A General Framework for Equivariant Neural Networks on Reductive Lie Groups

TL;DR

This paper introduces a universal framework for constructing neural networks that are equivariant to any reductive Lie group, enabling symmetry-respecting models across various scientific disciplines.

Contribution

It generalizes existing equivariant architectures to all reductive Lie groups and provides a software library for easy implementation in diverse applications.

Findings

Successfully applied to top quark decay tagging (Lorentz group)

Achieved shape recognition with orthogonal group equivariance

Demonstrated broad applicability and high performance

Abstract

Reductive Lie Groups, such as the orthogonal groups, the Lorentz group, or the unitary groups, play essential roles across scientific fields as diverse as high energy physics, quantum mechanics, quantum chromodynamics, molecular dynamics, computer vision, and imaging. In this paper, we present a general Equivariant Neural Network architecture capable of respecting the symmetries of the finite-dimensional representations of any reductive Lie Group G. Our approach generalizes the successful ACE and MACE architectures for atomistic point clouds to any data equivariant to a reductive Lie group action. We also introduce the lie-nn software library, which provides all the necessary tools to develop and implement such general G-equivariant neural networks. It implements routines for the reduction of generic tensor products of representations into irreducible representations, making it easy to apply our architecture to a wide range of problems and groups. The generality and performance of our approach are demonstrated by applying it to the tasks of top quark decay tagging (Lorentz group) and shape recognition (orthogonal group).