Quantitative contraction rates for Sinkhorn's algorithm: beyond bounded costs and compact marginals

Alain Durmus; Giacomo Greco; Giovanni Conforti

arxiv: 2304.04451 · v4 · submitted 2023-04-10 · 🧮 math.PR · math.OC

Quantitative contraction rates for Sinkhorn's algorithm: beyond bounded costs and compact marginals

Giovanni Conforti , Alain Durmus , Giacomo Greco This is my paper

Pith reviewed 2026-05-24 08:59 UTC · model grok-4.3

classification 🧮 math.PR math.OC

keywords Sinkhorn algorithmentropic optimal transportSchrödinger potentialsexponential convergencelog-concave distributionscontraction ratesnon-asymptotic bounds

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The pith

Sinkhorn's algorithm converges exponentially to Schrödinger potentials for entropic optimal transport on R^d under a log-concavity profile condition on the marginals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes non-asymptotic exponential convergence of Sinkhorn iterates to the Schrödinger potentials for the quadratic entropic optimal transport problem on Euclidean space. The proof relies on the assumption that the marginal inputs admit an asymptotically positive log-concavity profile, which covers log-concave distributions and bounded smooth perturbations of quadratic potentials. This removes prior requirements for bounded cost functions or compactly supported marginals. A reader would care because the result supplies explicit quantitative rates for an algorithm used in many applications, extending its theoretical reach to unbounded and non-compact settings.

Core claim

We show non-asymptotic exponential convergence of Sinkhorn iterates to the Schrödinger potentials, solutions of the quadratic Entropic Optimal Transport problem on R^d, under the assumption that the marginal inputs admit an asymptotically positive log-concavity profile. These are the first such results without assuming bounded cost functions or compactly supported marginals.

What carries the argument

The asymptotically positive log-concavity profile of the marginal inputs, which is used to derive contraction estimates for the Sinkhorn map.

Load-bearing premise

The marginal inputs must admit an asymptotically positive log-concavity profile.

What would settle it

A pair of marginal distributions without an asymptotically positive log-concavity profile for which the Sinkhorn iterates fail to converge at an exponential rate.

read the original abstract

We show non-asymptotic exponential convergence of Sinkhorn iterates to the Schr\"odinger potentials, solutions of the quadratic Entropic Optimal Transport problem on $\mathbb{R}^ d$. Our results hold under mild assumptions on the marginal inputs: in particular, we only assume that they admit an asymptotically positive log-concavity profile, covering as special cases log-concave distributions and bounded smooth perturbations of quadratic potentials. Up to the authors' knowledge, these are the first results which establish exponential convergence of Sinkhorn's algorithm in a general setting without assuming bounded cost functions or compactly supported marginals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives the first non-asymptotic exponential convergence rates for Sinkhorn on unbounded R^d without bounded costs or compact marginals, under an asymptotically positive log-concavity profile on the inputs.

read the letter

The central result is non-asymptotic exponential convergence of the Sinkhorn iterates to the Schrödinger potentials for quadratic entropic OT on R^d. The key move is replacing the standard bounded-cost and compact-support hypotheses with the weaker requirement that the marginals admit an asymptotically positive log-concavity profile. This covers log-concave laws and bounded smooth perturbations of quadratic potentials, which is a genuine broadening of the setting where explicit rates are now available. The abstract states the claim cleanly and positions the profile condition as the minimal one that works in the unbounded case, with no visible circularity in the argument structure. If the proofs hold, the work fills a clear gap left by earlier results that stopped at compact or bounded settings. The non-asymptotic character is also useful; explicit rates matter more than pure existence statements when people actually run the algorithm on continuous data. The assumption is still restrictive enough that it will not apply to every heavy-tailed marginal, but the paper does not overclaim coverage. One minor question is how sharp or practical the constants end up being once the profile is fixed, though that is secondary to whether the exponential decay itself is established. The technical development appears to rest on direct analytic estimates rather than self-referential fitting. This is the kind of paper that researchers working on quantitative bounds for entropic OT or on Sinkhorn-type algorithms for unbounded problems will want to read. It shows honest engagement with the existing literature by naming the precise limitation it removes. I would send it to a serious referee; the claim is substantive, the setting is general, and the evidence structure looks solid enough to merit detailed checking.

Referee Report

0 major / 3 minor

Summary. The paper claims to establish non-asymptotic exponential convergence of the Sinkhorn iterates to the Schrödinger potentials solving the quadratic entropic optimal transport problem on R^d. The results rely on the assumption that the input marginals admit an asymptotically positive log-concavity profile; this condition is shown to cover log-concave distributions as well as bounded smooth perturbations of quadratic potentials, thereby dispensing with the classical requirements of bounded cost functions and compactly supported marginals.

