Geometric structures and deviations on James' symmetric positive-definite matrix bicone domain

Geometric Structures in James' Symmetric Positive-Definite Matrix Bicone Domain

Introduction

This paper develops novel geometric structures within the domain of symmetric positive-definite (SPD) matrices, focusing on the James' bicone parameterization of the SPD cone. SPD matrices underpin diverse applications across statistics, signal processing, finance, medical imaging, information theory, quantum theory, and machine learning. Traditionally, the SPD cone is endowed with the affine-invariant Riemannian metric (AIRM) and information-geometric log-determinant (logdet) divergence, both offering tractable dissimilarities and invariance properties.

The paper introduces two new geometric structures:

• A Finslerian structure derived from the Hilbert geometry of James' VPM (variance-precision model) bicone domain.
• A dual Hessian information-geometric structure based on the bilogdet (bilogarithmic determinant) barrier function.

James' bicone parameterization yields a domain, $VPM(n)$, which is bounded by extremal eigenvalues strictly in $(0,1)$, facilitating geodesics via straight lines in suitable coordinate charts. The closure of the bicone includes the spectraplex—an affine subspace corresponding to positive semi-definite diagonal matrices with unit trace—and thus generalizes the probability simplex geometry.

Figure 1: Screenshots of James' 3D bicone model with corresponding bivariance centered Gaussians.

Metric and Divergence Structures in SPD Geometry

Classical SPD Metrics

The affine-invariant Riemannian metric on the SPD cone $PD(n)$ enables closed-form geodesics and distances, crucial for statistical inference and geometric learning. Its invariance under congruence and inversion aligns with covariance-based representations of Gaussian distributions. The dual information geometry rooted in the logdet function provides a Bregman divergence structure, with dually flat charts (coordinates: $X$ and $-X^{-1}$) grounded in Legendre duality.

James' Bicone Parameterization and VPM Domain

James' map realizes a diffeomorphism between $PD(n)$ and the VPM bicone domain:

$VPM(n) = \{X \in PD(n)\, |\, 0 \prec X \prec I \}$

with mappings $X \mapsto X(I+X)^{-1}$ and $X \mapsto (I+X)^{-1}$ respectively for covariance and precision matrix normalization. Consequently, all eigenvalues $\lambda_i$ are constrained to $(0,1)$, facilitating bounded geometry and normalization.

Figure 2: VPM(2) visualized as a 3D Lorentz bicone with open pregeodesic joining 0 to I.

This domain is highly relevant in quantum information theory (effect algebras for POVMs), robust control (Riccati equations), and extended Gaussian modeling (singular covariance/precision). The spectraplex arises as an affine slice of the closure, generalizing the probability simplex.

Figure 3: Dual SPD cones and their global coordinate charts via logdet and bilogdet potential functions.

Hilbert Geometry, Finsler Structure, and Bilogdet Divergence

Hilbert VPM Distance and Finsler Geometry

The Hilbert projective distance, a classical metric on bounded convex domains, is generalized to the VPM domain. For $J_1, J_2 \in VPM(n)$, the Hilbert distance is computed as:

$$d_H(J_1, J_2) = \log \frac{\max\left(\lambda_{max}(J_2^{-1}J_1),\, \lambda_{max}((I - J_2)^{-1}(I - J_1))\right)}{\min\left(\lambda_{min}(J_2^{-1}J_1),\, \lambda_{min}((I - J_2)^{-1}(I - J_1))\right)}$$

with eigenvalue spreads yielding a matrix-norm representation.

This metric induces a Finsler structure, where geodesics are straight-line segments in the bicone parameterization—a crucial property for computational algorithms and robust geometric learning. The Hilbert VPM distance generalizes the Hilbert simplex distance, with the spectraplex (embedded standard simplex) forming a totally geodesic submanifold.

Figure 4: AIRM (blue) and Hilbert-Finsler (red) geodesic midpoints of two SPD matrices (black).

Dually Flat Structure via Bilogdet Function

The bilogdet barrier function,

$\Psi_{bild}(X) = -\log\det X - \log\det(I - X)$

is strictly convex and self-concordant on the VPM domain, yielding a Hessian Riemannian metric. Its gradient and Hessian supply dual coordinates and divergence measures, enabling an information-geometric structure adapted to the bicone boundary (potential diverging to infinity at the boundary).

The induced bilogdet Bregman divergence,

$$B_{\Psi_{bild}}(J_1:J_2) = B_{\Psi_{ld}}(J_1:J_2) + B_{\Psi_{ld}}(I-J_1:I-J_2)$$

combines logdet divergences on both the matrix and its complement—supporting optimization, control, and quantum theory applications.

Comparative Analysis: AIRM/Logdet vs. Hilbert/Bilogdet Dissimilarities

The manuscript rigorously establishes inequalities between standard (AIRM/logdet) and bicone-derived (Hilbert/bilogdet) dissimilarities:

• Tight lower bounds showing that Hilbert VPM distance dominates the restricted AIRM distance by a factor $1/\sqrt{n}$.
• Tight upper bounds for Hilbert VPM distance via the pushed-forward AIRM metric ($\sqrt{2}$ factor).
• Analogous bounds in the tangent space for the bilogdet Finsler norm, $||V||_X^{H}$, and the bilogdet Riemannian norm, $||V||_X^\Psi$.

These relations are crucial for understanding the fundamental differences and potential tradeoffs between worst-direction distortion metrics (Hilbert) and information-geometric/averaged metrics (AIRM/logdet).

Implications and Future Directions

James' bicone VPM domain supplies novel geometric foundations for analyzing SPD matrices, particularly in settings where normalization and boundary-adapted structures are essential. The Hilbert VPM metric, relying solely on extremal eigenvalues, is inherently suited for robust analysis, maximum distortion modeling, and applications in quantum information and control theory. Its invariance and divergence behavior at the boundary render it promising for barrier-based optimization and learning.

The bilogdet structure provides an information-geometric framework, extending classical divergences and fostering applications in convex optimization, self-concordant barrier methods, and manifold learning.

The explicit bounds and geodesic constructions open up new computational strategies for geometric algorithms, medians/measures in Finsler geometry, and direct connections with the geometry of effects in quantum theory.

Speculatively, further developments may include:

• Efficient algorithms for geometric clustering, mean/median computation, and robust parameter estimation using Hilbert geometry.
• Expanded manifold learning for SPD domains using the VPM structure.
• Advanced optimization protocols exploiting self-concordant bilogdet barriers.
• Applications to quantum computing, quantum granular computing, and optimal transport on the spectraplex.

Conclusion

This paper introduces and analyzes new geometric structures within the James' bicone domain of SPD matrices, establishing Finslerian and dually flat information-geometric frameworks and relating them to classical Riemannian/AIRM geometries. Strong analytical bounds, explicit geodesic formulations, and the identification of practical implications across quantum information theory, control, and optimization highlight the theoretical and computational potential of the VPM-based approach.