Nonconvex Optimization for Robust Tensor Completion from Grossly Sparse Observations

In this paper, we consider the robust tensor completion problem for recovering a low-rank tensor from limited samples and sparsely corrupted observations, especially by impulse noise. A convex relaxation of this problem is to minimize a weighted combination of tubal nuclear norm and the \(\ell _1\)-norm data fidelity term. However, the \(\ell _1\)-norm may yield biased estimators and fail to achieve the best estimation performance. To overcome this disadvantage, we propose and develop a nonconvex model, which minimizes a weighted combination of tubal nuclear norm, the \(\ell _1\)-norm data fidelity term, and a concave smooth correction term. Further, we present a Gauss–Seidel difference of convex functions algorithm (GS-DCA) to solve the resulting optimization model by using a linearization technique. We prove that the iteration sequence generated by GS-DCA converges to the critical point of the proposed model. Furthermore, we propose an extrapolation technique of GS-DCA to improve the performance of the GS-DCA. Numerical experiments for color images, hyperspectral images, magnetic resonance imaging images and videos demonstrate that the effectiveness of the proposed method.