Robust Low-Rank Tensor Recovery With Regularized Redescending M-Estimator

This paper addresses the robust low-rank tensor recovery problems. Tensor recovery aims at reconstructing a low-rank tensor from some linear measurements, which finds applications in image processing, pattern recognition, multitask learning, and so on. In real-world applications, data might be contaminated by sparse gross errors. However, the existing approaches may not be very robust to outliers. To resolve this problem, this paper proposes approaches based on the regularized redescending M-estimators, which have been introduced in robust statistics. The robustness of the proposed approaches is achieved by the regularized redescending M-estimators. However, the nonconvexity also leads to a computational difficulty. To handle this problem, we develop algorithms based on proximal and linearized block coordinate descent methods. By explicitly deriving the Lipschitz constant of the gradient of the data-fitting risk, the descent property of the algorithms is present. Moreover, we verify that the objective functions of the proposed approaches satisfy the Kurdyka-Łojasiewicz property, which establishes the global convergence of the algorithms. The numerical experiments on synthetic data as well as real data verify that our approaches are robust in the presence of outliers and still effective in the absence of outliers.