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 perturbation analysis for the truncated singular value decomposition for real matrices. In the first part, we provide 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 no less than $r$. 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. In the second part of the paper, 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 proposed error bound holds universally for perturbations with arbitrary magnitude.

中文翻译：

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截断奇异值分解的扰动扩展和误差界限

截断奇异值分解是奇异值分解的简化版本，其中仅保留少数最大的奇异值。本文介绍了实矩阵截断奇异值分解的扰动分析。在第一部分中，我们为 $r$ 阶的奇异值截断提供了扰动扩展。我们扩展奇异子空间分解的扰动结果，推导出截断算子关于秩不小于 $r$ 的矩阵的一阶扰动展开式。观察到当矩阵具有精确的秩 $r$ 时可以大大简化一阶展开，我们进一步表明奇异值截断允许关于秩 $r$ 矩阵的简单二阶微扰展开。在论文的第二部分，我们在扰动秩-$r$ 矩阵的截断奇异值分解的线性近似上引入了第一已知误差界限。我们的界限仅取决于未扰动矩阵的最小奇异值和扰动矩阵的范数。有趣的是，虽然已知奇异子空间对加性噪声极其敏感，但所提出的误差界限普遍适用于任意幅度的扰动。