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 improves 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：本文的主要贡献是对设计矩阵乘法算法的激光方法提出了一种新的改进变体。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$\omega$的下界，还提升了[Le Gall and Urrutia SODA'2018]关于矩形矩阵乘法的已知界。特别地，$\omega$的新上界为$\omega \le 2.371552$（从$\omega \le 2.371866$改进而来）。对于对偶矩阵乘法指数$\alpha$（定义为使$\omega(1, \alpha, 1) = 2$的最大$\alpha$），我们获得了$\alpha \ge 0.321334$的改进（从$\alpha \ge 0.31389$改进而来）。在矩形矩阵乘法的其他各类指数上也获得了类似改进。