Symmetric Positive Semi-definite Riemannian Geometry with Application to Domain Adaptation

In this paper, we present new results on the Riemannian geometry of symmetric positive semi-definite (SPSD) matrices. First, based on an existing approximation of the geodesic path, we introduce approximations of the logarithmic and exponential maps. Second, we present a closed-form expression for Parallel Transport (PT). Third, we derive a canonical representation for a set of SPSD matrices. Based on these results, we propose an algorithm for Domain Adaptation (DA) and demonstrate its performance in two applications: fusion of hyper-spectral images and motion identification.