Plane rotation-based EVD updating schemes for efficient subspace tracking

Abstract

We present new algorithms based on plane rotations for tracking the eigenvalue decomposition (EVD) of a time-varying data covariance matrix. These algorithms directly produce eigenvectors in orthonormal form and are well suited for the application of subspace methods to nonstationary data. After recasting EVD tracking as a simplified rank-one EVD update problem, computationally efficient solutions are obtained in two steps. First, a new kind of parametric perturbation approach is used to express the eigenvector update as an unimodular orthogonal transform, which is represented in exponential matrix form in terms of a reduced set of small, unconstrained parameters. Second, two approximate decompositions of this exponential matrix into products of plane (or Givens) rotations are derived, one of which being previously unknown. These decompositions lead to new plane rotation-based EVD-updating schemes (PROTEUS), whose main feature is the use of plane rotations for updating the eigenvectors, thereby preserving orthonormality. Finally, the PROTEUS schemes are used to derive new EVD trackers whose convergence and numerical stability are investigated via simulations. One algorithm can track all the signal subspace EVD components in only O(LM) operations, where L and M, respectively, denote the data vector and signal subspace dimensions while achieving a performance comparable to an exact EVD approach and maintaining perfect orthonormality of the eigenvectors. The new algorithms show no signs of error buildup.