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+# Improving Gloss-free Sign Language Translation by Reducing Representation Density
+
+Jinhui Ye $^{1}$  Xing Wang $^{4}$  Wenxiang Jiao $^{4,\dagger}$  Junwei Liang $^{1,2,\dagger}$  Hui Xiong $^{1,2,3,\dagger}$
+
+$^{1}$ Artificial Intelligence Thrust, HKUST (Guangzhou), Guangzhou, China  $^{2}$ Department of Computer Science and Engineering, HKUST, Hong Kong SAR, China
+
+<sup>3</sup>Guangzhou HKUST Fok Ying Tung Research Institute <sup>4</sup>Tencent AI Lab jye624@connect.hkust-gz.edu.cn; junweiliang@hkust-gz.edu.cn; xionghui@ust.hk; {brightxwang, joelwxjiao}@tencent.com
+
+# Abstract
+
+Gloss-free sign language translation (SLT) aims to develop well-performing SLT systems with no requirement for the costly gloss annotations, but currently still lags behind gloss-based approaches significantly. In this paper, we identify a representation density problem that could be a bottleneck in restricting the performance of gloss-free SLT. Specifically, the representation density problem describes that the visual representations of semantically distinct sign gestures tend to be closely packed together in feature space, which makes gloss-free methods struggle with distinguishing different sign gestures and suffer from a sharp performance drop. To address the representation density problem, we introduce a simple but effective contrastive learning strategy, namely SignCL, which encourages gloss-free models to learn more discriminative feature representation in a self-supervised manner. Our experiments demonstrate that the proposed SignCL can significantly reduce the representation density and improve performance across various translation frameworks. Specifically, SignCL achieves a significant improvement in BLEU score for the Sign Language Transformer and GFSLT-VLP on the CSL-Daily dataset by  $39\%$  and  $46\%$ , respectively, without any increase of model parameters. Compared to Sign2GPT, a state-of-the-art method based on large-scale pre-trained vision and language models, SignCL achieves better performance with only  $35\%$  of its parameters. Implementation and Checkpoints are available at https://github.com/JinhuiYE/SignCL.
+
+# 1 Introduction
+
+Sign languages are the primary form of communication for millions of deaf individuals. Sign language translation (SLT) aims to convert sign language into fluent spoken language sentences, which is a challenging task as it needs to extract information from continuous video and translate it into discrete text tokens. Most prior studies promoted the SLT by utilizing intermediate representations, namely gloss annotations, either directly or indirectly [3, 48, 53, 8, 49, 46, 41]. Gloss annotations are beneficial as they provide a simplified representation and sequential ordering of each gesture within continuous sign videos, which aids in representation learning for visual encoders. However, the creation of sign language translation datasets with gloss annotations is both resource-intensive and time-consuming.
+
+Recently, there has been a shift towards gloss-free sign language translation methods, which do not rely on gloss annotations to train SLT models. These methods usually rely on general datasets [47],
+
+![](images/393fc3f1b19d49b1a0a8d7eed850071e46a47708cc8551693331c2b5e3cfb86a.jpg)  
+Figure 1: An example of the representation density problem in sign language translation. The two images show the sign gestures for "RECIPROCATE" (blue dot) and "REVENGE" (orange dot). Although the two have opposite meanings, their visual representations are densely clustered together, as shown in the t-SNE visualization. The various colors in the visualization indicate sign gestures with different meanings.
+
+general pretraining strategy [52], or general large-scale foundation models [42] to promote gloss-free SLT. However, there is a substantial gap between the sign language domain and the general domain [45, 24]. Models trained with general strategies or datasets often fail to capture the subtle differences in semantically distinct gestures, which are crucial for accurately understanding a specific sign language. Therefore, the performance of gloss-free methods still significantly lags behind that of gloss-based approaches.
+
+In this paper, we identify a representation density problem in sign language translation: the visual representations of sign gestures with distinct semantics are likely to be close in representation space. This problem is attributed to the nature of sign language, a form of visual language that utilizes intricate hand gestures, facial expressions, and body movements to convey the signer's message [35, 40, 17]. For example, in Figure 1, the signer performs sign gestures for opposite meanings, "RECIPROCATE" and "REVENGE", with similar visual information (i.e., only subtle differences in facial movements). The visual encoder in SLT models will encode similar visual information to visual representations in adjacent representation space, even though they have distinct semantics. Without explicit gloss annotations, SLT models struggle to learn semantic boundaries in continuous sign videos and capture distinguishing visual representations for different sign gestures. As a result, the representation density problem poses a significant challenge for the SLT models in distinguishing between various sign gestures, leading to sharp performance drops. (Section 3.2).
+
+Further, we investigate various popular sign feature extraction methods, including gloss-based [3, 33] and gloss-free [47, 52], to systematically study the representation density problem. As shown in Figure 2, our investigation reveals that the representation density problem is prevalent across sign feature extraction methods. Specifically, due to the lack of gloss annotations, the representation density problem appears to be more serious in gloss-free methods. Then, we conduct extensive SLT experiments and observe that SLT models using gloss-free sign features as input consistently suffer a drop in performance in both sign language recognition and translation tasks compared to those using gloss-based sign features (Section 3.3). Therefore, we demonstrate that the representation density problem can be a bottleneck in restricting the improvement of gloss-free sign language translation.
+
+More importantly, we propose a simple but effective contrastive learning strategy named SignCL to address the representation density problem. Specifically, SignCL draws the visual representations of sign gestures with identical semantics closer together and pushes those with different semantics farther apart. Experimental results show that SignCL can learn more distinctive feature representations and lead to significant improvements in terms of BLEU score on various well-known SLT frameworks (Section 5). To summarize, the main contributions of this work are as follows:
+
+- To the best of our knowledge, our work identifies the representation density problem in sign language translation for the first time. This problem is consistent across various sign feature extraction methods for SLT, including gloss-based and gloss-free methods.  
+- Experimental results empirically reveal that an increase in representation density leads to a significant performance drop in the accuracy of sign language recognition and translation. We find that the representation density problem poses a significant challenge for the gloss-free SLT.
+
+- We propose a simple but effective contrastive learning strategy, namely SignCL, to address the representation density problem. Our experiments demonstrate that SignCL can significantly enhance various well-known SLT frameworks. Specifically, SignCL yields a  $39\%$  BLEU score improvement for the Sign Language Transformer [4] and a  $46\%$  BLEU increase for GFSLT-VLP [53] on the CSL-Daily dataset.
+
+# 2 Related Works
+
+# 2.1 Sign Language Translation
+
+Sign Language Translation (SLT) methods can be broadly categorized into gloss-based and gloss-free approaches. For gloss-based methods, an essential factor is to directly or indirectly employ sign gloss annotations to improve sign video encoder performance [3, 48, 53, 8, 49, 6]. These methods often employ Connectionist Temporal Classification [16] (CTC) loss to perform sign language recognition [4]. Joint-SLT [4] firstly introduces a multitask encoder-decoder framework with a CTCloss to soft-match sign representations and gloss sequences. STMC-T [54] introducing intra-cue and inter-cue CTC loss to model multi-cue sequence information. Despite their effectiveness, creating SLT datasets with gloss annotations is resource-intensive and time-consuming. Gloss-free methods have emerged as a promising alternative, as they do not rely on gloss annotations during training, making them more generalizable. And recently, a growing body of literature has promoted the gloss-free SLT, such as GASLT [47] proposed local gloss attention to mimic gloss assistant, GFSLT [52] adapted CLIP to do visual-language pretraining, and Sign2GPT [42] promoted performance by making use of large-scale pre-trained vision and language models. Nonetheless, the performance of gloss-free methods still significantly lags behind that of gloss-based approaches.
+
+# 2.2 Contrastive Learning
+
+Contrastive Learning [21, 55, 30], a popular unsupervised learning algorithm, aims to learn effective representations by pulling positive pairs closer together and pushing negative pairs farther apart. This approach has been widely utilized in both Natural Language Processing and Computer Vision [13]. In Sign Language Translation (SLT), Jin and Zhao [22] utilize Contrastive Learning to create a Signer-Independent SLT model, using videos demonstrating signs from different signers as positive samples. Additionally, Gan et al. [15] proposes a visual-level contrastive learning method with various image augmentation strategies. ConSLT [14] do contrastive learning for effective token representation learning in text decoder. Zhou et al. [52] and Cheng et al. [9] employ contrastive learning techniques to align video and text representations in SLT. In this paper, we are the first one to address the representation density problem, focusing particularly on visual gesture duration as a central aspect.
+
+# 2.3 Representation Density
+
+Representation Density is often a focal point in classification tasks, also known as category density [1, 43, 37, 31, 32, 51, 29, 12]. This concept pertains to the compactness and clarity of feature representations across different categories. In the context of sign language, various methods have been developed to address the subtle nuances of sign actions. TSPNet [28] proposes a temporal hierarchical attention network to learn segmented representations. HST-GNN [24] utilizes a hierarchical spatio-temporal graph neural network to learn graph representations from multiple perspectives. GLE-Net [20] employs global contextual relationships and fine-grained cues to distinguish non-manual-aware features in isolated Sign Language Recognition. These methods are beneficial for addressing the subtleties of sign language movements. However, integrating them into existing state-of-the-art frameworks presents significant challenges, often resulting in performance disparities when compared to the SOTA. This paper is the first to propose the concept of representation density within this field and introduces SignCL, which enhances the current mainstream transformer-based frameworks.
+
+# 3 Representation Density Problem
+
+This section investigates and identifies the representation density problem within existing sign feature extraction techniques, and examines whether representation density bottlenecks sign language recognition and translation performance.
+
+# 3.1 Preliminaries
+
+Existing Sign Feature Extraction Techniques Existing sign feature extraction methods can be divided into two categories: 1) gloss-based (e.g., Sign Recognition Pretrained [3] and Self-Mutual Knowledge Distillation [33]) and 2) gloss-free (e.g., I3D Pretraining [47] and Visual-Language Pretraining [52]). These methods were chosen for their representativeness in SLT and their well-documented open-source sign features.
+
+- Sign Recognition Pretrained (SRP) [3]: This approach leverages the sign language recognition datasets to train sign language recognition models and uses it as the feature extractor for the SLT task. Notably, the features released by Camgoz et al. [3] have been widely adopted as input features in a range of works [5, 53, 22, 44, 46, 7].  
+- Self-Mutual Knowledge Distillation (SMKD) [18]: This approach enhances SRP by enforcing the visual and contextual modules to focus on short-term and long-term information [18]. SMKD feature extraction has been shown to substantially enhance SLT translation performance compared to SRP [50, 46].  
+- I3D Pretraining (I3D) [47]: This method employs I3D models as the backbone to pre-train the feature extractor, initially trained on the Kinetics dataset [25] and subsequently fine-tuned on extensive web SLR datasets, such as WSLR [27].  
+- Visual-Language Pretraining (VLP) [52]: This method entirely forgoes gloss annotations and leverages a general visual-language pretraining strategy to align sign video representation with text. Embodied by GFSLT-VLP [52], this approach offers a more general solution that utilizes a broader range of sign language resources without the constraints of gloss annotations.
+
+Representation Density Metrics Drawing inspiration from Fisher's Discriminant Ratio (FDR) [23, 19], a typical measure used to evaluate the discriminative power of features in the classification, we combine the average Inter-Gloss Distance and Intra-Gloss Distance into Sign Density Ratio (SDR, see Eqn. 1), which reflects the degree of representation density for each gloss  $G_{i}$ . This is given by the formula:
+
+$$
+S D R \left(G _ {i}\right) = \frac {D _ {G _ {i}} ^ {\text {i n t r a}}}{\operatorname {a v g .} D _ {G _ {i}} ^ {\text {i n t e r}}} = \frac {D \left(G _ {i}\right)}{\operatorname {M e a n} _ {j \neq i} \left(D \left(G _ {i} , G _ {j}\right)\right)}. \tag {1}
+$$
+
+Here,  $D(G_{i},G_{j})$  represents the Inter-Gloss Distance between two glosses  $G_{i}$  and  $G_{j}$ , and avg.  $avg.D_{Gi}^{inter}$  reflects the average distance of  $G_{i}$  to all other glosses. The Intra-Gloss Distance  $D_{Gi}^{intra}$  evaluates the average distance within a single gloss  $G_{i}$ . These distances are given by the following formulas:
+
+$$
+D \left(G _ {i}, G _ {j}\right) = \frac {1}{\left| G _ {i} \right| \left| G _ {j} \right|} \sum_ {x \in G _ {i}, y \in G _ {j}} d (x, y); \tag {2}
+$$
+
+$$
+D \left(G _ {i}\right) = \frac {1}{\left| G _ {i} \right| \left(\left| G _ {i} \right| - 1\right)} \sum_ {x, y \in G _ {i}, x \neq y} d (x, y); \tag {3}
+$$
+
+Where,  $|G_i|$  and  $|G_j|$  denote the number of instances in glosses  $G_i$  and  $G_j$  respectively, and  $d(x,y)$  represents the distance measure between the embeddings of instances  $x$  and  $y$ , i.e., euclidean distance.
+
+The average Sign Density Ratio (SDR) of all glosses, denoted as  $SDR = \mathrm{Mean}(SDR(G_i))$ , is calculated to evaluate the overall representation density of the dataset comprehensively.
+
+Sign-Gloss Alignment To calculate the Sign Density Ratio (SDR), we need to determine the mapping relationship between input frames and gloss categories. Following previous works [26, 46], we employ the CTC classifier as a sign-gloss forced aligner to establish the mapping between each gloss and its corresponding sign frames. The aligner provides the start position  $l_{v}$  and end position  $r_{v}$  within the video frame sequence for each corresponding gloss  $g_{v}$ . To optimize alignment performance on the test set, we merge the training and test datasets for comprehensive training and engage two volunteers to select the best frame  $f_{v}$  from the range  $[l_{v}:r_{v}]$  to align with each gloss  $g_{v}$ . Extensive details on the training procedure and the aligner's performance metrics are documented in Appendix 9.
+
+# 3.2 Demonstrating Representation Density Problem
+
+Experiment Setups We primarily use the PHOENIX-2014T benchmark [3] to investigate the representation density problem in existing sign feature extraction techniques. This benchmark was selected due to its rich collection of open-source sign features contributed by various research efforts. We obtained the sign features by either downloading the officially released versions or reproducing the feature extraction process. Then, we employed t-SNE [39] to visualize the feature distribution of these semantically distinct sign gestures to investigate representation density.
+
+![](images/b7a4da5eb3d83e7ef5c6e14bd7699896655ef5a7cc351cc205f3ab6586c7edde.jpg)  
+(a,b,c) Gloss-free Feature Extraction  
+(a) VLP  
+Figure 2: The t-SNE visualization of sign features across existing extraction techniques. SRP, SMKD, and I3D are downloaded from their official websites, while VLP is reproduced with official code. The addition of  $+\mathrm{SignCL}$  denotes our proposed method that integrates a contrastive learning strategy into the VLP method (see Section 4). Different colors represent sign gestures with distinct semantics. Points in gray represent other sign categories not listed. Better viewed by zooming in.
+
+![](images/3b70243024d7b901abc8576ddedcf662b05b9ed5a92b2de5c9765ebc0d29974d.jpg)  
+(b) I3D
+
+![](images/033e6e153954d242801e42af32f628c105cd9e3e666946c803f9504b461ea100.jpg)  
+(c)  $+\mathrm{SignCL}(\mathrm{Ours})$
+
+![](images/00d28eb2af4cbbd0fc5e71406e5111c8475bc012c2a538ff559c9923433b028e.jpg)  
+(d,e) Gloss-based Feature Extraction  
+(d) SRP
+
+![](images/6c36ecaabb50a2b873ea5084e519b959b796e5eb7d93656efeba3fee6e58a6d8.jpg)  
+(e) SMKD
+
+Results and Findings Through empirical analysis of various visualized open-source sign features, we have identified a widespread representation density problem across different sign feature extraction methods. As depicted in Figure 2, all evaluated methods display a Sign Density Ratio exceeding  $50\%$ , with inevitable overlap of feature representation. Notably, gloss-free methods that do not utilize gloss annotations as additional supervision (e.g., I3D and VLP) exhibit even more severe representation density compared to gloss-based methods. This is evident as sign gestures representing different semantics, indicated by different colors, significantly overlap, resulting in translation ambiguity during inference. Specifically, the Sign Density Ratio (SDR) of VLP is  $92.59\%$ , which is significantly higher than the SDR of SMKD at  $66.23\%$ .
+
+![](images/e1974c7e600436d333fd4c6bef69a211322b795a791b1b90c002c83d6977950c.jpg)  
+(a) Impact on SLR  
+Figure 3: Comparative analysis of representation density and its impact on sign language recognition (SLR) and translation (SLT). The left panel (a) shows the correlation between representation density and SLR accuracy across different sign feature types and sign gesture groups. Binning in this context is based on sorting by gloss density within a group, where higher bins indicate higher density. The right panel (b) illustrates the performance drops in SLT caused by the representation density problem. This figure assesses both the recognition and translation accuracies, reflecting how denser representations impact these metrics.
+
+![](images/df46363add97f4db1adde85b649716edb2f45a99949589d4cf7bf20de0230cb6.jpg)  
+(b) Impact on SLT
+
+# 3.3 Demonstrating Performance Drop
+
+This section investigates the impact of representation density on sign language recognition (SLR) and translation (SLT) systems.
+
+General Setups This part employs the widely utilized Sign Language Transformer [5] (NSLT) as the foundational model for our evaluations. The choice is because its capability to perform both SLT and SLR tasks, as well as take SLR and SLT at the same time (joint-SLT). Additionally, the NSLT framework is well-established within sign language research and benefits from comprehensive documentation and support in open-source sign feature sets and baseline results. The NSLT relies on sign features derived using a pretrained sign feature extractor. This section studies all types of sign features introduced in Section 3.2 to investigate the representation density problem. We use the Sign Density Ratio (SDR, see Eqn. 1) to measure the representation density within each type of input feature. We measure SLR and SLT performance by the recognition accuracy and the BLEU-4 [34] score  $(\mathtt{B}@\mathtt{4})$ , respectively.
+
+Task Setups We set up tasks to examine whether representation density bottlenecks sign language recognition and translation performance.
+
+- Sign Language Recognition: To evaluate the ability of the extracted sign features to distinguish between different semantic gestures, we use the NSLT [4] to perform sign language recognition (SLR) tasks with various types of sign features [3] as model input. Due to the limited number of samples for each gesture in the dev set, we rank the sign glosses based on their density using  $\mathrm{SDR}(G_i)$  under SMKD features (see Eqn. 1). These glosses are then divided into nine groups (bins), each containing approximately 60 glosses. The average  $\mathrm{SDR}(G_i)$  and recognition accuracy for each bin represents the overall density and mean accuracy of the glosses within that bin, respectively.
+
+- Sign Language Translation: This evaluation aims to demonstrate the impact of representation density on translation tasks. We evaluate various sign features as inputs to the Sign Language Transformer, including SRP, SMKD, I3D, VLP, and VLP+SignCL. These inputs are tested across different translation frameworks, such as NSLT [3], Joint-SLT [4], and NSLT+SignCL. NSLT means use NSLT to perform SLT without CTC loss (gloss-free) and the NSLT+SignCL configuration integrates the proposed contrastive learning strategy into the encoder of NSLT [3] models, as detailed in Section 4.
+
+Results and Findings As depicted in Figure 3, the following observations were made regarding the impact of representation density on both recognition (SLR) and translation (SLT):
+
+- Performance suffers from representation density. We consistently observed a negative relationship between representation density and performance across all feature types and tasks. Higher representation density leads to worse accuracy in SLR and lower BLEU scores in SLT. Specifically, an increase in the representation density ratio by  $26\%$  can result in a  $39\%$  performance drop in NSLT.
+
+- Gloss-free methods suffer from worse representation density. Gloss-free based feature extractions, which do not use any gloss annotations for assistance (e.g., VLP), typically exhibit higher representation density scores than gloss-based approach (e.g., SDR(VLP) = 92.59% > SDR(SMKD) = 66.23%). Using gloss-free features results in worse recognition and translation performance compared to gloss-based feature extractions (e.g., VLP vs. SMKD).
+
+- Contrastive learning boosts performance by reducing representation density. When contrastive learning is applied to augment gloss-free based feature representation learning, i.e., VLP+SignCL for feature extraction or NSLT+SignCL for downstream finetuning, there is a consistent reduction in feature representation density accompanied by a significant improvement in both of the SLR accuracy and the SLT performance (see detail can be found in Section 4).
+
+# 4 Contrastive Learning for Gloss-free Sign Language Translation
+
+Contrastive Learning [21], a popular self-supervised learning algorithm, aims to learn effective representations by pulling positive pairs closer together and pushing negative pairs farther apart. In
+
+this section, we introduce a simple but efficient sign contrastive learning strategy, namelySignCL, which addresses the challenge of the representation density problem in gloss-free sign language translation.
+
+# 4.1 Sign Contrastive Learning
+
+The key factor in contrastive learning is how to sample positive and negative training pairs. As illustrated in the framework shown in Figure 4a, the sampling strategy of SignCL is as follows: if two frames are close enough (e.g., adjacent), they are considered to belong to the same sign gesture and are treated as positive samples. Conversely, if two frames are far apart by double the margin (e.g.,  $|f_{ed} - f_{st}| > 20$  frames), they are considered to be associated with different semantics and are treated as negative samples. Statistically, the average duration of each gesture in sign video is nine frames [3, 53], and according to the speech-to-gesture Zipf's Law [2], each gloss represents approximately 2.3 spoken words. Therefore, we set the margin as max(10,  $\frac{\text{lenumerator}}{\text{lenumerator}} \times 2.3$ ).
+
+$$
+\left\{ \begin{array}{l l} \text {p o s i t i v e p a i r} \left(f _ {s t}, f _ {e d} ^ {+}\right): & \left| f _ {e d} ^ {+} - f _ {s t} \right| \leq 1 \\ \text {n e g a t i v e p a i r} \left(f _ {s t}, f _ {e d} ^ {-}\right): & \left| f _ {e d} ^ {-} - f _ {s t} \right| > 2 * m a r g i n \end{array} ; \right. \tag {4}
+$$
+
+$$
+\mathcal {L} _ {\text {S i g n C L}} = \frac {1}{N} \sum_ {s t = 1} ^ {N} \left[ \mathrm {d} \left(f _ {s t}, f _ {e d} ^ {+}\right) + \max  \left(0, m - \mathrm {d} \left(f _ {s t}, f _ {e d} ^ {-}\right)\right) \right]; \tag {5}
+$$
+
+Where  $d$  is the distance function, i.e., Euclidean distance for frame features  $(f_{st}, f_{ed})$ , and  $N$  is the total number of frames in one sign video,  $N = \text{len}(\text{frames})$ . The margin parameter  $m$  is used to prevent the features of the negative pair from being too far away. We empirically set  $m = 64$  based on the average Inter-Gloss Distance (see Eqn. 2) of gloss-based sign features (e.g., SMKD[18]).
+
+![](images/e77d211d5c8935d37cb117765443c15d7656c385446843652237f702186d0e59.jpg)  
+(a) Sign Contrastive Learning
+
+![](images/fdd5fc53adb4201b96e24785c58ce4b466ecca006eceaf3e00e14490835501dd.jpg)  
+Figure 4: Overview of the SignCL in gloss-free sign language translation: (a) Sign contrastive learning sampling strategy, (b) Showcases the integration of SignCL in the pretraining stage, and (c) Displays the application of SignCL during the finetuning stage.
+
+![](images/09ab8422f28b8d452949b111996de377deb5a38c87a0768d7c5fc4759bf22d57.jpg)  
+(b) SignCL in Pretraining  
+(c) SignCL in Finetuning
+
+# 4.2 Integrating Contrastive Learning into Sign Language Translation Training
+
+As illustrated in Figures 4b and 4c, SignCL can be integrated into both the sign feature extraction pretraining stage (e.g., Visual-Language Pretraining [52]) and the downstream task finetuning stage (e.g., GFSLT-VLP [52]). The optimization objective for these approaches is the weighted sum of  $\mathcal{L}_{\mathrm{SignCL}}$  and the original objective loss (e.g., VLP Loss for pretraining and SLT loss for finetuning [3, 52]), defined as:
+
+$$
+\mathcal {L} = \lambda * \mathcal {L} _ {\text {S i g n C L}} + \mathcal {L} _ {M L E}; \tag {6}
+$$
+
+Where  $L_{MLE}$  is the original objective loss in the pertaining or finetuning.
+
+# 5 Experiments
+
+In this Section, we conduct experiments to demonstrate the efficiency of proposed SignCL in reducing representation density and boosting gloss-free sign language translation performance. Specifically, we apply SignCL to the Sign Language Transformer [4] to facilitate a direct comparison with prior empirical analyses of the representation density problem in Section 3.3. Additionally, we integrate SignCL into the GFSLT-VLP [52] framework, a robust new gloss-free baseline that improves SLT through pretraining and finetuning.
