Persistent Homology via Finite Topological Spaces
Selçuk Kayacan
TL;DR
The paper addresses the robustness and interpretability of persistent homology when derived from metric data by introducing a functorial pipeline that passes from finite metrics to finite topological spaces and then to posets and crosscut complexes. This yields well-defined persistence modules even without inclusion relations among complexes, and allows poset-level simplifications that preserve persistent invariants. A density-based filtration provides a concrete instantiation with stability guarantees, proven via interleaving arguments and the Isometry Theorem. Overall, the framework decouples metric-driven topology from homology, enabling combinatorial, potentially more scalable analyses while preserving key persistent descriptors.
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 simplicial complexes via crosscut constructions, we obtain persistence modules without requiring inclusion relations between the resulting complexes. We show that standard poset-level simplifications preserve persistent invariants and prove stability of the resulting persistence diagrams under perturbations of the input metric in a density-based instantiation.