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

arXiv: https://arxiv.org/abs/2508.14369 · HTML: https://mlsanity.com/paper/2508.14369/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 35 references that are in the map.
- First posted on arXiv 20 Aug 2025.

## Builds on (most cited first, 9 of 9)

- [Hilbert geometry of the Siegel disk: The Siegel-Klein disk model](https://mlsanity.com/paper/2004.08160/) (2020)
- [Non-linear Embeddings in Hilbert Simplex Geometry](https://mlsanity.com/paper/2203.11434/) (2022)
- [Differential geometry with extreme eigenvalues in the positive semidefinite cone](https://mlsanity.com/paper/2304.07347/) (2023)
- [Voronoi Diagrams in the Hilbert Metric](https://mlsanity.com/paper/2112.03056/) (2021)
- [On representing the positive semidefinite cone using the second-order cone](https://mlsanity.com/paper/1610.04901/) (2016)
- [Simplicial faces of the set of correlation matrices](https://mlsanity.com/paper/1801.00749/) (2018)
- [A Category for unifying Gaussian Probability and Nondeterminism](https://mlsanity.com/paper/2204.14024/) (2022)
- [Delaunay Triangulations in the Hilbert Metric](https://mlsanity.com/paper/2312.05987/) (2023)
- [On The Heine-Borel Property and Minimum Enclosing Balls](https://mlsanity.com/paper/2412.17138/) (2024)

## Built on by (most cited first, 2 of 2)

- [Classifiers in High Dimensional Hilbert Metrics](https://mlsanity.com/paper/2601.13410/) (2026)
- [Geometric structures and deviations on James' symmetric positive-definite matrix bicone domain](https://mlsanity.com/paper/2603.02483/) (2026)

## Abstract (arXiv)

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.