  It is proven that encoding images and videos through Symmetric Positive
Definite (SPD) matrices, and considering the Riemannian geometry of the
resulting space, can lead to increased classification performance. Taking into
account manifold geometry is typically done via embedding the manifolds in
tangent spaces, or Reproducing Kernel Hilbert Spaces (RKHS). Recently, it was
shown that embedding such manifolds into a Random Projection Spaces (RPS),
rather than RKHS or tangent space, leads to higher classification and
clustering performance. However, based on structure and dimensionality of the
randomly generated hyperplanes, the classification performance over RPS may
vary significantly. In addition, fine-tuning RPS is data expensive (as it
requires validation-data), time consuming, and resource demanding. In this
paper, we introduce an approach to learn an optimized kernel-based projection
(with fixed dimensionality), by employing the concept of subspace clustering.
As such, we encode the association of data points to the underlying subspace of
each point, to generate meaningful hyperplanes. Further, we adopt the concept
of dictionary learning and sparse coding, and discriminative analysis, for the
optimized kernel-based projection space (OPS) on SPD manifolds. We validate our
algorithm on several classification tasks. The experiment results also
demonstrate that the proposed method outperforms state-of-the-art methods on
such manifolds.
