Visualization with Voronoi tessellation and moving output units in Self-Organizing map of the real-number system

The Self-Organizing map (SOM) proposed by T. Kohonen is a method to produce a low-dimensional representation from high-dimensional input data automatically, where output units are restrictedly placed on grid points. We propose real-number SOM (RSOM), where output units are freely placed on the real-number coordinates plane and visualized as a Voronoi diagram. RSOM is a natural extension of the conventional SOM because Voronoi tessellation for the output units on the square grid generates square regions on the output plane, the same as the conventional SOM. We propose two methods of moving with preserving topology of the input data and several visualization method such as minimum spanning tree, variable boundary width and spherical RSOM. We illustrate moving methods decrease errors in results of simulation.