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August 1997 About the multidimensional competitive learning vector quantization algorithm with constant gain
Catherine Bouton, Gilles Pagès
Ann. Appl. Probab. 7(3): 679-710 (August 1997). DOI: 10.1214/aoap/1034801249

Abstract

The competitive learning vector quantization (CLVQ) algorithm with constant step $\varepsilon > 0$--also known as the Kohonen algorithm with 0 neighbors--is studied when the stimuli are i.i.d. vectors. Its first noticeable feature is that, unlike the one-dimensional case which has $n!$ absorbing subsets, the CLVQ algorithm is "irreducible on open sets" whenever the stimuli distribution has a path-connected support with a nonempty interior. Then the Doeblin recurrence (or uniform ergodicity) of the algorithm is established under some convexity assumption on the support. Several properties of the invariant probability measure $\nu^{\varepsilon}$ are studied, including support location and absolute continuity with respect to the Lebesgue measure. Finally, the weak limit set of $\nu^{\varepsilon}$ as $\varepsilon \to 0$ is investigated.

Citation

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Catherine Bouton. Gilles Pagès. "About the multidimensional competitive learning vector quantization algorithm with constant gain." Ann. Appl. Probab. 7 (3) 679 - 710, August 1997. https://doi.org/10.1214/aoap/1034801249

Information

Published: August 1997
First available in Project Euclid: 16 October 2002

zbMATH: 0892.60082
MathSciNet: MR1459266
Digital Object Identifier: 10.1214/aoap/1034801249

Subjects:
Primary: 60J20
Secondary: 60F99 , 60J10

Keywords: Markov chain , neural networks , uniform ergodicity , Vector quantization

Rights: Copyright © 1997 Institute of Mathematical Statistics

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Vol.7 • No. 3 • August 1997
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