An associated bundle approach to the Bures--Wasserstein geometry of fixed rank covariance matrices

2025/07/08 by Leonardo Marconi, Marconi, Leonardo
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Random Matrices and Applications #Noncommutative and Quantum Gravity Theories

paper · pdf · doi:10.48550/arxiv.2507.05873

Abstract

The Bures--Wasserstein geometry of covariance matrices provides a canonical distance on the statistical manifold of centred Gaussian measures and lies at the intersection of information geometry, quantum information, and optimal transport. The space of covariance matrices admits a natural stratified structure whose strata consist of fixed-rank covariance matrices. In this paper we focus on the rank-k stratum \Sym⁺(n,k) and revisit its geometry through the diffeomorphic associated-bundle model \Sym⁺(n,k)≅\St(n,k)×_O(k)\Sym⁺(k). Working in this bundle picture, we (i) derive a system of differential equations for Bures--Wasserstein geodesics, (ii) prove that the fibers are totally geodesic and (iii) establish a one-to-one correspondence between Grassmannian logarithms and Bures--Wasserstein logarithms on \Sym⁺(n,k), and hence between minimizing geodesics in the two spaces. This alternative viewpoint clarifies the role of the underlying base \Gr(k,n) in the Bures--Wasserstein geometry of low-rank covariance matrices and sets the stage for further investigations into structured covariance models.

Citations

• Bures-Wasserstein minimizing geodesics between covariance matrices of different ranks
• O(n)-invariant Riemannian metrics on SPD matrices
• A Novel Geometric Framework on Gram Matrix Trajectories for Human Behavior Understanding
• On the Bures-Wasserstein distance between positive definite matrices
• Regression on fixed-rank positive semidefinite matrices: a Riemannian\n approach

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