General Riemannian SOM

Abstract: Kohonen's Self-Organizing Maps (SOMs) have proven to be a successful data-reduction method to identify the intrinsic lower-dimensional sub-manifold of a data set that is scattered in the higher-dimensional feature space. Motivated by the possibly non-Euclidian nature of the feature space and of the intrinsic geometry of the data set, we extend the definition of classic SOMs to obtain the General Riemannian SOM (GRiSOM). We additionally provide an implementation as a proof-of-concept for geometries with constant curvature. We furthermore perform the analytic and numerical analysis of the stability limits of certain (GRi)SOM configurations covering the different possible regular tessellation of the map space in each geometry. A deviation between the numerical and analytic stability limit has been observed for the square and hexagonal Euclidean maps for very small neighbourhoods in the map space as well as agreement in case of longer-ranged relations between the map nodes.