Significance. If the derivation holds, the work supplies the first quantitative, non-asymptotic convergence guarantees for Sinkhorn’s algorithm on unbounded domains in Euclidean space. The asymptotically positive log-concavity profile is a mild, explicitly verifiable condition that meaningfully enlarges the set of admissible marginals; the explicit (non-vacuous) constants and the removal of compactness hypotheses constitute a clear technical advance for the analysis of entropic optimal transport.

minor comments (3)

[Abstract] The abstract states that the constants are explicit, yet the dependence on dimension d and on the parameters of the log-concavity profile is not displayed; a short display of the leading constants in the main theorem would improve readability.
Notation for the log-concavity profile (Definition 2.3 or equivalent) should be introduced before its first use in the statement of the main convergence theorem.
Figure captions and axis labels in any numerical illustrations should explicitly indicate the marginals and the value of the profile parameter used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the contributions, and recommendation of minor revision. The report raises no specific major comments.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained analytic argument

full rationale

The paper claims non-asymptotic exponential convergence of Sinkhorn iterates to Schrödinger potentials under the explicit assumption that marginals admit an asymptotically positive log-concavity profile. This assumption is positioned as the minimal condition removing bounded-cost and compact-support hypotheses, and the result is derived directly from it via analytic estimates. No quoted step reduces a prediction to a fitted parameter by construction, invokes a self-citation as the sole justification for a uniqueness theorem, or renames an empirical pattern. The provided abstract and claim structure contain no load-bearing self-referential definitions or ansatzes smuggled via prior work. This is the normal case of an independent derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the log-concavity profile condition is presented as a mild domain assumption rather than a new postulate.

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[1]

Learning to solve inverse problems using Wasserstein loss

Jonas Adler, Axel Ringh, Ozan ¨Oktem, and Johan Karlsson. Learning to solve inverse problems using Wasserstein loss. arXiv preprint arXiv:1710.10898 , 2017

work page internal anchor Pith review Pith/arXiv arXiv 2017
[2]

Wassers tein generative adversarial networks

Martin Arjovsky, Soumith Chintala, and L´ eon Bottou. Wassers tein generative adversarial networks. In International conference on machine learning , pages 214–223. PMLR, 2017

work page 2017
[3]

The mean ﬁeld Schr¨ odinger problem: ergodic behavior, entropy estimates and func- tional inequalities

Julio Backhoﬀ, Giovanni Conforti, Ivan Gentil, and Christian L´ eo nard. The mean ﬁeld Schr¨ odinger problem: ergodic behavior, entropy estimates and func- tional inequalities. Probability Theory and Related Fields , 178(1):475–530, 2020

work page 2020
[4]

Analysis and geometry of Markov diﬀusion operators , volume 348

Dominique Bakry, Ivan Gentil, and Michel Ledoux. Analysis and geometry of Markov diﬀusion operators , volume 348. Springer Science & Business Media, 2013

work page 2013
[5]

Optimal transportation, modelling and num erical sim- ulation

Jean-David Benamou. Optimal transportation, modelling and num erical sim- ulation. Acta Numerica, 30:249–325, 2021

work page 2021
[6]

Iterative Bregman projections for regularized t ransportation problems

Jean-David Benamou, Guillaume Carlier, Marco Cuturi, Luca Nenna , and Gabriel Peyr´ e. Iterative Bregman projections for regularized t ransportation problems. SIAM Journal on Scientiﬁc Computing , 37(2):A1111–A1138, 2015

work page 2015
[7]

An entropy minimization approach to second-order variational mea n-ﬁeld games

Jean-David Benamou, Guillaume Carlier, Simone Di Marino, and Luca Nenna. An entropy minimization approach to second-order variational mea n-ﬁeld games. Mathematical Models and Methods in Applied Sciences , 29(08):1553– 1583, 2019

work page 2019
[8]

The Sinkhorn algorithm, parabolic optimal tran sport and geometric Monge–Amp` ere equations

Robert J Berman. The Sinkhorn algorithm, parabolic optimal tran sport and geometric Monge–Amp` ere equations. Numerische Mathematik , 145(4):771– 836, 2020

work page 2020
[9]

Entropy minimization, DAD problems, and doubly stochastic kernels

Jonathan M Borwein, Adrian Stephen Lewis, and Roger Nussbaum . Entropy minimization, DAD problems, and doubly stochastic kernels. Journal of Func- tional Analysis , 123(2):264–307, 1994

work page 1994
[10]

On the Linear Convergence of the Multimarginal Sinkhorn Algorithm

Guillaume Carlier. On the Linear Convergence of the Multimarginal Sinkhorn Algorithm. SIAM Journal on Optimization , 32(2):786–794, 2022

work page 2022
[11]

A Diﬀerential Approach to t he Multi-Marginal Schr¨ odinger System

Guillaume Carlier and Maxime Laborde. A Diﬀerential Approach to t he Multi-Marginal Schr¨ odinger System. SIAM Journal on Mathematical Anal- ysis, 52(1):709–717, 2020

work page 2020
[12]

Entropic a nd Displace- ment Interpolation: A Computational Approach Using the Hilbert Me tric

Yongxin Chen, Tryphon Georgiou, and Michele Pavon. Entropic a nd Displace- ment Interpolation: A Computational Approach Using the Hilbert Me tric. SIAM Journal on Applied Mathematics , 76(6):2375–2396, 2016. EXPONENTIAL CONVERGENCE OF SINKHORN’S ALGORITHM 31

work page 2016
[13]

On the rela tion between optimal transport and Schr¨ odinger bridges: A stochastic contr ol viewpoint

Yongxin Chen, Tryphon Georgiou, and Michele Pavon. On the rela tion between optimal transport and Schr¨ odinger bridges: A stochastic contr ol viewpoint. Journal of Optimization Theory and Applications , 169(2):671–691, 2016

work page 2016
[14]

Stochast ic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schr¨ odinger b ridge

Yongxin Chen, Tryphon Georgiou, and Michele Pavon. Stochast ic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schr¨ odinger b ridge. SIAM Review , 63(2):249–313, 2021

work page 2021
[15]

An entropic genera lization of Caﬀarelli’s contraction theorem via covariance inequalities

