Approximation by finitely supported measures

TL;DR

This paper investigates how quickly a compactly supported probability measure on a Riemannian manifold can be approximated by finitely supported measures in Wasserstein distance, connecting to quantization and centroidal Voronoi tessellations.

Contribution

It provides an analysis of the asymptotic speed of approximation in Wasserstein distance for measures on Riemannian manifolds, linking quantization and tessellation problems.

Findings

Established asymptotic rates of approximation in Wasserstein distance.

Connected quantization and centroidal Voronoi tessellations to measure approximation.

Provided theoretical bounds for approximation speed.

Abstract

Given a compactly supported probability measure on a Riemannian manifold, we study the asymptotic speed at which it can be approximated (in Wasserstein distance of any exponent p) by finitely supported measure. This question has been studied under the names of ``quantization of distributions'' and, when p=1, ``location problem''. When p=2, it is linked with Centroidal Voronoi Tessellations.