Paper ID sheet UCLINMA2018.06
 Title

Quotient geometry with simple geodesics for the manifold of fixedrank positivesemidefinite matrices
 Authors
 Estelle Massart, P.A. Absil
 Abstract

This paper explores the wellknown identification of the manifold of rank p positivesemidefinite matrices of size n with the quotient of the set of fullrank nbyp 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 Buresâ€“Wasserstein metric on the manifold of positivedefinite 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
 positivesemidefinite matrices; lowrank; Riemannian quotient manifold; geodesics; Riemannian logarithm; data fitting
 Status
 SIAM J. Matrix Anal. Appl., 41(1), 171â€“198
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 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 the view the equivalence class as a subset of the total space.)
 Page 175, equation (4): replace "\zeta" by "\eta" (2 times).
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