Theoretical Aspects of Group Equivariant Neural Networks

TL;DR

This paper explores the theoretical foundations of group equivariant neural networks, detailing their mathematical basis, applications to 3D networks, and recent theoretical results that characterize their structure.

Contribution

It provides an exposition of the mathematical machinery behind group equivariant networks and discusses recent theoretical characterizations of their convolutional structure.

Findings

Group equivariant networks reduce sample and model complexity.

Recent theorems characterize equivariance as convolutional structure.

Applications include SO(3) and SE(3) equivariant networks.

Abstract

Group equivariant neural networks have been explored in the past few years and are interesting from theoretical and practical standpoints. They leverage concepts from group representation theory, non-commutative harmonic analysis and differential geometry that do not often appear in machine learning. In practice, they have been shown to reduce sample and model complexity, notably in challenging tasks where input transformations such as arbitrary rotations are present. We begin this work with an exposition of group representation theory and the machinery necessary to define and evaluate integrals and convolutions on groups. Then, we show applications to recent SO(3) and SE(3) equivariant networks, namely the Spherical CNNs, Clebsch-Gordan Networks, and 3D Steerable CNNs. We proceed to discuss two recent theoretical results. The first, by Kondor and Trivedi (ICML'18), shows that a neural network is group equivariant if and only if it has a convolutional structure. The second, by Cohen et al. (NeurIPS'19), generalizes the first to a larger class of networks, with feature maps as fields on homogeneous spaces.