On the Efficiency of Sinkhorn-Knopp for Entropically Regularized Optimal Transport

Abstract

The Sinkhorn--Knopp (SK) algorithm is a cornerstone method for matrix scaling and entropically regularized optimal transport (EOT). Despite its empirical efficiency, existing theoretical guarantees to achieve a target marginal accuracy $\varepsilon$ deteriorate severely in the presence of outliers, bottlenecked either by the global maximum regularized cost $\eta\|C\|_\infty$ (where $\eta$ is the regularization parameter and $C$ the cost matrix) or the matrix's minimum-to-maximum entry ratio $\nu$. This creates a fundamental disconnect between theory and practice. In this paper, we resolve this discrepancy. For EOT, we introduce the novel concept of well-boundedness, a local bulk mass property that rigorously isolates the well-behaved portion of the data from extreme outliers. We prove that governed by this fundamental notion, SK recovers the target transport plan for a problem of dimension $n$ in $O(\log n - \log \varepsilon)$ iterations, completely independent of the regularized cost $\eta\|C\|_\infty$. Furthermore, we show that a virtually cost-free pre-scaling step eliminates the dimensional dependence entirely, accelerating convergence to a strictly dimension-free $O(\log(1/\varepsilon))$ iterations. Beyond EOT, we establish a sharp phase transition for general $(\boldsymbol{u},\boldsymbol{v})$-scaling governed by a critical matrix density threshold. We prove that when a matrix's density exceeds this threshold, the iteration complexity is strictly independent of $\nu$. Conversely, when the density falls below this threshold, the dependence on $\nu$ becomes unavoidable; in this sub-critical regime, we construct instances where SK requires $\Omega(n/\varepsilon)$ iterations.