Geometric statistics with subspace structure preservation for SPD matrices

Abstract: We present a geometric framework for the processing of SPD-valued data that preserves subspace structures and is based on the efficient computation of extreme generalized eigenvalues. This is achieved through the use of the Thompson geometry of the semidefinite cone. We explore a particular geodesic space structure in detail and establish several properties associated with it. Finally, we review a novel inductive mean of SPD matrices based on this geometry. Keywords: convex cones, differential geometry, geodesics, geometric statistics, positive definite matrices, matrix means, Thompson metric 1. INTRODUCTION means a significant increase in computational complexity, particularly for larger matrices.