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Convergence of the one-dimensional Kohonen algorithm

Part of: Sequential methods

Published online by Cambridge University Press:  01 July 2016

Michel Benaïm ,
Jean-Claude Fort  and
Gilles Pagès
Michel Benaïm*
Affiliation:
Université Paul Sabatier, Toulouse
Jean-Claude Fort*
Affiliation:
Université Nancy I and SAMOS-Paris I
Gilles Pagès*
Affiliation:
Universités Paris 12 and Paris 6, URA 224
*
∗ Postal address: Laboratoire de Statistique et Probabilitiés, Université Paul Sabatier, 118 Route de Narbonne, 31062, Toulouse, France.
∗∗ Postal address: Faculté des Sciences, Université Nancy 1, F-54506 Vaudœuvre-les-Nancy, Cedex, France.
∗∗∗ Laboratoire de Probabilités, Tour 56 3e étage, Université Pierre et Marie Curie, 4 Place Jussieu, F-75252 Paris Cedex 05, France. Email address: gpa@ccr.jussieu.fr
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Abstract

We show in a very general framework the a.s. convergence of the one-dimensional Kohonen algorithm–after self-organization–to its unique equilibrium when the learning rate decreases to 0 in a suitable way. The main requirement is a log-concavity assumption on the stimuli distribution which includes all the usual (truncated) probability distributions (uniform, exponential, gamma distribution with parameter ≥ 1, etc.). For the constant step algorithm, the weak convergence of the invariant distributions towards equilibrium as the step goes to 0 is established too. The main ingredients of the proof are the Poincaré-Hopf Theorem and a result of Hirsch on the convergence of cooperative dynamical systems.

Keywords

Kohonen algorithmstochastic algorithmcooperative dynamical system

MSC classification

Primary: 62L20: Stochastic approximation
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 3 , September 1998 , pp. 850 - 869
Copyright
Copyright © Applied Probability Trust 1998 

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