[2603.02483] Geometric structures and deviations on James' symmetric positive-definite matrix bicone domain

Statistics > Machine Learning arXiv:2603.02483 (stat) [Submitted on 3 Mar 2026] Title:Geometric structures and deviations on James' symmetric positive-definite matrix bicone domain Authors:Jacek Karwowski, Frank Nielsen View a PDF of the paper titled Geometric structures and deviations on James' symmetric positive-definite matrix bicone domain, by Jacek Karwowski and Frank Nielsen View PDF HTML (experimental) 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 diagona...