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Regularization of Barycenters in the Wasserstein Space Cazelles, Elsa


The concept of barycenter in the Wasserstein space corresponds to define a notion of Fréchet mean of a set of probability measures. However, depending on the data at hand, such barycenters may be irregular. We thus introduce a convex regularization of Wasserstein barycenters for random measures supported on $\mathbb{R}^d$. We prove the existence and uniqueness of such barycenters for a large class of regularizing functions. A stability result of regularized barycenters in terms of Bregman distance associated to the convex regularization term is also given. This allows to compare the case of data made of $n$ probability measures $\nu_1,\ldots,\nu_n$ with the more realistic setting where we have only access to a dataset of random variables $(X_{i,j})_{1\leq i\leq n; 1\leq j\leq p_i}$ organized in the form of $n$ experimental units such that $X_{i,1},\ldots,X_{i,p_i}$ are iid observations sampled from the measure $\nu_i$ for each $1\leq i\leq n$. We also analyse the convergence of the regularized empirical barycenter of a set o $n$ iid random probability measures towards its population counterpart, and we discuss its rate of convergence. This approach is shown to be appropriate for the statistical analysis of discrete or absolutely continuous random measures.

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