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+# CAMEX: CURVATURE-AWARE MERGING OF EXPERTS
+
+Dung V. Nguyen $^{1*}$  Minh H. Nguyen $^{1*}$  Luc Q. Nguyen $^{2*}$  Rachel S.Y. Teo $^{3}$  Tan M. Nguyen $^{3\dagger}$  Linh Duy Tran $^{2\dagger}$
+
+$^{1}$ Faculty of Mathematics and Informatics, Hanoi University of Science and Technology  
+$^{2}$ Viettel AI, Viettel Group  
+$^{3}$ Department of Mathematics, National University of Singapore  
+{dung.nv232215M, minh.nh232331M}@sis.hust.edu.vn  
+{lucnq1, linhtd15}@viettel.com.vn  
+rachel.tsy@u.nus.edu, tanmn@nus.edu.sg
+
+# ABSTRACT
+
+Existing methods for merging experts during model training and fine-tuning predominantly rely on Euclidean geometry, which assumes a flat parameter space. This assumption can limit the model's generalization ability, especially during the pre-training phase, where the parameter manifold might exhibit more complex curvature. Curvature-aware merging methods typically require additional information and computational resources to approximate the Fisher Information Matrix, adding memory overhead. In this paper, we introduce CAMEx (Curvature-Aware Merging of Experts), a novel expert merging protocol that incorporates natural gradients to account for the non-Euclidean curvature of the parameter manifold. By leveraging natural gradients, CAMEx adapts more effectively to the structure of the parameter space, improving alignment between model updates and the manifold's geometry. This approach enhances both pre-training and fine-tuning, resulting in better optimization trajectories and improved generalization without the substantial memory overhead typically associated with curvature-aware methods. Our contributions are threefold: (1) CAMEx significantly outperforms traditional Euclidean-based expert merging techniques across various natural language processing tasks, leading to enhanced performance during pre-training and fine-tuning; (2) we introduce a dynamic merging architecture that optimizes resource utilization, achieving high performance while reducing computational costs, facilitating efficient scaling of large language models; and (3) we provide both theoretical and empirical evidence to demonstrate the efficiency of our proposed method. The code is publicly available at: https://github.com/kpup1710/CAMEx.
+
+# 1 INTRODUCTION
+
+Sparse Mixture of Experts (SMoE) (Jacobs et al., 1991; Shazeer et al., 2017) is currently a core component for constructing foundation and large language models (LLMs), whose parameters count can rise up to billions and trillions (Devlin et al., 2019; Yang et al., 2019; Liu et al., 2019; Raffel et al., 2020; Nguyen et al., 2022b; Fedus et al., 2022; Wei et al., 2022; Teo & Nguyen, 2025). Nevertheless, (Hoffmann et al., 2024; Kaplan et al., 2020) recognized a scaling law that underpins the LLM's evolution, which is larger models require exponentially more computational resources and data to continue improving, and without sufficient scaling in all dimensions, performance gains may plateau. Thus, identifying and implementing efficient methodologies for the sustainable scaling of LLMs is imperative. SMoE addresses this challenge by sparsely activating parameters of large models, which can boost model performance with only minor losses in computational efficiency. The methodology is integrated chiefly into feedforward layers of transformers, processing tokens by selectively activating a small number of experts and hence trimming down the computing memory and FLOPS (Fedus et al., 2022; Lepikhin et al., 2021).
+
+Since its debut in (Shazeer et al., 2017), SMoE has gone through numerous explorations and advancements in routing mechanism development and expert architecture design. (Dai et al., 2022) proposes a two-phase
+
+training strategy for stabilizing the gate function so that the expert's selection of one token does not fluctuate between different inference times. (Zhou et al., 2022) changes the perspective of the router to experts with experts choice routing, ensuring a balancing load between experts. (Teo & Nguyen, 2024) incorporates heavy-ball momentum in SMoE to enhance the model's stability and robustness. (Chi et al., 2022) and (Chen et al., 2023) address the concern of representation collapse in SMoE by proposing cosine scoring and a fixed random initialized router, respectively. Some other works view the routing mechanism as a reinforcement learning, clustering, or optimal transport problem (Nielsen et al., 2025). In terms of expert design orientation, (Rajbhandari et al., 2022) and (Dai et al., 2024) introduce the concept of shared experts wherein each token is processed by a fixed expert and another selected through gating, achieving two experts engagement per layer without increasing the communication cost beyond that of top-1 gating. (Muqeeth et al., 2024) proposes to merge experts by taking the weighted mean of the expert's parameters with respect to router scores. This methodology is then extended in (He et al., 2023), (Zhong et al., 2024), and (Li et al., 2024) for causal language modeling pretraining and fine-tuning tasks.
+
+Among existing rigorous research on SMoE, our work focuses on the experts merging lines of research. Specifically, we systemically integrate natural gradient into task-specific merging protocol for SMoE. To the best of our knowledge, the current merging protocol applied for SMoE still deems the parameter space of the expert's parameters as Euclidean ones. Nevertheless, it has been shown that the space of neural network parameters brings the characteristic of the Riemannian manifold (Amari, 1998). Therefore, it is natural for us to make an effort in such a direction for merging experts. Although some existing works on merging models have already leveraged the Fisher Information Matrix (Matena & Raffel, 2022; Jin et al., 2023), we find that they require large computational space and complicated steps to perform well. In contrast, our merging protocol is simple and straightforward to implement while still taking into account the curvature of the parameters manifold. We discover the superior performance of curvature-aware merging in our method compared to the regular merging procedure applied to SMoE. Our main contributions are three-fold:
+
+1. We introduce a novel rapid and efficient merging technique named Curvature-Aware Merging of Experts (CAMEx) for SMOE that includes information about the curvature of the expert's parameters manifold.  
+2. We propose a new architecture based on CAMEx, which dynamicalizes the merging protocol along with parameters reduction. Our empirical experiments prove the dominant performance of this architecture on pre-training tasks.  
+3. We theoretically prove that our CAMEx obtains better alignment between experts and the training task domain.
+
+We empirically demonstrate that 1) our proposed merging method can add in rapidness of convergence speed for pre-training and 2) when combined with other merging protocols, it can boost the model's performance on a variety of practical tasks, including language modeling, text classification, question answering, and image classification.
+
+# 2 CURVATURE-AWARE MERGING OF EXPERTS
+
+This section aims to give an overview of model merging methods and their integration into SMoE architecture. We then introduce our curvature-aware merging protocol stamping from the natural gradient. Finally, we perform a theoretical analysis to support our proposal.
+
+# 2.1 BACKGROUND: EXPERT MERGING IN SPARSE MIXTURE OF EXPERTS
+
+It is convenient to recall the concept of SMOE and a few well-known experts merging methods for SMOE. From this point to the rest of our paper, let us use the notations summarized in Table 1.
+
+Sparse Mixture of Experts. A SMOE layer processes the tokens series as follows:
+
+$$
+\left\{ \begin{array}{l} \mathbf {y} _ {t} = \sum_ {i \in S _ {t}} \mathbf {G} (t, i) \cdot \mathbf {E} _ {i} \mathbf {h} _ {t} \\ \mathbf {G} (t, \cdot) = \operatorname {s o f t m a x} \left(\mathbf {W} _ {g} \mathbf {h} _ {t}\right) \\ S _ {t} = \operatorname {t o p - k} (\mathbf {G} (t, \cdot)) \end{array} \right. \tag {1}
+$$
+
+SMEAR. (Muqeeth et al., 2024) introduces the ensemble of expert parameters through weighted average computation with the factors being the router scores.
+
+![](images/7faf8dddef9ec462dd717afa4f83e4f5fea4d1d3fa42d3ae5c531dcdc6fa86d1.jpg)  
+Figure 1: Overview of CAMEx for a causal language modeling SMoE. The experts are merged through the router scores and the curvature-matrix M. During the merging protocol, we can generate the masks for the domain-vectors, denoted as  $\gamma_{i}$ , such as Ties or Dare. We follow the causal segmenting pipeline from (Zhong et al., 2024) to achieve both memory efficiency and causal information constraints. Note that stop gradient operator is applied for the first segment router scores.
+
+Table 1: Notations and Definitions.  
+
+<table><tr><td>Symbol</td><td>Description</td><td>Symbol</td><td>Description</td></tr><tr><td>T</td><td>Number of tokens</td><td>k</td><td>Number of selected experts</td></tr><tr><td>N</td><td>Total number of experts</td><td>Ei∈Rd×h</td><td>Weights for theith expert</td></tr><tr><td>h∈RT×d</td><td>Input tokens or hidden states</td><td>G(·,·)∈RT×N</td><td>Gating function output</td></tr><tr><td>St</td><td>Set of top-k experts for token ht</td><td>α∈[0,1]</td><td>Rescalling factor</td></tr></table>
+
+Task-specific merging in SMoE. Our work will follow the scheme of task-specific merging (Ilharco et al., 2023). In such a setting, we assume the existence of  $N$  pre-trained models parameterized by  $\theta_{i}$ , each of which was pre-trained on a different task. We then define the task-vector for each pre-trained model through the merged model  $\theta_{m}$  as  $\tau_{i} = \theta_{i} - \theta_{m}$ . The merging protocol will be performed by Eqn. Merg. Under the context of SMoE, each expert learns to handle a particular subset of the input space or specializes in a specific type of feature or pattern (Jacobs et al., 1991; Dai et al., 2024). We believe it is more suitable to reference this technique as domain-specific merging. We, therefore, will rename the tensors  $\tau_{i} = \mathbf{E}_{i} - \mathbf{E}_{m}$  as domain-vector. Additionally, to take the router information into account, we will define the formulation for domain-specific merging in a SMoE layer as follows:
+
+$$
+\hat {\mathbf {E}} _ {m} = \mathbf {E} _ {m} + \alpha \sum_ {i = 1} ^ {N - 1} s _ {i} \tau_ {i} \tag {2}
+$$
+
+where  $s_i$  denotes the score of the router for the  $i$ th expert. We want to note that with  $0 < \alpha < 1$ , domain-specific merging aligns with soft merging.
+
+# 2.2 BACKGROUND: OTHER MODEL MERGING METHODS
+
+In this section, we discuss other recent and widely-adopted model merging methods outside the context of SMOE that we will combine with our curvature-aware merging method in our experiments in Section 3.
+
+TIES merging. (Yadav et al., 2023) improves upon task arithmetic by removing interference between the task vectors. Specifically, TIES zeros out entries in a given task vector with low magnitude and resolves sign conflicts across different task vectors.
+
+DARE merging. Different from TIES, DARE merging randomly zeros out the neurons like a Dropout layer (Yu et al., 2024).
+
+Fisher merging. Existing work on Fisher merging suffers from computational complexity since computing and inverting the Fisher Information Matrix, especially for large neural networks, is often intractable. Even when using approximations like diagonal or block-diagonal Fisher matrices, these methods can
+
+still be computationally expensive and challenging to apply at scale. Furthermore, the accuracy of Fisher approximations, such as diagonal or block-diagonal, can be problematic (Matena & Raffel, 2022).
+
+# 2.3 GRADIENT INTERPRETATION OF MODELS MERGING
+
+We want to emphasize the alignments between the paradigm of gradient descent and model merging. For this, we denote  $\theta \in \mathbb{R}^N$ ,  $\mathcal{L}(\theta)$ , and  $\eta$  as the model's parameters, the empirical loss function, and the learning rate, respectively. During the training process of a deep learning model, the parameters are updated following the gradient descent formula:
+
+$$
+\theta_ {n + 1} = \theta_ {n} + \eta (- \nabla \mathcal {L} (\theta_ {n})) \tag {GD}
+$$
+
+In the aspect of deep models merging, we also have an update rule in a similar manner, which is
+
+$$
+\hat {\theta} ^ {m} = \theta^ {m} + \alpha \sum_ {i = 1} ^ {n} \underbrace {\left(\theta^ {i} - \theta^ {m}\right)} _ {\substack {\text {gradient-like} \\ \text {update direction}}} \tag{Merg}
+$$
+
+where  $\theta^m$  denotes the merged model's parameters, and  $\theta^i$  denotes the parameters of the  $i$ th expert. Here, we interpret  $\theta^i$  as the optimal parameters of the model for a specific task or domain, and then the update rule gives us a direction toward optimizing for all tasks.
+
+However, it has been pointed out by (Amari, 1998) that the parameter space structure of deep learning models has Riemannian characteristics. Therefore, a more natural gradient updating scheme was proposed,
+
+$$
+\theta_ {n + 1} = \theta_ {n} + \eta \underbrace {G \left(\theta_ {n}\right) (- \nabla \mathcal {L} \left(\theta_ {n}\right))} _ {\text {n a t u r a l g r a d i e n t}} \tag {NGD}
+$$
+
+In this formula,  $G(\theta_n) \in \mathbb{R}^{N \times N}$  denotes the Riemannian metric tensor (Amari, 1998; Amari & Douglas, 1998), which characterizes the intrinsic curvature of a particular manifold in  $N$ -dimensional space (Martens, 2020) or sometimes, the inversed Fisher Information Matrix. The same ideology was introduced for merging large language models in Fisher merging (Matena & Raffel, 2022) and Regmean (Jin et al., 2023). However, both methods suffer from the bottleneck in the computation cost of approximating the Fisher Information. Moreover, these methods are challenging to apply in sparse layers of SMOE since they would introduce huge latency, FLOPS, and memory for computing and storing matrices whose number of entries is proportional to a number of expert parameters.
+
+# 2.4 EXPERTS MERGING WITH CURVATURE-AWARE
+
+In this section, for the sake of conciseness, we focus on the language modeling task; a similar methodology can be applied to other tasks, such as classification. We introduce an efficient way to merge experts within SMoE layers, based on the causal segmenting approach proposed by (Zhong et al., 2024). The goal of the causal segment routing strategy is to enhance the efficiency of expert merging operations while maintaining the autoregressive nature of language models. More details about this algorithm can be found in Appendix C.1 and Algorithm 1. We then perform the following merging protocols:
+
+$$
+\hat {\mathbf {E}} _ {m} ^ {l} = \mathbf {E} _ {m} ^ {l} + \alpha \sum_ {i = 1} ^ {N - 1} \mathbf {M} _ {i} \cdot \left(s _ {i} ^ {l} * \tau_ {i} ^ {l}\right) \tag {CA-Merg}
+$$
+
+where  $\mathbf{M}_I\in \mathbb{R}^{d_{in}d_{out}\times d_{in}d_{out}}$  denote the curvature matrix which performs matrix product with the gradient-like component. The curvature of the parameters manifold will be learned through these tensors while optimizing the empirical loss. This approach has also proven its effectiveness in meta-learning for few-shot classification (Park & Oliva, 2019). We further explore the computing efficiency of merging experts by proposing a novel dynamic merging formula
+
+$$
+\left\{ \begin{array}{l l} \mathbf {E} _ {m} ^ {l + 1} & = \mathbf {E} _ {m} ^ {l} + \frac {\alpha}{N - 1} \sum_ {i = 1} ^ {N - 1} \mathbf {M} _ {i} \cdot \tau_ {i} ^ {l} \\ \hat {\mathbf {E}} _ {m} ^ {l + 1} & = \mathbf {E} _ {m} ^ {l + 1} + \alpha \sum_ {i = 1} ^ {N - 1} \mathbf {M} _ {i} \cdot \left(s _ {i} ^ {l + 1} * \tau_ {i} ^ {l + 1}\right) \end{array} \right. \tag {Dynamic-Merg}
+$$
+
+![](images/1107dd3083907b1e8d49bb9bd02842f7bb34bc7a42a546dfe4c8339850b9c186.jpg)  
+Figure 2: Overall architecture of different SMOE layers. The figure presents the vanilla SMOE layer on the left, the merging expert layer in the middle, and our proposed dynamic merging SMOE layer on the right. Our architecture reduces the number of parameters compared to the other two, while maintaining the same number of activated neurons per layer. Importantly, despite the dynamic merging mechanism, our architecture preserves the same number of experts at each layer as the other SMOE architectures, ensuring comparable model capacity, i.e., the number of activated parameters per layer.
+
+![](images/b43777e0ad53701b0919530c2e4866e1c7855a24abc2e1b5de17c69f6fd78ac9.jpg)
+
+![](images/ecf96ebe6ca84547c59c6dec4164f86e35915e0e1c3ea1421f45d882233bd416.jpg)
+
+The architecture corresponding to this recurrent representation can be found in Figure 2. The architecture contains a global expert that traverses through the SMoE layers by the updating rule in Eqn. Dynamic-Merg. Not only will this allow a notable reduction in model size and GFLOPS, but it also ensures the number of experts in each SMoE is the same as in the full-expert setting, where each layer has the same number of experts. We refer to Appendix F for a step-by-step walkthrough of key equations in CAMEx.
+
+# 2.5 EFFICENCY
+
+Parameter efficient approximation of curvature matrix. Storing and computing a curvature matrix requires a whopping memory and time complexity of  $O(n^{4})$  and  $O(n^{4})$ , respectively. This is infeasible even for a simple SMoE layer, as one layer can contain many experts. To mitigate this problem, we follow (Martens & Grosse, 2015) and approximate the curvature matrix using the Kronecker product. It has been proven by (Hameed et al., 2022) that we can approximate an arbitrary matrix using a finite sum of Kronecker products. For a curvature matrix  $\mathbf{M}_i \in \mathbb{R}^{d_{in}d_{out} \times d_{in}d_{out}}$ , we present the rank-1 approximation as below:
+
+$$
+\mathbf {M} _ {i} \approx \mathbf {M} _ {i} ^ {\text {i n}} \otimes \mathbf {M} _ {i} ^ {\text {o u t}} \tag {3}
+$$
+
+with  $\mathbf{M}_i^{in} \in \mathbb{R}^{d_{in} \times d_{in}}$  and  $\mathbf{M}_i^{out} \in \mathbb{R}^{d_{out} \times d_{out}}$ . Still, this form of approximation is too large to compute and store during training time, so we further decompose  $\mathbf{M}_i^{in}$  and  $\mathbf{M}_i^{out}$  using Kronecker product because of the efficient computation using tensor algebra. This form of approximation reduces the number of parameters added and only puts negligible memory and computational overhead to the training process at the cost of additional  $O(n)$  memory complexity and  $O(n^{2.5})$  computational complexity. Although this might limit the representative capacity of the curvature matrix, we empirically find that the performance of our method still surpasses other merging methods.
+
+Efficient test-time inference with reparameterization. We focus on the case where  $\alpha = 1$ . To further optimize the computation of curvature-aware merging, we embed the curvature matrices into the domain-vectors using the following reparameterization trick:
+
+$$
+\mathbf {E} _ {i} ^ {\prime} \leftarrow \mathbf {E} _ {m} + \mathbf {M} _ {i} \cdot \tau_ {i} \tag {4}
+$$
+
+In this case, the merging formula at test time becomes:
+
+$$
+\hat {\mathbf {E}} _ {m} = \mathbf {E} _ {m} + \sum_ {i = 1} ^ {N - 1} s _ {i} \cdot \left(\mathbf {E} _ {i} ^ {\prime} - \mathbf {E} _ {m}\right) = \mathbf {E} _ {m} + \sum_ {i = 1} ^ {N - 1} \mathbf {M} _ {i} \cdot \left(s _ {i} \cdot \tau_ {i}\right)
+$$
+
+Thus, during inference, we avoid storing the curvature matrices and recomputing their product with domain vectors, reducing the total FLOPs. This explains the computational efficiency seen in Section 3.
+
+# 2.6 THEORETICAL ANALYSIS
+
+In this section, we investigate how optimizing the curvature matrix  $\mathbf{M}$  in Eqn. CA-Merg can improve the generalization of the expert merging process for downstream tasks. We first recall the formulation for domain-specific merging
+
+$$
+\hat {\mathbf {E}} _ {m} = \mathbf {E} _ {m} + \alpha \sum_ {j = 1} ^ {N - 1} s _ {j} \cdot \tau_ {j} \tag {5}
+$$
+
+We begin by calculating the gradient of the downstream task loss function, denoted by  $\mathcal{L}$ , with respect to  $\mathbf{M}_j$  at a specific  $l$ th layer as follows:
+
+$$
+\frac {\partial \mathcal {L}}{\partial \mathbf {M} _ {j}} = \frac {\partial \mathcal {L}}{\partial \hat {\mathbf {E}} _ {m}} \cdot \frac {\partial \hat {\mathbf {E}} _ {m}}{\partial \mathbf {M} _ {j}} = \frac {\partial \mathcal {L}}{\partial \hat {\mathbf {E}} _ {m}} \cdot (\alpha s _ {j} * \tau_ {j}) = \alpha s _ {j} * \frac {\partial \mathcal {L}}{\partial \hat {\mathbf {E}} _ {m}} \cdot (\mathbf {E} _ {j} - \mathbf {E} _ {m}) \tag {6}
+$$
+
+This is the outer product of the gradients of the task loss and the domain vectors. It is important to note the connection with the Fisher Information Matrix. For a downstream task, if we define the loss function as the negative log-likelihood, such as in a supervised classification task  $\mathcal{L}(\theta) = \mathbb{E}_{(x,y)\sim p}[-\log_{\theta}p(y|x)]$ , then the empirical Fisher Information Matrix can be defined as
+
+$$
+\mathbf {F} = \mathbb {E} _ {(x, y) \sim p} [ \nabla_ {\theta} \log_ {\theta} p (y | x) \nabla_ {\theta} \log_ {\theta} p (y | x) ^ {\top} ]
+$$
+
+Next, we consider how the gradient of the curvature matrix can contribute to better performance. Given a time step  $t$ , the standard gradient descent from an initial point  $\mathbf{M}_j$  with learning rate  $\beta$  yields the following update:
+
+$$
+\mathbf {M} _ {j} ^ {t + 1} = \mathbf {M} _ {j} ^ {t} - \beta * \frac {\partial \mathcal {L}}{\partial M _ {j} ^ {t}} = \mathbf {M} _ {j} ^ {t} - \alpha \beta * s _ {j} ^ {t} * \frac {\partial \mathcal {L}}{\partial \hat {\mathbf {E}} _ {m} ^ {t}} \cdot (\mathbf {E} _ {j} ^ {t} - \mathbf {E} _ {m} ^ {t}) \tag {7}
+$$
+
+We assume standard gradient descent for simplicity, but the argument extends to other advanced gradient algorithms, such as momentum (Polyak, 1964; Sutskever et al., 2013; Nguyen et al., 2020; He et al., 2020; Wang et al., 2022; Nguyen et al., 2022a) and ADAM (Kingma & Ba, 2015; Loshchilov & Hutter, 2019). We then apply  $\mathbf{M}_j$  to the merging process in Eqn. CA-Merg and get
+
+$$
+\hat {\mathbf {E}} _ {m} = \quad \mathbf {E} _ {m} + \alpha \sum_ {j = 1} ^ {N - 1} s _ {j} ^ {t + 1} * \mathbf {M} _ {j} ^ {t} \cdot \tau_ {j} ^ {t + 1} - \alpha^ {2} \beta \sum_ {j = 1} ^ {N - 1} s _ {j} ^ {t} s _ {j} ^ {t + 1} * \left(\tau_ {j} ^ {t} ^ {\top} \cdot \tau_ {j} ^ {t + 1}\right) \cdot \frac {\partial \mathcal {L}}{\partial \hat {\mathbf {E}} _ {m} ^ {t}} \tag {8}
+$$
+
+domain-specific merging with curvature-aware
+
+The detail for the derivation can be found in Appendix E. We can see that the first term in Eqn. 8 is the classic domain-specific merging formula with the guidance of the learned curvature. Furthermore, the second term contains the direction from the task loss gradient and the inner product between domain-vectors from two consecutive gradient steps. If  $\mathbf{M}_j = \mathbf{I} \forall j$ , this term can be seen as an auxiliary signal from task loss of the previous update step guiding the merging direction. The term  $s_j^t s_j^{t+1} \cdot (\tau_j^{t^\top} \tau_j^{t+1})$  modeling the agreement of the merging direction between updating steps: if there are conflicts between current and the previous updating direction, then this signal will be alleviated, thus dampening the harm to the merging direction of the current step; otherwise if they show strong agreement, this amplifies the impact of the updating direction toward minimizing the task loss with respect to the previous step, thus accelerate the training process while implicitly helping current merging direction with additional experience.
+
+On the other hand, we can rewrite Eqn. 8 as follows:
+
+$$
+\hat {\mathbf {E}} _ {m} = \mathbf {E} _ {m} + \alpha \sum_ {j = 1} ^ {N - 1} s _ {j} ^ {t + 1} * \mathbf {M} _ {j} ^ {t} \cdot \tau_ {j} ^ {t + 1} - \alpha^ {2} \beta \sum_ {j = 1} ^ {N - 1} s _ {j} ^ {t} s _ {j} ^ {t + 1} * \underbrace {\left(\tau_ {j} ^ {t ^ {\top}} \frac {\partial \mathcal {L}}{\partial \hat {\mathbf {E}} _ {m} ^ {t}}\right) \cdot \tau_ {j} ^ {t + 1}} _ {\text {g r a d i e n t m a t c h i n g}} \tag {9}
+$$
+
+We now have the inner-product between the gradient of the task loss and the domain-vector. This can be interpreted as the matching between the update from the task loss gradient and the domain-specific direction. We then have the updated domain-specific direction for each expert whose weighting factors are calculated by the inner-product. Therefore, we are performing a soft nearest distance voting to find the experts that agree the most with the task loss and enhance the merged experts with the corresponding domain-vector.
+
+Table 2: Performance of T5-base variants on the fine-tuning tasks for GLUE. All SMoE variants have 8 experts per layer. We follow (Devlin et al., 2019) in conducting experiments on the GLUE benchmark. Our curvature-aware methods outperform all baselines across tasks, while maintaining the same number of parameters and FLOPs as the SMoE models.  
+
+<table><tr><td>Methods</td><td>Params</td><td>TFLOPs</td><td>SST-2</td><td>MRPC</td><td>CoLA</td><td>QQP</td><td>STSB</td><td>QNLI</td><td>RTE</td><td>MNLI</td></tr><tr><td>Vanilla</td><td>220M</td><td>4.65</td><td>93.34</td><td>89.70</td><td>58.06</td><td>88.76</td><td>89.06</td><td>92.34</td><td>74.36</td><td>86.36</td></tr><tr><td>SMoE</td><td>1.0B</td><td>4.65</td><td>94.26</td><td>90.87</td><td>56.78</td><td>88.69</td><td>89.44</td><td>92.07</td><td>70.75</td><td>86.38</td></tr><tr><td>Domain-Specific</td><td>1.0B</td><td>4.65</td><td>93.57</td><td>90.19</td><td>58.07</td><td>88.77</td><td>89.40</td><td>92.51</td><td>72.56</td><td>86.40</td></tr><tr><td>Ties</td><td>1.0B</td><td>4.65</td><td>93.92</td><td>91.44</td><td>58.54</td><td>86.47</td><td>88.58</td><td>91.87</td><td>75.54</td><td>86.39</td></tr><tr><td>Dare</td><td>1.0B</td><td>4.65</td><td>93.80</td><td>89.46</td><td>58.33</td><td>88.72</td><td>89.13</td><td>92.29</td><td>73.64</td><td>86.20</td></tr><tr><td>Domain-specific-CA</td><td>1.0B</td><td>4.65</td><td>93.80</td><td>91.16</td><td>58.57</td><td>88.86</td><td>89.47</td><td>92.60</td><td>74.72</td><td>86.44</td></tr><tr><td>Dare-CA</td><td>1.0B</td><td>4.65</td><td>94.49</td><td>91.15</td><td>58.56</td><td>88.76</td><td>89.56</td><td>92.80</td><td>78.70</td><td>86.34</td></tr><tr><td>Ties-CA</td><td>1.0B</td><td>4.65</td><td>94.61</td><td>92.49</td><td>60.06</td><td>88.83</td><td>89.54</td><td>91.89</td><td>75.81</td><td>86.45</td></tr></table>
+
+# 3 EXPERIMENTAL RESULTS
+
+We perform evaluations on four major tasks, including language modeling, text classification, question answering, and image classification. For language modeling, we use the Wikitext-2 and Wikitext-103 (Merit et al., 2016) benchmarks. For text classification, we employ a subset of the GLUE (Wang et al., 2019) benchmark, a collection of eight diverse tasks designed to test different aspects of language understanding. For question answering, we employ two famous benchmarks: SQuAD (Rajpurkar et al., 2016) and WikiQA (Yang et al., 2015). Finally, the ImageNet-1k (Deng et al., 2009) dataset is chosen for image classification evaluation.
+
+We choose GPT-2 (Radford et al., 2019) small and SwinTransformer small (Liu et al., 2021) as our backbones for language modeling and image classification, respectively. Regarding GLUE and question-answering tasks, T5 base (Raffel et al., 2020) is chosen.
+
+Our experimental results confirm that the proposed merging method accelerates pre-training convergence and, when combined with other merging protocols, enhances model performance across tasks and settings. All results are averaged over 5 runs with different random seeds. Detailed information on the datasets, models, training procedures, and hyperparameters is provided in Appendix B and Appendix D. For additional experiments on different routers and merging methods, we refer to Appendix H.1, and H.2.
+
+![](images/556b0bec9ffea96e050d6e7c07f540f29522154f9601303ea6afdf2de6799977.jpg)  
+Figure 3: Perplexity of GPT2-small variants starting at the tenth epoch.
+
+# 3.1 TRAINING AND EVALUATION DETAILS
+
+We fix the number of epochs for all models on each task. For each text-related task, we first undertake a comprehensive hyper-parameter search. This encompasses batch sizes from  $\{8, 16, 32, 64\}$ , learning rates from  $\{3e - 4, 1e - 4, 3e - 5, 1e - 5\}$ , to pinpoint the optimal fine-tuned models. Regarding image classification tasks, a batchsize of 96 was chosen for all models. In addition, we choose AdamW (Loshchilov & Hutter, 2019) as the default optimizer and conduct all experiments on NVIDIA A100 GPUs. We compare our proposal to three merging baselines, including domain-specific, Ties, and Dare merging. There exists prior works on merging methods with the aid of the Fisher Information Matrix, such as (Matena & Raffel, 2022), which rely on access to a validation set used to compute the Fisher matrix or fine-tune hyperparameters. To eliminate the need for a validation set, (Jin et al., 2023) proposes storing and transmitting inner product matrices derived from the training data for each task, which are of the same size as the original model. However, this approach becomes costly for large models, as storage and transmission demands increase linearly with model size and the number of tasks as well as the number of experts. Therefore, we choose baselines that are needless of extra information and computational cost to perform comparisons. More details about theoretical comparison between CAMEx and Fisher-based
+
+merging methods can be found in Appendix A. We want to note that our merging protocol can be easily integrated into other works such as merge then compress protocol (Li et al., 2024).
+
+# 3.2 RESULTS
+
+In the following tables, the row with our method's results is highlighted in grey. Results with the best and second best performance are written in bold and underline, respectively. In addition, methods with the postfix "-CA" denote the curvature-aware version of the corresponding baseline.
+
+In Table 2, the results demonstrate that CA-augmented models consistently outperform their non-CA counterparts. Ties-CA achieves the highest scores on SST-2 (94.61), MRPC (92.49), CoLA (60.06), and MNLI (86.45), showing considerable improvements over both the vanilla and standard Ties models. Similarly, Dare-CA performs best on RTE (78.70), surpassing Dare (73.64), indicating that CA improves performance on smaller datasets and tasks with higher variability. Furthermore, Domain-specific-CA exceeds the non-CA version on QNLI and MNLI, demonstrating the broader applicability of curvature-aware methods. We provided a significant t-test at Appendix G.
+
+In Table 3, the Domain-specific-CA and Dynamic outperform the Vanilla, SMoE, and Domain-specific baselines, with lower perplexity values. Domain-specific-CA achieves the lowest perplexity score of 21.50, showcasing superior performance in language modeling tasks when compared to all other methods. The Dynamic architecture follows closely with a perplexity of 21.55 while also reducing the parameter count by  $9\%$ , compared to the
+
+Table 3: Performance of GPT-2 small variants for the pre-training task on Wikitext-103.
+
+<table><tr><td>Methods</td><td>Perplexity↓</td><td>Params (M)</td><td>GFLOPS ↓</td></tr><tr><td>Vanilla</td><td>23.03</td><td>125</td><td>292.5</td></tr><tr><td>SMoE</td><td>22.42</td><td>522</td><td>292.5</td></tr><tr><td>Domain-specific</td><td>21.64</td><td>522</td><td>292.5</td></tr><tr><td>Domain-specific-CA</td><td>21.50</td><td>522</td><td>292.5</td></tr><tr><td>Dynamic</td><td>21.55</td><td>470</td><td>292.5</td></tr></table>
+
+other methods. This highlights the Dynamic architecture's efficiency in maintaining strong performance with fewer parameters, making it ideal for resource-constrained environments. Moreover, the Dynamic architecture is competitive with Domain-specific-CA and outperforms the rest in terms of convergence speed, which is shown in Figure 3.
+
+In Table 4, the Vanilla model reaches a perplexity of 21.84. Despite increasing parameters, SMoE only slightly improves to 21.60. Domain-specific, Ties, and Dare methods show small gains, with Ties reaching 21.45. However, curvature-aware (CA) methods outperform all others. Domain-specific-CA achieves the best perplexity at 21.06, followed by Ties-CA (21.11) and Dare-CA (21.42), each significantly improving over their non-CA counterparts. All models beyond Vanilla share the same computational cost, indicating that CA methods enhance performance without added complexity.
+
+Domain-specific-CA stands out, demonstrating the clear advantage of curvature-aware optimization.
+
+Table 4: Performance of GPT-2 small variants for the supervised fine-tuning task on Wikitext-2
+
+<table><tr><td>Methods</td><td>Perplexity↓</td><td>Params (M)</td><td>GFLOPS ↓</td></tr><tr><td>Vanilla</td><td>21.84</td><td>125</td><td>292.5</td></tr><tr><td>SMoE</td><td>21.60</td><td>522</td><td>292.5</td></tr><tr><td>Domain-specific</td><td>21.56</td><td>522</td><td>292.5</td></tr><tr><td>Ties</td><td>21.45</td><td>522</td><td>292.5</td></tr><tr><td>Dare</td><td>21.60</td><td>522</td><td>292.5</td></tr><tr><td>Domain-specific-CA</td><td>21.06</td><td>522</td><td>292.5</td></tr><tr><td>Dare-CA</td><td>21.42</td><td>522</td><td>292.5</td></tr><tr><td>Ties-CA</td><td>21.11</td><td>522</td><td>292.5</td></tr></table>
+
+In Table 5, the baseline models, including Vanilla, SMOE, and non-CA versions of Ties and Dare, achieve solid results but show diminishing improvements as model complexity increases. In contrast, our curvature-aware methods significantly outperform their counterparts. For instance, on the SQuAD dataset, Dare-CA achieves the highest Exact Match (EM) score of  $81.76\%$  and an F1 score of  $88.60\%$ , surpassing all other methods. Similarly, on WikiQA, Ties-CA attains the highest accuracy of  $96.55\%$ , with Dare-CA closely following at  $96.23\%$ .
+
+Table 5: Performance of T5-base variants on question answering tasks.
+
+<table><tr><td>Methods</td><td>Params</td><td>TFLOPs</td><td>SQuAD Em/F1</td><td>WikiQA Accuracy</td></tr><tr><td>Vanilla</td><td>222M</td><td>2.86</td><td>81.01/88.14</td><td>96.06</td></tr><tr><td>SMoE</td><td>1.0B</td><td>2.86</td><td>81.25/88.50</td><td>96.04</td></tr><tr><td>Domain-specific</td><td>1.0B</td><td>2.86</td><td>80.21/87.44</td><td>95.32</td></tr><tr><td>Ties</td><td>1.0B</td><td>2.86</td><td>80.76/88.11</td><td>95.87</td></tr><tr><td>Dare</td><td>1.0B</td><td>2.86</td><td>80.88/88.03</td><td>96.01</td></tr><tr><td>Domain-specific-CA</td><td>1.0B</td><td>2.86</td><td>80.44/87.69</td><td>95.72</td></tr><tr><td>Ties-CA</td><td>1.0B</td><td>2.86</td><td>81.52/88.60</td><td>96.55</td></tr><tr><td>Dare-CA</td><td>1.0B</td><td>2.86</td><td>81.76/88.60</td><td>96.23</td></tr></table>
+
+In Table 6, while Vanilla and SMOE exhibit solid accuracy scores, they are surpassed by the curvature-aware (CA) enhanced versions of the models. Notably, Ties-CA delivers the best top-1 accuracy at  $83.38\%$  and
+
+![](images/6ffdfa797d0906dccc425bdb551f9c337372da8700990da3d154cc3a9db6307f.jpg)  
+Figure 4: Impact of the  $\alpha$  parameter on Curvature-Aware method performance across NLP tasks. We observe that the scaling factors that are within the range [0.8,1] consistently improve model's performance.
+
+![](images/1fe84bb36e13279795716a83368e36ecae89b12a4df8890bdeb03217d5170f91.jpg)  
+Figure 5: Impact of the Kronecker rank of curvature matrix on model's performance. We observe that as the rank increases the performance drops and then saturates. However, we would like to note that this curve might change depending on the downstream tasks and the merging protocol.
+
+the highest top-5 accuracy at  $96.96\%$ , slightly edging out Dare-CA, which achieves  $83.38\%$  and  $96.94\%$ , respectively.
+
+# 4 ABLATION
+
+Impact of the scaling factor. The plot in Figure 4 illustrates the impact of the  $\alpha$  parameter on the performance of three curvature-aware (CA) model variants Domain-specific-CA, Ties-CA, and Dare-CA across three natural language processing tasks: STSB, MRPC, and RTE. The  $\alpha$  parameter ranges from 0.5 to 1.0. The overall trend suggests that increasing  $\alpha$  leads to better generalization, particularly for complex tasks such as RTE, where sentence-level entailment and similarity benefit from stronger curvature-aware representations. Moreover, across all tasks, the model reaches its peak performance when  $\alpha$  is inside the range [0.8,1]. This observation aligns with that indicated by (Yadav et al., 2023). For a more comprehensive analysis on the impact of  $\alpha$  and number of experts, we direct the readers to Appendix H.4.1 and H.4.2.
+
+Improved performance with higher Kronecker rank. Across all three tasks (STSB, MRPC, and RTE), the evaluation metrics tend to improve as the rank increases from 1 to 8. This indicates that higher-ranked models generally perform better, suggesting a positive correlation between rank and task performance. Notably, the Domain-specific-CA model consistently achieves high performance across all tasks, especially in STSB, where metrics approach 0.90. Although MRPC and RTE show slightly lower metrics, ranging from 0.50 to 0.75, there is a clear improvement in performance as
+
+rank increases, particularly in the lower-to-mid ranks. However, we observed a decline in performance for Ties-CA and Dare-CA as the rank increases. We hypothesize that this is due to the masking mechanism employed by these methods, which may interfere with the learning process of the curvature matrices.
+
+Table 6: Comparison of Accuracy for SwinTransformer small variants on ImageNet-1k.  
+
+<table><tr><td>Methods</td><td>Params (M)</td><td>GFLOPs</td><td>Acc@1</td><td>Acc@5</td></tr><tr><td>Vanilla</td><td>50</td><td>6.75</td><td>83.14</td><td>96.90</td></tr><tr><td>SMoE</td><td>157</td><td>6.75</td><td>83.15</td><td>96.71</td></tr><tr><td>Domain-specific</td><td>157</td><td>6.75</td><td>83.15</td><td>96.91</td></tr><tr><td>Ties</td><td>157</td><td>6.75</td><td>83.28</td><td>96.93</td></tr><tr><td>Dare</td><td>157</td><td>6.75</td><td>83.13</td><td>96.88</td></tr><tr><td>Domain-specific-CA</td><td>157</td><td>6.75</td><td>83.29</td><td>96.95</td></tr><tr><td>Ties-CA</td><td>157</td><td>6.75</td><td>83.38</td><td>96.96</td></tr><tr><td>Dare-CA</td><td>157</td><td>6.75</td><td>83.38</td><td>96.94</td></tr></table>
+
+Robustness against noise. Table 7 demonstrates that curvature-aware models offer superior performance on corrupted ImageNet datasets compared to both Vanilla and SMOE variants. Among the models, our best configuration (Ties-CA) stands out as the best performer, showcasing robustness to corruptions across all datasets. These results suggest that incorporating curvature-awareness can
+
+substantially improve model robustness in challenging conditions.
+
+Table 7: Comparison for Swin-Transformer small variants on corrupted ImageNet.  
+
+<table><tr><td>Methods</td><td>ImageNet-O</td><td>ImageNet-A</td><td>ImageNet-R</td></tr><tr><td>Vanilla</td><td>45.88</td><td>23.68/53.10</td><td>37.34/52.34</td></tr><tr><td>SMoE</td><td>43.34</td><td>23.72/53.15</td><td>38.02/55.17</td></tr><tr><td>Ours</td><td>50.69</td><td>25.45/54.24</td><td>38.37/55.42</td></tr></table>
+
+# 5 RELATED WORK
+
+Sparse Mixture-of-Experts (SMoE). As the demand for model scaling grows increasingly widespread, there is a pressing inquiry into efficient ways to optimize computing costs while minimizing the impact on model performance. To address this need, Sparse Mixture of Experts (SMoE) has emerged and undergone extensive research and exploration (Shazeer et al., 2017; Lepikhin et al., 2021; Fedus et al., 2022). Starting with (Shazeer et al., 2017), the integration of SMoE into transformer architectures followed shortly after with the works of (Lepikhin et al., 2021) and (Fedus et al., 2022). The principle of SMoE is based on a simple concept: scaling the horizontal dimension of models (i.e., the number of feedforward blocks) rather than the vertical dimension (i.e., the number of stacked layers). This allows the model to selectively activate units or parameters based on the input tokens, thereby optimizing resource usage while maintaining performance.
+
+SMoE Efficiency Bottlenecks and Emerging Solutions. While it remains controversial whether to use Top-1 or Top-K routing, some research has highlighted the potential performance gains from increasing the number of activated experts (Shazeer et al., 2017; Chen et al., 2023). Other studies have found redundancies among experts in MoE layers (Li et al., 2024; Lu et al., 2024a). Additionally, some work has proposed using low-rank experts (Wu et al., 2024b; Liu et al., 2024; Wu et al., 2024a) inspired by LoRA (Hu et al., 2022). Despite the varying research directions, these studies consistently show that training a robust SMoE requires substantial computational and memory resources. This has motivated researchers such as (Li et al., 2024), (He et al., 2023), and (Zhong et al., 2024) to merge experts within each MoE layer, reducing the number of experts to a single one and significantly improving training and inference efficiency.
+
+Model Merging with curvature-aware. Though numerous methods for merging models have been introduced and developed (Yadav et al., 2023; Cai et al., 2023; Ilharco et al., 2022; Matena & Raffel, 2022; Jin et al., 2022; Don-Yehiya et al., 2022; Rame et al., 2023; Tuan et al., 2024; Lu et al., 2024b), most of these works consider merging protocols in the Euclidean parameter space. However, it has been noted that the space of deep neural network models is a Riemannian one (Amari, 1998). (Matena & Raffel, 2022) and (Jin et al., 2022) were the first to fuse model weights while accounting for the Fisher Information. Despite their promising results, these methods require massive computation to approximate the inversion of the Fisher matrix. Moreover, the Fisher matrix has a size proportional to the dimension of the model parameters, which significantly increases memory usage. Consequently, these methods are challenging for directly integrating into SMoE layers to fuse expert weights.
+
+# 6 LIMITATION AND CONCLUSION
+
+In this work, we introduced CAMEx, a curvature-aware approach to expert merging in Mixture of Experts architectures. By leveraging natural gradients to account for the parameter manifold's curvature, CAMEx enhances model performance and reduces computational costs during both pre-training and fine-tuning, outperforming traditional Euclidean-based methods. Additionally, our dynamic merging architecture optimizes resource usage by incorporating a global expert across layers, thus minimizing model size without sacrificing accuracy. Despite the overall improvements, a minor limitation is that curvature-aware merging demonstrates reduced compatibility with Ties and Dare merging at higher Kronecker ranks. Future work could dive deeper into this phenomenon and extend CAMEx to other expert merging methods and explore curvature-aware approaches in broader neural network models to further enhance our dynamic architecture. This research lays the groundwork for developing more efficient and scalable models in large-scale machine learning.
+
+# ACKNOWLEDGMENTS
+
+Dung V. Nguyen was funded by the Master, PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), code VINIF.2023.ThS.024.  
+This research / project is supported by the National Research Foundation Singapore under the AI Singapore Programme (AISG Award No: AISG2-TC-2023-012-SGIL). This research / project is supported by the Ministry of Education, Singapore, under the Academic Research Fund Tier 1 (FY2023) (A-8002040-00-00, A-8002039-00-00). This research / project is also supported by the NUS Presidential Young Professorship Award (A-0009807-01-00) and the NUS Artificial Intelligence Institute-Seed Funding (A-8003062-00-00).  
+Reproducibility Statement: Source codes for our experiments are provided in the supplementary materials of the paper. The details of our experimental settings and computational infrastructure are given in Section 3 and the Appendix D. All datasets that we used in the paper are published, and they are easy to find in the Internet.  
+Ethics Statement: Given the nature of the work, we do not foresee any negative societal and ethical impacts of our work.
+
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+
+# Supplement to “CAMEx: Curvature-aware Merging of Experts”
+
+# Table of Contents
+
+# A Comprision of CAMEx and Fisher-based merging methods 16
+
+# B Additional Details on datasets 18
+
+B.1 Language Modeling on WikiText 18  
+B.2 Text classification on GLUE benchmark 18  
+B.3 Question-answering on SQuAD and WikiQA 18  
+B.4 Image Classification on Imagenet 18  
+B.5 Adversarial Examples and Out-of-distribution datasets 19
+
+# C Algorithm and implementation details 19
+
+C.1 Causal segmenting 19  
+C.2 Some implementations 20
+
+# D More Experiment Details 20
+
+# E Derivation 21
+
+# F Step-by-step walkthrough for key equations of CAMEx 21
+
+F.1 Key equation for merging of CAMEx 21  
+F.2 Key equation for merging in dynamic architecture 21  
+F.3 Curvature update 22
+
+# G Student's t-test for experiments on GLUE dataset 22
+
+# H Additional Experiments 23
+
+H.1 Integrating CAMEx into Twin-Merging 23  
+H.2 Experiments on Token-choice vs Expert-choice routing 24  
+H.3 Longer training for Wikitext-103 pre-training 24  
+H.4 More comprehensive ablation study on hyperparameters 25
+
+H.4.1 Ablation study on  $\alpha$  25  
+H.4.2 Ablation study on number of experts 25
+
+H.5 Results on large backbone (Phi-3) 26
+
+# A COMPRISION OF CAMEX AND FISHER-BASED MERGING METHODS
+
+The pipeline comparison between CAMEx and Fisher-based merging methods is shown in Figure 6. Both approaches aim to capture the curvature of the parameter space during the merging process. (Diagonal) Fisher Merging (Matena & Raffel, 2022) applies a diagonal approximation to the Fisher information matrix. In this work, they estimate the diagonal of the Fisher matrix as:
+
+$$
+\mathbf {F} = \underset {\mathbf {x} \sim \mathbf {D} _ {\mathrm {m}}} {\mathbb {E}} \left[ \underset {\mathbf {y} \sim \mathbf {p} _ {\theta} (\mathbf {y} | \mathbf {x})} {\mathbb {E}} \left[ \nabla_ {\theta} \log \mathbf {p} _ {\theta} (\mathbf {y} | \mathbf {x}) \nabla_ {\theta} \log \mathbf {p} _ {\theta} (\mathbf {y} | \mathbf {x}) ^ {\top} \right] \right], \tag {10}
+$$
+
+![](images/10be15a2d30fe7fdaf1e379196739c58db85fac58a209fcc2099a3311a5c7988.jpg)  
+a) CAMEx
+
+![](images/164d43a881b016801a725ac9bda385d08b233137c7235ca07f0d38cd77d1fc74.jpg)  
+b) Fisher Merging  
+Figure 6: CAMEx merging pipeline vs Fisher-based merging pipeline. Note that Fisher merging requires the storing of the  $\nabla_{E_i}\log p_{E_i}(y|x)$  for all experts and all  $x$  in training dataset. Furthermore, it has been pointed out that Fisher merging will have poor performance while using fewer examples to estimate the Fisher.
+
+The expectation over  $y$  can be estimated via sampling from  $p_{\theta}(y|x_i)$  or computed exactly when the number of classes is small. The closed-form solution for Fisher merging (without necessarily applying the diagonal approximation) is given by:
+
+$$
+\hat {\mathbf {E}} _ {\mathbf {m}} ^ {\mathrm {l}} = \left(\sum_ {\mathbf {m} = 1} ^ {\mathrm {M}} \mathbf {F} _ {\mathbf {m}} ^ {\mathrm {l}}\right) ^ {- 1} \left(\sum_ {\mathrm {i} = 1} ^ {\mathrm {N}} \mathbf {F} _ {\mathrm {i}} ^ {\mathrm {l}} \mathbf {E} _ {\mathrm {i}} ^ {\mathrm {l}}\right). \tag {11}
+$$
+
+Thus, to approximate the Fisher Information Matrix for SMOE models, Fisher merging requires storing  $\nabla_{E_i}\log p_{E_i}(y|x)$  for all experts and all  $x$  in the training dataset. Additionally, it has been noted that Fisher merging can suffer from poor performance when fewer examples are used to estimate the Fisher matrix (Matena & Raffel, 2022).
+
+In the case of our method (depicted in Figure 6a), by denoting  $\mathbf{M}_i$  as the curvature matrix of the  $i$ -th expert, CAMEx utilizes the formula for merging experts derived from the natural gradient descent update as:
+
+$$
+\hat {\mathbf {E}} _ {\mathbf {m}} ^ {\mathrm {l}} = \mathbf {E} _ {\mathbf {m}} ^ {\mathrm {l}} + \alpha \sum_ {\mathrm {i} = 1} ^ {\mathrm {N} - 1} \mathbf {M} _ {\mathrm {i}} \cdot \left(\mathbf {s} _ {\mathrm {i}} ^ {\mathrm {l}} * \tau_ {\mathrm {i}} ^ {\mathrm {l}}\right) \quad (\text {C A - M e r g})
+$$
+
+CAMEx implicitly implements the gradient-based matching between the task loss gradient and domain-vector of the corresponding expert to approximate the empirical Fisher through the dynamic of gradient descend update of  $\mathbf{M}_{\mathrm{i}}$ :
+
+$$
+\mathbf {M} _ {\mathbf {i}} ^ {\mathrm {t} + 1} = \mathbf {M} _ {\mathbf {i}} ^ {\mathrm {t}} - \beta * \frac {\partial \mathcal {L}}{\partial \mathbf {M} _ {\mathbf {i}} ^ {\mathrm {t}}} = \mathbf {M} _ {\mathbf {i}} ^ {\mathrm {t}} - \alpha \beta * \mathbf {s} _ {\mathbf {i}} ^ {\mathrm {t}} * \frac {\partial \mathcal {L}}{\partial \hat {\mathbf {E}} _ {\mathbf {m}} ^ {\mathrm {t}}} \cdot \left(\mathbf {E} _ {\mathbf {i}} ^ {\mathrm {t}} - \mathbf {E} _ {\mathbf {m}} ^ {\mathrm {t}}\right), \tag {12}
+$$
+
+where the term  $\frac{\partial\mathcal{L}}{\partial\hat{\mathbf{E}}_{\mathbf{m}}}\cdot (\mathbf{E}_{\mathbf{i}} - \mathbf{E}_{\mathbf{m}})$  represents the outer product of the gradients of the task loss and the domain vectors. This operation contributes to capturing the curvature of the expert parameter space,
+
+ensuring curvature awareness during the merging process. This approach eliminates the need to compute the inversion of the empirical Fisher Information Matrix, thereby reducing computational overhead while maintaining sensitivity to parameter space curvature.
+
+# B ADDITIONAL DETAILS ON DATASETS
+
+This section provides detailed information on the datasets and evaluation metrics used in the experiments in Section 3.
+
+# B.1 LANGUAGE MODELING ON WIkIText
+
+The WikiText-103 dataset consists of Wikipedia articles designed to capture long-range contextual dependencies. The training set includes approximately 28,000 articles, totaling around 103 million words. The validation and test sets have 218,000 and 246,000 words, respectively, spread across 60 articles per set, with each set comprising roughly 268,000 words. Our experiments follow the standard procedure described in (Merit et al., 2017).
+
+WikiText-2 is a smaller version of WikiText-103, containing 2 million tokens and a vocabulary of 33,000 words.
+
+# B.2 TEXT CLASSIFICATION ON GLUE BENCHMARK
+
+These tasks include MNLI (Williams et al., 2018), which assesses a model's ability to determine entailment between pairs of sentences; QQP (Quora, 2017) and MRPC (Dolan & Brockett, 2005), which focus on identifying sentence similarity and paraphrase detection; SST-2 (Socher et al., 2013) for sentiment analysis; CoLA (Warstadt et al., 2019) for grammaticality judgment; and QNLI (Wang et al., 2019) for question-answer classification. Additionally, STSB (Cer et al., 2017) evaluates the model's ability to measure sentence similarity, while RTE (Dagan et al., 2006) tests logical reasoning.
+
+# B.3 QUESTION-ANSWERING ON SQUAD AND WIKIQA
+
+SQuADv1.1 (Rajpurkar et al., 2016) (Stanford Question Answering Dataset) is a widely used benchmark for reading comprehension and question answering tasks. It contains over 100,000 question-answer pairs sourced from more than 500 Wikipedia articles. Each question is paired with a paragraph from the article, where the answer is a span of text extracted from the passage. The dataset consists of natural language questions that cover a wide range of topics, context paragraphs from Wikipedia, and answers marked by their start and end positions within the context. The primary task is to extract the correct answer span based on the posed question. Key features of the dataset include the need for exact span extraction, the large dataset size, and its task design focused on reading comprehension. Evaluation is typically done using Exact Match (EM), which measures the percentage of predictions that exactly match the ground-truth answers, and the F1 score, which measures the overlap between predicted and ground-truth answers by calculating the harmonic mean of precision and recall.
+
+WikiQA (Yang et al., 2015) is an open-domain question answering dataset designed for answer sentence selection tasks. It consists of natural language questions primarily extracted from search engine queries, with candidate sentences sourced from Wikipedia articles. Each candidate sentence is labeled as either a correct or incorrect answer for the given question. The dataset contains 3,047 questions and 29,258 candidate sentences. The main challenge is selecting the correct sentence from a set of candidates, unlike SQuADv1.1, where the task focuses on extracting a text span. Key features include its real-world query origins, the sentence selection task, and the open-domain nature, which requires models to identify relevant sentences from diverse topics. WikiQA is evaluated using Accuracy.
+
+# B.4 IMAGE CLASSIFICATION ON IMAGENET
+
+ImageNet-1k, the most widely utilized subset of the ImageNet dataset introduced by (Deng et al., 2009), comprises 1.28 million images for training and 50,000 images for validation, across 1,000 categories. Performance evaluation is typically based on top-1 and top-5 accuracy metrics.
+
+Algorithm 1 The Overall Procedures of CAMEx.  
+1: Initialize: A model  $\mathcal{M}$  with  $l$  SMOE layers, the total number of original experts  $N$ .  
+2: Let  $\mathsf{H}^t \in \mathbb{R}^{B \times L \times N}$  and  $\mathsf{T}^t \in \mathbb{R}^{B \times L \times d}$  denote the router logits and the sequence of tokens at intermediate layer  $t$ , respectively.  
+3: for layer  $t = 1, \dots, l$  do  
+4:  $K = L / S, T^l \gets \text{RESHAPE}(T, B * K, S, d)$   
+5:  $\mathsf{H}^l \gets \mathbf{G}(T^l)$   
+6:  $\mathsf{H}^l \gets \text{ROLLandDETACH}(\mathsf{H}^l)$   
+7: if TIES-MERGING then  
+8: for expert  $i = 1, \dots, N - 1$  do  
+9:  $\tau_i \gets \mathbf{E}_i - \mathbf{E}_m$   
+10:  $\gamma_i \gets sgn(\tau_i)$   
+11: end for  
+12:  $\gamma^m = sgn(\sum_{i=1}^{N-1} \tau_i)$   
+13: for expert  $i = 1, \dots, N - 1$  do  
+14:  $\tau_i^m \gets \gamma_i \wedge \gamma^m$   
+15:  $\tau_i \gets \tau_i \cdot \mathbf{M}_i$   
+16: end for  
+17: else  
+18: Generate mask for DARE-MERGING  
+19: end if  
+20:  $\mathbf{E}_m \gets \mathbf{E}_m + \gamma_m * \sum_{i=1}^{N-1} H_i^l * \tau_i$   
+Merge Experts
+
+# B.5 ADVERSARIAL EXAMPLES AND OUT-OF-DISTRIBUTION DATASETS
+
+ImageNet-A: The ImageNet-A dataset (Hendrycks et al., 2021b) contains real-world images specifically curated to fool ImageNet classifiers. It focuses on 200 classes, a subset of the 1,000 classes in ImageNet-1k. Errors made within these 200 classes are considered particularly significant, as they represent a wide variety of categories from ImageNet-1k.
+
+ImageNet-O: This dataset consists of examples adversarially filtered to challenge out-of-distribution (OOD) detectors on ImageNet (Hendrycks et al., 2021b). It includes images from the larger ImageNet-22k dataset but excludes those present in ImageNet-1k. The selected samples are those that a ResNet-50 model confidently misclassifies as an ImageNet-1k class, and the primary evaluation metric is the area under the precision-recall curve (AUPR).
+
+ImageNet-R: ImageNet-R contains a variety of artistic renditions of object classes found in the original ImageNet dataset (Hendrycks et al., 2021a). This dataset includes 30,000 artistic representations of images from 200 classes, selected from the ImageNet-1k subset. The dataset was created to challenge models with non-standard visual interpretations of the classes.
+
+# C ALGORITHM AND IMPLEMENTATION DETAILS
+
+# C.1 CAUSAL SEGMENTING
+
+Background of Causal Segmenting: A significant advancement in SMOE design centers on fully differentiable architectures that eliminate the need for additional loss terms to stabilize training. In (Muqeeth et al., 2024), a model was introduced that computes a weighted average of expert feed-forward networks (FFNs). For an input  $x$  with corresponding routing weights, the output is defined as:
+
+$$
+o _ {x} = \operatorname {F F N} \left(h _ {x}; \sum_ {i = 1} ^ {N} s _ {i} \cdot \mathbf {E} _ {i}\right), \quad \text {w h e r e} \quad s _ {i} = \operatorname {S o f t m a x} \left(\mathbf {G} \left(h _ {x}\right)\right) _ {i}.
+$$
+
+However, applying this approach to autoregressive language models is computationally costly, as the merged FFN must be computed for each token in the sequence, leading to costs that scale linearly with the number of experts. An alternative based on pooling—routing via the sequence's average representation, as
+
+follows:
+
+$$
+s _ {i} = \operatorname {S o f t m a x} \left(\mathbf {G} \left(\frac {\sum_ {j = 1} ^ {L} h _ {x _ {j}}}{L}\right)\right) _ {i}.
+$$
+
+This, however, disrupts the autoregressive property essential for pre-training. To address this, (Zhong et al., 2024) introduced causal segment routing. This technique merges FFNs in an MoE layer by utilizing information from the preceding segment to process the current segment. Specifically, given a training instance  $X$  consisting of  $L$  tokens (e.g.,  $L = 4096$ ), we divide the instance into  $N$  segments, each containing  $T$  (e.g.,  $T = 256$ ) consecutive tokens. For the  $k$ -th segment  $S_{k}$ , where  $k > 1$ , we compute the average of the hidden representations from the previous segment  $S_{k-1}$ , denoted as  $\bar{h}_{k-1}$ . By using the average hidden representation, the model can adapt to prompts of varying lengths during inference. The average hidden representation  $\bar{h}_{k-1}$  is then employed to determine the routing weights, leading to a merged expert  $\bar{\mathbf{E}}$ :
+
+$$
+\bar {h} _ {k - 1} = \frac {1}{T} \sum_ {x \in S _ {k - 1}} h _ {x}, \quad s _ {i} = \operatorname {S o f t m a x} \left(\mathbf {G} \left(\bar {h} _ {k - 1}\right)\right), \quad \bar {\mathbf {E}} = \sum_ {i} s _ {i} \cdot \mathbf {E} _ {\mathbf {i}}. \tag {13}
+$$
+
+The merged expert  $\bar{\mathbf{E}}$  is then used to process all tokens in the current segment  $S_{k}$ , i.e.,  $o_{x} = \mathrm{FFN}(h_{x};\bar{\mathbf{E}}), \forall x \in S_{k}$ . This approach ensures that the model's routing decisions rely exclusively on data from preceding positions. For the first segment  $S_{1}$ , the segment's own representation is used to compute the merging weights for its FFN. To prevent information leakage, a stop-gradient operation is applied to  $\mathbf{G}(\bar{h}_1)$ :
+
+$$
+\bar {h} _ {0} = \frac {1}{T} \sum_ {x \in S _ {0}} ^ {T} h _ {x} \tag {14}
+$$
+
+These tokens are then used to calculate the scores for the merging procedure
+
+$$
+s _ {0} = \operatorname {D E T A C H} \left(\mathbf {G} \left(\bar {h} _ {1}, k\right)\right) \quad (\text {R O L L a n d D E T A C H})
+$$
+
+$$
+s _ {i} = \mathbf {G} (\bar {h} _ {i - 1}), \quad i = 1, \dots , S - 1
+$$
+
+# C.2 SOME IMPLEMENTATIONS
+
+Implementation of Kronecker product We consider the case where experts are linear layers  
+```txt
+# Calculating domain-specific vectors  
+taus = weights - weight_m  
+# output_size = dim_out1 * dim_out2, input_size = dim_in1 * dim_in2  
+taus = taus.view(1, -1, dim_out1, dim_out2, dim_in1, dim_in2).repeat(rank, 1, 1, 1, 1, 1)  
+# Calculate Kronecker-product  
+taus = torch.einsum("rbij, rbjklm->rbiklm", curvel_out, taus)  
+taus = torch.einsum("rbik, rbjklm->rbjilm", curve2_out, taus)  
+taus = torch.einsum("rbil, rbjklm->rbjkim", curvel_in, taus)  
+taus = torch.einsum("rbim, rbjklm->rbjkli", curve2_in, taus)  
+# Summation along the Kronecker rank dimension and reshape  
+taus = taus.sum(0)  
+taus = taus.reshape(-1, output_size, input_size)
+```
+
+# D MORE EXPERIMENT DETAILS
+
+Supervised Fine-Tuning Hyper-Parameters Besides {batch size, learning rate, epoch counts} which vary for each task, we keep other hyper-parameters of supervised fine-tuning fixed for all tasks. These are shown in Table 8.
