AI 中文总结
本文研究随机候选点构造拟均匀设计，证明候选池规模需为$\Theta(N\log N)$以保证有界网格比，并分析最远点采样及其近似实现的性能。
AI 中文摘要
我们研究在满足双侧多项式球增长条件的紧致度量测度空间上拟均匀设计的随机构造。给定整数$N\le M$，我们首先抽取$M$个独立候选点，然后保留候选集最远点遍历中的前$N$个点。我们证明了网格比的概率非渐近界，并表明$M=\Theta(N\log N)$是获得有界网格比所需候选池规模的尖锐阶：$N\log(N/\delta)$的足够大倍数以至少$1-\delta$的概率足够，而若$M=o(N\log N)$，则对于从同一独立候选池中选择$N$个点的任何过程，网格比在概率上发散。若$M/(N\log M)\to\infty$，精确最远点采样（FPS）的网格比上界趋于$2$。我们还量化了近似FPS的影响，并分析了有界倍增维空间中直接和快速实现。最后，利用单个无限候选点流和递增的候选预算，我们构造了一个几乎必然拟均匀的嵌套序列；在超临界过采样和精确FPS下，或更一般地当近似因子趋于$1$时，对于欧几里得空间的紧致正体积子集，其网格比上极限等于$2$。
英文摘要
We study randomized constructions of quasi-uniform designs on a compact metric-measure space satisfying two-sided polynomial ball-growth conditions. Given integers $N\le M$, we first draw $M$ independent candidate points and then retain the first $N$ points of a farthest-point traversal of the candidate set. We prove probabilistic non-asymptotic bounds on the mesh ratio and show that $M=Θ(N\log N)$ is the sharp order of the candidate-pool size required for bounded mesh ratio: a sufficiently large multiple of $N\log(N/δ)$ suffices with probability at least $1-δ$, whereas, if $M=o(N\log N)$, the mesh ratio diverges in probability for every procedure that selects $N$ points from the same independent candidate pool. If $M/(N\log M)\to\infty$, the upper bound on the mesh ratio for the exact farthest-point sampling (FPS) tends to $2$. We also quantify the effect of approximate FPS and analyze direct and fast implementations in spaces of bounded doubling dimension. Finally, using a single infinite stream of candidate points and increasing candidate budgets, we construct an almost surely quasi-uniform nested sequence; under supercritical oversampling and exact FPS, or more generally when the approximation factors tend to $1$, its mesh-ratio limit superior equals $2$ for compact positive-volume subsets of Euclidean space.
Comments39 pages, 5 figures
