The cost function of the SOM, Equation 7, can be decomposed into two terms as follows (Lampinen and Oja, 1992; here a discrete version will be presented):
Here 
denotes the number of the data items which are closest to
the reference vector 
, and 
, where 
is the Voronoi region
corresponding to the reference vector . When deriving this
approximation is was assumed that 
for all i,
which holds exactly for toroidal maps when the kernel h has the same
shape for all i, and also away from the borders on a non-toroidal
map if the kernel differs from zero only locally.
The first term in Equation 9 corresponds to the cost
function of the K-means clustering algorithm, that is, the average
distance from the data points to the nearest cluster centroid. Here
the clusters are not defined in terms of their centroids, however, but
in terms of the reference vectors . This first term may be
interpreted as one way of measuring how accurately the map follows the
distribution of the input data.
The second term, on the other hand, may be interpreted as governing
the ordering of the reference vectors. In considering the second term
it may be of help to note that 
and will in
general be close to each other, since is the centroid of
the cluster defined by . At a fixed point of the SOM
algorithm 
, which is closer to the more the
neighborhood kernel 
is centered around c. To minimize the
second term the units that are close to each other on the map should
have similar reference vectors, since the value of the neighborhood
kernel is large. Units that lie farther away on the map may, on the
other hand, have quite dissimilar reference vectors, since the
neighborhood kernel is small and the distance thus does not make a
large contribution to the error.
