Computing active subspaces efficiently with gradient sketching

Active subspaces are an emerging set of tools for identifying and exploiting the most important directions in the space of a computer simulation´s input parameters; these directions depend on the simulation´s quantity of interest, which we treat as a function from inputs to outputs. To identify a function´s active subspace, one must compute the eigenpairs of a matrix derived from the function´s gradient, which presents challenges when the gradient is not available as a subroutine. We numerically study two methods for estimating the necessary eigenpairs using only linear measurements of the function´s gradient. In practice, these measurements can be estimated by finite differences using only two function evaluations, regardless of the dimension of the function´s input space.