Reducing Symmetry Increase in Equivariant Neural Networks

2026-08-12

AI Summary

This work addresses the issue of symmetry amplification in equivariant neural networks, which can impair their ability to represent geometric structures when processing symmetric inputs. The study provides the first rigorous proof that this phenomenon is bounded below by a limit determined by the structure of the feature space. Leveraging group representation theory and an analysis of equivariant mappings, the authors develop a computable framework to determine this infimum. Based on this theoretical foundation, they propose design principles for constructing features that mitigate detrimental symmetry amplification. Experiments on synthetic data and the QM9 dataset validate the theoretical predictions, demonstrating that the proposed approach effectively alleviates symmetry amplification and enhances model expressivity.

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.

Problem

• equivariant neural networks
• symmetry increase
• expressivity
• symmetric inputs
• feature space

Innovation

• equivariant neural networks
• symmetry increase
• feature space design
• symmetry reduction
• group representation