O(n)-invariant Riemannian metrics on SPD matrices - IDEX UCA JEDI Université Côte d'Azur Accéder directement au contenu
Article Dans Une Revue Linear Algebra and its Applications Année : 2022

O(n)-invariant Riemannian metrics on SPD matrices

Résumé

Symmetric Positive Definite (SPD) matrices are ubiquitous in data analysis under the form of covariance matrices or correlation matrices. Several O(n)-invariant Riemannian metrics were defined on the SPD cone, in particular the kernel metrics introduced by Hiai and Petz. The class of kernel metrics interpolates between many classical O(n)-invariant metrics and it satisfies key results of stability and completeness. However, it does not contain all the classical O(n)-invariant metrics. Therefore in this work, we investigate super-classes of kernel metrics and we study which key results remain true. We also introduce an additional key result called cometric-stability, a crucial property to implement geodesics with a Hamiltonian formulation. Our method to build intermediate embedded classes between O(n)-invariant metrics and kernel metrics is to give a characterization of the whole class of O(n)-invariant metrics on SPD matrices and to specify requirements on metrics one by one until we reach kernel metrics. As a secondary contribution, we synthesize the literature on the main O(n)-invariant metrics, we provide the complete formula of the sectional curvature of the affine-invariant metric and the formula of the geodesic parallel transport between commuting matrices for the Bures-Wasserstein metric.
Fichier principal
Vignette du fichier
main.pdf (595.8 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03338601 , version 1 (11-09-2021)
hal-03338601 , version 2 (13-09-2021)
hal-03338601 , version 3 (15-11-2022)

Identifiants

Citer

Yann Thanwerdas, Xavier Pennec. O(n)-invariant Riemannian metrics on SPD matrices. Linear Algebra and its Applications, inPress. ⟨hal-03338601v2⟩
278 Consultations
719 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More