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（SPD）矩阵形式进行数据分析与处理的微分几何方法，已在计算机视觉、医学成像和机器学习等多个领域取得了显著的成功应用。此类应用中的主流几何范式主要依赖于少数几种黎曼几何，这些几何与谱计算相关，而在高规模和高维度下计算代价高昂。本文提出了一条基于半定锥的Hilbert与Thompson几何，通过高效计算极端广义特征值来实现SPD值数据可扩展几何分析框架的路径。我们详细研究了基于Thompson几何的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，我们基于该几何定义了一种新的SPD矩阵迭代均值，并证明了其对于给定有限点集的存在性与唯一性。最后，我们阐述并证明了该均值满足的若干理想性质。