Low-rank tensor completion plays an important role in many applications such as image processing, computer vision, and machine learning. A widely used convex relaxation of this problem is to minimize the nuclear norm of the square deal matrix generated by reshaping a tensor. However, this approach can be substantially suboptimal. In order to seek a highly accurate solution, in this paper, we propose to use a family of nonconvex functions onto the singular values of the square deal matrix of the tensor to approximate the rank of the tensor. A proximal linearized minimization (PLM) algorithm is proposed to solve the resulting model. Furthermore, based on the Kurdyka-Łojasiewicz property, we show that the sequence generated by the PLM algorithm globally converges to a critical point of the objective function. Extensive numerical experiments including synthetic data, video data, and the extended Yale Face Database B show the effectiveness of the proposed model compared with several existing state-of-the-art models.

In this paper, we study the problem of low-rank tensor recovery from limited sampling with noisy observations for third-order tensors. A tensor nuclear norm method based on a convex relaxation of t...

In this paper, we propose and study a nonconvex data fitting term and a total variation regularization term for image restoration with impulse noise removal. The proposed model is different from existing image restoration models where the data fitting term is based on the $\ell_1$- or $\ell_2$- norm, and the regularization term is based on the total variation, $\ell_1$-norm, or some nonconvex functions. Theoretically, we analyze the properties of minimizers of the proposed objective function with the nonconvex data fitting term and the total variation regularization term. We show that minimizers can preserve piecewise constant regions or match with the data points perfectly. This property is particularly useful for impulse noise removal. The proposed image restoration model can be solved by the proximal linearized minimization algorithm, and the global convergence of the iterative algorithm can also be established according to Kurdyka--Łojasiewicz property. The performance of the proposed model is tested ...

The robust tensor recovery problem consists in reconstructing a tensor from a sample of entries corrupted by noise, which has attracted great interest in a wide range of practical situations such as image processing and computer vision. In this paper, we study robust tensor recovery for third-order tensors with different degradations, which aims to recover a tensor from partial observations corrupted by Gaussian noise and sparse noise simultaneously. In contrast to traditional approaches based on the tensor nuclear norm penalty for the low-rank component and the tensor ℓ ₁ norm penalty for the sparse component, we propose a nonlocal robust low-rank tensor recovery model with nonconvex regularization (NRTRM) to explore the global low-rankness and nonlocal self-similarity of the underlying tensor. The NRTRM method is first to extract similar patched-tubes to form a third-order sub-tensor. Then a class of nonconvex low-rank penalties and nonconvex sparse penalties are employed to explore the low-rank component and the sparse corruptions for such sub-tensor, respectively. Moreover, a proximal alternating linearized minimization algorithm is developed to solve the resulting model in each group and its convergence is established under very mild conditions. Extensive numerical experiments on both multispectral images and video datasets demonstrate the superior performance of NRTRM in comparison with several state-of-the-art methods.