Is affine invariance well defined on SPD matrices? A principled continuum of metrics

Abstract

Symmetric Positive Definite (SPD) matrices have been widely used in medical data analysis and a number of different Riemannian met-rics were proposed to compute with them. However, there are very few methodological principles guiding the choice of one particular metric for a given application. Invariance under the action of the affine transformations was suggested as a principle. Another concept is based on symmetries. However, the affine-invariant metric and the recently proposed polar-affine metric are both invariant and symmetric. Comparing these two cousin metrics leads us to introduce much wider families: power-affine and deformed-affine metrics. Within this continuum, we investigate other principles to restrict the family size.

References in corpus (3)

• Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging
• Statistical analysis of trajectories on Riemannian manifolds: Bird migration, hurricane tracking and video surveillance
• Power Euclidean metrics for covariance matrices with application to diffusion tensor imaging

Cited by in corpus (5)

• On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry
• Approximation and bounding techniques for the Fisher-Rao distances between parametric statistical models
• Covariance-modulated optimal transport and gradient flows
• Differential geometry with extreme eigenvalues in the positive semidefinite cone
• Projective Wishart Distributions