide your analysis in this format:1. Strategic principles to follow2. Specific moves to consider3. Moves to avoid4. Key patterns to watch for</td></tr></table>
+
+detailed solution to a math problem, identical to the passive setting. Second, the learner (student) reviews the solution and is prompted to identify specific confusions or follow-up questions based on the instructor's response. Finally, the instructor receives the student's clarifying question and provides a targeted explanation aimed at resolving the confusion. This interactive exchange is used as training data for the learner model, capturing a more realistic and adaptive instructional process. This procedure reflects a more realistic and adaptive instructional scenario. The full prompting sequence for this setting is shown in Table 16.
+
+We also present a concrete case study illustrating the benefits of the Interactive prompting approach, as shown in Table 19, Table 20, and Table 21. In this example, the learner poses meaningful and specific questions about the instructor's initial solution, such as clarifying the strategy behind number grouping, order of operations, and the existence of alternative solutions. The instructor's detailed responses systematically address these queries, clearly enhancing the instructional quality compared to the passive setting. This exchange highlights the distinct advantage of interactive clarification,
+
+where learner-generated questions provoke richer, more targeted explanations, ultimately improving learning effectiveness.
+
+Table 16: Interactive prompts for mathematical domain.
+
+# Interactive Step 1: Instructor Receives Problem
+
+# System Prompt:
+
+You are a mathematician and educator. Solve the following math problem with accurate, complete, and clear explanations. Break down your reasoning into a logical chain of steps, and provide the final answer only after completing the reasoning.
+
+# User Prompt:
+
+{question}
+
+# Interactive Step 2: Student Reviews Solution
+
+# System Prompt:
+
+You are a math student. You will read the math problem and a solution provided by your math teacher. If you have any questions or confusions about the solution, ask specific questions about those parts.
+
+# User Prompt:
+
+Math Problem: {question}
+
+Here is the solution provided by your teacher: {answer}
+
+If you have any questions or confusions about the solution, please list them.
+
+# Interactive Step 3: Instructor Addresses Confusions
+
+# System Prompt:
+
+You are a mathematician and educator dedicated to resolving confusions about math solutions. Provide clear, step-by-step explanations to logically address each confusion.
+
+# User Prompt Context:
+
+Initial Question: {question}
+
+Provided Solution: {answer}
+
+Student's Confusions: Confusions about the solution: {confusion}
+
+Please address the confusions.
+
+Table 17: Passive prompt for mathematical domain.
+
+# Passive Prompt for Mathematical Domain
+
+# System Prompt:
+
+You are a mathematician and educator. Solve the following math problem with accurate, complete, and clear explanations. Break down your reasoning into a logical chain of steps, and provide the final answer only after completing the reasoning.
+
+# User Prompt:
+
+{question}
+
+Table 18: Case study: passive prompt in mathematics.
+
+# Passive Prompt Math Problem and Solution:
+
+Prompt: You are a mathematician and educator. Solve the following math problem with accurate, complete, and clear explanations. Break down your reasoning into a logical chain of steps, and provide the final answer only after completing the reasoning.
+
+Question: If  $a, b$ , and  $c$  are positive integers satisfying  $ab + c = bc + a = ac + b = 41$ , what is the value of  $a + b + c$ ?
+
+Solution (Output): We have  $ab + c = 41$  and  $bc + a = 41$ , so  $ab + c = bc + a$ . Subtracting the two equations, we have  $ab - bc + c - a = 0 \Rightarrow b(a - c) + (c - a) = 0 \Rightarrow (b - 1)(a - c) = 0$ . Therefore,  $a = c$  or  $b = 1$ .
+
+If  $a = c$ , we have  $ab + c = a^2 + b = 41$ . We also have  $bc + a = ab + c = 41$ . If  $a^2 + b = 41$ , then  $a^2 \leq 41$ , so  $a \leq 6$ . If  $a = 6$ , then  $36 + b = 41 \Rightarrow b = 5$ . If  $a = c = 6$  and  $b = 5$ , then our three equations are satisfied, so  $a + b + c = 6 + 5 + 6 = \boxed{17}$ . If  $a < 6$ , then  $b \geq 6$ , so  $b > a$ . We also have  $a^2 + b = 41$ , so  $b = 41 - a^2 < 25$ . Therefore,  $6 \leq a < 6$ , so  $a = 6$ , and we have the same solution as before.
+
+If  $b = 1$ , we have  $ab + c = a + c = 41$  and  $bc + a = c + a = 41$ . Therefore,  $a + c = c + a$ , which is always true. However, we also have  $ac + b = ac + 1 = 41 \Rightarrow ac = 40$ . Therefore,  $a$  and  $c$  are factors of 40. The only factor pairs of 40 are (1, 40), (2, 20), (4, 10), and (5, 8). However, we are looking for positive integers  $a, b, c$  such that  $a + c = 41$ , so the only possible solution is  $a = 5$  and  $c = 8$ . Therefore,  $a + b + c = 5 + 1 + 8 = \boxed{14}$ . The answer is 14
+
+Table 19: Case study: interactive prompts for mathematical domain.
+
+# Interactive Step 1: Instructor Receives Problem
+
+# System Prompt:
+
+You are a mathematician and educator. Solve the following math problem with accurate, complete, and clear explanations. Break down your reasoning into a logical chain of steps, and provide the final answer only after completing the reasoning.
+
+# User Prompt (Question):
+
+Using the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 in order, and only using the operations of addition and subtraction, how can you make the expression equal 100? Parentheses can be used to change the order of operations.
+
+# Interactive Step 2: Student Reviews Solution & Asks Questions
+
+# System Prompt:
+
+You are a math student. You will read the math problem and a solution provided by your math teacher. If you have any questions or confusions about the solution, ask specific questions about those parts.
+
+# User Prompt:
+
+Math Problem: Using the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 in order, and only using the operations of addition and subtraction, how can you make the expression equal 100? Parentheses can be used to change the order of operations.
+
+Here is the solution provided by your teacher (Initial Answer): To solve the problem of making the expression using the numbers 1 through 9 in order, and only using addition and subtraction to equal 100, we need to strategically place the operations and parentheses. Let's break down the solution step-by-step.
+
+First, let's consider the structure of the problem. We need to use all the digits from 1 to 9 in order and use only addition and subtraction to reach 100. We will experiment with different placements of operations and parentheses to find a valid expression.
+
+One effective approach is to group numbers in a way that allows us to use the operations to balance the result to 100. Let's start by trying to use the larger numbers to get close to 100 and then adjust with smaller numbers.
+
+Consider the sequence:
+
+$$
+1 2 3 - 4 5 - 6 7 + 8 9
+$$
+
+Let's break this down: 1. Calculate  $123 - 45$ :
+
+$$
+1 2 3 - 4 5 = 7 8
+$$
+
+2. Next, calculate  $78 - 67$ :
+
+$$
+7 8 - 6 7 = 1 1
+$$
+
+3. Finally, add 89 to 11:
+
+$$
+1 1 + 8 9 = 1 0 0
+$$
+
+So, the expression  $123 - 45 - 67 + 89$  equals 100.
+
+Therefore, the final answer is:
+
+$$
+\boxed {1 2 3 - 4 5 - 6 7 + 8 9}
+$$
+
+If you have any questions or confusions about the solution, please list them (Student's Confusions): I understand the solution provided, but I have a few questions and points of confusion:
+
+1. **Grouping Numbers**: Why did we choose to group the numbers as 123, 45, 67, and 89? Is there a specific strategy or intuition behind this grouping?  
+2. **Order of Operations**: How do we ensure that the order of operations (addition and subtraction) is correctly applied when evaluating the expression? Specifically, how do the parentheses ensure the correct order?  
+3. **Alternative Solutions**: Are there other ways to group the numbers and use addition and subtraction to reach 100? If so, could you provide an example of another valid expression?  
+4. **General Strategy**: Is there a general method or algorithm to solve problems like this, where we need to use a sequence of numbers and operations to reach a specific target?  
+5. **Verification**: How can we verify that no other combination of addition and subtraction using the numbers 1 through 9 in order will also result in 100?
+
+Table 21: Case study: interactive prompts for mathematical domain - Part 1
+
+# Interactive Step 3: Instructor Addresses Confusions
+
+# System Prompt:
+
+You are a mathematician and educator dedicated to resolving confusions about math solutions.
+
+Provide clear, step-by-step explanations to logically address each confusion.
+
+# User Prompt Context:
+
+Initial Question: Using the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 in order, and only using the operations of addition and
+
+subtraction, how can you make the expression equal 100? Parentheses can be used to change the order of operations.
+
+Provided Solution (Initial Answer):
+
+To solve the problem of making the expression using the numbers 1 through 9 in order, and only using addition and subtraction
+
+to equal 100, we need to strategically place the operations and parentheses. Let's break down the solution step-by-step.
+
+First, let's consider the structure of the problem. We need to use all the digits from 1 to 9 in order and use only addition and
+
+subtraction to reach 100. We will experiment with different placements of operations and parentheses to find a valid expression.
+
+One effective approach is to group numbers in a way that allows us to use the operations to balance the result to 100. Let's
+
+start by trying to use the larger numbers to get close to 100 and then adjust with smaller numbers.
+
+Consider the sequence:
+
+$$
+1 2 3 - 4 5 - 6 7 + 8 9
+$$
+
+1. Calculate  $123 - 45 = 78$  
+2. Then  $78 - 67 = 11$  
+3. Finally  $11 + 89 = 100$
+
+Therefore the expression  $123 - 45 - 67 + 89$  equals 100, and the final answer is  $\boxed{123 - 45 - 67 + 89}$
+
+Table 21: Case study: interactive prompts for mathematical domain - Part 2  
+
+<table><tr><td>Student&#x27;s Confusions:
+1. The strategy behind grouping 123, 45, 67, 89.
+2. Ensuring the order of operations, especially the role of parentheses.
+3. Whether alternative solutions exist and an example.
+4. Whether a general algorithm exists for such problems.
+5. How to verify that no other combinations yield 100.</td></tr><tr><td>Instructor&#x27;s Response:
+1. Grouping Numbers (123 | 45 | 67 | 89).
+A reliable heuristic is to fuse the early digits into the largest possible chunk (here 123) so you begin near the target of 100.
+Next, look for a pair of medium-sized numbers whose difference is small (45 and 67 differ by 22) and a final adjustment term (±89) that brings the running total to the goal. This “large-number-first, cancel-with-a-pair, fine-tune” pattern dramatically shrinks the search space compared with blind trial-and-error.
+2. Order of Operations &amp; Parentheses.
+Because addition and subtraction are left-associative and equal-priority, 123 - 45 - 67 + 89 is automatically evaluated
+left-to-right, producing 100 without extra parentheses. Parentheses matter only when we deliberately override that rule—e.g. (123 - (45 - 67) + 89), which forces 45 - 67 first.
+3. Alternative Solutions.
+Multiple expressions satisfy the constraints, for example
+• 123 + 4 - 5 + 67 - 89 = 100
+• 12 + 3 - 4 + 5 + 6 + 78 = 100
+Each follows the same philosophy: create or destroy “mass” with big blocks, then make small tweaks.
+4. General Strategy / Algorithm.
+A systematic method is depth-first search with pruning:
+• At each digit boundary, choose to concatenate or to insert + / -.
+• Track a running total; abandon any branch whose partial total already overshoots a sensible bound.
+• There are 28 = 256 operator patterns and 28 = 256 digit-merge patterns, so the full tree has at most 65536 leaves—small for a computer and still manageable by hand with pruning.
+5. Verifying Uniqueness.
+Write a short program (or spreadsheet) that enumerates every merge / operator pattern; collect the ones equalling 100.
+You will find only a handful of distinct families and no hidden outliers, giving a rigorous guarantee that you have all solutions.
+Armed with these tactics—large-chunk heuristics, operator-precedence insights, a DFS-with-pruning template, and programmatic verification—you can tackle not only this puzzle but the entire class of “insert ± to hit a target” challenges with confidence.</td></tr></table>
+
+[GAME] You are Player 0 in Tak.
+
+Your goal is to connect two opposite edges of the board with your pieces to form a road... % --- Rules and examples have been abridged for brevity ---
+
+[Player 0] Given that the board is currently empty, I will begin by placing a flat stone... [place () {(0,0): [FO]}]
+
+[GAME] Player 0 placed a piece on  $(([(0,0)])$ . New board state:
+
+![](images/69780362093f296eb687a785049fb5f0dc48773baac3d63bf04c254bf698c7f6.jpg)
+
+[Player 1] For my move, I will place a wall stone to block potential pathways... [place () {(0,1): [W1]}]
+
+[GAME] Player 1 placed a piece on  $((0,1)]$ . New board state:
+
+![](images/5938466603763cf9e66b7c989f8ca7fe69b7b2361659eb5485e2decf1867fc92.jpg)
+
+$\%$  Intermediate moves are omitted for brevity
+
+[GAME] Player 1 placed a piece on  $(([(3,1)])$ . New board state:
+
+![](images/c06455fd1bb64db87daf6aaeee1bc5fba74c71096267608b5032d4a19ba98a92.jpg)
+
+![](images/096178d53c1b1d293fcd7acd647ef4d861ac5ce4494ffd52678d11401c4ad96b.jpg)
+
+![](images/85b6cfb94e726b8b21f5fa70b2c5c90829b9712dbe3dc937e85827e009090f3d.jpg)
+
+![](images/74d673bcbc58490194b68160449d73e9e2b326ee685a05732f8352481e19bc7b.jpg)
+
+![](images/d041c2cf2897b161cab94083c53df6e1702d779e0722db3a623aba759b5493a8.jpg)  
+Figure 11: Abridged game history log as presented to the agent. It includes the initial rules, followed by a turn-by-turn log of actions and resulting board states.
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
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+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
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\ No newline at end of file
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+# Unveiling the Power of Multiple Gossip Steps: A Stability-Based Generalization Analysis in Decentralized Training
+
+Qinglun Li $^{1}$ , Yingqi Liu $^{2}$ , Miao Zhang $^{1}$ , Xiaochun Cao $^{2}$ , Quanjun Yin $^{1,*}$ , Li Shen $^{2,3,*}$
+
+$^{1}$ State Key Laboratory of Digital Intelligent Modeling and Simulation,
+
+National University of Defense Technology, Changsha, 410073
+
+$^{2}$ School of Cyber Science and Technology, Shenzhen Campus of Sun Yat-sen University, China
+
+<sup>3</sup> Center for AI Theoretical Foundation and Systems, Shenzhen Loop Area Institute, China liqinglun@nudt.edu.cn, yin_quanjun@163.com, mathshenli@gmail.com
+
+# Abstract
+
+Decentralized training removes the centralized server, making it a communication-efficient approach that can significantly improve training efficiency, but it often suffers from degraded performance compared to centralized training. Multi-Gossip Steps (MGS) serve as a simple yet effective bridge between decentralized and centralized training, significantly reducing experiment performance gaps. However, the theoretical reasons for its effectiveness and whether this gap can be fully eliminated by MGS remain open questions. In this paper, we derive upper bounds on the generalization error and excess error of MGS using stability analysis, systematically answering these two key questions. 1). Optimization Error Reduction: MGS reduces the optimization error bound at an exponential rate, thereby exponentially tightening the generalization error bound and enabling convergence to better solutions. 2). Gap to Centralization: Even as MGS approaches infinity, a nonnegligible gap in generalization error remains compared to centralized mini-batch SGD  $(\mathcal{O}(T^{\frac{c\beta}{c\beta + 1}} / nm)$  in centralized and  $\mathcal{O}(T^{\frac{2c\beta}{2c\beta + 2}} / nm^{\frac{1}{2c\beta + 2}})$  in decentralized). Furthermore, we provide the first unified analysis of how factors like learning rate, data heterogeneity, node count, per-node sample size, and communication topology impact the generalization of MGS under non-convex settings without the bounded gradients assumption, filling a critical theoretical gap in decentralized training. Finally, promising experiments on CIFAR datasets support our theoretical findings.
+
+# 1 Introduction
+
+Recently, decentralized training [1, 2] has emerged as a promising alternative to centralized training, which suffers from challenges like high communication overhead [3], single point of failure [4], and privacy risks [5]. In contrast, decentralized training eliminates the central server, offering stronger privacy protection [6], faster model training [7, 2], and robustness to slow client devices [8], making it an increasingly popular method [4, 7].
+
+![](images/2b17b1e46d751d8c8580f257c38f293fe7a7f0eb118e5c7b2cc70b9fbfdf5a63.jpg)  
+Figure 1: Under ring topology, DSGD-MGS with 20 gossip steps still shows significant performance gaps versus Mini-batch SGD in both training loss and test accuracy (LeNet on CIFAR-10, Dir 0.3, 50 nodes).
+
+However, despite the aforementioned advan
+
+tages of decentralized training, some works [9, 10, 11] have pointed out that decentralized training methods underperform compared to centralized training methods in terms of model performance. Therefore, improving the performance of decentralized training models remains an important research question. Multiple Gossip Steps (MGS) [12, 13], as a simple yet effective method to enhance the performance of decentralized training models, has been experimentally proven to significantly improve the efficiency and performance of decentralized training [14, 15, 16, 17]. Even under communication compression, MGS continues to demonstrate its advantages in performance improvement [18].
+
+Despite the substantial empirical benefits of MGS, the underlying theoretical understanding of its efficacy and its potential to eliminate the performance gap with centralized training remain critical open questions. Specifically, two key issues need to be addressed:
+
+(1) Why is MGS effective in improving model performance?  
+(2) Can decentralized training ultimately match or even surpass the performance of centralized training by increasing the number of gossip steps?
+
+To answer these open questions, we aim to theoretically explain how MGS works and how it affects model generalization. Using stability analysis, we find upper limits for the generalization error and excess error of MGS, giving systematic theoretical answers to these two main questions.
+
+For Question 1, our theoretical analysis shows that MGS can reduce the optimization error bound at an exponential rate. This reduction in optimization error directly leads to an exponential reduction in the generalization error (as shown in Theorem 2, 3, and Remark 2), enabling the model to find better solutions. This relationship clearly explains why MGS effectively improves model performance. As illustrated in Figure 1, when the number of gossip steps is increased from 1 to 5, there is a significant reduction in the training loss (indicating reduced optimization error), and the test accuracy (measuring generalization) also shows a noticeable improvement. Furthermore, this improvement tends to diminish almost linearly as the number of gossip steps increases exponentially, consistent with the exponential decay in our theory findings.
+
+For Question 2, our further analysis shows that even with a very large number of gossip steps, a basic difference in generalization error remains between decentralized DSGD-MGS and centralized mini-batch SGD.
+
+Specifically, when the number of gossip steps becomes extremely large, the generalization error bound for DSGD-MGS becomes at most  $\mathcal{O}\big(T^{\frac{2c\beta}{2c\beta + 2}} / nm^{\frac{1}{2c\beta + 2}}\big)$ . However, this is still noticeably larger than the centralized mini-batch SGD bound of  $\mathcal{O}(T^{\frac{c\beta}{c\beta + 1}} / nm)$ , highlighting a lasting difference in how it scales with the number of clients  $m$  (because  $1 / m < 1 / m^{\frac{1}{2c\beta + 2}}$  when  $m > 1$ ). This theoretical observation reveals a basic constraint: decentralized training cannot fully achieve the generalization performance of centralized training solely by increasing the number of MGS steps. Experiments shown in Figure 1 support this conclusion, indicating that even with 20 gossip steps, DSGD-MGS still performs worse than centralized mini-batch SGD in the same settings.
+
+Moreover, we are the first to provide a theoretical framework to understand how critical factors, including learning rate, data heterogeneity, number of nodes, sample size per node, and communication topology, jointly influence the generalization performance of MGS (see Reamrk 2-9). Remarkably, we also eliminate the bounded gradient assumption in the non-convex condition. This work enhances our understanding of the challenges in decentralized learning and provides theoretical insights for hyperparameters to better model generalization. Finally, extensive experiments on CIFAR datasets further validate our theoretical results. The main contributions of this paper can be summarized as follows:
+
+- Theoretically elucidating the mechanism by which MGS enhances the generalization performance of decentralized training models through an exponential reduction in optimization error.  
+- Revealing that even with sufficient gossip communication, a theoretical gap in generalization error remains between MGS and centralized training, and this gap cannot be eliminated by MGS alone.  
+- Establishing, for the first time under non-convex and without the bound gradient assumption, a unified framework analyzing factors impacting the MGS generalization performance (i.e., learning rate, data heterogeneity, number of nodes, sample size, and topology), thereby addressing a significant gap in existing theoretical frameworks.
+
+- Validating our theoretical findings through empirical experiments on the CIFAR datasets.
+
+These findings provide new theoretical insights and practical implications for understanding and improving decentralized learning algorithms.
+
+# 2 Related Works
+
+This section reviews the current theoretical understanding and challenges in decentralized training, along with the evolution and impact of MGS. Moreover, at the end of each subsection, we highlight the existing gaps and open questions within these areas to position the contributions of this paper.
+
+Theoretical Analysis of D-SGD. Decentralized learning has attracted significant research interest due to its potential for enhanced privacy, communication efficiency, and scalability [7, 5, 6, 8]. Early theoretical studies primarily focused on the convergence analysis of D-SGD, examining the number of iterations or communication rounds needed to reach an  $\epsilon$ -accurate solution [7, 18, 19]. More recently, attention has shifted towards understanding the generalization performance of these algorithms. Sun et al. [20] were the first to analyze the generalization performance of D-SGD using uniform stability, later extending their results to asynchronous D-SGD [21]. However, these analyses assumed homogeneous data and bounded gradients. Zhu et al. [22] further studied the impact of communication topology on the generalization error of D-SGD, with their generalization bounds later improved by [11], but they also relied on the same assumptions. More recently, Ye et al. [23] analyzed the generalization behavior of D-SGD under heterogeneous data, but their analysis was limited to strongly convex loss functions. Overall, current D-SGD theories still lack a unified framework that comprehensively accounts for all key algorithm parameters (e.g., data heterogeneity, non-convex loss function, topology, etc.).
+
+MGS in Decentralized Training. Multiple Gossip Steps (MGS) [24, 12] is a technique that improves consensus by allowing multiple rounds of local communication. When integrated into decentralized algorithms, MGS not only enhances generalization performance but also accelerates convergence [25]. Additionally, Yuan et al. [19] showed that MGS can reduce the adverse effects of data heterogeneity, a finding supported by other studies [26, 16]. Li et al. [27] found that MGS can significantly improve algorithm accuracy. In the field of decentralized federated learning, Shi et al. [16] incorporated MGS into their DFedSAM algorithm, significantly improving its generalization performance experimentally. Notably, MGS alone can achieve optimal convergence rates in non-convex settings [19] without relying on more complex techniques like gradient tracking [28], quasi-global momentum [29], or adaptive momentum [30]. However, these studies have largely overlooked the question of why MGS is effective from a generalization perspective, with these advantages demonstrated mainly through empirical results, leaving a significant gap in the theoretical understanding of MGS.
+
+# 3 Background
+
+In this section, we first present some fundamental definitions required for stability analysis, including population risk, empirical risk, generalization error, excess error, and  $l_{2}$  on-average model stability. Subsequently, we introduce a key lemma that establishes the relationship between the generalization error bound and  $l_{2}$  on-average model stability.
+
+# 3.1 Stability and Generalization in Decentralized Learning
+
+We consider the general statistical learning setting, adapted to a decentralized framework with  $m$  agents $^2$ . Each agent  $k$  observes data points drawn from a local distribution  $\mathcal{D}_k$  with support  $\mathcal{Z}$ . The goal is to find a global model  $\theta \in \mathbb{R}^d$  that minimizes the population risk, defined as:
+
+$$
+R (\theta) \triangleq \frac {1}{m} \sum_ {k = 1} ^ {m} l _ {k} (\theta) \triangleq \frac {1}{m} \sum_ {k = 1} ^ {m} \mathbb {E} _ {Z \sim \mathcal {D} _ {k}} [ \ell (\theta ; Z) ],
+$$
+
+where  $\ell$  is some loss function. We denote by  $\theta^{\star}$  a global minimizer of the population risk, i.e.,  $\theta^{\star} \in \arg \min_{\theta} R(\theta)$ .
+
+Although the population risk  $R(\theta)$  is not directly computable, we can instead evaluate an empirical counterpart using  $m$  local datasets  $S \triangleq (S_{1},\ldots ,S_{m})$ , where  $S_{k} = \{Z_{1k},\dots,Z_{nk}\}$  represents the dataset of agent  $k$ , with each sample  $Z_{ik}$  drawn from the local distribution  $\mathcal{D}_k$ . For simplicity, we assume that each local dataset has the same size  $n$ , though our analysis can be extended to the heterogeneous case. The resulting empirical risk is given by:
+
+$$
+R _ {S} (\theta) \triangleq \frac {1}{m} \sum_ {k = 1} ^ {m} R _ {S _ {k}} (\theta) \triangleq \frac {1}{m n} \sum_ {k = 1} ^ {m} \sum_ {i = 1} ^ {n} \ell (\theta ; Z _ {i k}).
+$$
+
+One of the most well-known and extensively studied estimators is the empirical risk minimizer, defined as  $\widehat{\theta}_{\mathrm{ERM}} \triangleq \arg \min_{\theta} R_S(\theta)$ . However, in most practical scenarios, directly computing this estimator is infeasible. Instead, one typically employs a potentially random decentralized optimization algorithm  $A$ , which takes the full dataset  $S$  as input and returns an approximate minimizer  $A(S) \in \mathbb{R}^d$  for the empirical risk  $R_S(\theta)$ .
+
+In this setting, the expected excess risk  $R(A(S)) - R(\theta^{\star})$  can be upper-bounded by the sum of the (expected) generalization error  $(\epsilon_{\mathrm{gen}})$  and the (expected) optimization error  $(\epsilon_{\mathrm{opt}})$  [23, 11]:
+
+$$
+\mathbb {E} _ {A, S} \left[ R (A (S)) - R \left(\theta^ {\star}\right) \right] \leq \epsilon_ {\text {g e n}} + \epsilon_ {\text {o p t}} \tag {3.1}
+$$
+
+where  $\epsilon_{\mathrm{gen}} \triangleq \mathbb{E}_{A,S}[R(A(S)) - R_S(A(S))]$  and  $\epsilon_{\mathrm{opt}} \triangleq \mathbb{E}_{A,S}[R_S(A(S)) - R_S(\widehat{\theta}_{\mathrm{ERM}})]$ . This work focuses on controlling the expected generalization error  $\epsilon_{\mathrm{gen}}$ , for which a common approach is to use the stability analysis of the algorithm  $A$ .
+
+Contrary to a large body of works using the well-known uniform stability [31, 32], our analysis relies on the notion of on-average model stability [33], which has the advantage of removing the bounded gradient assumption [3, 34, 10] in our analysis, making the theoretical results more general. Below, we recall this notion, with a slight adaptation to the decentralized setting.
+
+Definition 1 ( $l_{2}$  on-average model stability). Let  $S = (S_{1},\ldots ,S_{m})$  with  $S_{k} = \{Z_{1k},\dots,Z_{nk}\}$  and  $\tilde{S} = (\tilde{S}_1,\dots,\tilde{S}_m)$  with  $\tilde{S}_k = \{\tilde{Z}_{1k},\dots,\tilde{Z}_{nk}\}$  be two independent copies such that  $Z_{ik}\sim \mathcal{D}_k$  and  $\tilde{Z}_{ik}\sim \mathcal{D}_k$ . For any  $i\in \{1,\dots,n\}$  and  $j\in \{1,\dots,m\}$ , let us denote by  $S^{(ij)} = (S_1,\dots,S_{j - 1},S_j^{(i)},S_{j + 1},\dots,S_m)$ , with  $S_j^{(i)} = \{Z_{1j},\dots,Z_{i - 1j},\tilde{Z}_{ij},Z_{i + 1j},\dots,Z_{nj}\}$ , the dataset formed from  $S$  by replacing the  $i$ -th element of the  $j$ -th agent's dataset by  $\tilde{Z}_{ij}$ . A randomized algorithm  $A$  is said to be  $l_{2}$  on-average model  $\varepsilon$ -stable if
+
+$$
+\mathbb {E} _ {S, \tilde {S}, A} \left[ \frac {1}{m n} \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {m} | | A (S) - A \left(S ^ {(i j)}\right) | | _ {2} ^ {2} \right] \leq \varepsilon^ {2}. \tag {3.2}
+$$
+
+A key aspect of on-average model stability is that it can directly be linked to the generalization error, as shown in the following lemma.
+
+Lemma 1 (Generalization via on-average model stability [33]). Let  $A$  be  $l_{2}$  on-average model  $\varepsilon$ -stable. Let  $\gamma > 0$ . Then, if  $\ell(\cdot; z)$  is nonnegative and is  $\beta$ -smoothness for all  $z \in \mathcal{Z}$ , we have
+
+$$
+\epsilon_ {g e n} \leq \frac {1}{2 m n \gamma} \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {m} \mathbb {E} _ {A, S} [ \| \nabla \ell (A (S); Z _ {i j}) \| ^ {2} ] + \frac {\beta + \gamma}{2 m n} \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {m} \mathbb {E} _ {A, \tilde {A}, S} [ \| A (S) - A (S ^ {(i j)}) \| ^ {2} ]
+$$
+
+In fact, we modified the proof of the lemma from Lei et al.[33], replacing the  $R_{S}(A(S))$  on the right-hand side with a gradient  $\mathbb{E}_{A,S}[\|\nabla\ell(A(S);Z_{ij})\|^2]$ . This adjustment better captures the impact of data heterogeneity on the generalization error. With this lemma, obtaining the desired generalization bound reduces to controlling the  $l_2$  on-average model stability of the decentralized algorithm  $A$ .
+
+# 3.2 Decentralized SGD with Multiple Gossip Steps
+
+In this paper, we focus on the widely-used Decentralized Stochastic Gradient Descent (D-SGD) algorithm [35, 7], which aims to find minimizers (or saddle points) of the empirical risk  $R_{S}(\theta)$  in a fully decentralized manner. This algorithm relies on peer-to-peer communication between agents, with a graph representing which pairs of agents (or nodes) are able to interact. Specifically, the communication topology is captured by a gossip matrix  $W \in [0,1]^{m \times m}$  (see Definition 2), where  $W_{jk} > 0$  indicates the weight that agent  $j$  assigns to messages from agent  $k$ , and  $W_{jk} = 0$  (no edge) implies that agent  $j$  does not receive messages from agent  $k$ .
