In Search of Projectively Equivariant Networks
This work addresses a theoretical limitation in equivariant neural networks for researchers in machine learning, but it is incremental as it builds on existing equivariance studies.
The authors tackled the problem of relaxing equivariance in neural networks to a projective sense, proposing a method to construct projectively equivariant networks and showing it is the most general possible for linear layers, with results demonstrated in two simple experiments.
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.