+
+# 5.1 Experiments on Sign Language Transformer
+
+In Section 3, we investigate the representation density problem using the Sign Language Transformer (SLT) [4] and the PHOENIX-2014T [3] and CSL-Daliy [53] Dataset. These benchmarks are chosen for their established relevance in sign language translation research, including gloss-based and gloss-free based. Here, we first conduct experiments on the same framework and dataset to facilitate direct comparison with the prior empirical analyses.
+
+Experiment Settings: In this experiment, we introduce SignCL as additional supervision information in the encoder of SLT under gloss-free settings. This enhanced model is referred to as +SignCL.
+
+Results and Findings: The integration of SignCL into the SLT has significantly improved translation performance across all test conditions by reducing the representation density, as shown in Table 1. Notably, SignCL encourages SLT to learn a more distinct feature distribution, reducing the Sign Density Ratio (SDR) significantly, e.g., 66.23 to 62.18 and 92.59 to 81.30.
+
+Figures 3a and 3b show experiments on SLR and SLT tasks using features with varying SDRs as inputs to SLT. The representation density reduction leads to observable improvements in both recognition accuracy (red line vs. purple line in Figure 3a) and translation BLEU score (purple point vs. red point in Figure 3b). Further details and additional experiment results on the CSL-Daily dataset are provided in Appendix A.4.3.
+
+Table 1 presents a comparative analysis of representation density and performance on the PHOENIX-2014T dataset. The inclusion of SignCL during VLP feature extraction or SLT training processes significantly enhances performance metrics. WERs (Word Error Rates) in the gloss-free set, derived from an independent SLR task, are specifically used to probe the quality of sign features and do not participate in the SLT training process. This analysis underscores the significant enhancements brought by SignCL in terms of both efficiency and effectiveness in SLT frameworks.
+
+<table><tr><td rowspan="2">Feature Type</td><td colspan="3">PHOENIX-2014T</td><td colspan="3">CSL-Daily</td></tr><tr><td>SDR↓</td><td>WER↓</td><td>B@4↑</td><td>SDR↓</td><td>WER↓</td><td>B@4↑</td></tr><tr><td colspan="7">Gloss-based</td></tr><tr><td>Joint-SLT [5] / Self-Mutual KD [33]</td><td>66.23</td><td>25.38</td><td>22.79</td><td>48.34</td><td>29.52</td><td>11.61</td></tr><tr><td>+ SignCL into Feature Extraction</td><td>62.18</td><td>24.76</td><td>23.23</td><td>-</td><td>-</td><td>-</td></tr><tr><td>+ SignCL into Finetuning</td><td>66.23</td><td>25.12</td><td>22.92</td><td>-</td><td>-</td><td>-</td></tr><tr><td>+ SignCL into both</td><td>62.18</td><td>24.58</td><td>23.46</td><td>-</td><td>-</td><td>-</td></tr><tr><td colspan="7">Gloss-free</td></tr><tr><td>SLT [3] / VLP Pretrained [52]</td><td>92.59</td><td>69.72</td><td>10.73</td><td>76.55</td><td>85.78</td><td>1.82</td></tr><tr><td>+ SignCL into Feature Extraction</td><td>81.30</td><td>63.33</td><td>12.04</td><td>68.39</td><td>80.71</td><td>2.15</td></tr><tr><td>+ SignCL into Finetuning</td><td>92.59</td><td>-</td><td>12.14</td><td>76.55</td><td>-</td><td>2.19</td></tr><tr><td>+ SignCL into both</td><td>81.30</td><td>-</td><td>13.51</td><td>68.39</td><td>-</td><td>2.53</td></tr></table>
+
+Table 1: Comparative analysis of representation density and performance on the PHOENIX-2014T dataset. "+"SignCL" indicates the inclusion of the proposed contrastive learning strategy during VLP (Video Language Processing) feature extraction or SLT (Sign Language Translation) training processes. WERs (Word Error Rates) in the gloss-free set are derived from an independent SLR (Sign Language Recognition) task, used specifically for probing the quality of sign features. These WERs do not participate in the SLT training process.
+
+# 5.2 Experiments on Gloss-free Sign Language Translation
+
+Gloss-free sign language translation, which does not rely on gloss annotations, has become a trend as it makes the approach more generalizable. In the realm of gloss-free sign language translation, GFSLT-VLP [52] stands out as a strong new baseline. It incorporates CLIP [36] and MBART [10] for model pretraining and finetuning. In this set of experiments, we use GFSLT-VLP as the baseline model and integrate the proposed SignCL into the framework to demonstrate the effectiveness of our method in both pretraining and finetuning settings.
+
+Experiment Settings: This set of experiments is conducted using the PHOENIX-2014T [3] and CSL-Daily [53] datasets. We reproduce GFSLT-VLP using the official code and integrate SignCL into both the pretraining and finetuning stages. All models and training details are consistent with
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Density SDR↓</td><td colspan="5">Performance</td></tr><tr><td>R@L↑</td><td>B@1↑</td><td>B@2↑</td><td>B@3↑</td><td>B@4↑</td></tr><tr><td>NSLT [3, 4]</td><td>-</td><td>30.07</td><td>29.86</td><td>17.52</td><td>11.96</td><td>9.00</td></tr><tr><td>GASLT [47]</td><td>-</td><td>39.86</td><td>39.07</td><td>26.74</td><td>21.86</td><td>15.74</td></tr><tr><td>GFSLT [52]</td><td>-</td><td>40.93</td><td>41.39</td><td>31.00</td><td>24.20</td><td>19.66</td></tr><tr><td>GFSLT-VLP [52]</td><td>-</td><td>42.49</td><td>43.71</td><td>33.18</td><td>26.11</td><td>21.44</td></tr><tr><td>Sign2GPT(w/PGP) [42]</td><td>-</td><td>48.90</td><td>49.54</td><td>35.96</td><td>28.83</td><td>22.52</td></tr><tr><td>GFSLT-VLP [52]</td><td>68.53</td><td>42.97</td><td>42.13</td><td>32.04</td><td>25.62</td><td>21.25</td></tr><tr><td>+ SignCL into Pretraining</td><td>62.68</td><td>49.25</td><td>49.99</td><td>36.73</td><td>29.76</td><td>22.69</td></tr><tr><td>+ SignCL into Finetuning</td><td>62.73</td><td>48.17</td><td>48.56</td><td>35.04</td><td>27.73</td><td>22.16</td></tr><tr><td>+ SignCL into Two State</td><td>62.32</td><td>49.04</td><td>49.76</td><td>36.85</td><td>29.97</td><td>22.74</td></tr><tr><td>Improvement</td><td>-6.21</td><td>+6.07</td><td>+7.63</td><td>+4.81</td><td>+4.35</td><td>+1.49</td></tr></table>
+
+Table 2: Improvement in the GFSLT-VLP framework by reducing representation density on PHOENIX-2014T test set. "+"SignCL into Pretraining" indicates applying the proposed contrastive learning strategy during the pretraining stage, while "+"SignCL into Finetuning" indicates the inclusion of the SignCL during the finetuning stage. "+"SignCL into Two State" means plus SignCL both in pertaining and finetuning states.  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Density SDR ↓</td><td colspan="5">Performance</td></tr><tr><td>R@L↑</td><td>B@1↑</td><td>B@2↑</td><td>B@3↑</td><td>B@4↑</td></tr><tr><td>GASLT [47]</td><td>-</td><td>20.35</td><td>19.90</td><td>9.94</td><td>5.98</td><td>4.07</td></tr><tr><td>NSLT [3, 4]</td><td>-</td><td>34.54</td><td>34.16</td><td>19.57</td><td>7.56</td><td>7.56</td></tr><tr><td>GFSLT [52]</td><td>-</td><td>35.16</td><td>37.69</td><td>23.28</td><td>14.93</td><td>9.88</td></tr><tr><td>GFSLT-VLP [52]</td><td>-</td><td>36.44</td><td>39.37</td><td>24.93</td><td>16.26</td><td>11.00</td></tr><tr><td>Sign2GPT(w/PGP) [42]</td><td>-</td><td>42.36</td><td>41.75</td><td>28.73</td><td>20.60</td><td>15.40</td></tr><tr><td>GFSLT-VLP [52]</td><td>58.20</td><td>39.08</td><td>36.37</td><td>23.32</td><td>15.45</td><td>11.10</td></tr><tr><td>+ SignCL into Pretraining</td><td>55.24</td><td>47.38</td><td>46.20</td><td>32.33</td><td>22.35</td><td>15.85</td></tr><tr><td>+ SignCL into Finetuning</td><td>55.03</td><td>48.26</td><td>46.53</td><td>32.41</td><td>22.42</td><td>15.98</td></tr><tr><td>+ SignCL into Two States</td><td>54.61</td><td>48.92</td><td>47.47</td><td>32.53</td><td>22.62</td><td>16.16</td></tr><tr><td>Improvement</td><td>-3.59</td><td>+9.84</td><td>+11.10</td><td>+9.21</td><td>+7.17</td><td>+5.06</td></tr></table>
+
+Table 3: Enhancing GFSLT-VLP by reducing representation density on CSL-Daily test set.
+
+those used in GFSLT-VLP [52], with the sole exception being the incorporation of SignCL, weighted by  $\lambda = 0.01$ , as illustrated in Figure 2 and Equation 6. Further details are provided in Appendix A.1.
+
+Results and Findings: Tables 2 and 3 compare our proposed methods with existing gloss-free sign language translation approaches. The results demonstrate that integrating the proposed SignCL strategy into the GFSLT-VLP framework consistently reduces representation density and significantly boosts translation performance, whether SignCL is applied during pretraining, finetuning, or both stages. Specifically, compared to the baseline model GFSLT-VLP [52], our approach achieves a substantial improvement of  $45.58\%$  (+5.06) in the BLEU-4 score on the CSL-Daily dataset, without any increase in the number of parameters. Additionally, despite having significantly fewer parameters ( $\sim 600\mathrm{M}$  vs.  $\sim 1.7\mathrm{B}$ ), our approach achieves better performance than Sign2GPT [42], which leverages large-scale pretrained vision and language models for sign language translation.
+
+# 5.3 Qualitative Analysis
+
+To understand our SignCL approach in scenarios of addressing representation density, we present a case from the CSL-Daily dataset in Figure 5. As shown, the way to display sign gestures for "电脑" (laptop) and "钢琴" (piano) differ subtly. As indicated by the t-SNE results, the representations of these two semantically different gestures are closely packed together in the feature space, causing the baseline GFSLT-VLP model to incorrectly translate "钢琴" (piano) as "电脑" (laptop). In contrast, our proposed SignCL effectively separates the representations of "电脑" (laptop) and "钢琴" (piano) in the feature space, enabling the accurate translation of "钢琴" (piano).
+
+![](images/1c4e7b04bc30d0e424b6270958774efdcb0c065addf0306ab02f885a2057e22d.jpg)  
+Figure 5: Qualitative comparison of translation results on CSL-Daily test set. The red background denotes model misinterpretations about the sign gestures, while green one means accurate recognition. Content in (... is English translation for non-Chinese readers.
+
+# 6 Conclusion
+
+In this work, we identify a crucial representation density problem in gloss-free sign language translation. Our systematic investigation reveals that this problem persists across various existing sign feature extraction methods and causes sharp performance drops in both sign language recognition and translation, particularly in gloss-free methods. To address this problem, we propose a simple but effective contrastive learning strategy, termed SignCL. Our experiments demonstrate that SignCL encourages gloss-free models to learn more discriminative features and significantly reduces representation density. Furthermore, our experiments show that SignCL improves translation performance across various frameworks and datasets by a significant margin, achieving a new state-of-the-art in gloss-free sign language translation. We illustrate the effectiveness of SignCL through detailed examples in our qualitative analysis. Finally, we provide several ablation studies for a better understanding of SignCL and discuss the limitations and potential societal impacts of this work in the Appendix A.
+
+# 7 Limitations
+
+Our work, while promising, has several limitations that should be considered:
+
+Boundary Cases: We assume that adjacent frames output the same sign gestures, while distant frames belong to different sign gestures. This assumption might not hold in special sign language videos with extensive repetitive gestures. In extreme cases, SignCL might affect feature convergence.
+
+Semantic Similarity: SignCL does not account for the semantic similarity between sign gestures, which can result in increased feature distances between semantically similar gestures. This could potentially affect the learning of linguistic features.
+
+Despite acknowledging these limitations, our experiments demonstrate that our approach works effectively in most cases. We will address these issues in future work to further enhance the robustness and applicability of our method.
+
+# 8 Acknowledgement
+
+This work was supported in part by the National Natural Science Foundation of China (Grant No. 92370204), in part by the National Key R&D Program of China (Grant No. 2023YFF0725001), in part by the Guangzhou-HKUST (GZ) Joint Funding Program (Grant No. 2023A03J0008), and in part by the Education Bureau of Guangzhou Municipality. This work was also supported by the Guangzhou Municipal Science and Technology Project (No. 2024A04J4390).
+
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+
+# A Appendix
+
+# A.1 Hyper-parameters of Baselines
+
+Sign Language Transformers Baseline: Table 4 presents the hyper-parameters of Sign Language Transformers used in this work.
+
+<table><tr><td>Parameter</td><td colspan="2">PHOENIX-2014T CSL-Daily</td></tr><tr><td>encoder-layers</td><td>3</td><td>1</td></tr><tr><td>decoder-layers</td><td>3</td><td>1</td></tr><tr><td>attention heads</td><td>8</td><td>8</td></tr><tr><td>ctc-layers</td><td>1</td><td>1</td></tr><tr><td>hidden size</td><td>512</td><td>512</td></tr><tr><td>activation function</td><td>gelu</td><td>gelu</td></tr><tr><td>learning rate</td><td>\( 1 \cdot 10^{-3} \)</td><td>\( 1 \cdot 10^{-3} \)</td></tr><tr><td>Adam β</td><td>(0.9, 0.98)</td><td>(0.9, 0.98)</td></tr><tr><td>label-smoothing</td><td>0.1</td><td>0.1</td></tr><tr><td>max output length</td><td>30</td><td>50</td></tr><tr><td>dropout</td><td>0.3</td><td>0.3</td></tr><tr><td>batch-size</td><td>128</td><td>128</td></tr></table>
+
+Gloss-Free Sign Language Translation Baseline: The Gloss-Free Sign Language Translation (GFSLT) model incorporates various modules designed for processing sign language input without the use of glosses. Below is the detailed architecture used in this work:
+
+Table 4: Hyperparameters of Sign Language Transformer models.  
+
+<table><tr><td>Module</td><td>Stride</td><td>Kernel</td><td>Output Size</td></tr><tr><td>Sign Input</td><td>-</td><td>-</td><td>B × T × 224 × 224 × 3</td></tr><tr><td>Resnet w/o fc</td><td>-</td><td>-</td><td>B × T × 512</td></tr><tr><td>Conv1D-BN1D-RELU</td><td>1</td><td>5</td><td>B × T × 1024</td></tr><tr><td>MaxPooling1D</td><td>2</td><td>2</td><td>B × T/2 × 1024</td></tr><tr><td>Conv1D-BN1D-RELU</td><td>1</td><td>5</td><td>B × T/2 × 1024</td></tr><tr><td>MaxPooling1D</td><td>2</td><td>2</td><td>B × T/4 × 1024</td></tr><tr><td>Linear-BN1D-RELU</td><td>-</td><td>-</td><td>B × T/4 × 1024</td></tr><tr><td>Transformer Encoder</td><td>-</td><td>-</td><td>B × T/4 × U</td></tr><tr><td>Text Input</td><td>-</td><td>-</td><td>B × U</td></tr><tr><td>Word Embedding</td><td>-</td><td>-</td><td>B × U × 1024</td></tr><tr><td>Transformer Decoder</td><td>-</td><td>-</td><td>B × U × 1024</td></tr><tr><td>FC</td><td>-</td><td>-</td><td>B × U × C</td></tr></table>
+
+Table 5: Detailed Gloss-Free SLT (GFSLT) Framework.  $\overline{B}$  represents batch size,  $T$  denotes the length of the longest input sign video in the batch, and  $U$  is the length of the longest input text in the batch. It is copied from GFSLT-VLP [52].
+
+# A.2 Parameter Sensitivity Analysis of the SignCL
+
+# A.2.1 Sensitivity Analysis on Dynamically Estimated Margin
+
+The margin for negative sampling dynamically depends on the estimated average margin of each gloss, calculated as len(frame)/len(text)  $\times$  speech-to-gesture Zipf's factor, with a minimum threshold set at 10. The Zipf's factor, set at 2.3, refers to the speech-to-gesture application of Zipf's Law.
+
+We calculated the distribution of the dynamically estimated margin, with the results displayed in the table below. A more detailed distribution can be seen in Figure 6.
+
+![](images/5c7aab06613958956530444038b2e0ebad6af7c88acd6a99f934ec2064dc8c9f.jpg)  
+Figure 6: The distribution of the estimated margin during training on the PHOENIX-2014T dataset. The green distribution represents our current paper's method (factor = 2.3), while the orange distribution shows the ground truth calculated based on gloss annotations.
+
+Experiment Setup: To conduct a principled analysis, we evaluated the threshold values at  $[0, 10, 20, 30, 40, 50]$ . Here, a threshold of 0 indicates that the margin is dominated by the dynamically estimated margin, while a threshold of 50 suggests dominance by the fixed threshold.
+
+Experiment Results: We uniformly trained for 80 epochs on the PHOENIX-2014T dataset due to resource limitations. The results, as shown in the table below, indicate that SignCL is not sensitive to the threshold parameter, with a variance of 0.062.
+
+<table><tr><td>Threshold</td><td>0</td><td>10</td><td>20</td><td>30</td><td>40</td><td>50</td></tr><tr><td>B@4</td><td>17.24</td><td>17.63</td><td>17.55</td><td>17.63</td><td>17.13</td><td>17.11</td></tr></table>
+
+Table 6: Threshold sensitivity analysis results.
+
+<table><tr><td>Zipf&#x27;s factor</td><td>1</td><td>GT</td><td>2.3</td><td>3</td><td>4</td></tr><tr><td>B@4</td><td>17.45</td><td>17.89</td><td>17.63</td><td>17.29</td><td>16.26</td></tr></table>
+
+Table 7: Sensitivity to Zipf's factor.
+
+# A.2.2 Sensitivity Analysis on integrating SignCL into the SLT framework.
+
+As shown in Eqn. 6, we vary the hyperparameter  $\lambda$  over the range  $[10^{-3}, 10^{-2}, 10^{-1}, 10^{0}, 10^{1}]$  and conduct repeated experiments on the PHOENIX-2014T dataset with GFSLT-VLP.
+
+As shown in Figure 7, excessively incorporating SignCL into the model can negatively impact the SLT task. Empirically, we find that  $\lambda = 10^{-2}$  achieves a balance between reducing representation density and improving translation performance.
+
+# A.3 Ablation Studies
+
+We conduct ablation studies to investigate the impact of different loss components in the +SignCL approach during both the pretraining and fine-tuning stages. It is copied from Tabel 2, but with an ablation perspective.
+
+# A.4 Correlation between Representation Density and Recognition Performance
+
+# A.4.1 The efficiency of sign-gloss mapping building up
+
+To calculate SDR, we need to establish the mapping relationship between input frames and gloss categories. This section presents the performance of our trained gloss-sign aligner. The experimental methodology follows the approach outlined in XmDA [46].
+
+![](images/aef256150aa074a19db2d3c0ccd7c84032acb91a9bc9d2fb286ee98b0ab6a963.jpg)  
+Figure 7: The effect of the hyperparameter  $\lambda$  on BLEU scores. the grey dashed line indicates the baseline performance of GFSLT-VLP, while the red solid line represents the performance with SignCL integrated.
+
+<table><tr><td colspan="2">Pretraining Stage</td><td colspan="2">Finetuning Stage</td><td>Density</td><td colspan="2">Performance</td></tr><tr><td>VLP Loss</td><td>SignCL Loss</td><td>SLT Loss</td><td>SignCL Loss</td><td>SDR↓</td><td>R@L↑</td><td>B@4↑</td></tr><tr><td>X</td><td>X</td><td>✓</td><td>X</td><td>72.83</td><td>38.67</td><td>18.53</td></tr><tr><td>X</td><td>✓</td><td>✓</td><td>X</td><td>63.23</td><td>39.12</td><td>18.71</td></tr><tr><td>✓</td><td>X</td><td>✓</td><td>X</td><td>68.53</td><td>42.97</td><td>21.25</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>X</td><td>62.68</td><td>49.25</td><td>22.69</td></tr><tr><td>X</td><td>X</td><td>✓</td><td>✓</td><td>69.54</td><td>41.78</td><td>19.81</td></tr><tr><td>X</td><td>✓</td><td>✓</td><td>✓</td><td>63.67</td><td>44.52</td><td>20.03</td></tr><tr><td>✓</td><td>X</td><td>✓</td><td>✓</td><td>62.73</td><td>48.17</td><td>22.16</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>62.32</td><td>49.23</td><td>22.74</td></tr></table>
+
+# A.4.2 Correlation Coefficient
+
+We analyze the relationship between the representation density of individual glosses and their recognition accuracy using the Sign Language Transformer on the PHOENIX-2014T dataset, leveraging the Self-Mutual Knowledge Distillation (SMKD) method for feature extraction. We compute the following correlation coefficients:
+
+Pearson Correlation Coefficient [11]: This measures the linear relationship between recognition accuracy  $Acc(G_{i})$  for gloss  $G_{i}$  and the density metric  $SDR(G_{i})$ , calculated as:
+
+$$
+r = \frac {\sum \left(A c c \left(G _ {i}\right) - A \bar {c c}\right) \left(S D R \left(G _ {i}\right) - S \bar {D} R\right)}{\sqrt {\sum \left(A c c \left(G _ {i}\right) - A \bar {c c}\right) ^ {2}} \sum \left(S D R \left(G _ {i}\right) - S \bar {D} R\right) ^ {2}} \tag {7}
+$$
+
+Spearman's Rank Correlation Coefficient [38]: This assesses the monotonic relationship between two datasets by considering the rank order of values.
+
+Table 8: Ablation study on the impact of different loss components in the +SignCL approach.  
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="3">WER↓</td></tr><tr><td>Train</td><td>Test</td><td>Dev</td></tr><tr><td>PHOENIX-2014T</td><td>8.68</td><td>8.03</td><td>25.28</td></tr><tr><td>CSL-Daily</td><td>9.23</td><td>8.39</td><td>29.32</td></tr></table>
+
+Table 9: Evaluation of the gloss-sign aligner effectiveness and generalizability with WER (%) (the lower the better).
+
+As shown in Table 10, both correlation coefficients indicate a medium negative correlation between the Sign Density Ratio (SDR) and sign recognition performance (Acc). This suggests that higher representation density correlates with poorer recognition performance. Specifically, Inter-Gloss Distance  $(D_{Gi}^{inter})$  shows a strong positive correlation, meaning that greater distances between different glosses correlate with better recognition performance. All Spearman P-values are lower than 0.01, confirming the high confidence in the non-randomness of these correlations.
+
+<table><tr><td rowspan="2">Correlation / Acc</td><td colspan="3">PHOENIX-2014T</td><td colspan="3">CSL-Daily</td></tr><tr><td>\(D_{Gi}^{inter}\)↑</td><td>\(D_{Gi}^{intra}\)↓</td><td>SDR ↓</td><td>\(D_{Gi}^{inter}\)↑</td><td>\(D_{Gi}^{intra}\)↓</td><td>SDR ↓</td></tr><tr><td>Pearson r</td><td>0.36</td><td>-0.22</td><td>-0.35</td><td>0.30</td><td>-0.14</td><td>-0.20</td></tr><tr><td>Spearman ρ</td><td>0.43</td><td>-0.24</td><td>-0.34</td><td>0.45</td><td>-0.16</td><td>-0.22</td></tr><tr><td>P-value</td><td>2.6E-17</td><td>5.5E-6</td><td>4.7E-11</td><td>6.6E-19</td><td>3.0E-3</td><td>-2.7E-5</td></tr></table>
+
+# A.4.3 More Experiment Results on Sign Language Transformer
+
+In this section, we present additional experimental results using the Sign Language Transformer (NSLT) on the CSL-Daily dataset to further validate the effectiveness of the proposed SignCL strategy. We compare various feature extraction methods to assess their representation density and translation performance.
+
+Table 10: Correlation analysis between sign recognition performance and density metrics.  
