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O(n)-invariant Riemannian metrics on SPD matrices

Abstract : 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.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03338601
Contributor : Yann Thanwerdas Connect in order to contact the contributor
Submitted on : Monday, September 13, 2021 - 12:55:14 PM
Last modification on : Wednesday, September 15, 2021 - 3:31:40 AM

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  • HAL Id : hal-03338601, version 2
  • ARXIV : 2109.05768

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Yann Thanwerdas, Xavier Pennec. O(n)-invariant Riemannian metrics on SPD matrices. 2021. ⟨hal-03338601v2⟩

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