Characterization of probability distribution convergence in Wasserstein distance by $L^{p}$-quantization error function

TL;DR

This paper characterizes probability measures through their $L^{p}$-quantization error functions, providing criteria for static identification and dynamic convergence in Wasserstein distance, with optimal conditions in Hilbert spaces.

Contribution

It introduces a geometrical criterion for quantization levels applicable in any norm and reduces the level to N=2 in Hilbert spaces, advancing measure characterization methods.

Findings

Established conditions for measure identification via quantization errors.

Proved that in Hilbert spaces, N=2 suffices for characterization.

Discussed the completeness of quantization error-based distances.

Abstract

We establish conditions to characterize probability measures by their $L^{p}$-quantization error functions in both $\mathbb{R}^{d}$ and Hilbert settings. This characterization is two-fold: static (identity of two distributions) and dynamic (convergence for the $L^p$-Wasserstein distance). We first propose a criterion on the quantization level $N$, valid for any norm on $\mathbb{R}^{d}$ and any order $p$ based on a geometrical approach involving the Vorono\"i diagram. Then, we prove that in the $L^2$-case on a (separable) Hilbert space, the condition on the level $N$ can be reduced to $N=2$, which is optimal. More quantization based characterization cases on dimension 1 and a discussion of the completeness of a distance defined by the quantization error function can be found in the end of this paper.