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

Abstract

Abstract. Indefinite approximations of positive semidefinite matrices arise in various data analysis applications involving covariance matrices and correlation matrices. We propose a method for restoring positive semidefiniteness of an indefinite matrix M 0 that constructs a convex linear combination S(α) = αM 1 + (1 − α)M 0 of M 0 and a positive semidefinite target matrix M 1 . In statistics, this construction for improving an estimate M 0 by combining it with new information in M 1 is known as shrinking. We make no statistical assumptions about M 0 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 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 indefinite approximations of correlation matrices shrinking can be at least an order of magnitude faster than computing the nearest correlation matrix.

Explicit Solutions to Correlation Matrix Completion Problems, with an Application to Risk Management and Insurance

“…Anther possible use of the MaxDet solution E is to compute the smallest α ∈ [0, 1] such that αE yields a positive semidefinite completion, or equivalently the smallest α ∈ [0, 1] such that (1−α) Σ +αΣ is positive semidefinite. This is precisely the method of shrinking [15] with initial matrix Σ and target matrix the MaxDet completion. The optimal α is 2 α * = 3.4908 × 10 −2 ; it gives α * E F = 3.0148 × 10 −2 and a completion with determinant 3.3809 × 10 −16 and eigenvalues This comparison emphasizes that the MaxDet completion is very different from the nearest correlation matrix, and that through shrinking it can yield a completion not much further from Σ than the nearest correlation matrix.…”

Bounds for the Distance to the Nearest Correlation Matrix

“…Let A ∈ R n×n be symmetric with unit diagonal and eigenvalues The next result gives a sharper upper bound than (3.5). The proof uses the idea of shrinking from Higham, Strabić, andŠego [17].…”

“…The rate of convergence of this method is at best linear but we have recently shown that the convergence can be accelerated significantly using Anderson acceleration [16]. The alternating projections method remains widely used in applications (see, for example, the references in [16], [17]), though a faster, globally quadratically convergent Newton algorithm was later developed by Qi and Sun [24], to which practical improvements were made by Borsdorf and Higham [6]. The algorithm of Borsdorf and Higham requires an eigendecomposition of a symmetric matrix on each iteration and so it costs at least 10n 3 /3 flops per iteration.…”

“…The information they provide enables a user to identify an invalid correlation matrix relatively cheaply and to decide whether to revisit its construction or proceed to compute a replacement directly from it. A natural replacement is the nearest correlation matrix itself; an alternative is to use shrinking [17].…”

Accuracy of approximate projection to the semidefinite cone

“…Projection of symmetric or Hermitian matrices to the positive semidefinite cone is a standard operation that arises frequently in scientific computing. A common, practical, example is restoring positive definiteness of partially unknown or corrupted correlation matrices [17] arising in e.g., economics [12], integrated circuit design [20] and wireless communications [11]. Further, more generic, examples include quasi-newton optimization methods [7, §4.2.2], incomplete matrix factorizations of sparse matrices [5, §15.11] and, finally, first order methods for solving semidefinite problems (SDPs) [2,19] which, as we proceed to explain, was the motivating example for this work.…”