Analysis of the convergence properties of topology preserving neural networks

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Abstract
The authors provide a rigorous treatment of the convergence of the topology preserving neural networks proposed by Kohonen for the one-dimensional case. The approach extends the original work by Kohonen on the convergence properties of such networks in several respects. First, the authors investigate the convergence of the neuron weights directly as compared to Kohonen's treatment of the dynamic behavior of the expectation values of the weights. Second, the problem is formulated for a more general case of selecting the neighborhood amplitude of interaction rather than the uniform amplitude. Third, the proof of convergence is based on the well-known Gladyshev theorem which uses Lyapunov's function method. The authors provide a step-by-step constructive proof which establishes the asymptotic convergence to a unique solution. This proof also provides the relation between the boundary neurons' weight vectors and the number of neurons in the network. The approach is then extended to the two-dimensional case and the result is stated in a theorem.