Stability of entropic optimal transport and Schrödinger bridges

Ghosal, Promit; Nutz, Marcel; Bernton, Espen

doi:10.1016/j.jfa.2022.109622

Cited by 17 publications

(5 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As concerns ðu 0 , w 0 Þ, due to the lack of regularity of the OT problem (1.4), even after the normalization (3.1) uniqueness of the Kantorovich potentials may fail. Some results in this direction are known for some specific examples, for instance in the Euclidean case if at least one marginal is absolutely continuous with respect to the Lebesgue measure and if its support is connected then uniqueness holds [50,Appendix B]. However here we are interested in the convergence of the Schr€ odinger map Id À 2Tru T to the Brenier map s (as introduced above in (1.9)), whose uniqueness instead holds true in our setting.…”

Section: Corrector Estimatesmentioning

confidence: 99%

“…Recently, there has been an increasing interest in the quantitative stability for the EOT problem, which is strongly linked to SP. A first stability estimate for EOT in a general setting has been established in [30]. There, the authors, without any integrability assumption, manage to prove a qualitative stability result by relying on a geometric notion inspired by the cyclical monotonicity property in Optimal Transport.…”

Section: Literature Reviewmentioning

confidence: 99%

See 1 more Smart Citation

Gradient estimates for the Schrödinger potentials: convergence to the Brenier map and quantitative stability

Chiarini

Conforti

Greco

et al. 2023

Communications in Partial Differential Equations

View full text Add to dashboard Cite

published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User

show abstract

Section: Corrector Estimatesmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

Gradient estimates for the Schrödinger potentials: convergence to the Brenier map and quantitative stability

Chiarini

Conforti

Greco

et al. 2023

Communications in Partial Differential Equations

View full text Add to dashboard Cite

show abstract

“…After the present paper was posted on arXiv, several relevant advances were made regarding this topic. We mention here Nutz and Wiesel [32], Bernton et al [7], Ghosal et al [23]. We stress that all the papers mentioned in this paragraph address a different problem: in Cuturi [17] and subsequent works, the requirement of an exact matching of the marginal distributions is maintained.…”

Section: Literature Reviewmentioning

confidence: 99%

“…The works by Bernton et al [7] and Ghosal et al [23] study geometric properties of minimisers of the entropic OT, by means of the concept of cyclical invariance. This is a counterpart to the characterisation, using c-cyclical monotonicity, of the geometry of optimal transport plans in the classical framework of OT.…”

Section: Literature Reviewmentioning

confidence: 99%

Entropy martingale optimal transport and nonlinear pricing–hedging duality

Doldi

Frittelli

2023

Finance Stoch

View full text Add to dashboard Cite

The objective of this paper is to develop a duality between a novel entropy martingale optimal transport (EMOT) problem and an associated optimisation problem. In EMOT, we follow the approach taken in the entropy optimal transport (EOT) problem developed in Liero et al. (Invent. Math. 211:969–1117, 2018), but we add the constraint, typical of martingale optimal transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures. In the associated problem, the objective functional, related via Fenchel conjugacy to the entropic term in EMOT, is no longer linear as in (martingale) optimal transport. This leads to a novel optimisation problem which also has a clear financial interpretation as a nonlinear subhedging problem. Our theory allows us to establish a nonlinear robust pricing–hedging duality which also covers a wide range of known robust results. We also focus on Wasserstein-induced penalisations and study how the duality is affected by variations in the penalty terms, with a special focus on the convergence of EMOT to the extreme case of MOT.

show abstract

“…Generalizations and extensions for discrete measures have been proved by ; . A growing body of work investigates the properties of the entropy regularized optimal transport problem from the perspective of probability and analysis, including its asymptotic properties as ϵ → 0 Nutz and Wiesel (2021); Eckstein and Nutz (2021); Ghosal et al (2021); Nutz and Wiesel (2022); Altschuler et al (2022); Berman (2020); , opening the door to further statistical applications of entropy regularised transport.…”

Section: Introductionmentioning

confidence: 99%

Statistical analysis of the optimal transport problem

González Sanz

View full text Add to dashboard Cite

Optimal transportation is a resource allocation problem present in fields such as economics, finance, physics or artificial intelligence. From a probabilistic point of view, the optimal transport cost endows the space of probability measures with a metric topology. In particular, this topology is equivalent to the weak topology of probability measures together with the convergence of moments. This makes the transport cost an appropriate tool for measuring discrepancies between distributions. On the other hand, the solution of the transport problem is known as optimal plan. That is, an unambiguous way to relate two distributions following an optimality criterion. This optimal plan, when deterministic, is called a transport map.However, in many cases the probability distribution is a theoretical, unattainable entity. It is only visible to the practitioner through its empirical version, i.e. a finite data set of size n. This work examines the asymptotic behaviour of the transport cost in its empirical version.In other words, we study the limits of the empirical cost and plans when the data grows to infinity. It is well-known that the empirical transport cost converges to the population one. Moreover, for continuous measures it does so at a rate that decreases with dimension. In this thesis we prove the consistency of the transport map using topology of set-valued maps. This leads, indirectly, to being able to state that the rate at which the fluctuations-difference between the expected empirical cost and the empirical cost itself-approximate zero is the parametric n − 1 2 , irrespective of the dimension. Moreover, these fluctuations multiplied by n 1 2 RésuméLe transport optimal est un problème d'allocation de ressources que l'on retrouve dans des domaines tels que l'économie, la finance, la physique et l'intelligence artificielle. D'un point de vue probabiliste, le coût de transport optimal dote l'espace des mesures de probabilité d'une topologie métrique. Cela fait du coût de transport un outil approprié pour mesurer les écarts entre les distributions. D'autre part, la solution du problème de transport est connue comme le plan optimal. C'est-à-dire une manière non ambiguë de mettre en relation deux distributions suivant un critère d'optimalité. Ce plan optimal, lorsqu'il est déterministe, s'appelle une application de transport. Cependant, la distribution de probabilité est souvent une entité théorique, irréalisable. Elle n'est visible pour le praticien qu'à travers sa version empirique, c'est-à-dire un ensemble de données fini de taille n. Ce document examine le comportement asymptotique du coût de transport dans sa version empirique. En d'autres termes, nous étudions les limites du coût empirique et de le plan lorsque les données croissent à l'infini. Les travaux précédents ont montré que le coût de transport empirique converge vers le coût théorique. De plus, pour les mesures continues, elle le fait à un taux qui diminue avec la dimension. Dans cette thèse, nous démontrons la cohérence de l'application de transp...

show abstract

Stability of entropic optimal transport and Schrödinger bridges

Cited by 17 publications

References 39 publications

Gradient estimates for the Schrödinger potentials: convergence to the Brenier map and quantitative stability

Gradient estimates for the Schrödinger potentials: convergence to the Brenier map and quantitative stability

Entropy martingale optimal transport and nonlinear pricing–hedging duality

Statistical analysis of the optimal transport problem

Contact Info

Product

Resources

About