持久Betti数的Bootstrap方法及其他稳定化统计量
Bootstrapping Persistent Betti Numbers and Other Stabilizing Statistics
AI总结：
本文研究了用于拓扑数据分析等稳定化统计量的多元Bootstrap方法，证明了平滑Bootstrap在标准非参数方法失效时能提供一致的置信区间估计，并放宽了无需连续底层密度等理论条件，最后通过模拟和SDSS宇宙网数据验证了其有效性。
摘要：
本文研究了针对一般稳定化统计量的多元Bootstrap方法，并特别应用于拓扑数据分析。现有的拓扑统计量极限定理在实际中难以用于构建置信区间，这促使了Bootstrap在此方面的应用。然而，标准的非参数Bootstrap在某些情况下无法直接提供渐近有效的置信区间。相反，平滑Bootstrap方法在这些场景下被证明能给出一致的估计。本工作与稳定化统计量领域的其他一般性结果相关，包括临界状态下Poisson过程与Binomial过程泛函的中心极限定理。所考虑的具体统计量包括$\mathbb R^d$上点集的Čech复形与Vietoris-Rips复形的持久Betti数，以及Euler特征和$k$-最近邻图的总边长。全文特别强调削弱建立Bootstrap一致性所需的必要条件，特别是无需假设连续的底层密度。文中提供了模拟研究以评估平滑Bootstrap在有限样本量下的表现，并进一步应用于Sloan Digital Sky Survey (SDSS)的宇宙网数据集。源代码可在github.com/btroycraft/stabilizing_statistics_bootstrap获取。
英文摘要：
The present contribution investigates multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. Existing limit theorems for topological statistics prove difficult to use in practice for the construction of confidence intervals, motivating the use of the bootstrap in this capacity. However, the standard nonparametric bootstrap does not directly provide for asymptotically valid confidence intervals in some situations. A smoothed bootstrap procedure, instead, is shown to give consistent estimation in these settings. The present work relates to other general results in the area of stabilizing statistics, including central limit theorems for functionals of Poisson and Binomial processes in the critical regime. Specific statistics considered include the persistent Betti numbers of Čech and Vietoris-Rips complexes over point sets in $\mathbb R^d$, along with Euler characteristics, and the total edge length of the $k$-nearest neighbor graph. Special emphasis is made throughout to weakening the necessary conditions needed to establish bootstrap consistency. In particular, the assumption of a continuous underlying density is not required. A simulation study is provided to assess the performance of the smoothed bootstrap for finite sample sizes, and the method is further applied to the cosmic web dataset from the Sloan Digital Sky Survey (SDSS). Source code is available at github.com/btroycraft/stabilizing_statistics_bootstrap.
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。
