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[2512.23348] Persistent Homology via Finite Topological Spaces
Summary
This paper presents a functorial framework for persistent homology using finite topological spaces, demonstrating stability in persistence diagrams and practical applications in real datasets.
Why It Matters
Persistent homology is crucial in topological data analysis, providing insights into the shape of data. This research enhances the understanding of persistent homology by introducing a framework that simplifies the process and ensures stability, making it more applicable in computational geometry and machine learning.
Key Takeaways
- Introduces a functorial framework for persistent homology based on finite topological spaces.
- Establishes stability of persistence diagrams under input metric perturbations.
- Demonstrates practical viability through real dataset implementations.
- Poset-level simplifications preserve persistent invariants.
- Offers a density-guided construction that respects neighborhood structures.
Mathematics > Algebraic Topology arXiv:2512.23348 (math) [Submitted on 29 Dec 2025 (v1), last revised 23 Feb 2026 (this version, v2)] Title:Persistent Homology via Finite Topological Spaces Authors:Selçuk Kayacan View a PDF of the paper titled Persistent Homology via Finite Topological Spaces, by Sel\c{c}uk Kayacan View PDF HTML (experimental) 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. Subjects: Algebraic Topology (math.AT); Computational Geometry (cs.CG); Machine Learning (cs.LG) MSC classes: 55N31 Cite as: arXiv:2512.23348 [math.AT] (or arXiv:2512.23348v2 [math.AT] for this version) h...
