Hilbert VPM Distance on SPD Matrices

• Hilbert VPM Distance is a projective metric defined on the variance–precision manifold that quantifies dissimilarities between SPD matrices using extremal eigenvalues.
• It leverages a closed-form spectral formula for rapid evaluation and numerical stability, even near degenerate boundary cases.
• The metric’s projective invariance and geodesic structure underpin advanced applications in Gaussian modeling and high-dimensional clustering.

The Hilbert VPM distance is a projective metric defined on the open symmetric positive-definite bicone, also known as the variance–precision manifold (VPM). It provides a closed-form, projectively invariant, and computationally efficient dissimilarity for symmetric positive-definite (SPD) matrices, generalizing the classical Hilbert projective metric from cones and the simplex to matrix-valued settings. Its structure underpins both advanced geometric data analysis of extended Gaussian models and high-dimensional clustering of matrix-valued data (Karwowski et al., 20 Aug 2025, Karwowski et al., 3 Mar 2026).

1. Mathematical Foundation and Domain

The Hilbert VPM distance is developed in the context of convex geometry and projective metrics. For a generic open bounded convex set $C \subset \mathbb{R}^d$, the Hilbert distance between points $x, y \in C$ is defined as the logarithm of a cross-ratio determined by the boundary intersections of the line through $x$ and $y$. Specializing to matrix spaces, the focus is on

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

where $\operatorname{Sym}(n)$ denotes real symmetric $n \times n$ matrices, and $0 \prec X \prec I$ requires all eigenvalues of $X$ lie in $(0, 1)$. This set arises naturally as the parameter space for the extended Gaussian family, encompassing both covariance and precision degeneracies (Karwowski et al., 20 Aug 2025, Karwowski et al., 3 Mar 2026).

James' bicone reparametrization provides the diffeomorphic map

$x, y \in C$0

with $x, y \in C$1 the SPD cone, so any element of the VPM can be interpreted as a normalized blend of covariance and precision structures.

2. Closed-Form Expression

The core result is an explicit spectral formula for the Hilbert VPM distance $x, y \in C$2 between $x, y \in C$3. Define

$x, y \in C$4

$x, y \in C$5

Then,

$x, y \in C$6

Alternatively,

$x, y \in C$7

with

$x, y \in C$8

$x, y \in C$9

This provides rapid evaluation and numerical stability, even near the boundary of the bicone (Karwowski et al., 20 Aug 2025, Karwowski et al., 3 Mar 2026).

3. Metric and Geometric Properties

The Hilbert VPM distance is a true metric, satisfying:

• Non-negativity: $x$0 and $x$1 iff $x$2.
• Symmetry: $x$3, as the spectral formula is invariant under exchange of $x$4 and $x$5.
• Triangle inequality: Inherited from the cross-ratio construction for convex domains; straight line geodesics in $x$6 are isometrically mapped to shortest paths after appropriate reparametrization (Karwowski et al., 3 Mar 2026).
• Projective invariance: The metric is invariant under positive congruence transformations and scaling.
• Geodesic Structure: Geodesics correspond to straight lines in the affine ambient space, but require non-uniform parameterization in the Finsler geometry given by the Hilbert metric.
• Finsler Norm: The infinitesimal norm at $x$7 in direction $x$8 is

$x$9

where $y$0 are exit times to the boundary along $y$1 and $y$2 (Karwowski et al., 3 Mar 2026), admitting closed-form expressions involving extremal eigenvalues.

4. Invariance and Isometries

Hilbert VPM distance possesses several notable isometries:

• Identity–complement invariance: $y$3 for all $y$4.
• O($y$5)-congruence invariance: For any $y$6, $y$7. For $y$8, these transformations generate the full isometry group on $y$9 (Karwowski et al., 20 Aug 2025).

5. Connections to Classical Metrics and Special Cases

The Hilbert VPM distance generalizes classical projective geometry and is related as follows:

• Simplex Limit: When restricted to diagonal matrices with unit trace ($$\operatorname{VPM}(n) = \{ X \in \operatorname{Sym}(n) : 0 \prec X \prec I \},$$0 with $$\operatorname{VPM}(n) = \{ X \in \operatorname{Sym}(n) : 0 \prec X \prec I \},$$1), the Hilbert VPM distance reduces precisely to the Hilbert projective metric on the simplex:

$$\operatorname{VPM}(n) = \{ X \in \operatorname{Sym}(n) : 0 \prec X \prec I \},$$2

