  Robust tensor recovery plays an instrumental role in robustifying tensor
decompositions for multilinear data analysis against outliers, gross
corruptions and missing values and has a diverse array of applications. In this
paper, we study the problem of robust low-rank tensor recovery in a convex
optimization framework, drawing upon recent advances in robust Principal
Component Analysis and tensor completion. We propose tailored optimization
algorithms with global convergence guarantees for solving both the constrained
and the Lagrangian formulations of the problem. These algorithms are based on
the highly efficient alternating direction augmented Lagrangian and accelerated
proximal gradient methods. We also propose a nonconvex model that can often
improve the recovery results from the convex models. We investigate the
empirical recoverability properties of the convex and nonconvex formulations
and compare the computational performance of the algorithms on simulated data.
We demonstrate through a number of real applications the practical
effectiveness of this convex optimization framework for robust low-rank tensor
recovery.
