Geometric Deep Learning and Equivariant Neural Networks

TL;DR

This paper surveys the mathematical foundations of geometric deep learning, focusing on group and gauge equivariant neural networks, their construction on manifolds, and applications like segmentation and object detection, emphasizing spherical networks and representation theory.

Contribution

It develops gauge and group equivariant CNNs on manifolds and homogeneous spaces, linking them through representation theory and illustrating their applications in deep learning tasks.

Findings

Gauge equivariant CNNs on arbitrary manifolds are developed.

Group equivariant layers are shown as intertwiners of induced representations.

Spherical networks utilize Fourier analysis with Wigner matrices and spherical harmonics.

Abstract

We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds $\mathcal{M}$ using principal bundles with structure group $K$ and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces $\mathcal{M}=G/K$, which are instead equivariant with respect to the global symmetry $G$ on $\mathcal{M}$. Group equivariant layers can be interpreted as intertwiners between induced representations of $G$, and we show their relation to gauge equivariant convolutional layers. We analyze several applications of this formalism, including semantic segmentation and object detection networks. We also discuss the case of spherical networks in great detail, corresponding to the case $\mathcal{M}=S^2=\mathrm{SO}(3)/\mathrm{SO}(2)$. Here we emphasize the use of Fourier analysis involving Wigner matrices, spherical harmonics and Clebsch-Gordan coefficients for $G=\mathrm{SO}(3)$, illustrating the power of representation theory for deep learning.