Persistent homology is a central methodology in topological data analysis that has been successfully implemented in many fields and is becoming increasingly popular and relevant. The output of persistent homology is a persistence diagram -- a multiset of points supported on the upper half plane -- that is often used as a statistical summary of the topological features of data. In this paper, we study the random nature of persistent homology and estimate the density of expected persistence diagrams from observations using wavelets; we show that our wavelet-based estimator is optimal. Furthermore, we propose an adaptive estimator that offers a sparse representation of the expected persistence diagram that achieves near-optimality.

翻译：持续同调是拓扑数据分析中的核心方法，已在众多领域成功应用并日益普及。其输出为持续图——支撑于上半平面的多点集——常被用作数据拓扑特征的统计摘要。本文通过研究持续同调的随机特性，利用小波从观测数据中估计期望持续图的密度，并证明我们提出的基于小波的估计量具有最优性。此外，我们设计了一种自适应估计方法，能够实现期望持续图的稀疏表示，且达到近乎最优的性能。