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Pré-Publication, Document De Travail Année : 2007

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

Résumé

Let $ r, s>0 $. For a given probability measure $P$ on $\mathbb{R}^d$, let $(\alpha_n)_{n \geq 1}$ be a sequence of (asymptotically) $L^r(P)$- optimal quantizers. For all $\mu \in \mathbb{R}^d $ and for every $\theta >0$, one defines the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$ by : $\forall n \geq 1, \ \alpha_n^{\theta, \mu} = \mu + \theta(\alpha_n - \mu) = \{ \mu + \theta(a- \mu), \ a \in \alpha_n \} $. In this paper, we are interested in the asymptotics of the $L^s$-quantization error induced by the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$. We show that for a wide family of distributions, the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$ is $L^s$-rate-optimal. For the Gaussian and the exponential distributions, one shows how to choose the parameter $\theta$ such that $(\alpha_n^{\theta, \mu})_{n \geq 1}$ satisfies the empirical measure theorem.
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Dates et versions

hal-00162075 , version 1 (12-07-2007)
hal-00162075 , version 2 (19-11-2007)

Identifiants

Citer

Abass Sagna. Universal L^s -rate-optimality of L^r-optimal quantizers by dilatation and contraction. 2007. ⟨hal-00162075v1⟩

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