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

Jacek Karwowski, Frank Nielsen

TL;DR

This work introduces two new structures, a Finslerian structure and a dual information-geometric structure, both derived from James'bicone reparameterization of the SPD domain, and discusses several applications of these Finsler/dual Hessian structures.

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.

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

TL;DR

This work introduces two new structures, a Finslerian structure and a dual information-geometric structure, both derived from James'bicone reparameterization of the SPD domain, and discusses several applications of these Finsler/dual Hessian structures.

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.
Paper Structure (49 sections, 39 theorems, 162 equations, 4 figures, 2 tables)

This paper contains 49 sections, 39 theorems, 162 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Riemannian geometry: Affine-invariant Riemannian metric
  3. Dual information geometry: Hessian metric
  4. James' bicone
  5. Proof:
  6. Proof:
  7. Contributions and paper outline
  8. Hilbert VPM distance and Finsler structure
  9. Preliminaries
  10. Proof:
  11. Proof:
  12. Hilbert distance as matrix spread
  13. Proof:
  14. Proof:
  15. Finsler norm and Finsler metric
  16. ...and 34 more sections

Key Result

Proposition 1

The eigenvalues $\lambda_i(V(X))\in(0,1)$ and $\lambda_i(P(X))\in(0,1)$, i.e., both $V(X)\in\operatorname{VPM}^\circ(n)$ and $P(X)\in\operatorname{VPM}^\circ(n)$.

Figures (4)

  • Figure 1: Screenshots of James' 3D bicone model with corresponding bivariance centered Gaussians.
  • Figure 2: $\operatorname{VPM}^\circ(2)$ can be visualized as a 3D Lorentz bicone, here shown on the left in a slanted view for better perception, and with the open pregeodesic joining $0$ to $I$, i.e. $\{0 \prec \alpha I\prec I \ :\ \alpha\in(0,1)\}$.
  • Figure 3: Both the dual SPD cones $\Theta=\operatorname{PD}(n)$ and $H=\mathrm{ND}(n)=-\operatorname{PD}(n)$ via the logdet potential function, and the bicone $\Xi=\operatorname{VPM}^\circ(n)$ and its dual $\operatorname{Sym}(n)$ are global coordinate charts of the SPD cone manifold $\mathcal{C}$.
  • Figure 4: AIRM (blue) and Hilbert-Finsler (red) geodesic midpoints of two SPD matrices (black).

Theorems & Definitions (56)

  • Proposition 1
  • Definition 1: James' map
  • Proposition 2: Differential of $\iota$
  • Lemma 1: Inverse formulas for $\iota$
  • Definition 2: Matrix norms
  • Lemma 2: Frobenius-operator norm inequality
  • Lemma 3: Range-$l_2$ inequality
  • Theorem 1: karwowski2025hilbertgeometrysymmetricpositivedefinite
  • Corollary 1: Lower bound
  • Example 1
  • ...and 46 more