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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.05873 |
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| _version_ | 1866918413718257664 |
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| author | Marconi, Leonardo |
| author_facet | Marconi, Leonardo |
| contents | 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 $\Sym^+(n,k)\cong\St(n,k)\times_{O(k)}\Sym^{+}(k)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05873 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An associated bundle approach to the Bures--Wasserstein geometry of fixed rank covariance matrices Marconi, Leonardo Differential Geometry 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 $\Sym^+(n,k)\cong\St(n,k)\times_{O(k)}\Sym^{+}(k)$. 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. |
| title | An associated bundle approach to the Bures--Wasserstein geometry of fixed rank covariance matrices |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2507.05873 |