Geodesic
Optimal quantization for the one–dimensional uniform distribution with Rényi-$\alpha$-entropy constraints
Kybernetika, Tome 46 (2010) no. 1, pp. 96-113 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We establish the optimal quantization problem for probabilities under constrained Rényi-$\alpha$-entropy of the quantizers. We determine the optimal quantizers and the optimal quantization error of one-dimensional uniform distributions including the known special cases $\alpha = 0$ (restricted codebook size) and $\alpha = 1$ (restricted Shannon entropy).
We establish the optimal quantization problem for probabilities under constrained Rényi-$\alpha$-entropy of the quantizers. We determine the optimal quantizers and the optimal quantization error of one-dimensional uniform distributions including the known special cases $\alpha = 0$ (restricted codebook size) and $\alpha = 1$ (restricted Shannon entropy).
Classification : 60E99, 62B10, 62H30, 94A17, 94A29
Keywords: optimal quantization; uniform distribution; Rényi-$\alpha $-entropy 
@article{KYB_2010_46_1_a6,
     author = {Kreitmeier, Wolfgang},
     title = {Optimal quantization for the one{\textendash}dimensional uniform distribution with {R\'enyi-}$\alpha$-entropy constraints},
     journal = {Kybernetika},
     pages = {96--113},
     year = {2010},
     volume = {46},
     number = {1},
     mrnumber = {2666897},
     zbl = {1187.94018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2010_46_1_a6/}
}
TY  - JOUR
AU  - Kreitmeier, Wolfgang
TI  - Optimal quantization for the one–dimensional uniform distribution with Rényi-$\alpha$-entropy constraints
JO  - Kybernetika
PY  - 2010
SP  - 96
EP  - 113
VL  - 46
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/KYB_2010_46_1_a6/
LA  - en
ID  - KYB_2010_46_1_a6
ER  - 
%0 Journal Article
%A Kreitmeier, Wolfgang
%T Optimal quantization for the one–dimensional uniform distribution with Rényi-$\alpha$-entropy constraints
%J Kybernetika
%D 2010
%P 96-113
%V 46
%N 1
%U http://geodesic.mathdoc.fr/item/KYB_2010_46_1_a6/
%G en
%F KYB_2010_46_1_a6
Kreitmeier, Wolfgang. Optimal quantization for the one–dimensional uniform distribution with Rényi-$\alpha$-entropy constraints. Kybernetika, Tome 46 (2010) no. 1, pp. 96-113. http://geodesic.mathdoc.fr/item/KYB_2010_46_1_a6/

[1] J. Aczél and Z. Daróczy: On Measures of Information and Their Characterizations. (Mathematics in Science and Engineering Vol. 115.) Academic Press, London 1975. | MR

[2] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty: Nonlinear Programming, Theory and Algorithms. Wiley, New York 1993. | MR

[3] C. Beck and F. Schlögl: Thermodynamics of Chaotic Systems. Cambridge University Press, Cambridge 1993. | MR

[4] M. Behara: Additive and Nonadditive Measures of Entropy. Wiley, New Delhi 1990. | MR | Zbl

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

[6] A. Gersho and R. M. Gray: Vector Quantization and Signal Compression. Kluwer, Boston 1992.

[7] S. Graf and H. Luschgy: The quantization of the Cantor distribution. Math. Nachr. 183 (1997), 113–133. | MR

[8] S. Graf and H. Luschgy: Foundations of Quantization for Probability Distributions. (Lecture Notes in Computer Science 1730.) Springer, Berlin 2000. | MR

[9] R. M. Gray and D. Neuhoff: Quantization. IEEE Trans. Inform. Theory 44 (1998), 2325–2383. | MR

[10] M. Grendar: Entropy and effective support size. Entropy 8 (2006), 169–174. | MR | Zbl

[11] A. György and T. Linder: Optimal entropy-constrained scalar quantization of a uniform source. IEEE Trans. Inform. Theory 46 (2000), 2704–2711. | MR

[12] A. György and T. Linder: On the structure of optimal entropy-constrained scalar quantizers. IEEE Trans. Inform. Theory 48 (2002), 416–427. | MR

[13] G. H. Hardy, J. E. Littlewood and G. Polya: Inequalities. Second edition. Cambridge University Press, Cambridge 1959.

[14] P. Harremoës and F. Topsøe: Inequalities between entropy and index of coincidence derived from information diagrams. IEEE Trans. Inform. Theory 47 (2001), 2944–2960. | MR

[15] M. Kesseböhmer and S. Zhu: Stability of quantization dimension and quantization for homogeneous Cantor measure. Math. Nachr. 280 (2007), 866–881. | MR

[16] W. Kreitmeier: Optimal quantization for dyadic homogeneous Cantor distributions. Math. Nachr. 281 (2008), 1307–1327. | MR

[17] F. Liese, D. Morales, and I. Vajda: Asymptotically sufficient partitions and quantizations. IEEE Trans. Inform. Theory 52 (2006), 5599–5606. | MR

[18] K. Pötzelberger and H. Strasser: Clustering and quantization by MSP-partitions. Statist. Decisions 19 (2001), 331–371. | MR