Asymptotics of optimal quantizers for some scalar distributions

  • Authors:

  • Jean-Claude Fort;Gilles Pagès

  • Affiliations:

  • Universitè de Nancy I, Institut E. Cartan, B.P. 239, F-54506 Vandæuvre-Lès-Nancy Cedex, France and SAMOS, Universitè de Paris I, U.F.R. 27, 90, F-75634 rue de Tolbiac, ...;Laboratoire de probabilitès et modèles alèatoires, UMR7599, Universitè Pierre et Marie Curie, Case 188, 4 place Jussieu, 75252 Cedex 05, Paris, France

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2002

  • SOM's mathematics

    Neural Networks - 2006 Special issue: Advances in self-organizing maps--WSOM'05

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Abstract

We obtain semi-closed forms for the optimal quantizers of some families of one-dimensional probability distributions. They yield the first examples of non-log-concave distributions for which uniqueness holds. We give two types of applications of these results. One is a fast computation of numerical approximations of one-dimensional optimal quantizers and their use in a multidimensional framework. The other is some asymptotics of the standard empirical measures associated to the optimal quantizers in terms of distribution function, Laplace transform and characteristic function. Moreover, we obtain the rate of convergence in the Bucklew & Wise Theorem and finally the asymptotic size of the Voronoi tessels.