+
+Table 8: Fine-tuning hyper-parameters of all models in Section 3  
+
+<table><tr><td>Hyper-Parameters</td><td>Values</td></tr><tr><td>Optimizer</td><td>ADAMW</td></tr><tr><td>Adam ε</td><td>1e-6</td></tr><tr><td>Adam β</td><td>(0.9, 0.98)</td></tr><tr><td>Warm-up steps</td><td>16</td></tr><tr><td>Weight decay</td><td>0.01</td></tr><tr><td>LR scheduler</td><td>LINEAR DECAY</td></tr><tr><td>Scaling factor α</td><td>1</td></tr><tr><td>Kronecker rank r</td><td>1</td></tr></table>
+
+# E DERIVATION
+
+This is the derivation for Eqn 8 in Section 2.6
+
+$$
+\begin{array}{l} \hat {\mathbf {E}} _ {m} = \mathbf {E} _ {m} + \alpha \sum_ {j = 1} ^ {N - 1} \mathbf {M} _ {j} ^ {t + 1} \cdot \left(s _ {j} ^ {t + 1} * \tau_ {j} ^ {t + 1}\right) (15) \\ = \mathbf {E} _ {m} + \alpha \sum_ {j = 1} ^ {N - 1} \left[ \mathbf {M} _ {j} ^ {t} - \alpha \beta * s _ {j} ^ {t} * \frac {\partial \mathcal {L}}{\partial \hat {\mathbf {E}} _ {m} ^ {t}} \cdot \tau_ {j} ^ {t} \right] \cdot \left(s _ {j} ^ {t + 1} * \tau_ {j} ^ {t + 1}\right) (16) \\ = \underbrace {\mathbf {E} _ {m} + \alpha \sum_ {j = 1} ^ {N - 1} s _ {j} ^ {t + 1} * \mathbf {M} _ {j} ^ {t} \cdot \tau_ {j} ^ {t + 1}} _ {\text {d o m a i n - s p e c i f i c m e r g i n g w i t h c u r v a t u r e - a w a r e}} - \alpha^ {2} \beta \sum_ {j = 1} ^ {N - 1} s _ {j} ^ {t} s _ {j} ^ {t + 1} * \left(\tau_ {j} ^ {t ^ {\top}} \cdot \tau_ {j} ^ {t + 1}\right) \cdot \frac {\partial \mathcal {L}}{\partial \hat {\mathbf {E}} _ {m} ^ {t}} (17) \\ \end{array}
+$$
+
+# F STEP-BY-STEP WALKTHROUGH FOR KEY EQUATIONS OF CAMEX
+
+F.1 KEY EQUATION FOR MERGING OF CAMEX
+
+$$
+\hat {\mathbf {E}} _ {\mathbf {m}} ^ {\mathrm {l}} = \mathbf {E} _ {\mathbf {m}} ^ {\mathrm {l}} + \alpha \sum_ {\mathrm {i} = 1} ^ {\mathrm {N} - 1} \mathbf {M} _ {\mathrm {i}} \cdot \left(\mathbf {s} _ {\mathrm {i}} ^ {\mathrm {l}} * \tau_ {\mathrm {i}} ^ {\mathrm {l}}\right) \quad (\text {C A - M e r g})
+$$
+
+In (CA-Merg) equation, we consider the merging of experts at  $l$ -th layer of the model.  $\mathbf{E}_{\mathbf{m}}^{1}$  denotes the "base" expert that is not included in the routing process.  $\tau_{i}^{l} = \mathbf{E}_{\mathbf{i}}^{1} - \mathbf{E}_{\mathbf{m}}^{1}$  denotes  $i$ -th the domain-vector that adapts the "base" expert to the corresponding domain. Finally,  $s_i^l$  denotes the score of the  $i$ -th domain vector w.r.t the input. We view the merging of experts as a optimization problem where  $\alpha * s_i^l$  acts as the adaptive learning rate. Therefore, it is straightforward to integrate natural gradient approach into this equation by introducing curvature matrices  $\mathbf{M}_{\mathrm{i}}$ . Due to the challenging tractability of the Fisher Matrix in the intermediate layers of deep models, we proposed to learn them empirically through backpropagation as indicated by Eqn. 6 in the main text and a similar approach using meta-learning (Park & Oliva, 2019).
+
+F.2 KEY EQUATION FOR MERGING IN DYNAMIC ARCHITECTURE
+
+$$
+\left\{ \begin{array}{l l} \mathbf {E} _ {\mathbf {m}} ^ {\mathrm {l + 1}} & = \mathbf {E} _ {\mathbf {m}} ^ {\mathrm {l}} + \frac {\alpha}{\mathbf {N} - \mathbf {1}} \sum_ {\mathrm {i} = 1} ^ {\mathrm {N} - 1} \mathbf {M} _ {\mathrm {i}} \cdot \tau_ {\mathrm {i}} ^ {\mathrm {l}} \\ \hat {\mathbf {E}} _ {\mathbf {m}} ^ {\mathrm {l + 1}} & = \mathbf {E} _ {\mathbf {m}} ^ {\mathrm {l + 1}} + \alpha \sum_ {\mathrm {i} = 1} ^ {\mathrm {N} - 1} \mathbf {M} _ {\mathrm {i}} \cdot \left(\mathbf {s} _ {\mathrm {i}} ^ {\mathrm {l + 1}} * \tau_ {\mathrm {i}} ^ {\mathrm {l + 1}}\right) \end{array} \right. \tag {Dynamic-Merg}
+$$
+
+The (Dynamic-Merg) system performs two steps which are calculating base expert for the next layer and performing (CA-Merge), respectively. For the first step, we eliminate the score and take the average of curvature-aware domain vector instead to avoid information leakage. The result then takes the role as the base expert for the next layer.
+
+# F.3 CURVATURE UPDATE
+
+In the main text, we try to give an explanation of how our method we update the curvature matrix with the curvature information of the parameters space. To achieve that, we first take the derivative of equation (CA-Merge) w.r.t the curvature matrix  $\mathbf{M}_1$ :
+
+$$
+\frac {\partial \hat {\mathbf {E}} _ {\mathbf {m}}}{\partial \mathbf {M} _ {\mathbf {j}}} = (\alpha s _ {j} * \tau_ {j}) = \alpha s _ {j} * \left(\mathbf {E} _ {\mathbf {j}} - \mathbf {E} _ {\mathbf {m}}\right) \tag {18}
+$$
+
+To evaluate the gradient of the task loss  $\mathcal{L}$  w.r.t  $\mathbf{M}_{\mathrm{i}}$  we apply the chain-rule:
+
+$$
+\frac {\partial \mathcal {L}}{\partial \mathbf {M} _ {\mathbf {j}}} = \frac {\partial \mathcal {L}}{\partial \hat {\mathbf {E}} _ {\mathbf {m}}} \cdot \frac {\partial \hat {\mathbf {E}} _ {\mathbf {m}}}{\partial \mathbf {M} _ {\mathbf {j}}} = \alpha s _ {j} * \frac {\partial \mathcal {L}}{\partial \hat {\mathbf {E}} _ {\mathbf {m}}} \cdot (\mathbf {E} _ {\mathbf {j}} - \mathbf {E} _ {\mathbf {m}}) \tag {19}
+$$
+
+# G STUDENT'S T-TEST FOR EXPERIMENTS ON GLUE DATASET
+
+We report the t-test results, beginning with the null hypothesis  $H_0$ : The performance between each pair of T5-Ties-CA vs T5-Ties, and T5 on GLUE SST-2, MRPC, CoLA, and MNLI are the same.. In this test, we choose the significant value to be 0.05.
+
+Table 9: Evaluation results on SST-2 with different random seeds.
+
+<table><tr><td>Index</td><td>Ties_CA</td><td>Ties</td><td>Vanilla</td></tr><tr><td>1</td><td>94.44</td><td>93.77</td><td>93.31</td></tr><tr><td>2</td><td>94.86</td><td>94.13</td><td>93.33</td></tr><tr><td>3</td><td>94.62</td><td>93.90</td><td>93.21</td></tr><tr><td>4</td><td>94.60</td><td>94.12</td><td>93.46</td></tr><tr><td>5</td><td>94.54</td><td>93.70</td><td>93.41</td></tr><tr><td>6</td><td>94.55</td><td>93.87</td><td>93.56</td></tr><tr><td>7</td><td>94.37</td><td>94.03</td><td>93.67</td></tr></table>
+
+Table 10: T-statistic and p-value when evaluating on SST-2.  
+
+<table><tr><td>Test</td><td>t-statistic</td><td>p-value</td></tr><tr><td>Ties-CA vs Vanilla</td><td>13.72</td><td>1.08e-8</td></tr><tr><td>Ties-CA vs Ties</td><td>7.36</td><td>8.74e-6</td></tr></table>
+
+Table 11: Evaluation results on MRPC with different random seeds.  
+
+<table><tr><td>Index</td><td>Ties_CA</td><td>Ties</td><td>Vanilla</td></tr><tr><td>1</td><td>92.35</td><td>91.35</td><td>89.85</td></tr><tr><td>2</td><td>92.61</td><td>91.30</td><td>89.65</td></tr><tr><td>3</td><td>92.55</td><td>91.40</td><td>89.74</td></tr><tr><td>4</td><td>92.40</td><td>91.55</td><td>89.49</td></tr><tr><td>5</td><td>92.54</td><td>91.62</td><td>89.76</td></tr><tr><td>6</td><td>92.44</td><td>91.77</td><td>89.85</td></tr><tr><td>7</td><td>92.33</td><td>91.43</td><td>89.62</td></tr></table>
+
+Table 12: T-statistic and p-value when evaluating on MRPC.  
+
+<table><tr><td>Test</td><td>t-statistic</td><td>p-value</td></tr><tr><td>Ties-CA vs Vanilla</td><td>42.91</td><td>1.67e-14</td></tr><tr><td>Ties-CA vs Ties</td><td>12.95</td><td>2.06e-8</td></tr></table>
+
+Table 13: Evaluation results on CoLA with different random seeds.  
+
+<table><tr><td>Index</td><td>Ties_CA</td><td>Ties</td><td>Vanilla</td></tr><tr><td>1</td><td>61.01</td><td>57.95</td><td>57.74</td></tr><tr><td>2</td><td>59.53</td><td>58.63</td><td>57.82</td></tr><tr><td>3</td><td>60.36</td><td>58.90</td><td>58.03</td></tr><tr><td>4</td><td>60.13</td><td>58.92</td><td>58.23</td></tr><tr><td>5</td><td>59.41</td><td>58.31</td><td>58.51</td></tr><tr><td>6</td><td>59.33</td><td>57.38</td><td>57.36</td></tr><tr><td>7</td><td>60.03</td><td>58.53</td><td>58.40</td></tr></table>
+
+Table 14: T-statistic and p-value when evaluating on CoLA.  
+
+<table><tr><td>Test</td><td>t-statistic</td><td>p-value</td></tr><tr><td>Ties-CA vs Vanilla</td><td>7.14</td><td>1.18e-5</td></tr><tr><td>Ties-CA vs Ties</td><td>5.16</td><td>2.00e-4</td></tr></table>
+
+Table 15: Evaluation results on MNLI with different random seeds.  
+
+<table><tr><td>Index</td><td>Ties_CA</td><td>Ties</td><td>Vanilla</td></tr><tr><td>1</td><td>86.52</td><td>86.25</td><td>86.22</td></tr><tr><td>2</td><td>86.45</td><td>86.32</td><td>86.31</td></tr><tr><td>3</td><td>86.37</td><td>86.39</td><td>86.36</td></tr><tr><td>4</td><td>86.59</td><td>86.46</td><td>86.41</td></tr><tr><td>5</td><td>86.32</td><td>86.53</td><td>86.50</td></tr><tr><td>6</td><td>86.54</td><td>86.38</td><td>86.34</td></tr><tr><td>7</td><td>86.47</td><td>86.41</td><td>86.34</td></tr></table>
+
+Table 16: T-statistic and p-value when evaluating on MNLI.  
+
+<table><tr><td>Test</td><td>t-statistic</td><td>p-value</td></tr><tr><td>Ties-CA vs Vanilla</td><td>2.29</td><td>0.04</td></tr><tr><td>Ties-CA vs Ties</td><td>1.49</td><td>0.16</td></tr></table>
+
+Based on the p-values in the tables above, we draw the following conclusions:
+
+- The T5-Ties-CA variant significantly outperforms T5-Ties and T5-Vanilla on SST-2, MRPC, and CoLA.  
+- While T5-Ties-CA does not statistically significantly outperform T5-Ties on MNLI, it still demonstrates significant improvement over T5-Vanilla.
+
+# H ADDITIONAL EXPERIMENTS
+
+# H.1 INTEGRATING CAMEX INTO TWIN-MERGING
+
+We expand our experiments to include a broader range of most recent merging expert methods. Specifically, we integrated our CAMEx method with the Twin-Merging approach (Lu et al., 2024b). Key distinctions between CAMEx and Twin-Merging lie in their core mechanisms:
+
+- Our method is a non-Euclidean merging method, which utilizes the curvature-aware matrix, whereas Twin-Merging is a model merging method, which relies on Euclidean merging.  
+- Our approach is specifically designed for finetuning, in contrast to Twin-Merging, which is intended for post-training.
+
+- Finally, our dynamic mechanism performs inter-layer to form the merged expert, unlike Twin-Merging, which uses within-layer pre-calculations for merging. To integrate our method with Twin-Merging, we first fine-tune the Curvature Aware model for a specific GLUE task. At test time, we apply the Twin-Merging algorithm to merge experts, referring to our approach as Twin-CA. Notably, we found Twin-Merging to be a simple yet powerful technique that is easy to implement and helps reduce memory usage during inference. We adhere to the original implementation settings, using a sparsity density value of 0.2.
+
+Table 17: Performance of Twin-Merging and its Curvature Aware (CA) variant on GLUE tasks.  
+
+<table><tr><td>Method</td><td>MRPC</td><td>RTE</td><td>STSB</td></tr><tr><td>Twin-Merging</td><td>91.97</td><td>72.20</td><td>88.56</td></tr><tr><td>Twin-CA (Ours)</td><td>92.30</td><td>74.73</td><td>89.55</td></tr></table>
+
+The results in Table 17 demonstrate the effectiveness of our CAMEx approach when integrated with the Twin-Merging mechanism on GLUE tasks, highlighting its strong potential for incorporation into more advanced merging techniques.
+
+# H.2 EXPERIMENTS ONTOKEN-CHOICE VS EXPERT-CHOICE ROUTING
+
+We also demonstrate our merging approach with the following routing mechanisms. We compare the baseline performance (i.e., the ties merging expert without Curvature Aware) under different routing mechanisms with the corresponding Curvature-Aware counterparts to see how different routing functions affect CAMEx performance. It is worth noting that Expert Choice routing is not compatible with the experts merging method, as discussed by Lory (Zhong et al., 2024) in their Subsection 5.3.:
+
+- Stable MoE routing (Dai et al., 2022).  
+- Naive routing (Shazeer et al., 2017).  
+X-MoE routing (Chi et al., 2022).
+
+Note that the Curvature Aware model leverages the segment routing strategy (the causal segmenting strategy) proposed in Lory (Zhong et al., 2024), enabling a direct comparison between our model and the expert choice method. The results in Table 18 suggest that Curvature-Aware merging benefit from more advanced routing strategies. The CA model consistently outperforms the baseline with Ties merging across all routing mechanisms. Additionally, we observe that both Naive routing CA and X-MoE routing CA deliver robust performance across GLUE tasks, while Stable MoE routing CA emerges as the most reliable choice overall.
+
+Table 18: Performance of T5-base variants on the finetuning GLUE tasks  
+
+<table><tr><td>Method</td><td>MRPC</td><td>RTE</td><td>STSB</td><td>SST-2</td></tr><tr><td>Expert Choice MoE</td><td>93.10</td><td>66.78</td><td>89.19</td><td>93.80</td></tr><tr><td>Stable MoE routing Ties</td><td>91.92</td><td>75.48</td><td>89.48</td><td>93.37</td></tr><tr><td>Stable MoE routing CA</td><td>92.96</td><td>78.76</td><td>89.64</td><td>94.63</td></tr><tr><td>Naive routing Ties</td><td>91.44</td><td>75.54</td><td>88.58</td><td>93.92</td></tr><tr><td>Naive routing CA</td><td>92.49</td><td>78.70</td><td>89.56</td><td>94.61</td></tr><tr><td>X-MoE routing Ties</td><td>91.99</td><td>75.29</td><td>88.42</td><td>93.26</td></tr><tr><td>X-MoE routing CA</td><td>92.79</td><td>78.20</td><td>89.26</td><td>94.38</td></tr></table>
+
+# H.3 LONGER TRAINING FOR WIKTEXT-103 PRE-TRAINING
+
+We conduct additional experiments by training for longer iterations on the Wikitext-103 dataset. The performance gaps between methods remain stable starting around epoch 40.
+
+As shown in Figure 7 the trends demonstrate consistent improvements of our method over the baseline, with the gap remaining significant even after prolonged training.
+
+![](images/bf998ba31a4ffaf7da23172e62f295015677a226c15f4bff1f8a7c6c98d92da8.jpg)  
+Figure 7: Performance of Vannila, Domain-specific, Domain-specific-CA under longer pre-training.
+
+# H.4 MORE COMPREHENSIVE ABLATION STUDY ON HYPERPARAMETERS
+
+# H.4.1 ABLATION STUDY ON  $\alpha$
+
+We extend the range of  $\alpha$  for the ablation study, specifically evaluating Dare-CA and Ties-CA with  $\alpha \in [0.1, 1.6]$ . The evaluation is conducted using 5 different seeds, and the results are averaged.
+
+![](images/93dd1bad4af454cbfd12bcc3916c61be5477a441f00bdb95deb0fcc96b471292.jpg)  
+Figure 8: Test performance of Curvature-Aware methods under varying settings of  $\alpha$ .
+
+![](images/4287857ce519fc149e647ff2019bb0684a9c82fb0cdee6f14fc3c1a7a4edb00c.jpg)
+
+![](images/b6f0d23f3df7987e2cc1bea5f6b650f3e6f4be2fa38b3daff406ce40202c4068.jpg)
+
+The results in Figure 8 lead to the following observations:
+
+- The performance of the models is suboptimal or even worse than the vanilla baseline when  $\alpha$  is either too small ( $\alpha \in [0.1, 0.4]$ ) or too large ( $\alpha > 1.1$ ).  
+- Dare-CA is more sensitive to the choice of  $\alpha$ , showing sharper improvements and declines across the range.  
+- Ties-CA exhibits more gradual changes, suggesting it is more robust to variations in  $\alpha$ . The optimal range for  $\alpha$  is [0.8,1.0].
+
+# H.4.2 ABLATION STUDY ON NUMBER OF EXPERTS
+
+We conduct additional studies on our method using different numbers of experts in the T5 backbone.
+
+The following conclusions can be drawn from Figures 9:
+
+- Increasing the number of experts generally improves accuracy up to a certain point: Accuracy improves as the number of experts increases, with the most significant gains occurring from 4 to 8 experts.
+
+![](images/a4aa5c4337d5a50e2dd15cec2cde2c4cd47addd6b97c066ed985b3ec026cfb34.jpg)  
+Figure 9: Test performance of Curvature-Aware methods under varying number of experts.
+
+![](images/08e02578293dfe2e2bb270e6fab29b4de2263ab513843f2db330db6979e610ca.jpg)
+
+![](images/b01e7295c490d410fbd268552e7285a98fde232427b9ccf92cf37dc47755dc9b.jpg)
+
+- After 12 experts, the accuracy either saturates or slightly decreases.  
+- We suggest using eight experts, as it provides a balanced trade-off between performance and efficiency.
+
+# H.5 RESULTS ON LARGE BACKBONE (PHI-3)
+
+To evaluate the effectiveness of CAMEx with large backbone, we experiment with Phi-3.
+
+Table 19: Performance of Phi-3-mini variants on the fine-tuning tasks for the GLUE benchmark.
+
+<table><tr><td>Model</td><td>Params</td><td>SST-2</td><td>MRPC</td><td>CoLA</td><td>STSB</td><td>RTE</td><td>QQP</td><td>QNLI</td><td>MNLI</td></tr><tr><td>Phi-3</td><td>3.8B</td><td>95.76</td><td>90.12</td><td>61.36</td><td>88.7</td><td>80.51</td><td>92.38</td><td>94.84</td><td>90.39</td></tr><tr><td>Phi3-Ties</td><td>7.4B</td><td>96.56</td><td>92.25</td><td>62.33</td><td>89.99</td><td>87.73</td><td>94.13</td><td>95.58</td><td>91.28</td></tr><tr><td>Phi3-Ties-CA</td><td>7.4B</td><td>97.23</td><td>94.04</td><td>63.43</td><td>90.27</td><td>88.09</td><td>94.80</td><td>95.82</td><td>92.13</td></tr></table>
+
+- The Ties-CA and Ties variants remarkably outperform the vanilla version, creating a substantial performance gap.  
+- Ties-CA further enhances the performance of Ties in all listed tasks.
+
+Thus, we believe that curvature awareness has potential for improving other language models (LMs).

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+# CARTS: ADVANCING NEURAL THEOREM PROVING WITH DIVERSIFIED TACTIC CALIBRATION AND BIAS-RESISTANT TREE SEARCH
+
+Xiao-Wen Yang $^{1,2}$ , Zhi Zhou $^{1}$ , Haiming Wang $^{3,5}$ , Aoxue Li $^{4}$ , Wen-Da Wei $^{1,2}$ , Hui Jin $^{4}$ , Zhenguo Li $^{4}$ , Yu-Feng Li $^{1,2*}$
+
+$^{1}$ National Key Laboratory for Novel Software Technology, Nanjing University
+
+$^{2}$ School of Artificial Intelligence, Nanjing University
+
+$^{3}$ Sun Yat-sen University  $^{4}$ Noah's Ark Lab, Huawei  $^{5}$ Moonshot AI
+
+# ABSTRACT
+
+Recent advancements in neural theorem proving integrate large language models with tree search algorithms like Monte Carlo Tree Search (MCTS), where the language model suggests tactics and the tree search finds the complete proof path. However, many tactics proposed by the language model converge to semantically or strategically similar, reducing diversity and increasing search costs by expanding redundant proof paths. This issue exacerbates as computation scales and more tactics are explored per state. Furthermore, the trained value function suffers from false negatives, label imbalance, and domain gaps due to biased data construction. To address these challenges, we propose CARTS (diversified tactic CALibration and bias-Resistant Tree Search), which balances tactic diversity and importance while calibrating model confidence. CARTS also introduce preference modeling and an adjustment term related to the ratio of valid tactics to improve the bias-resistance of the value function. Experimental results demonstrate that CARTS consistently outperforms previous methods achieving a pass@1 rate of  $49.6\%$  on the miniF2F-test benchmark. Further analysis confirms that CARTS improves tactic diversity and leads to a more balanced tree search. The code for our implementation is available at https://github.com/njuyxw/CARTS.
+
+# 1 INTRODUCTION
+
+Automated theorem proving (ATP) (Harrison et al., 2014) is an essential task of artificial intelligence (AI) with significant challenge. Recently, the development of large language models has brought new vitality and advancements to this field (Han et al., 2022; Jiang et al., 2023; Xin et al., 2024a). For example, AlphaProof (Deepmind, 2024; Trinh et al., 2024) solved four out of six problems from International Mathematical Olympiad (IMO), achieving the same level as a silver medalist in the competition. These advancements stem from the integration of language models and formal theorem proving systems (such as Lean (Moura & Ullrich, 2021) or Isabella (Paulson, 1994)), which model the theorem proving task as a Markov Decision Process (MDP) (Polu & Sutskever, 2020). The language model functions as a policy network that provides heuristic proof tactics, while tree search methods are utilized to explore correct sequence of steps that maximize the reward.
+
+Although the improvements of language models (Xin et al., 2024a) can significantly improve the performance of theorem proving, efficient tree search methods remains crucial for theorems with long and complex proof steps. Existing search techniques (Polu & Sutskever, 2020; Wang et al., 2023; Xin et al., 2024a) primarily rely on Best First Search (BFS) or Monte Carlo Tree Search (MCTS) (Kocsis & Szepesvári, 2006). While these methods can achieve impressive performance, they have two significant drawbacks. Firstly, the output sampling of auto-regressive language models frequently exhibits significant redundancy, often producing similar tactics. Although the language model generates a substantial number of tactics that differ at the character level, they share the same underlying semantics. For instance, both 'intro h' and 'intro H' can be generated
+
+![](images/177a3fd422b27a94b9f10236e51c506bcedc855a92612a44d7408c14e6c33bdc.jpg)  
+Figure 1: Overall Framework. Diversified tactic calibration can calibrate the model confidence and enhance the diversity of candidate tactics, thus mitigating ineffective exploration. The bias-resistant value function can adapt to the test data, provide more accurate scores for evaluating tactics, thus improving the efficiency of utilization.
+
+by the language model; however, they convey the same meaning in Lean4, as they both introduce a hypothesis into the proving context. This redundancy can also lead to an imbalance in the number of high-level proof strategies. For example, most of the tactics generated by the language model may focus on the strategy of proof by contradiction, while there are relatively few tactics involving mathematical induction. These will result in a huge amount of ineffective exploration during the tree search process, thus increasing search costs. This issue worsens as computation scales and more tactics are explored per state. Secondly, the construction of value function training data often relies on existing policy models to generate negative samples and thus introducing bias. On one hand, this construction may produce a substantial number of negative samples, potentially far exceeding the positive ones. This could result in label imbalance, causing the cross-entropy loss used in training the value function to easily converge to local optima. On the other hand, the samples generated by the policy model may contain false negatives, introducing noise into the dataset. Additionally, the domain gap between the training dataset (e.g., Mathlib) and the test dataset (e.g., IMO problems) exacerbates the bias of the value function during the inference stage. This leads to inaccurate evaluations of the current proof state's value, hindering effective exploitation during the tree search process. Overall, these two issues prevent existing tree search techniques from efficient exploration and effective exploitation, resulting in sub-optimal search performance.
+
+In order to solve these challenges, we propose diversified tactic CAlibration and bias-Resistant Tree Search (CARTS). Diversified tactic calibration involves reordering and rescoring multiple candidate tactics generated by a language model's sampling output. This approach balances importance and diversity, thereby enhancing exploration efficiency. We use the Maximal Marginal Relevance (MMR) algorithm (Peng et al., 2005) to achieve this, which is a classical method in the field of information retrieval. Meanwhile, we propose a bias-resistant value function. During the training stage, preference modeling is employed to construct the training dataset, and the Bradley-Terry model (Bradley & Terry, 1952) is utilized to train the value network. This approach addresses the issue of data imbalance and false negatives. During the inference stage, we introduce an adjustment term related to the ratio of valid tactics into the value function to mitigate the domain gap between the training and test dataset. This stems from a insight that if the number of valid tactics is limited, concerns
+
+may arise regarding the effectiveness of the current policy model, necessitating a reduced value for the current action. Bias-resistant value function can enhance the effectiveness of exploitation during the search process. The complete framework of CARTS is shown in Figure 1. We conducted sufficient experiments on the widely recognized theorem-proving benchmarks, namely miniF2F (Zheng et al., 2022) and ProofNet (Azerbayev et al., 2023) in Lean. Our proposed CARTS demonstrates superior performances compared to all other search methods when the policy network remains unchanged. We achieved a pass@1 success rate of  $49.6\%$  on the miniF2F-test dataset, which is the state-of-the-art performance among all one-step tree search methods.
+
+To summarize, this paper (i). proposes a diversified tactic calibration assisted monte carlo tree search to improve the exploration efficiency. (ii). proposes a bias-resistant value function to improve the exploitation effectiveness. (iii). demonstrates the effectiveness of the proposed method across different models and benchmarks in experiments.
+
+# 2 RELATED WORK
+
+Neural theorem proving. In recent years, the advancement of large language models has brought new progress to theorem proving (Li et al., 2024). GPT-f (Polu & Sutskever, 2020) is the first to utilize language models trained on proof data to predict candidate proof steps and employ search algorithms to discover the complete proof path. A series of subsequent studies have employed diverse language model techniques from various perspectives to enhance theorem proving performance. In terms of model training, PACT (Han et al., 2022) employs a set of self-supervised auxiliary tasks to train the model. Curriculum Learning (Polu et al., 2023) introduces curriculum expert iterations to update the network. Llemma (Azerbayev et al., 2024) continues pre-training the CodeLlama models (Roziere et al., 2023) on a math-focused corpus. AlphaGeometry (Trinh et al., 2024) integrates a transformer model trained on synthetic geometry data with a symbolic deduction engine to solve olympiad geometry problems. InterLM2-Math (Ying et al., 2024b) compiles a substantial collection of both formal and informal contest-level math problems (Ying et al., 2024a; Wu et al., 2024). First et al. (2023) incorporates a repair feedback mechanism in proof generation. This feedback is facilitated by an LLM fine-tuned on tuples consisting of incorrect proof, error message and correct proof. In terms of algorithmic design, Reprover (Yang et al., 2023) employs retrieval-augmented generation for proof generation. DSP (Jiang et al., 2023) initially uses informal hints to guide proofs by translating informal proofs into formal sketches, which are then completed with Isabelle's automated reasoning tactics. LEGOProver (Xin et al., 2024c) enhances DSP with a skill library that expands throughout the proof search. Lyra (Zheng et al., 2024) iterates on DSP by using error feedback to modify the formal sketch, employing automated reasoning tools to correct incorrect proofs of intermediate hypotheses. COPRA (Thakur et al., 2024) utilizes in-context learning agents to augment theorem proving. These methods employ different formal systems. In our paper, we focus on Lean, which has been verified to perform well on IMO-level tasks (Deepmind, 2024).
+
+Search methods for theorem proving. Neural theorem proving primarily consists of two categories: whole proof generation methods Xin et al. (2024a); Wang et al. (2024b) and tree search methods. Tree search methods are increasingly becoming the mainstream approach in recent years. A typical approach involves using Best First Search (BFS), as seen in methods like GPT-f (Polu & Sutskever, 2020), Reprover (Yang et al., 2023) and others (Lin et al., 2024; Welleck & Saha, 2023). In contrast, Thakur et al. (2024) employs depth-first search (DFS). Inspired by AlphaZero (Silver et al., 2018), many methods utilize MCTS, such as HyperTree Proof Search (Lample et al., 2022). There are also several improvements to the MCTS algorithm for theorem proving tasks. For instance, DT-Solver (Wang et al., 2023) uses virtual nodes and a proof-level value function to dynamically guide the MCTS search. Wang et al. (2024a) introduces a novel method that allows for the emergence of unproven lemmas during the search, which are subsequently proven recursively. The aforementioned methods are all one-step tree search techniques, which generates a single tactic at each step. Recently, multi-step tree search methods have been developed. For instance, DeepSeekProver-V1.5 (Xin et al., 2024b) employs MCTS to enhance the whole proof generation process, utilizing intrinsic rewards and discounted upper confidence bounds to guide exploration. Despite the success of these methods, challenges remain, namely the lack of diversity for searched proof paths and bias in the trained value function. This paper focuses on addressing these challenges by employing our diversified tactic calibration and bias-resistant tree search.
+
+Algorithm 1 Maximal Marginal Relevance for Diversified Tactic Calibration  
+Require: current state  $s$ , tactics set  $\{a_1, a_2, \dots, a_e\}$ , next state set  $\{s_1', s_2', \dots, s_e'\}$ , number of selected tactics  $k$ , parameter  $\lambda$ $S \gets \{s\}$ $A \gets \{\}$   
+while  $|A| < \min(k, e)$  do  
+[ a^* \gets \arg \max_{a_i} \left( \lambda \cdot v_{policy}(s, a_i) - (1 - \lambda) \cdot \max_{s_j' \in S} f_{enc}(s_i')^\top f_{enc}(s_j') \right) ]  
+Add  $a^*$  to  $A$   
+Add the corresponding next state  $s^*$  to  $S$   
+end while  
+return the expanded action set  $A$
+
+# 3 METHOD
+
+In this section, we present the details of our proposed method CARTS, which consists of two components. We begin by introducing the diversified tactic calibration (3.1), followed by giving details of our bias-resistant value function (3.2). Together, these two approaches enhance the effectiveness of exploration and exploitation during the search process.
+
+# 3.1 DIVERSIFIED TACTIC CALIBRATION
+
+In practice, the multiple candidate tactics generated by language models often exhibit redundancy. Diversified tactic calibration addresses this by calibrating candidate tactics' model confidence based on their intrinsic similarity. We implement the calibration using the Maximal Marginal Relevance (MMR) algorithm (Peng et al., 2005) and structure our method's framework through Monte Carlo Tree Search (Kocsis & Szeptsvári, 2006).
+
+The standard MCTS method used in theorem proving (Wang et al., 2023; Xin et al., 2024b) involves three steps: Selection, Expansion and Backpropagation. We incorporate diversified tactic calibration into the Expansion phase, resulting in our CARTS method, which comprises three steps: Selection, Calibration & Expansion, and Backpropagation. Details of each step are provided as follows.
+
+Selection. In the selection phase, the algorithm starts from the root node and traverses the tree down to a leaf node. It uses a tree policy to choose child nodes that balance exploration and exploitation. The tree policy at a tree node  $s$  selects an action  $a$  that maximizes the weighted upper confidence bound (WUCB) score, the WUCB score for each tree node  $s$  is formulated as follows:
+
+$$
+\mathrm {W U C B} (s, a) = \frac {W (s , a)}{N (s , a)} + w (s, a) \cdot \frac {\sqrt {N (s , \cdot)}}{N (s , a)} \tag {1}
+$$
+
+Here,  $N(s,a)$  denotes the count of how many times action  $a$  has been taken in state  $s$  and  $N(s,\cdot)$  the total number of times any action has been taken in state  $s$  during the whole search.  $W(s,a)$  denotes the total value accumulated. Unlike the PUCT score used in DT-Solver (Wang et al., 2023), which incorporates probabilities estimated by the language model, we introduce a weight  $w(s,a)$  that represents both importance (model confidence) and diversity for the tactic at the current state. We will detail the weights in the calibration phase.
+
+Calibration & Expansion. At this stage, multiple candidate tactics are generated from the language model, followed by verification through the Lean prover. Verified tactics that pass in Lean are calibrated and expanded into the search tree. Concretely, the proof generation model is designed to generate a one-step proof tactic  $a$  from a given proof state  $s$ , along with a conditional probability  $p(a|s)$ . Typically, we use beam search to sample a large collection of tactics (the quantity is  $E$ ) from the language model, which may result in much low probabilities for each tactic. This will lead to reduced exploration during the search process. To fix this, we apply a length penalty, defined as  $v_{policy}(s,a) = p^{\frac{1}{l}}(s|a)$ , where  $l$  is the token length of tactic  $a$ . This value reflects the model's confidence or the importance of the current action.
+
+After verification by Lean, only  $e$  tactics remain, denoted as  $\{a_1, a_2, \dots, a_e\}$ . Here, we do not directly expand these tactics into the search tree because they often exhibit significant redundancy. Therefore, we need to capture the similarity between these tactics and then reorder them for both diversity and importance. We can compare the similarities between two actions using a pre-trained and fixed sentence encoder. However, directly using action similarity poses a challenge: similar actions may lead to different next states, or dissimilar actions may result in the same next state. This undermines our goal of enhancing search diversity. Therefore, we assess next states similarities after executing these tactics. Formally, we denote the set of the next states as  $\{s_1', s_2', \dots, s_e'\}$ , where  $s_i'$  represents the next state of  $s$  following the execution of action  $a_i$ . We use a sentence encoder  $f_{enc}(\cdot)$ , which has been already pre-trained on a large-scale corpus, to accept the textualized state and outputs high-dimensional embeddings. Then we utilize the MMR algorithm to reorder the tactics and calibrate the model confidence. The algorithm iteratively selects items (e.g., tactics in our context) from a candidate set to maximize the following objective function:
+
+$$
+\operatorname {M M R} (s, a _ {i}) = \lambda \cdot v _ {\text {p o l i c y}} (s, a _ {i}) - (1 - \lambda) \cdot \max  _ {s _ {j} ^ {\prime} \in S} f _ {\text {e n c}} \left(s _ {i} ^ {\prime}\right) ^ {\top} f _ {\text {e n c}} \left(s _ {j} ^ {\prime}\right) \tag {2}
+$$
+
+Where,  $\lambda$  is a parameter that controls the trade-off between importance and diversity. It typically ranges from 0 to 1. The calculation of the MMR score serves a calibration, effectively penalizing tactics with low diversity, as represented by the second term in the formula. The algorithm begins with an initial state set  $S = \{s\}$  and an empty action set  $A$ . While the size of  $A$  is less than the predefined value  $k$ , which is smaller than  $E$ , the action with the highest MMR score is selected and added to  $A$ , along with its next state being added to  $S$ . Once  $k$  actions have been chosen, the set  $A$  is returned. If  $k < e$ , we select all the tactics and reorder them. The value  $k$  serves as a constraint on the maximum number of expansion nodes. It is worth noting that we add the current state  $s$  into the initial state set  $S$  to mitigate the recurrence of identical states. The reason this way is effective is that our algorithm assigns a lower MMR score to the actions when the next state closely resembles the current state. Algorithm 1 illustrates the complete process of diversified tactic calibration.
+
+After diversified tactic calibration, we obtain a small set of actions  $A$  that contains both importance and diversity. Then, these actions are treated as edges, with their corresponding next state as nodes, which are expanded into the current search tree. For each edge  $a_i$ , we assign a weight  $w(s, a_i) = \max \{0, \mathrm{MMR}(s, a_i)\}$ , utilized during the selection phase to assess the need for exploration. Unlike traditional MCTS (Kocsis & Szepesvári, 2006) or DT-Solver (Wang et al., 2023), our weights place greater emphasis on encouraging the exploration of tactics with high diversity.
+
+Backpropagation. At this stage, we update the statistics of the nodes and edges along the search trajectory. We have a bias-resistant value function  $V(s,a)$  which will be detailed in the next section, estimating the value of taking action  $a$  from the source node  $s$ . For a given trajectory, we use the value function to evaluate the value of the leaf node and accumulate this value along all edges in the path. Specifically, we update the weight of the edge recursively as follows:  $W(s_{t},a_{t}) + = V(s,a)$ , where  $s_t$  and  $a_{t}$  represent the node and edge at the trajectory. Additionally, we increment the visit count for the edge:  $N(s_{t},a_{t}) + = 1$ . This process ensures that the statistics reflect the outcomes of the simulations, allowing for improved selection in future iterations.
+
+# 3.2 BIAS-RESISTANT VALUE FUNCTION
+
+In MCTS-based methods, training a value function is crucial (Polu & Sutskever, 2020; Lample et al., 2022), typically involving the creation of positive and negative samples using the policy network on training data. Positive samples consist of correct actions (or trajectories (Wang et al., 2023)) from the dataset, while negative samples are those generated by the policy network that lead to undesirable states. Binary cross-entropy loss is then used to train the value network. Due to the hardness of verifying the correctness of actions not on the proof path, previous work (Polu & Sutskever, 2020) often treats these actions as negative samples, resulting in an excessive number of negative samples, some of which are even inaccurate. This makes binary loss unsuitable and biases the value function. Furthermore, the domain gap between the training and test datasets also contributes to biases. In this paper, we conduct debiasing during both training and inference stages, as detailed below.
+
+Training. To mitigate bias introduced by data collection, we first structure the dataset into preference pairs of positive and negative samples. We utilize an embedding model  $f_{enc}$  to effectively
+
+filter out noisy samples. Specifically, if  $f_{enc}(s')^\top f_{enc}(s'_pos) > \tau$ , we discard the action  $a$ .  $s'$  is the next state from a sampled negative action  $a$  at the current state  $s$ ,  $s'_pos$  is the correct next state and  $\tau$  is a threshold. This filtering ensures that the selected negative actions are more likely to be undesirable, thus reducing data noise. Moreover, we adopt the preference modeling framework to train our bias-resistant value function. We employ the Bradley-Terry (BT) (Bradley & Terry, 1952) model, a widely used technique for preference modeling. The BT model posits that the probability of action  $a_{pos}$  being preferred over action  $a_{neg}$  given state  $s$  is expressed as:
+
+$$
+\mathbb {P} \left(a _ {p o s} \succ a _ {n e g} \mid s\right) = \frac {\exp \left(V _ {\theta} (s , a _ {p o s})\right)}{\exp \left(V _ {\theta} (s , a _ {p o s})\right) + \exp \left(V _ {\theta} (s , a _ {n e g})\right)} \tag {3}
+$$
+
+Assuming access to the filtered dataset  $D = \{(s^{(i)},a_{pos}^{(i)},a_{neg}^{(i)})\}_{i = 1}^{N}$ , we can parametrize the value function  $V_{\theta}(s,a)$  and estimate the parameters  $\theta$  by minimizing the negative log-likelihood.
+
+Preference modeling offers several advantages. Firstly, by only providing the relative superiority among samples, false negative samples do not require further processing. This is because we can reasonably assume that the correct proof steps provided in the dataset are always optimal and align with human theorem proving's preferences. Additionally, since the dataset is presented in the form of preference pairs, this effectively oversamples (Shi et al., 2023) the positive pairs, alleviating the issue of class imbalance between positive and negative samples, as demonstrated in some studies (Zhang et al., 2024; Pattnaik et al., 2024).
+
+Inference. To mitigate the domain gap between the training and test datasets, we introduce an adjustment term into the value function during the inference stage. As previously mentioned, before calibration in CARTS, all  $E$  tactics should be processed through the Lean system to filter out  $e$  valid tactics. Intuitively, if the number of valid tactics is small, people will raise concerns about the capability of the current policy model, needing for a reduced reward for the current action. We define this reward adjustment as:  $\alpha = e / E$ , representing the ratio between the number of valid tactics and the total number of tactics generated by the language model at the current state. This adjustment term serves as a test-time adaptation to the test dataset. The final bias-resistant value function integrates both the trained value network and this adjustment term as:
+
+$$
+V (s, a) = \left\{ \begin{array}{l l} 0, & \text {i f} s ^ {\prime} \text {h a s n o c h i l d n o d e s}, \\ 1, & \text {e l s e i f} s ^ {\prime} \text {i s t h e p r o v e d s t a t e}, \\ \frac {1}{2} (\alpha + V _ {\theta} (s, a)), & \text {o t h e r w i s e}. \end{array} \right. \tag {4}
+$$
+
+Where  $s'$  is the next state. Unlike the intrinsic reward introduced by DeepSeek-Prover-V1.5 (Xin et al., 2024b), which only considers whether the search expands nodes, we consider both the expansion capability of the policy network and the generalizability of the value network, forming our final bias-resistant value function. The adjustment term can be interpreted as a form of test-time adaptation to the distribution of test data, thus can mitigate the domain gap.
+
+# 4 EXPERIMENTS
+
+In this section, we evaluate the theorem-proving performance of CARTS in Lean. We first describe the experimental setup, then present the main results, followed by an analysis of our method. Currently, theorem-proving methods are primarily categorized into two main types: whole-proof generation methods and tree search methods. Our approach is applicable exclusively to one-step tree search methods; therefore, we focus our comparison solely on this category.
+
+# 4.1 EXPERIMENTAL SETUP
+
+Datasets. We follow Internlm-math (Ying et al., 2024b; Wu et al., 2024) and DeepSeek-Prover-V1.5 (Xin et al., 2024b), utilizing miniF2F benchmark (Zheng et al., 2022) and ProofNet benchmark (Azerbayev et al., 2023) for our evaluation. We specifically use the test split of miniF2F same as (Xin et al., 2024b), which includes 244 problems ranging from basic algebra and number theory and also contains AIME and IMO challenging problems. ProofNet is a benchmark for undergraduate-level mathematics, comprising 371 formal problems derived from widely-used undergraduate pure mathematics textbooks. It covers topics such as real and complex analysis, abstract algebra, and
+
+Table 1: Results on the miniF2F-test for various models and search methods. The highest performance for each search method is highlighted in bold.  
+
+<table><tr><td>Model</td><td>Sample Budget</td><td>Search Method</td><td>miniF2F-test</td></tr><tr><td colspan="4">Tree Search Methods</td></tr><tr><td>COPRA(Code Llama) (Thakur et al., 2024)</td><td>500</td><td>DFS</td><td>5.7%</td></tr><tr><td>COPRA(GPT-3.5) (Thakur et al., 2024)</td><td>60</td><td>DFS</td><td>9.0%</td></tr><tr><td>COPRA(GPT-4) (Thakur et al., 2024)</td><td>60</td><td>DFS</td><td>26.6%</td></tr><tr><td>Llemma-7B (Azerbayev et al., 2024)</td><td>32 × 100</td><td>BFS</td><td>26.2%</td></tr><tr><td>Llemma-34B (Azerbayev et al., 2024)</td><td>32 × 100</td><td>BFS</td><td>25.8%</td></tr><tr><td>LLMStep (Welleck &amp; Saha, 2023)</td><td>32 × 100</td><td>BFS</td><td>27.9%</td></tr><tr><td>Curriculum Learning (Polu et al., 2023)</td><td>8 × 512</td><td>BFS</td><td>29.6%</td></tr><tr><td>InternLM2-Math-7B (Ying et al., 2024b)</td><td>32 × 100</td><td>BFS</td><td>30.3%</td></tr><tr><td>InternLM2-Math-Plus-7B (Ying et al., 2024a)</td><td>32 × 100</td><td>BFS</td><td>43.4%</td></tr><tr><td>DeepSeek-Prover-V1.5-SFT (Xin et al., 2024b)</td><td>3200</td><td>RMaxTS</td><td>53.5%</td></tr><tr><td>DeepSeek-Prover-V1.5-RL (Xin et al., 2024b)</td><td>3200</td><td>RMaxTS</td><td>55.0%</td></tr><tr><td rowspan="4">Reprover-Lean4 (229M) (Yang et al., 2023)</td><td rowspan="4">64 × 100</td><td>BFS</td><td>35.7%</td></tr><tr><td>MCTS</td><td>36.5%</td></tr><tr><td>DTSolver</td><td>36.0%</td></tr><tr><td>CARTS</td><td>37.7%</td></tr><tr><td rowspan="4">InternLM2-Math-Plus-1.8B (Ying et al., 2024b)</td><td rowspan="4">64 × 100</td><td>BFS</td><td>38.9%</td></tr><tr><td>MCTS</td><td>39.3%</td></tr><tr><td>DTSolver</td><td>38.5%</td></tr><tr><td>CARTS</td><td>41.0%</td></tr><tr><td rowspan="4">StepProver (7B) (Wu et al., 2024)</td><td rowspan="4">32 × 300</td><td>BFS</td><td>48.8%</td></tr><tr><td>MCTS</td><td>46.7%</td></tr><tr><td>DTSolver</td><td>46.3%</td></tr><tr><td>CARTS</td><td>49.6%</td></tr></table>
+
+topology. This benchmark presents a greater challenge than miniF2F, posing significant difficulties for theorem provers. Although the original versions of both benchmarks are Lean3, we have modified them to Lean4 for CARTS's evaluation, aligning with the development of the Lean community.
+
+Baselines models. We include baselines representing classical and state-of-the-art neural theorem proving in Lean. COPRA (Thakur et al., 2024) is an in-context learning agent that utilizes general language models to generate tactics for finding the final proof. Llemma (Azerbayev et al., 2024) is trained on extensive mathematical corpora. Additionally, we incorporate advanced models such as LLMStep (Welleck & Saha, 2023), Reprover (Yang et al., 2023), Curriculum Learning (Polu et al., 2023), InternLM2-Math (Ying et al., 2024b), and StepProver (Wu et al., 2024). All these models are based on one-step tree search methods. We also include DeepSeek-Prover-V1.5 (Xin et al., 2024b), which integrates whole proof generation and tree search. However, it is not suitable for our CARTS, and thus, we mark it in gray.
+
+Search methods. Baseline models employ various search methods, such as depth-first search (DFS) (Thakur et al., 2024), best first search (BFS) (Yang et al., 2023), and monte carlo tree search (MCTS). Additionally, DT-Solver extends MCTS using virtual nodes. We compared the performance of BFS, MCTS, DT-Solver and CARTS on Reprover (Yang et al., 2023), InternLM2-Math (Ying et al., 2024b), and InternLM2-StepProver (Wu et al., 2024). For MCTS and DT-Solver, we replace the value network with the intrinsic reward (Xin et al., 2024b) for simplicity. It is noteworthy that multiple tree search attempts with different seeds can be applied and ensemble (Polu & Sutskever, 2020; Lin et al., 2024; Xin et al., 2024a); however, due to computational cost limitation, we only compared the results for one single tree search attempt.
+
+Metrics. We evaluate the performance of various search methods using the pass@1 metric with a budget  $B$ . Similar to (Xin et al., 2024b), if  $B$  is a single value, it indicates the number of model
+
+Table 2: Results on the ProofNet for various models and search methods.  
+
+<table><tr><td>Model</td><td>Sample Budget</td><td>Search Method</td><td>ProofNet</td></tr><tr><td rowspan="3">Reprover-Lean4 (Yang et al., 2023)</td><td rowspan="3">64 × 100</td><td>BFS</td><td>11.1%</td></tr><tr><td>MCTS</td><td>11.7%</td></tr><tr><td>CARTS</td><td>11.9%</td></tr><tr><td rowspan="3">StepProver (Wu et al., 2024)</td><td rowspan="3">32 × 300</td><td>BFS</td><td>18.1%</td></tr><tr><td>MCTS</td><td>18.3%</td></tr><tr><td>CARTS</td><td>18.8%</td></tr></table>
+
+![](images/5e084df84361b50e22f54af89318b6c135ed1e3728e561e977d999f8906e74e3.jpg)  
+Figure 2: Improvement curve in pass@1 for the miniF2F test as the expansion budget varies. Left illustrates Reprover-Lean4 (Yang et al., 2023) and right illustrates StepProver (Xin et al., 2024b).
+
+![](images/c1d56fb70cd4d115b0629fa98c84749e582dc8a19a935091e7711341969716fb.jpg)
+
+generations used in tree expansions. If  $B = E \times T$ ,  $E$  represents the number of tactics generated per expansion, and  $T$  denotes the number of expansion iterations.
+
+Experimental details. In terms of parameter settings, for Reprover-Lean4 (Yang et al., 2023), we set  $\lambda = 0.8$ , and for InternLM2-Math-Plus-1.8B (Ying et al., 2024b) and StepProver (Wu et al., 2024), we set  $\lambda = 0.9$ . Additionally, we set  $k = 8$  for all models. We use the text embedding model intfloat/e5-small1-v2 (Wang et al., 2022) as the encoder  $f_{enc}$ . Details regarding data collection and training for the bias-resistant value function are presented in Appendix A.
+
+# 4.2 MAIN RESULTS
+
+In Table 1, we illustrate the pass@1 successful rate on the miniF2F-test benchmark of various models and search methods. Our proposed CARTS surpasses all compared search methods when the policy model is fixed. The results encompass different architectures and parameter sizes of language models, demonstrating that our method is more effective at search stage regardless of the policy model. Notably, we achieve a  $49.6\%$  success rate on the miniF2F-test, representing state-of-the-art performance among one-step tree search methods. Table 2 demonstrates the results on the ProofNet benchmark. Due to the dataset's complexity, current policy models exhibit low accuracy, which constrain the search performances. However, our proposed CARTS also achieves the highest performance compared to other search methods when the budget is fixed.
+
+Additionally, to demonstrate that CARTS has the superior search efficiency, we present a comparison of the pass@1 performance among four search methods on miniF2F-test when varying the expansion budget, as illustrated in Figure 2. It is evident that while the performance of different search methods is comparable under a low budget, our method exhibits a significant improvement as the expansion budget increases. Furthermore, we observed that the CARTS curve increases more rapidly relative to other search methods. These findings demonstrate that our approach enhances the efficiency and effectiveness of tree search.
+
+Table 3: Ablation study of various components of CARTS on the miniF2F-test. We compare the pass@1 performance across three models, with the highest performance highlighted in bold.  
+
+<table><tr><td rowspan="2">Search Method</td><td colspan="3">Model</td></tr><tr><td>Reprover-Lean4</td><td>InternLM2-Math-Plus-1.8B</td><td>StepProver</td></tr><tr><td>▷ Baselines:</td><td></td><td></td><td></td></tr><tr><td>BFS</td><td>35.7%</td><td>38.9%</td><td>48.8%</td></tr><tr><td>MCTS</td><td>36.5%</td><td>39.3%</td><td>46.7%</td></tr><tr><td>▷ Ablations:</td><td></td><td></td><td></td></tr><tr><td>Diversified tactic calibration</td><td>36.1%</td><td>41.0%</td><td>49.2%</td></tr><tr><td>Bias-resistant value function</td><td>36.9%</td><td>40.6%</td><td>46.3%</td></tr><tr><td>CARTS (without adjustment term)</td><td>37.3%</td><td>41.3%</td><td>49.2%</td></tr><tr><td>CARTS (Ours)</td><td>37.7%</td><td>41.0%</td><td>49.6%</td></tr></table>
+
+# 4.3 ANALYSIS
+
+In this section, we conduct a detailed analysis of CARTS' effectiveness through experiments. We include ablation studies and specifically investigate the length distribution of results produced by CARTS, supplemented by several case studies.
+
+Ablation for different components of CARTS. CARTS consists of two components: diversified tactic calibration and a bias-resistant value function. We isolated these components for analysis. Additionally, we compared the performance of CARTS without the adjustment term in the value function. The experimental results are shown in Table 3. The results indicate that both components enhance performance, and their combination further increases the pass rate, suggesting that CARTS can significantly improve the efficiency of tree search exploration. Our findings reveal that for InternLM2-Math-Plus-1.8B, CARTS without the adjustment term achieves superior results, whereas CARTS with the adjustment does not perform optimally. We believe this is because the adjustment term is merely a heuristic calibration; the policy model generates more unacceptable tactics, the less likely it is to generate accurate tactics. This is not always valid due to differences between the properties of the proof state. Nonetheless, experiments across multiple models suggest that it is generally effective for debiasing the value function.
+
+Ablation study for  $k$ . The parameter  $k$  limits the maximum expansion number of nodes after diversified tactic calibration. A smaller  $k$  significantly restricts the search space, potentially leading to performance degradation. Conversely, a larger  $k$  expands the search space, increasing the likelihood of identifying the correct tactic. However, this comes at the cost of substantially increased search time, thereby reducing overall efficiency. To assess its impact on CARTS, we test multiple values of  $k$  using Reprover-Lean4 on the miniF2F-test. The experimental results are shown in the left side of Figure 3. As  $k$  increases, the pass@1 rate initially improves, reaching a peak, and then gradually declines. This indicates that there is an optimal range for  $k$  that balances performance and search efficiency. Therefore, we recommend choosing a moderate  $k$ , such as  $k = 8$ , in practice.
+
+Analysis of the length distribution of discovered proofs. To comprehensively demonstrate the effectiveness of CARTS, we analyze the proof step lengths obtained by the Reprover-Lean4 model on the miniF2F-test. The results are presented on the right side of Figure 3. As illustrated, BFS yields a higher proportion of theorems with step lengths of one and two, but with none exceeding three steps. In contrast, CARTS achieves more theorems with lengths of three or greater, significantly outperforming both BFS and MCTS. This indicates that CARTS enhances search efficiency, enabling the exploration of longer proof chains within the same budget. Additionally, we find that CARTS discover two proofs with the length of five. We examine this case in detail in next paragraph.
+
+Case study. We conduct a comprehensive case study to closely examine the results of the CARTS search. Figure 4 demonstrates one theorem which is proved by CARTS on Reprover-Lean4 model.
+
+![](images/0ba34c77b357dab865ef9ee310eedbd107834b20b18aa648d116f3a6a3d0e511.jpg)  
+Figure 3: The left figure shows the ablation study for  $k$ , while the right figure illustrates the length distribution of proofs searched by CARTS. Both analyses are conducted on the miniF2F-test using the Reprover-Lean4 (Yang et al., 2023) model.
+
+![](images/0433be4143323af31d3776e0e02e5b4da9126fb380d4b9b43fe7b2ac64779ccf.jpg)  
+Figure 4: Case study. Proved theorem induction_12dvd4expnplp20 by CARTS in miniF2F-test of ReProver-Lean 4 (Yang et al., 2023). This theorem is proved in 77 expansions.
+
+```txt
+Case: induction_12dvd4expnplp20 theorem induction_12dvd4expnplp20 (n : N) : 12 | 4^(n + 1) + 20 := by norm_num [Nat.pow_succ] induction' n with n n_ih case zero => simpa using n_ih rw [pow_succ] omega
+```
+
+This case has proof path of five length. This theorem induction_12dvd4expnplp20 pertains to divisibility and its initial step involves simplification using norm_num [Nat.pow_succ]. However, the tactic generated by Repover-Lean4 exhibits a low confidence level, only ranking eight compared to other incorrect tactics. If the BFS approach is to be applied, it would necessitate the exploration of numerous redundant tactics before arriving at this specific action. Our experiments reveal that the CARTS improves the score of this tactic and increases its rank to six. Furthermore, we find that the score of the second tactic, namely induction' n with n n.i.h, improves from an initial rank of sixth to fourth. While the increase in the ranking of the correct tactic score is only by two, the exponential growth in the complexity of tree search highlights the significance of early exploration of the correct action at the root node. Therefore, these improvements in rankings play a crucial role in successfully proving this theorem within a limited number of expansion.
+
+# 5 CONCLUSION AND LIMITATION
+
+In this paper, we presented CARTS, a novel approach that combines diversified tactic calibration with a bias-resistant value function for enhancing the performance of neural theorem proving. Our diversified tactic calibration effectively mitigates the redundancy in language model generations, promoting a more diverse and balanced exploration of candidate tactics. The bias-resistant value function addresses the prevalent biases during both the training and inference stage, thereby refining the exploitation capabilities of the tree search by providing more accurate and reliable evaluations of tactics. Experimental results on miniF2F and ProofNet have demonstrated that CARTS significantly outperforms existing one-step tree search methods, achieving state-of-the-art results.
+
+While CARTS has proven effective, it also has limitations. Our method currently only supports one-step tree search, whereas multi-step tree search, such as DeepSeek-Prover-V1.5 (Xin et al., 2024b), has shown promise but is not supported by our approach. In the future, we will explore variants of CARTS that can be integrated into multi-step tree search methods.
+
+# ACKNOWLEDGMENT
+
+This research was supported by the Key Program of Jiangsu Science Foundation (BK20243012), Leading-edge Technology Program of Jiangsu Science Foundation (BK20232003), the Jiangsu Science Foundation (BG2024036) and the Fundamental Research Funds for the Central Universities (022114380023). We would like to thank the reviewers for their constructive suggestions.
+
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+
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+Huajian Xin, Z.Z. Ren, Junxiao Song, Zhihong Shao, Zhao Wanjia, Haocheng Wang, Bo Liu, Liyue Zhang, Xuan Lu, Qiushi Du, Wenjun Gao, Qihao Zhu, Dejian Yang, Zhibin Gou, Z.F. Wu, Fuli Luo, and Chong Ruan. DeepSeek-Prover-V1.5: Harnessing Proof Assistant Feedback for Reinforcement Learning and Monte-Carlo Tree Search. arXiv preprint arXiv:2408.08152, 2024b.  
