哪些采样密度适用于无界域上的谱聚类？
Which Sampling Densities are Suitable for Spectral Clustering on Unbounded Domains?
摘要：
考虑一个随机几何图，其顶点采样自$\mathbb R^d$上的概率测度，并研究其连通性。我们表明，图通常是断开的，除非采样密度有超指数衰减。在后一种设置中，我们为图形的半径参数确定一个渐近门限值，使得对于超过门限的半径值，图形的采样点保持某些浓度特性，而对于低于相同门限的半径值，图形断开。众所周知，点过程的性质与几何学习问题的分析密切相关，例如谱聚类。这项工作可以被看作是第一步，了解谱聚类的一致性时，概率测度有无界的支持。特别是，我们缩小了设置下，$\mathbb R^d$上的谱聚类算法可以预期实现一致性，一个足够快的衰减的采样密度（超指数）和一个足够慢的衰减半径参数值的函数$n$，采样点的数量。
英文摘要：
We consider a random geometric graph with vertices sampled from a probability measure supported on $\mathbb R^d$, and study its connectivity. We show the graph is typically disconnected, unless the sampling density has superexponential decay. In the later setting, we identify an asymptotic threshold value for the radius parameter of the graph such that, for radius values beyond the threshold, some concentration properties hold for the sampled points of the graph, while the graph is disconnected for radius values below the same threshold. Properties of point processes are well-known to be closely related to the analysis of geometric learning problems, such as spectral clustering. This work can be seen as a first step towards understanding the consistency of spectral clustering when the probability measure has unbounded support. In particular, we narrow down the setting under which spectral clustering algorithms on $\mathbb R^d$ may be expected to achieve consistency, to a sufficiently fast decay of the sampling density (superexponential) and a sufficiently slowly decaying radius parameter value as a function of $n$, the number of sampled points.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
