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arXiv · 2307.07970

New Bounds for Matrix Multiplication: from Alpha to Omega

Virginia Vassilevska Williams·Yinzhan Xu·Zixuan Xu·Renfei Zhou
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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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Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, Renfei Zhou. 2023-07-16. New Bounds for Matrix Multiplication: from Alpha to Omega. https://arxiv.org/abs/2307.07970

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