Brauer's Group Equivariant Neural Networks

TL;DR

This paper fully characterizes group equivariant neural networks for tensor powers of ^n under the orthogonal, special orthogonal, and symplectic groups, filling a gap in the machine learning literature.

Contribution

It provides a complete description of all linear equivariant layers for these groups and tensor spaces, including a basis of learnable matrices.

Findings

Characterization of equivariant layers for O(n), SO(n), Sp(n)

Explicit basis of learnable matrices in standard and symplectic bases

Fills a gap in the literature for these symmetry groups

Abstract

We provide a full characterisation of all of the possible group equivariant neural networks whose layers are some tensor power of $\mathbb{R}^{n}$ for three symmetry groups that are missing from the machine learning literature: $O(n)$, the orthogonal group; $SO(n)$, the special orthogonal group; and $Sp(n)$, the symplectic group. In particular, we find a spanning set of matrices for the learnable, linear, equivariant layer functions between such tensor power spaces in the standard basis of $\mathbb{R}^{n}$ when the group is $O(n)$ or $SO(n)$, and in the symplectic basis of $\mathbb{R}^{n}$ when the group is $Sp(n)$.