← 研踪 PaperTrailarXiv:2611.08963 · v1
Global Sinkhorn Convergence and Spectral Stability of Matrix Scaling
首次提交 2026-11-06 12:17
版本更新 2026-11-06 12:17
雷达观测 2026-11-09 10:16
摘要
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.
作者备注：71 pages
观测到的版本
相邻记录可能跳过中间版本；首次发现 v2 及以上也会标为版本更新。
v1 · 2026-11-06 12:17 · 最新观测版本
Global Sinkhorn Convergence and Spectral Stability of Matrix Scaling
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.
查看此版本解读 →查看此版本原文 ↗