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

TL;DR

This paper introduces efficient algorithms for solving positive semi-definite total least squares problems, including minimum rank and correlation matrix computation, with proven quadratic convergence and practical numerical validation.

Contribution

It proposes a new approach considering data errors, using optimization on Stiefel manifolds, and extends to general problems with demonstrated efficiency.

Findings

Algorithms show quadratic convergence.

Numerical results confirm efficiency.

Performance profiles demonstrate superiority.

Abstract

We have recently presented a method to solve an overdetermined linear system of equations with multiple right hand side vectors, where the unknown matrix is to be symmetric and positive definite. The coefficient and the right hand side matrices are respectively named data and target matrices. A more complicated problem is encountered when the unknown matrix is to be positive semi-definite. The problem arises in estimating the compliance matrix to model deformable structures and approximating correlation and covariance matrices in financial modeling. Several methods have been proposed for solving such problems assuming that the data matrix is unrealistically error free. Here, considering error in measured data and target matrices, we propose a new approach to solve a positive semi-definite constrained total least squares problem. We first consider solving the problem when the rank of the unknown matrix is known, by defining a new error formulation for the positive semi-definite total least squares problem and use of optimization methods on Stiefel manifolds. We prove quadratic convergence of our proposed approach. We then describe how to generalize our proposed method to solve the general positive semi-definite total least squares problem. We further apply the proposed approach to solve the 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 our comparative study.