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

Perturbation expansions and error bounds for the truncated singular value decomposition