A probabilistic framework for approximating functions in active subspaces

TL;DR

This paper introduces a probabilistic framework for approximating functions within active subspaces, enhancing the theoretical understanding and practical computation of low-dimensional representations of high-dimensional functions.

Contribution

It extends existing active subspace methods by developing a fully probabilistic approach to function approximation, accounting for randomness in the integral estimates.

Findings

Framework supports probabilistic analysis of approximations

Numerical example demonstrates practical applicability

Extends previous deterministic approaches

Abstract

This paper develops a comprehensive probabilistic setup to compute approximating functions in active subspaces. Constantine et al. proposed the active subspace method in (Constantine et al., 2014) to reduce the dimension of computational problems. It can be seen as an attempt to approximate a high-dimensional function of interest $f$ by a low-dimensional one. To do this, a common approach is to integrate $f$ over the inactive, i.e. non-dominant, directions with a suitable conditional density function. In practice, this can be done with a finite Monte Carlo sum, making not only the resulting approximation random in the inactive variable for each fixed input from the active subspace, but also its expectation, i.e. the integral of the low-dimensional function weighted with a probability measure on the active variable. In this regard we develop a fully probabilistic framework extending results from (Constantine et al., 2014, 2016). The results are supported by a simple numerical example.