TY - JOUR

T1 - Barriers for fast matrix multiplication from irreversibility

AU - Christandl, Matthias

AU - Vrana, Péter

AU - Zuiddam, Jeroen

PY - 2021

Y1 - 2021

N2 - 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. 

AB - 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. 

KW - Barriers

KW - Laser method

KW - Matrix multiplication exponent

UR - http://www.scopus.com/inward/record.url?scp=85070684439&partnerID=8YFLogxK

U2 - 10.4086/toc.2021.v017a002

DO - 10.4086/toc.2021.v017a002

M3 - Journal article

VL - 17

SP - 1

EP - 32

JO - Theory of Computing

JF - Theory of Computing

SN - 1557-2862

M1 - 2

T2 - 34th Computational Complexity Conference, CCC 2019

Y2 - 18 July 2019 through 20 July 2019

ER -