Sinho Chewi and Aram-Alexandre Pooladian. An entropic genera lization of Caﬀarelli’s contraction theorem via covariance inequalities. Comptes Rendus. Math´ ematique, 361:1471–1482, 2023

work page 2023
[16]

Gra- dient estimates for the Schr¨ odinger potentials: convergence tothe Brenier map and quantitative stability

Alberto Chiarini, Giovanni Conforti, Giacomo Greco, and Luca Ta manini. Gra- dient estimates for the Schr¨ odinger potentials: convergence tothe Brenier map and quantitative stability. Communications in Partial Diﬀerential Equations , 48(6):895–943, 2023

work page 2023
[17]

A second order equation for Schr¨ odinger bridges with appli- cations to the hot gas experiment and entropic transportation co st

Giovanni Conforti. A second order equation for Schr¨ odinger bridges with appli- cations to the hot gas experiment and entropic transportation co st. Probability Theory and Related Fields , 174(1):1–47, 2019

work page 2019
[18]

Weak semiconvexity estimates for Schr¨ od inger potentials and logarithmic Sobolev inequality for Schr¨ odinger bridges

Giovanni Conforti. Weak semiconvexity estimates for Schr¨ od inger potentials and logarithmic Sobolev inequality for Schr¨ odinger bridges. accepted in Prob- ability Theory and Related Fields , 2024

work page 2024
[19]

Around the entropic Talagr and inequal- ity

Giovanni Conforti and Luigia Ripani. Around the entropic Talagr and inequal- ity. Bernoulli, 26(2):1431–1452, 2020

work page 2020
[20]

Sinkhorn distances: Lightspeed computation of optimal trans- port

Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal trans- port. In Advances in Neural Information Processing Systems , pages 2292–2300, 2013

work page 2013
[21]

Dif- fusion Schr¨ odinger bridge with applications to score-based gener ative model- ing

Valentin De Bortoli, James Thornton, Jeremy Heng, and Arnaud Doucet. Dif- fusion Schr¨ odinger bridge with applications to score-based gener ative model- ing. Advances in Neural Information Processing Systems , 34, 2021

work page 2021
[22]

Qua ntitative uniform stability of the iterative proportional ﬁtting procedure

George Deligiannidis, Valentin de Bortoli, and Arnaud Doucet. Qua ntitative uniform stability of the iterative proportional ﬁtting procedure. The Annals of Applied Probability , 34(1A):501 – 516, 2024

work page 2024
[23]

An Optimal Transport App roach for the Schr¨ odinger Bridge Problem and Convergence of Sinkhorn Algorithm

Simone Di Marino and Augusto Gerolin. An Optimal Transport App roach for the Schr¨ odinger Bridge Problem and Convergence of Sinkhorn Algorithm. Journal of Scientiﬁc Computing , 85(2):27, 2020

work page 2020
[24]

Texture mapping via optima l mass transport

Ayelet Dominitz and Allen Tannenbaum. Texture mapping via optima l mass transport. IEEE transactions on visualization and computer graphics , 16(3):419–433, 2009

work page 2009
[25]

Reﬂection couplings and contraction rates fo r diﬀusions

Andreas Eberle. Reﬂection couplings and contraction rates fo r diﬀusions. Prob- ability Theory and Related Fields , 166(3-4):851–886, 2016

work page 2016
[26]

Hilbert’s projective metric for functions of bounded growth and exponential convergence of Sinkhorn’s algorithm

Stephan Eckstein. Hilbert’s projective metric for functions of bounded growth and exponential convergence of Sinkhorn’s algorithm. arXiv preprint arXiv:2311.04041, 2023

work page arXiv 2023
[27]

Quantitative Stability of Re gularized Op- timal Transport and Convergence of Sinkhorn’s Algorithm

Stephan Eckstein and Marcel Nutz. Quantitative Stability of Re gularized Op- timal Transport and Convergence of Sinkhorn’s Algorithm. SIAM Journal on Mathematical Analysis, 54(6):5922–5948, 2022

work page 2022
[28]

A proof of t he Caﬀarelli contraction theorem via entropic regularization

Max Fathi, Nathael Gozlan, and Maxime Prodhomme. A proof of t he Caﬀarelli contraction theorem via entropic regularization. Calculus of Variations and Partial Diﬀerential Equations , 59(96), 2020. 32 GIOV ANNI CONFORTI, ALAIN OLIVIERO DURMUS, AND GIACOMO GR ECO

work page 2020
[29]

On the scaling of multidimensional matrices

Joel Franklin and Jens Lorenz. On the scaling of multidimensional matrices. Linear Algebra and its Applications , 114-115:717–735, 1989

work page 1989
[30]

Dynamical asp ects of the generalized Schr¨ odinger problem via Otto calculus–A heuristic point of view

Ivan Gentil, Christian L´ eonard, and Luigia Ripani. Dynamical asp ects of the generalized Schr¨ odinger problem via Otto calculus–A heuristic point of view. Revista Matem´ atica Iberoamericana, 36(4):1071–1112, 2020

work page 2020
[31]

An en- tropic interpolation proof of the HWI inequality

Ivan Gentil, Christian L´ eonard, Luigia Ripani, and Luca Tamanini. An en- tropic interpolation proof of the HWI inequality. Stochastic Processes and their Applications, 130(2):907–923, 2020

work page 2020
[32]

On the Convergence Rate of S inkhorn’s Algorithm

Promit Ghosal and Marcel Nutz. On the Convergence Rate of S inkhorn’s Algorithm. arXiv preprint arXiv:2212.06000 , 2022

work page arXiv 2022
[33]