+
+The D-SGD with Multiple Gossip Steps (DSGD-MGS) algorithm performs multiple gossip updates during the communication phase of the D-SGD algorithm, while all other computational components remain identical to D-SGD, as detailed in Algorithm 1. Specifically, the main procedure at time  $t$  is divided into two steps:
+
+- Local Update Steps: Each node independently and uniformly draws a training sample  $Z_{I_k^t k}$  from its local dataset  $S_{k}$ . Based on the current model parameter  $\theta_{k}^{(t)}$ , it computes the gradient  $\nabla \ell (\theta_{k}^{(t)};Z_{I_{k}^{t}k})$  and performs gradient descent to obtain the initial point for Multiple Gossip Steps:  $\theta_{k}^{(t,0)} = \theta_{k}^{(t)} - \eta_{t}\nabla \ell (\theta_{k}^{(t)};Z_{I_{k}^{t}k})$ , where  $\eta_t$  denotes the step size.
+
+Algorithm 1 Decentralized SGD with MGS  
+1: Input: Initialize  $\forall k, \theta_k^{(0)} = \theta^{(0)} \in \mathbb{R}^d$ , iterations  $T$ , stepsizes  $\{\eta_t\}_{t=0}^{T-1}$ , weight matrix  $W$ , Multiple Gossip Steps  $Q$ .  
+2: for  $t = 0, \ldots, T-1$  do  
+3: for each node  $k = 1, \ldots, m$  in parallel do  
+4: Local Update Steps:  
+5: Sample  $I_k^t \sim \mathcal{U}\{1, \ldots, n\}$   
+6:  $\theta_k^{(t,0)} = \theta_k^{(t)} - \eta_t \nabla \ell(\theta_k^{(t)}; Z_{I_k^t k})$   
+7: Multiple Gossip Steps:  
+8: for  $q = 0$  to  $Q-1$  do  
+9:  $\theta_k^{(t,q+1)} = \sum_{l=1}^{m} W_{kl} \theta_l^{(t,q)}$   
+10: end for  
+11:  $\theta_k^{(t+1)} = \theta_k^{(t,Q)}$   
+12: end for  
+13: end for
+
+- Multiple Gossip Steps: Each node exchanges
+
+information with its neighbors through  $Q$  gossip averaging steps:  $\theta_k^{(t,q+1)} = \sum_{l=1}^m W_{kl} \theta_l^{(t,q)}$ . The resulting model parameter  $\theta_k^{(t,q+1)}$  is then used as the initial point  $\theta_k^{(t+1)}$  for the next Local Update Steps.
+
+# 4 Generalization Analysis
+
+In this section, we first introduce the Definition and Assumptions required for analyzing the generalization of the DSGD-MGS algorithm. We then present the upper bounds for the generalization error and excess error, followed by a detailed analysis of these bounds. Proofs for all Lemmas and Theorems can be found in the Appendix B.
+
+# 4.1 Definition and Assumption
+
+Definition 2 (Gossip Matrix). Let  $W \in [0,1]^{n \times n}$  be a symmetric doubly stochastic matrix. This means that  $W = W^{\top}$ , and both the row sums and column sums of  $W$  equal one, i.e.,  $W\mathbf{1} = \mathbf{1}$  and  $\mathbf{1}^{\top}W = \mathbf{1}^{\top}$ , where  $\mathbf{1}$  is the vector of all ones. The eigenvalues of  $W$  are ordered as  $1 = |\lambda_1(W)| > |\lambda_2(W)| \geq \dots \geq |\lambda_n(W)|$ . The spectral gap of  $W$ , denoted by  $\delta$ , is defined as  $\delta := 1 - |\lambda_2(W)| \in (0,1)$ .
+
+Assumption 1. ( $\beta$ -smoothness). The loss function  $\ell$  is  $\beta$ -smooth i.e.  $\exists \beta > 0$  such that  $\forall \theta, \theta' \in \mathbb{R}^d$ ,  $z \in \mathcal{Z}$ ,  $\|\nabla \ell(\theta; z) - \nabla \ell(\theta'; z)\|_2 \leq \beta\|\theta - \theta'\|_2$ .
+
+Assumption 2. (Bounded Stochastic Gradient Noise). There exists  $\sigma^2 > 0$  such that  $\mathbb{E}_{Z_{i,j}}\| \nabla \ell (\theta ;Z_{i,j}) - \nabla R_{\mathcal{S}_j}(\theta)\| ^2\leq \sigma^2$ , for any agent  $j\in [m]$  and  $\theta \in \mathbb{R}^d$ .
+
+Assumption 3. (Bounded Heterogeneity). There exists  $\xi^2 >0$  such that  $\frac{1}{m}\sum_{k = 1}^{m}\| \nabla R_{S_k}(\theta) - \nabla R_S(\theta)\| ^2\leq \xi^2$ , for any  $\theta \in \mathbb{R}^d$ .
+
+Using the property  $\beta$ -smoothness of  $\ell(\theta; z)$ , it is straightforward to show that  $\ell_k(\theta) = \mathbb{E}_{Z \sim \mathcal{D}_k}[\ell(\theta; Z)]$  and  $R_{S_k}(\theta) = \frac{1}{n} \sum_{i=1}^{n} \ell(\theta; Z_{ik})$  also satisfy the property  $\beta$ -smoothness.
+
+Remark 1. Definition 2 stipulates that the communication topology must be a doubly stochastic matrix, which appears in many decentralized optimization works [7, 34, 11, 18, 3]. Assumption 1 specifies that the loss function is smooth, which is often used in optimization and generalization studies under non-convex settings [36, 10, 37, 38, 39, 40]. Assumption 2 states that the stochastic gradients of the samples are bounded, and Assumption 3 bounds the heterogeneity of the data. These assumptions are frequently used in the convergence analysis of many works [36, 3, 16, 37], and we will employ them in this paper to analyze the stability and generalization of DSGD-MGS.
+
+# 4.2 Generalization Error and Excess Error of DSGD-MGS
+
+Due to its fully decentralized structure, DSGD-MGS produces  $m$  distinct outputs,  $A_{1}(S) \triangleq \theta_{1}^{(T)}, \ldots, A_{m}(S) \triangleq \theta_{m}^{(T)}$ , one for each agent. As a result, the stability and generalization anal
+
+ysis that follows will focus on these individual outputs, rather than a single global output  $A(S)$  as described in Section 3.1. Denote by  $A_{k}(S) = \theta_{k}^{(T)}$  and  $A_{k}(S^{(ij)}) = \theta_{k}^{(T)}(i,j)$ , the final iterates of agent  $k$  for DSGD-MGS run over two data sets  $S$  and  $S^{(ij)}$  that differ only in the  $i$ -th sample of agent  $j$ . To obtain a tighter upper bound for the non-convex case, we modify Lemma 1 by introducing a variable  $t_0$ , resulting in the following key lemma, which transforms the computation of the generalization error upper bound  $\epsilon_{\mathrm{gen}}$  into the computation of the stability upper bound.
+
+Lemma 2. Assume the loss function  $\ell(\cdot, z)$  is nonnegative and bounded in  $[0,1]$ , and that Assumptions 1 hold. For all  $i = 1, \ldots, n$  and  $j = 1, \ldots, m$ , let  $\{\theta_k^{(t)}\}_{t=0}^T$  and  $\{\tilde{\theta}_k^{(t)}(i,j)\}_{t=0}^T$ , the iterates of agent  $k = 1, \ldots, m$  for DSGD-MGS run on  $S$  and  $S^{(ij)}$  respectively. Then, for every  $t_0 \in \{0,1, \ldots, T\}$  we have:
+
+$$
+\begin{array}{l} \left. \left| \mathbb {E} _ {A, S} \left[ R \left(A _ {k} (S)\right) - R _ {S} \left(A _ {k} (S)\right) \right] \right| \right. \\ \leq \frac {t _ {0}}{n} + \underbrace {\frac {\gamma + \beta}{2 m n} \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {m} \mathbb {E} [ \delta_ {k} ^ {(T)} (i , j) | \delta^ {(t _ {0})} (i , j) = \mathbf {0} ]} _ {I _ {1}: l _ {2} o n - a v e r a g e m o d e l s t a b i l i t y} + \underbrace {\frac {1}{2 m n \gamma} \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {m} \mathbb {E} [ \| \nabla \ell (A _ {k} (S) ; Z _ {i j}) \| ^ {2} ]} _ {I _ {2}: R e l a t e d t o o p t i m i z a t i o n e r r o r} \\ \end{array}
+$$
+
+where  $\delta^{(t)}(i,j)$  is the vector containing  $\forall k = 1,\dots ,m,\delta_k^{(t)}(i,j) = \| \theta_k^{(t)} - \tilde{\theta}_k^{(t)}(i,j)\| _2^2$
+
+According to Lemma 2, to compute the generalization error  $\epsilon_{\mathrm{gen}}$ , We need to calculate the  $l_{2}$  on-average model stability  $(I_1)$  and the gradient related to the optimization error  $(I_2)$ . Below, we first provide the stability upper bound, followed by the optimization error upper bound.
+
+Upper bound of  $I_{1}$ : For a fixed couple  $(i,j)$ , we are first going to control the vector  $\Delta^{(t)} = \frac{1}{mn}\sum_{i,j}\Delta^{(t)}(i,j)$ , where  $\Delta^{(t)}(i,j) \triangleq \mathbb{E}[\delta^{(t)}(i,j)|\delta^{(t_0)}(i,j) = 0]$ . When it is clear from context, we simply write  $\tilde{\theta}_k^{(t)}(i,j) = \tilde{\theta}_k^{(t)}$ . Next, we provide the upper bound of the  $l_{2}$  on-average model stability for the DSGD-MGS algorithm.
+
+Theorem 1 (Stability for the DSGD-MGS). As in the conditions of Lemma 2, then the following holds:
+
+$$
+\frac {1}{m n} \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {m} \mathbb {E} [ \delta_ {k} ^ {(T)} (i, j) | \delta^ {(t _ {0})} (i, j) = \mathbf {0} ] \leq \frac {8 e \sqrt {2 \beta} c ^ {2}}{(1 + 2 c \beta) n m t _ {0}} \left(\frac {T}{t _ {0}}\right) ^ {2 c \beta}
+$$
+
+Upper bound of  $I_{2}$ : Let  $\bar{G} = \frac{1}{mn}\sum_{i,j}\mathbb{E}[\|\nabla\ell(\theta_k^{(T)};Z_{ij})\|^2]$ . According to the Assumptions 2 and 3, the following inequality holds:
+
+$$
+\begin{array}{l} \bar {G} = \frac {1}{m n} \sum_ {i, j} \mathbb {E} [ \| \nabla \ell (\theta_ {k} ^ {(T)}; Z _ {i j}) \| ^ {2} ] = \frac {1}{m n} \sum_ {i, j} \mathbb {E} [ \| \nabla \ell (\theta_ {k} ^ {(T)}; Z _ {i j}) \pm \nabla R _ {S _ {k}} (\theta_ {k} ^ {(T)}) \pm \nabla R _ {S} (\theta_ {k} ^ {(T)}) \| ^ {2} ] \\ \leq 3 \sigma^ {2} + 3 \xi^ {2} + 3 \mathbb {E} [ \| \nabla R _ {S} (\theta_ {k} ^ {(T)}) \| ^ {2} ] \\ \end{array}
+$$
+
+Since  $\ell$  satisfies the  $\beta$ -smoothness property, it is straightforward to show that  $R_{S}(\theta_{k}^{(T)})$  also satisfies the  $\beta$ -smoothness property. Consequently,  $R_{S}(\theta)$  also satisfies the self-bounding property in Lemma 3 (see the Appendix B), i.e.,  $\| \nabla R_S(\theta)\|^2 \leq 2\beta R_S(\theta)$ . Then, we have
+
+$$
+\bar {G} \leq 3 \sigma^ {2} + 3 \xi^ {2} + 6 \beta \mathbb {E} _ {S} \left[ R _ {S} \left(\theta_ {k} ^ {(T)}\right) \right] \tag {4.1}
+$$
+
+Next, we will focus on bounding  $\mathbb{E}_S[R_S(\theta_k^{(T)})]$ . According to the results from [18, Theorem 1] (see Lemma 2 in the Appendix B), we have the following theorem:
+
+Theorem 2 (Optimization error of DSGD-MGS). Let  $\Delta^2 \coloneqq \max_{\theta^* \in \mathcal{X}^*} \sum_{k=1}^{m} \|\nabla R_{S_k}(\theta^*)\|^2$ ,  $R_0 \coloneqq R_S(\theta^{(0)}) - R_S^*$ , where  $\mathcal{X}^* = \arg \min_{\theta} R_S(\theta)$  and  $R_S^* = R_S(\widehat{\theta}_{ERM})$ . Suppose Assumptions 1 and Polyak-Lojasiewicz (PL) condition (see Assumption 4 in the Appendix) hold. Define
+
+$$
+Q _ {0} := \log (\bar {\rho} / 4 6) / \log \left(1 - \frac {\delta \tilde {\gamma}}{2}\right), \bar {\rho} := 1 - \frac {\mu}{m \beta}, \tilde {\gamma} = \frac {\delta}{\delta^ {2} + 8 \delta + (4 + 2 \delta) \lambda_ {\max } ^ {2} (I - W)}.
+$$
+
+Then, if the nodes are initialized such that  $\theta_k^Q = 0$ , for any  $Q > Q_0$  after  $T$  iterations the iterates of DSGD-MGS with  $\eta_t = \frac{1}{\beta}$  satisfy
+
+$$
+\mathbb {E} _ {S} \left[ R _ {S} \left(\theta_ {k} ^ {(T)}\right) \right] - R _ {S} ^ {*} = \mathcal {O} \left(\frac {\Delta^ {2} e ^ {- \frac {\delta \bar {\gamma} Q}{4}}}{1 - \bar {\rho}} + \left[ 1 + \frac {\beta}{\mu \bar {\rho}} \left(1 + e ^ {- \frac {\delta \bar {\gamma} Q}{4}}\right) \right] R _ {0} \rho^ {T}\right). \tag {4.2}
+$$
+
+Here,  $\delta$  represents the spectral gap of  $W$ , and  $\rho \triangleq 1 - \delta = |\lambda_2(W)|$  is defined in definition 2.
+
+By combining Equation (4.1) with Theorem 2, we obtain the upper bound for  $\bar{G}$ .
+
+$$
+\bar {G} = \mathcal {O} \left(\sigma^ {2} + \delta^ {2} + R _ {S} ^ {*}\right) + \mathcal {O} \left(\frac {\beta \Delta^ {2} e ^ {- \frac {\delta \bar {\gamma} Q}{4}}}{1 - \bar {\rho}} + \left[ 1 + \frac {\beta}{\mu \bar {\rho}} \left(1 + e ^ {- \frac {\delta \bar {\gamma} Q}{4}}\right) \right] R _ {0} \beta \rho^ {T}\right) \tag {4.3}
+$$
+
+Generalization Bound for DSGD-MGS: With the above Theorem 1 & 2, we can derive the generalization error upper bound for DSGD-MGS.
+
+Theorem 3 (Generalization error of DSGD-MGS). Based on Lemma 2, Theorem 1 and Theorem 2, and assuming that Assumptions 1-3 hold, let the learning rate satisfy  $\eta_t \leq \frac{c}{t + 1}$  for some constant  $c > 0$ . We derive the following result by appropriately selecting  $t_0$  and  $\gamma$ :
+
+$$
+\begin{array}{l} | \mathbb {E} _ {A, S} [ R (A _ {k} (S)) - R _ {S} (A _ {k} (S)) ] | \\ \leq \frac {2 c \beta + 3}{(n (2 c \beta + 1)) ^ {\frac {2 c \beta + 2}{2 c \beta + 3}}} \left(\frac {2 \bar {G} e \sqrt {2 \beta} c ^ {2} T ^ {2 c \beta}}{m}\right) ^ {\frac {1}{2 c \beta + 3}} + \frac {2 c \beta + 2}{n (2 c \beta + 1)} \left(\frac {4 \beta e \sqrt {2 \beta} c ^ {2} T ^ {2 c \beta}}{m}\right) ^ {\frac {1}{2 c \beta + 2}} \\ \end{array}
+$$
+
+where the expression for  $\bar{G}$  is given in Equation (4.3).
+
+Remark 2 (Optimization Error Reduction). As shown in Theorem 3, the generalization error bound obtained via  $l_{2}$  on-average model stability is closely related to the optimization error  $\bar{G}$ . Analyzing the MGS-related terms reveals that increasing the number of MGS steps  $Q$  reduces  $\bar{G}$ , thereby tightening the generalization error bound. Moreover, a more detailed analysis shows that the reduction in the generalization error bound is exponential, specifically on the order of  $\mathcal{O}(e^{-\frac{\delta\gamma Q}{4}})$ , indicating that even a small increase in  $Q$  can lead to significant gains. This observation will also be validated in the experimental section 5.2.
+
+Remark 3 (Gap to Centralization). As indicated by Theorem 3, by letting  $Q$  approach infinity, we can derive the limiting generalization error bound, which helps address whether DSGD-MGS with sufficiently many steps can effectively approximate centralized mini-batch SGD. The answer is no, because the resulting bound is at most  $\mathcal{O}\left(T^{\frac{2c\beta}{2c\beta + 2}} / nm^{\frac{1}{2c\beta + 2}}\right)$ , which still differs in terms of node count  $m$  and per-node data size  $n$  from the bound  $\mathcal{O}\left(T^{\frac{c\beta}{c\beta + 1}} / mn\right)$  established for centralized mini-batch SGD based on uniform stability in [11, 41]. Therefore, this gap persists unless the number of nodes or the data size per node is significantly increased. As illustrated in Figure 1.
+
+Remark 4 (Related to the Optimization Error). Compared to prior works on the generalization error of D-SGD [42, 41, 11, 22], which rely on Lipschitz assumptions for the loss function, our approach removes this assumption, allowing for a more explicit connection between optimization error and generalization error. In those works, the Lipschitz assumption effectively absorbs optimization-related quantities (e.g., gradients) into a Lipschitz constant, obscuring this relationship. In contrast, our work removes the Lipschitz assumption, making the relationship between generalization and optimization errors more explicit. Our results show that reducing optimization error can also decrease generalization error to some extent, which explains the common observation that as training progresses, both the training error decreases and the model's performance on the validation set improves.
+
+Remark 5 (Influential Factors of the Generalization Error for DSGD-MGS). When the model, loss function, and dataset are fixed, parameters like the smoothness  $\beta$ , gradient noise  $\sigma$ , and data heterogeneity  $\delta$  are also fixed. In this case, to reduce the generalization error bound according to the upper bound in Theorem 3, the following strategies are effective: 1) Increase the data size per node  $n$ ; 2) Increase the number of nodes  $m$ ; 3) Increase the MGS step count  $Q$ ; 4) Reduce the distance between the optimal point and the initial point  $R_0$ ; 5) Use a communication topology with a larger spectral gap  $\delta$  (which implies a smaller  $\rho$ ); 6) Decrease the learning rate  $c$ . The first five are straightforward, while the sixth is recommended because the number of iterations  $T$  is usually large, making
+
+$T^{\frac{2c\beta}{2c\beta + 2}}$  the dominant term in the bound. Reducing  $c$  can significantly reduce this term. Additionally, if the choice of dataset is flexible, selecting one that is as close to i.i.d. as possible is beneficial, as a larger data heterogeneity parameter  $\xi$  will generally increase the generalization error bound.
+
+Remark 6 (Innovation in Generalization Error Bounds). Our work introduces  $l_{2}$  on-average model stability to deriving generalization error bounds for decentralized algorithms, characterized by the following key innovations: 1) Removal of Lipschitz Assumption: Unlike previous proofs based on uniform stability [41, 20, 11, 10, 9, 43], our approach removes the Lipschitz assumption on the loss function (which implicitly bounds the gradient), allowing the relationship between optimization error and generalization error to become more explicit. 2) Explicit Role of Optimization Error: We establish, for the first time, a direct connection between the optimization error and generalization error of the D-SGD algorithm, revealing that reducing the optimization error also decreases the generalization error, which aligns better with observed training dynamics. 3) Exponential MGS Benefit: Our bounds demonstrate that the impact of MGS on reducing generalization error is exponential, highlighting the significant gains achievable with a moderate number of MGS steps. 4) Quantification of Heterogeneity Impact: Ye et al.[23] were the first to theoretically reveal that data heterogeneity can degrade the generalization bound of the D-SGD algorithm under the strongly convex setting. Building on this, we take a further step by providing a precise characterization of how data heterogeneity affects generalization in the non-convex setting, filling a critical gap in existing theoretical analyses.
+
+Theorem 4 (Excess Error of DSGD-MGS). Under the same conditions and notation as Theorems 3 and 2, and based on the decomposition of excess error in Equation (3.1), the optimization error bound (Equation 4.2), and the generalization error bound (Theorem 3), we obtain the following upper bound for the excess error.
+
+$$
+\begin{array}{l} \mathbb {E} _ {A, S} \left[ R (A (S)) - R \left(\theta^ {\star}\right) \right] = \mathcal {O} \left(\frac {\Delta^ {2} e ^ {- \frac {\delta \gamma Q}{4}}}{1 - \bar {\rho}} + \left[ 1 + \frac {\beta}{\mu \bar {\rho}} \left(1 + e ^ {- \frac {\delta \gamma Q}{4}}\right) \right] R _ {0} \rho^ {T} \right. \tag {4.4} \\ + \frac {1}{n ^ {\frac {2 c \beta + 2}{2 c \beta + 3}}} \left(\frac {\bar {G} \beta^ {\frac {3}{2}} c ^ {3} T ^ {2 c \beta}}{m}\right) ^ {\frac {1}{2 c \beta + 3}} + \frac {1}{n} \left(\frac {\beta^ {\frac {3}{2}} c ^ {2} T ^ {2 c \beta}}{m}\right) ^ {\frac {1}{2 c \beta + 2}} \\ \end{array}
+$$
+
+Remark 7 (The difference of conclusions obtained from excess error and generalization error). Since the excess error can be decomposed as  $\mathbb{E}_{A,S}[R(A(S)) - R(\theta^{\star})] \leq \epsilon_{gen} + \epsilon_{opt}$ , most conclusions about the generalization error also apply to the excess error (see Remark 5). The only key difference lies in the choice of learning rate. For  $\epsilon_{gen}$ , a smaller learning rate (i.e., smaller  $c$ ) is preferred, as  $\epsilon_{gen}$  is dominated by the term  $\mathcal{O}(T^{\frac{2c\beta}{2c\beta + 2}})$ , meaning that reducing  $c$  significantly reduces this term and hence the generalization error. However, this is not the case for  $\epsilon_{opt}$ . Prior work on the convergence of D-SGD [7] shows that  $\epsilon_{opt} = \mathcal{O}\left(\frac{R_0}{T\eta}\right)$ , indicating that an excessively large learning rate increases  $\epsilon_{opt}$ , thereby undermining convergence. Thus, the choice of learning rate involves a trade-off between minimizing generalization error and maintaining convergence, a conclusion that will be confirmed in the Experimental Section A.2.
+
+Remark 8. (On the Technical Role of the PL Condition). Our analysis of the generalization error requires bounding the expected squared gradient norm at the final iterate, denoted as  $\bar{G}$ . However, establishing a tight upper bound for the final iterate's gradient in non-convex decentralized optimization remains a challenging frontier problem. While recent advances have been made in last-iterate convergence analysis (e.g., [44]), existing results either do not incorporate the MGS mechanism or provide bounds only on the function value gap, which are insufficient for directly bounding  $\bar{G}$ . To bridge this gap, we adopt the Polyak-Lojasiewicz (PL) condition. This is a standard approach in the literature (e.g., [34]) used to connect the squared gradient norm with the function value gap. This technical choice is deliberate and crucial, as a tight upper bound on the function value gap under the MGS setting is available [18]. Consequently, the PL condition enables us to derive some of the first fine-grained, MGS-aware generalization bounds that explicitly link the generalization error to key algorithmic hyperparameters, including the number of MGS steps ( $Q$ ), communication topology, and learning rate. This provides concrete, quantitative insights that significantly advance beyond high-level bounds, such as the classic  $\mathcal{O}(1 / T)$  analysis provided by L2-stability [33]. Therefore, the reliance on the PL condition reflects the current theoretical limits in non-convex last-iterate analysis rather than a fundamental limitation of our stability framework. Our framework is modular: should future research provide a direct, assumption-free upper bound for  $\bar{G}$  in the MGS setting, our generalization bounds can be immediately strengthened by replacing this component. A more detailed discussion is provided in the Appendix D.4.
+
+Remark 9. All the above discussions are also solid to  $\bar{\theta}^{(T)}\triangleq \frac{1}{m}\sum_{k = 1}^{m}\theta_k^{(T)}$ . In addition, our theoretical results apply to decentralized topologies other than the fully connected case. When the topology becomes fully connected, the iterative update reduces to the centralized setting. For detailed analysis, please refer to the Appendix. For detailed proof, please refer to Appendix D. Additionally, we provide a consensus error analysis to further illustrate the behavior of MGS in both finite and infinite regimes (detailed discussion provided in Appendix C). Furthermore, we extend our theoretical analysis to the case involving batch size  $b$ . The detailed proofs and analyses are provided in the Appendix D.2.
+
+# 5 Experiment
+
+In this section, we present extensive experiments to validate our theoretical findings. We first describe the experimental setup, followed by the empirical results and corresponding analysis. Due to space constraints, the experimental validation of excess error is presented in Appendix A.2. Furthermore, we conduct an in-depth exploration of the subtle relationship between mini-batch size and (Q) on the CIFAR-100 dataset, providing practitioners with insights for achieving higher performance. Detailed analyses and discussions can be found in the Appendix D.3.
+
+# 5.1 Empirical Setup
+
+We conduct experiments on the CIFAR-10 dataset [45] with a Dirichlet distribution (non-IID,  $\alpha = 0.3$ ) using LeNet to validate the excess error and generalization error of DSGD-MGS. To examine the impact of key hyperparameters, we follow the study by Hardt et al.[41] and investigate the weight distance  $(\sum_{i=1}^{n}\sum_{j=1}^{m}||\theta_j^{(t)} - \tilde{\theta}_j^{(t)}||_2^2)$  and the loss distance  $(R(\tilde{\theta}^{(t)}) - R_S(\tilde{\theta}^{(t)}))$  when replacing only one data point in the training dataset. We primarily validate the experimental performance of key parameters in the DSGD-MGS algorithm, such as communication topology, the number of MGS steps, and the total number of clients. For fairness, when exploring one parameter, all other parameters are kept at the same settings. Further implementation details are provided in Appendix A.1.
+
+# 5.2 Experimental Validation of Generalization Error.
+
+As shown in Figure 2, subplots (a) and (b) respectively illustrate the weight distance and loss distance for different parameter settings of the DSGD-MGS algorithm on the perturbed dataset. Overall, both weight distance and loss distance exhibit the same power-law behavior as our theoretical bound  $\mathcal{O}\big(T^{\frac{2c\beta}{2c\beta + 2}}\big)$  (see Theorem 3). Additionally, within each column of Figure 2 (corresponding to the same parameter setting), these two metrics follow similar trends, confirming the validity of Lemma 1 [33], which states that the generalization error can indeed be captured by the stability bound.
+
+![](images/0b37fdd1f0fa0efffcbb16fb306c71f91297023841d8e8776a473c9303a8395c.jpg)
+
+![](images/4072a9d29b988dc1954dcf75afd966f3232e0617b3b24545155cfbcec30cde54.jpg)  
+(a) Weight distance
+
+![](images/f18d971c359676074e7f61ab686c899de249e55468f1dfe60ac3d87c525bddcf.jpg)
+
+![](images/bd0f737c0f72583abfde9df3963bedb9358170f7bd3f5b1345e5a66bdab26224.jpg)
+
+![](images/7e39504338a2d55e1f48ba6414f880241efa41cfa358689bad1e269c29f22778.jpg)  
+Figure 2: A comparison of the  $l_{2}$  weight distance and Loss distance (i.e. test loss - train loss) for the DSGD-MGS algorithm on the CIFar10 dataset.
+
+![](images/72bf1a98b4613f955d9b9ad73d990db75d4a08c03c3b2a6a08acc3a69d29a362.jpg)  
+(b) Loss distance
+
+![](images/0ee57d92b88b359f343f4022438957fcb8d9084dab606eb08de7354dc13d48fb.jpg)
+
+![](images/bd61a94253cfb60b50e8442ad4852ea6b8a418dd23f2801301892bdd2dc656c9.jpg)
+
+From subplots (a) and (b) in Figure 2, we can observe the following patterns: 1) Using a communication topology with a smaller spectral gap (i.e., a larger  $\rho$  in Theorem 3) leads to lower generalization error. 2) Increasing the number of MGS effectively reduces the generalization error. For example, in terms of weight distance (Figure 2 (a)), setting  $MGS = 5$  reduces the weight distance to roughly half of that with  $MGS = 1$ . 3) Smaller learning rates help reduce generalization error, consistent with the findings in [10] on decentralized federated learning. 4) A larger client number (i.e.,  $m$  in Theorem 3) also helps reduce generalization error, reflecting a nearly linear speedup effect with the number of clients. Notably, these observations align well with our theoretical results (see Theorem 3 and Remark 5). This further validates the correctness of our theoretical analysis.
+
+# 6 Conclusion
+
+This paper is the first to establish the generalization error and excess error bounds for the DSGD-MGS algorithm in non-convex settings without the bounded gradients assumption. It addresses how MGS can exponentially reduce the generalization error bound and shows that even with a very large number of MGS steps, it cannot completely close the gap between decentralized and centralized training. Additionally, our theoretical results capture the impact of key factors like data heterogeneity  $\delta$ , communication topology spectrum  $\xi$ , Multiple Gossip Steps  $Q$ , client number  $m$ , and per-client data size  $n$ . Previous work has not unified the analysis of these critical parameters, and this paper fills that gap, offering both theoretical insights and experimental validation and significantly advancing the theoretical understanding of decentralized optimization.
