  Representations in the form of Symmetric Positive Definite (SPD) matrices
have been popularized in a variety of visual learning applications due to their
demonstrated ability to capture rich second-order statistics of visual data.
There exist several similarity measures for comparing SPD matrices with
documented benefits. However, selecting an appropriate measure for a given
problem remains a challenge and in most cases, is the result of a
trial-and-error process. In this paper, we propose to learn similarity measures
in a data-driven manner. To this end, we capitalize on the \alpha\beta-log-det
divergence, which is a meta-divergence parametrized by scalars \alpha and
\beta, subsuming a wide family of popular information divergences on SPD
matrices for distinct and discrete values of these parameters. Our key idea is
to cast these parameters in a continuum and learn them from data. We
systematically extend this idea to learn vector-valued parameters, thereby
increasing the expressiveness of the underlying non-linear measure. We conjoin
the divergence learning problem with several standard tasks in machine
learning, including supervised discriminative dictionary learning and
unsupervised SPD matrix clustering. We present Riemannian gradient descent
schemes for optimizing our formulations efficiently, and show the usefulness of
our method on eight standard computer vision tasks.
