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