  This paper studies the quotient geometry of bounded or fixed-rank correlation
matrices. We establish a bijection between the set of bounded-rank correlation
matrices and a quotient set of a spherical product manifold by an orthogonal
group. We show that it forms an orbit space, whose stratification is determined
by the rank of the matrices, and the principal stratum has a compatible
Riemannian quotient manifold structure. We show that any minimizing geodesic in
the orbit space has constant rank on the interior of the segment. We also
develop efficient Riemannian optimization algorithms for computing the distance
and weighted the Frechet mean in the orbit space. Moreover, we examine
geometric properties of the quotient manifold, including horizontal and
vertical spaces, Riemannian metric, injectivity radius, exponential and
logarithmic map, curvature, gradient and Hessian. Finally, we apply our
approach to a functional connectivity study using the Autism Brain Imaging Data
Exchange.
