A Normal Form for Matrix Multiplication Schemes

TL;DR

This paper introduces an algorithm to compute a normal form for matrix multiplication schemes, enabling efficient pairwise equivalence checks among many schemes by leveraging their symmetry group.

Contribution

The paper presents a novel algorithm to compute a normal form for matrix multiplication schemes, improving the efficiency of equivalence testing.

Findings

The normal form algorithm reduces computational complexity.

Efficiently identifies equivalent schemes in large sets.

Enhances analysis of matrix multiplication schemes' symmetry groups.

Abstract

Schemes for exact multiplication of small matrices have a large symmetry group. This group defines an equivalence relation on the set of multiplication schemes. There are algorithms to decide whether two schemes are equivalent. However, for a large number of schemes a pairwise equivalence check becomes cumbersome. In this paper we propose an algorithm to compute a normal form of matrix multiplication schemes. This allows us to decide pairwise equivalence of a larger number of schemes efficiently.