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Unsupervised Learning
To decide which neighboring neurons need to be modified, the network uses
a neighborhood function ∧ (rr) ; normally, the Gaussian Mexican hat function is
chosen as a neighborhood function. The neighborhood function is mathematically
represented as follows:
∧ (rr) = ee − dd2
2σσ 2
Here, σσ is a time-dependent radius of influence of a neuron and d is its distance from
the winning neuron. Graphically the function looks like a hat (hence its name), as
you can see in the following figure:
Figure 6: The "Gaussian Maxican hat" function, visualized in graph form
Another important property of the neighborhood function is that its radius reduces
with time. As a result, in the beginning, many neighboring neurons' weights are
modified, but as the network learns, eventually a few neurons' weights (at times,
only one or none) are modified in the learning process. The change in weight is given
by the following equation:
dddd = ηη ∧ (XX − WW)
The process is repeated for all the inputs for a given number of iterations. As the
iterations progress, we reduce the learning rate and the radius by a factor dependent
on the iteration number.
SOMs are computationally expensive and thus are not really useful for very
large datasets. Still, they are easy to understand, and they can very nicely find
the similarity between input data. Thus, they have been employed for image
segmentation and to determine word similarity maps in NLP [3].
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