The Nonconvex Tensor Robust Principal Component Analysis Approximation Model via the Weighted $$\ell _p$$ ℓ p -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 $$\ell _1$$ -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 $$\ell _1$$ -norm generally is too relaxing as an estimator for the $$\ell _0$$ -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 $$\ell _p$$ -norm to attain a tighter approximation. The resultant non-convex optimization model 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.