CONVERGENCE AND ORDERING OF KOHONENS BATCH MAP

The convergence and ordering of Kohonen's batch-mode self-organizing m ap with Heskes and Kappen's (1993) winner selection are proved. Selim and Ismail's (1984) objective function for k-means clustering is gener alized in the convergence proof of the self-organizing map. It is show n that when the neighborhood relation is doubly decreasing, order in t he map is preserved. An unordered map becomes ordered when a degenerat e state of ordering is entered, where the number of distinct winners i s one or two. One strategy to enter this state is to run the algorithm with a broad neighborhood relation.