Hostname: page-component-7479d7b7d-jwnkl Total loading time: 0 Render date: 2024-07-12T22:28:34.638Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence in Distribution of the One-Dimensional Kohonen Algorithms when the Stimuli are not Uniform

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Catherine Bouton  and
Gilles Pagès
Catherine Bouton*
Affiliation:
Université du Maine
Gilles Pagès*
Affiliation:
Université Paris I
*
* Postal address: Université du Maine, Route de Laval, BP 535, F-72107 Le Mans Cedex, France.
** Postal address: Université Paris I, U.F.R. 27, 90 rue de Tolbiac, F-75634 Paris Cedex 13, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the one-dimensional self-organizing Kohonen algorithm (with zero or two neighbours and constant step ε) is a Doeblin recurrent Markov chain provided that the stimuli distribution μ is lower bounded by the Lebesgue measure on some open set. Some properties of the invariant probability measure vε (support, absolute continuity, etc.) are established as well as its asymptotic behaviour as ε ↓ 0 and its robustness with respect to μ.

Keywords

MARKOV CHAINSDOEBLIN RECURRENCENEURAL NETWORKS

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 60F99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 1 , March 1994 , pp. 80 - 103
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Both authors are also affiliated to Laboratoire de Probabilités, URA 224, Université Pierre et Marie Curie, 4 place Jussieu, F-75252 Paris Cedex 05, France.

References

[1] Benveniste, A., Metivier, M. and Priouret, P. (1987) Algorithmes adaptatifs et approximations stochastiques. Masson, Paris.Google Scholar
[2] Bouton, C. and Pages, G. (1993) Self-organization and convergence of the one-dimensional Kohonen algorithm with non uniformly distributed stimuli. Stoch. Proc. Appl. 47.CrossRefGoogle Scholar
[3] Cottrell, M. and Fort, J. C. (1986) Etude d'un algorithme d'auto-organisation. Ann. Inst. H. Poincaré 23, 1–20 Google Scholar
[4] Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[5] Duflo, M. (1990) Méthodes récursives aléatoires. Masson, Paris.Google Scholar
[6] Kohonen, T. (1982) Analysis of a simple self-organizing process. Biol. Cybernet. 44, 135–140.CrossRefGoogle Scholar
[7] Kohonen, T. (1984) Self-organization and Associative Memory. Springer-Verlag, Berlin.Google Scholar
[8] Kushner, H. J. and Clark, D. S. (1978) Stochastic Approximation for Constrained and Unconstrained Systems. Applied Math. Science Series 26, Springer-Verlag, Berlin.Google Scholar
[9] Pages, G. (1992) Mosaïque de Voronoï, algorithmes de quantification de l'espace et intégration numérique. Preprint no 138, Lab. de Proba. de l'Univ. Paris 6.Google Scholar
[10] Revuz, D. (1975) Markov Chains. North-Holland, Amsterdam.Google Scholar
[11] Sarzeaud, O., Stéphan, Y. and Touzet, C. (1990) Application des cartes auto-organisatrices à la génération de maillages aux éléments finis. Proc. Neuro-Nîmes, 1, 81–96.Google Scholar