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.

• The paper introduces a closed-form Hilbert metric for the symmetric positive-definite bicone in the extended Gaussian family using four extreme eigenvalues.
• The derived metric demonstrates invariance under identity-complement and orthonormal conjugation, elucidating the unique isometry group of the non-symmetric cone.
• The study highlights the metric’s computational efficiency over traditional affine-invariant methods, facilitating advanced geometric computations in high-dimensional models.

Hilbert Geometry of the Symmetric Positive-Definite Bicone and Its Application to Extended Gaussian Families

Introduction and Motivation

The paper investigates the Hilbert geometry of the symmetric positive-definite bicone, which arises as the parameter space of the extended Gaussian family. The extended Gaussian family is constructed by closing the set of zero-centered Gaussian distributions under both covariance and precision degeneracies, resulting in a parameter space that is the intersection of two positive semi-definite (PSD) cones joined at their bases—a bicone. This geometric structure is central to understanding the behavior of Gaussian distributions with degenerate covariance or precision matrices, which are relevant in open stochastic systems and in the context of information geometry.

Figure 1: The parameter space of the extended Gaussian family forms a closed double cone.

The bicone structure generalizes the familiar symmetric positive-definite (SPD) cone, and in two dimensions, it can be visualized as a double Lorentz cone in $\mathbb{R}^3$. This geometric perspective enables the study of metric properties, invariance, and isometries within the extended Gaussian family, with implications for both theoretical analysis and practical algorithms.

Figure 2: The parameter space of (2) can be viewed as a double Lorentz cone in $\mathbb{R}^3$: Three views of the double Lorentz cone.

Hilbert and Birkhoff Geometries in the Bicone Domain

Hilbert geometry is defined on open bounded convex domains and provides a projective metric where straight lines are geodesics. The Hilbert metric $d_H$ is constructed using cross-ratios of boundary intersections along lines, and is independent of the choice of norm. Birkhoff projective geometry, defined on open regular cones, introduces a projective distance $d_B$ that is invariant under positive scaling and is closely related to the Hilbert metric when the domain is a cone slice.

For the SPD cone, the Birkhoff distance between matrices $P$ and $Q$ is given by

$$d_B(P, Q) = \log \frac{\lambda_{\max}(PQ^{-1})}{\lambda_{\min}(PQ^{-1})}$$

where only the extreme eigenvalues are required. In contrast, the affine-invariant Riemannian metric (AIRM) requires the full spectrum.

The variance-precision manifold (VPM), defined as

$VPM(n) := \{X \in Sym(n) : 0 \prec X \prec I\}$

is shown to be open, convex, and bounded. Its closure corresponds to matrices with eigenvalues in $[0,1]$, and it is invariant under both $X \mapsto I - X$ and orthonormal conjugation $X \mapsto U^\top X U$.

Closed-Form Hilbert Metric in the VPM

A central contribution is the derivation of a closed-form formula for the Hilbert metric in the VPM domain. For $A, B \in VPM(n)$,

$$d_H(A, B) = \log \frac{\max(\lambda_{\max}, \mu_{\max})}{\min(\lambda_{\min}, \mu_{\min})}$$

where

• $\lambda_{\min}, \lambda_{\max}$ are the minimal and maximal eigenvalues of $B^{-1}A$,
• $\mu_{\min}, \mu_{\max}$ are the minimal and maximal eigenvalues of $(I - B)^{-1}(I - A)$.

This formula leverages the Birkhoff characterization and the Loewner order, and only requires computation of four extreme eigenvalues, making it computationally efficient compared to AIRM.

Invariance and Isometry Characterization

The Hilbert metric in VPM is invariant under two transformations:

1. Identity-complement: $X \mapsto I - X$,
2. Orthonormal conjugation: $X \mapsto U^\top X U$ for $U \in O(n)$.

These are proven to be the only isometries of the VPM for $n \geq 2$, via projective geometry and cone automorphism arguments. The group of isometries is generated by these two operations, and the VPM cone is shown to be non-symmetric, distinguishing its isometry group from that of symmetric cones.

Figure 3: The parameter space of $(2)$ (viewed as a Lorentz bicone in $\mathbb{R}^3$): Three views of the boundary of the Lorentz bicone with $VPM_\epsilon$ (in blue) encapsulating $VPM$ (in black).

Comparison with Affine-Invariant Riemannian Metric

A detailed comparison between the Hilbert VPM distance and the AIRM is provided:

Feature	AIRM Distance	Hilbert VPM Distance
Eigenvalues used	All eigenvalues of $Q_1 Q_2^{-1}$	Extreme eigenvalues of $Q_1 Q_2^{-1}$ and $Mob(Q_1, Q_2)$
Invariance under inversion	Yes	No (replaced by identity-complement invariance)
Invariance under $GL(n)$	Yes	No
Invariance under $O(n)$	Yes	Yes

The Hilbert VPM metric is less restrictive in terms of invariance, but more efficient computationally, and is particularly suited for applications where only extreme eigenvalues are accessible or relevant.

Computational Geometry Applications

The Hilbert geometry of the VPM enables efficient implementation of computational geometry primitives, such as the smallest enclosing ball (SEB) problem. The SEB can be approximated using iterative geodesic-cut algorithms, leveraging the straight-line geodesics of Hilbert geometry.

Figure 4: Fine approximation of the smallest enclosing ball (SEB, in red) with respect to the Hilbert VPM distance in 2D of a set of 2D SPD matrices (or equivalently, 2D zero-centered Gaussians, in black).

The ability to enlarge the VPM domain to $VPM_\epsilon$ allows for bounded Hilbert distances even for degenerate matrices, which is advantageous in scenarios involving mixtures of degenerate covariance and precision matrices.

Implications and Future Directions

The geometric characterization of the extended Gaussian family via the VPM bicone and its Hilbert geometry has both theoretical and practical implications:

• Theoretical: Provides a rigorous framework for understanding degeneracies in Gaussian families, duality between covariance and precision, and the structure of isometries in non-symmetric cones.
• Practical: Enables efficient computation of distances and geometric primitives in high-dimensional statistical models, with potential applications in machine learning, signal processing, and diffusion tensor imaging.

Future work may include:

• Extension to full Gaussian families via Calvo-Oller embedding,
• Study of correlation matrix subdomains (elliptopes) within the VPM,
• Facial characterization of the VPM and its boundary structure,
• Algorithmic development for clustering, classification, and optimization in the extended Gaussian setting.

Conclusion

The paper provides a comprehensive study of the Hilbert geometry of the symmetric positive-definite bicone, establishing closed-form metrics, invariance properties, and isometry characterizations. These results deepen the understanding of extended Gaussian families and open avenues for efficient geometric computations in statistical and machine learning applications. The interplay between projective geometry, matrix analysis, and information geometry is leveraged to yield both theoretical insights and practical tools for handling degenerate Gaussian models.