Weight convergence and weight density of the multi-dimensional SOFM algorithm

In this paper, we analyze convergence properties of the self-organizing feature map (SOFM) with multidimensional input using Robbins-Monro stochastic approximation principle. It is shown that the SOFM algorithm optimizes a well defined energy function and converges almost truly (i.e. with probability one) if the input data is from a discrete stochastic distribution. For the case of multidimensional inputs generated from continuous distributions, it is shown that the weights of the SOFM algorithm converge almost truly to the centroids of the cells of a Voronoi partition of the input space if the neighborhood function satisfies some reasonable conditions. The density of the weight space in the equilibrium states is also investigated