Quotient Geometry with Simple Geodesics for the Manifold of Fixed-Rank Positive-Semidefinite Matrices
Estelle Massart, Pierre-Antoine Absil
Abstract
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 $p$th 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.