  Collaborative representation-based classification (CRC) has demonstrated
remarkable progress in the past few years because of its closed-form analytical
solutions. However, the existing CRC methods are incapable of processing the
nonlinear variational information directly. Recent advances illustrate that how
to effectively model these nonlinear variational information and learn
invariant representations is an open challenge in the community of computer
vision and pattern recognition To this end, we try to design a new algorithm to
handle this problem. Firstly, the second-order statistic, i.e., covariance
matrix is applied to model the original image sets. Due to the space formed by
a set of nonsingular covariance matrices is a well-known Symmetric Positive
Definite (SPD) manifold, generalising the Euclidean collaborative
representation to the SPD manifold is not an easy task. Then, we devise two
strategies to cope with this issue. One attempts to embed the SPD
manifold-valued data representations into an associated tangent space via the
matrix logarithm map. Another is to embed them into a Reproducing Kernel
Hilbert Space (RKHS) by utilizing the Riemannian kernel function. After these
two treatments, CRC is applicable to the SPD manifold-valued features. The
evaluations on four banchmarking datasets justify its effectiveness.
