About the multidimensional competitive learning vector quantization algorithm with constant gain

The competitive learning vector quantization (CLVQ) algorithm with constant step .>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 .. are studied, including support location and absolute continuity with respect to the Lebesgue measure. Finally, the weak limit set of .. as ..0 is investigated.