Self-organization in the SOM with a decreasing neighborhood function of any width

A proof of self-organization for a general one dimensional SOM (i.e., one dimensional array of neurons, one dimensional input) with a strictly monotonically decreasing neighborhood function of any width W is given. The proof is not dependent on any particular type of probability distribution of the input but rather minimum conditions that the distribution must satisfy are specified. For a total of N neurons the degree (n) of the SOM is defined here as n=N div W+1 when N mod W≠0 or else n=N/W. It is shown that a total of 2ⁿ intervals of nonzero probability on the support of the input, separated by distances which depend on parameters of the SOM are sufficient for self-organization