TY - JOUR
T1 - MINIMAX ESTIMATION OF SMOOTH DENSITIES IN WASSERSTEIN DISTANCE
AU - Niles-Weed, Jonathan
AU - Berthet, Quentin
N1 - Funding Information:
Funding. Part of this research was conducted while at the Institute for Advanced Study. JNW gratefully acknowledges its support. Most of this work was conducted while QB was at University of Cambridge and was supported in part by The Alan Turing Institute under the EPSRC Grant EP/N510129/1.
Publisher Copyright:
© Institute of Mathematical Statistics, 2022
PY - 2022/6
Y1 - 2022/6
N2 - We study nonparametric density estimation problems where error is measured in the Wasserstein distance, a metric on probability distributions popular in many areas of statistics and machine learning. We give the first minimax-optimal rates for this problem for general Wasserstein distances for two classes of densities: smooth probability densities on [0,1]d bounded away from 0, and sub-Gaussian densities lying in the Hölder class Cs, s ∈ (0,1). Unlike classical nonparametric density estimation, these rates depend on whether the densities in question are bounded below, even in the compactly supported case. Motivated by variational problems involving the Wasserstein distance, we also show how to construct discretely supported measures, suitable for computational purposes, which achieve the minimax rates. Our main technical tool is an inequality giving a nearly tight dual characterization of the Wasserstein distances in terms of Besov norms.
AB - We study nonparametric density estimation problems where error is measured in the Wasserstein distance, a metric on probability distributions popular in many areas of statistics and machine learning. We give the first minimax-optimal rates for this problem for general Wasserstein distances for two classes of densities: smooth probability densities on [0,1]d bounded away from 0, and sub-Gaussian densities lying in the Hölder class Cs, s ∈ (0,1). Unlike classical nonparametric density estimation, these rates depend on whether the densities in question are bounded below, even in the compactly supported case. Motivated by variational problems involving the Wasserstein distance, we also show how to construct discretely supported measures, suitable for computational purposes, which achieve the minimax rates. Our main technical tool is an inequality giving a nearly tight dual characterization of the Wasserstein distances in terms of Besov norms.
KW - Wasserstein distance
KW - density estimation
KW - high-dimensional statistics
KW - optimal transport
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U2 - 10.1214/21-AOS2161
DO - 10.1214/21-AOS2161
M3 - Article
AN - SCOPUS:85134678219
SN - 0090-5364
VL - 50
SP - 1519
EP - 1540
JO - Annals of Statistics
JF - Annals of Statistics
IS - 3
ER -