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DOI: 10.4230/LIPIcs.ITCS.2025.18

Finite Matrix Multiplication Algorithms from Infinite Groups

Authors: Jonah Blasiak, Henry Cohn, Joshua A. Grochow, Kevin Pratt, and Chris Umans

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

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. As part of this framework, we introduce "separating functions" as a necessary new design component, and show that when the underlying group is G = GL_n, these functions are polynomials with their degree being the key parameter. In particular, we show that a construction with "half-dimensional" subgroups and optimal degree would imply ω = 2. We then build up machinery that reduces the problem of constructing optimal-degree separating polynomials to the problem of constructing a single polynomial (and a corresponding set of group elements) in a ring of invariant polynomials determined by two out of the three subgroups that satisfy the Triple Product Property. This machinery combines border rank with the Lie algebras associated with the Lie subgroups in a critical way. We give several constructions illustrating the main components of the new framework, culminating in a construction in a special unitary group that achieves separating polynomials of optimal degree, meeting one of the key challenges. The subgroups in this construction have dimension approaching half the ambient dimension, but (just barely) too slowly. We argue that features of the classical Lie groups make it unlikely that constructions in these particular groups could produce nontrivial bounds on ω unless they prove ω = 2. One way to get ω = 2 via our new framework would be to lift our existing construction from the special unitary group to GL_n, and improve the dimension of the subgroups from (dim G)/2 - Θ(n) to (dim G)/2 - o(n).

Cite as

Jonah Blasiak, Henry Cohn, Joshua A. Grochow, Kevin Pratt, and Chris Umans. Finite Matrix Multiplication Algorithms from Infinite Groups. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 18:1-18:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{blasiak_et_al:LIPIcs.ITCS.2025.18,
  author =	{Blasiak, Jonah and Cohn, Henry and Grochow, Joshua A. and Pratt, Kevin and Umans, Chris},
  title =	{{Finite Matrix Multiplication Algorithms from Infinite Groups}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{18:1--18:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.18},
  URN =		{urn:nbn:de:0030-drops-226460},
  doi =		{10.4230/LIPIcs.ITCS.2025.18},
  annote =	{Keywords: Fast matrix multiplication, representation theory, infinite groups}
}

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DOI: 10.4230/LIPIcs.ITCS.2023.19

Matrix Multiplication via Matrix Groups

Authors: Jonah Blasiak, Henry Cohn, Joshua A. Grochow, Kevin Pratt, and Chris Umans

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract

In 2003, Cohn and Umans proposed a group-theoretic approach to bounding the exponent of matrix multiplication. Previous work within this approach ruled out certain families of groups as a route to obtaining ω = 2, while other families of groups remain potentially viable. In this paper we turn our attention to matrix groups, whose usefulness within this framework was relatively unexplored. We first show that groups of Lie type cannot prove ω = 2 within the group-theoretic approach. This is based on a representation-theoretic argument that identifies the second-smallest dimension of an irreducible representation of a group as a key parameter that determines its viability in this framework. Our proof builds on Gowers' result concerning product-free sets in quasirandom groups. We then give another barrier that rules out certain natural matrix group constructions that make use of subgroups that are far from being self-normalizing. Our barrier results leave open several natural paths to obtain ω = 2 via matrix groups. To explore these routes we propose working in the continuous setting of Lie groups, in which we develop an analogous theory. Obtaining the analogue of ω = 2 in this potentially easier setting is a key challenge that represents an intermediate goal short of actually proving ω = 2. We give two constructions in the continuous setting, each of which evades one of our two barriers.

Cite as

Jonah Blasiak, Henry Cohn, Joshua A. Grochow, Kevin Pratt, and Chris Umans. Matrix Multiplication via Matrix Groups. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{blasiak_et_al:LIPIcs.ITCS.2023.19,
  author =	{Blasiak, Jonah and Cohn, Henry and Grochow, Joshua A. and Pratt, Kevin and Umans, Chris},
  title =	{{Matrix Multiplication via Matrix Groups}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{19:1--19:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.19},
  URN =		{urn:nbn:de:0030-drops-175226},
  doi =		{10.4230/LIPIcs.ITCS.2023.19},
  annote =	{Keywords: Fast matrix multiplication, representation theory, matrix groups}
}

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Finite Matrix Multiplication Algorithms from Infinite Groups

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Matrix Multiplication via Matrix Groups

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