In this paper, we establish a useful set of formulae for the $\sin\Theta$ distance between the original and the perturbed singular subspaces. These formulae explicitly show that how the perturbation of the original matrix propagates into singular vectors and singular subspaces, thus providing a direct way of analyzing them. Following this, we derive a collection of new results on SVD perturbation related problems, including a tighter bound on the $\ell_{2,\infty}$ norm of the singular vector perturbation errors under Gaussian noise, a new stability analysis of the Principal Component Analysis and an error bound on the singular value thresholding operator. For the latter two, we consider the most general rectangular matrices with full matrix rank.

翻译：本文建立了一套关于原始奇异子空间与摄动奇异子空间之间 $\sin\Theta$ 距离的有效公式。这些公式明确展示了原始矩阵的摄动如何传播到奇异向量和奇异子空间中，从而为分析它们提供了一种直接方法。基于此，我们推导出了一系列有关奇异值分解 (SVD) 摄动问题的新结果，包括：高斯噪声下奇异向量摄动误差的 $\ell_{2,\infty}$ 范数的更紧界、主成分分析的新的稳定性分析，以及奇异值阈值算子的一个误差界。对于后两者，我们考虑了具有满矩阵秩的最一般矩形矩阵。