On Kohonen's ordering theorem for one-dimensional self-organized mappings

Abstract

We show that Kohonen's ordering theorem for one-dimensional arrays of neurons, fed with scalar input signals, is not valid for learning protocols in which synaptic changes are not infinitesimal. In fact, alterations in the ordering of the sequences of synaptic values are beyond the scope of Kohonen's infinitesimal analysis. He gives a new general version of the ordering theorem. He shows that, for the above-mentioned learning protocols, there are more cases than those found by Kohonen, in which the Lyapunov function D may increase. He also shows that, in many cases, the partial sequences of synaptic values may evolve towards the folded orderings in which D increases, thus increasing the probability of finding the final map trapped in a disordered state. To estimate the number of cases in which the final state is disordered, he has performed numerical simulations of Kohonen's processes with different randomly selected initial synaptic weights. He found that, when the network is trained repeatedly with the same learning set and using the same small neighbourhood taken as the hypothesis in the ordering theorem, ordered states are reached in about 30% of the trials. He also shows some numerical examples in which the network dynamics has, as attractor, either a fixed point or a limit-cycle of disordered states. Altogether, these results suggest that the ordering theorem does not provide a definitive explanation for the ordering phase of Kohonen's algorithm, even for the simple case of one-dimensional self-organized mappings.