Abstract:
Previous analyses of the the one dimensional SOM have revealed certain conditions which must be satisfied before probability one self-organisation of the neuron weights can be proved. One general proof relies on the fact that the neighbourhood function is such that every neuron weight is updated at each iteration. The result is two intervals of no n zero probability of the input signal are required to prove self-organisation. By reducing the width of the neighbourhood function every neuron is not necessaraily updated at each iteration. By introducing an extra condition on the support of the probability distribution of the input signal, probability one self-organisation of the neuron weights can be proved for a neighbourhood function whose width is greater than half the number of neurons. This extra condition is specified in terms of the need for a third interval of non zero probability. The proof is extendable to higher dimensions.