在线细化在减少差异方面的作用
The power of online thinning in reducing discrepancy
摘要：
考虑一个从$[0,1]^d$中均匀选取的独立点的无穷序列。在查看序列中的每个点之后，允许监督者保留它或拒绝它，并且这种选择可能取决于所有先前保留的点的位置。然而，监督员必须保持每两个连续点中的至少一个。我们将以这种方式生成的序列称为\emph{two-thinning}序列。这里，监督器的目的是控制点的经验分布的差异，即，在选择$n$点之后，减少体积$A$的任何轴平行超矩形内的点的数量与$nA$的最大偏差。我们的主要结果是一个显式的低复杂度的两个细化策略，保证$O(\log^{2d+1} n)$的差异，所有$n$与高概率（相比，$Θ(\sqrt{n\log\log n})$没有细化）。这个结果的情况下$d=1$回答了Benjamini的一个问题。我们还扩展了建设，以实现相同的渐近界（$1+β$）细化，设置中，拒绝只允许概率$β$独立的每个点。此外，我们提出了一个改进和简化的策略，我们推测，以保证差异的$O(\log^{d+1} n)$（相比$θ(\log^d n)$，最有名的建设低差异序列）。最后，我们为我们的猜想提供理论和经验证据，并提供模拟支持我们的建设应用程序的可行性。
英文摘要：
Consider an infinite sequence of independent, uniformly chosen points from $[0,1]^d$. After looking at each point in the sequence, an overseer is allowed to either keep it or reject it, and this choice may depend on the locations of all previously kept points. However, the overseer must keep at least one of every two consecutive points. We call a sequence generated in this fashion a \emph{two-thinning} sequence. Here, the purpose of the overseer is to control the discrepancy of the empirical distribution of points, that is, after selecting $n$ points, to reduce the maximal deviation of the number of points inside any axis-parallel hyper-rectangle of volume $A$ from $nA$. Our main result is an explicit low complexity two-thinning strategy which guarantees discrepancy of $O(\log^{2d+1} n)$ for all $n$ with high probability (compare with $Θ(\sqrt{n\log\log n})$ without thinning). The case $d=1$ of this result answers a question of Benjamini. We also extend the construction to achieve the same asymptotic bound for ($1+β$)-thinning, a set-up in which rejecting is only allowed with probability $β$ independently for each point. In addition, we suggest an improved and simplified strategy which we conjecture to guarantee discrepancy of $O(\log^{d+1} n)$ (compare with $θ(\log^d n)$, the best known construction of a low discrepancy sequence). Finally, we provide theoretical and empirical evidence for our conjecture, and provide simulations supporting the viability of our construction for applications.
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