The Geometry of Polynomial Group Convolutional Neural Networks

Abstract

We study polynomial group convolutional neural networks (PGCNNs) for an arbitrary finite group $G$. In particular, we introduce a new mathematical framework for PGCNNs using the language of graded group algebras. This framework yields two natural parameterizations of the architecture, based on Hadamard and Kronecker products, related by a linear map. We compute the dimension of the associated neuromanifold, verifying that it depends only on the number of layers and the size of the group. Furthermore, we show the general fiber of both parameterizations is trivial up to the regular group action and rescaling. Hence both parametrization maps are identifiable.

37 pages, Conjecture 4.8 in v1 is now proved as Proposition 4.8