A∞ Persistent Homology Estimates Detailed Topology from Pointcloud Datasets
Let X be a closed subspace of a metric space M. It is well known that, under mild hypotheses, one can estimate the Betti numbers of X from a finite set P⊂ M of points approximating X. In this paper, we show that one can also use P to estimate much more detailed topological properties of X. We achiev...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Consejo Superior de Investigaciones Científicas (CSIC) |
| Repositorio: | DIGITAL.CSIC. Repositorio Institucional del CSIC |
| OAI Identifier: | oai:digital.csic.es:10261/295746 |
| Acceso en línea: | http://hdl.handle.net/10261/295746 |
| Access Level: | acceso abierto |
| Palabra clave: | Persistent homology Persistent cohomology Bottleneck distance Interleaving distance Stability Functoriality Applied algebraic topology Topological data analysis Topological estimation Geometric estimation A∞-persistence A∞persistent homology A∞-coalgebra A∞-algebra Betti numbers Cup product Massey products Linking number Loop spaces Formal spaces |
| Sumario: | Let X be a closed subspace of a metric space M. It is well known that, under mild hypotheses, one can estimate the Betti numbers of X from a finite set P⊂ M of points approximating X. In this paper, we show that one can also use P to estimate much more detailed topological properties of X. We achieve this by proving the stability of A-persistent homology. In its most general case, this stability means that given a continuous function f: Y→ R on a topological space Y, small perturbations in the function f imply at most small perturbations in the family of A-barcodes. This work can be viewed as a proof of the stability of cup-product and generalized-Massey-products persistence. The technical key of this paper consists of figuring out a setting which makes A-persistence functorial. |
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