Extreme $L_p$ discrepancy, numerical integration and the curse of dimensionality

TL;DR

This paper investigates the extreme $L_p$ discrepancy as a measure of point set irregularity, revealing it suffers from the curse of dimensionality for all finite p values, unlike the case p=∞.

Contribution

It establishes a duality between extreme $L_p$ discrepancy and a specific integration problem, demonstrating the curse of dimensionality for all p in (1,∞).

Findings

Extreme $L_p$ discrepancy suffers from the curse of dimensionality for all p in (1,∞).

The problem is known to be tractable for p=∞.

The case p=1 remains an open problem.

Abstract

The classical notion of extreme $L_p$ discrepancy is a quantitative measure for the irregularity of distribution of finite point sets in the $d$-dimensinal unit cube. In this paper we find a dual integration problem whose worst-case error is exactly the extreme $L_p$ discrepancy of the underlying integration nodes. Studying this integration problem we show that the extreme $L_p$ discrepancy suffers from the curse of dimensionality for all $p \in (1,\infty)$. It is known that the problem is tractable for $p=\infty$; the case $p=1$ stays open.