QUOTIENT GEOMETRY WITH SIMPLE GEODESICS FOR THE MANIFOLD OF FIXED-RANK POSITIVE-SEMIDEFINITE MATRICES

This paper explores the well-known identification of the manifold of rank p positive-semidefinite matrices of size n with the quotient of the set of full-rank n-by-p matrices by the orthogonal group in dimension p. The induced metric corresponds to the Wasserstein metric between centered degenerate Gaussian distributions and is a generalization of the Bures-Wasserstein metric on the manifold of positive-definite matrices. We compute the Riemannian logarithm and show that the local injectivity radius at any matrix C is the square root of the pth largest eigenvalue of C. As a result, the global injectivity radius on this manifold is zero. Finally, this paper also contains a detailed description of this geometry, recovering previously known expressions by applying the standard machinery of Riemannian submersions.