Uniform decomposition of probability measures: quantization, classification, rate of convergence

TL;DR

This paper introduces a new elementary method for uniformly decomposing probability measures in any dimension, providing sharp bounds on the convergence rate of their finite approximations, extending previous one-dimensional results.

Contribution

It generalizes uniform decomposition techniques to higher dimensions and establishes sharp convergence bounds for optimal uniform approximations of probability measures.

Findings

Provides an elementary construction for uniform decomposition in any dimension.

Establishes upper bounds on the convergence rate of uniform approximations.

Bounds are sharp for generic probability measures.

Abstract

The study of finite approximations of probability measures has a long history. In (Xu and Berger, 2017), the authors focus on constrained finite approximations and, in particular, uniform ones in dimension $d=1$. The present paper gives an elementary construction of a uniform decomposition of probability measures in dimension $d\geq 1$. This decomposition is then used to give upper-bounds on the rate of convergence of the optimal uniform approximation error. These bounds appear to be the generalization of the ones obtained in (Xu and Berger, 2017) and to be sharp for generic probability measures.