Generalized bounds for active subspaces

TL;DR

This paper develops generalized bounds for the active subspace method applicable even when traditional probabilistic Poincaré inequalities are invalid, especially for unbounded constants, and explores exponential distributions in higher dimensions.

Contribution

It introduces a framework for deriving generalized error bounds in active subspaces, accommodating unbounded Poincaré constants and analyzing exponential distributions.

Findings

Explicit Poincaré constants for exponential distributions in multiple dimensions

Framework enabling control over error bounds with unbounded constants

Potential extensions to broader distribution classes

Abstract

In this article, we consider scenarios in which traditional estimates for the active subspace method based on probabilistic Poincar\'e inequalities are not valid due to unbounded Poincar\'e constants. Consequently, we propose a framework that allows to derive generalized estimates in the sense that it enables to control the trade-off between the size of the Poincar\'e constant and a weaker order of the final error bound. In particular, we investigate independently exponentially distributed random variables in dimension two or larger and give explicit expressions for corresponding Poincar\'e constants showing their dependence on the dimension of the problem. Finally, we suggest possibilities for future work that aim for extending the class of distributions applicable to the active subspace method as we regard this as an opportunity to enlarge its usability.