Almost-sharp $O(k^{-1} \log k)$ convergence rate for the Sinkhorn algorithm in the asymptotically scalable case

TL;DR

This paper establishes an almost-sharp convergence rate for the Sinkhorn algorithm in the scalable case, narrowing the gap between known lower and upper bounds.

Contribution

It proves an $O(k^{-1} ext{log} k)$ convergence rate for the Sinkhorn algorithm, improving upon previous bounds and extending prior analysis.

Findings

Convergence rate of $O(k^{-1} ext{log} k)$ in the scalable case.

Bridging the gap between lower bound $oldsymbol{ extOmega(k^{-1})}$ and upper bound $oldsymbol{O(k^{-1/2})}$.

Generalizes analysis for the positive case by Dvurechensky et al. (2018).

Abstract

We prove that the Sinkhorn algorithm converges at a rate of $O(k^{-1} \log k)$ in $\ell_1$-norm marginal error, in the asymptotically scalable case. This almost closes the gap between the lower bound $\Omega(k^{-1})$ (Qu et al., 2025) and the previously best known upper bound $O(k^{-1/2})$ (L\'eger, 2021), and generalizes the analysis for the positive case by Dvurechensky et al. (2018).