Recursive Eigenspace Decomposition, RISE, and Applications

Abstract

Many important signal modeling and spectrum estimation problems have been solved robustly and accurately using the eigenvalues and eigenvectors of a covariance matrix or the singular value decomposition (SVD) of a data matrix. Due to the high computational cost of these techniques, the standard approach is to fix a model order or matrix dimension (often on somewhat heuristic grounds) and solve the problem a single time. Typically, no comparison of solutions for different choices of model order or matrix dimension is attempted. In many situations, performance can be improved by examining solutions for different model orders, however, until recently this has not been considered due to the computational burden and a lack of means for comparing relative quality. A recently developed algorithm, the Recursive/Iterative Self-adjoint Eigenspace Decomposition (RISE), enables an efficient computation of the eigenvalues and eigenvectors of successively larger sized Hermitian matrices. A reverse version of RISE that decomposes successively smaller sized Hermitian matrices has also been formulated. Due to its generality, RISE is applicable to many signal processing problems where the order or structure of the model is reflected in the size of a covariance matrix, and it is desired to examine solutions for different model orders. Three such applications are described in this paper, coming from the diverse fields of model order estimation, sensor array processing, and pattern recognition.