Title: Probing Equivariance and Symmetry Breaking in Convolutional Networks
Venue: NeurIPS 2025
Accepted: yes
Keywords: convolution networks

geometric deep learning

equivariance

symmetry-breaking

generative modeling
Abstract: In this work, we explore the trade-offs of explicit structural priors, particularly group-equivariance. We address this through theoretical analysis and a comprehensive empirical study focusing on point clouds. To enable controlled and fair comparisons, we introduce \texttt{Rapidash}, a unified group convolutional architecture that allows for different variants of equivariant and non-equivariant models. Our results suggest that more constrained equivariant models outperform less constrained alternatives when aligned with the geometry of the task, and increasing representation capacity does not fully eliminate performance gaps. We see improved performance of models with equivariance and symmetry-breaking through tasks like segmentation, regression, and generation across diverse datasets. Explicit \textit{symmetry breaking} via geometric reference frames consistently improves performance, while \textit{breaking equivariance} through geometric input features can be helpful when aligned with task geometry. Our results provide task-specific performance trends that offer a more nuanced way for model selection.
Code available at [github.com/Sharvaree/EquivarianceStudy](https://github.com/Sharvaree/EquivarianceStudy)


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces an architecture with flexible symmetry parametrisation based on regular group convolutions on R^3xS_2.\n\nThe paper then investigates incorporating various levels of symmetry when solving a number of tasks that may benefit from it.\n\nThe paper draws a number of conclusions about the importance of symmetry in this architecture, data scaling, model capacity scaling, and symmetry breaking.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The evaluation if reasonably well conceived, although the mix of tasks could be wider to disambiguate effects over multiple datasets to disentangle and dataset specific effects.\n- The layout of hypotheses are clear\n\nWeaknesses:\n- The authors seem to miss a key question regarding equivariant models in their analysis, eluded to in the introduction. Namely, that when equivariance is removed  from an architecture it is typically replaced with data augmentation. Without this inclusion I find the conclusions of  RQ1-3 tricky to conclude properly.\n- The authors only look at a single architecture. The conclusions seem unlikely to hold across all classes of model to me. In particular, with irregular representation convolutions vs regular representation convolutions, the additional complexity in parametrisation and runtime cost seem likely to significantly change the results of the expreiments.\n\nQUESTIONS:\n1. Do the authors think that adding data augmentation to the non-equivariant or reduced equivariant architectures would make a difference to the results? If so, should this be included in the ablation studies?\n2. Do the author expect these conclusion to hold across other types of equivariant architectures? Open source code exist for a number of other equivariant architectures, e.g. https://github.com/QUVA-Lab/escnn, https://github.com/vgsatorras/egnn, which could easily be adapted to the reduced symmetry and symmetry breaking settings of the paper, should comparisons to these alterative architeture types be included to make the conclusions of the paper more substantial?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nWhile equivariant kernels ensure generalization to specified groups, this comes at the cost of potentially over constrained networks on imperfectly symmetric datasets, generally lower network capacity, and higher compute requirements for a fixed parameter count. This has led to several papers that investigate the utility of equivariance by design on disparate tasks and datasets, some of which find benefits, whereas others do not. As a result, the literature currently conflicts with itself in terms of relevance to real world datasets and a unified analysis is needed. \n\nThis submission specifically pertains to deep learning on point clouds and provides a group-equivariant framework that enables isolating various key decisions (e.g., input feature types, equivariance constraints, etc.) such that they can be studied piece-by-piece. The paper also presents some new theoretical results to clarify when and how symmetry breaking and equivariance are useful depending on the task. Lastly, experimentally, it covers a broad set of tasks and domains on toy small-scale datasets such as ShapeNet, QM9, etc.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths:\n- The problem space of \"*does equivariance by design matter*\" is gigantic in scope and this paper does an admirable job in trying to construct a unifying framework for empirical investigation across a wide set of tasks. \n- Section 2, background is very well done and accessible to a broad audience.\n- New theoretical results on when breaking equivariance and symmetry help in generalization (and when they do not).\n\n## Weaknesses:\n### Over-claimed scope and toy experiments:\nThe paper claims more within its scope than it actually does. The paper primarily pertains to point clouds in all of its empirical investigations (which are its core contribution), yet the framing is left as general-purpose as possible, which IMO can be misleading. As an example, [Gruver&Finzi](https://openreview.net/forum?id=JL7Va5Vy15J) show that many of the statements made here do not translate to large-scale natural image-based vision where large networks without inductive bias trained on large datasets with extensive augmentation are practically *more equivariant* than equivariant-by-design architectures. This paper should revise itself to clarify its scope and which of its findings have the potential to translate across fields.