On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups

TL;DR

This paper provides a rigorous theoretical framework for understanding how convolutional neural networks can be generalized to be equivariant under the action of any compact group, extending beyond translations.

Contribution

It proves that convolutional structure is both necessary and sufficient for equivariance to compact group actions in neural networks, using advanced mathematical tools.

Findings

Convolutional structure is necessary for equivariance under compact groups.

Derived new generalized convolution formulas.

Extended the understanding of equivariance beyond translations.

Abstract

Convolutional neural networks have been extremely successful in the image recognition domain because they ensure equivariance to translations. There have been many recent attempts to generalize this framework to other domains, including graphs and data lying on manifolds. In this paper we give a rigorous, theoretical treatment of convolution and equivariance in neural networks with respect to not just translations, but the action of any compact group. Our main result is to prove that (given some natural constraints) convolutional structure is not just a sufficient, but also a necessary condition for equivariance to the action of a compact group. Our exposition makes use of concepts from representation theory and noncommutative harmonic analysis and derives new generalized convolution formulae.