Investigating how ReLU-networks encode symmetries

TL;DR

This paper explores how ReLU neural networks encode symmetries through group equivariance, revealing that equivariance does not always imply layerwise equivariance and supporting this with theoretical and experimental evidence.

Contribution

It provides a theoretical analysis of the relationship between network equivariance and layerwise equivariance, and offers experimental validation on standard datasets.

Findings

Equivariance does not necessarily imply layerwise equivariance in ReLU networks.

Training CNNs for equivariance tends to produce layerwise equivariance.

Merging a network with a group-transformed version is easier than merging two different networks.

Abstract

Many data symmetries can be described in terms of group equivariance and the most common way of encoding group equivariances in neural networks is by building linear layers that are group equivariant. In this work we investigate whether equivariance of a network implies that all layers are equivariant. On the theoretical side we find cases where equivariance implies layerwise equivariance, but also demonstrate that this is not the case generally. Nevertheless, we conjecture that CNNs that are trained to be equivariant will exhibit layerwise equivariance and explain how this conjecture is a weaker version of the recent permutation conjecture by Entezari et al. [2022]. We perform quantitative experiments with VGG-nets on CIFAR10 and qualitative experiments with ResNets on ImageNet to illustrate and support our theoretical findings. These experiments are not only of interest for understanding how group equivariance is encoded in ReLU-networks, but they also give a new perspective on Entezari et al.'s permutation conjecture as we find that it is typically easier to merge a network with a group-transformed version of itself than merging two different networks.