Connecting Permutation Equivariant Neural Networks and Partition Diagrams

TL;DR

This paper reveals that permutation equivariant neural networks' weight matrices can be derived using Schur-Weyl duality and partition diagrams, providing a new diagrammatic approach for their computation.

Contribution

It introduces a novel connection between permutation equivariant neural networks and partition diagrams via Schur-Weyl duality, enabling a simple diagrammatic calculation method.

Findings

All weight matrices can be obtained from Schur-Weyl duality.

A diagrammatic method for computing weight matrices is developed.

The approach simplifies understanding of permutation equivariant neural networks.

Abstract

Permutation equivariant neural networks are often constructed using tensor powers of $\mathbb{R}^{n}$ as their layer spaces. We show that all of the weight matrices that appear in these neural networks can be obtained from Schur-Weyl duality between the symmetric group and the partition algebra. In particular, we adapt Schur-Weyl duality to derive a simple, diagrammatic method for calculating the weight matrices themselves.