Constructing principal manifolds in sparse data sets by self-organizing maps with self-regulating neighborhood width

We study the extraction of principal manifolds (PMs) in high-dimensional spaces with modified self-organizing feature maps. Our algorithm embeds a lower-dimensional lattice into a high-dimensional space without topology violations by tuning the neighborhood widths locally. Topology preservation, however, is not sufficient for determining PMs. It still allows for considerable deviations from the PM and is rather unreliable in the case of sparse data sets. These two problems are solved by the introduction of a new principle exploiting the specific dynamical properties of the first-order phase transition induced by dimensional conflicts