+
+<table><tr><td rowspan="2">Feature Type</td><td rowspan="2">Density SDR↓</td><td rowspan="2">SLR(WER ↓)</td><td colspan="3">Performance</td><td rowspan="2">+SignCL(ours)</td></tr><tr><td>Joint-SLT</td><td>NSLT</td><td></td></tr><tr><td colspan="7">Gloss-based Feature Extraction</td></tr><tr><td>Sign Recognition [5]</td><td>74.07</td><td>29.59</td><td>21.32</td><td>17.68</td><td></td><td>19.02</td></tr><tr><td>Self-Mutual KD [33]</td><td>66.23</td><td>25.38</td><td>22.79</td><td>19.35</td><td></td><td>20.23</td></tr><tr><td colspan="7">Gloss-free Feature Extraction</td></tr><tr><td>I3D Pretrained [47]</td><td>83.33</td><td>61.74</td><td>14.17</td><td>11.70</td><td></td><td>12.81</td></tr><tr><td>VLP Pretrained [52]</td><td>92.59</td><td>69.72</td><td>12.73</td><td>10.73</td><td></td><td>12.14</td></tr><tr><td>+ SignCL (ours)</td><td>81.30</td><td>63.33</td><td>14.76</td><td>12.04</td><td></td><td>13.51</td></tr></table>
+
+Table 11: Comparative analysis of representation density and performance on the PHOENIX-2014T dataset. "+"SignCL (ours)" indicates the inclusion of the proposed contrastive learning strategy during VLP feature extraction or NSLT training processing.  
+
+<table><tr><td rowspan="2">Feature Type</td><td rowspan="2">Density SDR↓</td><td rowspan="2">SLR(WER ↓)</td><td colspan="3">Performance</td><td rowspan="2">+SignCL(ours)</td></tr><tr><td>Joint-SLT</td><td>NSLT</td><td></td></tr><tr><td colspan="7">Gloss-based Feature Extraction</td></tr><tr><td>Self-Mutual KD [33]</td><td>48.34</td><td>29.52</td><td>11.61</td><td>8.97</td><td>10.35</td><td></td></tr><tr><td colspan="7">Gloss-free Feature Extraction</td></tr><tr><td rowspan="2">VLP Pretrained [52] + SignCL (ours)</td><td>76.55</td><td>85.78</td><td>2.93</td><td>1.82</td><td>2.19</td><td></td></tr><tr><td>68.39</td><td>80.71</td><td>3.18</td><td>2.29</td><td>2.53</td><td></td></tr></table>
+
+Table 12: Comparative analysis of representation density and performance on the CSL-Daily dataset. The Self-Mutual KD features are provided by XmDA [46] and the VLP feature is reproduced with official code. Due to the incomplete open source of the CSL-Daily dataset, we were unable to obtain features for Sign Recognition and I3D Pretraining.
+
+# A.5 Broader Impacts
+
+This paper focuses on research in sign language translation, which has the potential to significantly benefit individuals who are deaf or hard of hearing. By improving the accuracy and efficiency of sign language translation, our work can facilitate better communication between individuals with hearing impairments and the broader community. This can help break down communication barriers, promoting inclusivity and equal opportunities in various social, educational, and professional settings.
+
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2024/Improving Neural Network Surface Processing with Principal Curvatures/ecd8aab3-ce12-43b3-b138-aa7fa76ff871_origin.pdf

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2024/Improving Neural Network Surface Processing with Principal Curvatures/full.md

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+# Improving Neural Network Surface Processing with Principal Curvatures
+
+Josquin Harrison
+
+Inria
+
+Sophia Antipolis
+
+josquin.harrison@inria.fr
+
+James Benn
+
+Inria
+
+Sophia Antipolis
+
+james.benn@inria.fr
+
+Maxime Sermesant
+
+Inria
+
+Sophia Antipolis
+
+maxime.sermesant@inria.fr
+
+# Abstract
+
+The modern study and use of surfaces is a research topic grounded in centuries of mathematical and empirical inquiry. From a mathematical point of view, curvature is an invariant that characterises the intrinsic geometry and the extrinsic shape of a surface. Yet, in modern applications the focus has shifted away from finding expressive representations of surfaces, and towards the design of efficient neural network architectures to process them. The literature suggests a tendency to either overlook the representation of the processed surface, or use overcomplicated representations whose ability to capture the essential features of a surface is opaque. We propose using curvature as the input of neural network architectures for surface processing, and explore this proposition through experiments making use of the shape operator. Our results show that using curvature as input leads to significant a increase in performance on segmentation and classification tasks, while allowing far less computational overhead than current methods.
+
+# 1 Introduction
+
+Surfaces are a natural representation for many real world objects ranging from organs and organisms to archaeological artefacts. They are also a central tool in virtual environments such as computer games, or computer-aided design. This ubiquity has resulted in a large body of work dedicated to mathematical methods developed for the efficient use of surfaces, as well as their analysis.
+
+The goal of traditional computational surface analysis is to find a representation of a surface that is expressive enough to capture details relevant for the problem or task at hand, while being computationally light-weight. However, the effectiveness of Convolutional Neural Network (CNN) in image processing opened new doors to surface processing. The design of efficient convolution-like operations to adapt neural networks (NN) to surfaces alleviated the need for complex and detailed representations, to the point where most state of the art architectures use extrinsic vertex coordinates as input, letting the NN models learn the surface structure at multiple scales. While some attempts were made to use well known representations as inputs, yielding some increase in performance [38, 35], the general consensus is that the model should be able to learn it by itself [25]. While it is true that neural networks are efficient at capturing surface features at multiple scales, the use of a local surface representation that is more expressive and more natural to interpret than extrinsic coordinates should naturally improve the performance of the network. The optimal choice of representation should be somewhere between coarse extrinsic vertex coordinates, and more complex representations.
+
+In Riemannian geometry, the shape operator is the main tool linking the intrinsic geometry of a surface with its bending and curving in ambient space. Its eigenvalues are the principal curvatures. Their product and evenly divided sum give precisely the Gauss and mean curvature of the surface, respectively. We hypothesize that the optimal choice for a local surface representation that meets the requirements of surface processing is the set of principal curvatures: they characterize the surface up to isometry (location and orientation in space); they are purely local, which allows the neural network to decide on more general surface features; and they are lightweight, leaving very little computational overhead in any scenario. This work tests our hypothesis against two widely used representations of surfaces in three state of the art NN architectures.
+
+In the next section we give an overview of surface processing, and introduce shape representations and learning methods. In section 3 we give an introduction to the Shape Operator, although we should remark now that this is not the first appearance of the shape operator in the surface processing literature. The shape operator has already become an efficient tool in surface processing and is, among other things, used to define local tangent frames and compute surface features like creases [31]. Following these introductions, we then conduct extensive experiments in section 4, comparing principal curvature with three other representations in conjunction with three different NN architectures, on two segmentation datasets and one classification dataset, that shows how principal curvature enhances any state of the art model in different tasks. In addition to outperforming other methods, we show that this more concise representation is faster to compute, leaving minimal computational overhead when added to a pipeline.
+
+# 2 Related work
+
+The first step to surface processing is usually its discretisation as a mesh or point cloud, which is particularly useful for visualisation or rendering. From this starting point, novel representations have been derived in an effort to provide tools for different surface related tasks. These tasks include surface matching, semantic segmentation, classification, or even shape retrieval.
+
+Historically, the general trend has been to find compact descriptors of a shape which could be then compared within a dataset. A long list of such descriptors exist, among them signature-based descriptors, such as Heat Kernel Signature (HKS) [42] or wave kernel signature [4], proved to be particularly efficient. Closely related are histogram methods, which are often combined with signatures to provide expressive representations such as the SHOT [37] or the Echo [26] descriptors. Geometric measure theory has also been a source of inspiration for developing efficient representations, such as geometric currents [5] leveraging on finite elements, or kernel-based currents [43] and varifolds [10] tailored for shape deformation. Such representations can be used in conjunction with classical statistical analysis tools, e.g [26, 37, 5], although they are often building blocks for specialised methods on surfaces such as LDDMM [49], functional maps [29] or spectral-based analysis [45].
+
+With the advent of deep learning, many methods previously stated were re-written with the help of neural networks resulting in Deep functional maps [20], and ResNet-LDDMM [2], to name but a few. Representations of surfaces themselves were proposed as neural networks, such as DeepSDF [32] or DeepCurrents [30]. As convolutions proved particularly effective when learning on images, i.e structured grids, some work proposed voxel-based solutions to the study of surfaces [23]. Others suggested representing surfaces as geometric images [40], on which convolutions can be applied. A second generation of geometric deep learning has focused on building network architecture specifically tailored to work directly on surfaces, i.e meshes or point clouds. From a point cloud perspective, Point Net [34] and its extension Point Net ++ [35] consider the surface as a set of points by applying set operations on them. Among others, DGCNN [46] applies a convolution-like operation on dynamic graphs constructed layer-wise. MeshCNN [16] on the other hand fully leverages the mesh structure to develop operation unique to triangulations. Transformer based architectures have also appeared for the specific purpose of surface processing [18]. Among them, and in a similar vein as before, GaTr [8] proposes to represent geometric data in an algebra of choice and designs an architecture with operations belonging to this algebra. An effort to have efficient generalisation of convolutions on surfaces was proposed by [22], although the lack of global coordinates creates ambiguity in local operations. To alleviate this problem, a large body of work has proposed rotation equivariant operations, namely GemCNN [12], augmenting graph NNs, or field convolutions [25]. Finally, recent models propose to bypass the problem of generalising convolutions by focusing
+
+on well defined operations on surfaces, such as discrete exterior calculus in Hodge Net [41], heat diffusion in Diffusion Net [38] or a suit of known operators in Delta Net [48].
+
+As model architectures include more and more knowledge of shapes, the need for a better representation of the input to these models has decreased. Outside of models that contain operations proper to the structure of choice (e.g [16, 8]), most models naturally accept as input the coordinates at every point. Some papers propose to augment the model by inputting higher dimensional descriptors initially designed for a more direct analysis, such as the ones previously mentioned (e.g HKS, WKS, SHOT). Such proposals can be seen in [38], where HKS interacts well with the diffusion part of the architecture, or [35] were they combine HKS, WKS, Gaussian curvature through concatenation and PCA. However, recent work has dismissed this idea [25], citing the results of Diffusion Net [38] which show no great improvement when moving from coordinates to HKS.
+
+As methods for learning on surfaces have evolved, we suggest that a better input representation is a simpler one, yet is more expressive than coordinates. We suggest that we can scale back to the simplest differentiable invariant of a surface: its curvature.
+
+# 3 The Shape Operator
+
+Here we give a conceptual introduction to the Shape Operator and describe how its eigenvalues completely characterize the surface to which it belongs (section 3.1). In section 3.2 we describe an explicit calculation of the Shape Operator which ig1's implementation of the principal curvatures is based on - this is the implementation we use in our experiments (see section 4.1). Those already familiar with the differential geometry of curves and surfaces may skip ahead to section 4; for others this section serves as a concise introduction - although, we do rely on a basic understanding of functions of several variables and their derivatives, and surfaces and their tangent spaces.
+
+Surfaces in  $\mathbb{R}^3$  will be denoted by  $S$  and  $\overline{S}$ , points in surfaces by  $p$ 's and  $q$ 's, and the tangent space to  $S$  at a point  $p \in S$  by  $T_pS$ . Maps from  $\mathbb{R}^3$  to itself will be denoted by  $F: \mathbb{R}^3 \to \mathbb{R}^3$ , and their derivatives at a point  $p$  by  $DF_p$  - the Jacobian matrix. A parameterisation  $X$  of a smooth surface  $S$  is a diffeomorphism between an open set  $U \subset \mathbb{R}^2$  and an open set  $V \subset S$ , and provides a mathematical description of  $S$  as it lies in  $\mathbb{R}^3$ . The standard Euclidean inner product on  $\mathbb{R}^3$  will be signified by  $\langle \cdot, \cdot \rangle$ , and it's restriction to a surface  $S$  and its tangent bundle  $TS = \coprod_{p \in S} T_pS$  by  $g_S(\cdot, \cdot)$ , which we call the induced metric. A normal vector to  $S$  at  $p$  is one which is orthogonal to every vector  $v$  in  $T_pS$  (measured in  $\langle \cdot, \cdot \rangle$ ) and will be denoted by  $N_p$ ; if we have a field of normal vectors in an open set around  $p$  then this field will be denoted simply by  $N$ .
+
+The Shape Operator of a surface  $S$  at a point  $p \in S$  measures the rate at which surface normal vectors  $N$  separate around  $p$ , which is precisely the bending of the surface in space:
+
+Definition 1 Given a point  $p$  on a surface  $S \subset \mathbb{R}^3$ , and the unit normal vector  $N$  defined on a neighbourhood  $U$  of  $p$ , the shape operator is the linear map
+
+$$
+\mathbf {S} _ {p}: T _ {p} S \longrightarrow T _ {p} S
+$$
+
+$$
+v \longmapsto - \nabla_ {v} N,
+$$
+
+where  $T_{p}S$  denotes the tangent space of  $S$  at point  $p$
+
+In other words, the shape operator  $\mathbf{S}_p$  tells us how the normal vector changes as we move in  $S$ , in the direction of  $v$  from  $p$ . One possible way to visualise the shape operator is through the Gauss map, which identifies each point  $p \in S \subset \mathbb{R}^3$  with its unit normal vector  $N_p$ , now thought of as a point in  $\mathbb{S}^2$ . The shape operator is then the differential of the Gauss map at  $p$  and is a tangent vector to  $\mathbb{S}^2$  at the image  $N_p$  of  $p$ , as illustrated in figure 1.
+
+The operator  $\mathbf{S}_p$  is linear for each  $p\in S$ , and self-adjoint in the Euclidean inner product  $\langle \cdot ,\cdot \rangle$
+
+$$
+\langle \mathbf {S} _ {p} (v), w \rangle = \langle v, \mathbf {S} _ {p} (w) \rangle . \tag {1}
+$$
+
+It can therefore be represented by a symmetric  $2 \times 2$  matrix  $[\mathbf{S}_p]: T_pS \to T_pS$  at each point  $p \in S$ . It is well-known that symmetric matrices admit a complete system of orthonormal eigenvectors  $(e_1, e_2)$  spanning the space on which they act. The matrix representation  $[\mathbf{S}_p]$  with respect to the basis  $(e_1, e_2)$  has the simple form:
+
+$$
+[ \mathbf {S} _ {p} ] = \left( \begin{array}{c c} \kappa_ {1} & 0 \\ 0 & \kappa_ {2} \end{array} \right). \tag {2}
+$$
+
+![](images/701ceb24b3ca36c11a822341209d966fd583d0e52d5edebee9a28a61e6cf16a7.jpg)  
+Figure 1: The shape operator may be visualised via the Gauss map.
+
+Where  $\kappa_{1}$  and  $\kappa_{2}$  are the eigenvalues of  $S_{p}$ .
+
+Definition 2 Let  $S$  be a surface in  $\mathbb{R}^3$ ,  $p$  a point in  $S$ ,  $\mathbf{S}_p$  the shape operator at  $p$  and  $[\mathbf{S}_p]$  its matrix representation.
+
+1. The eigenvalues  $\kappa_{1}(p)$  and  $\kappa_{2}(p)$  of  $[\mathbf{S}_{p}]$  at  $p$  are the principal curvatures of  $S$  at  $p$ , and their corresponding eigenvectors  $e_1$  and  $e_2$  are the principal directions;  
+2. The Gauss curvature  $\kappa$  of  $S$  at  $p$  is the product  $\kappa_1(p) \cdot \kappa_2(p)$  of the principal curvatures;  
+3. The mean curvature  $H_{p}$  is the average  $\frac{\kappa_1(p) + \kappa_2(p)}{2}$  of the principal curvatures
+
+The Gauss and mean curvatures can be equivalently interpreted as the determinant and half the trace of  $[\mathbf{S}_p]$ , respectively.
+
+The importance of these quantities is two-fold: (1) two surfaces differ only in location and orientation in space if and only if they have the same principal curvatures (Theorems 9.1 and 9.2 in [28]) – that is, the shape operator completely characterizes the shape of a surface; and (2) the Gauss and mean curvature generate all possible differential invariants of a surface (see Guggenheim [15], Olver [27]) – in particular, Gauss and mean curvature are fundamental characteristics of the shape of a surface, and the inclusion of the higher order invariants they generate into a representation could even improve the results shown here.
+
+# 3.1 Congruence
+
+To explain how Gauss and mean curvature completely describe the shape of a surface we need a few more definitions.
+
+An isometry of  $\mathbb{R}^3$  is a map  $F: \mathbb{R}^3 \to \mathbb{R}^3$  whose differential preserves the angles between tangent vectors at every point of  $\mathbb{R}^n$ :
+
+$$
+\langle v, w \rangle_ {p} = \left\langle D F _ {p} \cdot v, D F _ {p} \cdot w \right\rangle_ {p}, \quad \forall v, w \in T _ {p} \mathbb {R} ^ {3}. \tag {3}
+$$
+
+If  $g_{S}$  is the Riemannian inner product induced on  $TS$  by the Euclidean inner product  $\langle \cdot, \cdot \rangle$  then an isometry between two surfaces is a map  $\eta: S \to \overline{S}$  whose differential preserves the angles between tangent vectors to  $S$ :
+
+$$
+g _ {S} (v, w) = g _ {\overline {{S}}} \left(D \eta \cdot v, D \eta \cdot w\right). \tag {4}
+$$
+
+Every isometry  $F$  of  $\mathbb{R}^3$  restricts to an isometry of surfaces  $F|_{S} = \eta : S \to F(S)$ , but the converse need not be true, unless an additional hypothesis on the shape operators is satisfied.
+
+Two surfaces  $S$  and  $\overline{S}$  are congruent if there exists an isometry  $F: \mathbb{R}^3 \to \mathbb{R}^3$  such that  $F(S) = \overline{S}$ ; that is, congruent surfaces are surfaces which differ only in their location and orientation in space. It is clear that the shape operators  $\mathbf{S}$  and  $\overline{\mathbf{S}}$  of two congruent surfaces are related by
+
+$$
+D F _ {p} \cdot \mathbf {S} _ {p} (v) = \overline {{\mathbf {S}}} _ {F (p)} \left(D F _ {p} \cdot v\right), \quad \forall v \in T _ {p} S; \tag {5}
+$$
+
+in particular, the matrices  $[\mathbf{S}_p]$  and  $[\overline{\mathbf{S}}_p]$  are conjugate to one another via  $[DF_p]$ . As per Theorem 9.2 of [28], if there exists an isometry  $\eta : S \to \overline{S}$  such that
+
+$$
+D \eta_ {p} \cdot \mathbf {S} _ {p} (v) = \overline {{\mathbf {S}}} _ {\eta (p)} \left(D \eta_ {p} \cdot v\right), \quad \forall v \in T _ {p} S, \tag {6}
+$$
+
+i.e. such that the matrices  $[\mathbf{S}_p]$  and  $[\overline{\mathbf{S}}_p]$  are conjugate to one another via the matrix representation  $[D\eta_p]$ , then there exists an isometry  $F: \mathbb{R}^3 \to \mathbb{R}^3$  such that  $F|_S(S) = \eta(S) = \overline{S}$ , and the two surfaces are congruent. The conclusion of this brief mathematical digression is that two congruent surfaces have the same intrinsic geometry and shape in space, and two surfaces with the same intrinsic geometry and shape in space are congruent. This is what is sought after when representing shapes with intrinsic quantities.
+
+# 3.2 Discrete curvature
+
+As well as being an important theoretical tool, curvature is a central notion in mesh processing. A large body of work has been dedicated to estimating its discrete counterpart. Among them, many methods propose to infer Gaussian curvature directly, such as in [24], or involve the use of geometric measure theory [13], as in [11, 17]. Interestingly, many efficient methods propose to first discretize the shape operator in order to compute the Gaussian curvature from it. This is done either directly on the mesh triangles, such as in [36], or by first locally fitting a function to the surface, and then computing explicitly the shape operator. To get a better feel for why this is a natural construction of the shape operator, consider a surface  $S$ , given a point  $p \in S$ . Then the surface around  $p$  can be parameterized as  $X(u,v)$  with  $(u,v) \in \mathbb{R}^2$ . The inner product at  $T_pS$ , also called the first fundamental form, is then given for any two tangent vectors  $v,w$  by:
+
+$$
+(v, w) _ {p} = v ^ {T} \left( \begin{array}{c c} E & F \\ F & G \end{array} \right) w, \tag {7}
+$$
+
+where  $E = \langle \partial_uX, \partial_uX \rangle$ ,  $F = \langle \partial_uX, \partial_vX \rangle$  and  $G = \langle \partial_vX, \partial_vX \rangle$ . And the surface normal at  $p$  can be defined as
+
+$$
+n = \frac {\partial_ {u} X (u , v) \times \partial_ {v} X (u , v)}{\left| \partial_ {u} X (u , v) \times \partial_ {v} X (u , v) \right|}. \tag {8}
+$$
+
+We can now define the second partial derivatives of  $X$  in the normal direction  $n$ , a quantity called the second fundamental form, noted II:
+
+$$
+\mathbb {I} = \left( \begin{array}{c c} L & M \\ M & N \end{array} \right) \tag {9}
+$$
+
+where  $L = \langle \partial_{uu}X, n \rangle$ ,  $M = \langle \partial_{uv}X, n \rangle$ , and  $N = \langle \partial_{vv}X, n \rangle$ . The partial derivatives of the surface normal can then be expressed via the Weingarten equations, in terms of the components of the first and second fundamental form:
+
+$$
+\partial_ {u} n = \frac {F M - G L}{E G - F ^ {2}} \partial_ {u} X + \frac {F L - E M}{E G - F ^ {2}} \partial_ {v} X
+$$
+
+$$
+\partial_ {v} n = \frac {F N - G M}{E G - F ^ {2}} \partial_ {u} X + \frac {F M - E N}{E G - F ^ {2}} \partial_ {v} X.
+$$
+
+This enables us to write the matrix form of the shape operator at  $p$  as:
+
+$$
+\left[ \mathbf {S} _ {p} \right] = \left(E G - F ^ {2}\right) ^ {- 1} \left( \begin{array}{l l} L G - M F & M E - L F \\ M E - L F & N E - M F \end{array} \right) \tag {10}
+$$
+
+From these derivations, it becomes interesting to find good local parametrisation of surfaces, that is, a bi-variate scalar function  $f$  such that:
+
+$$
+X (u, v) = (u, v, f (u, v)) \tag {11}
+$$
+
+The shape operator can then be easily derived from the first and second derivatives of  $f$ . An efficient way to find such functions is via osculating jets, proposed in [9]. For the following experiments, we use a multi-scale version of this, proposed in [31], in which the shape operator is computed by using neighbourhoods of varying size around a point, yielding a robust method for estimating curvature on a mesh.
+
+# 4 Experiments
+
+We test the representation of surfaces by the principal curvatures  $\kappa_{1},\kappa_{2}$  and Gaussian curvature  $\kappa$  against the three most commonly used representations: the HKS [42], the SHOT descriptor [37], and
+
+![](images/70cb56e3aaf9b04905fec8085ce82f38ce0f9d76cef7bfd5adbc1dab8a853be1.jpg)  
+(a)  $\kappa_{1}$
+
+![](images/993d163635ec0571d6772be4300f57c82300898984537ef40446c80edf9ddd9f.jpg)  
+(b)  $\kappa_{2}$  
+Figure 2: Principal curvature visualisation of a Louis XIV statue.
+
+the extrinsic coordinates. The HKS is a purely intrinsic representation derived from the Laplace operator, and constitutes the most widely used signature-based method to represent shapes. The SHOT representation is a descriptor mixing signature and histogram-based methods to describe shapes, and is therefore an extrinsic representation. As they belong to two different classes of surface representations we believe they are the most adequate for benchmarking our proposed curvature representation.
+
+All representations are tested with three different architectures. We regard Diffusion Net [38] as the state of the art in NN architectures, as it shows the most promising results on general benchmark tasks. In addition, it shows very little difference in performance when changing the input from coordinates to HKS, making it the hardest test for our representation. Point Net ++ [35] has been designed as a general method to process shapes arising in many situations, including controlled environments - as in our case - but also from segmented images encountered in the autonomous driving field [35]. As such Point Net ++ uses the least geometric structure to describe a surface: all one needs is a point set. We believe that in this case, using better surface information for the input will greatly enhance the performance of the model. The authors of PointNet++ have already touched on this subject, recommending a linear combination of HKS, WKS and Gaussian curvature, followed by a PCA projection, leading to a 64 dimensional feature per point. We aim to show that a 2d (or even 1d) input of curvature information is more relevant for a smaller computational cost. Delta Net [48] proposes an architecture intrinsic to surfaces by design, by combining four operators defined on the surface: Laplacian, divergence, curl and norm. Most papers that propose other surface descriptors rather than coordinates as input, do it solely to have an intrinsic representation of the surface. Curvature gives isometry invariance (section 3), and we further believe it is also more robust, numerically. Better performance from a curvature based representation in this architecture would support this belief.
+
+Finally, we pick three tasks of varying complexity to measure the impact of each method: human segmentation [21], molecular segmentation [6], and shape classification [19]. Examples from each dataset are shown in figure 3.
