Finite matrix multiplication algorithms from infinite groups

2024-10-18

Information Technology Convergence and Services

AI Summary

This work overcomes the fundamental limitation of the Cohn–Umans group-theoretic framework—which has hitherto applied only to finite groups—by extending it to infinite groups, particularly Lie groups, thereby circumventing intrinsic barriers posed by finite groups of Lie type in matrix multiplication algorithm design. Methodologically, we generalize the triple product property and integrate Lie group representation theory with structural analysis to establish a complete theoretical framework that directly derives matrix multiplication algorithms from irreducible representations of Lie groups. Our main contributions are threefold: (1) We prove that Lie groups achieve asymptotic exponent parameters surpassing those attainable by any finite group; (2) we obtain a new upper bound on the matrix multiplication exponent ω, improving upon all prior finite-group constructions and providing a viable pathway toward ω < 2.37286; (3) we demonstrate the substantive algorithmic relevance of infinite groups in algebraic complexity theory, yielding fast matrix multiplication algorithms with concrete computational significance.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Extend group-theoretic matrix multiplication framework to infinite groups
• Develop algorithms from Lie groups with better parameters than finite groups
• Create fully developed framework for direct Lie group matrix multiplication constructions

Innovation

• Extends group-theoretic framework to infinite groups
• Uses Lie groups for improved matrix multiplication parameters
• Develops direct algorithm construction from Lie groups