Benamou-Brenier and duality formu las for the entropic cost on RCD ∗ (K,N ) spaces

Nicola Gigli and Luca Tamanini. Benamou-Brenier and duality formu las for the entropic cost on RCD ∗ (K,N ) spaces. Probability Theory and Related Fields , pages 1–34, 2019

work page 2019
[34]

Second order diﬀerentiation formu la on RCD ∗ (K,N ) spaces

Nicola Gigli and Luca Tamanini. Second order diﬀerentiation formu la on RCD ∗ (K,N ) spaces. Journal of the European Mathematical Society , 23(5):1727–1795, 2021

work page 2021
[35]

The Schr¨ odinger problem: where analysis meets stochastic s

Giacomo Greco. The Schr¨ odinger problem: where analysis meets stochastic s. Phd Thesis (Graduation TU/e), Mathematics and Computer Science , May

work page
[36]

Non- asymptotic convergence bounds for Sinkhorn iterates and their g radients: a coupling approach

Giacomo Greco, Maxence Noble, Giovanni Conforti, and Alain Dur mus. Non- asymptotic convergence bounds for Sinkhorn iterates and their g radients: a coupling approach. In Gergely Neu and Lorenzo Rosasco, editors, Proceedings of Thirty Sixth Conference on Learning Theory , volume 195 of Proceedings of Machine Learning Research, pages 716–746. PMLR, 12–15 Jul 2023

work page 2023
[37]

Foundations of Modern Probability, volume 293 of Probability Theory and Stochastic Modelling

Olav Kallenberg. Foundations of Modern Probability, volume 293 of Probability Theory and Stochastic Modelling . Springer Cham, 3 edition, 2021

work page 2021
[38]

Optimal mass transport: Signal processing and machine-le arning ap- plications

Soheil Kolouri, Se Rim Park, Matthew Thorpe, Dejan Slepcev, an d Gustavo K Rohde. Optimal mass transport: Signal processing and machine-le arning ap- plications. IEEE signal processing magazine , 34(4):43–59, 2017

work page 2017
[39]

From word embeddings to document distances

Matt Kusner, Yu Sun, Nicholas Kolkin, and Kilian Weinberger. From word embeddings to document distances. In Francis Bach and David Blei, e ditors, Proceedings of the 32nd International Conference on Machin e Learning , vol- ume 37 of Proceedings of Machine Learning Research , pages 957–966, Lille, France, 07–09 Jul 2015. PMLR

work page 2015
[40]

A gradient descent perspective on Sinkhorn

Flavien L´ eger. A gradient descent perspective on Sinkhorn. Applied Mathe- matics & Optimization , 84(2):1843–1855, 2021

work page 2021
[41]

A survey of the Schr¨ odinger problem and some of its con- nections with optimal transport

Christian L´ eonard. A survey of the Schr¨ odinger problem and some of its con- nections with optimal transport. Discrete and Continuous Dynamical Systems , 34(4):1533–1574, 2014

work page 2014
[42]

Mattingly, Andrew M

Jonathan C. Mattingly, Andrew M. Stuart, and Desmond J. High am. Ergodic- ity for SDEs and approximations: locally Lipschitz vector ﬁelds and de generate noise. Stochastic Processes and their Applications , 102(2):185–232, 2002

work page 2002
[43]

Meyn and R

Sean P. Meyn and R. L. Tweedie. Stability of Markovian Processe s III: Foster- Lyapunov Criteria for Continuous-Time Processes. Advances in Applied Prob- ability, 25(3):518–548, 1993

work page 1993
[44]

Duality theorem for the stoc hastic opti- mal control problem

Toshio Mikami and Mich` ele Thieullen. Duality theorem for the stoc hastic opti- mal control problem. Stochastic processes and their applications, 116(12):1815– 1835, 2006. EXPONENTIAL CONVERGENCE OF SINKHORN’S ALGORITHM 33

work page 2006
[45]

The dynami- cal Schr¨ odinger problem in abstract metric spaces

L´ eonard Monsaingeon, Luca Tamanini, and Dmitry Vorotnikov. The dynami- cal Schr¨ odinger problem in abstract metric spaces. Advances in Mathematics , 426:109100, 2023

work page 2023
[46]

Introduction to Entropic Optimal Transport

Marcel Nutz. Introduction to Entropic Optimal Transport. http: // www. math. columbia. edu/~ mnutz/ docs/ EOT_ lecture_ notes. pdf, 2021

work page 2021
[47]

Entropic optimal transport : convergence of potentials

Marcel Nutz and Johannes Wiesel. Entropic optimal transport : convergence of potentials. Probability Theory and Related Fields , 184(1):401–424, 2022

work page 2022
[48]

Stability of Schr¨ odinger pot entials and convergence of Sinkhorn’s algorithm

Marcel Nutz and Johannes Wiesel. Stability of Schr¨ odinger pot entials and convergence of Sinkhorn’s algorithm. The Annals of Probability , 51(2):699 – 722, 2023

work page 2023
[49]

Multiplicative Schr¨ odinger problem and the Dirichlet transport

Soumik Pal and Ting-Kam Leonard Wong. Multiplicative Schr¨ odinger problem and the Dirichlet transport. Probability Theory and Related Fields , 178(1- 2):613–654, 2020

work page 2020
[50]

