Computing Active Subspaces Efficiently with Gradient Sketching

TL;DR

This paper introduces gradient sketching methods to efficiently compute active subspaces in high-dimensional simulations using only linear measurements of gradients, enabling practical dimension reduction without explicit gradient access.

Contribution

It proposes two novel gradient sketching techniques for estimating eigenpairs of gradient-derived matrices, reducing computational cost in high-dimensional settings.

Findings

Gradient sketching methods accurately identify active subspaces.

Finite difference approximations require only two function evaluations.

Methods are effective regardless of input space dimension.

Abstract

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.