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Exploration of Balanced Metrics on Symmetric Positive Definite Matrices

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Abstract

Symmetric Positive Definite (SPD) matrices have been usedin many fields of medical data analysis. Many Riemannian metrics havebeen defined on this manifold but the choice of the Riemannianstructurelacks a set of principles that could lead one to choose properly the met-ric. This drives us to introduce the principle of balanced metrics that re-late the affine-invariant metric with the Euclidean and inverse-Euclideanmetric, or the Bogoliubov-Kubo-Mori metric with the Euclidean and log-Euclidean metrics. We introduce two new families of balanced metrics,the mixed-power-Euclidean and the mixed-power-affine metrics and wediscuss the relation between this new principle of balanced metrics and the concept of dual connection in information geometry.
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Dates and versions

hal-02158525 , version 1 (09-09-2019)

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Yann Thanwerdas, Xavier Pennec. Exploration of Balanced Metrics on Symmetric Positive Definite Matrices. GSI 2019 - 4th conference on Geometric Science of Information, Aug 2019, Toulouse, France. pp.484--493, ⟨10.1007/978-3-030-26980-7_50⟩. ⟨hal-02158525⟩
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