  Principal Component Analysis (PCA) and Kernel Principal Component Analysis
(KPCA) are fundamental methods in machine learning for dimensionality
reduction. The former is a technique for finding this approximation in finite
dimensions and the latter is often in an infinite dimensional Reproducing
Kernel Hilbert-space (RKHS). In this paper, we present a geometric framework
for computing the principal linear subspaces in both situations as well as for
the robust PCA case, that amounts to computing the intrinsic average on the
space of all subspaces: the Grassmann manifold. Points on this manifold are
defined as the subspaces spanned by $K$-tuples of observations. The intrinsic
Grassmann average of these subspaces are shown to coincide with the principal
components of the observations when they are drawn from a Gaussian
distribution. We show similar results in the RKHS case and provide an efficient
algorithm for computing the projection onto the this average subspace. The
result is a method akin to KPCA which is substantially faster. Further, we
present a novel online version of the KPCA using our geometric framework.
Competitive performance of all our algorithms are demonstrated on a variety of
real and synthetic data sets.
