Abstract
The Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider the fundamental question of how quickly the empirical measure obtained from n independent samples from μ approaches μ in the Wasserstein distance of any order. We prove sharp asymptotic and finite-sample results for this rate of convergence for general measures on general compact metric spaces. Our finite-sample results show the existence of multi-scale behavior, where measures can exhibit radically different rates of convergence as n grows.
Original language | English (US) |
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Pages (from-to) | 2620-2648 |
Number of pages | 29 |
Journal | Bernoulli |
Volume | 25 |
Issue number | 4 A |
DOIs | |
State | Published - 2019 |
Keywords
- Optimal transport
- Quantization
- Wasserstein metrics
ASJC Scopus subject areas
- Statistics and Probability