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

Indefinite estimates of positive semidefinite matrices arise in many data analysis applications involving covariance matrices and correlation matrices. We develop a method for restoring positive semidefiniteness of an indefinite estimate based on the process of shrinking, which finds a convex linear combination $S(\alpha) = \alpha M_1 + (1-\alpha)M_0$ of the original matrix $M_0$ and a target positive semidefinite matrix $M_1$. We describe three \alg s for computing the optimal shrinking parameter $\alpha_* = \min \{\alpha \in [0,1] : \mbox{$S(\alpha)$ is positive semidefinite}\}$. 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 $M_0$ can be used to construct a natural choice of the target matrix $M_1$. We treat in detail a problem variant in which a positive semidefinite leading principal submatrix of $M_0$ 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 estimates of correlation matrices shrinking can be at least an order of magnitude faster than computing the nearest correlation matrix