Bures–Wasserstein Metric

• The Bures–Wasserstein metric is a closed-form distance defined between nonnegative covariance operators derived from centered Gaussian measures.
• It underpins statistical methods such as Fréchet mean estimation and functional PCA while accommodating both finite and infinite-dimensional settings.
• Its explicit geometric structure, including log/exp maps and tangent spaces, enables robust applications in dynamic functional connectivity and operator-valued data analysis.

The Bures–Wasserstein metric is a canonical, closed-form metric on the space of nonnegative, self-adjoint, trace-class operators (in finite dimensions, symmetric positive definite or positive semidefinite matrices), arising as the 2-Wasserstein distance between centered Gaussian measures. This metric provides both the geometric structure for a broad class of operator-valued statistical models and forms the basis for methodologies in functional data analysis for covariance-valued flows, especially in infinite-dimensional or time-varying contexts. It enables rigorous definitions and algorithms for means, covariances, and principal component analysis of random processes whose observations are covariance operators.

1. Mathematical Formulation of the Bures–Wasserstein Metric

Given two nonnegative definite, self-adjoint, trace-class operators $F, G$ on a separable Hilbert space $\mathbb{H}$, the Bures–Wasserstein metric is defined as the 2-Wasserstein distance between the centered Gaussian measures %%%%2%%%%, $\mu_G = \mathcal{N}(0, G)$: $\Pi(F, G) := W_2(\mu_F, \mu_G).$ This admits the explicit formula

$$\boxed{ \Pi(F, G) = \sqrt{ \operatorname{tr}(F) + \operatorname{tr}(G) - 2 \operatorname{tr} \left( (G^{1/2} F G^{1/2})^{1/2} \right) } }$$

which holds for both finite and (with technical care) infinite-dimensional settings.

The unique (where defined) optimal transport map from $\mu_F$ to $\mu_G$ is: $$T_F^G = F^{-1/2} (F^{1/2} G F^{1/2})^{1/2} F^{-1/2}.$$

2. Riemannian-like Geometry and Stratification

The space of covariance operators equipped with the Bures–Wasserstein metric, denoted here as $(\mathcal{K}_{\mathbb{H}}, \Pi)$, exhibits stratified, manifold-like geometry:

• Tangent spaces at $F \in \mathcal{K}$:

$$T_F = \overline{ \{ \lambda \cdot (T - \operatorname{Id}) : \lambda \geq 0,\, T\, \text{optimal map} \} }$$

with inner product

$$\langle \Gamma, \Gamma' \rangle_{T_F} = \operatorname{tr}( \Gamma F \Gamma' ).$$

• Exponential/logarithm maps:

$$\exp_F(\Gamma) = (\Gamma + \operatorname{Id}) F (\Gamma + \operatorname{Id}), \qquad \log_F(G) = T_F^G - \operatorname{Id}.$$

• Geodesics: For $F_0, F_1$, the constant-speed geodesic is:

$$F_\lambda = [\lambda T_{F_0}^{F_1} + (1-\lambda)\operatorname{Id}]\, F_0\, [\lambda T_{F_0}^{F_1} + (1-\lambda)\operatorname{Id}], \quad \lambda \in [0,1].$$

The geometry is stratified due to possible rank-deficiency of operators, and all regular (full-rank) covariances in finite dimensions form a Riemannian manifold.

3. Covariance Flows and Functional Data Analysis

A covariance flow is a measurable path

$$\mathcal{F}: [0,1] \to \mathcal{K}, \qquad t \mapsto F_t,$$

where $F_t$ is a covariance operator at time $t$. The space of continuous flows is $\mathcal{F}_C$.

The metric for flows extends pointwise Bures–Wasserstein to the $L^2$ sense: $$d(\mathcal{F}, \mathcal{G}) = \left( \int_0^1 \Pi(F_t, G_t)^2 dt \right)^{1/2}.$$ This lifts the operator metric to sample paths, making $(\mathcal{F}_C, d)$ a metric space suitable for functional data analysis.

