Self-Organizing Maps and Unsupervised Classification - Neural Networks: Methodology and Applications - page 404

Fig. 7.13. Visualization of natural boundaries of the partition of the data set into
two subsets. The neurons without any observation allocation are shown as black
dots

7.3.3 Kohonen's Algorithm

The original SOM algorithm, as suggested by T. Kohonen, stems from the
dynamic clustering version that was described above. We will now describe
its specific features. As for k -means, a stochastic version of SOM is available.
It su ces to observe that, during the minimization phase, it is not necessary
to terminate the minimization process and compute the global minimum of
J som ( χ,W ) for a given allocation function χ : one just has to make it decrease.

Therefore, relation w c = r∈C K ( δ ( c,r )) Z r / r∈C K ( δ ( c,r )) n r may be re-
placed by a simple gradient descent step. Thus, at iteration t and for neuron
c , one has

µ t ∂J som
∂ w t− 1

w c = w t− 1

c −

,

c

where µ t is the gradient step at iteration t ,

=2
z i ∈A

∂J som
∂ w c

K T ( δ ( c,χ ( z i ))) ( z i − w c ) .