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Bi-Filtration and Stability of TDA Mapper for Point Cloud Data by Bungula, Wako; Darcy, Isabel is a scholarly article available to read on EtoBox.

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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, \textbf{$\epsilon$} and \textbf{MinPts}. If \textbf{MinPts = 1} then DBSCAN is equivalent to single linkage clustering with cutting height \textbf{$\epsilon$}. We show that if DBSCAN clustering is used with \textbf{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 \textbf{MinPts = 1} or \textbf{2} as the cover size increases, \textbf{$\epsilon$} increases, and/or \textbf{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 \textbf{$\delta$}, the persistent homology of the mapper graph

Author

Bungula, Wako; Darcy, Isabel

Published

2024

Language

EN