Tight Nonasymptotic Local Convergence of Sinkhorn-Knopp

2026-08-12

AI Summary

This work addresses the long-standing lack of theoretical characterization of the non-asymptotic local convergence of the Sinkhorn–Knopp algorithm for matrix scaling. We establish, for the first time, a non-asymptotic local linear convergence rate that aligns with the asymptotic Jacobian analysis, thereby revealing the algorithm’s polynomial-time complexity and local suboptimality under standard connectivity conditions. Building on this analysis, we propose an accelerated variant and improve the complexity of first-order matrix scaling for dense matrices from $O(n^{7/3}/\varepsilon^{2/3})$ to $O(n^{9/4}/\sqrt{\varepsilon})$, significantly advancing the theoretical efficiency frontier.

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 $O(\tfrac{n^{9/4}}{\sqrt{\varepsilon}})$.

Problem

• Sinkhorn-Knopp
• matrix scaling
• local convergence
• nonasymptotic analysis
• doubly stochastic

Innovation

• Sinkhorn-Knopp algorithm
• matrix scaling
• nonasymptotic convergence
• local linear convergence
• accelerated variants