The content delves into the stability and universality of the bottleneck distance in extended persistence diagrams. It discusses the construction of relative interlevel set homology, Mayer–Vietoris pyramids, and their implications. The analysis includes detailed definitions, proofs, and applications in topological data analysis.
The article presents a comprehensive study on extended persistence diagrams, focusing on stability under perturbations and establishing universality through rigorous constructions. It provides insights into the core concepts of relative interlevel set homology and its applications in topological data analysis.
Key points include defining lifts of points in M to construct functions with specific persistence diagrams, proving homotopy invariance, Mayer–Vietoris sequences, excision properties, exact sequences of pairs, additivity principles, dimension considerations for order-preserving affine maps, and realizing any given extended persistence diagram as relative interlevel set homology.
The discussion is structured around mathematical proofs and theoretical frameworks that underpin the stability and universality of the bottleneck distance in extended persistence diagrams.
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