Paper ID sheet UCL-INMA-2018.06


Quotient geometry with simple geodesics for the manifold of fixed-rank positive-semidefinite matrices

Estelle Massart, P.-A. Absil
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.
Key words
positive-semidefinite matrices; low-rank; Riemannian quotient manifold; geodesics; Riemannian logarithm; data fitting
SIAM J. Matrix Anal. Appl., 41(1), 171–198
The errata refers to the publisher's version (which is identical to the local copy).