Matrix multiplication algorithms from group orbits

TL;DR

This paper introduces a method to construct symmetric matrix multiplication algorithms using group orbits and representation theory, recovering known algorithms and proposing new ones with potential for future improvements.

Contribution

It presents a novel approach leveraging group actions and representation theory to derive symmetric matrix multiplication algorithms for various sizes.

Findings

Recovered Strassen's algorithm in a symmetric form

Developed new algorithms for larger matrix sizes

Provided a transparent proof of Strassen's algorithm using lattices

Abstract

We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of orbits under some finite group action. We show how to use the representation theory of the corresponding group to derive simple constraints on the decomposition, which we solve by hand for n=2,3,4,5, recovering Strassen's algorithm (in a particularly symmetric form) and new algorithms for larger n. While these new algorithms do not improve the known upper bounds on tensor rank or the matrix multiplication exponent, they are beautiful in their own right, and we point out modifications of this idea that could plausibly lead to further improvements. Our constructions also suggest further patterns that could be mined for new algorithms, including a tantalizing connection with lattices. In particular, using lattices we give the most transparent proof to date of Strassen's algorithm; the same proof works for all n, to yield a decomposition with $n^3 - n + 1$ terms.