Global Sinkhorn Convergence and Spectral Stability of Matrix Scaling

Abstract

We give a global spectral bound for the Sinkhorn--Knopp (SK) algorithm. Let $A$ be a row-stochastic nonnegative $n\times n$ matrix admitting a finite doubly stochastic scaling $B$. Define \[ g=1-s_2(B)^2>0, \qquad D=\min\left\{\|(u,v)\|_\infty: B=\operatorname{diag}(e^u)A\operatorname{diag}(e^{-v})\right\}. \] From the standard initialization, SK achieves total marginal $\ell_1$ error $\varepsilon$ in \[ O\!\left(\frac{D+\log n+\log(1/\varepsilon)}{g}\right) \] iterations. A scale-invariant potential contracts globally according to the spectrum of $B$. The linear dependence on $D$ is necessary in the fixed-gap regime: we construct strictly positive inputs with constant input and endpoint spectral gaps, $D=Θ(n)$, and an $Ω(n)$ iteration lower bound at error $n^{-2}$. Our upper bound gives $O(n)$ iterations on this family. We next give spectral certificates for scalability and short scaling distance. For a row-stochastic input $A$ with positive column sums $c_j$, set \[ K=\frac{\max_jc_j}{\min_jc_j},\qquad σ=s_2\!\left(A\operatorname{diag}(c)^{-1/2}\right). \] The input-only condition $\sqrt K\,σ<1$ certifies exact scalability and polynomial scaling distortion, with $D=O(\log n)$ for fixed $K$ and a fixed positive spectral margin. Given a finite scaling $B$, the weaker condition $\sqrt K\,σs_2(B)<1$ gives the same structural bounds. With a constant endpoint gap, these bounds imply logarithmic SK iteration complexity. Boundary constructions show that the strict spectral inequalities cannot uniformly be replaced by nonstrict ones.