Discrepancy of stratified samples from partitions of the unit cube

Abstract

We extend the notion of jittered sampling to arbitrary partitions and study the discrepancy of the related point sets. Let $\mathbf{\Omega}=(\Omega_1,\ldots,\Omega_N)$ be a partition of $[0,1]^d$ and let the $i$th point in $\mathcal{P}$ be chosen uniformly in the $i$th set of the partition (and stochastically independent of the other points), $i=1,\ldots,N$. For the study of such sets we introduce the concept of a uniformly distributed triangular array and compare this notion to related notions in the literature. We prove that the expected ${\mathcal{L}_p}$-discrepancy, $\mathbb{E} {\mathcal{L}_p}(\mathcal{P}_{\mathbf{\Omega}})^p$, of a point set $\mathcal{P}_\mathbf{\Omega}$ generated from any equivolume partition $\mathbf{\Omega}$ is always strictly smaller than the expected ${\mathcal{L}_p}$-discrepancy of a set of $N$ uniform random samples for $p>1$. For fixed $N$ we consider classes of stratified samples based on equivolume partitions of the unit cube into convex sets or into sets with a uniform positive lower bound on their reach. It is shown that these classes contain at least one minimizer of the expected ${\mathcal{L}_p}$-discrepancy. We illustrate our results with explicit constructions for small $N$. In addition, we present a family of partitions that seems to improve the expected discrepancy of Monte Carlo sampling by a factor of 2 for every $N$.