Preprint
Bi-Filtration and Stability of TDA Mapper for Point Cloud Data
ArXiv.org
Cornell University
09/25/2024
DOI: 10.48550/arxiv.2409.17360
Abstract
Carlsson, Singh and Memoli’s TDA mapper takes a point cloud dataset and outputs a graph that depends on several parameter choices. Dey, Memoli, and Wang developed Multiscale Mapper for abstract topological spaces so that parameter choices can be analyzed via persistent homology. However, when applied to actual data, one does not always obtain filtrations of mapper graphs. DBSCAN, one of the most common clustering algorithms used in the TDA mapper software, has two parameters, ϵ and MinPts. If MinPts = 1 then DBSCAN is equivalent to single linkage clustering with cutting height ϵ. We show that if DBSCAN clustering is used with MinPts > 2, a filtration of mapper graphs may not exist except in the absence of free-border points; but such filtrations exist if DBSCAN clustering is used with MinPts = 1 or 2 as the cover size increases, ϵ increases, and/or MinPts decreases. However, the 1-dimensional filtration is unstable. If one adds noise to a data set so that each data point has been perturbed by a distance at most δ, the persistent homology of the mapper graph of the perturbed data set can be significantly different from that of the original data set. We show that we can obtain stability by increasing both the cover size and ϵ at the same time. In particular, we show that the bi-filtrations of the homology groups with respect to cover size and ϵ between these two datasets are 2δ-interleaved
Details
- Title: Subtitle
- Bi-Filtration and Stability of TDA Mapper for Point Cloud Data
- Creators
- Wako BungulaIsabel Darcy
- Resource Type
- Preprint
- Publication Details
- ArXiv.org
- DOI
- 10.48550/arxiv.2409.17360
- ISSN
- 2331-8422
- Publisher
- Cornell University; Ithaca, New York
- Language
- English
- Date posted
- 09/25/2024
- Academic Unit
- Mathematics
- Record Identifier
- 9984721133002771
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