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

Alberto Chiarini; Giovanni Conforti; Giacomo Greco; Luca Tamanini

doi:10.1080/03605302.2023.2215527

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

Alberto Chiarini, Giovanni Conforti, Giacomo Greco, Luca Tamanini (Corresponding author)

Applied Analysis

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Abstract

We show convergence of the gradients of the Schrödinger potentials to the (uniquely determined) gradient of Kantorovich potentials in the small-time limit under general assumptions on the marginals, which allow for unbounded densities and supports. Furthermore, we provide novel quantitative stability estimates for the optimal values and optimal couplings for the Schrödinger problem (SP), that we express in terms of a negative order weighted homogeneous Sobolev norm. The latter encodes the linearized behavior of the 2-Wasserstein distance between the marginals. The proofs of both results highlight for the first time the relevance of gradient bounds for Schrödinger potentials, that we establish here in full generality, in the analysis of the short-time behavior of Schrödinger bridges. Finally, we discuss how our results translate into the framework of quadratic Entropic Optimal Transport, that is a version of SP more suitable for applications in machine learning and data science.

Original language	English
Pages (from-to)	895-943
Number of pages	49
Journal	Communications in Partial Differential Equations
Volume	48
Issue number	6
DOIs	https://doi.org/10.1080/03605302.2023.2215527
Publication status	Published - 2023

Bibliographical note

Funding Information:
While this work was written, AC was associated to INdAM and the group GNAMPA. GC acknowledges funding from the grant SPOT (ANR-20-CE40-0014). GG acknowledges support from NWO Research Project 613.009. “Analysis meets Stochastics: Scaling limits in complex systems.” LT acknowledges support from the PRIN 2017 (Prot. 2017TEXA3H) “Gradient flows, Optimal Transport and Metric Measure Structures.”

Funding Information:
This study was funded by Agence Nationale de la Recherche; Ministero dell’Istruzione, dell’Università e della Ricerca; Nederlandse Organisatie voor Wetenschappelijk Onderzoek. While this work was written, AC was associated to INdAM and the group GNAMPA. GC acknowledges funding from the grant SPOT (ANR-20-CE40-0014). GG acknowledges support from NWO Research Project 613.009. “Analysis meets Stochastics: Scaling limits in complex systems.” LT acknowledges support from the PRIN 2017 (Prot. 2017TEXA3H) “Gradient flows, Optimal Transport and Metric Measure Structures.”

Funding

While this work was written, AC was associated to INdAM and the group GNAMPA. GC acknowledges funding from the grant SPOT (ANR-20-CE40-0014). GG acknowledges support from NWO Research Project 613.009. “Analysis meets Stochastics: Scaling limits in complex systems.” LT acknowledges support from the PRIN 2017 (Prot. 2017TEXA3H) “Gradient flows, Optimal Transport and Metric Measure Structures.” This study was funded by Agence Nationale de la Recherche; Ministero dell’Istruzione, dell’Università e della Ricerca; Nederlandse Organisatie voor Wetenschappelijk Onderzoek. While this work was written, AC was associated to INdAM and the group GNAMPA. GC acknowledges funding from the grant SPOT (ANR-20-CE40-0014). GG acknowledges support from NWO Research Project 613.009. “Analysis meets Stochastics: Scaling limits in complex systems.” LT acknowledges support from the PRIN 2017 (Prot. 2017TEXA3H) “Gradient flows, Optimal Transport and Metric Measure Structures.”

Keywords

Curvature lower bounds
entropic regularization
gradient estimates
optimal transport
quantitative stability
Schrödinger potentials
Schrödinger problem

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10.1080/03605302.2023.2215527

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title = "Gradient estimates for the Schr{\"o}dinger potentials: convergence to the Brenier map and quantitative stability",

abstract = "We show convergence of the gradients of the Schr{\"o}dinger potentials to the (uniquely determined) gradient of Kantorovich potentials in the small-time limit under general assumptions on the marginals, which allow for unbounded densities and supports. Furthermore, we provide novel quantitative stability estimates for the optimal values and optimal couplings for the Schr{\"o}dinger problem (SP), that we express in terms of a negative order weighted homogeneous Sobolev norm. The latter encodes the linearized behavior of the 2-Wasserstein distance between the marginals. The proofs of both results highlight for the first time the relevance of gradient bounds for Schr{\"o}dinger potentials, that we establish here in full generality, in the analysis of the short-time behavior of Schr{\"o}dinger bridges. Finally, we discuss how our results translate into the framework of quadratic Entropic Optimal Transport, that is a version of SP more suitable for applications in machine learning and data science.",

