How Jellyfish Characterise Alternating Group Equivariant Neural Networks

TL;DR

This paper characterizes all neural networks that are equivariant to the alternating group $A_n$, providing a basis for their linear layers and extending the approach to local symmetries.

Contribution

It offers a complete mathematical characterization of $A_n$-equivariant neural networks and generalizes the method to local symmetry cases.

Findings

Basis of matrices for $A_n$-equivariant layers identified

Full characterization of $A_n$-equivariant neural networks provided

Extension to local symmetry equivariance described

Abstract

We provide a full characterisation of all of the possible alternating group ($A_n$) equivariant neural networks whose layers are some tensor power of $\mathbb{R}^{n}$. In particular, we find a basis of matrices for the learnable, linear, $A_n$-equivariant layer functions between such tensor power spaces in the standard basis of $\mathbb{R}^{n}$. We also describe how our approach generalises to the construction of neural networks that are equivariant to local symmetries.