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Asymptotic Optimality of Scalar Gersho Quantizers

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Abstract

In his famous paper (Gersho, IEEE Trans. Inf. Theory 25(4):373–380, 1979), Gersho stressed that the codecells of optimal quantizers asymptotically make an equal contribution to the distortion of the quantizer. Motivated by this fact, we investigate in this paper quantizers in the scalar case, where each codecell contributes with exactly the same portion to the quantization error. We show that such quantizers of Gersho type—or Gersho quantizers for short—exist for nonatomic scalar distributions. As a main result, we prove that Gersho quantizers are asymptotically optimal.

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References

  1. Abaya, E.F., Wise, G.L.: Some notes on optimal quantization. In: Proceedings of the International Conference on Communications, Denver, Colorado. IEEE Press, New York (1981)

    Google Scholar 

  2. Abaya, E.F., Wise, G.L.: On the existence of optimal quantizers. IEEE Trans. Inf. Theory 28(6), 937–940 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bucklew, J.A., Wise, G.L.: Multidimensional asymptotic quantization theory with rth power distortion measures. IEEE Trans. Inf. Theory 28(2), 239–247 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bucklew, J.A.: Two results on the asymptotic performance of quantizers. IEEE Trans. Inf. Theory 30(2), 341–348 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. Delattre, S., Fort, J.-C., Pagès, G.: Local distortion and μ-mass of the cells of one dimensional asymptotically optimal quantizers. Commun. Stat., Theory Methods 33(5), 1087–1117 (2004)

    Article  MATH  Google Scholar 

  6. Fort, J.-C., Pagès, G.: Asymptotics of optimal quantizers for some scalar distributions. J. Comput. Appl. Math. 146(2), 253–275 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gersho, A.: Asymptotically optimal block quantization. IEEE Trans. Inf. Theory 25(4), 373–380 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gersho, A., Gray, R.M.: Vector Quantization and Signal Compression. Kluwer Academic, Boston (1992)

    Book  MATH  Google Scholar 

  9. Graf, S., Luschgy, H.: The quantization of the Cantor distribution. Math. Nachr. 183, 113–133 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Graf, S., Luschgy, H.: Foundations of Quantization for Probability Distributions. Lecture Notes, vol. 1730. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  11. Graf, S., Luschgy, H.: The point density measure in the quantization of self-similar probabilities. Math. Proc. Camb. Philos. Soc. 138(3), 513–531 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gray, R., Neuhoff, D.: Quantization. IEEE Trans. Inf. Theory 44(6), 2325–2383 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kesseböhmer, M., Zhu, S.: Stability of quantization dimension and quantization for homogeneous Cantor measures. Math. Nachr. 280(8), 866–881 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kreitmeier, W.: Optimal quantization for dyadic homogeneous Cantor distributions. Math. Nachr. 281(9), 1307–1327 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kreitmeier, W.: Optimal vector quantization in terms of Wasserstein distance. J. Multivar. Anal. 102(8), 1225–1239 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kreitmeier, W., Linder, T.: Entropy density and mismatch in high-rate scalar quantization with Rényi entropy constraint. IEEE Trans. Inf. Theory 58, 4105–4116 (2012)

    Article  MathSciNet  Google Scholar 

  17. Linder, T.: On asymptotically optimal companding quantization. Probl. Control Inf. Theory 20(6), 475–484 (1991)

    MathSciNet  MATH  Google Scholar 

  18. Lloyd, S.P.: Least squares quantization in PCM. IEEE Trans. Inf. Theory 28(2), 129–137 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Panter, P.F., Dite, W.: Quantization in pulse-count modulation with nonuniform spacing of levels. Proc. IRE 39, 44–48 (1951)

    Article  Google Scholar 

  20. Pollard, D.: Quantization and the method of k-means. IEEE Trans. Inf. Theory 28(2), 199–205 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tóth, L.F.: Sur la représentation d’une population infinie par un nombre fini d’éléments. Acta Math. Acad. Sci. Hung. 10(3–4), 299–304 (1959)

    Article  MATH  Google Scholar 

  22. Trushkin, A.V.: Monotony of Lloyd’s method II for log-concave density and convex error weighting function. IEEE Trans. Inf. Theory 30(2), 380–383 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zador, P.: Development and evaluation of procedures for quantizing multivariate distributions. Ph.D. thesis, Stanford University (1963)

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Acknowledgements

The author is indebted to the referee for his/her valuable suggestions and comments, which improved the quality and presentation of this paper. The author was supported through a grant from the German Research Foundation (DFG).The author was supported through a grant from the German Research Foundation (DFG).

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  1. Department of Informatics and Mathematics, University of Passau, Innstraße 33, 94032, Passau, Germany

    Wolfgang Kreitmeier

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  1. Wolfgang Kreitmeier
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Correspondence to Wolfgang Kreitmeier.

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Communicated by Emmanuel J. Candes.

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Kreitmeier, W. Asymptotic Optimality of Scalar Gersho Quantizers. Constr Approx 38, 365–396 (2013). https://doi.org/10.1007/s00365-013-9214-2

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  • DOI: https://doi.org/10.1007/s00365-013-9214-2

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