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In the k-means based kernel SOM, proposed by MacDonald and Fyfe (2000), the update of the mean is based on a soft learning algorithm

mi(t + 1) = mi(t) + Λ[φ(x) − mi(t)]

where Λ is the normalized winning frequency of the i-the mean and is defined as

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where ζ is the winning counter and is often defined as a Gaussian function between the indexes of the two neurons. (Reference: https://personalpages.manchester.ac.uk/staff/hujun.yin/pubs/NN-SI-2006-Yin.pdf

I want to know how the normalized winning frequency is calculated. If possible, can someone link a reference that explains the notation?

Anshuman Jayaprakash
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1 Answers1

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This is standard SOM procedure. ζ is the neighborhood function. It's just normalized so that its values are between 0 and 1.

The neighborhood function, sometimes aka kernel, is often a function that depends on distance of unit from the winning unit (i), and that often shrinks over time. The influence that the input sample has in the SOM is determined by that equation. In the paper, it is defined as a bell-shaped function (i.e. a Gaussian). Think of it as the influence region of the input sample x.

I'd recommend you to read Kohonen's original paper about SOM. See equation 4: https://sci2s.ugr.es/keel/pdf/algorithm/articulo/1990-Kohonen-PIEEE.pdf

Some examples of neighborhood functions: https://github.com/ivallesp/somnium/blob/master/somnium/neighborhood.py

ivallesp
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  • Thank you very much. Could you please clarify whether the denominator is the summation of neighbourhood function of a particualar observation ,or if it is the summation of neighbourhood function using winning neuron and other neurons as arguments.In other words, does n refer to iteration or is n refering to the other surrounding neurons in the som grid? – Anshuman Jayaprakash Mar 11 '24 at 12:36
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    n is the index that iterates across all neurons. This denominator is the sum of the result of the neighborhood function of the winning neuron against the rest, i.e. the sum of all distances. – ivallesp Mar 12 '24 at 00:45