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

This paper explores the well-known identiï¬cation of the manifold of rank p positivesemideï¬nitematricesofsizenwiththequotientofthesetoffull-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-deï¬nitematrices. 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.