Bures-Wasserstein metric

The Bures-Wasserstein metric is an alternative geometry on the SPD manifold arising from optimal transport between zero-mean Gaussians: the squared distance is trace(P) + trace(Q) - 2 trace((P^{1/2} Q P^{1/2})^{1/2}). Unlike the affine-invariant metric it is not affine-invariant (only orthogonally invariant), and it does not diverge to infinity as a matrix approaches singularity, giving it different, sometimes gentler, behavior near the boundary of the cone.

It has drawn interest in BCI and in domain adaptation, since its barycenter is the Wasserstein barycenter used by optimal-transport transfer methods, as a computationally distinct alternative to AIRM and log-Euclidean, with its own closed-form geodesics and an iterative barycenter. Which metric is best is empirical and dataset-dependent; there is no universally dominant SPD metric.