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.

翻译：对称正定（SPD）矩阵数据集在众多科学领域中占据核心地位，包括信号处理、统计学、金融学、计算机视觉、信息论以及机器学习等。SPD矩阵的集合构成一个锥体，可视为底层SPD流形的全局坐标图。在该SPD锥流形上可定义丰富的微分几何结构。其中最广泛使用的几何框架包括仿射不变黎曼结构和对偶信息几何对数行列式障碍结构，它们分别对应不同的相异性度量（分别为距离与散度）。本文基于James对SPD域的双锥重参数化方法，引入两种新结构：芬斯勒结构和对偶信息几何结构。这些结构确保测地线在适当的坐标系中对应为直线。封闭的双锥域包含谱单形（具有单位迹的半正定对角矩阵集合）作为其仿射子空间，并证明希尔伯特VPM距离可推广已在机器学习中广泛应用的希尔伯特单形距离。最后，我们探讨了这些芬斯勒/对偶海森结构的多项应用，并给出了新旧相异性度量之间的若干不等式关系。