1. A∞ Persistent Homology Estimates Detailed Topology from Pointcloud Datasets
- Author
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Anastasios Stefanou, Francisco Belchí, Engineering and Physical Sciences Research Council (UK), University of Southampton, Consejo Superior de Investigaciones Científicas (España), Ministerio de Economía y Competitividad (España), and National Science Foundation (US)
- Subjects
Topological data analysis (TDA) ,Cup product ,Betti number ,Stability (learning theory) ,Applied algebraic topology ,Formal spaces ,Topology ,Theoretical Computer Science ,Persistent cohomology ,Combinatorics ,Discrete Mathematics and Combinatorics ,Betti numbers ,Persistent homology ,Finite set ,Mathematics ,Interleaving distance ,Massey products ,Loop spaces ,Topological estimation ,A∞-persistence ,Continuous function (set theory) ,A∞-coalgebra ,Function (mathematics) ,Metric space ,Bottleneck distance ,Computational Theory and Mathematics ,Functoriality ,Geometric estimation ,Linking number ,Geometry and Topology ,Stability ,A∞persistent homology ,A∞-algebra ,Subspace topology - Abstract
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., We would like to thank Justin Curry for valuable feedback on previous versions of this paper. F. Belchí was partially supported by the EPSRC grant EPSRC EP/N014189/1 (Joining the dots) to the University of Southampton and by the Spanish State Research Agency through the María de Maeztu Seal of Excellence to IRI (MDM-2016-0656). A. Stefanou was partially supported by the National Science Foundation through grants CCF-1740761 (TRIPODS TGDA@OSU) and DMS-1440386 (Mathematical Biosciences Institute at the Ohio State University).
- Published
- 2022