A near-stationary subspace for ridge approximation

TL;DR

This paper explores ridge approximation for high-dimensional response surfaces, demonstrating that active subspaces are near-stationary solutions and proposing heuristics for effective ridge function fitting, with applications to engineering models.

Contribution

It introduces a theoretical foundation showing active subspaces are near-stationary for ridge approximation and proposes heuristics for fitting ridge functions in high dimensions.

Findings

Active subspaces are near-stationary for the least-squares problem.

A heuristic using active subspaces can effectively fit ridge functions.

Application to an airfoil drag model demonstrates practical utility.

Abstract

Response surfaces are common surrogates for expensive computer simulations in engineering analysis. However, the cost of fitting an accurate response surface increases exponentially as the number of model inputs increases, which leaves response surface construction intractable for high-dimensional, nonlinear models. We describe ridge approximation for fitting response surfaces in several variables. A ridge function is constant along several directions in its domain, so fitting occurs on the coordinates of a low-dimensional subspace of the input space. We review essential theory for ridge approximation---e.g., the best mean-squared approximation and an optimal low-dimensional subspace---and we prove that the gradient-based active subspace is near-stationary for the least-squares problem that defines an optimal subspace. Motivated by the theory, we propose a computational heuristic that uses an estimated active subspace as an initial guess for a ridge approximation fitting problem. We show a simple example where the heuristic fails, which reveals a type of function for which the proposed approach is inappropriate. We then propose a simple alternating heuristic for fitting a ridge function, and we demonstrate the effectiveness of the active subspace initial guess applied to an airfoil model of drag as a function of its 18 shape parameters.