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Reducing Symmetry Increase in Equivariant Neural Networks

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.7Download Report PDF
Equivariant Neural NetworksSymmetry IncreaseCompact GroupIsotropy SubgroupOrbit TypeCurie’s Principle
Abstract:

Equivariant Neural Networks (ENNs) have empowered numerous applications in scientific fields. Despite their remarkable capacity for representing geometric structures, ENNs suffer from degraded expressivity when processing symmetric inputs: the output representations are invariant to transformations that extend beyond the input's symmetries. The mathematical essence of this phenomenon is that a symmetric input, after being processed by an equivariant map, experiences an increase in symmetry. While prior research has documented symmetry increase in specific cases, a rigorous understanding of its underlying causes and general reduction strategies remains lacking. In this paper, we provide a detailed and in-depth characterization of symmetry increase together with a principled framework for its reduction: (i) For any given feature space and input symmetry group, we prove that the increased symmetry admits an infimum determined by the structure of the feature space; (ii) Building on this foundation, we develop a computable algorithm to derive this infimum, and propose practical guidelines for feature design to prevent harmful symmetry increases. (iii) Under standard regularity assumptions, we demonstrate that for most equivariant maps, our guidelines effectively reduce symmetry increase. To complement our theoretical findings, we provide visualizations and experiments on both synthetic datasets and the real-world QM9 dataset. The results validate our theoretical predictions.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper contributes a rigorous mathematical framework for understanding and reducing symmetry increase in equivariant neural networks, proving the existence of a symmetry infimum determined by feature space structure and developing a computable algorithm to derive it. Within the taxonomy, it occupies the 'Symmetry Increase Analysis and Reduction Frameworks' leaf under 'Theoretical Foundations and Characterization of Symmetry Phenomena'. Notably, this leaf contains only the original paper itself—no sibling papers—indicating this is a relatively sparse and underexplored research direction focused specifically on formal characterization of symmetry increase phenomena.

The taxonomy reveals neighboring work in 'Equivariance Relaxation and Subgroup Constraints' (two papers on relaxing equivariance to subgroups) and 'Symmetry Breaking Methods' (five papers on explicit or probabilistic breaking mechanisms). While these adjacent directions address related challenges—managing unwanted symmetries or enabling lower-symmetry outputs—they differ fundamentally in approach. The original paper provides theoretical foundations for understanding why symmetry increases occur, whereas neighboring leaves focus on architectural mechanisms or training procedures to break symmetries. The taxonomy's scope notes clarify that formal characterization of increase belongs here, while breaking mechanisms without such characterization belong elsewhere.

Among seventeen candidates examined across three contributions, none were found to clearly refute any claimed novelty. The characterization of symmetry infimum examined three candidates with zero refutations; the computable algorithm examined four candidates with zero refutations; the theoretical guarantee examined ten candidates with zero refutations. This suggests that within the limited search scope of top-K semantic matches and citation expansion, no prior work appears to provide the same combination of formal symmetry increase characterization, infimum derivation, and reduction framework. The contributions addressing theoretical guarantees received the most scrutiny but still showed no overlapping prior work among examined candidates.

Based on the limited literature search of seventeen candidates, the work appears to occupy a novel position providing formal mathematical foundations for a phenomenon documented but not rigorously characterized in prior work. The absence of sibling papers in its taxonomy leaf and zero refutations across all contributions suggest substantive theoretical novelty, though this assessment is constrained by the search scope and does not constitute an exhaustive field survey.

Taxonomy

Core-task Taxonomy Papers
13
3
Claimed Contributions
17
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: Reducing symmetry increase in equivariant neural networks. Equivariant architectures are designed to respect known symmetries in data, but a subtle challenge arises when network layers inadvertently increase symmetry beyond what the problem requires, leading to reduced expressiveness or ambiguous representations. The field has organized around several complementary directions. Theoretical foundations examine how and why symmetry increase occurs, characterizing the mathematical conditions under which equivariant operations preserve or expand symmetry groups. Symmetry breaking methods develop explicit techniques—ranging from learnable breaking mechanisms to probabilistic approaches—that allow networks to selectively reduce unwanted symmetries while maintaining beneficial equivariances. Computational efficiency and architectural optimization focus on designing layers and operations that balance equivariance constraints with practical scalability, often leveraging sparse representations or efficient group convolutions. Specialized equivariant architectures tailor these principles to specific domains such as molecular modeling, image analysis, or graph learning, where particular symmetry groups (e.g., SO(3), SE(3)) dominate. Recent work has explored diverse strategies for managing symmetry. Some studies introduce relaxed equivariance frameworks that soften strict constraints to improve flexibility, as seen in Relaxed Equivariant GNNs[7] and Relaxed Equivariance Multitask[6], while others propose explicit symmetry breaking sets or probabilistic mechanisms (Symmetry Breaking Sets[3], Probabilistic Symmetry Breaking[5]) to disambiguate representations. Domain-specific innovations like SO3 to SO2[1] and SO3 Vessel Segmentation[4] demonstrate how reducing from higher to lower symmetry groups can enhance task performance. The original paper, Reducing Symmetry Increase[0], sits within the theoretical and analytical branch, focusing on formal frameworks for understanding and mitigating symmetry increase. Its emphasis on rigorous characterization contrasts with more application-driven works like SO3 Vessel Segmentation[4], yet complements method-oriented papers such as Symmetry Breaking Sets[3] by providing foundational insights that guide when and how symmetry reduction should be applied.

Claimed Contributions

Characterization of symmetry increase with a unique symmetry infimum

The authors establish a mathematical foundation showing that for any feature space and input symmetry group, the symmetry increase phenomenon has a lower bound called the symmetry infimum. This infimum is uniquely determined by the algebraic structure of the feature space itself.

3 retrieved papers
Computable algorithm for deriving the symmetry infimum

The authors propose an algorithm that computes the symmetry infimum through orbit type analysis. This algorithm enables practical guidelines for feature design that can predict and prevent harmful symmetry increases in equivariant neural networks.

4 retrieved papers
Theoretical guarantee for symmetry reduction under regularity conditions

The authors prove that under standard regularity assumptions like the manifold hypothesis, their framework effectively reduces symmetry increase for most equivariant maps. The output symmetry becomes exactly the predicted infimum, preventing loss of orientational information.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Within the taxonomy built over the current TopK core-task papers, the original paper is assigned to a leaf with no direct siblings and no cousin branches under the same grandparent topic. In this retrieved landscape, it appears structurally isolated, which is one partial signal of novelty, but still constrained by search coverage and taxonomy granularity.

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Characterization of symmetry increase with a unique symmetry infimum

The authors establish a mathematical foundation showing that for any feature space and input symmetry group, the symmetry increase phenomenon has a lower bound called the symmetry infimum. This infimum is uniquely determined by the algebraic structure of the feature space itself.

Contribution

Computable algorithm for deriving the symmetry infimum

The authors propose an algorithm that computes the symmetry infimum through orbit type analysis. This algorithm enables practical guidelines for feature design that can predict and prevent harmful symmetry increases in equivariant neural networks.

Contribution

Theoretical guarantee for symmetry reduction under regularity conditions

The authors prove that under standard regularity assumptions like the manifold hypothesis, their framework effectively reduces symmetry increase for most equivariant maps. The output symmetry becomes exactly the predicted infimum, preventing loss of orientational information.

Table of Contents