The ℓ_∞ Perturbation of HOSVD and Low Rank Tensor Denoising
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The higher order singular value decomposition (HOSVD) of tensors is a generalization of matrix SVD. The perturbation analysis of HOSVD under random noise is more delicate than its matrix counterpart. Recent progress has been made in Richard and Montanari (2014), Zhang and Xia (2017) and Liu et al. (2017) demonstrating that minimax optimal singular spaces estimation and low rank tensor recovery in ℓ_2-norm can be obtained through polynomial time algorithms. In this paper, we analyze the HOSVD perturbation under Gaussian noise based on a second order method, which leads to an estimator of singular vectors with sharp bound in ℓ_∞-norm. A low rank tensor denoising estimator is then proposed which achieves a fast convergence rate characterizing the entry-wise deviations. The advantages of these ℓ_∞-norm bounds are displayed in applications including high dimensional clustering and sub-tensor localizations.
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