Hilbert geometry of the symmetric positive-definite bicone: Application to the geometry of the extended Gaussian family

TL;DR

This paper explores the Hilbert geometry of the symmetric positive semi-definite matrix bicone, providing a closed-form Hilbert metric and examining its invariance properties, with potential applications to extended Gaussian distributions.

Contribution

It introduces the Hilbert geometry of the extended Gaussian family’s parameter space, a symmetric positive semi-definite bicone, and derives a closed-form Hilbert metric for it.

Findings

Closed-form formula for the Hilbert metric distance

Analysis of invariance properties of the geometry

Discussion of applications to extended Gaussian distributions

Abstract

The extended Gaussian family is the closure of the Gaussian family obtained by completing the Gaussian family with the counterpart elements induced by degenerate covariance or degenerate precision matrices, or a mix of both degeneracies. The parameter space of the extended Gaussian family forms a symmetric positive semi-definite matrix bicone, i.e. two partial symmetric positive semi-definite matrix cones joined at their bases. In this paper, we study the Hilbert geometry of such an open bounded convex symmetric positive-definite bicone. We report the closed-form formula for the corresponding Hilbert metric distance and study exhaustively its invariance properties. We also touch upon potential applications of this geometry for dealing with extended Gaussian distributions.