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

G-equivariant convolutional neural networks (GCNNs) is a geometric deep learning model for data defined on a homogeneous G-space M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}$$\end{document}. GCNNs are designed to respect the global symmetry in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}$$\end{document}, thereby facilitating learning. In this paper, we analyze GCNNs on homogeneous spaces M=G/K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M} = G/K$$\end{document} in the case of unimodular Lie groups G and compact subgroups K≤G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \le G$$\end{document}. 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.