A novel topological-data-analytical (TDA) method is proposed to distinguish, from noise, small holes surrounded by high-density regions of a probability density function. The proposed method is robust against additive noise and outliers. Traditional TDA tools, like those based on the distance filtration, often struggle to distinguish small features from noise, because both have short persistences. An alternative filtration, called the Robust Density-Aware Distance (RDAD) filtration, is proposed to prolong the persistences of small holes of high-density regions. This is achieved by weighting the distance function by the density in the sense of Bell et al. The concept of distance-to-measure is incorporated to enhance stability and mitigate noise. The persistence-prolonging property and robustness of the proposed filtration are rigorously established, and numerical experiments are presented to demonstrate the proposed filtration's utility in identifying small holes.

翻译：提出了一种新颖的拓扑数据分析方法，用于从噪声中区分被概率密度函数高密度区域包围的小孔洞。该方法对加性噪声和离群点具有鲁棒性。传统的拓扑数据分析工具（如基于距离滤形的方法）常因小特征与噪声均具有短持久性而难以区分二者。本文提出了一种替代性滤形——鲁棒密度感知距离（RDAD）滤形，通过延长高密度区域中小孔洞的持久性来解决这一问题。该滤形通过引入Bell等人的密度加权距离函数实现，并融合了距离到测度的概念以增强稳定性并抑制噪声。严格证明了所提滤形的持久性延长特性与鲁棒性，并通过数值实验展示了其在识别小孔洞中的实用性。