This paper is focused on the statistical analysis of probability measures @math on @math that can be viewed as independent realizations of an underlying stochastic process. We consider the situation of practical importance where the random measures @math are absolutely continuous with densities @math that are not directly observable. In this case, instead of the densities, we have access to datasets of real random variables @math organized in the form of @math experimental units, such that @math are iid observations sampled from a random measure @math for each @math . In this setting, we focus on first-order statistics methods for estimating, from such data, a meaningful structural mean measure. For the purpose of taking into account phase and amplitude variations in the observations, we argue that the notion of Wasserstein barycenter is a relevant tool. The main contribution of this paper is to characterize the rate of convergence of a (possibly smoothed) empirical Wasserstein barycenter towards its population counterpart in the asymptotic setting where both @math and @math may go to infinity. The optimality of this procedure is discussed from the minimax point of view with respect to the Wasserstein metric. We also highlight the connection between our approach and the curve registration problem in statistics. Some numerical experiments are used to illustrate the results of the paper on the convergence rate of empirical Wasserstein barycenters.
In the one dimensional case ( @math ), computing the Wasserstein barycenter of a finite set of probability measures simply amounts to averaging (in the usual way) their quantile functions. In statistics, this approach has been referred to as quantile synchronization @cite_14 . In the presence of phase variability in the data, quantile synchronization is known to be an appropriate alternative to the usual Euclidean mean of densities to compute a structural mean density that is more consistent with the data. Various asymptotic properties of quantile synchronization are studied in @cite_14 in a statistical model and asymptotic setting similar to that of this paper with @math . However, other measures of risk than the one in this paper are considered in @cite_14 , but the optimality of the resulting convergence rates of quantile synchronization is not discussed.
@cite_14
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We define a notion of barycenter for random probability measures in the Wasserstein space. We study the population barycenter in terms of existence and uniqueness. Using a duality argument, we give a precise characterization of the population barycenter for compactly supported measures, and we make a connection between averaging in the Wasserstein space and taking the expectation of optimal transport maps. Then, the problem of estimating this barycenter from @math independent and identically distributed random probability measures is considered. To this end, we study the convergence of the empirical barycenter proposed in Agueh and Carlier (2011) to its population counterpart as the number of measures @math tends to infinity. To illustrate the benefits of this approach for data analysis and statistics, we show the usefulness of averaging in the Wasserstein space for curve and image warping. In this setting, we also study the rate of convergence of the empirical barycenter to its population counterpart for some semi-parametric models of random densities.