A probabilistic framework for approximating functions in active subspaces

Abstract: This paper develops a comprehensive probabilistic setup to compute approximating functions in active subspaces. Constantine et al. proposed the active subspace method in (Constantine et al., 2014) to reduce the dimension of computational problems. It can be seen as an attempt to approximate a high-dimensional function of interest $f$ by a low-dimensional one. To do this, a common approach is to integrate $f$ over the inactive, i.e. non-dominant, directions with a suitable conditional density function. In practice, this can be done with a finite Monte Carlo sum, making not only the resulting approximation random in the inactive variable for each fixed input from the active subspace, but also its expectation, i.e. the integral of the low-dimensional function weighted with a probability measure on the active variable. In this regard we develop a fully probabilistic framework extending results from (Constantine et al., 2014, 2016). The results are supported by a simple numerical example.