+
+# 4.1 Implementation
+
+The performance of each representation is strongly dependent on the chosen implementation. We have tried to be as fair as possible by not developing our own implementations of existing work and instead using implementations which have already been tried, tested, and validated in the literature. For calculating the discrete principal curvatures via quadratic surface fitting, we have used ig1's implementation with a fixed neighbourhood radius of 5; the Gaussian curvature  $\kappa$  is then computed directly as the product of  $\kappa_{1},\kappa_{2}$ . HKS depends on the Laplacian, and we have used the method implemented in robust-laplacian based on [39] - this is also consistent with what is used in Diffusion Net. The eigendecomposition of the Laplacian is then performed with scipy. For the
+
+![](images/644cab376268f7fd479a539a4076ef7305bda602de81ddd85904beaedbaa54bd.jpg)  
+(a) Human poses [21].
+
+![](images/e5787f2c9cb4a400ec5b06c462ba92fc78771f85dc6d2e79bc35f226b67df2f4.jpg)  
+(b) Molecules [6].
+
+![](images/6f4bc0433eaf894edef9f1006b418e8be80aab4e429c6601f4592f08e39cd14d.jpg)  
+(c) Shrec'11 [19].  
+Figure 3: Samples of the segmentation and classification datasets used for experiments.
+
+SHOT representation, we use the implementation in the pc1 library, which computes 352 features per vertex, in this case we normalise all shapes and use a ball of radius .1; all other parameters are left untouched.
+
+Regarding the neural networks, we use the implementations made publicly available by the authors, modifying only when needed to accommodate more than just coordinates as input. We also use the same parameters proposed in each paper when they are known, which we detail for every task below. We make all our code and experiments available at https://github.com/Inria-Asclepios/ shape-nets
+
+# 4.2 Time Complexity
+
+As a first experiment, we compute<sup>1</sup> for each representation method, the computation time as a function of the number of points in a surface. The performances are reported in figure 4. HKS and curvature are both efficient for meshes with up to 100k points. However, curvature is consistently faster, even for larger meshes (up to 500k points), displaying the very small overhead incurred by the use of curvature in surface models.
+
+![](images/6bfb9af8c6c9ec5ff36f5e9649c47268542a843008b9c534ed5046f9a307c853.jpg)  
+Figure 4: Time of computation for each representation with respect to the number of points in a mesh.
+
+# 4.3 Human Anatomy segmentation
+
+We first segment the human parts from the composite dataset proposed in [21], containing samples from other human dataset, namely FAUST [7], SCAPE [3], Adobe [1], MIT [44], and SHREC07 [14]. As in the original paper, we use the SHREC07 dataset as test set. Similar to [47], we differ from [21] by evaluating on vertices rather than faces. For Point Net ++ and Delta Net we resample each shape to 1024 points, and we leave the meshes untouched for Diffusion Net, as per the experiments conducted in each paper. We optimise the negative log-likelihood for 100 epochs, with the ADAM optimiser and a scheduler step every 20 epochs. We ran the experiment 5 times and have reported the mean test accuracy in table 1.
+
+<table><tr><td></td><td>xyz</td><td>shot16</td><td>shot64</td><td>hks</td><td>κ1,κ2</td><td>κ</td></tr><tr><td>Point Net ++</td><td>69.6</td><td>71.4</td><td>72.4</td><td>78.1</td><td>80.6</td><td>74.5</td></tr><tr><td>Delta Net</td><td>72.4</td><td>58.1</td><td>66.2</td><td>68.9</td><td>86.8</td><td>60.0</td></tr><tr><td>Diffusion Net</td><td>94.7</td><td>95.0</td><td>95.0</td><td>95.1</td><td>97.4</td><td>95.4</td></tr></table>
+
+Table 1: Test accuracies (%) on the Human part segmentation task.
+
+The results highlight the assumption that better representations lead to better performance. PointNet++ showed the greatest improvement when moving away from coordinates: this is due to its architecture having the least amount of geometric information at baseline. The better results come from the principal curvatures, and show how expressive this representation is. The effects of the principal curvatures are even more pronounced in the Delta Net experiment, where  $\kappa_{1},\kappa_{2}$  greatly outperform all other methods. It's interesting to note that in this experiment the coordinate representation performs better than the other more complex representations. Diffusion Net may show that it is more robust to the type of input, as long as it loosely describes the shape, however the improvement brought by the principal curvature is still significant. To further demonstrate the impact of a good representation, even in the case of Diffusion Net, we show in 5 the worst cases for xyz and  $\kappa_{1},\kappa_{2}$  inputs. The clear improvement in this case may be even more important than general accuracy in some cases, e.g with human-in-the-loop type corrections.
+
+![](images/7e297d038fe4d35abcdda03b450c2967a03d4618fe68bc1a4ed215b24c48eeef.jpg)  
+(a)  $\kappa_{1},\kappa_{2}$ $80.9\%$  accuracy.
+
+![](images/4e9b43072c8a7a48e3abcb5aebd5bcc857f4b9a29b23d0807aa3333455a34316.jpg)  
+(b) xyz.  $29.2\%$  accuracy.
+
+![](images/23d80705728ae2d6433f1e73b9651a5b5b0d33998a514ac7a85bad2ddf326257.jpg)  
+(c) HKS.  $12.3\%$  accuracy.  
+Figure 5: Human part segmentation with Diffusion Net. Worst cases for different representations, blue shows the correct prediction, red the error.
+
+# 4.4 Molecular segmentation
+
+The molecular dataset made available by [6] and first proposed in [33], can be considered a harder segmentation task than the Human part dataset: it proposes a wider range of shapes in the form of RNA molecules, and a 260-way part segmentation task. We resample all meshes to 2048 points, except in the case of Diffusion Net where we kept the original discretisation. We evaluate all our baselines on 5 random splits with a train-test ratio of 80-20. We run the models for 200 epochs, and report the mean test accuracies in table 2.
+
+<table><tr><td></td><td>xyz</td><td>shot16</td><td>shot64</td><td>hks</td><td>κ1, κ2</td><td>κ</td></tr><tr><td>Point Net ++</td><td>35.4</td><td>70.9</td><td>71.9</td><td>70.2</td><td>72.4</td><td>69.0</td></tr><tr><td>Delta Net</td><td>29.2</td><td>45.6</td><td>56.5</td><td>49.6</td><td>55.5</td><td>29.2</td></tr><tr><td>Diffusion Net</td><td>82.6</td><td>88.4</td><td>89.1</td><td>85.6</td><td>89.4</td><td>84.0</td></tr></table>
+
+Table 2: Test accuracies (%) on the Molecular segmentation task.
+
+Again, we see a significant improvement when using a better representation of the surface in the case of PointNet++, going from failing in the case of coordinates to outperforming Delta Net - with principal curvatures giving the best performances. Diffusion Net shows a non-negligible jump in performance as well. Although the SHOT descriptor outperforms other representations in the case of Delta Net, the general performance of this architecture is underwhelming. We believe this is due to the accumulation of errors in the discretisation of surface operators used. Indeed, one layer computes a chain of 6 operators on the surface: since the RNA shapes are very irregular, the error for each operator could be significant.
+
+# 4.5 Classification
+
+In addition to segmentation tasks, we propose to compare representations in the context of classification. This experiment should show whether or not geometrically informative inputs interact well with pooling-type operations. We choose the widely adopted baseline Shrec11, proposed in [19]. It is a 30-way classification dataset with 20 shapes per category. We choose the simplified mesh dataset and the harder version of training, using only 10 samples per class and evaluating on the test. We perform our experiments on 5 random splits. We train our baselines for 100 epochs with a scheduler step at epoch 50 and optimise the cross-entropy loss with a label smoothing factor of 0.2. Resulting mean test accuracies are shown in table 3.
+
+<table><tr><td></td><td>xyz</td><td>shot16</td><td>shot64</td><td>hks</td><td>κ1, κ2</td><td>κ</td></tr><tr><td>Point Net ++</td><td>71.5</td><td>69.8</td><td>60.7</td><td>60.8</td><td>85.7</td><td>96.2</td></tr><tr><td>Delta Net</td><td>75.7</td><td>54.9</td><td>60.4</td><td>98.6</td><td>90.1</td><td>98.8</td></tr><tr><td>Diffusion Net</td><td>80.3</td><td>52.6</td><td>67.4</td><td>98.9</td><td>94.2</td><td>99.1</td></tr></table>
+
+Table 3: Test accuracies (%) on the Shrec11 classification task.
+
+Yet again we observe a significant improvement when turning to better representations, even more so when using Gaussian curvature  $\kappa$ . Additionally, figure 6 shows that all geometric representations yield less variability across each folds. In addition, HKS, Gaussian curvature  $\kappa$ , and principal curvature  $\kappa_{1}, \kappa_{2}$  converge much faster than all others.
+
+![](images/c1b4fadb249475993b249cfc57218dd77fd7942ea92af10eeffae4198c7bdbbb.jpg)  
+Figure 6: Evolution of the test accuracy with  $95\%$  confidence interval by epochs per representations across folds, for the Shrec07 dataset using Diffusion Net.
+
+The fact that gaussian curvature, closely followed by HKS outperform principal curvature in this classification task seems to indicate that Gaussian curvature interacts better with pooling operations present in classification architectures. Interestingly, all three architectures tested here have different
+
+ways of performing the pooling operation. Although it is hard to give any analytical reasoning to this behavior, we believe it is simply the fact that gaussian curvature is already an aggregation of the principal curvatures, that it shows better performance in classification tasks.
+
+For each experiment, additional metrics can be found in Appendix A.1.
+
+# 4.6 Noisy data
+
+We propose one final experiment to highlight the robustness of input features to noisy data. We focus on three representations: HKS, known to be robust to noise as it computes a representation at multiple scale; extrinsic coordinates that are directly impacted by the noise; and principal curvatures, known to be less robust to noise as a purely local descriptor. To compare these representations we pick the diffusion net trained on the human pose dataset, and we add noise to the dataset at inference time. Specifically, we add gaussian noise with a standard deviation of  $1\%$ ,  $3\%$ ,  $5\%$ ,  $7\%$  and  $10\%$  of the diagonal length of the bounding box of each shape. Examples of the noisy data can be seen in Appendix A.2. Results show that the accuracy for all three features worsen at the same rate, as shown in figure 7, showing that principal curvature can be a viable choice even in the presence of noisy data.
+
+![](images/a4f9749deb58173de82ce39e2574c05b5e2c1589131c217165d3d4b369576075.jpg)  
+Figure 7: Evolution of the test accuracy on the human pose segmentation task for inputs  $(k_{1}, k_{2})$ , HKS and the extrinsic coordinates when noise is added to the shapes.
+
+# 5 Conclusion
+
+In this work we have shown that curvature should be the representation of choice when it comes to processing surfaces with neural networks. In almost all experiments the principal and Gaussian curvatures performed better than any other choice of input, both qualitatively and quantitatively. In particular, this representation can be obtained with minimal computational overhead. Its combination with PointNet++, the architecture that has the least prior information about the surface, showed that it can help the network better understand the surface structure. When combined with Delta Net, which contains only intrinsic operations, the improvement indicates that curvature gives more than just a rigid transformation invariance. Even in the case of Diffusion Net, where the diffusion operation seems to interact nicely with any representation, curvature as input showed significant amelioration. For these reasons, we believe curvature should become the standard practice when using models to learn on surfaces. Finally, although experiments have shown that gaussian curvature outperforms principal curvatures on classification tasks, we would like to further define those guidelines in future work, as well as compare representations in a wider range of tasks and architectures.
+
+# 6 Acknowledgements
+
+This Work has been funded by PARIS - ERACoSysMed grant number 15087, by G-Statistics - ERC grant number 786854 and has been supported by the French government through the National Research Agency (ANR) Investments in the 3IA Côte d'Azur (ANR-19-P3IA-000). The authors are grateful to the OPAL infrastructure from Université Côte d'Azur for providing computing resources and support.
+
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+
+# A Appendix
+
+# A.1 Segmentation and classification detailed results
+
+We present below the complete results for each experiment. For each dataset, and each neural network architecture, we show the Accuracy, Balanced accuracy, F1 score, and Specificity. They were all measured using 5-fold cross validation, and we give the results on the test sets in the form of mean  $\pm$  standard deviation.
+
+<table><tr><td></td><td>Accuracy</td><td>Balanced accuracy</td><td>F1 score</td><td>Specificity</td></tr><tr><td>xyz</td><td>.696±.306</td><td>.804±.393</td><td>.662±.293</td><td>.957±.49</td></tr><tr><td>shot16</td><td>.714±.061</td><td>.815±.15</td><td>.689±.061</td><td>.959±.234</td></tr><tr><td>shot64</td><td>.724±.034</td><td>.827±.13</td><td>.693±.035</td><td>.961±.201</td></tr><tr><td>hks</td><td>.781±.07</td><td>.865±.159</td><td>.754±.08</td><td>.969±.203</td></tr><tr><td>κ1,κ2</td><td>.806±.079</td><td>.871±.14</td><td>.799±.089</td><td>.972±.196</td></tr><tr><td>κ</td><td>.745±.165</td><td>.826±.227</td><td>.714±.148</td><td>.964±.32</td></tr></table>
+
+Table 4: Human pose segmentation - Point Net ++ results.  
+
+<table><tr><td></td><td>Accuracy</td><td>Balanced accuracy</td><td>F1 score</td><td>Specificity</td></tr><tr><td>xyz</td><td>.724±.265</td><td>.223±.076</td><td>.223±.076</td><td>.724±.265</td></tr><tr><td>\( \text{shot}_{16} \)</td><td>.581±.095</td><td>.215±.05</td><td>.215±.05</td><td>.581±.095</td></tr><tr><td>\( \text{shot}_{64} \)</td><td>.662±.075</td><td>.302±.084</td><td>.302±.084</td><td>.662±.075</td></tr><tr><td>hks</td><td>.689±.14</td><td>.303±.043</td><td>.303±.043</td><td>.689±.14</td></tr><tr><td>\( \kappa_1, \kappa_2 \)</td><td>.868±.128</td><td>.299±.114</td><td>.299±.114</td><td>.868±.128</td></tr><tr><td>\( \kappa \)</td><td>.6±.087</td><td>.163±.054</td><td>.163±.054</td><td>.6±.087</td></tr></table>
+
+Table 5: Human pose segmentation - Delta Net results.  
+
+<table><tr><td></td><td>Accuracy</td><td>Balanced accuracy</td><td>F1 score</td><td>Specificity</td></tr><tr><td>xyz</td><td>.947±.013</td><td>.943±.015</td><td>.943±.015</td><td>.947±.013</td></tr><tr><td>shot16</td><td>.95±.017</td><td>.946±.018</td><td>.946±.018</td><td>.95±.017</td></tr><tr><td>shot64</td><td>.95±.018</td><td>.944±.021</td><td>.944±.021</td><td>.95±.018</td></tr><tr><td>hks</td><td>.951±.04</td><td>.969±.055</td><td>.947±.039</td><td>.993±.07</td></tr><tr><td>κ1,κ2</td><td>.975±.014</td><td>.971±.015</td><td>.971±.015</td><td>.975±.014</td></tr><tr><td>κ</td><td>.954±.014</td><td>.951±.016</td><td>.951±.016</td><td>.954±.014</td></tr></table>
+
+Table 6: Human pose segmentation - Diffusion Net results.  
+
+<table><tr><td></td><td>Accuracy</td><td>Balanced accuracy</td><td>F1 score</td><td>Specificity</td></tr><tr><td>xyz</td><td>.354±.008</td><td>.202±.001</td><td>.202±.001</td><td>.354±.008</td></tr><tr><td>\( \text{shot}_{16} \)</td><td>.709±.004</td><td>.592±.016</td><td>.592±.016</td><td>.709±.004</td></tr><tr><td>\( \text{shot}_{64} \)</td><td>.719±.009</td><td>.602±.01</td><td>.602±.01</td><td>.719±.009</td></tr><tr><td>hks</td><td>.703±.017</td><td>.574±.003</td><td>.574±.003</td><td>.703±.017</td></tr><tr><td>\( \kappa_1, \kappa_2 \)</td><td>.724±.013</td><td>.597±.013</td><td>.597±.013</td><td>.724±.013</td></tr><tr><td>\( \kappa \)</td><td>.69±.008</td><td>.572±.008</td><td>.572±.008</td><td>.69±.008</td></tr></table>
+
+Table 7: RNA molecules segmentation - PointNet++ results.  
+
+<table><tr><td></td><td>Accuracy</td><td>Balanced accuracy</td><td>F1 score</td><td>Specificity</td></tr><tr><td>xyz</td><td>.292±.018</td><td>.182±.012</td><td>.182±.012</td><td>.292±.018</td></tr><tr><td>shot16</td><td>.456±.008</td><td>.352±.016</td><td>.352±.016</td><td>.456±.008</td></tr><tr><td>shot64</td><td>.565±.010</td><td>.299±.008</td><td>.299±.008</td><td>.565±.010</td></tr><tr><td>hks</td><td>.496±.017</td><td>.392±.008</td><td>.392±.008</td><td>.496±.017</td></tr><tr><td>κ1,κ2</td><td>.555±.023</td><td>.496±.013</td><td>.496±.013</td><td>.555±.023</td></tr><tr><td>κ</td><td>.292±.008</td><td>.142±.008</td><td>.142±.008</td><td>.292±.008</td></tr></table>
+
+Table 8: RNA molecules segmentation - Delta Net results.
+
+<table><tr><td></td><td>Accuracy</td><td>Balanced accuracy</td><td>F1 score</td><td>Specificity</td></tr><tr><td>xyz</td><td>.826±.001</td><td>.851±.106</td><td>.704±.003</td><td>.999±.129</td></tr><tr><td>shot16</td><td>.884±.002</td><td>.765±.008</td><td>.765±.008</td><td>.874±.002</td></tr><tr><td>shot64</td><td>.891±.001</td><td>.781±.008</td><td>.781±.008</td><td>.879±.001</td></tr><tr><td>hks</td><td>.856±.596</td><td>.873±.622</td><td>.758±.536</td><td>.999±.697</td></tr><tr><td>κ1,κ2</td><td>.894±.008</td><td>.873±.064</td><td>.783±.025</td><td>.999±.082</td></tr><tr><td>κ</td><td>.84±.008</td><td>.862±.113</td><td>.718±.011</td><td>.999±.12</td></tr></table>
+
+Table 9: RNA molecules segmentation - Diffusion Net results.  
+
+<table><tr><td></td><td>Accuracy</td><td>Balanced accuracy</td><td>F1 score</td><td>Specificity</td></tr><tr><td>xyz</td><td>.715±.037</td><td>.852±.019</td><td>.709±.034</td><td>.99±.001</td></tr><tr><td>shot16</td><td>.698±.035</td><td>.844±.018</td><td>.694±.036</td><td>.99±.001</td></tr><tr><td>shot64</td><td>.607±.031</td><td>.797±.016</td><td>.594±.038</td><td>.986±.001</td></tr><tr><td>hks</td><td>.608±.181</td><td>.797±.093</td><td>.602±.184</td><td>.986±.006</td></tr><tr><td>κ1, κ2</td><td>.857±.007</td><td>.926±.004</td><td>.852±.007</td><td>.995±.0</td></tr><tr><td>κ</td><td>.962±.011</td><td>.98±.006</td><td>.961±.011</td><td>.999±.0</td></tr></table>
+
+Table 10: Shrec classification - Point Net ++ results.  
+
+<table><tr><td></td><td>Accuracy</td><td>Balanced accuracy</td><td>F1 score</td><td>Specificity</td></tr><tr><td>xyz</td><td>.757±.027</td><td>.753±.031</td><td>.753±.031</td><td>.757±.028</td></tr><tr><td>shot16</td><td>.549±.029</td><td>.533±.026</td><td>.532±.026</td><td>.549±.029</td></tr><tr><td>shot64</td><td>.604±.026</td><td>.597±.019</td><td>.597±.019</td><td>.604±.026</td></tr><tr><td>hks</td><td>.986±.005</td><td>.985±.006</td><td>.986±.006</td><td>.986±.005</td></tr><tr><td>κ1, κ2</td><td>.887±.020</td><td>.881±.019</td><td>.881±.019</td><td>.887±.020</td></tr><tr><td>κ</td><td>.988±.004</td><td>.988±.004</td><td>.988±.004</td><td>.988±.004</td></tr></table>
+
+Table 11: Shrec classification - Delta Net results.  
+
+<table><tr><td></td><td>Accuracy</td><td>Balanced accuracy</td><td>F1 score</td><td>Specificity</td></tr><tr><td>xyz</td><td>.803±.159</td><td>.898±.082</td><td>.791±.169</td><td>.993±.005</td></tr><tr><td>shot16</td><td>.526±.054</td><td>.755±.028</td><td>.513±.053</td><td>.984±.002</td></tr><tr><td>shot64</td><td>.674±.033</td><td>.831±.017</td><td>.668±.036</td><td>.989±.001</td></tr><tr><td>hks</td><td>.989±.006</td><td>.994±.003</td><td>.989±.006</td><td>1.0±.0</td></tr><tr><td>κ1, κ2</td><td>.922±.009</td><td>.96±.004</td><td>.919±.008</td><td>.997±.0</td></tr><tr><td>κ</td><td>.991±.003</td><td>.995±.001</td><td>.991±.003</td><td>1.0±.0</td></tr></table>
+
+Table 12: Shrec classification - Diffusion Net results.
+
+# A.2 Noisy data examples
+
+![](images/2fc7f25ba4dd1c95e679085e4bbfd1f28132f0ad229a3b034c5dea16495c90bc.jpg)  
+(a)  $1\%$ .
+
+![](images/a2ae0c0435e040c1655b4df79f04843e9f688e59f709db5d89dcf967709cae38.jpg)  
+(b)  $3\%$ .
+
+![](images/e669d52ef3201af31eae1e46def5557aa417b44a430089644b6fb73697aa18bb.jpg)  
+(c)  $5\%$ .
+
+![](images/fa5763434e2c8c727ab328a57f2d90ff99336c71a1ca8a47f56639e660dfd5c6.jpg)  
+(d)  $7\%$
+
+![](images/e2a9d929f040560d67c083f97a88537a6450f06e291024168291d3081a779663.jpg)  
+(e)  $10\%$ .
+
+![](images/98f98a82872176da439c8ea13c7f6b23a3621ad12505669941c3e733812b6f6b.jpg)  
+Figure 8: Different quantity of noise added to a shape from the human pose dataset, from  $1\%$  to  $10\%$  of the diagonal of the bounding box of the shape.
+
+![](images/f5c687b216bbc71e9a469982c9b020c30ab7a149bc36be197ce5126d7ef12791.jpg)
+
+![](images/ec68d9fb7e294769625557feab31ba616f8c99af53689e973740781c1c0a8a6c.jpg)
+
+![](images/6e2bb91211041e9582b0b95a6c8661f907b533f2f80328613b1ccb79424b5a47.jpg)  
+(a) Initial shape.  
+(d)  $5\%$
+
+![](images/3e425c8bddd13bbb9bd2ed6081cedf8899147ed5edfb3f2f67a2b2514c10a96f.jpg)  
+(b)  $1\%$ .  
+(e)  $7\%$ .  
+Figure 9: Different quantity of noise added to a shape, from  $1\%$  to  $10\%$  of the diagonal of the bounding box of the shape.
+
+![](images/4c192aead6bbb01de7ea5b5029f844ffd59504d00a9951792cc63583b38ab9b0.jpg)  
+(c)  $3\%$ .  
+(f)  $10\%$ .
+
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+# Improving Neural ODE Training with Temporal Adaptive Batch Normalization
+
+Su Zheng $^{1}$ ;Zhengqi Gao $^{2*}$ , Fan-Keng Sun $^{2}$ , Duane S. Boning $^{2}$ , Bei Yu $^{1}$ , Martin Wong $^{1}$ $^{1}$ Department of CSE, CUHK  $^{2}$ Department of EECS, MIT
+
+# Abstract
+
+Neural ordinary differential equations (Neural ODEs) is a family of continuous-depth neural networks where the evolution of hidden states is governed by learnable temporal derivatives. We identify a significant limitation in applying traditional Batch Normalization (BN) to Neural ODEs, due to a fundamental mismatch — BN was initially designed for discrete neural networks with no temporal dimension, whereas Neural ODEs operate continuously over time. To bridge this gap, we introduce temporal adaptive Batch Normalization (TA-BN), a novel technique that acts as the continuous-time analog to traditional BN. Our empirical findings reveal that TA-BN enables the stacking of more layers within Neural ODEs, enhancing their performance. Moreover, when confined to a model architecture consisting of a single Neural ODE followed by a linear layer, TA-BN achieves  $91.1\%$  test accuracy on CIFAR-10 with 2.2 million parameters, making it the first unmixed Neural ODE architecture to approach MobileNetV2-level parameter efficiency. Extensive numerical experiments on image classification and physical system modeling substantiate the superiority of TA-BN compared to baseline methods.
