  We propose a representation of Gaussian processes (GPs) based on powers of
the integral operator defined by a kernel function, we call these stochastic
processes integral Gaussian processes (IGPs). Sample paths from IGPs are
functions contained within the reproducing kernel Hilbert space (RKHS) defined
by the kernel function, in contrast sample paths from the standard GP are not
functions within the RKHS. We develop computationally efficient non-parametric
regression models based on IGPs. The main innovation in our regression
algorithm is the construction of a low dimensional subspace that captures the
information most relevant to explaining variation in the response. We use ideas
from supervised dimension reduction to compute this subspace. The result of
using the construction we propose involves significant improvements in the
computational complexity of estimating kernel hyper-parameters as well as
reducing the prediction variance.
