next up previous contents
Next: Important properties Up: Mathematical treatment Previous: Convergence:

Energy function:

As opposed to many other neural algorithms, the original SOM algorithm cannot be derived from an energyfunction [7]. However, in the case of a discrete data set and a fixed neighborhood kernel, the SOM has been shown to have an energy function [35]:


This resembles the energy function of the k-means vector quantization algorithm, except that the SOM takes into account the distance of vector tex2html_wrap_inline3156 from every map unit weighted by the neighborhood kernel. The energy function can be further divided into two parts [24], cf. [16]:,


where tex2html_wrap_inline3266 is the number of data items in the Voronoi region tex2html_wrap_inline3182 of tex2html_wrap_inline3162 , and tex2html_wrap_inline3272 is their centroid tex2html_wrap_inline3274 . The first term equals the energy function of the k-means algorithm and corresponds here to the vector quantization quality of the map. The second term is minimized when nearby map units have weight vectors close to each other in the input space. Thus, it corresponds to the ordering of the map.

Juha Vesanto
Tue May 27 12:40:37 EET DST 1997