Geodesic distance

A geodesic is the shortest curve between two points that stays on the manifold; its length is the geodesic distance, the manifold analogue of a straight-line Euclidean distance. Under the affine-invariant metric the geodesic between SPD matrices P and Q has the closed form gamma(t) = P^{1/2} (P^{-1/2} Q P^{-1/2})^t P^{1/2} for t in [0,1], and the distance is || log(P^{-1/2} Q P^{-1/2}) ||_F.

Using geodesic rather than Euclidean distance is what makes covariance comparisons meaningful: the geodesic path never leaves the SPD cone, and the distance inherits affine invariance. Different metrics (affine-invariant, log-Euclidean, Bures-Wasserstein) define different geodesics and therefore different distances and means; the choice trades invariance properties against computational cost.