keywords = "Curvature lower bounds, entropic regularization, gradient estimates, optimal transport, quantitative stability, Schr{\"o}dinger potentials, Schr{\"o}dinger problem",

author = "Alberto Chiarini and Giovanni Conforti and Giacomo Greco and Luca Tamanini",

note = "Funding Information: While this work was written, AC was associated to INdAM and the group GNAMPA. GC acknowledges funding from the grant SPOT (ANR-20-CE40-0014). GG acknowledges support from NWO Research Project 613.009. “Analysis meets Stochastics: Scaling limits in complex systems.” LT acknowledges support from the PRIN 2017 (Prot. 2017TEXA3H) “Gradient flows, Optimal Transport and Metric Measure Structures.” Funding Information: This study was funded by Agence Nationale de la Recherche; Ministero dell{\textquoteright}Istruzione, dell{\textquoteright}Universit{\`a} e della Ricerca; Nederlandse Organisatie voor Wetenschappelijk Onderzoek. While this work was written, AC was associated to INdAM and the group GNAMPA. GC acknowledges funding from the grant SPOT (ANR-20-CE40-0014). GG acknowledges support from NWO Research Project 613.009. “Analysis meets Stochastics: Scaling limits in complex systems.” LT acknowledges support from the PRIN 2017 (Prot. 2017TEXA3H) “Gradient flows, Optimal Transport and Metric Measure Structures.” ",

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T2 - convergence to the Brenier map and quantitative stability

AU - Chiarini, Alberto

AU - Conforti, Giovanni

AU - Greco, Giacomo

AU - Tamanini, Luca

N1 - Funding Information: While this work was written, AC was associated to INdAM and the group GNAMPA. GC acknowledges funding from the grant SPOT (ANR-20-CE40-0014). GG acknowledges support from NWO Research Project 613.009. “Analysis meets Stochastics: Scaling limits in complex systems.” LT acknowledges support from the PRIN 2017 (Prot. 2017TEXA3H) “Gradient flows, Optimal Transport and Metric Measure Structures.” Funding Information: This study was funded by Agence Nationale de la Recherche; Ministero dell’Istruzione, dell’Università e della Ricerca; Nederlandse Organisatie voor Wetenschappelijk Onderzoek. While this work was written, AC was associated to INdAM and the group GNAMPA. GC acknowledges funding from the grant SPOT (ANR-20-CE40-0014). GG acknowledges support from NWO Research Project 613.009. “Analysis meets Stochastics: Scaling limits in complex systems.” LT acknowledges support from the PRIN 2017 (Prot. 2017TEXA3H) “Gradient flows, Optimal Transport and Metric Measure Structures.”

PY - 2023

Y1 - 2023

N2 - We show convergence of the gradients of the Schrödinger potentials to the (uniquely determined) gradient of Kantorovich potentials in the small-time limit under general assumptions on the marginals, which allow for unbounded densities and supports. Furthermore, we provide novel quantitative stability estimates for the optimal values and optimal couplings for the Schrödinger problem (SP), that we express in terms of a negative order weighted homogeneous Sobolev norm. The latter encodes the linearized behavior of the 2-Wasserstein distance between the marginals. The proofs of both results highlight for the first time the relevance of gradient bounds for Schrödinger potentials, that we establish here in full generality, in the analysis of the short-time behavior of Schrödinger bridges. Finally, we discuss how our results translate into the framework of quadratic Entropic Optimal Transport, that is a version of SP more suitable for applications in machine learning and data science.

AB - We show convergence of the gradients of the Schrödinger potentials to the (uniquely determined) gradient of Kantorovich potentials in the small-time limit under general assumptions on the marginals, which allow for unbounded densities and supports. Furthermore, we provide novel quantitative stability estimates for the optimal values and optimal couplings for the Schrödinger problem (SP), that we express in terms of a negative order weighted homogeneous Sobolev norm. The latter encodes the linearized behavior of the 2-Wasserstein distance between the marginals. The proofs of both results highlight for the first time the relevance of gradient bounds for Schrödinger potentials, that we establish here in full generality, in the analysis of the short-time behavior of Schrödinger bridges. Finally, we discuss how our results translate into the framework of quadratic Entropic Optimal Transport, that is a version of SP more suitable for applications in machine learning and data science.

KW - Curvature lower bounds

KW - entropic regularization

KW - gradient estimates

KW - optimal transport

KW - quantitative stability

KW - Schrödinger potentials

KW - Schrödinger problem

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U2 - 10.1080/03605302.2023.2215527

DO - 10.1080/03605302.2023.2215527

M3 - Article

AN - SCOPUS:85161823590

SN - 0360-5302

VL - 48

SP - 895

EP - 943

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

IS - 6

ER -

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

Abstract

Bibliographical note

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Keywords

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