The Geometry of Polynomial Group Convolutional Neural Networks

We prove that polynomial group convolutional neural networks (PGCNNs), for an arbitrary finite group $G$, are identifiable. In order to achieve this, we introduce a new mathematical framework for PGCNNs using the language of graded group algebras. This framework yields two natural parametrizations 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. The identifiability of PGCNNs follows from showing that the general fiber of both parametrizations is trivial up to the regular group action and rescaling.