Universal Ls-rate-optimality of Lr-optimal quantizers by dilatation and contraction
Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Pierre et Marie Curie, Case 188, 4 place Jussieu, 75252 Cedex 05, Paris, France; firstname.lastname@example.org
Revised: 20 February 2008
We investigate in this paper the properties of some dilatations or contractions of a sequence (αn)n≥1 of Lr-optimal quantizers of an -valued random vector defined in the probability space with distribution . To be precise, we investigate the Ls-quantization rate of sequences when or s ∈ (r, +∞) and . 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.
Mathematics Subject Classification: 60G15 / 60G35 / 41A52
Key words: Rate-optimal quantizers / empirical measure theorem / dilatation / Lloyd algorithm
© EDP Sciences, SMAI, 2009