Paper ID sheet UCL-INMA-2018.06

Title

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

Authors

Estelle Massart, P.-A. 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 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

Status

SIAM J. Matrix Anal. Appl., 41(1), 171–198, 2020

Download

• Publisher's version: doi:10.1137/18M1231389
• Local copy
• Preprint

BibTeX entry

@article{MasAbs2020,
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},
}
  

Errata

The errata refers to the publisher's version (which is identical to the local copy).

• Page 174, line after (1): replace "$\pi(Y)$" by "$\pi^{-1}(\pi(Y))$" or equivalently by "$Y\mathcal{O}_p$". (This is a pedantic correction. Since the codomain of \pi is the quotient space, $\pi(Y)$ an equivalence class viewed as an element of the quotient space, whereas we need here to view the equivalence class as a subset of the total space.)
• Page 175, equation (4): replace "$\zeta$" by "$\eta$" (2 times).

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.