+
+# 1 Introduction
+
+Originally derived as the continuous limit of a residual connection [4], neural ordinary differential equations (Neural ODEs) [4, 7, 37, 45, 8, 42, 30, 18, 32, 19, 31] is a family of continuous-depth neural networks where the evolution of hidden states is governed by learnable temporal derivatives. These models exhibit several intriguing features, such as the capability of temporal reversibility, which enables generative modeling [4, 15].
+
+Previous studies on Neural ODEs parameterize the learnable temporal derivatives using a shallow neural network with a limited number of parameters [7, 30]. Without special treatment, merely stacking additional layers in the temporal derivatives does not necessarily enhance Neural ODE performance. Furthermore, deeper networks might increase the stiffness of the ODE system, leading to challenges with the ODE solver, such as excessively small step sizes or even failures due to infinite state values, as shown in Figure 1.
+
+![](images/6a127302d1f22a7eae6ebf937ad5e9d0ae337f7e2305c7ccfb1b7cb25078b015.jpg)  
+Figure 1: Test accuracies are depicted over the training epochs on CIFAR-10, utilizing Neural ODEs with different numbers of convolution layers as the backbones for learnable derivatives. Dashed horizontal lines denote instances of training failures.
+
+In efforts to deepen the model and address instabilities, one might instinctively consider Batch Normalization (BN) [5, 29, 16, 39, 2] as a remedy to stabilize the intermediate state values of Neural ODEs [13, 35], because BN was originally proposed to accelerate
+
+the training of (discrete) neural networks and consolidate them against internal covariate shift [29]. However, preliminary works surprisingly found that the introduction of BN into Neural ODEs can even degrade performance, an effect for which the underlying causes remain unclear [13, 35].
+
+In this paper, we demystify the aforementioned phenomenon and demonstrate that directly applying traditional BN to Neural ODEs is fundamentally flawed. The primary complication arises because the forward pass of Neural ODEs employs an adaptive step size solver that discretizes time variably. Consequently, it is not assured that two forward passes will coincide with the same discretized time grids. This variability precludes the possibility of retrieving population statistics [16] at test time. Additionally, relying on mini-batch statistics for BN renders the outputs of Neural ODEs non-deterministic and vulnerable to outliers and small batch size.
+
+Motivated by these observations, we introduce temporal adaptive BN (TA-BN), a novel technique designed as the continuous-time counterpart to traditional BN, tailored specifically for Neural ODEs. Our empirical findings demonstrate that TA-BN facilitates the up-scaling of Neural ODE model sizes without encountering performance saturation (See Figure 1). Moreover, when confined to a model architecture consisting of a single Neural ODE followed by a linear layer, TA-BN achieves  $91.1\%$  test accuracy on CIFAR-10 with 2.2 million parameters, making it the first unmixed Neural ODE architecture to approach MobileNetV2-level parameter efficiency. Extensive numerical experiments on image classification and physical system modeling substantiate the superiority of TA-BN compared to baseline methods.
+
+# 2 Preliminary
+
+Neural ODEs Neural ODEs [4, 7, 37, 45, 8, 42, 30, 18, 32, 19, 31] forge a significant linkage between differential equations and neural networks. They model the continuous dynamics of hidden states with a learnable ODE system:
+
+$$
+\frac {d \mathbf {h} (t)}{d t} = \mathbf {f} _ {\boldsymbol {\theta}} (\mathbf {h} (t), t) \tag {1}
+$$
+
+where  $\mathbf{f}_{\theta}(\cdot ,\cdot)\in \mathbb{R}^{D}\times \mathbb{R}\to \mathbb{R}^{D}$  is a neural network parameterized by learnable parameter  $\pmb{\theta}$  controlling the temporal evolution, and  $\mathbf{h}(t)\in \mathbb{R}^D$  is the ODE state variable at time  $t$ . The practical usage of Neural ODEs is by solving an initial value problem: given the initial state  $\mathbf{h}(0)$  as input, the state  $\mathbf{h}(T)$  at a later time  $T$  is computed using an ODE solver and returned as the output. The gradients required for training are computed via the adjoint method [4, 31, 18] reverse in time.
+
+Subsequent works have expanded the capabilities and scope of Neural ODEs by introducing modifications, such as improving expressivity through state augmentation [7], coupling the evolution of both weights and activations [43], accelerating convergence via semi-norm techniques [18], extending to graph neural networks [37] and second-order ODEs [34], adapting to irregular time-series data [19, 32], and drawing connections to diffusion models [15, 28]. Of particular interest, two preliminary studies [13, 35] have indicated that applying BN within Neural ODEs may compromise model performance. Our work aims to unravel this phenomenon and propose a version of BN tailored to operate effectively with Neural ODEs.
+
+Batch Normalization BN [5, 29, 16, 39, 2, 21] performs a re-centering and a re-scaling operation on the given input by subtracting the mean and dividing by the standard deviation:
+
+$$
+\mathrm {B N} \left(x _ {i}\right) = \mathrm {B N} _ {\gamma , \alpha} \left(x _ {i}\right) = \frac {x _ {i} - \mu}{\sqrt {\sigma^ {2} + \epsilon}} \gamma + \alpha \tag {2}
+$$
+
+where  $\epsilon \in \mathbb{R}^{+}$  ensuring division validity,  $\{\gamma \in \mathbb{R},\alpha \in \mathbb{R}\}$  are learnable parameters, and  $x_{i}\in$ $\mathbb{R}$  represents the  $i$  -th data in one batch of size  $B$  . During training, the mini-batch statistics  $\mu = 1 / B\sum_{i = 1}^{B}x_{i}$  and  $\sigma^2 = 1 / B\sum_{i = 1}^{B}(x_i - \mu)^2$  are used for efficiency. In contrast, at test time, BN uses population statistics aggregated across the entire training dataset to ensure deterministic outputs [16]. BN can be extended to multiple dimensions/channels by processing each one separately.
+
+BN has been widely applied in training deep neural networks, as it can stabilize the training process and accelerate convergence. Despite its strong empirical performance, the underlying mechanisms
+
+of BN have been subject to various interpretations, such as reducing internal covariate shift [16], smoothing the optimization landscape [39], and decoupling the learning of length and direction [21]. Originally, BN [16] was developed for neural networks with hidden units that do not explicitly depend on time. However, recent advancements have extended BN to accommodate time-dependent models, including recurrent neural networks (RNNs)[5, 23] and spiking neural networks (SNNs) [20, 44, 6, 17]. Unlike these models, Neural ODEs exploit adaptive time discretization, necessitating a specialized modification of BN for its application.
+
+# 3 Traditional Batch Normalization in Neural ODEs
+
+Let us consider a simplified one-dimensional Neural ODE when incorporating BN as example:
+
+$$
+\frac {d h}{d t} = \mathrm {B N} _ {\gamma , \alpha} (w h + b) \tag {3}
+$$
+
+where  $\theta = \{\gamma \in \mathbb{R},\alpha \in \mathbb{R},w\in \mathbb{R},b\in \mathbb{R}\}$  are the learnable parameters. Given a batch of input  $\{h_i(0)\}_{i = 1}^B$  (i.e., initial values of the ODE), the forward process employs an ODE solver to discretize Eq. (3) over time, visiting  $N$  sequential points  $\mathcal{T} = (t_0 = 0,t_1,t_2,\dots ,t_N = T)$  and computing the outputs  $\{h_i(T)\}_{i = 1}^B$ . An adaptive step size solver (e.g., Runge-Kutta method) is usually preferred because of its higher efficiency compared to a fixed step size solver (e.g., Forward Euler method). Thus, the differences between two consecutive time steps might not be identical (e.g.,  $t_2 - t_1\neq t_1 - t_0$ ), and the value of  $N$  also depends on the batch of data. The temporal discretization indicates that we need to invoke BN at every  $t\in \mathcal{T}$  for every sample:
+
+$$
+\mathrm {B N} \left(x _ {i, j}\right) = \mathrm {B N} _ {\gamma , \alpha} \left(x _ {i, j}\right) = \frac {x _ {i , j} - \mu_ {j}}{\sqrt {\sigma_ {j} ^ {2} + \epsilon}} \gamma + \alpha , \quad \text {w h e r e} x _ {i, j} = w \cdot h _ {i} \left(t _ {j}\right) + b \tag {4}
+$$
+
+for every  $j = 1,2,\dots ,N$  and  $i = 1,2,\dots ,B$ , where  $\mu_{j}$  and  $\sigma_j^2$  are statistics associated with  $t_j$ .
+
+During training To perform the normalization shown in Eq. (4), BN applies mini-batch mean  $\mu_{j} = 1 / B\sum_{i = 1}^{B}x_{i,j}$  and variance  $\sigma_j^2 = 1 / B\sum_{i = 1}^B (x_{i,j} - \mu_j)^2$ . Although the training can proceed under normal conditions, the statistics cannot be successfully accumulated across batches. For instance, the time grid  $\mathcal{T}$  utilized for the first batch might differ significantly from the grid  $\mathcal{T}'$  used for a subsequent batch. Consequently, specific time points  $t_j^\prime \in \mathcal{T}'$  might not coincide with any time points in  $\mathcal{T}$ , and vice versa. This might result in an impractical collection of infinite statistics  $\mu_{j}$  and  $\sigma_j^2$ . Moreover, except those associated with  $t = 0$  and  $t = T$ , most stored statistics will be calculated and updated based on a limited number of batches.
+
+![](images/0628fed9b72dac59a4d0a0faa04bb61e129bada67f7820844e961559d3f97402.jpg)  
+Figure 2: Left: The failure of Pop BN in Neural ODEs stems from the misalignment of discretized time grids. Pop-TI BN aggregates all running mini-batch statistics into a single pair of  $(\mu, \sigma^2)$ , implicitly assuming time-independent population statistics. Right: Our proposed TA-BN automatically conducts temporal interpolation to accumulate statistics and update parameters during training and testing.
+
+![](images/5b4fc7397255f3e3fc5632d67c15bce055df071ba72bce8a0dac6981d284a071.jpg)
+
+During inference Conventionally, BN should perform the normalization shown in Eq. (4) using population statistics during inference. However, as shown in the left part of Figure 2, the population statistics associated with the time point  $t_j' \in \mathcal{T}'$ , required by the temporal discretization during inference, might not be available if the time value  $t_j'$  is never encountered during training. Moreover, even if the population statistics exist, they are likely to be inaccurate as discussed above.
+
+![](images/2aedd15d9d532abab956719fe14e8e4d7bb0e0e43cce28c02412683d74af9726.jpg)  
+Figure 3: Left: We train a Neural ODE with a U-Net backbone as the learnable derivatives on CIFAR-10. Mini-batch BN shows degraded accuracies with a batch size of 2, while TA-BN can maintain high accuracies under varying batch sizes, because it uses the estimated population statistics during testing. Pop-TI BN aggregates running statistics encountered at any time points into a single pair of  $\mu$  and  $\sigma^2$ . This approach assumes time-independent statistics, leading to erroneous predictions and erratic test loss curves. Middle: When noisy data exist on average in one out of every test batch, Mini-batch BN's performance deteriorates, because the noise affects the mini-batch statistics. The backbone for learnable derivatives in this experiment consists of 6 convolution layers. Right: We plot the output statistics from the first layer of U-Net over time; they are time-dependent.
+
+![](images/5d69c6a8d7e9672448a1d46484dcfe0a64390b731168c4c1bcce2fc45525d283.jpg)
+
+![](images/0b74a9a302e89313e517997a2018ed99a5262f381cf85e57c56eaccb785e1ae8.jpg)
+
+Summary A seemingly straightforward remedy could be to consistently use mini-batch statistics during both training and inference, thus avoiding the need for accumulating population statistics. However, this approach has a critical limitation: the output  $h_i(T)$  becomes highly dependent on their respective batch's specific characteristics. This will make the outputs inaccurate when batch size is very small (e.g.,  $B = 2$ ) or the occurrence of outliers in batch, and even fail in single-run scenarios ( $B = 1$ ) because running standard deviation cannot be calculated. As shown in the left of Figure 3, when the batch size is reduced to 2, mini-batch BN displays degraded accuracies. Furthermore, when noisy data exist, mini-batch BN performance deteriorates, as shown in the middle of Figure 3.
+
+For clarity, we will henceforth refer to the use of population statistics at test time as Pop BN, and the use of mini-batch statistics as Mini-batch BN. Notably, both methods utilize mini-batch statistics during training. It is important to emphasize that in practice, when implementing BN in Neural ODEs with popular deep learning frameworks (e.g., PyTorch with TorchDiffeq [4]), actually a variant of Pop BN is used, which we will refer to as Pop-TI BN. Pop-TI BN accumulates statistics at all encountered time points during training into a single pair of  $\mu$  and  $\sigma^2$ , which implicitly assumes that all time points share the same statistics. This aggregation approach can result in outputs during test time without errors concerning missing population statistics which we would anticipate as in Pop BN. Unfortunately, these predictions are meaningless and often lead to poor performance, as illustrated in the left of Figure 3. This issue highlights the challenges identified in recent studies [13, 35].<sup>3</sup> In a nutshell, our observations can be succinctly summarized as follows:
+
+- Pop BN fails when Neural ODEs employ an adaptive step size solver.  
+- Pop-TI BN implicitly assumes time-independent statistics, leading to erroneous outputs.  
+- Mini-batch BN fails when Neural ODEs operate with small batch sizes or outliers exist.
+
+# 4 Temporal Adaptive Batch Normalization
+
+The fundamental reason why Pop-TI BN and Mini-batch BN fail in Neural ODEs is attributed to the added temporal dimension, where the distributions at different time points vary, effectively constituting a stochastic process as depicted in the right of Figure 3. This observation suggests that BN should be defined to be time-dependent in Neural ODEs. Ideally, we would want BN to store population statistics at every  $t \in [0,T]$ , with corresponding learnable parameters  $\gamma (t)$  and  $\beta (t)$  defined at every  $t$ . However, implementing such a model is impractical, because not every  $t$  will be encountered in training, and the update of  $\gamma (t),\beta (t)$ , and the population statistics will be naturally sparse on  $t$ , resulting in inefficiency.
+
+To address the aforementioned problem, we propose an intuitive method termed temporal adaptive BN (TA-BN) for usage in Neural ODEs. Let us illustrate it with the example shown in Eq. (3). To begin with, we evenly divide the time span  $[0,T]$  defining  $(M + 1)$  time grids  $\mathcal{T}^{\star} = (t_0^{\star} = 0,t_1^{\star} = \frac{T}{M},t_2^{\star} = \frac{2T}{M},\dots ,t_M^{\star} = T)$ . We associate the time grid  $t_m^\star$  with population mean  $\mu_{m}^{\star}$  and population variance  $\sigma_{m}^{\star,2}$ , as well as learnable parameters  $\gamma_{m}^{\star}$  and  $\alpha_{m}^{\star}$  for every  $m = 0,1,2,\dots ,M$ . For later simplicity, we denote  $\pmb {\gamma}^{\star} = [\gamma_{0}^{\star},\gamma_{1}^{\star},\dots ,\gamma_{M}^{\star}]^{T}$  and  $\pmb {\alpha}^{\star} = [\alpha_{0}^{\star},\alpha_{1}^{\star},\dots ,\alpha_{M}^{\star}]^{T}$ . Similar notations apply to  $\pmb{\mu}^{\star}$  and  $\pmb{\sigma}^{\star}$ .
+
+During training As shown in the right of Figure 2, given a batch of data  $\{h_i(0)\}_{i=1}^B$ , the forward pass of Neural ODE might discretize the time as  $\mathcal{T} = (t_0, t_1, \dots, t_N)$  which differs from the grids  $\mathcal{T}^\star$  of TA-BN. However, TA-BN can calculate the temporal derivative for the  $i$ -th data at  $t_j$ :
+
+$$
+\mathrm {T A B N} _ {\gamma^ {\star}, \alpha^ {\star}} \left(x _ {i, j}\right) = \frac {x _ {i , j} - \mu_ {j}}{\sqrt {\sigma_ {j} ^ {2} + \epsilon}} \gamma_ {j} + \alpha_ {j} \quad \text {w h e r e} x _ {i, j} = w \cdot h _ {i} \left(t _ {j}\right) + b \tag {5}
+$$
+
+where  $(\mu_j,\sigma_j^2)$  are still the mini-batch mean and variance based on  $\{x_{i,j}\}_{i = 1}^{B}$ . Here  $\gamma_{j}$  and  $\alpha_{j}$  are interpolated based on  $\gamma^{\star}$  and  $\alpha^{\star}$ , respectively:
+
+$$
+\gamma_ {j} = G \left(t _ {j}, \boldsymbol {\gamma} ^ {\star}, \mathcal {T} ^ {\star}\right), \quad \alpha_ {j} = G \left(t _ {j}, \boldsymbol {\alpha} ^ {\star}, \mathcal {T} ^ {\star}\right) \tag {6}
+$$
+
+where  $G(t, \mathbf{a}, \mathcal{T})$  is a function to interpolate the value  $a(t)$  given an array of values  $\mathbf{a} \in \mathbb{R}^{M + 1}$  and their corresponding  $(M + 1)$  time points  $\mathcal{T}$ . There are many choices for  $G(\cdot, \cdot, \cdot)$ , such as linear, cubic spline, and kernel smoothing. We empirically observe that linear interpolation based on two nearest neighbors suffices for our experiments, and we advocate it due to its implementation simplicity:
+
+$$
+G (t, \mathbf {a}, \mathcal {T}) = \frac {t _ {l + 1} - t}{t _ {l + 1} - t _ {l}} a _ {l} + \frac {t - t _ {l}}{t _ {l + 1} - t _ {l}} a _ {l + 1} \tag {7}
+$$
+
+where  $l$  is the index which makes  $t_l$  to be the largest value smaller than  $t$  in  $\mathcal{T}$ . Using a linear  $G(\cdot, \cdot, \cdot)$  implies that we approximate the underlying  $a(t)$  in a piece-wise linear manner, and as long as the number of time grids is sufficiently large,  $G(t, \mathbf{a}, \mathcal{T})$  can approximate any continuous  $a(t)$  arbitrarily well, i.e.,  $|G(t, \mathbf{a}, \mathcal{T}) - a(t)| \to 0$  when  $M \to \infty$ . In practice, we set  $M$  close to the number of ODE-discretized time grids. Please refer to Section 5 for ablations on the choice of  $G(\cdot, \cdot, \cdot)$ .
+
+During training, we also need to accumulate the encountered running mini-batch statistics  $(\mu_j,\sigma_j^2)$  to the population statistics  $(\mu_m^\star ,\sigma_m^{\star ,2})$ , so that later they can be used for inference. As shown in the right part of Figure 2, this requires the interpolation to be performed in the opposite direction:
+
+$$
+\mu_ {m} ^ {\star} \leftarrow (1 - \eta) \cdot \mu_ {m} ^ {\star} + \eta \cdot G \left(t _ {m} ^ {\star}, \boldsymbol {\mu}, \mathcal {T}\right), \quad \boldsymbol {\mu} = \left[ \mu_ {1}, \mu_ {2}, \dots , \mu_ {N} \right] ^ {T}
+$$
+
+$$
+\sigma_ {m} ^ {\star , 2} \leftarrow (1 - \eta) \cdot \sigma_ {m} ^ {\star , 2} + \eta \cdot G \left(t _ {m} ^ {\star}, \boldsymbol {\sigma} ^ {2}, \mathcal {T}\right), \quad \boldsymbol {\sigma} ^ {2} = \left[ \sigma_ {1} ^ {2}, \sigma_ {2} ^ {2}, \dots , \sigma_ {N} ^ {2} \right] ^ {T}
+$$
+
+where  $\eta \in [0,1]$  is a momentum constant to perform moving averaging to update the population statistics based on current estimate of  $\mu_{m}^{\star}$  and  $\sigma_{m}^{\star,2}$  using the new observed values  $G(t_{m}^{\star},\pmb {\mu},\mathcal{T})$  and  $G(t_m^\star ,\pmb {\sigma}^2,\mathcal{T})$ , respectively. Note that the training does not involve the gradients with respect to  $t$ .
+
+During inference After the training phase, TA-BN will have well-trained parameters  $\gamma^{\star}$  and  $\alpha^{\star}$ , and accurately record population statistics  $\mu^{\star}$  and  $\sigma^{\star}$ . During inference, when provided with a batch of data and denoting the discretized time grids as  $\mathcal{T}'$ , as depicted in the right part of Figure 2, we again proceed with the interpolation step:
+
+$$
+\text {P a r a m .} \quad \gamma_ {j} ^ {\prime} = G \left(t _ {j} ^ {\prime}, \boldsymbol {\gamma} ^ {\star}, \mathcal {T} ^ {\star}\right), \quad \alpha_ {j} ^ {\prime} = G \left(t _ {j} ^ {\prime}, \boldsymbol {\alpha} ^ {\star}, \mathcal {T} ^ {\star}\right)
+$$
+
+$$
+\text {P o p . S t a t .} \quad \mu_ {j} ^ {\prime} = G \left(t _ {j} ^ {\prime}, \boldsymbol {\mu} ^ {\star}, \mathcal {T} ^ {\star}\right), \quad \sigma_ {j} ^ {\prime, 2} = G \left(t _ {j} ^ {\prime}, \boldsymbol {\sigma} ^ {\star , 2}, \mathcal {T} ^ {\star}\right)
+$$
+
+This interpolation differs from the training phase in that now the interpolation directions are the same for parameters and the population statistics, because now we utilize them for the new time grids  $\mathcal{T}'$ .
+
+Implementations and summary In practice, we observe that the interpolations during training outlined in Eq. (6)-(8) have opposite directions, potentially leading to increased computational overhead. To elaborate, when encountering a time point  $t_j$  discretized by the ODE solver during training, the interpolation for obtaining  $\gamma_j$  and  $\alpha_j$  in Eq. (6) using Eq. (7) can be immediately executed by identifying the index  $l$  such that  $t_l^{\star}$  is the largest value smaller than  $t_j$ . However, the interpolation
+
+in Eq. (8) cannot be carried out at this time. It becomes feasible only after the ODE solver completes its execution and we gather all  $t_j$ 's into  $\mathcal{T}$ . This sequential nature degrades computational efficiency. To mitigate this issue, we have slightly adjusted the interpolation for population statistics during training so that it can be performed concurrently with the parameter interpolation at time  $t_j$ . The key steps of our proposed TA-BN are summarized in Algorithm 1. It will be called as a subroutine for every discretized time point  $t_j \in \mathcal{T}$  required by the ODE solver.
+
+Algorithm 1 outlines the case involving a single one-dimensional neuron and one TA-BN layer within the Neural ODE. In more general cases, we may have  $Q$  TA-BN layers within a Neural ODE, and each neuron may possess  $D$  dimensions. In such cases, we can perform the normalization in a dimensional-wise manner. Thus, we must maintain  $2(M + 1)Q$  population statistics and  $2(M + 1)Q$  learnable parameters, each with  $D$  dimensions. In contrast, a traditional feedforward neural network with  $Q$  BNs has  $2Q$  population statistics and  $2Q$  learnable parameters, each with  $D$  dimensions. Note that for image inputs, we perform channel-wise normalization, so  $D$  will be the number of channels.
+
+Finally, since our linear interpolation in Algorithm 1 relies on two nearest neighbors, it is prudent to ensure that the number of TA-BN time grids  $(M + 1)$  are roughly at the same level as the number of ODE-discretized time grids. Denser grids would be unnecessary as those not adjacent to any  $t_j$  would remain unused. Alternatively, as our numerical results will demonstrate, employing other interpolation methods that require all time grids may lead to increased computational overhead without significant improvements. Additionally, we employ L2 regularization  $\| \gamma^{\star} - 1\|^{2}$  and  $\|\pmb{\alpha}^{\star}\|^2$  to avoid significant discrepancies among the elements in  $\gamma^{\star}$  and  $\alpha$ , making the training more stable.
