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One-dimensional Kohonen maps are super-stable with exponential rate

Part of: Mathematical biology in general Artificial intelligence (68Txx) Multivariate analysis Linear inference, regression Markov processes

Published online by Cambridge University Press:  01 July 2016

Robert M. Burton  and
David C. Plaehn
Robert M. Burton*
Affiliation:
Oregon State University
David C. Plaehn*
Affiliation:
Oregon State University
*
∗ Postal address: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331, USA.
∗ Postal address: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331, USA.
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Abstract

Kohonen self-organizing interval maps are considered. In this model a linear graph is embedded randomly into the unit interval. At each time a point is chosen randomly according to a fixed distribution. The nearest vertex and some of its nearby neighbors are moved closer to the point. These models have been proposed as models of learning in the audio-cortex. The models possess not only the structure of a Markov chain, but also the added structure of a random dynamical system. This structure is used to show that for a large class of these models, in a strong way, the initial conditions are unimportant and only the dynamics govern the future. A contractive condition is proven in spite of the fact that the maps are not continuous. This, in turn, shows that the Markov chain is uniformly ergodic.

Keywords

Self-organizationKohonenneural networkMarkov chainrandom transformationsuper-stability

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 92B20: Neural networks, artificial life and related topics 68T10: Pattern recognition, speech recognition
Secondary: 62J20: Diagnostics 62H30: Classification and discrimination; cluster analysis
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 2 , June 1999 , pp. 367 - 393
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

Research supported by AFOSR grant 90-0251.

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