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The central theorem in persistent homology states how the data retained in persistence diagrams/barcodes is stable under small deformations of the initial data.
Specifically, the original form of the stability theorem (Cohen, Steiner, Edelsbrunner & Harer 2007) applies to persistence modules given by the connected components of the sub-level sets of a continuous function on some data set
(equipped with the evident inclusions) and states that as the function is deformed to another continuous function , the bottleneck distance (CSEH07, p. 3)
between the corresponding persistence diagrams is bounded by the supremum norm of the difference between the two functions:
Various generalizations of this stability result exist, notably the algebraic stability theorem (CCGGO09).
From Botnan & Lesnick 18, p. 2:
The algebraic stability theorem is perhaps the central theorem in the theory of persistent homology; it provides the core mathematical justification for the use of persistent homology in the study of noisy data. The theorem is used, in one form or another, in nearly all available results on the approximation, inference, and estimation of persistent homology.
The stability theorem originates in:
The algebraic stability theorem:
Frédéric Chazal, David Cohen-Steiner, Marc Glisse, Leonidas J. Guibas, Steve Y. Oudot, Proximity of persistence modules and their diagrams, SCG ‘09: Proceedings of the twenty-fifth annual symposium on Computational geometry (2009) 237–246 doi:10.1145/1542362.1542407
Ulrich Bauer, Michael Lesnick, Induced Matchings and the Algebraic Stability of Persistence Barcodes, Journal of Computational Geometry 6 2 (2015) 162-191 arXiv:1311.3681, doi:10.20382/jocg.v6i2a9
Further developments:
Generalization to zigzag persistence modules:
Refinement to persistent homotopy:
Andrew J. Blumberg, Michael Lesnick, Universality of the Homotopy Interleaving Distance arXiv:1705.01690
J. F. Jardine, Data and homotopy types arXiv:1908.06323
Version for persistent cohomotopy:
Last revised on May 23, 2022 at 12:37:43. See the history of this page for a list of all contributions to it.