A forward-backward training algorithm for parallel, self-organizing hierachical neural networks (PSHNN's) is described. Using linear algebra, it is shown that the forward-backward training of an n-stage PSHNN until convergence is equivalent to the pseudo-inverse solution for a single, total network designed in the leastsquares sense with the total input vector consisting of the actual input vector and its additional nonlinear transformations. These results are also valid when a single long input vector is partitioned into smaller length vectors. A number of advantages achieved are small modules for easy and fast learning, parallel implementation of small modules during testing, faster convergence rate, better numerical error-reduction, and suitability for learning input nonlinear transformations by other neural networks. The backpropagation (BP) algorithm is proposed for learning input nonlinearities. Better performance in terms of deeper minimum of the error function and faster convergence rate is achieved when a single BP network is replaced by a PSHNN of equal complexity in which each stage is a BP network of smaller complexity than the single BP network.
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