矩阵乘法的新界：从 Alpha 到 Omega
New Bounds for Matrix Multiplication: from Alpha to Omega
AI总结：
本文改进 laser 方法，在 Duan 等人 2023 年技术基础上引入新要素，将矩阵乘法指数 $ω$ 降至 2.371552，并改进矩形矩阵乘法的对偶指数 $α$ 等多项界。
摘要：
本文的主要贡献是提出一种用于设计矩阵乘法算法的、经过改进的新型 laser 方法变体。该新方法建立在 [Duan, Wu, Zhou, FOCS 2023] 的近期技术之上，并引入若干新要素，不仅给出了矩阵乘法指数 $ω$ 的改进界，还改进了 [Le Gall and Urrutia, SODA 2018] 关于矩形矩阵乘法的已知界。具体而言，$ω$ 的新界为 $ω\le 2.371552$（由 $ω\le 2.371866$ 改进而来）。对于对偶矩阵乘法指数 $α$（定义为使 $ω(1,α,1)=2$ 成立的最大 $α$），我们得到改进 $α\ge 0.321334$（由 $α\ge 0.31389$ 改进而来）。对于矩形矩阵乘法的其他各种指数，也获得了类似改进。
英文摘要：
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 $ω$, but also improve the known bounds on rectangular matrix multiplication by [Le Gall and Urrutia, SODA 2018]. In particular, the new bound on $ω$ is $ω\le 2.371552$ (improved from $ω\le 2.371866$). For the dual matrix multiplication exponent $α$ defined as the largest $α$ for which $ω(1,α,1)=2$, we obtain the improvement $α\ge 0.321334$ (improved from $α\ge 0.31389$). Similar improvements are obtained for various other exponents for multiplying rectangular matrices.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
