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.
@article{1034801249,
author = {Bouton, Catherine and Pag\`es, Gilles},
title = {About the multidimensional competitive learning vector
quantization algorithm with constant gain},
journal = {Ann. Appl. Probab.},
volume = {7},
number = {1},
year = {1997},
pages = { 679-710},
language = {en},
url = {http://dml.mathdoc.fr/item/1034801249}
}
Bouton, Catherine; Pagès, Gilles. About the multidimensional competitive learning vector
quantization algorithm with constant gain. Ann. Appl. Probab., Tome 7 (1997) no. 1, pp. 679-710. http://gdmltest.u-ga.fr/item/1034801249/