Research On Graph Signal Reconstruction Method Based On Matrix Completion

Data reconstruction is the process of recovering the original data from partial known data,which is one of the core tasks of information processing.It plays an important role in the fields of optical communication,image processing,speech processing and so on,and has attracted more and more attention.With the rapid development of the new generation of information technology and the substantial of emerging network applications,the structure of data is becoming more and more complex,such as sensor network data,social network data,Three-Dimensional(3D)model data.Data reconstruction technology provides basic support for the effective analysis and processing of these data,and vigorously promote the level of social intelligent development.The key of reconstruction is to make full and reasonable use of the characteristics of data.At present,matrix completion has become one of the mainstream methods because of the simplicity of signal model design and solution.However,the traditional matrix completion,such as low-rank matrix completion,does not fully consider the complex characteristic(such as the irregular spatial distribution)in the data,so it is difficult to describe and effectively use the correlation information among data elements.In recent years,matrix completion in the field of GSP can effectively characterize and utilize the complex characteristics of data,and be used to reconstruct the irregularly distributed,high-dimensional and dynamically changing data(such as network data,3D point cloud data),providing a new research idea for data reconstruction.Therefore,it is of great practical significance to study the signal model construction and algorithm design of matrix completion in GSP.Based on the matrix completion theory,this paper aiming to overcome the shortcomings of the existing GSP model based matrix completion methods in the aspects of signal model construction and algorithm design,to improve the performance of graph signal(data)reconstruction.A low-rank and global smoothness based graph signal reconstruction algorithm,a graph convolution network proximal-unrolling based piecewise smooth graph signal reconstruction method,and a low-rank and graph-time smoothness based timevarying graph signal reconstruction method are proposed.The above methods are used to effectively reconstruct the missing data and lay the foundation for subsequent tasks based on data analysis.The main work and contributions of this paper are as follows:1.Faced with the problem of slow convergence of graph signal reconstruction algorithm based on nuclear norm regularization,by making full use of the advantages of(parallel)truncated Jacobi method,the GNTJM and the GNPTJM are proposed to solve the GMCR model,to improve the convergence rate of reconstruction.Both GNTJM and GNPTJM make use of the gradient and second derivative information of the GMCR model,thus reducing the iterations.At the same time,the two proposed algorithms can avoid explicit singular value decomposition and reduce the demand on storage space because they use the estimated singular matrix composed of sparse factors.The computational complexity of the two proposed methods is analyzed.The experiments are carried out on the global sea level pressure and Jester data sets,and the results verify that the proposed algorithms have faster convergence under the low cost of reconstruction accuracy compared with the algorithms based on the first derivative information.2.Aiming at the problem that the current reconstruction methods do not consider the local smoothness of graph signals,which leads to low accuracy.A graph signal model based on global-local characterization is constructed by making full use of the local characteristics of graph signals,in which low-rank and piecewise smoothness represent the global and local correlation of graph signals respectively.Based on this model,by taking advantage of the powerful feature information learning ability of the graph convolutional network,Pro-Un LRPS is proposed to improve the accuracy of reconstruction.Pro-Un LRPS uses the denoising convolutional network which encodes prior information of data to replace the proximal operator in the ADMM iterative framework,avoiding the singular value decomposition and reducing the computational complexity.Meanwhile,in the iterative process,the low-rank,the piecewise smoothness of the graph signal,and the feature information learned from the graph convolutional network are utilized,to achieve more accurate and faster reconstruction.Finally,the convergence and computational complexity of Pro-Un LRPS are analyzed.Experimental results on various data sets show that the proposed method has better reconstruction performance than the existing matrix completion methods.3.In view of the problem that the existing reconstruction methods does not make full use of the correlation in time-varying data(represented by time-varying graph signal),which leads to low accuracy.By analyzing and utilizing the global and local space-time characteristics of time-varying graph signals,a graph signal model combining the constraints of global and local space-time correlation is designed,to achieve a more comprehensive characterization and description of time-varying graph signal.Aiming at this model,the LRGTS is proposed.LRGTS utilizes the ability of H-P filtering technology to effectively suppress the variance of time series fluctuations to capture the second-order time difference smoothness information,and effectively fuses the global and local space-time correlations of graph signals to improve the reconstruction performance.Finally,the computational complexity of LRGTS is analyzed.Simulation results on several real data sets,such as global sea-level pressure and surface relative humidity,show that the proposed LRGTS has higher accuracy than the existing matrix completion methods.