Flip Graphs with Symmetry and New Matrix Multiplication Schemes

TL;DR

This paper enhances the flip graph algorithm for discovering matrix multiplication schemes by incorporating symmetry, enabling more efficient searches and new schemes for multiplying 5x5 and 6x6 matrices with fewer multiplications.

Contribution

It introduces a symmetry-aware version of the flip graph algorithm, reducing search complexity and producing new matrix multiplication schemes.

Findings

New 5x5 matrix multiplication scheme with 93 multiplications.

New 6x6 matrix multiplication scheme with 153 multiplications.

Symmetry facilitates lifting solutions from F2 to Z.

Abstract

The flip graph algorithm is a method for discovering new matrix multiplication schemes by following random walks on a graph. We introduce a version of the flip graph algorithm for matrix multiplication schemes that admit certain symmetries. This significantly reduces the size of the search space, allowing for more efficient exploration of the flip graph. The symmetry in the resulting schemes also facilitates the process of lifting solutions from $F_2$ to $\mathbb{Z}$. Our results are new schemes for multiplying $5\times 5$ matrices using $93$ multiplications and $6\times 6$ matrices using $153$ multiplications over arbitrary ground fields.