A gradient flow approach to quantization of measures

TL;DR

This paper introduces a gradient flow framework for measure quantization in one dimension, establishing a continuous limit and proving stability and convergence under certain regularity conditions.

Contribution

It develops a novel gradient flow approach to measure quantization, connecting discrete particle systems with a continuous limit in $L^2$ space.

Findings

Proves uniform stability of the quantization process.

Establishes quantitative convergence between discrete and continuous models.

Provides conditions under which the models reliably approximate each other.

Abstract

In this paper we study a gradient flow approach to the problem of quantization of measures in one dimension. By embedding our problem in $L^2$, we find a continuous version of it that corresponds to the limit as the number of particles tends to infinity. Under some suitable regularity assumptions on the density, we prove uniform stability and quantitative convergence result for the discrete and continuous dynamics.