Asymptotic behavior of self-organizing maps with nonuniform stimuli distribution

Abstract

Here the almost sure convergence of one-dimensional Kohonen's algorithm in its general form, namely, the 2k-neighbor setting with a nonuniform stimuli distribution, is proved. We show that the asymptotic behavior of the algorithm is governed by a cooperative system of differential equations which is irreducible. The system of differential equations possesses an asymptotically stable equilibrium, a compact subset of whose domain of attraction will be visited by the state variable $X^n$ infinitely often. The assumptions on the stimuli distribution and the neighborhood functions are weakened, too.