Constructive quantization: approximation by empirical measures

TL;DR

This paper investigates how well empirical measures approximate probability measures in Wasserstein distance, providing bounds and formulas that demonstrate the optimality of empirical quantization under certain conditions.

Contribution

It introduces refined bounds and a high-resolution formula for empirical measure approximation, establishing optimality in Wasserstein distance under weak assumptions.

Findings

Refined upper and lower bounds for approximation error

A high-resolution formula for empirical measure approximation

Universal Pierce type estimate based on moments

Abstract

In this article, we study the approximation of a probability measure $\mu$ on $\mathbb{R}^{d}$ by its empirical measure $\hat{\mu}_{N}$ interpreted as a random quantization. As error criterion we consider an averaged $p$-th moment Wasserstein metric. In the case where $2p<d$, we establish refined upper and lower bounds for the error, a high-resolution formula. Moreover, we provide a universal estimate based on moments, a so-called Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.