Model And Algorithm Of Nonconvex Rank Approximation For Low-rank Tensor Completion

With the rapid development of sensor and information technology,quick access to a large number of high-dimensional data is becoming more and more convenient.However,these original high-dimensional data are often inevitably interfered by some factors during the collection,storage and transmission process,and loss some information.Therefore,it is important to develop efficient algorithms to recover missing information from limited observations.In this paper,we focus on this type of low-rank tensor completion problem,aiming to effectively recover high-dimensional target tensors from relatively few noisy observations.For this problem,a widely used convex relaxation method is to minimize the sum of the nuclear norms of unfolding matrices of a tensor,but this method treats each singular value equally,without taking into account the physical meaning repre-sented by singular values in the actual data.In order to overcome the shortcomings of this method,in this paper,we propose a general nonconvex rank approximation method based on tensor Tucker rank.A series of nonconvex functions are applied to the singular values of unfolding matrices of a tensor.We impose different degrees of constraints on different singular values to achieve a better approximation of the tensor Tucker rank,while retaining the main components of the data.It is worth noting that the proposed model is a general nonconvex model,which is applicable to many nonconvex functions.In order to solve this problem,we propose a proximal linearized minimization algorithm,and a alternating direction method of multipliers is used to solve the subproblem of the proximal linearized minimization algorithm.Extensive numerical experimental results show that our method can recover the tar-get tensor more effectively than the existing methods,even when the sample rate is very low.