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

TL;DR

This paper introduces new geometric structures on the SPD matrix domain derived from James' bicone reparameterization, enabling straight-line geodesics and broadening applications in machine learning and related fields.

Contribution

The work presents two novel geometric frameworks, a Finslerian and a dual information-geometric structure, on the SPD bicone domain, extending existing differential-geometric tools.

Findings

Hilbert VPM distance generalizes Hilbert simplex distance

New structures ensure geodesics are straight lines in specific coordinates

Applications include inequalities between new and traditional dissimilarities

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.