Universal L s -rate-optimality of L r -optimal quantizers by dilatation and contraction
ESAIM: Probability and Statistics, Tome 13 (2009), pp. 218-246.

We investigate in this paper the properties of some dilatations or contractions of a sequence (α n ) n1 of L r -optimal quantizers of an d -valued random vector XL r () defined in the probability space (Ω,𝒜,) with distribution X =P. To be precise, we investigate the L s -quantization rate of sequences α n θ,μ =μ+θ(α n -μ)={μ+θ(a-μ),aα n } when θ + ,μ,s(0,r) or s(r,+) and XL s (). We show that for a wide family of distributions, one may always find parameters (θ,μ) such that (α n θ,μ ) n1 is L s -rate-optimal. For the gaussian and the exponential distributions we show the existence of a couple (θ ,μ ) such that (α θ ,μ ) n1 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 (α θ ,μ ) n1 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.

DOI : 10.1051/ps:2008008
Classification : 60G15, 60G35, 41A52
Mots clés : rate-optimal quantizers, empirical measure theorem, dilatation, Lloyd algorithm
@article{PS_2009__13__218_0,
     author = {Sagna, Abass},
     title = {Universal $L^s$-rate-optimality of $L^r$-optimal quantizers by dilatation and contraction},
     journal = {ESAIM: Probability and Statistics},
     pages = {218--246},
     publisher = {EDP-Sciences},
     volume = {13},
     year = {2009},
     doi = {10.1051/ps:2008008},
     mrnumber = {2518547},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2008008/}
}
TY  - JOUR
AU  - Sagna, Abass
TI  - Universal $L^s$-rate-optimality of $L^r$-optimal quantizers by dilatation and contraction
JO  - ESAIM: Probability and Statistics
PY  - 2009
SP  - 218
EP  - 246
VL  - 13
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2008008/
DO  - 10.1051/ps:2008008
LA  - en
ID  - PS_2009__13__218_0
ER  - 
%0 Journal Article
%A Sagna, Abass
%T Universal $L^s$-rate-optimality of $L^r$-optimal quantizers by dilatation and contraction
%J ESAIM: Probability and Statistics
%D 2009
%P 218-246
%V 13
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps:2008008/
%R 10.1051/ps:2008008
%G en
%F PS_2009__13__218_0
Sagna, Abass. Universal $L^s$-rate-optimality of $L^r$-optimal quantizers by dilatation and contraction. ESAIM: Probability and Statistics, Tome 13 (2009), pp. 218-246. doi : 10.1051/ps:2008008. http://www.numdam.org/articles/10.1051/ps:2008008/

[1] S. Delattre, S. Graf, H. Luschgy and G. Pagès, Quantization of probability distributions under norm-based distribution measures. Statist. Decisions 22 (2004) 261-282. | MR | Zbl

[2] J.C. Fort and G. Pagès, Asymptotics of optimal quantizers for some scalar distributions. J. Comput. Appl. Math. 146 (2002) 253-275. | MR | Zbl

[3] J.H. Friedman, J.L. Bentley and R.A. Finkel, An Algorithm for Finding Best Matches in Logarithmic Expected Time, ACM Trans. Math. Software 3 (1977) 209-226. | Zbl

[4] A. Gersho and R. Gray, Vector Quantization and Signal Compression, 6th edition. Kluwer, Boston (1992). | Zbl

[5] S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Lect. Notes Math. 1730. Springer, Berlin (2000). | MR | Zbl

[6] S. Graf, H. Luschgy and G. Pagès, Distorsion mismatch in the quantization of probability measures, ESAIM: PS 12 (2008) 127-153. | Numdam | MR

[7] J. Mcnames, A Fast Nearest-Neighbor algorithm based on a principal axis search tree, IEEE Trans. Pattern Anal. Machine Intelligence 23 (2001) 964-976.

[8] G. Pagès, Space vector quantization method for numerical integration, J. Comput. Appl. Math. 89 (1998) 1-38. | Zbl

[9] G. Pagès, H. Pham and J. Printems, An Optimal markovian quantization algorithm for multidimensional stochastic control problems, Stochastics and Dynamics 4 (2004) 501-545. | MR | Zbl

[10] G. Pagès, H. Pham and J. Printems, Optimal quantization methods and applications to numerical problems in finance, Handbook on Numerical Methods in Finance (S. Rachev, ed.), Birkhauser, Boston (2004) 253-298. | MR | Zbl

Cité par Sources :