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

• URL: http://arxiv.org/abs/2604.03787v1
• Date: Sat, 04 Apr 2026 16:24:19 GMT
• Title: On the Efficiency of Sinkhorn-Knopp for Entropically Regularized Optimal Transport
• Abstract summary: The Sinkhorn--Knopp algorithm is a cornerstone method for matrix scaling and optimal transport.<n>We introduce the novel concept of well-boundedness, a local bulk mass property.<n>We show that a virtually cost-free pre-scaling step eliminates the dimensional dependence entirely.
• Score: 10.787490135016155
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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