The autoassociative neural network in signal analysis: I. The data dimensionality reduction and its geometric interpretation

In the present paper, we show that the network ability in correctly reproducing as output the given input after a passage through the bottleneck layer (which by definition should have fewer nodes than either input or output layers) could be conceived as a topological mapping between abstract spaces. Apart from the less critical choice of the number of nodes in the mapping and demapping layers, the topological mapping will be successful - and the AANN will be able to perform the required data reconstruction - provided that the number of nodes of the bottleneck layer is related to the dimensionality d of the abstract projection space. We show how to obtain a numerical estimate d* for the real dimension d. This numerical estimate will firmly base the choice of the number of nodes f of the bottleneck layer, thus avoiding the usual troubling trial-and-error procedure. The power of the proposed approach is demonstrated firstly on a few geometrical cases and then on the analysis of nuclear transients simulated by the classic Chernick's model.