Computing Representations for Lie Algebraic Networks

TL;DR

This paper introduces a new algorithm for constructing Lie group representations from algebra structure constants, a software toolkit for building equivariant neural networks, and a benchmark for relativistic object classification, advancing symmetry-aware learning.

Contribution

It presents a novel method to find Lie group representations without explicit group representations, along with software and a new benchmark dataset for equivariant neural networks.

Findings

Algorithm successfully finds Lie group representations from structure constants.

Software enables easy construction of Lie group-equivariant neural networks.

Model achieves Poincaré group equivariance for relativistic object tracking.

Abstract

Recent work has constructed neural networks that are equivariant to continuous symmetry groups such as 2D and 3D rotations. This is accomplished using explicit Lie group representations to derive the equivariant kernels and nonlinearities. We present three contributions motivated by frontier applications of equivariance beyond rotations and translations. First, we relax the requirement for explicit Lie group representations with a novel algorithm that finds representations of arbitrary Lie groups given only the structure constants of the associated Lie algebra. Second, we provide a self-contained method and software for building Lie group-equivariant neural networks using these representations. Third, we contribute a novel benchmark dataset for classifying objects from relativistic point clouds, and apply our methods to construct the first object-tracking model equivariant to the Poincar\'e group.