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Log-Euclidean Lie Groups

Olivier Bisson, Xavier Pennec

TL;DR

This work develops a rigorous log-Euclidean Lie group framework for manifolds diffeomorphic to vector spaces, with $\mathcal{S}^+(n)$ and $\mathrm{Cor}^+(n)$ as central examples. It proves that all log-Euclidean metrics on equal-dimension manifolds are isometrically isomorphic and provides explicit isometries between SPD and correlation geometries via the diffeomorphisms $\log$, $\mathrm{Log}$, and $\log^{\bullet}$. A principal-bundle viewpoint yields a quotient geometry where the canonical section yields an orthogonal metric splitting, and the quotient metric aligns with the off-log metric on correlation matrices; however, for $n\ge 3$ no metric pair makes $\mathrm{Cor}^+(n)$ an isometric embedding in $\mathcal{S}^+(n)$. Altogether, the paper unifies disparate log-Euclidean metrics and supplies concrete, structure-preserving maps for geometric statistics on SPD and correlation manifolds.

Abstract

We develop a self-contained comprehensive theory of log-Euclidean Lie groups: smooth manifolds diffeomorphic to finite-dimensional vector spaces; extending existing results for symmetric positive-definite (SPD) matrices S+(n) and full-rank correlation matrices Cor+(n). We specify an isometric isomorphism theorem, showing that all log-Euclidean metrics on manifolds of identical dimension are isomorphically Riemannian isometric. Explicit isometries are constructed, linking SPD and correlation matrix manifolds. We further characterize quotient log-Euclidean metrics within a principal-bundle framework. We show that for any n $\ge$ 3, there is no choice of log-Euclidean metrics on S+(n) and on Cor+(n) for which the inclusion i\,: Cor+(n) $\rightarrow$ S+(n) becomes an isometric embedding. These results unify several disparate metrics in geometric statistics.

Log-Euclidean Lie Groups

TL;DR

This work develops a rigorous log-Euclidean Lie group framework for manifolds diffeomorphic to vector spaces, with and as central examples. It proves that all log-Euclidean metrics on equal-dimension manifolds are isometrically isomorphic and provides explicit isometries between SPD and correlation geometries via the diffeomorphisms , , and . A principal-bundle viewpoint yields a quotient geometry where the canonical section yields an orthogonal metric splitting, and the quotient metric aligns with the off-log metric on correlation matrices; however, for no metric pair makes an isometric embedding in . Altogether, the paper unifies disparate log-Euclidean metrics and supplies concrete, structure-preserving maps for geometric statistics on SPD and correlation manifolds.

Abstract

We develop a self-contained comprehensive theory of log-Euclidean Lie groups: smooth manifolds diffeomorphic to finite-dimensional vector spaces; extending existing results for symmetric positive-definite (SPD) matrices S+(n) and full-rank correlation matrices Cor+(n). We specify an isometric isomorphism theorem, showing that all log-Euclidean metrics on manifolds of identical dimension are isomorphically Riemannian isometric. Explicit isometries are constructed, linking SPD and correlation matrix manifolds. We further characterize quotient log-Euclidean metrics within a principal-bundle framework. We show that for any n 3, there is no choice of log-Euclidean metrics on S+(n) and on Cor+(n) for which the inclusion i\,: Cor+(n) S+(n) becomes an isometric embedding. These results unify several disparate metrics in geometric statistics.

Paper Structure

This paper contains 16 sections, 51 theorems, 169 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Log-Euclidean Lie groups
  3. Preliminaries
  4. Definitions and Elementary Properties
  5. Riemannian Manifolds: SPD and Full-Rank Correlation Matrices
  6. The Exponential Diffeomorphism
  7. The Off-Log Diffeomorphism
  8. The Log-Scaling Diffeomorphism
  9. The Isomorphic Isometry Theorem
  10. Isometry between $\left(\mathcal{S}^+(n-1), g^{\mathrm{LE}}\right)$ and $\left(\mathrm{Cor}^+(n), g^{\mathrm{OL}}\right)$
  11. Isometry between $\left(\mathcal{S}^+(n-1), g^{\mathrm{LE}}\right)$ and $\left(\mathrm{Cor}^+(n), g^{\mathrm{LS}}\right)$
  12. Isometry between $\left(\mathrm{Cor}^+(n), g^{\mathrm{OL}}\right)$ and $\left(\mathrm{Cor}^+(n), g^{\mathrm{LS}}\right)$
  13. Quotient of Log-Euclidean Lie groups
  14. Isomorphic sections
  15. Riemannian submersion and isometric embeddings
  16. ...and 1 more sections

Key Result

Theorem 2.3

Let $G$ and $H$ be Lie groups with Lie algebras $\mathfrak g$ and $\mathfrak h$ respectively, and let $\phi\colon G \to H$ be a Lie group homomorphism. Then:

Theorems & Definitions (109)

  • Definition 2.1: Lie groups
  • Definition 2.2: Lie algebra
  • Theorem 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Theorem 2.6
  • Definition 2.7: $1$-parameter subgroup
  • Definition 2.8: Lie exponential map
  • Theorem 2.9
  • Definition 2.10: $\mathrm{Ad}$-invariance
  • ...and 99 more