ORTHOGONAL EIGENSUBSPACE ESTIMATION USING NEURAL NETWORKS

In this paper, we present a neural network (NN) approach for simultane ously estimating all or some of the orthogonal eigenvectors of a symme tric nonindefinite matrix corresponding to its repeated minimum (in ma gnitude) eigenvalue. This problem has its origin in the constrained mi nimization framework and has extensive applications in signal processi ng. We recast this problem into the NN framework by constructing an ap propriate energy function which the NN minimizes. The NN is of feedbac k type with the neurons having sigmoidal activation function. The prop osed approach is analyzed to characterize the nature of the minimizers of the energy function. The main result is that ''the matrix W is a minimizer of the energy function if and only if the columns of W are the orthogonal eigenvectors with a given norm corresponding to the sma llest eigenvalue of the given matrix.'' Further, all minimizers are gl obal minimizers. Bounds on the integration time-step that is required to numerically solve the system of differential equations (which defin e the dynamics of the NN) have also been derived. Results of computer simulations are presented to support our analysis.