On approximating the shape of one dimensional functions

TL;DR

This paper introduces a novel method for accurately approximating the shape of one-dimensional functions derived from high-dimensional functions evaluated via low discrepancy sequences, using polynomial smoothing and integration rules.

Contribution

It proposes the first method to approximate one-dimensional functions from LDS evaluations, combining integration rules with polynomial smoothing for improved efficiency.

Findings

The method converges to the true one-dimensional function under certain conditions.

It is computationally more efficient than grid-based approaches.

The approach effectively captures the shape of the functions in high-dimensional settings.

Abstract

Consider an $s$-dimensional function being evaluated at $n$ points of a low discrepancy sequence (LDS), where the objective is to approximate the one-dimensional functions that result from integrating out $(s-1)$ variables. Here, the emphasis is on accurately approximating the shape of such \emph{one-dimensional} functions. Approximating this shape when the function is evaluated on a set of grid points instead is relatively straightforward. However, the number of grid points needed increases exponentially with $s$. LDS are known to be increasingly more efficient at integrating $s$-dimensional functions compared to grids, as $s$ increases. Yet, a method to approximate the shape of a one-dimensional function when the function is evaluated using an $s$-dimensional LDS has not been proposed thus far. We propose an approximation method for this problem. This method is based on an $s$-dimensional integration rule together with fitting a polynomial smoothing function. We state and prove results showing conditions under which this polynomial smoothing function will converge to the true one-dimensional function. We also demonstrate the computational efficiency of the new approach compared to a grid based approach.