  Principal component analysis (PCA) is a commonly used pattern analysis method
that maps high-dimensional data into a lower-dimensional space maximizing the
data variance, that results in the promotion of separability of data. Inspired
by the principle of PCA, a novel type of learning discriminative matrix
information geometry (MIG) detectors in the unsupervised scenario are
developed, and applied to signal detection in nonhomogeneous environments.
Hermitian positive-definite (HPD) matrices can be used to model the sample
data, while the clutter covariance matrix is estimated by the geometric mean of
a set of secondary HPD matrices. We define a projection that maps the HPD
matrices in a high-dimensional manifold to a low-dimensional and more
discriminative one to increase the degree of separation of HPD matrices by
maximizing the data variance. Learning a mapping can be formulated as a
two-step mini-max optimization problem in Riemannian manifolds, which can be
solved by the Riemannian gradient descent algorithm. Three discriminative MIG
detectors are illustrated with respect to different geometric measures, i.e.,
the Log-Euclidean metric, the Jensen--Bregman LogDet divergence and the
symmetrized Kullback--Leibler divergence. Simulation results show that
performance improvements of the novel MIG detectors can be achieved compared
with the conventional detectors and their state-of-the-art counterparts within
nonhomogeneous environments.
