Perturbation analysis for two-sided (or complete) orthogonal decompositions

Two-sided (or complete) orthogonal decompositions are good alternatives to the singular value decomposition (SVD) because they can yield good approximations to the fundamental subspaces associated with a numerically rank-deficient matrix. In this paper we derive perturbation bounds for the subspaces associated with a general two-sided orthogonal decomposition of a numerically rank-deficient matrix. The results imply the subspaces are only slightly more sensitive to perturbations than singular subspaces, provided the norm of the off-diagonal blocks of the middle matrices are sufficiently small with respect to the size of the perturbation. We consider regularizing the solution to the ill-conditioned least squares problem by truncating the decomposition and present perturbation theory for the minimum norm solution of the resulting least squares problem. The main results can be specialized to well known SVD-based perturbation bounds for singular subspaces as well as the truncated least squares solution.