Critical points in uniform quantization of probability densities

Abstract: I will consider the problem of approximating a given diffuse measure on a domain of R^d with an atomic one, with fixed number N of atoms. Two variants of this problem are studied, when the masses of the atoms are free (which consists in finding an optimal Voronoi tessellation) or when they are prescribed to be equal to 1/N (where, instead, Laguerre cells pop up). Both problems can be attacked via an iterated algorithm originally due to Lloyd, where atoms are moved at each step to the barycenters of the corresponding cell. Despite the underlying optimization problem being non-convex, this converges in practice to reasonable solutions. The goal of the talk is to suggest some reasons why this happens, studying properties of critical points and of stable critical points in the case of the uniform weights 1/N. The results that I will present are obtained in various works in collaboration with Q. Mérigot, with our former student C. Sarrazin, and with A. Figalli.