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Hessian stability and convergence rates for entropic and Sinkhorn potentials via semiconcavity

Giacomo Greco, Luca Tamanini

TL;DR

This work addresses second-order quantitative stability in entropic optimal transport by bounding the gradient and Hessian of entropic potentials under perturbations of the second marginal. It develops a stochastic-analytic framework based on semiconcavity and Schrödinger bridges to connect stability of entropic plans with the stability of dual potentials, and it derives exponential convergence rates for the Hessians of Sinkhorn iterates with polynomial dependence on the regularization parameter $T$. The results are stated in general form and then specialized to compact-support and log-concave marginals, yielding explicit constants and rates; the analysis also discusses relaxing absolute-continuity assumptions via regularization. Overall, the paper advances understanding of stability and convergence in unbounded settings, with implications for the numerical stability and speed of Sinkhorn-type algorithms in high dimensions.

Abstract

In this paper we determine quantitative stability bounds for the Hessian of entropic potentials, \ie, the dual solution to the entropic optimal transport problem. To the authors' knowledge this is the first work addressing this second-order quantitative stability estimate in general unbounded settings. Our proof strategy relies on semiconcavity properties of entropic potentials and on the representation of entropic transport plans as laws of forward and backward diffusion processes, known as Schrödinger bridges. Moreover, our approach allows to deduce a stochastic proof of quantitative stability estimates for entropic transport plans and for gradients of entropic potentials as well. Finally, as a direct consequence of these stability bounds, we deduce exponential convergence rates for gradient and Hessian of Sinkhorn iterates along Sinkhorn's algorithm, a problem that was still open in unbounded settings. Our rates have a polynomial dependence on the regularization parameter.

Hessian stability and convergence rates for entropic and Sinkhorn potentials via semiconcavity

TL;DR

This work addresses second-order quantitative stability in entropic optimal transport by bounding the gradient and Hessian of entropic potentials under perturbations of the second marginal. It develops a stochastic-analytic framework based on semiconcavity and Schrödinger bridges to connect stability of entropic plans with the stability of dual potentials, and it derives exponential convergence rates for the Hessians of Sinkhorn iterates with polynomial dependence on the regularization parameter . The results are stated in general form and then specialized to compact-support and log-concave marginals, yielding explicit constants and rates; the analysis also discusses relaxing absolute-continuity assumptions via regularization. Overall, the paper advances understanding of stability and convergence in unbounded settings, with implications for the numerical stability and speed of Sinkhorn-type algorithms in high dimensions.

Abstract

In this paper we determine quantitative stability bounds for the Hessian of entropic potentials, \ie, the dual solution to the entropic optimal transport problem. To the authors' knowledge this is the first work addressing this second-order quantitative stability estimate in general unbounded settings. Our proof strategy relies on semiconcavity properties of entropic potentials and on the representation of entropic transport plans as laws of forward and backward diffusion processes, known as Schrödinger bridges. Moreover, our approach allows to deduce a stochastic proof of quantitative stability estimates for entropic transport plans and for gradients of entropic potentials as well. Finally, as a direct consequence of these stability bounds, we deduce exponential convergence rates for gradient and Hessian of Sinkhorn iterates along Sinkhorn's algorithm, a problem that was still open in unbounded settings. Our rates have a polynomial dependence on the regularization parameter.

Paper Structure

This paper contains 14 sections, 18 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Exponential convergence of Hessian of Sinkhorn's iterates
  3. Schrödinger bridge point of view
  4. Literature review
  5. Preliminaries
  6. Entropic and gradients' stability
  7. Proofs of the main results
  8. Quantitative stability estimates of Hessian
  9. Explicit computations for $C^{\varphi^\nu}$ and $\Lambda(\varphi^\nu_0)$
  10. Marginal $\rho$ with compact support
  11. Log-concavity of $\rho$
  12. Explicit computations for the stability of Hessians
  13. Marginal $\nu$ with compact support
  14. Log-concavity of $\nu$

Key Result

Theorem 1.1

Assume ass:basic and ass:mu. We have up to multiplicative constants that depend polynomially only on $\rho,\,\nu,\,T$ (and not on $\mu$), and are explicit. Moreover, if we specify our result to the following settings we deduce that

Theorems & Definitions (35)

  • Remark 1
  • Theorem 1.1: Informal main result
  • Remark 2
  • Theorem 1.2
  • Theorem 2.1
  • Lemma 1
  • proof
  • Corollary 1
  • proof : Proof of \ref{['thm:entropic_stab']}
  • Corollary 2
  • ...and 25 more