Barriers for fast matrix multiplication from irreversibility

Research output: Contribution to journalJournal articleResearchpeer-review


Determining the asymptotic algebraic complexity of matrix multiplication, succinctly represented by the matrix multiplication exponent ω, is a central problem in algebraic complexity theory. The best upper bounds on ω, leading to the state-of-the-art ω≤2.37.., have been obtained via Strassen's laser method and its generalization by Coppersmith and Winograd. Recent barrier results show limitations for these and related approaches to improve the upper bound on ω. 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 ω=2. In quantitative terms, we prove that the best upper bound achievable is at least twice 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. 

Original languageEnglish
Article number2
JournalTheory of Computing
Pages (from-to)1-32
Publication statusPublished - 2021
Event34th Computational Complexity Conference, CCC 2019 - New Brunswick, United States
Duration: 18 Jul 201920 Jul 2019


Conference34th Computational Complexity Conference, CCC 2019
CountryUnited States
CityNew Brunswick
SponsorMicrosoft Research

    Research areas

  • Barriers, Laser method, Matrix multiplication exponent

Number of downloads are based on statistics from Google Scholar and

No data available

ID: 279887419