Yating Liu — Page 3 | Search arXiv

arXiv2020

Convergence rate of optimal quantization grids and application to empirical measure

We study the convergence rate of the optimal quantization for a probability measure sequence $(μ_{n})_{n\in\mathbb{N}^{*}}$ on $\mathbb{R}^{d}$ converging in the Wasserstein distance in two aspects: the first one is the convergence rate of optimal quantizer $x^{(n)}\in(\mathbb{R}^{d})^{K}$ of $μ_{n}$ at level $K$; the other one is the convergence rate of the distortion function valued at $x^{(n)}$, called the "performance" of $x^{(n)}$. Moreover, we also study the mean performance of the optimal quantization for the empirical measure of a distribution $μ$ with finite second moment but possibly unbounded support. As an application, we show that the mean performance for the empirical measure of the multidimensional normal distribution $\mathcal{N}(m, Σ)$ and of distributions with hyper-exponential tails behave like $\mathcal{O}(\frac{\log n}{\sqrt{n}})$. This extends the results from [BDL08] obtained for compactly supported distribution. We also derive an upper bound which is sharper in the quantization level $K$ but suboptimal in $n$ by applying results in [FG15].

math.ST↗

arXiv2019

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

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 Voronoï 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.

math.PR↗