Connection between Self-Organizing Maps and Metric Multidimensional Scaling

The self-organizing map (SOM) and some of its variants such as visualization induced SOM (ViSOM) have been seen to yield similar results to multidimensional scaling (MDS). However the exact connection has yet been established, though similar topographic mapping results have been shown in several studies. In this paper we provide a review on the SOM and its cost function and topological measures. Then we examine its relationship with MDS from their cost functions in the aspect of data visualization. The SOM is shown to produce a quantized, qualitative scaling and while the ViSOM a quantitative or metric scaling. The SOM can also be regarded as a generalized MDS to relate two metric spaces by forming a topological mapping between them. The connection between MDS and principal manifolds is also discussed.