Modified Active Subspaces Using the Average of Gradients

Abstract

The active subspace is useful in reducing the dimensionality of the input space of a function. This article proposes a modification of the active subspace that can substantially improve the dimension reduction, concerning explained variance of the function. For general quadratic functions, the modified active subspace minimizes an upper bound on the variance explained by its complement. A two-dimensional example demonstrates that the modification is effective for functions that have both strong linear trend and quadratic trend. Finally, the numerical examples empirically demonstrate the improvements that result from using the modified active subspace instead of the active subspace.