+Huajian Xin, Haiming Wang, Chuanyang Zheng, Lin Li, Zhengying Liu, Qingxing Cao, Yinya Huang, Jing Xiong, Han Shi, Enze Xie, et al. LEGO-Prover: Neural Theorem Proving with Growing Libraries. In Proceedings of the International Conference on Learning Representations, 2024c.  
+Kaiyu Yang, Aidan Swope, Alex Gu, Rahul Chalamala, Peiyang Song, Shixing Yu, Saad Godil, Ryan Prenger, and Anima Anandkumar. LeanDojo: Theorem Proving with Retrieval-Augmented Language Models. In Proceedings of the International Conference on Neural Information Processing Systems, 2023.  
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+Huaiyuan Ying, Shuo Zhang, Linyang Li, Zhejian Zhou, Yunfan Shao, Zhaoye Fei, Yichuan Ma, Jiawei Hong, Kuikun Liu, Ziyi Wang, et al. InternLM-Math: Open Math Large Language Models Toward Verifiable Reasoning. arXiv preprint arXiv:2402.06332, 2024b.  
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+Kunhao Zheng, Jesse Michael Han, and Stanislas Polu. MiniF2F: A Cross-System Benchmark for Formal Olympiad-Level Mathematics. In Proceedings of the International Conference on Learning Representations, 2022.
+
+# A DETAILS OF BIAS-RESISTANT VALUE FUNCTION
+
+Data Construction To train our bias-resistant value function on Lean4, we utilized two data sources. The first source is the random portion of the leandrojoenchmark (Yang et al., 2023), which is derived from mathlib4 (commit: 3ce43c18f614b76e161f911b75a3e1ef641620ff). The second source is Lean-Github (Wu et al., 2024), a recently released dataset compiled from over 100 Lean 4 repositories. We employed InternLM2-Math-Plus-7B (Ying et al., 2024b) to generate eight candidate tactics for each proof state in the dataset. Tactics that differed from the correct tactic in the generations are used as negative samples, which we then paired with positive samples to form preference pairs. However, as discussed in the main text, this dataset contains significant noise, with positive and negative samples being semantically similar, as illustrated in Figure 5. To address this, we filtered the samples based on their similarity, using the embedding model intfloat/e5-small-v2 (Wang et al., 2022) and a threshold  $\tau = 0.85$ . After filtering, the dataset consists of 1,797,951 (about 1800K) training preference pairs and 2,000 test preference pairs.
+
+```txt
+STATE:   
+case a.h   
+m : Type u   
+n : Type v   
+ $\alpha$  :Type w   
+ $\mathrm{inst}^{+4}$  :DecidableEq n   
+ $\mathrm{inst}^{+3}$  :Fintype n   
+ $\mathrm{inst}^{+2}$  :DecidableEq m   
+ $\mathrm{inst}^{+1}$  :Fintype m   
+ $\mathrm{inst}+$  : CommRing  $\alpha$    
+A : Matrix n n  $\alpha$    
+i j : n   
+ $\vdash \forall (x:\mathrm{Perm}~n),$  （20  $x\in \mathrm{univ}\rightarrow$  ↑↑(↑sign x）*Ii1:n,updateRow A I (Pi(single j 1)（↑x i_1)i_1= ↑↑(↑sign x）*Ii1:n,updateColumn A j(Pi.single i 1)(↑x i_1)i_1   
+POSITIVE ACTION:   
+intro  $\sigma_{-}$    
+NEGATIVE ACTION:   
+intros  $\sigma_{-}$
+```
+
+Before Filtering
+
+```txt
+STATE:   
+case a.h   
+m : Type u   
+n : Type v   
+ $\alpha$  :Type w   
+ $\mathrm{inst}^{+4}$  :DecidableEq n   
+ $\mathrm{inst}^{+3}$  :Fintype n   
+ $\mathrm{inst}^{+2}$  :DecidableEq m   
+ $\mathrm{inst}^{+1}$  :Fintype m   
+ $\mathrm{inst}+$  : CommRing  $\alpha$    
+A : Matrix n n  $\alpha$    
+i j : n   
+ $\vdash \forall (x:\mathrm{Perm}~n),$  （20  $x\in \mathrm{univ}\rightarrow$  ↑(sign x）*Ii1:n,updateRow A I (Pi.single j 1)(↑x i_1)i_1= ↑(sign x)*IIi_1:n,updateColumn A j(Pi.single i 1)(↑x i_1)i_1   
+POSITIVE ACTION:   
+intro  $\sigma_{-}$    
+NEGATIVE ACTION:   
+simp only [updateRow_apply, updateColumn_apply]
+```
+
+After Filtering
+
+Figure 5: This illustrates an example of our dataset organization: the left side contains unfiltered data with noise, while the right side shows the samples after filtering.
+
+Training Details We utilized the microsoft/phi-1.5 model (Li et al., 2023), which contains 1.3 billion parameters. Phi-1.5 has been trained on a large-scale code corpus, endowing it with a strong understanding of formal languages. We fine-tuned the model using LoRA (Hu et al., 2021) for one training epoch, using the learning rate as  $1 \times 10^{-4}$ . After training, the model attained an accuracy of  $94.10\%$  on the test dataset. The training loss curve and validation accuracy curve are illustrated in Figure 6.
+
+# B EFFECTIVENESS OF BIAS-RESISTANT VALUE FUNCTION
+
+In this section, we supplement experiments to demonstrate the effectiveness of our bias-resistant value function and draw the following conclusions: 1. The Bradley-Terry (BT) model effectively addresses the issue of class imbalance in training the value function. 2. Our analysis underscores the critical importance of data filtering and provides a way for determining the hyperparameter  $\tau$ .
+
+![](images/b89efd81fa2b306c61d834ec6a138487aaa78354503845c540513144905a7444.jpg)  
+Figure 6: The training loss curve and the validation accuracy curve.
+
+![](images/bbec211477f3dd1fc4237828236ef974772a0a8d34a77056bdc47602cdfeff0d.jpg)
+
+# B.1 ANALYSIS BETWEEN BT AND CE
+
+We reconstruct the pairwise dataset and employ various models to validate the effectiveness of the BT modeling approach for the value function. Unlike the main text, where both leandojoenchmark (Yang et al., 2023) and Lean-Github (Wu et al., 2024) are utilized as data sources, this analysis exclusively uses leandojoenchmark for training, with its test split serving as the evaluation dataset. Furthermore, data from Lean-Github are employed for Out-of-Distribution (OOD) evaluation. To enable training of Cross-Entropy (CE) loss, we convert the pair-wise datasets into the binary classification datasets. The training data exhibits an imbalance between positive and negative samples (approximately  $1:5$ ), while we ensured that both test sets remains class-balanced. Detailed statistics of the dataset are presented in Table 4.
+
+Table 4: Statistics of our constructed dataset.  
+
+<table><tr><td rowspan="2">Split</td><td colspan="2">Count</td></tr><tr><td>For BT</td><td>For CE</td></tr><tr><td>Train</td><td>694,600</td><td>861,186</td></tr><tr><td>Test</td><td>3,465</td><td>6,930</td></tr><tr><td>Test-OOD</td><td>3,499</td><td>6,998</td></tr></table>
+
+We conducted training on three models to compare the differences between BT and CE, including Qwen-2.5-0.5B (Qwen Team, 2024), Llama3.2-1B (Dubey et al., 2024), and Llama3.2-3B (Dubey et al., 2024). We evaluated the accuracy (Acc) on Test and the out-of-distribution accuracy (Acc-OOD) on Test-OOD. All experiments are full fine-tuning for one epoch with a learning rate of 1e-5. The experimental results are presented in Table 5.
+
+The experimental results show that BT significantly outperforms CE in terms of in-domain accuracy, highlighting the advantages of using BT for modeling. Furthermore, we observe a slight performance improvement in BT as model size increases; however, the improvements remains relatively small. We attribute this to the limited dataset size, which hinders the emergence of clear scaling properties. In addition, we observe that CE shows poor performance in scenarios with a significant domain gap, and its accuracy is almost the same as random guessing. In contrast, the BT model achieves approximately a  $5\%$  improvement in such cases, although its overall performance remains limited. We believe this is due to the significant domain gap between the leandojoenchmark and Lean-Github datasets, making the task inherently challenging. (Note: In our main text, both datasets were included in the training set to mitigate the gap with miniF2F.)
+
+Table 5: Results between BE and CE of different models.  
+
+<table><tr><td>Model</td><td>Method</td><td>Acc (%)</td><td>Acc-OOD (%)</td></tr><tr><td rowspan="2">Qwen-2.5-0.5B (Qwen Team, 2024)</td><td>CE</td><td>63.2</td><td>50.9</td></tr><tr><td>BT</td><td>76.5</td><td>55.2</td></tr><tr><td rowspan="2">Llama3.2-1B (Dubey et al., 2024)</td><td>CE</td><td>67.4</td><td>52.2</td></tr><tr><td>BT</td><td>76.6</td><td>54.6</td></tr><tr><td rowspan="2">Llama3.2-3B (Dubey et al., 2024)</td><td>CE</td><td>63.0</td><td>50.7</td></tr><tr><td>BT</td><td>77.7</td><td>56.0</td></tr></table>
+
+# B.2 ANALYSIS OF DATA FILTERING
+
+In Section 3.2, we introduced a filtering step for the data constructed. Here we conduct experiments to analyze the necessity of this step. Specifically, we perform an ablation study on the filtering process using the Qwen-2.5-0.5B model (Qwen Team, 2024), comparing the model's performance before and after applying filtering. The results in Table 6 demonstrate that filtering improves the model's performance. We have presented examples of noise introduced by the absence of filtering in Figure 5, further highlighting the necessity of this step.
+
+Table 6: Ablation study of data filtering.  
+
+<table><tr><td>Method</td><td>Acc</td><td>Acc-OOD</td></tr><tr><td>Before filtering</td><td>76.2</td><td>54.9</td></tr><tr><td>After filtering</td><td>76.5</td><td>55.2</td></tr></table>
+
+The choice of  $\tau$  represents a trade-off: setting it too high introduces excessive noisy data, while setting it too low reduces the number of training samples. To provide insight into the selection of  $\tau$ , we analyze the similarity distribution of positive and negative samples before data filtering, as shown in Figure 7. The results reveal a bimodal distribution. Higher similarity values indicate a greater likelihood of noisy samples, leading us to hypothesize that one of the components corresponds to noise. Using Gaussian Mixture Model (GMM) analysis, we identify an approximate threshold of 0.87 to distinguish noisy samples. Consequently, the selection of 0.85 for  $\tau$  in the main text is considered reasonable.
+
+![](images/f0da40cadb94af8af7bdc7c977105ad894120dade1f30e1c4ce45633db245742.jpg)  
+Figure 7: The distribution of similarities between positive and negative samples.
+
+# C EXAMPLE SOLUTIONS
+
+We show several typical theorems proven by CARTS below.
+
+Case 1:mathd_algebra_148  
+```txt
+theorem mathd_algebra_148 (c :  $\mathbb{R}$ ) (f :  $\mathbb{R} \to \mathbb{R}$ )  
+ $(h_0: \forall x, f x = c * x^3 - 9 * x + 3)$ $(h_1: f 2 = 9): c = 3 :=$   
+by  
+rcases eq_or_ne c 1 with hc | hc  
+cases c  
+all_goals simp_all [h_0]  
+on_goal 1 => norm_num at h_1  
+linarith only [h_0, h_1]
+```
+
+Figure 8: Case 1. Proved theorem mathd_algebra_148 by CARTS in mini-F2F-test of ReProver-Lean 4 (Yang et al., 2023). This theorem is proved in 68 expansions.
+
+Case 2: mathd_numbertheory_234  
+```txt
+theorem mathd_numbertheory_234  
+(a b : N)  
+(h0 : 1 ≤ a ∧ a ≤ 9 ∧ b ≤ 9)  
+(h1 : (10 * a + b) ^ 3 = 912673) :  
+a + b = 16 :=  
+by  
+simp only [Nat.one_le_iff_ne_zero] at h0  
+obtain <h2, h3, h4> := h0  
+interval_cases a <;> interval_cases b <;> simp_all
+```
+
+Figure 9: Case 2. Proved theorem mathd_numbertheory_234 by CARTS in mini-F2F-test of InternLM2-Plus-1.8B (Ying et al., 2024b). This theorem is proved in 12 expansions.
+
+Case 3: mathd_numbertheory_135  
+```txt
+theorem mathd_numbertheory_135  
+(n A B C : N)  
+(h0 : n =3^17 +3^10)  
+(h1 : 11 | (n +1))  
+(h2 : [A,B,C].Pairwise (∅·))  
+(h3 : {A,B,C} ⊆Fsubset.Icc 0 9)  
+(h4 : Odd A ∧Odd C)  
+(h5 : ¬3 | B)  
+(h6 : Nat.digits 10 n = [B,A,B,C,C,A,C,B,A] : 100 *A +10 *B +C =129 := by  
+rw [h0] at h1  
+simp [h0] at h6  
+linarith [h6]
+```
+
+Figure 10: Case 3. Proved theorem mathd_numbertheory_135 by CARTS in mini-F2F-test of InternLM2-Plus-1.8B (Yang et al., 2023). This theorem is proved in 38 expansions.
+
+Case 4: mathd_numbertheory_314  
+```txt
+theorem mathd_numbertheory_314  
+(r n : N)  
+(h0 : r = 1342 % 13)  
+(h1 : 0 < n)  
+(h2 : 1342 | n)  
+(h3 : n % 13 < r) :  
+6710 ≤ n :=  
+by  
+obtain {k, hk} := h2  
+contrapose! h1  
+rw [hk] at h1  
+have h2 : k < 5 := by linarith  
+interval_cases k <> simp_all
+```
+
+Figure 11: Case 4. Proved theorem mathd_numbertheory_314 by CARTS in mini-F2F-test of StepProver (Wu et al., 2024). This theorem is proved in 159 expansions.
+
+Case 5: artin exercise_13_6_10  
+```txt
+theorem exercise_13_6_10 {K : Type*} [Field K] [Fintype K^x] :  
+(Π x : K^x, x) = -1 :=  
+by  
+have h : Πx : K^x, x = Πx : K^x, x^{-1-1} := by congr  
+simp only [inv_inv] at h  
+haveI : DecidableEq K := Classical.decEq K  
+apply FiniteField.prod_univ.units_id_eq_neg_one
+```
+
+Figure 12: Case 5. Proved theorem artin exercise_13_6_10 by CARTS in ProofNet of StepProver (Wu et al., 2024). This theorem is proved in 119 expansions.

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ICLR/2025/CAT-3DGS_ A Context-Adaptive Triplane Approach to Rate-Distortion-Optimized 3DGS Compression/full.md

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+# CAT-3DGS: A CONTEXT-ADAPTIVE TRIPLANE APPROACH TO RATE-DISTORTION-OPTIMIZED 3DGS COMPRESSION
+
+Yu-Ting Zhan\*, Cheng-Yuan Ho\*, Hebi Yang\*, Yi-Hsin Chen\*, Jui Chiu Chiang\* Yu-Lun Liu\*, Wen-Hsiao Peng\*
+
+$^{1}$ National Yang Ming Chiao Tung University, Taiwan  
+$^{2}$ National Chung Cheng University, Taiwan  
+{dotori25.i12, kelvinhe0218.cs12}@nycu.edu.tw  
+{mrrrIMAGE32.cs13, yhchen12101.cs09}@nycu.edu.tw  
+rachel@ccu.edu.tw  
+{yulunliu,wpeng}@cs.nycu.edu.tw  
+* Contributed equally
+
+![](images/11e08a552584a1b607fa89d0dc78a67e9c5d8038559ca4883ad9994af0d0fe77.jpg)
+
+![](images/76adf2827442ddd000a15c0049bb2e01820e24e4633659e7ce1a9ee33ca6f659.jpg)  
+(e) Rate-distortion Comparison
+
+![](images/48dae74c1c3d52537443098ef7001562aa4227dda77f61da1c24dfa96c0e77e5.jpg)
+
+![](images/5810a9661368869dc52b988178a388dafa7ad341ba74d6c215a68fdb7d6afe73.jpg)  
+(c) View Frequency-aware Masking
+
+![](images/a38c0dafa53ccefb1aa4fa2d59833b8c86d223397a3bc3ed0011d33f71119aa8.jpg)  
+(d) PCA-guided Triplanes  
+Figure 1: Comparison of the proposed CAT-3DGS and HAC (Chen et al., 2024). CARM: Channelwise Autoregressive Models. SARM: Spatial Autoregressive Models.
+
+# ABSTRACT
+
+3D Gaussian Splitting (3DGS) has recently emerged as a promising 3D representation. Much research has been focused on reducing its storage requirements and memory footprint. However, the needs to compress and transmit the 3DGS representation to the remote side are overlooked. This new application calls for rate-distortion-optimized 3DGS compression. How to quantize and entropy encode sparse Gaussian primitives in the 3D space remains largely unexplored. Few early attempts resort to the hyperprior framework from learned image compression. But, they fail to utilize fully the inter and intra correlation inherent in Gaussian primitives. Built on ScaffoldGS, this work, termed CAT-3DGS, introduces a context-adaptive triplane approach to their rate-distortion-optimized coding. It features multi-scale triplanes, oriented according to the principal axes of Gaussian primitives in the 3D space, to capture their inter correlation (i.e. spatial correlation) for spatial autoregressive coding in the projected 2D planes. With these triplanes serving as the hyperprior, we further perform channel-wise autoregressive coding to leverage the intra correlation within each individual Gaussian prim
+
+itive. Our CAT-3DGS incorporates a view frequency-aware masking mechanism. It actively skips from coding those Gaussian primitives that potentially have little impact on the rendering quality. When trained end-to-end to strike a good rate-distortion trade-off, our CAT-3DGS achieves the state-of-the-art compression performance on the commonly used real-world datasets.
+
+# 1 INTRODUCTION
+
+3D Gaussian Splatting (3DGS) (Kerbl et al., 2023) has emerged as a promising representation for 3D scenes. It lends itself to novel view synthesis particularly within differentiable rendering frameworks. Unlike Neural Radiance Fields (NeRF) (Mildenhall et al., 2021), which require many sampling points per pixel for volumetric rendering, 3DGS, as a rasterization-based method, uses 3D Gaussians as geometric primitives, achieving greater efficiency for real-time rendering and the state-of-the-art rendering quality.
+
+Despite these advantages, the redundancy inherent in the 3DGS representation has prompted new research directions. Many prior works have been focused on a compact representation of Gaussian primitives. This category of methods aim at minimizing the parameter count or quantizing parameters to save storage space and memory footprint. These techniques include pruning insignificant Gaussian primitives (Lee et al., 2024; Fan et al., 2023; Girish et al., 2024; Ali et al., 2024; Wang et al., 2024a), vector quantizing their attributes (Lee et al., 2024; Fan et al., 2023; Navaneet et al., 2023; Niedermayr et al., 2024; Wang et al., 2024a; Morgenstern et al., 2023), developing compact latent representations for attributes (Girish et al., 2024), and representing sparse Gaussian primitives in a more structural way (Lu et al., 2024; Ren et al., 2024; Sun et al., 2024). However, most of them overlook the needs to transmit the compressed 3DGS representation to the remote side, which calls for entropy coding and rate-distortion-optimized compression.
+
+Recently, the rate-distortion-optimized compression for 3DGS started to attract attention. Unlike the compact 3DGS representation, this new school of thought (Wang et al., 2024a; Liu et al., 2024; Chen et al., 2024; Wang et al., 2024b) aims to strike an optimized trade-off between the compressed bit rate and rendering image quality in an end-to-end manner. Built on the vanilla 3DGS representation, RDO-Gaussian (Wang et al., 2024a) adopts the entropy-constrained vector quantization to quantize the attributes (e.g. opacity, scales, rotations and colors) of each Gaussian primitive. Instead of performing vector quantization, HAC (Chen et al., 2024) and ContextGS (Wang et al., 2024b) turn to the ScaffoldGS (Lu et al., 2024) representation to perform scalar quantization with respect to the latent features of these attributes, a technique analogous to the well-established transform coding plus scalar quantization for image/video compression. To entropy encode the quantized features, both introduce the hyperprior from learned image compression (Ballé et al., 2018) to model their coding probabilities. In formulating the hyperprior, HAC (Chen et al., 2024) draws inspiration from BiRF (Shin & Park, 2024) to create multi-scale binary hash grids (Figure 1 (a)), whereas ContextGS learns a separate feature as the hyperprior for each individual Gaussian primitive. Both assume the components of the hyperprior are independent and identically distributed, in coding the hyperprior. Notably, ContextGS organizes Gaussian primitives in the 3D space in a hierarchical manner in order to benefit from the contextual coding of the quantized latent features. In this regard, CompGS (Liu et al., 2024) shares parallels with ContextGS.
+
+This work introduces a novel rate-distortion-optimized compression framework for 3DGS (Figure 1 (b)). First, motivated by the tensor decomposition (Fridovich-Keil et al., 2023), we project the unorganized Gaussian primitives in the 3D space onto a set of multi-scale triplanes. These triplanes, oriented according to the principal components of the Gaussian primitives (Figure 1 (d)), serve as the hyperprior for coding their attributes in the latent space. Because they capture largely the inter correlation (i.e. spatial correlation) between the Gaussian primitives in the 3D space, we are able to encode efficiently the triplane-based hyperprior by spatial autoregressive models. This design aspect differs significantly from most existing techniques, in which the hyperprior is assumed to be factorial. Second, given that our triplane-based hyperprior has exploited much of the inter correlation between the Gaussian primitives, we decouple their coding dependency and encode their latent features independently by channel-wise autoregressive models. This avoids the challenge of having to organize sparse Gaussian primitives in the 3D space to leverage their inter correlation. Moreover, the intra correlation within each individual Gaussian primitive is explored for coding. Lastly, we
+
+![](images/e652584b42dc29b4dc68000cc7260dd4eb72639bbaa49586bca46ac32f1ad349.jpg)  
+Figure 2: Taxonomy of the rate-distortion-optimized 3DGS compression.
+
+develop a view frequency-aware masking mechanism, skipping from coding the Gaussian primitives that contribute little to the rendering quality (Figure 1 (c)). To sum up, our contributions include:
+
+- A triplane-based hyperprior that leverages the inter correlation (i.e. spatial correlation) between Gaussian primitives in the 3D space for efficient spatial autoregressive coding.  
+- A channel-wise autoregressive model with uneven slice partition that exploits the intra correlation within each individual Gaussian primitive to further improve coding efficiency.  
+- A view frequency-aware masking mechanism that evaluates the significance of Gaussian primitives based on their impact on the rendering quality to skip less critical ones from coding.
+
+With these novel elements, our scheme, called CAT-3DGS, is able to achieve the state-of-the-art rate-distortion performance on several commonly used real-world datasets (Figure 1 (e)).
+
+# 2 RELATED WORK
+
+The related work can be divided into two main categories: the compact 3DGS representation and the rate-distortion-optimized 3DGS compression.
+
+Compact 3DGS Representation. This category of methods, being a weak form of compression, aims to make more compact the 3DGS representation by pruning, quantizing, or structuring Gaussian primitives. Typical methods that involve pruning include Compact3DGS (Lee et al., 2024), LightGaussian (Fan et al., 2023), EAGLES (Girish et al., 2024), and Trimming the fat (Ali et al., 2024). Compact3DGS features a learnable binary mask and a mask loss to suppress less critical Gaussian primitives during training. In contrast, LightGaussian and EAGLES adopt a post-processing strategy to remove less significant Gaussians based on score-based criteria. In a similar vein, Trimming the fat (Ali et al., 2024) performs pruning progressively. Other methods that involve structuring sparse Gaussian primitives are ScaffoldGS (Lu et al., 2024), OctreeGS (Ren et al., 2024), and F3DGS (Sun et al., 2024). For instance, ScaffoldGS takes an anchor-based approach, where each anchor represents a group of Gaussian primitives whose attributes are represented collectively by a latent feature vector.
+
+Rate-distortion-optimized 3DGS Compression. This emerging research area targets the generation and coding of Gaussian primitives in an end-to-end and rate-distortion-optimized manner. Figure 2 presents a taxonomy for methods in this category, including RDO-Gaussian (Wang et al., 2024a), CompGS (Liu et al., 2024), ContextGS (Wang et al., 2024b), and HAC (Chen et al., 2024). Unlike the compact 3DGS representation, these compression techniques involve entropy coding the quantized Gaussian primitives. One central theme is how to predict the probability distributions of the coding features and/or attributes. To this end, some (Chen et al., 2024; Liu et al., 2024; Wang et al., 2024b) borrow the idea of hyperprior from learned image compression to model the distributions of the latent features of Gaussian primitives. One exception is RDO-Gaussian (Wang et al., 2024a), which applies entropy-constrained vector quantization to the Gaussian attributes. As opposed to HAC (Chen et al., 2024) and CompGS (Liu et al., 2024), ContextGS (Wang et al., 2024b) additionally introduces contextual coding in the latent space to leverage the inter correlation between Gaussian primitives. In common, all these schemes consider the hyperprior to be factorial. From Figure 2, our CAT-3DGS represents a novel attempt that makes use of both inter and intra
+
+correlation for coding Gaussian primitives. In terms of the use of inter correlation, it differs from ContextGS (Wang et al., 2024b) and CompGS (Liu et al., 2024) in performing spatial autoregressive coding in the hyperprior domain, which is made possible with our triplane-based hyperprior. More than that, it makes full use of the intra correlation within each individual Gaussian primitive by performing channel-wise autoregressing coding in the latent domain, which is first proposed for 3DGS compression.
+
+# 3 PRELIMINARY
+
+ScaffoldGS (Lu et al., 2024) builds upon 3DGS (Kerbl et al., 2023) and introduces a storage-efficient, anchor-based representation of 3D Gaussian primitives. Instead of directly storing a large number of Gaussian primitives and their attributes, ScaffoldGS introduces the notion of anchor points, with each representing a cluster of Gaussian primitives. The attributes of a Gaussian primitive include its 3D position  $\mu^g$ , scale  $s$ , rotation  $r$ , spherical harmonic coefficients  $c$ , and opacity  $\alpha$ . Likewise, each anchor is characterized by its position  $x$ , latent feature  $f$ , scaling factor  $l$ , and  $K$  learnable offsets  $\{O_i\}_{i=1}^K$ . The latent feature  $f$  encodes the attributes of the Gaussian primitives attached to the same anchor, effectively reducing the data redundancy. The learnable offsets indicate their relative positions with respect to that of the anchor.
+
+With ScaffoldGS, rendering a 2D image involves decoding the view-dependent attributes for all the Gaussians primitives from an anchor representation according to the anchor feature  $\pmb{f}$  and camera position  $\pmb{x}_c$ :
+
+$$
+\left\{\boldsymbol {c} _ {i}, \boldsymbol {r} _ {i}, \boldsymbol {s} _ {i}, \boldsymbol {\alpha} _ {i} \right\} _ {i = 1} ^ {K} = F _ {S} (\boldsymbol {f}, \boldsymbol {\sigma} _ {c}, \vec {\boldsymbol {d}} _ {c}), \tag {1}
+$$
+
+where  $\sigma_{c} = \| (\pmb{x} - \pmb{x}_{c})\|_{2}$ ,  $\vec{d_c} = \pmb{x} - \pmb{x}_c / \| \pmb{x} - \pmb{x}_c\|_2$ , and  $F_{S}$  is an MLP decoder. The position  $\mu_i^g$  of a Gaussian in the cluster is evaluated by adding the anchor position  $\pmb{x}$  to the offsets  $O_{i}$  regularized by the scaling factor  $l$ , as follows:
+
+$$
+\left\{\boldsymbol {\mu} _ {i} ^ {g} \right\} _ {i = 1} ^ {K} = \boldsymbol {x} + \left\{\boldsymbol {O} _ {i} \right\} _ {i = 1} ^ {K} \cdot \boldsymbol {l}. \tag {2}
+$$
+
+Given these parameters, the rendering process proceeds similarly to 3DGS (Kerbl et al., 2023).
+
+# 4 PROPOSED METHOD: CAT-3DGS
+
+Based on ScaffoldGS, this work (termed CAT-3DGS) introduces a content-adaptive triplane approach to coding the anchors' attributes (i.e. the latent feature  $f \in \mathbb{R}^{50}$ , scaling factor  $l \in \mathbb{R}^6$ , and offsets  $\{O_i \in \mathbb{R}^3\}_{i=1}^K$ ) in an end-to-end, rate-distortion-optimized fashion. Our CAT-3DGS adopts a hyperprior framework to model the probability distributions of the anchors' attributes. Because of the unordered and sparse nature of the anchor points, which collectively form an unorganized point cloud in the 3D space, we project them onto the multi-scale, dense triplanes oriented according to the principal components of the anchor points. As such, our triplane-based hyperprior organizes the projected anchor points in an ordered way on the 2D triplanes. This enables us to use spatial autoregressive models to exploit their inter correlation (i.e. spatial correlation) for better entropy coding the hyperprior itself and thus the anchors' attributes. In comparison, the 3D hash-based grid hyperprior (Chen et al., 2024), although compact, is not able to exploit such inherent inter correlation due to the pseudo random mapping between the dense grid points and their hyperprior representations in the hash table. In addition, CAT-3DGS features a channel-wise contextual coding scheme to leverage the intra correlation among the components of individual latent features  $f$  for their coding. Lastly, we incorporate a view frequency-aware masking mechanism to skip from coding those Gaussian primitives that contribute little to the rendering quality in different views.
+
+# 4.1 SYSTEM OVERVIEW
+
+Figure 3 illustrates our CAT-3DGS framework. The encoding of a 3D scene begins with the generation of the anchor points characterized by their positions  $\pmb{x} \in \mathbb{R}^3$  and attributes, including the latent feature  $\pmb{f} \in \mathbb{R}^{50}$ , offsets  $\{O_i \in \mathbb{R}^3\}_{i=1}^K$  and scaling  $l \in \mathbb{R}^6$ . Given the geometry information  $\pmb{x}$  of the anchor points, we formulate multi-scale, dense triplanes by conducting a principal component analysis. These triplanes consist of regularly structured grid points, which are quantized and coded by our lightweight spatial autoregressive models (Sec. 4.2). They serve the purpose of
+
+![](images/54a1de50ca8f002c5c5de8911f25e747dd8a45cb98f8c9cb30644879399df742.jpg)  
+Figure 3: Illustration of our CAT-3DGS framework. CARM: Channel-wise Autoregressive Models. SARM: Spatial Autoregressive Models.
+
+the hyperprior, and are queried and decoded to arrive at the coding distributions of the learned attributes associated with an anchor point according to its position  $\pmb{x}$  (Sec. 4.2). Considering that the latent features  $\pmb{f}$  collectively constitute a large portion of the compressed bitstream, they each are encoded recursively by channel-wise autoregressive coding (Sec. 4.4). During the learning process, our view frequency-aware masking mechanism is incorporated to mask out the Gaussian primitives which have a minimal impact on the rendering quality (Sec. 4.5). With CAT-3DGS, the information to be compressed in the bitstream include (a) the triplanes, (b) the anchors' attributes and positions, (c) the binary mask, (d) the network weights of the MLP decoder  $F_{s}$ , spatial autoregressive model  $F_{ARM}$ , hyperprior decoder  $F_{tri}$ , and channel-wise autoregressive model  $F_{ch}$ . The anchors' positions  $\pmb{x}$  and the network weights are signaled in 16-bit and 32-bit floating-point formats, respectively. The binary mask is entropy encoded.
+
+The rendering of a 2D image proceeds in much the same way as Scaffold-GS (Lu et al., 2024). Specifically, we use an MLP decoder  $F_{S}$  to decode the coded latent feature  $\hat{f}$  of an anchor point to obtain the attributes (i.e. color, opacity, rotation, scale) of all the Gaussian primitives belonging to the same anchor. Accordingly, the coded offsets  $\{\hat{O}_i\}_{i=1}^K$  and scaling  $\hat{l}$  are combined with the anchor's position  $x$  to reconstruct their positions. The same decoding process is repeated for the remaining anchor points needed to render the 2D image of a specific viewpoint.
+
+# 4.2 TRIPLANE-BASED HYPERPRIOR
+
+Our triplane-based hyperprior aims to learn the prior distributions on the attributes (the latent features  $f$ , offsets  $\{O_i\}_{i=1}^K$  and scaling  $l$ ) of the anchor points. Conceptually, a triplane is composed of three 2D planes, denoted as  $\mathcal{P}_c$ ,  $c \in \{xy, yz, zx\}$ , of the same 2D spatial resolution and channel dimension (See Figure 1 (b)). The notion of triplanes originates from decomposing a dense, 3D grid, which is costly to represent, into three 2D planes, which are storage friendly yet with more restricted expressiveness. Each grid point in these 2D planes is a learnable parameter. Our CAT-3DGS learns multiple triplanes of various resolutions to capture both coarse and fine detail. We thus augment  $\mathcal{P}_c$  with an upsampling scale  $r$  as  $\mathcal{P}_{r,c} \in \mathbb{R}^{ch \times rB \times rB}$ , where  $ch$  denotes the number of channels,  $r$  are integers denoting the upsampling scales and  $B$  denotes the spatial resolution of the triplane at the lowest scale (i.e.  $r = 1$ ).
+
+To retrieve the hyperprior  $h(\pmb{x})$  for an anchor point  $\pmb{x}$  in the 3D space, we project  $\pmb{x}$  onto each 2D plane  $\mathcal{P}_{r,c}$ , with the projected 2D coordinates given by  $\pi_{r,c}(\pmb{x})$ . When  $\pi_{r,c}(\pmb{x})$  is fractional, we interpolate between the nearest integer grid points with an interpolation kernel  $\psi$  to get the triplane feature. In symbols, we have  $\psi(\mathcal{P}_{r,c}, \pi_{r,c}(\pmb{x}))$ . The same process is repeated for every combination of permissible  $r$  and  $c$ , with the resulting triplane features concatenated to formulate the hyperprior
+
+$h(\pmb {x})$  ..
+
+$$
+h (\boldsymbol {x}) = \bigcup_ {r} \bigcup_ {c \in \{x y, y z, z x \}} \psi \left(\mathcal {P} _ {r, c}, \pi_ {r, c} (\boldsymbol {x})\right). \tag {3}
+$$
+
+In doing so, we notice that  $\pmb{x}$  may potentially be unbounded. However, the spatial resolutions of the triplanes must be bounded, because these triplanes need to be signaled in the bitstream. With their finite spatial resolutions, a contraction function (Barron et al., 2022b) is applied in order to fit the potentially unbounded  $\pmb{x}$  to our bounded triplanes:
+
+$$
+\operatorname {c o n t r a c t} (\boldsymbol {x}) = \left\{ \begin{array}{l l} \boldsymbol {x} & \text {i f} \| \boldsymbol {x} \| \leq 1 \\ (2 - \frac {1}{\| \boldsymbol {x} \|}) \left(\frac {\boldsymbol {x}}{\| \boldsymbol {x} \|}\right) & \text {i f} \| \boldsymbol {x} \| > 1. \end{array} \right. \tag {4}
+$$
+
+Careful examination of Eq. (4) reveals that  $\pmb{x}$  needs to be normalized in order to minimize the impact of the non-linear scaling applied to  $\pmb{x}$  with  $\| \pmb{x} \| > 1$  while maximizing the usage of the grid points in each triplane to represent those  $\pmb{x}$  with  $\| \pmb{x} \| \leq 1$ . To this end, we conduct a principal component analysis (PCA) with respect to the positions  $\pmb{x}$  of the anchor points, performing a linear transformation  $T(\pmb{x})$  of  $\pmb{x}$  before it is contracted with Eq. (4). More specifically,  $T(\pmb{x})$  is given by
+
+$$
+\boldsymbol {x} ^ {\prime} = T (\boldsymbol {x}) = \operatorname {c o n t r a c t} \left(\frac {\boldsymbol {R} ^ {\boldsymbol {x}} \left(\boldsymbol {x} - \boldsymbol {\mu} ^ {\boldsymbol {x}}\right)}{\boldsymbol {\sigma} ^ {\boldsymbol {x}}}\right), \tag {5}
+$$
+
+where  $\pmb{\mu}^{\pmb{x}}\in \mathbb{R}^{3}$  is the mean vector,  $\sigma^x\in \mathbb{R}^3$  are the variances along the three principal axes, and  $\pmb{R}^{\pmb{x}}\in \mathbb{R}^{3\times 3}$  is the PCA rotation matrix. As illustrated in Figure 1 (d), this transformation centers the point cloud of the anchor points, allowing most of the central anchor points to be linearly scaled. We finally substitute  $T(\pmb {x})$  into Eq. (3) to evaluate the hyperprior  $h(x^{\prime})$
+
+To entropy encode (or decode) the quantized attributes  $\hat{\pmb{a}}\in \{\hat{f}_1,\{\hat{O}_i\} ,\hat{l}\}$  ) of an anchor, an MLP  $F_{tri}$  is used to decode  $h(\pmb{x}^{\prime})$  to predict their means, variances, and quantization step size. That is,  $(\pmb {\mu},\pmb {\sigma},q) = F_{tri}(h(\pmb{x}^{\prime}))$  . Notably, each of these attributes is assumed to follow a Gaussian distribution, with their coding probabilities given by
+
+$$
+p \left(\hat {\boldsymbol {a}} \mid h \left(\boldsymbol {x} ^ {\prime}\right)\right) = \int_ {\hat {\boldsymbol {a}} - \frac {q}{2}} ^ {\hat {\boldsymbol {a}} + \frac {q}{2}} \mathcal {N} (\boldsymbol {\mu}, \boldsymbol {\sigma}) d \boldsymbol {a}. \tag {6}
+$$
+
+The acute reader may have observed that only part of the latent feature  $\hat{f}$  is involved in Eq. (6). The coding of the remaining part (i.e.  $\hat{f}_2, \hat{f}_3, \ldots$ ) will be elaborated in Sec. 4.4.
+
+# 4.3 SPATIAL AUTOREGRESSIVE MODELS (SARM) FOR TRIPLANE CODING
+
+The triplane-based hyperprior must be encoded into the bitstream. Observing that the grid points in each 2D plane capture to a large extent the inter correlation (i.e. spatial correlation) between the anchor points in the 3D space, we introduce spatial autoregressive models for triplane coding. Currently, a dedicated autoregressive model  $F_{ARM}$  is learned and shared for 2D planes  $\mathcal{P}_{r,c}$  of the same orientation  $c \in \{xy, yz, zx\}$  without regard to its upsampling scale  $r$ . Thus, a total of three  $F_{ARM}$ , one for each orientation, are learned. Moreover, the 2D plane  $\mathcal{P}_{r,c}$  has  $ch$  channels. These channels are encoded (and decoded) independently of each other with the same  $F_{ARM}$  to strike a balance between complexity and coding efficiency. In fact, all the channels from these 2D planes  $\mathcal{P}_{r,c}$  can be encoded (and decoded) in parallel.
+
+To entropy encode (and decode) a grid point  $y_{i,j,k}$  in the channel  $k$  of the 2D plane  $\mathcal{P}_{r,c}$ ,  $F_{ARM}$  formulates the context  $c_{i,j,k}$  by referring to the previously decoded grid points of the same channel in the neighborhood specified by  $c_{i,j,k} = [\hat{y}_{i-2:i-1,j-2:j+2,k};\hat{y}_{i,j-2:j-1,k}]$  (Figure 3 (e)). It outputs the parameters  $(\mu_{i,j,k},\sigma_{i,j,k}) = F_{ARM}(c_{i,j,k})$  of a Laplace distribution that models the distribution of  $y_{i,j,k}$ . The coding probability of the quantized grid point  $\hat{y}_{i,j,k}$  is then given by
+
+$$
+p \left(\hat {y} _ {i, j, k} \mid c _ {i, j, k}\right) = \int_ {\hat {y} _ {i, j, k} - \frac {Q}{2}} ^ {\hat {y} _ {i, j, k} + \frac {Q}{2}} \operatorname {L a p l a c e} \left(\mu_ {i, j, k}, \sigma_ {i, j, k}\right) d y _ {i, j, k}, \tag {7}
+$$
+
+where  $Q = 1 / 16$  is the quantization step size. This small quantization step is chosen to make the training more stable when the training process transitions from the non-quantization-aware training to the quantization-aware training.
+
+# 4.4 CHANNEL-WISE AUTOREGRESSIVE MODELS (CARM) FOR FEATURE CODING
+
+The coding of the latent features  $\pmb{f}$  deserves additional effort as they normally represent a considerable portion of the compressed bitstream. For efficient feature coding, we leverage the intra correlation among the components of a feature  $\pmb{f}$ . We divide every individual feature  $\pmb{f}$  into  $M$  slices along the channel dimension, followed by introducing a channel-wise autoregressive models for coding these slices. The coding of slice  $\pmb{f}_{m}$  with  $m > 1$  is able to benefit from referring to the previous coded slices, i.e.  $\{\pmb{f}_i|i < m\}$ , and the hyperprior. In symbols, we have the coding probability of the quantized slice  $\hat{\pmb{f}}_m$  as
+
+$$
+p \left(\hat {\boldsymbol {f}} _ {\boldsymbol {m}} \mid h \left(\boldsymbol {x} ^ {\prime}\right), \hat {\boldsymbol {f}} _ {\boldsymbol {0}: \boldsymbol {m} - 1}\right) = \int_ {\hat {\boldsymbol {f}} _ {\boldsymbol {m}} - \frac {q}{2}} ^ {\hat {\boldsymbol {f}} _ {\boldsymbol {m}} + \frac {q}{2}} N \left(\boldsymbol {\mu} + \boldsymbol {\mu} _ {\boldsymbol {c h}}, \boldsymbol {\sigma} + \boldsymbol {\sigma} _ {\boldsymbol {c h}}\right) d \boldsymbol {f}, \tag {8}
+$$
+
+where  $(\pmb{\mu}, \pmb{\sigma}, q) = F_{tri}(h(\pmb{x}'))$  and  $(\pmb{\mu}_{ch}, \pmb{\sigma}_{ch}) = F_{ch}(\hat{\pmb{f}}_{0:m-1})$ . In our design, the slices are unevenly partitioned, with further details provided in Sec. 5.3.
+
+# 4.5 VIEW FREQUENCY-AWARE MASKING
+
+Our view frequency-aware masking is designed to distinguish between Gaussian primitives in terms of their potential contribution to the rendering quality. As observed in Compact-3DGS (Lee et al., 2024) and HAC (Chen et al., 2024), using a learnable binary mask  $M_{n,k}$  to mask out the  $k$ -th Gaussian primitive of the  $n$ -th anchor can be effective in reducing the number of Gaussian primitives to be signaled, thereby saving the storage space and transmission bandwidth. This masking mechanism is usually implemented as  $M_{n,k} = \mathbb{1}(\text{sigmoid}(m_{n,k}) > \epsilon)$ , where  $m_{n,k}$  is a learnable parameter and  $\epsilon$  is a global hyperparameter shared across every Gaussian primitive to determine its existence. Although effective, this blind approach may risk removing some critical Gaussian primitives. A question that arises naturally is whether we could prioritize Gaussian primitives in the masking process according to their potential contribution to the rendering quality. We observe that during training, some Gaussian primitives are more frequently used in rendering the training views. As such, we attach to each Gaussian primitive a weight  $p_{n,k}$  that reflects its relative frequency of being used in rendering these training views. We adopt  $p_{n,k}$  in our masking function:
+
+$$
+M _ {n, k} = \mathbb {1} (\operatorname {s i g m o i d} \left(m _ {n, k}\right) \cdot p _ {n, k} > \epsilon). \tag {9}
+$$
+
+With the same  $\epsilon$  applied to every Gaussian primitive, a higher  $p_{n,k}$  requires sigmoid  $(m_{n,k})$  to approach zero more closely in order to skip the corresponding Gaussian primitive, making the task more difficult. As a result, more Gaussian primitives that are critical to rendering the training views are retained. For this scheme to work well, the basic premise is that the distribution of training views should be similar to that of test views. We argue that this is true to some extent because when their distributions differ significantly, there is little guarantee of the rendering quality in those test views.
+
+# 4.6 TRAINING OBJECTIVES
+
+The training of CAT-3DGS involves minimizing the rate-distortion cost  $L_{\mathrm{Scaffold}} + \lambda_r L_{\mathrm{rate}}$  together with a masking loss  $\lambda_m L_m$ :
+
+$$
+L = L _ {\text {S c a f f o l d}} + \lambda_ {r} L _ {\text {r a t e}} + \lambda_ {m} L _ {m}, \tag {10}
+$$
+
+where we follow Scaffold-GS Lu et al. (2024) to evaluate  $L_{\mathrm{Scaffold}}$ , which includes the distortion between the original and rendered images as well as a regularization term imposed on the scales  $s$  of Gaussian primitives.  $L_{\mathrm{rate}}$  indicates the number of bits needed to signal the hyperprior and the anchors' attributes:
+
+$$
+L _ {\text {r a t e}} = \frac {1}{N (5 0 + 6 + 3 K)} \left(L _ {\text {r a t e}} ^ {\mathcal {A}} + \lambda_ {\text {t r i}} L _ {\text {r a t e}} ^ {\mathcal {P}}\right), \tag {11}
+$$
+
+where  $L_{\mathrm{rate}}^{\mathcal{A}} = -\sum_{\hat{\boldsymbol{a}}} \log_2 p(\hat{\boldsymbol{a}})$  is the estimated bit rate of the anchors' attributes,  $L_{\mathrm{rate}}^{\mathcal{P}} = -\sum_{\hat{y}} \log_2 p(\hat{y})$  is the triplanes' bit rate, and  $N(50 + 6 + 3K)$  is the total number of parameters of anchors' attributes. In Eq. (10),  $L_{m}$  is the mask loss adopted from Compact3DGS (Lee et al., 2024) to regularize the view frequency-aware masking:
+
+$$
+L _ {m} = \sum_ {n = 1} ^ {N} \sum_ {k = 1} ^ {K} \operatorname {s i g m o i d} \left(m _ {n, k}\right). \tag {12}
+$$
+
+![](images/2e9353af3f73b6199ea4a3138a28dc725c767ea74e1dad5b9056072e7fd3f0cf.jpg)
+
+![](images/9494937e076e8a14804f5296371dc0634b46c71c6a7a360920d95195eba6423a.jpg)
+
+![](images/41a20c246ceef9a98ac99eda98a3a6ab8b1cf5c187cb40262dc82d1cfa115d1f.jpg)  
+Total Size (MB)
+
+![](images/b84dbbf744f14e5b42dc50e221f48c17c5ed697238b0736a0d0722cd609ab82d.jpg)
+
+![](images/86834a2cfb12826e98bf474d328301002b4a439f22ac56de83b7a3cbe75e376f.jpg)
+
+![](images/a5a3c25f156fc3db81947c06d5b67d463c65bf29f0fe05308879516bd8330f58.jpg)  
+Figure 4: Rate-distortion comparison of our CAT-3DGS, HAC, ContextGS, RDO-Gaussian, CompGS, and several other compact 3DGS representations (normally without entropy coding and visualized as rate-distortion points).
+
+![](images/757637bf06aa12644b9622f9e9d62d8974f454aa6c55b5715cdc525e97b15b52.jpg)  
+(a) Reference image (The scene bicycle from Mip-NeRF 360 dataset)
+
+![](images/3a1543551326717d51c490e63c9cca5f1590cfafe158b2dfc18518db24064cfc.jpg)
+
+![](images/784e88286d16aebad63e8e25bc82913545734aa1dc39654622ee76bbaecc390c.jpg)
+
+![](images/ad3c898b74e0a3bbf725c34f27fd718c658fb60fd758c329dcce2fd1e3fb0b22.jpg)
+
+![](images/408c66baff4d09086abefade2b93790e41fbd178eec53524b7959ae195cb1cd8.jpg)  
+(a) Reference image (The scene amstderdam from BungeeNeRF dataset)  
+Figure 5: Qualitative results of our CAT-3DGS, HAC and ScaffoldGS.
+
+![](images/a6269882964d0c43855cf02ebb2f3b1151edc40f665ee80bca99e811b8d3283d.jpg)  
+(27.10 dB/243 MB)
+
+![](images/5b2afc3c0188db40a398d96be173471ffad110d269c7587965446345ea16f46f.jpg)  
+(b)ScaffoldGS
+
+![](images/3988a0bc7f1d12561da4acd23230df5fb36539104f9b3a82d8540ee7375e1593.jpg)  
+(c) HAC  
+(27.25 dB / 31.84 MB)
+
+![](images/5090031fbcc916cab7201fb2a7d09b052f090b1510c9359f3009f3cf9f161ff8.jpg)  
+(d) Ours (CAT-3DGS)  
+(27.57 dB / 31.98 MB)
+
+In particular,  $\lambda_{m} = \max (10^{-3}, 0.3 \cdot \lambda_{r})$  changes with the rate parameter  $\lambda_{r}$ . Further details about this design aspect are provided in Appendix B.
+
+# 5 EXPERIMENTAL RESULTS
+
+# 5.1 IMPLEMENTATION DETAILS
+
+This part summarizes some crucial implementation details for reproducibility. First, the spatial resolution  $B$  of the triplane at the lowest scale ( $r = 1$ ) is determined in proportional to the number of anchor points obtained after 10k training iterations. The choices of the other hyperparameters include: the channel number  $ch = 72$ ,  $\epsilon = 0.01$  (0.0004 for BungeeNeRF) for the view frequency-aware masking,  $M = 4$  with uneven slices (5, 10, 15, 25) for the channel-wise autoregressive coding. The rate parameter  $\lambda_r$  ranges from 0.002 to 0.04, and from 0.001 to 0.02 for BungeeNeRF. Lastly, our triplanes have only two scales; that is,  $r = 1, 2$ .
+
+# 5.2 RATE-DISTORTION COMPARISON
+
+Baselines. For comparison, our baseline methods include (1) the vanilla 3DGS, (2) ScaffoldGS (our base model), and (3) four rate-distortion-optimized approaches—namely, HAC (Chen et al., 2024), RDO-Gaussian (Wang et al., 2024a), CompGS (Liu et al., 2024), and ContextGS (Wang et al., 2024b). Notably, ContextGS is a concurrent work of our CAT-3DGS. Due to the emerging nature of the rate-distortion-optimized 3DGS compression, there are only few early attempts. We thus also include for comparison several compact 3DGS techniques without joint rate-distortion-optimized training (Lee et al., 2024; Fan et al., 2023; Niedermayr et al., 2024; Navaneet et al., 2023; Morgenstern et al., 2023; Girish et al., 2024; Ali et al., 2024). Generally, these techniques do not consider entropy coding. They are visualized as individual rate-distortion points in our rate-distortion plots.
+
+![](images/b181df05afd0c86317d996f741d8395b47330d4ab9d9a34111bb4254fbf98675.jpg)  
+Figure 6: (a) Rate-distortion curves comparing our triplanes and the binary hash grids in HAC. (b) Rate-distortion curves comparing our SARM with the factorized model (FM).
+
+![](images/d18254c3e212845f821870c4b2636eef39f55d0dbeb9d85c242d44c5e399a353.jpg)
+
+![](images/9344360c906bc14619205a5c6ed217bc6859307b5b91b1e0f120849862b8c651.jpg)
+
+![](images/bef38a3763ae14118090e5bd3130b54dc6d1c8fd8ea5884520eb76825787eef2.jpg)
+
+Datasets. We follow the common test protocol to test our CAT-3DGS on real-world scenes, including Mip-NeRF 360 (Barron et al., 2022a), Tanks & Temples (Knapitsch et al., 2017), Deep Blending (Hedman et al., 2018) and BungeeNeRF (Xiangli et al., 2022). For comparison, we choose the same scenes from each dataset as those used in the prior works (Lu et al., 2024; Chen et al., 2024).
+
+Metrics. We compare the rate-distortion performance of the competing methods by visualizing their rate-distortion plots. The quality metric is PSNR measured in the RGB domain. The bit rate is the file size of the compressed bitstream obtained by performing entropy encoding. When reporting the rate-distortion results for a dataset, we take the average of the per-sequence PSNRs and file sizes. In our ablation study, we additionally report the BD-rate saving (Bjontegaard, 2001) to single out the contribution of individual components. In particular, the BD-rate saving is evaluated for each test scene and averaged across all the scenes in the dataset. Negative values suggest rate saving at the same quality level as compared to the anchor (a chosen baseline method) and vice versa. Note that evaluating the BD-rate requires at least 4 rate-distortion points and largely overlapping distortion intervals. We thus use it only in our ablation study.
+
+Compression Results. In Figure 4, our CAT-3DGS outperforms the competing methods (particularly those rate-distortion-optimized ones) across all the datasets, achieving the state-of-the-art rate-distortion performance. On the Mip-NeRF 360 dataset, our CAT-3DGS achieves (at its second highest rate point)  $78 \times$  and  $26 \times$  rate reductions than 3DGS and ScaffoldGS, respectively, while achieving slightly higher PSNR by  $0.16 \mathrm{~dB}$ . Figure 5 offers the subjective quality comparison among ScaffoldGS, HAC, and our CAT-3DGS. On the bicycle scene, our CAT-3DGS achieve a  $57\%$  size reduction while showing similar visual quality to HAC. Likewise, on the amsterdam scene, it achieves  $0.32 \mathrm{~dB}$  higher PSNR and better subjective quality than HAC, but with a similar file size.
+
+# 5.3 ABLATION EXPERIMENTS
+
+We conduct ablation experiments on the Mip-NeRF360 dataset for its diverse scenes.
+
+Triplanes versus Binary Hash Grids. This study investigates the benefits of our triplane-based hyperprior. Based on the HAC framework, we change its hyperprior from the binary hash grids to our multi-scale triplanes with spatial autoregressive coding. The remaining components and training procedure are the same as HAC. From Figure 6 (a), our triplane-based hyperprior achieves an  $18\%$  BD-rate saving. It highlights the advantage of the triplane representations, which are able to capture the spatial correlation of the anchor points and enable more efficient entropy coding with spatial autoregressive models.
+
+Spatial Autoregressive Models (SARM) versus Factorized Models (FM). Based on CAT-3DGS, this ablation study replaces our SARM with FM. The latter assumes that the triplane-based hyperprior has independent and identically distributed components. For fair comparison, 3 FMs (one for each plane orientation) are trained and used for coding the triplanes. The result on Mip-NeRF360 indicates that our SARM achieves  $19\%$  BD-rate saving as compared to FM. The result suggests the strong spatial correlation in the triplane-based hyperprior. Figure 6 (b) offers a breakdown analysis, showing that SARM benefits not only the coding of the triplane-based hyperprior, but more importantly that of the anchors' attributes, which constitute the major portion of the compressed bitstream.
+
+![](images/2c041c799a0cb62d39f78edb1098f89156afbfb8384f5a1e5ab215a163160733.jpg)  
+Figure 7: Breakdown analysis of different coding parts w/ and w/o our CARM for the bicycle scene.
+
+Table 1: The impact of the slice number and partition in our CARM on compression performance The results are obtained with Mip-NeRF 360.  
+
+<table><tr><td>M Slices</td><td>Channels per Slice</td><td>BD-rate</td></tr><tr><td>1</td><td>50</td><td>0</td></tr><tr><td>2</td><td>25, 25</td><td>-6.3</td></tr><tr><td>2</td><td>15, 35</td><td>-8.9</td></tr><tr><td>4</td><td>12, 12, 13, 13</td><td>-8.9</td></tr><tr><td>4</td><td>5, 10, 15, 20</td><td>-11.9</td></tr></table>
+
+Table 2: Comparison of decoding time and rendering throughput: Ours (CAT-3DGS) vs. HAC.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Scene</td><td rowspan="2">Anchor Count (K)</td><td rowspan="2">Base Resolution B</td><td colspan="3">Decoding Time (s) ↓</td><td rowspan="2">Rendering Speed (FPS) ↑</td></tr><tr><td>Triplane</td><td>Anchor Attributes</td><td>Total</td></tr><tr><td rowspan="2">Ours</td><td>room</td><td>36.5</td><td>64</td><td>11.4</td><td>2.2</td><td>13.6</td><td>127.0</td></tr><tr><td>amsterdam</td><td>533.0</td><td>128</td><td>47.4</td><td>17.0</td><td>64.4</td><td>83.4</td></tr><tr><td rowspan="2">HAC</td><td>room</td><td>228.8</td><td>N/A</td><td>N/A</td><td>-</td><td>6.6</td><td>103.1</td></tr><tr><td>amsterdam</td><td>483.5</td><td>N/A</td><td>N/A</td><td>-</td><td>11.3</td><td>77.9</td></tr></table>
+
+Channel-wise Autoregressive Models (CARM). Figure 7 presents two pie charts to single out the contribution of our CARM on the bicycle scene. As shown, CARM reduces the compressed size of the latent features from  $5.7\mathrm{MB}$  to  $3.4\mathrm{MB}$ , amounting to a  $40\%$  rate reduction in this part of the bitstream. We also observe a similar trend in the other scenes. More results are provided in Appendix F. From Table 1, more slices lead to improved coding performance. Unevenly-partitioned slices perform slightly better than evenly-partitioned slices. With uneven slices, we found that more essential information is packed in the first or two slices, an effect that is similar to energy compaction and is much desirable for coding purposes. More discussions are presented in Appendix C.
+
+View Frequency-Aware Masking. We conduct another ablation study that disables the view frequency-aware masking in our CAT-3DGS. In other words, we remove the weight  $p_{n,k}$  in the masking function. Doing so results in a  $16\%$  BD-rate drop on Mip-NeRF360. The removal of Gaussian primitives less critical to the rendering quality helps reduce the bit rate. More results are provided in Appendix D.
+
+Decoding Time and Rendering Throughput. Table 2 compares the decoding time (seconds) and rendering throughput (frames per second) of our CAT-3DGS and HAC for two scenes, amsterdam in BungeeNeRF (Xiangli et al., 2022) and room in Mip-NeRF360 (Barron et al., 2022b). This information is collected on one NVIDIA V100. CAT-3DGS has much higher decoding time than HAC due to the use of autoregressive models. However, in terms of rendering throughput, our CAT-3DGS is faster than HAC. This is because our frequency-aware masking effectively reduces the number of anchors. The decoding time of CAT-3DGS can be further improved by making full use of the parallelism in decoding triplanes. Recall that all the 2D planes in our triplane-based hyerprior are independently decodable and so are the channels in each 2D plane (Sec. 4.3). Currently, only the channel parallelism is used in our implementation. The decoding of different anchor attributes is parallelizable to some extent. The coding dependency is in the channel dimension and not between the anchors' attributes. Last but not least, the 3DGS system normally has decoding and rendering as two decoupled processes. The Gaussian primitives are decoded first, followed by rendering images in different views. The decoding is generally less time sensitive and has little impact on the rendering process, which is the same as ScaffoldGS (Lu et al., 2024) with our CAT-3DGS.
+
+# 6 CONCLUSIONS
+
+This work presents a novel rate-distortion-optimized 3DGS compression framework. In an effort to leverage the inter correlation between Gaussian primitives for coding, it features PCA-guided triplanes as the hyperprior and incorporates a spatial autoregressive model for their coding. Furthermore, a channel-wise autoregressive model is introduced for the first time to explore the intra correlation within each individual Gaussian primitive for coding. When combined with a view frequency-aware masking mechanism, these features lead to the state-of-the-art coding performance.
+
+# ACKNOWLEDGMENTS
+
+This work is supported by National Science and Technology Council, Taiwan under the Grant NSTC 113-2634-F-A49-007-, MediaTek, and National Center for High-performance Computing, Taiwan.
+
+# REFERENCES
+
+Muhammad Salman Ali, Maryam Qamar, Sung-Ho Bae, and Enzo Tartaglione. Trimming the fat: Efficient compression of 3d gaussian splats through pruning. arXiv preprint arXiv:2406.18214, 2024.  
+Johannes Balle, David Minnen, Saurabh Singh, Sung Jin Hwang, and Nick Johnston. Variational image compression with a scale hyperprior, 2018. URL https://arxiv.org/abs/1802.01436.  
+Jonathan T Barron, Ben Mildenhall, Dor Verbin, Pratul P Srinivasan, and Peter Hedman. Mip-nerf 360: Unbounded anti-aliased neural radiance fields. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 5470-5479, 2022a.  
+Jonathan T. Barron, Ben Mildenhall, Dor Verbin, Pratul P. Srinivasan, and Peter Hedman. Mipnerf 360: Unbounded anti-aliased neural radiance fields, 2022b. URL https://arxiv.org/abs/2111.12077.  
+Gisle Bjontegaard. Calculation of average psnr differences between rd-curves. ITU SG16 Doc. VCEG-M33, 2001.  
+Yihang Chen, Qianyi Wu, Jianfei Cai, Mehrtash Harandi, and Weiyao Lin. Hac: Hash-grid assisted context for 3d gaussian splatting compression. arXiv preprint arXiv:2403.14530, 2024.  
+Zhiwen Fan, Kevin Wang, Kairun Wen, Zehao Zhu, Dejia Xu, and Zhangyang Wang. Lightgaussian: Unbounded 3d gaussian compression with  $15\mathrm{x}$  reduction and  $200+$  fps, 2023.  
+Sara Fridovich-Keil, Giacomo Meanti, Frederik Warburg, Benjamin Recht, and Angjoo Kanazawa. K-planes: Explicit radiance fields in space, time, and appearance, 2023. URL https:// arxiv.org/abs/2301.10241.  
