Equivariant log-extrinsic statistics on symmetric cones
Résumé
Symmetric cones, and their determinant-one hypersurfaces, form an important class of geometric spaces for data science, encompassing covariance matrices as well as Lorentz cones and hyperbolic spaces. Classical approaches to statistical analysis on these cones typically rely on Riemannian machinery. Fréchet means are computed via iterative optimization, or approximated by flattening the geometry with Euclidean surrogates, such as the log‑Euclidean approach. We propose an alternative framework that leads to closed-form expressions, preserves equivariance, while only relying on elementary linear algebra, making it straightforward to implement. We first introduce the log‑extrinsic mean via the orbital decomposition of the cone into its determinant‑one surface and determinant component, yielding a closed‑form expression. A key result of the paper shows that under common symmetry assumptions, the equivariance of the construction makes the log‑extrinsic and Riemannian Fréchet means coincide. This provides a direct, non‑iterative way to estimate Fréchet means. The same decomposition gives rise to a family of log‑extrinsic Gaussian distributions naturally paired with log-extrinsic means. Together, these constructions provide coherent building blocks for a consistent statistical framework on symmetric cones.
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