On the Wasserstein distance between mutually singular measures

Giuseppe Buttazzo; Guillaume Carlier; Maxime Laborde

doi:10.1515/acv-2017-0036

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On the Wasserstein distance between mutually singular measures

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Published/Copyright: January 17, 2018

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From the journal Advances in Calculus of Variations Volume 13 Issue 2

Abstract

We study the Wasserstein distance between two measures μ,ν which are mutually singular. In particular, we are interested in minimization problems of the form

W⁢(μ,𝒜)=inf⁡{W⁢(μ,ν):ν∈𝒜},

where μ is a given probability and 𝒜 is contained in the class μ⊥ of probabilities that are singular with respect to μ. Several cases for 𝒜 are considered; in particular, when 𝒜 consists of L1 densities bounded by a constant, the optimal solution is given by the characteristic function of a domain. Some regularity properties of these optimal domains are also studied. Some numerical simulations are included, as well as the double minimization problem

min⁡{P⁢(B)+k⁢W⁢(A,B):|A∩B|=0,|A|=|B|=1},

where k>0 is a fixed constant, P⁢(A) is the perimeter of A, and both sets A,B may vary.

Keywords: Wasserstein distance; singular measures; perimeter penalization

MSC 2010: 49J45; 49M29; 49Q20; 49A99; 49B99

Communicated by Gianni Dal Maso

Funding source: Agence Nationale de la Recherche

Award Identifier / Grant number: ANR-16-CE40-0014

Funding statement: The first author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica. The second author gratefully acknowledges support from the ANR through the project MAGA (ANR-16-CE40-0014) and from Inria, équipe-projet commune MOKAPLAN. The third author acknowledges the hospitality of the Dipartimento di Matematica at Pisa University and the support from the Fondation Sciences Mathématiques de Paris. The three authors are grateful to S. Di Marino for fruitful discussions and for sharing .

Acknowledgements

The three authors are grateful to S. Di Marino for fruitful discussions and for sharing [8].

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Received: 2017-06-29

Revised: 2017-10-19

Accepted: 2017-12-21

Published Online: 2018-01-17

Published in Print: 2020-04-01

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Articles in the same Issue

https://doi.org/10.1515/acv-2017-0036

Keywords for this article

Wasserstein distance; singular measures; perimeter penalization