The Self-Organizing Map [Kohonen, 1982,Kohonen, 1995,Kohonen et al., 1996a] is an unsupervised non-parametric regression process to represent nonlinear, high-dimensional data on a low-dimensional illustrative display. The input data points are mapped to SOM units on a usually one or two-dimensional grid. The mapping is learned from the training data samples by a simple stochastic learning process, where the SOM units are adjusted by small steps with respect to the feature vectors that are extracted from the data and presented one after another (in a random order).
The most familiar version of the algorithm
[Kohonen, 1990b]
uses the minimum Euclidean distance criterion
for D-dimensional stochastic sample 
to find the winner unit c (the best-matching unit, BMU)
of the total of N SOM units.
The criterion works so that the weight vector 
of the BMU
fulfills the condition
 |
(1) |
The update of the SOM weights corresponding to 
and c
from the equation (1) is
![\begin{displaymath}
\mbox{\boldmath$m$}_i(t+1) = \mbox{\boldmath$m$}_i(t) + h_{c...
... - \mbox{\boldmath$m$}_i(t)] \:,\:
\forall i = 1, \ldots, N \:,\end{displaymath}](img20.gif){kind=small} |
(2) |
is a discrete time index.
The value of the neighborhood function hci(t)
can be, for example,
the learning rate ![$\alpha(t) \in ]0,1[ \: $](img22.gif){kind=small}
,
if the array distance between
and is smaller than
the neighborhood radius r(t)
and zero otherwise
[Kohonen, 1995].
The desired form of convergence and characteristics of the result is ensured
by the gradual decrease of 
and r(t) while t increases.
For faster convergence and simpler computations, the learning rate control can be eliminated by using the iterative batch adaptation [Kohonen, 1992,Kohonen, 1993,Kohonen, 1995,Mulier and Cherkassky, 1995]:
 |
(3) |