Equivariance via Minimal Frame Averaging for More Symmetries and Efficiency

TL;DR

This paper introduces Minimal Frame Averaging (MFA), a novel mathematical framework that achieves exact equivariance efficiently across various groups, enhancing symmetry encoding in machine learning tasks.

Contribution

MFA provides a minimal, provably exact equivariance method that extends to more groups and improves efficiency over existing approaches.

Findings

MFA achieves exact equivariance efficiently.

MFA outperforms sampling-based methods in diverse tasks.

MFA extends to groups like Lorentz and unitary groups.

Abstract

We consider achieving equivariance in machine learning systems via frame averaging. Current frame averaging methods involve a costly sum over large frames or rely on sampling-based approaches that only yield approximate equivariance. Here, we propose Minimal Frame Averaging (MFA), a mathematical framework for constructing provably minimal frames that are exactly equivariant. The general foundations of MFA also allow us to extend frame averaging to more groups than previously considered, including the Lorentz group for describing symmetries in space-time, and the unitary group for complex-valued domains. Results demonstrate the efficiency and effectiveness of encoding symmetries via MFA across a diverse range of tasks, including $n$-body simulation, top tagging in collider physics, and relaxed energy prediction. Our code is available at https://github.com/divelab/MFA.