Partitions for stratified sampling

TL;DR

This paper introduces a novel method for creating partitions of the unit cube for stratified sampling, improving discrepancy properties and flexibility for arbitrary sample sizes using hyperplanes orthogonal to the main diagonal.

Contribution

It presents a new construction of equivolume partitions with hyperplanes orthogonal to the main diagonal, applicable to any number of samples, and optimizes discrepancy through numerical methods.

Findings

Constructed equivolume partitions for arbitrary N

Achieved improved discrepancy bounds

Optimized discrepancy numerically using black-box methods

Abstract

Classical jittered sampling partitions $[0,1]^d$ into $m^d$ cubes for a positive integer $m$ and randomly places a point inside each of them, providing a point set of size $N=m^d$ with small discrepancy. The aim of this note is to provide a construction of partitions that works for arbitrary $N$ and improves straight-forward constructions. We show how to construct equivolume partitions of the $d$-dimensional unit cube with hyperplanes that are orthogonal to the main diagonal of the cube. We investigate the discrepancy of such point sets and optimise the expected discrepancy numerically by relaxing the equivolume constraint using different black-box optimisation techniques.