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.

翻译：缩略单值单值分解是单值分解的缩略版, 仅保留几个最大的单值。 本文为真实矩阵的缩略单值分解提供了一个新颖的扰动分析。 首先, 我们描述单值单值调解析的振动扩展。 第二, 我们扩展单子空间分解的扰动结果, 以得出被调转的分解操作员对一个等级大于或等于美元的信息总库的首级分解扩张。 观察当矩阵准确排位为美元时, 第一阶扩展可以大大简化。 我们进一步显示, 单值调解析会承认一个简单的第二顺序分解扩展, 用于单值调单值调, 以得出单调单值分解析结果, 以得出一个等级高于或等于美元的信息总值的矩阵。 我们的捆绑定仅取决于一个最小的单值, 单位矩阵的扩展可以大大简化。 我们的分流矩阵和透析的常规的分解析度, 与我们所认识的反复度分析结果一起, 最终显示一个已知的精确度。