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The higher order singular value decomposition (HOSVD) of tensors is ageneralization of matrix SVD. The perturbation analysis of HOSVD under randomnoise is more delicate than its matrix counterpart. Recently, polynomial timealgorithms have been proposed where statistically optimal estimates of thesingular subspaces and the low rank tensors are attainable in the Euclideannorm. In this article, we analyze the sup-norm perturbation bounds of HOSVD andintroduce estimators of the singular subspaces with sharp deviation bounds inthe sup-norm. We also investigate a low rank tensor denoising estimator anddemonstrate its fast convergence rate with respect to the entry-wise errors.The sup-norm perturbation bounds reveal unconventional phase transitions forstatistical learning applications such as the exact clustering in highdimensional Gaussian mixture model and the exact support recovery in sub-tensorlocalizations. In addition, the bounds established for HOSVD also elaborate theone-sided sup-norm perturbation bounds for the singular subspaces of unbalanced(or fat) matrices.

The Sup-norm Perturbation of HOSVD and Low Rank Tensor Denoising