Restoring Definiteness via Shrinking, with an Application to Correlation Matrices with a Fixed Block

Indefinite approximations of positive semidefinite matrices arise in many data analysis applications involving covariance matrices and correlation matrices. We propose a method for restoring positive semidefiniteness of an indefinite matrix M0 that constructs a convex linear combination S(α) = αM1 + (1− α)M0 of M0 and a positive semidefinite target matrix M1. In statistics, this construction for improving an estimate M0 by combining it with new information in M1 is known as shrinking. We make no statistical assumptions about M0 and define the optimal shrinking parameter as α∗ = min{α ∈ [0, 1] : S(α) is positive semidefinite}. We describe three algorithms for computing α∗. One algorithm is based on the bisection method, with the use of Cholesky factorization to test definiteness, a second employs Newton’s method, and a third finds the smallest eigenvalue of a symmetric definite generalized eigenvalue problem. We show that weights that reflect confidence in the individual entries of M0 can be used to construct a natural choice of the target matrix M1. We treat in detail a problem variant in which a positive semidefinite leading principal submatrix of M0 remains fixed, showing how the fixed block can be exploited to reduce the cost of the bisection and generalized eigenvalue methods. Numerical experiments show that when applied to indefinite approximations of correlation matrices shrinking can be at least an order of magnitude faster than computing the nearest correlation matrix.