Brauer's Group Equivariant Neural Networks

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 $Sp(n)$.

arXiv commentICML 2023 OralPoster; 22 pages