The $\ell_\infty$ Perturbation of HOSVD and Low Rank Tensor Denoising

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 `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 `∞-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 `∞-norm bounds are displayed in applications including high dimensional clustering and sub-tensor localizations.