An Adaptive Multidimensional Scaling and Principled Nonlinear Manifold

The self-organizing map (SOM) and its variant, the visualization induced SOM (ViSOM), have been linked with principal manifolds. They have also been shown to yield similar results to multidimensional scaling (MDS). However the exact connection has not 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, 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 preservation, to adaptively construct a metric local scaling and extract a 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.