Research On Theories And Applications Of Nonconvex Low-tubal-rank Tensor Model

With the wide applications of compressed sensing and low-rank matrix model in the fields of computer vision and machine learning,low-rank tensor model has received more and more attentions from scholars.At present,the conventional method for solving low-rank tensor model is to replace tensor rank with tensor nuclear norm,and then the low-rank tensor model is transformed into its convex relaxation model.However,in many cases there is a big difference between them,the calculation results of convex relaxation model are difficult to meet the accuracy requirements,so researchers begin to study non-convex relaxation model of the original model.However,in current researches,the non-convex function in non-convex relaxation model makes the correlation theories complicated,causing difficulties in estimating the applicability and usage limitations of non-convex relaxation model in real problems.In this paper,we conduct in-depth researches on two non-convex functions widely used in current researches,i.e.,weighted nuclear norm and Schatten-p norm,as well as on low-tubal-rank tensor model which has better effects in image and video restorations.On the one hand,through analyzing the properties of two non-convex functions and comparing the calculation results with other relevant models,we give the usage limitations of two non-convex relaxation models in real problems.On the other hand,to further expand the applicability of low-tubal-rank tensor model,we propose to relax it by general concave functions.More details are as follows:1.To improve the restoration effects of damaged images and videos by low-tubalrank tensor model,we propose to relax low-tubal-rank tensor completion model and tensor robust principal component analysis model by using weighted tensor nuclear norm.Firstly,we use algorithms WTNN-TC and WTNN-TRPCA to solve the two models,then give the convergence analysis of algorithm WTNN-TC according to the properties of weighted tensor nuclear norm.At last,through simulation experiments,we confirm that algorithms WTNN-TC and WTNN-TRPCA consider the requirements of reconstruction accuracy and calculation time.2.Aiming at the restriction of giving weighting factors before using non-convex relaxation model with weighted tensor nuclear norm,we propose to relax the low-tubalrank tensor completion model by tensor Schatten-p norm.According to the properties of tensor Schatten-p norm,tensor completion model relaxed by tensor Schatten-p norm is transformed into non-convex relaxation model with weighted tensor nuclear norm,the supergradient of tensor Schatten-p norm is its weighting factor,and we use algorithm Lp-RWTN to solve the model.The experimental results show that algorithm Lp-RWTN outperforms other related algorithms in restoring damaged images and videos.3.Because the way of regulating weighting factors by non-convex relaxation model with tensor Schatten-p norm is too single,it is difficult to meet the practical application needs of low-tubal-rank tensor model.The block diagonal matrix functions induced by general concave functions are proposed to relax low-tubal-rank tensor completion model and tensor robust principal component analysis model.We use algorithms RWTN-TC and RWTN-TRPCA to solve the two models.The experimental results of restoring damaged images and videos show that algorithms RWTN-TC and RWTN-TRPCA are superior to other related algorithms in reconstruction accuracy and computation time.