Singular vector perturbation is an important topic in numerical analysis and statistics. The main goal of this paperis 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 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 .
In this paper , we establish a useful set of formulae for the \sin\Theta 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 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 .
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Singular vector perturbation is an important topic in numerical analysis and statistics. The main goal of this paperis to provide a useful tool to tackle matrix perturbation problems. Explicitly
In this paper
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formula for the sin\Theta angles between the perturbed and the original
set of formulae for the \sin\Theta distance between the original and the perturbed
R
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 thisformula to derive a one-sided version of the sin\Theta theorem, as well as improve the
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
R
error. Following this, we proceed to show that two other popular stability problems (i.e., the stability
errors under Gaussian noise, a new stability analysis
R
the stability of the
an error bound on the
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) can be solved with the help of these new results
. For the latter two, we consider the most general rectangular matrices with full matrix rank