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Mathematical characterizations


Rigorous mathematical treatment of the SOM algorithm has turned out to be extremely difficult in general (reviews have been provided by Kangas, 1994; and Kohonen, 1995c). In the case of a discrete data set and a fixed neighborhood kernel, however, there exists a potential function for the SOM, namely [Kohonen, 1991, Ritter and Schulten, 1988]


where the index c depends on the tex2html_wrap_inline2163 and the reference vectors tex2html_wrap_inline2175 (cf. Eq. 5).

The learning rule of the SOM, Equation 6, corresponds to a gradient descent step in minimizing the sample function


obtained by selecting randomly a sample tex2html_wrap_inline2341 at iteration t. The learning rule then corresponds to a step in the stochastic approximation of the minimum of Equation 7, as discussed by Kohonen (1995c).

Note: In Equation 7 the index c is a function of all the reference vectors, which implies that it may change when the gradient descent step is taken. Locally, if the index tex2html_wrap_inline2369 does not change for any tex2html_wrap_inline2163 , the gradient step is valid, however.

Sami Kaski
Mon Mar 31 23:43:35 EET DST 1997