The Algorithm Research For Structure Matrix Completion

The matrix filling problem is one of the hot topics in matrix analysis,optimization,and image processing in recent years.It mainly studies how to accurately complete these missing elements through the known partial elements in the absence of sampling matrix elements.In the end,the incomplete sampling matrix is completed.In practice,the sampling matrix sometimes have special structure,such as symmetric matrix,Toeplitz matrix,and the two structural matrices play an important role in communication engineering and power systems,especially in the field of signal and image processing.Therefore,the filling problem of symmetric matrix and Toeplitz matrix is studied in this paper.In the process of symmetric matrix filling,the symmetric matrix is decomposed by a simple matrix decomposation,and the step length is searched using an inexact linear search method.The non-convex filling algorithm of the symmetric matrix is designed,and the ratio-nality and effectiveness of the algorithm are explained by theoretical analysis and numerical experiments.In the algorithm of Toeplitz matrix filling,three new manifold approximation algorithms are proposed based on the existing mean value algorithm.A low rank matrix is obtained by applying the least-squares approximation to the sampling matrix in the left sin-gular vector space.The unknown part of the low rank matrix is combined with the known part of the sampling matrix to form a new matrix.The new matrix is modified by l1 norm,l?norm and median value.Finally find iterative matrices of Toeplitz structure.Then the new algorithm for preserving structured Toeplitz matrices is find.Since the generated iterative matrices are all Toeplitz matrices,So the fast singular value decomposition method is used.The computational complexity of the algorithm can be reduced by fast singular value decom-position and the efficiency of the algorithm can be improved.Theoretically,the convergence of the algorithm is proved.Experimentally,numerical experiments are also performed by taking different sampling densities to show the effectiveness of the four guaranteed structure algorithms.And these four algorithms are compared.