On the mean speed of convergence of empirical and occupation measures in Wasserstein distance

By Emmanuel Boissard, Thibaut Le Gouic. 2011

145 citations recorded by the source corpus.

In this work, we provide non-asymptotic bounds for the average speed of convergence of the empirical measure in the law of large numbers, in Wasserstein distance. We also consider occupation measures of ergodic Markov chains. One motivation is the approximation of a probability measure by finitely supported measures (the quantization problem). It is found that rates for empirical or occupation measures match or are close to previously known optimal quantization rates in several cases. This is notably highlighted in the example of infinite-dimensional Gaussian measures.

Subfields: Applied Mathematics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ab8b88eedffa7799d45b9fcfe0cffb308ebbb93b

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Citations in the corpus

Empirical optimal transport on countable metric spaces: Distributional limits and statistical applications (2017, methodology): “There are several refinements of these results, e.g., Boissard and Gouic (2014) and Fournier and Guillin (2014).”
Simple Bounds for the Convergence of Empirical and Occupation Measures in 1-Wasserstein Distance (2011, methodology): “As proved in [5], Appendix 1, the Gaussian measure μ satisfies a T2(2σ (2)) inequality, hence a T1 inequality with the same constant.”
Free transport-entropy inequalities for non-convex potentials and application to concentration for random matrices (2012, result): “These results have been extended in [Boi11] and [BLG11].”
On Optimal Matching of Gaussian Samples (2019, background): “When p > 1, the general investigations of [15, 9, 20] mainly based on dyadic decompositions yield the following typical conclusions.”
Constructive quantization: Approximation by empirical measures (2011, background): “Interestingly, there has been progress on this class of problems in several aspects [2,4] parallel to our research.”
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.”
Statistical Aspects of Wasserstein Distances (2018, methodology): “The version given here can be found in Weed & Bach (2017) and extends Boissard & Le Gouic (2014).”
Convergence and concentration of empirical measures under Wasserstein distance in unbounded functional spaces (2018, background): “For general distributions, [7, 12, 16] establish sharp upper bounds of EWp(μ̂, μ) in finite dimensional Euclidean spaces.”