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Consistent estimation of a population barycenter in the Wasserstein space

Abstract : We define a notion of barycenter 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 and taking the expectation of optimal transport maps with respect to a fixed reference measure. We also discuss the usefulness of this approach for warping problems in statistics and data analysis. In this setting, the problem of estimating a population barycenter from n independent and identically distributed random probability measures is also considered. In particular, we study the convergence of the empirical barycenter proposed in Agueh and Carlier (2011) to its population counterpart as the number of measures n tends to infinity.
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https://hal.archives-ouvertes.fr/hal-00763668
Contributor : Jérémie Bigot <>
Submitted on : Monday, March 9, 2015 - 12:35:44 PM
Last modification on : Wednesday, June 9, 2021 - 10:00:09 AM
Long-term archiving on: : Wednesday, June 10, 2015 - 2:30:12 PM

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  • HAL Id : hal-00763668, version 4
  • ARXIV : 1212.2562

Citation

Jérémie Bigot, Thierry Klein. Consistent estimation of a population barycenter in the Wasserstein space. 2013. ⟨hal-00763668v4⟩

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