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

Abstract : 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 and probably be asymptotically $L^s$-optimal.
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Cited literature [6 references]

https://hal.archives-ouvertes.fr/hal-00162075
Contributor : Abass Sagna <>
Submitted on : Monday, November 19, 2007 - 5:12:16 PM
Last modification on : Wednesday, December 9, 2020 - 3:16:27 PM
Long-term archiving on: : Tuesday, September 21, 2010 - 3:20:08 PM

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• HAL Id : hal-00162075, version 2
• ARXIV : 0707.1808

### Citation

Abass Sagna. Universal L^s -rate-optimality of L^r-optimal quantizers by dilatation and contraction. 2007. ⟨hal-00162075v2⟩

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