HAL CCSD
Convergence rate of optimal quantization grids and application to empirical measure
Liu, Yating
Pagès, Gilles
Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)) ; Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)
International audience
ISSN: 1532-4435
EISSN: 1533-7928
Journal of Machine Learning Research
Microtome Publishing
hal-03890795
https://cnrs.hal.science/hal-03890795
https://cnrs.hal.science/hal-03890795
Journal of Machine Learning Research, 2022, 21 (1), pp.3352-3387
ARXIV: 1811.08351
info:eu-repo/semantics/altIdentifier/arxiv/1811.08351
en
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
info:eu-repo/semantics/article
Journal articles
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].
2022-02