Explicit Geodesic Projection Distance on the Statistical Manifold of Multivariate Elliptical Distributions

Geodesic projection distances on statistical manifolds have been widely applied across various research fields. Nevertheless, the existing closed-form solution is only available for the Gaussian manifold. Multivariate elliptical distributions (MEDs) include the Gaussian, generalized Gaussian, Student's t distribution, contaminated Gaussian, and other heavy-tailed models. Despite their widespread practical applicability, explicit geodesic projection solutions for MED manifolds remain unexplored. This paper derives the explicit geodesic projection distance from an arbitrary point on the MED manifold equipped with the Fisher metric onto its commonly used submanifold with a fixed mean vector. The core theoretical contribution lies in a novel symmetry-exploitation technique proposed to tackle the highly nonlinear and complex geodesic equations, and this methodological framework is readily extendable to other statistical manifolds. The derived results substantially advance the state-of-the-art by generalizing existing theories from the Gaussian manifold to the broader family of MEDs.