Subspace tracking with full rank updates

Abstract

A new method of trading the principal invariant subspaces of a time-varying covariance matrix is proposed. The method addresses the case encountered frequently in applications where the covariance is updated by a full rank matrix at each time step; it is not assumed that the covariance changes by rank-one updates. In contrast to subspace tracking algorithms that exploit the algebraic structure that rank-one updates provide, the proposed algorithm uses the geometric structure of full rank updates. The main idea is to determine the time derivative of the subspace using the subspace tracking equation (introduced here), perform an optimization line search in this direction, then finish with a (truncated) Newton, method comparable to Rayleigh quotient iteration. The algorithm is performed on the constraint surface (Grassmann manifold) of matrices with orthonormal columns. The convergence rate and cost of this method are compared with other subspace tracking and standard eigenvalue decomposition (EVD) algorithms using the example problem of tracking the time-varying clutter interference of a rotating sensor array. Simulations indicate that the proposed method is cheaper than a full EVD, but its superlinear convergence rate and higher overhead make it about 25% more costly than a linearly convergent subspace iteration method. Nevertheless, the proposed method's generality makes it appropriate for nonlinear eigenvalue problems and other time-varying problems where linear EVD algorithms cannot be applied.