An Algorithm for Generalized Principal Curves with Adaptive Topology in Complex Data Sets

Generalized principal curves are capable of representing complex data structures as they may have branching points or may consist of disconnected parts. For their construction using an unsupervised learning algorithm the templates need to be structurally adaptive. The present algorithm meets this goal by a combination of a competitive Hebbian learning scheme and a self-organizing map algorithm. Whereas the Hebbian scheme captures the main topological features of the data, in the map the neighborhood widths are automatically adjusted in order to suppress the noisy dimensions. It is noteworthy that the procedure which is natural in prestructured Kohonen nets could be carried over to a neural gas algorithm which does not use an initial connectivity. The principal curve is then given by an averaging procedure over the critical uctuations of the map exploiting noise-induced phase transitions in the neural gas.