On the expected L2-discrepancy of jittered sampling

TL;DR

This paper derives explicit formulas for the expected L2-discrepancy of jittered sampling in high-dimensional unit cubes, accounting for stratification and projections, advancing understanding of sampling uniformity.

Contribution

It provides the first closed-form expressions for the expected L2-discrepancy of jittered samples in any dimension, including projections, based on recent stratified sampling formulas.

Findings

Closed-form formula for expected L2-discrepancy of jittered samples.

Derived a similar formula for Hickernell L2-discrepancy considering projections.

Advances theoretical understanding of stratified sampling uniformity.

Abstract

For $m, d \in \mathbb{N}$, a jittered sample of $N=m^d$ points can be constructed by partitioning $[0,1]^d$ into $m^d$ axis-aligned equivolume boxes and placing one point independently and uniformly at random inside each box. We utilise a formula for the expected $\mathcal{L}_2-$discrepancy of stratified samples stemming from general equivolume partitions of $[0,1]^d$ which recently appeared, to derive a closed form expression for the expected $\mathcal{L}_2-$discrepancy of a jittered point set for any $m, d \in \mathbb{N}$. As a second main result we derive a similar formula for the expected Hickernell $\mathcal{L}_2-$discrepancy of a jittered point set which also takes all projections of the point set to lower dimensional faces of the unit cube into account.