+
+Limitation. The theoretical findings in this paper depend on the properties of the last iteration of D-SGD in optimization theory, which is an emerging area yet to be explored. This paper derives the properties of the function value at the last iteration under the PL-condition. Future work can further explore the properties of the loss function gradient at the last iteration under non-convex conditions.
+
+# Acknowledgment
+
+This work is supported by STI 2030—Major Projects (No. 2021ZD0201405), NSFC Grant (No. 62576364), Shenzhen Basic Research Project (Natural Science Foundation) Basic Research Key Project (NO. JCYJ20241202124430041) and National Natural Science Foundation of China (No.62025604, No.62261160653). Miao Zhang is supported by the National Natural Science Foundation of China (NO. 62403484).
+
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+# Unveiling the Spatial-temporal Effective Receptive Fields of Spiking Neural Networks
+
+Jieyuan Zhang $^{1}$ , Xiaolong Zhou $^{1}$ , Shuai Wang $^{1}$  Wenjie Wei $^{1}$ , Hanwen Liu $^{1}$ , Qian Sun $^{1}$ , Malu Zhang $^{1,3*}$ , Yang Yang $^{1}$ , Haizhou Li $^{2,3}$
+
+<sup>1</sup>University of Electronic Science and Technology of China,
+
+2The Chinese University of Hong Kong, Shenzhen,
+
+3Shenzhen Loop Area Institute
+
+# Abstract
+
+Spiking Neural Networks (SNNs) demonstrate significant potential for energy-efficient neuromorphic computing through an event-driven paradigm. While training methods and computational models have greatly advanced, SNNs struggle to achieve competitive performance in visual long-sequence modeling tasks. In artificial neural networks, the effective receptive field (ERF) serves as a valuable tool for analyzing feature extraction capabilities in visual long-sequence modeling. Inspired by this, we introduce the Spatio-Temporal Effective Receptive Field (ST-ERF) to analyze the ERF distributions across various Transformer-based SNNs. Based on the proposed ST-ERF, we reveal that these models suffer from establishing a robust global ST-ERF, thereby limiting their visual feature modeling capabilities. To overcome this issue, we propose two novel channel-mixer architectures: multi-layer-perceptron-based mixer (MLPixer) and splash-and-reconstruct block (SRB). These architectures enhance global spatial ERF through all timesteps in early network stages of Transformer-based SNNs, improving performance on challenging visual long-sequence modeling tasks. Extensive experiments conducted on the Meta-SDT variants and across object detection and semantic segmentation tasks further validate the effectiveness of our proposed method. Beyond these specific applications, we believe the proposed ST-ERF framework can provide valuable insights for designing and optimizing SNN architectures across a broader range of tasks. The code is available at EricZhang1412/Spatial-temporal-ERF.
+
+# 1 Introduction
+
+Spiking Neural Networks (SNNs) [1, 2] have emerged as a prominent research focus, characterized by binary spike activation that offers high sparsity, event-driven processing [3, 4], and biological plausibility [5]. Recent advances in encoding schemes [6, 7, 8], training methodologies [9, 10], and neuromorphic hardware [11, 12, 13] have enabled SNNs to achieve remarkable success in diverse tasks, including image processing [14, 15, 16], point/event analysis [17, 18], language understanding [19, 20, 21], and speech processing [22, 23, 24]. Nonetheless, SNNs still struggle to achieve performance comparable to their Artificial Neural Networks (ANNs) counterparts in visual long-sequence modeling tasks.
+
+Compared to conventional image classification, visual long-sequence modeling tasks [25, 26] demand spatially dense outputs with prediction scales several orders of magnitude higher. This paradigm requires architectures capable of modeling long-range spatial dependencies, which are essential for
+
+![](images/d38546ee475ed4c24eab4351ec0c522fe916e28b56f0c1dba093f48400ac2afd.jpg)  
+Figure 1: (a) The human visual field. (b) ERF in ANNs. (c) ST-ERF in SNNs. It extends the ERF to the temporal dimension, thus facilitating a comprehensive analysis of feature extraction in SNNs.
+
+achieving competitive performance [25]. The Transformer [27] architecture introduces a self-attention mechanism that enables effective modeling of long-range spatial dependencies [28, 29, 30]. Motivated by this, recent studies have proposed various Transformer-based SNNs [31, 32, 33, 34], achieving notable progress in visual long-sequence tasks [15]. However, simply combining Transformers with SNNs may lead to suboptimal designs without fully considering the intrinsic spatio-temporal dynamics of spiking neurons. To bridge this gap, a more structured and interpretable framework is required to examine how SNNs model spatio-temporal dependencies. In this context, receptive field (RF) analysis offers a concrete lens through which their feature extraction capacity and attention allocation can be theoretically characterized.
+
+In neuroscience, the RF represents the region of sensory input that can modulate a neuron's activity [35]. Borrowing this concept, the deep learning community defines a neuron's RF as the region of the input that can influence its output [36], with its size determined by network topology. However, this topology-based definition treats all positions within the RF equally, ignoring the learnable weights that determine the actual contribution of each input location. To refine this problem, researchers proposed the effective receptive field (ERF) [37] to quantify input features' contributions to output features via gradient analysis. Unlike topology-based RF, gradient-based ERF provides a more faithful characterization of the network's feature extraction patterns. However, such framework cannot be directly applied to SNNs due to the intrinsic spatio-temporal dynamics of spiking neurons. Therefore, we introduce the Spatial-Temporal Effective Receptive Field (ST-ERF) framework, which quantifies input feature contributions across spatio-temporal locations to characterize SNNs' feature extraction patterns. By jointly modeling temporal dependencies and spatial relationships, ST-ERF facilitates a comprehensive analysis of information processing in SNNs.
+
+Based on the proposed ST-ERF, we analyze various Transformer-based SNNs and identify that existing models fail to establish effective global receptive fields across all timesteps. This limitation stems from the prevalent use of convolutional channel-mixers, which inherently introduce locality bias [38]. Despite facilitating efficient local feature extraction and long-range sparse modeling, this architectural design fundamentally constrains the long-range dense spatial interactions necessary for effective visual long-sequence modeling in SNNs. Building on these insights, we propose two novel channel-mixer architectures: multi-layer-perceptron-based mixer (MLPixer) and splash-and-reconstruct block (SRB). These designs use pixel-wise MLPs to keep spatial features separate when mixing channels, which reduces locality bias and improves the global receptive field in early stages of Transformer-based SNNs. Extensive experiments demonstrate the effectiveness of our methods on visual long-sequence tasks. Specially, on COCO 2017 object detection and ADE20K semantic segmentation, our Meta-SDT-Base [33] with SRB achieves  $48.9\%$ $\mathrm{AP}_{50}^{\mathrm{b}}$  and  $43.7\%$  mIoU, respectively, while maintaining a smaller model size. These results surpass state-of-the-art Transformer-based SNNs, thereby validating our ST-ERF analysis and further advancing SNNs in visual long-sequence modeling. Our main contributions are listed as follows:
+
+- We propose the ST-ERF framework, extending the traditional ERF concept to the temporal dimension with rigorous mathematical formalization. ST-ERF systematically quantifies how input features at different spatial and temporal locations contribute to output features. This provides a theoretical tool for understanding and optimizing feature extraction in SNNs.  
+- We analyze various Transformer-based SNNs using the ST-ERF framework, revealing a critical limitation that existing models fail to establish a global ERF across all timesteps. To overcome this issue, we introduce two novel channel mixer designs: MLPixer and SRB, enabling Transformer-based SNNs to fully exploit their global modeling potential of long-range dependencies.  
+- We conduct extensive experiments on visual long-sequence modeling tasks and demonstrate that our method achieves superior performance. For instance, our Meta-SDT-Base with SRB achieves  $48.9\% \mathrm{AP}_{50}^{\mathrm{b}}$  on COCO 2017 detection and  $43.7\%$  mIoU on ADE20K segmentation. It significantly outperforms existing state-of-the-art Transformer-based SNNs while using a smaller model size. These results strongly support the validity of our ST-ERF theoretical analysis and demonstrate the effectiveness of the proposed architectural designs.
+
+# 2 Related Works
+
+# 2.1 Receptive Field in Neural Networks
+
+The human visual system perceives the external world through the visual fields of both eyes. As illustrated in Figure 1(a), each eye covers a specific region of the visual space, and these regions partially overlap in the center to enable binocular vision. Neurons along the visual pathway respond selectively to stimuli within their RFs. Over the past decades, the RF theory has profoundly influenced our understanding of how the brain filters and integrates visual information across spatial locations [39]. Inspired by the RF theory, deep neural networks adopt a similar principle by characterizing hierarchical ERFs that capture progressively abstract representations of input data. As shown in Figure 1(b), the ERF [37] formalizes this process by analyzing how spatial stimuli contribute to network activations. ERF has motivated extensive research at different architectural levels, from understanding basic operators [40] to designing higher-level modules and network structures such as Adaptive Receptive Fields [41], RF-Next [42], and AutoRF [43]. RF-based analysis has also driven advances in computational efficiency for lightweight architectures, influencing the development of CNN-based MobileNet variants [44, 45, 46] and MLP-based networks such as MLP-Mixer [47] and TSMixer [48]. Building on these insights, this work extends the ERF concept to SNNs, offering a theoretical framework for analyzing and optimizing their spatio-temporal feature extraction processes.
+
+# 2.2 Visual Long-sequence Modeling in SNNs
+
+Visual long-sequence modeling refers to tasks that require multiple predictions per image, rather than a single-label classification [49]. These tasks mainly include detection, segmentation, video understanding, and so on [25]. As these tasks involve modeling complex spatial and temporal dependencies, they demand architectures capable of capturing long-range contextual information. Transformer has become the dominant paradigm for visual long-sequence modeling owing to its global self-attention mechanism and flexible scalability. However, such models still suffer from high computational costs, primarily due to the quadratic complexity of self-attention, dense prediction requirements, and high-resolution inputs [50]. Recently, leveraging the sparse spike-driven nature of SNNs has emerged as a promising direction to mitigate these computational costs. Spike-driven Transformer series [51, 33, 52] adapt the standard Metaformer into an SNN framework for object detection and semantic segmentation, demonstrating the feasibility of SNNs in dense prediction tasks. Spike2Former [15] integrates normalized integer leaky-and-integrated firing (NI-LIF) neurons and spike-driven deformable attention to achieve competitive performance on segmentation benchmarks while maintaining low energy consumption. Despite these advancements, SNNs still lag behind ANNs in visual long-sequence modeling. This underscores the need for deeper investigation into SNNs' spatio-temporal bottlenecks and architectural optimization.
+
+# 3 Theoretical Analysis of Spatio-temporal Effective Receptive Field
+
+In this section, we first introduce the concept of ERF in conventional ANNs. Subsequently, we extend this conventional ERF into the temporal dimension to characterize the ST-ERF in SNNs. Finally, we introduce a loss-derived method to efficiently compute ST-ERF in SNNs.
+
+# 3.1 ERF in ANNs
+
+The concept of the ERF has been widely adopted to analyze how input features contribute to network activations and how such influences are distributed within the RF [40, 41]. Under the assumption of a single channel per layer, Luo et al. [37] mathematically characterized how each input feature contributes to the output of a neural network layer. It can be defined as follows:
+
+$$
+\operatorname {E R F} _ {(i, j)} \left[ y _ {(m, n)}; \mathbf {x} \right] = \frac {\partial y _ {(m , n)}}{\partial x _ {(i , j)}}, \tag {1}
+$$
+
+where  $\mathbf{x} \in \mathbb{R}_1$  is the input feature and  $\mathbf{y} \in \mathbb{R}_2$  is the output feature. In this manner, the ERF measures the partial derivative of an output feature  $y_{(m,n)} \in \mathbf{y}$  with respect to each input feature  $x_{(i,j)} \in \mathbf{x}$  within a given layer. As illustrated in Figure 1(b), the ERF of a given network  $\mathcal{F}(\cdot; \theta)$  describes the input regions that contribute to a particular output activation.
+
+As shown in Eq. (1), the ERF can be computed at any output location. However, most studies evaluate the ERF at the central output feature  $y_{(0,0)}$  by assigning a unit gradient to this location [53, 54]. This practice establishes a centered and symmetric reference, ensuring stable and comparable visualization results. In this work, we also follow the setting of [37] and adopt the ERF at the central output feature  $y_{(0,0)}$  as the evaluation metric.
+
+# 3.2 ST-ERF in SNNs
+
+Due to the inherent temporal dynamics, SNNs require additional consideration of the input at each timestep. To address this, we formally define the ST-ERF (i.e.,  $\mathrm{ERF}^{(\mathcal{S},\mathcal{T})}$ ). Firstly, we redefine the mapping relationship of SNNs. Consider a SNN layer with learnable parameters  $\theta$  that maps input spike features  $\mathbf{x}[1:T] \in \hat{\mathbb{R}}_1$  to output spike features  $\mathbf{y}[1:T] \in \hat{\mathbb{R}}_2$ :
+
+$$
+\mathbf {y} [ 1: T ] = \mathcal {F} (\mathbf {x} [ 1: T ]; \theta), \mathcal {F}: \mathbb {R} _ {1} \rightarrow \mathbb {R} _ {2}. \tag {2}
+$$
+
+Its ERF needs to account not only for the accumulation across spatial dimensions but also for that across temporal dimensions. Specifically,  $\mathrm{ERF}^{(\mathcal{S},\mathcal{T})}\in \hat{\mathbb{R}}_1$  can be expressed as:
+
+$$
+\operatorname {E R F} _ {(i, j)} ^ {(\mathcal {S}, \mathcal {T})} [ y _ {(m, n)} [ t ], \tau ; \mathbf {x} ] = \frac {\partial y _ {(m , n)} [ t ]}{\partial x _ {(i , j)} [ t - \tau ]}, 1 \leq t \leq T, 0 \leq \tau \leq t - 1. \tag {3}
+$$
+
+Accordingly,  $\mathrm{ERF}^{(S,\mathcal{T})}$  quantifies how much each input feature  $x_{(i,j)}[t - \tau ]\in \mathbf{x}$  at a previous timestep  $t - \tau$  contributes to a specific output feature  $y_{(m,n)}[t]\in \mathbf{y}$ . Based on this definition, the spatial ERF (i.e.,  $\mathrm{ERF}^{(S)})$  can be seen as the weighted average of the ST-ERFs over all timesteps:
+
+$$
+\operatorname {E R F} _ {(i, j)} ^ {(S)} [ y _ {(m, n)}; \mathbf {x} ] = \frac {1}{T} \sum_ {t = 1} ^ {T} \sum_ {\tau = 0} ^ {t - 1} w (t, \tau) \cdot \operatorname {E R F} _ {(i, j)} ^ {(S, \mathcal {T})} [ y _ {(m, n)} [ t ]; \mathbf {x}, \tau ], \tag {4}
+$$
+
+where  $w(t,\tau)$  represents the relative contribution of the input with delay  $\tau$  at time  $t$  to the output. The specific form of  $w(t,\tau)$  depends on the neuronal dynamics and network architecture. For example, in Leaky Integrate-and-Fire (LIF) neurons, inputs closer to the current time step may have a higher influence due to the decay of membrane potential over time.
+
+The temporal ERF (i.e.,  $\mathrm{ERF}^{(\mathcal{T})}$ ) can be seen as the integration over the spatial dimensions of ST-ERF to indicate the contribution of inputs at different timesteps to the final output:
+
+$$
+\operatorname {E R F} ^ {(T)} [ \tau ; \mathbf {x} ] = \sum_ {i, j} \sum_ {m, n} \operatorname {E R F} _ {(i, j)} ^ {(S, T)} [ y _ {(m, n)} [ T ]; \mathbf {x}, \tau ]. \tag {5}
+$$
+
+As shown in Figure 1(c), we visualize an example of the ST-ERF. Similar to conventional ERF analysis, we focus on the center of the feature map at a specific timestep (e.g., the final timestep in Fig. 1(c)) to analyze the spatio-temporal feature representations in an SNN. Depending on the purpose of analysis, one may investigate the spatial distribution of the ST-ERF at a given timestep (spatial ERF) or its temporal distribution across one or more layers (temporal ERF).
+
+# 3.3 Loss-Derived Calculation for ST-ERFs
+
+Based on Eq. (3), computing the ST-ERF in SNNs requires evaluating first-order derivatives of outputs with respect to all input features. To obtain the ST-ERF conveniently, we introduce the loss-derived calculation method to efficiently compute using PyTorch's Automatic Differentiation functionality. Consider a SNN with input spike features  $s^{\ell -1}$ , output spike features  $s^\ell$  at the  $\ell$ -th layer, and an arbitrary loss function  $\mathcal{L}$ . The spatial ERF of SNNs can be easily obtained by calculating the average of the gradient of the loss with respect to input features at position  $(i,j)$  across all timesteps  $T$ . Specifically, it can be computed as follows:
+
+$$
+\operatorname {E R F} _ {(i, j)} ^ {(\mathcal {S})} \left[ s _ {(0, 0)} ^ {\ell} \right] = \frac {1}{T} \sum_ {t = 1} ^ {T} \frac {\partial s _ {(0 , 0)} ^ {\ell} [ t ]}{\partial s _ {(i , j)} ^ {\ell - 1} [ t ]} = \frac {1}{T} \sum_ {t = 1} ^ {T} \frac {\partial \mathcal {L}}{\partial s _ {(i , j)} ^ {\ell - 1} [ t ]}, \text {w h e n} \forall t, \frac {\partial \mathcal {L}}{\partial s _ {(\hat {i} , \hat {j})} ^ {\ell} [ t ]} = \left\{ \begin{array}{l l} 1, & \hat {i} = 0, \hat {j} = 0, \\ 0, & \text {o t h e r w i s e} \end{array} . \right. \tag {6}
+$$
+
+The temporal ERF of SNNs can be obtained by calculating the sum of the gradient of the loss function with respect to input features at timestep  $T - \tau$  across all spatial positions. It can be computed as:
+
+$$
+\operatorname {E R F} ^ {(\mathcal {T})} [ \tau ] = \sum_ {i, j} \sum_ {\hat {i}, \hat {j}} \frac {\partial s _ {(\hat {i} , \hat {j})} ^ {\ell} [ T ]}{\partial s _ {(i , j)} ^ {\ell - 1} [ T - \tau ]} = \sum_ {i, j} \frac {\partial \mathcal {L}}{\partial s _ {(i , j)} ^ {\ell - 1} [ T - \tau ]}, \quad \text {w h e n} \forall \hat {i}, \hat {j}, \frac {\partial \mathcal {L}}{\partial \mathbf {s} _ {(\hat {i} , \hat {j})} ^ {\ell} [ T ]} = 1. \tag {7}
+$$
+
+Proof can be found in Appendix A. We refer to the conditions in Eq. (6) and (7) as gradient stimuli. Based on this proposition, we could easily obtain the spatial and temporal ERF with automatic back-propagation without an explicit loss function.
+
+# 4 Problem Analysis on Transformer-based SNNs using ST-ERF
+
+In this section, we use the ST-ERF framework to analyze existing Transformer-based SNNs and identify their limitations in visual long-sequence modeling tasks.
+
+# 4.1 Different ST-ERF Behaviors in Transformer-based SNNs
+
+We apply the ST-ERF framework to analyze Transformer-based SNNs' spatial ERF behaviors across all timesteps. Specifically, we compared two groups of architectures (a: ViT-like architecture group and b: Meta-architecture group) with their ANN counterparts to investigate the differences in the formation of their spatial ERFs. For the loss-derived calculation, we set the central patch across all channels and timesteps in the output tensor as the gradient stimuli (uniform values of 1), then perform automatic back-propagation. Each experiment comprised 60 iterations using randomly sampled input tensors under standard normal distribution  $(\mu = 0, \sigma^2 = 1)$ . Note that we average the ST-ERF over all timesteps to obtain a clear visualization.
+
+![](images/f607718efcb4b1a00f0b2798b632571e2af8ee7c95fab3c1067481e6484d3c7a.jpg)  
+(a)
+
+![](images/c2cf130bbb6ac6d617fb9d879c19765d55f76d003a95e163c5d72272cfaa9d25.jpg)
+
+![](images/0306a9635d8bbf6e5670263be2c99b57d4321cd5649b0cd5e302f23cdbf39b30.jpg)  
+(b)  
+Figure 2: Comparison of spatial ERF with ANN Vision Transformers and ST-ERF with different Transformer-based SNNs. (a) ViT-like architecture comparison group: ViT-B and SDT-V1. (b) Meta-architecture comparison group: CAFormer-s18 and its counterpart Meta-SDT-S.
+
+![](images/fb741a849b29ddb3f33ac7531cf11fd40ef5048e5bbd1da64a694c6f84b43ac9.jpg)
+
+The comparison of ViT-like architectures is illustrated in Figure 2(a). Compared with the classic ViT-B, SDT-V1 exhibits a more centrally concentrated yet markedly narrower spatial ERF. This
+
+observation suggests that the architectural modifications in SDT-V1 may restrict the receptive field's spatial extent, thereby enhancing its attention on local spatial dependencies. In fact, SDT-V1 adopts a fundamentally different strategy from the vanilla ViT in the patch splitting stage. Specifically, SDT-V1 employs the Spike Patch Splitting (SPS) module, consisting a Patch Splitting Module (PSM) to linearly project the input image and a Relative Position Embedding (RPE) [55] block to generate the latent position information. The SPS module incorporates multiple convolutional layers at the early stage of the network, facilitating low-level spatial features extraction from input images.
+
+The comparison of meta-architectures is illustrated in Figure 2(b). Although Meta-SDT exhibits ERF behaviors similar to those of its ANN counterparts, it also struggles to maintain long-range feature attention. This limitation can be attributed to the additional employment of convolutional layers, which tend to emphasize localized features rather than global spatial contexts. Compared with CAFormer, Meta-SDT introduces the Re-parameterization Convolution (RepConv)[56] to perform the linear projection of queries, keys, and values[51]. This design enhances local feature extraction, yet it inherently constrains the model's capacity to aggregate information across distant spatial regions. Together, these findings suggest that the convolutional operations enhances local feature sensitivity but poses challenges for maintaining long-range spatial coherence in Transformer-based SNNs.
+
+# 4.2 Visual Long-sequence Modeling Needs Global ST-ERF
+
+Visual long-sequence modeling tasks often involve dense predictions across an entire image, requiring the processing of thousands of input tokens [25]. Therefore, capturing long-range dependencies and global context is crucial for achieving accurate and robust representations [57]. Prior studies have found that vision models with global receptive fields often excel at segmentation and detection, for instance when using self-attention mechanisms as in Transformer architectures [57, 58]. In contrast, architectures lacking global context integration tend to struggle. While early convolutional layers excel at extracting low-level structural patterns [59], their locality inherently limits the capacity to capture long-range dependencies, making them suboptimal for visual long-sequence tasks.
+
+However, despite the need for global spatial awareness in visual long-sequence modeling, Transformers-based SNNs still fail to achieve a truly global ST-ERF. As discussed above, they tend to focus heavily on the center and expand to limited size. This contrasts sharply with the expected behavior required for visual long-sequence modeling tasks, where the weak global ST-ERF limits information aggregation and consequently degrades performance on such scenarios [57].
+
+# 5 Methods
+
+In this section, we propose two novel channel-mixing designs, MLPixer and SRB, which enable Transformer-based SNNs to more effectively capture long-range dependencies. Furthermore, we integrate these modules into the Meta-SDT architecture to enhance performance on visual long-sequence modeling tasks.
+
+# 5.1 Design of Channel Mixer Block
+
+To enhance the global modeling capability of SNN in visual long-sequence tasks, we propose two novel channel mixer designs. The first is the multi-layer perceptron-based mixer (MLPixer), which employs a two-layer MLP structure to more effectively extract global features. It is defined as follows:
+
+$$
+\operatorname {M L P i x e r} (\mathbf {X}) = \operatorname {B N} \left(\operatorname {M L P} \left(\mathbb {S N} \{\operatorname {B N} (\operatorname {M L P} \{\mathbb {S N} (\mathbf {X}) \}) \}\right)\right), \tag {8}
+$$
+
+where  $\mathbf{X} \in \mathbb{R}^{T \times B \times N \times D}$  denotes the input of channel mixers in the Transformer block.  $\mathbb{SN}(\cdot)$  denotes a spiking neuron layer that transforms the input sequence into the spike trains.  $\mathrm{MLP}(\cdot)$  denotes a single-layer fully connected (FC) operation, and  $\mathrm{BN}(\cdot)$  denotes batch normalization.
+
+Compared with vanilla channel mixers [60, 51] that rely on convolution operations, the MLPixel employs a two-layer MLP operation to mix features across channels. This design reduces reliance on convolutional operations, mitigating the ERF's bias toward a Gaussian-like central concentration and enabling SNNs to capture long-range dependencies more effectively.
+
+![](images/bb7cba2ddef8df4511ca62eee764baef316fcef4b6efbd6789b046f411181561.jpg)  
+(a)
+
+![](images/635ab91e6f1ce8c30c1a9c95e444af9796812ff591eeb9505bc67ad2b5a57398.jpg)  
+(b)
+
+![](images/97de41efa7a6d8fc1fee259b0d2996445ffe92d02c3c7d7d2edc37a1ba3f7ebf.jpg)  
+(c)  
+Figure 3: Comparison between the original channel mixer design and our proposed methods, along with their ST-ERF. For clearer visualization, the ST-ERF is averaged over timesteps. (a) Vanilla convolution-based channel mixer. (b) Proposed MLPixer architecture. (c) Proposed SRB architecture. Obviously, the vanilla convolution-based channel mixer exhibits a limited ST-ERF, whereas our MLPixer and SRB modules achieve a more global ST-ERF. Moreover, due to the reduced use of convolutions, MLPixer exhibits an even broader effective receptive field.
+
+Building on this, we further propose the SRB module. It replaces only the second convolution in the channel mixer with a single-layer MLP operation. Specifically, the SRB is defined as follows:
+
+$$
+\operatorname {S R B} (\mathbf {X}) = \operatorname {B N} \left(\operatorname {M L P} \left(\mathbb {S N} \{\operatorname {B N} \left(\operatorname {C o n v} \{\mathbb {S N} (\mathbf {X}) \}\right) \}\right)\right). \tag {9}
+$$
+
+Here,  $\mathrm{Conv}(\cdot)$  denotes a  $1\times 1$  convolution operation. In this manner, SRB module reduces additional parameters while maintaining performance. To validate the effectiveness of our approach, we visualize the ERFs of the Conv-based mixer, the MLPixer, and the SRB modules.
+
+As shown in Figure 3(a), the vanilla convolution-based channel mixer exhibits a limited ST-ERF. In contrast, the proposed MLPixer and SRB modules demonstrate a more global ST-ERF. Furthermore, the comparison between Figure 3(b) and Figure 3(c) further demonstrates that MLPixer exhibits a more global ERF. This stems from reduced use of convolutions and further suggests that MLPs provide stronger global modeling capacity than convolutional operators. We will validate the proposed module on visual long-sequence modeling tasks in the experiment section.
+
+![](images/93138ff337eef6427570f1f6348133a5ec4f6aa2fc73ffbf7997926686587fbc.jpg)  
+Figure 4: The overall architecture of Meta-SDT, which typically comprises four hierarchical stages. The first two stages use convolution-based SNN blocks, while the latter two adopt Transformer-SNN blocks. To strengthen the global modeling capacity of SNNs, we introduce two novel channel mixer architectures, MLPixer and SRB, to replace the convolution-based SNN blocks in the first two stages.
+
+# 5.2 Overall Architecture
+
+To further validate the effectiveness of our approach, we integrate the proposed SRB and MLPixer modules into the CAFormer [60] and Meta-SDT [33] architectures. As shown in Figure 4, these architectures adopt a multi-stage design, where the first two stages consist of Conv-based SNN blocks, and the latter two stages comprise Transformer-SNN blocks. In this work, we replace only the operations in the first two stages. Specifically, the first two stages are represented as:
+
+$$
+\mathbf {X} ^ {\prime} = \mathbf {X} + \operatorname {S S C} (\mathbf {X}), \mathbf {X} ^ {\prime \prime} = \mathbf {X} ^ {\prime} + \operatorname {M i x e r} _ {\epsilon} \left(\mathbf {X} ^ {\prime}\right). \tag {10}
+$$
+
+Here,  $\mathrm{Mixer}_{\epsilon}(\cdot)$  is the channel mixer. In this work, we implement the approach using both the MLPixer module and the SRB module. Similar to the vanilla channel mixer, our method adopts an up-projection followed by a down-projection with a nonlinear activation in between, where  $\epsilon > 1$  represents the intermediate dimensional expansion ratio.  $\mathrm{SSC}(\cdot)$  is the spike-driven separable convolution block as token mixer, it is defined as follows:
+
+$$
+\operatorname {S S C} (\mathbf {X}) = \operatorname {P W C o n v} _ {2} (\mathbb {S N} (\operatorname {D W C o n v} (\mathbb {S N} (\operatorname {P W C o n v} _ {1} (\mathbb {S N} (\mathbf {X})))))). \tag {11}
+$$
+
+$\mathrm{PWConv}_1(\cdot)$  and  $\mathrm{PWConv}_2(\cdot)$  are pointwise convolutions,  $\mathrm{DWConv}(\cdot)$  is depthwise convolution.  $\mathbb{SN}(\cdot)$  denotes the spiking neuron layer. To maintain the spike-driven characteristics of the network, we implement membrane-shortcut residual connection mechanism. Furthermore, Transformer-SNN blocks are utilized in Stage 3 and Stage 4, following the same configuration as that of Meta-SDT-V3 [33]. We will further verify the effectiveness of the proposed method in the experimental section.
+
+# 6 Experiments
+
+In this section, we validate the effectiveness of our method through visualization and experimental analysis. First, we examine the changes in the ST-ERF after integrating the proposed modules into Meta-SDT, showing that our method achieves stronger global spatial receptive fields across all stages. Second, we evaluate its performance improvement on long-sequence modeling tasks, including object detection and semantic segmentation. Finally, we further investigate the method on complex event modeling tasks to assess its applicability in more challenging scenarios.
+
+# 6.1 ST-ERF Behavior in Transformer-based SNNs
+
+In order to study the impact of our proposed block on the receptive field of Meta-SDT, we compared temporal-averaged spatial ERFs between our two Meta-SDT variants with previous models. We initialized the central spatial feature across all channels and timesteps in the output tensor as uniform gradient stimuli (value = 1), and propagated the gradients backward through the network. Each experiment consisted of 60 iterations with input tensors randomly drawn from a standard normal distribution  $(\mu = 0, \sigma^2 = 1)$ .
