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

TL;DR

This work tackles the challenge of distinguishing small topological holes that are embedded in high-density regions from noise. It introduces the Robust Density-Aware Distance ($RDAD$) filtration, a scale-invariant, density-weighted topological filtration enhanced with distance-to-measure ($DTM$) for robustness, and provides both population and empirical formulations. The authors prove fundamental properties: persistence prolongation for high-density regions, scale invariance, and stability under additive noise and outliers, complemented by a bootstrapping scheme for feature significance. Empirical validation on synthetic Voronoi-like data and real cellular-tower locations demonstrates improved detection of small holes, with an open-source implementation enabling practical use in diverse domains.

Abstract

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.

Figures (14)

• Figure 1: Datasets with small topological features.
• Figure 2: The distance filtration and its persistence diagrams. In the first subplot is a sample of points near a circle. Unions of balls centered at these points with different radii are shown the subsequent subplots. The last subplot shows the persistence diagrams of these unions of balls. The red diamond points correspond to the dimension-0 diagram, and the blue circular points correspond to the dimension-1 diagram. The point marked by dashed lines near the diagonal corresponds to the marked polygon in the third subplot and is filled in the fifth subplot. The marked blue point that is far away from the diagonal corresponds to the main loop that is formed in the fourth subplot and is filled in the second last subplot.
• Figure 3: Contour plots and persistence diagrams of different filtrations for the "David and Goliath" two-square dataset.
• Figure 4: Sample points of the "Antman" two-square dataset, and the persistence diagram of the empirical DAD filtration for this dataset
• Figure 5: Sample points of corrupted "Antman" two-square datasets (by outliers and by additive noise).

Theorems & Definitions (32)

• Definition 1: Population RDAD
• Definition 2: Population DAD
• Definition 3: Empirical DAD and RDAD
• Theorem 1
• Corollary 2
• Corollary 3
• Proposition 4: Scale invariance
• Theorem 5: Stability
• Corollary 6
• Proposition 7: RDAD as an Approximation of DAD