Low-rank and sparse structures have been frequently exploited in matrix recovery and robust PCA problems. In this paper, we develop an alternating directional method and its variant equipped with the non-monotone search procedure for solving a non-convex optimization model of low-rank and sparse matrix recovery problems, where the concerned matrix with incomplete data is separable into a low-rank part and a sparse part. The main idea is to use the alternating minimization method for the low-rank matrix part, and to use the non-monotone line search technique for the sparse matrix part to iteratively update, respectively. To some extent, the non-monotone strategy relaxes the single-step descent into a multi-step descent and then greatly improves the performance of the alternating directional method. Theoretically, we prove the global convergence of the two proposed algorithms under some mild conditions. Finally, the comparison of numerical experiments shows that the alternate directional method with non-monotone technique is more effective than the original monotone method and the previous method. The efficiency and effectiveness of the proposed algorithms are demonstrated by solving some instances of random incomplete matrix recovery problems and some problems of the background modeling in video processing.

In this paper, we put up with a new algorithm for tensor completion problems that include missing slices or row/column fibers, where embedding a structured tensor by a multi-way delay-embedding transform (MDT) makes the tensor to be completed have a special structure. The main idea is to employ a tensor completion algorithm based on the tensor ring rank, constructing latent tensor ring factors with a structure that approximates the original tensor starting from the tensor structure. It is also proved that the sequence generated by the new algorithm converges to the optimal solution. Finally, the feasibility of the proposed algorithm is verified by experiments. Compared with other completed algorithms based on tensor ring rank, the completed accuracy is improved, up to 30%.

In this paper, a new hybrid singular value thresholding with diagonal-modify algorithm based on the augmented Lagrange multiplier (ALM) method was proposed for low-rank matrix recovery, in which only part singular values were treated by a hybrid threshold operator with diagonal-update, and which allowed the algorithm to make use of simple arithmetic operation and keep the computational cost of each iteration low. The new algorithm decreased the complexity of the singular value decomposition and shortened the computing time. The convergence of the new algorithm was discussed. Finally, numerical experiments shown that the new algorithm greatly improved the solving efficiency of a matrix recovery problem and saved the calculation cost, and its effect was obviously better than that of the other algorithms mentioned in experiments.