Volume 22, 2018
|Page(s)||35 - 57|
|Published online||11 October 2018|
Characterization of barycenters in the Wasserstein space by averaging optimal transport maps★
Institut de Mathématiques de Bordeaux et CNRS (UMR 5251), Université de Bordeaux,
2 ENAC – Ecole Nationale de l’Aviation Civile, 31400 Toulouse, France
3 Institut de Mathématiques de Toulouse et CNRS (UMR 5219), Université de Toulouse, 31400 Toulouse, France
* Coressponding author: firstname.lastname@example.org
Accepted: 5 November 2017
This paper is concerned by the study of barycenters for random probability measures in the Wasserstein space. Using a duality argument, we give a precise characterization of the population barycenter for various parametric classes of random probability measures with compact support. In particular, we make a connection between averaging in the Wasserstein space as introduced in Agueh and Carlier [SIAM J. Math. Anal. 43 (2011) 904–924], and taking the expectation of optimal transport maps with respect to a fixed reference measure. We also discuss the usefulness of this approach in statistics for the analysis of deformable models in signal and image processing. In this setting, the problem of estimating a population barycenter from n independent and identically distributed random probability measures is also considered.
Mathematics Subject Classification: Primary 62G05 / secondary 49J40
Key words: Wasserstein space / empirical and population barycenters / Fréchet mean / convergence of random variables / optimal transport / duality / curve and image warping / deformable models
The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-JCJC-SIMI1 DEMOS. We would like to also thank Jérôme Bertrand for fruitful discussions on Wasserstein spaces and the optimal transport problem. Finally, we thank the referees and the associate editor for their comments and suggestions.
© EDP Sciences, SMAI 2018
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