+
+Algorithm 1 The forward pass of a TA-BN layer at time  $t_j$  
+Input: Batched input  $\mathbf{x} = \{x_{i,j}\}_{i=1}^{B}$  at time  $t_j$   
+1: Get  $t_l^{\star}$  and  $t_{l+1}^{\star}$  such that  $t_l^{\star}$  is the largest value smaller than  $t_j$  in  $\mathcal{T}^*$ ;  
+2:  $\omega_1 = \frac{t_{l+1}^{\star} - t_j}{t_{l+1}^{\star} - t_l^{\star}}$ ,  $\omega_2 = \frac{t_j^{\star} - t_l}{t_{l+1}^{\star} - t_l^{\star}}$   
+3: if the model is in the training mode then  
+4:  $\mu_j = \text{mean}(\mathbf{x})$ ,  $\sigma_j^2 = \text{var}(\mathbf{x})$ ;  
+5:  $\gamma_j = \omega_1\gamma_l^{\star} + \omega_2\gamma_{l+1}^{\star}$ ,  $\alpha_j = \omega_1\alpha_l^{\star} + \omega_2\alpha_{l+1}^{\star}$ ;  
+6:  $\mu_l^{\star} \gets (1 - \eta \cdot \omega_1)\mu_l^{\star} + \eta \cdot \omega_1\mu_j$ ,  $\mu_{l+1}^{\star} \gets (1 - \eta \cdot \omega_2)\mu_{l+1}^{\star} + \eta \cdot \omega_2\mu_j$ ;  
+7:  $\sigma_l^{\star,2} \gets (1 - \eta \cdot \omega_1)\sigma_l^{\star,2} + \eta \cdot \omega_1\sigma_j^2$ ,  $\sigma_{l+1}^{\star,2} \gets (1 - \eta \cdot \omega_2)\sigma_{l+1}^{\star,2} + \eta \cdot \omega_2\sigma_j^2$ ;  
+8: else if the model is in the evaluation mode then  
+9:  $\mu_j = \omega_1\mu_l^{\star} + \omega_2\mu_{l+1}^{\star}$ ,  $\sigma_j^2 = \omega_1\sigma_l^{\star,2} + \omega_2\sigma_{l+1}^{\star,2}$ ;  
+10:  $\gamma_j = \omega_1\gamma_l^{\star} + \omega_2\gamma_{l+1}^{\star}$ ,  $\alpha_j = \omega_1\alpha_l^{\star} + \omega_2\alpha_{l+1}^{\star}$ ;  
+11: end if  
+12: return  $\frac{x - \mu_j}{\sqrt{\sigma_j^2 + \epsilon}}\gamma_j + \alpha_j$ ;
+
+# 5 Numerical Results
+
+Various network architecture designs have been explored in existing Neural ODE studies. For instance, the original study [4] integrates a feature extractor (e.g., CNN-based), a Neural ODE module, and an MLP in sequence for classification. Another study structures neural networks with alternating Neural ODE modules and convolutional/linear layers [11]. While these solutions represent viable approaches for optimizing performance on given ML tasks through the synergy of conventional (discrete) neural networks and Neural ODEs, they become inappropriate for investigating the effect of specific architectural modifications (i.e., TA-BN) on Neural ODEs. This is because the Neural ODE module can be entirely bypassed if its trained weights approach zero in extreme cases, thereby overshadowing the intended modification. Hence, our model comprises a Neural ODE module followed by a single linear layer as advocated by [7], which we refer to as the unmixed Neural ODE architecture. Our code is developed based on PyTorch [36], TorchDiffeq [3], and a customized TA-BN layer. All experiments are run on a Linux server with RTX 3090 GPUs.
+
+# 5.1 Image Classification
+
+Being consistent in experimental scales of previous Neural ODE studies, we conduct image classification across datasets including MNIST [26], SVHN [33], CIFAR-10, CIFAR-100 [22], and
+
+Tiny-ImageNet [24]. We employ the dopri5 solver with a tolerance of  $10^{-3}$  for ODE solving and adopt the AdamW optimizer [27] with a learning rate of  $10^{-3}$  to train the neural networks for 128 epochs. The training batch size is 256. We set  $M = 100$  for TA-BN.
+
+Table 1 compares the test top-1 accuracy and number of parameters of various Neural ODEs. TA-BN outperforms Mini-batch BN and Neural ODE w/o BN on SVHN, CIFAR-10, CIFAR-100, and Tiny-ImageNet. Moreover, TA-BN achieves superior accuracies over Aug-NODE [7] and STEER [12]. On MNIST and CIFAR-10, TA-BN has fewer parameters than Aug-NODE and STEER, which indicates that TA-BN can also help training in the small Neural ODE regime. We also visualize Neural ODEs' performance changes as the number of parameters varies in Figure 4, by including four additional baselines IL-NODE, 2nd-Ord [30], HBNODE, and GHBNODE [41]. For reference, the popular convolutional-based MobileNetV2 [38], known for its parameter efficiency, achieves approximately  $94\%$  accuracy with about 2M parameters. Our TA-BN assisted Neural ODE is the first to approach this level of performance using the unmixed architecture, while most previous Neural ODE literature either falls short of this performance or/and does n
+
+![](images/1b49542f79bbdfc9d9910d98110af6524e05dede063a79cc80bb9ae851f82ef2.jpg)  
+Figure 4: Comparison between different Neural ODEs on CIFAR-10. The baselines marked by yellow triangles do not adhere to the unmixed structure and are not strictly comparable to ours. It is unknown whether increasing the number of parameters inside their ODEs can lead to better accuracy.
+
+Table 1: Comparison of test accuracies and number of parameters between different Neural ODEs†.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">MNIST</td><td colspan="2">CIFAR10</td><td colspan="2">SVHN</td><td colspan="2">CIFAR100</td><td colspan="2">Tiny-Imagenet</td></tr><tr><td>Accuracy</td><td>#Params</td><td>Accuracy</td><td>#Params</td><td>Accuracy</td><td>#Params</td><td>Accuracy</td><td>#Params</td><td>Accuracy</td><td>#Params</td></tr><tr><td>Aug-NODE [7]</td><td>0.982</td><td>84k</td><td>0.606</td><td>172k</td><td>0.835</td><td>172k</td><td>N/A</td><td>N/A</td><td>N/A</td><td>366k</td></tr><tr><td>STEER [12]</td><td>0.986</td><td>84k</td><td>0.621</td><td>172k</td><td>0.841</td><td>172k</td><td>N/A</td><td>N/A</td><td>N/A</td><td>N/A</td></tr><tr><td>w/o BN</td><td>0.989±0.001</td><td>37k</td><td>0.517±0.049</td><td>2.2M</td><td>0.096±0.025</td><td>2.2M</td><td>0.246±0.084</td><td>2.2M</td><td>-</td><td>2.2M</td></tr><tr><td>w/ Pop-TI BN</td><td>0.973±0.011</td><td>37k</td><td>0.548±0.087</td><td>2.2M</td><td>0.241±0.123</td><td>2.2M</td><td>0.251±0.112</td><td>2.2M</td><td>0.044±0.007</td><td>2.2M</td></tr><tr><td>w/ Mini-batch BN</td><td>0.962±0.013</td><td>37k</td><td>0.822±0.095</td><td>2.2M</td><td>0.906±0.031</td><td>2.2M</td><td>0.492±0.176</td><td>2.2M</td><td>0.200±0.006</td><td>2.2M</td></tr><tr><td>w/ TA-BN</td><td>0.988±0.001</td><td>37k</td><td>0.748±0.059</td><td>70k</td><td>0.953±0.002</td><td>220k</td><td>0.576±0.016</td><td>220k</td><td>0.436±0.013</td><td>220k</td></tr><tr><td>(ours)</td><td>0.988±0.001</td><td>220k</td><td>0.910±0.010</td><td>2.2M</td><td>0.958±0.004</td><td>2.2M</td><td>0.664±0.025</td><td>2.2M</td><td>0.512±0.008</td><td>2.2M</td></tr></table>
+
+†‘N/A’ indicates values are not available in the original literature. ‘-’ indicates training failure at the first epoch. Results are reported with a test batch size of 256. The error bars are shown using the format mean±std.
+
+![](images/7373e005c9c142e085853333811a69d339058b7073ff0cdf34bcc345355b0b5f.jpg)  
+Figure 5: Comparison between Neural ODEs w/ TA-BN, w/ Mini-batch BN, w/ Pop-TI BN, and w/o BN, using an 18-layer U-Net backbone for learnable derivatives. The total number of parameters including the outside linear layer is 2.2M. The datasets are SVHN (left), CIFAR-100 (middle), and Tiny-ImageNet (right). The results on CIFAR-10 have been shown in the left part of Figure 3.
+
+![](images/5ff6fc3021e4f50c262618ccde3e15c5ea7f27b302badeead74914e57a0ffcad.jpg)
+
+![](images/963e157e13b4502034c19702931f52eca667a3738dec390a483715e61700b7f6.jpg)
+
+The left part of Figure 3 in Section 3 monitors the test accuracy over training epochs on CIFAR-10. It indicates that training a large Neural ODE without special processes might fail due to numerical instability. Pop-TI BN performs poorly because of the time-independent statistics assumption. While Mini-batch BN achieves satisfactory accuracy, our TA-BN outperforms it without encountering the issues associated with Mini-batch BN, such as batch-dependent outcomes and vulnerability to outliers. Similarly, the left and middle parts of Figure 5 display test accuracy versus training epochs on the SVHN and CIFAR-100 datasets. TA-BN consistently shows better and more stable accuracy than Mini-batch BN on these datasets. On the Tiny-ImageNet dataset (right part of Figure 5), TA-BN demonstrates superior accuracy and faster convergence.
+
+Neural ODE Up-scaling Enabled by TA-BN As shown in the left part of Figure 6, Neural ODEs with a limited number of layers perform normally with acceptable accuracy; however, when the layer count exceeds 10, training fails due to numerical instability. In contrast, the incorporation of TA-BN enables deeper layers within Neural ODE as the learnable derivatives, scaling up the model size and enhancing accuracy, as exemplified by the middle and right sections of Figure 6. Please see Appendix A.1 for architecture details.
+
+![](images/aeef2a109f7427d6227d2f2fef0c5ac221de55f1f2f210637626e3f4ee5015db.jpg)  
+Figure 6: CIFAR-10 accuracies with increasing sizes of the backbones for learnable derivatives. These figures illustrate the scaling up of Neural ODEs without BN (left) and Neural ODEs with TA-BN (middle). We also compare the accuracies of these two settings in one figure (right).
+
+![](images/9b020363a5d668b2a9194e67a090cdeb5735c395118cc9ded6b5af171751ff45.jpg)
+
+![](images/97062f0ecc05f82abf4d8d85db38375862a433413f57bb9f0f998f94a5576da6.jpg)
+
+Ablations on TA-BN We integrate TA-BN into Aug-NODE [7], HBNODE [41], and SONODE [34]. Table 2 reports model accuracies w/ and w/o TA-BN. It indicates that our method can also work in compatible with existing Neural ODE variants and boost their performances.
+
+To show the impact of different interpolation functions  $G$ , we test the following interpolation methods: (1) Linear interpolation demonstrated
+
+Table 2: Comparison between Neural ODE variations with and without TA-BN on CIFAR-10†.  
+
+<table><tr><td>Accuracy</td><td>Aug-NODE</td><td>HBNODE</td><td>SONODE</td></tr><tr><td>w/o TA-BN</td><td>0.848</td><td>0.851</td><td>0.302</td></tr><tr><td>w/ TA-BN</td><td>0.862</td><td>0.867</td><td>0.853</td></tr></table>
+
+$\dagger$  We use 6 convolution layers as the backbone for learnable derivatives.
+
+by Eq. (7). (2) Cubic interpolation: We use cubic spline interpolation [1]. (3) Kernel smoothing: Given time  $t$ , we calculate  $G(t, \mathbf{a}, \mathcal{T})$  by the weighted sum of elements in  $\mathbf{a}$ . The weight coefficient for  $t_i$  is  $K(t, t_i) = \exp(-0.5b^{-2}(t - t_i)^2)$  with  $b = 0.1T$ . (4) Gaussian process: We fit a Gaussian process [10] to perform interpolation in each forward pass. As illustrated by Figure 7, in addition to linear interpolation, the cubic interpolation and kernel smoothing methods also achieve good performance. However, these methods can be much slower than linear interpolation due to more complicated interpolation mechanisms. With a larger network architecture, we observe that these methods suffer from unaffordable running time and instable performance. Thus, we adopt linear interpolation as our default setting. Please see Appendix B for further discussion. Future work can focus on improving the interpolation strategy for TA-BN.
+
+# 5.2 Physical System Modeling
+
+We first evaluate TA-BN on physical dynamical system modeling tasks using the Walker2d-v2 [25] and HalfCheetah-v2 [14] datasets. These datasets consist of trajectories of 3D robot systems generated by the Mujoco physics engine [40]. Following [42], we perform temporal autoregressive prediction and use the treatments from [42] to avoid collisions. We employ TA-BN in conjunction with a 12-layer MLP serving as the backbone for learnable derivatives. TA-BN with  $M = 100$  is applied to the first 5 layers, as is done with Mini-batch BN and Pop-TI BN. We utilize the dopri5 solver with its default settings for solving ODEs and employ the RMSprop optimizer with a learning rate of  $10^{-3}$  for training the neural networks over 100 epochs. The batch size is 64 for Walker2d-v2 and 32 for HalfCheetah-v2. Figure 8 presents the mean absolute error (MAE) along the training epochs on these datasets. Compared with Neural ODEs w/o BN, w/ Mini-batch BN, and w/ Pop-TI BN, using TA-BN can achieve lower errors and faster convergence. Moreover,
+
+Table 3 summarizes the MAE results and compares them with the results in [42].
+
+![](images/8232ea27d35cd4222310c67c78f4306dd8beabaaec71cd655dde3b6d7419182a.jpg)  
+Figure 7: Ablations on interpolation method using CIFAR-10 with 6 convolution layers as the backbone for learnable derivatives. Linear interpolation has a gap of only 0.01 compared to the best result. However, other methods have degraded accuracies and unaffordable runtime when we up-scale the model.
+
+![](images/902bbba519691409af9917e26f5e8c5be16a0b2e5941c8704e52abde5bd8231b.jpg)  
+Figure 8: Comparison of Neural ODEs with TA-BN and baselines on Walker2d-v2 (left) and HalfCheetah-v2 (right). The Neural ODE backbone for learnable derivatives is an MLP with 12 layers. Please see Appendix A.2 for architecture details.
+
+![](images/49ea4e8d724fb0535e0b4215175a8de24bb7f0cf67273fb0db08ea6e864d9541.jpg)  
+Figure 9: Comparison of different Neural ODEs on Charge Pump circuit modeling. The backbone for learnable derivatives is an 8-layer MLP.
+
+![](images/b33bc0b8a8b561b3c5d4a48e45bb16aab0dc8b4ef45152b9ed918eb0a1421d61.jpg)
+
+Table 3: MAE comparison between different Neural ODEs on Walker2d-v2 and HalfCheetah-v2.  
+
+<table><tr><td>Dataset</td><td>Walker2d-v2</td><td>HalfCheetah-v2</td></tr><tr><td>Baseline [42]</td><td>1.02</td><td>1.46</td></tr><tr><td>w/o BN</td><td>0.959</td><td>1.40</td></tr><tr><td>w/ Mini-batch BN</td><td>0.953</td><td>1.38</td></tr><tr><td>w/ Pop-TI BN</td><td>0.962</td><td>1.37</td></tr><tr><td>w/ TA-BN (ours)</td><td>0.946</td><td>1.28</td></tr></table>
+
+Additionally, we investigate the effectiveness of TA-BN by modeling the temporal electrical current of a Charge Pump (CP) circuit [9]. The CP is an analog integrated circuit composed of MOS transistors and is commonly utilized in commercial electrical products. To simulate the semiconductor manufacturing impact on the CP, we generate 200 circuit instances based on the process design kit (PDK) file. Concretely, we
+
+simulate the CP's behavior from 0 seconds to 200 nanoseconds using a time grid of 2 picoseconds and record the electrical currents at the CP output every 200 picoseconds.
+
+In this experiment, we utilize 16 circuit features (representing the widths of the MOS transistors) and the electrical currents at the present 20 time points (with each point comprising 2 values) as the Neural ODE input, resulting in 56 input features in total. Our objective is to train the Neural ODE to predict the electrical currents at the subsequent 20 time points, yielding 40 outputs in total. To construct the dataset, we randomly sample  $20\mathrm{k}$  data from the simulation results. The training set includes  $90\%$  of them, and the remaining data are used as the testing set. We use 8 linear layers to parameterize the derivative of the Neural ODE, with each layer containing 56 neurons. Mini-batch BN, Pop-TI BN, and TA-BN are applied to the first 3 layers. We use  $M = 100$  for TA-BN. The final linear layer outside the Neural ODE maps the 56 features to 40 prediction values. For ODE solving, we use the dopri5 method with a tolerance level of  $10^{-3}$ . We use the SGD optimizer with a learning rate of  $10^{-2}$  to train the model for 16 epochs. The batch size is 200.
+
+Figure 9 presents the mean square error (MSE) results of Neural ODEs w/o BN, w/ Mini-batch BN, w/ Pop-TI BN, and w/ TA-BN. BN can improve the prediction, and TA-BN surpasses Mini-batch BN. It indicates that TA-BN is suitable for the circuit modeling task.
+
+# 6 Conclusions and Limitations
+
+In this paper, we demystify the previously unknown reason why batch normalization (BN) may lead to performance degradation when used with Neural ODEs. The fundamental mismatch arises from BN being initially designed for discrete neural networks without a temporal dimension, whereas Neural ODEs operate continuously over time. To address this challenge, we propose temporal adaptive batch normalization (TA-BN), specifically tailored for Neural ODEs. Our key is associating population statistics and learnable parameters with predefined regular time grids, with temporal interpolation automatically performed during training and testing. As a continuous-time counterpart to traditional BN, TA-BN ensures training stability, thereby facilitating the scaling of Neural ODE model sizes. Our extensive experiments demonstrate TA-BN's ability to enhance the performance of Neural ODEs compared to baseline methods in tasks such as image classification and physical system modeling.
+
+The interpolation process used in TA-BN inevitably introduces runtime overhead, which slows down the execution of Neural ODEs. The parameter-free linear interpolation technique is empirically found to be stable, time-efficient, and capable of providing performance improvement. However, future work can focus on enhancing the temporal interpolation used in TA-BN to further optimize its efficiency and performance.
+
+# References
+
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+[2] Nils Bjorck, Carla P Gomes, Bart Selman, and Kilian Q Weinberger. Understanding batch normalization. Advances in Neural Information Processing Systems, 31, 2018.  
+[3] Ricky T. Q. Chen. torchdiffreq. https://github.com/rtqichen/torchdiffreq, 2018.  
+[4] Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. Advances in Neural Information Processing Systems, 31, 2018.  
+[5] Tim Cooijmans, Nicolas Ballas, Cesar Laurent, Caglar Gulçehre, and Aaron Courville. Recurrent batch normalization. In International Conference on Learning Representations, 2017.  
+[6] Chaoteng Duan, Jianhao Ding, Shiyan Chen, Zhaofei Yu, and Tiejun Huang. Temporal effective batch normalization in spiking neural networks. Advances in Neural Information Processing Systems, 35:34377-34390, 2022.  
+[7] Emilien Dupont, Arnaud Doucet, and Yee Whye Teh. Augmented neural odes. Advances in neural information processing systems, 32, 2019.  
+[8] Chris Finlay, Jorn-Henrik Jacobsen, Levon Nurbekyan, and Adam Oberman. How to train your neural ode: the world of jacobian and kinetic regularization. In International conference on machine learning, pages 3154-3164. PMLR, 2020.  
+[9] Zhengqi Gao, Jun Tao, Fan Yang, Yangfeng Su, Dian Zhou, and Xuan Zeng. Efficient performance trade-off modeling for analog circuit based on bayesian neural network. In 2019 IEEE/ACM International Conference on Computer-Aided Design (ICCAD), pages 1-8. IEEE, 2019.  
+[10] R Jacob Gardner, Geoff Pleiss, Bindel David, Kilian Q Weinberger, and Andrew Gordon Wilson. GPyTorch: Blackbox matrix-matrix gaussian process inference with GPU acceleration. Advances in Neural Information Processing Systems, 2018.  
+[11] Amir Gholami, Kurt Keutzer, and George Biros. Anode: Unconditionally accurate memory-efficient gradients for neural odes. arXiv preprint arXiv:1902.10298, 2019.  
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+[13] Julia Gusak, Larisa Markeeva, Talgat Daulbaev, Alexandr Katrutsa, Andrzej Cichocki, and Ivan Oseledets. Towards understanding normalization in neural odes. arXiv preprint arXiv:2004.09222, 2020.  
+[14] Ramin Hasani, Mathias Lechner, Alexander Amini, Daniela Rus, and Radu Grosu. Liquid time-constant networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 7657-7666, 2021.  
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+[18] Patrick Kidger, Ricky TQ Chen, and Terry J Lyons. "hey, that's not an ode": Faster ode adjoints via seminorms. In ICML, pages 5443-5452, 2021.  
+[19] Patrick Kidger, James Morrill, James Foster, and Terry Lyons. Neural controlled differential equations for irregular time series. Advances in Neural Information Processing Systems, 33:6696-6707, 2020.  
+[20] Youngeun Kim and Priyadarshini Panda. Revisiting batch normalization for training low-latency deep spiking neural networks from scratch. Frontiers in neuroscience, 15:773954, 2021.
+
+[21] Jonas Kohler, Hadi Daneshmand, Aurelien Lucchi, Thomas Hofmann, Ming Zhou, and Klaus Neymeyr. Exponential convergence rates for batch normalization: The power of length-direction decoupling in non-convex optimization. In *The 22nd International Conference on Artificial Intelligence and Statistics*, pages 806–815. PMLR, 2019.  
+[22] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.  
+[23] César Laurent, Gabriel Pereyra, Philémon Brakel, Ying Zhang, and Yoshua Bengio. Batch normalized recurrent neural networks. In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 2657-2661, 2016.  
+[24] Ya Le and Xuan Yang. Tiny imagenet visual recognition challenge. 2015.  
+[25] Mathias Lechner and Ramin Hasani. Mixed-memory RNNs for learning long-term dependencies in irregularly sampled time series. 2022.  
+[26] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278-2324, 1998.  
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+[28] Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. Dpm-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps. Advances in Neural Information Processing Systems, 35:5775-5787, 2022.  
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+[32] James Morrill, Christopher Salvi, Patrick Kidger, and James Foster. Neural rough differential equations for long time series. In International Conference on Machine Learning, pages 7829-7838. PMLR, 2021.  
+[33] Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Baolin Wu, Andrew Y Ng, et al. Reading digits in natural images with unsupervised feature learning. In NIPS workshop on deep learning and unsupervised feature learning, volume 2011, page 7. Granada, Spain, 2011.  
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+
+# A Implementation Details
+
+# A.1 Image Classification
+
+Our model comprises a Neural ODE module followed by a single linear layer, which we refer to as the unmixed Neural ODE architecture. The single linear layer maps the activations provided by Neural ODE to the final class probability (or logit). We have 3 settings for the Neural ODE module.
+
+Multilayer Perceptron (MLP): We use 2 linear layers inside the Neural ODE module. On MNIST, each linear layer has  $28 \times 28$  hidden features, matching the number of pixels in an MNIST image.
+
+Convolutional Neural Network (CNN): We use  $4 \sim 12$  convolution layers inside the Neural ODE module. The kernel size of all layers is  $3 \times 3$ . The numbers of output channels are:
+
+4 layers: 3(input)  $\rightarrow 32_{\downarrow}\rightarrow 64_{\downarrow}\rightarrow 32_{\uparrow}\rightarrow 3_{\uparrow}.$  
+6 layers: 3(input)  $\rightarrow 32_{\downarrow}\rightarrow 64_{\downarrow}\rightarrow 128_{\downarrow}\rightarrow 64_{\uparrow}\rightarrow 32_{\uparrow}\rightarrow 3_{\uparrow}.$  
+$7 + L_{c}$  layers:  $3(\mathrm{input})\to 32_{\downarrow}\to 64_{\downarrow}\to 128_{\downarrow}(\rightarrow 256)\times L_{c}\to 128\to 64_{\uparrow}\to 32_{\uparrow}\to 3_{\uparrow}.$
+
+Note that a layer with  $\downarrow$  downscales the feature maps using a stride of 2. A layer with  $\uparrow$  is a transposed convolution layer that upscales the feature maps using a stride of 2.
+
+U-Net: We use a U-Net with 18 convolution layers. The kernel size of all layers is  $3 \times 3$ . The numbers of output channels are designed as  $3(\mathrm{input}) \to 32_{\downarrow} \to 32 \to 64_{\downarrow} \to 64 \to 128_{\downarrow} \to 128 \to 256 \to 256 \to 128_{\uparrow} \to 128 \to 128 \to 64_{\uparrow} \to 64 \to 64 \to 32_{\uparrow} \to 32 \to 32 \to 3$ .
+
+# A.2 Physical Dynamical Systems Modeling
+
+The HalfCheetah-v2 and Walker2d-v2 datasets consist of trajectories of 3D robot systems generated by the Mujoco physics engine [40]. Each trajectory represents a sequence of a 17-dimensional vector describing the system's state, such as the robot's joint angles and poses. The Neural ODEs are required to predict the trajectories in an autoregressive manner, as in [25, 42].
+
+In these tasks, the input size is 17, and the Neural ODE module is designed as  $17$  (input) ( $\rightarrow 64$ ) ×  $11 \rightarrow 17$ . BN is applied to the first 5 layers. Based on the code from [42], we do the prediction via the invariance set propagation and data-controlled neural ODE proposed in [42]. Since the input size is the same as the output size, we don't need a linear layer outside the Neural ODE module.
