%0 Journal Article %T Convergence rate of optimal quantization grids and application to empirical measure %+ Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)) %A Liu, Yating %A Pagès, Gilles %< avec comité de lecture %@ 1532-4435 %J Journal of Machine Learning Research %I Microtome Publishing %V 21 %N 1 %P 3352–3387 %8 2022-02 %D 2022 %Z 1811.08351 %Z Mathematics [math]/Probability [math.PR]Journal articles %X We study the convergence rate of the optimal quantization for a probability measure sequence $(\mu_{n})_{n\in\mathbb{N}^{*}}$ 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 $\mu_{n}$ 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 $\mu$ 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 $\mathcal{N}(m, \Sigma)$ 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]. %G English %L hal-03890795 %U https://cnrs.hal.science/hal-03890795 %~ CNRS %~ INSMI %~ LPSM %~ SORBONNE-UNIVERSITE %~ SORBONNE-UNIV %~ SU-SCIENCES %~ UNIV-PARIS %~ UNIVERSITE-PARIS %~ UP-SCIENCES %~ SU-TI %~ ALLIANCE-SU