Optimal low-rank approximation to a correlation matrix

Abstract

Low-rank approximation of a correlation matrix is a constrained minimization problem that can be translated into a minimization–maximization problem by the method of Lagrange multiplier. In this paper, we solve the inner maximization problems with a single spectral decomposition, and the outer minimization problems with gradient-based descending methods. An in-depth analysis is done to characterize the solutions of the inner maximization problem for the case when they are non-unique. The well-posedness of the Lagrange multiplier problem and the convergence of the descending methods are rigorously justified. Numerical results are presented.