Abstract: Discovering Active Subspaces for High-Dimensional Computer Models
Computer models are commonly used in physics and engineering to aid in the understanding of complex physical systems. These models often have tens, hundreds or even thousands of input variables, which can lead to a variety of challenges in practice. Active subspace methods are a powerful tool for analysis of such models and can be used to form low dimensional approximations to the computer model, perform sensitivity analysis, accelerate posterior sampling and more. When the number of inputs is large, the standard Monte Carlo approach to estimating the active subspace can be inefficient and inflexible. Recent approaches based on a Gaussian process (GP) surrogate are appealing, but can become computationally infeasible for a large number of inputs. In this paper, we derive a closed form solution for fundamental active subspace calculations ---the expected outer product of the gradient--- for a large class of functions. A special case of this function class coincides precisely with the models constructed by the multivariate adaptive regression splines (MARS) algorithm. Since MARS and Bayesian MARS specialize in handling a large number of input parameters both efficiently and accurately, this leads to effective strategies for estimation of the active subspace in practice. Our closed form solution is derived with respect to a broad class of prior distributions and we show that the estimator can be preferable to Monte Carlo and the GP based estimator when the number of input parameters is large. We demonstrate the effectiveness of our proposed estimator using two important real-world examples, showing how to handle complicated settings such as constrained input spaces, calibration posteriors and a priori covariance structure.