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

arXiv: https://arxiv.org/abs/2603.02483 · HTML: https://mlsanity.com/paper/2603.02483/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 45 references that are in the map.
- First posted on arXiv 3 Mar 2026.

## Builds on (most cited first, 10 of 10)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matrices](https://mlsanity.com/paper/1507.02772/) (2015)
- [From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices](https://mlsanity.com/paper/1407.1120/) (2014)
- [Parallel Transport on the Cone Manifold of SPD Matrices for Domain Adaptation](https://mlsanity.com/paper/1807.10479/) (2018)
- [Minimal Expected Regret in Linear Quadratic Control](https://mlsanity.com/paper/2109.14429/) (2021)
- [Non-linear Embeddings in Hilbert Simplex Geometry](https://mlsanity.com/paper/2203.11434/) (2022)
- [Differential geometry with extreme eigenvalues in the positive semidefinite cone](https://mlsanity.com/paper/2304.07347/) (2023)
- [A Category for unifying Gaussian Probability and Nondeterminism](https://mlsanity.com/paper/2204.14024/) (2022)
- [Hilbert geometry of the symmetric positive-definite bicone: Application to the geometry of the extended Gaussian family](https://mlsanity.com/paper/2508.14369/) (2025)
- [Foundations of Quantum Granular Computing with Effect-Based Granules, Algebraic Properties and Reference Architectures](https://mlsanity.com/paper/2511.22679/) (2025)

## Abstract (arXiv)

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.