Truncation Dimension for Function Approximation

TL;DR

This paper investigates the approximation of high-dimensional functions in weighted anchored spaces, demonstrating that under certain conditions, the effective dimensionality needed for accurate approximation remains surprisingly low.

Contribution

It introduces the concept of epsilon-truncation dimension and shows it remains small for functions with fast decaying weights and moderate error demands.

Findings

Truncation dimension is very small for fast decaying weights.

Effective approximation is feasible with low-dimensional algorithms.

Results hold for error demands up to approximately 10^{-5}.

Abstract

We consider approximation of functions of $s$ variables, where $s$ is very large or infinite, that belong to weighted anchored spaces. We study when such functions can be approximated by algorithms designed for functions with only very small number ${\rm dim^{trnc}}(\varepsilon)$ of variables. Here $\varepsilon$ is the error demand and we refer to ${\rm dim^{trnc}}(\varepsilon)$ as the $\varepsilon$-truncation dimension. We show that for sufficiently fast decaying product weights and modest error demand (up to about $\varepsilon \approx 10^{-5}$) the truncation dimension is surprisingly very small.