Upper and lower risk bounds for estimating the Wasserstein barycenter of random measures on the real line

Abstract

This paper is focused on the statistical analysis of probability measures ν 1 , . . . , ν n on R that can be viewed as independent realizations of an underlying stochastic process. We consider the situation of practical importance where the random measures ν i are absolutely continuous with densities f i that are not directly observable. In this case, instead of the densities, we have access to datasets of real random variables (X i,j ) 1≤i≤n; 1≤j≤pi organized in the form of n experimental units, such that X i,1 , . . . , X i,pi are iid observations sampled from a random measure ν i for each 1 ≤ i ≤ n. 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 n and min 1≤i≤n p i 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.

Statistical data analysis in the Wasserstein space

“…Moreover, as shown in [13], the population Wasserstein barycenter µ P is the unique a.c. measure with quantile function…”

“…Techniques based on optimal transport for data science have thus recently received an increasing interest in mathematical and computational statistics [8,[10][11][12][13][14][15]28,29,44,45,47,49,51,54,58,62,66,67], machine learning [4,6,22,23,32,[36][37][38]40,55,57], image processing and computer vision [7,17,24,30,31,41,52,59,60] or computational biology [56].…”

“…In this setting, the following upper bound on the expected quadratic risk E W 2 2 μ n,p , µ P has been obtained in [13].…”

“…Hence, if J 2 (µ P ) < +∞, it follows thatμ n,p converges to zero at the parametric rate O( 1 n + 1 p ). A detailed discussion on the derivation of upper bounds on the rate of converge ofμ n,p is proposed in [13] with some results on minimax optimality of such bounds.…”

Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics

“…Finally, note that while possibly suboptimal in some cases, the rates provided by Theorems 2.1 and 2.5, combined with examples 2.4, 2.6 and discussions of paragraph 3.2, provide up to our knowledge the first rates for the Wasserstein barycenter problem at this level of generality. An exception is the Wasserstein space over the real line (studied, for instance, in [BGKL18]) which happens to be isometric to a convex subset of a Hilbert space as can be deduced for instance from combining statement (iii) of Proposition 3.5 in [Stu03] and Proposition 4.1 in [Klo10].…”