Any-Subgroup Equivariant Networks via Symmetry Breaking

TL;DR

This paper introduces a flexible neural network architecture, ASEN, capable of being equivariant to multiple symmetry groups simultaneously by modulating auxiliary inputs, enabling better generalization across diverse geometric data.

Contribution

The authors propose a novel approach to achieve subgroup equivariance in neural networks through symmetry-breaking inputs, relaxing the computational difficulty of exact symmetry enforcement.

Findings

ASEN outperforms separate equivariant models on graph and image tasks.

Single ASEN model achieves superior multitask and transfer learning results.

Theoretical guarantees of universality if the base model is universal.

Abstract

The inclusion of symmetries as an inductive bias, known as equivariance, often improves generalization on geometric data (e.g. grids, sets, and graphs). However, equivariant architectures are usually highly constrained, designed for symmetries chosen a priori, and not applicable to datasets with other symmetries. This precludes the development of flexible, multi-modal foundation models capable of processing diverse data equivariantly. In this work, we build a single model -- the Any-Subgroup Equivariant Network (ASEN) -- that can be simultaneously equivariant to several groups, simply by modulating a certain auxiliary input feature. In particular, we start with a fully permutation-equivariant base model, and then obtain subgroup equivariance by using a symmetry-breaking input whose automorphism group is that subgroup. However, finding an input with the desired automorphism group is computationally hard. We overcome this by relaxing from exact to approximate symmetry breaking, leveraging the notion of 2-closure to derive fast algorithms. Theoretically, we show that our subgroup-equivariant networks can simulate equivariant MLPs, and their universality can be guaranteed if the base model is universal. Empirically, we validate our method on symmetry selection for graph and image tasks, as well as multitask and transfer learning for sequence tasks, showing that a single network equivariant to multiple permutation subgroups outperforms both separate equivariant models and a single non-equivariant model.