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

• URL: http://arxiv.org/abs/2508.14369v1
• Date: Wed, 20 Aug 2025 02:57:02 GMT
• Title: Hilbert geometry of the symmetric positive-definite bicone: Application to the geometry of the extended Gaussian family
• Abstract summary: The parameter space of the extended Gaussian family forms a symmetric positive semi-definite matrix bicone.<n>We study the Hilbert geometry of such an open bounded convex symmetric positive-definite bicone.<n>We touch upon potential applications of this geometry for dealing with extended Gaussian distributions.
• Score: 5.985204759362746
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Geometric Gaussian Approximations of Probability Distributions [25.021782278452005]
  We study the expressivity of geometric Gaussian approximations.<n>We provide a constructive proof that such approximations are universal.<n>We discuss whether a common diffeomorphism can be found to obtain uniformly high-quality geometric Gaussian approximations.
  arXiv  Detail & Related papers  (2025-07-01T09:54:43Z)
• Geometric Jensen-Shannon Divergence Between Gaussian Measures On Hilbert Space [5.985204759362746]
  This work studies the Geometric Jensen-Shannon divergence, based on the notion of geometric mean of probability measures, in the setting of Gaussian measures on an infinite-dimensional Hilbert space.<n>We present a closed form expression for this divergence that directly generalizes the finite-dimensional version.
  arXiv  Detail & Related papers  (2025-06-12T08:53:16Z)
• Topological nature of edge states for one-dimensional systems without symmetry protection [46.87902365052209]
  We numerically verify and analytically prove a winding number invariant that correctly predicts the number of edge states in one-dimensional, nearest-neighbor (between unit cells)<n>Our winding number is invariant under unitary or similarity transforms.
  arXiv  Detail & Related papers  (2024-12-13T19:44:54Z)
• Product Geometries on Cholesky Manifolds with Applications to SPD Manifolds [65.04845593770727]
  We present two new metrics on the Symmetric Positive Definite (SPD) manifold via the Cholesky manifold. Our metrics are easy to use, computationally efficient, and numerically stable.
  arXiv  Detail & Related papers  (2024-07-02T18:46:13Z)
• Wigner's Theorem for stabilizer states and quantum designs [0.6374763930914523]
  We describe the symmetry group of the stabilizer polytope for any number $n$ of systems and any prime local dimension $d$. In the qubit case, the symmetry group coincides with the linear and anti-linear Clifford operations. We extend an observation of Heinrich and Gross and show that the symmetries of fairly general sets of Hermitian operators are constrained by certain moments.
  arXiv  Detail & Related papers  (2024-05-27T18:00:13Z)
• Symmetry-restricted quantum circuits are still well-behaved [45.89137831674385]
  We show that quantum circuits restricted by a symmetry inherit the properties of the whole special unitary group $SU(2n)$. It extends prior work on symmetric states to the operators and shows that the operator space follows the same structure as the state space.
  arXiv  Detail & Related papers  (2024-02-26T06:23:39Z)
• Intrinsic Bayesian Cramér-Rao Bound with an Application to Covariance Matrix Estimation [49.67011673289242]
  This paper presents a new performance bound for estimation problems where the parameter to estimate lies in a smooth manifold. It induces a geometry for the parameter manifold, as well as an intrinsic notion of the estimation error measure.
  arXiv  Detail & Related papers  (2023-11-08T15:17:13Z)
• Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles [55.41644538483948]
  We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the transformations symmetry and the corresponding generators. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.
  arXiv  Detail & Related papers  (2023-01-13T16:25:25Z)
• Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry [40.23168342389821]
  We propose a novel generalization of the Bures-Wasserstein geometry, which we call the GBW geometry. We provide a rigorous treatment to study various differential geometric notions on the proposed novel generalized geometry. We also present experiments that illustrate the efficacy of the proposed GBW geometry over the BW geometry.
  arXiv  Detail & Related papers  (2021-10-20T10:03:06Z)
• q-Paths: Generalizing the Geometric Annealing Path using Power Means [51.73925445218366]
  We introduce $q$-paths, a family of paths which includes the geometric and arithmetic mixtures as special cases. We show that small deviations away from the geometric path yield empirical gains for Bayesian inference.
  arXiv  Detail & Related papers  (2021-07-01T21:09:06Z)
• Degeneracy and hidden symmetry -- an asymmetric quantum Rabi model with an integer bias [0.0]
  We investigate the hidden symmetry of the asymmetric quantum Rabi model (AQRM) with a half-integral bias (ibQRM$_ell$) The existence of such symmetry has been widely believed to cause the degeneration of the spectrum, that is, the crossings on the energy curves. In this paper we propose a conjectural relation between the symmetry and degeneracy for the ibQRM$_ell$ given explicitly.
  arXiv  Detail & Related papers  (2021-06-16T16:17:11Z)
• Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels [0.0]
  We show how to compute marginals of the probability density functions on a random matrix type of symmetric spaces. We also show how the probability density functions are a particular case of diffusion kernels of the Karlin-McGregor type, describing non-intersecting processes in the Weyl chamber of Lie groups.
  arXiv  Detail & Related papers  (2020-11-27T11:41:29Z)
• Hilbert-space geometry of random-matrix eigenstates [55.41644538483948]
  We discuss the Hilbert-space geometry of eigenstates of parameter-dependent random-matrix ensembles. Our results give the exact joint distribution function of the Fubini-Study metric and the Berry curvature. We compare our results to numerical simulations of random-matrix ensembles as well as electrons in a random magnetic field.
  arXiv  Detail & Related papers  (2020-11-06T19:00:07Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.