Perturbation expansions and error bounds for the truncated singular value decomposition

Truncated singular value decomposition is a reduced version of the in which only few largest values are retained. This paper presents novel perturbation analysis for truncated real matrices. First, we describe expansions truncation order r . We extend results subspace to derive first-order expansion operator about matrix with rank greater than or equal Observing that can be greatly simplified when has exact , further show admits simple second-order rank- matrix. Second, introduce first-known error bound on linear approximation perturbed Our depends least unperturbed and norm Intriguingly, while subspaces known extremely sensitive additive noises, newly established holds universally perturbations arbitrary magnitude. Finally, demonstrate an application our mean squared associated TSVD-based denoising solution.