Self-organisation in Kohonen's SOM

Self-organisation in Kohonen's self-organising map (SOM) is analysed by considering the neuron weights to be a Markov process. While many works exist which analyse the one-dimensional SOM, the aim of the study is to demonstrate probability one convergence of the neuron weights to an organised configuration in one- and also in higher-dimensional SOMs. A proof of self-organisation is given for the one-dimensional case for a general type of probability distribution satisfying conditions given in terms of the parameters of the network. A modified version of the SOM algorithm is described which has an absorbing organised configuration, even in higher dimensions. Probability one convergence to this configuration is demonstrated. The higher-dimensional SOM is also analysed and it is shown for certain conditions that the first entry time of the neuron weights into a predefined organised state is finite with probability one. Copyright (C) 1996 Elsevier Science Ltd