In Search of Projectively Equivariant Networks

TL;DR

This paper introduces a novel approach to neural network equivariance, relaxing strict equivariance to a projective sense, and demonstrates its theoretical generality and practical potential through simple experiments.

Contribution

It proposes a method to construct projectively equivariant networks by modifying linear group representations, expanding the scope of equivariance in neural networks.

Findings

The approach is theoretically the most general for linear layers.

Constructed networks exhibit projective equivariance.

Experimental validation demonstrates the method's feasibility.

Abstract

Equivariance of linear neural network layers is well studied. In this work, we relax the equivariance condition to only be true in a projective sense. We propose a way to construct a projectively equivariant neural network through building a standard equivariant network where the linear group representations acting on each intermediate feature space are "multiplicatively modified lifts" of projective group representations. By theoretically studying the relation of projectively and linearly equivariant linear layers, we show that our approach is the most general possible when building a network out of linear layers. The theory is showcased in two simple experiments.