Detection of Small Holes by the Scale-Invariant Robust Density-Aware Distance (RDAD) Filtration

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 whose mass is concentrated near a manifold (or more generally, a CW complex) embedded in a high-dimensional Euclidean space. The proposed method is robust against additive noise and outliers. In particular, sample points are allowed to be perturbed away from the manifold. Traditional TDA tools, like those based on the distance filtration, often struggle to distinguish small features from noise, because of their short persistence. An alternative filtration, called Robust Density-Aware Distance (RDAD) filtration, is proposed to prolong the persistence of small holes surrounded by high-density regions. This is achieved by weighting the distance function by the density in the sense of Bell et al. Distance-to-measure is incorporated to enhance stability and mitigate noise due to the density estimation. The utility of the proposed filtration in identifying small holes, as well as its robustness against noise, are illustrated through an analytical example and extensive numerical experiments. Basic mathematical properties of the proposed filtration are proven.