+
+The results are illustrated in Figure 5. Surprisingly, we found that Spikformer exhibits diffuse receptive fields across all stages. The SDT-V1, Meta-SDT, and QKFormer demonstrate markedly centered distribution that gradually expand as the network deepens, all manifesting a Gaussian-like effect. Additionally, we observed dissipation of spatial ERF in SDT-V1 during the final stage. In contrast, our proposed two Meta-SDT variants establish robust global spatial receptive fields in the early stages. The MLPixer-SDT establishes a strong global spatial ERF in Stage 1. As the network deepens, its spatial ERF selectively contracts toward specific regions. The ERF behavior in SRB-SDT is slightly different, as it only begins to form a preliminary spatial ERF at Stage 2, and this distribution continues to evolve with increasing network depth.
+
+# 6.2 Performance in Visual Long-sequence Modeling Tasks
+
+We selected two challenging datasets to evaluate performance on classic visual long-sequence modeling tasks: object detection and instance segmentation on COCO 2017, and semantic segmentation on the ADE20K dataset. We choose the Meta-SDT(v3) [33] as the baseline and construct Meta-SDT variants with MLPixer(ε4), MLPixer(ε6) and SRB(ε4).
+
+Performance on COCO 2017 We evaluate the efficacy of the MLPixer and SRB on Meta-SDT and select the classic and large-scale COCO [61] dataset as our benchmark for evaluation. Following
+
+![](images/6cf891a7a7a8add05d685d678088e9ec720558a8b4c58a88e6a231bc5979c0cf.jpg)  
+Figure 5: Comparison of temporal-averaged spatial ERF evolution across stages. From top to bottom are Stage 1 through Stage 4. Spikformer shows diffuse receptive fields across all stages. SDT-V1, Meta-SDT, and QKFormer exhibit more centered spatial distributions that gradually expand as depth increases. Our two Meta-SDT variants establish global spatial receptive fields in the early stages.
+
+the previous work [51, 33, 52], we use the MMdetection [62] codebase with a spiking version and then deploy our model. We employ the Meta-SDT [33] with two variants as the backbone network to extract features, along with fine-tuning Mask R-CNN [63] for object detection and instance segmentation. All backbone networks are pretrained on ImageNet-1K [64], while the incremental layers are initialized following [65]. During fine-tuning, we strictly obey the  $1 \times$  training schedule.
+
+The comparison results of object detection and instance segmentation are shown in Table 1. Under the same training schedule, both the MLPixer and SRB variants outperforms the baseline across all metrics. More specifically, the SRB variant exceeds the performance of SDTv3-T and SDTv3-B by  $10.42\%$  and  $4.26\%$  on the  $\mathrm{AP}_{50}^{\mathrm{b}}$  metric, while maintaining almost the same model size. In conclusion, our approach demonstrates efficacy in object detection and instance segmentation, setting a new benchmark for COCO dataset in the SNN domain.
+
+Performance on ADE20K We evaluate the performance of MLPixer and SRB on the semantic segmentation task using the challenging ADE20K dataset [66]. Similar to the COCO experiments, we utilize the spiking version of MMSegmentation [67] as our codebase and employ the Meta-SDT [33] with two variants as the backbone network. We fine-tune the Semantic FPN framework [68] for semantic segmentation. Backbone networks are initialized with ImageNet-1K pre-trained weights [64], and new layers follow the initialization scheme of [65]. All models are strictly obey the same training schedule for 160k iterations.
+
+As shown in Table 2, both MLPixer and SRB variants surpass the baseline in terms of mIoU. The SRB variant improves performance by  $3.3\%$  and  $2.6\%$  over SDTv3-T and SDTv3-B, respectively, while reducing parameters by 0.3M and 1.2M. The MLPixer  $(\epsilon 4)$  variant achieves the largest parameter reduction of 0.6M and 2.4M, with comparable or superior accuracy to SDTv3-T and SDTv3-B. These results highlight the effectiveness of the proposed modules in enhancing semantic segmentation on ADE20K.
+
+# 6.3 Performance in Complex Event Modeling Tasks
+
+Event-based Tracking We evaluate the effectiveness of two channel mixers in the context of event-based tracking, a highly challenging yet practically significant application domain for SNNs. Our experiments follow the SDTrack pipeline [18], which employs the Global Trajectory Prompt
+
+<table><tr><td>Arch.</td><td>#T</td><td>#P</td><td>APb</td><td>APb50</td><td>APb75</td><td>APm</td><td>APm50</td><td>APm75</td></tr><tr><td>SDTv3-T[33]</td><td>4</td><td>25M</td><td>15.2</td><td>35.5</td><td>10.2</td><td>15.2</td><td>33.0</td><td>12.3</td></tr><tr><td>MLPixer(€4)</td><td>4</td><td>24M</td><td>16.2</td><td>37.0</td><td>11.5</td><td>15.2</td><td>32.9</td><td>12.5</td></tr><tr><td>MLPixer(€6)</td><td>4</td><td>25M</td><td>17.5</td><td>38.5</td><td>13.2</td><td>16.2</td><td>34.5</td><td>13.5</td></tr><tr><td>SRB(€4)</td><td>4</td><td>25M</td><td>18.2</td><td>39.2</td><td>13.8</td><td>17.5</td><td>34.8</td><td>14.3</td></tr><tr><td>SDTv3-B[33]</td><td>4</td><td>39M</td><td>21.7</td><td>46.9</td><td>17.0</td><td>20.1</td><td>41.8</td><td>17.5</td></tr><tr><td>MLPixer(€4)</td><td>4</td><td>36M</td><td>22.9</td><td>47.6</td><td>19.2</td><td>21.0</td><td>43.4</td><td>18.3</td></tr><tr><td>MLPixer(€6)</td><td>4</td><td>39M</td><td>25.1</td><td>48.8</td><td>22.5</td><td>21.9</td><td>43.5</td><td>19.6</td></tr><tr><td>SRB(€4)</td><td>4</td><td>37M</td><td>25.8</td><td>48.9</td><td>22.8</td><td>22.5</td><td>43.9</td><td>20.4</td></tr></table>
+
+Table 1: Object detection and instance segmentation with Mask R-CNN on COCO val2017, using ImageNet-1K pre-training and  $1\times$  training schedule.  
+
+<table><tr><td>Arch.</td><td>Ch. Mixer</td><td>#T</td><td>Param.(M)</td><td>mIoU(%)</td></tr><tr><td rowspan="2">SDTv3</td><td>C2d-k3(ε4)</td><td>4</td><td>6.5 BASE</td><td>34.9 BASE</td></tr><tr><td>MLPix.(ε4)</td><td>4</td><td>5.9 (↓0.6)</td><td>34.9 (↑0.0)</td></tr><tr><td rowspan="2">-T[33]</td><td>MLPix.(ε6)</td><td>4</td><td>6.6 (↑0.1)</td><td>35.9 (↑1.0)</td></tr><tr><td>SRB(ε4)</td><td>4</td><td>6.2 (↓0.3)</td><td>38.2 (↑3.3)</td></tr><tr><td rowspan="2">SDTv3</td><td>C2d-k3(ε4)</td><td>4</td><td>20.4 BASE</td><td>41.1 BASE</td></tr><tr><td>MLPix.(ε4)</td><td>4</td><td>18.0 (↓2.4)</td><td>42.0 (↑0.9)</td></tr><tr><td rowspan="2">-B[33]</td><td>MLPix.(ε6)</td><td>4</td><td>20.7 (↑0.3)</td><td>43.4 (↑2.3)</td></tr><tr><td>SRB(ε4)</td><td>4</td><td>19.2 (↓1.2)</td><td>43.7 (↑2.6)</td></tr></table>
+
+Table 2: Segmentation results on ADE20K based on different mixer block, using ImageNet-1K pre-training and 160k iter.
+
+method to convert event streams into event frames. We strictly adhere to the original training protocol, modifying only the backbone by replacing SDTrack with our proposed SDTrack+MLPixer or SDTrack+SRB variants. As presented in Table 3, extensive experiments on the FE108 [69] and VisEvent [70] datasets demonstrate that our architectures surpass the original SDTrack in several key metrics. These results confirm that both the MLPixer and SRB designs preserve the Transformers-based SNNs' performance, yet highlight opportunities for further improvement in subsequent temporal benchmarks.
+
+Table 3: Performance comparison on event-based object tracking, a challenging yet important application for SNNs. Evaluation is conducted on two benchmark datasets, FE108 and VisEvent.  
+
+<table><tr><td rowspan="2">Architecture</td><td rowspan="2">Timesteps</td><td rowspan="2">Param. (M)</td><td colspan="2">FE108 [69]</td><td colspan="2">VisEvent [70]</td></tr><tr><td>AUC(%)</td><td>PR(%)</td><td>AUC(%)</td><td>PR(%)</td></tr><tr><td>SD-Track(Tiny) [18]</td><td>4 × 1</td><td>19.61</td><td>56.7</td><td>89.1</td><td>35.4</td><td>48.7</td></tr><tr><td>+MLPixer (ε = 4)</td><td>4 × 1</td><td>20.21</td><td>57.1</td><td>89.2</td><td>33.7</td><td>47.3</td></tr><tr><td>+MLPixer (ε = 6)</td><td>4 × 1</td><td>22.99</td><td>57.9</td><td>90.1</td><td>34.5</td><td>48.9</td></tr><tr><td>+SRB (ε = 4)</td><td>4 × 1</td><td>21.43</td><td>58.2</td><td>88.5</td><td>33.8</td><td>48.0</td></tr></table>
+
+# 7 Conclusion
+
+This paper presents ST-ERF as a novel framework for analyzing the spatial-temporal modeling behaviors in SNNs from a new perspective. Through this analysis, an inherent limitation in current Transformer-based SNN models is identified when applied to visual long-sequence modeling tasks. To address this limitation, two channel-mixer architectures, MLPixer and SRB, are proposed. Visualization of ST-ERF demonstrates that both modules enhance the global receptive field. Extensive experiments on long-sequence modeling tasks, including object detection and semantic segmentation, show that MLPixer and SRB improve overall performance, with SRB achieving an optimal balance between accuracy and model size. Furthermore, the study investigates complex event modeling tasks to assess the applicability of MLPixer and SRB in more challenging scenarios. Overall, the proposed ST-ERF framework offers valuable insights for the design and optimization of SNN architectures across a wide range of tasks.
+
+# Acknowledgments
+
+This work is supported in part by the National Natural Science Foundation of China (No. 62220106008 and 62271432), in part by the Shenzhen Science and Technology Program (Shenzhen Key Laboratory, Grant No. ZDSYS20230626091302006) and in part by the Program for Guangdong Introducing Innovative, Entrepreneurial Teams, Grant No. 2023ZT10X044, and in part by the State Key Laboratory of Brain Cognition and Brain-inspired Intelligence Technology, Grant No. SKLBI-K2025010. This work was partially supported by UESTC Kunpeng&Ascend Center of Cultivation.
+
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+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: Our abstract and introduction clearly describe our contribution, the algorithm, and the experimental results.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: This paper discusses the limitations of the work in Appendix C.
+
+Guidelines:
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+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+# Answer: [Yes]
+
+Justification: We mentioned our proposition and properties of ST-ERF and provided a whole set of proofs in Appendix A.
+
+# Guidelines:
+
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+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+# Answer: [Yes]
+
+Justification: We provide a detailed description of our model architecture and present all the training details, including dataset processing methods and hyperparameter settings in Appendix B.
+
+# Guidelines:
+
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+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
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+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
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+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: We provide the full details in Appendix B.
+
+Guidelines:
+
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+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: Although we did provide mean  $\pm$  std range to show several experiments' numerical range, we need to clarify that the numerical experiments are focused on the validation of ST-ERF properties, so we provided several independent trials with different input samples and different network initialization. The mean value of performance (e.g. the fitted curve) is solid enough to clarify our theories.
+
+Guidelines:
+
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+
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
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+Answer: [Yes]
+
+Justification: This paper provides sufficient information on the computer resources.
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+Guidelines:
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+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
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+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [NA]
+
+Justification: This work is a foundational research and not tied to particular societal applications.
+
+# Guidelines:
+
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+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: This paper poses no such risks.
+
+# Guidelines:
+
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+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
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+Justification: Yes, the paper properly credits the creators or original owners of assets (e.g., code, data, models) and explicitly mentions and respects the relevant licenses and terms of use.
+
+# Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
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+
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+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
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+
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+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
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+Answer: [NA]
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+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
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+Answer: [NA]
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+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: The proposed method in this research does not involve LLMs as any important, original, or non-standard components.
+
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+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# Appendix
+
+# A Proof of 3.3
+
+Proof. Leaky Integrate-and-Fire (LIF) model [5, 2] can be described by the following equations:
+
+$$
+\mathbf {v} ^ {\ell} [ t ] = \mathbf {h} ^ {\ell} [ t - 1 ] + f \left(\mathbf {w} ^ {\ell}, \mathbf {x} ^ {\ell - 1} [ t - 1 ]\right), \quad (\text {C h a r g i n g}) \tag {12}
+$$
+
+$$
+\mathbf {s} ^ {\ell} [ t ] = \boldsymbol {\Theta} (\mathbf {v} ^ {\ell} [ t ] - \vartheta), \quad \text {(F i r i n g f u n c t i o n)}, \tag {13}
+$$
+
+$$
+\mathbf {h} ^ {\ell} [ t ] = \left\{ \begin{array}{l l} \beta \mathbf {v} ^ {\ell} [ t ] - \vartheta \mathbf {s} ^ {\ell} [ t ], & \text {s o f t r e s e t} \\ \mathbf {v} ^ {\ell} [ t ] (1 - \mathbf {s} ^ {\ell} [ t ]), & \text {h a r d r e s e t} \end{array} \right. \quad (\text {L e a k - a n d - r e s e t f u n c t i o n}), \tag {14}
+$$
+
+where  $\beta$  is the decay constant,  $t$  is the time step,  $\mathbf{w}^{\ell}$  is the weight matrix of layer  $\ell$ ,  $f(\cdot)$  is the operation that stands for convolution (Conv) or fully connected (FC),  $\mathbf{x}$  is the input, and  $\Theta (\cdot)$  denotes the Heaviside step function. When the membrane potential  $\mathbf{v}$  exceeds the firing threshold  $\vartheta$ , the LIF neuron will trigger a spike s; otherwise, it remains inactive. After spike emission, the neuron invokes the reset mechanism, where the soft reset function is employed.  $\mathbf{h}$  is the membrane potential following the reset function.
+
+For the back-propagation of this neuron, we introduce the training process of SNN gradient descent and the parameter update method of spatio-temporal back-propagation (STBP) [9, 71]. The accumulated gradients of loss  $\mathcal{L}$  with respect to weights  $\mathbf{w}$  at layer  $\ell$  can be calculated as:
+
+$$
+\frac {\partial \mathcal {L}}{\partial \mathbf {w} ^ {\ell}} = \sum_ {t = 1} ^ {T} \frac {\partial \mathcal {L}}{\partial \mathbf {s} ^ {\ell + 1} [ t ]} \frac {\partial \mathbf {s} ^ {\ell + 1} [ t ]}{\partial \mathbf {v} ^ {\ell + 1} [ t ]} \left(\frac {\partial \mathbf {v} ^ {\ell + 1} [ t ]}{\partial \mathbf {w} ^ {\ell}} + \sum_ {\tau <   t} \prod_ {i = t - 1} ^ {\tau} \left(\frac {\partial \mathbf {v} ^ {\ell + 1} [ i + 1 ]}{\partial \mathbf {v} ^ {\ell + 1} [ i ]} + \frac {\partial \mathbf {v} ^ {\ell + 1} [ i + 1 ]}{\partial \mathbf {s} ^ {\ell + 1} [ i ]} \frac {\partial \mathbf {s} ^ {\ell + 1} [ i ]}{\partial \mathbf {v} ^ {\ell + 1} [ i ]}\right) \frac {\partial \mathbf {v} ^ {\ell + 1} [ \tau ]}{\partial \mathbf {w} ^ {\ell}}\right), \tag {15}
+$$
+
+where  $\mathbf{s}^{\ell}[t]$  and  $\mathbf{v}^{\ell}[t]$  represent the output spikes and membrane potential of the neuron in layer  $\ell$ , at time  $t$ . Moreover, notice that  $\frac{\partial \mathbf{s}^{\ell}[t]}{\partial \mathbf{v}^{\ell}[t]}$  is non-differentiable. To overcome this problem, Wu et al. [9] propose the surrogate function to make only the neurons whose membrane potentials close to the firing threshold receive nonzero gradients during back-propagation.
+
+In this paper, we use the rectangle function, which has been shown to be effective in gradient descent and may be calculated by:
+
+$$
+\frac {\partial \mathbf {s} ^ {\ell} [ t ]}{\partial \mathbf {v} ^ {\ell} [ t ]} = \frac {1}{a} \operatorname {s i g n} \left(\left| \mathbf {v} ^ {\ell} [ t ] - \vartheta \right| <   \frac {a}{2}\right), \tag {16}
+$$
+
+where  $a$  is a defined coefficient for controlling the width of the gradient window.
+
+To compute  $\sum_{t=1}^{T} \frac{\partial \mathbf{s}_{(0,0)}^{\ell}[t]}{\partial \mathbf{s}_{(i,j)}^{\ell-1}[t]}$ , we follow the chain rule with an arbitrary loss  $\mathcal{L}$ . Consider  $\sum_{t=1}^{T} \frac{\partial \mathcal{L}}{\partial \mathbf{s}_{(i,j)}^{\ell-1}[t]$ :
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \frac {\partial \mathcal {L}}{\partial \mathbf {s} _ {(i , j)} ^ {\ell - 1} [ t ]} = \sum_ {t = 1} ^ {T} \sum_ {\hat {i}, \hat {j}} \frac {\partial \mathcal {L}}{\partial \mathbf {s} _ {(\hat {i} , \hat {j})} ^ {\ell} [ t ]} \frac {\partial \mathbf {s} _ {(\hat {i} , \hat {j})} ^ {\ell} [ t ]}{\partial \mathbf {s} _ {(i , j)} ^ {\ell - 1} [ t ]} \\ = \sum_ {\hat {i}} \sum_ {\hat {j}} \sum_ {t = 1} ^ {T} \frac {\partial \mathcal {L}}{\partial \mathbf {s} _ {(\hat {i} , \hat {j})} ^ {\ell} [ t ]} \frac {\partial \mathbf {s} _ {(\hat {i} , \hat {j})} ^ {\ell} [ t ]}{\partial \mathbf {s} _ {(i , j)} ^ {\ell - 1} [ t ]} \tag {17} \\ = \sum_ {\hat {i} \neq 0} \sum_ {\hat {j} \neq 0} \sum_ {t = 1} ^ {T} \frac {\partial \mathcal {L}}{\partial \mathbf {s} _ {(\hat {i} , \hat {j})} ^ {\ell} [ t ]} \frac {\partial \mathbf {s} _ {(\hat {i} , \hat {j})} ^ {\ell} [ t ]}{\partial \mathbf {s} _ {(i , j)} ^ {\ell - 1} [ t ]} + \sum_ {t = 1} ^ {T} \frac {\partial \mathcal {L}}{\partial \mathbf {s} _ {(0 , 0)} ^ {\ell} [ t ]} \frac {\partial \mathbf {s} _ {(0 , 0)} ^ {\ell} [ t ]}{\partial \mathbf {s} _ {(i , j)} ^ {\ell - 1} [ t ]}. \\ \end{array}
+$$
+
+When the following conditions are met:
+
+$$
+\forall t \in T, \frac {\partial \mathcal {L}}{\partial \mathbf {s} _ {(\hat {i} , \hat {j}) [ t ]} ^ {\ell}} = \left\{ \begin{array}{l l} 1 & \hat {i} = 0, \hat {j} = 0, \\ 0 & \text {o t h e r w i s e} \end{array} . \right. \tag {18}
+$$
+
+We can get:
+
+$$
+\sum_ {t = 1} ^ {T} \frac {\partial \mathcal {L}}{\partial \mathbf {s} _ {(i , j)} ^ {\ell - 1} [ t ]} = \sum_ {t = 1} ^ {T} \frac {\partial \mathbf {s} _ {(0 , 0)} ^ {\ell} [ t ]}{\partial \mathbf {s} _ {(i , j)} ^ {\ell - 1} [ t ]}, \quad \frac {1}{T} \sum_ {t = 1} ^ {T} \frac {\partial \mathcal {L}}{\partial \mathbf {s} _ {(i , j)} ^ {\ell - 1} [ t ]} = \frac {1}{T} \sum_ {t = 1} ^ {T} \frac {\partial \mathbf {s} _ {(0 , 0)} ^ {\ell} [ t ]}{\partial \mathbf {s} _ {(i , j)} ^ {\ell - 1} [ t ]}. \tag {19}
+$$
+
+The spatial ERF at position  $(i,j)$  can thus be calculated by summing the gradients of the loss with respect to all timesteps.
+
+For the temporal ERF, we need to compute  $\sum_{i,j} \frac{\partial s_{(0,0)}^{\ell}[T]}{\partial s_{(i,j)}^{\ell-1}[T-\tau]}$ . We consider  $\sum_{i,j} \frac{\partial \mathcal{L}}{\partial \mathbf{s}_{(i,j)}^{\ell-1}[T-\tau]}$ . By applying the chain rule:
+
+$$
+\sum_ {i, j} \frac {\partial \mathcal {L}}{\partial \mathbf {s} _ {(i , j)} ^ {\ell - 1} [ T - \tau ]} = \sum_ {i, j} \sum_ {\hat {i}, \hat {j}} \frac {\partial \mathcal {L}}{\partial \mathbf {s} _ {(\hat {i} , \hat {j})} ^ {\ell} [ T ]} \frac {\partial \mathbf {s} _ {(\hat {i} , \hat {j})} ^ {\ell} [ T ]}{\partial \mathbf {s} _ {(i , j)} ^ {\ell - 1} [ T - \tau ]} \tag {20}
+$$
+
+When the following conditions are met:
+
+$$
+\forall \hat {i}, \hat {j}, \frac {\partial \mathcal {L}}{\partial \mathbf {s} _ {(\hat {i} , \hat {j})} ^ {\ell} [ T ]} = 1 \tag {21}
+$$
+
+We can simplify:
+
+$$
+\sum_ {i, j} \frac {\partial \mathcal {L}}{\partial s _ {(i , j)} ^ {\ell - 1} [ T - \tau ]} = \sum_ {\hat {i}, \hat {j}} \sum_ {i, j} \frac {\partial s _ {(\hat {i} , \hat {j})} ^ {\ell} [ T ]}{\partial s _ {(i , j)} ^ {\ell - 1} [ T - \tau ]} \tag {22}
+$$
+
+The temporal ERF at delay  $\tau$  can thus be calculated by summing the gradients of the loss with respect to all spatial positions at timestep  $T - \tau$ .
+
+# B Details in Detection and Segmentation Experiments
+
+On ImageNet-1K pretraining, we employ three scales of Meta-SDT with three different channel mixer design ((1): Conv-Mixer; (2): MLPixer; (3): SRB) in Table 4 and utilize the hyper-parameters in Table 5 to pre-train models in our paper for further fine-tuning on COCO 2017 and ADE20K datasets. Note that  $\epsilon$  represents the channel expand ratio (CHW  $\rightarrow$ $\epsilon$  CHW  $\rightarrow$  CHW).
+
+For COCO 2017 dataset, We utilize the MMDetection [72] framework to implement the existing models and our method. The object detection and instance segmentation framework strictly follows Mask R-CNN, with a training schedule of  $1 \times (12$  epochs). We use a total batch size of 4/GPU, utilize the AdamW optimizer with a learning rate of  $1 \times 10^{-4}$  and a weight decay of 0.05. Images are resized and cropped into  $1333 \times 800$  for training and testing and maintain the ratio. Random horizontal flipping and resize with a ratio of 0.5 was applied for augmentation during training. This pre-training fine-tuning method is a commonly used strategy in ANNs.
+
+For ADE20K dataset, we utilize the MMSegmentation [73] framework. The training configuration strictly encompasses for 160,000 iterations. The batch size is set to 4/GPU, and the AdamW optimizer is used. The learning rate and weight decay parameters are tuned to  $2 \times 10^{-4}$  and 0.05, respectively. To speed up training, we warm up the model for 1.5k iterations with a linear decay schedule. All the experiments are conducted on 4 NVIDIA-A100 80GB GPUs.
+
+Table 4: Configurations of different Meta-SDT Variants.  
+
+<table><tr><td>Stage</td><td># Tokens</td><td colspan="3">Layer Specification</td><td>Tiny</td><td>Medium</td><td>Base</td></tr><tr><td rowspan="14">1</td><td rowspan="7">H/2×W/2</td><td rowspan="2" colspan="2">Downsampling</td><td>Conv</td><td colspan="3">7x7 stride 2</td></tr><tr><td>Dim</td><td>16</td><td>24</td><td>32</td></tr><tr><td rowspan="5">Conv-based SNN block</td><td rowspan="2">SepConv</td><td>DWConv</td><td colspan="3">7x7 stride 1</td></tr><tr><td>MLP ratio</td><td colspan="3">2</td></tr><tr><td rowspan="3">Channel Mixer</td><td>(1)Conv+Conv</td><td colspan="3">ε = 4</td></tr><tr><td>(2)MLP+MLP</td><td colspan="3">ε = 4/6</td></tr><tr><td>(3)MLP+Conv</td><td colspan="3">ε = 4</td></tr><tr><td rowspan="7">H/4×W/4</td><td rowspan="2" colspan="2">Downsampling</td><td>Conv</td><td colspan="3">3x3 stride 2</td></tr><tr><td>Dim</td><td>32</td><td>48</td><td>64</td></tr><tr><td rowspan="5">Conv-based SNN block</td><td rowspan="2">SepConv</td><td>DWConv</td><td colspan="3">7x7 stride 1</td></tr><tr><td>MLP ratio</td><td colspan="3">2</td></tr><tr><td rowspan="3">Channel Mixer</td><td>(1)Conv+Conv</td><td colspan="3">ε = 4</td></tr><tr><td>(2)MLP+MLP</td><td colspan="3">ε = 4/6</td></tr><tr><td>(3)MLP+Conv.</td><td colspan="3">ε = 4</td></tr><tr><td rowspan="8">2</td><td rowspan="8">H/8×W/8</td><td rowspan="2" colspan="2">Downsampling</td><td>Conv</td><td colspan="3">3x3 stride 2</td></tr><tr><td>Dim</td><td>64</td><td>96</td><td>128</td></tr><tr><td rowspan="6">Conv-based SNN block</td><td rowspan="2">SepConv</td><td>DWConv</td><td colspan="3">7x7 stride 1</td></tr><tr><td>MLP ratio</td><td colspan="3">2</td></tr><tr><td rowspan="3">Channel Mixer</td><td>(1)Conv+Conv</td><td colspan="3">ε = 4</td></tr><tr><td>(2)MLP+MLP</td><td colspan="3">ε = 4/6</td></tr><tr><td>(3)MLP+Conv,</td><td colspan="3">ε = 4</td></tr><tr><td colspan="2"># Blocks</td><td colspan="3">2</td></tr><tr><td rowspan="5">3</td><td rowspan="5">H/16×W/16</td><td rowspan="2" colspan="2">Downsampling</td><td>Conv</td><td colspan="3">3x3 stride 2</td></tr><tr><td>Dim</td><td>128</td><td>192</td><td>256</td></tr><tr><td rowspan="3">Transformer-based SNN block</td><td>SDSA</td><td>RepConv</td><td colspan="3">3x3 stride 1</td></tr><tr><td>Channel MLP</td><td>MLP ratio</td><td colspan="3">4</td></tr><tr><td colspan="2"># Blocks</td><td colspan="3">6</td></tr><tr><td rowspan="5">4</td><td rowspan="5">H/16×W/16</td><td rowspan="2" colspan="2">Downsampling</td><td>Conv</td><td colspan="3">3x3 stride 1</td></tr><tr><td>Dim</td><td>192</td><td>240</td><td>360</td></tr><tr><td rowspan="3">Transformer-based SNN block</td><td>SDSA</td><td>RepConv</td><td colspan="3">3x3 stride 1</td></tr><tr><td>Channel MLP</td><td>MLP ratio</td><td colspan="3">4</td></tr><tr><td colspan="2"># Blocks</td><td colspan="3">2</td></tr></table>
+
+Table 5: Hyper-parameters for pre-training on ImageNet-1K  
+
+<table><tr><td>Hyper-parameter</td><td>Settings</td><td>Hyper-parameter</td><td>Settings</td></tr><tr><td>Model size</td><td>T/M/B</td><td>Timestep</td><td>4</td></tr><tr><td>Epochs</td><td>200</td><td>Resolution</td><td>224*224</td></tr><tr><td>Batch size</td><td>1568</td><td>Optimizer</td><td>LAMB</td></tr><tr><td>Base learning rate</td><td>6e-4</td><td>Learning rate decay</td><td>Cosine</td></tr><tr><td>Warmup eopchs</td><td>10</td><td>Weight decay</td><td>0.05</td></tr><tr><td>Random augment</td><td>9/0.5</td><td>Mixup</td><td>None</td></tr><tr><td>Cutmix</td><td>None</td><td>Label smoothing</td><td>0.1</td></tr></table>
+
+# C Limitations
+
+This work presents several avenues for future exploration, such as how neuronal dynamics parameters influence ST-ERF in more dynamic and diverse SNNs. Given that one of SNN's major successes stems from its inherent membrane potential memory update mechanism, this represents a particularly worthwhile direction for deeper investigation. We will further explore the interactions between spiking neurons' neurodynamics and the networks' temporal response in the future. Nevertheless, this work provides a viable analytical framework for understanding SNN model behavior, with practical implications for architectural design across various levels of SNNs.
