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

2026-04-04

AI Summary

This work addresses the limitations of existing convergence bounds for the Sinkhorn–Knopp algorithm in the presence of outliers, which heavily depend on the regularization parameter or element-wise ratios and thus poorly reflect practical performance. To overcome this, we introduce the notion of “well-boundedness” to characterize the intrinsic quality of the dominant data structure and combine it with a pre-scaling technique to effectively isolate the influence of outliers. Building on this framework, we uncover a density-threshold-driven phase transition phenomenon in matrix scaling and establish a novel convergence analysis. Under the well-boundedness condition, the algorithm achieves ε-accuracy in only O(log(1/ε)) iterations, providing the first rigorous convergence guarantee that is independent of problem dimension, regularization cost, and outlier contamination.

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 $η\|C\|_\infty$ (where $η$ is the regularization parameter and $C$ the cost matrix) or the matrix's minimum-to-maximum entry ratio $ν$. 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 $η\|C\|_\infty$. 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 $ν$. Conversely, when the density falls below this threshold, the dependence on $ν$ becomes unavoidable; in this sub-critical regime, we construct instances where SK requires $Ω(n/\varepsilon)$ iterations.

Problem

• Sinkhorn-Knopp algorithm
• entropically regularized optimal transport
• outliers
• iteration complexity
• matrix scaling

Innovation

• Sinkhorn-Knopp algorithm
• entropically regularized optimal transport
• well-boundedness
• matrix scaling
• iteration complexity