发表机构
Pathway（Pathway公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文扩展了OpenAI的矩阵乘法分析至矩形情形，证明相关指数界，并应用于全对最短路径，将运行时间改进至$O(n^{2.4999})$。
AI 中文摘要
在这篇注记中，我们将OpenAI最近一项矩阵乘法结果背后的分析扩展到矩形乘积，并证明对于$0\le k\le \frac{1}{2}$有$\omega(1,k,1)\le 2$，对于$k\ge \frac{1}{2}$有$\omega(1,k,1)\le 1+k+\frac{1}{4k}$。特别地，$\omega(1,\frac{1}{2},1)=2$，且对偶指数满足$\alpha\ge \frac{1}{2}$。我们使用了该工作中的共享腿熵不等式和多项式乘法退化，保留了双腿对称性和每个扇区的方向。对数平均产生齐次辅助轮廓。它们的幂次版本具有共同的渐近斜率，对其截距进行约束给出了谱约束$b\le 4a(1-a)$。通过张量谱对偶性，这得到了矩形曲线。作为应用，Zwick算法用于有向无权图中的全对最短路径，运行时间为$O(n^{2.5})$。将矩形界与Alman和Vassilevska Williams的$(\min,+)$-乘积改进相结合，进一步得到$O(n^{2.4999})$的运行时间。
英文摘要
In this note, we extend the analysis underlying a recent matrix-multiplication result by OpenAI to rectangular products and prove that $ω(1,k,1)\le 2$ for $0\le k\le \frac{1}{2}$ and $ω(1,k,1)\le 1+k+\frac{1}{4k}$ for $k\ge \frac{1}{2}$. In particular, $ω(1,\frac{1}{2},1)=2$ and the dual exponent satisfies $α\ge \frac{1}{2}$. We use the shared-leg entropy inequality and polynomial-multiplication degenerations from that work, retaining two-leg symmetry and the orientation of each sector. Logarithmic averaging produces homogeneous auxiliary profiles. Their powered versions have a common asymptotic slope, and bounding their intercepts gives the spectral constraint $b\le 4a(1-a)$. This yields the rectangular curve by tensor-spectrum duality. As an application, Zwick's algorithm for all-pairs shortest paths in directed unweighted graphs runs in $O(n^{2.5})$ time. Combining the rectangular bound with the $(\min,+)$-product improvement of Alman and Vassilevska Williams further gives $O(n^{2.4999})$ running time.
