Research And Application Of Reconstruction Algorithm Of Low Rank Matrix And Tensor

With the rapid development of modern sensor and storage technology,high-order data analysis has been widely applied to signal processing and engineering control.As a new tool for high-order data analysis and processing,low-rank matrix reconstruction has become a research hotspot in machine learning,computer vision and data mining,and matrix rank minimization has strong global constraints,which can better characterize the sparsity of two-dimensional matrices.However,the actual high-dimensional data to be analyzed often have more complex data structures,such as color images,multi-spectral images and video sequences.Traditional data representations,such as vectors and matrices,cannot describe the multilinear structure of these high-dimensional data very well.Tensors,as high-order generalizations of vectors and matrices,can better express the intrinsic geometric structure of high-order data.In practical applications,high-dimensional data generally have lower intrinsic dimension,that is,they are low rank or approximate low rank.In the process of obtaining high-dimensional data,some data elements may be lost.Low Rank Tensor Completion(LRTC)is to study how to estimate and reconstruct those missing elements using the information of known data elements.In recent years,the nuclear norm minimization method of low rank tensor complementation has become a hot topic for scholars.However,the existing tensor nuclear norm model itself can not accurately characterize the low-rank properties of tensors.In addition,their algorithms usually need a large number of iterations to solve,and each iteration needs one or more singular value decomposition(SVD),resulting in a very high time complexity of the algorithm.Based on the research of existing models and algorithms,this paper analyses the shortcomings of existing tensor completion algorithms,and deeply discusses and studies the improvement of tensor completion models and the optimization of algorithms.The main work and innovations of this paper are as follows:First,based on the capped nuclear norm model,a low-rank tensor completion algorithm based on the capped nuclear norm is proposed.The new algorithm model replaces the traditional tensor nuclear norm model with the capped nuclear norm regularization term,and then constructs a new tensor complementation model.Based on the proposed capped nuclear norm tensor completion model,and as a constraint objective function,the algorithm combines the Majorization Minimization(MM)optimization framework to iteratively solve the original objective function,which can be used for adaptive tensor completion.Second,based on the improved truncated nuclear norm model,the truncated nuclear norm model in matrix complementation model is further extended to three-dimensional tensor complementation model,and a new low-rank tensor complementation model based on truncated nuclear norm regularization is proposed.By adjusting the residual result generated by each iteration in advance,we can better reconstruct the color image and video.Third,in order to solve the problem that existing algorithms need to calculate large-scale matrix SVD in each iteration,which result in high time complexity,a skinny tensor-Singular Value Decomposition(t-SVD)technique is adopted to solve the proposed model efficiently.Fourth,verification on given different image and video data sets shows that the proposed algorithm performs better than the tensor completion algorithms proposed by SiLRTC,HaLRTC,Tmac and TNN in recent years,and the time complexity of the proposed algorithm is also better than the majority of comparison algorithms.