The Nonconvex Tensor Robust Principal Component Analysis Approximation Model via theWeighted lp-Norm Regularization

Tensor robust principal component analysis (TRPCA), which aims to recover the underlying low-rank multidimensional datasets from observations corrupted by noise and/or outliers, has been widely applied to various fields. The typical convex relaxation of TRPCA in literature is to minimize a weighted combination of the tensor nuclear norm (TNN) and the l(1)-norm. However, owing to the gap between the tensor rank function and its lower convex envelop (i.e., TNN), the tensor rank approximation by using the TNN appears to be insufficient. Also, the l(1)-norm generally is too relaxing as an estimator for the l(0)-norm to obtain desirable results in terms of sparsity. Different from current approaches in literature, in this paper, we develop a new non-convex TRPCA model, which minimizes a weighted combination of non-convex tensor rank approximation function and the weighted l(p)-norm to attain a tighter approximation. The resultant non-convex optimizationmodel can be solved efficiently by the alternating direction method of multipliers (ADMM). We prove that the constructed iterative sequence generated by the proposed algorithm converges to a critical point of the proposed model. Numerical experiments for both image recovery and surveillance video background modeling demonstrate the effectiveness of the proposed method.