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Convergence in distribution of the multidimensional Kohonen algorithm

  • Here we consider the Kohonen algorithm with a constant learning rate as a Markov process evolving in a topological space. it is shown that the process is an irreducible and aperiodic T-chain, regardless of the dimension of both data space and network and the special shape of the neighborhood function. Moreover the validity of Deoblin's condition is proved. These imply the convergence in distribution of the process to a finite invariant measure with a uniform geometric rate. In addition we show the process is positive Harris recurrent, which enables us to use statistical devices to measure its centrality and variability as the time goes to infinity.

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Metadaten
Author:Ali A. Sadeghi
URN (permanent link):urn:nbn:de:hbz:386-kluedo-6023
Serie (Series number):Berichte der Arbeitsgruppe Technomathematik (AGTM Report) (196)
Document Type:Preprint
Language of publication:English
Year of Completion:1999
Year of Publication:1999
Publishing Institute:Technische Universität Kaiserslautern
Date of the Publication (Server):2000/04/03
Tag:Markov process; multidimensional Kohonen algorithm; neural networks; stochastic stability; uniform ergodicity
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Jxx Markov processes / 60J05 Discrete-time Markov processes on general state spaces
60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Jxx Markov processes / 60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]
92-XX BIOLOGY AND OTHER NATURAL SCIENCES / 92Bxx Mathematical biology in general / 92B20 Neural networks, artificial life and related topics [See also 68T05, 82C32, 94Cxx]
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011