Exploring the Lazy Witness Complex for Efficient Persistent Homology in Large-Scale Data

Abstract

In this paper, topological data analysis (TDA) techniques have been explored, with a focus on the selection of the Witness Complex and Persistent Homology of some nested families of Lazy Witness Complex as approximations for analyzing complex datasets. The Witness Complex was chosen for its efficiency and scalability, as it constructs a simplicial complex using landmark points, reducing computational load compared to methods like the Vietoris-Rips and Čech complexes. This makes it suitable for large, high-dimensional datasets, accurately representing the dataset's intrinsic geometry even with varying data densities. Persistent Homology was then reviewed with the aim of calculating it on some nested families of the Witness Complex. Subsequently, the nested families of the Lazy Witness Complex were introduced mathematically, with an example of the entire construction process for a well-known point cloud dataset. For this purpose, 50 points were generated randomly from a circle, and persistent diagrams of the point cloud data were analyzed to understand and compare the behavior among the approximations of the Witness Complex after choosing 10 landmarks using the Maxmin method. Since the families are nested, the filtration process became faster for each successive family, thus reducing computational complexity. For all three cases , the persistent barcodes indicated the same shape of the dataset. This study may help in choosing the suitable family of the Witness Complex over Persistent Homology to balance computational feasibility with topological accuracy, enabling efficient handling of large datasets while preserving important topological features. This approach allows for extracting meaningful insights from complex data while effectively managing computational resources.