Duan, Wu and Zhou (FOCS 2023) recently obtained the improved upper bound on the exponent of square matrix multiplication $\omega<2.3719$ by introducing a new approach to quantify and compensate the ``combination loss" in prior analyses of powers of the Coppersmith-Winograd tensor. In this paper we show how to use this new approach to improve the exponent of rectangular matrix multiplication as well. Our main technical contribution is showing how to combine this analysis of the combination loss and the analysis of the fourth power of the Coppersmith-Winograd tensor in the context of rectangular matrix multiplication developed by Le Gall and Urrutia (SODA 2018).

翻译：Duan、Wu 和 Zhou (FOCS 2023) 近期通过引入一种新方法，对先前分析 Coppersmith-Winograd 张量幂次时的“组合损失”进行量化与补偿，将方形矩阵乘法指数 $\omega<2.3719$ 的上界改进至更优。本文展示如何利用这一新方法改进矩形矩阵乘法的指数。我们的主要技术贡献在于，将组合损失分析框架与 Le Gall 及 Urrutia (SODA 2018) 在矩形矩阵乘法中提出的 Coppersmith-Winograd 张量四次幂分析相结合。