\ No newline at end of file
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+# Unveiling the Uncertainty in Embodied and Operational Carbon of Large AI Models through a Probabilistic Carbon Accounting Model
+
+Xiaoyang Zhang $^{1}$ , Fang He $^{1}$ , Yang Deng $^{1}$ , and Dan Wang $^{2}$
+
+$^{1}$ Department of Computing, Hong Kong Polytechnic University  
+ $^{2}$ Division of Environment and Sustainability, Hong Kong University of Science and Technology  
+{xiaoyang.zhang, fangf.he, yang2.deng}@connect.polyu.hk, wangdan@ust.hk
+
+# Abstract
+
+The rapid growth of large AI models has raised significant environmental concerns due to their substantial carbon footprint. Existing carbon accounting methods for AI models are fundamentally deterministic and fail to account for inherent uncertainties in embodied and operational carbon emissions. Our work aims to investigate the effect of these uncertainties on embodied and operational carbon footprint estimates for large AI models. We propose a Probabilistic Carbon Accounting Model (PCAM), which quantifies uncertainties in the carbon accounting of large AI models. We develop parameter models to quantify key components (processors, memory, storage) in the carbon footprint of AI models. To characterize the distribution of the parameters, we develop a carbon dataset by aggregating related data from various sources. Then, we generate the probabilistic distribution of the parameters from the collected dataset. We compare the performance of PCAM with LLMCarbon, the state-of-the-art carbon accounting method for large AI models. PCAM achieves  $\leq 7.44\%$  error compared to LLMCarbon's  $\leq 108.51\%$ .
+
+# 1 Introduction
+
+Large AI models have demonstrated high efficacy across various tasks. However, as model parameters and training datasets expand, the computational demands for training and deploying large-scale models grow significantly, leading to notable carbon emissions. For example, recent studies reveal that training advanced language models like Google's T5 generates  $40\%$  more carbon emissions than a transcontinental flight between San Francisco and New York [1]. The AI community is increasingly focused on developing methodologies to tackle social issues [2, 3, 4], particularly in promoting decarbonization and achieving carbon neutrality by 2050 [5, 6, 7, 8]. Carbon accounting, i.e., to estimate the carbon footprint of a product, is crucial for understanding the environmental impact of various operations and for developing strategies to reduce carbon footprints. By quantifying emissions, organizations can set reduction targets, comply with regulations, and demonstrate their commitment to sustainability. The carbon footprint of large AI models encompasses two components: operational carbon (i.e., emissions associated with electricity consumed) and embodied carbon (i.e., emissions associated with AI hardware manufacturing).
+
+While existing studies have initiated carbon accounting for AI models [5, 1], current methodologies exhibit two fundamental limitations. First, they predominantly focus on operational carbon, while neglecting or applying a simple model to account for embodied carbon. They utilize coarse-grained average embodied carbon values for one hardware class (e.g., 14nm CPU) directly from Life Cycle Assessment (LCA) reports, potentially introducing substantial inaccuracies. Second, all existing
+
+carbon models for AI models are single-deterministic and fail to capture the inherent uncertainty in the carbon footprint of AI models. The uncertainty in carbon accounting for AI models comes from (1) geotemporal differences in manufacturing, whereby identical device classes fabricated in different regions and periods embody distinct emissions due to different electricity carbon intensity during production; (2) efficiency evolution over time, as yield improvements, and fabrication energy efficiency gains; and (3) operating context variability, since the operational carbon of AI models can vary significantly depending on the location and time of the operation due to spatiotemporal variations in the carbon intensity of electricity.
+
+In this paper, we propose a Probabilistic Carbon Accounting Model (PCAM), which can accurately estimate the carbon footprint of large AI models by capturing the uncertainty in the carbon modeling. Specifically, we develop parameter models for each hardware component (e.g., processors, memory, storage) in the carbon footprint of large AI models. To construct the distribution of the parameters, we make an effort to develop a hardware and electricity dataset by aggregating data from diverse sources, including Environmental, Social, and Governance (ESG) reports from hardware manufacturers, power grid operator statistics, industry reports, and peer-reviewed research publications. Then, we implement a simple yet effective dual-stage distribution modeling (i.e., converting the related collected data into frequency histograms followed by Kernel Density Estimation (KDE)), to generate the continuous probability density functions of the parameters.
+
+We compare the performance of PCAM to LLMCarbon, the state-of-the-art carbon modeling method for large AI models. The evaluation is conducted based on four representative large AI models (XLM, T5, GPT-3, and Switch) [5, 9]. We compare the performance at key distribution percentiles (5th, 10th, 50th, 90th, and 95th) of PCAM's probability density function outputs against the ground truth based on the collected carbon dataset. Our PCAM outperforms LLMCarbon in both embodied and operational carbon on all large AI models. The results of PCAM show differences with the ground truth of only  $\leq 2.67\%$  for embodied carbon and  $\leq 7.44\%$  for operation carbon, demonstrating significantly higher accuracy compared to LLMCarbon ( $\leq 23.02\%$  and  $\leq 108.51\%$ , respectively).
+
+Our contributions can be summarised as follows:
+
+- We propose a new Probabilistic Carbon Accounting Model (PCAM) for large AI models that can generate distribution-based outputs of carbon footprint instead of singular values, allowing AI developers to make risk-aware and sustainable decisions.  
+- We characterize the uncertainty in the carbon footprint of large AI models across key hardware components by generating the distributions of AI hardware-related parameters through a simple yet effective dual-stage distribution modeling from the collected dataset based on KDE.  
+- We make an effort to develop a carbon dataset  ${}^{1}$  containing AI hardware-related parameters (e.g., yield, etc.) across different technology nodes from various technology reports, ESG reports, LCA reports, and carbon intensity of electricity data across regions from power grid operators.
+
+# 2 Background
+
+# 2.1 Large AI models carbon footprint
+
+The carbon footprint of large AI models includes two components, i.e., operational carbon and embodied carbon. The operational carbon of large AI models refers to the emissions stemming from the electricity consumption in the model training or inference. The embodied carbon arises from the manufacturing processes of the hardware used to execute the AI models. The development of large AI models requires not only substantial electricity but also extensive computing hardware resources. For example, training and deploying large AI models require high-performance GPUs (e.g., NVIDIA A100 [10]) and ML accelerators (e.g., Google TPU [11]). These devices are typically manufactured with large processor chips using advanced technologies (e.g., 5nm), which results in significant carbon emissions. This refers to the embodied carbon of large AI models, which is especially notable in the models operated using the latest high-performance devices. As the proportion of green energy in the power grid increases and more and more data centers adopt carbon-free energy, the running
+
+Table 1: Carbon emission factors (g/kWh) for energy sources [14]  
+
+<table><tr><td>Emission factors</td><td>Oil</td><td>Coal</td><td>Natural gas</td><td>Nuclear</td><td>Wind</td><td>Solar</td><td>Hydro</td><td>Geothermal</td><td>Biomass</td><td>Other</td></tr><tr><td>Life-cycle emissions</td><td>650</td><td>820</td><td>490</td><td>12</td><td>11</td><td>45</td><td>24</td><td>38</td><td>230</td><td>700</td></tr></table>
+
+carbon of large AI models will be greatly reduced, resulting in embodied carbon accounting for a non-negligible part of the large AI models' carbon footprint [12, 13].
+
+# 2.2 Carbon emission factor and carbon intensity
+
+The carbon emission factor (in g/kWh) is defined as the quantity of carbon emission per unit of electricity generated by a specific energy source. The carbon emission factors of brown sources (e.g., coal, gas, etc.) are much higher than those of green sources (e.g., wind, solar, etc.), as Table 1 shows.
+
+The carbon intensity of electricity is the carbon emission rate (in  $g/kWh$ ) when the electricity is generated, i.e., the total amount of carbon emitted (Gram) as against the electricity generation (Kilowatt - Hour). It is the weighted average of carbon emissions by each energy source due to the electricity generated by them. Mathematically, the carbon intensity of electricity generated at any time is as follows:
+
+$$
+\text {C a r b o n I n t e n s i t y} = \frac {\sum e f ^ {k} \times E ^ {k}}{\sum E ^ {k}} \tag {1}
+$$
+
+where  $ef^k$  is the carbon emission factor and  $E^k$  is the electricity generated by energy source  $k$ .
+
+# 3 Methodology
+
+# 3.1 Limitation of existing modeling techniques
+
+The existing carbon modeling techniques for AI models can be summarized as equation 2, where  $C_{model}$  denotes the total carbon footprint of the AI models, including embodied carbon ( $EC_{model}$ ) and operational carbon ( $OC_{model}$ ),  $t$  is the time the model occupies the hardware system,  $T$  is the lifetime of the hardware system,  $E$  is the electricity consumed by the hardware system, and  $CI$  is the carbon intensity of electricity.
+
+$$
+C _ {\text {m o d e l}} = E C _ {\text {m o d e l}} + O C _ {\text {m o d e l}} = t / T \cdot E C _ {\text {s y s t e m}} + E \cdot C I \tag {2}
+$$
+
+These existing modeling techniques for AI models' carbon footprint are fundamentally deterministic and estimate the carbon emission model as a single deterministic value. These methods typically rely on lifecycle assessment (LCA) reports that generalize embodied carbon estimates ( $EC_{system}$ ) across entire product categories (e.g., 28nm CPUs, LPDDR4 DRAM). However, this paradigm fails to account for critical spatial, temporal, and manufacturing variability that substantially impacts actual carbon emissions. Specifically, the existing carbon models for AI models fail to capture: (1) geotemporal differences in manufacturing, whereby the device instances in identical device classes are fabricated in different regions and periods, e.g., in the winter in Ireland (Intel) or in the summer in Taiwan (TSMC), embody distinct emissions due to different electricity carbon intensity during production; It is known that carbon intensity depends on the electricity generation process, which has spatial and temporal dynamics; (2) efficiency evolution over time, arising from continuous advancements in process yield and the energy efficiency of fabrication methods; and (3) operating context variability, since the operational carbon of AI models can vary significantly depending on the when and where the AI models depoly because of spatiotemporal variations in the carbon intensity of electricity. These characteristics introduce uncertainty into carbon estimates for AI models; however, current carbon accounting methods for AI models fail to account for these uncertainties.
+
+# 3.2 Embodied carbon modeling
+
+We extend the existing carbon modeling methodology of AI models by giving parameter models for each hardware component and characterizing the uncertainty. Specifically, we analyze the parameter
+
+![](images/cf16d49924fce353aa6cfccb9669cf6bf454e3bf5b70be351d95cbce874e8a30.jpg)  
+Figure 1: Histograms of hourly carbon intensity data (monthly data in China) from five major IC production regions between 2021-2023 with their individual kernel density estimates (KDEs).
+
+![](images/7322d793dda3063be789e2212c597fee5135f5119481c770136afee7666d3df8.jpg)
+
+![](images/0135e0063f3628172d45091633f1ea7f791efa8f21ac39943c28fa78901d0857.jpg)
+
+![](images/130c7a0466441cd90edb739ac7f3374bc09131465b9df81b67f71af7ba520b4a.jpg)
+
+![](images/71d04b1b2a74ec6b9eaa8d67dd39c8367d38f557a95b5f5f84292381036948d5.jpg)
+
+![](images/e35833f988da1d0eccc6ba743155ce9fc848acbbe99ec953f5e7e74a36b94dbd.jpg)  
+Figure 2: The distribution of carbon Intensity of processors, memory (DRAM), and storage (DRAM) fabrication 2021-2023.
+
+![](images/92d92824ebec946459558fb49996ad29288036151fb0039785c1797f949cd1e4.jpg)
+
+value evolution in the fabrication process by collecting data from various sources (e.g., ESG reports, technology reports, power grid operators, etc.). We leverage KDE to generate the distributions of the parameters based on the data we collect. KDE is a standard non-parametric estimation technique used to estimate the probability density function of a random variable by applying a Gaussian kernel weighting technique to data [15]. It is used when the behavior of a sparse data sample is extrapolated to resample data to a larger sample size.
+
+# 3.2.1 Embodied carbon of processors
+
+We model the embodied carbon of large AI models caused by processors as follows: (1) the operation time of large AI models on processors, denoted as  $t_p$ , and the processor's lifetime, denoted as  $T_p$ . (2) the carbon emissions from electricity used during the fabrication process, calculated by multiplying the Electricity consumption Per unit of die Size (EPS) by the carbon intensity of electricity during manufacturing  $CI_m$ ; (3) the emissions associated with the raw materials, denoted as  $MPS$  (Material Per Size); and (4) the carbon released by gas (e.g., fluorinated compounds) in the manufacturing process denoted as  $GPS$  (Gas Per Size); (5) the die size of the processor, denoted as  $DizeSize$ ; (6) fab yield denoted as  $Y$ . Then, we can model the embodied carbon of processors for large AI models as follows:
+
+$$
+E C _ {\text {m o d e l}} ^ {p} = \frac {t _ {p} \cdot \text {D i e S i z e}}{T _ {p} \cdot \tilde {Y}} \cdot \left(C \tilde {I} _ {m} \cdot E \tilde {P} S + G P S + M P S\right) \tag {3}
+$$
+
+Here, we denote  $\tilde{Y}$  as the probabilistic model of yield  $Y$ . Similarly, we have  $C\tilde{I}_m$  and  $E\tilde{P}S$ . We will give the detailed uncertainty characterization of the three parameters in the following.
+
+Carbon intensity distribution in the manufacturing process.
+
+$CI_{m}$  denotes the carbon intensity of electricity consumed during semiconductor manufacturing, a critical parameter determined by the energy source mix (e.g., solar, wind, coal, etc.) employed in electricity generation. This metric exhibits spatiotemporal variability due to fluctuations in both manufacturing schedules and geographic locations. The inherent uncertainty in carbon intensity primarily comes from temporal variations throughout annual cycles, driven by seasonal patterns in renewable energy generation and fluctuating electricity demand. When the manufacturing location is specified, regional fabrication facilities demonstrate distinct carbon intensity distribution derived from historical carbon intensity data from 2021 to 2023, as illustrated in Figure 1.
+
+Table 2: Global wafer fabrication capacity by technology by regions in 2020.  
+
+<table><tr><td>Regions</td><td>US</td><td>China</td><td>Taiwan</td><td>Korea</td><td>Japan</td><td>Others</td></tr><tr><td>&lt;10nm Logic</td><td>0%</td><td>0%</td><td>69%</td><td>31%</td><td>0%</td><td>0%</td></tr><tr><td>10-22nm Logic</td><td>28%</td><td>6%</td><td>40%</td><td>9%</td><td>5%</td><td>12%</td></tr><tr><td>28nm+ Logic</td><td>8%</td><td>33%</td><td>30%</td><td>5%</td><td>10%</td><td>14%</td></tr><tr><td>Memory(DRAM)</td><td>3%</td><td>18%</td><td>20%</td><td>52%</td><td>7%</td><td>0%</td></tr><tr><td>Storage(NAND)</td><td>3%</td><td>26%</td><td>4%</td><td>30%</td><td>30%</td><td>7%</td></tr></table>
+
+Memory can usually be manufactured by multiple manufacturers (e.g., SK Hynix, Samsung, Micron, etc.), unlike processor manufacturers, which have a monopoly (e.g., most GPUs are manufactured by TSMC). For the scenario involving uncertain manufacturing locations, the region selection can be modeled as a discrete random variable. The probability distribution across geographical regions is weighted according to their respective contributions to global IC production capacity for specific process nodes [16]. As detailed in Table 2 from industry reports [17], these capacity allocations serve as probabilistic weights for regional selection. The composite probability distribution for each process node is constructed through a mixture modeling approach, incorporating the following components: (1) regional KDE: non-parametric distributions developed using historical carbon intensity data (2021-2023) for each major production region; (2) capacity-weighted sampling: a Monte Carlo sampling strategy where regional selection probabilities correspond to normalized production capacities; (3) composite distribution formation: aggregation of weighted regional samples through kernel density smoothing, as visualized in Figure 2. This enables probabilistic modeling of carbon intensity while accounting for both geographical production distributions and temporal energy mix variations.
+
+Yield Distribution. Semiconductor manufacturing yield is defined as the ratio of defect-free semiconductor dies on a wafer to the total number of dies on that wafer. The parameter exhibits inherent uncertainty that stems primarily from temporal variations in defect density observed across fabrication facilities. To systematically investigate this phenomenon, we employ TSMC's historical defect density data across four distinct process nodes [18], coupled with the application of the Poisson yield model [19] for temporal yield computation. Following empirical data collection, we generate yield histograms and subsequently apply KDE [20] to derive comprehensive probability density functions characterizing yield distributions per unit die size, as visually presented in Figure 3.
+
+![](images/aa82c0ef730385efe2eb5a4a9b1772aaf66cf2185ff455692c99408d3f4015d0.jpg)  
+Figure 3: The Kernel Density Estimation of Yield  $\tilde{Y}$
+
+![](images/f010ff725e5cd0f75a8c467e3209eb9816a00eb96196162741739c0fc9b53255.jpg)  
+Figure 4: The kernel density estimation of  $E\tilde{P}S$  across different technology nodes.
+
+EPS Distribution. The inherent uncertainty in  $EPS$  is from temporal fluctuations in process energy efficiencies across different fabrication stages. We build the EPS distributions based on the EPS values from the STEC model [13], the ACT model [12], the imec model [21], and the annual energy efficiency improvements data from across the TSMC ESG reports [22] across various technology process nodes (e.g., 5nm, 7nm). To construct the distributions, the analytical procedure involved three key transformations: First, energy efficiency metrics for each process node are temporally normalized relative to their baseline year values. Second, raw  $EPS$  values are adjusted through division by these normalized efficiency coefficients to account for technological advancements. Finally, we implement a dual-stage distribution modeling approach, i.e., converting the processed data into frequency histograms followed by KDE to derive continuous probability density functions of  $\tilde{EPS}$ , as visualized in Figure 4.
+
+# 3.2.2 Embodied carbon of memory
+
+We model the embodied carbon of large AI models caused by memory as follows: (1) the operation time of large AI models on memory  $t_m$  and the memory's lifetime  $T_m$ ; (2) the carbon released by the electricity consumed during the process of manufacturing the memory. This is calculated by multiplying the electricity consumption per unit of size ( $EPS$ ) by the carbon intensity of a power grid ( $CI_m$ ) then divided by bit density ( $BD$ , in  $GB/mm^2$ ); (3) the carbon released independent of electricity, such as raw materials, distribution, and packaging, denoted as  $\alpha_m$ ; and (4) the capacity of memory  $C_m$ . Then, we can model the embodied carbon of memory for large AI models as follows:
+
+$$
+E C _ {\text {m o d e l}} ^ {m} = t _ {m} / T _ {m} \cdot C _ {m} \cdot \left(C \tilde {I} _ {m} \cdot E \tilde {P} S / B D + \alpha_ {m}\right) \tag {4}
+$$
+
+Here, the distribution of  $C\tilde{I}_m$  and  $E\tilde{P}S$  for memory is similar to that in processors. We develop the distribution based on the data we collect from Hynix [23] as Figure 2 shows.
+
+# 3.2.3 Embodied carbon of storage
+
+We model the embodied carbon of large AI models caused by storage as follows: (1) the operation time of large AI models on storage  $t_s$  and the storage's lifetime  $T_s$ ; (2) the carbon emissions from the process of manufacturing the storage, which is determined by multiplying the electricity consumed per unit of Giga-Byte during the manufacturing process (EPG) by the carbon intensity  $CI_m$ ; (3) the carbon released independent of electricity such as raw materials, denoted as  $\alpha_S$ .  $\alpha_S$  can be found in the industry reports [24]; and (4) the capacity of storage, denoted as  $C_s$ . Then, we can model the embodied carbon of storage for large AI models and omit the distribution derivation as follows:
+
+$$
+E C _ {\text {m o d e l}} ^ {s} = t _ {s} / T _ {s} \cdot C _ {s} \cdot \left(C \tilde {I} _ {m} \cdot E P G + \alpha_ {S}\right) \tag {5}
+$$
+
+# 3.3 Operational carbon
+
+The operational carbon footprint of large AI models  $(OC_{model})$  refers to the carbon emissions caused by the electricity used during operation. It is determined by the electricity consumed by the model  $(E_o)$  and the carbon intensity of the electricity  $(CI_o)$  as follows:
+
+$$
+O C _ {\text {m o d e l}} = E _ {o} \cdot \hat {C I} _ {o} \tag {6}
+$$
+
+where  $\hat{C}I_{o}$  represents the distribution of the carbon intensity of that electricity. This uncertainty comes from the uncertainty of the time and location of model training or inference.  $E_{o}$  can be recorded by hardware during the model operation, or it can be estimated by the following equations:
+
+$$
+E _ {o} = \sum_ {i \in H a r d w a r e S e t} \left(P _ {i} \cdot e f f _ {i} \cdot n _ {i} \cdot t _ {i}\right) \cdot P U E \tag {7}
+$$
+
+where  $P_{i}$  represents the peak power of hardware  $i$ ;  $eff_{i}$  denotes the efficiency of hardware  $i$  that can be estimated by the hardware efficiency model [1];  $n_{i}$  is the number of hardware  $i$ ; and  $t_{i}$  refers to the execution time for hardware  $i$  that can be estimated according to the existing flop model [1]. These hardware units include CPUs, GPUs, memory, storage, and others.
+
+# 4 Evaluation
+
+# 4.1 Setup
+
+To comprehensively evaluate PCAM for the carbon footprint of large AI models, we make an effort to develop a hardware and electricity dataset. This dataset includes hardware-related parameters (e.g., EPS, EPG, MPS, BD, etc.) and electricity-related parameters (e.g., CI), which is constructed by aggregating data from diverse sources, including ESG reports from hardware manufacturers, power grid operator statistics, industry reports, and peer-reviewed research publications
+
+Table 3: The hardware information about XLM.  
+
+<table><tr><td>Hardware</td><td>Number</td><td>Die Size/Unit</td><td>Technology Nodes</td></tr><tr><td>GPU</td><td>512</td><td>8.15 cm2</td><td>12nm</td></tr><tr><td>CPU</td><td>64</td><td>1.47 cm2</td><td>16nm</td></tr><tr><td>Storage(SSD)</td><td>64</td><td>32TB</td><td>Seagate Nytro 3332</td></tr><tr><td>Memory(DRAM)</td><td>64</td><td>256GB</td><td>10nm DDR4</td></tr></table>
+
+For performance evaluation, we conduct a comparative analysis between PCAM and LLMCarbon, the current state-of-the-art carbon model for large AI models. The evaluation is performed using published training data from four representative large AI models (XLM, T5, GPT-3, and Switch) [5, 9]. We compare the performance at key distribution percentiles (5th, 10th, 50th, 90th, and 95th) by resampling PCAM's probability distribution outputs against the ground truth based on the collected carbon dataset. The evaluation focuses on the training processes of large AI models, as the accounting methodology is identical for both training and inference. Please note that the carbon footprint can not be measured by sensors due to its inherent characteristic. The "ground truth" in the following evaluation is the accounting result based on the carbon dataset we collected without KDE.
+
+# 4.2 Embodied carbon evaluation
+
+We conduct a comparative analysis of embodied carbon accounting between PACM and LLMcarbon, utilizing the published XLM training data [5]. Notably, to the best of our knowledge, the XLM dataset represents the only publicly available information detailing hardware infrastructure associated with embodied carbon for large AI models training. The hardware configuration, comprising 64 servers with detailed specifications presented in Table 3, is employed for XLM's training process. The training duration lasts 20.4 days, with all hardware components assumed to have a standard operational lifetime of 5 years [5].
+
+The embodied carbon accounting performance of PCAM and LLMCarbon is compared across different hardware components and statistical percentiles, as detailed in Table 4. For total embodied carbon, PCAM demonstrates great alignment with ground truth values across all percentiles, maintaining a maximum deviation of only  $2.67\%$  at the 10th percentile and achieving remarkable precision at the median  $(0.48\%)$  deviation). In contrast, LLMCarbon shows substantially larger deviations ranging from  $6.03\%$  at the median to  $23.02\%$  at the
+
+Table 4: The comparison between LLMCarbon and PCAM on embodied carbon accounting.  
+
+<table><tr><td rowspan="2">Models</td><td colspan="5">Total Embodied Carbon (kg) at each Percentile</td></tr><tr><td>5th</td><td>10th</td><td>Median</td><td>90th</td><td>95th</td></tr><tr><td>Ground Truth</td><td>229.17</td><td>236.27</td><td>279.75</td><td>331.32</td><td>352.15</td></tr><tr><td>PCAM</td><td>235.2</td><td>242.74</td><td>281.09</td><td>339.95</td><td>359.93</td></tr><tr><td>PCAM Δ</td><td>2.56%</td><td>2.67%</td><td>0.48%</td><td>2.54%</td><td>2.16%</td></tr><tr><td>LLMCarbon</td><td></td><td></td><td>297.71</td><td></td><td></td></tr><tr><td>LLMCarbonΔ</td><td>23.02%</td><td>20.64%</td><td>6.03%</td><td>11.29%</td><td>18.29%</td></tr><tr><td colspan="6">CPU Embodied Carbon (kg) at each Percentile</td></tr><tr><td>Ground Truth</td><td>1.14</td><td>1.18</td><td>1.49</td><td>2.04</td><td>2.18</td></tr><tr><td>PCAM</td><td>1.16</td><td>1.19</td><td>1.48</td><td>2.01</td><td>2.12</td></tr><tr><td>PCAMΔ</td><td>1.72%</td><td>0.81%</td><td>0.52%</td><td>1.45%</td><td>2.61%</td></tr><tr><td>LLMCarbon</td><td></td><td></td><td>1.55</td><td></td><td></td></tr><tr><td>LLMCarbonΔ</td><td>26.50%</td><td>23.89%</td><td>4.08%</td><td>31.47%</td><td>40.25%</td></tr><tr><td colspan="6">GPU Embodied Carbon (kg) at each Percentile</td></tr><tr><td>Ground Truth</td><td>98.69</td><td>101.36</td><td>133.76</td><td>189.11</td><td>211.05</td></tr><tr><td>PCAM</td><td>96.97</td><td>99.42</td><td>132.29</td><td>187.62</td><td>209.39</td></tr><tr><td>PCAMΔ</td><td>1.77%</td><td>1.95%</td><td>1.11%</td><td>0.79%</td><td>0.79%</td></tr><tr><td>LLMCarbon</td><td></td><td></td><td>141.68</td><td></td><td></td></tr><tr><td>LLMCarbonΔ</td><td>66.85%</td><td>28.46%</td><td>5.59%</td><td>33.48%</td><td>48.96%</td></tr><tr><td colspan="6">DRAM Embodied Carbon (kg) at each Percentile</td></tr><tr><td>Ground Truth</td><td>7.97</td><td>8.15</td><td>10.39</td><td>14.47</td><td>15.3</td></tr><tr><td>PCAM</td><td>7.83</td><td>8.07</td><td>10.26</td><td>14.34</td><td>15.21</td></tr><tr><td>PCAMΔ</td><td>1.79%</td><td>0.99%</td><td>1.27%</td><td>0.91%</td><td>0.59%</td></tr><tr><td>LLMCarbon</td><td></td><td></td><td>10.85</td><td></td><td></td></tr><tr><td>LLMCarbonΔ</td><td>26.54%</td><td>24.88%</td><td>4.24%</td><td>33.36%</td><td>41.01%</td></tr><tr><td colspan="6">SSD Embodied Carbon (kg) at each Percentile</td></tr><tr><td>Ground Truth</td><td>105.99</td><td>110.23</td><td>123.62</td><td>145.17</td><td>147.74</td></tr><tr><td>PCAM</td><td>108.25</td><td>111.47</td><td>122.76</td><td>147.05</td><td>149.65</td></tr><tr><td>PCAMΔ</td><td>2.09%</td><td>1.11%</td><td>0.70%</td><td>1.28%</td><td>1.28%</td></tr><tr><td>LLMCarbon</td><td></td><td></td><td>125.64</td><td></td><td></td></tr><tr><td>LLMCarbonΔ</td><td>15.64%</td><td>12.27%</td><td>1.61%</td><td>15.54%</td><td>17.59%</td></tr></table>
+
+5th percentile, demonstrating that the single deterministic model fails to capture uncertainty in embodied carbon accounting.
+
+Component-wise analysis shows that PCAM outperforms LLMCarbon at each component. For CPU carbon accounting, PCAM errors are below  $2.62\%$  versus LLMCarbon's around  $4\% - 40\%$  deviations. GPU calculations show PCAM's robust performance with less than  $2\%$  deviations across all percentiles, contrasted by LLMCarbon's substantial overestimations reaching around  $67\%$  at the 5th percentile. The DRAM and SSD comparison follows similar trends, with PCAM errors remaining
+
+Table 5: The comparison between LLMCarbon and PCAM on different large AI models for operational carbon accounting.  
