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, 2020
BibTeX entry

author = {Massart, Estelle and Absil, P.-A.},
title = {Quotient Geometry with Simple Geodesics for the Manifold of Fixed-Rank Positive-Semidefinite Matrices},
journal = {SIAM Journal on Matrix Analysis and Applications},
volume = {41},
number = {1},
pages = {171-198},
year = {2020},
doi = {10.1137/18M1231389},
The errata refers to the publisher's version (which is identical to the local copy).
Further information
Note that, in Proposition 4.4 and Proposition 4.5, as shown in Examples 5.6 of arXiv:2204.09928v1, there are cases where $\mathrm{EXP}_{\pi(Y_1)} \xi$ is not equal to $\mathrm{Exp}_{\pi(Y_1)} \xi$ (see Definition 3.1 for "$\mathrm{EXP}$" and "$\mathrm{Exp}$"). In other words, there are cases where $\bar\xi_{Y_1}$ is not in $\mathcal{D}_{Y_1}$. However, $\mathrm{EXP}$ and $\mathrm{Exp}$ coincide on the shortest elements of the premiage of $\mathrm{EXP}$, i.e., $\mathrm{EXP}_{\pi(Y_1)} \xi^* = \mathrm{Exp}_{\pi(Y_1)} \xi^*$. The case where the shortest element is unique leads to Theorem 4.7. The non-unique case leads to an extension of Theorem 4.7 given in points 5 and 6 of Theorem 5.3 in arXiv:2204.09928v1.