+Sharath Girish, Kamal Gupta, and Abhinav Shrivastava. Eagles: Efficient accelerated 3d gaussians with lightweight encodings. European Conference on Computer Vision, 2024.  
+Peter Hedman, Julien Philip, True Price, Jan-Michael Frahm, George Drettakis, and Gabriel Bros-tow. Deep blending for free-viewpoint image-based rendering. ACM Transactions on Graphics (ToG), 37(6):1-15, 2018.  
+Bernhard Kerbl, Georgios Kopanas, Thomas Leimkuhler, and George Drettakis. 3d gaussian splatting for real-time radiance field rendering. ACM Trans. Graph., 42(4):139-1, 2023.  
+Arno Knapitsch, Jaesik Park, Qian-Yi Zhou, and Vladlen Koltun. Tanks and temples: Benchmarking large-scale scene reconstruction. ACM Transactions on Graphics (ToG), 36(4):1-13, 2017.  
+Joo Chan Lee, Daniel Rho, Xiangyu Sun, Jong Hwan Ko, and Eunbyung Park. Compact 3d gaussian representation for radiance field. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 21719-21728, 2024.  
+Xiangrui Liu, Xinju Wu, Pingping Zhang, Shiqi Wang, Zhu Li, and Sam Kwong. Compgs: Efficient 3d scene representation via compressed gaussian splatting. arXiv preprint arXiv:2404.09458, 2024.  
+Tao Lu, Mulin Yu, Linning Xu, Yuanbo Xiangli, Limin Wang, Dahua Lin, and Bo Dai. Scaffolds: Structured 3d gaussians for view-adaptive rendering. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 20654-20664, 2024.  
+Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. Communications of the ACM, 65(1):99-106, 2021.
+
+Wieland Morgenstern, Florian Barthel, Anna Hilsmann, and Peter Eisert. Compact 3d scene representation via self-organizing gaussian grids. arXiv preprint arXiv:2312.13299, 2023.  
+KL Navaneet, Kossar Pourahmadi Meibodi, Soroush Abbasi Koohpayegani, and Hamed Piriavash. Compact3d: Compressing gaussian splat radiance field models with vector quantization. arXiv preprint arXiv:2311.18159, 2023.  
+Simon Niedermayr, Josef Stumpfegger, and Rüdiger Westermann. Compressed 3d gaussian splatting for accelerated novel view synthesis. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 10349-10358, June 2024.  
+Kerui Ren, Lihan Jiang, Tao Lu, Mulin Yu, Linning Xu, Zhangkai Ni, and Bo Dai. Octreegs: Towards consistent real-time rendering with lod-structured 3d gaussians. arXiv preprint arXiv:2403.17898, 2024.  
+Seungjoo Shin and Jaesik Park. Binary radiance fields. Advances in neural information processing systems, 36, 2024.  
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+Yufei Wang, Zhihao Li, Lanqing Guo, Wenhan Yang, Alex C Kot, and Bihan Wen. Contextgs: Compact 3d gaussian splattering with anchor level context model. arXiv preprint arXiv:2405.20721, 2024b.  
+Yuanbo Xiangli, Linning Xu, Xingang Pan, Nanxuan Zhao, Anyi Rao, Christian Theobalt, Bo Dai, and Dahua Lin. Bungeenerf: Progressive neural radiance field for extreme multi-scale scene rendering. In European conference on computer vision, pp. 106-122. Springer, 2022.
+
+# A TRAINING PROCESS
+
+![](images/2d5211b3f0bedbf4caf78557a5fa81608df213e68def99f7cd369d8ed50c89eb.jpg)  
+Figure 8: Detailed Training Process of our CAT-3DGS.
+
+Figure 8 depicts our detailed training procedure. It includes the following training stages.
+
+Anchor Spawning In this stage, we adopt ScaffoldGS to ensure a stable start of both anchor attribute training and anchor spawning. However, to simulate the quantization effect, we introduce noise to the attributes of the anchor points starting at iteration 3000. To prevent the generation of an excessive number of anchors, we disable anchor growing during iterations 3000 to 4000.
+
+Triplane-based Hyperprior After adding noise to the attributes of the anchor points during iterations 3000 to 10,000, we begin using the triplane as a hyperprior to learn the distribution of anchors' attributes starting from the 10,000th iteration.
+
+Spatial Autoregressive Models for Triplane Coding We start triplane coding at iteration 15,000. Specifically, we warm up the spatial autoregressive model while freezing the other learnable parameters between iterations 15,000 and 16,000.
+
+# B RATE-AWARE MASK TRADE-OFF (RMT)
+
+![](images/8c6372663933d5048812f8895f1d728a23e035a6947fbf7ec668c2e9236005eb.jpg)  
+(a)  
+Figure 9: Rate-distortion comparison w/ and w/o our rate-aware mask trade-off. RMT: Rate-aware Mask Trade-off.
+
+![](images/f7f316e8eae19d0dbe133ea6bfe5468af6f585adb6c5be1a99890a83858a736b.jpg)  
+(b)
+
+![](images/f4142b63aa6e161ab4534e9ee82b424bb8a64e35241dd5a2ccd59f1571a61a8b.jpg)
+
+Based on the observation that the number of anchors can be further reduced while maintaining similar PSNR, we accordingly relate the mask hyperparameter  $\lambda_{m}$  to the rate hyperparameter  $\lambda_{r}$  using the relationship  $\lambda_{m} = \max(10^{-3}, 0.3 \cdot \lambda_{r})$ . This implies that, at higher bit rates,  $\lambda_{r}$  decreases and more offsets (i.e. Gaussian primitives) and anchors are kept. Conversely, at lower bit rates,  $\lambda_{r}$  increases and more offsets and anchors are removed.
+
+In Figure 9 (a), we evaluate the performance with a fixed mask trade-off set to 0.0005 (labeled "w/o RMT") and compare it with the rate-aware mask trade-off approach (labeled "w/ RMT") on Mip-
+
+NeRF 360. The rate-aware mask trade-off has greater coding performance gain, particularly at lower bit rates. For further details regarding this gain, we select the treehill scene from Mip-NeRF 360, as shown in Figure 9 (b). Our method with RMT demonstrates that, at the lowest rate point, the number of anchor points is only one-third of that "w/o RMT", and the total size achieves a  $60\%$  reduction, while maintaining similar PSNR.
+
+# C THE IMPACT OF SLICE PARTITIONING ON OUR CARM
+
+Table 3: Analysis the number of bits per channel within a slice in different cases.  
+
+<table><tr><td>Method</td><td>slices</td><td>Channels per slice</td><td>Size per channel within one slice (kB)</td><td>Size for f (MB)</td><td>Total size (MB)</td><td>PSNR</td></tr><tr><td>w/o CARM</td><td>1</td><td>(50)</td><td>(8.91)</td><td>0.44</td><td>1.69</td><td>30.78</td></tr><tr><td>w/ CARM (even)</td><td>4</td><td>(12, 12, 13, 13)</td><td>(22.2, 6.38, 4.65, 3.98)</td><td>0.45</td><td>1.74</td><td>30.85</td></tr><tr><td>w/ CARM (uneven)</td><td>4</td><td>(5, 10, 15, 20)</td><td>(17.22, 9.61, 5.35, 3.64)</td><td>0.33</td><td>1.59</td><td>30.89</td></tr></table>
+
+In this section, we further analyze the number of bits per channel within a slice in the cases of w/o CARM, w/ CARM (even), and w/ CARM (uneven), as shown in Table 3. We observe that when only one slice is used, indicating CARM is disabled, the kilo bytes per channel is 8.91. However, when CARM is enabled, the first slice uses 22.2 kilo bytes per channel for the even partition case and 17.22 kilo bytes per channel for the uneven case. The data suggest that the earlier slices contain more information, resulting in larger sizes. This also indicates that the richer information in the earlier slices can help the coding of the subsequent slices, allowing them to use fewer bits. As for the even and uneven cases, we observe that the uneven partition yields a better result, reducing the size of  $f$  by 0.12MB compared to the even case.
+
+# D VIEW FREQUENCY-AWARE MASKING
+
+![](images/3ed2a9cfefb6aecbecd1e669f5adc6c0295da97c1d83e80f0114909c270daf4e.jpg)  
+Figure 10: Rate-distortion comparison w/ and w/o our view frequency-aware masking. VFM: View Frequency-aware Masking.
+
+In Figure 10, we compare CAT-3DGS (denoted by "w/ VFM") with a variant (denoted by "w/o VFM") that removes the weight  $p_{n,k}$  in the masking function. The view frequency-aware masking has a significant effect at low bit rates, reducing the total size by  $29\%$  at the lowest rate point while PSNR drops by only 0.1dB. At the highest rate point, it reduces the total size by  $14\%$  while still maintaining similar PSNR. This indicates that the view frequency-aware masking can retain more important points while pruning a larger number of relatively less important ones, even when most of the anchors are masked out.
+
+# E BITSTREAM OF EACH COMPONENT
+
+Our bit stream consists of seven components: the anchor positions  $\mathbf{a}$ , three anchor attributes (features  $\mathbf{f}$ , offsets  $\{O_i\}$ , and scaling factors  $l$ ), a set of triplanes  $\mathcal{P}$ , binary masks  $M$ , and MLPs  $F_S$ ,  $F_{tri}$ ,  $F_{ch}$ ,  $F_{ARM}$ . After applying our triplane hyperprior and channel-wise autoregressive model, the size of the attributes has been significantly reduced. Additionally, due to the spatial autoregressive
+
+model, the triplane size occupies only a small portion. The rate-aware mask tradeoff further allows us to reduce the number of anchors that need to be compressed at low bit rates, thereby reducing the total size.
+
+Table 4: Bitstream of each component. The result is for the scene treehill on Mip-NeRF 360 dataset.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Number of Anchors (K)</td><td colspan="7">Bitstream of Each Component (MB)</td><td rowspan="2">Total size (MB)</td><td colspan="2">Fidelity</td></tr><tr><td>Position</td><td>Feature</td><td>Scaling</td><td>Offsets</td><td>Masks</td><td>MLPs</td><td>Triplane</td><td>PSNR</td><td>SSIM</td></tr><tr><td>treehill (high-rate)</td><td>516.3</td><td>3.10</td><td>7.75</td><td>2.47</td><td>3.19</td><td>0.53</td><td>0.35</td><td>0.11</td><td>17.5</td><td>23.28</td><td>0.646</td></tr><tr><td>treehill (low-rate)</td><td>85.1</td><td>0.51</td><td>0.33</td><td>0.28</td><td>0.18</td><td>0.06</td><td>0.35</td><td>0.04</td><td>1.74</td><td>22.71</td><td>0.558</td></tr></table>
+
+# F ADDITIONAL BREAKDOWN RESULTS FOR CARM
+
+![](images/b1ebecf56d78940b7b355db1ff3e20a7daa867d3b84c2d51579e6bb6a741a43e.jpg)  
+Figure 11: The breakdown analyses of different coding parts w/ and w/o our CARM for the flower and bonsai scenes, respectively.
+
+![](images/f36329b0bf6444bf747184719700cfcac2f2b15c2561369890576f316711ee8a.jpg)
+
+Figure 11 provides additional breakdown results for flower and bonsai scenes. The results confirm again that our CARM effectively reduces the compressed size of the latent features associated with anchors' attributes.
+
+# G THE IMPACT OF CARM ON DECODING TIME
+
+Table 5: Comparison of the decoding time w/ and w/o our CARM.  
+
+<table><tr><td rowspan="2">Scene</td><td rowspan="2">Method</td><td colspan="3">Decoding Time (s)</td></tr><tr><td>Triplane</td><td>Anchor Attributes</td><td>Total</td></tr><tr><td rowspan="2">room</td><td>CAT-3DGS (w/ CARM)</td><td>11.4</td><td>2.2</td><td>13.6</td></tr><tr><td>CAT-3DGS w/o CARM</td><td>11.3</td><td>2.1</td><td>13.4</td></tr><tr><td rowspan="2">amsterdam</td><td>CAT-3DGS (w/ CARM)</td><td>47.4</td><td>17.0</td><td>64.4</td></tr><tr><td>CAT-3DGS w/o CARM</td><td>47.2</td><td>14.9</td><td>62.1</td></tr></table>
+
+We compare the decoding time of our schemes w/ and w/o CARM in Table 5. The results indicate that CARM has a negligible impact on the decoding time.
+
+# H COMPLETE BREAKDOWN OF QUANTITATIVE RESULTS
+
+Quantitative results For a more comprehensive data presentation, we present detailed information on the rate-distortion curves, as shown in Figure 4, and the quantitative results are shown in Table 6.
+
+Per-scene Results of Our CAT-3DGS Framework The detailed results of our approach for MipNeRF 360 dataset (Barron et al., 2022a) are presented in Table 7.
+
+The detailed results of our approach for BungeeNeRF dataset (Xiangli et al., 2022) are presented in Table 8.
+
+The detailed results of our approach for Tank&Temples dataset (Knapitsch et al., 2017) are presented in Table 9.
+
+The detailed results of our approach for Deep Blending dataset (Hedman et al., 2018) are presented in Table 10.
+
+Per-scene Results of the Baseline Models Per-scene results for all datasets from our two baseline models, ScaffoldGS (Lu et al., 2024) and HAC (Chen et al., 2024) are also provided in Table 15 and Table 11 12 13 14, respectively.
+
+Table 6: The Quantitative results of our CAT-3DGS and other approaches. 3DGS (Kerbl et al., 2023) and ScaffoldGS (Lu et al., 2024) are baseline methods, which are presented in the first section. Approaches in the second section are compact representation, while the third section covers rate-distortion-optimized compression approaches. For comparison, we also provide two results of different size and fidelity tradeoffs by adjusting  $\lambda_r$ . The size is measured in megabytes (MB).
+
+<table><tr><td>Datasets</td><td colspan="4">Mip-NeRF360</td><td colspan="4">Tank&amp;Temples</td><td colspan="4">DeepBlending</td><td colspan="4">BungeeNeRF</td></tr><tr><td>Methods</td><td>psnr ↑</td><td>ssim ↑</td><td>lpips ↓</td><td>size ↓</td><td>psnr ↑</td><td>ssim ↑</td><td>lpips ↓</td><td>size ↓</td><td>psnr ↑</td><td>ssim ↑</td><td>lpips ↓</td><td>size ↓</td><td>psnr ↑</td><td>ssim ↑</td><td>lpips ↓</td><td>size ↓</td></tr><tr><td>3DGS(SIGGRAPH&#x27;23)</td><td>27.49</td><td>0.813</td><td>0.222</td><td>744.7</td><td>23.69</td><td>0.844</td><td>0.178</td><td>431.0</td><td>29.42</td><td>0.899</td><td>0.247</td><td>663.9</td><td>24.87</td><td>0.841</td><td>0.205</td><td>1616</td></tr><tr><td>ScaffoldGS (CVPR&#x27;24)</td><td>27.50</td><td>0.806</td><td>0.252</td><td>253.9</td><td>23.96</td><td>0.853</td><td>0.177</td><td>86.50</td><td>30.21</td><td>0.906</td><td>0.254</td><td>66.00</td><td>26.62</td><td>0.865</td><td>0.241</td><td>183.0</td></tr><tr><td>EAGLES(ECCV&#x27;24)</td><td>27.15</td><td>0.808</td><td>0.238</td><td>68.89</td><td>23.41</td><td>0.840</td><td>0.200</td><td>34.00</td><td>29.91</td><td>0.910</td><td>0.250</td><td>62.00</td><td>25.24</td><td>0.843</td><td>0.221</td><td>117.1</td></tr><tr><td>LightGaussian</td><td>27.00</td><td>0.799</td><td>0.249</td><td>44.54</td><td>22.83</td><td>0.822</td><td>0.242</td><td>22.43</td><td>27.01</td><td>0.872</td><td>0.308</td><td>33.94</td><td>24.52</td><td>0.825</td><td>0.255</td><td>87.28</td></tr><tr><td>Compact3DGS (CVPR&#x27;24)</td><td>27.08</td><td>0.798</td><td>0.247</td><td>48.80</td><td>23.32</td><td>0.831</td><td>0.201</td><td>39.43</td><td>29.79</td><td>0.901</td><td>0.258</td><td>43.21</td><td>23.36</td><td>0.788</td><td>0.251</td><td>82.60</td></tr><tr><td>Compressed3D (CVPR&#x27;24)</td><td>26.98</td><td>0.801</td><td>0.238</td><td>28.80</td><td>23.32</td><td>0.832</td><td>0.194</td><td>17.28</td><td>29.38</td><td>0.898</td><td>0.253</td><td>25.30</td><td>24.13</td><td>0.802</td><td>0.245</td><td>55.79</td></tr><tr><td>Morgenstern et al.</td><td>26.01</td><td>0.772</td><td>0.259</td><td>23.90</td><td>22.78</td><td>0.817</td><td>0.211</td><td>13.05</td><td>28.92</td><td>0.891</td><td>0.276</td><td>8.40</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Navaneet et al.</td><td>27.16</td><td>0.808</td><td>0.228</td><td>50.30</td><td>23.47</td><td>0.840</td><td>0.188</td><td>27.97</td><td>29.75</td><td>0.903</td><td>0.247</td><td>42.77</td><td>24.63</td><td>0.823</td><td>0.239</td><td>104.3</td></tr><tr><td>Trimming the fat</td><td>27.13</td><td>0.798</td><td>0.248</td><td>20.057</td><td>23.68</td><td>0.831</td><td>0.210</td><td>8.555</td><td>29.42</td><td>0.897</td><td>0.267</td><td>12.49</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CompGS (high-rate)</td><td>27.26</td><td>0.802</td><td>0.239</td><td>16.5</td><td>23.70</td><td>0.835</td><td>0.205</td><td>9.61</td><td>29.33</td><td>0.900</td><td>0.270</td><td>10.4</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CompGS (low-rate)</td><td>26.79</td><td>0.791</td><td>0.258</td><td>11.0</td><td>23.105</td><td>0.815</td><td>0.235</td><td>5.89</td><td>28.99</td><td>0.900</td><td>0.280</td><td>7.00</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>RDO-Gaussian (high-rate)</td><td>27.05</td><td>0.802</td><td>0.239</td><td>23.46</td><td>23.34</td><td>0.835</td><td>0.195</td><td>12.02</td><td>29.63</td><td>0.902</td><td>0.252</td><td>18.00</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>RDO-Gaussian (low-rate)</td><td>24.43</td><td>0.683</td><td>0.406</td><td>1.71</td><td>22.09</td><td>0.755</td><td>0.318</td><td>1.32</td><td>28.38</td><td>0.872</td><td>0.331</td><td>1.22</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>HAC (ECCV&#x27;24)(high-rate)</td><td>27.77</td><td>0.811</td><td>0.230</td><td>21.87</td><td>24.40</td><td>0.853</td><td>0.177</td><td>11.24</td><td>30.34</td><td>0.906</td><td>0.258</td><td>6.35</td><td>27.05</td><td>0.868</td><td>0.217</td><td>26.16</td></tr><tr><td>HAC (ECCV&#x27;24)(low-rate)</td><td>26.11</td><td>0.759</td><td>0.312</td><td>5.96</td><td>23.11</td><td>0.809</td><td>0.238</td><td>3.68</td><td>28.62</td><td>0.888</td><td>0.302</td><td>2.64</td><td>24.56</td><td>0.768</td><td>0.327</td><td>9.74</td></tr><tr><td>Ours (high-rate)</td><td>27.77</td><td>0.809</td><td>0.241</td><td>12.35</td><td>24.41</td><td>0.853</td><td>0.189</td><td>6.93</td><td>30.29</td><td>0.909</td><td>0.269</td><td>3.56</td><td>27.35</td><td>0.886</td><td>0.183</td><td>26.59</td></tr><tr><td>Ours (low-rate)</td><td>25.82</td><td>0.730</td><td>0.362</td><td>1.72</td><td>22.97</td><td>0.786</td><td>0.293</td><td>1.42</td><td>28.53</td><td>0.878</td><td>0.336</td><td>0.93</td><td>25.19</td><td>0.808</td><td>0.279</td><td>10.14</td></tr></table>
+
+Table 7: Results of Our approach for each scene from Mip-NeRF 360 dataset (Barron et al., 2022a).  
+
+<table><tr><td>λr</td><td>Scenes</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>SIZE↓</td></tr><tr><td rowspan="10">0.002</td><td>bicycle</td><td>25.04</td><td>0.735</td><td>0.280</td><td>21.42</td></tr><tr><td>bonsai</td><td>32.98</td><td>0.947</td><td>0.192</td><td>6.06</td></tr><tr><td>counter</td><td>29.66</td><td>0.915</td><td>0.197</td><td>6.33</td></tr><tr><td>flower</td><td>21.41</td><td>0.580</td><td>0.375</td><td>18.92</td></tr><tr><td>garden</td><td>27.49</td><td>0.847</td><td>0.152</td><td>18.64</td></tr><tr><td>kitchen</td><td>31.34</td><td>0.926</td><td>0.135</td><td>6.92</td></tr><tr><td>room</td><td>31.95</td><td>0.924</td><td>0.212</td><td>4.11</td></tr><tr><td>stump</td><td>26.79</td><td>0.766</td><td>0.271</td><td>11.23</td></tr><tr><td>treehill</td><td>23.28</td><td>0.646</td><td>0.359</td><td>17.50</td></tr><tr><td>AVG</td><td>27.77</td><td>0.809</td><td>0.241</td><td>12.35</td></tr><tr><td rowspan="10">0.004</td><td>bicycle</td><td>25.16</td><td>0.738</td><td>0.280</td><td>16.56</td></tr><tr><td>bonsai</td><td>32.74</td><td>0.944</td><td>0.198</td><td>4.79</td></tr><tr><td>counter</td><td>29.51</td><td>0.911</td><td>0.203</td><td>4.98</td></tr><tr><td>flower</td><td>21.37</td><td>0.576</td><td>0.383</td><td>14.28</td></tr><tr><td>garden</td><td>27.30</td><td>0.840</td><td>0.164</td><td>14.55</td></tr><tr><td>kitchen</td><td>30.97</td><td>0.922</td><td>0.140</td><td>5.36</td></tr><tr><td>room</td><td>31.73</td><td>0.921</td><td>0.221</td><td>3.22</td></tr><tr><td>stump</td><td>26.87</td><td>0.766</td><td>0.278</td><td>8.50</td></tr><tr><td>treehill</td><td>23.27</td><td>0.642</td><td>0.370</td><td>13.20</td></tr><tr><td>AVG</td><td>27.66</td><td>0.807</td><td>0.249</td><td>9.49</td></tr><tr><td rowspan="10">0.01</td><td>bicycle</td><td>25.03</td><td>0.728</td><td>0.302</td><td>9.40</td></tr><tr><td>bonsai</td><td>31.67</td><td>0.932</td><td>0.217</td><td>2.94</td></tr><tr><td>counter</td><td>28.89</td><td>0.898</td><td>0.225</td><td>3.15</td></tr><tr><td>flower</td><td>21.25</td><td>0.564</td><td>0.403</td><td>8.13</td></tr><tr><td>garden</td><td>26.83</td><td>0.816</td><td>0.210</td><td>8.65</td></tr><tr><td>kitchen</td><td>30.37</td><td>0.911</td><td>0.160</td><td>3.32</td></tr><tr><td>room</td><td>31.24</td><td>0.911</td><td>0.243</td><td>2.06</td></tr><tr><td>stump</td><td>26.62</td><td>0.752</td><td>0.309</td><td>4.45</td></tr><tr><td>treehill</td><td>23.22</td><td>0.629</td><td>0.400</td><td>7.17</td></tr><tr><td>AVG</td><td>27.24</td><td>0.793</td><td>0.274</td><td>5.47</td></tr><tr><td rowspan="10">0.015</td><td>bicycle</td><td>24.80</td><td>0.714</td><td>0.323</td><td>6.59</td></tr><tr><td>bonsai</td><td>31.25</td><td>0.927</td><td>0.227</td><td>2.35</td></tr><tr><td>counter</td><td>28.51</td><td>0.886</td><td>0.244</td><td>2.45</td></tr><tr><td>flower</td><td>21.12</td><td>0.552</td><td>0.419</td><td>6.13</td></tr><tr><td>garden</td><td>26.56</td><td>0.803</td><td>0.231</td><td>6.82</td></tr><tr><td>kitchen</td><td>29.90</td><td>0.902</td><td>0.174</td><td>2.59</td></tr><tr><td>room</td><td>30.90</td><td>0.902</td><td>0.262</td><td>1.60</td></tr><tr><td>stump</td><td>26.38</td><td>0.736</td><td>0.333</td><td>3.25</td></tr><tr><td>treehill</td><td>23.08</td><td>0.616</td><td>0.423</td><td>5.13</td></tr><tr><td>AVG</td><td>26.94</td><td>0.782</td><td>0.293</td><td>4.10</td></tr><tr><td rowspan="10">0.03</td><td>bicycle</td><td>24.38</td><td>0.676</td><td>0.368</td><td>3.35</td></tr><tr><td>bonsai</td><td>29.99</td><td>0.905</td><td>0.260</td><td>1.58</td></tr><tr><td>counter</td><td>27.63</td><td>0.860</td><td>0.285</td><td>1.49</td></tr><tr><td>flower</td><td>20.74</td><td>0.519</td><td>0.455</td><td>3.32</td></tr><tr><td>garden</td><td>25.81</td><td>0.755</td><td>0.304</td><td>3.85</td></tr><tr><td>kitchen</td><td>28.76</td><td>0.878</td><td>0.216</td><td>1.53</td></tr><tr><td>room</td><td>30.19</td><td>0.887</td><td>0.292</td><td>1.05</td></tr><tr><td>stump</td><td>25.64</td><td>0.690</td><td>0.389</td><td>1.83</td></tr><tr><td>treehill</td><td>22.84</td><td>0.580</td><td>0.472</td><td>2.44</td></tr><tr><td>AVG</td><td>26.22</td><td>0.750</td><td>0.338</td><td>2.27</td></tr><tr><td rowspan="10">0.04</td><td>bicycle</td><td>24.02</td><td>0.646</td><td>0.399</td><td>2.36</td></tr><tr><td>bonsai</td><td>29.44</td><td>0.892</td><td>0.277</td><td>1.30</td></tr><tr><td>counter</td><td>27.14</td><td>0.845</td><td>0.306</td><td>1.23</td></tr><tr><td>flower</td><td>20.44</td><td>0.496</td><td>0.477</td><td>2.42</td></tr><tr><td>garden</td><td>25.40</td><td>0.727</td><td>0.337</td><td>3.01</td></tr><tr><td>kitchen</td><td>28.16</td><td>0.862</td><td>0.244</td><td>1.20</td></tr><tr><td>room</td><td>29.73</td><td>0.879</td><td>0.307</td><td>0.88</td></tr><tr><td>stump</td><td>25.30</td><td>0.666</td><td>0.415</td><td>1.35</td></tr><tr><td>treehill</td><td>22.71</td><td>0.558</td><td>0.499</td><td>1.74</td></tr><tr><td>AVG</td><td>25.82</td><td>0.730</td><td>0.362</td><td>1.72</td></tr></table>
+
+Table 8: Results of Our approach for each scene from BungeeNeRF dataset (Xiangli et al., 2022).  
+
+<table><tr><td>λr</td><td>Scenes</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>SIZE↓</td></tr><tr><td rowspan="7">0.001</td><td>amsterdam</td><td>27.57</td><td>0.908</td><td>0.148</td><td>31.98</td></tr><tr><td>bilbao</td><td>28.25</td><td>0.897</td><td>0.165</td><td>25.38</td></tr><tr><td>hollywood</td><td>25.16</td><td>0.815</td><td>0.254</td><td>24.58</td></tr><tr><td>pompidou</td><td>25.82</td><td>0.864</td><td>0.214</td><td>28.83</td></tr><tr><td>quebec</td><td>30.59</td><td>0.943</td><td>0.144</td><td>21.72</td></tr><tr><td>rome</td><td>26.74</td><td>0.887</td><td>0.178</td><td>27.06</td></tr><tr><td>AVG</td><td>27.35</td><td>0.886</td><td>0.184</td><td>26.59</td></tr><tr><td rowspan="7">0.002</td><td>amsterdam</td><td>27.52</td><td>0.903</td><td>0.159</td><td>27.33</td></tr><tr><td>bilbao</td><td>28.15</td><td>0.892</td><td>0.176</td><td>22.03</td></tr><tr><td>hollywood</td><td>25.12</td><td>0.808</td><td>0.268</td><td>20.95</td></tr><tr><td>pompidou</td><td>25.70</td><td>0.857</td><td>0.224</td><td>24.26</td></tr><tr><td>quebec</td><td>30.23</td><td>0.938</td><td>0.154</td><td>17.84</td></tr><tr><td>rome</td><td>26.57</td><td>0.881</td><td>0.190</td><td>23.17</td></tr><tr><td>AVG</td><td>27.22</td><td>0.880</td><td>0.195</td><td>22.60</td></tr><tr><td rowspan="7">0.003</td><td>amsterdam</td><td>27.28</td><td>0.897</td><td>0.169</td><td>24.47</td></tr><tr><td>bilbao</td><td>28.13</td><td>0.888</td><td>0.181</td><td>19.41</td></tr><tr><td>hollywood</td><td>25.05</td><td>0.801</td><td>0.280</td><td>18.72</td></tr><tr><td>pompidou</td><td>25.69</td><td>0.854</td><td>0.232</td><td>21.52</td></tr><tr><td>quebec</td><td>30.04</td><td>0.934</td><td>0.163</td><td>15.89</td></tr><tr><td>rome</td><td>26.42</td><td>0.875</td><td>0.200</td><td>20.81</td></tr><tr><td>AVG</td><td>27.10</td><td>0.875</td><td>0.204</td><td>20.14</td></tr><tr><td rowspan="7">0.006</td><td>amsterdam</td><td>26.96</td><td>0.882</td><td>0.194</td><td>19.38</td></tr><tr><td>bilbao</td><td>27.72</td><td>0.876</td><td>0.202</td><td>15.63</td></tr><tr><td>hollywood</td><td>24.68</td><td>0.778</td><td>0.307</td><td>14.90</td></tr><tr><td>pompidou</td><td>25.21</td><td>0.839</td><td>0.251</td><td>17.22</td></tr><tr><td>quebec</td><td>29.53</td><td>0.925</td><td>0.178</td><td>13.28</td></tr><tr><td>rome</td><td>25.78</td><td>0.856</td><td>0.223</td><td>16.87</td></tr><tr><td>AVG</td><td>26.65</td><td>0.859</td><td>0.226</td><td>16.21</td></tr><tr><td rowspan="7">0.01</td><td>amsterdam</td><td>26.32</td><td>0.863</td><td>0.215</td><td>15.96</td></tr><tr><td>bilbao</td><td>27.26</td><td>0.861</td><td>0.222</td><td>12.73</td></tr><tr><td>hollywood</td><td>24.41</td><td>0.757</td><td>0.325</td><td>12.71</td></tr><tr><td>pompidou</td><td>24.86</td><td>0.825</td><td>0.267</td><td>14.33</td></tr><tr><td>quebec</td><td>28.84</td><td>0.914</td><td>0.197</td><td>10.68</td></tr><tr><td>rome</td><td>25.13</td><td>0.834</td><td>0.245</td><td>13.86</td></tr><tr><td>AVG</td><td>26.14</td><td>0.842</td><td>0.245</td><td>13.38</td></tr><tr><td rowspan="7">0.02</td><td>amsterdam</td><td>25.65</td><td>0.833</td><td>0.253</td><td>12.33</td></tr><tr><td>bilbao</td><td>26.30</td><td>0.829</td><td>0.258</td><td>9.74</td></tr><tr><td>hollywood</td><td>23.80</td><td>0.714</td><td>0.361</td><td>9.12</td></tr><tr><td>pompidou</td><td>23.78</td><td>0.789</td><td>0.298</td><td>10.75</td></tr><tr><td>quebec</td><td>27.76</td><td>0.891</td><td>0.225</td><td>8.28</td></tr><tr><td>rome</td><td>23.85</td><td>0.790</td><td>0.283</td><td>10.63</td></tr><tr><td>AVG</td><td>25.19</td><td>0.808</td><td>0.280</td><td>10.14</td></tr></table>
+
+Table 9: Results of Our approach for each scene from Tank & Temples dataset (Knapitsch et al., 2017).  
+
+<table><tr><td>λr</td><td>Scenes</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>SIZE↓</td></tr><tr><td rowspan="3">0.002</td><td>train</td><td>22.68</td><td>0.820</td><td>0.221</td><td>6.28</td></tr><tr><td>truck</td><td>26.14</td><td>0.885</td><td>0.157</td><td>7.57</td></tr><tr><td>AVG</td><td>24.41</td><td>0.853</td><td>0.189</td><td>6.93</td></tr><tr><td rowspan="3">0.004</td><td>train</td><td>22.45</td><td>0.817</td><td>0.226</td><td>5.06</td></tr><tr><td>truck</td><td>25.98</td><td>0.882</td><td>0.162</td><td>5.88</td></tr><tr><td>AVG</td><td>24.22</td><td>0.850</td><td>0.194</td><td>5.47</td></tr><tr><td rowspan="3">0.01</td><td>train</td><td>22.31</td><td>0.802</td><td>0.249</td><td>3.43</td></tr><tr><td>truck</td><td>25.68</td><td>0.871</td><td>0.184</td><td>3.74</td></tr><tr><td>AVG</td><td>23.99</td><td>0.837</td><td>0.217</td><td>3.58</td></tr><tr><td rowspan="3">0.015</td><td>train</td><td>22.23</td><td>0.794</td><td>0.262</td><td>2.71</td></tr><tr><td>truck</td><td>25.55</td><td>0.866</td><td>0.195</td><td>3.00</td></tr><tr><td>AVG</td><td>23.89</td><td>0.830</td><td>0.228</td><td>2.86</td></tr><tr><td rowspan="3">0.03</td><td>train</td><td>22.01</td><td>0.769</td><td>0.295</td><td>1.80</td></tr><tr><td>truck</td><td>24.97</td><td>0.842</td><td>0.238</td><td>1.89</td></tr><tr><td>AVG</td><td>23.49</td><td>0.806</td><td>0.266</td><td>1.85</td></tr><tr><td rowspan="3">0.04</td><td>train</td><td>21.42</td><td>0.750</td><td>0.315</td><td>1.47</td></tr><tr><td>truck</td><td>24.51</td><td>0.822</td><td>0.271</td><td>1.38</td></tr><tr><td>AVG</td><td>22.97</td><td>0.786</td><td>0.293</td><td>1.42</td></tr></table>
+
+Table 10: Results of Our approach for each scene from Deep Blending (Hedman et al., 2018).  
+
+<table><tr><td>λr</td><td>Scenes</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>SIZE↓</td></tr><tr><td rowspan="3">0.002</td><td>drjohnson</td><td>29.67</td><td>0.906</td><td>0.266</td><td>4.27</td></tr><tr><td>playroom</td><td>30.90</td><td>0.911</td><td>0.272</td><td>2.85</td></tr><tr><td>AVG</td><td>30.29</td><td>0.909</td><td>0.269</td><td>3.56</td></tr><tr><td rowspan="3">0.004</td><td>drjohnson</td><td>29.51</td><td>0.904</td><td>0.272</td><td>3.43</td></tr><tr><td>playroom</td><td>30.81</td><td>0.909</td><td>0.279</td><td>2.21</td></tr><tr><td>AVG</td><td>30.16</td><td>0.906</td><td>0.275</td><td>2.82</td></tr><tr><td rowspan="3">0.01</td><td>drjohnson</td><td>29.23</td><td>0.898</td><td>0.288</td><td>2.12</td></tr><tr><td>playroom</td><td>30.23</td><td>0.903</td><td>0.295</td><td>1.46</td></tr><tr><td>AVG</td><td>29.73</td><td>0.900</td><td>0.292</td><td>1.79</td></tr><tr><td rowspan="3">0.015</td><td>drjohnson</td><td>29.06</td><td>0.892</td><td>0.299</td><td>1.76</td></tr><tr><td>playroom</td><td>29.91</td><td>0.897</td><td>0.309</td><td>1.21</td></tr><tr><td>AVG</td><td>29.48</td><td>0.894</td><td>0.304</td><td>1.48</td></tr><tr><td rowspan="3">0.03</td><td>drjohnson</td><td>28.52</td><td>0.879</td><td>0.323</td><td>1.20</td></tr><tr><td>playroom</td><td>29.19</td><td>0.887</td><td>0.331</td><td>0.85</td></tr><tr><td>AVG</td><td>28.85</td><td>0.883</td><td>0.327</td><td>1.03</td></tr><tr><td rowspan="3">0.04</td><td>drjohnson</td><td>28.31</td><td>0.874</td><td>0.332</td><td>1.09</td></tr><tr><td>playroom</td><td>28.74</td><td>0.881</td><td>0.340</td><td>0.76</td></tr><tr><td>AVG</td><td>28.53</td><td>0.878</td><td>0.336</td><td>0.93</td></tr></table>
+
+Table 11: Results of HAC (Chen et al., 2024) for each scene from Mip-NeRF 360 dataset (Barron et al., 2022a).  
+
+<table><tr><td>λe</td><td>Scenes</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>SIZE↓</td></tr><tr><td rowspan="10">0.001</td><td>bicycle</td><td>25.11</td><td>0.742</td><td>0.259</td><td>39.15</td></tr><tr><td>bonsai</td><td>32.97</td><td>0.948</td><td>0.180</td><td>12.72</td></tr><tr><td>counter</td><td>29.74</td><td>0.918</td><td>0.184</td><td>10.44</td></tr><tr><td>flower</td><td>21.27</td><td>0.575</td><td>0.377</td><td>27.55</td></tr><tr><td>garden</td><td>27.46</td><td>0.849</td><td>0.139</td><td>32.17</td></tr><tr><td>kitchen</td><td>31.63</td><td>0.930</td><td>0.122</td><td>12.07</td></tr><tr><td>room</td><td>31.90</td><td>0.926</td><td>0.198</td><td>7.85</td></tr><tr><td>stump</td><td>26.59</td><td>0.763</td><td>0.264</td><td>25.26</td></tr><tr><td>treehill</td><td>23.26</td><td>0.648</td><td>0.345</td><td>29.65</td></tr><tr><td>AVG</td><td>27.77</td><td>0.811</td><td>0.230</td><td>21.87</td></tr><tr><td rowspan="10">0.002</td><td>bicycle</td><td>25.10</td><td>0.742</td><td>0.262</td><td>33.14</td></tr><tr><td>bonsai</td><td>32.70</td><td>0.945</td><td>0.184</td><td>10.51</td></tr><tr><td>counter</td><td>29.65</td><td>0.915</td><td>0.189</td><td>8.88</td></tr><tr><td>flower</td><td>21.32</td><td>0.576</td><td>0.377</td><td>23.73</td></tr><tr><td>garden</td><td>27.43</td><td>0.847</td><td>0.143</td><td>27.52</td></tr><tr><td>kitchen</td><td>31.46</td><td>0.928</td><td>0.125</td><td>10.05</td></tr><tr><td>room</td><td>31.87</td><td>0.925</td><td>0.201</td><td>6.47</td></tr><tr><td>stump</td><td>26.59</td><td>0.761</td><td>0.268</td><td>21.75</td></tr><tr><td>treehill</td><td>23.34</td><td>0.647</td><td>0.350</td><td>24.83</td></tr><tr><td>AVG</td><td>27.72</td><td>0.809</td><td>0.233</td><td>18.54</td></tr><tr><td rowspan="10">0.004</td><td>bicycle</td><td>25.05</td><td>0.742</td><td>0.264</td><td>27.54</td></tr><tr><td>bonsai</td><td>32.28</td><td>0.942</td><td>0.189</td><td>8.56</td></tr><tr><td>counter</td><td>29.35</td><td>0.911</td><td>0.195</td><td>7.26</td></tr><tr><td>flower</td><td>21.26</td><td>0.572</td><td>0.381</td><td>19.59</td></tr><tr><td>garden</td><td>27.28</td><td>0.842</td><td>0.151</td><td>22.69</td></tr><tr><td>kitchen</td><td>31.16</td><td>0.923</td><td>0.131</td><td>8.05</td></tr><tr><td>room</td><td>31.55</td><td>0.921</td><td>0.208</td><td>5.53</td></tr><tr><td>stump</td><td>26.58</td><td>0.762</td><td>0.269</td><td>18.11</td></tr><tr><td>treehill</td><td>23.30</td><td>0.645</td><td>0.356</td><td>20.04</td></tr><tr><td>AVG</td><td>27.53</td><td>0.807</td><td>0.238</td><td>15.26</td></tr><tr><td rowspan="10">0.01</td><td>bicycle</td><td>24.79</td><td>0.733</td><td>0.284</td><td>17.80</td></tr><tr><td>bonsai</td><td>31.27</td><td>0.933</td><td>0.208</td><td>5.16</td></tr><tr><td>counter</td><td>28.68</td><td>0.898</td><td>0.220</td><td>4.54</td></tr><tr><td>flower</td><td>21.18</td><td>0.561</td><td>0.400</td><td>12.15</td></tr><tr><td>garden</td><td>26.76</td><td>0.822</td><td>0.188</td><td>14.53</td></tr><tr><td>kitchen</td><td>30.51</td><td>0.914</td><td>0.149</td><td>4.85</td></tr><tr><td>room</td><td>31.20</td><td>0.912</td><td>0.234</td><td>3.27</td></tr><tr><td>stump</td><td>26.54</td><td>0.752</td><td>0.296</td><td>11.21</td></tr><tr><td>treehill</td><td>23.13</td><td>0.628</td><td>0.392</td><td>12.27</td></tr><tr><td>AVG</td><td>27.12</td><td>0.795</td><td>0.263</td><td>9.53</td></tr><tr><td rowspan="10">0.02</td><td>bicycle</td><td>24.60</td><td>0.717</td><td>0.306</td><td>14.97</td></tr><tr><td>bonsai</td><td>30.51</td><td>0.922</td><td>0.225</td><td>4.24</td></tr><tr><td>counter</td><td>27.78</td><td>0.878</td><td>0.248</td><td>3.30</td></tr><tr><td>flower</td><td>20.84</td><td>0.537</td><td>0.425</td><td>8.90</td></tr><tr><td>garden</td><td>26.34</td><td>0.800</td><td>0.222</td><td>10.87</td></tr><tr><td>kitchen</td><td>29.86</td><td>0.902</td><td>0.169</td><td>3.69</td></tr><tr><td>room</td><td>30.51</td><td>0.900</td><td>0.256</td><td>2.56</td></tr><tr><td>stump</td><td>26.20</td><td>0.730</td><td>0.326</td><td>8.68</td></tr><tr><td>treehill</td><td>23.03</td><td>0.610</td><td>0.418</td><td>9.25</td></tr><tr><td>AVG</td><td>26.63</td><td>0.777</td><td>0.288</td><td>7.38</td></tr><tr><td rowspan="10">0.035</td><td>bicycle</td><td>24.16</td><td>0.694</td><td>0.333</td><td>11.59</td></tr><tr><td>bonsai</td><td>29.63</td><td>0.909</td><td>0.242</td><td>3.72</td></tr><tr><td>counter</td><td>27.19</td><td>0.864</td><td>0.269</td><td>2.68</td></tr><tr><td>flower</td><td>20.57</td><td>0.513</td><td>0.449</td><td>7.17</td></tr><tr><td>garden</td><td>25.86</td><td>0.780</td><td>0.252</td><td>8.90</td></tr><tr><td>kitchen</td><td>29.09</td><td>0.886</td><td>0.191</td><td>3.08</td></tr><tr><td>room</td><td>29.94</td><td>0.889</td><td>0.276</td><td>2.15</td></tr><tr><td>stump</td><td>25.77</td><td>0.707</td><td>0.358</td><td>7.03</td></tr><tr><td>treehill</td><td>22.82</td><td>0.591</td><td>0.443</td><td>7.32</td></tr><tr><td>AVG</td><td>26.11</td><td>0.759</td><td>0.312</td><td>5.96</td></tr></table>
+
+Table 12: Results of HAC (Chen et al., 2024) for each scene from BungeeNeRF dataset (Xiangli et al., 2022).  
+
+<table><tr><td>λe</td><td>Scenes</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>SIZE↓</td></tr><tr><td rowspan="7">0.001</td><td>amsterdam</td><td>27.25</td><td>0.886</td><td>0.190</td><td>31.84</td></tr><tr><td>bilbao</td><td>27.98</td><td>0.886</td><td>0.190</td><td>24.38</td></tr><tr><td>hollywood</td><td>24.59</td><td>0.772</td><td>0.319</td><td>23.41</td></tr><tr><td>pompidou</td><td>25.58</td><td>0.851</td><td>0.236</td><td>29.19</td></tr><tr><td>quebec</td><td>30.30</td><td>0.934</td><td>0.163</td><td>21.23</td></tr><tr><td>rome</td><td>26.61</td><td>0.876</td><td>0.203</td><td>26.91</td></tr><tr><td>AVG</td><td>27.05</td><td>0.868</td><td>0.217</td><td>26.16</td></tr><tr><td rowspan="7">0.002</td><td>amsterdam</td><td>27.13</td><td>0.880</td><td>0.202</td><td>27.14</td></tr><tr><td>bilbao</td><td>28.02</td><td>0.880</td><td>0.205</td><td>20.91</td></tr><tr><td>hollywood</td><td>24.43</td><td>0.763</td><td>0.330</td><td>20.09</td></tr><tr><td>pompidou</td><td>25.27</td><td>0.842</td><td>0.249</td><td>24.85</td></tr><tr><td>quebec</td><td>29.98</td><td>0.929</td><td>0.175</td><td>17.90</td></tr><tr><td>rome</td><td>26.28</td><td>0.866</td><td>0.219</td><td>23.07</td></tr><tr><td>AVG</td><td>26.85</td><td>0.860</td><td>0.230</td><td>22.33</td></tr><tr><td rowspan="7">0.003</td><td>amsterdam</td><td>26.95</td><td>0.873</td><td>0.214</td><td>24.41</td></tr><tr><td>bilbao</td><td>27.82</td><td>0.872</td><td>0.218</td><td>18.76</td></tr><tr><td>hollywood</td><td>24.27</td><td>0.753</td><td>0.342</td><td>17.87</td></tr><tr><td>pompidou</td><td>25.34</td><td>0.837</td><td>0.255</td><td>22.49</td></tr><tr><td>quebec</td><td>29.67</td><td>0.924</td><td>0.185</td><td>16.15</td></tr><tr><td>rome</td><td>25.98</td><td>0.855</td><td>0.231</td><td>20.83</td></tr><tr><td>AVG</td><td>26.67</td><td>0.852</td><td>0.241</td><td>20.08</td></tr><tr><td rowspan="7">0.004</td><td>amsterdam</td><td>26.80</td><td>0.865</td><td>0.224</td><td>22.49</td></tr><tr><td>bilbao</td><td>27.65</td><td>0.864</td><td>0.231</td><td>17.14</td></tr><tr><td>hollywood</td><td>24.25</td><td>0.748</td><td>0.347</td><td>16.55</td></tr><tr><td>pompidou</td><td>25.16</td><td>0.829</td><td>0.266</td><td>20.40</td></tr><tr><td>quebec</td><td>29.33</td><td>0.918</td><td>0.192</td><td>15.06</td></tr><tr><td>rome</td><td>25.68</td><td>0.845</td><td>0.243</td><td>19.30</td></tr><tr><td>AVG</td><td>26.48</td><td>0.845</td><td>0.250</td><td>18.49</td></tr><tr><td rowspan="7">0.008</td><td>amsterdam</td><td>26.13</td><td>0.839</td><td>0.258</td><td>17.22</td></tr><tr><td>bilbao</td><td>26.81</td><td>0.842</td><td>0.262</td><td>14.01</td></tr><tr><td>hollywood</td><td>23.83</td><td>0.713</td><td>0.377</td><td>12.90</td></tr><tr><td>pompidou</td><td>24.48</td><td>0.807</td><td>0.289</td><td>16.10</td></tr><tr><td>quebec</td><td>28.60</td><td>0.906</td><td>0.216</td><td>11.71</td></tr><tr><td>rome</td><td>24.71</td><td>0.811</td><td>0.276</td><td>15.34</td></tr><tr><td>AVG</td><td>25.76</td><td>0.820</td><td>0.280</td><td>14.55</td></tr><tr><td rowspan="7">0.02</td><td>amsterdam</td><td>25.03</td><td>0.788</td><td>0.308</td><td>11.92</td></tr><tr><td>bilbao</td><td>25.84</td><td>0.799</td><td>0.308</td><td>9.08</td></tr><tr><td>hollywood</td><td>22.92</td><td>0.640</td><td>0.425</td><td>8.38</td></tr><tr><td>pompidou</td><td>23.16</td><td>0.759</td><td>0.332</td><td>10.43</td></tr><tr><td>quebec</td><td>27.15</td><td>0.872</td><td>0.262</td><td>8.18</td></tr><tr><td>rome</td><td>23.25</td><td>0.750</td><td>0.327</td><td>10.47</td></tr><tr><td>AVG</td><td>24.56</td><td>0.768</td><td>0.327</td><td>9.74</td></tr></table>
+
+Table 13: Results of HAC (Chen et al., 2024) for each scene from Tank & Temples dataset (Knapitsch et al., 2017).  
+
+<table><tr><td>λe</td><td>Scenes</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>SIZE↓</td></tr><tr><td rowspan="3">0.001</td><td>truck</td><td>26.02</td><td>0.883</td><td>0.147</td><td>12.42</td></tr><tr><td>train</td><td>22.78</td><td>0.823</td><td>0.207</td><td>10.07</td></tr><tr><td>AVG</td><td>24.40</td><td>0.853</td><td>0.177</td><td>11.24</td></tr><tr><td rowspan="3">0.004</td><td>truck</td><td>25.88</td><td>0.878</td><td>0.158</td><td>9.26</td></tr><tr><td>train</td><td>22.19</td><td>0.815</td><td>0.216</td><td>6.94</td></tr><tr><td>AVG</td><td>24.04</td><td>0.846</td><td>0.187</td><td>8.10</td></tr><tr><td rowspan="3">0.01</td><td>truck</td><td>25.69</td><td>0.874</td><td>0.172</td><td>7.04</td></tr><tr><td>train</td><td>22.13</td><td>0.807</td><td>0.230</td><td>5.26</td></tr><tr><td>AVG</td><td>23.91</td><td>0.841</td><td>0.201</td><td>6.15</td></tr><tr><td rowspan="3">0.02</td><td>truck</td><td>25.36</td><td>0.863</td><td>0.189</td><td>5.59</td></tr><tr><td>train</td><td>22.02</td><td>0.795</td><td>0.250</td><td>3.97</td></tr><tr><td>AVG</td><td>23.69</td><td>0.829</td><td>0.219</td><td>4.78</td></tr><tr><td rowspan="3">0.035</td><td>truck</td><td>24.96</td><td>0.853</td><td>0.208</td><td>5.04</td></tr><tr><td>train</td><td>21.96</td><td>0.783</td><td>0.266</td><td>3.17</td></tr><tr><td>AVG</td><td>23.46</td><td>0.818</td><td>0.237</td><td>4.10</td></tr><tr><td rowspan="3">0.045</td><td>truck</td><td>24.81</td><td>0.847</td><td>0.216</td><td>4.64</td></tr><tr><td>train</td><td>21.42</td><td>0.771</td><td>0.279</td><td>2.72</td></tr><tr><td>AVG</td><td>23.11</td><td>0.809</td><td>0.248</td><td>3.68</td></tr></table>
+
+Table 14: Results of HAC (Chen et al., 2024) for each scene from Deep Blending dataset (Hedman et al., 2018).  
+
+<table><tr><td>λe</td><td>Scenes</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>SIZE↓</td></tr><tr><td rowspan="3">0.001</td><td>playroom</td><td>30.84</td><td>0.906</td><td>0.262</td><td>5.03</td></tr><tr><td>drjohnson</td><td>29.85</td><td>0.906</td><td>0.255</td><td>7.67</td></tr><tr><td>AVG</td><td>30.34</td><td>0.906</td><td>0.258</td><td>6.35</td></tr><tr><td rowspan="3">0.002</td><td>playroom</td><td>30.66</td><td>0.905</td><td>0.265</td><td>4.12</td></tr><tr><td>drjohnson</td><td>29.69</td><td>0.905</td><td>0.258</td><td>6.51</td></tr><tr><td>AVG</td><td>30.17</td><td>0.905</td><td>0.262</td><td>5.32</td></tr><tr><td rowspan="3">0.004</td><td>playroom</td><td>30.44</td><td>0.902</td><td>0.272</td><td>3.15</td></tr><tr><td>drjohnson</td><td>29.53</td><td>0.903</td><td>0.265</td><td>5.55</td></tr><tr><td>AVG</td><td>29.98</td><td>0.902</td><td>0.269</td><td>4.35</td></tr><tr><td rowspan="3">0.01</td><td>playroom</td><td>30.31</td><td>0.906</td><td>0.279</td><td>2.58</td></tr><tr><td>drjohnson</td><td>29.34</td><td>0.901</td><td>0.275</td><td>4.23</td></tr><tr><td>AVG</td><td>29.83</td><td>0.903</td><td>0.277</td><td>3.40</td></tr><tr><td rowspan="3">0.02</td><td>playroom</td><td>29.87</td><td>0.900</td><td>0.292</td><td>2.23</td></tr><tr><td>drjohnson</td><td>28.81</td><td>0.893</td><td>0.288</td><td>3.65</td></tr><tr><td>AVG</td><td>29.34</td><td>0.897</td><td>0.290</td><td>2.94</td></tr><tr><td rowspan="3">0.035</td><td>playroom</td><td>29.26</td><td>0.893</td><td>0.302</td><td>2.01</td></tr><tr><td>drjohnson</td><td>27.97</td><td>0.883</td><td>0.302</td><td>3.26</td></tr><tr><td>AVG</td><td>28.62</td><td>0.888</td><td>0.302</td><td>2.64</td></tr></table>
+
+Table 15: Results of ScaffoldGS (Lu et al., 2024) for all evaluated datasets.  
+
+<table><tr><td>Datasets</td><td>Scenes</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>SIZE↓</td></tr><tr><td rowspan="10">Mip-NeRF360</td><td>bicycle</td><td>24.50</td><td>0.705</td><td>0.306</td><td>248.00</td></tr><tr><td>bonsai</td><td>32.70</td><td>0.946</td><td>0.185</td><td>258.00</td></tr><tr><td>counter</td><td>29.34</td><td>0.914</td><td>0.191</td><td>194.00</td></tr><tr><td>flower</td><td>21.14</td><td>0.566</td><td>0.417</td><td>253.00</td></tr><tr><td>garden</td><td>27.17</td><td>0.842</td><td>0.146</td><td>271.00</td></tr><tr><td>kitchen</td><td>31.30</td><td>0.928</td><td>0.126</td><td>173.00</td></tr><tr><td>room</td><td>31.93</td><td>0.925</td><td>0.202</td><td>133.00</td></tr><tr><td>stump</td><td>26.27</td><td>0.784</td><td>0.284</td><td>493.00</td></tr><tr><td>treehill</td><td>23.19</td><td>0.642</td><td>0.410</td><td>262.00</td></tr><tr><td>AVG</td><td>27.50</td><td>0.806</td><td>0.252</td><td>253.89</td></tr><tr><td rowspan="3">Tank&amp;Temples</td><td>truck</td><td>25.77</td><td>0.883</td><td>0.147</td><td>107.00</td></tr><tr><td>train</td><td>22.15</td><td>0.822</td><td>0.206</td><td>66.00</td></tr><tr><td>AVG</td><td>23.96</td><td>0.853</td><td>0.177</td><td>86.50</td></tr><tr><td rowspan="3">DeepBlending</td><td>playroom</td><td>30.62</td><td>0.904</td><td>0.258</td><td>63.00</td></tr><tr><td>drjohnson</td><td>29.80</td><td>0.907</td><td>0.250</td><td>69.00</td></tr><tr><td>AVG</td><td>30.21</td><td>0.906</td><td>0.254</td><td>66.00</td></tr><tr><td rowspan="7">BungeeNeRF</td><td>amsterdam</td><td>27.16</td><td>0.898</td><td>0.188</td><td>223.00</td></tr><tr><td>bilbao</td><td>26.60</td><td>0.857</td><td>0.257</td><td>178.00</td></tr><tr><td>hollywood</td><td>24.49</td><td>0.787</td><td>0.318</td><td>155.00</td></tr><tr><td>pompidou</td><td>24.94</td><td>0.839</td><td>0.271</td><td>209.00</td></tr><tr><td>quebec</td><td>30.28</td><td>0.936</td><td>0.190</td><td>159.00</td></tr><tr><td>rome</td><td>26.23</td><td>0.873</td><td>0.225</td><td>174.00</td></tr><tr><td>AVG</td><td>26.62</td><td>0.865</td><td>0.241</td><td>183.00</td></tr></table>

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ICLR/2025/CBMA_ Improving Conformal Prediction through Bayesian Model Averaging/full.md

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+# CBMA: IMPROVING CONFORMAL PREDICTION THROUGH BAYESIAN MODEL AVERAGING
+
+# Pankaj Bhagwat
+
+Department of Mathematical and Statistical Sciences, University of Alberta, AB, Canada pbhagwat@ualberta.ca
+
+# Linglong Kong
+
+Department of Mathematical and Statistical Sciences, University of Alberta, AB, Canada lkong@ualberta.ca
+
+# Bai Jiang
+
+Department of Mathematical and Statistical Sciences, University of Alberta, AB, Canada beil@ualberta.ca
+
+# ABSTRACT
+
+Conformal prediction has emerged as a popular technique for facilitating valid predictive inference across a spectrum of machine learning models, under minimal assumption of exchangeability. Recently, Hoff (2023) showed that full conformal Bayes provides the most efficient prediction sets (smallest by expected volume) among all prediction sets that are valid at the  $(1 - \alpha)$  level if the model is correctly specified. However, a critical issue arises when the Bayesian model itself may be mis-specified, resulting in prediction set that might be suboptimal, even though it still enjoys the frequentist coverage guarantee. To address this limitation, we propose an innovative solution that combines Bayesian model averaging (BMA) with conformal prediction. This hybrid not only leverages the strengths of Bayesian conformal prediction but also introduces a layer of robustness through model averaging. Theoretically, we prove that the resulting prediction set will converge to the optimal level of efficiency, if the true model is included among the candidate models. This assurance of optimality, even under potential model uncertainty, provides a significant improvement over existing methods, ensuring more reliable and precise uncertainty quantification.
+
+# 1 INTRODUCTION
+
+Conformal prediction is a distribution-free uncertainty quantification method that generates prediction sets with valid coverage under minimal assumptions of exchangeability (Vovk et al., 2005). By building on existing machine learning techniques, conformal prediction uses available data to establish valid prediction set  $C_{\alpha}(X_{n + 1})$  for new instances  $X_{n + 1}$  with coverage guarantee  $P(Y_{n + 1}\in C_{\alpha}(X_{n + 1}))\geq 1 - \alpha$ . Prediction sets that include the ground truth with high probability are essential for high-stakes applications, such as autonomous vehicles (Bojarski et al., 2016) and clinical settings (Esteva et al., 2017). However, it is also preferable for the prediction sets  $C_{\alpha}(X_{n + 1})$  to be efficient i.e. as small as possible, as smaller sets are more informative. Recently, Hoff (2023) investigated the optimal efficiency of full conformal Bayes prediction sets under a correctly specified Bayesian model, showing that they yield prediction sets with the minimum expected volume at a given target coverage level  $(1 - \alpha)$ . In addition, Fong & Holmes (2021) introduced scalable methods for generating full conformal Bayes prediction sets applicable to any Bayesian model, highlighting the advantages of conformal Bayesian predictions over traditional Bayesian posterior predictive sets, especially in the presence of model misspecification.
+
+However, constructing or selecting the correct Bayesian model to obtain optimally efficient conformal prediction sets is a challenging task. Although conformal prediction sets provide finite-sample coverage guarantees for any model, they may fail to achieve volume optimality when the underlying
+
+model is misspecified. This challenge underscores a broader limitation of traditional conformal prediction methods, which typically rely on a fixed machine learning model to generate prediction sets with a predetermined marginal coverage level. Given the vast array of possible predictive models for a given problem, and the fact that conformal methods yield valid sets for all of them, identifying the most appropriate model becomes inherently difficult. This issue of model uncertainty has long been a persistent and underexplored problem in the conformal prediction literature.
+
+Our Contributions: In this paper, we provide an innovative solution in the form of Conformal Bayesian model averaging (CBMA) to the challenging issue of constructing efficient conformal prediction sets when there is model uncertainty in the Bayesian framework. Our proposed CBMA prediction method seamlessly combines conformity scores of each Bayesian model in order to construct a single conformal prediction set. This paves the way for incorporating model averaging procedures into conformal prediction framework which is currently lacking in the literature. To the best of our knowledge, our CBMA approach is the first method which combines conformity scores from diverse models to construct valid conformal prediction sets and requires no data splitting (data efficient). Our choice of conformity scores in the form of posterior predictive densities is also optimal and leads to the most efficient prediction set when the underlying Bayesian model is true (Hoff, 2023). Our CBMA method can construct full conformal prediction sets given samples for model parameters of each model from their posterior distributions and posterior model probabilities, which can be obtained as the output of BMA. Theoretically, we also prove the optimal efficiency achieved by conformal prediction sets constructed using CBMA method as the sample size increases. Such guarantee of optimal efficiency even under model uncertainty provides an improvement over existing methods. Finally, our method incorporates both data and model uncertainty into the construction of prediction sets which enhances the reliability of the predictions.
