Nonconvex Robust Low-Rank Tensor Reconstruction via an Empirical Bayes Method

Low-rank tensor reconstruction has attracted a great deal of research interest in signal processing, image processing and machine learning. To deal with outliers that are now ubiquitous in many modern applications, a representative approach is to decompose and reconstruct the data into a low-rank component and a sparse outlier component. As surrogate functions of the matrix rank and the sparsity that are non-convex and discontinuous, the nuclear norm and the $\ell _1$ norm are often exploited for recovering higher order tensors and removing outliers, respectively. However, the nuclear norm and the $\ell _1$ norm are loose approximations of the matrix rank and the sparsity, respectively. Furthermore, the tensor nuclear norm is not guaranteed to be the tightest convex envelope of a multilinear rank. In this paper, we develop a new method for robust tensor reconstruction. By capitalizing on the empirical Bayes method, whose performance has been shown to act as state-of-the-art to the performance of standard sparse and low-rank matrix recovery, we formulate the objective function of the empirical Bayes method for robust tensor reconstruction, and then adjust the objective function to facilitate algorithm derivation, while keeping desirable attributes including concavity and symmetry of the objective function of the empirical Bayes method. We demonstrate the superior performance of the proposed algorithm in comparison with state-of-the-art alternatives by conducting experiments on both synthetic data and real data.