+
+<table><tr><td rowspan="2" colspan="2">AI models</td><td rowspan="2">Carbon Models</td><td colspan="5">Operational Carbon in the Spatial Dimension (ton)</td><td colspan="5">Operational Carbon in the Temporal Dimension (ton)</td></tr><tr><td>5th per.</td><td>10th per.</td><td>Median</td><td>90th per.</td><td>95th per.</td><td>5th per.</td><td>10th per.</td><td>Median</td><td>90th per.</td><td>95th per.</td></tr><tr><td rowspan="5">XLM</td><td rowspan="5">Training Day: 20.4; PUE: 1.1; Ave. Power: 342 kw; Num. of device: 512</td><td>Ground Truth</td><td>2.87</td><td>4.93</td><td>28.19</td><td>56.77</td><td>62.74</td><td>33.41</td><td>34.67</td><td>39.78</td><td>45.07</td><td>46.41</td></tr><tr><td>LLMCarbon</td><td></td><td></td><td>30.09</td><td></td><td></td><td></td><td></td><td>38.86</td><td></td><td></td></tr><tr><td>LLMCarbon Δ</td><td>90.46%</td><td>83.62%</td><td>6.31%</td><td>88.67%</td><td>108.51%</td><td>16.18%</td><td>13.02%</td><td>2.37%</td><td>13.07%</td><td>16.43%</td></tr><tr><td>PCAM</td><td>2.79</td><td>5.13</td><td>29.08</td><td>57.13</td><td>63.71</td><td>33.59</td><td>34.19</td><td>39.07</td><td>45.51</td><td>46.03</td></tr><tr><td>PCAM Δ</td><td>2.87%</td><td>3.90%</td><td>3.06%</td><td>0.63%</td><td>1.52%</td><td>0.54%</td><td>1.40%</td><td>1.82%</td><td>0.97%</td><td>0.83%</td></tr><tr><td rowspan="5">T5</td><td rowspan="5">Training Day: 20; PUE: 1.12; Ave. Power: 310 kw; Num. of device: 512</td><td>Ground Truth</td><td>2.6</td><td>4.46</td><td>25.51</td><td>51.36</td><td>56.77</td><td>30.23</td><td>31.37</td><td>35.99</td><td>40.78</td><td>41.99</td></tr><tr><td>LLMCarbon</td><td></td><td></td><td>27.23</td><td></td><td></td><td></td><td></td><td>37.07</td><td></td><td></td></tr><tr><td>LLMCarbon Δ</td><td>90.45%</td><td>83.62%</td><td>6.32%</td><td>88.62%</td><td>108.48%</td><td>16.19%</td><td>13.03%</td><td>2.91%</td><td>13.06%</td><td>16.41%</td></tr><tr><td>PCAM</td><td>2.42</td><td>4.74</td><td>25.68</td><td>51.19</td><td>57.13</td><td>30.91</td><td>31.98</td><td>36.85</td><td>40.13</td><td>42.83</td></tr><tr><td>PCAM Δ</td><td>7.44%</td><td>5.91%</td><td>0.66%</td><td>0.33%</td><td>0.63%</td><td>2.20%</td><td>1.91%</td><td>2.33%</td><td>1.62%</td><td>1.96%</td></tr><tr><td rowspan="5">GPT3</td><td rowspan="5">Tarinin Day: 14.8; PUE:1.1; Ave. Power: 330 kw; Num. of device:10K</td><td>Ground Truth</td><td>39.36</td><td>67.53</td><td>385.49</td><td>776.22</td><td>857.85</td><td>456.83</td><td>474.09</td><td>543.94</td><td>616.31</td><td>634.55</td></tr><tr><td>LLMCarbon</td><td></td><td></td><td>411.49</td><td></td><td></td><td></td><td></td><td>559.09</td><td></td><td></td></tr><tr><td>LLMCarbon Δ</td><td>90.43%</td><td>83.59%</td><td>6.32%</td><td>88.64%</td><td>108.47%</td><td>16.19%</td><td>13.03%</td><td>2.71%</td><td>13.07%</td><td>16.41%</td></tr><tr><td>PCAM</td><td>37.75</td><td>69.71</td><td>390.21</td><td>769.02</td><td>869.37</td><td>449.87</td><td>469.99</td><td>535.68</td><td>605.95</td><td>626.12</td></tr><tr><td>PCAM Δ</td><td>4.26%</td><td>3.13%</td><td>1.21%</td><td>0.94%</td><td>1.33%</td><td>1.55%</td><td>0.87%</td><td>1.54%</td><td>1.71%</td><td>1.35%</td></tr><tr><td rowspan="5">Switch</td><td rowspan="5">Tarinin Day: 27; PUE:1.1; Ave. Power: 245 kw; Num. of device: 1K</td><td>Ground Truth</td><td>5.33</td><td>9.14</td><td>52.21</td><td>105.13</td><td>116.19</td><td>61.87</td><td>64.21</td><td>73.67</td><td>83.47</td><td>85.94</td></tr><tr><td>LLMCarbon</td><td></td><td></td><td>55.73</td><td></td><td></td><td></td><td></td><td>74.82</td><td></td><td></td></tr><tr><td>LLMCarbon Δ</td><td>90.44%</td><td>83.60%</td><td>6.32%</td><td>88.64%</td><td>108.49%</td><td>16.19%</td><td>13.02%</td><td>1.54%</td><td>13.07%</td><td>16.42%</td></tr><tr><td>PCAM</td><td>5.01</td><td>9.49</td><td>52.23</td><td>105.04</td><td>118.09</td><td>61.18</td><td>64.88</td><td>73.08</td><td>83.95</td><td>85.08</td></tr><tr><td>PCAM Δ</td><td>6.39%</td><td>3.69%</td><td>0.04%</td><td>0.09%</td><td>1.61%</td><td>1.13%</td><td>1.03%</td><td>0.81%</td><td>0.57%</td><td>1.01%</td></tr></table>
+
+below  $1.8\%$  in DRAM  $(2.1\%)$  in SSD) versus LLMCarbon's around  $25\% -40\%$  in DRAM (around  $2\% -18\%$  in SSD) deviations.
+
+Our analysis reveals that GPUs exhibit a substantially wider absolute uncertainty range (98.69 - 211.05 kg,  $\Delta = 114.36$  kg) in embodied carbon compared to the cumulative variance of other hardware components, establishing them as the dominant contributor to total embodied carbon variability. This disproportionate impact stems from two key manufacturing characteristics: First, GPU production employs advanced semiconductor fabrication processes that require significantly higher energy, making its embodied carbon particularly sensitive to variations in the carbon intensity of electricity. Second, the combination of larger die sizes and complex architectures results in lower yields with greater variation, amplifying the uncertainty in the embodied carbon of GPUs.
+
+Overall, this significant variance underscores critical limitations in conventional deterministic carbon accounting models like LLMCarbon and highlights the necessity of probabilistic modeling that explicitly accounts for parameter uncertainties in carbon modeling. These findings challenge the validity of current sustainability certification paradigms that rely on deterministic embodied carbon accounting. We propose that comprehensive environmental impact assessments for large AI models should adopt distribution-aware evaluation methodologies and report confidence intervals alongside central estimates to enable more informed decision-making in sustainable large AI models development.
+
+# 4.3 Operation carbon evaluation
+
+Table 5 presents the operation carbon evaluation results of PCAM for the four large AI models, XLM, T5, GPT3, and Switch, based on their published training information [9]. The training information is listed in Table 5, where "avg. power (W)" conveys the average system power per computing device (e.g., GPU, TPU, DRAM, etc.); "Num. of device" is the total number of computing devices. We estimate the electricity consumption of large AI models based on the training data and subsequently calculate their operational carbon footprint.
+
+To comprehensively evaluate model performance, we conducted comparative analyses across spatial and temporal dimensions. In the spatial analysis, we maintain a fixed training date while considering global geographical variations (across 90 regions). Conversely, the temporal analysis fixed the training location (the USA) while accounting for time-dependent factors (across 2021-2023 years). Overall, PCAM outperforms LLMCarbon  $(0.09\% -7.44\%$  vs.  $1.54\% -108.51\%$  deviations) in the operational carbon accounting for each AI model. Besides, the absolute error of the LLMCarbon is much larger than that of PCAM, e.g.,  $372.13t$  vs.  $3.6t$  for GPT3 under the 5th percentile in the spatial dimension.
+
+Besides, spatial variations demonstrate particularly pronounced differences. For example, the deviations of LLMCarbon under the spatial dimension are from  $90.43\%$  (5th percentile) to  $108.51\%$  (95th percentile), while that is from  $16.18\%$  to  $16.43\%$  under the temporal dimension.
+
+![](images/61499c59d7464ce066a1012afc927eb39dd04c4365cf14cd812625a204b7c2ab.jpg)  
+Figure 5: The comparison between LLMCarbon and PCAM on operational carbon accounting in the US, IT, and DE.
+
+This contrast highlights that geographical factors introduce greater uncertainty in operational carbon accounting than temporal variations.
+
+# 5 Case Study
+
+PCAM delivers practical value by enabling risk-aware decision-making through carbon budget overrun probability assessments, ensuring realistic sustainability targets. It facilitates sensitivity analysis to pinpoint key uncertainty drivers, guiding operational and hardware prioritization while aiding manufacturers in green production optimization. For operational choices, PCAM helps developers schedule computing tasks based on temporal and regional grid carbon intensity. It also supports multi-source hardware comparisons by evaluating embodied carbon distributions to inform procurement. Furthermore, PCAM encourages industry-wide engagement by quantifying supply chain impacts and promoting transparency. Its probabilistic reporting enhances credibility for compliance while aligning with carbon-aware computing trends by providing confidence bounds for auditing and long-term planning.
+
+Figure 5 provides a visual comparison of operational carbon estimates between PCAM and LLM-Carbon for XLM training in three regions: California (US), Germany (DE), and Italy (IT). While LLMCarbon suggests the US produces lower emissions than IT and IT has lower emissions than DE, PCAM reveals more nuanced probabilistic outcomes: there is an  $81.66\%$  probability of IT's operational carbon footprint exceeding that of the US and a  $33.06\%$  probability of surpassing DE's emissions. This probabilistic approach enables PCAM to provide multi-dimensional insights, empowering AI developers to make more informed sustainability decisions.
+
+# 6 Related Work
+
+The current methodologies for carbon footprint estimation of large AI models predominantly concentrate on operational carbon, which is calculated as the product of electricity consumed and carbon intensity. Most studies predominantly focus on tracking or estimating the electricity consumed. A subset of studies has developed software-integrated instrumentation to monitor real-time CPU/GPU power utilization during inference or training phases [25, 26, 27, 28], while alternative approaches derive energy estimates through the parameters of and hardware (e.g., thermal design power) [29, 30]. However, these methodologies systematically neglect the embodied carbon accounting associated with AI hardware infrastructure, which is non-negligible in the carbon footprint of large AI models, as power grids decarbonize and data centers increasingly adopt zero-carbon energy sources.
+
+Regarding embodied carbon modeling, existing frameworks such as SustainableAI [5] estimated the embodied carbon of large AI models through manufacturer-reported emission factors for hardware components, which is an average value. LLMCarbon [1] adopts a deterministic parametric model to estimate the embodied carbon for processors and average values for storage and memory in ESG
+
+reports. However, these approaches lead to oversimplified modeling constrained by two critical limitations: (1) fail to capture the effect of temporal and spatial specificity in hardware instance (e.g., 28nm CPU made in winter in China vs. 28nm CPU made in summer in the US); (2) fail to capture the effect of heterogeneous manufacturing configurations within device class (e.g., yield and fabrication efficiency improvement for certain processors). These characteristics lead to uncertainty in carbon estimates in large AI models, but current large AI models' carbon models fail to capture this uncertainty.
+
+# 7 Limitations and Discussions
+
+This study examines the carbon footprint of large AI models across four life-cycle stages: (1) Hardware manufacturing, referring to emissions generated during the production of core computing components; (2) Hardware transport, encompassing emissions from shipping hardware to end-users; (3) Operational use, comprising emissions resulting from software execution, primarily due to electricity consumption; and (4) End-of-life processing, involving emissions associated with hardware recycling. Among these, the operational use phase corresponds to the operational carbon of large AI models, while the remaining stages contribute to their embodied carbon. Hardware manufacturing and operational use are the dominant sources of emissions, for instance, over  $97\%$  in the case of a Dell R740 server [31]. In contrast, transport and end-of-life stages make negligible contributions and are thus excluded from the current scope. Within hardware manufacturing, we focus specifically on emissions from computer hardware production, such as those arising from material extraction, energy use, and chemical processing, while ancillary factors like infrastructure, cooling systems, and human labor are not considered in this study. We acknowledge that these omitted factors also play a significant role in the overall embodied carbon footprint. A full quantification of their associated uncertainties requires interdisciplinary collaboration and is reserved for future work.
+
+PCAM only considers the uncertainty of some parameters due to the lack of relevant carbon data. Inaccuracy comes from using averaged/aggregated data in various aspects (e.g., raw material, supply chain, packaging, etc.). Intrinsically, carbon data is unique in that it cannot be directly measured by sensors (unlike electricity) and can only be calculated through accounting models, which leads to the fact that perfect carbon modeling is difficult. The increasing focus on carbon emissions considerations among hardware producers drives substantial enhancements in carbon data quality. This trend leads to more accurate and reliable accounting models for the carbon footprint of large AI models.
+
+# 8 Conclusion
+
+In this paper, we propose PCAM, a Probabilistic Carbon Accounting Model for large AI Models, which systematically captures and quantifies the uncertainties inherent in both embodied and operational carbon emissions. Unlike existing deterministic approaches such as LLMCarbon, PCAM provides distribution-based carbon footprint estimates, enabling AI developers and policymakers to make risk-aware decisions. By moving beyond point estimates to probabilistic carbon accounting, PCAM supports more robust environmental impact assessments, facilitates compliance with emerging carbon regulations, and promotes transparency in AI sustainability reporting. Future work will expand PCAM to incorporate additional lifecycle stages and supply chain factors, fostering a more holistic and actionable framework for carbon-aware AI development. To facilitate the decarbonization of large AI models, we are open-sourcing the carefully curated carbon dataset compiled in this study, thereby providing the community with the necessary tools to quantify, understand, and mitigate the environmental impact of AI model development. We strongly call for collective action across the industry to enhance the transparency of carbon-related data, including the carbon intensity of power grids and hardware-related data. Establishing a culture of open and standardized emissions reporting is fundamental to driving systemic decarbonization. It enables the establishment of credible baselines, pinpoints optimization hotspots, and paves the way for accountable and verifiable progress across the entire AI ecosystem.
+
+# Acknowledgements
+
+Dan Wang's work is supported in part by RGC GRF 15200321, 15201322, 15230624, 15239925, ITC ITF-ITS/056/22MX, ITS/052/23MX, and PolyU 1-CDKK, G-SAC8, K-ZYAP.
+
+# References
+
+[1] Ahmad Faiz, Sotaro Kaneda, Ruhan Wang, Rita Osi, Prateek Sharma, Fan Chen, and Lei Jiang. Llmcarbon: Modeling the end-to-end carbon footprint of large language models. In The Twelfth International Conference on Learning Representations. ICLR, 2024.  
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+
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+
+Answer: [NA].
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+
+Table 6: The comparison between PCAM and STEC on the embodied carbon accounting of XLM.  
+
+<table><tr><td rowspan="2">Models</td><td colspan="5">Total Embodied Carbon (kg) at each Percentile</td></tr><tr><td>5th</td><td>10th</td><td>Median</td><td>90th</td><td>95th</td></tr><tr><td>Ground Truth</td><td>229.17</td><td>236.27</td><td>279.75</td><td>331.32</td><td>352.15</td></tr><tr><td>PCAM</td><td>235.20</td><td>242.74</td><td>281.09</td><td>339.95</td><td>359.93</td></tr><tr><td>PCAMΔ</td><td>2.56%</td><td>2.67%</td><td>0.48%</td><td>2.54%</td><td>2.16%</td></tr><tr><td>STEC</td><td>283.01</td><td>286.00</td><td>294.90</td><td>303.02</td><td>305.01</td></tr><tr><td>STECΔ</td><td>23.49%</td><td>21.05%</td><td>5.42%</td><td>8.55%</td><td>13.39%</td></tr><tr><td>LLMCarbon</td><td></td><td></td><td>297.71</td><td></td><td></td></tr><tr><td>LLMCarbonΔ</td><td>23.02%</td><td>20.64%</td><td>6.03%</td><td>11.29%</td><td>18.29%</td></tr></table>
+
+# Appendix
+
+We provide additional experiments to compare PCAM with STEC [13], which is an embodied carbon accounting model for computer systems. STEC incorporates spatiotemporal variability in the carbon intensity of electricity used during hardware manufacturing. However, it remains a fundamentally deterministic model and does not probabilistically account for other key sources of uncertainty, such as geotemporal manufacturing capacity distribution or efficiency evolution over time (e.g., yield and fabrication efficiency improvements). To enable a direct quantitative comparison with PCAM at the AI model level, we extended the STEC methodology by scaling its hardware-level carbon accounting using the foundational equation (Eq. 2 from the main text) and generating a distribution of embodied carbon based on the regional electricity carbon intensity frequency data. PCAM outperforms STEC consistently across all percentiles, with errors under  $3\%$  compared to STEC's under  $24\%$ , as Table 6 shows.
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+# User-Instructed Disparity-aware Defocus Control
+
+Yudong Han $^{1}$  Yan Yang $^{2}$  Hao Yang $^{1}$  Liyuan Pan $^{1,3\boxtimes}$
+
+<sup>1</sup>Beijing Institute of Technology, Beijing, China
+
+$^{2}$ BDSI, Australian National University, Canberra, Australia
+
+<sup>3</sup> Yangtze Delta Region Academy of Beijing Institute of Technology, Jiaxing, China hanyudong.sdu@gmail.com, yan.yang@anu.edu.au, {hao.yang, liyuan.pan}@bit.edu.cn
+
+![](images/d6dc3100b90d789948b496b1816a24594c75ea57d86a0ee89c305e6ae0e905fa.jpg)  
+Figure 1: An example of refocusing using our method. The input and refocused images are on the left and right. Two regions are highlighted with red and blue boxes, and their zoomed-in views are displayed in the center.
+
+# Abstract
+
+In photography, an All-in-Focus (AiF) image may not always effectively convey the creator's intent. Professional photographers manipulate Depth of Field (DoF) to control which regions appear sharp or blurred, achieving compelling artistic effects. For general users, the ability to flexibly adjust DoF enhances creative expression and image quality. In this paper, we propose UiD, a User-Instructed DoF control framework, that allows users to specify refocusing regions using text, box, or point prompts, and our UiD automatically simulates in-focus and out-of-focus (OoF) regions in the given images. However, controlling defocus blur in a single-lens camera remains challenging due to the difficulty in estimating depth-aware aberrations and the suboptimal quality of reconstructed AiF images. To address this, we leverage dual-pixel (DP) sensors, commonly found in DSLR-style and mobile cameras. DP sensors provide a small-baseline stereo pair in a single snapshot, enabling depth-aware aberration estimation. Our approach first establishes an invertible mapping between OoF and AiF images to learn spatially varying defocus kernels and the disparity features. These depth-aware kernels enable bidirectional image transformation—deblurring out-of-focus (OoF) images into all-in-focus (AiF) representations, and conversely reburbling AiF images into OoF outputs—by seamlessly switching between the kernel and its inverse form. For user-guided refocusing, we first generate masks based on user prompts using SAM, which modulates disparity features in closed form, allowing dynamic kernel re-estimation for reburbling. This achieves user-controlled refocusing effects. Extensive experiments on both common datasets and the self-collected dataset demonstrate that UiD offers superior flexibility and quality in DoF manipulation imaging.
+
+# 1 Introduction
+
+Modern cameras utilise sophisticated compound lenses designed to focus light onto the sensor. However, adjusting the aperture has another effect: a larger aperture (lower  $f$ -number) introduces depth-dependent blur, rendering only specific regions in focus [20]. This depth of field (DoF) effect is often intentionally used by photographers to direct viewer attention to a subject (e.g., macro photography) or create artistic compositions (e.g., tilt-shift photography) [2]. Unlike professional photographers, who possess the expertise to control DoF effectively, general users often struggle with achieving the desired DoF composition. Therefore, a framework that enables users to flexibly manipulate sharp and blurred regions in their captured images using simple text or point-based inputs to create visually striking effects would be a valuable application.
+
+Given images with undesired DoF regions captured by general users, effectively controlling depth-dependent blur involves several key steps: (i) estimating the all-in-focus (AiF) image; (ii) computing the corresponding depth/defocus map of the undesired image; (iii) generating a mask to designate regions for in-focus and out-of-focus (OoF); and (iv) adjusting the defocus map with the mask for refocusing. Inspired by [52], which employs dual-pixel (DP) sensors [8] to synthetically generate blur on human subjects, we integrate DP sensors into our framework for flexible, user-instructed refocusing.
+
+DP sensors are widely used in DSLR and smartphone cameras [22]. These sensors divide each pixel into two photodiodes, capturing separate left and right views of a scene, effectively forming a stereo system with a tiny baseline [39]. The depth information inherently encoded in the DP pair also contains defocus information [6], which makes DP image pairs highly beneficial for tasks such as defocus deblurring, refocusing, and bokeh effect synthesis [52]. Therefore, in this work, we focus on DP refocusing by incorporating user-specified prompts to identify desired focus regions, thereby enabling our network to effectively address all key steps within a single unified framework.
+
+Different from existing work [52]—which synthesizes DSLR-style images from smartphone-captured data by directly treating the input DP image as the AiF image during refocusing—our task is more challenging. First, to estimate the latent AiF image and the defocus map, we jointly optimize defocus deblurring and reblurring. In contrast to earlier methods [5, 45, 56, 57] that treat deblurring and reblurring as independent processes or rely on ground truth depth maps for reblurring, we propose an invertible deblurring and reblurring framework that learns the robust defocus map and the spatially-varying associated blur kernels to assist the final refocusing step. Next, to facilitate mask generation, we employ flexible user inputs—such as point or box prompts—and models like SAM [43, 62] to generate masks that guide the refocusing process. Finally, to ensure that the generated mask effectively controls the depth-aware blur, we use the blur kernels learned by the invertible deblurring and reblurring framework to produce the final refocused image.
+
+Specifically, we first analyze the relationship between DP disparity, the circle of confusion (CoC, or defocus map), and spatially varying defocus blur, which leads to a closed-form solution for controlling the disparity of a DP pair to achieve refocusing. This insight enables our network to perform controllable refocusing at test time simply by specifying regions of interest—even without explicit training on refocused images or reliance on depth annotations. Building on this understanding, we perform both invertible deblurring and reblurring within a unified framework, enabling the network to learn and regularize the mapping between blurred DP inputs and the sharp image in a self-consistent manner via learned disparity features. These learned disparity features are then converted into CoC features, which are used to construct spatially varying deblurring kernels. To ensure consistency, these kernels are further constrained to perform reblurring when inverted, thereby enabling effective simulation of the DP model. During testing, we first restore an all-in-focus image from the input DP pair and then modulate its disparity using a user-specified mask to generate the target CoC features, guiding the reblurring process for spatially controllable refocusing. In addition to comprehensively evaluating the robustness and flexibility of the proposed method in real-world refocusing scenarios and to enrich community benchmarking, we further construct a dual-pixel real-world evaluation dataset. The detailed evaluations and visualizations will be provided in the supplementary material. Our contributions are as follows:
+
+- A framework that enables general users to flexibly manipulate sharp and blurred regions using simple text or point-based inputs.  
+- An invertible deblurring and reblurring framework that learns the defocus map and associated blur kernels in close alignment with the mathematical formulation of the DP model.
+
+- Extensive experiments, including a self-collected dual-pixel real-world dataset, on defocus deblurring and image refocusing, demonstrate the superior performance of our method.
+
+# 2 Related Work
+
+Defocus Deblurring. Defocus blur refers to the blur effect when observed objects fall outside the depth-of-field (DoF) of a camera lens. To recover lost details from such blurred images, one widely-embraced scheme [1, 26, 58] is a two-stage pipeline: (1) estimating an explicit defocus map that accurately reflects the level of blur, followed by (2) performing deblurring guided by this map. Earlier approaches often relied on hardware assistance [10, 12, 17] to obtain a depth map as a proxy for defocus map in deblurring tasks. To improve flexibility, several studies have explored using pre-trained monocular depth estimators to infer depth maps from single images, which are then used as guidance for defocus-aware deblurring [40, 41]. While these methods offer greater accessibility, they often struggle to generalize beyond their training datasets. To enhance depth estimation accuracy for defocus map recovery, other researchers have incorporated additional cues, such as dual-pixel (DP) data [52]. DP images provide defocus-related disparity information that significantly improves the guidance quality during the deblurring process [42]. Wadhwa et al. [52] propose the first framework that uses DP sensors to enlarge the blur effect in the background to synthesize the DSLR-style images, and assume the foreground faces are in focus. Different from the settings in [52], in this paper, our UiD leverages the disparity cues encoded in DP image pairs to flexibly control the blur in a scalable and effective manner.
+
+Image Refocus. Achieving accurate refocusing in post-capture images is challenging, as it requires both deblurring sharp regions and synthesizing realistic blur effects that align with the desired DoF [45]. Several studies [16, 36, 54] have explored software- and hardware-based approaches to enhance refocusing performance. Prior work [36, 54] often relies on specialized hardware to capture light field information, enabling post-capture focus control. However, light field cameras typically suffer from limited spatial resolution, making it difficult to capture intricate scene dynamics. An alternative solution [7, 48] is to capture a focus stack and leverage multi-frame information to estimate the desired focus distance. However, capturing long-term focus stacks inevitably leads to scene evolution, restricting refocusing to static environments [5]. Another line of research focuses on single-image refocusing, where the typical framework first deblurs the image to obtain an all-in-focus (AiF) representation, followed by reblurring. A recent representative study, RefocusGAN [45], trains a two-stage conditional GAN [35] for sequential deblurring and reblurring, enabling flexible single-image refocusing using a focus control vector. The most recent work,  $\mathrm{DC}^2$  [5], adopts a dual-camera setup, where the captured wide-frame and ultra-wide-frame images are first aligned, and a built-in depth sensor is leveraged to guide the image refocusing process. In this work, we follow the deblur-and-reblur paradigm and develop a disparity-aware invertible framework to overcome the limitations of inaccessible hardware (e.g., built-in depth sensors), the constraints of capturing a focus stack, and the challenge of achieving both defocus deblurring and selective reblurring.
+
+# 3 User-Instructed Disparity-aware Defocus Control
+
+Motivation. In the DP image pair, two images  $\mathbf{B}_l$  and  $\mathbf{B}_r$  are generated by light rays from the real scene passing through the left and right sub-apertures into two juxtaposed half pixels. The disparity  $\mathbf{D}$  for a DP pair  $(\mathbf{B}_l,\mathbf{B}_r)$  at location  $(i,j)$  is approximately,
+
+$$
+\mathbf {D} (i, j) = f \times B / \mathbf {Z} (i, j) + \operatorname {c o n s t} \approx f \times B / \mathbf {Z} (i, j), \tag {1}
+$$
+
+where  $f$  is the focal distance,  $B$  denotes the shift between two sub-apertures, and  $\mathbf{Z}(i,j)$  denotes the scene depth. DP disparity is strongly correlated with the degree of defocus blur [42]. Specifically, the blur radius at location  $(i,j)$  is given by,
+
+$$
+\mathbf {C} (i, j) = \mathcal {J} (f) \cdot \mathbf {D} (i, j), \tag {2}
+$$
+
+where  $\mathcal{J}(f)$  is a function of the focus distance  $f$ , controlling the strength of the blur.  $\mathbf{C}$  is known as CoC. When refocusing to objects indicated by a binary mask  $\mathbf{M}$ , its blur radius follows a linear relation with respect to the disparity offset from the target focal disparity. The refocused blur radius at  $(i,j)$  becomes,
+
+$$
+\mathbf {C} (i, j) = \mathcal {J} (f) \cdot (\mathbf {D} (i, j) - d _ {t r g}), \tag {3}
+$$
+
+where the target focal disparity  $d_{trg}$  is computed as the average disparity within the masked region:
+
+$$
+d _ {t r g} = \frac {1}{| \Omega |} \sum_ {\left(i ^ {\prime}, j ^ {\prime}\right) \in \Omega : \mathbf {M} \left(i ^ {\prime}, j ^ {\prime}\right) = 1} \mathbf {D} \left(i ^ {\prime}, j ^ {\prime}\right), \tag {4}
+$$
+
+with  $|\Omega|$  denoting the number of pixels where  $\mathbf{M}(i', j') = 1$ . This formulation enables spatially adaptive refocusing to user-specified regions of interest defined by the mask.
+
+Overview. Our key idea is to learn a disparity-aware feature space that adheres to Eq. 2, enabling a blur-aware representation suitable for both deblurring and refocusing. Leveraging standard DP defocus-deblurring datasets containing DP pairs  $(\mathbf{B}_l,\mathbf{B}_r)$  and their corresponding sharp images I, we train an invertible network that jointly models deblurring and reblurring (i.e., red, orange, and green arrow in Figure 2). During training, a CoC-based feature  $\mathbf{F}_{coc}$  is computed in the feature space by combining a disparity feature and a blur strength feature according to Eq. 2. This feature is used to generate spatially varying deblurring kernels  $\mathbf{K}$  that restore the input DP pair  $(\mathbf{B}_l,\mathbf{B}_r)$  into a sharp image  $\hat{\mathbf{I}}$ . Reblurring is then performed by inverting these kernels, using  $\mathbf{K}^{-1}$  to degrade the sharp image  $\hat{\mathbf{I}}$  back into a blurred DP pair. At test time, we first
+
+![](images/e56091368d75a98eb2808351db330a14cdee9ffd812bc61cb3754b94f85e0716.jpg)  
+Figure 2: Illustration of overall framework. The arrow with red color and green color denotes the deblurring pathway, and the arrow with orange color and green color denotes the reblurring pathway. These two pathways are collaboratively trained with disparity-aware shared kernel learning (i.e., green arrow). The arrow with gray color indicates the user-instructed mask generation, which is only activated during inference time. The dotted box with yellow color denotes the early fusion paradigm.  $\mathcal{F}_m$  can be multimodal encoder or MLLM [29-31], proj is the projector, and  $\mathcal{F}_{sam}^v$  denotes the visual encoder of SAM.
+
+apply the deblurring process to  $(\mathbf{B}_l,\mathbf{B}_r)$  to recover the sharp image  $\hat{\mathbf{I}}$ . To perform refocusing, the disparity feature obtained during deblurring is modulated according to Eq. 4, resulting in a target CoC feature  $\mathbf{F}_{coc}^{trg}$ , computed using the user-provided mask  $\mathbf{M}^{trg}$  and blur level as in Eq. 2. The mask can be derived from point or box prompts by leveraging the SAM-like model [43, 62]. This target feature guides the reburbling process, transforming the sharp image  $\hat{\mathbf{I}}$  into a refocused DP image. An overview of the network architecture is shown in Figure 2, where deblurring, reburbling, and refocusing operations are carried out in a shared feature space learned by an encoder  $\mathcal{F}_e(\cdot ,\cdot)$  and a decoder  $\mathcal{F}_d(\cdot)$ .
