Square-root Householder subspace tracking

Abstract

A class of singular value decomposition (SVD)-type subspace trackers based on the overdetermined row-Householder principle is introduced. These algorithms are maximally fast with a dominant operations count of 3Nr multiplications per time update. They can be regarded as square-root forms of previously introduced conventional fast subspace trackers and offer interesting features such as perfectly orthonormal basis estimates, lowest dynamic range requirements, and highest numerical robustness and stability. Several variants of the method are proposed and studied experimentally. Mathematics Subject Classification (2000)15A03 · 15A18 · 47A15 · 47A75 · 54B05

Updating the principal angle decomposition

“…2. Based on the solution of this optimization problem, we effectively establish the basis for a new family of subspace trackers, namely, the square-root Householder subspace trackers [17]. To see this, consider the approximant of the corresponding covariance matrix:…”

“…Details about this exciting new class of square-root Householder subspace trackers, including a deeper discussion of the underlying optimization problem E(t) ! = min, its fast sequential solution and maximally fast forms, will be presented in [17].…”

“…This solution is very powerful although it does not solve the underlying minimization problem exactly. For details on this subject, see [17]. Equation-array (75a-g) is a psuedocode of this elementary square-root Householder subspace tracker for updating the rectangular low-rank Cholesky factor S(t) in time.…”