Quantization of Measures and Gradient Flows: a Perturbative Approach in the 2-Dimensional Case

TL;DR

This paper investigates the optimality of hexagonal lattices for measure quantization in the plane, showing they are strict minimizers and that gradient flows can rapidly evolve perturbed configurations to the optimal lattice.

Contribution

It introduces a perturbative approach demonstrating the asymptotic optimality of hexagonal lattices and analyzes the gradient flow dynamics for convergence.

Findings

Hexagonal lattices are strict minimizers of the energy as the number of points grows.

Gradient flows can exponentially quickly evolve perturbed configurations to the optimal lattice.

The analysis provides a mathematical justification for the asymptotic optimality of hexagonal lattices.

Abstract

In this paper we study a perturbative approach to the problem of quantization of measures in the plane. Motivated by the fact that, as the number of points tends to infinity, hexagonal lattices are asymptotically optimal from an energetic point of view (see [Morgan, Bolton: Amer. Math. Monthly 109 (2002), 165-172]), we consider configurations that are small perturbations of the hexagonal lattice and we show that: (1) in the limit as the number of points tends to infinity, the hexagonal lattice is a strict minimizer of the energy; (2) the gradient flow of the limiting functional allows us to evolve any perturbed configuration to the optimal one exponentially fast. In particular, our analysis provides a solid mathematical justification of the asymptotic optimality of the hexagonal lattice among its nearby configurations.