A short proof on the rate of convergence of the empirical measure for the Wasserstein distance
Vincent Divol
TL;DR
The paper proves that the expected Wasserstein distance between the empirical measure μ_n and a density-bounded measure μ on the d-dimensional flat torus decays as n^{-1/d} for d≥3, with a (log n)^{1/2} n^{-1/2} rate at d=2 and n^{-1/2} at d=1. The approach combines a kernel-smoothing step with a bound in a pointed negative Sobolev norm, leveraging a Fourier-multiplier framework and a Rosenthal-type bound to control fluctuations. The main result is complemented by a p=2 simplification and an extension to densities with regularity s, which yields faster rates that align with minimax lower bounds. The work highlights how density bounds and Fourier-analytic methods can yield sharp convergence rates for Wasserstein distances on a torus, circumventing boundary complications.
Abstract
We provide a short proof that the Wasserstein distance between the empirical measure of a n-sample and the estimated measure is of order n^-(1/d), if the measure has a lower and upper bounded density on the d-dimensional flat torus.