+
+# 2 PRELIMINARIES
+
+We consider a collection of i.i.d. observations  $Z_{1:n} = \{(X_i,Y_i)\}_{i=1}^n$ , where each observation  $(X_i,Y_i) \in \mathbb{R}^d \times \mathbb{R}$  is a covariate-response pair. We begin by summarizing full conformal prediction framework (Vovk et al., 2005), conformal Bayes (Fong & Holmes, 2021) and Bayesian model averaging (BMA) (Raftery et al., 1997).
+
+# 2.1 FULL CONFORMAL PREDICTION
+
+In the full conformal prediction method, candidate labels  $y_{n + 1}$  are included in  $C_{\alpha}(X_{n + 1})$  if the resulting pair  $(X_{n + 1},y_{n + 1})$  is sufficiently similar to the examples in  $Z_{1:n}$ . A conformity score  $\sigma_{i}$
+
+$$
+\sigma_ {i} := \sigma (Z _ {1: n + 1}; Z _ {i}),
+$$
+
+takes as input a set of data points  $Z_{1:n+1} = Z_{1:n} \cup Z_{n+1}$  where  $Z_{n+1} = \{(X_{n+1}, y_{n+1})$ , and computes how similar the data point  $Z_i$  is for  $i = 1, \ldots, n+1$ . The key property of any conformity score is that it must be exchangeable in the first argument, meaning the conformity core for  $Z_i$  remains unchanged under permutation of  $Z_{1:n+1}$ . We refer to a conformity score with this property as a valid conformity score. Assuming  $Z_{1:n+1}$  is exchangeable,  $\sigma_{1:n+1}$  will also be exchangeable, and its rank will be uniformly distributed among  $\{1, \ldots, n+1\}$  (assuming continuous  $\sigma_{1:n+1}$ ). This implies that the rank of  $\sigma_{n+1}$  is a valid p-value. For a given value  $Y_{n+1} = y$  (with  $X_{n+1}$  known), we can denote the rank of  $\sigma_{n+1}$  among  $\sigma_{1:n+1}$  as
+
+$$
+r (y) = \frac {1}{n + 1} \sum_ {i = 1} ^ {n + 1} \mathbf {1} (\sigma_ {i} \leq \sigma_ {n + 1}).
+$$
+
+For miscoverage level  $\alpha$ , the full conformal predictive set,  $C_{\alpha}(X_{n + 1}) = \{y\in \mathcal{V}:\bar{r} (y) > \alpha \}$ , ensures the desired frequentist coverage  $\mathbf{P}\left(Y_{n + 1}\in C_{\alpha}(X_{n + 1})\right)\geq 1 - \alpha$ , where  $\mathbf{P}$  is over  $Z_{1:n + 1}$ . A formal proof can be found in Vovk et al. (2005), Chapter 8.7. For continuous  $\sigma_{1:n + 1}$ , Lei et al. (2018), Theorem 1 show that the conformal predictive set does not significantly over-cover.
+
+# 2.2 FULL CONFORMAL BAYES
+
+In the Bayesian prediction, given the model likelihood  $p_{\theta}(y|x)$  and prior on parameters,  $\pi(\theta)$  for  $\theta \in \mathbb{R}^p$ , the Bayesian posterior predictive distribution for the response at new instance
+
+$X_{n + 1} = x_{n + 1}$  conditioned on the observed data  $Z_{1:n}$  represents subjective and coherent beliefs. As noted by Draper (1995) and Fraser (2011), Bayesian predictive sets such as Highest posterior density predictive credible sets are poorly calibrated in the frequentist sense. This has consequences for the robustness of such approaches and trust in using Bayesian models to aid decisions. Conformal inference for calibrating Bayesian models was previously suggested in Vovk et al. (2005); Wasserman (2011). In a Bayesian model, it is natural to consider the posterior predictive density as the conformity score:
+
+$$
+\sigma_ {i} = p \left(Y _ {i} \mid X _ {i}, Z _ {1: n + 1}\right) = \int p _ {\theta} \left(y _ {i} \mid x _ {i}\right) \pi \left(\theta \mid Z _ {1: n + 1}\right) d \theta
+$$
+
+This is a valid conformity score,  $\sigma_{i}$  is indeed invariant to the permutation of  $Z_{1:n + 1}$  due to the form of  $\pi (\theta |Z_{1:n + 1})\propto \pi (\theta)\prod_{i = 1}^{n + 1}p_{\theta}(y_i|x_i)$ . This method is referred to as Conformal Bayes in Fong & Holmes (2021).
+
+# 2.3 BMA: BAYESIAN MODEL AVERAGING
+
+In detail discussion on BMA can be found in the seminal work of Hoeting et al. (1999), Wasserman (2000), Raftery et al. (1997). Let  $X \in \mathbb{R}^d$  denote the vector consisting of all predictors under consideration and  $Y \in \mathbb{R}$  denote the response variable. In the supervised learning tasks, our goal is to construct a predictive model for  $Y$  based on  $X$  via some model  $\mathcal{M}$ . The Bayesian model  $\mathcal{M}$  in such scenario can be represented as
+
+$$
+Y | X, \theta , \mathcal {M} \sim p _ {\theta} (y | x); \theta | \mathcal {M} \sim \pi (\theta),
+$$
+
+where  $\theta$  denotes the parameters of the model  $\mathcal{M}$ ,  $\pi(\theta)$  denotes the prior on the parameters and  $p_{\theta}(y|x)$  denotes the likelihood function under the model  $\mathcal{M}$ .
+
+Now suppose, we have several competitive models  $\mathcal{M}_k$  with parameters  $\theta_k \in \Theta_k$  leading to model space  $\mathcal{M} = \{\mathcal{M}_1, \dots, \mathcal{M}_K\}$ . The Bayesian model for each model is described as:
+
+$$
+Y | X, \theta_ {k}, \mathcal {M} _ {k} \sim p _ {\theta_ {k}} (y | x); \quad \theta_ {k} | \mathcal {M} _ {k} \sim \pi_ {k} (\theta_ {k}).
+$$
+
+If  $\Delta$  is a quantity of interest, say predicted values at a covariate  $x\in \mathbb{R}^d$ , then the expected value of  $\Delta$  given the data  $Z_{1:n}$  is obtained by first finding the posterior expectation of  $\Delta$  under each model, and then weighting each expectation by the posterior probability of the model in the BMA framework:
+
+$$
+\mathbf {E} [ \Delta | Z _ {1: n} ] = \sum_ {k = 1} ^ {K} p (\mathcal {M} _ {k} | Z _ {1: n}) \mathbf {E} [ \Delta | Z _ {1: n}, \mathcal {M} _ {k} ].
+$$
+
+Similarly, the posterior distribution for  $\Delta$  can be represented as a mixture distribution over all models,
+
+$$
+p (\Delta | Z _ {1: n}) = \sum_ {k = 1} ^ {K} p \left(\mathcal {M} _ {k} \mid Z _ {1: n}\right) p \left(\Delta \mid Z _ {1: n}, \mathcal {M} _ {k}\right) \tag {1}
+$$
+
+where  $p(\Delta |Z_{1:n},\mathcal{M}_k)$  is the posterior distribution of  $\Delta$  under model  $\mathcal{M}_k$ . In equation 1 the posterior probability of model  $\mathcal{M}_k$  is given by
+
+$$
+p(\mathcal{M}_{k}|Z_{1:n}) = \frac{m(Z_{1:n}|\mathcal{M}_{k})p(\mathcal{M}_{k})}{\sum_{k = 1}^{K}m(Z_{1:n}|\mathcal{M}_{k})p(\mathcal{M}_{k})},
+$$
+
+where  $p(\mathcal{M}_k)$  is the prior probability that  $\mathcal{M}_k$  is the true model and  $m(Z_{1:n}|\mathcal{M}_k)$  is the marginal likelihood under model  $\mathcal{M}_k$ ,
+
+$$
+{m (Z _ {1: n} | \mathcal {M} _ {k})} {= \int \prod_ {i = 1} ^ {n} p _ {\theta_ {k}} (Y _ {i} | X _ {i}) \pi_ {k} (\theta_ {k}) d \theta_ {k}}
+$$
+
+obtained by integrating over the prior distributions for model specific parameters. Note that all conditional expressions in this section denote condition on  $Z_{1:n} = \{(X_i,Y_i)\}_{i=1}^n$ .
+
+# 2.4 PROBLEM SETUP
+
+Our objective is to construct a conformal prediction set  $C_{\alpha}(X_{n + 1}) \subseteq \mathbb{R}$  that provides a range of possible responses for a test point  $X_{n + 1}$  based on a collection of i.i.d. observations  $Z_{1:n} = \{(X_i,Y_i)\}_{i = 1}^n$ . We focus on Bayesian prediction based on the training data  $Z_{1:n}$ . Suppose we have  $K$  competing models for this data, denoted as  $\mathcal{M}_1,\dots ,\mathcal{M}_K$ , forming a model space  $\mathcal{M} = \{\mathcal{M}_1,\ldots ,\mathcal{M}_K\}$ . For each model  $\mathcal{M}_k$  with  $k = 1,2,\ldots ,K$ , the data likelihood is specified as  $p_{\theta_k}(y_i\mid x_i)$ , where  $\theta_{k}$  represents the parameters of model  $\mathcal{M}_k$ . Let  $\pi_k(\theta_k)$  denote the prior distribution on the parameters  $\theta_{k}$  of model  $\mathcal{M}_k$ .
+
+The key question is: with multiple candidate models in  $\mathcal{M}$ , how can we combine their predictions to create a single conformal prediction set? Additionally, we aim to construct prediction sets using a transductive conformal approach, rather than relying on data-splitting inductive methods, which are often less efficient in terms of information usage. We propose to average conformity scores from each model to construct combined conformity score. This task presents several challenges including the issue of the same dataset being utilized to train individual models, the conformity scores are not independent. Also, note that, we need to combine conformity scores in a fashion that assigns higher weights to the conformity scores of the true model, if it is included in the model space. Thus, weights used for averaging should be data dependent.
+
+# 2.5 RELATED WORKS
+
+Conformal Bayes has been a topic of research interest for a long time. But some recent advances in terms of optimal efficiency of such prediction sets and scalable methods for the construction of full conformal sets are noteworthy. Hoff (2023) demonstrated that the log predictive densities are the optimal conformity measure to construct conformal prediction sets (minimum expected volume) when the underlying Bayesian model is true. Fong & Holmes (2021) recently provided a scalable Full conformal Bayes algorithm to construct a Bayes conformal set efficiently. They also explored the benefits of conformal Bayes prediction over Bayesian predictive sets when the model is misspecified. But, the conformal prediction set they construct may be inefficient. We address this issue by incorporating model averaging technique of Bayesian model averaging (BMA) into the conformal framework. BMA has been proposed as a natural Bayesian solution to account for model uncertainty (Hoeting et al., 1999; Wasserman, 2000; Raftery et al., 1997). Some recent works proposing model aggregation strategies to combine conformal prediction sets or p-values include Gasparin & Ramdas (2024a), Gasparin & Ramdas (2024b), and Yang & Kuchibhotla (2024). Gasparin & Ramdas (2024b) proposed conformal set aggregation by majority vote strategy, and proved that a merged set suffers a loss in coverage i.e. merged conformal set will have at least  $1 - 2\alpha$  marginal coverage if the individual sets have  $1 - \alpha$  marginal coverage. Furthermore, size of such merged sets is shown to be no larger than the maximum size among the given sets. However, they do not show if the merged set will achieve optimal efficiency under certain conditions. Yang & Kuchibhotla (2024) provide efficiency first conformal prediction method but this approach suffers from a coverage loss in the aggregation step. Many existing works are available for aggregating inductive conformal predictors which require data splitting (Vovk, 2015; Linusson et al., 2017; Carlsson et al., 2014; Toccaceli & Gammerman, 2019) and are based on combining p-values which may not be well-calibrated. Linusson et al. (2020) proposed the use of out-of-bag ensemble conformal predictors. But, their theoretical validity of coverage is only approximate and evaluated solely based on empirical studies. Few methods attempt to merge conformal scores from various models under the assumption that distinct data are employed to train each model (Gauraha & Spjuth, 2021), thereby constructing conformal sets accordingly. Overall, common limitations of existing methods include all or some of the following: the requirement data splitting or access to hold-out set for calibration, which may not be feasible for small sample sizes, the lack of coverage in the aggregation step, not targeted to achieve efficient prediction sets (instead focus on efficient set aggregation which may not lead to narrower prediction sets), and no theoretical guarantees of coverage and efficiency. To overcome these pitfalls of the existing approaches, we propose CBMA framework which (1) provide theoretical guarantee of efficiency of conformal prediction sets under model uncertainty, (2) allows the use of full data for all model fits and computing conformity scores for each model as well as aggregation weights are data and prior dependent, which are computed using the same data. Thus, there is no efficiency loss by data splitting at any step, and (3) no coverage loss in the aggregation process.
+
+# 3 CBMA: CONFORMAL BAYESIAN MODEL AVERAGING
+
+In this section, we describe our CBMA approach to construct conformal predictive sets. For each model  $\mathcal{M}_k$  in the model space  $\mathcal{M} = \{\mathcal{M}_1,\dots ,\mathcal{M}_K\}$ , we consider Bayesian prediction using training data  $Z_{1:n}$  for an outcome of interest  $Y_{i}\in \mathbb{R}$  and covariates  $X_{i}\in \mathbb{R}^{d}$ . Given a model likelihood  $p_{\theta_k}(y_i|x_i)$  and prior on parameters,  $\pi_k(\theta_k)$  for  $\theta_{k}\in \Theta_{k}$ , the posterior predictive distribution for the response at a new  $X_{n + 1} = x_{n + 1}$  takes on the form
+
+$$
+p _ {\mathcal {M} _ {k}} (y | x _ {n + 1}, Z _ {1: n}) = \int p _ {\theta_ {k}} (y | x _ {n + 1}) \pi_ {k} (\theta_ {k} | Z _ {1: n}) \mathrm {d} \theta_ {k},
+$$
+
+where  $\pi_k(\theta_k|Z_{1:n})$  is the Bayesian posterior and  $\pi_k(\theta_k|Z_{1:n})\propto \pi_k(\theta_k)\prod_{i = 1}^n p_{\theta_k}(y_i|x_i)$ . We first fit  $K$  candidate models using BMA. Following Fong & Holmes (2021), full conformal Bayes prediction sets are constructed using posterior predictive densities as conformity measures:
+
+$$
+\sigma_ {i} ^ {\mathcal {M} _ {k}} = p _ {\mathcal {M} _ {k}} (Y _ {i} | X _ {i}, Z _ {1: n + 1})
+$$
+
+This is a valid conformity score,  $\sigma_{i}^{\mathcal{M}_{k}}$  is indeed invariant to the permutation of  $Z_{1:n + 1}$  due to the form of  $\pi (\theta |Z_{1:n})$ . For each model  $\mathcal{M}_k$ , we employ the Add-One-In importance sampling strategy as described by Fong & Holmes (2021) once we obtain posterior samples  $\theta_k^{1:T}\sim \pi_k(\theta_k|Z_{1:n})$  for a large  $T$  using MCMC. Specifically, for  $Y_{n + 1} = y$  and  $\theta_k^{1:T}\sim \pi_k(\theta_k|Z_{1:n})$ , we can compute:
+
+$$
+\hat {\sigma} _ {i} ^ {\mathcal {M} _ {k}} = \hat {p} _ {\mathcal {M} _ {k}} \left(Y _ {i} \mid X _ {i}, Z _ {1: n + 1}\right) = \sum_ {t = 1} ^ {T} \tilde {w} _ {k} ^ {(t)} p _ {\theta_ {k} ^ {(t)}} \left(Y _ {i} \mid X _ {i}\right), \tag {2}
+$$
+
+where weights  $\tilde{w}_k^{(t)}$  are of the form
+
+$$
+w _ {k} ^ {(t)} = p _ {\theta_ {k} ^ {(t)}} (y | x _ {n + 1}), \quad \tilde {w} _ {k} ^ {(t)} = \frac {w _ {k} ^ {(t)}}{\sum_ {t = 1} ^ {T} w _ {k} ^ {(t)}}.
+$$
+
+We can also compute posterior model probabilities  $p(\mathcal{M}_k|Z_{1:n})$  based on the observations  $Z_{1:n}$  so that we don't need to refit the models.
+
+Finally, we aggregate individual model conformity scores  $\sigma_{i}^{\mathcal{M}_{k}}$  to construct a weighted combination  $\sigma_{i}^{CBMA}$  defined as:
+
+$$
+\sigma_ {i} ^ {C B M A} = \sum_ {k = 1} ^ {K} q _ {k} \left(Z _ {1: n}, Z _ {n + 1}\right) \sigma_ {i} ^ {\mathcal {M} _ {k}}, \tag {3}
+$$
+
+where
+
+$$
+q _ {k} \left(Z _ {1: n}, Z _ {n + 1}\right) = \frac {p \left(\mathcal {M} _ {k} \mid Z _ {1 : n}\right) p _ {\mathcal {M} _ {k}} \left(y \mid x _ {n + 1} , Z _ {1 : n}\right)}{\sum_ {k = 1} ^ {K} p \left(\mathcal {M} _ {k} \mid Z _ {1 : n}\right) p _ {\mathcal {M} _ {k}} \left(y \mid x _ {n + 1} , Z _ {1 : n}\right)}. \tag {4}
+$$
+
+The aggregation weights are proportional to the posterior probabilities of the models and predictive posterior density of new test instance  $(y, x_{n+1})$  conditioned on the observed data  $Z_{1:n}$ . Note that, we can easily obtain both the terms using posterior samples for parameters based on  $Z_{1:n}$  and BMA also provides posterior model probabilities in the output. Now combined score  $\sigma_i^{CBMA}$  can be computed as
+
+$$
+\hat {\sigma} _ {i} ^ {C B M A} = \frac {\sum_ {k = 1} ^ {K} \hat {p} \left(\mathcal {M} _ {k} \mid Z _ {1 : n}\right) \left(\frac {1}{T} \sum_ {t = 1} ^ {T} w _ {k} ^ {(t)}\right) \hat {\sigma} _ {i} ^ {\mathcal {M} _ {k}}}{\sum_ {k = 1} ^ {K} \hat {p} \left(\mathcal {M} _ {k} \mid Z _ {1 : n}\right) \frac {1}{T} \sum_ {t = 1} ^ {T} w _ {k} ^ {(t)}}. \tag {5}
+$$
+
+Note that because of the data dependent weights  $q(Z_{1:n}, Z_{n+1})$  in the construction of  $\sigma_i^{CBMA}$ , it is not clear if it is a valid conformity score in order to construct conformal prediction set. In Lemma 1, we show that  $\sigma_i^{CBMA}$  is indeed a conformity measure satisfying the exchangeability criterion. Then, we construct full conformal prediction sets with coverage guarantee. We call this method as Conformal Bayesian model averaging (CBMA). We summarize our method in Algorithm 1.
+
+Algorithm 1 CBMA: Conformal Bayesian Model averaging  
+Input: Observed data is  $Z_{1:n}$ ; test point  $X_{n+1}$ ; Specify miscoverage level  $\alpha$ ; for every model  $\mathcal{M}_k \in \mathcal{M}$ , Model likelihood  $p_{\theta_k}(y|x)$  and prior  $\pi_k(\theta_k)$ .  
+Output: Return  $C_\alpha^{BMA}(X_{n+1})$ , Individual sets  $C_\alpha^{M_k}(X_{n+1})$   
+(Run the usual BMA)  
+for  $k \in 1:K$  do  
+    Obtain posterior samples  $\theta_k^{1:T} \sim \pi_k(\theta_k|Z_{1:n})$  through MCMC  
+    Compute  $\hat{p}(\mathcal{M}_k|Z_{1:n})$   
+end for  
+for  $y \in Y_{Grid}$  do  
+    Compute  $\sigma_{1:n}^{M_k}$  and  $\sigma_{n+1}^{M_k}$  using (2).  
+    Store the rank,  $r_k(y)$ , of  $\sigma_{n+1}^{M_k}$  among  $\sigma_{1:n+1}^{M_k}$ .  
+    Compute  $\sigma_{1:n}^{CBMA}$  and  $\sigma_{n+1}^{CBMA}$ , using (3) and (4).  
+    Store the rank,  $r_{BMA}(y)$ , of  $\sigma_{n+1}^{CBMA}$  among  $\sigma_{1:n}^{CBMA}$ .  
+end for  
+Return: Set  $C_\alpha^{CBMA}(X_{n+1}) = \{y \in Y_{Grid}: r_{CBMA}(y) > \alpha\}$ ; Individual Sets  $C_\alpha^{M_k}(X_{n+1}) = \{y \in Y_{Grid}: r_k(y) > \alpha\}$
+
+# 4 THEORETICAL PROPERTIES
+
+In the Lemma 1, we show that aggregated score  $\sigma_{i}^{CBMA}$  in (7) is indeed a valid conformity score. Our proof of Lemma 1 relies on an alternative representation for  $\sigma_{i}^{CBMA}$  which can be interpreted as the posterior predictive density under the hierarchical Bayesian model:
+
+$$
+Y | X, \theta_ {k}, \mathcal {M} _ {k} \sim p _ {\theta_ {k}} (y | x) \quad \theta_ {k} | \mathcal {M} _ {k} \sim \pi_ {k} (\theta_ {k}); \quad \mathcal {M} _ {k} \sim p (\mathcal {M} _ {k}). \tag {6}
+$$
+
+Lemma 1 The aggregated score  $\sigma_{i}^{CBMA}$  can be rewritten as
+
+$$
+\sigma_ {i} ^ {C B M A} = \sum_ {k = 1} ^ {K} p \left(\mathcal {M} _ {k} \mid Z _ {1: n + 1}\right) p _ {\mathcal {M} _ {k}} \left(Y _ {i} \mid X _ {i}, Z _ {1: n + 1}\right) = \sum_ {k = 1} ^ {K} p \left(\mathcal {M} _ {k} \mid Z _ {1: n + 1}\right) \sigma_ {i} ^ {\mathcal {M} _ {k}}. \tag {7}
+$$
+
+This is the posterior predictive density under the hierarchical model (6) and hence, is a valid conformity measure. Here  $p(\mathcal{M}_k|Z_{1:n + 1})$  is the posterior model probability of  $\mathcal{M}_k$  conditional on  $Z_{1:n + 1}$ .
+
+Proof: We provide the details of obtaining alternative representation in (7) for aggregated scores given in (3) in Appendix A.1. Hierarchical model (6) is fully Bayesian with two kinds of parameters: the parameters  $\theta_{k}$  of each of the model  $\mathcal{M}_k$  and the models  $\mathcal{M}_k$ 's themselves. Thus, we note that for i.i.d. observations  $Z_{1:n+1} = \{(Y_i, X_i)\}_{i=1}^{n+1}$  from the hierarchical model (6) are exchangeable. Also, the conformity measure considered is the posterior predictive density function, which is invariant to the permutation of  $Z_{1:n+1}$ . This follows by observing that the likelihood under each model is  $\prod_{i=1}^{n+1} p_{\theta_k}(Y_i | X_i)$  is invariant under the permutations of  $Z_{1:n+1}$ . Thus, this is a valid conformity measure.
+
+Once we have established  $\sigma_{i}^{CBMA}$  as a valid conformity measure, we can define the conformal prediction set as
+
+$$
+r _ {C B M A} (y) = \frac {\sum_ {i = 1} ^ {n + 1} \mathbf {1} \left(\sigma_ {i} ^ {C B M A} \leq \sigma_ {n + 1} ^ {C B M A}\right)}{n + 1}, \quad C _ {\alpha} ^ {C B M A} \left(X _ {n + 1}\right) = \left\{y \in \mathcal {Y}: r _ {C B M A} (y) > \alpha \right\}. \tag {8}
+$$
+
+Theorem 1 Assume that  $Z_{1:n+1} = \{(Y_i, X_i)\}_{i=1}^{n+1}$  are exchangeable, and the conformity measure  $\sigma_i^{CBMA}$  as in (7) is invariant to the permutation of  $Z_{1:n+1}$ . We then have
+
+$$
+\mathbf {P} \left(Y _ {n + 1} \in C _ {\alpha} ^ {C B M A} (X _ {n + 1})\right) \geq 1 - \alpha ,
+$$
+
+where,  $C_{\alpha}(X_{n+1})$  is defined in (8) and  $\mathbf{P}$  is over  $Z_{1:n+1}$ . Moreover, if the conformity measures  $\{\sigma_i^{CBMA}\}_{i=1}^n$  are almost surely distinct, then we have
+
+$$
+\mathbf {P} \left(Y _ {n + 1} \in C _ {\alpha} ^ {C B M A} \left(X _ {n + 1}\right)\right) \leq 1 - \alpha + \frac {1}{n + 1}.
+$$
+
+Our next result demonstrates the convergence of CBMA prediction sets to the conformal prediction sets based on the true model as the sample size increases, if the true model is included in the model space  $\mathcal{M}$ . The intuition behind this result is grounded in the established observation that BMA approaches the true model as the sample size  $n$  becomes large, provided the true model is within the considered model space (Le & Clarke, 2022). This convergence ensures that the CBMA prediction sets become increasingly accurate and reliable, mirroring the properties of conformal prediction sets derived from the true model. The detailed proof of this result is provided in the Appendix A.2.
+
+Theorem 2 Suppose true model  $\mathcal{M}_{\text{true}}$  is in the model space  $\mathcal{M} = \{\mathcal{M}_i\}_{i=1}^k$ . Under the hypothesis of Theorem 3 in Le & Clarke (2022), as  $n \to \infty$ , under  $\mathcal{M}_{\text{true}}$ , we have  $q_k(Z_{1:n}, Z_{n+1}) \to 1$  in probability when  $\mathcal{M}_k = \mathcal{M}_{\text{true}}$  and  $q_k(Z_{1:n}, Z_{n+1}) \to 0$  in probability when  $\mathcal{M}_k \neq \mathcal{M}_{\text{true}}$ . Thus, for any  $i$ , the conformity score  $\sigma_i^{CBMA}$  converges in probability to conformity score  $\sigma_i^{M_{\text{true}}}$  under true model. Furthermore, let  $V_{CBMA}(X_{new}, Z_{1:n})$  denote the volume of the CBMA prediction set and  $V_{true}(X_{new}, Z_{1:n})$  denote the volume of the optimal conformal prediction set under the true model  $\mathcal{M}_{\text{true}}$ . Then,  $\lim_{n \to \infty} |\mathbb{E}V_{CBMA}(X_{new}, Z_{1:n}) - \mathbb{E}V_{true}(X_{new}, Z_{1:n})| = 0$ .
+
+Remark 1 Theorem 2 implies that, as the sample size  $n$  becomes large, our CBMA prediction set will be similar to conformal Bayes prediction set based on true model. Thus, the expected volume of the CBMA prediction set converges to the expected volume of the conformal Bayes prediction set based on the true model. It is important to note that the expected volume of the conformal prediction set under the true model is optimal in terms of minimum expected volume (Hoff, 2023). Therefore, as  $n$  grows, our CBMA prediction set becomes as efficient as the optimal conformal prediction set derived from the true model. This convergence ensures that the CBMA prediction set not only aligns with the true model but also maintains efficiency, making it a powerful tool for practical applications where model uncertainty is present.
+
+Remark 2 In the case of small sample sizes, inferring the true model for the data generation process is challenging and often impossible. In such scenarios, our method can enhance the performance of predictions by leveraging the improved predictive power of BMA. Note that BMA has been shown to outperform individual models in terms of log scores (Raftery et al., 1997).
+
+Remark 3 When the true model is not in the model space, we note that  $q_{k}(Z_{1:n}, Z_{n+1}) \to 1$  in probability as  $n \to \infty$  for  $\mathcal{M}_{k^*}$  such that  $D_{KL}(p_{\mathcal{M}_{true}}||p_{\mathcal{M}_{k^*}}) = \inf_{k=1,\dots,K} D_{KL}(p_{\mathcal{M}_{true}}||p_{\mathcal{M}_k})$ , where  $D_{KL}(\cdot|\cdot)$  denotes the Kullback-Leibler (KL) divergence. This follows from general posterior concentration results (White, 1982; Berk, 1966; Ramamoorthi et al., 2015). Thus, if model space is large enough to contain a model in the KL divergence neighborhood of the true model, our CBMA approach will construct near-optimal conformal prediction sets.
+
+# 5 SIMULATION STUDY
+
+We evaluate the performance of our conformal Bayesian model averaging prediction method in two experimental settings:  $(i)$  when true model is a part of the model space, and  $(ii)$  when true model is not included in the model space. In the former case, our numerical results show that the performance of our methodology is similar to that of conformal Bayes and Bayes predictive sets based on true model. This is in accordance with the results obtained by Fong & Holmes (2021) and Hoff (2023) but our method also incorporates model uncertainty adding an extra layer of robustness to the predictive inference and elevate reliability. In the second scenario, we note that our methodology leads to construction of valid prediction sets with smaller average size as compared to conformal Bayes prediction sets and Bayesian sets for individual models. This demonstrates the advantage offered by our proposed method over conformal Bayes by Fong & Holmes (2021). The details of our simulation experiments are given in the Appendix A. All codes and results are provided in the supplementary material.
+
+# 5.1 QUADRATIC MODEL
+
+We draw independent and identically distributed (i.i.d.) samples  $x_{i} \sim \text{Unif}(0,1)$ , for  $i = 1, \ldots, n$ . The response  $y$  is then generated through following model:
+
+$$
+y _ {i} = \beta_ {0} + \beta_ {1} x _ {i} + \beta_ {2} x _ {i} ^ {2} + \epsilon_ {i};
+$$
+
+$$
+\beta_ {0} \sim \mathcal {N} (0, 0. 2 5), \beta_ {1} \sim \mathcal {N} (1, 0. 2 5), \beta_ {2} \sim \mathcal {N} (0. 5, 0. 2 5), \epsilon \sim \mathcal {N} (0, 0. 2).
+$$
+
+We consider three models: Model 1  $(\mathcal{M}_1)$  represents the true model, Model 2  $(\mathcal{M}_2:y_i = \beta_0 + \beta_1x_i + \epsilon_i)$  is a linear model derived by omitting the quadratic term from Model 1, and Model 3  $(\mathcal{M}_3:y_i = \beta_0 + \beta_2x_i^2 +\epsilon_i)$  includes only an intercept and a quadratic term. In this experiment, we set the miscoverage level  $\alpha = 0.20$  . We use  $40\%$  of the total sample size  $n$  as the test set with the remaining  $60\%$  forming the training set. The summary of the results of the different prediction sets is given in Table 1 (for sample sizes  $n = 100$  and  $n = 200$  ), provided in A.4.1.
+
+We observe that, all conformal prediction sets (which are intervals here) achieve the target coverage in all cases. Also, as observed by Fong & Holmes (2021), conformal Bayes prediction sets have shorter lengths as compared to their corresponding Bayesian prediction sets, for both true and misspecified models. Our proposed CBMA method provides prediction sets which have similar average lengths as that of conformal Bayes based on true model. Note that CBMA also incorporates model uncertainty as opposed to conformal Bayes, hence, leads to reliable predictive inference and provides more precise uncertainty quantification. Furthermore, as shown in Figure 1, CBMA achieves optimal mean length, equal to CB under model  $\mathcal{M}_1$ , even at smaller sample sizes  $n = 100, 200$ , demonstrating practical utility of our CBMA approach.
+
+![](images/d067dd57f712caf91b533f3b037302e97efee9d204102bbde96adff32746b1d9.jpg)  
+Figure 1: Quadratic model: Comparison of CBMA and CBM1 for different sample sizes.
+
+# 5.2 APPROXIMATION USING HERMITE POLYNOMIALS
+
+In this section, we consider a data generating mechanisms similar to Lu & Su (2015). Let  $X_{i} \sim \mathrm{Weibull}(1,1)$  are i.i.d. The data is generated from the following model
+
+$$
+Y _ {i} = \theta \frac {\exp \left\{X _ {i} \right\}}{1 + \exp \left\{X _ {i} \right\}} + \epsilon_ {i}.
+$$
+
+Here, we consider  $\epsilon_{i} = (0.01 + X_{i})\zeta_{i}$  and  $\zeta_i\sim \mathcal{N}(0,1)$ . This is a scenario of heteroskedasticity. In this study, we employ Hermite polynomials to approximate the unknown function  $\theta \frac{\exp\{X_i\}}{1 + \exp\{X_i\}}$ . We build a model space consisting of  $K$  models with the covariates  $X_{ij},j = 1,2,\ldots$
+
+$$
+X _ {i j} = (X _ {i} - \bar {X}) ^ {j - 1} \exp \left\{- \frac {(X _ {i} - \bar {X}) ^ {2}}{2 s _ {X} ^ {2}} \right\}, j = 1, 2, \ldots ,
+$$
+
+where  $\bar{X}$  and  $s_X^2$  represent the sample mean and standard deviation of the set  $X_{i}$ , respectively. In our simulation experiment, we set  $1 - \alpha = 0.8$ . We use  $40\%$  of the total sample size as the test set, with the remaining  $60\%$  forming the training set ( $n_{train} = 0.60 \times n$ ). The number of models considered is  $K = 11$ . We consider nested model sequence, i.e. for  $k = 1, \ldots, K$ , model  $\mathcal{M}_k$  includes covariates  $X_{i1}, \ldots, X_{ik}$ . We summarize our results based on  $E = 50$  experiments for samples size  $n = 100$  in Figure 2 and Figure 5 (in A.4.2). The results for other studies for  $n = 100, \theta = 10$  and  $n = 50, \theta = 1$  are reported in Table 5 and Table 6 in Appendix A.4.2. We note that our CBMA method provides sets (here, prediction sets are in the form of intervals) for which average length is smaller than all other prediction sets. Also, note that, most of Bayes prediction sets overcover and hence have larger sizes. Conformal Bayes as well as our CBMA prediction sets have valid target coverage. But, on average, CBMA prediction sets have shorter length than CB prediction sets. These results demonstrate the empirical efficiency of CBMA prediction sets over CB prediction sets.
+
+![](images/d53ac5ad5a118c507414a20dca4e11a01c0289d8d104529efe6eaee39027a961.jpg)  
+Figure 2: Approximation using Hermite polynomials: Comparison of mean length for prediction sets obtained with different methods: CBMA (proposed method), BMA, individual Bayes prediction sets (in red), and individual conformal Bayes (CB) prediction sets (in blue). Here, we report results based on  $E = 50$  number of experiments. We have set the target coverage as  $(1 - \alpha) = 0.80$ , sample size  $n = 100$  and  $\theta = 1$ .
+
+# 5.3 REAL DATA EXAMPLE: CALIFORNIA HOUSING DATA
+
+We demonstrate our method using the California Housing dataset. The dataset is available in sklearn, and consists of  $n = 20640$  subjects, where the continuous response variable denotes the median house value for California districts, expressed in hundreds of thousands of dollars (100,000), and  $d = 8$  covariates, including median income and median house age in block group, average number of rooms per household, block group population. This dataset was derived from the 1990 U.S. census. We first draw random samples of sizes  $n = 50, 100$  or 150 from this dataset for our study. We then standardize all covariates and the response to have mean 0 and standard deviation 1. We consider four different models for these experiments by considering only two covariates in the model. Thus, the models we consider are
+
+$$
+\mathcal {M} _ {k}: y = \beta_ {0} + \beta_ {1} X _ {k} + \beta_ {1} X _ {k + 1} + \epsilon , k = 1, 2, 3, 4.
+$$
+
+The priors we consider on the parameters are  $\beta_{i} \sim N(0,5)$ ,  $\epsilon \sim N_{+}(0,1)$ , half-normal distribution. Among all four models,  $\mathcal{M}_1$  seems more appropriate as it contains important covariates median income and median house age in block group, as opposed to other covariates. We fit these four models to obtain MCMC samples from the posterior distribution and compute marginal likelihoods for each model. To check coverage, we repeatedly divide our dataset into a training and test dataset for 50 repeats, with  $40\%$  of the dataset in the test split. We adopt the setup for simulations similar to Fong & Holmes (2021) in their section 4.1, so that we can compare the performance of our
+
+CBMA method with individual conformal Bayes. We set the miscoverage level at  $\alpha = 0.2$ . To illustrate the time complexity, we note that, for  $n = 150$ , the average times for constructing conformal Bayes prediction sets were  $0.806(0.004)$ ,  $0.793(0.002)$ ,  $0.786(0.002)$  and  $0.794(0.003)$  for four models, Bayes credible sets took average times of  $0.447(0.003)$ ,  $0.444(0.002)$ ,  $0.441(0.002)$  and  $0.449(0.004)$ . Further, to construct Bayes credible sets using BMA, average additional time was  $0.091(0.005)$  and our CBMA method, additional average time was  $0.128(0.007)$ . We report average lengths and coverages of various prediction sets (prediction intervals in this case) in Figure 3 and 6. We note that, as suggested earlier, model  $\mathcal{M}_1$  fits data well as compared to other three models, we expect to have conformal Bayes prediction sets based on this model will be the shortest among all four models. As indicated by our Theorem 2 and Remark 3, our CBMA intervals achieve the shortest lengths as the sample size becomes sufficiently large enough. In our results, we can see CBMA achieves shortest lengths even at finite sample size of  $n = 150$ . All conformal Bayes approaches as well as CBMA achieves, on average, the target coverage of  $1 - \alpha = 0.80$ . On the other hand, Bayes models over cover in almost all cases.
+
+![](images/1e6513fcb486a356e12a0835020ebbc671632cbf36a686f955b50144b8ec0712.jpg)  
+Figure 3: California Housing data: comparison of mean lengths of intervals using Bayes prediction, conformal Bayes (CB) for all four models, CBMA and BMA for different sample sizes.
+
+![](images/0dde38f208ad5b0e41e8b1a4293e9730f626b0e21d728fcd83c39ca2accfa315.jpg)
+
+![](images/92266f607f61c1c1c123664ed950bd91a0fee4e6e2b5cd48e8d6cf00998e0ba3.jpg)
+
+# 6 CONCLUSIONS
+
+In this work, we identified a challenging issue with the traditional conformal prediction framework: selecting a model in advance in order to construct conformal prediction sets may result in suboptimal prediction sets. In addressing this issue of model uncertainty within the conformal prediction framework, we offer a natural Bayesian solution by integrating Bayesian model averaging in conformal prediction framework. Our proposed CBMA framework seamlessly amalgamates conformity scores from individual Bayesian models to construct a unified conformal prediction set, thus introducing the incorporation of model averaging procedures—a notable advancement currently overlooked from current literature on conformal prediction theory. Additionally, we present an efficient algorithm for constructing CBMA prediction sets, which essentially represent full conformal predictions. Theoretically, we establish the optimal efficiency of our CBMA prediction sets as sample sizes increase, underpinning the efficacy of our approach. Overall, our method holds particular value in real-world scenarios where model uncertainty is a prevalent concern, providing a robust solution within the conformal prediction framework.
+
+However, our approach does have limitations due to the inherent characteristics of MCMC algorithms. The error in these sampling methods can affect prediction sets constructed using CBMA. As noted in Fong & Holmes (2021), we can also expect some robustness to conformal sets as they are based on ranks and not actual values. Furthermore, since we are averaging over multiple models, we expect that the errors in one of the models' training can be masked on averaging.
+
+# ACKNOWLEDGMENTS
+
+Bei Jiang and Linglong Kong were partially supported by grants from the Canada CIFAR AI Chairs program, the Alberta Machine Intelligence Institute (AMII), and Natural Sciences and Engineering Council of Canada (NSERC), and Linglong Kong was also partially supported by grants from the
+
+Canada Research Chair program from NSERC. The authors would also like to thank the anonymous reviewers for their constructive comments that improved the quality of this article.
+
+# REFERENCES
+
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+
+# A APPENDIX
+
+# A.1 PROOF OF LEMMA 1:
+
+(Representation for the conformity measure  $\sigma_{i}^{CBMA}$  in (7)) Under hierarchical Bayesian model in (6), the posterior predictive density for new text point  $(y, x_{n+1})$  can be obtained as follows:
+
+$$
+\begin{array}{l} p (Y _ {i} | X _ {i}, Z _ {1: n + 1}) = \sum_ {k = 1} ^ {K} p (\mathcal {M} _ {k} | Z _ {1: n + 1}) p _ {\mathcal {M} _ {k}} (Y _ {i} | X _ {i}, Z _ {1: n + 1}) \\ = \sum_ {k = 1} ^ {K} \frac {m (Z _ {1 : n + 1} | \mathcal {M} _ {k}) p (\mathcal {M} _ {k})}{\sum_ {k ^ {\prime} = 1} ^ {K} m (Z _ {1 : n + 1} | \mathcal {M} _ {k ^ {\prime}}) p (\mathcal {M} _ {k ^ {\prime}})} p _ {\mathcal {M} _ {k}} (Y _ {i} | X _ {i}, Z _ {1: n + 1}) \\ = \frac {\sum_ {k = 1} ^ {K} m \left(Z _ {1 : n + 1} \mid \mathcal {M} _ {k}\right) p \left(\mathcal {M} _ {k}\right) p _ {\mathcal {M} _ {k}} \left(Y _ {i} \mid X _ {i} , Z _ {1 : n + 1}\right)}{\sum_ {k = 1} ^ {K} m \left(Z _ {1 : n + 1} \mid \mathcal {M} _ {k}\right) p \left(\mathcal {M} _ {k}\right)} \\ = \frac {\sum_ {k = 1} ^ {K} p (\mathcal {M} _ {k}) \frac {m (Z _ {1 : n} | \mathcal {M} _ {k})}{m (Z _ {1 : n} | \mathcal {M} _ {k})} m (Z _ {1 : n + 1} | \mathcal {M} _ {k}) p _ {\mathcal {M} _ {k}} (Y _ {i} | X _ {i} , Z _ {1 : n + 1})}{\sum_ {k = 1} ^ {K} p (\mathcal {M} _ {k}) \frac {m (Z _ {1 : n} | \mathcal {M} _ {k})}{m (Z _ {1 : n} | \mathcal {M} _ {k})} m (Z _ {1 : n + 1} | \mathcal {M} _ {k})} \\ = \frac {\sum_ {k = 1} ^ {K} p (\mathcal {M} _ {k}) p _ {\mathcal {M} _ {k}} (y | x _ {n + 1} , Z _ {1 : n}) m (Z _ {1 : n} | \mathcal {M} _ {k}) p _ {\mathcal {M} _ {k}} (Y _ {i} | X _ {i} , Z _ {1 : n + 1})}{\sum_ {k = 1} ^ {K} p (\mathcal {M} _ {k}) p _ {\mathcal {M} _ {k}} (y | x _ {n + 1} , Z _ {1 : n}) m (Z _ {1 : n} | \mathcal {M} _ {k})} \\ = \frac {\sum_ {k = 1} ^ {K} p (\mathcal {M} _ {k} | Z _ {1 : n}) p _ {\mathcal {M} _ {k}} (y | x _ {n + 1} , Z _ {1 : n}) p _ {\mathcal {M} _ {k}} (Y _ {i} | X _ {i} , Z _ {1 : n + 1})}{\sum_ {k = 1} ^ {K} p (\mathcal {M} _ {k} | Z _ {1 : n}) p _ {\mathcal {M} _ {k}} (y | x _ {n + 1} , Z _ {1 : n})}. \\ \end{array}
+$$
+
+# A.2 PROOF OF THEOREM 2:
+
+Theorem 3 in Le & Clarke (2022) establishes conditions under which the posterior model probabilities converge to their true values. Below, we summarize key assumptions necessary for proving the theorem, which are also detailed in Le & Clarke (2022).
+
+1. For each  $k = 1,\dots ,K$  , let  $\theta_{k}\in \Theta_{k}$  , an open subset of  $\mathbb{R}^{d_k}$  
+2. The densities  $p_{\theta_k}(y_i|x_i)$  are measurable in  $y$  for all  $\theta_k$ ,  $x$  and continuous in  $\theta_k$ . All second partial derivatives of  $p_{\theta_k}(y_i|x_i)$  w.r.t.  $\theta_k$  exist and are continuous for all  $x, y$ , and maybe passed under the integral sign in  $\int p_{\theta_k}(y_i|x_i) dy$ .  
+3. There exists a function  $K(\cdot)$  such that  $E_{\theta_k}K(Y) < \infty$  and each element of the matrix  $\left(\frac{\partial^2}{\partial\theta_{k,l}\partial\theta_{k,j}}\log p_{\theta_k}(y_i|x_i)\right)_{l,j = 1,\ldots ,d_j}$  is absolute value by  $K(y)$  uniformly for  $\theta_{k}$  in an open neighbourhood of  $\theta_{k,0}$ .  
+4. Assume that, for any  $x_{i}$ ,
+
+$$
+I _ {i} (\theta_ {k, 0} | x _ {i}) = \left(- E _ {\theta_ {k, 0}} \frac {\partial^ {2}}{\partial \theta_ {k , l} \partial \theta_ {k , j}} \log p _ {\theta_ {k}} (y _ {i} | x _ {i})\right) _ {l, j = 1, \dots , d _ {k}}
+$$
+
+is continuous on an open set around the true value  $\theta_{k,0}$  and is positive definite at  $\theta_{k,0}$ .
+
+5. For any  $\theta_{k,0} \in \Theta_k$  and any  $x_i$ ,  $p_{\theta_k}(y_i|x_i) = p_{\theta_{k,0}}(y_i|x_i)$  a.e. in  $y$  implies  $\theta_k = \theta_{k,0}$ .
+
+6. The average  $(1/n)\sum I_i(\theta_{k,0}|x_i)$  is invertible for any  $n$  and the  $d_k$ -vector  $\Psi(\theta_{k,0}|x,y) = \left(\nabla_{\theta_k}\log p_{\theta_{k,0}}(y_i|x_i)\right)^T$  satisfies
+
+$$
+\sum E \| \frac {1}{n} \left(\sum I _ {i} \left(\theta_ {k, 0} | x _ {i}\right)\right) ^ {- 1 / 2} \Psi \left(\theta_ {k, 0} | x, Y\right) \| ^ {3} = o \left(\frac {1}{n ^ {3 / 2}}\right).
+$$
+
+7. Priors  $\pi_k(\theta_k)$  are continuous for  $\theta_{k}\in \Theta_{k}$
+
+Under these conditions, the posterior model probabilities satisfy
+
+$$
+p \left(\mathcal {M} _ {k} \mid Z _ {1: n + 1}\right)\rightarrow 1 \text {o r} 0 \text {i n p r o b a b i l i t y a s} n \rightarrow \infty ,
+$$
+
+when  $\mathcal{M}_k$  corresponds to the true model  $\mathcal{M}_{\mathrm{true}}$  or an incorrect model, respectively. This result directly implies
+
+$$
+\sigma_ {i} ^ {C B M A} \rightarrow \sigma_ {i} ^ {\mathcal {M} _ {\text {t r u e}}}, \quad \text {i n p r o b a b i l i t y , a s} n \rightarrow \infty .
+$$
+
+Consequently, the conformal Bayesian model averaging (CBMA) prediction set indicator satisfies
+
+$$
+\mathbf {1} (y \in C _ {\alpha} ^ {C B M A} (X _ {n + 1})) \rightarrow \mathbf {1} (y \in C _ {\alpha} ^ {\mathcal {M} _ {\text {t r u e}}} (X _ {n + 1})) \text {i n p r o b a b i l i t y a s} n \rightarrow \infty .
+$$
+
+Thus, the CBMA algorithm (Algorithm 1) produces prediction sets that asymptotically converge to the conformal Bayes prediction set  $C_{\alpha}^{\mathcal{M}\mathrm{true}}(X_{n + 1})$  based on the true model  $\mathcal{M}_{\mathrm{true}}$ .
+
+# A.3 EXPERIMENTS
+
+# A.4 COMPLEXITY OF CONSTRUCTING CBMA INTERVALS
+
+Given the posterior samples and posterior model probabilities, obtaining individual models' conformity scores has a complexity of  $\mathcal{O}(n_{\mathrm{grid}} \times \mathrm{posterior~samples} \times n_{\mathrm{train}})$  (Fong & Holmes, 2021). The aggregation step (requires dot product of  $K$  weights and  $K$  conformity scores) has a linear overhead of  $\mathcal{O}(K)$ , where  $K$  is the number of models.
+
+# A.4.1 QUADRATIC MODEL
+
+We now examine the experimental results for the quadratic model, as discussed in Section 5. Figure 4 presents the posterior model probabilities (PMP) for all three models under consideration across  $E = 50$  repetitions with a sample size of  $n = 100$ . The results indicate that Model 1 frequently attains the highest posterior model probability. However, Model 2 also exhibits a substantial number of instances where it achieves a higher PMP. This variability suggests that pre-selecting either Model 1 or Model 2 for constructing conformal prediction sets may lead to suboptimal uncertainty quantification. In contrast, our CBMA method leverages predictions from multiple models, ensuring a more comprehensive integration of model uncertainty. This leads to more efficient and reliable uncertainty quantification. Additionally, Table 2 presents descriptive statistics from these repeated experiments. Notably, CBMA prediction set lengths exhibit a symmetric distribution, whereas CB.M1 prediction sets display a negatively skewed length distribution. This suggests that CBMA results in smaller prediction sets in many instances compared to CB.M1, enhancing efficiency without compromising coverage.
+
+# EXECUTION TIMES:
+
+Given the posterior samples obtained from MCMC, we evaluate the computational efficiency of constructing Conformal Bayes (CB) prediction sets and Bayes prediction sets. Table 3 reports the mean execution times (with standard errors in parentheses) for constructing these intervals across different models and sample sizes. Additionally, Table 4 presents the execution times for CBMA and BMA methods. Notably, CBMA incurs higher computational costs compared to BMA, reflecting the additional complexity introduced by model averaging in the conformal framework. However, these times remain negligible relative to the broader MCMC sampling process, making CBMA a computationally feasible approach for uncertainty quantification.
+
+![](images/680dd2ed209511e47d6be0caad8a2c8d43caf5c9471ba24514cd96f2f679cd2a.jpg)  
+Figure 4: Posterior model probabilities (PMP) for all three models considered in our experiment with quadratic model with sample size  $n = 100$ .
+
+Table 1: Quadratic model: Comparison of coverage and length for prediction sets obtained with different methods: CBMA (proposed method), BMA, individual Bayes prediction sets BayesM1, BayesM2, BayesM3, individual conformal Bayes prediction sets CBM1, CBM2, CBM3. Here, we report mean and standard error (SE) based on  $E = 50$  number of experiments. We have set the target coverage as  $(1 - \alpha) = 0.80$ .  
+
+<table><tr><td>Method</td><td>CBMA</td><td>BMA</td><td>BayesM1</td><td>BayesM2</td><td>BayesM3</td><td>CBM1</td><td>CBM2</td><td>CBM3</td></tr><tr><td colspan="9">Coverage</td></tr><tr><td>n = 100</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mean</td><td>0.800</td><td>0.811</td><td>0.813</td><td>0.803</td><td>0.804</td><td>0.801</td><td>0.799</td><td>0.804</td></tr><tr><td>SE</td><td>0.010</td><td>0.010</td><td>0.010</td><td>0.010</td><td>0.009</td><td>0.010</td><td>0.011</td><td>0.010</td></tr><tr><td>n = 200</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mean</td><td>0.807</td><td>0.811</td><td>0.813</td><td>0.813</td><td>0.805</td><td>0.812</td><td>0.815</td><td>0.805</td></tr><tr><td>SE</td><td>0.007</td><td>0.007</td><td>0.007</td><td>0.007</td><td>0.007</td><td>0.007</td><td>0.007</td><td>0.008</td></tr><tr><td colspan="9">Length</td></tr><tr><td>n = 100</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mean</td><td>0.535</td><td>0.537</td><td>0.537</td><td>0.545</td><td>0.579</td><td>0.534</td><td>0.547</td><td>0.577</td></tr><tr><td>SE</td><td>0.008</td><td>0.007</td><td>0.007</td><td>0.007</td><td>0.009</td><td>0.008</td><td>0.009</td><td>0.010</td></tr><tr><td>n = 200</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mean</td><td>0.522</td><td>0.522</td><td>0.522</td><td>0.532</td><td>0.558</td><td>0.524</td><td>0.536</td><td>0.559</td></tr><tr><td>SE</td><td>0.006</td><td>0.004</td><td>0.004</td><td>0.005</td><td>0.006</td><td>0.006</td><td>0.007</td><td>0.007</td></tr></table>
+
+# A.4.2 APPROXIMATION USING HERMITE POLYNOMIALS
+
+The results of studies for  $n = 100$ , 50,  $\theta = 10$  and  $n = 50$ ,  $\theta = 1$  are reported in Table 5 and Table 6. We note that CBMA sets have shorter lengths on average in all these experiments.
+
+Table 2: Descriptive statistics for average lengths of prediction sets obtained with different methods: CBMA, BMA, individual Bayes prediction sets Bayes M1, Bayes M2, Bayes M3, individual conformal Bayes prediction sets CB M1, CB M2, CB M3. Here, we report Median, Mean, Standard error (SE), Lower and upper quartiles  $(Q_{1}$  and  $Q_{3})$ , Inter-Quantile range (IQR) based on  $T = 50$  number of experiments. We have set  $\alpha = 0.1$ .  
+
+<table><tr><td>Prediction</td><td>CBMA</td><td>BMA</td><td>Bayes M1</td><td>Bayes M2</td><td>Bayes M3</td><td>CB M1</td><td>CB M2</td><td>CB M3</td></tr><tr><td>n = 100</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Median</td><td>0.684</td><td>0.697</td><td>0.695</td><td>0.715</td><td>0.739</td><td>0.702</td><td>0.688</td><td>0.736</td></tr><tr><td>Q1</td><td>0.64</td><td>0.651</td><td>0.652</td><td>0.666</td><td>0.706</td><td>0.643</td><td>0.64</td><td>0.666</td></tr><tr><td>Q3</td><td>0.736</td><td>0.738</td><td>0.74</td><td>0.756</td><td>0.788</td><td>0.735</td><td>0.755</td><td>0.782</td></tr><tr><td>IQR</td><td>0.096</td><td>0.087</td><td>0.088</td><td>0.09</td><td>0.082</td><td>0.092</td><td>0.115</td><td>0.116</td></tr><tr><td>Mean</td><td>0.686</td><td>0.697</td><td>0.696</td><td>0.709</td><td>0.745</td><td>0.687</td><td>0.696</td><td>0.732</td></tr><tr><td>SE</td><td>0.01</td><td>0.008</td><td>0.008</td><td>0.009</td><td>0.008</td><td>0.009</td><td>0.01</td><td>0.011</td></tr><tr><td>n = 200</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Median</td><td>0.676</td><td>0.676</td><td>0.675</td><td>0.69</td><td>0.706</td><td>0.671</td><td>0.681</td><td>0.709</td></tr><tr><td>Q1</td><td>0.643</td><td>0.644</td><td>0.647</td><td>0.664</td><td>0.683</td><td>0.64</td><td>0.646</td><td>0.681</td></tr><tr><td>Q3</td><td>0.711</td><td>0.699</td><td>0.699</td><td>0.711</td><td>0.745</td><td>0.711</td><td>0.71</td><td>0.74</td></tr><tr><td>IQR</td><td>0.068</td><td>0.055</td><td>0.052</td><td>0.047</td><td>0.062</td><td>0.071</td><td>0.064</td><td>0.059</td></tr><tr><td>Mean</td><td>0.673</td><td>0.676</td><td>0.675</td><td>0.691</td><td>0.714</td><td>0.672</td><td>0.682</td><td>0.708</td></tr><tr><td>SE</td><td>0.007</td><td>0.006</td><td>0.006</td><td>0.006</td><td>0.007</td><td>0.007</td><td>0.006</td><td>0.008</td></tr></table>
+
+<table><tr><td>Model</td><td>n = 100</td><td>n = 200</td></tr><tr><td>CBM1</td><td>11.198 (0.071)</td><td>25.470 (0.067)</td></tr><tr><td>CBM2</td><td>11.631 (0.075)</td><td>28.040 (0.222)</td></tr><tr><td>CBM3</td><td>11.637 (0.140)</td><td>25.950 (0.222)</td></tr><tr><td>BayesM1</td><td>4.425 (0.038)</td><td>8.380 (0.040)</td></tr><tr><td>BayesM2</td><td>4.641 (0.037)</td><td>9.496 (0.113)</td></tr><tr><td>BayesM3</td><td>4.661 (0.061)</td><td>8.956 (0.110)</td></tr></table>
+
+Table 3: Mean times (SE) for CB and Bayes prediction intervals for models  $(\mathcal{M}_1,\mathcal{M}_1,\mathcal{M}_1)$ .  
+
+<table><tr><td>Model</td><td>n = 100</td><td>n = 200</td></tr><tr><td>CBMA</td><td>0.02397 (0.0014346)</td><td>0.09902 (0.0039647)</td></tr><tr><td>BMA</td><td>0.00077 (0.0000435)</td><td>0.00123 (0.0000492)</td></tr></table>
+
+Table 4: Additional mean times (SE) for CBMA and BMA prediction intervals.
+
+# A.5 REAL DATA EXAMPLE: CALIFORNIA HOUSING DATA
+
+Figure 6 presents the mean coverage probabilities for different methods, with a confidence level of  $1 - \alpha = 0.8$ . The results indicate that Bayes prediction intervals overcover, providing overly conservative intervals. In contrast, Conformal Bayes (CB) and CBMA consistently achieve the target coverage, ensuring both reliability and efficiency. Additionally, CBMA, by integrating model uncertainty, results in more adaptive and stable prediction intervals compared to individual CB models.
+
+Furthermore, we compare our CBMA set aggregation step with majority vote conformal set aggregation method proposed by Gasparin & Ramdas (2024b). While this method is designed to effectively aggregate sets while preserving the coverage guarantee, it does not necessarily aim to minimize the size of the merged set. In contrast, our CBMA approach offers the additional advantage of allowing the weights in the aggregation step to be learned directly from the same data, addressing a key limitation of existing model aggregation methods. We conducted a direct evaluation by comparing the lengths of CBMA-aggregated intervals to those obtained from the majority vote strategy using individual conformal Bayes prediction sets. For this comparison, we applied Corollary 4.1 from Gasparin & Ramdas (2024b), accounting for the dependency among our individual sets. In our example, the mean ratios of the lengths of intervals from the majority vote procedure relative to
+
+![](images/789e93244440f9e9971d862f958fcd1ec36b90cc86c12e6e673e8aae5bf33f9e.jpg)  
+Figure 5: Approximation using Hermite polynomials: Comparison of mean coverage for prediction sets obtained with different methods: CBMA (proposed method), BMA, individual Bayes prediction sets (in red), and individual conformal Bayes (CB) prediction sets (in blue). Here, we report results based on  $E = 50$  number of experiments. We have set the target coverage as  $(1 - \alpha) = 0.80$ , sample size  $n = 100$  and  $\theta = 1$ .
+
+![](images/e43f32c9ec22226e8856ad03aaef41e6649813ed86514472b048444618df0349.jpg)  
+Figure 6: California Housing data: comparison of mean coverages of intervals using Bayes prediction, conformal Bayes (CB) for all four models, CBMA and BMA for different sample sizes. Here, we set  $1 - \alpha = 0.8$ .
+
+![](images/877311088f025c8d60c15728e860f9b5036cf10da01d22df0473ce820ae34a6d.jpg)
+
+![](images/02eaa633d862fec5b54dd7645dade058a28a80d77cf0d472ef509e12dfe1df36.jpg)
+
+those from CBMA were observed to be 1.0765 (0.0253), 1.1196 (0.0151), and 1.1458 (0.0133) for sample sizes  $n = 50$ ,  $n = 100$ , and  $n = 150$ , respectively. These results show that our CBMA approach gives shorter intervals as compared to majority vote aggregation strategy.
+
+Table 5: Approximation using Hermite polynomials: Comparison of coverage and length for prediction sets obtained with different methods: CBMA (proposed method), BMA, individual Bayes prediction sets (BayesM1-BayesM11), and individual conformal Bayes prediction sets (CBM1-CBM11). Here, we report mean and standard error (SE) based on  $E = 50$  number of experiments. We have set the target coverage as  $(1 - \alpha) = 0.80$  and  $\theta = 1$ .  
