Minimax perturbation bounds of the low-rank matrix under Ky Fan norm

This paper considers the minimax perturbation bounds of the low-rank matrix under Ky Fan norm. We first explore the upper bounds via the best rank-$ r $ approximation $ \hat{A}_r $ of the observation matrix $ \hat{A} $. Next, the lower bounds are established by constructing special matrix groups to...

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Bibliographic Details
Published inAIMS mathematics Vol. 7; no. 5; pp. 7595 - 7605
Main Authors Qi, Xinyu, Wang, Jinru, Shao, Jiating
Format Journal Article
LanguageEnglish
Published AIMS Press 01.01.2022
Subjects
ky fan norm
low-rank matrix estimation
perturbation theory
singular value decomposition
sinθ distance
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ISSN2473-6988
2473-6988
DOI10.3934/math.2022426

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Summary:This paper considers the minimax perturbation bounds of the low-rank matrix under Ky Fan norm. We first explore the upper bounds via the best rank-$ r $ approximation $ \hat{A}_r $ of the observation matrix $ \hat{A} $. Next, the lower bounds are established by constructing special matrix groups to show the upper bounds are tight on the low-rank matrix estimation error. In addition, we derive the rate-optimal perturbation bounds for the left and right singular subspaces under Ky Fan norm $ \sin\Theta $ distance. Finally, some simulations have been carried out to support our theories.
ISSN:2473-6988
2473-6988
DOI:10.3934/math.2022426