Convergence of the one-dimensional Kohonen algorithm

We show in a very general framework the a.s. convergence of the one-dimensional Kohonen algorithm-after self-organization-to its unique equilibrium when the learning rate decreases to 0 in a suitable way. The main requirement is a log-concavity assumption on the stimuli distribution which includes all the usual (truncated) probability distributions (uniform, exponential, gamma distribution with parameter greater than or equal to 1, etc.). For the constant step algorithm, the weak convergence of the invariant distributions towards equilibrium as the step goes to 0 is established too. The main ingredients of the proof are the Poincare-Hopf Theorem and a result of Hirsch on the convergence of cooperative dynamical systems.