  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.
