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.
Mots-clés : Wasserstein space, empirical and population barycenters, Fréchet mean, convergence of random variables, optimal transport, duality, curve and image warping, deformable models
@article{PS_2018__22__35_0,
author = {Bigot, J\'er\'emie and Klein, Thierry},
title = {Characterization of barycenters in the {Wasserstein} space by averaging optimal transport maps},
journal = {ESAIM: Probability and Statistics},
pages = {35--57},
publisher = {EDP-Sciences},
volume = {22},
year = {2018},
doi = {10.1051/ps/2017020},
mrnumber = {3872127},
zbl = {1409.62049},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2017020/}
}
TY - JOUR AU - Bigot, Jérémie AU - Klein, Thierry TI - Characterization of barycenters in the Wasserstein space by averaging optimal transport maps JO - ESAIM: Probability and Statistics PY - 2018 SP - 35 EP - 57 VL - 22 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2017020/ DO - 10.1051/ps/2017020 LA - en ID - PS_2018__22__35_0 ER -
%0 Journal Article %A Bigot, Jérémie %A Klein, Thierry %T Characterization of barycenters in the Wasserstein space by averaging optimal transport maps %J ESAIM: Probability and Statistics %D 2018 %P 35-57 %V 22 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2017020/ %R 10.1051/ps/2017020 %G en %F PS_2018__22__35_0
Bigot, Jérémie; Klein, Thierry. Characterization of barycenters in the Wasserstein space by averaging optimal transport maps. ESAIM: Probability and Statistics, Tome 22 (2018), pp. 35-57. doi : 10.1051/ps/2017020. http://www.numdam.org/articles/10.1051/ps/2017020/
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