+
+# 3.1 Deblurring
+
+Feature Encoding. We restore the DP pair  $(\mathbf{B}_l, \mathbf{B}_r)$  in feature space by first embedding them into a base feature representation  $\mathbf{F}_B$  using an encoder  $\mathcal{F}_e(\cdot, \cdot)$ . In parallel, a disparity-related feature  $\mathbf{F}_G$  is extracted using a stereo matching network  $\mathbf{G}(\cdot, \cdot)$  applied to the DP pair. Within  $\mathbf{G}(\cdot, \cdot)$ , we first use dual visual encoder for feature extraction, forming  $\mathbf{F}_{l,b}$  and  $\mathbf{F}_{r,b}$ . For each pixel in  $\mathbf{F}_{l,b}$ , we search for its counterpart  $\mathbf{F}_{r,b}$  within scope  $d$  horizontally, where  $d$  is defined as the disparity index. We calculate all the possible disparity using widely-used matching costs following [50], where the left and right features concatenated to form the cost volume,
+
+$$
+\mathbf {C} _ {d} (x, y) = \left[ \mathbf {F} _ {l, b} (x, y), \mathbf {F} _ {r, b} (x - d, y) \right], \tag {5}
+$$
+
+where  $x$  and  $y$  are the pixel coordinates. Then, all the obtained disparity feature  $\mathbf{C}_d$  are concatenated, forming  $\mathcal{C} = [\mathbf{C}_1, \mathbf{C}_d, \dots, \mathbf{C}_D]$ , where  $D$  denotes the maximum disparity scope. Afterwards, two cascaded 3D convolutional networks are sequentially employed for aggregating the geometry information stored in the cost volumes  $\mathcal{C}$ , learning the approximate disparity clues in each location. Finally, a lightweight regression block is used on the top of network to compute the expected disparity feature  $\mathbf{F}_G$ .
+
+While disparity or depth is fundamental to refocus control, directly manipulating the feature space spanned by  $\mathbf{F}_G$  based on these geometric cues is infeasible. In other words, the lack of an interpretable
+
+correspondence between feature dimensions and defocus blur prevents meaningful, controllable editing. Therefore, we thus further normalize the DP disparity feature and scale it as follow,
+
+$$
+\mathbf {F} _ {D} = 2 \cdot \operatorname {T a n h} \left(\operatorname {N o r m} \left(\mathbf {F} _ {G}\right) / \rho\right) - 1, \tag {6}
+$$
+
+where  $\mathrm{Tanh}(\cdot)$  denotes the hyperbolic tangent function,  $\mathrm{Norm}(\cdot)$  is the  $\ell_2$  normalization along with channel dimension, and  $\rho$  is a temperature parameter that sharp the distribution of the output. By this way, we condense the depth information conveyed in  $\mathbf{F}_G$  into a one-channel disparity map  $\mathbf{F}_D$ , which serves as a direct interface for refocus control. Recognizing that the DP pair also encodes focal length-dependent blur information, we extract this information from feature  $\mathbf{F}_G$  using an encoder  $\mathcal{J}(\cdot)$ . Following Eq. 2, the initial CoC-related feature can be rewritten as,
+
+$$
+\mathbf {F} _ {\text {i n i t}} = \mathcal {J} \left(\mathbf {F} _ {G}\right) \cdot \mathbf {F} _ {D}. \tag {7}
+$$
+
+Next, we apply a decoder-encoder structure to refine  $\mathbf{F}_{init}$ . The refined feature is projected into RGB space via a projection head  $\mathcal{P}_{rgb}(\cdot)$ , and projected back to the feature space through a second projection head  $\mathcal{P}_{feat}(\cdot)$ , with a gating residual connection [47] for numerical stability,
+
+$$
+\mathbf {F} _ {c o c} = \mathbf {R} \odot \mathbf {F} _ {G} + \mathcal {P} _ {\text {f e a t}} \left(\mathcal {P} _ {r g b} \left(\mathbf {F} _ {\text {i n i t}}\right)\right), \tag {8}
+$$
+
+where  $\mathbf{R}$  denotes the learnable gate (see supplementary material for details). This decoder-encoder structure enables auxiliary supervision either through ground-truth CoC annotations or via self-supervised CoC estimation applied to the RGB-projected output  $\mathcal{P}_{rgb}(\mathbf{F}_{init})$ .
+
+Feature Restoration. By maintaining a set of  $N$  learnable blur kernels  $\{\mathbf{K}_n\}_{n=1}^N$ , we use the CoC-related feature  $\mathbf{F}_{coc}$  to query spatially-varying deblurring kernels from the set, and apply the queried kernels to restore the feature  $\mathbf{F}_B$ . Note that we keep  $\mathbf{F}_{coc}$  and  $\mathbf{F}_B$  as the same spatial size. Specifically, for each pixel  $(i,j)$ , a spatially-varying kernel  $\mathbf{K}_{i,j}$  is constructed as a weighted sum of the kernel set,
+
+$$
+\mathbf {K} _ {i, j} = \sum_ {n = 1} ^ {N} \mathbf {a} _ {i, j} (n) \cdot \mathbf {K} _ {n}, \quad \mathbf {a} _ {i, j} = \operatorname {S o f t m a x} \left(\mathcal {P} _ {k} \left(A v g \left(\left\{\mathbf {K} _ {n} \right\} _ {n = 1} ^ {N}\right) \odot \mathcal {P} _ {c o c} \left(\mathbf {F} _ {c o c} (i, j)\right)\right)\right), \tag {9}
+$$
+
+where  $\mathbf{a}_{i,j}$  is a soft attention vector over the  $N$  kernels. The function  $Avg(\cdot)$  computes the average of each kernel  $\mathbf{K}_n$  (resulting in a scalar per kernel), forming a  $N$ -dim vector representation. This vector is element-wise multiplied  $\odot$  with the CoC-projected feature  $\mathcal{P}_{coc}(\mathbf{F}_{coc}(i,j))$ , and passed through a linear layer  $\mathcal{P}_k(\cdot)$  followed by a Softmax function to produce the attention weights. The learnable projections  $\mathcal{P}_k(\cdot)$  and  $\mathcal{P}_{coc}(\cdot)$  are both implemented as linear layers.
+
+We then convolve  $\mathbf{F}_B$  with queried kernels for each pixel  $(i,j)$ , and refine the feature by using forward mapping of invertible block  $Inv(\cdot)$ , to get restored feature  $\hat{\mathbf{F}}_I$ ,
+
+$$
+\hat {\mathbf {F}} _ {I} (i, j) = \operatorname {I n v} \left(\sum_ {\delta i, \delta j} \mathbf {F} _ {B} (i + \delta i, j + \delta j) \cdot \mathbf {K} _ {i, j} (\delta i, \delta j)\right), \tag {10}
+$$
+
+where  $(\delta i, \delta j)$  is offset of the receptive field of the spatial kernel  $\mathbf{K}_{i,j}$ . Notably, the special design of architecture theoretically guarantees that Jacobian determinant  $\left|\frac{\partial\hat{\mathbf{F}}_I}{\partial\mathbf{F}_B}\right|$  is non-zero, ensuring that invertibility of overall network, irrespective of the specific form of the components within  $Inv(\cdot)$ . The details can be referred to supplementary materials.
+
+Feature Decoding. We decode the  $\hat{\mathbf{F}}_I(i,j)$  to RGB space for calculating an all-in-focus restoration with a decoder  $\mathcal{F}_d(\cdot)$  by  $\hat{\mathbf{I}} = \mathcal{F}_d(\hat{\mathbf{F}}_I)$ .
+
+# 3.2 Reblurring
+
+Our reblurring process is analogous to the deblurring procedure, but it employs the inverse of the queried kernels. For an all-in-focus image  $\mathbf{I}$ , we first obtain feature  $\mathbf{F}_I$  by  $\mathbf{F}_I = \mathcal{F}_e(\mathbf{F}_I,\mathbf{F}_I)$ . Then, we reverse the computation performed by the invertible block  $Inv(\cdot)$  by applying its corresponding inverse operation  $Inv'(\cdot)$ . Simultaneously, for each pixel  $(i,j)$ , we calculate the inverse of the deblurring kernel  $\mathbf{K}_{i,j}$  with the Fourier transform  $\mathcal{F}(\cdot)$ ,
+
+$$
+\mathbf {K} ^ {- 1} = \mathcal {F} ^ {- 1} \left(\frac {1}{\mathcal {F} (\mathbf {K})}\right), \tag {11}
+$$
+
+outputting an inverse kernel  $\mathbf{K}_{i,j}^{-1}$  for reblurring, where  $\mathcal{F}^{-1}(\cdot)$  denotes the inverse Fourier transform. The reblurred feature is calculated as,
+
+$$
+\hat {\mathbf {F}} _ {B} (i, j) = \sum_ {\delta i, \delta j} I n v ^ {\prime} \left(\mathbf {F} _ {I}\right) (i + \delta i, j + \delta j) \cdot \mathbf {K} _ {i, j} ^ {- 1} (\delta i, \delta j). \tag {12}
+$$
+
+The resulting reblurred feature map  $\hat{\mathbf{F}}_B$  is then decoded into an image  $\hat{\mathbf{B}}$  using a decoder  $\mathcal{F}_d(\cdot)$ . The goal is to synthesize the center view of the DP pair  $\mathbf{B} = \frac{1}{2}\mathbf{B}_l + \frac{1}{2}\mathbf{B}_r$ .
+
+# 3.3 Refocusing
+
+During testing, to perform refocusing on a DP pair  $(\mathbf{B}_l,\mathbf{B}_r)$  using a user-provided focus mask  $\mathbf{M}^{trg}$ , we first restore an all-in-focus image  $\hat{\mathbf{I}}$  following the forward mapping of our invertible network (Eq. 10). We then reblur the image by modulating the original disparity feature  $\mathbf{F}_D$  by
+
+$$
+\mathbf {F} _ {D} ^ {t r g} (i, j) = \mathbf {F} _ {D} (i, j) - \frac {1}{| \Omega |} \sum_ {\left(i ^ {\prime}, j ^ {\prime}\right) \in \Omega : \mathbf {M} ^ {t r g} \left(i ^ {\prime}, j ^ {\prime}\right) = 1} \mathbf {F} _ {D} \left(i ^ {\prime}, j ^ {\prime}\right), \tag {13}
+$$
+
+This operation shifts the disparity feature such that the selected region appears in focus, effectively implementing a disparity-aware refocusing strategy, as outlined in Eq. 4. We then calculate an updated CoC feature  $\mathbf{F}_{coc}^{trg}$  based on Eq. 8, and repeat the kernel querying process defined in Eq. 9. These kernels are applied in the reblurring process according to Eq. 12, ultimately producing the refocused feature to decode the target refocused image, denoted as  $\mathbf{B}^{trg}$ .
+
+Training-free Instructed SAM. Given a user instruction  $\mathcal{T}$  specifying the region of interest, the instructed Segment Anything Model (SAM) generates a target mask  $\mathbf{M}^{trg}$  semantically aligned with the prompt. A key challenge lies in obtaining a high-quality segmentation mask to effectively guide the refocusing process. While end-to-end joint training of the instructed SAM with our deblur-and-reblur framework is a natural solution, it suffers from two critical drawbacks: (1) the segmentation mask, when suboptimal in early training stages, dynamically affects the refocusing network and may produce erroneous gradients, impeding optimal convergence; and (2) fine-tuning SAM from scratch with deblur-and-reblur framework entails significant computational overhead, making the overall process resource-intensive. To circumvent these issues, we first forgo the joint training tactics. Instead, we first training our deblur-and-reblur network using a mask-free manner, which means that it does not require masks during training. Mask are only provided at test time, where the user inputs a binary mask to specify the region of interest for refocusing. Therefore, the degradation on mask would not affect the dynamics and robustness of our model training. Given the inaccessibility of clear reference images during testing, we adopt an early fusion strategy to deeply exploit the rich semantic cues conveyed by both the user prompt and the input image, enabling more precise mask generation. This approach demonstrates superior performance compared to the conventional late fusion paradigm [62]. Specifically, it takes the restored image  $\hat{\mathbf{I}}$  that generated via forward mapping in Eq. 10, prompt  $\mathcal{T}$ , and a random initialized  $[cls]$  query as input, and using multimodal transformer-based encoder  $\mathcal{F}_m(\cdot)$  to enhance the interaction between modalities,
+
+$$
+\mathbf {q} _ {\text {c l s}} = \mathcal {F} _ {m} (\tilde {\mathbf {I}}, \mathcal {T}, \mathbf {q} _ {\text {c l s}}), \tag {14}
+$$
+
+the output [cls] query  $\mathbf{q}_{cls}$  encodes the compact multimodal clues, and then fed into the mask decoder together with the SAM-extracted image features  $\mathbf{F}_{sam}$ , enabling accurate spatial alignment towards robust mask generation,
+
+$$
+\mathbf {M} _ {t r g} = \mathcal {F} _ {g} \left(\mathbf {q} _ {c l s}, \mathbf {F} _ {s a m}\right). \tag {15}
+$$
+
+Furthermore, we localize the region of interest using structured, template-based prompts that combine number, object color, category, and spatial position, such as "a brown dog on the chair", to enhance spatial grounding precision. For those still imperfect masks, we further apply post-processing techniques such as dilation or erosion operations for refinement. The detailed implementation of  $\mathcal{F}_m(\cdot)$  and  $\mathcal{F}_g(\cdot)$  can be referred in supplementary material.
+
+# 3.4 Loss
+
+Our network is trained with a combination of objectives to supervise deblurring, reblurring, and CoC estimation. Specifically, we use a deblurring loss  $\mathcal{L}_{deb}(\mathbf{I},\hat{\mathbf{I}})$  and a reblurring loss  $\mathcal{L}_{reb}(\mathbf{B},\hat{\mathbf{B}})$  based
+
+on multi-scale setting following [57], a MSE-based CoC regression loss  $\mathcal{L}_{coc}(\mathbf{C},\mathcal{P}_{rgb}(\mathbf{F}_{int}))$ , and a MSE-based CoC gradient loss  $\mathcal{L}_{coc}^{\nabla}(\mathbf{C},\mathcal{P}_{rgb}(\mathbf{F}_{int}))$  to enhance edge precision. The total loss is,
+
+$$
+\mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {\text {d e b}} (\mathbf {I}, \hat {\mathbf {I}}) + \mathcal {L} _ {\text {r e b}} (\mathbf {B}, \hat {\mathbf {B}}) + \lambda_ {\text {c o c}} \cdot \mathcal {L} _ {\text {c o c}} (\mathbf {C}, \mathcal {P} _ {\text {r g b}} (\mathbf {F} _ {\text {i n i t}})) + \lambda_ {\text {g r a d}} \cdot \mathcal {L} _ {\text {c o c}} ^ {\nabla} (\mathbf {C}, \mathcal {P} _ {\text {r g b}} (\mathbf {F} _ {\text {i n i t}})), \tag {16}
+$$
+
+where each  $\lambda$  controls the relative weight of the corresponding loss term.
+
+# 4 Experiment
+
+Dataset. For defocus deblurring task, we evaluate our method on widely-used DPD-blur [2] dataset and recent DP5K [27] dataset. For refocusing task, we use three datasets for evaluation, DP5K dataset, DPD-disp dataset, and our self-collected DP dataset. Our model is trained using the training splits of DPD-blur and DP5K. Refer to our supplementary material for detailed dataset configurations and evaluation protocols.
+
+Evaluation Metrics. We evaluate the model with using standard metrics, i.e., peak signal-to noise ratio (PSNR) [19], structural similarity (SSIM) [55], relative error (RMSE rel) [37], mean absolute error (MAE), and learned perceptual image patch similarity (LPIPS) [61].
+
+Implementation Detail. For training, we use the AdamW optimizer [24] with  $\beta_{1} = 0.9$ ,  $\beta_{2} = 0.999$ , a learning rate of  $3 \times 10^{-4}$ , and a weight decay of  $10^{-6}$ . A cosine annealing learning rate [32] scheduler with warmup is employed, where the cycle steps, warmup steps, and minimum learning rate are set to 200, 100, and  $6 \times 10^{-5}$ , respectively. For the DPD-blur dataset, the model is trained for 40k iterations with a batch size of 4. For the DP5K dataset, we train the model for 64k iterations with a batch size of 6. We elaborate our network architecture and more hyper-parameter setting in supplementary material.
+
+# 4.1 Experimental Results
+
+As mentioned Sec. 3, image refocusing task is decomposed into two key stages: (1) defocus deblurring, and then (2) refocusing. In this section, we conduct extensive experiments to evaluate the effectiveness of each part in two stages. First, we perform quantitative and qualitative evaluations of both deblurring and refocusing, comparing their performance with several strong baselines. Then, we provide a detailed component-wise analysis of our refocusing framework.
+
+# 4.1.1 Quantitative and Qualitative Results on Image Refocus
+
+Setting. Image refocus takes a user-specified mask and an all-in-focus image as input, the goal is to generate a new image where the masked region remains sharp while other areas exhibit controlled defocus blur. Therefore, obtaining an all-in-focus image from blurred image is the first step to achieve refocusing, and its restored quality greatly determine refocus performance.
+
+Table 1: Deblurring quantitative comparisons on DPD-blur dataset [2] with several SOTA baselines.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Computational Cost</td><td colspan="3">Quantitative Metrics</td></tr><tr><td>Params (M)</td><td>Flops (G)</td><td>PSNR↑</td><td>SSIM↑</td><td>MAE(10-1)↓</td><td>MSE rel(10-1)↓</td></tr><tr><td colspan="7">single-image defocus deblurring</td></tr><tr><td>EBDB</td><td>-</td><td>-</td><td>23.45</td><td>0.683</td><td>0.49</td><td>0.67</td></tr><tr><td>DMENet</td><td>26.71</td><td>4787</td><td>23.41</td><td>0.714</td><td>0.51</td><td>0.67</td></tr><tr><td>RDPD</td><td>24.28</td><td>901</td><td>25.39</td><td>0.772</td><td>0.40</td><td>0.53</td></tr><tr><td>IFAN</td><td>10.48</td><td>794</td><td>25.99</td><td>0.804</td><td>0.37</td><td>0.50</td></tr><tr><td>BAMBNet</td><td>4.50</td><td>1804</td><td>26.40</td><td>0.821</td><td>0.36</td><td>0.47</td></tr><tr><td>DeepRFT</td><td>9.60</td><td>3682</td><td>25.71</td><td>0.801</td><td>0.37</td><td>0.51</td></tr><tr><td>Restormer</td><td>26.13</td><td>4458</td><td>26.66</td><td>0.833</td><td>0.35</td><td>0.46</td></tr><tr><td colspan="7">dual-pixel defocus deblurring</td></tr><tr><td>DPDNet</td><td>31.03</td><td>3150</td><td>25.13</td><td>0.786</td><td>0.41</td><td>0.55</td></tr><tr><td>DDDNet</td><td>6.40</td><td>1661</td><td>25.36</td><td>0.768</td><td>0.41</td><td>0.54</td></tr><tr><td>K3DN</td><td>5.00</td><td>1033</td><td>26.84</td><td>0.829</td><td>0.35</td><td>0.46</td></tr><tr><td>Ours</td><td>3.12</td><td>495</td><td>26.89</td><td>0.829</td><td>0.33</td><td>0.47</td></tr></table>
+
+![](images/db30a5bfbfa680b53f76844e81dc5942ac27d8699d64c27dce4d56a98c7b1f83.jpg)  
+Blurry Image
+
+![](images/bbb4cd81c91e6e15383fba79332af82b7ae10895a0bdd277f3a7f1ad9f110f40.jpg)
+
+![](images/0ede06f5552b8d8f5bff51b9df795ad278a2dbc02aeb36fb18ab8932c69f3d8e.jpg)
+
+![](images/33a9c53bb8c0a135f1ebd4a1a1f69cf506141c21e19632e514106c74c1924087.jpg)
+
+![](images/831ea68acb39162366691c51c235970dd1fdb1030d8a84dea1c6c47e1d243a7d.jpg)  
+Blurry Image  
+Figure 3: Qualitative comparison on the DP5K-test dataset [3]. We present the blurry image in the first column, and boxed regions with red color are zoomed results.
+
+![](images/19d51154c59c9a1a42152009db69182308350fed2fc2a732aa88bc35dc82c56f.jpg)  
+Blurry Image  
+Restormer [60]
+
+![](images/004e039df5205d38724ed2dae5c52bd094c4462864ff3f6772b75dca18db0dbd.jpg)  
+GT Image
+
+![](images/9f9cadf1b455d3bbf8f55b00cf8679e80e1d756cb1a64e24cd24211bd9ec55f1.jpg)  
+DDDNet [37]  
+Ours
+
+![](images/8188cd0ebbebdf2825c9eb82c8c626d840fb3534454e0a694280608a30b6ac03.jpg)  
+Blurry Image
+
+![](images/bdc8c45370436efe817bb76b1da2204f82f3896b0c2b35f83fc74ee9ba371edb.jpg)  
+K3DN [58]
+
+![](images/7e41d057d372fbe7514819eb18ab380511407e0a705dcb40fdb432bef4d06935.jpg)
+
+![](images/62da0eb0c220e12aee9ab5ada564932846eb237a4ae94d0b3fd3991f475a8872.jpg)  
+Restormer [60]
+
+![](images/3750945a19875aec9ab08c093113fa9dd7d7234a9a21105d278aa0bae128adf2.jpg)  
+GT Image  
+K3DN [58]
+
+![](images/e1175564f74b3a721dea2430f526646371771715ad5c4db765b4bba881d021e8.jpg)  
+DDDNet [37]  
+Ours
+
+Table 2: Comparison of refocusing performance on DP5K-test. † denotes the model version adapted for refocusing  
+
+<table><tr><td>Methods</td><td>PSNR↑</td><td>SSIM↑</td><td>MAE(10-1)↓</td><td>LPIPS↓</td></tr><tr><td>Omni-Kernel†</td><td>25.32</td><td>0.816</td><td>0.47</td><td>23.5</td></tr><tr><td>K3DN†</td><td>30.76</td><td>0.943</td><td>0.30</td><td>14.4</td></tr><tr><td>Ours</td><td>31.21</td><td>0.953</td><td>0.33</td><td>13.8</td></tr></table>
+
+Table 3: Comparison of refocusing performance on DPD-disp for OOD evaluation. † denotes the model version adapted for refocusing.  
+
+<table><tr><td>Methods</td><td>PSNR↑</td><td>SSIM↑</td><td>MAE(10-1)↓</td><td>LPIPS↓</td></tr><tr><td>Omni-Kernel†</td><td>18.75</td><td>0.692</td><td>0.79</td><td>31.4</td></tr><tr><td>K3DN†</td><td>21.72</td><td>0.765</td><td>0.67</td><td>29.7</td></tr><tr><td>Ours</td><td>21.93</td><td>0.772</td><td>0.56</td><td>25.1</td></tr></table>
+
+Methods. To evaluate the deblurring ability of our framework, we compared with two primary categories of defocus deblurring approaches. (1) Single-image defocus deblurring: KPAC [49], RDPD [4], DRBNet [44], DeepRFT [34], IFAN [26], RAMBNet [28], and Restormer [59]. (2) Dual-pixel defocus deblurring: DPDNet [2], DDDNet [38], K3DN [57]. For refocusing, BokehMe [40] and Dual-Camera [5] are most closely related to our approach. However, they neither release training code nor pretrained models. Thus, we manually modify two SOTA methods on deblurring task, K3DN and Omni-Kernel [11], to enable their refocusing functionality. Refer to our supplementary materials for details.
+
+Evaluation. The deblurring comparisons with state-of-the-art methods are given in Table 1 and Table 5. The result reveals that we achieve superior performance, while having smallest computation cost (495 GFlops). Figure 3 showcases the deblurring results of two cases against several state-of-the-art baselines. Our method is able to restore severely blurred regions, especially textual region, compared to existing models. For refocusing, we report the qualitative comparison in Table 2 and
+
+![](images/9c635304be4d57295220a183ab40e74bd87632203fa5d8ee06f6d433a00aa090.jpg)  
+$\mathbf{F}_D = -0.6$
+
+![](images/5640ee2ad9a0b9227b12023e8ecaf74a03afbbe0a2b8ee337e6f0563df302f81.jpg)  
+$\mathbf{F}_D = -\mathbf{0.4}$
+
+![](images/5e26cb968217d0c680036fdcff76e9f220af5ff640a6da4c1081dedad2858530.jpg)  
+$\mathbf{F}_D = -\mathbf{0.2}$
+
+![](images/7d8c40059d959a39570ff9f1ccfeac026e203288ab12f8ba377a617bbace6274.jpg)  
+$\mathbf{F}_D = \mathbf{0}$  0
+
+![](images/4e7e6da1eb3007931ca4ce1e8086482502ee695358943b93a2f7c1de24b57d3c.jpg)
+
+![](images/5f7f7407f10fea64af9ed7293cf7f5c5f29d39dd368d5adc3c5f466daf2a5178.jpg)  
+$\mathbf{F}_D = \mathbf{0}\cdot \mathbf{2}$  
+Figure 4: Visualization of blur effect by setting  $\{\mathbf{F}_D(i,j)|(i,j)\in \Omega_{book}\} = 0$  when performing reburbling using Eq. 12. We take the all-in-focus and a user-specified mask (a mask referring to a unfolding book in this case) as input, and observe the blur variation with different  $\mathbf{F}_D$ . The top-right figure shows the overall PSNR variation trend with different  $\mathbf{F}_D$ .
+
+![](images/c5f89fab2b02af234e0ef9d8c20a038d66382331291bf2bd38e454615b231f34.jpg)  
+$\mathbf{F}_D = \mathbf{0.4}$
+
+![](images/786117b7358b2db53011bd7b6dd9f73573c7814bb103c0d5ae25707b3d146e9d.jpg)  
+$\mathbf{F}_D = \mathbf{0.6}$
+
+![](images/785565ced76517380bee5635da3bed21eebe2ecf51619f0d2d9f6fbdaeeb60a4.jpg)  
+$\mathbf{F}_D = \mathbf{0}\cdot \mathbf{8}$
+
+![](images/f8b341a73eae48403aca086fa6369ad3a415da3cec6fee917f90d80ba72209e4.jpg)  
+All-in-Focus Image
+
+Table 4: The effect of joint deblurringreblurring training on performance.  
+
+<table><tr><td>Model Variants</td><td>PSNR↑</td><td>SSIM↑</td><td>MAE(10-1)↓</td></tr><tr><td colspan="4">Deblurring Performance</td></tr><tr><td>w Lreb + Ldeb</td><td>26.89</td><td>0.829</td><td>0.35</td></tr><tr><td>w Ldeb</td><td>26.80</td><td>0.823</td><td>0.35</td></tr><tr><td colspan="4">Reblurring Performance</td></tr><tr><td>w Lreb + Ldeb</td><td>29.18</td><td>0.875</td><td>0.22</td></tr><tr><td>w Lreb</td><td>29.06</td><td>0.863</td><td>0.28</td></tr></table>
+
+Table 5: Comparison of deblurring performance on DP5K-test. The best is bold with black.  
+
+<table><tr><td>Methods</td><td>PSNR↑</td><td>SSIM↑</td><td>RMSE_rel(10-1)↓</td><td>MAE(10-1)↓</td></tr><tr><td>DDDNet</td><td>25.59</td><td>0.777</td><td>0.525</td><td>0.58</td></tr><tr><td>IFAN</td><td>28.23</td><td>0.875</td><td>0.387</td><td>0.44</td></tr><tr><td>DeepRFT</td><td>30.24</td><td>0.915</td><td>0.307</td><td>0.37</td></tr><tr><td>BAMBNet</td><td>30.54</td><td>0.917</td><td>0.297</td><td>0.34</td></tr><tr><td>Restormer</td><td>30.71</td><td>0.922</td><td>0.291</td><td>0.33</td></tr><tr><td>K3DN</td><td>30.82</td><td>0.923</td><td>0.291</td><td>0.32</td></tr><tr><td>Ours</td><td>30.72</td><td>0.926</td><td>0.280</td><td>0.30</td></tr></table>
+
+![](images/f20d2d3beab28633c4de8bbbfbfa3ffac39ea687c077ce0e2fe8317b1f08e1e1.jpg)  
+Blurry Image B
+
+![](images/0e6d621aad8df72cdc1495b20e07fea13b887cca7145befcd9d446fa1c9be153.jpg)  
+Deblurred Image I
+
+![](images/cb79f8e95e9617c521d18a940f5a8b3623e2ad9a2a9100c87840b60d0741ea29.jpg)  
+Reblur using  $\mathbf{F}_D$
+
+![](images/4ae1e7763fa259ee48cf3a04fd53397d0183fce913eaba116c6a3f311a873539.jpg)  
+Refocus using  $\mathbf{F}_D^{trg}$
+
+![](images/8d98bf9735bc37847b6063a08f245877b2d1730ed29e2fbca8721ffe2ef07da9.jpg)  
+Zoom in
+
+![](images/62b5ad567f84613180fe2629589cf6d881e97e3e1b8d03691a71e08c5794334b.jpg)  
+$\mathbf{F}_D$
+
+![](images/8da933361db186c5a1da9d831ac4297012c35764e14a44f7ff48579cdf32100b.jpg)  
+$P_{rgb}$  (Finit)
+
+![](images/48083d503fdcc05f26ad0f64fca93e6444f4a20e1cbceca08014e53c60fb09a4.jpg)  
+$\mathbf{F}_D^{trg}$  
+Figure 5: Visualization on immediate process in refocusing. Notably,  $\mathbf{F}_D$  and  $\mathbf{F}_D^{trg}$  denotes the disparity map used in reburring and refocusing, and  $P_{rgb}(\mathbf{F}_{init})$  and  $P_{rgb}^{trg}(\mathbf{F}_{init})$  are predicted CoC map from their corresponding disparity map (Eq. 8), which both exhibits satisfactory consistency.
+
+![](images/a54256b7d75f5b45ab46c37d50fc22a8519bf9fd9051fd3109ff125e672b89af.jpg)  
+$P_{rgb}^{trg}(\mathbf{F}_{init})$  
+Table 3. We also show two cases from DPD-disp dataset to evaluate the out-of-distribution (OOD) generalization capability of our model, as shown in Figure 6.
+
+![](images/e2f1f9c4940008eae2b36067daa92f9073fb37d6d940bda87b443b104338ef94.jpg)  
+$\mathbf{F}_D^{trg} - \mathbf{F}_D$
+
+# 4.1.2 Component-wise Analysis
+
+Effect of Joint Deblurring and Reblurring Training. Joint deblurring-reblurring training enables robust blur kernel estimation and exhibits more flexible refocusing. As can be seen in the Table 4, model with both  $\mathcal{L}_{reb}$  and  $\mathcal{L}_{deb}$  supervision achieves substantial advantages compared to that only with  $\mathcal{L}_{reb}$  or  $\mathcal{L}_{deb}$ . This observation indicates that one process not only does not disturb another branch but also effectively help to optimize the learned blur kernel, thereby enhancing the representation capability learned by our network. Using the shared kernel in deblurring process, user-specified refocusing process can be simply achieved by using the with inverse kernel obtained in Eq. 11, which exhibits satisfactory flexibility and viability.
