Orthogonal eigensubspace estimation using neural networks

Abstract

We present a neural network (NN) approach for simultaneously estimating all or some of the orthogonal eigenvectors of a symmetric nonindefinite matrix corresponding to its repeated minimum (in magnitude) eigenvalue. This problem has its origin in the constrained minimization framework and has extensive applications in signal processing. We recast this problem into the NN framework by constructing an appropriate energy function which the NN minimizes. The NN is of feedback type with the neurons having sigmoidal activation function. The proposed 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 smallest eigenvalue of the given matrix". Further, all minimizers are global minimizers. Bounds on the integration time-step that is required to numerically solve the system of differential equations (which define the dynamics of the NN) have also been derived. Results of computer simulations are presented to support our analysis.