Bounds for the Distance to the Nearest Correlation Matrix

In a wide range of practical problems correlation matrices are formed in such a way that, while symmetry and a unit diagonal are assured, they may lack semidefiniteness. We derive a variety of new upper bounds for the distance from an arbitrary symmetric matrix to the nearest correlation matrix. The bounds are of two main classes: those based on the eigensystem and those based on a modified Cholesky factorization. Bounds from both classes have a computational cost of O(n3) flops for a matrix of order n but are much less expensive to evaluate than the nearest correlation matrix itself. For unit diagonal A with |aij | ≤ 1 for all i 6= j the eigensystem bounds are shown to overestimate the distance by a factor at most 1 + n √ n. We show that for a collection of matrices from the literature and from practical applications the eigensystem-based bounds are often good order of magnitude estimates of the actual distance; indeed the best upper bound is never more than a factor 5 larger than a related lower bound. The modified Cholesky bounds are less sharp but also less expensive, and they provide an efficient way to test for definiteness of the putative correlation matrix. Both classes of bounds enable a user to identify an invalid correlation matrix relatively cheaply and to decide whether to revisit its construction or to compute a replacement, such as the nearest correlation matrix.