\n\nMore broadly, the paper does not get to the true tension of the field: if equivariant architectures worked at scale across tasks, they would be used at scale across tasks by the community. There are some theoretical results provided, but all of the experiments are performed on simple toy datasets such as ShapeNet and QM9 wherein it is very easy to show great results that then do not translate to real tasks. I believe that the theoretical results should be validated on at least one large-scale real-world dataset with *well tuned* baselines used as a point of comparison.\n\n### Unclear contributions:\nIt is hard to say definitively what to take away from the paper. The theoretical framework, Rapidash, is a multiscale extension of Ponita in [Bekkers, 2024](https://arxiv.org/pdf/2310.02970) (I suppose this naming scheme makes sense if you have a pokedex :), but this is never explicitly stated in the paper and it is claimed as an entirely new framework. However, many of its theoretical properties are inherited from Ponita. As a result, a significant portion of the methods section recaps properties that are originally specific to Ponita (e.g., Sec 3.1) and it is hard to isolate what is truly specific to Rapidash. Also, this raises the burden of empirical completeness to make a strong contribution, in my opinion.\n\n### Missing analysis of augmentation and regularization:\nThe paper's analysis misses key components of performance such as augmentation types and strengths and regularization via techniques such as weight decay. Augmentation is not mentioned within the paper and in practice is key to networks learning equivariance in practice ([Gruver&Finzi](https://openreview.net/forum?id=JL7Va5Vy15J)). It is plausible that the corresponding non-equivariant networks require significantly more regularization and/or augmentation strength (with more training steps) to lower-the-margin-to or even exceed their equivariant counterparts on these toy point cloud datasets.\n\nPlease detail the types, strengths, and probabilities of the augmentations and regularizations used, and whether they were held identical for both equivariant and non-equivariant networks. If they were, please perform a few experiments where the non-equivariant networks are well tuned.\n\n### Unclear relationship between the research questions\nThe research questions posed in the methods section come off as a bag of loosely-related topics that are of independent interest within the field, each of which could be its own paper. For example, the transition into Section 3.2 is very abrupt as it goes into detail about a paper by Lawrence et al 2025 proposing a probabilistic interpretation of symmetry breaking that is (a) previously unintroduced (b) hard to parse and translate to the context of this paper without being very familiar with Lawrence, et al.\n\nIn the revision, please make explicit what ties these research questions together and provide necessary background in Section 2 instead of in the Methods, which should primarily cover what is new and/or specific to this paper.\n\n### Presentation: \n- The actual methodological contribution (Rapidash) is only described in the appendix, but the main text of the paper writes at length about its properties. It is hard to follow if you don't know what the method actually is. The description should be moved into the main text and a lot of space can be made by removing unnecessary portions including L65--68 and L233--245 (which just restates the intro paragraph).\n- The paper's opening paragraph doesn't contextualize the state of the field well. It comes off as just listing papers without outlining cohesive themes across papers. The latter would be much more accessible to readers who are not following the monthly back-and-forths on this contentious topic.\n\n## Minor comments that do not impact my rating:\n- Related work of interest: In the interest of unification, Gruver & Finzi 2023 should absolutely be discussed in this paper and how its findings on images could relate to the findings on point clouds here. Not essential but potentially interesting in the context of discussing Rapidash, there are works on position-orientation deep learning in medical imaging (e.g., in vessel analysis and diffusion MRI) that could be worth briefly discussing for better contextualization.\n- L20: Please clarify that it's AlphaFold3 and not other versions of AF that did use equivariant architectures.\n- Fig2: The radar plot + seemingly random text snippets make the figure very hard to parse. Please split into two separate subfigures.\n- L218: Weight sharing in CNNs was introduced quite a bit earlier than LeCun2010 :) \n- The connections made to tensor field networks can be left at a sentence with the rest moved to the appendix. There's already too much packed into the main text of the paper as is.\n\nQUESTIONS:\nPlease correct me if I misunderstood any central points and I would be happy to reconsider my rating. Currently, I would very likely raise my score if:\n- The paper clearly revises itself to narrow its scope to point cloud analysis and to describe which of its findings have the potential to translate across fields.\n- The empirical investigations are expanded to include at least one large-scale real-world dataset with *well tuned* baselines used as a point of comparison (with extensive augmentation and/or regularization). It is understandable if this is not doable within the time allocated to the rebuttal, but in that case, please provide a rigorous discussion regarding the choice of datasets and what we can extrapolate to real data.\n- It is clearly disambiguated which of the theoretical results are truly specific to this paper and which are inherited from previous work.\n\nLIMITATIONS:\nThe discussion section includes 2 sentences on being compute bound in terms of running larger scale experiments and mentions limiting analysis to convolutional architectures. While I don't think that these are the primary limitations of this work, these are thoughtful inclusions that should be appreciated.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces a unified architecture, Rapidash, which unifies non-equivariant and equivariant models, and assess the effect of models with different structural priors, including forms of symmetry/equivariance breaking, to different tasks.