Sampling discretization error of integral norms for function classes with small smoothness

TL;DR

This paper investigates the absolute error sampling discretization of integral norms for function classes with small smoothness, linking entropy numbers to discretization errors and providing new bounds for multivariate functions.

Contribution

It introduces a general result connecting entropy numbers to discretization errors and applies it to derive new bounds for classes of functions with mixed smoothness.

Findings

Entropy numbers dominate discretization errors in the uniform norm.

New error bounds are established for multivariate function classes with small smoothness.

The approach bridges supervised learning theory and numerical integration.

Abstract

We consider infinitely dimensional classes of functions and instead of the relative error setting, which was used in previous papers on the integral norm discretization, we consider the absolute error setting. We demonstrate how known results from two areas of research -- supervised learning theory and numerical integration -- can be used in sampling discretization of the square norm on different function classes. We prove a general result, which shows that the sequence of entropy numbers of a function class in the uniform norm dominates, in a certain sense, the sequence of errors of sampling discretization of the square norm of this class. Then we use this result for establishing new error bounds for sampling discretization of the square norm on classes of multivariate functions with mixed smoothness.