Quotient geometry of bounded or fixed rank correlation matrices

Abstract

This paper studies the quotient geometry of bounded or fixed-rank correlation matrices. The set of bounded-rank correlation matrices is in bijection with a quotient set of a spherical product manifold by an orthogonal group. We show that it admits an orbit space structure and its stratification is determined by the rank of the matrices. Also, the principal stratum has a compatible Riemannian quotient manifold structure. We develop efficient Riemannian optimization algorithms for computing the distance and the weighted Frechet mean in the orbit space. We prove that any minimizing geodesic in the orbit space has constant rank on the interior of the segment. Moreover, we examine geometric properties of the quotient manifold, including horizontal and vertical spaces, Riemannian metric, injectivity radius, exponential and logarithmic map, gradient and Hessian.