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

TL;DR

The paper establishes dimension-robust rates for the convergence of empirical measures $\mu_N$ to $\mu$ in negative Sobolev spaces $W^{-\alpha,p}$ by employing a heat-kernel based norm. For $\alpha p > (p-1)d$, it proves a Gaussian-tailed, $O(N^{-p/2})$-type moment bound, with explicit constants, and extends the results to Gaussian-regularized measurements when the regime fails. The core methodology combines subgaussian bounds for heat-kernel evaluations, maximal-function techniques, and McDiarmid-type concentration to yield tight, dimension-free convergence rates. The paper further provides precise second-moment formulas in the Hilbert-space case $p=2$ and analyzes how these rates adapt under Gaussian regularization and $\varepsilon\to 0$ limits. Overall, it offers a coherent framework for high-dimensional empirical measure convergence outside the Wasserstein metric, highlighting the utility of negative Sobolev norms for dimension-insensitive probabilistic guarantees.

Abstract

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

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Theorem 2.1
• proof : Proof of Theorems \ref{['t:mainIntro1']} and \ref{['t:mainRegularized']}
• Proposition 2.2
• Proposition 3.1
• Lemma 3.2
• Lemma 3.3