Gradient-based dimension reduction of multivariate vector-valued functions

TL;DR

This paper introduces a gradient-based technique for identifying dominant input directions in high-dimensional vector-valued functions, enabling effective low-dimensional approximations in uncertainty quantification.

Contribution

It generalizes active subspaces to vector-valued functions and provides a mathematical framework for dimension reduction using gradient information.

Findings

Gradient-based methods effectively identify dominant directions.

Norm choice influences low-dimensional approximation quality.

Method extends active subspaces to vector-valued functions.

Abstract

Multivariate functions encountered in high-dimensional uncertainty quantification problems often vary most strongly along a few dominant directions in the input parameter space. We propose a gradient-based method for detecting these directions and using them to construct ridge approximations of such functions, in the case where the functions are vector-valued (e.g., taking values in $\mathbb{R}^n$). The methodology consists of minimizing an upper bound on the approximation error, obtained by subspace Poincar\'e inequalities. We provide a thorough mathematical analysis in the case where the parameter space is equipped with a Gaussian probability measure. The resulting method generalizes the notion of active subspaces associated with scalar-valued functions. A numerical illustration shows that using gradients of the function yields effective dimension reduction. We also show how the choice of norm on the codomain of the function has an impact on the function's low-dimensional approximation.