{"ID":2880072,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.14369","arxiv_id":"2508.14369","title":"Hilbert geometry of the symmetric positive-definite bicone: Application to the geometry of the extended Gaussian family","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.","short_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 sem...","url_abs":"https://arxiv.org/abs/2508.14369","url_pdf":"https://arxiv.org/pdf/2508.14369v1","authors":"[\"Jacek Karwowski\",\"Frank Nielsen\"]","published":"2025-08-20T02:57:02Z","proceeding":"cs.CG","tasks":"[\"cs.CG\",\"cs.LG\",\"math.PR\"]","methods":"[]","has_code":false}