Algorithm Research And Application Of Matrix And Tensor Recovery

Recently, compressed sensing(CS) has attracted more and more attention forboth its scientific challenges and its wide applications, through which certain sig-nals and images are able to be accurately recovered from far fewer measurementsor samples with high probability. Based on the above, utilizing the matrix torepresent the vector, matrix recovery problem can be got, in which low-rank ma-trix recovery problem gets more attention and a series of efective algorithms hasbeen proposed to solve the problem. With the rapid development of the sensortechnology, multimedia technology, computer network and communication tech-nology, high-dimensional data with more complex structure such as face images,surveillance videos and multispectral images are becoming very ubiquitous acrossmany areas of science and engineering. The low n-rank tensor recovery problemis an interesting extension of the CS. This problem consists of finding a tensor ofminimum n-rank subject to linear equality constraints and has been proposed inmany areas such as data mining, machine learning and computer vision. In thispaper, operator splitting technique and convex relaxation technique are adaptedto transform the low n-rank tensor recovery problem into a convex, unconstrainedoptimization problem, in which the objective function is the sum of a convex s-mooth function with Lipschitz continuous gradient and a convex function on a setof matrices. Furthermore, in order to solve the unconstrained nonsmooth convexoptimization problem, an accelerated proximal gradient algorithm is proposed.Then, some computational techniques are used to improve the algorithm. At theend of this paper, some preliminary numerical results demonstrate the poten-tial value and application of the tensor as well as the efciency of the proposedalgorithm.