RIEMANNIAN STRUCTURE FOR CORRELATION MATRICES

In this paper we present a new approach to viewing the set of non-degenerate correlation matrices Corr(n) as a manifold and provide an optimization procedure using its new found Riemannian structure. First we give a proof that Corr(n) is a quotient submanifold of the symmetric positive-definite matrices SPD(n) obtained via a Lie group action of positive diagonal matrices Diag(+)(n). With this structure Corr(n) naturally inherits a Riemannian metric from SPD(n) and therefore enables us to develop a Riemannian-based Newton's method on Corr(n). We subsequently compare this Newton method to other optimization methods on Corr(n).