分层抽样的划分
Partitions for stratified sampling
AI总结：
本文提出一种适用于任意N的d维单位立方体等体积划分构造方法，利用正交于主对角线的超平面，并通过黑盒优化技术放宽等体积约束以数值优化点集的期望差异。
摘要：
经典抖动抽样将$[0,1]^d$划分为$m^d$个立方体（$m$为正整数），并在每个立方体内随机放置一个点，从而生成大小为$N=m^d$且差异较小的点集。本文旨在提供一种适用于任意$N$的划分构造方法，并改进直接构造。我们展示了如何利用正交于立方体主对角线的超平面，构造$d$维单位立方体的等体积划分。我们研究了此类点集的差异，并通过使用不同的黑盒优化技术放宽等体积约束，对期望差异进行了数值优化。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
