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.
The results of this paper are very much connected with those in @cite_2 where a new framework is developed for the registration of multiple point processes on the real line for the purpose of separating amplitude and phase variation in such data. In @cite_2 , consistent estimators of the structural mean of multiple point processes are obtained by the use of smoothed Wasserstein barycenters with an appropriate choice of kernel smoothing. Also, rates of convergence of such estimators are derived for the Wasserstein metric. The statistical analysis of multiple point processes is very much connected to the study of repeated observations organized in samples from independent subjects or experimental units. Therefore, some of our results in this paper on smoothed empirical Wasserstein barycenters are built upon the work in @cite_2 . Nevertheless, novel contributions include the derivation of an exact formula to compute the risk of non-smoothed Wasserstein barycenters in the case of samples of equal size, and new upper bounds on the rate of convergence of the Wasserstein risk of non-smoothed and smoothed empirical Wasserstein barycenters, together with a discussion of their optimality from the minimax point of view.
@cite_2
1639961155 2297520515 1629559917 1479849574
This paper proposes a new definition of the averaging of discrete probability distributions as a barycenter over the Monge-Kantorovich optimal transport space. To overcome the time complexity involved by the numerical solving of such problem, the original Wasserstein metric is replaced by a sliced approximation over 1D distributions. This enables us to introduce a new fast gradient descent algorithm to compute Wasserstein barycenters of point clouds. This new notion of barycenter of probabilities is likely to find applications in computer vision where one wants to average features defined as distributions. We show an application to texture synthesis and mixing, where a texture is characterized by the distribution of the response to a multi-scale oriented filter bank. This leads to a simple way to navigate over a convex domain of color textures. We develop a canonical framework for the study of the problem of registration of multiple point processes subjected to warping, known as the problem of separation of amplitude and phase variation. The amplitude variation of a real random function Y(x) : x is an element of [0, 1] corresponds to its random oscillations in the y-axis, typically encapsulated by its (co) variation around a mean level. In contrast, its phase variation refers to fluctuations in the x-axis, often caused by random time changes. We formalise similar notions for a point process, and nonparametrically separate them based on realisations of i.i.d. copies Pi(i) of the phase-varying point process. A key element in our approach is to demonstrate that when the classical phase variation assumptions of Functional Data Analysis (FDA) are applied to the point process case, they become equivalent to conditions interpretable through the prism of the theory of optimal transportation of measure. We demonstrate that these induce a natural Wasserstein geometry tailored to the warping problem, including a formal notion of bias expressing over-registration. Within this framework, we construct nonparametric estimators that tend to avoid over-registration in finite samples. We show that they consistently estimate the warp maps, consistently estimate the structural mean, and consistently register the warped point processes, even in a sparse sampling regime. We also establish convergence rates, and derivev root n-consistency and a central limit theorem in the Cox process case under dense sampling, showing rate optimality of our structural mean estimator in that case. We present new algorithms to compute the mean of a set of empirical probability measures under the optimal transport metric. This mean, known as the Wasserstein barycenter, is the measure that minimizes the sum of its Wasserstein distances to each element in that set. We propose two original algorithms to compute Wasserstein barycenters that build upon the subgradient method. A direct implementation of these algorithms is, however, too costly because it would require the repeated resolution of large primal and dual optimal transport problems to compute subgradients. Extending the work of Cuturi (2013), we propose to smooth the Wasserstein distance used in the definition of Wasserstein barycenters with an entropic regularizer and recover in doing so a strictly convex objective whose gradients can be computed for a considerably cheaper computational cost using matrix scaling algorithms. We use these algorithms to visualize a large family of images and to solve a constrained clustering problem. 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.