CONVERGENCE OF THE ONE-DIMENSIONAL KOHONEN ALGORITHM

We show in a very general framework the a.s. convergence of the one-di mensional Kohonen algorithm-after self-organization-to its unique equi librium when the learning rate decreases to 0 in a suitable way. The m ain requirement is a log-concavity assumption on the stimuli distribut ion 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 conve rgence of the invariant distributions towards equilibrium as the step goes to 0 is established too. The main ingredients of the proof are th e Poincare-Hopf Theorem and a result of Hirsch on the convergence of c ooperative dynamical systems.