Lyapunov stability analysis of the quantization error for DCS neural networks

In this paper we show that the quantization error for Dynamic Cell Structures (DCS) Neural Networks (NN) as defined by Bruske and Sommer provides a measure of the Lyapunov stability of the weight centers of the neural net. We also show, however, that this error is insufficient in itself to verify that DCS neural networks provide stable topological representation of a given fixed input feature manifold. While it is true that DCS generates a topology preserving feature map, it is unclear when and under what circumstances DCS will have achieved an accurate representation. This is especially important in safety critical systems where it is necessary to understand when the topological representation is complete and accurate. The stability analysis here shows that there exists a Lyapunov function for the weight adaptation of the DCS NN system applied to a fixed feature manifold. The Lyapunov function works in parallel during DCS learning, and is able to provide a measure of the effective placement of neural units during the NN´s approximation. It does not, however, guarantee the formation of an accurate representation of the feature manifold. Simulation studies from a selected CMU-Benchmark involving the use of the constructed Lyapunov function indicate the existence of a Globally Asymptotically Stable (GAS) state for the placement of neural units, but an example is given where the topology of the constructed network fails to mirror that of the input manifold even though the quantization error continues to decrease monotonically.