Equivariant log-extrinsic statistics on symmetric cones

Author

Trindade, Gabriel and Chevallier, Emmanuel and Nielsen, Frank and Nicolet, Andre

Abstract

Symmetric cones, such as the space of positive-definite matrices or Lorentz cones, are fundamental to statistical modeling in fields ranging from neuroimaging to radar. Standard approaches, however, face a dilemma: intrinsic Riemannian methods require costly iterative optimization, while Euclidean approximations sacrifice desirable equivariance properties. We introduce a log-extrinsic framework that resolves this trade-off, utilizing the orbital decomposition of the cone to treat the determinant and the determinant-one surface independently. This approach yields an equivariant, closed-form mean that requires only elementary linear algebra, rendering it highly scalable to high-dimensional problems. We prove that under natural symmetry assumptions, this log-extrinsic mean coincides with the Riemannian Fréchet mean. Finally, we define an exponential family of log-extrinsic Gaussian distributions naturally paired with this mean. The result is a coherent statistical toolbox that preserves key geometric symmetries while remaining computationally tractable.