Theories And Numerical Methods For Low Rank Approximations Of Several Structured Matrices

The low rank approximation of structured matrices and its related optimization problems are one of the important research topics in the field of numerical algebra and nonlinear optimization,and it has been widely used in scientific computing such as signal processing,image processing,model reduction and computer vision.This paper systematically studied theoretical and numerical methods for several kinds of low rank approximation problems of structured matrix.In Chapter 2,we studied the low rank approximation problem of the Hankel matrix(?) By using the relationship between the rank function of the matrix and its singular value,the rank constraint is transformed into an equality constraint.And the penalty function method and Ky Fan's minimax value theorem are used to deal with the equality constraint,so that the original problem is transformed into a bivariate optimization problem with orthogonal constraint.The alternating least squares algorithm is constructed to solve the transformed problem.Numerical results show that the new method is feasible and effective.In Chapter 3,we studied the regularized low rank approximation problem of the non-negative matrix(?) With the aid of the full rank decomposition of matrix,we transformed the original problem into a non-negative matrix decomposition problem.The alternating least squares algorithm and the projected gradient algorithm are proposed to solve the transformed non-negative matrix decomposition problem.Numerical results show that the method is feasible.In Chapter 4,we studied the graph matching problem over permutation matrix set(?) Based on the permutation matrix set is the intersection of doubly stochastic matrices and orthogonal matrices,we will start from the orthogonal matrix set.The penalty method,inexact accelerated augmented Lagrange function method are constructed to solve the graph matching problem.We showed results of numerical experiments to support that these presented algorithms are all effective.