In Search of Projectively Equivariant Networks

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. Hereby, we introduce the topic of projective equivariance to the machine learning audience. We theoretically study the relation of projectively and linearly equivariant linear layers. We find that in some important cases, surprisingly, the two types of layers coincide. We also propose a way to construct a projectively equivariant neural network, which boils down to building a standard equivariant network where the linear group representations acting on each intermediate feature space are lifts of projective group representations. Projective equivariance is showcased in two simple experiments. Code for the experiments is provided in the supplementary material.