Finite matrix multiplication algorithms from infinite groups
Abstract
The Cohn-Umans (FOCS '03) group-theoretic framework for matrix multiplication produces fast matrix multiplication algorithms from three subsets of a finite group satisfying a simple combinatorial condition (the Triple Product Property). The complexity of such an algorithm then depends on the representation theory of . 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.
Cite
@article{arxiv.2410.14905,
title = {Finite matrix multiplication algorithms from infinite groups},
author = {Jonah Blasiak and Henry Cohn and Joshua A. Grochow and Kevin Pratt and Chris Umans},
journal= {arXiv preprint arXiv:2410.14905},
year = {2025}
}
Comments
40 pages; only minor updates from previous version