The Partition Principle Revisited: Non-Equal Volume Designs Achieve Minimal Expected Star Discrepancy

TL;DR

This paper introduces non-equal volume partitions for stratified sampling, demonstrating they achieve lower expected star discrepancy than traditional jittered sampling, with explicit bounds and theoretical advantages for high-dimensional integration.

Contribution

The paper establishes a new partition principle showing non-equal volume partitions outperform jittered sampling in reducing expected star discrepancy, with explicit bounds and theoretical insights.

Findings

Non-equal volume partitions yield lower expected star discrepancy.

Explicit upper bounds improve upon jittered sampling bounds.

Theoretical foundation for high-dimensional numerical integration.

Abstract

We study the expected star discrepancy under a newly designed class of non-equal volume partitions. The main contributions are twofold. First, we establish a strong partition principle for the star discrepancy, showing that our newly designed non-equal volume partitions yield stratified sampling point sets with lower expected star discrepancy than classical jittered sampling. Specifically, we prove that $\mathbb{E}(D^{*}_{N}(Z)) < \mathbb{E}(D^{*}_{N}(Y))$, where $Y$ and $Z$ represent jittered sampling and our non-equal volume partition sampling, respectively. Second, we derive explicit upper bounds for the expected star discrepancy under our non-equal volume partition models, which improve upon existing bounds for jittered sampling. Our results provide a theoretical foundation for using non-equal volume partitions in high-dimensional numerical integration.