  In this paper, we develop an approach to exploiting kernel methods with
manifold-valued data. In many computer vision problems, the data can be
naturally represented as points on a Riemannian manifold. Due to the
non-Euclidean geometry of Riemannian manifolds, usual Euclidean computer vision
and machine learning algorithms yield inferior results on such data. In this
paper, we define Gaussian radial basis function (RBF)-based positive definite
kernels on manifolds that permit us to embed a given manifold with a
corresponding metric in a high dimensional reproducing kernel Hilbert space.
These kernels make it possible to utilize algorithms developed for linear
spaces on nonlinear manifold-valued data. Since the Gaussian RBF defined with
any given metric is not always positive definite, we present a unified
framework for analyzing the positive definiteness of the Gaussian RBF on a
generic metric space. We then use the proposed framework to identify positive
definite kernels on two specific manifolds commonly encountered in computer
vision: the Riemannian manifold of symmetric positive definite matrices and the
Grassmann manifold, i.e., the Riemannian manifold of linear subspaces of a
Euclidean space. We show that many popular algorithms designed for Euclidean
spaces, such as support vector machines, discriminant analysis and principal
component analysis can be generalized to Riemannian manifolds with the help of
such positive definite Gaussian kernels.
