Quotient Geometry with Simple Geodesics for the Manifold of Fixed-Rank Positive-Semidefinite Matrices

Massart, Estelle;Absil, Pierre-Antoine
(2020) SIAM Journal on Matrix Analysis and Applications — Vol. 41, n° 1, p. 171-198 (2020)

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This paper explores the well-known identification of the manifold of rank p positivesemidefinitematricesofsizenwiththequotientofthesetoffull-rankn-by-pmatricesbytheorthogonal 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 manifoldofpositive-definitematrices. WecomputetheRiemannianlogarithm,andshowthatthelocal injectivity radiusat anymatrix C isthesquareroot ofthe pth largesteigenvalue of C. Asa result, the globalinjectivityradiusonthismanifoldiszero. Finally,thispaperalsocontainsadetaileddescription of this geometry, recovering previously known expressions by applying the standard machinery of Riemannian submersions.
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Massart, E., & Absil, P.-A. (2020). Quotient Geometry with Simple Geodesics for the Manifold of Fixed-Rank Positive-Semidefinite Matrices. SIAM Journal on Matrix Analysis and Applications, 41(1), 171-198. https://doi.org/10.1137/18m1231389 (Original work published 2020)