An exact sin$Θ$ formula for matrix perturbation analysis and its applications

In this paper, we establish an exact expression of the sin$\Theta$ angle for the singular subspace perturbation. This expression clearly characterizes the linear dependence of the sin$\Theta$ angle on the perturbation matrix. It serves as a tool that allows us to study a set of related problems in a more systematical way. In particular, we derive a useful perturbation bound for the Principal Component Analysis, which seems to be missing in the literature. We derive the perturbation bound for the singular value thresholding operator applied to full-rank matrices. We derive an improved $\ell_{2}\rightarrow \ell_{\infty}$ bound on singular vectors, which matches the best-known results for eigenvectors derived under more restricted conditions. We also derive a one-sided version of the classical sin$\Theta$ theorem.