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On a Class of Partitions with Lower Expected Star Discrepancy and Its Upper Bound than Jittered Sampling

Xiaoda Xu, Jun Xian

TL;DR

The paper addresses reducing the expected star discrepancy of sampling in the unit cube by designing a new family of convex equivolume partitions parameterized by an angle $\theta$. It combines geometric partition design with a probabilistic variance framework and $\delta$-covers to derive strong bounds, proving a strong partition principle: $\mathbb{E}[D_N^*(Z)] \leq \mathbb{E}[D_N^*(Y)] < \mathbb{E}[D_N^*(X)]$, and providing explicit upper bounds that improve upon jittered sampling, with an optimal choice at $\theta=\arctan(1/2)$ yielding $P(\theta)=-2/45$. The results resolve Open Question 2 from KP21a and offer refined bounds of the form $\mathbb{E}[D_N^*(Z)] \leq \frac{\sqrt{2d + \frac{2P(\theta)}{3^{d-2}N^{2-1/d}}} + 1}{N^{1/2 + 1/(2d)}}$, along with a simple bound for random sampling. The methodology blends geometric analysis of the partition, Bernstein concentration, and discretization via $\delta$-covers, enabling tighter, explicit discrepancy bounds and advancing the theory of partition-based sampling in high dimensions with potential impacts on quasi-MMC and numerical integration. The work also points to future directions including non-convex partitions, broader sampling schemes, and empirical validation.

Abstract

We investigate the expected star discrepancy under a newly designed class of convex equivolume partition models. The main contributions are two-fold. First, we establish a strong partition principle for the star discrepancy, showing that our newly designed partitions yield stratified sampling point sets with lower expected star discrepancy than both classical jittered sampling and simple random sampling. Specifically, we prove that $\mathbb{E}(D^{*}_{N}(Z))\leq\mathbb{E}(D^{*}_{N}(Y))<\mathbb{E}(D^{*}_{N}(X))$, where $X$, $Y$, and $Z$ represent simple random sampling, jittered sampling, and our new partition sampling, respectively. Second, we derive explicit upper bounds for the expected star discrepancy under our partition models, which improve upon existing bounds for jittered sampling. Our results resolve Open Question 2 posed in Kiderlen and Pausinger (2021) regarding the strong partition principle for star discrepancy.

On a Class of Partitions with Lower Expected Star Discrepancy and Its Upper Bound than Jittered Sampling

TL;DR

The paper addresses reducing the expected star discrepancy of sampling in the unit cube by designing a new family of convex equivolume partitions parameterized by an angle . It combines geometric partition design with a probabilistic variance framework and -covers to derive strong bounds, proving a strong partition principle: , and providing explicit upper bounds that improve upon jittered sampling, with an optimal choice at yielding . The results resolve Open Question 2 from KP21a and offer refined bounds of the form , along with a simple bound for random sampling. The methodology blends geometric analysis of the partition, Bernstein concentration, and discretization via -covers, enabling tighter, explicit discrepancy bounds and advancing the theory of partition-based sampling in high dimensions with potential impacts on quasi-MMC and numerical integration. The work also points to future directions including non-convex partitions, broader sampling schemes, and empirical validation.

Abstract

We investigate the expected star discrepancy under a newly designed class of convex equivolume partition models. The main contributions are two-fold. First, we establish a strong partition principle for the star discrepancy, showing that our newly designed partitions yield stratified sampling point sets with lower expected star discrepancy than both classical jittered sampling and simple random sampling. Specifically, we prove that , where , , and represent simple random sampling, jittered sampling, and our new partition sampling, respectively. Second, we derive explicit upper bounds for the expected star discrepancy under our partition models, which improve upon existing bounds for jittered sampling. Our results resolve Open Question 2 posed in Kiderlen and Pausinger (2021) regarding the strong partition principle for star discrepancy.
Paper Structure (27 sections, 5 theorems, 33 equations)

This paper contains 27 sections, 5 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Our Contributions
  4. Technical Overview
  5. Preliminaries
  6. Jittered Sampling and Basic Partitions
  7. New Partition Design
  8. $\delta$-Covers and Discretization
  9. Main Results
  10. Comparison of Expected Star Discrepancy
  11. Explicit Upper Bounds
  12. Proofs of Main Results
  13. Preliminary Lemmas
  14. Proof of Theorem \ref{['thm:main1']}
  15. Step 1: Relating Star Discrepancy to Variance
  16. ...and 12 more sections

Key Result

Theorem 3.1

Let $m, d \in \mathbb{N}$ with $m \geq d \geq 2$, and $0 \leq \theta \leq \pi/2$. Set $N = m^d$. Let: Then:

Theorems & Definitions (10)

  • Definition 2.1: Jittered Sampling
  • Definition 2.2: $\delta$-Cover
  • Theorem 3.1: Strong Partition Principle for Star Discrepancy
  • Remark 3.2
  • Theorem 3.3: Upper Bounds for New Partitions
  • Remark 3.4
  • Corollary 3.5: Simple Random Sampling Bound
  • Lemma 4.1: Bernstein's Inequality
  • Lemma 4.2: $L_2$-Discrepancy Difference
  • proof : Proof Sketch of Lemma \ref{['lem:L2_diff']}