Homogeneous vector bundles and $G$-equivariant convolutional neural networks

TL;DR

This paper analyzes G-equivariant CNNs on homogeneous spaces, showing that homogeneous vector bundles provide a natural framework and establishing criteria for G-equivariant layers as convolutions, with implications for geometric deep learning.

Contribution

It introduces homogeneous vector bundles as the natural setting for G-equivariant CNNs and derives a precise criterion for G-equivariant layers to be expressed as convolutions.

Findings

Homogeneous vector bundles naturally model G-equivariant CNNs.

A reproducing kernel Hilbert space criterion for G-equivariant layers.

Bandwidth criterion enhances understanding for specific groups.

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 $K \leq G$. We demonstrate that homogeneous vector bundles is the natural setting for GCNNs. We also use reproducing kernel Hilbert spaces to obtain a precise criterion for expressing $G$-equivariant layers as convolutional layers. This criterion is then rephrased as a bandwidth criterion, leading to even stronger results for some groups.