  Ridge functions have recently emerged as a powerful set of ideas for
subspace-based dimension reduction. In this paper we begin by drawing parallels
between ridge subspaces, sufficient dimension reduction and active subspaces,
contrasting between techniques rooted in statistical regression and those
rooted in approximation theory. This sets the stage for our new algorithm that
approximates what we call a Gaussian ridge function---the posterior mean of a
Gaussian process on a dimension-reducing subspace---suitable for both
regression and approximation problems. To compute this subspace we develop an
iterative algorithm that optimizes over the Stiefel manifold to compute the
subspace, followed by an optimization of the hyperparameters of the Gaussian
process. We demonstrate the utility of the algorithm on two analytical
functions, where we obtain near exact ridge recovery, and a turbomachinery case
study, where we compare the efficacy of our approach with three well-known
sufficient dimension reduction methods: SIR, SAVE, CR. The comparisons motivate
the use of the posterior variance as a heuristic for identifying the
suitability of a dimension-reducing subspace.
