Fast matrix multiplication is one of the most fundamental problems in algorithm research. The exponent of the optimal time complexity of matrix multiplication is usually denoted by $\omega$. This paper discusses new ideas for improving the laser method for fast matrix multiplication. We observe that the analysis of higher powers of the Coppersmith-Winograd tensor [Coppersmith & Winograd 1990] incurs a "combination loss", and we partially compensate for it using an asymmetric version of CW's hashing method. By analyzing the eighth power of the CW tensor, we give a new bound of $\omega<2.371866$, which improves the previous best bound of $\omega<2.372860$ [Alman & Vassilevska Williams 2020]. Our result breaks the lower bound of $2.3725$ in [Ambainis, Filmus & Le Gall 2015] because of the new method for analyzing component (constituent) tensors.

翻译：快速矩阵乘法是算法研究中最基础的问题之一。矩阵乘法最优时间复杂度的指数通常记为$\omega$。本文讨论了改进激光方法以实现快速矩阵乘法的新思路。我们观察到，对Coppersmith-Winograd张量[Coppersmith & Winograd 1990]的高次幂分析会产生“组合损失”，并利用CW哈希方法的非对称版本部分补偿了这一损失。通过分析CW张量的八次幂，我们给出了新的界$\omega<2.371866$，优于此前的最佳界$\omega<2.372860$[Alman & Vassilevska Williams 2020]。由于采用了分析分量（构成）张量的新方法，我们的结果突破了[Ambainis, Filmus & Le Gall 2015]中$2.3725$的下界。