Research On Low-Tubal-Rank Tensor Recovery Based On Prior And Non-Convex Methods

With the rapid development of emerging information technology such as big da-ta and artificial intelligence,various industries continue to produce high-dimensional data represented by tensors,such as images,video,network traffic data,and so on.Low-rank tensor recovery has become a popular method to process and analyze these tensor data with high dimension,large scale,complex structure and rich intrinsic information.In recent years,relying on the newly proposed algebraic framework of tensor singular value decomposition,low-tubal-rank tensor recovery is more efficient in mining and rcconstructing potential low rank properties of rcal tensor data.It has quickly become the focus of scholars’ attention in the field of low-rank tensor recovery.Focusing on the poor actual optimization effect of the tensor nuclear norm minimization method,which is most well known in the study of low-tubal-rank ten-sor recovery,in this paper,the low-tubal-rank tensor recovery and its specific to the low-tubal-rank tensor completion are deeply and systematically studied from two aspects of further coupling other prior information and adopting non-convex tubal rank relaxation.The main research results are as follows:Aiming at the specific problem of low-tubal-rank tensor completion,considering that the tensor nuclear norm minimization model only models the whole low-tubal-rankness of the underlying tensor and ignores other available prior information in the real scene,and inspired by one kind of structured prior information which is common in the missing part of the survey data,we proposed a low-tubal-rank ten-sor completion model with coupled structured missing prior information.In terms of theory,the established model has less recovery error than the tensor nuclear nor-m minimization model,and,it can be recovered with high probability as long as the underlying tensor satisfies the low-tubal-rankness in the whole part and the s-parsity in the missing part.We design an efficient solution algorithm by using the alternating direction multiplier algorithm framework and verified the validity of the proposed model and the correctness of the theory on the simulation data and inter-est point recommendation data,then applied it to the removal of salt and pepper noise in color images.Compared with the correlation tensor modeling method,our method greatly improves the image quality of denoising results without increasing the running time.It achieves a very competitive denoising effect.Aiming at the general problem of low-tubal-rank tensor recovery,considering that the tensor nuclear norm minimization method ignores the difference between the singular values of the underlying tensor and there is still a large gap in the approximation degree of the tensor tubal rank,and inspired by the advantages of the non-convex relaxation method in balancing the low rank property and effective calculation,we proposed a generalized non-convex tensor tubal rank minimization model.We defined a generalized nonconvex relaxation function whose approxima-tion to the tensor tubal rank is more compact than the tensor nuclear norm,and established the exact and robust recovery theory guarantee of the proposed gen-eralized non-convex tensor tubal rank minimization model by developing the null space properties in the tensor case.At the same time,we proved its the null s-pace condition is strictly weaker than that of the tensor nuclear norm minimization model.It is proved creatively that the non-convex relaxation method has the the-oretical advantage than the convex relaxation method.In the aspect of algorithm,We designed an iteratively reweighted tensor nuclear norm algorithm by using the framework of majorization minimization algorithm and carried out a series of in-depth convergence analysis.Especially,we introduced the famous KL property to prove the global convergence and convergence rate of the algorithm.Finally,exper-iments on simulated data verify the recovery advantages and convergence properties of the proposed non-convex algorithm.We also applied the proposed algorithm to the completion problem of color images,face images,video and network traffic data,whose resulting effects are much better than that of the related algorithm.