A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups

TL;DR

This paper introduces a general algorithm for constructing equivariant multilayer perceptrons for any matrix group, enabling broader application of symmetry-aware neural networks across diverse domains.

Contribution

It provides a universal method for designing equivariant layers for arbitrary matrix groups, including previously unaddressed groups like O(1,3), O(5), Sp(n), and the Rubik's cube group.

Findings

Outperforms non-equivariant baselines in experiments

Successfully constructs equivariant networks for complex groups

Enables applications in particle physics and dynamical systems

Abstract

Symmetries and equivariance are fundamental to the generalization of neural networks on domains such as images, graphs, and point clouds. Existing work has primarily focused on a small number of groups, such as the translation, rotation, and permutation groups. In this work we provide a completely general algorithm for solving for the equivariant layers of matrix groups. In addition to recovering solutions from other works as special cases, we construct multilayer perceptrons equivariant to multiple groups that have never been tackled before, including $\mathrm{O}(1,3)$, $\mathrm{O}(5)$, $\mathrm{Sp}(n)$, and the Rubik's cube group. Our approach outperforms non-equivariant baselines, with applications to particle physics and dynamical systems. We release our software library to enable researchers to construct equivariant layers for arbitrary matrix groups.