Applications And Algorithms Of Low Rank Tensor

Tensor completion aims at restore the original tensor by partial observed tensor el-ements,but if in the absence of any prior information,the problem is NP Hard.This is because there are an infinite number of ways to fill in unobserved elements based on the available information.But in fact,the tensor being observed is unique and definite.Therefore,in order to better solve such problems,we often use the low-rank or sparse prior information of most tensor data to help solve the filling problem.With the increase of people's requirements for the clarity of data,we pursue a more accurate tensor filling model.This paper explores the tensor filling model and studies the tensor filling model based on tensor singular value decomposition(t-SVD)and tensor ring decomposition(TR)respectively.The main research contents are as follows:(1)For the tensor filling model based on t-SVD,this paper proposes a tensor com-pletion model using both low-rank and sparsity information of tensors by improving the t-SVD model.In this model,the truncated nuclear norm of tensor is used to describe the low rank of tensor data to be completd,which can solve the problem that the nuclear norm of tensor attenuated too much to large singular values.Secondly,considering that the ten-sor data is usually sparse when it has low rank property,the sparsity of the tensor in the discrete cosine transform(DCT)is used and the l₁norm is used to describe the sparsity.The proposed model is solved by the Alternating Multiplier Direction of method(ADMM)method,and the proposed model is applied to the filling of image and video tensor data.The numerical results and visualization results both prove that the proposed model has better completion performance than the similar methods.(2)For the Tensor completion model based on TR,this paper improves the TR-rank based tensor completion model and proposes three non-convex tensor completion models.The first model uses the?norm instead of the tensor nuclear norm,because the norm re-duces the attenuation of large singular values,the model can approximate the rank function better.The second model uses the Laplace norm to replace the tensor nuclear norm,which can compress the larger singular values less,so that important information can be retained better.The third model uses truncated nuclear norm to replace tensor nuclear norm.The starting point of this method is to retain the larger singular value in front after singular value decomposition,and directly minimize the smaller singular value in the back,so as to retain important information to the greatest extent.The three models proposed based on TR rank are all solved by ADMM framework.In the experimental process,in order to make better use of the strong representation of TR,the visual data tensor(VDT)method is used to transform the low-dimensional tensor into the high-order tensor,so as to improve the visual and numerical results of image filling.