Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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QUANTIZATION FOR A PROBABILITY DISTRIBUTION GENERATED BY AN INFINITE ITERATED FUNCTION SYSTEM

  • Received : 2021.08.06
  • Accepted : 2022.01.19
  • Published : 2022.07.31
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Abstract

Quantization for probability distributions concerns the best approximation of a d-dimensional probability distribution P by a discrete probability with a given number n of supporting points. In this paper, we have considered a probability measure generated by an infinite iterated function system associated with a probability vector on ℝ. For such a probability measure P, an induction formula to determine the optimal sets of n-means and the nth quantization error for every natural number n is given. In addition, using the induction formula we give some results and observations about the optimal sets of n-means for all n ≥ 2.

Keywords

Acknowledgement

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

References

  1. E. F. Abaya and G. L. Wise, Some remarks on the existence of optimal quantizers, Stat. Probab. Lett. 2 (1984), no. 6, 349-351. https://doi.org/10.1016/0167-7152(84)90045-2
  2. D. Comez and M. K. Roychowdhury, Quantization for uniform distributions of Cantor dusts on ℝ2, Topology Proc. 56 (2020), 195-218.
  3. Q. Du, V. Faber, and M. Gunzburger, Centroidal Voronoi tessellations: applications and algorithms, SIAM Rev. 41 (1999), no. 4, 637-676. https://doi.org/10.1137/S0036144599352836
  4. A. Gersho and R. M. Gray, Vector Quantization and Signal Compression, Kluwer Academy publishers: Boston, 1992.
  5. R. M. Gray, J. C. Kieffer, and Y. Linde, Locally optimal block quantizer design, Inform. and Control 45 (1980), no. 2, 178-198. https://doi.org/10.1016/S0019-9958(80)90313-7
  6. S. Graf and H. Luschgy, The quantization of the Cantor distribution, Math. Nachr. 183 (1997), 113-133. https://doi.org/10.1002/mana.19971830108
  7. S. Graf and H. Luschgy, Foundations of quantization for probability distributions, Lecture Notes in Mathematics, 1730, Springer-Verlag, Berlin, 2000. https://doi.org/10.1007/BFb0103945
  8. R. M. Gray and D. L. Neuhoff, Quantization, IEEE Trans. Inform. Theory 44 (1998), no. 6, 2325-2383. https://doi.org/10.1109/18.720541
  9. A. Gyorgy and T. Linder, On the structure of optimal entropy-constrained scalar quantizers, IEEE Trans. Inform. Theory 48 (2002), no. 2, 416-427. https://doi.org/10.1109/18.978755
  10. J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713-747. https://doi.org/10.1512/iumj.1981.30.30055
  11. R. D. Mauldin and M. Urbanski, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3) 73 (1996), no. 1, 105-154. https://doi.org/10.1112/plms/s3-73.1.105
  12. M. Moran, Hausdorff measure of infinitely generated self-similar sets, Monatsh. Math. 122 (1996), no. 4, 387-399. https://doi.org/10.1007/BF01326037
  13. L. Roychowdhury, Optimal quantization for nonuniform Cantor distributions, J. Interdiscip. Math. 22 (2019), 1325-1348. https://doi.org/10.1080/09720502.2019.1698401
  14. M. K. Roychowdhury, Quantization and centroidal Voronoi tessellations for probability measures on dyadic Cantor sets, J. Fractal Geom. 4 (2017), no. 2, 127-146. https://doi.org/10.4171/JFG/47
  15. M. K. Roychowdhury, Optimal quantization for the Cantor distribution generated by infinite similutudes, Israel J. Math. 231 (2019), no. 1, 437-466. https://doi.org/10.1007/s11856-019-1859-5
  16. P. L. Zador, Asymptotic quantization error of continuous signals and the quantization dimension, IEEE Trans. Inform. Theory 28 (1982), no. 2, 139-149. https://doi.org/10.1109/TIT.1982.1056490