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Approximation of Optimal Transport problems with marginal moments constraints

Aurélien Alfonsi 1, 2 Rafaël Coyaud 1, 2 Virginie Ehrlacher 1, 3 Damiano Lombardi 4
2 MATHRISK - Mathematical Risk Handling
Inria de Paris, ENPC - École des Ponts ParisTech, UPEM - Université Paris-Est Marne-la-Vallée
3 MATHERIALS - MATHematics for MatERIALS
CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique, Inria de Paris
4 COMMEDIA - COmputational Mathematics for bio-MEDIcal Applications
Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
Abstract : Optimal Transport (OT) problems arise in a wide range of applications, from physics to economics. Getting numerical approximate solution of these problems is a challenging issue of practical importance. In this work, we investigate the relaxation of the OT problem when the marginal constraints are replaced by some moment constraints. Using Tchakaloff's theorem, we show that the Moment Constrained Optimal Transport problem (MCOT) is achieved by a finite discrete measure. Interestingly, for multimarginal OT problems, the number of points weighted by this measure scales linearly with the number of marginal laws, which is encouraging to bypass the curse of dimension. This approximation method is also relevant for Martingale OT problems. We show the convergence of the MCOT problem toward the corresponding OT problem. In some fundamental cases, we obtain rates of convergence in O(1/n) or O(1/n 2) where n is the number of moments, which illustrates the role of the moment functions. Last, we present algorithms exploiting the fact that the MCOT is reached by a finite discrete measure and provide numerical examples of approximations.
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Contributor : Virginie Ehrlacher <>
Submitted on : Tuesday, May 14, 2019 - 11:26:22 AM
Last modification on : Friday, April 10, 2020 - 5:28:36 PM

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  • HAL Id : hal-02128374, version 1
  • ARXIV : 1905.05663

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Aurélien Alfonsi, Rafaël Coyaud, Virginie Ehrlacher, Damiano Lombardi. Approximation of Optimal Transport problems with marginal moments constraints. 2019. ⟨hal-02128374⟩

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