@article {68,
	title = {Universal $L^s$-rate-optimality of $L^r$-optimal quantizers by dilatation and contraction},
	journal = {ESAIM: Probability and Statistics},
	volume = {13},
	year = {2009},
	pages = {218-246},
	abstract = {We investigate in this paper the properties of some dilatations or contractions of a sequence $(\alpha _{n})_{n \ge 1}$ 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 )=\lbrace \mu + \theta (a-\mu ), \ a \in \alpha _n \rbrace $ 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 $(\alpha _n^{\theta ,\mu })_{n \ge 1}$ 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 $(\alpha ^{\theta ^{\star },\mu ^{\star }})_{n \ge 1}$ 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 $(\alpha ^{\theta ^{\star },\mu ^{\star }})_{n \ge 1}$ 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. },
	keywords = {dilatation, empirical measure theorem, Lloyd algorithm, rate-optimal quantizers},
	author = {Abass Sagna}
}
