Nerves, fibers and homotopy groups
Journal of Combinatorial Theory Series A
Discrete & Computational Geometry
Vines and vineyards by updating persistence in linear time
Proceedings of the twenty-second annual symposium on Computational geometry
Stability of Persistence Diagrams
Discrete & Computational Geometry
Proximity of persistence modules and their diagrams
Proceedings of the twenty-fifth annual symposium on Computational geometry
Technical Section: Fast construction of the Vietoris-Rips complex
Computers and Graphics
Zigzag persistent homology in matrix multiplication time
Proceedings of the twenty-seventh annual symposium on Computational geometry
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Topological persistence is, by now, an established paradigm for constructing robust topological invariants from point cloud data: the data are converted into a filtered simplicial complex, the complex gives rise to a persistence module, and the module is described by a persistence diagram. In this paper, we study the geometry of the spaces of persistence modules and diagrams, with special attention to Cech and Rips complexes. The metric structures are determined in terms of interleaving maps between modules and matchings between diagrams. We show that the relationship between the Cech and Rips complexes is governed by certain `coherence' conditions on the corresponding families of interleavings or matchings.