An exact sinΘ formula for matrix perturbation analysis and its applications
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Singular vector perturbation is an important topic in numerical analysis and statistics. The main goal of this paper is to provide a useful tool to tackle matrix perturbation problems. Explicitly, we establish a useful formula for the sinΘ angles between the perturbed and the original singular subspaces. This formula is expressed in terms of the perturbation matrix and therefore characterizes how the singular vector perturbation is induced by the additive noise. We then use this formula to derive a one-sided version of the sinΘ theorem, as well as improve the bound on the ℓ_2,∞ 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.
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