In recent years, matrix completion has become one of the main concepts in data science. Truncated nuclear norm regularization (TNNR) approximation of the rank function is an example of the favorite approaches in matrix completion that performs better than other approximations like the nuclear norm. In all TNNR based methods, the observed data is considered to be exact, so they do not give an appropriate solution for inexact observed data. But, in real applications, in addition to missing data, observed data may be inaccurate. In this paper, we proposed a novel method based on TNNR that can deal with missing data and inexact observed data by adding some terms in the objective and constraint of the matrix completion problem. Experimental results confirm the quality of our proposed method in comparison with the other state of the art matrix completion and low-rank approximation techniques.

Image restoration is one of the main parts of image processing. Mathematically, this problem can be modeled as a large-scale structured ill-posed linear system. Ill-posedness of this problem results in low-convergence rate of iterative solvers. For speeding up the convergence, preconditioning usually is used. Despite the existing preconditioners for image restoration, which are constructed based on approximations of the blurring matrix, in this paper, we propose a novel preconditioner with a different viewpoint. Here, we show that image restoration problem can be modeled as a tensor contractive linear equation. This modeling enables us to propose a new preconditioner based on an approximation of the blurring tensor operator. Due to the particular structure of the blurring tensor for zero boundaries, we show that the truncated higher order singular value decomposition of the blurring tensor is obtained very fast and so could be used as a preconditioner. Experimental results confirm the efficiency of this new preconditioner in image restoration and its outperformance in comparison with the other well-known preconditioners.