Self-organization in the one-dimensional SOM with a decreasing neighborhood

A proof of self-organization for a general, standard, one-dimensional SOM and a monotonically decreasing neighborhood function such that W < or = N, is given, where N is the total number of neurons and W the width of the neighborhood function. It is the generalization of already existing proofs for two specific cases with W > or = N/2. Lebesgue continuity of the distribution of the input is not a requirement. The order p of the SOM, where p = [N/W] + 1, if N mod W not = 0 or else p = NIW, is fundamental to the proof. A total of 2p basic intervals of non-zero probability on the support of the input are sufficient for self-organization. These basic intervals are separated by minimum distances which depend on parameters of the SOM (e.g. gain, neighborhood function). The result gives a relationship which in a practical situation can be used to determine, for a given number of neurons and neighborhood function, the minimum number of discrete data points which can guarantee self-organization.