On the contraction properties of Sinkhorn semigroups
File(s)
Accepted version
Author(s)
Akyildiz, O Deniz
Moral, Pierre Del
Miguez, Joaquin
Type
Journal Article
Abstract
We develop a novel stability theory for Sinkhorn semigroups based on Lyapunov techniques and quantitative contraction coefficients, and establish exponential convergence of Sinkhorn iterations on weighted Banach spaces. This operator-theoretic framework yields explicit exponential decay rates of Sinkhorn iterates toward Schrödinger bridges with respect to a broad class of -divergences and Kantorovich-type distances, including relative entropy, squared Hellinger integrals, -divergences, weighted total variation norms, and Wasserstein distances. To the best of our knowledge, these results provide the first systematic contraction inequalities of this kind for entropic transport and the Sinkhorn algorithm.
We further introduce Lyapunov contraction principles under minimal regularity assumptions, leading to quantitative exponential stability estimates for a large family of Sinkhorn semigroups. The framework applies to models with polynomially growing potentials and heavy-tailed marginals on general normed spaces, as well as to more structured boundary state-space models, including semicircle transitions and Beta, Weibull, and exponential marginals, together with semi-compact settings. Finally, our approach extends naturally to statistical finite mixtures of such models, including kernel-based density estimators arising in modern generative modeling.
We further introduce Lyapunov contraction principles under minimal regularity assumptions, leading to quantitative exponential stability estimates for a large family of Sinkhorn semigroups. The framework applies to models with polynomially growing potentials and heavy-tailed marginals on general normed spaces, as well as to more structured boundary state-space models, including semicircle transitions and Beta, Weibull, and exponential marginals, together with semi-compact settings. Finally, our approach extends naturally to statistical finite mixtures of such models, including kernel-based density estimators arising in modern generative modeling.
Date Issued
2026-12-01
Date Acceptance
2026-01-17
Citation
Foundations of Data Science, 2026, 11, pp.150-206
ISSN
2639-8001
Publisher
American Institute of Mathematical Sciences
Start Page
150
End Page
206
Journal / Book Title
Foundations of Data Science
Volume
11
Issue
0
Copyright Statement
Copyright © 2026 American Institute of Mathematical Sciences. This is the author’s accepted manuscript made available under a CC-BY licence in accordance with Imperial’s Research Publications Open Access policy (www.imperial.ac.uk/oa-policy)
License URL
Subjects
2020 Mathematics Subject Classification. Primary 37M25
49Q22
47H09
60J20; secondary: 60J05
94A17 Entropic optimal transport
Sinkhorn semigroups
Schrödinger bridges
contraction coefficients
weighted total variation norms
Lyapunov functions
Kantorovich and entropy criteria
Wasserstein semi-distances
Publication Status
Published
Date Publish Online
2026-02-03