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On the contraction properties of Sinkhorn semigroups

O. Deniz Akyildiz, Pierre del Moral, Joaquin Miguez

TL;DR

This work provides a Lyapunov-based contraction theory for Sinkhorn semigroups in weighted Banach spaces, proving exponential convergence of entropically regularized transport toward Schrödinger bridges under broad divergence and distance criteria. By introducing drift-minorization conditions and a suite of contraction coefficients (Birkhoff, Dobrushin, entropy), it yields quantitative exponential decay rates in Wasserstein and various $\Phi$-divergences, and relative-entropy for the bridge distributions. The framework applies to a wide range of models, including polynomial-growth and heavy-tailed marginals, boundary and semi-compact spaces, linear-Gaussian settings, and finite mixtures, demonstrating stability across unbounded costs and noncompact marginals. These results advance the theoretical understanding of entropic OT and Sinkhorn algorithms in non-compact settings and provide practical guidance for using entropic transport in generative modeling and statistical estimation. To the authors' knowledge, the contraction inequalities established here are among the first of their kind for Sinkhorn semigroups in entropic transport.

Abstract

We develop a novel semigroup stability analysis based on Lyapunov techniques and contraction coefficients to prove exponential convergence of Sinkhorn equations on weighted Banach spaces. This operator-theoretic framework yields exponential decays of Sinkhorn iterates towards Schrödinger bridges with respect to general classes of $φ$-divergences and Kantorovich-type criteria, including the relative entropy, squared Hellinger integrals, $α$-divergences as well as weighted total variation norms and Wasserstein distances. To the best of our knowledge, these contraction inequalities are the first results of this type in the literature on entropic transport and the Sinkhorn algorithm. We also provide Lyapunov contractions principles under minimal regularity conditions that allow to provide quantitative exponential stability estimates for a large class of Sinkhorn semigroups. We apply this novel framework in a variety of situations, ranging from polynomial growth potentials and heavy tailed marginals on general normed spaces to more sophisticated boundary state space models, including semi-circle transitions, Beta, Weibull, exponential marginals as well as semi-compact models. Last but not least, our approach also allows to consider statistical finite mixture of the above models, including kernel-type density estimators of complex data distributions arising in generative modeling.

On the contraction properties of Sinkhorn semigroups

TL;DR

This work provides a Lyapunov-based contraction theory for Sinkhorn semigroups in weighted Banach spaces, proving exponential convergence of entropically regularized transport toward Schrödinger bridges under broad divergence and distance criteria. By introducing drift-minorization conditions and a suite of contraction coefficients (Birkhoff, Dobrushin, entropy), it yields quantitative exponential decay rates in Wasserstein and various -divergences, and relative-entropy for the bridge distributions. The framework applies to a wide range of models, including polynomial-growth and heavy-tailed marginals, boundary and semi-compact spaces, linear-Gaussian settings, and finite mixtures, demonstrating stability across unbounded costs and noncompact marginals. These results advance the theoretical understanding of entropic OT and Sinkhorn algorithms in non-compact settings and provide practical guidance for using entropic transport in generative modeling and statistical estimation. To the authors' knowledge, the contraction inequalities established here are among the first of their kind for Sinkhorn semigroups in entropic transport.

Abstract

We develop a novel semigroup stability analysis based on Lyapunov techniques and contraction coefficients to prove exponential convergence of Sinkhorn equations on weighted Banach spaces. This operator-theoretic framework yields exponential decays of Sinkhorn iterates towards Schrödinger bridges with respect to general classes of -divergences and Kantorovich-type criteria, including the relative entropy, squared Hellinger integrals, -divergences as well as weighted total variation norms and Wasserstein distances. To the best of our knowledge, these contraction inequalities are the first results of this type in the literature on entropic transport and the Sinkhorn algorithm. We also provide Lyapunov contractions principles under minimal regularity conditions that allow to provide quantitative exponential stability estimates for a large class of Sinkhorn semigroups. We apply this novel framework in a variety of situations, ranging from polynomial growth potentials and heavy tailed marginals on general normed spaces to more sophisticated boundary state space models, including semi-circle transitions, Beta, Weibull, exponential marginals as well as semi-compact models. Last but not least, our approach also allows to consider statistical finite mixture of the above models, including kernel-type density estimators of complex data distributions arising in generative modeling.

Paper Structure

This paper contains 47 sections, 30 theorems, 408 equations.

Table of Contents

  1. Introduction
  2. Entropic optimal transport
  3. Background
  4. Contributions
  5. Outline of the article
  6. Some basic notation
  7. Statement of main results
  8. Sinkhorn bridges
  9. Sinkhorn semigroups
  10. Some stability theorems
  11. Illustrations and comments
  12. Contraction of Markov operators
  13. Metrics and divergences
  14. Hilbert projective metric
  15. Integral operator norms
  16. ...and 32 more sections

Key Result

Theorem 2.2

Assume that the transition potential $W$ is uniformly bounded. In this case, for any $\Phi$-entropy criteria $D_{\Phi}$ there is some parameters $\rho\in ]0,1[$ and $c>0$ such that for any $n\geq 0$ we have the contraction inequalities The above inequalities remain valid when we replace $(\pi_{2n},\nu_V)$ by $(\nu_V,\pi_{2n})$. Similar estimates hold for $(\lambda_U,\pi_{2n+1})$. There also exist

Theorems & Definitions (37)

  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 4.1
  • Corollary 4.2
  • Theorem 4.4
  • Corollary 4.5
  • Remark 4.6
  • Theorem 4.8
  • Remark 4.11
  • ...and 27 more