Yinlin Ye

Hongbing Zhang, Xinyi Liu, Hongtao Fan et al.

Low-rank tensor completion (LRTC) is an important problem in computer vision and machine learning. The minimax-concave penalty (MCP) function as a non-convex relaxation has achieved good results in the LRTC problem. To makes all the constant parameters of the MCP function as variables so that futherly improving the adaptability to the change of singular values in the LRTC problem, we propose the bivariate equivalent minimax-concave penalty (BEMCP) theorem. Applying the BEMCP theorem to tensor singular values leads to the bivariate equivalent weighted tensor $Γ$-norm (BEWTGN) theorem, and we analyze and discuss its corresponding properties. Besides, to facilitate the solution of the LRTC problem, we give the proximal operators of the BEMCP theorem and BEWTGN. Meanwhile, we propose a BEMCP model for the LRTC problem, which is optimally solved based on alternating direction multiplier (ADMM). Finally, the proposed method is applied to the data restorations of multispectral image (MSI), magnetic resonance imaging (MRI) and color video (CV) in real-world, and the experimental results demonstrate that it outperforms the state-of-arts methods.

Hongbing Zhang, Xinyi Liu, Hongtao Fan et al.

Tensor sparse modeling as a promising approach, in the whole of science and engineering has been a huge success. As is known to all, various data in practical application are often generated by multiple factors, so the use of tensors to represent the data containing the internal structure of multiple factors came into being. However, different from the matrix case, constructing reasonable sparse measure of tensor is a relatively difficult and very important task. Therefore, in this paper, we propose a new tensor sparsity measure called Tensor Full Feature Measure (FFM). It can simultaneously describe the feature information of each dimension of the tensor and the related features between two dimensions, and connect the Tucker rank with the tensor tube rank. This measurement method can describe the sparse features of the tensor more comprehensively. On this basis, we establish its non-convex relaxation, and apply FFM to low rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA). LRTC and TRPCA models based on FFM are proposed, and two efficient Alternating Direction Multiplier Method (ADMM) algorithms are developed to solve the proposed model. A variety of real numerical experiments substantiate the superiority of the proposed methods beyond state-of-the-arts.

Hongbing Zhang, Xinyi Liu, Hongtao Fan et al.

The low rank tensor completion (LRTC) problem has attracted great attention in computer vision and signal processing. How to acquire high quality image recovery effect is still an urgent task to be solved at present. This paper proposes a new tensor $L_{2,1}$ norm minimization model (TLNM) that integrates sum nuclear norm (SNN) method, differing from the classical tensor nuclear norm (TNN)-based tensor completion method, with $L_{2,1}$ norm and Qatar Riyal decomposition for solving the LRTC problem. To improve the utilization rate of the local prior information of the image, a total variation (TV) regularization term is introduced, resulting in a new class of tensor $L_{2,1}$ norm minimization with total variation model (TLNMTV). Both proposed models are convex and therefore have global optimal solutions. Moreover, we adopt the Alternating Direction Multiplier Method (ADMM) to obtain the closed-form solution of each variable, thus ensuring the feasibility of the algorithm. Numerical experiments show that the two proposed algorithms are convergent and outperform compared methods. In particular, our method significantly outperforms the contrastive methods when the sampling rate of hyperspectral images is 2.5\%.