SOM Algorithm Research Articles

The Softmap Algorithm

A new unsupervised competitive learning rule is introduced, called the Self-organizing free-topology map (Softmap) algorithm, for nonparametric density estimation. The receptive fields of the formal neurons are overlapping, radially-symmetric kernels, the radii of which are adapted to the local input density together with the weight vectors which define the kernel centers. A fuzzy code membership function is introduced in order to encompass, in a novel way, the presence of overlapping receptive fields in the competitive learning scheme. Furthermore, a computationally simple heuristic is introduced for determining the overall degree of smoothness of the resulting density estimate. Finally, the density estimation performance is compared to that of the variable kernel method, VBAR and Kohonen‘s SOM algorithm.

Self-Organization as an Iterative Kernel Smoothing Process

Kohonen's self-organizing map, when described in a batch processing mode, can be interpreted as a statistical kernel smoothing problem. The batch SOM algorithm consists of two steps. First, the training data are partitioned according to the Voronoi regions of the map unit locations. Second, the units are updated by taking weighted centroids of the data falling into the Voronoi regions, with the weighing function given by the neighborhood. Then, the neighborhood width is decreased and steps 1, 2 are repeated. The second step can be interpreted as a statistical kernel smoothing problem where the neighborhood function corresponds to the kernel and neighborhood width corresponds to kernel span. To determine the new unit locations, kernel smoothing is applied to the centroids of the Voronoi regions in the topological space. This interpretation leads to some new insights concerning the role of the neighborhood and dimensionality reduction. It also strengthens the algorithm's connection with the Principal Curve algorithm. A generalized self-organizing algorithm is proposed, where the kernel smoothing step is replaced with an arbitrary nonparametric regression method.