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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.
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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 

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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

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