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

We prove that the Sinkhorn algorithm converges at a rate of $O(k^{-1}\log k)$ in ${\mathrm{\ell }}_1$-norm marginal error, in the asymptotically scalable case. This almost closes the gap between the lower bound $\mathrm{\Omega }(k^{-1})$ (Qu et al., 2025) and the previously best known upper bound $O(k^{-1\mathrm{/}2})$ (Léger, 2021), and generalizes the analysis for the positive case by Dvurechensky et al. (2018).