Convergence of Empirical Measures for i.i.d. samples in $W^{-{\alpha}, p}$

TL;DR

This paper investigates how quickly empirical measures from i.i.d. samples converge to the true measure in negative Sobolev spaces, providing explicit rates and bounds depending on the space parameters.

Contribution

It establishes convergence rates of empirical measures in $W^{- ext{alpha}, p}$ spaces, including explicit bounds and Gaussian tail estimates, for different parameter regimes.

Findings

Convergence rate of $N^{-p/2}$ in certain Sobolev spaces.

Explicit constant $C_d$ depending on dimension.

Gaussian tail bounds for the convergence.

Abstract

Given $N$ i.i.d. samples from a probability measure $\mu$ on $\mathbf{R}^d$, we study the rate of convergence of the empirical measure $\mu_N \to \mu$ in the negative Sobolev space $W^{-\alpha, p}$. When $W^{-\alpha, p}$ contains point measures (i.e. when $\alpha p > (p-1)d$), we show $\mathbf{E} \| \mu_N - \mu \|_{W^{-\alpha, p}}^p \leq C_d / N^{p/2}$ for an explicit dimensional constant $C_d$, and obtain a Gaussian tail bound. When $0 < \alpha p \leq d(p-1)$, we prove a similar result for Gaussian regularizations.