Homogeneous vector bundles and G-equivariant convolutional neural networks

Abstract

AbstractG-equivariant convolutional neural networks (GCNNs) is a geometric deep learning model for data defined on a homogeneous G-space $$\mathcal {M}$$ M . GCNNs are designed to respect the global symmetry in $$\mathcal {M}$$ M , thereby facilitating learning. In this paper, we analyze GCNNs on homogeneous spaces $$\mathcal {M} = G/K$$ M = G / K in the case of unimodular Lie groups G and compact subgroups $$K \le G$$ K ≤ 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.