抖动采样的一个精确离散界
A Sharp Discrepancy Bound for Jittered Sampling
摘要：
对于$m, d \in {\mathbb N}$，通过将单位立方体$[0,1)^d$划分为大小相等的$m^d$轴对齐立方体，然后在每个立方体中独立均匀地放置一个点，来构造在$[0,1)^d$中具有$N = m^d$点的抖动采样点集$P$。我们发现，有常数$c \ge 0$和$C$，使所有$d$和所有$m \ge d$的预期非归一化的星的抖动采样点集的差异满足\[c \,dm^{\frac{d-1}{2}} \sqrt{1 + \log(\tfrac md)} \le {\mathbb E} D^*(P) \le C\, dm^{\frac{d-1}{2}} \sqrt{1 + \log(\tfrac md)}.\]这种差异是因此小于一个因素的$Θ\big(\sqrt{\frac{1+\log(m/d)}{m/d}}\,\big)$比一个均匀分布的随机点集的$m^d$点。这一结果改进了Pausinger和Steinerberger给出的抖动采样差异的上限和下限（Journal of Complexity（2016））。它还消除了$m$与$d$相比足够大的渐近要求。
英文摘要：
For $m, d \in {\mathbb N}$, a jittered sampling point set $P$ having $N = m^d$ points in $[0,1)^d$ is constructed by partitioning the unit cube $[0,1)^d$ into $m^d$ axis-aligned cubes of equal size and then placing one point independently and uniformly at random in each cube. We show that there are constants $c \ge 0$ and $C$ such that for all $d$ and all $m \ge d$ the expected non-normalized star discrepancy of a jittered sampling point set satisfies \[c \,dm^{\frac{d-1}{2}} \sqrt{1 + \log(\tfrac md)} \le {\mathbb E} D^*(P) \le C\, dm^{\frac{d-1}{2}} \sqrt{1 + \log(\tfrac md)}.\] This discrepancy is thus smaller by a factor of $Θ\big(\sqrt{\frac{1+\log(m/d)}{m/d}}\,\big)$ than the one of a uniformly distributed random point set of $m^d$ points. This result improves both the upper and the lower bound for the discrepancy of jittered sampling given by Pausinger and Steinerberger (Journal of Complexity (2016)). It also removes the asymptotic requirement that $m$ is sufficiently large compared to $d$.
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