Before revision:  We then use thisformula to derive a one-sided version of the sin\Theta theorem, as well as improve the bound on the 2,\infty norm of the singular vector perturbation error. Following this, we proceed to show that two other popular stability problems (i.e., the stability of the Principal Component Analysis and the stability of the singular value thresholding operator ) can be solved with the help of these new results .
After revision:  We then use thisformula to derive a one-sided version of the sin\Theta theorem, as well as improve the bound on the 2,\infty norm of the singular vector perturbation errors under Gaussian noise, a new stability analysis of the Principal Component Analysis and the stability of the singular value thresholding operator ) can be solved with the help of these new results .