nLab
persistent homology

Idea

Persistent homology is a homology theory adapted to a computational context, for instance, in analysis of large data sets. It keeps track of homology classes which stay ‘persistent’ when the approximate image of a space gets refined to higher resolutions.

More detail

We suppose given a ‘data cloud of samples’, $P\subset \mathbb{R}^m$, from some space $X$, yielding a simplicial complex $S_\rho(X)$ for each $\rho \gt 0$ via one of the family of simplicial complex approximation methods that are listed below (TO BE ADDED). For these, the important idea to retain is that if $\rho \lt \rho^\prime$, then

$$S_\rho(X) \hookrightarrow S_{\rho^\prime}(X),$$

so we get a ‘filtration structure’ on the complex.

The idea of persistent homology is to look for features that persist for some range of parameter values. Typically a feature, such as a hole, will initially not be observed, then will appear, and after a range of values of the parameter it will disappear again. A typical feature will be a Betti number of the complex, $S_\rho(X)$, which then will vary with the parameter $\rho$.

Software

One can compute intervals for homological features algorithmically over field coefficients and software packages are available for this purpose. See for instance Perseus. The principal algorithm is based on bringing the filtered complex to its canonical form by upper-triangular matrices from (Barannikov1994, §2.1)

Related concepts

• persistence module

• well group

• topological data analysis

• computational topology

References

General

Introduction and survey

• Robert Ghrist, Barcodes: The Persistent Topology of Data, Bull. Amer. Math. Soc. 45 (2008), 61-75 (doi:10.1090/S0273-0979-07-01191-3, pdf)

• Gunnar Carlsson, Topology and data, Bull. Amer. Math. Soc. 46 (2009), no. 2, 255-308 (doi:10.1090/S0273-0979-09-01249-X)

• Gunnar Carlsson, Persistent Homology and Applied Homotopy Theory, in: Handbook of Homotopy Theory, CRC Press, 2019 (arXiv:2004.00738)

See also

• Tim Porter, Observing Information: Applied Computational Topology 2008 (Slides)

• Bei Wang, Topological Data Analysis, Lecture 2010 (pdf)

• Wikipedia, Persistent homology

Bar-codes were discovered under the name of canonical forms invariants of filtered complexes in

• Serguei Barannikov, Framed Morse complex and its invariants, pdf Advances in Soviet Mathematics 21 93–115 (1994)

Other references:

• A. Zomorodian, Gunnar Carlsson, Computing persistent homology, Discrete Comput. Geom. 33, 249–274 (2005) (doi:10.1007/s00454-004-1146-y)

• Gunnar Carlsson, V. de Silva, Zigzag persistence (arXiv:0812.0197)

• Robert J. Adler, Omer Bobrowski, Matthew S. Borman, Eliran Subag, Shmuel Weinberger, Persistent homology for random fields and complexes Institute of Mathematical Statistics Collections 6:124–143, 2010 (arxiv/1003.1001)

• Paweł Dłotko, Hubert Wagner, Computing homology and persistent homology using iterated Morse decomposition (arxiv/1210.1429)

• Robert MacPherson, Benjamin Schweinhart, Measuring shape with topology, J. Math. Phys. 53, 073516 (2012) (doi:10.1063/1.4737391)

• Robert J. Adler, Omer Bobrowski, Shmuel Weinberger, Crackle: the persistent homology of noise, arxiv/1301.1466

• D. Le Peutrec, N. Nier, C. Viterbo, Precise Arrhenius Law for p-forms: The Witten Laplacian and Morse–Barannikov Complex, Annales Henri Poincaré 14 (3): 567–610 (2013) doi

• Ulrich Bauer, Michael Kerber, Jan Reininghaus, Clear and compress: computing persistent homology in chunks, arxiv/1303.0477

• Sara Kališnik, Persistent homology and duality, 2013 (pdf, pdf)

• Francisco Belchí Guillamón, Aniceto Murillo Mas, A-infinity persistence, arxiv/1403.2395

• João Pita Costa, Mikael Vejdemo Johansson, Primož Škraba, Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology, arxiv/1409.8613

• Genki Kusano, Kenji Fukumizu, Yasuaki Hiraoka, Persistence weighted Gaussian kernel for topological data analysis, arxiv/1601.01741

• Heather A. Harrington, Nina Otter, Hal Schenck, Ulrike Tillmann, Stratifying multiparameter persistent homology, arxiv/1708.07390

• H. Edelsbrunner, D. Morozov, Persistent homology: theory and practice pdf

The following paper uses persistent homology to single out features relevant for training neural networks:

• Jean-Baptiste Bardin, Gard Spreemann, Kathryn Hess, Topological exploration of artificial neuronal network dynamics, arxiv/1810.01747

Variants

Discussion of persistent Cohomotopy as an improvement over homological well groups:

• Peter Franek, Marek Krčál, On Computability and Triviality of Well Groups, Discrete Comput Geom (2016) 56: 126 (arXiv:1501.03641, doi:10.1007/s00454-016-9794-2)

• Peter Franek, Marek Krčál, Persistence of Zero Sets, Homology, Homotopy and Applications, Volume 19 (2017) Number 2 (arXiv:1507.04310, doi:10.4310/HHA.2017.v19.n2.a16)

• Peter Franek, Marek Krčál, Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory, talk at ACAT meeting 2015 (pfd slides)

• Peter Franek, Marek Krčál, Hubert Wagner, Solving equations and optimization problems with uncertainty, J Appl. and Comput. Topology (2018) 1: 297 (arxiv:1607.06344, doi:10.1007/s41468-017-0009-6)

See also

• Robert MacPherson, Amit Patel, Persistent Local Systems (arXiv:1805.02539)

category: topology, applications