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Journal Articles Linear Algebra and its Applications Year : 2023

O(n)-invariant Riemannian metrics on SPD matrices

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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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Dates and versions

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

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Yann Thanwerdas, Xavier Pennec. O(n)-invariant Riemannian metrics on SPD matrices. Linear Algebra and its Applications, 2023, 661, pp.163-201. ⟨10.1016/j.laa.2022.12.009⟩. ⟨hal-03338601v3⟩
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