On the rate of convergence in Wasserstein distance of the empirical measure

By Nicolas G. Fournier, Arnaud Guillin. Probability theory and related fields, 2013

1,435 citations recorded by the source corpus.

Let μN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _N$$\end{document} be the empirical measure associated to a N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document}-sample of a given probability distribution μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} on Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document}. We are interested in the rate of convergence of μN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _N$$\end{document} to μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}, when measured in the Wasserstein distance of order p>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>0$$\end{document}. We provide some satisfying non-asymptotic Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}-bounds and concentration inequalities, for any values of p>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>0$$\end{document} and d≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 1$$\end{document}. We extend also the non asymptotic Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}-bounds to stationary ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-mixing sequences, Markov chains, and to some interacting particle systems.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/32d55bb86cdb1bc374767383c77f97d8c5bce0ea

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Citations in the corpus

A review of the mean field limits for Vlasov equations (2014, background): “Even more precise concentration estimates were recently obtained in [63].”
Minimax Statistical Learning and Domain Adaptation with Wasserstein Distances (2017, methodology): “Here, Wp(μn, νm) can be computed from unlabeled data by solving a finite-dimensional linear program [8], and the following convergence result of Fournier and Guillin [15] implies that, with high probability, both Wp(μ, μn) and Wp(ν, νm) rapidly converge to zero as n,m → ∞:”
Performative Prediction (2020, background): “To do so, we leverage the fact that, under mild regularity conditions, the empirical distribution Dn given by n samples drawn i.i.d. from a true distribution D is with high probability close to D in earth mover’s distance (Fournier & Guillin, 2015).”
Quantifying Distributional Model Risk via Optimal Transport (2016, neutral): “sporting mass between probability measures, and have been widely used in probability theory and its applications (see, for example, [43,44,53,2] for a massive collection of classical applications and [7,25,40,54,21,15,50,51,22] for a sample of growing list of new applications). Relative entropy (or)”
Distributionally Robust Logistic Regression (2015, methodology): “We now exploit a recent measure concentration result charac terizing the speed at whicĥ PN converges toP with respect to the Wasserstein distance [26] in order to der ive out-of-sample performance guarantees for distributionally robust logistic reg ression.”
Plugin estimation of smooth optimal transport maps (2021, background): “The two recent works of Hütter and Rigollet (2021) and Gunsilius (2021) establish L2(P ) convergence rates for transport map estimators.”
Central limit theorems for empirical transportation cost in general dimension (2017, background): “On the other hand EW2 2 (Pn, Q) is of order n−2/d if d ≥ 5 (see Theorem 1 and subsequent comments in [11]) and we cannot have √ W22(Pn, Q) → 0 (otherwise we would conclude that EW2 2 (Pn, Q) = o(n−1/2).”
Sampling of probability measures in the convex order and approximation of Martingale Optimal Transport problems (2017, neutral): “Then, by Theorem 2 of Fournier and Guillin [8], there are constants c, C > 0 depending on %, d, α, γ, Eα,γ such that ∀x ∈ (0, 1),P(W%(μ, μI) > x) = P(W % % (μ, μI) > x) ≤ C exp(−cIxd∨(2%)).”
Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions (2021, background): “4 below) relies on a sharp upper bound of W2 obtained in [19, 22] (see also [14, Section 5.”
Robust Wasserstein profile inference and applications to machine learning (2016, neutral): “To begin with, as a direct approach towards choosing the size of uncertainty δ, one can perhaps use a suitable concentration inequality that measures such rate of convergence in terms of Wasserstein distances (see, for example, [17], and references therein).”
Distributionally Robust Optimization: A Review (2019, background): “statement on the discrepancy between empirical distribution and the true unknown distribution, established in Fournier and Guillin [105].”
Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance (2017, methodology): “Nevertheless, the techniques employed in [16,21] are very similar to those employed in [7], and we follow the same approach.”