Quantitative Concentration Inequalities for Empirical Measures on Non-compact Spaces

By François Bolley, Arnaud Guillin, Cédric Villani. 2005

287 citations recorded by the source corpus.

We establish quantitative concentration estimates for the empirical measure of many independent variables, in transportation distances. As an application, we provide some error bounds for particle simulations in a model mean field problem. The tools include coupling arguments, as well as regularity and moment estimates for solutions of certain diffusive partial differential equations.

Subfields: Applied Mathematics, Statistics and Probability

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Citations in the corpus

Computational Optimal Transport (2018, methodology): “This approach is detailed in Bonneel et al. [2015], following ideas from Marc Bernot.”
Theoretical Analysis of Domain Adaptation with Optimal Transport (2016, methodology): “Following [4], it can be specialized for the case of W1 as follows5 Theorem 1 ([4], Theorem 1.”
Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance (2017, background): “The arguments of [8] rely on transportation inequalities such as the celebrated Bobkov-Götze inequality [5].”
A review of the mean field limits for Vlasov equations (2014, background): “8 of that article, see also [67], [24] for similar estimates) Proposition 7.”
Distances between transition probabilities of diffusions and applications to nonlinear Fokker–Planck–Kolmogorov equations (2016, unknown): “, [1], [2], [16], [17], [18], [20], [21], [23], [30], and [33].”
Joint distribution optimal transportation for domain adaptation (2017, unknown): “Theorem E.1 (from [35], Theorem 1.1.)”
Wasserstein Distance Guided Representation Learning for Domain Adaptation (2017, methodology): “…domain distributions has been proved, while the proof of using empirical measures on the source and target domain samples can be further proved according to Theorem 2.1 in (Bolley, Guillin, and Villani 2007) as the same way in (Redko, Habrard, and Sebban 2016) and this proof is provided in the…”
Particles approximations of Vlasov equations with singular forces : Propagation of chaos (2011, methodology): “But, the convergence may be quantified in Wasserstein distance, and recently upper bounds on the large deviations ofW1(μN , f) were obtained by Bolley, Guillin and Villani [10] and Boissard [8].”
A weighted k-nearest neighbor density estimate for geometric inference (2011, methodology): “The empirical measure μ n based on X 1 , . . . , X n is deﬁned, for any Borel set A ⊂ R d , This empirical distribution is known to provide a suitable approximation of μ with respect to the Wasserstein distance (Bolley, Guillin, and Villani [7]).”
Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations (2015, background): “Modern measure concentration results from statistics guarantee that the unknown data-generating distribution P belongs to the Wasserstein ambiguity set around P̂N with confidence 1− β if its radius is a polynomial of log(1/β)/N for sufficiently large N [11, 19].”