  Off-the-shelf Gaussian Process (GP) covariance functions encode smoothness
assumptions on the structure of the function to be modeled. To model complex
and non-differentiable functions, these smoothness assumptions are often too
restrictive. One way to alleviate this limitation is to find a different
representation of the data by introducing a feature space. This feature space
is often learned in an unsupervised way, which might lead to data
representations that are not useful for the overall regression task. In this
paper, we propose Manifold Gaussian Processes, a novel supervised method that
jointly learns a transformation of the data into a feature space and a GP
regression from the feature space to observed space. The Manifold GP is a full
GP and allows to learn data representations, which are useful for the overall
regression task. As a proof-of-concept, we evaluate our approach on complex
non-smooth functions where standard GPs perform poorly, such as step functions
and robotics tasks with contacts.
