The prevailing statistical approach to analyzing persistence diagrams is concerned with filtering out topological noise. In this paper, we adopt a different viewpoint and aim at estimating the actual distribution of a random persistence diagram, which captures both topological signal and noise. To that effect, Chazel and Divol (2019) proved that, under general conditions, the expected value of a random persistence diagram is a measure admitting a Lebesgue density, called the persistence intensity function. In this paper, we are concerned with estimating the persistence intensity function and a novel, normalized version of it -- called the persistence density function. We present a class of kernel-based estimators based on an i.i.d. sample of persistence diagrams and derive estimation rates in the supremum norm. As a direct corollary, we obtain uniform consistency rates for estimating linear representations of persistence diagrams, including Betti numbers and persistence surfaces. Interestingly, the persistence density function delivers stronger statistical guarantees.

翻译：分析持续同调图的主流统计方法侧重于滤除拓扑噪声。本文采用不同视角，旨在估计随机持续同调图的实际分布，该分布同时包含拓扑信号与噪声。为此，Chazel与Divol（2019）证明在一般条件下，随机持续同调图的期望值是一种可测度，该测度存在勒贝格密度，称为持续同调强度函数。本文聚焦于估计持续同调强度函数及其新型归一化版本——持续同调密度函数。我们基于持续同调图的独立同分布样本提出一类核估计器，并推导其在一致范数下的估计速率。作为直接推论，我们获得了用于估计持续同调图线性表示（包括贝蒂数与持续同调曲面）的一致收敛速率。值得关注的是，持续同调密度函数给出了更强的统计保证。