Algorithms And Implementations Of Symmetric Matrix And Symmetric Positive Semi-definite Matrix Reconstruction

Matrix reconstruction has gradually become a hot research topic and attracted more and more attention of researchers in recent years,because it is widely used in many fields.Matrix reconstruction is mainly divided into three parts:matrix recovery,matrix completion and Low-rank Representation.Matrix reconstruction algorithm is very rich.But most of the algorithms are based on singular value decomposition,which takes a lot of time.So it brings a lot of difficulties to the reconstruction of the large matrix.On the other hand,most of the algorithms are for general matrix reconstruction,which ignore the structural properties of the matrix.However,in many practical applications,the sampling matrix usually has a special structure.For example,in the image acquisition the acquired data is often symmetric matrix,and symmetric positive semi-definite matrices is everywhere in the field of mathematical analysis and other fields.Due to the extensive applications of this two kinds of matrixes,it is very meaningful to optimize the algorithm of matrix reconstruction.According to the structure characteristics of symmetric matrix and symmetric positive semi-definite matrix,the algorithm is modified based on the augmented Lagrange multiplication.A large number of numerical experiments show that the modified algorithm not only has a very good stability and convergence but also has a faster speed which is faster 10 times than original algorithm.And the main results are obtained as follows:(1)In the practical applications,the symmetric matrix elements maybe lost the authenticity and symmetry because of interference.In order to achieve the purpose of protecting structure,mainly consider the following two factors:firstly,symmetric matrix's singular decomposition complexity is much smaller than the common matrix's singular decomposition complexity.Secondly,in order to better consistent with the original matrix structure.So the modified method based on the augmented Lagrange multiplier firstly makes the matrix symmetric.A large number of numerical experiments show that the modified algorithm has reduced a large number of CPU running time.In particular,when the ratio of the interfered elements is large,the modified algorithm has better convergence.(2)According to the characteristics of symmetric positive semi-definite matrix,we proposed a modified algorithm for positive semi-definite matrix completion,which is based on the augmented Lagrange multiplier,and verify the convergence of the modified algorithm.The singular decomposition is replaced by eigenvalue decomposition in modified algorithm,which makes the eigenvalues non-negative.A large number of numerical experiments show that the modified algorithm reduces the number of iterations as well as the running time under the same convergence precision.So it certainly improves the efficiency.(3)A modified algorithm for the recovery of the positive semi-definite matrix is proposed,according to the structural characteristics of the positive semi-definite matrix.The modified algorithm is based on the augmented Lagrange multiplier,in which the singular decomposition is replaced by eigenvalue decomposition.Through the symmetry and the selection of positive eigenvalue we can keep the matrix having a positive semi-definite structure.Secondly,the convergence of the modified algorithm is analyzed,and the effectiveness is also verified by a large number of numerical experiments.In particular,when the ratio of the interfered elements is large,the modified algorithm has better convergence.