Geodesic of the Quotient-Affine Metrics on Full-Rank Correlation Matrices

TL;DR

This paper develops the fundamental Riemannian geometry tools for the quotient-affine metric on full-rank correlation matrices, enabling intrinsic analysis in neuroscientific applications.

Contribution

It provides explicit formulas for the metric, geodesics, Levi-Civita connection, and curvature of the quotient-affine metric on correlation matrices.

Findings

Derived explicit geodesic expressions.

Computed Levi-Civita connection and curvature.

Facilitates intrinsic statistical analysis on correlation matrices.

Abstract

Correlation matrices are used in many domains of neurosciences such as fMRI, EEG, MEG. However, statistical analyses often rely on embeddings into a Euclidean space or into Symmetric Positive Definite matrices which do not provide intrinsic tools. The quotient-affine metric was recently introduced as the quotient of the affine-invariant metric on SPD matrices by the action of diagonal matrices. In this work, we provide most of the fundamental Riemannian operations of the quotient-affine metric: the expression of the metric itself, the geodesics with initial tangent vector, the Levi-Civita connection and the curvature.