On the Geometry of Low Rank Correlation Matrices

Abstract

Correlation matrices of bounded rank form a stratified space whose factor representations lie in a product of spheres. Using the orbit-type description of the rank stratification, we prove that padding factors preserves the Riemannian geometry of each fixed-rank stratum and the intrinsic length metric of every rank-bounded subset. In the reduced model, a Sylvester equation describes the mechanical connection and its horizontal lifts. We derive sectional-curvature formulas and bounds, characterize zero-curvature directions, and determine the connection's holonomy groups. In particular, a flat rank-two connection can have nontrivial dihedral holonomy. For geodesics subject to a rank bound, rank cannot decrease in the interior of a minimizing segment with maximal-rank endpoints; the local excess over the unconstrained quotient distance is controlled by a dimension-independent cubic estimate. Finally, we compute the exact change of measure between quotient Riemannian volume and Euclidean surface volume in the off-diagonal entries. Coarea yields explicit factor densities for both normalized uniform laws.