TY - JOUR
T1 - Empirical measures and random walks on compact spaces in the quadratic Wasserstein metric
AU - Borda, Bence
N1 - Publisher Copyright:
© Association des Publications de l’Institut Henri Poincaré, 2023.
PY - 2023
Y1 - 2023
N2 - Estimating the rate of convergence of the empirical measure of an i.i.d. sample to the reference measure is a classical problem in probability theory. Extending recent results of Ambrosio, Stra and Trevisan on 2-dimensional manifolds, in this paper we prove sharp asymptotic and nonasymptotic upper bounds for the mean rate in the quadratic Wasserstein metric W2 on a d-dimensional compact Riemannian manifold. Under a smoothness assumption on the reference measure, our bounds match the classical rate in the optimal matching problem on the unit cube due to Ajtai, Komlós, Tusnády and Talagrand. The i.i.d. condition is relaxed to stationary samples with a mixing condition. As an example of a nonstationary sample, we also consider the empirical measure of a random walk on a compact Lie group. Surprisingly, on semisimple groups random walks attain almost optimal rates even without a spectral gap assumption. The proofs are based on Fourier analysis, and in particular on a Berry–Esseen smoothing inequality for W2 on compact manifolds, a result of independent interest with a wide range of applications.
AB - Estimating the rate of convergence of the empirical measure of an i.i.d. sample to the reference measure is a classical problem in probability theory. Extending recent results of Ambrosio, Stra and Trevisan on 2-dimensional manifolds, in this paper we prove sharp asymptotic and nonasymptotic upper bounds for the mean rate in the quadratic Wasserstein metric W2 on a d-dimensional compact Riemannian manifold. Under a smoothness assumption on the reference measure, our bounds match the classical rate in the optimal matching problem on the unit cube due to Ajtai, Komlós, Tusnády and Talagrand. The i.i.d. condition is relaxed to stationary samples with a mixing condition. As an example of a nonstationary sample, we also consider the empirical measure of a random walk on a compact Lie group. Surprisingly, on semisimple groups random walks attain almost optimal rates even without a spectral gap assumption. The proofs are based on Fourier analysis, and in particular on a Berry–Esseen smoothing inequality for W2 on compact manifolds, a result of independent interest with a wide range of applications.
KW - Berry–Esseen inequality
KW - Heat kernel
KW - Lie group
KW - Occupation measure
KW - Optimal transportation
KW - Riemannian manifold
UR - http://www.scopus.com/inward/record.url?scp=85150333474&partnerID=8YFLogxK
U2 - 10.1214/22-AIHP1322
DO - 10.1214/22-AIHP1322
M3 - Article
AN - SCOPUS:85150333474
SN - 0246-0203
VL - 59
SP - 2017
EP - 2035
JO - Annales de l'institut Henri Poincare (B) Probability and Statistics
JF - Annales de l'institut Henri Poincare (B) Probability and Statistics
IS - 4
ER -