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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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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

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

First available in Project Euclid: 26 October 2010

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Mathematical Reviews number (MathSciNet)

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

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

Copyright © 2010, Institute of Mathematical Statistics


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.

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