论文标题
矩阵扰动分析及其应用的精确$ \sinθ$公式
An exact $\sinΘ$ formula for matrix perturbation analysis and its applications
论文作者
论文摘要
在本文中，我们为原始子空间和扰动的单数子空间之间的$ \sinθ$距离建立了一套有用的公式。这些公式明确表明，原始矩阵的扰动如何传播到奇异的向量和奇异子空间中，从而提供了一种直接的分析方式。此后，我们获得了有关SVD扰动相关问题的新结果的集合，包括在$ \ ell_ {2，\ infty} $上绑定了奇异矢量扰动错误的$ \ ell_ {2，\ infty} $ norm，在高斯噪声下遇到的错误，对主要组件分析的新稳定性分析以及对单差值的误差限制了奇异的值。对于后两个，我们认为最通用的矩形矩阵具有完整的矩阵等级。
In this paper, we establish a useful set of formulae for the $\sinΘ$ 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.
