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.

翻译：本文的主要贡献在于提出了一种用于设计矩阵乘法算法的激光方法的新改进变体。基于[Duan, Wu, Zhou, FOCS 2023]的最新技术，新方法引入了若干新要素，不仅改进了矩阵乘法指数$\omega$的界，还改进了[Le Gall和Urrutia, SODA 2018]中关于矩形矩阵乘法的已知界。具体而言，$\omega$的新界为$\omega\le 2.371552$（从$\omega\le 2.371866$改进而来）。对于对偶矩阵乘法指数$\alpha$（定义为满足$\omega(1,\alpha,1)=2$的最大$\alpha$），我们获得了改进$\alpha \ge 0.321334$（从$\alpha \ge 0.31389$改进而来）。对于矩形矩阵相乘的其他各种指数，也获得了类似改进。