Representing images and videos with Symmetric Positive Definite (SPD) matrices and considering the Riemannian geometry of the resulting space has proven beneficial for many recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices –especially of high-dimensional ones– comes at a high cost that limits the applicability of existing techniques. In this paper we introduce an approach that lets us handle high-dimensional SPD matrices by constructing a lower-dimensional, more discriminative SPD manifold. To this end, we model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. In particular, we search for a projection that yields a low-dimensional manifold with maximum discriminative power encoded via an affinity-weighted similarity measure based on metrics on the manifold. Learning can then be expressed as an optimization problem on a Grassmann manifold. Our evaluation on several classification tasks shows that our approach leads to a significant accuracy gain over state-of-the-art methods.