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

Erich Novak, Friedrich Pillichshammer

TL;DR

The paper develops a duality between extreme $L_p$ discrepancy and a corresponding dual integration problem in a new function space $F_{d,q}$ defined via box-indicator representations. It proves that the worst-case integration error with linear rules equals the generalized extreme $L_p$ discrepancy and establishes a precise equivalence between minimal discrepancy and minimal worst-case error across topologies and weight choices. The main contribution is showing that, for $p\in(1,\infty)$, extreme $L_p$ discrepancy suffers from the curse of dimensionality (exponential growth in $d$ for the inverse discrepancy), while the case $p=\infty$ remains tractable; the case $p=1$ remains open. The proof combines a spline-interpolation approach for the worst-case function with a multivariate product construction to derive quantitative lower bounds, thereby linking information complexity to discrepancy growth and clarifying high-dimensional behavior of QMC-type methods in this generalized setting.

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.

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

TL;DR

The paper develops a duality between extreme discrepancy and a corresponding dual integration problem in a new function space defined via box-indicator representations. It proves that the worst-case integration error with linear rules equals the generalized extreme discrepancy and establishes a precise equivalence between minimal discrepancy and minimal worst-case error across topologies and weight choices. The main contribution is showing that, for , extreme discrepancy suffers from the curse of dimensionality (exponential growth in for the inverse discrepancy), while the case remains tractable; the case remains open. The proof combines a spline-interpolation approach for the worst-case function with a multivariate product construction to derive quantitative lower bounds, thereby linking information complexity to discrepancy growth and clarifying high-dimensional behavior of QMC-type methods in this generalized setting.

Abstract

The classical notion of extreme discrepancy is a quantitative measure for the irregularity of distribution of finite point sets in the -dimensinal unit cube. In this paper we find a dual integration problem whose worst-case error is exactly the extreme discrepancy of the underlying integration nodes. Studying this integration problem we show that the extreme discrepancy suffers from the curse of dimensionality for all . It is known that the problem is tractable for ; the case stays open.
Paper Structure (14 sections, 8 theorems, 134 equations, 2 figures)

This paper contains 14 sections, 8 theorems, 134 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Extreme $L_p$ discrepancy
  3. A related dual integration problem
  4. Representation-space definition via box indicators.
  5. Integration problem and worst-case error.
  6. Minimal discrepancy and minimal worst-case error.
  7. Initial error and worst-case function.
  8. Inverse discrepancy and information complexity.
  9. The curse of dimensionality
  10. The proof of Theorem \ref{['thm:main3']}
  11. Spline interpolation for the worst-case function.
  12. The proof of Theorem \ref{['thm:main3']}.
  13. Appendix
  14. Appendix

Key Result

Theorem 2

Let $p,q \in [1,\infty ]$ be Hölder conjugates. For every point set $\mathcal{P}=\{{\boldsymbol x}_1,\ldots,{\boldsymbol x}_N\}$ in $[0,1)^d$ and arbitrary corresponding real weights $\mathcal{A}=\{c_1,\ldots,c_N\}$ we have

Figures (2)

  • Figure 1: The quantity $C_p$ for $p \in (1,20]$.
  • Figure 2: The function $F_p(y)$, $y \in [0,1]$ for $p \in \{2,8\}$ (first row -- the maximum is attaind in $y=\tfrac{1}{2}$) and $p \in \{20,50\}$ (second row -- the maximum is not attained in $y=\frac{1}{2}$, rather in $y=\tfrac{1}{2}$ a local minimum is attained).

Theorems & Definitions (19)

  • Definition 1: Representation-space definition of $F_{d,q}$
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2: Discrepancy-integration duality
  • proof
  • Corollary 1
  • Proposition 3
  • proof
  • Corollary 2
  • ...and 9 more