Low-Rank Tensor (Matrix) Completion Algorithm With Applications

With the rapid development of information technology,the huge value of massive data has attracted the attention of many researchers.Data completion has become a hot research topic in the fields of computer vision,artificial intelligence and optimization.In data completion one is given a partially observed data,using its prior information to recover the missing data.The mainstream data completion methods include matrix com-pletion and tensor completion method.Most of the existing matrix completion methods need to calculate the singular value decomposition of the matrix,which is computation-ally expensive.In addition,in practical applications,the data we need to recover is often multi-dimensional,and traditional matrix completion methods cannot make good use of the multi-dimensional structure of these data.As an extension of the matrix completion method,the tensor completion method has been applied in many fields,such as internet traffic data recovery and image recovery.This paper mainly studies the matrix completion and tensor completion method.The main work of this paper is summarized as follows:(1)An adaptive Frank-Wolfe algorithm is proposed for solving the matrix comple-tion problem.The Nesterov's acceleration technique is used to speed up the standard Frank-Wolfe in the proposed algorithm,which also including a rank reduction step.So,the adaptive Frank-Wolfe algorithm improves the convergence rate of the standard Frank-Wolfe algorithm and reduces the average cost of iteration.Numerical experimental results show that the proposed algorithm is more efficient in achieving similar or better predic-tion performance than the original Frank-Wolfe algorithm or some related methods in the literature.(2)We propose an internet traffic data recovery model by combining tensor decom-position based on tensor product(T-product)and total variation(TV)regularization.Low-rank constraints can capture the global structure of data well,but cannot effectively cap-ture the local smoothness of internet traffic data.The proposed model makes full use of the global structure information and local smoothness of internet traffic data to realize the recovery of the internet traffic data.Aiming at the model,an effective proximal alternat-ing minimization(PAM)algorithm is proposed,and the global convergence of the PAM algorithm is established.Numerical experiments are carried out on Abilene and G'EANT datasets,and the results show that the method has good performance.(3)On the basis of the previous work,we propose an image data restoration model by combining tensor decomposition based on generalized T_u-product and TV regularization.The tensor tubal rank based on T-product only considers one dimension of the tensor and ignores the information of the other two dimensions.The tensor multi-tubal rank based on T_u-product balances the information of the three dimensions of the tensor,making full use of the multi-dimensional characteristics of the image data.The model is also solved by the PAM algorithm,and numerical experiments are performed on real image datasets.The results show that the method has good performance.