4. Statistical Structures: Means, Covariances, Karhunen-Loève Expansions

(a) Fréchet Mean Flow

The Fréchet mean flow $\mathcal{M}$ minimizes expected squared distance: $$\mathcal{M} = \arg\min_{\mathcal{G} \in \mathcal{F}_C} \mathbb{E}[ d^2(\mathcal{F}, \mathcal{G}) ]$$ which reduces to pointwise minimization: $$M_t = \arg\min_{G \in \mathcal{K}} \mathbb{E}[ \Pi^2(G, F_t) ].$$

(b) Covariance of Random Flows and Principal Components

The logarithmic process,

$$\log_{\mathcal{M}} \mathcal{F}(t) := T_{\mathcal{M}(t)}^{F(t)} - \operatorname{Id},$$

lives in the tangent bundle along $\mathcal{M}$. The tangent bundle,

$$\mathscr{T}_{\mathcal{M}} = \Big\{ V : [0,1] \to T_{\mathcal{M}(t)}, \int_0^1 \| V(t) \|_{T_{\mathcal{M}(t)}}^2 dt < \infty \Big\},$$

inherits its inner product from the operator geometry. The covariance operator of the random log-process is defined as

$$\mathcal{C} = \mathbb{E}\big[ (\log_{\mathcal{M}} \mathcal{F}) \otimes (\log_{\mathcal{M}} \mathcal{F}) \big].$$

Principal components follow from the spectral decomposition of $\mathcal{C}$, yielding a Karhunen-Loève expansion for covariance flows.

5. Estimation, Consistency, and Functional PCA

From i.i.d. sample flows $\mathcal{F}_1, \dots, \mathcal{F}_n$, the empirical Fréchet mean flow $\widehat{\mathcal{M}}$ is computed pointwise. The empirical covariance of the log-processes yields estimators for $\mathcal{C}$.

Theoretical consistency and convergence rates:

• For integral (in time) metrics: $O_p(1/\sqrt{n})$ rates are established.
• In finite-dimensional (matrix) settings, with regularity, uniform rates are sharper, e.g.,

$$\sup_{t \in [0,1]} \mathbb{E}[ \Pi(\mathcal{M}(t), \widehat{\mathcal{M}}_n(t))^2 ] = O(n^{-1}).$$

Functional principal component analysis (PCA) is achieved by embedding the tangent bundle into a common Hilbert space, e.g., via $J_F(U) = U F^{1/2}$, making classical linear PCA tools applicable to the covariance log-processes.

Estimation steps, including gradient descent for the Fréchet mean, are robust to discretization in both time and operator spaces.

6. Applications and Finite vs. Infinite-Dimensional Considerations

Finite-dimensional simplification: In the matrix case, invertibility is standard, and the log, exponential, and tangent bundle structure are globally well-defined and computationally tractable. Convergence rates and estimation procedures simplify, with explicit gradient formulas available.

Application areas: The Bures–Wasserstein geometry for flows is directly applicable to:

• Dynamic functional connectivity analysis (e.g., in fMRI).
• Functional time series (e.g., spectral density operator flows).
• Modern functional data contexts involving operator-valued random processes.

Demonstrative examples include geodesic interpolations between covariances, synthetic flows, and real data from neuroimaging or demographic statistics.

Summary Table

Aspect	Infinite-Dimensional Setting	Finite-Dimensional Simplification
Metric	Bures–Wasserstein $\Pi$	Same formula (matrix case)
Log/Exp Map	May not be globally defined	Globally defined for invertible matrices
Tangent Spaces	Vary pointwise, carefully constructed	Tangent spaces equivalent for regular case
Optimal Maps	May be only densely defined or unbounded	Always defined and bounded
Inference	Requires embedding for tangent comparisons	All structures aligned; optimal rates
Statistical Tasks	Mean/covariance estimation, K-L expansion	Simpler algorithms, explicit gradients

7. Implications and Methodological Significance

The Bures–Wasserstein geometry provides a rigorous and practical framework for operator-valued statistical analysis, particularly for random and dynamic data where observations are covariance operators or matrix-valued flows. The explicit geometric machinery enables:

• Precise definition of means and covariances for operator-valued random elements.
• A functional PCA procedure respecting the nonlinear geometry of the sample space.
• Robust inference methodologies for both estimation and hypothesis testing.

By exploiting the intrinsic structure of the space of covariance operators—stratified, with Riemannian-like features and computable exponential/logarithmic maps—these techniques generalize and unify linear procedures for principal component analysis and mean estimation to a broad class of non-Euclidean data structures.

Conclusion

The Bures–Wasserstein metric allows the extension of foundational statistical concepts to the nonlinear metric space of covariance operators and their flows. Through its explicit geometry and closed-form expressions, it supports efficient and principled methodologies for mean, covariance, and principal components, providing the basis for modern operator- and matrix-valued functional data analysis in both finite and infinite dimensions.