Application Of Tensor Decomposition And Smoothness Regularity To Low-Rank Tensor Completion

In the era of big data,high-dimensional data,such as Internet data and traffic flow data,is generated all the time.Due to data collection difficulties or equipment failures,data incompleteness is often caused,which greatly affects the analysis and application of data,and the missing data needs to be filled.Since high-dimensional data in many prac-tical problems usually have low-rank characteristics,low-rank tensor filling has received widespread attention in recent years,and it has been widely used in the fields of visual data recovery,face recognition,and traffic data recovery.The main work of this paper is as follows:（1）Tensor Train（TT）decomposition can characterize the correlation between dif-ferent modules of higher-order tensors and is parallelizable.Many data in the real world have smoothness in space or time,we can use the Toeplitz（-1,1,0）matrix as the smooth-ness constraint,and obtain a sparse tensor after its transformation,and further use the₁norm as a measure of sparsity,and establish a tensor low-rank filling model based on TT decomposition and ₁norm smoothness regularity.Furthermore,the near-alternating minimal method（PAM）is designed to solve the low-rank filling model,and the global convergence results of the algorithm are given,and numerical experiments show that the proposed model and solution algorithm are effective.（2）The tensor-tensor product（T product）can obtain tensor singular value decompo-sition similar to the matrix singular value decomposition,and derive the corresponding tensor tube rank and tensor multi-rank.In order to avoid singular value decomposition,the tensor T-integral solution is selected to characterize the low-rank features of the tensor.Considering that smoothness features may exist on each module in real data,this paper es-tablishes three tensor low-rank decomposition models with TV-Tikhonov regularity,and also uses Toeplitz（-1,1,0）as the transformation matrix for smoothness constraints.The PAM algorithm is used to solve the model.Similarly,the global convergence results of the algorithm are given,and numerical experiments show that the proposed model and solution algorithm are effective.