Title :
Subspace tracking with full rank updates
Author :
Smith, Steven T.
Author_Institution :
Lincoln Lab., MIT, Lexington, MA, USA
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.
Keywords :
Newton method; airborne radar; array signal processing; convergence of numerical methods; covariance matrices; direction-of-arrival estimation; eigenvalues and eigenfunctions; optimisation; phased array radar; radar clutter; radar tracking; search problems; Grassmann manifold; Rayleigh quotient iteration; airborne rotating phased array radar; algebraic structure; constraint surface; cost; eigenvalue decomposition; full rank matrix; full rank updates; geometric structure; linearly convergent subspace iteration method; nonlinear eigenvalue problems; optimization line search; orthonormal columns; overhead; principal invariant subspaces; simulations; subspace tracking algorithms; subspace tracking equation; superlinear convergence rate; time derivative; time-varying clutter interference; time-varying covariance matrix; truncated Newton method; Clutter; Convergence; Costs; Covariance matrix; Eigenvalues and eigenfunctions; Equations; Interference constraints; Matrix decomposition; Optimization methods; Sensor arrays;
Conference_Titel :
Signals, Systems & Computers, 1997. Conference Record of the Thirty-First Asilomar Conference on
Conference_Location :
Pacific Grove, CA, USA
Print_ISBN :
0-8186-8316-3
DOI :
10.1109/ACSSC.1997.680553