发表机构
Matroid, Inc.（Matroid 公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出一种新的渐近矩阵乘法构造，使矩阵乘法指数ω<2.37115924，改进了此前的界，通过结合共享递归参数族与条件优化解决相关障碍，给出精确上界并提供验证工具。
AI 中文摘要
我们提出了一种新的渐近矩阵乘法构造，其指数ω<2.37115924，因此该算法可在有理数、实数或复数域上以O(n^2.37115924)次算术运算完成两个n×n矩阵的乘法。该构造在Coppersmith-Winograd张量的组合损失框架内，改进了Dupont等人给出的2.371177的界。我们的方法结合了共享递归参数族与条件优化过程，解决了两个障碍：低质量分量处的弱梯度，以及保留指数中的竞争瓶颈。一种依赖质量和概率的重缩放支持在212820个坐标上的拟牛顿优化，而六控制校正则平衡了每个相关阶段的三个提取作用。正有理熵对偶因子与独立的普通字符串验证器将所得构造转化为精确证书。经验证的上界表达式位于[2.3711592385931075066123, 2.3711592385931075066124]区间内。完整的见证及仅使用Python标准库的离线验证器随本文一同提供。
英文摘要
We present a new asymptotic matrix multiplication construction yielding $ω<2.37115924$, and therefore an algorithm that multiplies two $n\times n$ matrices in $O(n^{2.37115924})$ arithmetic operations over the rationals, reals, or complex numbers. The construction improves the bound $2.371177$ of Dupont et al. within the combination-loss framework for the Coppersmith--Winograd tensor. Our approach combines a shared recursive parameter family with a conditioned optimization procedure that addresses two obstacles: weak gradients at low-mass constituents and competing bottlenecks in the retained exponent. A mass- and probability-dependent rescaling supports quasi-Newton optimization in $212\,820$ coordinates, while a six-control correction balances the three extraction roles at each relevant stage. Positive rational entropy-dual factors and an independent ordinary-string verifier turn the resulting construction into an exact certificate. The verified upper-bound expression lies in $[2.3711592385931075066123,\,2.3711592385931075066124]$. The complete witness and an offline verifier using only the Python standard library accompany the paper.
