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 
from every map unit weighted by
the neighborhood kernel. The energy function can be further
divided into two parts [24], cf. [16]:,
where 
is the number of data items in the Voronoi region 
of
, and 
is their centroid
. 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.
