Self-Organizing Maps and Unsupervised Classification - Neural Networks: Methodology and Applications - page 397

Fig. 7.8. Basic principle of self-organizing map modeling of data space. A label c ,
which is selected among P neurons of map C , is associated to any observation z
of the data set D , using the allocation function χ ( χ ( z i = c )); that label allows the
definition of the reference vector w c

7.3.2 The Batch Optimization Algorithm for Topological Maps

In the present section, we describe the minimization of the cost function
J som ( χ,W ). The only difference between the k -means and the self-organizing
map algorithm is the difference between the two cost functions. When T is
kept constant, the minimization of J som may be written in the dynamic clus-
tering formalism (see the section that is devoted to k -means). Here, just as
in the previous section, that formalism provides a proof of convergence of the
algorithm to a local minimum of the cost function.
When T is kept fixed, the minimization of J som is performed iteratively.
Each iteration has two phases. The first phase is an allocation phase and the
second phase is a minimization phase where the cost function that is associated
to the current partition is minimized:

Allocation phase. J som ( χ,W ) is minimized with respect to the allocation
function χ . The set W of reference vectors is kept fixed during that phase.

•

The expression of J som ( χ,W )andof d T z i , w χ ( z i ) show that the best
allocation function is defined for each observation z by

2 =argmax
r∈C

χ T ( z ) = arg max
r∈C

d T ( z , w r ) .

K T ( δ ( c,r )) z − w c

c∈C

That phase allows defining an allocation function χ and the associated
partition of data space D . Then the closest reference vector with respect
to the weighted distance d T is allocated to each observation.

Minimization phase. J som ( χ,W ) is minimized with respect to the reference
vector set W . That minimization is performed while freezing the alloca-
tion function χ that was previously computed. Since J som is convex with
respect to the parameters from W , the minimization can be performed by

•