Persistent Homology via Finite Topological Spaces
This provides a new mathematical foundation for persistent homology that could benefit researchers in topological data analysis.
The authors tackled the problem of computing persistent homology without requiring inclusion relations between complexes by proposing a functorial framework based on finite topological spaces and posets, demonstrating stability under metric perturbations and practical viability on real datasets.
We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are continuous identity maps. By passing functorially to posets and to order complexes, we obtain persistence modules without requiring inclusion relations between the resulting complexes. We show that standard poset-level simplifications preserve persistent invariants and establish stability of the resulting persistence diagrams under perturbations of the input metric in a basic density-based instantiation, illustrating how stability arguments arise naturally in our framework. We further introduce a concrete density-guided construction, designed to be faithful to anchor neighborhood structure at each scale, and demonstrate its practical viability through an implementation tested on real datasets.