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Barriers for Fast Matrix Multiplication from Irreversibility

Authors Matthias Christandl , Péter Vrana , Jeroen Zuiddam



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Matthias Christandl
  • Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark
Péter Vrana
  • Department of Geometry, Budapest University of Technology and Economics, Egry József u. 1., 1111 Budapest, Hungary
  • MTA-BME Lendület (Momentum) Research group
Jeroen Zuiddam
  • Institute for Advanced Study, 1 Einstein Drive, Princeton 08540, NJ, USA

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Matthias Christandl, Péter Vrana, and Jeroen Zuiddam. Barriers for Fast Matrix Multiplication from Irreversibility. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.CCC.2019.26

Abstract

Determining the asymptotic algebraic complexity of matrix multiplication, succinctly represented by the matrix multiplication exponent omega, is a central problem in algebraic complexity theory. The best upper bounds on omega, leading to the state-of-the-art omega <= 2.37.., have been obtained via the laser method of Strassen and its generalization by Coppersmith and Winograd. Recent barrier results show limitations for these and related approaches to improve the upper bound on omega. We introduce a new and more general barrier, providing stronger limitations than in previous work. Concretely, we introduce the notion of "irreversibility" of a tensor and we prove (in some precise sense) that any approach that uses an irreversible tensor in an intermediate step (e.g., as a starting tensor in the laser method) cannot give omega = 2. In quantitative terms, we prove that the best upper bound achievable is lower bounded by two times the irreversibility of the intermediate tensor. The quantum functionals and Strassen support functionals give (so far, the best) lower bounds on irreversibility. We provide lower bounds on the irreversibility of key intermediate tensors, including the small and big Coppersmith - Winograd tensors, that improve limitations shown in previous work. Finally, we discuss barriers on the group-theoretic approach in terms of "monomial" irreversibility.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Matrix multiplication exponent
  • barriers
  • laser method

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