Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields including computer vision, medical imaging, and machine learning. The dominant geometric paradigm for such applications has consisted of a few Riemannian geometries associated with spectral computations that are costly at high scale and in high dimensions. We present a route to a scalable geometric framework for the analysis and processing of SPD-valued data based on the efficient computation of extreme generalized eigenvalues through the Hilbert and Thompson geometries of the semidefinite cone. We explore a particular geodesic space structure based on Thompson geometry in detail and establish several properties associated with this structure. Furthermore, we define a novel iterative mean of SPD matrices based on this geometry and prove its existence and uniqueness for a given finite collection of points. Finally, we state and prove a number of desirable properties that are satisfied by this mean.

翻译：以对称正定矩阵形式进行数据分析和处理的微分几何方法已在计算机视觉、医学影像和机器学习等多个领域取得显著成功。此类应用的主流几何范式包括若干黎曼几何方法，这些方法需进行高计算代价的谱分解，在规模较大或维度较高时尤为昂贵。本文基于半定锥的希尔伯特几何与汤普森几何，通过高效计算极端广义特征值，提出了一条可扩展的对称正定矩阵数据分析与处理几何框架。我们详细探讨了基于汤普森几何的一种特定测地线空间结构，并建立了与该结构相关的若干性质。此外，基于该几何结构定义了一种新的对称正定矩阵迭代均值，并证明了其在给定有限点集下的存在性与唯一性。最后，我们阐述并证明了该均值满足的一系列理想性质。