Convergence of one-dimensional self-organizing map

Analyzes the convergence property of the one-dimensional self-organizing map (SOM). The key of the proof is the application of Ljung's theorem [1977]. With the aid of the theorem, the authors can conclude that convergence of the one dimensional self-organizing map is almost certain if the following conditions are fulfilled, (i) the map is initial in order, (ii) the neighborhood interacting function (NIF) is non-increasing outward throughout the neighborhood interacting set (NIS) and (iii) the input distribution is stationary. Note that these conditions are less restrictive than those obtained previously in two folds: (i) there is no limit on the size of the NIS and (ii) the input distribution is not required to be uniform.<<ETX>>