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 estimator that offers a sparse representation of the expected persistence diagram that achieves near-optimality. We demonstrate the utility of our contributions in a machine learning task in the context of dynamical systems.

翻译：持续同调是拓扑数据分析中的核心方法，已在多个领域成功应用并日益成为重要工具。持续同调的输出为持续图——一种支撑在上半平面的多点集，常作为数据拓扑特征的统计摘要。本文研究持续同调的随机性质，利用小波从观测数据中估计期望持续图的密度；我们证明了所提出的小波估计量具有最优性。此外，我们提出了一种能够稀疏表示期望持续图且达到近最优性的估计量。通过动力系统背景下的机器学习任务，我们验证了该方法的实用性。