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

arXiv: https://arxiv.org/abs/2204.07821 · HTML: https://mlsanity.com/paper/2204.07821/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 85 references that are in the map.
- First posted on arXiv 16 Apr 2022.

## Builds on (most cited first, 17 of 17)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [Flows for simultaneous manifold learning and density estimation](https://mlsanity.com/paper/2003.13913/) (2020)
- [Persistence Homology of Networks: Methods and Applications](https://mlsanity.com/paper/1907.08708/) (2019)
- [Robust Topological Inference: Distance To a Measure and Kernel Distance](https://mlsanity.com/paper/1412.7197/) (2014)
- [Persistent homology detects curvature](https://mlsanity.com/paper/1905.13196/) (2019)
- [Interactive Visualization of 2-D Persistence Modules](https://mlsanity.com/paper/1512.00180/) (2015)
- [DTM-based Filtrations](https://mlsanity.com/paper/1811.04757/) (2018)
- [Topological Time Series Analysis](https://mlsanity.com/paper/1812.05143/) (2018)
- [Stability of 2-Parameter Persistent Homology](https://mlsanity.com/paper/2010.09628/) (2020)
- [Building a Balanced k-d Tree in O(kn log n) Time](https://mlsanity.com/paper/1410.5420/) (2014)
- [Persistent Homology of Geospatial Data: A Case Study with Voting](https://mlsanity.com/paper/1902.05911/) (2019)
- [Computing the Multicover Bifiltration](https://mlsanity.com/paper/2103.07823/) (2021)
- [Generalized Persistence Algorithm for Decomposing Multi-parameter Persistence Modules](https://mlsanity.com/paper/1904.03766/) (2019)
- [Persistence Terrace for Topological Inference of Point Cloud Data](https://mlsanity.com/paper/1705.02037/) (2017)
- [A Family of Density-Scaled Filtered Complexes](https://mlsanity.com/paper/2112.03334/) (2021)

## Abstract (arXiv)

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.