Extensible Grids: Uniform Sampling on a Space Filling Curve
可扩展网格：空间填充曲线上的均匀采样AI 解读 ↓
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可扩展网格：空间填充曲线上的均匀采样
- 研究问题
- 能否通过空间填充曲线生成任意样本规模的积分点，在保留网格采样误差阶的同时支持逐步扩展样本？
- 方法与思路
- 作者将一维均匀点经 Hilbert 曲线映射到多维空间，分析确定性、随机分层与扰乱 van der Corput 采样的误差。
- 材料陈述的贡献
- 摘要称该方法可达到网格相同误差阶且允许任意样本规模，并对 Lipschitz 及特定分段 Lipschitz 函数给出积分误差界。
- 阅读关注点
- 应核对不同维度下的误差结论、随机化机制与可扩展性，检查不连续集合的几何条件及不同填充曲线的比较范围。
- 阅读边界
- 仅据所给 Crossref 摘要版本，不能推断刊本变化。误差阶依赖维度、平滑性和不连续集合条件，不能推广到任意被积函数或有限样本优势。
原文摘要
Summary We study the properties of points in [0,1]d generated by applying Hilbert's space filling curve to uniformly distributed points in [0, 1]. For deterministic sampling we obtain a discrepancy of O(n−1/d) for d⩾2. For random stratified sampling, and scrambled van der Corput points, we derive a mean-squared error of O(n−1−2/d) for integration of Lipschitz continuous integrands, when d⩾3. These rates are the same as those obtained by sampling on d-dimensional grids and they show a deterioration with increasing d. The rate for Lipschitz functions is, however, the best possible at that level of smoothness and is better than plain independent and identically distributed sampling. Unlike grids, space filling curve sampling provides points at any desired sample size, and the van der Corput version is extensible in n. We also introduce a class of piecewise Lipschitz functions whose discontinuities are in rectifiable sets described via Minkowski content. Although these functions may have infinite variation in the sense of Hardy and Krause, they can be integrated with a mean-squared error of O(n−1−1/d). It was previously known only that the rate was o(n−1). Other space filling curves, such as those due to Sierpinski and Peano, also attain these rates, whereas upper bounds for the Lebesgue curve are somewhat worse, as if the dimension were log2(3) times as high.