Self-Organizing Maps on non-euclidean Spaces

Abstract

This chapter proposes a new type of self-organizing map (SOM) that is based on discretizations of curved, non-euclidean spaces. As an introductory example, it briefly discusses “spherical SOMs” on tesselations of the sphere for the display of directional data. It describes the construction of “hyperbolic SOMs” using regular tesselations of the hyperbolic plane, which is a non-euclidean space characterized by constant negative gaussian curvature. While the applications of the SOM are extremely wide-spread, the majority of uses still follow the original motivation of the SOM to create dimension-reduced “feature maps” for various uses, most prominently the data visualization. The approach is motivated by the recent observation that the geometry of hyperbolic spaces possesses very favorable properties for the mapping of hierarchical data. The chapter concludes with some initial simulation results illustrating some properties of the hyperbolic SOM and discusses a number of issues for future research.