Homogeneous vector bundles and $G$-equivariant convolutional neural networks
This work provides theoretical foundations for geometric deep learning models, addressing symmetry in data for researchers in machine learning and mathematics, but it is incremental as it builds on existing GCNN frameworks.
The paper tackles the problem of analyzing G-equivariant convolutional neural networks (GCNNs) on homogeneous spaces by establishing homogeneous vector bundles as the natural framework and using reproducing kernel Hilbert spaces to derive a criterion for expressing equivariant layers as convolutional layers, resulting in a bandwidth criterion that strengthens results for certain groups.
$G$-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.