Stable Subspace Tracking Algorithm Based on a Signed URV Decomposition

Subspace estimation and tracking are of fundamental importance in many signal processing algorithms. The class of “Schur subspace estimators” provides a complete parametrization of all “principal subspace estimates,” defined as the column spans of corresponding low-rank matrix approximants that lie within a specified 2-norm distance of a given matrix. The parametrization is found in terms of a two-sided hyperbolic decomposition (Hyperbolic URV, or HURV), which can be computed using hyperbolic rotations. Unfortunately, such rotations are commonly associated with numerical instabilities.In this paper, we present a numerically stable, non-iterative algorithm to compute the HURV, called the Signed URV (SURV) algorithm. We show that this algorithm implicitly imposes certain constraints on the HURV such that important norm bounds that guarantee stability are satisfied. The constraints also restrict the parametrization of the subspace estimate such that it becomes close to the principal subspace provided by the SVD (which is a special case within this class). The complexity of the algorithm is of the same order as that of a QR update. Updating and downdating are of the same complexity and are both numerically stable. SURV is proven to provide rank estimates consistent with the SVD with the same rank threshold. It can replace an SVD where only subspace estimation is needed. Typical applications would e.g. be the detection of the number of signals in array signal processing, and subspace estimation for source separation and interference mitigation, such as the first step in MUSIC and ESPRIT-type algorithms. Simulation results demonstrate the numerical stability and confirm that this algorithm provides exact rank estimates and good principal subspace estimates as compared to the SVD.