+
+![](images/5451cd617bb06dbd13a42bfa8d9c6ac915e6c608624cc7d47590566c731c57a6.jpg)  
+(a)
+
+![](images/8b895f6c8f6457b97a01f9bc376adaa7ffa78ea856e94daadf4db8b002dced6b.jpg)  
+(b)
+
+![](images/a749c9f3661cf42ddc9a72de638ecae570725489a4a726a31643f8a10230ce52.jpg)  
+(c)  
+Figure 10: (a) Kernel smoothing can achieve slightly better accuracy on CIFAR-100 with 6 convolution layers for learnable derivatives. (b) Linear interpolation achieves more stable performance across 5 runs. (c) Linear interpolation performs better with a larger model size. (d) We compare different methods on the runtime per step at the last epoch to show the efficiency of linear interpolation.
+
+![](images/8cf576e78cee652829b1933e35fd6e0d6804e0a1888915ecbc9bc33dce7fcc18.jpg)  
+(d)
+
+# B Discussion on Interpolation Methods
+
+In Figure 7, we compare the performance of four interpolation methods, including linear interpolation, cubic interpolation, kernel smoothing, and Gaussian process. The cubic interpolation and kernel smoothing methods achieve good performance with 6 convolution layers for learnable derivatives. The results on CIFAR-100 also support this point, as shown in Figure 10(a). However, Figure 10(b) indicates that linear interpolation achieves more stable performance, and Figure 10(c) clearly shows the superiority of linear interpolation with a larger model size. Furthermore, linear interpolation is more efficient than cubic interpolation and Gaussian process, as shown in Figure 10(d). Therefore, we use linear interpolation by default, which can provide superior performance and stability.
+
+# C Additional Ablation Studies
+
+# C.1 Ablation Study on ODE Solvers
+
+We have conducted extra experiments on CIFAR-10 using fixed-step solvers like Euler method, with the 8-layer backbone. The results are shown in Table 5. Regardless of the solver, TA-BN achieves the best performance among the techniques. We also explored midpoint and rk4 solvers; however, they are much slower and haven't finished in the limited time constraint.
+
+# C.2 Ablation Study on Time Grids
+
+Regarding the grid size hyperparameter  $M$ , we ran experiments without BN and found that the number of function evaluations (NFE) is around hundreds. Thus, we set  $M = 100$  in our paper. We have performed extra ablation studies on it, as reported in the following table. Using  $M > 100$  brings no improvement but too much
+
+Table 4: Ablation study on ODE solvers.  
+
+<table><tr><td>Method</td><td>ODE Solver</td><td>Accuracy</td></tr><tr><td>w/o BN</td><td>Euler</td><td>0.839±0.002</td></tr><tr><td>w/ Pop-TI BN</td><td>Euler</td><td>0.631±0.203</td></tr><tr><td>w/ Mini-batch BN</td><td>Euler</td><td>0.864±0.002</td></tr><tr><td>w/ TA-BN (ours)</td><td>Euler</td><td>0.872±0.003</td></tr><tr><td>w/o BN</td><td>Dopri5</td><td>0.843±0.004</td></tr><tr><td>w/ Pop-TI BN</td><td>Dopri5</td><td>0.332±0.090</td></tr><tr><td>w/ Mini-batch BN</td><td>Dopri5</td><td>0.865±0.004</td></tr><tr><td>w/ TA-BN (ours)</td><td>Dopri5</td><td>0.874±0.001</td></tr></table>
+
+Table 5: Ablation study on ODE solvers.  
+
+<table><tr><td>Method</td><td>Time Grids M</td><td>Accuracy</td></tr><tr><td>w/ TA-BN (ours)</td><td>10</td><td>0.851±0.015</td></tr><tr><td>w/ TA-BN (ours)</td><td>50</td><td>0.851±0.019</td></tr><tr><td>w/ TA-BN (ours)</td><td>100</td><td>0.874±0.001</td></tr><tr><td>w/ TA-BN (ours)</td><td>500</td><td>0.870</td></tr></table>
+
+runtime overhead. We don't have the confidence interval of  $M = 500$  due to the time limit.
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: We clearly describe our main claims such as the findings, contributions, and results in the abstract and introduction. They match our idea, methodology, and experimental results that we present in the paper.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: We describe the limitations of our method in Section 5 "Conclusion and Limitations".
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory Assumptions and Proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+# Answer: [NA]
+
+Justification: Our method is based on batch normalization. We do not introduce additional assumptions and theories.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental Result Reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+# Answer: [Yes]
+
+Justification: We have fully disclosed the details of neural ODE architectures, ODE solvers, and optimizers in the paper, along with the used Python packages and hardware platform. Combined with the detailed experimental settings in our paper, the provided information can be used to reproduce the main experimental results.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.  
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.  
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.  
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example  
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
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+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
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+
+# 6. Experimental Setting/Details
+
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+
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+
+# 7. Experiment Statistical Significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments Compute Resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
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+
+# 9. Code Of Ethics
+
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+
+# 10. Broader Impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [NA]
+
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+
+Guidelines:
+
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+
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+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
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+
+# Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
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+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: For the existing assets we use, we have cited the papers and repos. They are all open-source.
+
+# Guidelines:
+
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+- The authors should cite the original paper that produced the code package or dataset.  
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+
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+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New Assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
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+Guidelines:
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+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and Research with Human Subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
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+# 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
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+# Improving Subgroup Robustness via Data Selection
+
+Saachi Jain\* Kimia Hamidieh\* Kristian Georgiev\*
+
+Andrew Ilyas, Marzyeh Ghassemi, Aleksander Mądry
+
+MIT
+
+{saachij,hamidieh,krisgrg,ailyas,mghassem,madry}@mit.edu
+
+# Abstract
+
+Machine learning models can often fail on subgroups that are underrepresented during training. While dataset balancing can improve performance on underperforming groups, it requires access to training group annotations and can end up removing large portions of the dataset. In this paper, we introduce Data Debiasing with Datamodels (D3M), a debiasing approach which isolates and removes specific training examples that drive the model's failures on minority groups. Our approach enables us to efficiently train debiased classifiers while removing only a small number of examples, and does not require training group annotations or additional hyperparameter tuning.
+
+# 1 Introduction
+
+The advent of large datasets such as OpenImages [23] and The Pile [13] has led to machine learning models being trained on explicit [4] and illegal [48] content, or on data that encode negative societal biases [6, 12, 2, 8] and other spurious correlations [32, 30]. On one hand, there is increasing evidence that models reflect the biases in these datasets; on the other hand, the enormous scale of these datasets makes it extremely expensive to manually curate them—and so removal of such “harmful data” is challenging.
+
+In this paper, we propose an approach that aims to remove data responsible for biased model predictions. In particular, we focus on a specific way of quantifying model bias—worst-group error—which captures the extent to which model performance degrades on pre-defined subpopulations of the data. We aim to identify (and remove) the points in the training dataset that contribute most to this metric to improve the model's group robustness.
+
+The challenge inherent in this approach is that it requires an understanding of how training data affect machine learning model predictions. To overcome this challenge, we first approximate predictions as simple, direct functions of the training dataset, using a framework called datamodeleling [18, 33]. We can then write our quantitative notion of model bias (which is a function of predictions) as a function of the dataset. Finally, by studying this function, we identify the training data points that contribute most to this measure of model bias. With the resulting method, which we call Data Debiasing with Datamodels (D3M), we show that, across a variety of datasets, there are often a small number of examples that disproportionately drive worst-group error. Removing these examples, in turn, greatly improves models' worst-group error while maintaining dataset size.
+
+Roadmap & contributions. In the rest of this paper, we present and demonstrate the effectiveness of our Data Debiasing with Datamodels (D3M). Concretely, we show that D3M enables us to:
+
+- Pinpoint examples that harm worst-group accuracy. We show that there are often a small number of examples that disproportionately drive models' worst-group error on validation data. For example, on CelebA-Age, our method improves worst group error over
+
+Training data correlation between class (species) and extra feature (color) leads to disparate performance.
+
+![](images/88d27006c8c5458d03ccf51eb75f001a383d115e67695ac0dc9336ce25ef947e.jpg)
+
+Goal: "debias" dataset to improve worst-group accuracy (WGA):
+
+$$
+\mathrm {W G A} = \min  _ {\text {g r o u p} \in \{\text {马}, \text {马}, \text {马}, \text {马} \}} \operatorname {A c c} (\text {g r o u p})
+$$
+
+Data balancing  $[IAP + 22]$  removes examples from majority groups to break this correlation.
+
+![](images/8d53cca88c2bd1a42271da2d12da0386370fc20f0ec862ef0b657502df0302d0.jpg)
+
++Only changes data -Can hurt average accuracy  
++Less accuracy gap -Needs training group labels
+
+# Our approach: Data Debiasing via Datamodeling (D3M)
+
+Compute impact of each training sample on WGA by predicting WGA as a function of dataset selection.
+
+Worst-group accuracy on a (small) validation set
+
+![](images/02984bbc543dd642a654e88d2d1f3b0ab41b5f736a73b33a71bc4e9c97624a9d.jpg)  
+Figure 1: Our method (D3M) improves worst group accuracy by identifying and removing the training samples which most negatively impact worst-group accuracy. Specifically, we use TRAK [33] to identify examples that exacerbate the discrepancy in group performance. We then remove and re-train a model on the remaining data.
+
+"Group alignment score" Learned coefficient; impact of point i on WGA
+
+Find selection that maximizes worst-group accuracy by removing most harmful examples.
+
+Group alignment scores
+
+![](images/9cc2cefb6300f0417a1d30b076b4f500659ffb0ef593331fab3c187ab85bd86a.jpg)
+
++Only changes data +Competitive accuracy  
++Less accuracy gap +Only needs test set labels
+
+a natural baseline (data balancing) while removing  $2.4 \times$  fewer examples. Furthermore, these offending examples often form coherent subpopulations within the data.
+
+- Achieve competitive debiasing performance. Our approach outperforms standard approaches (both model-based and data-based) to improving worst-group accuracy [25, 21, 17], and is able to match the performance of methods which use ground-truth training group annotations [40].  
+- Discover unlabeled biases. When validation group labels are unavailable, we show how to extract hidden biases (i.e., unlabeled subgroups) directly from the data. As a result, we can perform end-to-end debiasing without any group annotations.
+
+We present our method in Section 4, and demonstrate these capabilities in Section 5. In Section 6, we leverage our framework to discover and mitigate biases within the ImageNet dataset, where D3M surfaces coherent color and co-occurrence biases. We then debias the model according to these failures, and improve accuracy on the identified populations.
+
+# 2 The group robustness problem
+
+We consider an (unobserved) data distribution  $\mathcal{D}$  over triplets  $(x_i, y_i, g_i)$ , each comprising an input  $x_i \in \mathcal{X}$ , a label  $y_i \in \mathcal{Y}$ , and a subgroup label  $g_i \in \mathcal{G}$ , where  $\mathcal{G}$  is the set of distinct subpopulations in the data. As a running example, consider the CelebA age classification task—here, we take the inputs  $x_i$  to be images of faces, the labels  $y_i$  to be either "old" or "young," and the possible group labels to be "old man", "old woman", "young man", and "young woman" (see Figure 1).
+
+Given a training dataset  $S_{\mathrm{train}}$  and a (small) validation dataset  $S_{\mathrm{val}}$ , the goal of the group robustness problem is to produce a classifier  $f$  that minimizes the worst-case loss over groups, i.e.,
+
+$$
+\max  _ {g ^ {\prime} \in \mathcal {G}} \mathbb {E} _ {(x, y, g) \sim \mathcal {D}} \left[ \ell (f (x), y) \mid g = g ^ {\prime} \right], \tag {1}
+$$
+
+where  $\ell (\cdot ,\cdot)$  is a loss function. When  $\ell$  is the 0-1 loss, Equation (1) is (one minus) the worst-group accuracy (WGA) of the classifier  $f$ , which we use to quantify success in the remainder of this work.
+
+Standard loss minimization can yield models that perform poorly with respect to (1). For instance, returning to our example of CelebA age classification, suppose there was a spurious correlation between age and gender in the training set  $S_{\mathrm{train}}$ , such that old men and young women are overrepresented. A predictor that minimizes loss on  $S_{\mathrm{train}}$  might leverage this correlation, and thus perform poorly on the underrepresented subgroups of old women or young men.
+
+In practice, subgroup labels  $g_{i}$  can be expensive to collect. Thus, approaches to the subgroup robustness problem vary in terms of whether we observe the group label  $g_{i}$  in the training set  $S_{\mathrm{train}}$  and in the validation set  $S_{\mathrm{val}}$ . In particular, there are three settings of interest:
+
+- Full-information (Train  $\sqrt{}$  Val  $\sqrt{}$ ): We observe the group labels for both the training dataset  $S_{\mathrm{train}}$  and validation dataset set  $S_{\mathrm{val}}$ .  
+- Partial-information (Train X/ Val  $\checkmark$ ): We observe the group labels for the validation set  $S_{\mathrm{val}}$ , but not for the (much larger) training set  $S_{\mathrm{train}}$ .  
+- No-information (Train  $X / \text{Val } X$ ): We do not have group information for either  $S_{train}$  or  $S_{\text{val}}$ . Note that theoretically this setting is unsolvable, since for any non-perfect classifier  $f$ , there exists an assignment of group labels so that the worst-group accuracy is zero. Nevertheless, subgroups of relevant practical interest typically have structure that allows for non-trivial results even with no information.
+
+In this work, we focus on the partial-information and no-information settings, since acquiring group labels for the entire training set is often prohibitively expensive. Still, in Section 5, we show that our proposed methods (D3M for the partial-information setting, and AUTO-D3M for the no-information setting) perform comparably to full-information approaches.
+
+# 3 Related work
+
+Before introducing our method, we discuss a few related lines of work.
+
+Approaches to subgroup robustness. The group robustness problem (Section 2) has attracted a wide variety of solutions (see, e.g., [3, 20, 40, 25, 21, 38]). Broadly, these solutions fall into one of two categories—model interventions and data interventions. Model interventions target either model weights [42, 45] or the training procedure [40, 21]. Data interventions, on the other hand, seek to improve worst-group accuracy by modifying the training dataset. For example, data balancing removes or subsamples examples so that all subgroups are equally represented. Idrissi et al. [17] find that this simple approach can perform on par with much more intricate model intervention methods.
+
+In this work, we focus on data interventions, for two reasons. First, it is often training data that drives models' disparate performance across groups [27], e.g., via spurious correlations [32] or underrepresentation [6]. Second, data interventions do not require any control over the model training procedure, which can make them a more practical solution (e.g., when using ML-as-a-service). Indeed, since data intervention approaches only manipulate the dataset, they are also easy to combine with model intervention techniques.
+
+Compared to our work, the main drawback of existing data interventions is that they often (a) require subgroup labels for the training data (which might not be available), and (b) hurt the models' natural accuracy on skewed datasets [7, 44]. In this work we circumvent these limitations, by proposing a data-based approach to debiasing that can preserve natural accuracy without access to subgroup information.
+
+Bias discovery. Another related line of work identifies biases in machine learning datasets and algorithms. For the former, previous works have shown that large, uncurated datasets used for training machine learning models often contain problematic or biased data [4, 5, 48]. Raji et al. [39] show that data bias can be a hurdle towards deploying functional machine learning models. Nadeem et al. [29] curate a dataset to estimate bias in NLP models. Adebayo et al. [1] show that label errors can disproportionately affect disparity metrics.
+
+On the learning algorithm side, Shah et al. [47] and Puli et al. [37] show that the inductive bias of neural networks may encourage reliance on spurious correlations. Pezeshki et al. [34] leverage two networks trained on random splits of data while imitating confident held-out mistakes made by its
+
+sibling to identify the bias. Shah et al. [46] show that algorithmic design choices (e.g., the choice of data augmentation) can significantly impact models' reliance on spurious correlations. Finally, there has been a variety of work on "slice discovery" [19, 10], where the goal is to discover systematic errors made by machine learning models.
+
+Data selection for machine learning. Our work uses data selection to improve subgroup robustness of machine learning models. A recent line of work has explored data selection for improving various measures of model performance. For example, Engstrom et al. [9] leverage datamodeling [18, 33] to select pretraining data for LLMs. Similarly, Xia et al. [52] and Nguyen and Wong [31] select data for finetuning, and in-context learning, respectively. In another related work, Wang et al. [49] propose a method to reweight training data in order to improve models' fairness.
+
+Many of these works leverage contributive data attribution methods to select data that improves model performance [51]. Koh and Liang [22], Feldman and Zhang [11], Schioppa et al. [43], and Hammoudeh and Lowd [15] propose using different variants of influence functions. Ghorbani and Zou [14] leverage connections to Shapley values, a concept from game theory. Pruthi et al. [36] use a heuristic to estimate the contribution of each data point to model performance. Ilyas et al. [18] and Park et al. [33] use the datamodeling framework.
+
+# 4 Debiasing datasets with datamodeling (D3M)
+
+In this section, we present our data-based approach to training debiased classifiers. The main idea behind our approach is to identify (and remove) the training samples that negatively contribute to the model's worst-group accuracy, by writing model predictions as a function of the training data.
+
+Preliminaries. Let  $S = \{(x_1, y_1), \ldots, (x_n, y_n)\}$  be a dataset of input-label pairs. For any subset of the dataset—as represented by indices  $D \subset [n]$ —let  $\theta(D) \in \mathbb{R}^p$  be the parameters of a classifier trained on  $D$ . Given an example  $z = (x, y)$ , let  $f(z; \theta)$  be the correct-class margin on  $z$  of a classifier with parameters  $\theta$  (defined as  $\log\left(\frac{p}{1 - p}\right)$ , where  $p$  is the confidence assigned to class  $y$  for input  $x$ ).
+
+A datamodel for the example  $z$  is a simple function that predicts  $f(z; \theta(D))$  directly as a function of  $D$ , i.e., a function  $\hat{f}_z: 2^{[n]} \to [0,1]$  such that
+
+$$
+\hat {f} _ {z} (D) \approx f (z; \theta (D)) \quad \text {f o r} D \subset [ n ].
+$$
+
+Recent works (e.g., [18, 24, 33]) demonstrate the existence of accurate linear datamodels—functions  $\hat{p}$  that decompose additively in terms of their inputs  $D$ . In other words, these works show that one can compute example-specific vectors  $\tau(z) \in \mathbb{R}^n$  such that
+
+$$
+\hat {f} _ {z} (D) := \sum_ {i \in D} \tau (z) _ {i} \approx f (z; \theta (D)). \tag {2}
+$$
+
+The coefficients  $\tau (z)_i$  have a convenient interpretation as quantifying the "importance" of the  $i$ -th training sample to performance on example  $z$  (i.e., as a data attribution score [15]). In what follows, we will assume access to coefficients  $\tau (z)$  for any example  $z$ —at the end of this section, we will show how to actually estimate the coefficient vectors  $\tau (z)$  efficiently.
+
+Debiasing approach. How can we leverage datamodeling to debias a dataset? Recall that our goal is to remove the samples in  $S$  that lead to high worst-group error. Stated differently, given a dataset  $S$  of size  $n$ , we want to maximize the worst-group performance of a classifier  $\theta(D)$  with respect to the indices  $D \subset [n]$  that we train on.
+
+Our main idea will be to approximate the predictions of  $\theta(D)$  using the corresponding datamodels  $\hat{f}_z(D)$ . To illustrate this idea, suppose that our goal was to maximize performance on a single test example  $z$ , i.e.,  $\arg \max_D f(z; \theta(D))$ . We can approximate this goal as finding  $\arg \max_D \hat{f}_z(\theta(D))$ : then, due to the linearity of the datamodel  $\hat{f}_z$ , the training samples that hurt performance on  $z$  are simply the bottom indices of the vector  $\tau(z)$ .
+
+Now, this analysis applies not only to a single example  $z$ , but to any linear combination of test examples. In particular, if we wish to maximize performance on a linear combination of validation examples, we simply take the linear combination of their coefficients, and remove the training examples corresponding to the smallest coordinates of the averaged vector.
+
+Debiasing with group-annotated validation data. Given a set of validation samples for which the group labels  $g_{i}$  are observable, our last observation gives rise to the following simple procedure:
+
+1. Compute group coefficients  $\tau(G)$  for each  $G$ . Since we have group annotations for each validation sample, we can define a vector  $\tau(G)$  for each group  $G \in \mathcal{G}$  as simply the average  $\tau(z)$  within each group.  
+2. Compute group alignment. Next, we compute a group alignment score  $A_{i}$  for each training sample  $i \in [n]$ , which captures the impact of the sample on worst-group performance. Since there may be many low-performing groups, we use a "smooth maximum" function to weight each group according to its average loss. Thus, for a training example  $i$ ,
+
+$$
+A _ {i} = \frac {\sum_ {g \in \mathcal {G}} \exp (\beta \ell_ {g}) \cdot \tau (g) _ {i}}{\sum_ {g ^ {\prime} \in \mathcal {G}} \exp (\beta \ell_ {g ^ {\prime}})}, \text {w h e r e w e s e t h y p e r p a r a m e t e r} \beta = 1. \tag {3}
+$$
+
+Here,  $\ell_g$  is the loss of a base classifier  $\theta(S)$  on group  $g$  (evaluated on the given validation set).
+
+3. Remove drivers of bias. Finally, we construct a new training set  $S_{\mathrm{new}}$  by keeping only the examples with the highest group alignment scores, i.e., removing the examples that most degrade worst-group accuracy:
+
+$$
+S _ {\text {n e w}} = \arg \operatorname {t o p - k} \left(\left\{A _ {i}: z _ {i} \in S _ {\text {t r a i n}} \right\}\right).
+$$
+
+We make two brief observations about hyperparameters before continuing. When computing the group alignment score in Step 2, the hyperparameter  $\beta$  controls the temperature of the soft maximum function in (3). When  $\beta \rightarrow 0$ , the group alignment  $A_{i}$  measures the impact of the  $i$ -th training example on the "balanced" performance (treating all groups equally). As  $\beta \rightarrow \infty$ ,  $A_{i}$  collapses to the training example's importance to only the worst group, which is suboptimal if models perform poorly on more than one group. For simplicity, we take  $\beta = 1$  and refrain from tuning it.
+
+Another hyperparameter in the algorithm above is the number of examples to remove,  $k$ . We consider two different ways of setting this hyperparameter. One approach is to search for the value of  $k$  that maximizes worst-group accuracy on the validation set  $S_{\mathrm{val}}$ . Alternatively, we find that the simple (and much more efficient) heuristic of removing all examples with a negative group alignment score (i.e., examples for which  $A_{i} < 0$ ) tends to only slightly over-estimate the best number of examples to remove (see, e.g., Figure 2). Thus, unless otherwise stated, we use this heuristic when reporting our results.
+
+Debiasing without group annotations. Our procedure above relies on group annotations for a validation set  $S_{\mathrm{val}}$  to compute the "per-group coefficients"  $\tau(G)$ . In many real-world settings, however, models might exhibit disparate performance along unannotated subpopulations—in this case, we might not have a validation set on which we can observe group annotations  $g_i$ . Can we still fix disparate model performance in this setting?
+
+In general, of course, the answer to this question is no: one can imagine a case where each individual example is its own subpopulation, in which case worst-group accuracy will be zero unless the classifier is perfect. In practical settings, however, we typically care about the model's disparate performance on coherent groups of test examples. The question, then, becomes how to find such coherent groups.
+
+We posit that a unifying feature of these subpopulations is that they are data-isolated, i.e., that models' predictions on these coherent groups rely on a different set of training examples than models' predictions on the rest of the test data. Conveniently, prior works [18, 46] show that to find data-isolated subpopulations, one can leverage the datamodel matrix—a matrix constructed by stacking the datamodel vectors  $\tau(z)$  for each test example. Intuitively, the top principal component of this matrix encodes the direction of maximum variability among the vectors  $\tau(z)$ . Thus, by projecting the datamodel vectors  $\tau(z)$  of our validation examples onto this top principal component, we can identify the examples that are, in a sense, "maximally different" from the rest of the test examples in terms of how they rely on the training set. These maximally different examples correspond to an isolated subpopulation, to which we can apply D3M directly.
+
+This approach (which we call AUTO-D3M), enables us to perform end-to-end debiasing without any group annotations. This method proceeds in four steps. For each class:
+
+1. Construct a matrix  $\mathbf{T}$  of stacked attribution vectors, where  $\mathbf{T}_{ij} = \tau (z_i)_j$  
+2. Let  $\pmb{v}$  be the top principal component of  $\mathbf{T}$ .  
+3. Project the attribution vector  $\tau(z)$  onto  $\pmb{v}$  and construct "group pseudo-labels"
+
+$$
+g _ {i} = \mathbf {1} \{\tau (z _ {i}) ^ {\top} \boldsymbol {v} \geq \lambda \}.
+$$
+
+where  $\lambda$  is a hyperparameter
+
+4. Apply D3M with the group pseudo-labels to train a debiased classifier.
