Upgrading Subgroup Triple Product Property Triples

TL;DR

This paper introduces methods to enlarge TPP triples in group-theoretic matrix multiplication, enhancing the potential for bounding the matrix multiplication exponent more tightly.

Contribution

It presents new upgrade and reduction techniques for TPP triples, focusing on subgroup triples to improve the search for efficient matrix multiplication bounds.

Findings

Enlargement of TPP triples by up to 100% after one upgrade step.

Effective reduction methods for non-upgradable TPP triples.

Potential to improve bounds on matrix multiplication exponent.

Abstract

In 2003 COHN and UMANS introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets of a group $G$ satisfying the Triple Product Property (TPP) as a means to bound the exponent $\omega$ of matrix multiplication. Recently, Hedtke and Murthy discussed several methods to find TPP triples. Because the search space for subset triples is too large, it is only possible to focus on subgroup triples. We present methods to upgrade a given TPP triple to a bigger TPP triple. If no upgrade is possible we use reduction methods (based on random experiments and heuristics) to create a smaller TPP triple that can be used as input for the upgrade methods. If we apply the upgrade process for subset triples after one step with the upgrade method for subgroup triples we achieve an enlargement of the triple size of 100 % in the best case.