Homogeneous vector bundles and G-equivariant convolutional neural networks

Abstract

G-equivariant convolutional neural networks (GCNNs) is a geometric deep learning model for data defined on a homogeneous G-space mathcal {M}. GCNNs are designed to respect the global symmetry in mathcal {M}, thereby facilitating learning. In this paper, we analyze GCNNs on homogeneous spaces mathcal {M} = G/K in the case of unimodular Lie groups G and compact subgroups K le G. We demonstrate that homogeneous vector bundles are the natural setting for GCNNs. We also use reproducing kernel Hilbert spaces (RKHS) to obtain a sufficient criterion for expressing G-equivariant layers as convolutional layers. Finally, stronger results are obtained for some groups via a connection between RKHS and bandwidth.