Convergence in distributions of the multi-dimensional Kohonen algorithm
type: Item
instance of: scholarly article
zbMATH Open document ID: 0983.60061
author: Ali A. Sadeghi
published in: Journal of Applied Probability
publication date: 2002-04-21T00:00:00Z
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 60J20
Mathematics Subject Classification ID: 92B20
zbMATH DE Number: 1625613
zbMATH Keywords: self-organizing map
zbMATH Keywords: artificial intelligence applications
zbMATH Keywords: neural network
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1239/jap/996986649
OpenAlex ID: W1990634878
DOI: 10.1239/JAP/996986649
title: Convergence in distributions of the multi-dimensional Kohonen algorithm
review text: \textit{T. Kohonen} [Biol. Cybern. 43, 59-69 (1982; Zbl 0466.92002)] introduced the self-organizing map (SOM) as a simple model of a process of self-organization between different areas of the cortex and different sensory inputs. See also \textit{T. Kohonen} [``Self-organizing maps'' (1994; Zbl 0827.68092)] and the author [``A mathematical study of self-organizing neural networks'' (Shaker, Aachen)]. The simplicity of its learning algorithm immediately made the map interesting for many artificial intelligence applications, e.g., clustering, feature mapping and building topology preserving maps. Besides the feedforward neural networks, which are designed for supervised learning, SOM is the most well known neural network with a reasonable capability for unsupervised data analysis.NEWLINENEWLINENEWLINEIn the present paper, the author considers the Kohonen algorithm with a constant learning rate as a Markov process evolving in a topological space. Despite the fact that this algorithm is neither weak nor strong Feller, he shows that it is a \(T\)-chain, regardless of the dimensionalities of both data space and the special shape of the neighborhood function. Furthermore, in the case of a multidimensional setting, which is practically important, the author shows that the chain is \(\Psi\)-irreducible and aperiodic. He then shows that Doeblin's condition is valid, which ensures the convergence in distribution of the process to an invariant probability measure with a geometric rate. Finally, he shows that the process is Harris recurrent, which enables the use of statistical devices to measure the centrality and variability of the invariant probability measure.
reviewed by: Andreas N. Philippou
description: scientific article; zbMATH DE number 1625613