Efficient Algorithms for Positive Semi-Definite Total Least Squares Problems, Minimum Rank Problem and Correlation Matrix Computation

摘要

We have recently presented a method to solve an overdetermined linear systemof equations with multiple right hand side vectors, where the unknown matrix isto be symmetric and positive definite. The coefficient and the right hand sidematrices are respectively named data and target matrices. A more complicatedproblem is encountered when the unknown matrix is to be positive semi-definite.The problem arises in estimating the compliance matrix to model deformablestructures and approximating correlation and covariance matrices in financialmodeling. Several methods have been proposed for solving such problems assumingthat the data matrix is unrealistically error free. Here, considering error inmeasured data and target matrices, we propose a new approach to solve apositive semi-definite constrained total least squares problem. We firstconsider solving the problem when the rank of the unknown matrix is known, bydefining a new error formulation for the positive semi-definite total leastsquares problem and use of optimization methods on Stiefel manifolds. We provequadratic convergence of our proposed approach. We then describe how togeneralize our proposed method to solve the general positive semi-definitetotal least squares problem. We further apply the proposed approach to solvethe minimum rank problem and the problem of computing correlation matrix.Comparative numerical results show the efficiency of our proposed algorithms.Finally, the Dolan-More performance profiles are shown to summarize ourcomparative study.