Neural Computing

| January 6, 2015

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The purpose of this assignment is to test the ability of the Kohonen Self- Organizing Feature Map (SOFM) network to represent a distribution of input vectors xi ∈ Rn in some space. It is generally assumed that the vectors are ‘features’ extracted from a data set. The aim of the SOFM is to capture the main components of their statistical distribution, so that it might be used as the front-end of a pattern recognition system, or even as a data compres- sion. It works by imposing a certain topology on the representative vectors uj , 1 ≤ j ≤ N , such that at each iteration only the neighbours of a given unit are modified. How the topology is to be chosen is a moot point – Kohonen himself tended to justify it as being given by the ‘architecture’ of the learning system; for example, the cortex is essentially two dimensional in its layout, so one obvious topology is a two dimensional neighbour system. That is the one we are using in the tests. In fact, it is the classic example that Kohonen used to illustrate the SOFM: a set of vectors vi is drawn uniformly from the unitsquare,ie. vi ∈(0,1]×(0,1]. AsetofN=M2 pointsuj ∈R2 isused in the SOFM. Initially, these points are picked at random. On each iteration i, the following update rule is used:
If j = argmink∥vi − uk∥2 then 1. uj = uj + λi(vi − uj)
1
2.uk=uk+γi(vi−uk), k∈Nj
where Nj is the set of neighbours of node j, which in the case of the 2 − D
grid is those units having 2 − D co-ordinates satisfying
d(j, k) = (x(j) − x(k))2 + (y(j) − y(k))2 ≤ R2 (1)
ie. within a certain radius of unit j. The relation between unit number and 2 − D index depends on the size of the array:
x(j) = ⌊j/M⌋, y(j) = j mod M
The constant λ is a learning coefficient, which controls the rate at which the units change, while γ has a similar role for the neighbours: it can be used to encourage neighbours to be similar to the given unit, in a sort of mutual excitation, or it can discourage them, in a form of mutual inhibition.
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