Dimension reduction techniques have long been an important topic in
statistics, and active subspaces (AS) have received much attention this past
decade in the computer experiments literature. The most common approach towards
estimating the AS is to use Monte Carlo with numerical gradient evaluation.
While sensible in some settings, this approach has obvious drawbacks. Recent
research has demonstrated that active subspace calculations can be obtained in
closed form, conditional on a Gaussian process (GP) surrogate, which can be
limiting in high-dimensional settings for computational reasons. In this paper,
we produce the relevant calculations for a more general case when the model of
interest is a linear combination of tensor products. These general equations
can be applied to the GP, recovering previous results as a special case, or
applied to the models constructed by other regression techniques including
multivariate adaptive regression splines (MARS). Using a MARS surrogate has
many advantages including improved scaling, better estimation of active
subspaces in high dimensions and the ability to handle a large number of prior
distributions in closed form. In one real-world example, we obtain the active
subspace of a radiation-transport code with 240 inputs and 9,372 model runs in
under half an hour