An Adaptive Multidimensional Scaling and Principled Nonlinear Manifold

Abstract

The self-organizing map (SOM) and some of its variants such as visualization induced SOM (ViSOM) have been shown to yield similar results to multidimensional scaling (MDS). However the exact connection has yet been established. In this paper we first examine their relationship with (generalized) MDS from their cost functions in the aspect of data visualization and dimensionality reduction. The SOM is shown to produce a quantized, qualitative or nonmetric scaling and while the ViSOM is a quantitative metric scaling. Then we propose a way to use the core principle of the ViSOM, i.e. local distance preserving, to adaptively and incrementally construct a metric local scaling and to extract nonlinear manifold. Comparison with other methods such as ISOMAP and LLE has been made, especially in mapping highly nonlinear subspaces. The advantages over other methods are also discussed.