The geometry of mixed-Euclidean metrics on symmetric positive definite matrices

Several Riemannian metrics and families of Riemannian metrics were defined on the manifold of Symmetric Positive Definite (SPD) matrices. Firstly, we formalize a common general process to define families of metrics: the principle of deformed metrics. We relate the recently introduced family of alpha-Pro crustes metrics to the general class of mean kernel metrics by providing a sufficient condition under which elements of the former belong to the latter. Secondly, we focus on the principle of balanced bilinear forms that we recently introduced. We give a new sufficient condition under which the balanced bilinear form is a metric. It allows us to introduce the Mixed-Euclidean (ME) metrics which generalize the Mixed-Power Euclidean (MPE) metrics. We unveal their link with the (u, v)-divergences and the (alpha, beta)-divergences of information geometry and we provide an explicit formula of the Riemann curvature tensor. We show that the sectional curvature of all ME metrics can take negative values and we show experimentally that the sectional curvature of all MPE metrics but the log-Euclidean, power-Euclidean and power-affine metrics can take positive values.(c) 2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Automated summary

Several Riemannian metrics and families of Riemannian metrics are defined on the manifold of Symmetric Positive Definite (SPD) matrices. The principle of deformed metrics is used to relate the alpha-Pro crustes metrics to the mean kernel metrics. The principle of balanced bilinear forms is introduced to define the Mixed-Euclidean (ME) metrics, which generalize the Mixed-Power Euclidean (MPE) metrics and have links with (u, v)-divergences and (alpha, beta)-divergences of information geometry.