Dimensionality Reduction of Self Organising Maps

Question

I've probably read any article on dimensionality reduction of Self Organising Maps but just couldn't fully comprehend this process.

My understanding so far is:

SOM are two-layer networks, whose layers (first - input layer, second - output layer) are connected via a weight matrix.

Dimensionality reduction is the process of reducing the dimensionality of input data to a lower (e.g. one or two) dimensionality, which will later be displayed by the Kohonen network (output layer).

Now the tough part begins..

Let's imagine we do have a 3 x 3 grid (9 points) and we want to represent the color space (3 dimension: red, green, blue).

What does happen next?

• (1) Does dimensionality reduction mean that the weight vectors which are assigned to any node in the output layer can be less then the dimensionality of the input samples?

or

• (2) Does dimensionality reduction only means that the high dimensionality of the sample data is reduced to the dimensionality of the output layer (in our case 2 dimensions - because of 3 x 3). However, any neuron of the output layer carries a weight vector whose dimensionality equals the dimensionality of the sample data?

To project that on our example

• To (1) - Our 3D-input space (red, green, blue) is reduced to (red, green).
• To (2) - Our 3D-input space is mapped onto a 2D-map, where any of the 9 nodes is carrying a vector which represents (red, green, blue)

I really hope that my explanation is comprehensive and you get my problem.
Thank you a lot!

Answer 1

SOM can be thought of as a single layer of neurons. The inputs connect to the neurons, and the neurons are in a 2D sheet. Each neuron will contain a weight vector that corresponds to a value of red, green and blue. For dimensionality reduction, you can use the position of the best matching neuron instead of the input value. So, that corresponds to (2) above.

Number (1) doesn't make much sense to me, because if you lose the (blue) then won't you lose information?

This tutorial is good, and uses the color map as its example: http://www.ai-junkie.com/ann/som/som1.html

Kohonen also used the colors as an example in his recent MATLAB book (pages 59-64), available at http://docs.unigrafia.fi/publications/kohonen_teuvo