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The paper is well-written, provides an extensive study that ranges from theoretical analysis, empirical analysis, to practical architecture and empirical results on well-known data sets. The scope of this evaluation strengthens conclusions that are drawn.\n- The paper builds upon a solid geometric framework established in prior work.\n- Proposes a concrete architecture, Rapidash, that is useful to better understand and empirically evaluate geometric priors in the context of deep learning.\n- Insightful results and findings that benefit the community to better understand how models with structural geometric priors can benefit practical applications. In particular, the work offers important insights in the effect of incorporating symmetry breaking or relaxations have on predictive performance\n\nWeaknesses:\n- Although the analysis is quite extensive, one minor point of critique is that it does not seem to consider the difference between training and test error as particularly relevant. However, this distinction is crucial for understanding the difference between fitting (symmetry-restricted hypotheses) and the ability to generalize (having the right inductive bias).\n- The large result tables can be daunting at times and might benefit from clearer explanations in the text. In particular, it would be helpful to explicitly state the expected generalization behavior and explain how the results align with, or deviate from, those expectations. I recognize this may be a matter of personal preference and could also be constrained by page limits, so I won’t weigh it heavily in my overall assessment.\n\nQUESTIONS:\nSymmetries reduce the hypothesis space to functions that respect the symmetry (hence potentially limiting train fit capacity), but allow better generalization (test data fit) if the symmetries match the underlying data distribution, with symmetry breaking/relaxations being something in between.\n\nQ1: Given the relevance of distinguishing train/test data, would it be possible to report both train and test loss. Currently it is not always clear whether MSE is on train or test data (assuming it’s the test metric for CMU), but it would be helpful if both can be provided? Adding these as an Appendix should be enough.\n\nQ2: Related to this, results of the paper are particularly strong because the representational capacity is taken into account, ensuring that symmetric models do not merely generalize better due to undercapacity. The paper mentions that saturating performance suggests that ‘the limits of their architecturally defined hypothesis space for geometric expressivity’. Although true, it seems this could be easily verified by checking if models achieve low train error.\n\nLIMITATIONS:\n- The paper is very well-written. The main point of critique is the limited assessment of the train/test error difference, which seems highly relevant given that the paper focuses on measuring generalization capabilities in the context of structural geometric priors. \n- Secondly, the evaluation and interpretation of the results could be clearer and more precise. The paper presents results in large tables, which would benefit from additional exposition and discussion.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the tradeoffs of utilizing various types of equivariant and unconstrained models for learning a broad range of 3D tasks. Due to the nature of the 3D tasks, the authors focus on SE(3) symmetries and introduce a scalable group convolutional architecture (called Rapidash) that allows them to control the level of equivariance and symmetry breaking applied to the models in a unified way. Using Rapidash, the paper investigates how different levels of equivariant constraints affect the performance of the models, how the size of the models impacts the various equivariant architectures, and how the performance of these architectures scales with different dataset sizes. Additionally, the authors investigate how applying a symmetry-breaking mechanism can affect the performance of the models. After defining the specific questions that they aim to answer, they perform an extensive evaluation of the different options on various 3D tasks and provide an in-depth discussion interpreting their empirical observations.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The topic of the paper is very timely, as understanding the benefits and the tradeoffs of utilizing equivariant networks is something that the geometric deep learning community is currently actively interested in. Additionally, the authors do a good job in the introduction of the paper of providing readers with the context surrounding the research questions this work aims to answer.\n- The proposed architecture is very flexible and can be of interest to the community even beyond the investigation performed in this work.\n- The experimental evaluation is extensive, showing how the different combinations of equivariant tasks and symmetry-breaking mechanisms can affect performance. This can be of great use to practitioners who want to decide the degrees of symmetries they want to apply to their specific task.\n\nWeaknesses:\n- While the evaluation of different degrees of equivariance is extensive, there is no discussion or comparison with the use of data augmentation, which is one of the most common ways of incorporating known symmetries into the learning process.\n- The paper focuses on symmetries that are subgroups of SE(3). It is not clear if these conclusions generalize to symmetries of groups that are fundamentally different than SE(3).\n\nQUESTIONS:\n- What is the role of augmentations when investigating the benefit of using equivariant models? Are augmentations and equivariance exclusive of each other or complementary? Additionally, when combined with the information-theoretic arguments regarding the benefits of symmetry breaking, what is the role of augmentations there?\n- Is it reasonable to expect that these results and conclusions generalize to different types of groups? What would be some necessary conditions for these results to generalize?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-05"}