Phase transition of the Sinkhorn-Knopp algorithm

TL;DR

This paper analyzes the iteration complexity of the Sinkhorn-Knopp matrix scaling algorithm, establishing a phase transition at density 1/2 with tight bounds for both upper and lower iteration limits.

Contribution

It provides the first tight bounds on the iteration count of Sinkhorn-Knopp, revealing a phase transition at density 1/2 for normalized matrices.

Findings

Nearly doubly stochastic matrices are produced in O(log n - log ε) iterations for matrices with density > 1/2.

A tight lower bound of Ω(n^{1/2}/ε) iterations is established for positive matrices.

A phase transition at density γ=1/2 is identified in the algorithm's performance.

Abstract

The matrix scaling problem, particularly the Sinkhorn-Knopp algorithm, has been studied for over 60 years. In practice, the algorithm often yields high-quality approximations within just a few iterations. Theoretically, however, the best-known upper bound places it in the class of pseudopolynomial-time approximation algorithms. Meanwhile, the lower-bound landscape remains largely unexplored. Two fundamental questions persist: what accounts for the algorithm's strong empirical performance, and can a tight bound on its iteration count be established? For an $n\times n$ matrix, its normalized version is obtained by dividing each entry by its largest entry. We say that a normalized matrix has a density $\gamma$ if there exists a constant $\rho > 0$ such that one row or column has exactly $\lceil \gamma n \rceil$ entries with values at least $\rho$, and every other row and column has at least $\lceil \gamma n \rceil$ such entries. For the upper bound, we show that the Sinkhorn-Knopp algorithm produces a nearly doubly stochastic matrix in $O(\log n - \log \varepsilon)$ iterations and $\widetilde{O}(n^2)$ time for all nonnegative square matrices whose normalized version has a density $\gamma > 1/2$. Such matrices cover both the algorithm's principal practical inputs and its typical theoretical regime, and the $\widetilde{O}(n^2)$ runtime is optimal. For the lower bound, we establish a tight bound of $\widetilde{\Omega}\left(n^{1/2}/\varepsilon\right)$ iterations for positive matrices under the $\ell_2$-norm error measure. Moreover, for every $\gamma < 1/2$, there exists a matrix with density $\gamma$ for which the algorithm requires $\Omega\left(n^{1/2}/\varepsilon\right)$ iterations. In summary, our results reveal a sharp phase transition in the Sinkhorn-Knopp algorithm at the density threshold $\gamma = 1/2$.