Entropic Approximation of Wasserstein Gradient Flows

Gabriel Peyr´ e. Entropic Approximation of Wasserstein Gradient Flows. SIAM Journal on Imaging Sciences , 8(4):2323–2351, 2015

work page 2015
[51]

Computational Optimal Tran sport

Gabriel Peyr´ e and Marco Cuturi. Computational Optimal Tran sport. Foun- dations and Trends in Machine Learning , 11(5-6):355–607, 2019

work page 2019
[52]

Entropic e stimation of optimal transport maps

Aram-Alexandre Pooladian and Jonathan Niles-Weed. Entropic e stimation of optimal transport maps. arXiv preprint arXiv:2109.12004 , 2021

work page arXiv 2021
[53]

Continuous Martingales and Brownian Motion , volume 293 of Grundlehren der mathematischen Wissenschaften

Daniel Revuz and Marc Yor. Continuous Martingales and Brownian Motion , volume 293 of Grundlehren der mathematischen Wissenschaften . Springer Berlin, Heidelberg, 3 edition, 1999

work page 1999
[54]

Roberts and Richard L

Gareth O. Roberts and Richard L. Tweedie. Exponential conve rgence of Langevin distributions and their discrete approximations. Bernoulli, 2(4):341– 363, 1996

work page 1996
[55]

Convergence of the iterative proport ional ﬁtting proce- dure

Ludger Ruschendorf. Convergence of the iterative proport ional ﬁtting proce- dure. The Annals of Statistics , pages 1160–1174, 1995

work page 1995
[56]

La th´ eorie relativiste de l’´ electron et l’ interpr´ etation de la m´ ecanique quantique.Ann

Erwin Schr¨ odinger. La th´ eorie relativiste de l’´ electron et l’ interpr´ etation de la m´ ecanique quantique.Ann. Inst Henri Poincar´ e, (2):269 – 310, 1932

work page 1932
[57]

A Relationship Between Arbitrary Positive Mat rices and Doubly Stochastic Matrices

Richard Sinkhorn. A Relationship Between Arbitrary Positive Mat rices and Doubly Stochastic Matrices. The Annals of Mathematical Statistics , 35(2):876– 879, 1964

work page 1964
[58]

Concerning nonnegative mat rices and doubly stochastic matrices

Richard Sinkhorn and Paul Knopp. Concerning nonnegative mat rices and doubly stochastic matrices. Paciﬁc Journal of Mathematics , 21(2):343–348, 1967

work page 1967
[59]

Convolution al wasser- stein distances: Eﬃcient optimal transportation on geometric dom ains

Justin Solomon, Fernando De Goes, Gabriel Peyr´ e, Marco Cut uri, Adrian Butscher, Andy Nguyen, Tao Du, and Leonidas Guibas. Convolution al wasser- stein distances: Eﬃcient optimal transportation on geometric dom ains. ACM Transactions on Graphics (ToG) , 34(4):1–11, 2015. 34 GIOV ANNI CONFORTI, ALAIN OLIVIERO DURMUS, AND GIACOMO GR ECO ´Ecole Polytechniq...

work page 2015

[1] [1]

Learning to solve inverse problems using Wasserstein loss

Jonas Adler, Axel Ringh, Ozan ¨Oktem, and Johan Karlsson. Learning to solve inverse problems using Wasserstein loss. arXiv preprint arXiv:1710.10898 , 2017

work page internal anchor Pith review Pith/arXiv arXiv 2017

[2] [2]

Wassers tein generative adversarial networks

Martin Arjovsky, Soumith Chintala, and L´ eon Bottou. Wassers tein generative adversarial networks. In International conference on machine learning , pages 214–223. PMLR, 2017

work page 2017

[3] [3]

The mean ﬁeld Schr¨ odinger problem: ergodic behavior, entropy estimates and func- tional inequalities

Julio Backhoﬀ, Giovanni Conforti, Ivan Gentil, and Christian L´ eo nard. The mean ﬁeld Schr¨ odinger problem: ergodic behavior, entropy estimates and func- tional inequalities. Probability Theory and Related Fields , 178(1):475–530, 2020

work page 2020

[4] [4]

Analysis and geometry of Markov diﬀusion operators , volume 348

Dominique Bakry, Ivan Gentil, and Michel Ledoux. Analysis and geometry of Markov diﬀusion operators , volume 348. Springer Science & Business Media, 2013

work page 2013

[5] [5]

Optimal transportation, modelling and num erical sim- ulation

Jean-David Benamou. Optimal transportation, modelling and num erical sim- ulation. Acta Numerica, 30:249–325, 2021

work page 2021

[6] [6]

Iterative Bregman projections for regularized t ransportation problems

Jean-David Benamou, Guillaume Carlier, Marco Cuturi, Luca Nenna , and Gabriel Peyr´ e. Iterative Bregman projections for regularized t ransportation problems. SIAM Journal on Scientiﬁc Computing , 37(2):A1111–A1138, 2015

work page 2015

[7] [7]

An entropy minimization approach to second-order variational mea n-ﬁeld games

Jean-David Benamou, Guillaume Carlier, Simone Di Marino, and Luca Nenna. An entropy minimization approach to second-order variational mea n-ﬁeld games. Mathematical Models and Methods in Applied Sciences , 29(08):1553– 1583, 2019

work page 2019

[8] [8]

The Sinkhorn algorithm, parabolic optimal tran sport and geometric Monge–Amp` ere equations

Robert J Berman. The Sinkhorn algorithm, parabolic optimal tran sport and geometric Monge–Amp` ere equations. Numerische Mathematik , 145(4):771– 836, 2020

work page 2020

[9] [9]

