  The kernel matrix used in kernel methods encodes all the information required
for solving complex nonlinear problems defined on data representations in the
input space using simple, but implicitly defined, solutions. Spectral analysis
on the kernel matrix defines an explicit nonlinear mapping of the input data
representations to a subspace of the kernel space, which can be used for
directly applying linear methods. However, the selection of the kernel subspace
is crucial for the performance of the proceeding processing steps. In this
paper, we propose a component analysis method for kernel-based dimensionality
reduction that optimally preserves the pair-wise distances of the class means
in the feature space. We provide extensive analysis on the connection of the
proposed criterion to those used in kernel principal component analysis and
kernel discriminant analysis, leading to a discriminant analysis version of the
proposed method. Our analysis also provides more insights on the properties of
the feature spaces obtained by applying these methods.
