SELF-ORGANIZATION AS AN ITERATIVE KERNEL SMOOTHING PROCESS

Kohonen's self-organizing map, when described in a batch processing mo de, can be interpreted as a statistical kernel smoothing problem. The batch SOM algorithm consists of two steps. First, the training data ar e partitioned according to the Voronoi regions of the map unit locatio ns. Second, the units are updated by taking weighted centroids of the data falling into the Voronoi regions, with the weighing function give n by the neighborhood Then, the neighborhood width is decreased and st eps 1, 2 are repeated. The second step can be interpreted as a statist ical kernel smoothing problem where the neighborhood function correspo nds to the kernel and neighborhood width corresponds to kernel span. T o determine the new unit locations, kernel smoothing is applied to the centroids of the Voronoi regions in the topological space. This inter pretation leads to some new insights concerning the role of the neighb orhood and dimensionality reduction. It also strengthens the algorithm 's connection with the Principal Curve algorithm. A generalized self-o rganizing algorithm is proposed, where the kernel smoothing step is re placed with an arbitrary nonparametric regression method.