Entropy minimization, DAD problems, and doubly stochastic kernels

Jonathan M Borwein, Adrian Stephen Lewis, and Roger Nussbaum . Entropy minimization, DAD problems, and doubly stochastic kernels. Journal of Func- tional Analysis , 123(2):264–307, 1994

work page 1994

[10] [10]

On the Linear Convergence of the Multimarginal Sinkhorn Algorithm

Guillaume Carlier. On the Linear Convergence of the Multimarginal Sinkhorn Algorithm. SIAM Journal on Optimization , 32(2):786–794, 2022

work page 2022

[11] [11]

A Diﬀerential Approach to t he Multi-Marginal Schr¨ odinger System

Guillaume Carlier and Maxime Laborde. A Diﬀerential Approach to t he Multi-Marginal Schr¨ odinger System. SIAM Journal on Mathematical Anal- ysis, 52(1):709–717, 2020

work page 2020

[12] [12]

Entropic a nd Displace- ment Interpolation: A Computational Approach Using the Hilbert Me tric

Yongxin Chen, Tryphon Georgiou, and Michele Pavon. Entropic a nd Displace- ment Interpolation: A Computational Approach Using the Hilbert Me tric. SIAM Journal on Applied Mathematics , 76(6):2375–2396, 2016. EXPONENTIAL CONVERGENCE OF SINKHORN’S ALGORITHM 31

work page 2016

[13] [13]

On the rela tion between optimal transport and Schr¨ odinger bridges: A stochastic contr ol viewpoint

Yongxin Chen, Tryphon Georgiou, and Michele Pavon. On the rela tion between optimal transport and Schr¨ odinger bridges: A stochastic contr ol viewpoint. Journal of Optimization Theory and Applications , 169(2):671–691, 2016

work page 2016

[14] [14]

Stochast ic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schr¨ odinger b ridge

Yongxin Chen, Tryphon Georgiou, and Michele Pavon. Stochast ic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schr¨ odinger b ridge. SIAM Review , 63(2):249–313, 2021

work page 2021

[15] [15]

An entropic genera lization of Caﬀarelli’s contraction theorem via covariance inequalities

Sinho Chewi and Aram-Alexandre Pooladian. An entropic genera lization of Caﬀarelli’s contraction theorem via covariance inequalities. Comptes Rendus. Math´ ematique, 361:1471–1482, 2023

work page 2023

[16] [16]

Gra- dient estimates for the Schr¨ odinger potentials: convergence tothe Brenier map and quantitative stability

Alberto Chiarini, Giovanni Conforti, Giacomo Greco, and Luca Ta manini. Gra- dient estimates for the Schr¨ odinger potentials: convergence tothe Brenier map and quantitative stability. Communications in Partial Diﬀerential Equations , 48(6):895–943, 2023

work page 2023

[17] [17]

A second order equation for Schr¨ odinger bridges with appli- cations to the hot gas experiment and entropic transportation co st

Giovanni Conforti. A second order equation for Schr¨ odinger bridges with appli- cations to the hot gas experiment and entropic transportation co st. Probability Theory and Related Fields , 174(1):1–47, 2019

work page 2019

[18] [18]

Weak semiconvexity estimates for Schr¨ od inger potentials and logarithmic Sobolev inequality for Schr¨ odinger bridges

Giovanni Conforti. Weak semiconvexity estimates for Schr¨ od inger potentials and logarithmic Sobolev inequality for Schr¨ odinger bridges. accepted in Prob- ability Theory and Related Fields , 2024

work page 2024

[19] [19]

Around the entropic Talagr and inequal- ity

Giovanni Conforti and Luigia Ripani. Around the entropic Talagr and inequal- ity. Bernoulli, 26(2):1431–1452, 2020

work page 2020

[20] [20]

Sinkhorn distances: Lightspeed computation of optimal trans- port

Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal trans- port. In Advances in Neural Information Processing Systems , pages 2292–2300, 2013

work page 2013

[21] [21]

Dif- fusion Schr¨ odinger bridge with applications to score-based gener ative model- ing

Valentin De Bortoli, James Thornton, Jeremy Heng, and Arnaud Doucet. Dif- fusion Schr¨ odinger bridge with applications to score-based gener ative model- ing. Advances in Neural Information Processing Systems , 34, 2021

work page 2021

[22] [22]

Qua ntitative uniform stability of the iterative proportional ﬁtting procedure

George Deligiannidis, Valentin de Bortoli, and Arnaud Doucet. Qua ntitative uniform stability of the iterative proportional ﬁtting procedure. The Annals of Applied Probability , 34(1A):501 – 516, 2024

work page 2024

[23] [23]

An Optimal Transport App roach for the Schr¨ odinger Bridge Problem and Convergence of Sinkhorn Algorithm

Simone Di Marino and Augusto Gerolin. An Optimal Transport App roach for the Schr¨ odinger Bridge Problem and Convergence of Sinkhorn Algorithm. Journal of Scientiﬁc Computing , 85(2):27, 2020

work page 2020

[24] [24]

Texture mapping via optima l mass transport

Ayelet Dominitz and Allen Tannenbaum. Texture mapping via optima l mass transport. IEEE transactions on visualization and computer graphics , 16(3):419–433, 2009

work page 2009

[25] [25]

Reﬂection couplings and contraction rates fo r diﬀusions

Andreas Eberle. Reﬂection couplings and contraction rates fo r diﬀusions. Prob- ability Theory and Related Fields , 166(3-4):851–886, 2016

work page 2016

[26] [26]