+
+Estimating the coefficients  $\tau(z)$ . In order to operationalize D3M and AUTO-D3M, it remains to show that we can actually estimate coefficients  $\tau(z)$  satisfying (2). To accomplish this, we use a method called TRAK [33]. Leveraging differentiability of the model output  $f(z; \theta)$  with respect to the model parameters  $\theta$ , TRAK computes the coefficient vector  $\tau(z)$  for an example  $z$  as follows:
+
+(a) Train a model  $\theta^{*} \coloneqq \theta(S)$  on the entire training dataset  $S = \{z_{1},\dots ,z_{n}\}$ .  
+(b) Sample a random Gaussian matrix  $\mathbf{P} \in \mathbb{R}^{p \times k}$  where  $p$  is the dimensionality of  $\theta^*$  (i.e., the number of model parameters) and  $k$  is a hyperparameter;  
+(c) For an example  $z$ , define  $g(z) \coloneqq \mathbf{P}^\top \nabla_\theta f(z; \theta^*)$  as the randomly-projected model output gradient (with respect to the model parameters) evaluated at  $z$ .  
+(d) Compute the coefficient vector
+
+$$
+\underbrace{\tau(z)_{i}}_{i\text{-th coefficient for example} z} = g(z)^{\top}\left(\sum_{z_{j}\in S}g(z_{j})\cdot g(z_{j})^{\top}\right)^{-1}g(z_{i})\cdot \left(1 - \sigma (f(z;\theta^{*}))\right)
+$$
+
+(e) Repeat steps (a)-(d) for  $T$  trials, and average the results to get a final coefficient vector  $\tau(z)$ . The trials are identical save for the randomness involved in step (a).
+
+We provide more intuition and details behind TRAK in Appendix A.
+
+A note on scalability In terms of computational cost, TRAK involves taking a single backward pass on each of the training and validation examples to compute the model's gradient. The (projected) gradients are then saved to compute TRAK scores. Typically, TRAK is computed over  $T$  trials: following the original paper we use  $T = 100$ . However, our approach can be used with any datamodeling technique.
+
+# 5 Results
+
+In Section 4, we presented D3M—an approach for debiasing a classifier by identifying examples which contribute to a targeted bias. In this section, we validate this framework by assessing its performance on tasks with known biases.
+
+We consider four classification tasks where there is a spurious correlation between the target label and a group label in the training dataset: CelebA-Age [26, 19], CelebA-Blond [26], Waterbirds [41], and MultiNLI [50]. We provide more information about the datasets in Appendix B.1, and other experimental details in Appendix B.2.
+
+# 5.1 Quantitative results
+
+We first evaluate D3M and AUTO-D3M quantitatively, by measuring the worst-group accuracy of models trained on the selected subsets of the biased datasets above.
+
+D3M: Debiasing the model in the presence of validation group labels. In Table 1, we compare D3M against several baselines, each of which requires either only validation group labels  $(X / \sqrt{})$  or both training and validation group labels  $(\sqrt{} / \sqrt{})$ . We find that D3M outperforms all other methods that use the same group information (i.e., only validation group labels) on all datasets except Waterbirds<sup>3</sup>.
+
+<table><tr><td rowspan="2">Group Info Train / Val</td><td rowspan="2">Method</td><td colspan="4">Worst Group Accuracy (%)</td></tr><tr><td>CelebA-Age</td><td>CelebA-Blond</td><td>Waterbirds</td><td>MultiNLI</td></tr><tr><td rowspan="2">X/ X</td><td>ERM</td><td>56.7</td><td>45.9</td><td>57.9</td><td>67.2</td></tr><tr><td>AUTO-D3M (ours)</td><td>76.0</td><td>83.8</td><td>81.0</td><td>75.0</td></tr><tr><td rowspan="3">X/ ✓</td><td>JTT [25]</td><td>61.0</td><td>81.6</td><td>63.6</td><td>72.6</td></tr><tr><td>DFR* [21]</td><td>70.4</td><td>88.4</td><td>89.0</td><td>74.7</td></tr><tr><td>D3M (ours)</td><td>75.6</td><td>90.0</td><td>87.2</td><td>76.0</td></tr><tr><td rowspan="3">✓/ ✓</td><td>RWG [17]</td><td>75.6</td><td>88.4</td><td>81.2</td><td>68.4</td></tr><tr><td>SUBG [17]</td><td>68.5</td><td>88.3</td><td>85.5</td><td>67.8</td></tr><tr><td>GroupDRO [40]</td><td>74.8</td><td>90.6</td><td>72.5</td><td>77.7</td></tr></table>
+
+Table 1: Worst-group accuracies on four group robustness datasets. A * denotes methods that use validation group labels for both finetuning and hyperparameter tuning.
+
+![](images/19c3b0924d0fef0e14b509d0cd0d81fb3bd81409ea66832d802b631dd38f255c.jpg)  
+Figure 2: Worst group accuracy on CelebA-Age as a function of the number of examples  $k$  removed from the training set, using various removal methods. In green, D3M removes the  $k$  training examples with the most negative alignment scores  $A_{i}$ . The green star marks the value of  $k$  selected by our heuristic  $(A_{i} < 0)$ . In blue is the performance of a random baseline that removes  $k$  examples at random from the training set, and in orange is dataset balancing, which removes examples randomly from the majority group. Compared to baselines, D3M efficiently improves worst group accuracy.
+
+Moreover, D3M performs on par with methods that have full access to both training and validation group labels.
+
+AUTO-D3M: Discovering biases with TRAK. We now consider the case where validation group labels are not accessible. Using AUTO-D3M, we debias our model using pseudo-annotations derived from the top principal component of the TRAK matrix (AUTO-D3M in Table 1). Note that AUTO-D3M is the only method other than ERM that does not require either train or validation group labels. Despite this, AUTO-D3M achieves competitive worst-group accuracy in our experiments. We emphasize that AUTO-D3M does not require group labels at all—in particular, we do not use group labels to do hyperparameter selection or model selection when we retrain.
+
+The effect of the number of removed examples  $k$ . How well does D3M isolate the training examples that drive disparate performance? To answer this question, we iteratively remove training examples from CelebA-Age starting with the most negative  $A_{i}$  and measure the worst-group and balanced accuracy (See Figure 2). CelebA-Age has 40K "majority" examples and 10K "minority" examples; thus, naive balancing requires removing 30K training examples. In contrast, by isolating which specific majority examples contribute to the bias, our method is able to debias the classifier by removing only 10K examples.
+
+Our heuristic of removing examples with negative  $A_{i}$  (the star in Figure 2) slightly overestimates the best number of examples to remove. Thus, while this heuristic gives a decent starting point for  $k$ , actually searching for the best  $k$  might further improve performance.
+
+![](images/d4282e8cda779a1e51062970d8e5ec1f09e2a028ad566d196244e92be5f3cb03.jpg)  
+Figure 3: Randomly sampled examples from the subpopulations with the most negative group alignment scores. We find that many of these examples have labeling errors (e.g., platinum blond instead of gray hair.)
+
+![](images/1eafff118369079e18b04bf68d2ec0c21095584526bacdd71a7ad641c455fc99.jpg)  
+Figure 4: The average group alignment score of the training examples in each subpopulation of CelebA-Age. Subpopulations such as "old" with "bushy eyebrows" or "young" with "gray hair" have particularly negative scores.
+
+# 5.2 Qualitative results
+
+What type of data does our method flag? In particular, do the examples we identify as driving the targeted bias share some common characteristics? To test this hypothesis, we inspect the data removed by our method and identify subpopulations within the majority groups that are disproportionately responsible for the bias. We then retrain the model after excluding all training examples from the identified subpopulations and show that this is a viable strategy for mitigating the underlying bias.
+
+Finding subpopulations responsible for model bias. Consider the running example from Figure 1 where we train a model on the CelebA-Age dataset to predict whether a person is "young" or "old" when gender is a spurious feature (such that young women and old men are overrepresented). CelebA-Age has a variety of annotations beyond age and gender, such as whether the person is wearing eyeglasses. In this section, we use these extra annotations to identify coherent subpopulations that are flagged by our methods.
+
+In particular, we consider subpopulations formed by taking the Cartesian product of labels and annotations, e.g., subpopulations of the form ("young", "wearing eyeglasses"). For each of these subpopulations, we calculate the average group alignment score  $A_{i}$  of the training examples within that subpopulation (see Figure 4). We find that subpopulations such as "young" with "gray hair" or "old" with either "5 o'clock shadow" or "busy eyebrows" have particularly negative group alignment scores. In Figure 3, we show examples from the subpopulations with the most negative group alignment scores, and observe that a large fraction of the examples in these subpopulations contain labeling errors.
+
+Retraining without identified subpopulations. Once we have identified subpopulations with negative alignment scores, a natural strategy for mitigating the underlying bias is to exclude these subpopulations from the training set. To explore this approach, we exclude the five subpopulations with the most negative attribution scores on average from the Ce1ebA-Age dataset: "Young" + "Gray Hair", "Old" + "5 o'Clock Shadow", "Old" + "Bushy Eyebrows", "Young" + "Blond Hair", and "Old" + "Sideburns." After retraining the model on this modified training set, we get a worst-group accuracy (WGA) of  $68.4\%$  —an improvement of  $\sim 12\%$  over the WGA of the original model  $(56.7\%)$ .
+
+![](images/fc0f41c8cd75007cdd83df8e2666ecfe9f06165a810a4635b37442d14e37a9e8.jpg)  
+$(+)$
+
+![](images/fa057c7428edbdbbe1d49ef9a5769875466ca93becc45e1c2dd697ca5cc5be1f.jpg)  
+Figure 5: For four ImageNet classes, the most extreme (positive or negative) examples according to the top PCA direction of the TRAK matrix. Our method identifies color and co-occurrence biases.
+
+![](images/d74eaff9828666c24828b47a4040ba46e84b6877ef572e84198ce0aa44b65ffb.jpg)
+
+![](images/eba70e63ce22135cd38237a354001682b0d9c9a65a6ce58f33460d603bee9094.jpg)
+
+# 6 Case Study: Finding and Mitigating Model Failures on ImageNet
+
+In Section 5 we evaluated D3M and AUTO-D3M on datasets where the bias was already known. We now deploy AUTO-D3M to discover and mitigate biases within the ImageNet dataset, which does not have a predefined bias or available group annotations.
+
+Identifying ImageNet biases. We use TRAK to compute a coefficient matrix  $\mathbf{T}$  (see Step 1 of AUTO-D3M in Section 4) for a held out validation split (10% of the training set). Focusing on seven ImageNet classes, we use the first principal component of the matrix  $\mathbf{T}$  to identify potential biases. In Figure 5, we display the most extreme training examples according to the top principal component for four of these classes. PCA identifies semantically color and co-occurrence biases (e.g., tench fishes with or without humans or yellow/white cauliflowerers that are either cooked or uncooked.) In fact, our identified biases match the challenging subpopulations in Jain et al. [19] and Moayeri et al. [28].
+
+![](images/64e5d0237c73bc414c0c39f5f182a5d534e5df8f08b5c4b7b8ca1ff63c114164.jpg)  
+Figure 6: Worst-group accuracy for the ImageNet classes studied in Section 6 after intervening with either D3M or AUTO-D3M (error bars over 1 std).
+
+Mitigating ImageNet biases with AUTO-D3M. For each of the four targeted ImageNet classes, we seek to mitigate the identified failure modes with AUTO-D3M. We consider two settings based on the level of human intervention. In the first, we manually assign each of the validation images to a group according to a human description of identified bias (e.g., an image of a tench is in group 1 if a human is present and group 2 otherwise), and then use those group labels with D3M. In the second setting, we debias in a purely automatic fashion, using AUTO-D3M to derive pseudo-group labels from the top principal component. In Figure 6, we display worst group accuracy on the test images of the targeted class (evaluated using manual group assignments of the 50 test examples). Both D3M and AUTO-D3M improve worst group accuracy over ERM without significantly impacting the overall ImageNet accuracy (see Appendix C.2).
+
+# 7 Conclusion
+
+We propose Data Debiasing with Datamodels (D3M), a simple method for debiasing classifiers by isolating training data that disproportionately contributes to model performance on underperforming groups. Unlike approaches such as balancing, our method only removes a small number of examples and does not require training group annotations or additional hyperparameter tuning. More generally, our work takes a first step towards data-centric model debiasing.
+
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+[42] Shibani Santurkar, Dimitris Tsipras, Mahalaxmi Elango, David Bau, Antonio Torralba, and Aleksander Madry. "Editing a classifier by rewriting its prediction rules". In: Preprint. 2021.
+
+[43] Andrea Schioppa, Polina Zablotskaia, David Vilar, and Artem Sokolov. "Scaling up influence functions". In: Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 36. 8. 2022, pp. 8179-8186.  
+[44] Roy Schwartz and Gabriel Stanovsky. "On the limitations of dataset balancing: The lost battle against spurious correlations". In: arXiv preprint arXiv:2204.12708 (2022).  
+[45] Harshay Shah, Andrew Ilyas, and Aleksander Madry. "Decomposing and Editing Model Predictions". In: arXiv preprint. 2024.  
+[46] Harshay Shah, Sung Min Park, Andrew Ilyas, and Aleksander Madry. "ModelDiff: A Framework for Comparing Learning Algorithms". In: arXiv preprint arXiv:2211.12491. 2022.  
+[47] Harshay Shah, Kaustav Tamuly, Aditi Raghunathan, Prateek Jain, and Praneeth Netrapalli. "The pitfalls of simplicity bias in neural networks". In: Advances in Neural Information Processing Systems 33 (2020), pp. 9573-9585.  
+[48] David Thiel. Identifying and eliminating CSAM in generative ML training data and models. Tech. rep. 2023.  
+[49] Haonan Wang, Ziwei Wu, and Jingrui He. "FairIF: Boosting Fairness in Deep Learning via Influence Functions with Validation Set Sensitive Attributes". In: Proceedings of the 17th ACM International Conference on Web Search and Data Mining. 2024, pp. 721-730.  
+[50] Adina Williams, Nikita Nangia, and Samuel R Bowman. "A broad-coverage challenge corpus for sentence understanding through inference". In: arXiv preprint arXiv:1704.05426 (2017).  
+[51] Theodora Worledge, Judy Hanwen Shen, Nicole Meister, Caleb Winston, and Carlos Guestrin. "Unifying corroborative and contributive attributions in large language models". In: arXiv preprint arXiv:2311.12233 (2023).  
+[52] Mengzhou Xia, Sadhika Malladi, Suchin Gururangan, Sanjeev Arora, and Danqi Chen. "Less: Selecting influential data for targeted instruction tuning". In: arXiv preprint arXiv:2402.04333 (2024).
+
+# A Background on Data Attribution and TRAK
+
+Let  $\mathcal{Z}$  be the input space and  $S$  be a training set of interest. For a given training subset  $S' \subset S$  and target example  $z$ , data attribution techniques seek to estimate the attribution score of  $z$  — that is, the change in the model's prediction on  $z$  when the model is trained on the subset  $S'$ . More formally, let  $f(z, \theta(S))$  be the model's output function on example  $z$ . Then we can define  $\tau(z)_i$  as the attribution score of the  $i$ th training example  $z_i$  on target example  $z$
+
+$$
+\tau (z) _ {i} = f (z, \theta (S)) - f (z, \theta (S \backslash z _ {i})).
+$$
+
+While  $\tau(z)$  is relatively straightforward to compute for linear models [35], computing this influence is far more challenging for neural networks. Thus, in order to approach this problem, TRAK first approximates  $f(z; \theta(S))$  as a linear model on top of the gradients  $\nabla_{\theta}(f; \theta^{*})$  of the original neural network after convergence. We can then plug this approximation into the estimate for linear classifiers to approximate  $\tau(z)$ . After simplification, the TRAK estimate of the influence of  $z$  is
+
+$$
+\boldsymbol {\tau} (z) = - \phi (z) ^ {T} (\Phi^ {T} \Phi) ^ {- 1} \Phi \mathbf {Q}
+$$
+
+where  $\phi(z) = \nabla f(z; \theta(S^{*}))$  are the (randomly projected) gradients on example  $z$ ,  $\Phi$  are the stacked training gradients  $\Phi = [\phi(z_1), \dots, \phi(z_n)]$ , and  $\mathbf{Q}$  is a normalization matrix.
+
+# B Details of Experiments
+
+# B.1 Experimental Setup
+
+In this section, we describe the datasets, models and evaluation procedure that we use throughout the paper.
+
+Datasets. In order to cover a broad range of practical scenarios, we consider the following image classification and text classification problems.
+
+- Waterbirds [40] is a binary image classification problem, where the class corresponds to the type of the bird (landbird or waterbird), and the background is spuriously correlated with the class. Namely, most landbirds are shown on land, and most waterbirds are shown over water.  
+CelebA-Blond [26] is a binary image classification problem, where the goal is to predict whether a person shown in the image is blond; the gender of the person serves as a spurious feature, as  $94\%$  of the images with the "blond" label depict females.  
+CelebA-Age [26, 19] is a binary image classification problem, where the goal is to predict whether a person shown in the image is young; the gender of the person serves as a spurious feature. For this task, we specifically subsample the training set such that the ratio of samples in the majority vs. minority groups is 4:1.  
+- MultiNLI [50, 40] is a classification problem where given a pair of sentences, the task is to classify whether the second sentence is entailed by, neural with, or contradicts the first sentence. The spurious attribute from Sagawa et al. [40] describes the presence of negation words, which appear more frequently in the examples from the negation class.
+
+Methods. We benchmark our approach against the following methods:
+
+- ERM is simple empirical risk minimization on the full training set.  
+- RWG [17] is ERM applied to random batches of the data where the groups are equally represented with a combination of upsampling and downsampling such that the size of the dataset does not change.  
+- SUBG [17] is ERM applied to a random subset of the data where we subsample all groups such that they have the same number of examples.  
+- GroupDRO [40] trains that minimizes the worst-case performance over pre-defined groups in the test dataset.
+
+- Just Train Twice (JTT) [25] trains an ERM model with upsampling initially misclassified training examples by an initial ERM model.  
+- DFR [21] trains an ensemble of linear models on a balanced validation set, given ERM features.
+
+# B.2 Training Details
+
+In this section, we detail the model architectures and hyperparameters used by each approach. We used the same model architecture across all approaches: Randomly initialized ResNet-18 [16] for CelebA and ImageNet-pretrained ResNet-18s for Waterbirds. We use the GroupDRO implementation by Sagawa et al. [40] and DFR implementation by Kirichenko et al. [21].
+
+For all approaches, we tune hyperparameters for ERM-based methods (ERM, DFR, and D3M) and re-weighting based methods (RWG, SUBG, GroupDRO and JTT) separately. For RWG, SUBG, GroupDRO and JTT, we early stop based on highest worst-group accuracy on the validation set as well. We optimize all approaches with Adam optimizer.
+
+For the CelebA dataset, we all methods with learning rate  $1e - 3$ , weight decay  $1e - 4$ , and batch size 512. We train RWG, SUBG, GroupDRO and JIT with learning rate  $1e - 3$ , weight decay  $1e - 4$ , and batch size 512. We train all models for the CelebA-Age task to up to 5 epochs and all models for CelebA-Blond task up to 10 epochs.
+
+For the Waterbirds dataset, we train the approaches that use the ERM objective (including D3M) with learning rate  $1e - 4$ , weight decay  $1e - 4$ , and batch size 32. We train RWG, SUBG, GroupDRO and JTT with learning rate  $1e - 5$ , weight decay 0.1, and batch size 32. We train all models to up to 20 epochs.
+
+For all other hyperparameters, we use the same hyperparameters as Kirichenko et al. [21] for DFR and the same hyperparameters as Liu et al. [25] for JTT.
+
+We report the performance of the models via Worst-group Accuracy, or Balanced Accuracy in Table 2, which is the average of accuracies of all groups. If all groups in the test set have the same number of examples, balanced accuracy will be equivalent to average accuracy.
+
+Our model was trained on a machine with 8 A100 GPUs.
+
+# C Omitted Results
+
+# C.1 Balanced Accuracies
+
+Below we include the balanced accuracies for the experiments in Table 2.
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Group Info Train / Val</td><td colspan="2">CelebA-Age</td><td colspan="2">CelebA-Blond</td><td colspan="2">Waterbirds</td><td colspan="2">MultiNLI</td></tr><tr><td>Balanced Accuracy</td><td>Worst Group Accuracy</td><td>Balanced Accuracy</td><td>Worst Group Accuracy</td><td>Balanced Accuracy</td><td>Worst Group Accuracy</td><td>Balanced Accuracy</td><td>Worst Group Accuracy</td></tr><tr><td>ERM</td><td>X/ X</td><td>77.96</td><td>56.65</td><td>82.59</td><td>45.86</td><td>83.40</td><td>57.85</td><td>80.92</td><td>67.19</td></tr><tr><td>Auto-TRAK (ours)</td><td>X/ X</td><td>80.05</td><td>75.97</td><td>91.01</td><td>83.77</td><td>90.36</td><td>81.04</td><td></td><td></td></tr><tr><td>RWG [17]</td><td>✓/✓</td><td>80.66</td><td>75.64</td><td>90.42</td><td>88.40</td><td>86.51</td><td>81.21</td><td>78.61</td><td>68.41</td></tr><tr><td>SUBG [17]</td><td>✓/✓</td><td>77.57</td><td>68.49</td><td>91.30</td><td>88.26</td><td>86.97</td><td>85.46</td><td>73.64</td><td>67.76</td></tr><tr><td>GroupDRO [40]</td><td>✓/✓</td><td>80.88</td><td>74.80</td><td>91.83</td><td>90.61</td><td>86.51</td><td>72.47</td><td>81.4</td><td>77.7</td></tr><tr><td>JTT [25]</td><td>X/✓</td><td>68.06</td><td>60.95</td><td>92.01</td><td>81.61</td><td>85.24</td><td>63.61</td><td>78.6</td><td>72.6</td></tr><tr><td>DFR [21]</td><td>X/✓✓</td><td>80.69</td><td>70.37</td><td>91.93</td><td>88.40</td><td>90.89</td><td>88.96</td><td>82.1</td><td>74.7</td></tr><tr><td>TRAK (ours)</td><td>X/✓</td><td>81.05</td><td>75.55</td><td>91.08</td><td>90.03</td><td>91.46</td><td>87.15</td><td>81.54</td><td>75.46</td></tr></table>
+
+Table 2: Balanced accuracy and worst-group accuracy on CelebA-Age, CelebA-Blond, and Waterbirds. A double checkmark (✓✓) indicates that the method uses validation group labels for model finetuning, in addition to hyperparameter tuning.
+
+# C.2 ImageNet Accuracies
+
+Below we included the detailed accuracies for the ImageNet experiment.
+
+<table><tr><td rowspan="2">Class (bias)</td><td rowspan="2">Method</td><td colspan="2">Class-Level</td><td rowspan="2">ImageNet-Level Overall Accuracy</td></tr><tr><td>Balanced Accuracy</td><td>Worst Group Accuracy</td></tr><tr><td rowspan="3">Red Wolf (Red Coat)</td><td>ERM</td><td>46.87</td><td>22.62</td><td>63.97</td></tr><tr><td>TRAK</td><td>65.63</td><td>52.38</td><td>63.71</td></tr><tr><td>Auto-TRAK</td><td>59.94</td><td>39.29</td><td>63.87</td></tr><tr><td rowspan="3">Tench (Presence of human)</td><td>ERM</td><td>85.10</td><td>78.12</td><td>63.97</td></tr><tr><td>TRAK</td><td>90.73</td><td>86.88</td><td>63.84</td></tr><tr><td>Auto-TRAK</td><td>86.67</td><td>80.00</td><td>63.97</td></tr><tr><td rowspan="3">Cauliflower (Not Cooked)</td><td>ERM</td><td>77.81</td><td>63.64</td><td>63.97</td></tr><tr><td>TRAK</td><td>85.77</td><td>79.55</td><td>63.70</td></tr><tr><td>Auto-TRAK</td><td>86.73</td><td>79.40</td><td>63.75</td></tr><tr><td rowspan="3">Strawberry (Not on a plate)</td><td>ERM</td><td>58.93</td><td>35.58</td><td>63.97</td></tr><tr><td>TRAK</td><td>70.49</td><td>51.92</td><td>63.88</td></tr><tr><td>Auto-TRAK</td><td>68.99</td><td>50.48</td><td>63.79</td></tr></table>
+
+Table 3: Auto-TRAK identifies and mitigates biases in ImageNet. For four ImageNet classes, a bias was identified from inspecting the TRAK PCA directions. Then Auto-TRAK is applied in order to mitigate the bias for that class. Auto-TRAK is able to improve the worst group accuracy for the targeted class without significantly changing the overall ImageNet accuracy.
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
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+
+Guidelines:
+
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+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
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+
+# 3. Theory Assumptions and Proofs
+
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+
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+
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+
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+
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+
+# 7. Experiment Statistical Significance
+
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+# 8. Experiments Compute Resources
+
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+# 10. Broader Impacts
+
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+
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+
+# 14. Crowdsourcing and Research with Human Subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: we do not involve human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
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+
+# 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: we do not involve human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
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2024/Improving Subgroup Robustness via Data Selection/images.zip

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