  Representing images and videos with Symmetric Positive Definite (SPD)
matrices, and considering the Riemannian geometry of the resulting space, has
been shown to yield high discriminative power in many visual 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 algorithms
able to handle high-dimensional SPD matrices by constructing a
lower-dimensional SPD manifold. To this end, we propose to model the mapping
from the high-dimensional SPD manifold to the low-dimensional one with an
orthonormal projection. This lets us formulate dimensionality reduction as the
problem of finding a projection that yields a low-dimensional manifold either
with maximum discriminative power in the supervised scenario, or with maximum
variance of the data in the unsupervised one. We show that learning can be
expressed as an optimization problem on a Grassmann manifold and discuss fast
solutions for special cases. Our evaluation on several classification tasks
evidences that our approach leads to a significant accuracy gain over
state-of-the-art methods.
