Convergence theorems for the kohonen feature mapping algorithms with VLRPs

The convergence of the Kohonen feature mapping algorithm with vanishing learning rate parameters (VLRPs) is considered, which includes the simple competitive learning algorithm as a special case. A few examples show that the learning fails to converge to “global minima,” in general. Then, we present a novel approach which enables us to find out a new family of VLRPs such that the corresponding learning algorithm converges to the set of “global minima” with probability one. The new VLRPs is a generalization of the well-known rate parameters used in the simulated annealing. A numerical example is also included to confirm our theoretical approach. We believe that this discovery is of importance for a large class of learning algorithms in neural networks and statistics.