Sinkhorn-Knopp的紧非渐近局部收敛性

中文摘要 AI 辅助

我们重新研究用于矩阵缩放问题的Sinkhorn-Knopp（SK）算法。尽管已有大量关于SK及其变体全局收敛性的文献，但其局部线性收敛行为仍鲜为人知。我们通过对SK进行首次非渐近局部分析来填补这一空白，该分析与现有基于渐近雅可比的论证所得速率相匹配。我们证明，在特定连通性条件下，SK是用于双随机矩阵缩放的多项式时间算法。利用所开发的工具，我们展示了SK的局部次优性并提供了加速变体。最后，对于稠密矩阵，我们将现有一阶矩阵缩放算法的复杂度从$O(\tfrac{n^{7/3}}{\varepsilon^{2/3}})$提升至$O(\tfrac{n^{9/4}}{\sqrt{\varepsilon}})$。

英文摘要

We revisit the Sinkhorn-Knopp (SK) algorithm for the matrix scaling problem. Despite extensive literature on the global convergence of SK and its variants, its local linear convergence behavior remains less understood. We address this gap by providing the first nonasymptotic local analysis of SK that matches the rate obtained from existing asymptotic Jacobian-based arguments. We show that under certain connectivity conditions, SK is a polynomial-time algorithm for doubly stochastic matrix scaling. With the developed tools, we showcase the local suboptimality of SK and provide accelerated variants. Finally, for dense matrices, we improve the complexity of existing first-order matrix scaling algorithms from $O(\tfrac{n^{7/3}}{\varepsilon^{2/3}})$ to $O(\tfrac{n^{9/4}}{\sqrt{\varepsilon}})$.

发表机构

• Stanford University（斯坦福大学）
• Columbia University（哥伦比亚大学）

机构由 AI 辅助整理，请以论文原文为准。