Empirical optimal transport on countable metric spaces: Distributional limits and statistical applications

Abstract

We derive distributional limits for empirical transport distances between probability measures supported on countable sets. Our approach is based on sensitivity analysis of optimal values of infinite dimensional mathematical programs and a delta method for non-linear derivatives. A careful calibration of the norm on the space of probability measures is needed in order to combine differentiability and weak convergence of the underlying empirical process. Based on this we provide a sufficient and necessary condition for the underlying distribution on the countable metric space for such a distributional limit to hold. We give an explicit form of the limiting distribution for ultra-metric spaces. Finally, we apply our findings to optimal transport based inference in large scale problems. An application to nanoscale microscopy is given.MSC subject classification Primary 60F05, 60B12, 62E20; Secondary 90C08, 90C31, 62G10

Fréchet means and Procrustes analysis in Wasserstein space

“…When d = 1, the problem is well understood, owing to the flat geometry of Wasserstein space (Panaretos & Zemel [61]). When d > 1, however, the Wasserstein space has non-negative curvature, and one encounters the classical difficulties of non-Euclidean statistics, augmented by the infinite dimensionality and discrete measurement of the problem (see Anderes et al [9], Sommerfeld & Munk [71] and Tameling et al [73] for challenges involved in the discrete setting).…”

Central limit theorems for empirical transportation cost in general dimension

“…We refer to [19] for a more detailed account about the history of the problem. The problem has received a renewed interest in the last few years, both in the setup P = Q (see [3,25,39]) or for general P and Q (see [36] and [41] for finitely and countably supported probabilities, [19] for the case p = 2 and general probabilities and dimension and [18,6] for dimension d = 1 and general costs).…”

Fair Learning: an optimal transport based approach

“…Both papers focus on the case where the law underlying the empirical measure and the target measure are equal (in the setup of Theorem 5.2.1, the case F = G). With the more specific goal of CLT's for empirical transportation costs, Sommerfeld and Munk [2018] considers the case when the underlying probabilities are finitely supported, while Tameling et al [2017] covers probabilities with countable support. The approach in these two cases relies on Hadamard directional differentiability of the dual form of the finite (or countable) linear program associated to optimal transportation.…”

“…For the metric W 2 (or a trimmed version of it) some limiting results for (6.1.1) were given in for one-dimensional data. More recently, Sommerfeld and Munk [2018] handles d-dimensional data and general p, but it is constrained to the case when P and Q are finitely supported (extensions to probabilities with countable support are given in Tameling et al [2017]). The picture is less complete in the case of continuous distributions.…”

“…For one-dimensional data and quadratic cost p = 2 some limiting results for (6.1.1) were given in for the metric W 2 (or a trimmed version of it). More recently, Sommerfeld and Munk [2018] handles both general cost and dimension for P and Q finitely supported, with later extensions to countable support in Tameling et al [2017]. In general, few asymptotic results are avalaible in the case of continuous probabilities.…”