Title: Equivariance by Contrast: Identifiable Equivariant Embeddings from Unlabeled Finite Group Actions
Venue: NeurIPS 2025
Accepted: yes
Keywords: equivariance

identifiability

representation learning

group theory
Abstract: We propose Equivariance by Contrast (EbC) to learn equivariant embeddings from observation pairs $(\mathbf{y}, g \cdot \mathbf{y})$, where $g$ is drawn from a finite group acting on the data. Our method jointly learns a latent space and a group representation in which group actions correspond to invertible linear maps—without relying on group-specific inductive biases. We validate our approach on the infinite dSprites dataset with structured transformations defined by the finite group $G:= (R_m \times \mathbb{Z}_n \times \mathbb{Z}_n)$, combining discrete rotations and periodic translations. The resulting embeddings exhibit high-fidelity equivariance, with group operations faithfully reproduced in latent space. On synthetic data, we further validate the approach on the non-abelian orthogonal group $O(n)$ and the general linear group $GL(n)$. We also provide a theoretical proof for identifiability. While broad evaluation across diverse group types on real-world data remains future work, our results constitute the first successful demonstration of general-purpose encoder-only equivariant learning from group action observations alone, including non-trivial non-abelian groups and a product group motivated by modeling affine equivariances in computer vision.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a framework for learning embeddings that are equivariant with respect to the action of a finite group on the data. This is achieved by leveraging observed pairs consisting of a data point and its transformation under a fixed group element.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths.** The supplementary material is provided and includes both the theoretical proofs and additional experimental details for the conducted studies. However, at this stage, I am unable to offer a substantive evaluation of the strengths of the work, due to the concerns outlined in the Weaknesses section below.\n\n**Weaknesses.** \n\n*1.* The paper contains numerous typographical errors, unclear notations, and ambiguous conceptual explanations, which significantly reduce the overall clarity and hinder the readability of the work. It is essential that the authors carefully address these issues in a revised version. Below, I list several examples from the first two sections - though this list is not exhaustive - and many more issues are readily apparent in the experimental sections.\n\n- Line 5-7: The notation $R_m, \\mathbb{Z}_n$ are not clear. What are periodic translations? \n- Line 9-10: non-Abelian -> nonabelian or non-abelian\n- Line 10: generalized linear group -> general linear group\n- Line 18: allows to study -> allows us to study\n- Line 19-22: Only when a group acts on a vector space (of finite dimensions), we have the group representation.\n- The caption of Figure 2 is non-informative since it has too many typos in crucial equations. What does \"y'_i = gy\", or \"x_i = R(g)x_i\" mean? \n- Line 21, 62, 64, etc.: \\mapsto -> \\rightarrow\n- Line 34: questions -> question\n- Line 38: perturbations[4] -> perturbations [4] \n- Line 58-59: \"which are grouped 59 into n + 1 pairs undergoing\":  What does \"pairs\" mean here?\n- Line 66: Information of  -> Information about\n- Line 66: what is \"u\"? Is it \"c\"?\n- Line 70: process Assume -> process. Assume\n- Line 77: the second \\phi(y_i') should be \\phi'(y_i')\n- Line 79: What is R(\\Phi, \\Phi')? The paper only defines R(g) or \\hat{R}.\n- Equation (3): What is S here?\n- Line 84: where used -> where we use\n- Line 97: What is Eq. (X)?\n- Line 98: GL should not be in italic.\n- Line 99: a -> an\n- Line 112: Does \"Assume that p_\\phi = p_{id}\" mean you assume that \\phi is the identity map?\n- Line 114: I am not sure why the original vector relates to the equation right after.\n\n*2.* The discussion of related work is insufficiently thorough and lacks critical engagement with relevant prior literature.\n\n*3.* The experiments are conducted solely on the dSprites dataset. Given that the authors reference a wide range of real-world applications - including biology, neuroscience, drug discovery, and computer vision - the paper should include at least one task or dataset from these domains to substantiate these claims.\n\nQUESTIONS:\nThe current state of the paper makes it very difficult to evaluate the work effectively. I understand the time and content constraints of the rebuttal phase; however, I believe it is both appropriate and necessary to request that the authors revise Section 2 - which outlines the proposed problem and methodology - as this revision is essential for enabling a meaningful re-evaluation of the paper. \n\nIt is important to note that this does not imply that only Section 2 requires revision. Rather, gaining a clearer understanding of the proposed method would facilitate a more informed and constructive assessment of the rest of the paper.\n\nLIMITATIONS:\n.", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 3, "CLARITY": 1, "IMPACT": 1, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors present an algorithm to learn a representation of the group acting on a dataset as well as the latent space of the dataset. The author assume that the data $x_i $is only observable through a transformation $y_i = f(x_i)$. The authors address the problem by learning a mapping $\\phi \\approx f^{-1}$ (up to a linear transformation on x) and a matrix $R$ being a representation of the group action. They use a likelihood model as objective function to find $\\phi$ and $R$. $\\phi$ is represented with a neural network.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n- The introduction is well-written.\n- The concepts of section 2 are introduced with enough clarity to be understood.