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September  2018, 38(9): 4555-4570. doi: 10.3934/dcds.2018199

## Least upper bound of the exact formula for optimal quantization of some uniform Cantor distributions

 School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 West University Drive, Edinburg, TX 78539-2999, USA

Received  November 2017 Revised  May 2018 Published  June 2018

Fund Project: The research of the author was supported by U.S. National Security Agency (NSA) Grant H98230-14-1-0320

The quantization scheme in probability theory deals with finding a best approximation of a given probability distribution by a probability distribution that is supported on finitely many points. Let $P$ be a Borel probability measure on $\mathbb R$ such that $P = \frac 12 P\circ S_1^{-1}+\frac 12 P\circ S_2^{-1},$ where $S_1$ and $S_2$ are two contractive similarity mappings given by $S_1(x) = rx$ and $S_2(x) = rx+1-r$ for $0<r<\frac 12$ and $x∈ \mathbb R$. Then, $P$ is supported on the Cantor set generated by $S_1$ and $S_2$. The case $r = \frac 13$ was treated by Graf and Luschgy who gave an exact formula for the unique optimal quantization of the Cantor distribution $P$ (Math. Nachr., 183 (1997), 113-133). In this paper, we compute the precise range of $r$-values to which Graf-Luschgy formula extends.

Citation: Mrinal Kanti Roychowdhury. Least upper bound of the exact formula for optimal quantization of some uniform Cantor distributions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4555-4570. doi: 10.3934/dcds.2018199
##### References:
 [1] E. F. Abaya and G. L. Wise, Some remarks on the existence of optimal quantizers, Statistics & Probability Letters, 2 (1984), 349-351. doi: 10.1016/0167-7152(84)90045-2. [2] C. P. Dettmann and M. K. Roychowdhury, Quantization for uniform distributions on equilateral triangles, Real Analysis Exchange, 42 (2017), 149-166. doi: 10.14321/realanalexch.42.1.0149. [3] Q. Du, V. Faber and M. Gunzburger, Centroidal voronoi tessellations: Applications and algorithms, SIAM Review, 41 (1999), 637-676. doi: 10.1137/S0036144599352836. [4] A. Gersho and R. M. Gray, Vector Quantization and Signal Compression, Kluwer Academy publishers: Boston, 1992. doi: 10.1007/978-1-4615-3626-0. [5] R. M. Gray, J. C. Kieffer and Y. Linde, Locally optimal block quantizer design, Information and Control, 45 (1980), 178-198. doi: 10.1016/S0019-9958(80)90313-7. [6] S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics, 1730, Springer, Berlin, 2000. doi: 10.1007/BFb0103945. [7] S. Graf and H. Luschgy, The quantization of the cantor distribution, Math. Nachr., 183 (1997), 113-133. doi: 10.1002/mana.19971830108. [8] R. M. Gray and D. L. Neuhoff, Quantization, IEEE Transactions on Information Theory, 44 (1998), 2325-2383. doi: 10.1109/18.720541. [9] A. Gyögy and T. Linder, On the structure of optimal entropy-constrained scalar quantizers, IEEE Transactions on Information Theory, 48 (2002), 416-427. doi: 10.1109/18.978755. [10] J. Hutchinson, Fractals and self-similarity, Indiana Univ. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055. [11] D. Pollard, Quantization and the Method of $k$-Means, IEEE Transactions on Information Theory, 28 (1982), 199-205. doi: 10.1109/TIT.1982.1056481. [12] M. K. Roychowdhury, Optimal quantizers for some absolutely continuous probability measures, Real Analysis Exchange, 43 (2017), 105-136. [13] M. K. Roychowdhury, Quantization and centroidal Voronoi tessellations for probability measures on dyadic Cantor sets, Journal of Fractal Geometry, 4 (2017), 127-146. doi: 10.4171/JFG/47. [14] P. L. Zador, Asymptotic quantization error of continuous signals and the quantization dimension, IEEE Transactions on Information Theory, 28 (1982), 139-149. doi: 10.1109/TIT.1982.1056490. [15] R. Zam, Lattice Coding for Signals and Networks: A Structured Coding Approach to Quantization, Modulation, and Multiuser Information Theory, Cambridge University Press, 2014.

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##### References:
 [1] E. F. Abaya and G. L. Wise, Some remarks on the existence of optimal quantizers, Statistics & Probability Letters, 2 (1984), 349-351. doi: 10.1016/0167-7152(84)90045-2. [2] C. P. Dettmann and M. K. Roychowdhury, Quantization for uniform distributions on equilateral triangles, Real Analysis Exchange, 42 (2017), 149-166. doi: 10.14321/realanalexch.42.1.0149. [3] Q. Du, V. Faber and M. Gunzburger, Centroidal voronoi tessellations: Applications and algorithms, SIAM Review, 41 (1999), 637-676. doi: 10.1137/S0036144599352836. [4] A. Gersho and R. M. Gray, Vector Quantization and Signal Compression, Kluwer Academy publishers: Boston, 1992. doi: 10.1007/978-1-4615-3626-0. [5] R. M. Gray, J. C. Kieffer and Y. Linde, Locally optimal block quantizer design, Information and Control, 45 (1980), 178-198. doi: 10.1016/S0019-9958(80)90313-7. [6] S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics, 1730, Springer, Berlin, 2000. doi: 10.1007/BFb0103945. [7] S. Graf and H. Luschgy, The quantization of the cantor distribution, Math. Nachr., 183 (1997), 113-133. doi: 10.1002/mana.19971830108. [8] R. M. Gray and D. L. Neuhoff, Quantization, IEEE Transactions on Information Theory, 44 (1998), 2325-2383. doi: 10.1109/18.720541. [9] A. Gyögy and T. Linder, On the structure of optimal entropy-constrained scalar quantizers, IEEE Transactions on Information Theory, 48 (2002), 416-427. doi: 10.1109/18.978755. [10] J. Hutchinson, Fractals and self-similarity, Indiana Univ. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055. [11] D. Pollard, Quantization and the Method of $k$-Means, IEEE Transactions on Information Theory, 28 (1982), 199-205. doi: 10.1109/TIT.1982.1056481. [12] M. K. Roychowdhury, Optimal quantizers for some absolutely continuous probability measures, Real Analysis Exchange, 43 (2017), 105-136. [13] M. K. Roychowdhury, Quantization and centroidal Voronoi tessellations for probability measures on dyadic Cantor sets, Journal of Fractal Geometry, 4 (2017), 127-146. doi: 10.4171/JFG/47. [14] P. L. Zador, Asymptotic quantization error of continuous signals and the quantization dimension, IEEE Transactions on Information Theory, 28 (1982), 139-149. doi: 10.1109/TIT.1982.1056490. [15] R. Zam, Lattice Coding for Signals and Networks: A Structured Coding Approach to Quantization, Modulation, and Multiuser Information Theory, Cambridge University Press, 2014.
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