Dimension reduction issues in self-organizing maps (SOM) Self organizing maps are claimed to be able to visualize/cluster high-dimensional data in a smaller dimensional space.  I have some difficulties in understanding this statement.
Consider a six-dimensional data set; the codebook vector/reference vector is also six-dimensional.  According to the SOM algorithm, updating these reference vectors is also conducted in the six-dimensional vector space.  If we are considering a two dimensional map, how should I understand the map between the six-dimensional data space and two-dimensional map space?
 A: You can think of the SOM as a grid, where the reference vectors are just placed. However, while the elements of the reference vector define the vectors orientation in the input space, they have no direct relation to the placement on the SOM.
During the competitive training of the SOM for each input vector, the winning node is determined, as its reference vector has the smallest (typically Euclidian) distance to the input vector. This reference vector is then adjusted towards the input vector. You can imagine  that, as if the input vector pulls the reference vector in the input space towards itself. The "strength" of pulling is determined by the training rate, which decreases monotonically over time.
Now comes the trick, which makes the SOM topology preserving: The reference vector of the winning node will adjust all reference vectors on the SOM towards itself. The "strength" of pulling is decreasing over the distance on the SOM. Hence, after having performed one training cycle, the reference vectors close to each other in input space will be also located close to each other on the SOM.