\n\n# Weaknesses\n- The proof of Theorem 1 is given in Appendix A.3 while the main text presents an \"informal\" version if the theorem. Unfortunately, the appendix does not present a formal version of the theorem. \n- The empirical evidence of Theorem 1 and Corollary 2 on a synthetic dataset, without noise, in section 4 is not convincingly explained.\n- In a context where one seeks to learn the group acting on a dataset, it does not seem realistic to assume that both $Y$ and $Y'$ are known.\n-In the subsection \"Metrics\", $\\hat{h}$ is not defined.\n- Experiments on $SO(n), O(n), GL(n)$, are limited to $n=3$.\n\nQUESTIONS:\n- The model of Eq. (2) assumes that the same group action (i.e., the same matrix R) is applied to the complete dataset X. Why do the authors make this assumption? It would seem more realistic that different actions are applied on different data points.\n- Why using the terms \"content\" and \"style\" in section 2?\n- The authors do not detail how they deal with different (1000) group actions in their experiments. Indeed, the method on section 2 is only described in the case where a single group action is applied on the whole dataset.\n- In the section \"Metrics\", the quantities of Eq. (8)-(10) use $h$, which should be unknown since $h=\\phi\\cdot f$  and $f$ is unknown. How are these quantities computed in practice?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers the learning of equivariance by unsupervised/contrastive means.  The key mathematical result is on identifiability in this setting, by extending results of Roeder et al. a little bit.  The algorithmic approach follows the mathematical development and is demonstrated to work in one exemplary setting.\n\nSTRENGTHS AND WEAKNESSES:\nSTRONG: The problem of symmetry discovery is important in numerous settings, and the findings here for such a problem are intriguing.  Note that the setting is where there are batches of samples under the same transformation.  The paper hits all the points of formulation, theory, algorithm, and basic experimental demonstration.\n\n\nWEAK: The seemingly non-native English phrasing, e.g. \"Group theory allows to study this structure\" -> \"Group theory allows studying this structure\" could be improved. \n\nEven though Thm 1 is the main result, the technical diversity conditions are omitted in the main text: to this reviewer, this feels like a significant shortcoming that is easily rectified by moving them up from Appendix A.\n\nThe mathematical advance over Roeder et al [14] seems marginal.\n\nI am not an expert in the motivating examples, but the batched data needed for this approach seems like a very strong assumption.\n\nQUESTIONS:\nClarifying the differences between the present work and other past work on learning group-theoretic structure would be very helpful in clarifying the contributions here.  For example [Dehmamy et al., \"Automatic symmetry discovery with lie algebra convolutional network,\" NeurIPS 2021], [Yang, et al., \"Generative adversarial symmetry discovery,\" ICML 2023], or [Yu et al., \"Information lattice learning,\" JAIR 2023].  See also references therein and thereto.  Can you do that?  It would help with your claim about \"group representation learning from unlabeled observational data is feasible at all\".\n\nLIMITATIONS:\nyes, they have addressed limitations.  no discussion of potential negative societal impact.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a method for learning representation of a group from data through contrastive learning approach with theoretical guarantees and verifies on synthetic and 2D image datasets.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n- The paper is very well written, with clear motivation and exposition.\n- Figure 3 provides an intuitive visual explanation of the proposed method, making the main idea easy to understand.\n- Theoretical results (e.g., Theorem 1) demonstrate identifiability of the group action and latent space under mild assumptions, which is uncommon for self-supervised equivariant learning methods.\n- The approach is simple and efficient, relying on contrastive learning with an encoder-only pipeline and on-the-fly least-squares fitting, avoiding the complexity of generative models.\n- Strong empirical results: Experiments cover a diverse set of group structures (e.g., SO(3), O(3), GL(3), and product groups), demonstrating the generality of the method. Achieves near-perfect latent reconstruction and group prediction accuracy, outperforming relevant baselines on synthetic data.\n\n**Weaknesses:**\n\n- Data efficiency is not systematically studied; there is no analysis on how performance scales with dataset size, batch size, or the number of negative samples, which is especially relevant given the theoretical assumptions on required samples. I would suggest including experiments sweeping dataset size and batch/negative counts to measure convergence and data efficiency.\n- Comparison to baselines is somewhat limited; recent related approaches such as Park et al. (ICML 2022), Homomorphism Auto-Encoder (Keurti et al., ICML 2023), and other group-equivariant learning methods are not included or discussed in detail.\n- The key assumption (that all n+1 samples in a mini-batch share the same, unknown group element) is not tested or validated on real-world datasets; it is unclear how well the method handles weaker or approximate group structure in practical settings.\n- Application to real-world data is limited; all experiments are on synthetic datasets (e.g., Infinite-dSprites), with no demonstration on realistic or noisy sequence data.\n\n**Additional References Suggested:**\n\n- Park et al., “Learning Symmetric Embeddings for Equivariant World Models,” ICML 2022.\n- Keurti et al., “Homomorphism Auto-Encoder,” ICML 2023.\n- Winter et al., “Learning Invariant and Equivariant Representations in Unsupervised Group Settings,” 2022.\n\nQUESTIONS:\n- Figure 2: The encoder in Caption is f, and there is another encoder \\phi after the samples, which is a bit confusing. What is Q in the Figure 2?\n- I have less background in neuroscience and biology, and I’d appreciate some concrete examples on this.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-10"}