Convergence and concentration of empirical measures under Wasserstein distance in unbounded functional spaces

By Jing Lei. Bernoulli, 2018

140 citations recorded by the source corpus.

We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence on the dimensionality. Our method also covers the important case of Gaussian processes in separable Hilbert spaces, with rate-optimal upper bounds for functional data distributions whose coordinates decay geometrically or polynomially. Moreover, our bounds of the expected value can be combined with mean-concentration results to yield improved exponential tail probability bounds for the Wasserstein error of empirical measures under a Bernstein-type tail condition.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/68cefaa8634fe28fdaf429efb608c655582e2dde

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Larry A. Wasserman
2. Sivaraman Balakrishnan
3. Tudor Manole
4. Jonathan Niles-Weed

In Statistics and Probability

1. Sivaraman Balakrishnan
2. Larry A. Wasserman
3. Tudor Manole
4. Jonathan Niles-Weed
5. Mathias Drton
6. Tommaso Cai
7. E.J. Candes
8. Alessio Farcomeni

Citations in the corpus

• Plugin estimation of smooth optimal transport maps (2021, background): “Characterizing the convergence rate of the empirical measure under the Wasserstein distance is a classical problem (Dudley, 1969; Boissard and Le Gouic, 2014; Fournier and Guillin, 2015; Weed and Bach, 2019; Lei, 2020) which immediately leads to upper bounds on the convergence rate of the empirical ”