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New Bounds for Matrix Multiplication: from Alpha to Omega

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arxiv 2307.07970 v2 pith:JPFRAFQD submitted 2023-07-16 cs.DS cs.CC

Virginia Vassilevska Williams , Yinzhan Xu , Zixuan Xu , Renfei Zhou This is my paper · ORCID
classification cs.DScs.CC
keywords alphaomegamatrixmultiplicationimprovedboundboundsexponent
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abstract

The main contribution of this paper is a new improved variant of the laser method for designing matrix multiplication algorithms. Building upon the recent techniques of [Duan, Wu, Zhou, FOCS 2023], the new method introduces several new ingredients that not only yield an improved bound on the matrix multiplication exponent $\omega$, but also improve the known bounds on rectangular matrix multiplication by [Le Gall and Urrutia, SODA 2018]. In particular, the new bound on $\omega$ is $\omega\le 2.371552$ (improved from $\omega\le 2.371866$). For the dual matrix multiplication exponent $\alpha$ defined as the largest $\alpha$ for which $\omega(1,\alpha,1)=2$, we obtain the improvement $\alpha \ge 0.321334$ (improved from $\alpha \ge 0.31389$). Similar improvements are obtained for various other exponents for multiplying rectangular matrices.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Asymptotic tensor rank is characterized by polynomials

    cs.CC 2024-11 conditional novelty 8.0 of 10

    Sublevel sets of asymptotic tensor rank are Zariski-closed, making the parameter well-ordered in value, complete over the complex numbers, and computable from above.

  2. Towards Faster Feasible Matrix Multiplication by Trilinear Aggregation

    cs.DS 2025-08 reject novelty 6.0 of 10

    The paper claims a 44x44 base-case matrix multiplication algorithm with exponent 2.773203, beating Pan's 2.773372, but the proof of the key lemma is flawed.

  3. Finding Large Sets Without Arithmetic Progressions of Length Three: An Empirical View and Survey II

    math.CO 2025-01 reject novelty 4.0 of 10

    For n up to 186 the paper tabulates exact maximum sizes of 3-free subsets, but its headline large-n comparison uses a sphere variant that is not actually 3-free.

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