标题： MaxTDA：拓扑数据分析中最大持久性的稳健统计推断 显示英文标题

标题： MaxTDA: Robust Statistical Inference for Maximal Persistence in Topological Data Analysis

摘要： 持久同调是拓扑数据分析（TDA）中的一个领域，能够揭示数据中不同维度的孔洞（连通分支、环、空腔等）。这些孔洞的部分特征由它们在不同尺度上的持久性来刻画。噪声数据可能会导致许多并非真正拓扑信号的额外孔洞。已经提出了各种稳健的TDA技术来减少噪声孔洞的数量，然而，这些稳健方法往往也会削弱拓扑信号。 本文介绍了一种名为Maximal TDA（MaxTDA）的统计框架，解决了TDA中的一个局限性，即稳健推断技术系统性地低估了显著同调特征的持久性。MaxTDA通过结合核密度估计与通过拒绝采样的水平集阈值化，生成了最大化持久性特征的一致估计量，在保持对噪声和异常值的稳健性的同时尽量减少了偏差。我们确立了抽样过程的一致性和最大持久性估计器的稳定性。该框架还通过从限定估计偏差概率的分位数组建拒绝带，实现了对拓扑特征的统计推断。在需要精确量化统计意义上显著的拓扑特征以揭示复杂数据集中潜在结构属性的应用中，MaxTDA尤为有价值。数值模拟实验涵盖了多种数据集，包括一个来自系外行星天文学的例子，突显了MaxTDA在恢复真实拓扑信号方面的有效性。 显示英文摘要

摘要： Persistent homology is an area within topological data analysis (TDA) that can uncover different dimensional holes (connected components, loops, voids, etc.) in data. The holes are characterized, in part, by how long they persist across different scales. Noisy data can result in many additional holes that are not true topological signal. Various robust TDA techniques have been proposed to reduce the number of noisy holes, however, these robust methods have a tendency to also reduce the topological signal. This work introduces Maximal TDA (MaxTDA), a statistical framework addressing a limitation in TDA wherein robust inference techniques systematically underestimate the persistence of significant homological features. MaxTDA combines kernel density estimation with level-set thresholding via rejection sampling to generate consistent estimators for the maximal persistence features that minimizes bias while maintaining robustness to noise and outliers. We establish the consistency of the sampling procedure and the stability of the maximal persistence estimator. The framework also enables statistical inference on topological features through rejection bands, constructed from quantiles that bound the estimator's deviation probability. MaxTDA is particularly valuable in applications where precise quantification of statistically significant topological features is essential for revealing underlying structural properties in complex datasets. Numerical simulations across varied datasets, including an example from exoplanet astronomy, highlight the effectiveness of MaxTDA in recovering true topological signals.