+
+![](images/cd57befc22e2df4214dfddd2fc8b106b5f75811cbf97e1a9a02097f43111022a.jpg)
+
+![](images/97e3837a2e28764ecc53d3b227e1dc820c013836c3361f36362bc1e263ddb4c1.jpg)
+
+![](images/f28b922b6a93648926d599192628a4292dc32cc1f23b7a8a71145985434170ce.jpg)
+
+![](images/8c169e4fe31c9a454cf4b2b0eab503d78bdfb54b6cb34a78e028bc5b688e3811.jpg)
+
+![](images/d6e9a2f1b4f6d8465f049122fe90b47bdb8a92268e8b4c258b99476fa18ea882.jpg)
+
+![](images/12021b10e496c4d08f455a1741c3549479b03e388225253285dcc4fb7e1fa95f.jpg)
+
+![](images/8794dff584e65dfc7945dbd5ba208f88af0e561f2bf5c24e532328df620e5948.jpg)
+
+![](images/08d7313879a0c69d155eec92d51d3b48dd650fdad5cad32960ed84290332ed48.jpg)  
+Blurry Image B
+
+![](images/f02f5e55dde11b078d8a32d49d73d0074ebb1c6d1d21aa8904651b855aa0111f.jpg)
+
+![](images/2c76578cf401ed44700543a836eb598913a46120098889d345f8e8ebfa82bd08.jpg)
+
+![](images/569eff974134df313bb10d173531310e9b9deb49ab0af875d6756eedb62e48a2.jpg)
+
+![](images/1a87d346fca3fba4c8af0a2adddbe93911b4a616644411841f16eabd86ec4814.jpg)
+
+![](images/23acfb915f41f315182d242ff27ec487a37ef836482569c2a547d4e03733b242.jpg)
+
+![](images/659b25b1ba6cff0e3b5c4d6037b874c6e97210e7f3ce2e190da06f7e951b945a.jpg)
+
+![](images/1cb940fa31e9ecab4d036ea321f715bc5610d14fd611651190360c249d32bf4c.jpg)  
+Deblurred Image I  
+Refused Image  
+Figure 6: Visualization of refocusing on DPD-disp dataset. The bounding box is highlighted for clear observation, and region with red box refers to the user-specified region. The images in the first row refers to blurred image  $\mathbf{B}$  (unified by  $\mathbf{B} = \frac{1}{2} (\mathbf{B}_l + \mathbf{B}_r)$ ), images in the second row are deblurred image recovered by our model (Eq. 10), and the last row showcases the refocused images.
+
+![](images/992d972e6668e46ce9af1c0dbd100c30c2d4c4f9eb9e7ae98d3660cfd63defd5.jpg)
+
+![](images/6a3a9f0d3fe90ecbbe12ebd535b50a2aecfb2fa1971459567c313c1fa05943e1.jpg)
+
+![](images/c50dc412eac4b85e3b86266e30b13212f5e8c1dbc1c586defd4d4144ab75179a.jpg)
+
+![](images/8f558848e2e7d1e9c0fb6137a8fca3e5d14b41e849422ebe24902510b24df81f.jpg)  
+Refused Image
+
+![](images/ce2c0cd9b52782869f9b8e2d21455714229e03858093175a477e57df9025240e.jpg)
+
+![](images/ff043daef2f7d4cf65a7ff5166ec9433537936ed9a40f5c45a2f462770fe8e82.jpg)
+
+Visualization of Kernel Selection Behavior. We provide the visualization map of retrieved kernel index in kernel pool of different pixel locations to reveal deeper understanding. We first talk from the learned disparity feature  $\mathbf{F}_G$ , because the retrieved kernel intrinsically relies on the disparity feature. As shown in the second row in Figure 7, we use K-means clustering [23] to group the similar disparity feature with different cluster numbers,  $c = 6,10$  and then visualize the clustered group index. We can observe that our learned  $\mathbf{F}_G$  conforms with the actual depth or blur cues in  $\mathbf{B}$ . Done well with  $\mathbf{F}_G$ , we verify the reliability of using  $\mathbf{F}_G$  for kernel selection. As shown in kernel selection of Figure 7, the distribution of retrieved kernel index via Eq. 9 tends to approximate the blur distribution of the original blurred image. This observation further demonstrates the feasibility of our design: when performing refocusing, specifying the mask of focus would specify the disparity thereby further determining the kernel selection.
+
+Necessity of Disparity Alignment towards Refocusing Based on Eq. 2, we conclude that CoC map and disparity map provide the approximate indication clues for blur kernel estimation. Particularly, a pixel  $(i,j)$  with  $\mathbf{C}(i,j) = \mathbf{D}(i,j) = 0$  indicates to be well-focused. In practice, we empirically observe that training network only with disparity feature  $\mathbf{F}_{init}$  as blur kernel guidance enables faster convergence while achieving better deblurring and reblurring performance than that only using CoC feature. However, indirect supervision of  $\mathbf{F}_{init}$  through joint training on deblurring and reblurring tasks may not adequately capture explicit blur cues. Ideally, regions with  $\mathbf{F}_D = 0$  should exhibit perfect focus. As illustrated in Figure 4, in the absence of CoC map supervision, the disparity value corresponds to minimum PSNR happen to shift. With the aid of CoC supervision, the book is observed to be visu
+
+ally well-focused exactly.  $\{\mathbf{F}_D(i,j)|(i,j)\in \Omega_{book}\} = 0$  corresponds to minimum PSNR 23.6dB compared with ground-truth blurred image. It indicates that an accurate correspondence between disparity and blur kernel or defocus blur can be explicitly calibrated with the supervision of CoC map C. Moreover, we visualize the correspondence between  $\mathbf{F}_D$  and predicted CoC map, known as  $\mathcal{P}_{rgb}(\mathbf{F}_{init})$ , in Figure 5.
+
+Ablation of Components in Invertible Mapping. To investigate the impact of different invertible component designs towards the robustness of the visual representation, we tailor several variants. As shown in Table 6, model  $v1$  denotes adopting the  $1 \times 1$  convolution for channel shuffle [13], model  $v2$  adopts the residual invertible network similar to [18], and model  $v3$  incorporates affine transformation operations for more stable invertibility. Among them, model  $v3$  achieves the best performance. It demonstrates that robust invertible mapping enables more effec
+
+tive kernel learning for refocusing. More details of these variants are provided in the supplementary material.
+
+![](images/89ebba75f647529516410ebc7793e46526d5106f469648de1227c13a349da2c5.jpg)  
+Blurry Image B
+
+![](images/dbc9b109568d4eb37abb29bb94fd34f8057eb030ee50a7cd3bb1a3c92f921417.jpg)  
+Kernel Selection
+
+![](images/3cfe9f75938f110e88600ba074a42fd0c6bb3cd794f4f4edf34c93559aee5f9d.jpg)  
+Clustering on  $\mathbf{F}_G$  (c=10 and c=6)  
+In of kernel selection, a cavity feature  $\mathbf{F}_G$ .
+
+Table 6: The ablation of joint training on both deblurring and reblurring performance.  
+
+<table><tr><td>Variants</td><td>PSNR↑</td><td>SSIM↑</td><td>MAE(10-1)↓</td></tr><tr><td>v1 [13]</td><td>25.56</td><td>0.795</td><td>0.52</td></tr><tr><td>v2 [13]</td><td>26.21</td><td>0.810</td><td>0.35</td></tr><tr><td>v3 [18]</td><td>26.89</td><td>0.829</td><td>0.31</td></tr></table>
+
+# 5 Conclusion and Future Work
+
+Without bells and whistles, we re-frame the task of image refocus as two interconnected subtasks—joint reburring and deblurring using the shared kernel, and the disparity feature extracted from dual-pixel (DP) image pairs is leveraged as an indicator for blur kernel estimation. Specifically, our work experimentally explores several key aspects: (1) We investigate the potential benefits of using DP image pairs to achieve effective defocus control. (2) We mark the first attempt to demonstrate the feasibility of using joint deblurring and reburring as a proxy of image refocusing, which is a non-trivial endeavor. (3) We analyze the behavior of kernel selection and test its sensitivity to adjustment of disparity. (4) We incorporate an early-fusion SAM into our deblur-and-reblur framework to accurately identify the region of interest, enabling robust mask generation and flexible defocus control. In future work, we aim to adapt our method to the diffusion framework to further explore its practical applications and commercial potential.
+
+# Acknowledgments and Disclosure of Funding
+
+This work was supported by the National Natural Science Foundation of China under grant No. 62302045, and the Beijing Institute of Technology Special-Zone.
+
+# References
+
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+
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+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+# Answer: [Yes]
+
+Justification: We state all experimental details in Experiment section in the main paper and Supplementary Material A.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+# Answer: [Yes]
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
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+
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
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+Guidelines:
+
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+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Guidelines:
+
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+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
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+
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+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [Yes]
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
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+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
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+
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+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Guidelines:
+
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+
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
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+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
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+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [Yes]
+
+Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# Abstract of Appendix
+
+This appendix provides more implementation details (Appendix A), details of curated DP image dataset (Appendix B), adaptation for refocusing functionality (Appendix C), comparison with existing methods (Appendix D), and more visualization results and analysis (Appendix E).
+
+# A More Implementation Details
+
+Network Architecture. Our method adopts a 4-stage UNet-style architecture for progressive feature refinement. As mathematically formulated in Eq. 9 and Eq. 10 of main paper, each stage has two cascaded parameter-independent modules: (i) disparity-aware feature convolution, followed by (ii) invertible block  $\text{Inv}(\cdot)$  and  $\text{Inv}^{-1}(\cdot)$ . For (i), we use the disparity feature  $\mathbf{F}_{\text{coc}}(i,j)$  to retrieve a kernel  $\mathbf{K}_{i,j}$  from kernel pool  $\{\mathbf{K}_n\}_{n=1}^N$ , and then a point-wise convolution [51] with  $\mathbf{K}_{i,j}$  or  $\mathbf{K}_{i,j}^{-1}$  is employed for feature refinement to obtain  $\mathbf{U}^l$  or  $\mathbf{U}^{l+1}$ , and the architecture of (ii) is elaborated in Table 7. Notably, we parameterize learnable matrix  $\mathbf{W}$  directly in its LU decomposition [25] for efficiency, i.e.,  $\mathbf{W} = \mathbf{PL}(\mathbf{Q} + \mathrm{diag}(\mathbf{s}))$ , where  $\mathbf{P}$  is a permutation matrix,  $\mathbf{L}$  is a lower triangular matrix with ones on the diagonal,  $\mathbf{Q}$  is upper triangular matrix with zeros on the diagonal, and  $\mathbf{s}$  is a vector.
+
+Towards  $\mathcal{F}_m(\cdot)$  and  $\mathcal{F}_g(\cdot)$  in instructed SAM, we primarily adopt BEIT-3 [53] as  $\mathcal{F}_m(\cdot)$ , which implements a multi-way Transformer architecture. The input text is tokenized using the XLM-RobertaTokenizer [9]. Within each encoder block, image and text tokens are fused through cross-modal attention mechanisms and subsequently processed by separate Feed-Forward Networks (FFNs). Following the ViT paradigm [14], we extract the compact multimodal representation by taking the output of the  $[cls]$  token. Specifically, the original prompt encoder generates sparse embeddings of shape  $B\times P\times D$ , where  $B$  denotes the batch size,  $P$  is the number of input points or boxes, and  $D$  represents the embedding dimension. Following [62], we replace these spatial prompts with the projected multimodal embeddings from the multimodal encoder  $\mathcal{F}_m(\cdot)$ . If there are point or box inputs, these are concatenated with an empty sparse embedding tensor and perform alignment with the image feature extracted from  $\mathcal{F}_{sam}^v (\cdot)$ , subsequently fed into the mask decoder.
+
+Table 7: Illustration of invertible block operations  $\text{Inv}(\cdot)$  and  $\text{Inv}^{-1}(\cdot)$  in our method. In the forward mapping, the input and output to each block are denoted as  $\mathbf{U}^l$  and  $\mathbf{U}^{l+1}$ . During the backward mapping, only  $\text{Plus}(+)$  and  $\text{Multiply}(\odot)$  needs to be inverted.  $\phi_1, \phi_2, \phi_3$  and  $\phi_4$  do not need to be inverted, which can be any neural networks.
+
+<table><tr><td>#</td><td>Forward Operation Inv(·)</td><td>Backward Operation Inv-1(·)</td><td>Specification</td></tr><tr><td>R0</td><td>\(\tilde{\mathbf{U}}^l=\mathbf{PL}(\mathbf{Q}+\mathrm{diag}(\mathbf{s}))\mathbf{U}^l\)</td><td>\(\tilde{\mathbf{U}}^{l+1}=(\mathbf{Q}+\mathrm{diag}(\mathbf{s}))^{-1}\mathbf{P}^{-1}\mathbf{L}^{-1}\mathbf{U}^{l+1}\)</td><td>\(\mathbf{L}(\mathbf{Q}+\mathrm{diag}(\mathbf{s}))\) denotes the matrix by LU decomposition [25], and \(\mathbf{P}\) is a permutation matrix.</td></tr><tr><td>R1</td><td>\(\mathbf{U}_a^l,\mathbf{U}_b^l=\mathrm{Split}(\tilde{\mathbf{U}}^l)\)</td><td>\(\mathbf{U}_a^{l+1},\mathbf{U}_b^{l+1}=\mathrm{Split}(\tilde{\mathbf{U}}^{l+1})\)</td><td>Split(·) denotes the channel-wise split.</td></tr><tr><td>R2</td><td>\(\mathbf{U}_a^{l+1}=\mathbf{U}_a^l\odot\exp(\phi_1(\mathbf{U}_b^l))+\phi_2(\mathbf{U}_b^l)\)</td><td>\(\mathbf{U}_b^l=(\mathbf{U}_b^{l+1}-\phi_4(\mathbf{U}_a^{l+1}))/\exp(\phi_3(\mathbf{U}_a^{l+1}))\)</td><td rowspan="2">\(\phi_1,\phi_2,\phi_3\) and \(\phi_4\) can be any neural networks. ⊙ is the multiply operation.</td></tr><tr><td>R3</td><td>\(\mathbf{U}_b^{l+1}=\mathbf{U}_b^l\odot\exp(\phi_3(\mathbf{U}_a^{l+1}))+\phi_4(\mathbf{U}_a^{l+1})\)</td><td>\(\mathbf{U}_a^l=(\mathbf{U}_a^{l+1}-\phi_2(\mathbf{U}_b^l))/\exp(\phi_1(\mathbf{U}_b^l))\)</td></tr><tr><td>R4</td><td>\(\mathbf{U}^{l+1}=\mathrm{Concat}(\mathbf{U}_a^{l+1},\mathbf{U}_b^{l+1})\)</td><td>\(\mathbf{U}^l=\mathrm{Concat}(\mathbf{U}_a^l,\mathbf{U}_b^l)\)</td><td>Concat(·) is the channel-wise concatenation.</td></tr></table>
+
+As mentioned in Table 6 of main paper, we adopt two variants  $v1$  and  $v2$  to investigate the impact of different invertible blocks towards visual representation learning. Their architectures are illustrated in Table 8 and Table 9, respectively.
+
+Table 8: The invertible block of variant  $v1$ .  
+
+<table><tr><td>#</td><td>Forward Operation</td><td>Backward Operation</td><td>Specification</td></tr><tr><td>R0</td><td>Ul+1 = PL(Q + diag(s))Ul</td><td>Ul = (Q + diag(s))-1L-1P-1Ul+1</td><td>L(Q + diag(s)) denotes the matrix by LU decomposition [25], and P is a permutation matrix.</td></tr></table>
+
+Hyper-parameter Setting. Our  $\mathcal{L}_{deb}$  and  $\mathcal{L}_{reb}$  both use a combination of Multi-Scale Charbonnier loss  $\mathcal{L}_{char}$  [63], Multi-Scale Edge loss  $\mathcal{L}_{edge}$  [63], and Multi-Scale Frequency loss  $\mathcal{L}_{freq}$  [34], i.e.,  $\mathcal{L}_{reb} = \mathcal{L}_{deb} = \mathcal{L}_{char} + \lambda_1\mathcal{L}_{edge} + \lambda_2\mathcal{L}_{freq}$ . We set  $\lambda_{1} = 5\times 10^{-2}$  and  $\lambda_{2} = 1\times 10^{-2}$ . Regarding
+
+Table 9: The invertible block of variant  $v2$ .  
+
+<table><tr><td>#</td><td>Forward Operation</td><td>Backward Operation</td><td>Specification</td></tr><tr><td>R0</td><td>\(\tilde{\mathbf{U}}^l=\mathbf{PL}(\mathbf{Q}+\mathrm{diag}(\mathbf{s}))\mathbf{U}^l\)</td><td>\(\tilde{\mathbf{U}}^{l+1}=(\mathbf{Q}+\mathrm{diag}(\mathbf{s}))^{-1}\mathbf{L}^{-1}\mathbf{P}^{-1}\mathbf{U}^{l+1}\)</td><td>\(\mathbf{L}(\mathbf{Q}+\mathrm{diag}(\mathbf{s}))\) denotes the matrix by LU decomposition [25], and \(\mathbf{P}\) is a permutation matrix.</td></tr><tr><td>R1</td><td>\(\mathbf{U}_a^l,\mathbf{U}_b^l=\mathrm{Split}(\tilde{\mathbf{U}}^l)\)</td><td>\(\mathbf{U}_a^{l+1},\mathbf{U}_b^{l+1}=\mathrm{Split}(\tilde{\mathbf{U}}^{l+1})\)</td><td>Split(·) denotes the channel-wise split.</td></tr><tr><td>R2</td><td>\(\mathbf{U}_a^{l+1}=\mathbf{U}_a^l+\phi_1(\mathbf{U}_b^l)\)</td><td>\(\mathbf{U}_b^l=\mathbf{U}_b^{l+1}-\phi_3(\mathbf{U}_a^{l+1})\)</td><td rowspan="2">φ1, φ2, φ3 and φ4 can be any neural networks.</td></tr><tr><td>R3</td><td>\(\mathbf{U}_b^{l+1}=\mathbf{U}_b^l+\phi_2(\mathbf{U}_a^{l+1})\)</td><td>\(\mathbf{U}_a^l=\mathbf{U}_a^{l+1}-\phi_4(\mathbf{U}_b^l)\)</td></tr><tr><td>R4</td><td>\(\mathbf{U}^{l+1}=\mathrm{Concat}(\mathbf{U}_a^{l+1},\mathbf{U}_b^{l+1})\)</td><td>\(\mathbf{U}^l=\mathrm{Concat}(\mathbf{U}_a^l,\mathbf{U}_b^l)\)</td><td>Concat(·) is the channel-wise concatenation.</td></tr></table>
+
+other two loss supervision  $\lambda_{coc}$  and  $\lambda_{grad}$ , we set  $\lambda_{coc} = 0.5$  due to that a excessive large  $\lambda_{coc}$  would overwhelm the useful cues in  $\mathbf{F}_{init}$  learned from reblurring and deblurring task, and  $\lambda_{grad} = 0.5$  to reserve the high-frequency information and sharp the edge in restored image.
+
+Design of Gate Vector R in Eq. 7. A computed in Eq. 8 of main paper,
+
+$$
+\mathbf {F} _ {c o c} = \mathbf {R} \odot \mathbf {F} _ {G} + \mathcal {P} _ {\text {f e a t}} \left(\mathcal {P} _ {r g b} \left(\mathbf {F} _ {\text {i n i t}}\right)\right), \tag {17}
+$$
+
+where  $\mathbf{R}$  serves as the gating vector to balance the learning between the vanilla disparity feature  $\mathbf{F}_G$  and CoC-aligned feature  $\mathcal{P}_{rgb}(\mathbf{F}_{init})$ . Specifically, it is formulated as,
+
+$$
+\mathbf {R} = \operatorname {T a n h} \left(\mathcal {P} _ {G} \left(\mathbf {F} _ {G}\right) + \mathcal {P} _ {C} \left(\mathcal {P} _ {\text {f e a t}} \left(\mathcal {P} _ {\text {r g b}} \left(\mathbf {F} _ {\text {i n i t}}\right)\right)\right)\right), \tag {18}
+$$
+
+where  $\mathcal{P}_G(\cdot)$  and  $\mathcal{P}_C(\cdot)$  are two simple linear neural layers. We empirically observe that using the gate  $\mathbf{R}$  could strengthen the gradient of  $\mathbf{F}_{init}$ , and adaptively balance the contribution of  $\mathbf{F}_{init}$  and  $\mathbf{F}_G$  to feature vector  $\mathbf{F}_{coc}$ .
+
+# B Details of Curated DP image Dataset
+
+In this paper, We present a real-world DP dataset consisting of 10 high-quality pairs with a resolution of  $6720 \times 4480$  pixels, for refocusing evaluation. The DP image is captured by Canon EOS 5D MarkIV<sup>1</sup>. The camera sensor features two independent photodiodes embedded within each pixel, enabling phase-detection autofocus for rapid focusing. During image capture, the left and right sub-pixels combine their outputs to generate the final view. After collection, we use software Digital Photo Professional<sup>2</sup> to split the DP view from the captured center one. In our dataset, the image pairs are captured using aperture settings corresponding to  $f / 2.8$ , which results in the greatest DoF and thus most defocus blur. As shown in Figure 8, we select two objects  $o_1$  and  $o_2$  as focal points to form a image pair  $(\mathbf{B}_{o1}, \mathbf{B}_{o2})$ , and each of pair  $\mathbf{B}$  contains a dual-pixel image pair  $(\mathbf{B}_{.,l}, \mathbf{B}_{.,r})$ . When performing evaluation, we take one dual-pixel image pair as the input, and regard another image (center view) as the target image.
+
+Notably, considering that when two objects are far apart, focusing on different objects may introduce noticeable misalignment of the captured scene. To address this issue, we adopt the following strategies during the captured process: (1) Restrict the distance between the two focus targets within a certain range to allow only minimal and acceptable shifts. (2) Manually crop and align the two images when slight misalignment still occur. (3) First adopting classical image matching techniques [33, 46] for pixel matching, and then compute corresponding quantitative metrics. All the datasets will be released.
+
+# C Adaptation of Refocusing Functionality
+
+We manually modify two SOTA methods on deblurring task, K3DN and Omni-Kernel [11], for refocusing adaption. For K3DN (architecturally similar to our approach), we replace our 4-level UNet with K3DN's backbone while preserving only R0 (Table 7) to maintain essential reversibility. For Omni-Kernel that has more complex architecture, we simply concatenate disparity feature and DP image features in a channel-wise manner, and take them as input for target image approximation.
+
+![](images/88859e6b7dcaea5d89afc6b4de2bf38ac135dbda1a50cd8c737ef143dba06575.jpg)  
+B01J
+
+![](images/5203ec5c9f6c6970fdc0bbea7843f365b6482ec9eb0b837703f24d13db61f8ac.jpg)  
+B01r
+
+![](images/8902b2d67b280d2dbfeb1e6a37661db7994cabb428a06107c562354c1913009e.jpg)  
+B02.1
+
+![](images/fb461f772a4cfc3a599f7d47ec1e96c739e1983b54f30a3a5a327e9cb763539e.jpg)  
+B02r  
+Figure 8: Three examples of our collected DP image pairs. The red and green circles indicate the focal points, while the orange bounding box highlights the zoomed-in regions.
+
+![](images/f1ca3f8d6446393ea945683a05fb9b7f999bc3fab7ee98fe5f0ed1201c87987f.jpg)
+
+# D Comparison with Existing methods
+
+Compared to existing deblur-and-reblur frameworks (e.g., RefocusGAN), our key improvements are:
+
+- Reversible Block with Shared Kernel: We establish an invertible connection between deblurring and reblurring with shared kernel learning, which strictly follows the mathematical formulation in Eq. 11 while achieving better deblurring and reblurring performance.  
+- Unified Network Architecture: We reuse one network parameter to achieve refocusing task (deblur-and-reblur), significantly reducing the total number of parameters and improving training efficiency.
+
+To highlight our contribution, we explain the necessity of invertibility in what follows. The network's invertibility ensures that reblurring and deblurring can be jointly trained while sharing the blur kernel, better establishing one-to-one correspondence between disparity feature and blur kernel. Intuitively, an effective disparity-aware blur kernel should be capable of both restoring an image (deblurring) and blurring it (reblurring). This paves a crucial path for the subsequent refocusing, enabling accurate kernel retrieval for defocus control by modifying the disparity (refer to Figure 7 for illustration). When we empirically adopt a traditional encoder-decoder architecture, the correspondence between the kernel and disparity becomes difficult to learn effectively, leading to unintended control effects.
+
+# E More Visualization Results and Analysis
+
+![](images/34442170926c0a4d9e6fd9ceb32dd5c3cedc910f03b01d41cebaee7c84fbcd53.jpg)  
+Blurry Image B
+
+![](images/e6dccc8809e3052d03525abe3c72fa0487371be841e74fb2e5f72d0af9093fcf.jpg)  
+Kernel Selection Preference  
+Figure 9: Kernel selection preference when performing refocusing from all-in-focus image. Accumulated Weight refers to assigned weight score of each blur kernel  $\mathbf{K}_n$  accumulated across all the layer in invertible network.
+
+![](images/78d9b0912a7dec05353ce60bbacda7b9b5515e45b30339a6f785abac07dcec02.jpg)  
+Blurry Image B
+
+![](images/b42f383b812b1b7bd428f5be4c0e70a5d8348858f6c4bd5b51c71dbada61cf0b.jpg)  
+Kernel Selection Preference
+
+Kernel Setting. We visualize the kernel selection preference of two cases in Figure 9 for deeper behavior revelation. It can be observed that most kernels has a smaller accumulated weight while only a small part of kernels are activated for blur indication. In these two cases, the commonly used kernels lies in index \{56, 57, 65, 73, 86\}, and kernels with index \{3, 17, 22, 80\} are sample-specific. We also employ the hard selection of blur kernel, which can be differentiably achieved by Gumbel-Softmax [21]. However, we empirically observe that hard selection achieves the suboptimal deblurring and reblurring performance. We posit that this issue arises because hard selection induces
+
+Table 10: Performance with different number of pre-defined kernel  $\{\mathbf{K}_n\}_{n = 1}^N$  
+
+<table><tr><td>N</td><td>32</td><td>48</td><td>96</td><td>128</td></tr><tr><td colspan="5">Deblurring Performance</td></tr><tr><td>PSNR↑</td><td>25.92</td><td>26.14</td><td>26.89</td><td>26.82</td></tr><tr><td colspan="5">Reblurring Performance</td></tr><tr><td>PSNR↑</td><td>28.32</td><td>28.46</td><td>29.18</td><td>29.18</td></tr></table>
+
+Table 11: Comparison of refocusing performance on self-collected dataset.  
+
+<table><tr><td>Variants</td><td>PSNR↑</td><td>SSIM↑</td><td>MAE(10-1)↓</td><td>LPIPS↓</td></tr><tr><td>Omni-Kernel</td><td>19.50</td><td>0.690</td><td>0.56</td><td>32.94</td></tr><tr><td>K3DN</td><td>19.52</td><td>0.698</td><td>0.52</td><td>31.91</td></tr><tr><td>Ours</td><td>19.68</td><td>0.707</td><td>0.52</td><td>30.87</td></tr></table>
+
+an initial bias toward specific kernels during the early stages of training, which subsequently hinders the optimization process for kernel selection. Additionally, we further investigate how different number of pre-defined kernels affect model performance. As shown in Table 10, Both insufficient and excessive numbers of kernels impair model performance. Fewer kernels cannot adequately cover the necessary blur range across the dataset, while too many introduce excessive non-trainable parameters that compromise model robustness.
+
+![](images/b4b9a5d97384531fc7df7851230467018dac00d69ba45fe42d15c7a2944c7215.jpg)  
+Figure 10: Visualization comparison of refocusing on self-collected DP image pairs. The red point indicates the focal object, and the yellow bounding box highlights the zoomed-in regions.
+
+More Refocusing Results on Self-Collected DP image pair. We give the quantitative results on our self-collected dataset in Table 11, our method achieves the consistent superior results across all the metrics. As shown in Figure 10, we take the first case as the example, the sheep is focused initially, and our goal is to transfer the focal point from sheep to its right bottle. We compare our method with two SOTA baselines K3DN and Omni-Kernel, the comparison results show that our refocusing result is more scene-realistic.
+
+Evaluation on Mask Quality. To evaluate the mask quality from our adopted SAM, we manually annotate two objects for each image of our collected images as the groundtruth. We compare the average mIoU metrics [15] on different versions. From Table 12, we observe that early fusion achieves the clear-cut performance compared with late fusion. Furthermore, our structured template-based prompt achieves better segmentation result.
+
+Performance Variation with Different Blur Degree. Besides, we also exploit instructed SAM segmentation results towards the image inputs with different degree of defocus blur and our restored image  $\hat{\mathbf{I}}$ . We sample ten groups of sample from DP5K-test and each of which has 5 cases with different  $f$ -number, 1.8, 2.8, 4.0, and 5.6. We use the SAM with early fusion to generate the mask, and manually annotate the region with interest as the ground-truth. As
+
+can be seen from Table 13, we empirically observe that our used SAM could functions well when handling the image with large defocus blur (i.e.,  $f - 1.8$ ), and using our restored image as the SAM input could achieve the best result. Beyond that, with the blur degree decreasing, the segmentation result becomes saturated.
+
+Table 12: Comparison of segmentation performance with different prompts. † and * denote the early-fusion and late-fusion version, respectively. Ours denotes the template-based prompt.  
+
+<table><tr><td>3-Click†</td><td>Vanilla Prompt†</td><td>Ours*</td><td>Ours†</td></tr><tr><td>93.6</td><td>87.5</td><td>81.6</td><td>91.8</td></tr></table>
+
+Table 13: Segmentation performance with different degrees of defocus blur.  
+
+<table><tr><td>f-1.8</td><td>f-2.8</td><td>f-4.0</td><td>f-5.6</td><td>I</td></tr><tr><td>85.8</td><td>90.3</td><td>90.8</td><td>91.1</td><td>91.8</td></tr></table>
\ No newline at end of file
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