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

Leonardo Marconi

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 $\Sym^+(n,k)\cong\St(n,k)\times_{O(k)}\Sym^{+}(k)$. 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.

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

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- stratum and revisit its geometry through the diffeomorphic associated-bundle model . 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 , and hence between minimizing geodesics in the two spaces. This alternative viewpoint clarifies the role of the underlying base in the Bures--Wasserstein geometry of low-rank covariance matrices and sets the stage for further investigations into structured covariance models.

Paper Structure

This paper contains 18 sections, 10 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fiber bundles
  4. The Grassmannian as a principal fiber bundle
  5. The Bures--Wasserstein geometry of the fixed-rank stratum
  6. The diffeomorphism
  7. Tangent spaces
  8. Pullback metric
  9. Geodesics
  10. ODE system for geodesic equations and totally geodesic fibers
  11. Logarithm bijection
  12. Discussion and perspectives
  13. Appendix
  14. Proof of Proposition \ref{['prop. tangent spaces of associated bundle']}.
  15. Proof of Lemma \ref{['lemma:local to global diff']}.
  16. ...and 3 more sections

Key Result

Proposition 1

Let $\pi:P \to B$ be a principal $G$-bundle and $E=P \times_G F$ an associated bundle with fiber $F$ and left action $L_g f=g \cdot f$. Then the tangent spaces of $E$ are given by where $\mathfrak{g}$ is the Lie algebra of $G$ and is the infinitesimal action of $\xi$ on $F$.

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Proposition 2
  • ...and 19 more