  Data encoded as symmetric positive definite (SPD) matrices frequently arise
in many areas of computer vision and machine learning. While these matrices
form an open subset of the Euclidean space of symmetric matrices, viewing them
through the lens of non-Euclidean Riemannian geometry often turns out to be
better suited in capturing several desirable data properties. However,
formulating classical machine learning algorithms within such a geometry is
often non-trivial and computationally expensive. Inspired by the great success
of dictionary learning and sparse coding for vector-valued data, our goal in
this paper is to represent data in the form of SPD matrices as sparse conic
combinations of SPD atoms from a learned dictionary via a Riemannian geometric
approach. To that end, we formulate a novel Riemannian optimization objective
for dictionary learning and sparse coding in which the representation loss is
characterized via the affine invariant Riemannian metric. We also present a
computationally simple algorithm for optimizing our model. Experiments on
several computer vision datasets demonstrate superior classification and
retrieval performance using our approach when compared to sparse coding via
alternative non-Riemannian formulations.