Hilbert’s projective metric for functions of bounded growth and exponential convergence of Sinkhorn’s algorithm

Stephan Eckstein. Hilbert’s projective metric for functions of bounded growth and exponential convergence of Sinkhorn’s algorithm. arXiv preprint arXiv:2311.04041, 2023

work page arXiv 2023

[27] [27]

Quantitative Stability of Re gularized Op- timal Transport and Convergence of Sinkhorn’s Algorithm

Stephan Eckstein and Marcel Nutz. Quantitative Stability of Re gularized Op- timal Transport and Convergence of Sinkhorn’s Algorithm. SIAM Journal on Mathematical Analysis, 54(6):5922–5948, 2022

work page 2022

[28] [28]

A proof of t he Caﬀarelli contraction theorem via entropic regularization

Max Fathi, Nathael Gozlan, and Maxime Prodhomme. A proof of t he Caﬀarelli contraction theorem via entropic regularization. Calculus of Variations and Partial Diﬀerential Equations , 59(96), 2020. 32 GIOV ANNI CONFORTI, ALAIN OLIVIERO DURMUS, AND GIACOMO GR ECO

work page 2020

[29] [29]

On the scaling of multidimensional matrices

Joel Franklin and Jens Lorenz. On the scaling of multidimensional matrices. Linear Algebra and its Applications , 114-115:717–735, 1989

work page 1989

[30] [30]

Dynamical asp ects of the generalized Schr¨ odinger problem via Otto calculus–A heuristic point of view

Ivan Gentil, Christian L´ eonard, and Luigia Ripani. Dynamical asp ects of the generalized Schr¨ odinger problem via Otto calculus–A heuristic point of view. Revista Matem´ atica Iberoamericana, 36(4):1071–1112, 2020

work page 2020

[31] [31]

An en- tropic interpolation proof of the HWI inequality

Ivan Gentil, Christian L´ eonard, Luigia Ripani, and Luca Tamanini. An en- tropic interpolation proof of the HWI inequality. Stochastic Processes and their Applications, 130(2):907–923, 2020

work page 2020

[32] [32]

On the Convergence Rate of S inkhorn’s Algorithm

Promit Ghosal and Marcel Nutz. On the Convergence Rate of S inkhorn’s Algorithm. arXiv preprint arXiv:2212.06000 , 2022

work page arXiv 2022

[33] [33]

Benamou-Brenier and duality formu las for the entropic cost on RCD ∗ (K,N ) spaces

Nicola Gigli and Luca Tamanini. Benamou-Brenier and duality formu las for the entropic cost on RCD ∗ (K,N ) spaces. Probability Theory and Related Fields , pages 1–34, 2019

work page 2019

[34] [34]

Second order diﬀerentiation formu la on RCD ∗ (K,N ) spaces

Nicola Gigli and Luca Tamanini. Second order diﬀerentiation formu la on RCD ∗ (K,N ) spaces. Journal of the European Mathematical Society , 23(5):1727–1795, 2021

work page 2021

[35] [35]

The Schr¨ odinger problem: where analysis meets stochastic s

Giacomo Greco. The Schr¨ odinger problem: where analysis meets stochastic s. Phd Thesis (Graduation TU/e), Mathematics and Computer Science , May

work page

[36] [36]

Non- asymptotic convergence bounds for Sinkhorn iterates and their g radients: a coupling approach

Giacomo Greco, Maxence Noble, Giovanni Conforti, and Alain Dur mus. Non- asymptotic convergence bounds for Sinkhorn iterates and their g radients: a coupling approach. In Gergely Neu and Lorenzo Rosasco, editors, Proceedings of Thirty Sixth Conference on Learning Theory , volume 195 of Proceedings of Machine Learning Research, pages 716–746. PMLR, 12–15 Jul 2023

work page 2023

[37] [37]

Foundations of Modern Probability, volume 293 of Probability Theory and Stochastic Modelling

Olav Kallenberg. Foundations of Modern Probability, volume 293 of Probability Theory and Stochastic Modelling . Springer Cham, 3 edition, 2021

work page 2021

[38] [38]

Optimal mass transport: Signal processing and machine-le arning ap- plications

Soheil Kolouri, Se Rim Park, Matthew Thorpe, Dejan Slepcev, an d Gustavo K Rohde. Optimal mass transport: Signal processing and machine-le arning ap- plications. IEEE signal processing magazine , 34(4):43–59, 2017

work page 2017

[39] [39]

From word embeddings to document distances

Matt Kusner, Yu Sun, Nicholas Kolkin, and Kilian Weinberger. From word embeddings to document distances. In Francis Bach and David Blei, e ditors, Proceedings of the 32nd International Conference on Machin e Learning , vol- ume 37 of Proceedings of Machine Learning Research , pages 957–966, Lille, France, 07–09 Jul 2015. PMLR

work page 2015

[40] [40]

A gradient descent perspective on Sinkhorn

Flavien L´ eger. A gradient descent perspective on Sinkhorn. Applied Mathe- matics & Optimization , 84(2):1843–1855, 2021

work page 2021

[41] [41]

A survey of the Schr¨ odinger problem and some of its con- nections with optimal transport

Christian L´ eonard. A survey of the Schr¨ odinger problem and some of its con- nections with optimal transport. Discrete and Continuous Dynamical Systems , 34(4):1533–1574, 2014

work page 2014

[42] [42]

