2026-09-30
Oxford and Sony CSL give a closed-form Hilbert metric on the extended-Gaussian VPM bicone using four extreme eigenvalues; for n≥2, isometries are orthogonal conjugations and X↦I−X.
Zero-mean Gaussians can be parameterized by a covariance Σ or by a precision P=Σ^{-1}. The usual parameter space is the symmetric positive-definite cone. Degenerate covariances live on lower-dimensional affine subspaces. Degenerate precisions show up as non-determinism in open stochastic systems, and as improper uniform priors in Bayesian work. James (1973) compressed eigenvalues from [0,∞] into [0,1] via L=Σ(I+Σ)^{-1}, producing the variance-precision model: a closed bicone, the intersection of two partial PSD cones. The interior is VPM(n)={X∈Sym(n): 0≺X≺I}.
The default metric on SPD is AIRM, which needs a full eigendecomposition. Hilbert geometry lives on any open bounded convex set, with straight lines as geodesics. Birkhoff's projective distance on a cone only looks at extreme eigenvalues. This paper equips the interior of the VPM with Hilbert geometry, writes a closed-form distance, and classifies the isometry group.
VPM(n) is the set of n×n symmetric matrices whose eigenvalues all sit in (0,1). It is the intersection of PD and I−PD, hence open, convex, and bounded: A ≼ I implies the Frobenius norm is at most √n.
Hilbert distance is the log of a cross-ratio. Draw the line through two points, hit the boundary at x' and y', then take log(||x−y'|| ||x'−y|| / ||x'−x|| ||y−y'||). The authors skip computing the intersection points. They map A to the affine slice (A, I−A), build the cone over that slice, and use Birkhoff distance. Loewner order supplies the stretch-ratio bounds. The closed form is
dH(A,B)=log[ max(λmax(B^{-1}A), λmax((I−B)^{-1}(I−A))) / min(λmin(B^{-1}A), λmin((I−B)^{-1}(I−A))) ]
Read it as: look at the stretch ratio on the covariance side, B^{-1}A, and on the complementary (precision) side. Take the largest and smallest eigenvalues on each side, keep the most extreme pair among those four numbers, then take a log ratio. The precision-side term is the matrix Möbius map (I−A)^{-1}(I−B).
Two maps preserve the distance. X↦I−X swaps covariance and precision. Orthogonal conjugation X↦U^T X U leaves the spectrum alone. For n≥2 these generate the whole isometry group. The cone over VPM is not a symmetric cone: its dual is cut out by the trace of negative eigenvalues, so the shapes do not match. Walsh's theorem then says isometries equal collineations. Collineations lift to linear cone automorphisms. Linear automorphisms of the SPD cone are GL(n) conjugations; the argument then forces GL down to O(n), after possibly precomposing with I−X.
The closed form is the main theorem. Table 1 contrasts it with AIRM:
| Property | AIRM | Hilbert VPM |
| Eigenvalues needed | all n of Q1 Q2^{-1} | one extreme pair of Q1 Q2^{-1} and one of the Möbius map, four in total |
| Inversion-type invariance | X↦X^{-1} | X↦I−X |
| Congruence invariance | all of GL(n) | O(n) only |
No downstream task numbers. The discussion gives three operational extensions. Enlarge the domain to VPMε={−εI ⪯ X ⪯ (1+ε)I}: distances to degenerate Gaussians become finite, and grow smaller as ε grows, by monotonicity of Hilbert distance on nested domains. Non-centered Gaussians go through the Calvo–Oller embedding, writing (μ,Σ) as an (n+1)-block matrix Σμ^+ and landing in VPM(n+1). Straight Hilbert geodesics let the Bădoiu–Clarkson geodesic-cut iteration run as-is; the authors show a smallest-enclosing-ball demo on VPM(2), Figure 2(b).
For work on SPD manifolds, Gaussian mixtures, or diffusion tensor imaging, this is a cheaper distance than AIRM: four extreme eigenvalues instead of a full spectrum. Straight-line geodesics make Voronoi, Delaunay, and enclosing-ball primitives available off the shelf. Degenerate covariance and degenerate precision sit in one bounded convex body, so extended Gaussians get a single metric space instead of ad-hoc singular handling on the cone boundary.
This is geometry, not a new model. There is no classification or generation experiment. Against AIRM, the saving is a full eigendecomposition; the cost is weaker invariance, orthogonal conjugations only, not general invertible linear maps. The workshop paper is 19 pages with proofs inline. Extra material, code, and figures sit at arXiv:2508.14369.
On the open VPM, Hilbert distance to the boundary is infinite, so truly degenerate Gaussians remain unreachable until VPMε is developed in full. The paper treats centered Gaussians only; the non-centered case stops at an embedding formula. Isometry classification starts at n≥2; dimension one collapses to the known Hilbert distance on (0,1), log((1−x)y/(x(1−y))). DTI and enclosing balls are pointers, with no error table against log-Euclidean or AIRM. Birkhoff distance on the SPD cone has a contraction property; this paper does not check whether that contraction survives on the bicone. The application layer is still a sketch.