Group Cross-Correlations with Faintly Constrained Filters

TL;DR

This paper introduces weaker constraints on group convolutional filters that reduce node count and address limitations of previous methods, extending applicability to non-compact stabilizers and non-transitive group actions.

Contribution

We propose novel, less restrictive filter constraints for group convolutions, enabling broader group action applications and resolving previous incompatibility issues.

Findings

Reduced node requirements for group convolutional layers

Extended applicability to non-compact stabilizers

Generalized results to non-transitive group actions

Abstract

Group convolutional layers with respect to some group $G$ are modeled by convolutions or cross-correlations with a filter, and they provide the fundamental building block for group convolutional neural networks. For entirely unconstrained filters and $G$ a non-abelian group, any hidden layer of such a network requires as many nodes as vertices in a fine enough discretization of $G$. In order to reduce the necessary number of nodes, certain constraints on filters were proposed in the literature. We propose weaker constraints retaining this benefit while also resolving an incompatibility previous constraints have for group actions with non-compact stabilizers. Moreover, we generalize previous results to group actions that are not necessarily transitive, and we weaken the common assumption that $G$ is unimodular.