Finite matrix multiplication algorithms from infinite groups

TL;DR

This paper extends the Cohn-Umans group-theoretic framework for matrix multiplication algorithms from finite groups to infinite groups, especially Lie groups, enabling new constructions and algorithms that were impossible in finite settings.

Contribution

It develops a comprehensive framework for deriving matrix multiplication algorithms directly from Lie group structures, expanding the scope beyond finite groups.

Findings

Constructed matrix multiplication algorithms from Lie groups.

Identified parameters in Lie groups that outperform finite group methods.

Established a theoretical foundation for infinite group-based algorithms.

Abstract

The Cohn-Umans (FOCS '03) group-theoretic framework for matrix multiplication produces fast matrix multiplication algorithms from three subsets of a finite group $G$ satisfying a simple combinatorial condition (the Triple Product Property). The complexity of such an algorithm then depends on the representation theory of $G$. In this paper we extend the group-theoretic framework to the setting of infinite groups. In particular, this allows us to obtain constructions in Lie groups, with favorable parameters, that are provably impossible in finite groups of Lie type (Blasiak, Cohn, Grochow, Pratt, and Umans, ITCS '23). Previously the Lie group setting was investigated purely as an analogue of the finite group case; a key contribution in this paper is a fully developed framework for obtaining bona fide matrix multiplication algorithms directly from Lie group constructions.