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Persistent homology for random fields and complexes

Robert J. Adler, Omer Bobrowski, Matthew S. Borman, Eliran Subag, and Shmuel Weinberger

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Abstract

We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices and, in most detail, the algebraic topology of the excursion sets of random fields.

Chapter information

Source
James O. Berger, T. Tony Cai and Iain M. Johnstone, eds., Borrowing Strength: Theory Powering Applications – A Festschrift for Lawrence D. Brown (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2010), 124-143

Dates
First available in Project Euclid: 26 October 2010

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1288099016

Digital Object Identifier
doi:10.1214/10-IMSCOLL609

Subjects
Primary: 60G15: Gaussian processes 55N35: Other homology theories
Secondary: 60G55: Point processes 62H35: Image analysis

Keywords
persistent homology barcodes Betti numbers Euler characteristic random fields Gaussian processes manifold learning random complexes Gaussian kinematic formula

Rights
Copyright © 2010, Institute of Mathematical Statistics

Citation

Adler, Robert J.; Bobrowski, Omer; Borman, Matthew S.; Subag, Eliran; Weinberger, Shmuel. Persistent homology for random fields and complexes. Borrowing Strength: Theory Powering Applications – A Festschrift for Lawrence D. Brown, 124--143, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2010. doi:10.1214/10-IMSCOLL609. https://projecteuclid.org/euclid.imsc/1288099016.


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