• Previous Article
    Existence of torsion-low maximal isotopies for area preserving surface homeomorphisms
  • DCDS Home
  • This Issue
  • Next Article
    Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity
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 and Continuous Dynamical Systems, 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. DuV. 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. GrayJ. 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.

show all references

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. DuV. 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. GrayJ. 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.

[1]

James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667

[2]

Eunju Hwang, Kyung Jae Kim, Bong Dae Choi. Delay distribution and loss probability of bandwidth requests under truncated binary exponential backoff mechanism in IEEE 802.16e over Gilbert-Elliot error channel. Journal of Industrial and Management Optimization, 2009, 5 (3) : 525-540. doi: 10.3934/jimo.2009.5.525

[3]

María Isabel Cortez, Samuel Petite. Realization of big centralizers of minimal aperiodic actions on the Cantor set. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2891-2901. doi: 10.3934/dcds.2020153

[4]

Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185

[5]

Hanqing Jin, Shige Peng. Optimal unbiased estimation for maximal distribution. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 189-198. doi: 10.3934/puqr.2021009

[6]

Nicolai Haydn, Sandro Vaienti. The limiting distribution and error terms for return times of dynamical systems. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 589-616. doi: 10.3934/dcds.2004.10.589

[7]

Zengjing Chen, Yuting Lan, Gaofeng Zong. Strong law of large numbers for upper set-valued and fuzzy-set valued probability. Mathematical Control and Related Fields, 2015, 5 (3) : 435-452. doi: 10.3934/mcrf.2015.5.435

[8]

Xing Huang, Michael Röckner, Feng-Yu Wang. Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3017-3035. doi: 10.3934/dcds.2019125

[9]

Erika Asano, Louis J. Gross, Suzanne Lenhart, Leslie A. Real. Optimal control of vaccine distribution in a rabies metapopulation model. Mathematical Biosciences & Engineering, 2008, 5 (2) : 219-238. doi: 10.3934/mbe.2008.5.219

[10]

Alessia Marigo. Optimal traffic distribution and priority coefficients for telecommunication networks. Networks and Heterogeneous Media, 2006, 1 (2) : 315-336. doi: 10.3934/nhm.2006.1.315

[11]

Marcin Studniarski. Finding all minimal elements of a finite partially ordered set by genetic algorithm with a prescribed probability. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 389-398. doi: 10.3934/naco.2011.1.389

[12]

Giulia Cavagnari. Regularity results for a time-optimal control problem in the space of probability measures. Mathematical Control and Related Fields, 2017, 7 (2) : 213-233. doi: 10.3934/mcrf.2017007

[13]

Meng Wu, Jiefeng Yang. The optimal exit of staged investment when consider the posterior probability. Journal of Industrial and Management Optimization, 2017, 13 (2) : 1105-1123. doi: 10.3934/jimo.2016064

[14]

Xia Han, Zhibin Liang, Yu Yuan, Caibin Zhang. Optimal per-loss reinsurance and investment to minimize the probability of drawdown. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021145

[15]

Yu Yuan, Zhibin Liang, Xia Han. Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs. Journal of Industrial and Management Optimization, 2022, 18 (2) : 933-967. doi: 10.3934/jimo.2021003

[16]

Nicolai T. A. Haydn, Kasia Wasilewska. Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2585-2611. doi: 10.3934/dcds.2016.36.2585

[17]

Jonas Eriksson. A weight-based characterization of the set of correctable error patterns under list-of-2 decoding. Advances in Mathematics of Communications, 2007, 1 (3) : 331-356. doi: 10.3934/amc.2007.1.331

[18]

Kai Liu, Zhi Li. Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3551-3573. doi: 10.3934/dcdsb.2016110

[19]

Ahmad Ahmad Ali, Klaus Deckelnick, Michael Hinze. Error analysis for global minima of semilinear optimal control problems. Mathematical Control and Related Fields, 2018, 8 (1) : 195-215. doi: 10.3934/mcrf.2018009

[20]

Chang-Feng Wang, Yan Han. Optimal assignment of principalship and residual distribution for cooperative R&D. Journal of Industrial and Management Optimization, 2012, 8 (1) : 127-139. doi: 10.3934/jimo.2012.8.127

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (202)
  • HTML views (131)
  • Cited by (4)

Other articles
by authors

[Back to Top]