Tight Nonasymptotic Local Convergence of Sinkhorn-Knopp

• URL: http://arxiv.org/abs/2608.11760v2
• Date: Thu, 13 Aug 2026 02:05:14 GMT
• Title: Tight Nonasymptotic Local Convergence of Sinkhorn-Knopp
• Abstract summary: We revisit the Sinkhorn-Knopp algorithm for the matrix scaling problem.<n>We show that under certain conditions, SK is a convergence-time algorithm for doubly matrix scaling.
• Score: 21.37855446714896
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: 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 $O(\tfrac{n^{9/4}}{\sqrt{\varepsilon}})$.

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