On Tractability of Approximation for a Special Space of Functions

TL;DR

This paper investigates the approximation of a specific class of multivariate functions, demonstrating low variable dependence and tractability under various norms, even with high evaluation costs.

Contribution

It establishes the small active variable count and tractability results for the approximation problem in a special function space, extending understanding of computational complexity.

Findings

Functions have a small number of active variables.

Approximation problems are strongly polynomially or quasi-polynomially tractable.

Results hold even when evaluation costs grow exponentially with active variables.

Abstract

We consider approximation problems for a special space of d variate functions. We show that the problems have small number of active variables, as it has been postulated in the past using concentration of measure arguments. We also show that, depending on the norm for measuring the error, the problems are strongly polynomially or quasi-polynomially tractable even in the model of computation where functional evaluations have the cost exponential in the number of active variables.