Learning nonlinear principal manifolds by self-organising maps

Abstract

This chapter provides an overview on the self-organised map (SOM) in the context of manifold mapping. It first reviews the background of the SOM and issues on its cost function and topology measures. Then its variant, the visualisation induced SOM (ViSOM) proposed for preserving local metric on the map, is introduced and reviewed for data visualisation. The relationships among the SOM, ViSOM, multidimensional scaling, and principal curves are analysed and discussed. Both the SOM and ViSOM produce a scaling and dimension-reduction mapping or manifold of the input space. The SOM is shown to be a qualitative scaling method, while the ViSOM is a metric scaling and approximates a discrete principal curve/surface. Examples and applications of extracting data manifolds using SOM-based techniques are presented.