On multidimensional scaling and the embedding of self-organising maps.

The self-organising map (SOM) and its variant, visualisation induced SOM (ViSOM), have been known to yield similar results to multidimensional scaling (MDS). However, the exact connection has not been established. In this paper, a review on the SOM and its cost function and topological measures is provided first. We then examine the exact scaling effect of the SOM and ViSOM from their objective functions. The SOM is shown to produce a qualitative, nonmetric scaling, while the local distance-preserving ViSOM produces a quantitative or metric scaling. Their relationship with the principal manifold is also discussed. The SOM-based methods not only produce topological or metric scaling but also provide a principal manifold. Furthermore a growing ViSOM is proposed to aid the adaptive embedding of highly nonlinear manifolds. Examples and comparisons with other embedding methods such as Isomap and local linear embedding are also presented.