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

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 $\hat{I}\text{c◯}(k^{-1})$ (Qu et al., 2025) and the previously best known upper bound $O(k^{-1/2})$ (Léger, 2021), and generalizes the analysis for the positive case by Dvurechensky et al. (2018).

20 pages. v3: fix typos in proof of Lemma 2.5