文章基本信息

标题：Barriers for Fast Matrix Multiplication from Irreversibility
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作者：Matthias Christandl ; Péter Vrana ; Jeroen Zuiddam 等
期刊名称：Theory of Computing
印刷版ISSN：1557-2862
电子版ISSN：1557-2862
出版年度：2021
卷号：17
期号：1
页码：1-32
DOI：10.4086/toc.2021.v017a002
语种：English
出版社：University of Chicago
摘要：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.
关键词：algebraic complexity; matrix multiplication; barriers; tensors; tensor rank