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Generalized Discrepancy of Random Points

Erich Novak, Friedrich Pillichshammer

TL;DR

This work investigates generalized $L_p$-discrepancy for the Sobolev space $F_{d,q}$ by replacing uniform sampling with product densities and nonnegative weights, analyzed through the probabilistic method. For $p=2$ the authors solve the associated variational problem exactly, obtaining explicit optimal densities and improving the bound by a factor of $(4/3)^{d/2}$; for general $p$ they obtain sharp asymptotics with an optimized base $igl(\tfrac{p+2}{p+1}\bigr)^{1/2}$ in the exponent, significantly outperforming uniform-sampling bounds, especially at small $p$. Despite these gains, the curse of dimensionality remains for all $p\ge1$, suggesting similar limitations for deterministic $L_1$-discrepancy. The results frame the change of measure as a form of importance sampling in $F_{d,q}$ and provide guidance on when randomization yields meaningful gains versus when dimensionality imposes fundamental barriers.

Abstract

We study the $L_p$-discrepancy of random point sets in high dimensions, with emphasis on small values of $p$. Although the classical $L_p$-discrepancy suffers from the curse of dimensionality for all $p \in (1,\infty)$, the gap between known upper and lower bounds remains substantial, in particular for small $p \ge 1$. To clarify this picture, we review the existing results for i.i.d.\ uniformly distributed points and derive new upper bounds for \emph{generalized} $L_p$-discrepancies, obtained by allowing non-uniform sampling densities and corresponding non-negative quadrature weights. Using the probabilistic method, we show that random points drawn from optimally chosen product densities lead to significantly improved upper bounds. For $p=2$ these bounds are explicit and optimal; for general $p \in [1,\infty)$ we obtain sharp asymptotic estimates. The improvement can be interpreted as a form of importance sampling for the underlying Sobolev space $F_{d,q}$. Our results also reveal that, even with optimal densities, the curse of dimensionality persists for random points when $p\ge 1$, and it becomes most pronounced for small $p$. This suggests that the curse should also hold for the classical $L_1$-discrepancy for deterministic point sets.

Generalized Discrepancy of Random Points

TL;DR

This work investigates generalized -discrepancy for the Sobolev space by replacing uniform sampling with product densities and nonnegative weights, analyzed through the probabilistic method. For the authors solve the associated variational problem exactly, obtaining explicit optimal densities and improving the bound by a factor of ; for general they obtain sharp asymptotics with an optimized base in the exponent, significantly outperforming uniform-sampling bounds, especially at small . Despite these gains, the curse of dimensionality remains for all , suggesting similar limitations for deterministic -discrepancy. The results frame the change of measure as a form of importance sampling in and provide guidance on when randomization yields meaningful gains versus when dimensionality imposes fundamental barriers.

Abstract

We study the -discrepancy of random point sets in high dimensions, with emphasis on small values of . Although the classical -discrepancy suffers from the curse of dimensionality for all , the gap between known upper and lower bounds remains substantial, in particular for small . To clarify this picture, we review the existing results for i.i.d.\ uniformly distributed points and derive new upper bounds for \emph{generalized} -discrepancies, obtained by allowing non-uniform sampling densities and corresponding non-negative quadrature weights. Using the probabilistic method, we show that random points drawn from optimally chosen product densities lead to significantly improved upper bounds. For these bounds are explicit and optimal; for general we obtain sharp asymptotic estimates. The improvement can be interpreted as a form of importance sampling for the underlying Sobolev space . Our results also reveal that, even with optimal densities, the curse of dimensionality persists for random points when , and it becomes most pronounced for small . This suggests that the curse should also hold for the classical -discrepancy for deterministic point sets.

Paper Structure

This paper contains 13 sections, 2 theorems, 93 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. The function space setting.
  4. Integration and linear algorithms.
  5. Worst-case error and generalized discrepancy.
  6. Information complexity.
  7. The probabilistic method in discrepancy theory.
  8. Main questions.
  9. Asymptotic analysis.
  10. Change of measure for $p=2$
  11. Change of the measure for general $p$
  12. Stability, and open problems
  13. Open problems.

Key Result

Theorem 1

For the optimal density $\varrho_d^{\ast}$ we have We have $\varrho_d^{\ast}=\varrho^{\ast,\otimes d}$, where $\varrho^{\ast}$ is given in optdenp2d1.

Figures (3)

  • Figure 1: Comparison of $\alpha_{\text{old}}^2$, $\alpha_{\text{new}}^2$ and $c_p$.
  • Figure 2: Optimal density $\varrho^*$ for $p=1$ and $p=2$.
  • Figure 3: Optimal densities $\varrho^*$ for $p=10$ and $p=100$.

Theorems & Definitions (10)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 2
  • proof
  • Remark 6
  • Remark 7