For a probability measure $P$ on $\R^d$ and $n\!\! \in \! \N$ consider $e_n = \inf \displaystyle \int \min_{a \in \alpha} V(\| x-a \| )dP(x)$ where the infimum is taken over all subsets $\alpha$ of $\R^d$ with $\mbox{card} (\alpha) \leq n$ and $V$ is a nondecreasing function. Under certain conditions on $V$, we derive the precise $n$-asymptotics of $e_n$ for nonsingular and for (singular) self-similar distributions $P$ and we find the asymptotic performance of optimal quantizers using weighted empirical measures.
Publié le : 2004-10-11
Classification:
High-rate vector quantization,
norm-difference distortion,
empirical measure,
weak convergence,
local distortion,
point density measure,
60E99, 94A29, 28A80,
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
@article{hal-00003057,
author = {Delattre, Sylvain and Graf, Siegfried and Luschgy, Harald and Pages, Gilles},
title = {Quantization of probability distributions under norm-based distortion measures},
journal = {HAL},
volume = {2004},
number = {0},
year = {2004},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00003057}
}
Delattre, Sylvain; Graf, Siegfried; Luschgy, Harald; Pages, Gilles. Quantization of probability distributions under norm-based distortion measures. HAL, Tome 2004 (2004) no. 0, . http://gdmltest.u-ga.fr/item/hal-00003057/