About the multidimensional competitive learning vector quantization algorithm with constant gain

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.