+
+<table><tr><td rowspan="3">Method</td><td colspan="4">Length</td><td colspan="4">Coverage</td></tr><tr><td colspan="2">n=100</td><td colspan="2">n=50</td><td colspan="2">n=100</td><td colspan="2">n=50</td></tr><tr><td>Mean</td><td>SE</td><td>Mean</td><td>SE</td><td>Mean</td><td>SE</td><td>Mean</td><td>SE</td></tr><tr><td>BayesM1</td><td>3.485</td><td>0.118</td><td>3.574</td><td>0.144</td><td>0.858</td><td>0.011</td><td>0.862</td><td>0.014</td></tr><tr><td>BayesM2</td><td>3.492</td><td>0.121</td><td>3.574</td><td>0.146</td><td>0.858</td><td>0.011</td><td>0.854</td><td>0.013</td></tr><tr><td>BayesM3</td><td>3.410</td><td>0.118</td><td>3.537</td><td>0.148</td><td>0.858</td><td>0.011</td><td>0.856</td><td>0.013</td></tr><tr><td>BayesM4</td><td>3.384</td><td>0.118</td><td>3.461</td><td>0.142</td><td>0.856</td><td>0.01</td><td>0.853</td><td>0.013</td></tr><tr><td>BayesM5</td><td>3.348</td><td>0.116</td><td>3.389</td><td>0.141</td><td>0.853</td><td>0.01</td><td>0.846</td><td>0.013</td></tr><tr><td>BayesM6</td><td>3.333</td><td>0.116</td><td>3.344</td><td>0.139</td><td>0.848</td><td>0.01</td><td>0.845</td><td>0.013</td></tr><tr><td>BayesM7</td><td>3.297</td><td>0.115</td><td>3.290</td><td>0.137</td><td>0.844</td><td>0.01</td><td>0.838</td><td>0.014</td></tr><tr><td>BayesM8</td><td>3.208</td><td>0.108</td><td>3.240</td><td>0.143</td><td>0.837</td><td>0.011</td><td>0.827</td><td>0.015</td></tr><tr><td>BayesM9</td><td>3.138</td><td>0.103</td><td>3.182</td><td>0.141</td><td>0.837</td><td>0.011</td><td>0.828</td><td>0.016</td></tr><tr><td>BayesM10</td><td>3.087</td><td>0.099</td><td>3.131</td><td>0.141</td><td>0.830</td><td>0.010</td><td>0.829</td><td>0.016</td></tr><tr><td>BayesM11</td><td>3.039</td><td>0.094</td><td>3.067</td><td>0.141</td><td>0.830</td><td>0.010</td><td>0.829</td><td>0.015</td></tr><tr><td>CBM1</td><td>2.689</td><td>0.097</td><td>2.864</td><td>0.128</td><td>0.801</td><td>0.011</td><td>0.819</td><td>0.017</td></tr><tr><td>CBM2</td><td>2.693</td><td>0.098</td><td>2.857</td><td>0.131</td><td>0.806</td><td>0.012</td><td>0.811</td><td>0.016</td></tr><tr><td>CBM3</td><td>2.689</td><td>0.101</td><td>2.883</td><td>0.132</td><td>0.799</td><td>0.012</td><td>0.808</td><td>0.016</td></tr><tr><td>CBM4</td><td>2.667</td><td>0.094</td><td>2.877</td><td>0.137</td><td>0.798</td><td>0.012</td><td>0.819</td><td>0.015</td></tr><tr><td>CBM5</td><td>2.679</td><td>0.095</td><td>2.884</td><td>0.136</td><td>0.795</td><td>0.012</td><td>0.816</td><td>0.015</td></tr><tr><td>CBM6</td><td>2.705</td><td>0.093</td><td>2.867</td><td>0.120</td><td>0.797</td><td>0.012</td><td>0.818</td><td>0.015</td></tr><tr><td>CBM7</td><td>2.703</td><td>0.095</td><td>2.873</td><td>0.120</td><td>0.796</td><td>0.013</td><td>0.818</td><td>0.015</td></tr><tr><td>CBM8</td><td>2.691</td><td>0.094</td><td>2.823</td><td>0.121</td><td>0.794</td><td>0.013</td><td>0.818</td><td>0.016</td></tr><tr><td>CBM9</td><td>2.694</td><td>0.093</td><td>2.763</td><td>0.115</td><td>0.798</td><td>0.012</td><td>0.821</td><td>0.016</td></tr><tr><td>CBM10</td><td>2.704</td><td>0.094</td><td>2.740</td><td>0.112</td><td>0.799</td><td>0.012</td><td>0.822</td><td>0.015</td></tr><tr><td>CBM11</td><td>2.692</td><td>0.092</td><td>2.753</td><td>0.114</td><td>0.800</td><td>0.012</td><td>0.818</td><td>0.014</td></tr><tr><td>BMA</td><td>3.122</td><td>0.103</td><td>3.130</td><td>0.134</td><td>0.831</td><td>0.011</td><td>0.829</td><td>0.014</td></tr><tr><td>CBMA</td><td>2.634</td><td>0.091</td><td>2.683</td><td>0.118</td><td>0.792</td><td>0.013</td><td>0.808</td><td>0.015</td></tr></table>
+
+Table 6: Approximation using Hermite polynomials: Comparison of mean Coverage and Length for prediction sets obtained with different methods: CBMA (proposed method), BMA, individual Bayes prediction sets (BayesM1 - BayesM11), individual conformal Bayes prediction sets CBM1, ..., CBM11. We have set the target coverage as  $(1 - \alpha) = 0.8$ . Here,  $\theta = 10, K = 11, n = 100$ .  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Length</td><td colspan="2">Coverage</td></tr><tr><td>Mean</td><td>SE</td><td>Mean</td><td>SE</td></tr><tr><td>B1</td><td>4.958</td><td>0.101</td><td>0.861</td><td>0.01</td></tr><tr><td>B2</td><td>3.780</td><td>0.121</td><td>0.862</td><td>0.011</td></tr><tr><td>B3</td><td>3.634</td><td>0.121</td><td>0.856</td><td>0.011</td></tr><tr><td>B4</td><td>3.499</td><td>0.122</td><td>0.855</td><td>0.01</td></tr><tr><td>B5</td><td>3.465</td><td>0.121</td><td>0.852</td><td>0.01</td></tr><tr><td>B6</td><td>3.432</td><td>0.121</td><td>0.847</td><td>0.01</td></tr><tr><td>B7</td><td>3.406</td><td>0.120</td><td>0.843</td><td>0.01</td></tr><tr><td>B8</td><td>3.282</td><td>0.109</td><td>0.835</td><td>0.011</td></tr><tr><td>B9</td><td>3.189</td><td>0.103</td><td>0.835</td><td>0.011</td></tr><tr><td>B10</td><td>3.141</td><td>0.099</td><td>0.834</td><td>0.011</td></tr><tr><td>B11</td><td>3.094</td><td>0.096</td><td>0.831</td><td>0.011</td></tr><tr><td>CB1</td><td>3.989</td><td>0.083</td><td>0.812</td><td>0.011</td></tr><tr><td>CB2</td><td>2.922</td><td>0.091</td><td>0.805</td><td>0.010</td></tr><tr><td>CB3</td><td>2.726</td><td>0.092</td><td>0.798</td><td>0.011</td></tr><tr><td>CB4</td><td>2.708</td><td>0.094</td><td>0.803</td><td>0.011</td></tr><tr><td>CB5</td><td>2.707</td><td>0.091</td><td>0.799</td><td>0.011</td></tr><tr><td>CB6</td><td>2.703</td><td>0.088</td><td>0.800</td><td>0.011</td></tr><tr><td>CB7</td><td>2.722</td><td>0.090</td><td>0.799</td><td>0.011</td></tr><tr><td>CB8</td><td>2.722</td><td>0.090</td><td>0.800</td><td>0.011</td></tr><tr><td>CB9</td><td>2.716</td><td>0.090</td><td>0.801</td><td>0.011</td></tr><tr><td>CB10</td><td>2.733</td><td>0.098</td><td>0.801</td><td>0.012</td></tr><tr><td>CB11</td><td>2.718</td><td>0.098</td><td>0.800</td><td>0.012</td></tr><tr><td>BMA</td><td>3.163</td><td>0.099</td><td>0.834</td><td>0.010</td></tr><tr><td>CBMA</td><td>2.683</td><td>0.089</td><td>0.799</td><td>0.011</td></tr></table>

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+# CFD: LEARNING GENERALIZED MOLECULAR REPRESENTATION VIA CONCEPT-ENHANCED FEEDBACK DISENTANGLEMENT
+
+Among Wu, Cheng Deng*
+
+School of Electronic Engineering, Xidian University, Xi'an, China
+
+amwu@xidian.edu.cn, chdeng@mail.xidian.edu.cn
+
+# ABSTRACT
+
+To accelerate biochemical research, e.g., drug and protein discovery, molecular representation learning (MRL) has attracted much attention. However, most existing methods follow the closed-set assumption that training and testing data share identical distribution, which limits their generalization abilities in out-of-distribution (OOD) cases. In this paper, we explore designing a new disentangled mechanism for learning generalized molecular representation that exhibits robustness against distribution shifts. And an approach of Concept-Enhanced Feedback Disentanglement (CFD) is proposed, whose goal is to exploit the feedback mechanism to learn distribution-agnostic representation. Specifically, we first propose two dedicated variational encoders to separately decompose distribution-agnostic and spurious features. Then, a set of molecule-aware concepts are tapped to focus on invariant substructure characteristics. By fusing these concepts into the disentangled distribution-agnostic features, the generalization ability of the learned molecular representation could be further enhanced. Next, we execute iteratively the disentangled operations based on a feedback received from the previous output. Finally, based on the outputs of multiple feedback iterations, we construct a self-supervised objective to promote the variational encoders to possess the disentangled capability. In the experiments, our method is verified on multiple real-world molecular datasets. The significant performance gains over state-of-the-art baselines demonstrate that our method can effectively disentangle generalized molecular representation in the presence of various distribution shifts. The source code will be released at https://github.com/AmingWu/MoleculeCFD.
+
+# 1 INTRODUCTION
+
+With the rejuvenation of deep neural networks, to advance the development of biochemical research, e.g., drug discovery (Sanchez-Fernandez et al., 2022) and protein design (Ingraham et al., 2019), molecular representation learning (MRL) has attracted growing attention, which aims to transform molecules into low-dimensional and dense vectors (Gilmer et al., 2017; Yang et al., 2019; Rong et al., 2020; Fang et al., 2022; Yang et al., 2022; Zhuang et al., 2023). By means of the learned molecular representation, many downstream tasks could be addressed effectively, including drug property prediction (Wang et al., 2024b), binding affinity analysis (Karimi et al., 2019), and search of antibiotics (Stokes et al., 2020), etc.
+
+Though MRL has achieved significant progress, most existing methods often follow the closed-set assumption, i.e., the training and testing data share the same distribution. However, in practical biochemical research, molecules may exhibit many variants and originate from different distributions (Ji et al., 2022; Mendez et al., 2019). Taking the virtual screening (Lim et al., 2019) as an example, the model may be affected by the distribution shift in the molecule itself, e.g., size (Ji et al., 2022) or scaffold (Wu et al., 2018) changes. Furthermore, some unpredictable events like COVID-19 may bring new target molecules from unknown distributions. Obviously, these scenarios pose significant
+
+![](images/cf078d01b5c795736812c40e4ef3cbd99dadabb4ab50da4d8b51b56cee1ec134.jpg)  
+Figure 1: Three different mechanisms for disentangling generalized features. (a) and (b) are commonly used disentangled methods (Zhuang et al., 2023). (a) first uses a subgraph generator to separate the input, then each subgraph is encoded, respectively. (b) first encodes the input, then the representation is separated by a scorer (Zhuang et al., 2023). (c) is our proposed mechanism for learning generalized representation, i.e., Concept-Enhanced Feedback Disentanglement, which exploits the iteration manner to perform variational disentangled operations based on the feedback received from the previous iteration's output. The combination of all decomposed generalized features is taken as the final generalized representation for downstream tasks.
+
+![](images/da6882649b8f4be25ddb5ea28ab5fb52643e94042b7c32db3920b23ad0ee6681.jpg)
+
+challenges for closed-set assumption based MRL methods. To this end, improving the generalization of molecular representation is meaningful for reducing the impact of distribution shifts.
+
+Existing methods (Ahuja et al., 2021; Yang et al., 2024) mainly focus on learning generalized representation for regular Euclidean data (e.g., images). Instead, to preserve rich structural information (Gilmer et al., 2017), molecules are often represented as graphs, a typical non-Euclidean data, where atoms and bonds are separately taken as nodes and edges. Thus, the methods (Ahuja et al., 2021; Yang et al., 2024) corresponding to Euclidean data could not be directly applied to molecular graph data. Particularly, the work (Ji et al., 2022) points out that existing OOD methods (Sagawa et al., 2019; Sun & Saenko, 2016; Zhang, 2017) fail to significantly improve the performance of MRL tasks against distribution shifts. In this paper, we explore designing a disentangled mechanism for learning generalized molecular representation.
+
+Recently, some studies (Chen et al., 2022; Li et al., 2022b; Yang et al., 2022) have attempted to improve the generalization of molecular representation, which could be classified into two types, i.e., First-Separation-Then-Encoding and First-Encoding-Then-Separation (Zhuang et al., 2023). Specifically, as shown in Fig. 1 (a), the first type first divides the graph into generalized and spurious subgraphs and then encodes each part separately. However, this kind of method usually relies on the separation quality. For certain molecules involving complex structures, since there is no high-quality decomposition, the performance of these methods may be weakened severely. To this end, the work (Zhuang et al., 2023) introduces a "First-Encoding-Then-Separation" mechanism. As shown in Fig. 1 (b), it first employs a Graph Neural Network (GNN) (Kipf & Welling, 2017; Hamilton et al., 2017) to encode the molecule. Then, another GNN is utilized to calculate the score of molecule representation, which is taken as the critical factor for separating generalized and spurious features. However, for extremely complex and entangled molecules, e.g., biomacromolecules, only leveraging the calculated score could not effectively obtain expected separation results. Meanwhile, this method (Zhuang et al., 2023) fails to focus on certain important substructures containing invariant characteristics, affecting its performance in real-world molecules.
+
+In Fig. 1 (c), we propose a new disentangled mechanism, i.e., Concept-Enhanced Feedback Disentanglement (CFD), which still follows the "First-Encoding-Then-Separation" strategy and exploits the feedback mechanism (Zamir et al., 2017) to learn generalized representation. Concretely, given an input molecule, we first exploit a GNN to encode its representation. Then, two dedicated variational encoders are designed to separately decompose distribution-agnostic generalized features and spurious features. Next, a series of concepts are learned to capture certain critical substructures involving invariant molecule characteristics. By fusing these concepts into the decomposed distribution-agnostic features, the generalization ability could be further strengthened. Later, we iteratively perform the above disentangled operations based on a feedback received from the previous iteration's output.
+
+Finally, based on the outputs of multiple feedback iterations, a self-supervised objective is constructed to encourage the variational encoders to own the disentangled capability. Extensive experimental results on multiple molecular datasets demonstrate the superiorities of our method.
+
+The contributions are summarized as follows:
+
+(1) We propose a paradigm of feedback disentanglement that utilizes the iteration manner to decompose generalized molecular representation, improving the robustness against distribution shifts.  
+(2) We propose a concept mining module focusing on certain critical substructures involving invariant characteristics. By fusing these concepts into the disentangled representation, the generalization ability could be further enhanced.  
+(3) We conduct extensive experiments on a diverse set of real-world molecule datasets (Zhuang et al., 2023) with various distribution shifts. The significant performance gains over multiple baselines demonstrate that our method can disentangle generalized molecular representation effectively. Moreover, to further verify the ability of capturing important substructure, our method is evaluated on a new task, i.e., Molecule's Ground-State Conformation Prediction (Xu et al., 2024). We observe that plugging our method into Transformer architecture (Xu et al., 2024) could still improve the prediction performance, indicating the superiorities of our method.
+
+# 2 RELATED WORK
+
+OOD Generalization. Recent research (Wang et al., 2022a; Robey et al., 2021; Zhou et al., 2024) has indicated that existing machine learning methods usually suffer from significant performance degradation under distribution shifts, motivating a surge of works on OOD generalization. Particularly, most existing methods (Evuru et al., 2024; Ratner et al., 2017; Feder et al., 2024) attempt to improve the generalization from the data and representation perspectives. Particularly, an effective strategy is to utilize data augmentation to improve the generalization. Zhou et al. (Zhou et al., 2020) utilize domain information for creating an additive noise to increase the diversity of training data distribution while preserving the semantic information of data. Xu et al. (Xu et al., 2021a) exploit the classical Fourier transformation to obtain the augmented images with diverse styles, which enhances the generalization ability effectively. Besides, by simply mixing up different styles of input data, the work (Zhou et al., 2021) could strengthen the generalization performance significantly.
+
+Another effective strategy is to learn invariant representations (Chang et al., 2020; Creager et al., 2021). In general, the learned invariant representation should satisfy such a principle: 1) sufficiency: shows sufficient predictive power for the target, 2) invariance: contributes to optimal performance for the downstream tasks across all environments. Therefore, some works (Mahajan et al., 2021; Lv et al., 2022; Wang et al., 2022b) explore learning invariant representation from the causal perspective. Furthermore, feature disentanglement (Li et al., 2022a; Zhang et al., 2022; Wu et al., 2021) is also a commonly used mechanism to decompose distribution-agnostic representations. Though these methods have been demonstrated to be effective on many tasks, they often pay attention to regular Euclidean data (e.g., images). However, molecules are usually represented as graphs, which belong to a typical non-Euclidean data. At this time, directly employing the above methods to process graph data does not achieve satisfactory performance. Thus, it is meaningful to design a proper mechanism for learning generalized molecular representation.
+
+OOD Generalization on Molecular Representation. Recently, in order to promote the application of biochemical research, learning generalized molecular representation has attracted growing attention. Some methods (Yang et al., 2022; Chen et al., 2022; Fan et al., 2022) follows the "first-separationthen-encoding" paradigm. And they first design a dedicated module to disentangle the original input graph into two different parts. Then, two independent GNN are utilized to encode these two parts, respectively. However, this kind of methods are prone to relying on the separation results. For certain molecules with complex structures, these methods (Yang et al., 2022; Chen et al., 2022; Fan et al., 2022) could not achieve the expected performance. To this end, Zhuang et al. (Zhuang et al., 2023) propose a new strategy, i.e., "first-encoding-then-separation". Concretely, they first employ a GNN to encode molecules. Then, a disentangled module is designed to perform representation decomposition. Though this method has been shown to be effective, for certain biomacromolecules, the proposed score module could not disentangle invariant representations successfully, limiting its application in real-world molecules. In this paper, we explore a new mechanism, i.e., Concept-Enhanced Feedback
+
+![](images/d8d53569087bcc04c0c2acdfcf423e93220b2f3cecb1d782bc296f86a6e915d3.jpg)  
+Figure 2: The details of our method for disentangling generalized molecule representation. Our method mainly contains two components: (1) Feedback Disentanglement, and (2) Molecule Concept Extraction. Particularly, based on the current extracted representation, two variational encoders are designed to separately obtain spurious features and generalized features. By maximizing the distance between two estimated distributions, the decomposition ability of two variational encoders could be enhanced. Next, a series of concepts are learned to focus on certain important substructures involving invariant characteristics. By fusing these concepts into the disentangled features, the generalization ability could be further enhanced. Finally, by combining the generalized features from multiple iterations, the output could be taken as the final generalized representation for downstream tasks.
+
+Disentanglement, which aims to utilize the iteration manner for learning generalized molecular representation. Meanwhile, a series of concepts are mined to focus on certain critical substructures, which further enhances the generalization ability. Extensive experimental results on multiple real-world molecule datasets demonstrate that our method could indeed disentangle distribution-agnostic invariant molecular representation.
+
+# 3 CONCEPT-ENHANCED FEEDBACK DISENTANGLEMENT
+
+In this paper, we focus on disentangling generalized molecular representation to reduce the impact of distribution shifts. Here, we first define the OOD generalization problem for MRL.
+
+In general, a molecular graph can be represented as  $G = (V, E)$ , where  $V$  is the graph's node set corresponding to atoms and  $E$  denotes the graph's edge sets corresponding to bonds. The goal of MRL task is to predict the target label  $\mathcal{Y}$  given the associated input molecule  $\mathcal{G}$ . The OOD problem on MRL is defined as follows:
+
+$$
+\min  _ {f} \max  _ {e \in \xi} \mathbb {E} _ {\left(\mathcal {G} _ {i}, \mathcal {Y} _ {i}\right) \sim P \left(\mathcal {G}, \mathcal {Y} | e\right)} \left[ \mathcal {L} \left(f \left(\mathcal {G} _ {i}\right), \mathcal {Y} _ {i}\right) | e \right], \tag {1}
+$$
+
+where  $\xi$  represents the support of environments,  $f(\cdot)$  is the prediction model and  $\mathcal{L}(\cdot ,\cdot)$  indicates a loss function.  $\mathbb{E}_{(\mathcal{G}_i,\mathcal{Y}_i)\sim P(\mathcal{G},\mathcal{Y}|e)}[\mathcal{L}(f(\mathcal{G}_i),\mathcal{Y}_i)|e]$  is called the risk function under a given environment  $e$  (Krueger et al., 2021). The reason of resulting in the OOD problem on MRL lies in that the training data only cover very limited environments in  $\xi$  while the model is expected to perform well on all environments. Meanwhile, the OOD problem can be categorized into two cases, i.e., covariate and concept shift (Krueger et al., 2021; Widmer & Kubat, 1996). In covariate shift, the distribution of input differs. And the category space of input is the same. Instead, for concept shift, the distribution of input remains the same. But the category space between the training and testing data is inconsistent. In the experiments, we consider these two shift cases.
+
+# 3.1 VARIATIONAL DISENTANGLEMENT FOR MOLECULAR REPRESENTATION
+
+As shown in Fig. 2, our method follows "first-encoding-then-separation" mechanism (Zhuang et al., 2023) for learning generalized molecule representation. Different from this work (Zhuang et al., 2023), we leverage the feedback idea (Zamir et al., 2017) to iteratively extract distribution-agnostic representation, which is more applicable to molecules with highly complex structures. Moreover,
+
+considering that this work (Zhuang et al., 2023) fails to focus on critical molecular substructures, we propose to mine a series of concepts that involve invariant characteristics of certain molecule substructures, which could be used to further improve the generalization of the learned representation. Next, we will introduce the details of our method.
+
+Specifically, given an input molecule  $\mathcal{G}$ , we first utilize a GNN to encode it. The output is  $F$ :
+
+$$
+F = \operatorname {G N N} (\mathcal {G}) \in \mathbb {R} ^ {| \mathcal {V} | \times d}, \tag {2}
+$$
+
+where  $|\mathcal{V}|$  is the number of atoms in  $\mathcal{G}$ , and  $d$  is the dimensionality of the encoded features. Then, a spurious encoder (i.e.,  $W_{\mu}^{Spu}(\cdot)$  and  $W_{\sigma}^{Spu}(\cdot)$ ) and a generalized encoder (i.e.,  $W_{\mu}^{Gen}(\cdot)$  and  $W_{\sigma}^{Gen}(\cdot)$ ), each consisting of three fully-connected layers, are designed to separately estimate the corresponding distribution. The processes are shown as follows:
+
+$$
+\mu_ {t} ^ {S p u} = W _ {\mu} ^ {S p u} (Q _ {t}), \sigma_ {t} ^ {S p u} = W _ {\sigma} ^ {S p u} (Q _ {t}), \mu_ {t} ^ {G e n} = W _ {\mu} ^ {G e n} (Q _ {t}), \sigma_ {t} ^ {G e n} = W _ {\sigma} ^ {G e n} (Q _ {t}), \quad (3)
+$$
+
+where  $Q_{t} = \phi ([Z_{t - 1}^{Gen},Z_{t - 2}^{Gen}])$ .  $t = 0,\dots ,T$ . And  $Q_{0} = F$ ,  $Q_{1} = Z_{0}^{Gen}$ . '[.,.'] indicates the concatenation operation. '  $\phi (\cdot)$  represents a fully-connected layer that transforms channels.  $\mu_t^{Spu}\in \mathbb{R}^{|\mathcal{V}|\times d}$  and  $\mu_t^{Gen}\in \mathbb{R}^{|\mathcal{V}|\times d}$  are the estimated means for spurious and generalized features at the current iteration  $t$ .  $\sigma_t^{Spu}\in \mathbb{R}^{|\mathcal{V}|\times d}$  and  $\sigma_t^{Gen}\in \mathbb{R}^{|\mathcal{V}|\times d}$  represent the estimated variances for spurious and generalized features. Next, we separately perform a sampling operation to obtain the disentangled features  $Z_{t}^{Spu}\in \mathbb{R}^{|\mathcal{V}|\times d}$  and  $\tilde{Z}_{t}^{Gen}\in \mathbb{R}^{|\mathcal{V}|\times d}$ :
+
+$$
+Z _ {t} ^ {S p u} = D ^ {S p u} \left(\mu_ {t} ^ {S p u} + \epsilon \cdot \exp \left(\sigma_ {t} ^ {S p u}\right)\right), \quad \tilde {Z} _ {t} ^ {G e n} = D ^ {G e n} \left(\mu_ {t} ^ {G e n} + \epsilon \cdot \exp \left(\sigma_ {t} ^ {G e n}\right)\right), \tag {4}
+$$
+
+where  $D^{Spu}(\cdot)$  and  $D^{Gen}(\cdot)$  are two defined decoders, each consisting of three fully-connected layers.  $\epsilon$  denotes Gaussian noise sampled from  $\mathcal{N}(0,I)$ .
+
+To promote  $Z_{t}^{Spu}$  and  $\tilde{Z}_{t}^{Gen}$  to separately contain rich spurious and distribution-agnostic information, we first enlarge the distance between the two estimated distributions. Meanwhile, to encourage  $\tilde{Z}_{t}^{Gen}$  to involve plentiful invariant characteristics related to the input molecule, we further define a contrast loss. Finally, the loss  $\mathcal{L}_{vd}^{t}$  for variational disentanglement is defined as follows:
+
+$$
+\mathcal {L} _ {v d} ^ {t} = - \log \frac {\exp \left(\sin \left(\tilde {Z} _ {t} ^ {G e n} , F\right) / \tau\right)}{\exp \left(\sin \left(\tilde {Z} _ {t} ^ {G e n} , F\right) / \tau\right) + \exp \left(\sin \left(Z _ {t} ^ {S p u} , F\right) / \tau\right)} - \lambda \cdot K L \left(P _ {t} ^ {S p u}, P _ {t} ^ {G e n}\right), \tag {5}
+$$
+
+where  $\mathrm{sim}(\tilde{Z}_t^{Gen},F)$  denotes the average of the cosine similarity between all corresponding elements of  $\tilde{Z}_t^{Gen}$  and  $F$ .  $\tau$  and  $\lambda$  are two hyper-parameters, which are separately set to 1.0 and 0.01 in the experiments.  $P_{t}^{Spu} = \mathcal{N}(\mu_{t}^{Spu},\sigma_{t}^{Spu})$  and  $P_{t}^{Gen} = \mathcal{N}(\mu_{t}^{Gen},\sigma_{t}^{Gen})$ . Obviously, by minimizing the loss  $\mathcal{L}_{vd}^t$ , the gap between  $\tilde{Z}_t^{Gen}$  and  $Z_{t}^{Spu}$  could be enlarged from both semantic and distribution perspectives, which enhances the disentangled ability of our method.
+
+# 3.2 LEARNING MOLECULE CONCEPTS
+
+In general, the critical substructures involving invariant characteristics usually play an important role in downstream tasks. To this end, we explore mining a series of concepts that do not specialize to one particular type or class of molecule, which is instrumental in strengthening the generalization of the learned representation (Locatello et al., 2020).
+
+Concretely, the concepts are defined as  $C \sim \mathcal{N}(\mu_c, \sigma_c) \in \mathbb{R}^{K \times d}$ , where  $\mu_c$  and  $\sigma_c$  are learnable parameters.  $K$  indicates the number of concepts. Next, based on the disentangled representation  $\tilde{Z}_t^{Gen} \in \mathbb{R}^{|\mathcal{V}| \times d}$ , we calculate descriptors that represent molecule-level information:
+
+$$
+\mathcal {M} = \mathrm {K A N} \left(\tilde {Z} _ {t} ^ {\text {G e n}}\right), \quad \mathcal {H} _ {i} = \sum_ {j = 1} ^ {| \nu |} \frac {\exp \left(\mathcal {M} _ {j , i}\right)}{\sum_ {i = 1} ^ {K} \exp \left(\mathcal {M} _ {j , i}\right)} \left(\tilde {Z} _ {t, j} ^ {\text {G e n}} - C _ {i}\right), \tag {6}
+$$
+
+where  $i = 1,\dots ,K$ . Here, we employ one KAN layer (Liu et al., 2024) to transform  $\tilde{Z}_t^{Gen}$ , which outputs  $\mathcal{M} \in \mathbb{R}^{|V| \times K}$ . Different from MLPs having fixed activation functions on neurons, KANs have learnable activation functions (Liu et al., 2024), which is beneficial for improving the flexibility of the learned concepts.  $\mathcal{H} \in \mathbb{R}^{K \times d}$  represents the output descriptors.  $\tilde{Z}_{t,j}^{Gen} - C_i$  denotes the
+
+residual operation. By means of this operation, we can assign molecule substructure features to the corresponding concept  $C_i \in \mathbb{R}^d$ . Finally, the learned concepts are utilized to enhance the current representation  $\tilde{Z}_t^{Gen}$ . The corresponding processes are shown as follows:
+
+$$
+\tilde {\mathcal {H}} = \mathrm {K A N} \left(\mathcal {H} _ {r}\right), Z _ {t} ^ {G e n} = \mathrm {K A N} \left(\left[ \tilde {\mathcal {H}}, \tilde {Z} _ {t} ^ {G e n} \right]\right), \tag {7}
+$$
+
+where  $H_{r}\in \mathbb{R}^{1\times Kd}$  is the reshaped result of  $\mathcal{H}$ .  $\tilde{\mathcal{H}}\in \mathbb{R}^{1\times d}$ .  $Z_{t}^{Gen}\in \mathbb{R}^{|V|\times d}$  is the output generalized representation at the step  $t$ . By the concatenation operation, the learned concepts could be fused into the current decomposed representation, which is conducive to enhancing the generalization of the output representation  $Z_{t}^{Gen}$ . In the experiments,  $Z_{t}^{Gen}$  is utilized as the representation for computing the corresponding task loss, e.g., molecule property prediction, which promotes the learned concepts to capture plentiful task-specific invariant molecule characteristics.
+
+# 3.3 FEEDBACK SEPARATING GENERALIZED MOLECULAR REPRESENTATION
+
+Existing methods (Zhuang et al., 2023; Yang et al., 2022) for learning generalized molecular representation generally employ a one-step disentangled strategy, i.e., directly separating the input into generalized and spurious parts. However, in practice, we may encounter some unknown biomacromolecules containing more atoms and highly complex structures. At this time, using the original one-step mechanism could not obtain satisfactory disentangled results. To this end, we exploit the feedback idea (Zamir et al., 2017) to iteratively separate generalized molecular representation.
+
+As shown in Fig. 3, we assume that the number of feedback iterations is  $T$ . Given the current feature  $Q_{t} \in \mathbb{R}^{|\mathcal{V}| \times d}$  that is the concatenation of the previous two iteration outputs, we first leverage the above operations to disentangle  $Q_{t}$  into  $Z_{t}^{Spu}$  and  $Z_{t}^{Gen}$ , where  $t = 0, \dots, T$ . Then,  $Q_{t + 1} = \phi([Z_{t - 1}^{Gen}, Z_{t}^{Gen}])$  is taken as the
+
+![](images/7535352b33824830b0596e06d2b5110fa50251664cdf3fcd3058068ec7180f95.jpg)  
+Figure 3: Computation graph of One-Step vs Feedback Disentanglement. For the molecules with complex structures, using one-step disentanglement could not obtain the expected separation results. Instead, the feedback mechanism with multiple iteration steps is beneficial for decomposing complex structures.
+
+input of the current step to repeat the above disentangled operations. In the computation graph of feedback disentanglement (as shown in Fig. 3), we can observe that compared with One-Step disentanglement, the feedback strategy follows a hierarchical disentangled mechanism, which is instrumental in improving the ability of decomposing the molecules with complex structures. Finally, through multiple feedback iterations, we can obtain a series of generalized outputs, i.e.,  $(Z_0^{Gen}, Z_1^{Gen}, \dots, Z_T^{Gen})$ , and spurious outputs, i.e.,  $(Z_0^{Spu}, Z_1^{Spu}, \dots, Z_T^{Spu})$ . A self-supervised objective is constructed to close the distance between the output generalized features at each step and enlarge the distance between the decomposed generalized and spurious features at the same step, which strengthens the disentangled ability effectively:
+
+$$
+\mathcal {L} _ {f d} = - \log \left(\frac {1}{T \times T} \sum_ {i = 0} ^ {T} \sum_ {j = 0} ^ {T} \sin \left(Z _ {i} ^ {\text {G e n}}, Z _ {j} ^ {\text {G e n}}\right) _ {i \neq j} - \beta \cdot \frac {1}{T} \sum_ {i = 0} ^ {T} \sin \left(Z _ {i} ^ {\text {G e n}}, Z _ {i} ^ {\text {S p u}}\right)\right), \tag {8}
+$$
+
+where  $\beta$  is a hyper-parameter, which is set to 0.5 in the experiments. Finally, we take the concatenation result, i.e.,  $Z^{Gen} = \psi ([Z_0^{Gen},Z_1^{Gen},\dots ,Z_T^{Gen}])\in \mathbb{R}^{|\mathcal{V}|\times d}$ , as the last generalized representation for downstream tasks, where  $\psi (\cdot)$  indicates a fully-connected layer that transforms the channels.
+
+For specific downstream tasks, their goals are to provide the generalized representation  $Z^{Gen}$  with predictive capabilities. Since our method is to learn generalized molecule representations, it could be applied to different molecular tasks. During training, the choice of prediction loss function depends on the type of task. For classification loss, we employ the cross-entropy loss, while for regression tasks, we utilize the mean squared error loss. Taking the binary classification task as an example, the cross-entropy loss is computed between the predicted probability  $\hat{Y} = Prob(Z^{Gen})$  and the
+
+Table 1: Evaluation performance on GOOD benchmark. - denotes abnormal results caused by under-fitting declared in the leaderboard, and / denotes that the method cannot be applied to this dataset. For different distribution shifts, our method obtains superior generalization performance.  
+
+<table><tr><td rowspan="3">Method</td><td colspan="4">GOOD-HIV ↑</td><td colspan="4">GOOD-ZINC ↓</td><td colspan="4">GOOD-PCBA ↑</td></tr><tr><td colspan="2">scaffold</td><td colspan="2">size</td><td colspan="2">scaffold</td><td colspan="2">size</td><td colspan="2">scaffold</td><td colspan="2">size</td></tr><tr><td>covariate</td><td>concept</td><td>covariate</td><td>concept</td><td>covariate</td><td>concept</td><td>covariate</td><td>concept</td><td>covariate</td><td>concept</td><td>covariate</td><td>concept</td></tr><tr><td>ERM</td><td>69.55</td><td>72.48</td><td>59.19</td><td>61.91</td><td>0.1802</td><td>0.1301</td><td>0.2319</td><td>0.1325</td><td>17.11</td><td>21.93</td><td>17.75</td><td>15.60</td></tr><tr><td>IRM</td><td>70.17</td><td>71.78</td><td>59.94</td><td>(-)</td><td>0.2164</td><td>0.1339</td><td>0.6984</td><td>0.1336</td><td>16.89</td><td>22.37</td><td>17.68</td><td>15.82</td></tr><tr><td>VREx</td><td>69.34</td><td>72.21</td><td>58.49</td><td>61.21</td><td>0.1815</td><td>0.1287</td><td>0.2270</td><td>0.1311</td><td>17.10</td><td>21.65</td><td>17.80</td><td>15.85</td></tr><tr><td>GroupDRO</td><td>68.15</td><td>71.48</td><td>57.75</td><td>59.77</td><td>0.1870</td><td>0.1323</td><td>0.2377</td><td>0.1333</td><td>16.55</td><td>21.91</td><td>16.74</td><td>15.21</td></tr><tr><td>Coral</td><td>70.69</td><td>72.96</td><td>59.39</td><td>60.29</td><td>0.1769</td><td>0.1303</td><td>0.2292</td><td>0.1261</td><td>17.00</td><td>22.00</td><td>17.83</td><td>16.88</td></tr><tr><td>DANN</td><td>69.43</td><td>71.70</td><td>62.38</td><td>65.15</td><td>0.1746</td><td>0.1269</td><td>0.2326</td><td>0.1348</td><td>17.20</td><td>22.03</td><td>17.71</td><td>15.78</td></tr><tr><td>Mixup</td><td>70.65</td><td>71.89</td><td>59.11</td><td>62.80</td><td>0.2066</td><td>0.1391</td><td>0.2531</td><td>0.1547</td><td>16.52</td><td>20.52</td><td>17.42</td><td>13.71</td></tr><tr><td>DIR</td><td>68.44</td><td>71.40</td><td>57.67</td><td>74.39</td><td>0.3682</td><td>0.2543</td><td>0.4578</td><td>0.3146</td><td>16.33</td><td>23.82</td><td>16.04</td><td>16.80</td></tr><tr><td>GSAT</td><td>70.07</td><td>72.51</td><td>60.73</td><td>56.96</td><td>0.1418</td><td>0.1066</td><td>0.2101</td><td>0.1038</td><td>16.45</td><td>20.18</td><td>17.57</td><td>13.52</td></tr><tr><td>GREA</td><td>71.98</td><td>70.76</td><td>60.11</td><td>60.96</td><td>0.1691</td><td>0.1157</td><td>0.2100</td><td>0.1273</td><td>16.28</td><td>20.23</td><td>17.12</td><td>13.82</td></tr><tr><td>CAL</td><td>69.12</td><td>72.49</td><td>59.34</td><td>56.16</td><td>/</td><td>/</td><td>/</td><td>/</td><td>15.87</td><td>18.62</td><td>16.92</td><td>13.01</td></tr><tr><td>DisC</td><td>58.85</td><td>64.82</td><td>49.33</td><td>74.11</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td></tr><tr><td>MoleOOD</td><td>69.39</td><td>69.08</td><td>58.63</td><td>55.90</td><td>0.2752</td><td>0.1996</td><td>0.3468</td><td>0.2275</td><td>12.90</td><td>12.92</td><td>12.64</td><td>10.30</td></tr><tr><td>CIGA</td><td>69.40</td><td>71.65</td><td>61.81</td><td>73.62</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td><td>/</td></tr><tr><td>iMoLD</td><td>72.93</td><td>74.32</td><td>62.86</td><td>77.43</td><td>0.1410</td><td>0.1014</td><td>0.1863</td><td>0.1029</td><td>17.32</td><td>22.58</td><td>18.02</td><td>18.21</td></tr><tr><td>Ours (CFD)</td><td>76.42</td><td>77.83</td><td>64.14</td><td>79.28</td><td>0.1187</td><td>0.0765</td><td>0.1421</td><td>0.0852</td><td>19.78</td><td>25.64</td><td>19.18</td><td>20.03</td></tr></table>
+
+ground-truth label  $\mathcal{V}$  ..
+
+$$
+\mathcal {L} _ {p r e d} = \mathcal {Y} \log \hat {\mathcal {Y}} + (1 - \mathcal {Y}) \log (1 - \hat {\mathcal {Y}}), \tag {9}
+$$
+
+where  $Prob(\cdot)$  represents a function that computes probability. Finally, the learning objective can be defined as the weighted sum of the above losses:
+
+$$
+\mathcal {L} _ {\text {t a s k}} = \mathcal {L} _ {p r e d} + \alpha_ {1} \cdot \mathcal {L} _ {f d} + \alpha_ {2} \cdot \sum_ {t = 0} ^ {T} \mathcal {L} _ {v d} ^ {t}, \tag {10}
+$$
+
+where  $\alpha_{1}$  and  $\alpha_{2}$  are hyper-parameters, which are separately set to 0.1 and 0.01 in the experiments.
+
+# 4 EXPERIMENTS
+
+In this section, we conduct extensive experiments to answer the research questions. (1) Can our method CFD achieve better OOD generalization performance against baselines? (2) Does our method possess the ability to capture important substructures and improve the performance of molecular substructure prediction? (3) How does each component contribute to the final performance?
+
+# 4.1 EXPERIMENTAL SETUP
+
+Datasets. For OOD molecular representation learning, we follow the settings of the work (Zhuang et al., 2023) and employ two real-world datasets, i.e., GOOD (Gui et al., 2022) that is a systematic benchmark tailored specifically for graph OOD problems, and DrugOOD (Ji et al., 2022) that is a OOD benchmark for AI-aided drug discovery. Besides, for molecule's ground-state prediction, we follow the settings of the work (Xu et al., 2024) and utilize Molecule3D (Xu et al., 2021b) and QM9 (Ramakrishnan et al., 2014) to evaluate the ability of our method for focusing on substructures. In Appendix, we will introduce more details about these datasets.
+
+Implementation Details. Our method is used for learning specific molecule representation and could be plugged into existing methods (Zhuang et al., 2023; Xu et al., 2024). The number of the learned concepts is set to 12. The number of feedback iterations is set to 8. It is worth noting that during feedback, we do not introduce new parameters.
+
+Baselines. For OOD molecular representation learning, we mainly compare our method with some common approaches for non-Euclidean data, i.e., ERM (Vapnik, 2013), IRM (Arjovsky et al., 2019), VREx (Krueger et al., 2021), GroupDRO (Sagawa et al., 2019), Coral (Sun & Saenko, 2016), DANN (Ganin et al., 2016), Mixup (Zhang, 2017), and iMoLD (Zhuang et al., 2023). Meanwhile, our method is compared with some graph-specific algorithms, e.g., graph OOD algorithms such as CAL (Sui et al., 2022), DisC (Fan et al., 2022), MoleOOD (Yang et al., 2022), and CIGA (Chen et al., 2022), as well as interpretable graph learning methods, e.g., DIR (Wu et al., 2022), GSAT (Miao et al., 2022), and GREA (Liu et al., 2022).
+
+# 4.2 ANALYSIS OF OOD MOLECULAR REPRESENTATION LEARNING
+
+To evaluate the generalization ability of our method, our method is evaluated on GOOD benchmark (Gui et al., 2022). Table 1 shows the performance. We can observe that some graph-specific OOD methods could achieve effective performance on some synthetic datasets (e.g., to predict whether a specific motif is present in a synthetic graph). However, they do not perform well on molecules with complex data structures and distributions, which shows that our method could apply to molecules with complex structures and improve the generalization performance. Besides, the performance of iMoLD (Zhuang et al., 2023) outperforms MoleOOD (Yang et al., 2022), which shows the effectiveness of the strategy "first-encoding-then-separation". However, iMoLD (Zhuang et al., 2023) fails to capture the critical substructures that involve certain invariant characteristics, which may affect its performance. We can see that our method improves the performance significantly. This indicates that the feedback disentanglement could indeed decompose generalized features. Meanwhile, our method could be applied to complex molecules. This also shows that the learned concepts are beneficial for capturing important substructures, which further strengthens the generalization ability.
+
+Table 2 further evaluates our method on DrugOOD dataset (Ji et al., 2022). Particularly, the recent work MILI (Wang et al., 2024a) follows "first-separationthen-encoding" mechanism. And it first identifies privileged substructure and develops a dual-head GNN to achieve invariant representations of substructures. When the molecular substructures are accurately identified, the generalization performance of OOD molecule representation could be improved effectively. We can see that our method still outperforms iMoLD (Zhuang et al., 2023) and MILI (Wang et al., 2024a). This demonstrates that using the feedback iteration idea is beneficial for extracting general
+
+Table 2: Evaluation performance on DrugOOD dataset.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">IC50↑</td><td colspan="3">EC50↑</td></tr><tr><td>Assay</td><td>Scaffold</td><td>Size</td><td>Assay</td><td>Scaffold</td><td>Size</td></tr><tr><td>ERM</td><td>71.63</td><td>68.79</td><td>67.50</td><td>67.39</td><td>64.98</td><td>65.10</td></tr><tr><td>IRM</td><td>71.15</td><td>67.22</td><td>61.58</td><td>67.77</td><td>63.86</td><td>59.19</td></tr><tr><td>Coral</td><td>71.28</td><td>68.36</td><td>64.53</td><td>72.08</td><td>64.83</td><td>58.47</td></tr><tr><td>MixUp</td><td>71.49</td><td>68.59</td><td>67.79</td><td>67.81</td><td>65.77</td><td>65.77</td></tr><tr><td>DIR</td><td>69.84</td><td>66.33</td><td>62.92</td><td>65.81</td><td>63.76</td><td>61.56</td></tr><tr><td>GSAT</td><td>70.59</td><td>66.45</td><td>66.70</td><td>73.82</td><td>64.25</td><td>62.65</td></tr><tr><td>GREA</td><td>70.23</td><td>67.02</td><td>66.59</td><td>74.17</td><td>64.50</td><td>62.81</td></tr><tr><td>CAL</td><td>70.09</td><td>65.90</td><td>66.42</td><td>74.54</td><td>65.19</td><td>61.21</td></tr><tr><td>DisC</td><td>61.40</td><td>62.70</td><td>61.43</td><td>63.71</td><td>60.57</td><td>57.38</td></tr><tr><td>MoleOOD</td><td>71.62</td><td>68.58</td><td>65.62</td><td>72.69</td><td>65.74</td><td>65.51</td></tr><tr><td>CIGA</td><td>71.86</td><td>69.14</td><td>66.92</td><td>69.15</td><td>67.32</td><td>65.65</td></tr><tr><td>iMoLD</td><td>72.11</td><td>68.84</td><td>67.92</td><td>77.48</td><td>67.79</td><td>67.09</td></tr><tr><td>MILI</td><td>72.67</td><td>69.58</td><td>68.40</td><td>77.11</td><td>68.07</td><td>65.97</td></tr><tr><td>Ours (CFD)</td><td>73.86</td><td>70.02</td><td>69.73</td><td>78.32</td><td>69.13</td><td>67.62</td></tr></table>
+
+ized molecular characteristics. By fusing the concepts capturing plentiful substructure content, the generalization ability could be further enhanced.
+
+Table 3: Performance of molecule's ground-state prediction based on Molecule3D and QM9 datasets.  
+
+<table><tr><td rowspan="2"></td><td colspan="3">Validation</td><td colspan="3">Test</td></tr><tr><td>D-MAE↓</td><td>D-RMSE↓</td><td>C-RMSD↓</td><td>D-MAE↓</td><td>D-RMSE↓</td><td>C-RMSD↓</td></tr><tr><td colspan="7">(a) Molecule3D Random Split</td></tr><tr><td>RDKit DG</td><td>0.581</td><td>0.930</td><td>1.054</td><td>0.582</td><td>0.932</td><td>1.055</td></tr><tr><td>RDKit ETKDG</td><td>0.575</td><td>0.941</td><td>0.998</td><td>0.576</td><td>0.942</td><td>0.999</td></tr><tr><td>DeeperGCN-DAGNN (Xu et al., 2021b)</td><td>0.509</td><td>0.849</td><td>-</td><td>0.571</td><td>0.961</td><td>-</td></tr><tr><td>GINE (Hu et al., 2019)</td><td>0.590</td><td>1.014</td><td>1.116</td><td>0.592</td><td>1.018</td><td>1.116</td></tr><tr><td>GATv2 (Brody et al., 2021)</td><td>0.563</td><td>0.983</td><td>1.082</td><td>0.564</td><td>0.986</td><td>1.083</td></tr><tr><td>GPS (Rampášek et al., 2022)</td><td>0.528</td><td>0.909</td><td>1.036</td><td>0.529</td><td>0.911</td><td>1.038</td></tr><tr><td>GTMGC (Xu et al., 2024)</td><td>0.432</td><td>0.719</td><td>0.712</td><td>0.433</td><td>0.721</td><td>0.713</td></tr><tr><td>GTMGC + Ours</td><td>0.397</td><td>0.682</td><td>0.684</td><td>0.407</td><td>0.695</td><td>0.688</td></tr><tr><td colspan="7">(b) QM9</td></tr><tr><td>RDKit DG</td><td>0.358</td><td>0.616</td><td>0.722</td><td>0.358</td><td>0.615</td><td>0.722</td></tr><tr><td>RDKit ETKDG</td><td>0.355</td><td>0.621</td><td>0.691</td><td>0.355</td><td>0.621</td><td>0.689</td></tr><tr><td>GINE (Hu et al., 2019)</td><td>0.357</td><td>0.673</td><td>0.685</td><td>0.357</td><td>0.669</td><td>0.693</td></tr><tr><td>GATv2 (Brody et al., 2021)</td><td>0.339</td><td>0.663</td><td>0.661</td><td>0.339</td><td>0.659</td><td>0.666</td></tr><tr><td>GPS (Rampášek et al., 2022)</td><td>0.326</td><td>0.644</td><td>0.662</td><td>0.326</td><td>0.640</td><td>0.666</td></tr><tr><td>GTMGC (Xu et al., 2024)</td><td>0.262</td><td>0.468</td><td>0.362</td><td>0.264</td><td>0.470</td><td>0.367</td></tr><tr><td>GTMGC + Ours</td><td>0.223</td><td>0.434</td><td>0.305</td><td>0.218</td><td>0.442</td><td>0.309</td></tr></table>
+
+# 4.3 ANALYSIS OF MOLECULE'S GROUND-STATE PREDICTION
+
+The molecular ground-state conformation refers to the lowest energy state of a molecule on its potential energy surface (Xu et al., 2024). It belongs to substructures of original molecules, which
+
+![](images/f6ed14c0f53e5f5be54839263c7f0fad6e7b67906a00273f5aa75c27c19a24c4.jpg)  
+Ground Truth
+
+![](images/3344bdefec031823f8aafdf0e2d1108da6554c03a4263362be76815acba6d7e4.jpg)  
+Our Prediction
+
+![](images/99cb7108460ff7c4bb14971ccffec21256a0b9f60cdaf2d4eeaf089b185c19c8.jpg)  
+Ground Truth  
+Figure 4: The prediction results of the molecule's ground-state conformation. Compared with ground truth, our method could localize the critical substructures, improving the prediction accuracy.
+
+![](images/62629cc88de36eb915491ff3baf91fbc282949ccc91a77102b87e793a1e756e4.jpg)  
+Our Prediction
+
+play a crucial role in determining the physical, chemical, and biological properties of molecules. By evaluating this task, we can directly observe CFD's ability for capturing critical substructures. Here, our method is directly plugged into GTMGC (Xu et al., 2024) that is a transformer architecture. The training processes and optimization objectives are kept unchanged. In Table 3, we can observe that plugging our method could improve GTMGC's prediction performance effectively. This indicates that our method is able to capture critical substructure characteristics. Meanwhile, our method could be plugged into Transformer architecture, showing our method's flexibility.
+
+In Fig. 4, we provide two prediction examples from our method. We can observe that compared with ground truth results, our method could accurately capture those critical substructures, further demonstrating the effectiveness of our method.
+
+Table 4: Analysis of our CFD method. 'Feedback' and 'Concepts' separately indicate feedback disentanglement and molecule concept learning. 'Cov-Sold', 'Cet-Sold', 'Cov-Size', and 'Cet-Size' denote scaffold-covariate, scaffold-concept, size-covariate, and size-concept shifts, respectively.
+
+<table><tr><td>Feedback</td><td>Concepts</td><td>Cov-Sold</td><td>Cet-Sold</td><td>Cov-Size</td><td>Cet-Size</td></tr><tr><td>✓</td><td></td><td>0.1391</td><td>0.0923</td><td>0.1754</td><td>0.0983</td></tr><tr><td></td><td>✓</td><td>0.1685</td><td>0.1172</td><td>0.2046</td><td>0.1268</td></tr><tr><td>✓</td><td>✓</td><td>0.1187</td><td>0.0765</td><td>0.1421</td><td>0.0852</td></tr></table>
+
+# 4.4 ABLATION ANALYSIS
+
+To evaluate the effectiveness of each component, we take GOOD-ZINC (Gui et al., 2022) as the dataset to make an ablation analysis.
+
+Analysis of our method. As shown in Fig. 2, our method mainly consists of two components, i.e., feedback disentanglement and molecule concept mining. Meanwhile, these two modules could be taken as the independent module. In Table 4, we make an ablation experiment. 'Concepts' represents that the concept learning module is plugged into ERM (Vapnik,
+
+2013). We can see that for different shifts, using feedback disentanglement and learning molecule concepts could improve the performance significantly. This shows that our method could effectively separate generalized features from spurious features. Meanwhile, the mined concepts could capture critical substructure characteristics, further enhancing the disentangled features' generalization.
+
+Number of feedback iterations. To achieve the expected generalized representation, we employ multiple feedback iterations. In Table 5, we can see that the iteration number affects the generalization performance. For molecules with complex structures, using fewer iterations could not decompose the expected generalized representation effectively, which weakens the
+
+performance. Meanwhile, we observe that the performance change is very small when the iteration number is larger than 12 for the given dataset. For our method, the performance of using 8 feedback iterations is the best.
+
+Table 5: Analysis of feedback iteration number.  
+
+<table><tr><td>Iterations</td><td>Cov-Sold</td><td>Cet-Sold</td><td>Cov-Size</td><td>Cet-Size</td></tr><tr><td>1</td><td>0.1378</td><td>0.1012</td><td>0.1789</td><td>0.1021</td></tr><tr><td>4</td><td>0.1285</td><td>0.1008</td><td>0.1573</td><td>0.0963</td></tr><tr><td>8</td><td>0.1187</td><td>0.0765</td><td>0.1421</td><td>0.0852</td></tr><tr><td>12</td><td>0.1191</td><td>0.0799</td><td>0.1453</td><td>0.0844</td></tr><tr><td>16</td><td>0.1172</td><td>0.0805</td><td>0.1479</td><td>0.0867</td></tr></table>
+
+Table 6: Analysis of molecule concept number. Here, we only change the concept number. Other operations and settings are kept unchanged.  
+
+<table><tr><td>Number</td><td>Cov-Sold</td><td>Cet-Sold</td><td>Cov-Size</td><td>Cet-Size</td></tr><tr><td>8</td><td>0.1233</td><td>0.0892</td><td>0.1541</td><td>0.0927</td></tr><tr><td>12</td><td>0.1187</td><td>0.0765</td><td>0.1421</td><td>0.0852</td></tr><tr><td>16</td><td>0.1201</td><td>0.0743</td><td>0.1478</td><td>0.0874</td></tr></table>
+
+Number of molecule concepts. As shown in Fig. 2, we explore extracting a series of molecule concepts to focus on critical substructures. In Table 6, we can observe that the concept number affects the performance. Utilizing fewer concepts could not capture sufficient substructure characteristics
+
+effectively, which reduces the performance. For our method, when the concept number is set to 12, the generalization performance is the best. Besides, during mining molecule concepts (equation 6 and equation 7), we employ KAN (Liu et al., 2024) to improve the concept accuracy. Here, we find that replacing KAN with MLP results in performance degradation. For example, for Cov-Sold case, the performance is decreased by around  $0.6\%$ , indicating KAN's effectiveness.
+
+# 5 CONCLUSION
+
+In this paper, we propose a new method, i.e., Concept-Enhanced Feedback Disentanglement, for learning generalized molecular representation against distribution shifts. Particularly, by performing multiple feedback iterations, our method could progressively decompose expected features involving rich generalized information. Meanwhile, fusing the concepts that focus on substructures could further strengthen the generalization. Extensive experimental results on multiple datasets and tasks demonstrate that our method could indeed disentangle generalized molecular representations.
+
+# ACKNOWLEDGEMENT
+
+This work is supported in part by the National Key R&D Program of China (No. 2023YFC3305600), Joint Fund of Ministry of Education of China (8091B022149, 8091B02072404), National Natural Science Foundation of China (62472333, 62132016, 62171343), and Natural Science Basic Research Program of Shaanxi (2020JC-23).
+
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+
+# A APPENDIX
+
+To learn generalized molecular representation, this paper proposes a new method, i.e., Concept-Enhanced Feedback Disentanglement, aiming to use feedback decomposition and concepts involving critical substructures to improve the generalization of the learned representation. In the appendix, we provide more experimental details, additional analyses, and more prediction results.
+
+# B DATASETS AND METRICS
+
+Datasets. For molecular representation learning, we employ two benchmarks, i.e., GOOD (Gui et al., 2022) and DrugOOD (Ji et al., 2022). Particularly, we follow the work (Zhuang et al., 2023) to utilize three molecular datasets for the prediction task: (1) GOOD-HIV (Wu et al., 2018), where the purpose is binary classification to predict whether a molecule can inhibit HIV; (2) GOOD-ZINC (Gómez-Bombarelli et al., 2018), which is a regression dataset aimed at predicting molecular solubility; and (3) GOOD-PCBA (Wu et al., 2018), which includes 128 bioassays and forms 128 binary classification tasks. Each dataset contains two environment-splitting strategies (scaffold and size), and two shift types (covariate and concept) are applied per splitting outcome, resulting in a total of 12 distinct datasets (as shown in Table 1). Furthermore, DrugOOD (Ji et al., 2022) provides three environment-splitting strategies, including assay, scaffold, and size, and applies these three splitting to two measurements (IC50 and EC50). As a result, we obtain 6 datasets (as shown in Table 2), and each dataset contains a binary classification task for drug target binding affinity prediction
+
+For molecule's ground-state prediction, we follow the work (Xu et al., 2024) to utilize Molecule3D (Xu et al., 2021b) and QM9 (Wu et al., 2018) as benchmarks. Particularly, Molecule3D aims to use DNNs to predict the ground-state 3D geometries of molecules based solely on their 2D graph structure. And this large-scale dataset comprises approximately 4 million molecules, each with its own 2D molecular graph, ground-state 3D geometric structure, and four additional quantum properties. In this paper, we employ the random splitting according to the same distribution based on the molecule's core component. Besides, QM9 is a quantum chemistry dataset that provides geometry, energy, electronic, and thermodynamic properties for nearly 130,000 organic molecules with 9 heavy atoms, containing molecular most stable conformation.
+
+Metrics. For molecular representation learning, we report the ROC-AUC score for GOOD-HIV and DrugOOD datasets as the task is binary classification. For GOOD-ZINC, we use the Mean Average Error (MAE) since the task is regression. While for GOOD-PCBA, we use Average Precision (AP) averaged over all tasks as the evaluation metric due to extremely imbalanced classes.
+
+For molecule's ground-state prediction, we follow the work (Xu et al., 2024) to utilize three metrics, i.e., D-MAE, D-RMSD, and C-RMSD. Concretely, given a dataset with  $N$  interatomic distances, the Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) between the prediction  $\{d_i^*\}_{i=1}^N$  and the ground truth  $\{d_i\}_{i=1}^N$  are used to evaluate the performance at node-pair level:
+
+$$
+\mathrm {D} - \operatorname {M A E} \left(\left\{d _ {i} \right\} _ {i = 1} ^ {N}, \left\{d _ {i} ^ {*} \right\} _ {i = 1} ^ {N}\right) = \frac {1}{N} \sum_ {i = 1} ^ {N} \left| d _ {i} - d _ {i} ^ {*} \right|, \tag {11}
+$$
+
+$$
+\mathrm {D} - \operatorname {R M S E} \left(\left\{d _ {i} \right\} _ {i = 1} ^ {N}, \left\{d _ {i} ^ {*} \right\} _ {i = 1} ^ {N}\right) = \sqrt {\frac {1}{N} \sum_ {i = 1} ^ {N} \left(d _ {i} - d _ {i} ^ {*}\right) ^ {2}}. \tag {12}
+$$
+
+Additionally, C-RMSD between the ground truth  $\mathbf{G}$  and the prediction  $\mathbf{G}^*$  is computed as follows:
+
+$$
+\mathrm {C} - \operatorname {R M S D} \left(\mathbf {G}, \mathbf {G} ^ {*}\right) = \sqrt {\frac {1}{n} \sum_ {i = 1} ^ {n} \left\| g _ {i} - g _ {i} ^ {*} \right\| _ {2} ^ {2}}. \tag {13}
+$$
+
+# C THE CAUSE OF MOLECULAR DISTRIBUTION SHIFT
+
+For molecule data, there exist many kinds of distribution shifts. In this paper, we investigate three types of shift strategies, i.e., scaffold, size, and assay. Here, we give more explanations.
+
+![](images/0423499aad2542e3670c3916c18f7046af9416dca9169a2f54f2ebf529cb3d37.jpg)  
+Figure 5: More prediction results of molecule's ground-state conformation. We can observe that our method could accurately localize critical ground-state parts that determine the physical, chemical, and biological properties of the molecule in most cases, which demonstrates the effectiveness of our method for learning generalized molecule representation and capturing substructure characteristics.
+
+(1) Scaffold Environment: molecular scaffold is the fundamental structure of a molecule with desirable bioactive properties. Molecules with the same scaffold belong to the same environment. Distribution shift arises when there is a change in the molecular scaffold.  
+(2) Size Environment: the size of a molecule refers to the total number of atoms in the molecule. Molecular size is also an inherent structural characteristics of molecular graphs. The distribution shift occurs when size changes.  
+(3) Assay Environment: assay is an experimental method as an examination or determination for molecular characteristics. Due to variations in assay environments and targets, activity values measured by different assays often differ significantly. Samples tested within the same assay belong to a single environment, while a change in the assay leads to a distribution shift.
