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

ESAIM: Probability and Statistics (2009)

- Volume: 13, page 218-246
- ISSN: 1292-8100

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topSagna, Abass. "Universal Ls-rate-optimality of Lr-optimal quantizers by dilatation and contraction." ESAIM: Probability and Statistics 13 (2009): 218-246. <http://eudml.org/doc/250635>.

@article{Sagna2009,

abstract = {
We investigate in this paper the properties of some dilatations or contractions of a sequence (αn)n≥1 of Lr-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 Ls-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 ∈ (r, +∞) and $X \in L^s(\mathbb\{P\})$. We show that for a wide family of distributions, one may always find parameters (θ,µ) such that (αnθ,µ)n≥1 is Ls-rate-optimal. For the Gaussian and the exponential distributions we show the existence of a couple (θ*,µ*) such that (αθ*,µ*)n≥1 also satisfies the so-called Ls-empirical measure theorem. Our conjecture, confirmed by numerical experiments, is that such sequences are asymptotically Ls-optimal. In both cases the sequence (αθ*,µ*)n≥1 is incredibly close to Ls-optimality. However we show (see Rem. 5.4) that this last sequence is not Ls-optimal (e.g. when s = 2, r = 1) for the exponential distribution.
},

author = {Sagna, Abass},

journal = {ESAIM: Probability and Statistics},

keywords = {Rate-optimal quantizers; empirical measure theorem; dilatation; Lloyd algorithm; rate-optimal quantizers; Gaussian distribution; Lloyd's algorithm},

language = {eng},

month = {6},

pages = {218-246},

publisher = {EDP Sciences},

title = {Universal Ls-rate-optimality of Lr-optimal quantizers by dilatation and contraction},

url = {http://eudml.org/doc/250635},

volume = {13},

year = {2009},

}

TY - JOUR

AU - Sagna, Abass

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

JO - ESAIM: Probability and Statistics

DA - 2009/6//

PB - EDP Sciences

VL - 13

SP - 218

EP - 246

AB -
We investigate in this paper the properties of some dilatations or contractions of a sequence (αn)n≥1 of Lr-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 Ls-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 ∈ (r, +∞) and $X \in L^s(\mathbb{P})$. We show that for a wide family of distributions, one may always find parameters (θ,µ) such that (αnθ,µ)n≥1 is Ls-rate-optimal. For the Gaussian and the exponential distributions we show the existence of a couple (θ*,µ*) such that (αθ*,µ*)n≥1 also satisfies the so-called Ls-empirical measure theorem. Our conjecture, confirmed by numerical experiments, is that such sequences are asymptotically Ls-optimal. In both cases the sequence (αθ*,µ*)n≥1 is incredibly close to Ls-optimality. However we show (see Rem. 5.4) that this last sequence is not Ls-optimal (e.g. when s = 2, r = 1) for the exponential distribution.

LA - eng

KW - Rate-optimal quantizers; empirical measure theorem; dilatation; Lloyd algorithm; rate-optimal quantizers; Gaussian distribution; Lloyd's algorithm

UR - http://eudml.org/doc/250635

ER -

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