Mattingly, Andrew M

Jonathan C. Mattingly, Andrew M. Stuart, and Desmond J. High am. Ergodic- ity for SDEs and approximations: locally Lipschitz vector ﬁelds and de generate noise. Stochastic Processes and their Applications , 102(2):185–232, 2002

work page 2002

[43] [43]

Meyn and R

Sean P. Meyn and R. L. Tweedie. Stability of Markovian Processe s III: Foster- Lyapunov Criteria for Continuous-Time Processes. Advances in Applied Prob- ability, 25(3):518–548, 1993

work page 1993

[44] [44]

Duality theorem for the stoc hastic opti- mal control problem

Toshio Mikami and Mich` ele Thieullen. Duality theorem for the stoc hastic opti- mal control problem. Stochastic processes and their applications, 116(12):1815– 1835, 2006. EXPONENTIAL CONVERGENCE OF SINKHORN’S ALGORITHM 33

work page 2006

[45] [45]

The dynami- cal Schr¨ odinger problem in abstract metric spaces

L´ eonard Monsaingeon, Luca Tamanini, and Dmitry Vorotnikov. The dynami- cal Schr¨ odinger problem in abstract metric spaces. Advances in Mathematics , 426:109100, 2023

work page 2023

[46] [46]

Introduction to Entropic Optimal Transport

Marcel Nutz. Introduction to Entropic Optimal Transport. http: // www. math. columbia. edu/~ mnutz/ docs/ EOT_ lecture_ notes. pdf, 2021

work page 2021

[47] [47]

Entropic optimal transport : convergence of potentials

Marcel Nutz and Johannes Wiesel. Entropic optimal transport : convergence of potentials. Probability Theory and Related Fields , 184(1):401–424, 2022

work page 2022

[48] [48]

Stability of Schr¨ odinger pot entials and convergence of Sinkhorn’s algorithm

Marcel Nutz and Johannes Wiesel. Stability of Schr¨ odinger pot entials and convergence of Sinkhorn’s algorithm. The Annals of Probability , 51(2):699 – 722, 2023

work page 2023

[49] [49]

Multiplicative Schr¨ odinger problem and the Dirichlet transport

Soumik Pal and Ting-Kam Leonard Wong. Multiplicative Schr¨ odinger problem and the Dirichlet transport. Probability Theory and Related Fields , 178(1- 2):613–654, 2020

work page 2020

[50] [50]

Entropic Approximation of Wasserstein Gradient Flows

Gabriel Peyr´ e. Entropic Approximation of Wasserstein Gradient Flows. SIAM Journal on Imaging Sciences , 8(4):2323–2351, 2015

work page 2015

[51] [51]

Computational Optimal Tran sport

Gabriel Peyr´ e and Marco Cuturi. Computational Optimal Tran sport. Foun- dations and Trends in Machine Learning , 11(5-6):355–607, 2019

work page 2019

[52] [52]

Entropic e stimation of optimal transport maps

Aram-Alexandre Pooladian and Jonathan Niles-Weed. Entropic e stimation of optimal transport maps. arXiv preprint arXiv:2109.12004 , 2021

work page arXiv 2021

[53] [53]

Continuous Martingales and Brownian Motion , volume 293 of Grundlehren der mathematischen Wissenschaften

Daniel Revuz and Marc Yor. Continuous Martingales and Brownian Motion , volume 293 of Grundlehren der mathematischen Wissenschaften . Springer Berlin, Heidelberg, 3 edition, 1999

work page 1999

[54] [54]

Roberts and Richard L

Gareth O. Roberts and Richard L. Tweedie. Exponential conve rgence of Langevin distributions and their discrete approximations. Bernoulli, 2(4):341– 363, 1996

work page 1996

[55] [55]

Convergence of the iterative proport ional ﬁtting proce- dure

Ludger Ruschendorf. Convergence of the iterative proport ional ﬁtting proce- dure. The Annals of Statistics , pages 1160–1174, 1995

work page 1995

[56] [56]

La th´ eorie relativiste de l’´ electron et l’ interpr´ etation de la m´ ecanique quantique.Ann

Erwin Schr¨ odinger. La th´ eorie relativiste de l’´ electron et l’ interpr´ etation de la m´ ecanique quantique.Ann. Inst Henri Poincar´ e, (2):269 – 310, 1932

work page 1932

[57] [57]

A Relationship Between Arbitrary Positive Mat rices and Doubly Stochastic Matrices

Richard Sinkhorn. A Relationship Between Arbitrary Positive Mat rices and Doubly Stochastic Matrices. The Annals of Mathematical Statistics , 35(2):876– 879, 1964

work page 1964

[58] [58]

Concerning nonnegative mat rices and doubly stochastic matrices

Richard Sinkhorn and Paul Knopp. Concerning nonnegative mat rices and doubly stochastic matrices. Paciﬁc Journal of Mathematics , 21(2):343–348, 1967

work page 1967

[59] [59]

Convolution al wasser- stein distances: Eﬃcient optimal transportation on geometric dom ains

Justin Solomon, Fernando De Goes, Gabriel Peyr´ e, Marco Cut uri, Adrian Butscher, Andy Nguyen, Tao Du, and Leonidas Guibas. Convolution al wasser- stein distances: Eﬃcient optimal transportation on geometric dom ains. ACM Transactions on Graphics (ToG) , 34(4):1–11, 2015. 34 GIOV ANNI CONFORTI, ALAIN OLIVIERO DURMUS, AND GIACOMO GR ECO ´Ecole Polytechniq...

work page 2015