  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. Recently, polynomial time
algorithms have been proposed where statistically optimal estimates of the
singular subspaces and the low rank tensors are attainable in the Euclidean
norm. In this article, we analyze the sup-norm perturbation bounds of HOSVD and
introduce estimators of the singular subspaces with sharp deviation bounds in
the sup-norm. We also investigate a low rank tensor denoising estimator and
demonstrate its fast convergence rate with respect to the entry-wise errors.
The sup-norm perturbation bounds reveal unconventional phase transitions for
statistical learning applications such as the exact clustering in high
dimensional Gaussian mixture model and the exact support recovery in sub-tensor
localizations. In addition, the bounds established for HOSVD also elaborate the
one-sided sup-norm perturbation bounds for the singular subspaces of unbalanced
(or fat) matrices.
