Random Matrix Solution Of Nuclear Norm And Its Application In Image Processing

The main problem of low-rank matrix recovery is that when some elements in the matrix are destroyed,the damaged elements can be identified automatically.This problem has been applied in many fields such as signal processing,data analysis,computer vision and so on.In practical applications,the processing of high-dimensional data is inevitable.The degree of correlation and redundancy between data and the difficulty of analyzing and processing the data will change with the increase of the dimension of the data.Therefore,it is necessary to make better use of the sparsity and low rank of high-dimensional data more accurately,which is very important for the efficient collection,analysis and processing of large scale data.ADM is used to solve the convex optimization problem of low-rank matrix recovery.Each iteration of the sub problem,the optimization of the nuclear norm minimization is involved in the solution process,that is SVD of large-scale matrix.If the decomposition operation is performed directly,the cost of computation will be enormous.In this paper,the randomized matrix technique is applied to the optimization of the nuclear norm.Compared with the traditional direct calculation of the matrix,the randomized matrix algorithm has a lot of advantages.The main idea is to compress the main features of the original matrix into a low-dimensional approximation matrix,and then to calculate the approximation matrix.The dimensionality of the approximation matrix is much smaller than the original matrix,but it retains some important properties of the original matrix.Therefore,the computational efficiency is greatly improved,and the precision is considerable.Firstly,two improved algorithm of nuclear norm minimization is proposed,which is the prototype CUR decomposition and the faster CUR decomposition.CUR decomposition is the decomposition of the matrix into three parts,namely C,U,R,where the matrix C and R is obtained by column selection algorithm,and then the matrix inverse and matrix multiplication to obtain the cross matrix U.This matrix C(9)U(9)R is an approximation of the original matrix.The simulation results show that the computational efficiency of singular value decomposition can be improved by calculating the approximation matrix.Secondly,two randomzied algorithms of SVD are proposed,that is the prototype randomzied SVD algorithm and the faster randomzied SVD algorithm.The main idea is to reduce the dimensionality of the large-scale data by random sampling,using the random projection algorithm to obtain an approximation of the original data matrix,then do the corresponding matrix operation,and finally can be calculated with the original similar results.The simulation results show that the computational efficiency is greatly improved and memory storage space is greatly improved,and also has good accuracy.Finally,the randomzied SVD algorithm based on accelerated GPU is implemented,which can improve the computational efficiency of large-scale data matrix in practical application.Through simulation experiment,we analyse the characteristics of CPU and GPU,and the original random algorithm has been improved,so that it can better adapt to the GPU computing architecture.These improvement greatly improve the computational efficiency of nuclear norm minimization.