Exploration of Balanced Metrics on Symmetric Positive Definite Matrices

Yann Thanwerdas (1, 2) , Xavier Pennec (1, 2)

1 COMUE UCA - COMUE Université Côte d'Azur (2015-2019)
2 EPIONE - E-Patient : Images, données & mOdèles pour la médeciNe numériquE

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.

Mots clés: Dually flat connections; SPD matrices; Information geometry

Domaines: Géométrie différentielle [math.DG]

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⟩