Asymptotics for Optimal Empirical Quantization of Measures

TL;DR

This paper studies the asymptotic behavior of the minimal Wasserstein error when approximating probability measures in high dimensions using finite point sets, providing bounds, limits, and convergence rates for various cases.

Contribution

It establishes explicit asymptotic bounds, limit existence, and convergence speeds for the minimal empirical quantization error across different dimensions and measure types.

Findings

Asymptotic bounds for minimal error in high dimensions

Limit existence for certain measures and parameters

Convergence speed for measures with H"older densities

Abstract

We investigate the minimal error in approximating a general probability measure $\mu$ on $\mathbb{R}^d$ by the uniform measure on a finite set with prescribed cardinality $n$. The error is measured in the $p$-Wasserstein distance. In particular, when $1\le p<d$, we establish asymptotic upper and lower bounds as $n \to \infty$ on the rescaled minimal error that have the same, explicit dependency on $\mu$. In some instances, we prove that the rescaled minimal error has a limit. These include general measures in dimension $d = 2$ with $1 \le p < 2$, and uniform measures in arbitrary dimension with $1 \le p < d$. For some uniform measures, we prove the limit existence for $p \ge d$ as well. For a class of compactly supported measures with H\"older densities, we determine the convergence speed of the minimal error for every $p \ge 1$. Furthermore, we establish a new Pierce-type (i.e., nonasymptotic) upper estimate of the minimal error when $1 \le p < d$. In the initial sections, we survey the state of the art and draw connections with similar problems, such as classical and random quantization.