  Gaussian processes (GPs) are very widely used for modeling of unknown
functions or surfaces in applications ranging from regression to classification
to spatial processes. Although there is an increasingly vast literature on
applications, methods, theory and algorithms related to GPs, the overwhelming
majority of this literature focuses on the case in which the input domain
corresponds to a Euclidean space. However, particularly in recent years with
the increasing collection of complex data, it is commonly the case that the
input domain does not have such a simple form. For example, it is common for
the inputs to be restricted to a non-Euclidean manifold, a case which forms the
motivation for this article. In particular, we propose a general extrinsic
framework for GP modeling on manifolds, which relies on embedding of the
manifold into a Euclidean space and then constructing extrinsic kernels for GPs
on their images. These extrinsic Gaussian processes (eGPs) are used as prior
distributions for unknown functions in Bayesian inferences. Our approach is
simple and general, and we show that the eGPs inherit fine theoretical
properties from GP models in Euclidean spaces. We consider applications of our
models to regression and classification problems with predictors lying in a
large class of manifolds, including spheres, planar shape spaces, a space of
positive definite matrices, and Grassmannians. Our models can be readily used
by practitioners in biological sciences for various regression and
classification problems, such as disease diagnosis or detection. Our work is
also likely to have impact in spatial statistics when spatial locations are on
the sphere or other geometric spaces.
