Universal Ls-rate-optimality of Lr-optimal quantizers by dilatation and contraction

Issue		ESAIM: PS Volume 13, 2009


Page(s)		218 - 246
DOI		10.1051/ps:2008008
Published online		12 June 2009

ESAIM: PS, June 2009, Vol. 13, p. 218-246
DOI: 10.1051/ps:2008008

Universal L^s-rate-optimality of L^r-optimal quantizers by dilatation and contraction

Abass Sagna

Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Pierre et Marie Curie, Case 188, 4 place Jussieu, 75252 Cedex 05, Paris, France; sagna@ccr.jussieu.fr

Received July 12, 2007. Revised February 20, 2008. Published online 12 June 2009

Abstract
We investigate in this paper the properties of some dilatations or contractions of a sequence $(\alpha_{n})_{n \geq 1}$ of L^r-optimal quantizers of an $\mathbb{R} ^d$ -valued random vector $X \in L^r(\mathbb{P} )$ defined in the probability space $(\Omega,\mathcal{A},\mathbb{P} )$ with distribution $\mathbb{P} _{X} = P$ . To be precise, we investigate the L^s-quantization rate of sequences $\alpha_n^{\theta,\mu} = \mu + \theta(\alpha_n-\mu)=\{\mu + \theta(a-\mu), \ a \in \alpha_n \}$ when $\theta \in \mathbb{R} _{+}^{\star}, \mu \in \mathbb{R} , s \in (0,r)$ or $s \in (r,+\infty)$ and $X \in L^s(\mathbb{P} )$ . We show that for a wide family of distributions, one may always find parameters $(\theta,\mu)$ such that $(\alpha_n^{\theta,\mu})_{n \geq 1}$ is L^s-rate-optimal. For the Gaussian and the exponential distributions we show the existence of a couple $(\theta^{\star}, \mu^{\star})$ such that $(\alpha^{\theta^{\star},\mu^{\star}})_{n \geq 1}$ also satisfies the so-called L^s-empirical measure theorem. Our conjecture, confirmed by numerical experiments, is that such sequences are asymptotically L^s-optimal. In both cases the sequence $(\alpha^{\theta^{\star},\mu^{\star}})_{n \geq 1}$ is incredibly close to L^s-optimality. However we show (see Rem. 5.4) that this last sequence is not L^s-optimal (e.g. when s = 2, r = 1) for the exponential distribution.

Mathematics Subject Classification. 60G15, 60G35, 41A52

Key words: Rate-optimal quantizers, empirical measure theorem, dilatation, Lloyd algorithm

© EDP Sciences, SMAI 2009

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