Persistent Homology via Finite Topological Spaces

TL;DR

This paper introduces a novel functorial framework for persistent homology using finite topological spaces and posets, providing stability and practical tools for data analysis.

Contribution

It develops a new theoretical approach linking finite topologies and persistent homology, including stability results and a density-guided construction for real data applications.

Findings

Framework preserves persistent invariants under simplifications

Stability of persistence diagrams under metric perturbations

Practical implementation demonstrated on real datasets

Abstract

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.