Self-organization of a one-dimensional Kohonen network with quantized weights and inputs

doi:10.1016/0893-6080(94)90090-6

Neural Networks

Volume 7, Issue 9, 1994, Pages 1427-1439

https://doi.org/10.1016/0893-6080(94)90090-6 Get rights and content

Abstract

If a self-organizing neural network has to be implemented on a digital (or mixed analog and digital) circuit realization, with on-chip learning, all the input signals and the weight values have to be quantized. It is therefore crucial to study whether this quantization does not annihilate the self-organization property of the weights. This paper provides necessary and sufficient conditions on the parameters of a one-dimensional network, which ensure the organization of the weights for any one-dimensional input probability distribution. These results are rigorously proved using the Markovian formulation of Kohonen's algorithm.

References (15)

C. Bouton et al.
Self-organization and a.s. convergence of the one-dimensional Kohonen algorithm with non-uniformly distributed stimuli
Stochastic Processes and Their Applications
(1993)
T. Kohonen
Self-organizing maps: Optimization approaches
H. Ritter et al.
Topology-conserving maps for learning visuo-motor-coordination
Neural Networks
(1989)
M. Cottrell et al.
Etude d'un processus d'auto-organisation
Annales de l'Institut Henri Poincaré
(1987)
E. Erwin et al.
Self-organizing maps: Stationary states, metastability and convergence rate
Biological Cybernetics
(1992)
E. Erwin et al.
Self-organizing maps: Ordering, convergence properties and energy functions
Biological Cybernetics
(1992)
T. Kohonen
The “neural” phonetic typewriter
Computer
(1988)

There are more references available in the full text version of this article.

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Contributed articleSelf-organization of a one-dimensional Kohonen network with quantized weights and inputs

Abstract

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Contributed article
Self-organization of a one-dimensional Kohonen network with quantized weights and inputs