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.
Citation: Xinyu Qi, Jinru Wang, Jiating Shao. Minimax perturbation bounds of the low-rank matrix under Ky Fan norm[J]. AIMS Mathematics, 2022, 7(5): 7595-7605. doi: 10.3934/math.2022426
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.
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Xinyu Qi, Jinru Wang, Jiating Shao. Minimax perturbation bounds of the low-rank matrix under Ky Fan norm[J]. AIMS Mathematics, 2022, 7(5): 7595-7605. doi: 10.3934/math.2022426
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