In this paper, a regularization of Wasserstein barycenters for random measures supported on @math is introduced via convex penalization. The existence and uniqueness of such barycenters is first proved for a large class of penalization functions. The Bregman divergence associated to the penalization term is then considered to obtain a stability result on penalized barycenters. This allows the comparison of data made of @math absolutely continuous probability measures, within the more realistic setting where one only has access to a dataset of random variables sampled from unknown distributions. The convergence of the penalized empirical barycenter of a set of @math iid random probability measures towards its population counterpart is finally analyzed. This approach is shown to be appropriate for the statistical analysis of either discrete or absolutely continuous random measures. It also allows to construct, from a set of discrete measures, consistent estimators of population Wasserstein barycenters that are absolutely continuous.
* Statistical inference using optimal transport. The penalized barycenter problem is motivated by the nonparametric method introduced in @cite_2 for the classical density estimation problem from discrete samples. It is based on a variational regularization approach involving the Wasserstein distance as a data fidelity term. However, the adaptation of this work for the penalization of Wasserstein barycenter has not been considered so far.
@cite_2
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The work of M.B. has been supported by the German Science Foundation (DFG) through project Regularization with Singular Energies. C.B.S acknowledges the financial support provided by the Cambridge Centre for Analysis (CCA), the DFG Graduiertenkolleg 1023 Identification in Mathematical Models: Synergy of Stochastic and Numerical Methods and the project WWTF Five senses-Call 2006, Mathematical Methods for Image Analysis and Processing in the Visual Arts. Further, this publication is based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).