Second-Order Learning in Self-Organizing Maps

Kohonen's self-organizing map bears large potentials as a universal tool of nonlinear data analysis. From the practical point of view control parameters like the learning rate and the neighborhood width need special attention in order to exploit the possibilities of the approach. Our paper introduces second order learning methods which generalize the dynamics of Kohonen's learning algorithm in that control parameters are individually attributed to each neuron and adapted automatically. This is achieved by making use of the special properties of the map at phase transitions it undergoes when learning parameters cross critical values. We demonstrate by way of examples both the automatic control of the self-organization process itself, the extraction of principal manifolds, the mapping of hierarchically structured data, and provide also a version of the algorithm which proves feasible in the case of sparse data sets.