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