PARALLEL STRUCTURED NETWORKS FOR SOLVING A WIDE VARIETY OF MATRIX ALGEBRA PROBLEMS

In this paper, structured networks, which are multilayer feed-forward neural networks with linear neurons and with additional constraints on the weights of neurons, are developed for solving a wide variety of matrix algebra problems, including LU decomposition, matrix inversion, QR factorization, singular value decomposition, symmetric eigenproblem, and Schur decomposition. The basic idea of the structured network approaches is: first, represent a given matrix algebra problem by a structured network so that if the network matches a set of desired patterns, the weights of the linear neurons give the solution to the problem; then, train the structured network to match the desired patterns, and obtain the solution to the problem from the converged weights of neurons. Modified error back-propagation algorithms are developed for training the structured networks. Time complexities of these structured network approaches are analyzed. These approaches are tested for random matrices through Monte Carlo simulations. Finally, a general-purpose structured network is developed which can be programmed to solve all the matrix algebra problems considered in this paper. © 1992.