Hilbert geometry of the symmetric positive-definite bicone: Application to the geometry of the extended Gaussian family
This work addresses a theoretical geometry problem for statistical modeling, but it is incremental as it extends known Hilbert geometry concepts to a specific matrix structure.
The paper tackles the problem of modeling the extended Gaussian family, which includes degenerate covariance and precision matrices, by studying the Hilbert geometry of the symmetric positive-definite bicone. It reports a closed-form formula for the Hilbert metric distance and analyzes its invariance properties.
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.