Exploring Commutative Matrix Multiplication Schemes via Flip Graphs

TL;DR

This paper investigates new methods for discovering efficient commutative matrix multiplication algorithms using flip graphs, adapting techniques from non-commutative cases and demonstrating their effectiveness up to 5x5 matrices.

Contribution

It introduces two novel flip graph-based approaches for commutative matrix multiplication and a hybrid method, extending their application to matrices up to 5x5.

Findings

Recovered best known bounds for matrix multiplication up to 5x5

Demonstrated the potential of flip graph techniques in the commutative setting

Compared efficiency of different flip graph-based methods

Abstract

We explore new approaches for finding matrix multiplication algorithms in the commutative setting by adapting the flip graph technique: a method previously shown to be effective for discovering fast algorithms in the non-commutative case. While an earlier attempt to apply flip graphs to commutative algorithms saw limited success, we overcome both theoretical and practical obstacles using two strategies: one inspired by Marakov's algorithm to multiply 3x3 matrices, in which we construct a commutative tensor and approximate its rank using the standard flip graph; and a second that introduces a fully commutative variant of the flip graph defined via a quotient tensor space. We also present a hybrid method that combines the strengths of both. Across all matrix sizes up to 5x5, these methods recover the best known bounds on the number of multiplications and allow for a comparison of their efficiency and efficacy. Although no new improvements are found, our results demonstrate strong potential for these techniques at larger scales.