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Convergence in distribution of the multi-dimensional Kohonen algorithm

Part of: Markov processes Mathematical biology in general

Published online by Cambridge University Press:  14 July 2016

Ali A. Sadeghi
Ali A. Sadeghi*
Affiliation:
University of Calgary
*
∗ Postal address: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, Canada. Email address: sadeghi@math.ucalgary.ca
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Abstract

Here we consider the Kohonen algorithm with a constant learning rate as a Markov process evolving in a topological space. Despite the fact that the algorithm is not weak Feller, we show that it is a T-chain, regardless of the dimensionalities of both data space and network and the special shape of the neighborhood function. In addition for the practically important case of the multi-dimensional setting, it is shown that the chain is irreducible and aperiodic. We show that these imply the validity of Doeblin's condition, which in turn ensures the convergence in distribution of the process to an invariant probability measure with a geometric rate. Furthermore, it is shown that the process is positive Harris recurrent, which enables us to use statistical devices to measure the centrality and variability of the invariant probability measure. Our results cover a wide class of neighborhood functions.

Keywords

neural networksmulti-dimensional Kohonen algorithmMarkov processstochastic stabilityuniform ergodicity

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 92B20: Neural networks, artificial life and related topics
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 136 - 151
Copyright
Copyright © by the Applied Probability Trust 2001 

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