Convergences of the Kohonen maps: A dynamical system approach

Abstract

We present the now well-known ODE method in a general framework. We state a global convergence theorem when the ODE has no "pseu-docycle" which could apply in many cases. We note that the Kohonen algorithm in dimension 1 (units and stimuli) after self-organization is a cooperative dynamical system and prove the uniqueness of its equilibrium point. We then derive results of convergence (a.s and in distribution) for a very general class of stimuli distributions and neighbourhood functions.