Quantization of Probability Measures and the Import-Export Problem

The problem of quantization can be thought of as follows: given a probability distribution P and a number n, find a discrete probability, supported on at most n points, that is a “good approximation” of P. In this thesis, we introduce some of the most interesting results in the literature and present a few ones that are, to our knowledge, original. We focus in particular on asymptotic quantization, that is the problem of determining how good the best approximation is as n grows to infinity. In Chapter 1, we outline the basic objects and, in particular, define the Wasserstein distances on some spaces of measures, so that it makes sense to speak about the “closest” probability among those supported on at most n points. In Chapter 2, we prove a theorem by Bucklew, Wise and Zador, which determines precisely the order of convergence to zero for the quantization error. We also introduce a problem studied by Cohort and called random quantization. In Chapter 3, we find some convergence results for another, similar, problem: uniform quantization. In the latter, we minimize over the (smaller) set of measures whose atoms have a weight that is an integer multiple of 1/n. In Chapter 4, inspired by an article by Gigli and Figalli, we introduce the import-export problem. In their work, the two authors define new distances on spaces of probabilities over open bounded sets, similar to Wasserstein’s, but with a special role given to the boundary, that can be used as an infinite reserve of mass. We investigate the effects of this metric modification on quantization