+
+# D MORE ABLATION EXPERIMENTS OF HYPER-PARAMETERS
+
+In equation 10, we utilize two hyper-parameters, i.e.,  $\alpha_{1}$  and  $\alpha_{2}$ , to balance the two loss terms, i.e.,  $\mathcal{L}_{fd}$  and  $\mathcal{L}_{vd}$ . Here, based on GOOD-ZINC with Covariate-Scaffold shift, we make an ablation analysis. We only change the settings of hyper-parameters.
+
+Analysis of hyper-parameter  $\alpha_{1}$ . In equation 10, the goal of the loss term  $\mathcal{L}_{fd}$  is to encourage the feedback disentangled process to decompose generalized representation effectively. Thus, the setting of hyper-parameter  $\alpha_{1}$  is important. Here, we make an experiment. When  $\alpha_{1}$  is separately set to 0.5, 0.1, and 0.05, the corresponding performance is 0.1251, 0.1187, and 0.1239.
+
+Analysis of hyper-parameter  $\alpha_{2}$ . In equation 10, the loss term  $\mathcal{L}_{vd}^{t}$  is to promote the decomposed features to involve plentiful invariant characteristics related to the input molecule while enlarging the estimated distribution gap between the generalized and spurious features. Thus, the hyper-parameter  $\alpha_{2}$  is still an important hyper-parameter. Here, we make an experiment. When  $\alpha_{2}$  is separately set to 0.05, 0.01, and 0.005, the performance is 0.1248, 0.1187, and 0.1225.
+
+Input: Molecule Dataset  $\{G,Y\}$ , randomly initialized GNN, randomly initialized spurious encoder and generalized encoder, randomly initialized concept learning module, weight  $\alpha_{1}$  for the loss  $\mathcal{L}_{fd}$ , weight  $\alpha_{2}$  for the loss  $\mathcal{L}_{vd}$ .
+
+Output: Trained parameters.
+
+Algorithm 1 Training Process of Concept-Enhanced Feedback Disentanglement  
+while train do  
+Sample molecules from the given dataset  $\{G,Y\}$    
+Utilize a GNN to extract the representation  $F$    
+for  $t = 0$  do  $Q_{0} = F$  Use equation 3 to estimate corresponding means (i.e.,  $\mu_0^{Spu}$  and  $\mu_0^{Gen}$  ) and variances (i.e.,  $\sigma_0^{Spu}$  and  $\sigma_0^{Gen}$  ) for spurious and generalized features. Perform sampling and encoding operations to obtain  $Z_{0}^{Spu}$  and  $\tilde{Z}_0^{\mathrm{Gen}}$  based on equation 4. Learn molecule concepts  $C$  and calculate the generalized representation  $Z_{0}^{Gen}$  based on equation 6 and equation 7.   
+end   
+for  $t = 1$  do  $Q_{1} = Z_{0}^{Gen}$  Use equation 3 to estimate corresponding means (i.e.,  $\mu_1^{Spu}$  and  $\mu_1^{Gen}$  ) and variances (i.e.,  $\sigma_1^{Spu}$  and  $\sigma_1^{Gen}$  ) for spurious and generalized features. Perform sampling and encoding operations to obtain  $Z_{1}^{Spu}$  and  $\tilde{Z}_1^{Gen}$  based on equation 4. Learn molecule concepts  $C$  and calculate the generalized representation  $Z_{1}^{Gen}$  based on equation 6 and equation 7.   
+end   
+for  $t = 2,\ldots ,T$  do  $Q_{t} = \phi ([Z_{t - 1}^{Gen},Z_{t - 2}^{Gen}])$  Use equation 3 to estimate corresponding means (i.e.,  $\mu_t^{Spu}$  and  $\mu_t^{Gen}$  ) and variances (i.e.,  $\sigma_t^{Spu}$  and  $\sigma_t^{Gen}$  ) for spurious and generalized features. Perform sampling and encoding operations to obtain  $Z_{t}^{Spu}$  and  $\tilde{Z}_{t}^{Gen}$  based on equation 4. Learn molecule concepts  $C$  and calculate the generalized representation  $Z_{t}^{Gen}$  based on equation 6 and equation 7.   
+end Calculate the overall training objective  $\mathcal{L}_{\mathrm{task}}$  using equation 5, equation 8, equation 9. Update the trained parameters based on equation 10.   
+end
+
+Finally, Algorithm 1 shows the training processes of our method. For step  $t > = 2$ , the feedback is from the outputs of the previous two steps. By means of feedback disentanglement, our method could decompose generalized molecular representation effectively, which improves the performance of downstream tasks. In Fig. 5, we show more prediction results of molecule's ground-state conformation. We can see that our method could localize critical ground-state molecular substructures, which further demonstrates the superiorities of our method.
+
+# E MORE CHEMICAL INSPIRATION OF OUR METHOD
+
+This method is mainly based on the following chemical observations:
+
+(1) A chemical molecule is a complex composed of multiple atoms and bonds. The reason for the OOD problem is the structural change of molecules. For example, changes in molecular chirality can lead to deviations in molecular properties. Thus, learning robust molecule representation is important for molecule-based tasks, e.g., drug discovery.  
+(2) Lines 082-095 have indicated that existing methods could be classified into two types, i.e., First-Separation-Then-Encoding and First-Encoding-Then-Separation (as shown in Fig. 1 (a) and (b)). As indicated in Lines 272-283 and Fig. 3, these methods for learning generalized molecular representation generally employ a one-step disentangled strategy, i.e., directly separating the input into
+
+generalized and spurious parts. However, in practice, we may encounter some unknown biomacromolecules containing more atoms and highly complex structures, e.g., entanglement of multiple substructures. At this time, using the original one-step mechanism could not obtain satisfactory disentangled results. To this end, we exploit the feedback idea to iteratively and progressively separate generalized molecular representation.
+
+(3) In general, the molecule's ground-state conformation belongs to the substructures of molecules and determines their properties. Therefore, it is important to capture substructure-related characteristics. To this end, we define a series of concepts to build connections between molecule substructures and corresponding concepts, which further improves the performance.
+
+# F MORE EXPERIMENTS TO EVALUATE INVARIANCE
+
+Table 7: To further demonstrate the ability of disentangling invariance, we further evaluate our method on Two-Piece Graph Datasets Chen et al. (2024).  
+
+<table><tr><td>Datasets</td><td>{0.8, 0.9}</td><td>{0.7, 0.9}</td><td>Avg.</td></tr><tr><td>DisC Fan et al. (2022)</td><td>45.06</td><td>39.42</td><td>42.24</td></tr><tr><td>CIGA Chen et al. (2022)</td><td>57.87</td><td>43.62</td><td>50.75</td></tr><tr><td>GALA Chen et al. (2024)</td><td>76.42</td><td>72.50</td><td>74.46</td></tr><tr><td>Ours</td><td>78.26</td><td>74.32</td><td>76.29</td></tr></table>
+
+We follow the work Chen et al. (2024) and further evaluate our method on Two-Piece Graph Datasets Chen et al. (2024). Particularly, each dataset is generated from a variation of two-piece graph model, denoted as  $\{\mathbf{a},\mathbf{b}\}$ , where a refers to the invariant correlation strength and b refers to the spurious correlation strength. The results are shown in Table 7.
+
+We can observe that our method indeed improves the performance. Particularly, when the invariant correlation strength is weaker than the spurious correlation strength, the performance of our method outperforms state-of-the-art methods, further demonstrating that our method is indeed beneficial for disentangling invariant characteristics.
+
+# G COMPARISON BETWEEN RVQ AND MOLECULE CONCEPT LEARNING
+
+Our molecule concept learning module is significantly different from Residual Vector Quantization (RVQ) Zhuang et al. (2023). In essence, RVQ is to perform feature replacement. Our method is to build the connection to capture critical molecule substructures:
+
+(1) The motivation is different. RVQ aims to leverage the discretized continuous representation to improve the model generalization. As shown in Fig. 6, our method is to learn a series of concepts that could capture critical molecule structure information involving plentiful task-specific invariant molecule characteristics.  
+(2) The operations are different. RVQ first introduces a shared learnable codebook as a discrete latent space. For each node representation in molecular graph, RVQ looks up and fetches the nearest neighbor in the codebook and outputs it as the result. Meanwhile, a sum operation is further used to strengthen the representation ability. During training, similar to VQ-VAE Van Den Oord et al. (2017), RVQ employs the exponential moving average updates for the codebook.
+
+Differently, our concepts are first sampled from a learnable Gaussian distribution. And we employ KAN operation to promote the learning of concepts. Importantly, as shown in equation 6, the weighted residual operation is to align critical molecule substructure features to the corresponding concept, building the connection between molecule substructures and corresponding concepts. During training, we only utilize the task loss, e.g., molecule property prediction, to promote the learned concepts to capture plentiful task-specific invariant molecule characteristics.
+
+(3) Experimental results show the effectiveness of our method. Taking GOOD-ZINC as the example, for iMoLD Zhuang et al. (2023), we first remove the RVQ operation and keep other operations unchanged. We observe that removing RVQ does not affect the performance. Instead, our concept mining module could be plugged into iMoLD to further improve its performance. For the
+
+![](images/1c18ed2d2e24516cb7634ba77449dfde28d9a87478ae998a750b963923963974.jpg)  
+Figure 6: Molecule concepts (colorful stars) are learned from all molecule structures, which are not specific to certain molecule structures. By means of the residual operation (as shown in solid arrows), we can build the connection between molecule substructures and corresponding concepts, which is beneficial for aligning critical molecule substructure features to the corresponding concept.
+
+covariate case of scaffold, the performance is improved by 0.0142. Furthermore, the prediction results of the molecule's ground-state conformation in Fig. 4 and 5 also show that our learned concepts are beneficial for capturing critical substructures.
+
+# H FURTHER ANALYSIS OF THE LOSS FUNCTION
+
+equation 10 is the optimization objective, consisting of three terms, i.e.,  $\mathcal{L}_{pred}$ ,  $\mathcal{L}_{fd}$  (as shown in equation 8), and  $\mathcal{L}_{vd}^t$  (as shown in equation 5).
+
+Proposition 1.  $\mathcal{L}_{vd}^{t}$  mainly is a contrastive loss. Minimizing  $\mathcal{L}_{vd}^{t}$  could enlarge the gap between the disentangled  $\tilde{Z}_t^{Gen}$  and  $Z_{t}^{Spu}$ . Meanwhile,  $\tilde{Z}_t^{Gen}$  is promoted to retain plentiful input-related content.
+
+Analysis. The first term of  $\mathcal{L}_{vd}^t$  is related to the InfoNCE loss Oord et al. (2018). Concretely, we set  $\mathcal{B}_1 = \exp (\mathrm{sim}(\tilde{Z}_t^{Gen},F))$  and  $\mathcal{B}_2 = \exp (\mathrm{sim}(Z_t^{Spu},F))$ :
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \log \frac {\mathcal {B} _ {1}}{\mathcal {B} _ {1} + \mathcal {B} _ {2}} \right] = \mathbb {E} \left[ \log \mathcal {B} _ {1} \right] - \mathbb {E} \left[ \log \left(\mathcal {B} _ {1} + \mathcal {B} _ {2}\right) \right] \tag {14} \\ \leq \mathbb {E} [ \log \mathcal {B} _ {1} ] - \mathbb {E} [ \log \mathcal {B} _ {2} ] \\ \end{array}
+$$
+
+Obviously, maximizing equation 14 narrows the semantic distance between  $\tilde{Z}_t^{Gen}$  and the input  $F$  and enlarges the semantic gap between  $Z_{t}^{Spu}$  and  $F$ . Equally, this operation enlarges the gap between  $\tilde{Z}_{t}^{Gen}$  and  $Z_{t}^{Spu}$ , which is beneficial for strengthening the disentangled ability.
+
+Proposition 2. Minimizing  $\mathcal{L}_{fd}$  is instrumental in achieving the invariant representation  $Z^{Gen}$ .
+
+Analysis. In equation 8,  $\mathcal{L}_{fd}$  mainly consists of two terms, i.e.,  $\sum_{i=0}^{T} \sum_{j=0}^{T} \sin(Z_i^{Gen}, Z_j^{Gen})_{i \neq j}$  and  $\sum_{i=0}^{T} \sin(Z_i^{Gen}, Z_i^{Spu})$ :
+
+$$
+\sum_ {i = 0} ^ {T} \sum_ {j = 0} ^ {T} \sin \left(Z _ {i} ^ {\text {G e n}}, Z _ {j} ^ {\text {G e n}}\right) _ {i \neq j} - \sum_ {i = 0} ^ {T} \sin \left(Z _ {i} ^ {\text {G e n}}, Z _ {i} ^ {\text {S p u}}\right) \tag {15}
+$$
+
+Maximizing equation 15 first enlarges the gap between  $Z_{i}^{Gen}$  and  $Z_{i}^{Spu}$  at the  $i$ -th iteration, which is beneficial for enhancing the disentangled ability. Secondly,  $Z_{i}^{Gen}$  and  $Z_{j}^{Gen}$  separately represent disentangled generalized outputs at different iterations. Here, we assume that when taking a representation involving plentiful invariant information as the input, the disentangled output should contain less spurious information, i.e., the output is equal to the input. Therefore, narrowing the semantic distance is conducive to learning invariant representations across various environments.
+
+![](images/79746f91d82c0097596b0bce2ba4df8c75d79fe77da462ce73708b1409f28cc3.jpg)  
+Figure 7: Visualization examples of molecule concepts. Here, we directly project the captured concepts into the predicted molecules. The green regions represent the focused content.
+
+# I EXAMINING IMPORTANCE OF CONCEPTS ACROSS ITERATIONS
+
+Table 5 analyzes the impact of iteration number. We can observe that when the iteration number increases, inserting concepts could improve the performance effectively, which indicates the effectiveness of the learned concepts. To further analyze the impact of iteration number on concepts, we also make a new ablation experiment, i.e., removing the concepts at the 4th, and 8th iteration. The results are shown as follows:
+
+Table 8: Examining importance of concepts across iterations.  
+
+<table><tr><td>Iterations</td><td>Concepts</td><td>CovSold</td><td>CetSold</td><td>CovSize</td><td>CetSize</td></tr><tr><td>4</td><td>✓</td><td>0.1285</td><td>0.1008</td><td>0.1573</td><td>0.0963</td></tr><tr><td>4</td><td>-</td><td>0.1402</td><td>0.1117</td><td>0.1738</td><td>0.1102</td></tr><tr><td>8</td><td>✓</td><td>0.1187</td><td>0.0765</td><td>0.1421</td><td>0.0852</td></tr><tr><td>8</td><td>-</td><td>0.1189</td><td>0.0783</td><td>0.1449</td><td>0.0897</td></tr></table>
+
+We can observe that when the iteration number is small, inserting concepts could significantly affect the performance. Instead, when the iteration number is large, the impact of concepts is somewhat weak. The reason may be that when the iteration number is small, leveraging concepts is beneficial for capturing critical substructures, improving the performance. As the iteration number increases, concepts progressively pay attention to meaningful substructure characteristics, reducing the iteration impact on performance.
+
+# J MORE ANALYSIS OF MOLECULE CONCEPTS
+
+We perform an ablation experiment of the decoder that calculates task-related representation. We observe that for classification and regression tasks, using decoders with more layers leads to the performance degradation. The reason may be that this operation introduces more parameters, resulting in the overfitting risk.
+
+In Fig. 7, we show some visualization examples of molecule concepts. Since these concepts are learned from all molecule data, they are not specific to certain kinds of molecules, which is beneficial for improving the generalization of molecule concepts. For different predictions, we can observe that the learned concepts could indeed focus on critical molecule substructures, which strengthens the performance of molecule prediction.
+
+# K ANALYSIS OF COMPUTATIONAL COSTS
+
+We observe that introducing feedback iterations indeed increases computational costs. During training, based on the same batch size, the memory is increased by around 0.9GB. However, in Fig. 8, we observe that the convergence speed is significantly faster than the baseline Zhuang et al. (2023).
+
+During inference, compared with one-step disentanglement, the inference time of using 8 iterations is increased by around 0.06s. Although the inference time is longer, the disentangled invariant representations are more accurate, which is instrumental in improving the generalization ability. Particularly, for molecules with multiple atoms and various structures, we observe that employing
+
+![](images/e46f57019018a029984b24230dd1365e012f11eb6604833a8a2d7b362947bc6c.jpg)  
+Figure 8: The training loss of the baseline Zhuang et al. (2023) and our method.
+
+our method could significantly improve the performance of downstream tasks, further demonstrating the effectiveness of our method.
+
+# L QUANTIFY CONTRIBUTION OF FEEDBACK MECHANISM
+
+To quantify the contribution of feedback iterations, we make an ablation experiment of the iteration number. Here, we only change the iteration number. Other operations are kept unchanged. The results are shown in Table 5.
+
+Table 9: Quantify contribution of feedback mechanism on highly complex molecules.  
+
+<table><tr><td>1</td><td>4</td><td>8</td><td>12</td><td>16</td></tr><tr><td>0.3014</td><td>0.1867</td><td>0.1341</td><td>0.1279</td><td>0.1254</td></tr></table>
+
+We can observe that compared with one-step disentanglement, using multiple feedback iterations indeed improves the performance, indicating that our method is beneficial for decomposing invariant molecule representations. Besides, we also observe that when the iteration number is larger than 12, the performance improvement is limited. The reason is that for the current dataset, using 12 iterations is sufficient for disentangling highly complex molecules.
+
+Finally, we observe that complex molecules usually contain more atoms. Therefore, to further demonstrate the effectiveness of our method on highly complex molecules, we select those molecules with more than 12 atoms for additional evaluation. The results are shown in Table 9.
+
+We can see that for complex molecules, using one-step disentanglement could not decompose invariant characteristics effectively. Using multiple feedback iterations could indeed decouple the representations involving plentiful invariant information, which strengthens the generalization ability.
+
+# M PROTEIN-LIGAND BINDING-AFFINITY PREDICTION
+
+For OOD molecular representation learning, in Table 1 and 2, we show the corresponding results. Furthermore, in Table 3, our method is evaluated on an additional task, i.e., molecule's ground-state prediction. These results demonstrate that our method could effectively disentangle invariant molecule characteristics, which strengthens the generalization ability.
+
+Finally, to further demonstrate the superiorities of our method, we evaluate our method on protein-ligand binding-affinity prediction. Binding affinity is an important metric in biology and medicinal chemistry that measures the strength of a reversible association between biological macromolecules,
+
+such as proteins or DNA, and small-molecule ligands, such as drugs Boulougouri et al. (2024). Here, we follow the work Boulougouri et al. (2024) and employ the same datasets and metrics to verify our feedback disentanglement method. The results are shown in Table 10.
+
+Table 10: Analysis of protein-ligand binding-affinity prediction.  
+
+<table><tr><td>Method</td><td>RMSE</td><td>MAE</td><td>R</td></tr><tr><td>MAT</td><td>1.457</td><td>1.154</td><td>0.747</td></tr><tr><td>DimeNet</td><td>1.453</td><td>1.138</td><td>0.752</td></tr><tr><td>CMPNN</td><td>1.408</td><td>1.117</td><td>0.765</td></tr><tr><td>SR-BIND Boulougouri et al. (2024)</td><td>1.383</td><td>1.122</td><td>0.780</td></tr><tr><td>CFD</td><td>1.356</td><td>1.073</td><td>0.794</td></tr></table>
+
+Here, MAT, DimeNet, CMPNN are three GNN-based methods. RMSE, MAE, and R are three predefined metrics. We can observe that for the challenging protein-ligand binding-affinity prediction, employing our method still improves the performance for the given three metrics. This further demonstrates that for biological macromolecules, using feedback disentanglement is meaningful, which improves the generalization of the learned molecule representations.
+
+# N MORE INTERPRETATIONS OF KAN
+
+In general, molecules consist of multiple atoms and bonds. For certain biomacromolecules, their structures are entanglement of various substructures, which poses the challenge for learning molecule concepts that focus on critical substructures.
+
+Inspired by the Kolmogorov-Arnold representation theorem Kolmogorov (1961), KAN allows learning custom activations of the network. In this way, it is possible to analyze the contribution of individual components of the input, then providing a more transparent view of the network's decision-making process.
+
+Thus, using KAN could sufficiently convert the analyzed components to the corresponding concepts that involve plentiful molecule-aware substructure information, which is beneficial for strengthening the generalization of molecule representations.

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+# CFG++: MANIFOLD-CONSTRAINED CLASSIFIER FREE GUIDANCE FOR DIFFUSION MODELS
+
+Hyungjin Chung*, Jeongsol Kim*, Geon Yeong Park*, Hyelin Nam*, Jong Chul Ye KAIST
+
+*: Equal Contribution
+
+{hj.chung, jeongsol, pky3436, hyelin.nam, jong.ye}@kaist.ac.kr
+
+![](images/77b8b0bac992855e663a7d5729ca86c67ee5f2e0c716ddea0bad2b4d37258c94.jpg)  
+beautiful lady, freckles, big smile, blue eyes, short ginger hair, wearing a floral blue vest top, soft light, dark gray background  
+Figure 1: (Top) Comparison of T2I results by SDXL-Turbo for the prompt "kayak in the water, optical color, aerial view, rainbow". The CFG-guided image has significant artifacts, which are reduced in the CFG++ version. (Middle) DDIM Inversion results under CFG show noticeable artifacts at various CFG scales, which are significantly reduced by CFG++. (Bottom) The evolution of denoised estimates differs between CFG and CFG++. CFG exhibits sudden shifts and intense color saturation early in reverse diffusion, while CFG++ transitions smoothly from low to high-resolution.
+
+# ABSTRACT
+
+Classifier-free guidance (CFG) is a fundamental tool in modern diffusion models for text-guided generation. Although effective, CFG has notable drawbacks. For instance, DDIM with CFG lacks invertibility, complicating image editing; furthermore, high guidance scales, essential for high-quality outputs, frequently
+
+result in issues like mode collapse. This paper reveals that the problems may stem from the off-manifold phenomenon associated with CFG, rather than the diffusion models themselves. More specifically, inspired by the recent advancements of diffusion model-based inverse problem solvers (DIS), we reformulate text-guidance as an inverse problem with a text-conditioned score matching loss and develop CFG++, a novel approach that tackles the off-manifold challenges inherent in traditional CFG. CFG++ features a surprisingly simple fix to CFG, yet it offers significant improvements, including better sample quality for text-to-image generation, invertibility, smaller guidance scales, etc. Furthermore, CFG++ enables seamless interpolation between unconditional and conditional sampling at lower guidance scales, consistently outperforming traditional CFG at all scales. Moreover, CFG++ can be easily integrated into high-order diffusion solvers and naturally extends to distilled diffusion models. Experimental results confirm that our method significantly enhances performance in text-to-image generation, DDIM inversion, editing, and solving inverse problems, suggesting a wide-ranging impact and potential applications in various fields that utilize text guidance. Project Page: https://cfgpp-diffusion.github.io.
+
+# 1 INTRODUCTION
+
+Classifier-free guidance (CFG) (Ho & Salimans, 2021) forms the key basis of modern text-guided generation with diffusion models (Dhariwal & Nichol, 2021; Rombach et al., 2022). Nowadays, it is common practice to train a diffusion model with large-scale paired text-image data (Schuhmann et al., 2022), so that sampling (i.e. generating) a signal (e.g. image, video) from a diffusion model can either be done unconditionally from  $p_{\theta}(\pmb{x}|\varnothing) \equiv p_{\theta}(\pmb{x})$ , or conditionally from  $p_{\theta}(\pmb{x}|c)$ , where  $c$  is the text conditioning. Once trained, it seems natural that one would acquire samples from the conditional distribution by simply solving the probability-flow ODE or SDE sampling (Song et al., 2021b;a; Karras et al., 2022) with the conditional score function. In practice, however, it is observed that the conditioning signal is insufficient when used naively. To emphasize the guidance, one uses the guidance scale  $\omega > 1$ , where the direction can be defined by the direction from the unconditional score to the conditional score (Ho & Salimans, 2021).
+
+In modern text-to-image (T2I) diffusion models, the guidance scale  $\omega$  is typically set within the range of [5.0, 30], referred to as the moderately high range of CFG guidance (Chen et al., 2024; Podell et al., 2023). The insufficiency in guidance also holds for classifier guidance (Dhariwal & Nichol, 2021; Song et al., 2021b) so that a scale of 10 was used. While using a high guidance scale yields higher-quality images with better alignment to the condition, it is also prone to mode collapse, reduces sample diversity, and yields an inevitable accumulation of errors during the sampling process. One example is DDIM inversion (Dhariwal & Nichol, 2021), a pivotal technique for controllable synthesis and editing (Mokady et al., 2023), where running the inversion process with  $\omega > 1.0$  leads to significant compromise in the reconstruction performance (Mokady et al., 2023; Wallace et al., 2023). Another extreme example would be score distillation sampling (SDS) (Poole et al., 2022), where the guidance scale in the order of a few hundred is chosen. Using such a high guidance scale leads to better asset quality to some extent, but induces blurry and saturated results. Several research efforts have been made to mitigate this downside by exploring methods where using a smaller guidance scale suffices (Wang et al., 2024; Liang et al., 2023). Although recent progress in SDS-type methods has reduced the necessary guidance scale to a range that is similar to those of ancestral samplers, using a moderately large  $\omega$  is considered an inevitable choice.
+
+In this work, we aim to give an answer to this conundrum by revisiting the geometric view of diffusion models. In particular, inspired by the recent advances in diffusion-based inverse problem solvers (DIS) (Kadkhodaie & Simoncelli, 2021; Chung et al., 2023a; Song et al., 2023; Kim et al., 2024b; Chung et al., 2024), we reformulate the text guidance as an inverse problem with a text-conditioned score-matching loss and derive a reverse diffusion sampling strategy by utilizing decomposed diffusion sampling (DDS) (Chung et al., 2024). This results in a surprisingly simple fix of CFG to the sampling process without any computational overhead. The resulting process, which we call CFG++, works with a small guidance scale, typically  $\lambda \in [0.0,1.0]$ , that smoothly interpolates between unconditional and conditional sampling, with  $\lambda = 1.0$  having a similar effect as using CFG sampling with  $\omega \sim 12.5$  at 50 neural function evaluation (NFE). Furthermore, CFG++ reduces the inversion
+
+<table><tr><td colspan="2">Algorithm 1 Reverse Diffusion with CFG</td><td>Algorithm 2 Reverse Diffusion with CFG++</td></tr><tr><td colspan="2">Require: xT ~ N(0,Id), 0 ≤ ω ∈ R</td><td>Require: xT ~ N(0,Id), λ ∈ [0,1]</td></tr><tr><td colspan="2">1: for i = T to 1 do</td><td>1: for i = T to 1 do</td></tr><tr><td colspan="2">2:   \hat{\epsilon}_{c}^{\omega}(x_{t}) = \hat{\epsilon}_{\varnothing}(x_{t}) + ω[\hat{\epsilon}_{c}(x_{t}) - \hat{\epsilon}_{\varnothing}(x_{t})]</td><td>2:   \hat{\epsilon}_{c}^{\lambda}(x_{t}) = \hat{\epsilon}_{\varnothing}(x_{t}) + λ[\hat{\epsilon}_{c}(x_{t}) - \hat{\epsilon}_{\varnothing}(x_{t})]</td></tr><tr><td colspan="2">3:   \hat{x}_{c}^{\omega}(x_{t}) ← (x_{t} - \sqrt{1 - \bar{\alpha}_{t}}\hat{\epsilon}_{c}^{\omega}(x_{t}))/\sqrt{\bar{\alpha}_{t}}</td><td>3:   \hat{x}_{c}^{\lambda}(x_{t}) ← (x_{t} - \sqrt{1 - \bar{\alpha}_{t}}\hat{\epsilon}_{c}^{\lambda}(x_{t}))/\sqrt{\bar{\alpha}_{t}}</td></tr><tr><td colspan="2">4:   x_{t-1} = \sqrt{\bar{\alpha}_{t-1}}\hat{x}_{c}^{\omega}(x_{t}) + \sqrt{1 - \bar{\alpha}_{t-1}}\hat{\epsilon}_{c}^{\omega}(x_{t})</td><td>4:   x_{t-1} = \sqrt{\bar{\alpha}_{t-1}}\hat{x}_{c}^{\lambda}(x_{t}) + \sqrt{1 - \bar{\alpha}_{t-1}}\hat{\epsilon}_{\varnothing}(x_{t})</td></tr><tr><td colspan="2">5: end for</td><td>5: end for</td></tr><tr><td colspan="2">6: return x0</td><td>6: return x0</td></tr></table>
+
+Figure 2: Comparison between reverse diffusion process by CFG and CFG++. CFG++. proposes a simple but surprisingly effective fix: using  $\hat{\epsilon}_{\emptyset}(\pmb{x}_t)$  instead of  $\hat{\epsilon}_{c}^{\omega}(\pmb{x}_t)$  in updating  $\pmb{x}_{t-1}$ .
+
+error, enhancing and simplifying image reconstruction, as well as editing. Comparing CFG++ against CFG shows that we achieve consistently better sample quality for text-to-image (T2I) generation, significantly better DDIM inversion capabilities that lead to enhanced reconstruction and editing, and enabling the incorporation of CFG guidance to diffusion inverse solvers (DIS) (Chung et al., 2023a). While the applications of CFG++ that we show in this work are limited, we believe that our work will have a broad impact that can be applied to all applications that leverage text guidance through the traditional CFG.
+
+# 2 BACKGROUND
+
+Diffusion models. Diffusion models (Ho et al., 2020; Song et al., 2021b; Karras et al., 2022) are generative models designed to learn the reversal of a forward noising process. This process starts with an initial distribution  $p_0(\pmb{x})$  where  $\pmb{x} \in \mathbb{R}^n$ , and progresses towards the standard Gaussian distribution  $p_T(\pmb{x}) \approx \mathcal{N}(\pmb{0}, \pmb{I})$ , utilizing forward Gaussian perturbation kernels. Sampling from the data distribution can be performed by solving either the reverse stochastic differential equation (SDE) or the equivalent probability-flow ordinary differential equation (PF-ODE) (Song et al., 2021b). For example, under the choice  $p(\pmb{x}_t | \pmb{x}_0) = \mathcal{N}(\pmb{x}_0, \sigma_t^2 \pmb{I})$ , the generative PF-ODE reads
+
+$$
+d \boldsymbol {x} _ {t} = - \sigma_ {t} \nabla_ {\boldsymbol {x} _ {t}} \log p (\boldsymbol {x} _ {t}) d t = \frac {\boldsymbol {x} _ {t} - \mathbb {E} [ \boldsymbol {x} _ {0} | \boldsymbol {x} _ {t} ]}{\sigma_ {t}} d t, \quad \boldsymbol {x} _ {T} \sim p _ {T} (\boldsymbol {x} _ {T}), \tag {1}
+$$
+
+where Tweedie's formula (Efron, 2011)  $\mathbb{E}[\pmb{x}_0|\pmb{x}_t] = \pmb{x}_t + \sigma_t^2\nabla_{\pmb{x}_t}\log p(\pmb{x}_t)$  is applied to achieve the second equality. When aiming for a text-conditional diffusion model that can condition on arbitrary  $c$ , one can extend epsilon matching to a conditional one where the condition is dropped with a certain probability (Ho & Salimans, 2021) so that null conditioning with  $c = \varnothing$  is possible. The neural network architecture of  $\epsilon_{\theta}$  is designed so that the condition  $c$  can modulate the output through cross attention (Rombach et al., 2022). For simplicity in notation throughout the paper, we define  $\hat{\epsilon}_{c} := \epsilon_{\theta}(\pmb{x}_{t},\pmb{c})$  and  $\hat{\epsilon}_{\emptyset} := \epsilon_{\theta}(\pmb{x}_{t},\emptyset)$  by dropping  $\theta$  and  $\pmb{x}_t$ .
+
+Extending the result of Tweedie's formula to the unconditional case under the variance preserving (VP) framework of DDPMs (Ho et al., 2020), we have
+
+$$
+\mathbb {E} \left[ \boldsymbol {x} _ {0} \mid \boldsymbol {x} _ {t}, \varnothing \right] = \hat {\boldsymbol {x}} _ {\varnothing} \left(\boldsymbol {x} _ {t}\right) := \left(\boldsymbol {x} _ {t} - \sqrt {1 - \bar {\alpha} _ {t}} \hat {\epsilon} _ {\varnothing} \left(\boldsymbol {x} _ {t}\right)\right) / \sqrt {\bar {\alpha} _ {t}}. \tag {2}
+$$
+
+Leveraging (2), it is common to use DDIM sampling (Song et al., 2021b) to solve the conditional probability-flow ODE (PF-ODE) of the generative process. Specifically, a single iteration reads
+
+$$
+\hat {\boldsymbol {x}} _ {\varnothing} = \left(\boldsymbol {x} _ {t} - \sqrt {1 - \bar {\alpha} _ {t}} \hat {\epsilon} _ {\varnothing}\right) / \sqrt {\bar {\alpha} _ {t}} \tag {3}
+$$
+
+$$
+\boldsymbol {x} _ {t - 1} = \sqrt {\bar {\alpha} _ {t - 1}} \hat {\boldsymbol {x}} _ {\varnothing} + \sqrt {1 - \bar {\alpha} _ {t - 1}} \hat {\epsilon} _ {\varnothing}, \tag {4}
+$$
+
+where  $\hat{\pmb{x}}_{\emptyset} := \hat{\pmb{x}}_{\emptyset}(\pmb{x}_t) = \mathbb{E}[x_0|\pmb{x}_t,\emptyset]$  is the denoised signal by Tweedie's formula, and (4) corresponds to the renoising step<sup>1</sup>. This is repeated for  $t = T,T - 1,\dots ,1$
+
+For modern diffusion models, it is common to train a diffusion model in the latent space (Rombach et al., 2022) with the latent variable  $z$ . While most of our experiments are performed with latent diffusion models (LDM), as our framework holds both for pixel- and latent-diffusion models, we will simply use the notation  $x$  regardless of the choice.
+
+Classifier free guidance. For a conditional diffusion, Ho & Salimans (2021) considered the sharpened posterior distribution  $p^{\omega}(\pmb{x}|\pmb{c}) \propto p(\pmb{x})p(\pmb{c}|\pmb{x})^{\omega}$ . Using Bayes rule for some timestep  $t$ ,
+
+$$
+\nabla_ {\boldsymbol {x}} \log p ^ {\omega} (\boldsymbol {x} _ {t} | \boldsymbol {c}) = \nabla_ {\boldsymbol {x} _ {t}} \log p (\boldsymbol {x} _ {t}) + \omega \left(\nabla_ {\boldsymbol {x} _ {t}} \log p (\boldsymbol {x} _ {t} | \boldsymbol {c}) - \nabla_ {\boldsymbol {x} _ {t}} \log p (\boldsymbol {x} _ {t})\right) \tag {5}
+$$
+
+![](images/295117670bebf644f7b83d3ddc79b38fad845d8ace7283132e8ef2dfe6874db5.jpg)  
+(a)  
+Figure 3: Off-manifold phenomenon of CFG arise from: (a) the typical CFG scale  $\omega > 1.0$  which leads to extrapolation and deviation from the piecewise linear data manifold, and (b) CFG's renoising process, which introduces a nonzero offset  $\Delta^{\omega}$  from the correct manifold. CFG++ effectively mitigates all these artifacts.
+
+![](images/2684195f84afb1bdd1fcc7ad4cd6fca1e81f47cda3e114f23da221d9abd218a7.jpg)  
+(b)
+
+Parametrizing the score function with  $\epsilon_{\theta}$ , as in DDPM (Ho et al., 2020), we have
+
+$$
+\hat {\epsilon} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t}\right) := \hat {\epsilon} _ {\varnothing} \left(\boldsymbol {x} _ {t}\right) + \omega \left[ \hat {\epsilon} _ {c} \left(\boldsymbol {x} _ {t}\right) - \hat {\epsilon} _ {\varnothing} \left(\boldsymbol {x} _ {t}\right) \right] \tag {6}
+$$
+
+where we introduce a compact notation  $\hat{\epsilon}_c^{\omega}$  that guides the sampling from the sharpened posterior. When sampling with CFG guidance with DDIM sampling, one replaces  $\hat{\epsilon}_{\emptyset}$  with  $\hat{\epsilon}_c^{\omega}$  for both the Tweedie estimate (3) and the subsequent update step (4), leading to Algorithm 1.
+
+Diffusion model-based inverse problem solvers. Diffusion model-based inverse problem solvers (DIS) aims to perform posterior sampling from an unconditional diffusion model (Kadkhodaie & Simoncelli, 2021; Chung et al., 2023a; Song et al., 2023; Kim et al., 2024b). Specifically, for a given loss function  $\ell(\pmb{x})$  which often stems from the likelihood, the goal of DIS is to address the optimization problem  $\min_{\pmb{x} \in \mathcal{M}} \ell(\pmb{x})$ , where  $\mathcal{M}$  represents the clean data manifold sampled from the unconditional distribution  $p_0(\pmb{x})$ . It is essential to navigate in a way that minimizes cost while also identifying the correct clean manifold.
+
+Chung et al. (2023a) proposed diffusion posterior sampling (DPS), where the updated estimate from the noisy sample  $\boldsymbol{x}_t \in \mathcal{M}_t$  is constrained to stay on the same noisy manifold  $\mathcal{M}_t$ . This is achieved by computing the manifold constrained gradient (MCG) (Chung et al., 2022) on a noisy sample  $\boldsymbol{x}_t \in \mathcal{M}_t$  as  $\nabla_{\boldsymbol{x}_t}^{mcg}\ell(\boldsymbol{x}_t) \coloneqq \nabla_{\boldsymbol{x}_t}\ell(\hat{\boldsymbol{x}}_t)$ , where  $\hat{\boldsymbol{x}}_t$  is the denoised sample in (2) through Tweedie's formula (Efron, 2011). The resulting algorithm with DDIM (Song et al., 2021b) can be stated as follows:
+
+$$
+\boldsymbol {x} _ {t - 1} = \sqrt {\bar {\alpha} _ {t - 1}} \left(\hat {\boldsymbol {x}} _ {\varnothing} - \gamma_ {t} \nabla_ {\boldsymbol {x} _ {t}} \ell (\hat {\boldsymbol {x}} _ {\varnothing})\right) + \sqrt {1 - \bar {\alpha} _ {t - 1}} \hat {\boldsymbol {e}} _ {\varnothing}, \tag {7}
+$$
+
+where  $\gamma_{t} > 0$  denotes the step size. Under the linear manifold assumption (Chung et al., 2022; 2023a), this allows precise transition to  $\mathcal{M}_{t-1}$ . To mitigate the computational complexity and the instability of neural network backprop, Chung et al. (2024) shows that (7) can be equivalently represented as
+
+$$
+\boldsymbol {x} _ {t - 1} \simeq \sqrt {\bar {\alpha} _ {t - 1}} \left(\hat {\boldsymbol {x}} _ {\varnothing} - \gamma_ {t} \nabla_ {\hat {\boldsymbol {x}} _ {\varnothing}} \ell \left(\hat {\boldsymbol {x}} _ {\varnothing}\right)\right) + \sqrt {1 - \bar {\alpha} _ {t - 1}} \hat {\boldsymbol {\epsilon}} _ {\varnothing} \tag {8}
+$$
+
+under further assumptions on  $\mathcal{M}$ . This method, often referred to as the decomposed diffusion sampling (DDS), bypasses the computation of the score Jacobian, similar to Poole et al. (2022), making it stable and suitable for large-scale medical imaging inverse problems (Chung et al., 2024). In the following, we leverage the insight from DDS to propose an algorithm to improve upon the CFG algorithm.
+
+# 3 CFG++: MANIFOLD-CONSTRAINED CFG
+
+# 3.1 DERIVATION OF CFG++
+
+Instead of uncritically adopting the sharpened posterior distribution  $p^{\omega}(\boldsymbol{x}|\boldsymbol{c}) \propto p(\boldsymbol{x})p(\boldsymbol{c}|\boldsymbol{x})^{\omega}$  as introduced by Ho & Salimans (2021), we adopt a fundamentally different strategy by reformulating text-guidance as an optimization problem. Specifically, our focus is on identifying a loss function  $\ell(\boldsymbol{x})$  in (8) such that, when minimized under the condition set by the text, enables the reverse diffusion process to generate samples that increasingly satisfy the text condition progressively.
+
+One of the most significant contributions of this paper is to reveal that the text-conditioned score matching loss or alternatively, score distillation sampling (SDS) loss (Poole et al., 2022) is ideally
+
+suited for our purpose. Specifically, we are interested in solving the following inverse problem through diffusion models:
+
+$$
+\min  _ {\boldsymbol {x} \in \mathcal {M}} \ell_ {s d s} (x), \quad \ell_ {s d s} (\boldsymbol {x}) := \| \boldsymbol {\epsilon} _ {\theta} (\sqrt {\bar {\alpha} _ {t}} \boldsymbol {x} + \sqrt {1 - \bar {\alpha} _ {t}} \boldsymbol {\epsilon}, \boldsymbol {c}) - \boldsymbol {\epsilon} \| _ {2} ^ {2} \tag {9}
+$$
+
+This implies that our goal is to identify solutions on the clean manifold  $\mathcal{M}$  that optimally aligns with the text condition  $c$ .
+
+To avoid the Jacobian computation, in this paper, we attempt to solve (9) through DDS in (8). The resulting sampling process from reverse diffusion is then given by
+
+$$
+\boldsymbol {x} _ {t - 1} = \sqrt {\bar {\alpha} _ {t - 1}} \left(\hat {\boldsymbol {x}} _ {\varnothing} - \gamma_ {t} \nabla_ {\hat {\boldsymbol {x}} _ {\varnothing}} \ell_ {s d s} (\hat {\boldsymbol {x}} _ {\varnothing})\right) + \sqrt {1 - \bar {\alpha} _ {t - 1}} \hat {\epsilon} _ {\varnothing}. \tag {10}
+$$
+
+By using  $\pmb{x}_t = \sqrt{\bar{\alpha}_t} \pmb{x} + \sqrt{1 - \bar{\alpha}_t} \pmb{\epsilon}$  from the clean image  $\pmb{x} \in \mathcal{M}$ , we can equivalently write the loss as  $\ell_{sds}(\pmb{x}) = \frac{\alpha_t}{1 - \bar{\alpha}_t} \| \pmb{x} - \hat{\pmb{x}}_c \|^2$ , which leads to
+
+$$
+\boldsymbol {x} _ {t - 1} = \sqrt {\bar {\alpha} _ {t - 1}} \left(\hat {\boldsymbol {x}} _ {\varnothing} + \lambda \left(\hat {\boldsymbol {x}} _ {\boldsymbol {c}} - \hat {\boldsymbol {x}} _ {\varnothing}\right)\right) + \sqrt {1 - \bar {\alpha} _ {t - 1}} \hat {\epsilon} _ {\varnothing} \tag {11}
+$$
+
+where  $\lambda := \frac{2\bar{\alpha}_t}{1 - \bar{\alpha}_t}\gamma_t$ . Using the CFG notation  $\hat{\epsilon}_{\mathbf{c}}^{\lambda}(\boldsymbol{x}_t) := \hat{\epsilon}_{\emptyset}(\boldsymbol{x}_t) + \lambda [\hat{\epsilon}_{\mathbf{c}}(\boldsymbol{x}_t) - \hat{\epsilon}_{\emptyset}(\boldsymbol{x}_t)]$  and  $\hat{x}_{\emptyset} + \lambda (\hat{x}_{\mathbf{c}} - \hat{x}_{\emptyset}) = (\boldsymbol{x}_t - \sqrt{1 - \bar{\alpha}_t}\hat{\epsilon}_{\mathbf{c}}^{\lambda}(\boldsymbol{x}_t)) / \sqrt{\bar{\alpha}_t}$ , (11) can be equivalently represented as
+
+$$
+\hat {\boldsymbol {x}} _ {\boldsymbol {c}} ^ {\lambda} \left(\boldsymbol {x} _ {t}\right) = \left(\boldsymbol {x} _ {t} - \sqrt {1 - \bar {\alpha} _ {t}} \hat {\boldsymbol {\epsilon}} _ {\boldsymbol {c}} ^ {\lambda} \left(\boldsymbol {x} _ {t}\right)\right) / \sqrt {\bar {\alpha} _ {t}} \tag {12}
+$$
+
+$$
+\boldsymbol {x} _ {t - 1} = \sqrt {\bar {\alpha} _ {t - 1}} \hat {\boldsymbol {x}} _ {c} ^ {\lambda} (\boldsymbol {x} _ {t}) + \sqrt {1 - \bar {\alpha} _ {t - 1}} \hat {\epsilon} _ {\varnothing} (\boldsymbol {x} _ {t}) \tag {13}
+$$
+
+which is summarized in Algorithm 2. By examining Algorithm 1 and Algorithm 2, we observe that CFG and CFG++ are mostly the same, with a crucial difference in the renoising process. This surprisingly simple fix of utilizing the unconditional noise  $\hat{\epsilon}_{\emptyset}(\pmb{x}_t)$  instead of  $\hat{\epsilon}_{c}^{\omega}(\pmb{x}_t)$  leads to a smoother trajectory of generation, (Fig. 1 bottom) and generation with superior quality (Fig. 1 top).
+
+Although we focus our construction of the solver on DDIM for simplicity, note that DDIM is just one way of reverse sampling. There are other widely-used solvers such as Karras Euler and its variants (Karras et al., 2022), DPM-solver (Lu et al., 2022a;b), their ancestral variants $^{2}$ , etc. For most widely used solvers up to the second order, a single-step update of solving the unconditional PF-ODE can be represented as
+
+$$
+\boldsymbol {x} _ {i} = \hat {\boldsymbol {x}} _ {\emptyset} \left(\boldsymbol {x} _ {i - 1}\right) + a _ {i} \hat {\boldsymbol {x}} _ {\emptyset} \left(\boldsymbol {x} _ {i - 1}\right) + b _ {i} \hat {\boldsymbol {x}} _ {\emptyset} \left(\boldsymbol {x} _ {i - 2}\right) + c _ {i} \boldsymbol {x} _ {i - 1} + d _ {i} \epsilon , \quad \epsilon \sim \mathcal {N} (0, I), \tag {14}
+$$
+
+with  $d_{i} \neq 0$  when one uses an ancestral sampler. Note that the first right-hand side term in (14) corresponds to the denosing, whereas the rest terms describe the higher-order corrected version of the renoising process. As the goal of CFG++ is to optimize the denoising process under the text-guidance while keeping the renoising components equivalent to unconditional sampling, applying CFG++ to the general iteration in (14) simply leads to
+
+$$
+\begin{array}{l} \boldsymbol {x} _ {i} = \left(\hat {\boldsymbol {x}} _ {\emptyset} (\boldsymbol {x} _ {i - 1}) - \lambda \nabla_ {\hat {\boldsymbol {x}} _ {\emptyset} (\boldsymbol {x} _ {i - 1})} \ell_ {s d s} \left(\hat {\boldsymbol {x}} _ {\emptyset} (\boldsymbol {x} _ {i - 1})\right)\right) + a _ {i} \hat {\boldsymbol {x}} _ {\emptyset} (\boldsymbol {x} _ {i - 1}) + b _ {i} \hat {\boldsymbol {x}} _ {\emptyset} (\boldsymbol {x} _ {i - 2}) + c _ {i} \boldsymbol {x} _ {i - 1} + d _ {i} \epsilon \\ = \hat {x} _ {c} ^ {\lambda} \left(\boldsymbol {x} _ {i - 1}\right) + a _ {i} \hat {\boldsymbol {x}} _ {\varnothing} \left(\boldsymbol {x} _ {i - 1}\right) + b _ {i} \hat {\boldsymbol {x}} _ {\varnothing} \left(\boldsymbol {x} _ {i - 2}\right) + c _ {i} \boldsymbol {x} _ {i - 1} + d _ {i} \boldsymbol {\epsilon}, \boldsymbol {\epsilon} \sim \mathcal {N} (0, I). \tag {15} \\ \end{array}
+$$
+
+Moreover, CFG++ naturally extends to distilled diffusion models, such as SDXL-turbo (Sauer et al., 2023) and SDXL-lightning (Lin et al., 2024), where akin to (15), we use the conditional denoised estimate, but for the rest of the noise components in the Euler solver, we use the unconditional estimate. For details in how to apply CFG++ to various solvers, see Appendix A.
+
+# 3.2 GEOMETRY OF CFG++
+
+Mitigating off-manifold phenomenon. In Fig. 1 (bottom), we illustrate the evolution of the posterior mean through Tweedie's formula during the reverse diffusion process. Notably, in the early phases of reverse diffusion sampling under CFG, there is a sudden shift in the image and intense color saturation. Conversely, CFG++ is free from the undesirable off-manifold phenomenon. In the following, we investigate why this is the case.
+
+Note that the denoised estimate of CFG and CFG++ at time  $t$  can be equivalently represented as
+
+$$
+\hat {\boldsymbol {x}} _ {c} ^ {\lambda} \left(\boldsymbol {x} _ {t}\right) = (1 - \lambda) \hat {\boldsymbol {x}} _ {\varnothing} \left(\boldsymbol {x} _ {t}\right) + \lambda \hat {\boldsymbol {x}} _ {c} \left(\boldsymbol {x} _ {t}\right), \quad \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t}\right) = (1 - \omega) \hat {\boldsymbol {x}} _ {\varnothing} \left(\boldsymbol {x} _ {t}\right) + \omega \hat {\boldsymbol {x}} _ {c} \left(\boldsymbol {x} _ {t}\right). \tag {16}
+$$
+
+![](images/fd0626cb731546efdec0ccaa383a8a24c26d3e21ef71e9602985012716d41057.jpg)  
+Figure 4: Text-conditioned score matching loss throughout the reverse diffusion sampling for both CFG and CFG++ in SDXL. Avg. loss computed with 55 prompts from (Chen et al., 2024).
+
+$\lambda, \omega \in [0,1]$  facilitates an interpolation. However, with  $\omega > 1.0$ , CFG extrapolates beyond the unconditional and conditional estimates. Consequently, under the assumption that the clean manifold be piecewise linear (Chung et al., 2024), the conditional posterior mean estimates from CFG obtained with a guidance scale outside the range of  $[0,1]$ , can easily extend beyond the piecewise linear manifold. This may lead to the estimates potentially "falling off" the data manifold, as depicted by an orange arrow pointing downwards in Fig. 3(a). Thus, we select  $\lambda \in [0,1]$  as the guidance scale for CFG++ to ensure it remains an interpolation between the unconditional and conditional estimate, thus preventing it from falling off the clean data manifold. An additional source of the off-manifold phenomenon in CFG occurs during the transition from the clean manifold  $\mathcal{M}$  to the subsequent noisy manifold  $\mathcal{M}_{t-1}$ , due to a similar off-manifold phenomenon from the large guidance scale (i.e. extrapolation), illustrated in Fig. 3(b).
+
+Text-Image alignment. In Fig. 1 (top), and Fig. 10, we observe enhanced text-to-image alignment achieved with CFG++. This enhanced alignment capability is a natural consequence of CFG++, which directly minimizes the text-conditioned score-matching loss as shown in (9). In contrast, CFG indirectly seeks text alignment through the sharpened posterior distribution  $p^{\omega}(\boldsymbol{x}|\boldsymbol{c}) \propto p(\boldsymbol{x})p(\boldsymbol{c}|\boldsymbol{x})^{\omega}$ . Therefore, CFG++ inherently outperforms CFG in terms of text alignment due to its fundamental design principle. For example, Fig. 4 displays the normalized text-conditioned score matching loss, represented as  $(1 - \bar{\alpha}_t) \| \epsilon - \epsilon_\theta (\boldsymbol{x}_t, \boldsymbol{c}) \|^2 / \bar{\alpha}_t = \| \boldsymbol{x} - \hat{\boldsymbol{x}}_c \|^2$ , throughout the reverse diffusion sampling process for both CFG and CFG++. The loss plot associated with CFG shows fluctuations and maintains a noticeable gap compared to CFG++. Even after the completion of the reverse diffusion process. In fact, the fluctuation associated with CFG is also related to the off-manifold issue, and from Fig. 4 we can easily see that the off-manifold phenomenon is more dominant at early stage of reverse diffusion sampling. Conversely, the loss trajectory for CFG++ demonstrates a much smoother variation, particularly during the early stages of reverse diffusion.
+
+DDIM inversion. As discussed in (Song et al., 2021a), the denoising process for unconditional DDIM is approximately invertible, meaning that  $\boldsymbol{x}_t$  can generally be recovered from  $\boldsymbol{x}_{t-1}$ . Specifically, from (3) and (4), we have the following approximate inversion formula for unconditional DDIM:
+
+$$
+\hat {\boldsymbol {x}} _ {\varnothing} \left(\boldsymbol {x} _ {t}\right) = \left(\boldsymbol {x} _ {t - 1} - \sqrt {1 - \bar {\alpha} _ {t - 1}} \hat {\epsilon} _ {\varnothing} \left(\boldsymbol {x} _ {t}\right)\right) / \sqrt {\bar {\alpha} _ {t - 1}} \simeq \left(\boldsymbol {x} _ {t - 1} - \sqrt {1 - \bar {\alpha} _ {t - 1}} \hat {\epsilon} _ {\varnothing} \left(\boldsymbol {x} _ {t - 1}\right)\right) / \sqrt {\bar {\alpha} _ {t - 1}} \tag {17}
+$$
+
+$$
+\boldsymbol {x} _ {t} = \sqrt {\bar {\alpha} _ {t}} \hat {\boldsymbol {x}} _ {\emptyset} (\boldsymbol {x} _ {t}) + \sqrt {1 - \bar {\alpha} _ {t}} \hat {\epsilon} _ {\emptyset} (\boldsymbol {x} _ {t}) \simeq \sqrt {\bar {\alpha} _ {t}} \hat {\boldsymbol {x}} _ {\emptyset} (\boldsymbol {x} _ {t}) + \sqrt {1 - \bar {\alpha} _ {t}} \hat {\epsilon} _ {\emptyset} (\boldsymbol {x} _ {t - 1}) \tag {18}
+$$
+
+where the approximation arises from  $\hat{\epsilon}_{\emptyset}(\pmb{x}_t) \simeq \hat{\epsilon}_{\emptyset}(\pmb{x}_{t-1})$ . A similar inversion procedure has been employed for conditional DDIM inversion under CFG by assuming  $\hat{\epsilon}_c^{\omega}(\pmb{x}_t) \simeq \hat{\epsilon}_c^{\omega}(\pmb{x}_{t-1})$ , and replacing  $\hat{\epsilon}_{\emptyset}$  by  $\hat{\epsilon}_c^\omega$  as detailed in Algorithm 3.
+
+On the other hand, by examining (12) and (13), we can obtain the following approximate DDIM inversion formula under CFG++:
+
+$$
+\hat {\boldsymbol {x}} _ {c} ^ {\lambda} \left(\boldsymbol {x} _ {t}\right) \simeq \left(\boldsymbol {x} _ {t - 1} - \sqrt {1 - \bar {\alpha} _ {t - 1}} \hat {\epsilon} _ {\emptyset} \left(\boldsymbol {x} _ {t - 1}\right)\right) / \sqrt {\bar {\alpha} _ {t - 1}} \tag {19}
+$$
+
+$$
+\boldsymbol {x} _ {t} \simeq \sqrt {\bar {\alpha} _ {t}} \hat {\boldsymbol {x}} _ {c} ^ {\lambda} (\boldsymbol {x} _ {t}) + \sqrt {1 - \bar {\alpha} _ {t}} \hat {\boldsymbol {\epsilon}} _ {c} ^ {\lambda} (\boldsymbol {x} _ {t - 1}) \tag {20}
+$$
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">ω = 2.0, λ = 0.2</td><td colspan="3">ω = 5.0, λ = 0.4</td><td colspan="3">ω = 7.5, λ = 0.6</td><td colspan="3">ω = 9.0, λ = 0.8</td><td colspan="3">ω = 12.5, λ = 1.0</td></tr><tr><td>FID ↓</td><td>CLIP ↑</td><td>IR ↑</td><td>FID ↓</td><td>CLIP ↑</td><td>IR ↑</td><td>FID ↓</td><td>CLIP ↑</td><td>IR ↑</td><td>FID ↓</td><td>CLIP ↑</td><td>IR ↑</td><td>FID ↓</td><td>CLIP ↑</td><td>IR ↑</td></tr><tr><td>CFG (Ho &amp; Salimans, 2021)</td><td>13.84</td><td>0.298</td><td>-0.235</td><td>15.08</td><td>0.310</td><td>0.068</td><td>17.71</td><td>0.312</td><td>0.152</td><td>20.01</td><td>0.312</td><td>0.170</td><td>21.23</td><td>0.313</td><td>0.192</td></tr><tr><td>CFG++ (ours)</td><td>12.75</td><td>0.303</td><td>-0.218</td><td>14.95</td><td>0.310</td><td>0.071</td><td>17.47</td><td>0.312</td><td>0.156</td><td>19.34</td><td>0.313</td><td>0.208</td><td>20.88</td><td>0.313</td><td>0.194</td></tr></table>
+
+where we apply the approximation  $\hat{\epsilon}_{\mathbf{c}}^{\lambda}(\boldsymbol{x}_t) \simeq \hat{\epsilon}_{\mathbf{c}}^{\lambda}(\boldsymbol{x}_{t-1})$  alongside the usual assumption  $\hat{\epsilon}_{\emptyset}(\boldsymbol{x}_t) \simeq \hat{\epsilon}_{\emptyset}(\boldsymbol{x}_{t-1})$ . This formulation underpins the CFG++ guided DDIM inversion algorithm, presented in Algorithm 4.
+
+In practice, for small time step size the error  $\hat{\epsilon}_{\emptyset}(\pmb{x}_t) \simeq \hat{\epsilon}_{\emptyset}(\pmb{x}_{t-1})$  is relatively small, so the unconditional DDIM inversion by (17) and (18) lead to relatively insignificant errors. Unfortunately, the corresponding inversion from conditional diffusion under CFG is quite distorted as noted in (Mokady et al., 2023; Wallace et al., 2023). In fact, this distortion is originated from the inaccuracy of the approximation  $\hat{\epsilon}_{c}^{\omega}(\pmb{x}_{t}) \simeq \hat{\epsilon}_{c}^{\omega}(\pmb{x}_{t-1})$ . More specifically, even when  $\hat{\epsilon}_{\emptyset}(\pmb{x}_t) \simeq \hat{\epsilon}_{\emptyset}(\pmb{x}_{t-1})$ , the approximation error from the CFG is given by
+
+$$
+\begin{array}{l} \varepsilon_ {c f g} := \hat {\epsilon} _ {c} ^ {\omega} (\boldsymbol {x} _ {t}) - \hat {\epsilon} _ {c} ^ {\omega} (\boldsymbol {x} _ {t - 1}) = \left(\hat {\epsilon} _ {\varnothing} (\boldsymbol {x} _ {t}) - \hat {\epsilon} _ {\varnothing} (\boldsymbol {x} _ {t - 1})\right) + \omega \left(\delta \hat {\epsilon} _ {c} (\boldsymbol {x} _ {t}) - \delta \hat {\epsilon} _ {c} (\boldsymbol {x} _ {t - 1})\right) \\ \simeq \omega \left(\delta \hat {\epsilon} _ {c} \left(\boldsymbol {x} _ {t}\right) - \delta \hat {\epsilon} _ {c} \left(\boldsymbol {x} _ {t - 1}\right)\right), \tag {21} \\ \end{array}
+$$
+
+where  $\delta \hat{\epsilon}_c(\pmb{x}_t) \coloneqq \hat{\epsilon}_c(\pmb{x}_t) - \hat{\epsilon}_{\emptyset}(\pmb{x}_t)$  denotes the directional component from unconditional to the conditional diffusion. Since the directional component by the text guidance scale is not negligible, this error becomes significant for high guidance scale  $\omega$ . Accordingly, the guidance scale must be heavily downweighted in order for inversions on real world images to be stable, thus limiting the strength of edits. To mitigate this issue, the authors in (Mokady et al., 2023; Wallace et al., 2023) developed null text optimization and coupled transform techniques, respectively.
+
+On the other hand, under the usual DDIM assumption  $\hat{\epsilon}_{\emptyset}(\pmb{x}_t) \simeq \hat{\epsilon}_{\emptyset}(\pmb{x}_{t-1})$ , the approximation error of CFG++ mainly arises from (20), which is smaller than that of CFG since we have
+
+$$
+\left\| \boldsymbol {\varepsilon} _ {c f g + +} \right\| = \lambda \| \delta \hat {\boldsymbol {\epsilon}} _ {\boldsymbol {c}} (\boldsymbol {x} _ {t}) - \delta \hat {\boldsymbol {\epsilon}} _ {\boldsymbol {c}} (\boldsymbol {x} _ {t - 1}) \| <   \| \boldsymbol {\varepsilon} _ {c f g} \| \tag {22}
+$$
+
+thanks to  $\lambda < \omega$ . Therefore, CFG++ significantly improves the DDIM inversion as shown Fig. 1 (middle) for representative results and Sec. 4.2 for further discussions.
