Tractability properties of the discrepancy in Orlicz norms

TL;DR

This paper investigates the computational complexity of measuring discrepancy in high-dimensional spaces using Orlicz norms, revealing conditions under which the problem is tractable or weakly tractable, especially for exponential Orlicz spaces.

Contribution

It establishes the tractability properties of discrepancy in Orlicz norms, including polynomial tractability for $oldsymbol{\psi_oldsymbol{\alpha} ext{-norms}}$ of exponential Orlicz spaces, advancing understanding of high-dimensional discrepancy.

Findings

Discrepancy in Orlicz norms can be polynomially or weakly tractable.

$oldsymbol{\psi_oldsymbol{\alpha} ext{-norms}}$ of exponential Orlicz spaces are polynomially tractable.

The results delineate conditions for tractability in high-dimensional discrepancy problems.

Abstract

We show that the minimal discrepancy of a point set in the $d$-dimensional unit cube with respect to Orlicz norms can exhibit both polynomial and weak tractability. In particular, we show that the $\psi_\alpha$-norms of exponential Orlicz spaces are polynomially tractable.