Self-organization property of Kohonen's map with general type of stimuli distribution

Here the self-organization property of one-dimensional Kohonen's algorithm in its 2k-neighbor setting with a general type of stimuli distribution and non-increasing learning rate is considered. A new definition of the winner is given, which coincides with the usual definition in implementations of the algorithm. We prove that the probability of self-organization for all initial weights of neurons is uniformly positive. For the special case of a constant learning rate, it implies that the algorithm self-organizes with probability one. The conditions imposed on the neighborhood function, stimuli distribution and learning rate are quite general.