A new approach to the study of the manifold of fixed rank covariance matrices

The Bures-Wasserstein geometry of covariance matrices stands at the crossroad of statistics, information geometry, quantum information and optimal transport. The space of covariance matrices admits stratified structure with strata consisting on fixed rank covariance matrices. In this paper we focus on the rank-$k$ stratum $\textrm{Sym}(n,k)$ and revisit its geometry through a new diffeomorphic model $\textrm{Sym}(n,k) \cong \textrm{St}(n,k) \times_{O(k)} \textrm{Sym}^{+}(k)$. Working in this bundle picture, we (i) prove that the fibers are totally geodesic, (ii) derive a system of differential equations for the Bures-wasserstein geodesics (iii) exhibit several examples of closed-form solutions, and (iv) prove a one-to-one correspondence between Grassmannian and $\textrm{Sym}(n,k)$ logarithms and hence minimizing geodesics. The alternative approach we propose, clarifies the role of the underlying base $\textrm{Gr}(k,n)$ and sets the stage for further investigations into low-rank covariance matrices.