O(n)-invariant Riemannian metrics on SPD matrices

Symmetric Positive Definite (SPD) matrices are ubiquitous in data analysis under the form of covariance or correlation matrices. Several O(n)-invariant Riemannian metrics were defined on SPD cone, particular kernel introduced by Hiai and Petz. The class interpolates between many classical it satisfies key results stability completeness. However, does not contain all metrics. Therefore this work, we investigate super-classes study which remain true. We also introduce an additional result called cometric-stability, a crucial property to implement geodesics with Hamiltonian formulation. Our method build intermediate embedded classes is give characterization whole specify requirements one until reach As secondary contribution, synthesize literature main metrics, provide complete formula sectional curvature affine-invariant metric geodesic parallel transport commuting for Bures-Wasserstein metric.