论文标题
截短的奇异值分解的扰动扩展和误差界限
Perturbation expansions and error bounds for the truncated singular value decomposition
论文作者
论文摘要
截断的奇异值分解是奇异值分解的简化版本，其中仅保留了几个最大的奇异值。本文为真实矩阵的截断值分解提供了新的扰动分析。首先，我们描述了订单$ r $的单数值截断的扰动扩展。我们扩展了奇异子空间分解的扰动结果，以从矩阵大于或等于$ r $的矩阵中得出截短的操作员的一阶扰动扩展。观察到一阶扩展可以在矩阵具有精确的等级$ r $时大大简化，我们进一步表明，单数值截断符合了关于排名$ r $ $矩阵的简单二阶扰动扩展。其次，我们引入了第一个已知的误差，限制在截短的奇异值分解的线性近似值上。我们的界限仅取决于未渗透矩阵的最小奇异值和扰动矩阵的规范。有趣的是，虽然已知奇异子空间对加性噪声极为敏感，但新确定的误差绑定在任意幅度的扰动中普遍存在。最后，我们证明了结果的应用于与基于TSVD的基质denoising解决方案相关的平方误差的分析。
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.
