Regularization of barycenters in the Wasserstein space

The concept of barycenter in the Wasserstein space allows to define a notion of Fr\'echet mean of a set of probability measures. However, depending on the data at hand, such barycenters may be irregular. In this paper, 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 analyze the convergence of the regularized empirical barycenter of a set of $n$ 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. As an illustrative example, we focus on the analysis of probability measures supported on the real line. In this setting, we propose an efficient minimization algorithm based on accelerated gradient descent for the computation of regularized Wasserstein barycenters. This approach is finally illustrated with simulated and real data sets.