+
+# 4 EXPERIMENTAL RESULTS
+
+In this section, we design experiments to show the limitations of CFG and how CFG++ can effectively mitigate these downsides. The main experiments were conducted with SD v1.5 or SDXL with 50 NFE DDIM sampling. In this regime, we searched for the matching guidance values of  $\omega$  and  $\lambda$  for a fair comparison. We fix  $\lambda = 0.2, 0.4, 0.6, 0.8, 1.0$  and find the  $\omega$  values that produce the images that are of closest proximity in terms of LPIPS distance given the same seed. We found that the corresponding values were  $\omega = 2.0, 5.0, 7.5, 9.0, 12.5$ , respectively. Some of the experiments were also conducted with distilled model such as SDXL-turbo, lightning.
+
+# 4.1 TEXT-TO-IMAGE GENERATION
+
+Using the corresponding scales for  $\omega$  and  $\lambda$ , we directly compare the performance of the T2I task using SD v1.5 and SDXL. In Tab. 1, we report quantitative metrics using 10k images generated from COCO captions (Lin et al., 2014). Here, we observe a constant improvement of the FID metric across all guidance scales (also see Fig. 15 for an apples-to-apples comparison), with approximately the same level of CLIP similarity or better. Additionally, we evaluate Im
+
+Table 1: Quantitative evaluation of 50 NFE DDIM T2I with SD v1.5 on COCO 10k  
+
+<table><tr><td>Model</td><td>Metric</td><td>CFG</td><td>CFG++ (ours)</td></tr><tr><td rowspan="3">SDXL-Turbo</td><td>FID↓</td><td>59.67</td><td>59.21</td></tr><tr><td>CLIP↑</td><td>0.320</td><td>0.325</td></tr><tr><td>ImageReward↑</td><td>0.777</td><td>0.968</td></tr><tr><td rowspan="3">SDXL-Lightning</td><td>FID↓</td><td>56.11</td><td>55.19</td></tr><tr><td>CLIP↑</td><td>0.322</td><td>0.324</td></tr><tr><td>ImageReward↑</td><td>0.691</td><td>0.829</td></tr><tr><td rowspan="3">SD v1.5 (DPM++ 2M)</td><td>FID↓</td><td>32.72</td><td>32.58</td></tr><tr><td>CLIP↑</td><td>0.313</td><td>0.312</td></tr><tr><td>ImageReward↑</td><td>0.086</td><td>0.023</td></tr></table>
+
+Table 2: Quant. eval. on accelerated T2I sampling
+
+ageReward (IR) (Xu et al., 2024), trained on human judgment data for better correlation, and observe improvements in this metric as well. The improvements can also be clearly seen in Fig. 11 (SD v1.5), where the unnatural components of the generated images are corrected. Specifically, we see that unnatural depictions of human hands, and incorrect renderings of the text are corrected in CFG++, a
+
+![](images/f61eeb9d7ca8cffa68e78d886b8be92113ca19a9b650ef7d0aa59717dfa41bb6.jpg)  
+Figure 5: T2I using SDXL-  $\{$  turbo, lightning\}, 6 NFE, CFG vs CFG++.
+
+![](images/7678017abd44f59b3b4943dd3f1ee7e22d21bebafe40193f4ebc33fa3012f9e3.jpg)
+
+long-standing research question in and of its own (Podell et al., 2023; Pelykh et al., 2024; Chen et al., 2023). We find that the improvement gain from CFG++ is even more dramatic for distilled diffusion models such as SDXL-{turbo, lightning}. We quantify the results on the same 5k prompts with 6 NFE sampling. In Fig. 5, Fig. 12 and Fig. 13 we see significant boosts in the quality of the generated images, which is also depicted in the improvements seen in Tab. 2.
+
+To show the compatibility of CFG++ with higher-order solvers, we experiment with the DPM++ 2M (Lu et al., 2022b) solver using 20 NFE. We note that in such a low NFE regime, a CFG scale of 5.0 corresponds to a CFG++ scale of 1.0, and we have to slightly extrapolate the value of  $\lambda \geq 1.0$  to achieve stronger guidance results. In Tab. 2, we observe that CFG++ is broadly comparable to CFG. This outcome differs from the consistent improvement of CFG++ over CFG observed with 50 NFE DDIM sampling. We attribute this to two factors: (1) the overall image quality for 20 NFE DPM++ 2M is generally lower compared to 50 NFE DDIM, leading to noisier quantitative metrics, particularly for metrics like ImageReward that are sensitive to subtle quality changes; and (2) the difference between CFG and CFG++ tends to be less pronounced in low NFE regimes without distillation, as stronger guidance effects emerge with higher NFEs.
+
+# 4.2 DIFFUSION IMAGE INVERSION AND EDITING
+
+We further explore the effect of CFG++ on DDIM inversion (Dhariwal & Nichol, 2021), where the source image is reverted to a latent vector that can reproduce the original image through generative process. DDIM inversion is well-known to break down in the usual CFG setting as CFG magnifies the accumulated error of each inversion step (Mokady et al., 2023), violating local linearization assumptions (Wallace et al., 2023). We show that CFG++ mitigates this issue by improving the inversion and image editing capabilities. We evaluate our method following the experimental setups in Park et al. (2024) and Kim et al. (2024b).
+
+Diffusion Image Inversion. Using the matched set of scales for  $\omega$  and  $\lambda$ , we demonstrate the effect of CFG++ on the diffusion image inversion task. Specifically, we reconstruct the images after inversion and evaluate it through PSNR and RMSE, following the methodology of Wallace et al. (2023). In Fig. 6, we illustrate the reconstructed examples (6a) from a real image from COCO data set and computed metrics (6b) for 5k COCO-2014 (Lin et al., 2014) validation set. Both qualitative and quantitative evaluations demonstrate a consistent improvement on reconstruction performance induced by CFG++. Notably, DDIM inversion with CFG++ leads to a consistent reconstruction of the source image across all guidance scales, while DDIM inversion with CFG fails to reconstruct it, as shown in Fig. 16 for another real image.
+
+Image Editing. Fig. 6c compares the image editing result using CFG and CFG++ followed by image inversion with 5 NFEs. In the image editing stage, a word in the source text highlighted by green
+
+![](images/d4c95fb06444cb0f0136cd92ca32e045dc27a4f9cbc1b92c20d4f9cd308f7d07.jpg)  
+Figure 6: Inversion and editing results. (a) Reconstructed samples after inversion by CFG and CFG++. (b) Quantitative comparison between CFG and CFG++. for reconstruction. (c) Image editing comparison via SDXL.
+
+color is swapped with the target concept, and this modified text is used as the condition for sampling. CFG++ successfully edits the target concept while preserving other concepts, such as background. In particular, the water splashing in the background, which tends to disappear in the conventional CFG, is maintained through the inversion process by CFG++. This emphasizes CFG++.s superior ability to retain specific scene elements that are frequently lost in the standard CFG approach. Moreover, standard CFG's disrupted DDIM inversion leads to saturated and less faithful editing results, whereas CFG++. delivers precise and high-fidelity edits.
+
+# 4.3 TEXT-CONDITIONED INVERSE PROBLEMS
+
+Inverse problem involves restoring the original data  $\pmb{x}$  from a noisy measurement  $\pmb{y} = \mathcal{A}(\pmb{x}) + \pmb{n}$ , where  $\mathcal{A}$  represents an imaging operator introduces distortions (e.g., Gaussian blur) and  $\pmb{n}$  denotes the measurement noise. Diffusion inverse solvers (DIS) address this challenge by leveraging pretrained diffusion models as implicit priors and performing posterior sampling  $\pmb{x} \sim p(\pmb{x}|\pmb{y})$ . Methods that leverage latent diffusion have gained recent interest (Rout et al., 2024; Song et al., 2024), but leveraging texts for solving these problems remains relatively underexplored (Chung et al., 2023b; Kim et al., 2023). One of the main reasons for this is that naively using CFG on top of latent DIS leads to diverging samples (Chung et al., 2023b). Several heuristic modifications such as null prompt optimization (Kim et al., 2023) with modified sampling schemes were needed to mitigate this drawback. This naturally leads to the question: Is it possible to leverage CFG guidance as a plug-and-play component of existing solvers? Here, we answer this question with a positive by showing that CFG++ enables the incorporation of text prompts into a standard solver. Specifically, we focus on comparing the performance of PSLD (Rout et al., 2024) combined with CFG and CFG++ in solving linear inverse problems. This evaluation utilizes the FFHQ (Karras et al., 2019) 512x512 dataset and the text prompt "a high-quality photo of a face". Further details about the experimental settings can be found in the Appendix E.
+
+As shown in Tab. 3, our method mostly outperforms both the vanilla PSLD and PSLD with CFG. The superiority is also evident from the Fig. 7, where CFG++ consistently delivers high-quality reconstructions across all tasks. PSLD with CFG often suffers from artifacts and blurriness. Conversely, CFG++ achieves better fidelity, clearly distinguishing between faces and faithfully reproducing fine details like eyelids and hair texture. For more results, please refer to the Appendix F.
+
+![](images/fbf92fecbb1162c7f87e6d40e2ad538c00db1d5054a093e434a41ba2deac0675.jpg)  
+Figure 7: Qualitative comparison on various inverse problems using PSLD (Rout et al., 2024) under CFG and CFG++. For more results, please refer to the Appendix F.
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">SR (x8)</td><td colspan="3">Deblur (motion)</td><td colspan="3">Deblur (gauss)</td><td colspan="3">Inpaint</td></tr><tr><td>FID ↓</td><td>LPIPS ↓</td><td>PSNR ↑</td><td>FID ↓</td><td>LPIPS ↓</td><td>PSNR ↑</td><td>FID ↓</td><td>LPIPS ↓</td><td>PSNR ↑</td><td>FID ↓</td><td>LPIPS ↓</td><td>PSNR ↑</td></tr><tr><td>PSLD</td><td>46.24</td><td>0.413</td><td>24.41</td><td>97.51</td><td>0.500</td><td>21.83</td><td>41.65</td><td>0.388</td><td>26.88</td><td>10.27</td><td>0.053</td><td>30.15</td></tr><tr><td>PSLD + CFG</td><td>41.24</td><td>0.394</td><td>24.91</td><td>91.90</td><td>0.493</td><td>22.29</td><td>41.52</td><td>0.390</td><td>26.94</td><td>9.36</td><td>0.055</td><td>30.27</td></tr><tr><td>PSLD + CFG++ (ours)</td><td>36.58</td><td>0.385</td><td>24.87</td><td>65.67</td><td>0.482</td><td>21.93</td><td>39.85</td><td>0.400</td><td>26.90</td><td>9.78</td><td>0.052</td><td>30.31</td></tr></table>
+
+Table 3: Quantitative comparison (FID, LPIPS, PSNR) of PSLD, PSLD with CFG, and PSLD with CFG++ on Latent Diffusion Inverse Solver.
+
+# 5 RELATED WORKS AND DISCUSSIONS
+
+Kynkänniemi et al. (2024) observed that applying CFG guidance at the earlier stages of sampling always led to detrimental effects and drastically reduced the diversity of the samples. The guidance at the later stages of sampling had minimal effects. Drawing upon these observations, it was empirically shown that applying CFG only in the limited time interval near the middle led to the best performance. Similar observations were made in the SD community (Howard & Prashanth, 2022; Birch, 2023) in adjusting the guidance scale across  $t$ . These works try to empirically adjust the strength of the guidance, while we derive our trajectory in a principled manner. More recently, Bradley & Nakkiran (2024) observed that the CFG score is not a valid denoising direction, showing that in the asymptotic limit, CFG sampling can be considered as a specific type of predictor-corrector (PC) sampling, dubbed PCG. PCG shares a similar spirit with CFG++ in that for the mainstream of sampling, one defers from the use of mixed CFG score (we use unconditional score, PCG uses conditional score) to avoid invalid directions. While PCG derives an interesting connection of PC sampling with CFG, the proposed algorithm does not improve the sampling performance. In contrast, CFG++ improves the sample quality without additional computation overhead. Since CFG++ attempts to make adjustments of CFG using a linear combination of score functions, it is not surprising to see that CFG++ sampling can be achieved by setting a time-varying schedule. Specifically, by setting a time-varying schedule  $\omega_{t}$  as  $\omega_{t} = -\omega \left( \sqrt{1 - \bar{\alpha}_{t-1}} - \sqrt{\frac{1 - \bar{\alpha}_{t-1}}{\alpha_{t}}} \right)$ , one can achieve the same effect with CFG as CFG++. Details are shown in App. D. However, this specific choice of time-varying guidance scale has not been reported in the literature, as it is a schedule that is relatively complex to be drawn heuristically. We further show in Appendix C that such choice removes undesirable oscillatory behavior in the evolution of the posterior mean.
+
+# 6 CONCLUSION
+
+This paper revises the most widespread guidance method, classifier-free guidance (CFG), highlighting that a small and reasonable guidance scale, e.g.  $\lambda \in [0.0, 1.0]$ , might suffice successful guidance with a proper geometric correction. Observing that the original CFG suffers from off-manifold issues during sampling, we propose a simple but surprisingly effective fix. This change translates the conditional guidance from extrapolating to interpolating between unconditional and conditional sampling trajectories, leading to more interpretable guidance in contrast to the conventional CFG which relies on heuristic  $\omega$  scaling. Given that CFG++ mitigates off-manifold issues, it may be beneficial for other downstream applications requiring accurate latent denoised estimate representations, e.g. DIS, as demonstrated in our experiments.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by the National Research Foundation of Korea under Grant RS-2024-00336454 and by the Institute for Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (RS-2019-II190075, Artificial Intelligence Graduate School Program, KAIST).
+
+# REFERENCES
+
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+
+<table><tr><td>Algorithm 3 DDIM Inversion with CFG</td><td>Algorithm 4 DDIM Inversion with CFG++</td></tr><tr><td>Require: x0, 0 ≤ ω ∈ R1: for i = 0 to T - 1 do2:   \(\hat{\epsilon}_{c}^{\omega}\left(\boldsymbol{x}_{t}\right)=\hat{\epsilon}_{\varnothing}\left(\boldsymbol{x}_{t}\right)+\omega\left[\hat{\epsilon}_{c}\left(\boldsymbol{x}_{t}\right)-\hat{\epsilon}_{\varnothing}\left(\boldsymbol{x}_{t}\right)\right]\)3:   \(\hat{\boldsymbol{x}}_{c}^{\omega}\left(\boldsymbol{x}_{t}\right)\gets\left(\boldsymbol{x}_{t}-\sqrt{1-\bar{\alpha}_{t}}\hat{\epsilon}_{c}^{\omega}\left(\boldsymbol{x}_{t}\right)\right)/\sqrt{\bar{\alpha}_{t}}\)4:   \(\boldsymbol{x}_{t+1}=\sqrt{\bar{\alpha}_{t+1}}\hat{\boldsymbol{x}}_{c}^{\omega}\left(\boldsymbol{x}_{t}\right)+\sqrt{1-\bar{\alpha}_{t+1}}\hat{\epsilon}_{c}^{\omega}\left(\boldsymbol{x}_{t}\right)\)5: end for6: return xT</td><td>Require: x0, λ ∈ [0,1]1: for i = 0 to T - 1 do2:   \(\hat{\epsilon}_{c}^{\lambda}\left(\boldsymbol{x}_{t}\right)=\hat{\epsilon}_{\varnothing}\left(\boldsymbol{x}_{t}\right)+\lambda\left[\hat{\epsilon}_{c}\left(\boldsymbol{x}_{t}\right)-\hat{\epsilon}_{\varnothing}\left(\boldsymbol{x}_{t}\right)\right]\)3:   \(\hat{\boldsymbol{x}}_{c}^{\lambda}\left(\boldsymbol{x}_{t}\right)\gets\left(\boldsymbol{x}_{t}-\sqrt{1-\bar{\alpha}_{t}}\hat{\epsilon}_{\varnothing}\left(\boldsymbol{x}_{t}\right)\right)/\sqrt{\bar{\alpha}_{t}}\)4:   \(\boldsymbol{x}_{t+1}=\sqrt{\bar{\alpha}_{t+1}}\hat{\boldsymbol{x}}_{c}^{\lambda}\left(\boldsymbol{x}_{t}\right)+\sqrt{1-\bar{\alpha}_{t+1}}\hat{\epsilon}_{c}^{\lambda}\left(\boldsymbol{x}_{t}\right)\)5: end for6: return xT</td></tr></table>
+
+Figure 8: Comparison between DDIM inversion. CFG++ proposes a simple yet effective fix: using  $\hat{\epsilon}_{\varnothing}(\boldsymbol{x}_t)$  instead of  $\hat{\epsilon}_c^{\omega}(\boldsymbol{x}_t)$  in Tweedie's denoising step.
+
+# A EXTENSION OF CFG++ TO OTHER SOLVERS
+
+In this section, we discuss ways in which we can apply CFG++ to a more diverse set of ODE/SDE solvers. We consider solving the variance exploding (VE) PF-ODE as presented in (1), as reparametrization VP diffusion models can easily recover the VE formulation, as often implemented in widely used frameworks such as https://github.com/crowsonkb/k-diffusion. Following the notation in Lu et al. (2022b), we consider a sequence of timesteps  $\{t_i\}_{i=0}^M$ , where  $t_0 = T$  denotes the initial starting point of the reverse sampling (i.e. Gaussian noise).
+
+Euler (Karras et al., 2022) The construction is the same as in DDIM. We include it here for completeness. The update step reads
+
+$$
+\boldsymbol {x} _ {t _ {i + 1}} = \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i}}\right) + \frac {\boldsymbol {x} _ {t _ {i}} - \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i}}\right)}{\sigma_ {t _ {i}}} \cdot \sigma_ {t _ {i + 1}}, \tag {CFG}
+$$
+
+$$
+\boldsymbol {x} _ {t _ {i + 1}} = \hat {\boldsymbol {x}} _ {c} ^ {\lambda} \left(\boldsymbol {x} _ {t _ {i}}\right) + \frac {\boldsymbol {x} _ {t _ {i}} - \hat {\boldsymbol {x}} _ {\emptyset} \left(\boldsymbol {x} _ {t _ {i}}\right)}{\sigma_ {t _ {i}}} \cdot \sigma_ {t _ {i + 1}}, \quad (\mathrm {C F G} + +)
+$$
+
+Euler Ancestral Euler Ancestral sampler follows Euler sampler, but introduces stochasticity by taking larger steps and then adding a slight amount of noise. Adjusting with CFG++ is straightforward.
+
+$$
+\boldsymbol {x} _ {t _ {i + 1}} = \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i}}\right) + \frac {\boldsymbol {x} _ {t _ {i}} - \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i}}\right)}{\sigma_ {t _ {i}}} \cdot \left(\sigma_ {t _ {d _ {i}}} - \sigma_ {t _ {i}}\right) + \sigma_ {t _ {i}} \boldsymbol {\epsilon}, \tag {CFG}
+$$
+
+$$
+\boldsymbol {x} _ {t _ {i + 1}} = \hat {\boldsymbol {x}} _ {c} ^ {\lambda} \left(\boldsymbol {x} _ {t _ {i}}\right) + \frac {\boldsymbol {x} _ {t _ {i}} - \hat {\boldsymbol {x}} _ {\emptyset} \left(\boldsymbol {x} _ {t _ {i}}\right)}{\sigma_ {t _ {i}}} \cdot \left(\sigma_ {t _ {d _ {i}}} - \sigma_ {t _ {i}}\right) + \sigma_ {t _ {i}} \boldsymbol {\epsilon}, \quad (\text {C F G} + +)
+$$
+
+where  $t_i > t_{d_i} > t_{i + 1}$  and  $\epsilon \sim \mathcal{N}(0,\pmb {I})$
+
+DPM-solver++ 2M (Lu et al., 2022b) Define  $\sigma_t \coloneqq e^{-t}$ ,  $h_i \coloneqq t_i - t_{i-1}$ , and  $r_i \coloneqq h_{i-1} / h_i$ . After initializing  $\boldsymbol{x}_{t_0}$  with Gaussian noise, the first iteration is given by
+
+$$
+\boldsymbol {x} _ {t _ {1}} = \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {0}}\right) + e ^ {- h _ {1}} \left(\boldsymbol {x} _ {t _ {0}} - \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {0}}\right)\right) \tag {CFG}
+$$
+
+$$
+\boldsymbol {x} _ {t _ {1}} = \hat {\boldsymbol {x}} _ {c} ^ {\lambda} \left(\boldsymbol {x} _ {t _ {0}}\right) + e ^ {- h _ {1}} \left(\boldsymbol {x} _ {t _ {0}} - \hat {\boldsymbol {x}} _ {\emptyset} \left(\boldsymbol {x} _ {t _ {0}}\right)\right) \quad (\text {C F G} + +)
+$$
+
+The following iterations by using the standard CFG reads
+
+$$
+\boldsymbol {D} _ {i} = \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i - 1}}\right) + \frac {1}{2 r _ {i}} \left(\hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i - 1}}\right) - \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i - 2}}\right)\right), \tag {23}
+$$
+
+$$
+\boldsymbol {x} _ {t _ {i}} = e ^ {- h _ {i}} \boldsymbol {x} _ {t _ {i - 1}} - \left(e ^ {- h _ {i}} - 1\right) \boldsymbol {D} _ {i}. \tag {24}
+$$
+
+Rearranging (23), (24), we can rewrite the update steps as
+
+$$
+\boldsymbol {x} _ {t _ {i}} = \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i - 1}}\right) - e ^ {- h _ {i}} \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i - 1}}\right) + \frac {1 - e ^ {- h _ {i}}}{2 r _ {i}} \left(\hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i - 1}}\right) - \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i - 2}}\right)\right) + e ^ {- h _ {i}} \boldsymbol {x} _ {t _ {i - 1}}. \tag {25}
+$$
+
+Notice that in order to apply CFG++ to (25), we should only keep the first term as the conditional Tweedie, and use the unconditional estimates for the rest of the components. i.e.
+
+$$
+\boldsymbol {x} _ {t _ {i}} = \hat {\boldsymbol {x}} _ {c} ^ {\lambda} \left(\boldsymbol {x} _ {t _ {i - 1}}\right) - e ^ {- h _ {i}} \hat {\boldsymbol {x}} _ {\varnothing} \left(\boldsymbol {x} _ {t _ {i - 1}}\right) + \frac {1 - e ^ {- h _ {i}}}{2 r _ {i}} \left(\hat {\boldsymbol {x}} _ {\varnothing} \left(\boldsymbol {x} _ {t _ {i - 1}}\right) - \hat {\boldsymbol {x}} _ {\varnothing} \left(\boldsymbol {x} _ {t _ {i - 2}}\right)\right) + e ^ {- h _ {i}} \boldsymbol {x} _ {t _ {i - 1}} \tag {26}
+$$
+
+DPM-solver++ 2S (Lu et al., 2022b) With the same choices of  $\sigma_t, h_i$ , we additionally define the timesteps  $\{s_i\}_{i=1}^M$  with  $t_i > s_{i+1} > t_{i+1}$ . Further, let  $r_i = \frac{s_i - t_{i-1}}{t_i - t_{i-1}}$ . Using standard CFG, the iteration reads
+
+$$
+\boldsymbol {u} _ {i} = e ^ {- r _ {i} h _ {i}} \boldsymbol {x} _ {t _ {i - 1}} + \left(1 - e ^ {- r _ {i} h _ {i}}\right) \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i - 1}}\right) \tag {27}
+$$
+
+$$
+\boldsymbol {x} _ {t _ {i}} = \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i - 1}}\right) - e ^ {- h _ {i}} \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i - 1}}\right) + \frac {1 - e ^ {- h _ {i}}}{2 r _ {i}} \left(\hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {u} _ {i}\right) - \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i - 1}}\right)\right) + e ^ {- h _ {i}} \boldsymbol {x} _ {t _ {i - 1}} \tag {28}
+$$
+
+Applying the general transition rule from CFG to  $\mathrm{CFG + + }$  as introduced in (15), we can keep all of the Tweedie estimates to be unconditional, and only change the first term of (28), i.e.
+
+$$
+\boldsymbol {u} _ {i} = e ^ {- r _ {i} h _ {i}} \boldsymbol {x} _ {t _ {i - 1}} + \left(1 - e ^ {- r _ {i} h _ {i}}\right) \hat {\boldsymbol {x}} _ {\emptyset} \left(\boldsymbol {x} _ {t _ {i - 1}}\right) \tag {29}
+$$
+
+$$
+\boldsymbol {x} _ {t _ {i}} = \hat {\boldsymbol {x}} _ {\emptyset} \left(\boldsymbol {x} _ {t _ {i - 1}}\right) - e ^ {- h _ {i}} \hat {\boldsymbol {x}} _ {\emptyset} \left(\boldsymbol {x} _ {t _ {i - 1}}\right) + \frac {1 - e ^ {- h _ {i}}}{2 r _ {i}} \left(\hat {\boldsymbol {x}} _ {c} ^ {\lambda} (\boldsymbol {u} _ {i}) - \hat {\boldsymbol {x}} _ {\emptyset} \left(\boldsymbol {x} _ {t _ {i - 1}}\right)\right) + e ^ {- h _ {i}} \boldsymbol {x} _ {t _ {i - 1}}. \tag {30}
+$$
+
+For the ancestral version of DPM-solver++ 2S, we can follow the general transition rule, as was shown for Euler  $\rightarrow$  Euler Ancestral.
+
+# B EXTENSION OF CFG++ TO FLOW MATCHING
+
+Here, we generalize CFG++ with flow matching which subsumes diffusion models as specific instantiations. Let  $p_0$  and  $p_1$  be two data distributions on  $\mathbb{R}^n$ , and  $Q_{01}$  denotes a coupling of  $p_0$  and  $p_1$ . Then, the flow matching loss is defined as follows:
+
+$$
+\ell_ {\mathrm {F M}} (\theta ; Q _ {0 1}) = \mathbb {E} _ {(\boldsymbol {x} _ {0}, \boldsymbol {x} _ {1}) \sim Q _ {0 1}, t \sim \operatorname {u n i f} (0, 1)} \left[ \ell_ {\mathrm {M S E}} (\boldsymbol {x} _ {1} - \boldsymbol {x} _ {0}, \boldsymbol {v} _ {\theta} (\boldsymbol {x} _ {t}, t)) \right], \tag {31}
+$$
+
+where  $\pmb{v}_{\theta}:\mathbb{R}^n\times (0,1)\to \mathbb{R}^n$  refers to a velocity parameterized by  $\theta$ ,  $\ell_{\mathrm{MSE}}(\pmb {x},\pmb {y}) = \| \pmb {x} - \pmb {y}\| _2^2$ , and  $\pmb{x}_t = (1 - t)\pmb {x}_0 + t\pmb{x}_1$ . This objective can be reformulated in the form of denoising as follows:
+
+$$
+\ell_ {\mathrm {F M}} (\theta ; Q _ {0 1}) = \mathbb {E} _ {\left(\boldsymbol {x} _ {0}, \boldsymbol {x} _ {1}\right) \sim Q _ {0 1}, t \sim \operatorname {u n i f} (0, 1)} \left[ \frac {1}{t ^ {2}} \ell_ {\mathrm {M S E}} \left(\boldsymbol {x} _ {0}, \boldsymbol {D} _ {\theta} \left(\boldsymbol {x} _ {t}, t\right)\right) \right], \tag {32}
+$$
+
+where  $D_{\theta}(\pmb{x}_t,t) = \pmb{x}_t - t\pmb{v}_{\theta}(\pmb{x}_t,t)$  serves as a denoiser function. Then, one can derive an ODE which translates samples from  $p_0$  and  $p_1$  given a vector field  $\pmb{v}_{\theta}$ :
+
+$$
+d \boldsymbol {x} _ {t} = \boldsymbol {v} _ {\theta} (\boldsymbol {x} _ {t}, t) d t = \frac {\boldsymbol {x} _ {t} - \boldsymbol {D} _ {\theta} (\boldsymbol {x} _ {t} , t)}{t} d t, \quad t \in (0, 1), \tag {33}
+$$
+
+which resembles (1) with a choice of  $\sigma_t = t$ . For text-conditional flow, CFG can be applied to (33) by solving the following ODE (Kim et al., 2024a):
+
+$$
+d \boldsymbol {x} _ {t} = \left[ \boldsymbol {v} _ {\theta} (\boldsymbol {x} _ {t}, t, \varnothing) + \omega \left(\boldsymbol {v} _ {\theta} (\boldsymbol {x} _ {t}, t, \boldsymbol {c}) - \boldsymbol {v} _ {\theta} (\boldsymbol {x} _ {t}, t, \varnothing)\right) \right] d t = \tilde {\boldsymbol {v}} _ {c} ^ {\omega} (\boldsymbol {x} _ {t}) d t, \quad t \in (0, 1). \tag {34}
+$$
+
+We similarly define  $\tilde{\pmb{x}}_c^\omega (\pmb {x}_t) = \pmb {x}_t - t\tilde{v}_c^\omega (\pmb {x}_t)$ ,  $\tilde{\pmb{x}}_{\emptyset}(\pmb {x}_t) = \pmb {x}_t - t\pmb {v}_{\theta}(\pmb {x}_t,t,\emptyset) = \pmb {x}_t - t\tilde{\pmb{v}}_{\emptyset}(\pmb {x}_t)$ , and  $\tilde{\pmb{x}}_c^{\lambda}(\pmb {x}_t) = \pmb {x}_t - t\tilde{v}_c^{\lambda}(\pmb {x}_t)$ . Based on this ODE, we can reproduce results from a diverse set of solvers in Sec. A. For example, a single update step of Euler solver reads:
+
+$$
+\boldsymbol {x} _ {t _ {i + 1}} = \tilde {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i}}\right) + \frac {\boldsymbol {x} _ {t _ {i}} - \tilde {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t _ {i}}\right)}{t _ {i}} \cdot t _ {i + 1}, \tag {CFG}
+$$
+
+$$
+\boldsymbol {x} _ {t _ {i + 1}} = \tilde {\boldsymbol {x}} _ {c} ^ {\lambda} \left(\boldsymbol {x} _ {t _ {i}}\right) + \frac {\boldsymbol {x} _ {t _ {i}} - \tilde {\boldsymbol {x}} _ {\emptyset} \left(\boldsymbol {x} _ {t _ {i}}\right)}{t _ {i}} \cdot t _ {i + 1}. \quad (\mathrm {C F G} + +)
+$$
+
+This implies that  $\mathrm{CFG} + +$  is potentially compatible with various flow-based generative models.
+
+# C EVOLUTION OF THE POSTERIOR MEAN: CFG VS. CFG++
+
+Recall that one can equivalently view the evolution of the posterior mean  $\mathbb{E}[\pmb{x}_0|\pmb{x}_t]$  rather than the noisy variables  $\pmb{x}_t$ . In this context, we derive the sequential evolution of conditional posterior mean  $\hat{\pmb{x}}_c^\omega$  and  $\hat{\pmb{x}}_c^\lambda$  through time  $t$  to further understand the underlying behavior of the proposed sampling.
+
+Proposition 1. Let  $d\boldsymbol{z}(\boldsymbol{x}_t) \coloneqq \boldsymbol{z}(\boldsymbol{x}_t) - \boldsymbol{z}(\boldsymbol{x}_{t+1})$  denote the discrete time evolution of some random variable  $z$  at time  $t$ . Then, the evolution of  $\hat{\boldsymbol{x}}_c^\omega$  of CFG and  $\hat{\boldsymbol{x}}_c^\lambda$  of CFG++ is given by
+
+$$
+d \hat {\boldsymbol {x}} _ {c} ^ {\omega} (\boldsymbol {x} _ {t}) = \frac {\sqrt {1 - \bar {\alpha} _ {t}}}{\sqrt {\bar {\alpha} _ {t}}} d \hat {\epsilon} _ {\varnothing} (\boldsymbol {x} _ {t}) + \omega (\Delta (\boldsymbol {x} _ {t}, \boldsymbol {c}) - \Delta (\boldsymbol {x} _ {t + 1}, \boldsymbol {c})) \tag {35}
+$$
+
+$$
+d \hat {\boldsymbol {x}} _ {c} ^ {\lambda} (\boldsymbol {x} _ {t}) = \frac {\sqrt {1 - \bar {\alpha} _ {t}}}{\sqrt {\bar {\alpha} _ {t}}} \underbrace {d \hat {\epsilon} _ {\varnothing} (\boldsymbol {x} _ {t})} _ {\text {u n c o n d . s h i f t}} + \lambda \underbrace {\Delta (\boldsymbol {x} _ {t} , \boldsymbol {c})} _ {\text {c o n d . s h i f t}}, \tag {36}
+$$
+
+where  $\Delta (\pmb {x}_t,\pmb {c})\coloneqq \hat{\pmb{x}}_c(\pmb {x}_t) - \hat{\pmb{x}}_{\emptyset}(\pmb {x}_t)$
+
+Proof. We start by writing the iteration from  $t + 1 \rightarrow t$
+
+$$
+\boldsymbol {x} _ {t} = \sqrt {\bar {\alpha} _ {t}} \hat {\boldsymbol {x}} _ {c} ^ {\omega} (\boldsymbol {x} _ {t + 1}) + \sqrt {1 + \bar {\alpha} _ {t}} \hat {\epsilon} _ {c} ^ {\omega} (\boldsymbol {x} _ {t + 1}). \tag {37}
+$$
+
+The Tweedie estimate for the next step is then written as
+
+$$
+\begin{array}{l} \hat {\boldsymbol {x}} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t}\right) = \frac {\boldsymbol {x} _ {t} - \sqrt {1 - \bar {\alpha} _ {t}} \hat {\epsilon} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t}\right)}{\sqrt {\bar {\alpha} _ {t}}} (38) \\ = \frac {\sqrt {\bar {\alpha} _ {t}} \hat {x} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t + 1}\right) + \sqrt {1 - \bar {\alpha} _ {t}} \left(\hat {\epsilon} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t + 1}\right) - \hat {\epsilon} _ {c} ^ {\omega} \left(\boldsymbol {x} _ {t}\right)\right)}{\sqrt {\bar {\alpha} _ {t}}} (39) \\ = \hat {\boldsymbol {x}} _ {c} ^ {\omega} (\boldsymbol {x} _ {t + 1}) + \frac {\sqrt {1 - \bar {\alpha} _ {t}}}{\sqrt {\bar {\alpha} _ {t}}} \Big [ \hat {\epsilon} _ {\emptyset} (\boldsymbol {x} _ {t + 1}) - \hat {\epsilon} _ {\emptyset} (\boldsymbol {x} _ {t}) + \omega (\hat {\epsilon} _ {c} (\boldsymbol {x} _ {t + 1}) - \hat {\epsilon} _ {\emptyset} (\boldsymbol {x} _ {t + 1})) \\ \left. - \omega \left(\hat {\epsilon} _ {c} \left(\boldsymbol {x} _ {t}\right) - \hat {\epsilon} _ {\varnothing} \left(\boldsymbol {x} _ {t}\right)\right) \right]. (40) \\ \end{array}
+$$
+
+Using the relation  $\hat{\epsilon}_c^\omega (\pmb {x}_t) = -(\sqrt{\bar{\alpha}_t}\hat{\pmb{x}}_c^\omega (\pmb {x}_t) - \pmb {x}_t) / \sqrt{1 - \bar{\alpha}_t}$ , we have
+
+$$
+d \hat {\boldsymbol {x}} _ {c} ^ {\omega} (\boldsymbol {x} _ {t}) = \frac {\sqrt {1 - \bar {\alpha} _ {t}}}{\sqrt {\bar {\alpha} _ {t}}} d \hat {\epsilon} _ {\varnothing} (\boldsymbol {x} _ {t}) + \omega (\hat {\boldsymbol {x}} _ {c} (\boldsymbol {x} _ {t}) - \hat {\boldsymbol {x}} _ {\varnothing} (\boldsymbol {x} _ {t})) - \omega (\hat {\boldsymbol {x}} _ {c} (\boldsymbol {x} _ {t + 1}) - \hat {\boldsymbol {x}} _ {\varnothing} (\boldsymbol {x} _ {t + 1})). \tag {41}
+$$
+
+Similarly, for  $\mathrm{CFG} + +$
+
+$$
+\begin{array}{l} \hat {x} _ {c} ^ {\lambda} \left(\boldsymbol {x} _ {t}\right) = \frac {\boldsymbol {x} _ {t} - \sqrt {1 - \bar {\alpha} _ {t}} \hat {\epsilon} _ {c} ^ {\lambda} \left(\boldsymbol {x} _ {t}\right)}{\sqrt {\bar {\alpha} _ {t}}} (42) \\ = \frac {\sqrt {\bar {\alpha} _ {t}} \hat {x} _ {c} ^ {\lambda} \left(\boldsymbol {x} _ {t + 1}\right) + \sqrt {1 - \bar {\alpha} _ {t}} \left(\hat {\epsilon} _ {\varnothing} \left(\boldsymbol {x} _ {t + 1}\right) - \hat {\epsilon} _ {c} ^ {\lambda} \left(\boldsymbol {x} _ {t}\right)\right)}{\sqrt {\bar {\alpha} _ {t}}} (43) \\ = \hat {\boldsymbol {x}} _ {c} ^ {\lambda} (\boldsymbol {x} _ {t + 1}) + \frac {\sqrt {1 - \bar {\alpha} _ {t}}}{\sqrt {\bar {\alpha} _ {t}}} \left[ \hat {\epsilon} _ {\emptyset} (\boldsymbol {x} _ {t + 1}) - \hat {\epsilon} _ {\emptyset} (\boldsymbol {x} _ {t}) - \lambda \left(\hat {\epsilon} _ {c} (\boldsymbol {x} _ {t}) - \hat {\epsilon} _ {\emptyset} (\boldsymbol {x} _ {t})\right) \right]. (44) \\ \end{array}
+$$
+
+Hence,
+
+$$
+d \hat {\boldsymbol {x}} _ {c} ^ {\lambda} \left(\boldsymbol {x} _ {t}\right) = \frac {\sqrt {1 - \bar {\alpha} _ {t}}}{\sqrt {\bar {\alpha} _ {t}}} d \hat {\epsilon} _ {\varnothing} \left(\boldsymbol {x} _ {t}\right) + \lambda \left(\hat {\boldsymbol {x}} _ {c} \left(\boldsymbol {x} _ {t}\right) - \hat {\boldsymbol {x}} _ {\varnothing} \left(\boldsymbol {x} _ {t}\right)\right), \tag {45}
+$$
+
+Proposition 1 implies that the CFG++ update of  $\hat{x}_c^\lambda$  is decomposed into two shift terms: 1)  $d\hat{\epsilon}_{\emptyset}(\pmb{x}_t)$  represents the difference between consecutive unconditional scores (i.e. unconditional shift), and 2)  $\Delta (\pmb{x}_t,\pmb{c})$  denotes the direction of conditional guidance at time  $t$  (i.e. conditional shift). The conditional shift term is multiplied by a small interpolation factor  $\lambda$  in the case of CFG++, inducing a small nudge toward the condition.
+
+CFG  $\hat{x}_c^\omega$ , on the other hand, has the same unconditional shift, but has an oscillatory behavior for the conditional shift. The difference between CFG/CFG++ sampling arises from the unexpected additional shift from the previous timestep  $t + 1$  that exists for the CFG decomposition:  $-\Delta (\pmb{x}_{t + 1},\pmb{c})$ , and the scaling constant  $\omega$ . The initial conditional shift  $\omega \Delta (\pmb{x}_t,\pmb{c})$  with a large  $\omega$  pushes the trajectory off the manifold, but cancels some of its effects by subtracting the conditional shift from the previous step  $\omega \Delta (\pmb{x}_{t + 1},\pmb{c})$ . The compounded vector  $\Delta (\pmb{x}_t,\pmb{c}) - \Delta (\pmb{x}_{t + 1},\pmb{c})$  does induce a nudge closer to the condition but requires a large value of  $\omega$  to have a meaningful effect, and thus is hard to interpret.
+
+![](images/f402abbb98e4d2b06ac756e5de0c7975e99c227b75de94b3722a137607a61d54.jpg)  
+Figure 9: Continuous transition between CFG and CFG++ schedule
+
+# D CFG++ AS REWEIGHTED CFG
+
+One can rewrite the update step of CFG and  $\mathrm{CFG} + +$  succinctly as
+
+$$
+\begin{array}{l} \boldsymbol {x} _ {t - 1} = \sqrt {\bar {\alpha} _ {t - 1}} \hat {\boldsymbol {x}} _ {\boldsymbol {c}} ^ {\omega} + \sqrt {1 - \bar {\alpha} _ {t - 1}} \hat {\boldsymbol {\epsilon}} _ {\boldsymbol {c}} ^ {\omega^ {\prime}} (46) \\ = \sqrt {\bar {\alpha} _ {t - 1}} \left(\frac {\boldsymbol {x} _ {t} - \sqrt {1 - \bar {\alpha} _ {t}} \hat {\boldsymbol {\epsilon}} _ {\boldsymbol {c}} ^ {\omega} (\boldsymbol {x} _ {t})}{\sqrt {\bar {\alpha} _ {t}}} + \sqrt {1 - \bar {\alpha} _ {t - 1}} \hat {\boldsymbol {\epsilon}} _ {\boldsymbol {c}} ^ {\omega^ {\prime}}\right), (47) \\ \end{array}
+$$
+
+where for CFG,  $\omega' = \omega$ , and for CFG++,  $\omega' = 0$ . Note that we are using the same notation  $\omega$  here, but the guidance scale (represented as  $\lambda$  in the previous sections), is much smaller. One can derive
+
+$$
+\begin{array}{l} \boldsymbol {x} _ {t - 1} = \sqrt {\bar {\alpha} _ {t - 1}} \hat {\boldsymbol {x}} _ {\varnothing} - \frac {\omega \sqrt {1 - \bar {\alpha} _ {t}} (\hat {\boldsymbol {\epsilon}} _ {\boldsymbol {c}} - \hat {\boldsymbol {\epsilon}} _ {\varnothing})}{\sqrt {\alpha_ {t}}} + \sqrt {1 - \bar {\alpha} _ {t - 1}} \hat {\boldsymbol {\epsilon}} _ {\varnothing} + \sqrt {1 - \bar {\alpha} _ {t - 1}} \omega^ {\prime} (\hat {\boldsymbol {\epsilon}} _ {\boldsymbol {c}} - \hat {\boldsymbol {\epsilon}} _ {\varnothing}) (48) \\ = \left(\sqrt {\bar {\alpha} _ {t - 1}} \hat {\boldsymbol {x}} _ {\emptyset} + \sqrt {1 - \bar {\alpha} _ {t - 1}} \hat {\boldsymbol {\epsilon}} _ {\emptyset}\right) - \frac {\omega \sqrt {1 - \bar {\alpha} _ {t}} \left(\hat {\boldsymbol {\epsilon}} _ {\mathbf {c}} - \hat {\boldsymbol {\epsilon}} _ {\emptyset}\right)}{\sqrt {\alpha_ {t}}} + \sqrt {1 - \bar {\alpha} _ {t - 1}} \omega^ {\prime} \left(\hat {\boldsymbol {\epsilon}} _ {\mathbf {c}} - \hat {\boldsymbol {\epsilon}} _ {\emptyset}\right) (49) \\ = \boldsymbol {x} _ {t - 1, \varnothing} + \left(\hat {\epsilon} _ {c} - \hat {\epsilon} _ {\varnothing}\right) \underbrace {\omega^ {\prime} \sqrt {1 - \bar {\alpha} _ {t - 1}} - \omega \sqrt {\frac {1 - \bar {\alpha} _ {t - 1}}{\alpha_ {t}}}} _ {=: - \omega_ {t}} (50) \\ \end{array}
+$$
+
+Here, we see that the guidance scale used in the original CFG is a composition of two time-dependent functions. When composited with the same strength for  $\omega^{\prime}$  and  $\omega$ , as depicted in Fig. 9 (b), the guidance scale of CFG peaks at the earlier stages of sampling, drops down, then rises back up. The sudden peak in the earlier stages may explain the unnatural saturation in the earlier stages of sampling using CFG. In contrast, the guidance scale of CFG++ has a convex-like function, where the scale gradually increases to some value.
+
+By enforcing that the guidance scales integrate to 1 (i.e. enforce it as a PDF), we can generalize CFG/CFG++, and scrutinize the behavior when we interpolate between these two by setting  $0 < \omega' < \omega$ . The guidance scale functions for these interpolations are shown in Fig. 9. In Tab. 4, we see that the image quality measured by FID and ImageReward on COCO-1K data, gradually gets better as we transition from CFG++ to CFG.
+
+<table><tr><td></td><td>ω&#x27; = 0.0 (CFG++)</td><td>ω&#x27; = 0.4</td><td>ω&#x27; = 0.6</td><td>ω&#x27; = 0.8</td><td>ω&#x27; = 1.0 (CFG)</td></tr><tr><td>ImageReward ↑</td><td>-0.112</td><td>-0.125</td><td>-0.126</td><td>-0.153</td><td>-0.420</td></tr><tr><td>FID ↓</td><td>66.52</td><td>66.60</td><td>66.78</td><td>67.50</td><td>68.10</td></tr></table>
+
+# EXPERIMENTAL DETAILS
+
+# E.1 TEXT-CONDITIONED INVERSE PROBLEMS
+
+Problem settings. We evaluate our approach across following degradation types: 1) Super-resolution with a scale factor of x8, 2) Motion deblurring from an image convolved with a  $61 \times 61$  motion kernel, randomly sampled with an intensity value  $0.3^2$ , 3) Gaussian deblurring from an image convolved with a  $61 \times 61$  kernel with an intensity value 0.5, and 4) Inpainting from  $10 - 20\%$  free-form masking, as implemented in (Saharia et al., 2022). In inpainting evaluations, regions outside the mask are overlaid with the ground truth image.
+
+Datasets, Models. For evaluation, we use the FFHQ (Karras et al., 2019) 512x512 dataset and follow (Chung et al., 2023a) by selecting the first 1,000 images for testing. For the pre-trained latent diffusion model including baseline methods, we choose SD v1.5 trained on the LAION dataset. As a baseline for latent DIS, we consider PSLD (Rout et al., 2024). It enforces fidelity by projecting onto the subspace of  $\mathcal{A}$  during the intermediate step between decoding and encoding, in conjunction with the DPS (Chung et al., 2023a) loss term. For gradient updates in both vanilla PSLD and PSLD with CFG, we use static step sizes of  $\eta = 1.0$  and  $\gamma = 0.1$  as recommended in (Rout et al., 2024). For CFG scale  $\omega$ , we applied the corresponding scales that we found to be corresponded for CFG++ scale  $\lambda$ . Please refer to the Tab. 5 for the hyperparameters used for PSLD with CFG++.
+
+Table 4: Quantitative results of COCO-1k by interpolating between CFG and CFG++  
+
+<table><tr><td></td><td>SR(x8)</td><td>Deblur(motion)</td><td>Deblur(gauss)</td><td>Inpaint</td></tr><tr><td>η</td><td>1.3</td><td>0.4</td><td>1.0</td><td>1.0</td></tr><tr><td>γ</td><td>0.1</td><td>0.025</td><td>0.12</td><td>0.1</td></tr><tr><td>λ</td><td>0.1</td><td>0.2</td><td>0.2</td><td>0.6</td></tr><tr><td>ω</td><td>1.5</td><td>2.0</td><td>2.0</td><td>7.5</td></tr></table>
+
+Table 5: Hyperparameters for PSLD (Rout et al., 2024) with CFG++ and corresponding CFG scale  $\omega$ .
+
+# F FURTHER EXPERIMENTAL RESULTS
+
+# F.1 T2I
+
+Since the range of guidance scales for CFG and  $\mathrm{CFG + + }$  is different, we matched the guidance values of  $\omega$  and  $\lambda$  for comparison by computing the LPIPS distance using the same seed.
+
+Fig. 14 shows the clean estimates at each diffusion sampling time  $t$ , computed using Tweedie's formula, both with and without CFG. It is evident that the original CFG induces significant error, particularly at earlier times, leading to off-manifold samples. However, the CFG++ addresses this issue by adjusting the DDIM re-noising step and the guidance scale.
+
+# F.2 REAL IMAGE EDITING
+
+In Fig. 17-19, we provide qualitative comparison on real image editing via DDIM inversion with CFG and CFG++. For the DDIM inversion stage, we use "a photography of [source concept]" as conditioning prompt. For the sampling stage, we swap [source concept] to [target concept] and use it as conditioning prompt for generation. For example, in Fig. 17, we set [source concept] to "dog" and [target concept] to "cat". For all experiments, we set the guidance scale as  $\omega = 9.0$  and  $\lambda = 0.8$  as described in the main paper. The comparison demonstrates that CFG++ successfully edits the given image which was not possible by CFG. This results also support our claim on reduced error during DDIM inversion by CFG++.
+
+![](images/c5c5da0cf9cdacaf6d82c368ae6b440472ed43143dcaaad35b902ff400a06718.jpg)  
+Figure 10: Enhanced T2I results by SDXL ( $\omega = 9.0$ ,  $\lambda = 0.8$ ) with CFG++. Under CFG, the lion cub is not visible (top-left), the dog appears with two tails (top-right), the goggles have an unusual shape (bottom-left), and the tree trunk is folded (bottom-right). These artifacts are absent in those produced by CFG++.
+
+# F.3 TEXT-CONDITIONED INVERSE PROBLEMS
+
+In Fig. 20-23, we display additional qualitative comparison for text-conditioned inverse problem solver with CFG and CFG++. Experiments are conducted with FFHQ (512x512) validation set and CFG++ consistently leads to better reconstruction of true solution for various tasks.
+
+![](images/43f9dceac4c864899706001c392b99b40bf35dafca503d5ef500dbf48775e134.jpg)  
+C  
+"A man wearing a suit is taking a self portrait with a camera"
+
+![](images/0ab760f5709b6edab0443c51a141060ac6f534819dc93c442b4b8d419e992fb9.jpg)  
+... coffee shop with neon sign ... little girl with red hair...
+
+![](images/0c43f98f9fc8e085696ec537f47baee1a4ab7fbd15c3363e57b4ccae11afc314.jpg)  
+"A painter study hard to learn how to draw ..."
+
+![](images/de0ce697a356f7a6addbb94ab04f5c5adbeafc9bd3ed16a2f4ea5d81a8acd38e.jpg)  
+C+G  
+Figure 11: T2I using SD v1.5, CFG vs CFG++ ( $\omega = 9.0$ ,  $\lambda = 0.8$ ). Unnatural depictions of human hands, and incorrect renderings of the text by CFG are corrected in CFG++.
+
+![](images/4dcc44f0e0f387dde1fc659fcb9f46e6b37f7cbd6044d5d84f386bbc52fa5685.jpg)
+
+![](images/02a8332e094736370a8525b02fb987b7b0f0f6931b56df8cf650911e5a6c6400.jpg)
+
+![](images/2f42598885fdf6131954a8eaae30ba2654cb1878764bed347ba8ae2a39466241.jpg)  
+G  
+Large brick building with a clock tower at its height.
+
+![](images/5927ce992ed12addbf8782ebe91a15c222759e8435ffe63195e2369aa6bea64d.jpg)  
+"Spectacular Tiny World in the Transparent Jar..."  
+"A silhouette of a grand piano overlooking a dusky cityscape...."
+
+![](images/9c4a8836ea1c747686f178d3c7e277fd68015ecca6797122517797475de60c64.jpg)
+
+![](images/ec190092330ffc1c49d6658233d33fefbd8060c0ba8ccd671b67454ab67ed92f.jpg)  
+C  
+Figure 12: T2I using SDXL-Turbo, 6 NFE, CFG vs CFG++. The overall image quality and sophistication have improved with CFG++. DreamShaper XL was used for both images and metrics in main part.
+
+![](images/d82333471e9f86857663def14a26aac3749fcca49095c394355695a5c5e07211.jpg)
+
+![](images/f5d49b2ccdcf0e51e508415daa9475b5bd08bc0f0983c1a63f1e979345bb45ae.jpg)
+
+![](images/0787bf1ec2ae9c15d60c5279232d8cee2af68938e6b5deb749edf07262015e34.jpg)  
+CFG  
+"An alpaca made of colorful building blocks, cyberpunk."
+
+![](images/d5617606481151166454b3907e7aef3b8ab1f9ad3df5ff14261a007a5a8cd66a.jpg)  
+"Floating, colossal, futuristic statue in the sky, ..."  
+"A detailed oil painting of an old sea captain, steering his ship..."
+
+![](images/41c8af3ef8922d789e621178fd63d7439f576e6e2a0441e5b345b837dc115430.jpg)
+
+![](images/4d7ea86f07f30b65cb55172228be6005365f40f03d6ebdd224ec1681425a020a.jpg)  
+CFG++
+
+![](images/e254b517b5e0a1ea858920c540fc8281185e3fbe115b14d106c1589e97c73b43.jpg)  
+Figure 13: T2I using SDXL-Lightning, 6 NFE, CFG vs CFG++. The overall image quality and sophistication have improved with CFG++. Leosam's HelloWorld XL was used for both images and metrics in main part.
+
+![](images/4be77cb8a6e3ee9fe1c84dc44b4ac617fff028c67aff0e3bb2bde86824e073df.jpg)
+
+![](images/1be9181e2c5d4a4573ab9eb79e1dba34cc567ec95aff8729524d8dd7a585ec8f.jpg)
+
+![](images/70607ea791a1d244e8bf51e1e0a3df636184580ffee80ade0993852a0aa9978f.jpg)  
+"A small cactus with a happy face in the Sahara desert."
+
+![](images/34c129a8736ccbdfee6594c40d401ad46c40fb82f364ff142bbf22a9f6f55ca0.jpg)  
+"A baby painter trying to draw very simple picture, white background"
+
+![](images/1fc61bac2e86fd7e23f6d724305b57ff2bd41daec33212b14fe358eb6dd16eb2.jpg)  
+"A blue jay standing on a large basket of rainbow macarons."
+
+![](images/44aca842c6d87b9022b152d96e159d9d8819fc478046334d18560fcb6407544c.jpg)  
+"A transparent sculpture of a duck made out of glass. The sculpture is in front of a painting of a landscape."  
+"A 4k dsrl image of a lemur wearing a red magician hat and a blue coat performing magic tricks with cards in a garden."  
+Figure 14: The discrete evolution of the posterior mean in CFG and CFG++. Denoised estimates in latent space are decoded into pixel space at each timestep using SDXL.
+
+![](images/f0a6a1bef9dbe0401ed94a7c2db0acdb865b641741fb5a4d88b873cb0a5d94fb.jpg)  
+Figure 15: Interpolating behavior of T2I along various guidance scales using SD v1.5 with CFG and CFG++
+
+![](images/3caf42c6283eef7e91d1e8552d0b318c3cc550a285ea81fe07982cc213e791cf.jpg)  
+Figure 16: Real image inversion results at various CFG and CFG++ scales using SDv1.5. The image is from COCO dataset. We use the text prompt "a white dog is sleeping on a street and a bicycle" during the inversion.
+
+![](images/0537e52af76bf7a682a16afc7e26aea3bd405f29e9c49b16ee5c95c091bb0129.jpg)  
+Figure 17: Comparison on real image editing via DDIM inversion under CFG and CFG++ using SDXL. Cat  $\rightarrow$  Dog.
+
+![](images/f6e0ad219930af045be2f8e4132b84d493649bedee6ecc0994943ca84e074520.jpg)  
+Figure 18: Comparison on real image editing via DDIM inversion under CFG and CFG++ using SDXL. Horse  $\rightarrow$  Zebra.
+
+![](images/015449430438116565b644ec52976cffcb133088a30192166fba7b7ad251da00.jpg)  
+Figure 19: Comparison on real image editing via DDIM inversion under CFG and CFG++ using SDXL. Cat  $\rightarrow$  Cat with glasses.
+
+![](images/a8df25573ecc2fc9ef8cf0344c9e52b95c15aa1e385db63b5a8bb04a7ec35914.jpg)  
+Figure 20: Results of PSLD using CFG and CFG++ on the FFHQ dataset at Super-Resolution (x8).
+
+![](images/d910de075105585dee049493385b08c987c6eadfa3bf6dd1cacfe2b761ce91dd.jpg)  
+GT
+
+![](images/5057f127b50d9dce439dbab5f127d90b837d719cc551cf2626e5ec2beac81201.jpg)  
+Degraded
+
+![](images/e1cf6faee7f011b4256790ef2878daf112da2d89c20cee946818961873ddbf8d.jpg)  
+CFG
+
+![](images/e3847d01bc5a5544635b067ba7d0d73f46849f66a147ce1dde4745d35373c68d.jpg)  
+CFG++
+
+![](images/3101acb23e04e24c42cb1c8c530a49e7a83cba1bbc2f9856467980622e9c10c1.jpg)
+
+![](images/82defd4aa76e40081d56df3a36826f211434daa84c46e6ecb6e7c4e4fb752e67.jpg)
+
+![](images/b1cc3e7f4ff7e2df70ddacc0d9753a4e1b24ca5ce3f11d351e94c506ff665225.jpg)
+
+![](images/91276ccd05d5ca7bacaec94c9662a53151faeab99cb30f2a3a9f3dc1e4d707a2.jpg)
+
+![](images/88152732944b4a7da10ee1f9a36353ce4c2b1a4dcbff0585a2d0c55dc127bc91.jpg)
+
+![](images/7b453384d0934807d5aa3f7a27342cda45169ba2269996512a69c27b17565d5e.jpg)
+
+![](images/5fde88fa19b0bb282071913bf4109b770a3de922c7ede7628ce4589e769699df.jpg)
+
+![](images/88700ffaaf4789214a5a5f4efe511557f804691a1046f67f780d0c9e72ea9d53.jpg)
+
+![](images/5b4d778a62fdbef85fc358652d750a3b1b6d0346f099314a1167bae0bcbac5f7.jpg)  
+Figure 21: Results of PSLD using CFG and CFG++ on the FFHQ dataset at Motion Deblurring.
+
+![](images/a03defb0f6765c31ad232bc53c526bc84410fa417ab5b9ce3a3b1174e045e35a.jpg)
+
+![](images/2d704081abd48ff2985a4727a749ad58a3a8466c18a48879ed88d64a7041e69c.jpg)
+
+![](images/48c0548912bc4bf495a6afe5e9b3dbfb14b15e9b1749000a733c9f58ab07c567.jpg)
+
+![](images/a42e8b750e20e53e7ab30294ce62af66252a249c1230e019166f08d32a34401e.jpg)  
+GT
+
+![](images/9f3777330332a102aa39da198a149ca2e18685aa80d279847ed2e7de55ad2adf.jpg)  
+Degraded
+
+![](images/d5d5bc1690a1aabed41833d83111afcf327d60040533a96ef645c253305c75b5.jpg)  
+CFG
+
+![](images/59577c97dbbe08606f4b804f30bd57c1f909ab9aa185629400340c06e41aec48.jpg)  
+CFG++
+
+![](images/a2d817c5b4f5a781c3ef6b5418250c643ea3e4a7481a6556b84594f101ade404.jpg)
+
+![](images/346d9bdd5d7517e85f2d7e74bfea34cb33fe31443a8f0692937cb22845818d9f.jpg)
+
+![](images/0b3fda83aa5ae88a0ffc2850f0bb5a71b850421f3a3517ebd9033563a0f1e2f4.jpg)
+
+![](images/c258230241b0e563cdb9eee337534d8a475758cc256ef7d1d3115c82869e6fe3.jpg)
+
+![](images/7b83bba01b3dc8a7cef463bfed0979a8cf3f4dbda1db8728ab37829a45b4b9e7.jpg)
+
+![](images/bac634ea1bd029ba86707f920f317f774aecfca5a9fe388d73a66d06f48999c3.jpg)
+
+![](images/5f02a0d3bc171df9738c87ff7141129474f3548d87fdec560006e47d74391c8d.jpg)
+
+![](images/73b37f0d8881d8c49fe0ac629a0bc2bef6f68048e3b8c84604720ad194dcb3b2.jpg)
+
+![](images/35c008c787aee83ebef5d8627710031a79c9b1a705e658a96c1fd7beb33d63da.jpg)
+
+![](images/f1818c07403d8705354a1b7cae9e47cee03a6b8304cfdf046bce5bc0cc5bf014.jpg)
+
+![](images/afb54cef7d24abc1e32246c81679597bcf0f40d592cd94dc57426f6a73e4d5f4.jpg)  
+Figure 22: Results of PSLD using CFG and CFG++ on the FFHQ dataset at Gaussian Deblurring.
+
+![](images/2b173d0f6c1ec8714ac7b540f992d273eed1c4d9538c853f5c5c7d12244cb293.jpg)
+
+![](images/74e00e0ecf84b54729e6494b7cb5d31f2cc58cf74f9256a67606f9d4df85579b.jpg)
+
+![](images/ed94d6e8699bc76f7bf265b23582a25ea1d542dbf1d621528a5527f7ddb613a7.jpg)
+
+![](images/5ac3d13f239d31d0ac40ef3bac113a0a8114d70b97ea57af010f5ae854333f62.jpg)
+
+![](images/47103889e9a07c51d71a168b2acea03ce9e58fd64cdec864854d0b881fda3401.jpg)  
+Figure 23: Results of PSLD using CFG and CFG++ on the FFHQ dataset at Inpainting.

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