使用图割估计周长
Estimating perimeter using graph cuts
摘要：
研究了随机几何图的图割对集合周长的估计。对于$Ω\subset D = (0,1)^d$，利用$d \geq 2$，我们给出$n$随机i.i.d. $D$上在$Ω$中的成员身份已知的点。我们把样本看作是一个随机几何图，其连接距离为$\varepsilon>0$。我们估计$Ω$的周长（相对于$D$），适当地重新缩放，在$Ω$中的顶点和$D \backslash Ω$中的顶点之间的图切割。我们得到的偏差和方差估计的错误，这是最佳的缩放相对于$n$和$\varepsilon$。我们考虑两种缩放机制：密集（当顶点的平均度为$\infty$时）和稀疏（当度为$0$时）。在密集的制度有一个交叉的性质，近似在维度$d=5$：我们表明，在低维度$d=2,3,4$可以获得置信区间的近似误差，而在较高的维度只能获得误差估计用于测试的假设，即周长小于一个给定的数字。
英文摘要：
We investigate the estimation of the perimeter of a set by a graph cut of a random geometric graph. For $Ω\subset D = (0,1)^d$, with $d \geq 2$, we are given $n$ random i.i.d. points on $D$ whose membership in $Ω$ is known. We consider the sample as a random geometric graph with connection distance $\varepsilon>0$. We estimate the perimeter of $Ω$ (relative to $D$) by the, appropriately rescaled, graph cut between the vertices in $Ω$ and the vertices in $D \backslash Ω$. We obtain bias and variance estimates on the error, which are optimal in scaling with respect to $n$ and $\varepsilon$. We consider two scaling regimes: the dense (when the average degree of the vertices goes to $\infty$) and the sparse one (when the degree goes to $0$). In the dense regime there is a crossover in the nature of approximation at dimension $d=5$: we show that in low dimensions $d=2,3,4$ one can obtain confidence intervals for the approximation error, while in higher dimensions one can only obtain error estimates for testing the hypothesis that the perimeter is less than a given number.
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