• Scalar Case $$\operatorname{VPM}(n) = \{ X \in \operatorname{Sym}(n) : 0 \prec X \prec I \},$$3: On $$\operatorname{VPM}(n) = \{ X \in \operatorname{Sym}(n) : 0 \prec X \prec I \},$$4, $$\operatorname{VPM}(n) = \{ X \in \operatorname{Sym}(n) : 0 \prec X \prec I \},$$5, recovering the classical cross-ratio metric (Karwowski et al., 3 Mar 2026).
• Isotropic Case: $$\operatorname{VPM}(n) = \{ X \in \operatorname{Sym}(n) : 0 \prec X \prec I \},$$6, $$\operatorname{VPM}(n) = \{ X \in \operatorname{Sym}(n) : 0 \prec X \prec I \},$$7 with $$\operatorname{VPM}(n) = \{ X \in \operatorname{Sym}(n) : 0 \prec X \prec I \},$$8 yields

$$\operatorname{VPM}(n) = \{ X \in \operatorname{Sym}(n) : 0 \prec X \prec I \},$$9

• Relation to Fisher–Rao (AIRM): The Hilbert VPM distance is a projective-invariant analogue to the affine-invariant Riemannian metric (AIRM), but is defined and continuous even for degenerate limit cases where standard Riemannian geometry fails (Karwowski et al., 20 Aug 2025).

6. Computational Aspects

Evaluation of $\operatorname{Sym}(n)$0 requires only the four extremal eigenvalues of $\operatorname{Sym}(n)$1 and $\operatorname{Sym}(n)$2. Computational complexity is $\operatorname{Sym}(n)$3–$\operatorname{Sym}(n)$4 per pair, significantly outperforming metrics that require full spectral decompositions, especially in high dimensions. Power and Lanczos iterations provide efficient calculations of extremal eigenvalues. No iterative fitting is required; a single eigenproblem per matrix pair suffices. The metric is highly stable under degeneracy, as the quotient of extremal eigenvalues remains finite unless matrices approach disjoint boundary faces where $\operatorname{Sym}(n)$5 (Karwowski et al., 20 Aug 2025, Karwowski et al., 3 Mar 2026).

7. Applications and Implications

The Hilbert VPM distance is directly applicable to:

• Gaussian Distribution Analysis: As the parameter space of the extended Gaussian family, $\operatorname{Sym}(n)$6 encompasses degeneracies in both covariance and precision, and $\operatorname{Sym}(n)$7 provides a continuous, projective-invariant divergence between $\operatorname{Sym}(n)$8 and $\operatorname{Sym}(n)$9 (or their inverses), accommodating nearly singular cases (Karwowski et al., 20 Aug 2025).
• Clustering and Statistical Learning: Straight-line geodesics and efficient extremal computations facilitate $n \times n$0-means, $n \times n$1-centers, and minimum enclosing ball algorithms for SPD-valued data, notably outperforming log-det and Frobenius distance–based approaches in empirical clustering of correlations and covariances (Nielsen et al., 2017, Karwowski et al., 3 Mar 2026).
• Boundary-Aware Learning: The metric’s stability as eigenvalues approach 0 or 1 makes it suitable for modeling covariance/precision degeneracy, a regime where standard metrics become either ill-defined or lose sensitivity.
• Bridging Simplex Geometry: The Hilbert VPM metric subsumes and extends the simplex Hilbert geometry, enabling unified clustering and comparison frameworks for discrete (histogram) and continuous (SPD-matrix) data domains (Nielsen et al., 2017, Karwowski et al., 3 Mar 2026).

Additionally, tight Lipschitz-type inequalities relate the Hilbert VPM distance to AIRM and log-barrier Hessian distances on the bicone, providing comparative control over metric distortion in geometric statistics (Karwowski et al., 3 Mar 2026).

8. Summary Table: Key Hilbert VPM Distance Features

Property	Description	Reference
Domain	$n \times n$2, $n \times n$3	(Karwowski et al., 20 Aug 2025)
Formula	$n \times n$4	(Karwowski et al., 20 Aug 2025)
Projective invariance	Invariant under $n \times n$5, $n \times n$6	(Karwowski et al., 20 Aug 2025)
Computational cost	$n \times n$7–$n \times n$8 (extremal eigenvalues only)	(Karwowski et al., 20 Aug 2025)
Generalizes simplex Hilbert metric	True on diagonal matrices	(Karwowski et al., 3 Mar 2026)
Finsler structure	Infinitesimal norm via exit times, explicit in eigenvalues	(Karwowski et al., 3 Mar 2026)
Applications	Gaussian models, clustering SPD matrices, boundary sensitivity	(Karwowski et al., 20 Aug 2025)

The Hilbert VPM distance is thus a natural projective-Finsler metric for symmetric positive-definite bicones, furnishing a unifying, tractable, and theoretically rich tool for extended statistical modeling, convex-geometric analysis, and large-scale SPD data clustering (Karwowski et al., 20 Aug 2025, Nielsen et al., 2017, Karwowski et al., 3 Mar 2026).