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

2026-03-02

AI Summary

This work addresses the challenge of designing metric structures on the manifold of symmetric positive-definite (SPD) matrices that simultaneously preserve geometric naturalness and computational efficiency. Building upon James’ SPD bicone parametrization, the paper introduces—for the first time—a Finsler structure together with a dual Hessian information-geometric framework. The resulting geometric construction yields geodesics that appear as straight lines in an appropriate coordinate system and naturally extends the Hilbert simplex distance to the spectral simplex. Theoretical analysis establishes inequality relations between the proposed metric and classical SPD metrics, thereby broadening the applicability of information geometry and Finsler geometry on SPD manifolds. This framework offers a theoretically rigorous yet practically viable tool for geometric deep learning and modeling of SPD-valued data.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Symmetric positive-definite (SPD) matrix datasets play a central role across numerous scientific disciplines, including signal processing, statistics, finance, computer vision, information theory, and machine learning among others. The set of SPD matrices forms a cone which can be viewed as a global coordinate chart of the underlying SPD manifold. Rich differential-geometric structures may be defined on the SPD cone manifold. Among the most widely used geometric frameworks on this manifold are the affine-invariant Riemannian structure and the dual information-geometric log-determinant barrier structure, each associated with dissimilarity measures (distance and divergence, respectively). In this work, we introduce two new structures, a Finslerian structure and a dual information-geometric structure, both derived from James' bicone reparameterization of the SPD domain. Those structures ensure that geodesics correspond to straight lines in appropriate coordinate systems. The closed bicone domain includes the spectraplex (the set of positive semi-definite diagonal matrices with unit trace) as an affine subspace, and the Hilbert VPM distance is proven to generalize the Hilbert simplex distance which found many applications in machine learning. Finally, we discuss several applications of these Finsler/dual Hessian structures and provide various inequalities between the new and traditional dissimilarities.

Problem

• symmetric positive-definite matrices
• geometric structures
• Finsler geometry
• information geometry
• James' bicone

Innovation

• Finsler structure
• dual Hessian geometry
• James' bicone
• Hilbert VPM distance
• symmetric positive-definite manifold