Log-Euclidean Lie Groups
Olivier Bisson (EPIONE, UniCA), Xavier Pennec (EPIONE, UniCA)

TL;DR
This paper develops a comprehensive theory of log-Euclidean Lie groups, unifying various constructions for SPD and correlation matrices, and provides explicit isometries, metrics, and embeddings to facilitate matrix analysis and comparison.
Contribution
It introduces a unified framework for log-Euclidean Lie groups, deriving explicit isometries, metrics, and embeddings for SPD and correlation matrices, enhancing geometric understanding and matrix comparison.
Findings
Explicit Riemannian isometries linking log-Euclidean metrics.
Construction of a log-Euclidean metric making correlation matrices isometric.
Closed-form formulas for geodesics and orthogonal decompositions.
Abstract
We develop a self-contained theory of log-Euclidean Lie groups: smooth manifolds diffeomorphic to finite-dimensional vector spaces, equipped with the pullback of a constant Euclidean metric. This framework encompasses symmetric positive-definite (SPD) matrices S+(n) and full-rank correlation matrices Cor+(n), and explains why many seemingly different log-Euclidean constructions yield the same Riemannian geometry. We provide explicit Riemannian isometries (and Lie group isomorphisms) linking several log-Euclidean metrics on SPD and correlation matrix manifolds, and we characterize quotient log-Euclidean metrics in a principal-bundle setting. Finally, using the diagonal correction map underlying the off-log parametrization, we construct an explicit log-Euclidean metric on S+(n) for which the standard inclusion i\,: Cor+(n) S+(n) becomes an isometric (indeed, totally…
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Taxonomy
TopicsAdvanced Algebra and Geometry · Homotopy and Cohomology in Algebraic Topology · Geometric and Algebraic Topology
