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 \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}  .
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 sin\Theta 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 sin\Theta theorem, as well as improve the bound on the \ell_{2,\infty Principal Component Analysis and the stability of the singular value thresholding operator \rightarrow \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} ) can be solved with the help of these new results .
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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
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an exact expression of
a useful formula for
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angle for the singular subspace perturbation. This expression clearly characterizes the linear dependence
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
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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
theorem, as well as improve the bound on the \ell_{2,\infty
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, which seems to be missing in the literature. We derive the perturbation bound for the
and the stability of the
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applied to full-rank matrices. We derive an improved \ell_{2
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