This paper shows that the 2-neighbour Kohonen algorithm is self-organizing under pretty general assumptions on the stimuli distribution μ (supp(μc) contains a non-empty open set) and is a.s. convergent-in a weakened sense-as soon as μ admits a log-concave density. The 0-neighbour algorithm is shown to have similar converging properties. Some numerical simulations illustrate the theoretical results and a counter-example provided by a specific class of density functions. © 1993.
Bouton, C., & Pagès, G. (1993). Self-organization and a.s. convergence of the one-dimensional Kohonen algorithm with non-uniformly distributed stimuli. Stochastic Processes and Their Applications, 47(2), 249–274. https://doi.org/10.1016/0304-4149(93)90017-X