Zigzag Persistent Homology and Real-Valued Functions. Carlsson, G., family=Silva , g., & Morozov, D. In Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, of SCG '09, pages 247–256. ACM. ![Zigzag Persistent Homology and Real-Valued Functions [link]](https://bibbase.org/img/filetypes/link.svg "link")
Paper doi abstract bibtex We study the problem of computing zigzag persistence of a sequence of homology groups and study a particular sequence derived from the levelsets of a real-valued function on a topological space. The result is a local, symmetric interval descriptor of the function. Our structural results establish a connection between the zigzag pairs in this sequence and extended persistence, and in the process resolve an open question associated with the latter. Our algorithmic results not only provide a way to compute zigzag persistence for any sequence of homology groups, but combined with our structural results give a novel algorithm for computing extended persistence. This algorithm is easily parallelizable and uses (asymptotically) less memory.
@inproceedings{carlssonZigzagPersistentHomology2009,
location = {{New York, NY, USA}},
title = {Zigzag {{Persistent Homology}} and {{Real}}-Valued {{Functions}}},
isbn = {978-1-60558-501-7},
url = {http://doi.acm.org/10.1145/1542362.1542408},
doi = {10.1145/1542362.1542408},
abstract = {We study the problem of computing zigzag persistence of a sequence of homology groups and study a particular sequence derived from the levelsets of a real-valued function on a topological space. The result is a local, symmetric interval descriptor of the function. Our structural results establish a connection between the zigzag pairs in this sequence and extended persistence, and in the process resolve an open question associated with the latter. Our algorithmic results not only provide a way to compute zigzag persistence for any sequence of homology groups, but combined with our structural results give a novel algorithm for computing extended persistence. This algorithm is easily parallelizable and uses (asymptotically) less memory.},
booktitle = {Proceedings of the {{Twenty}}-Fifth {{Annual Symposium}} on {{Computational Geometry}}},
series = {{{SCG}} '09},
publisher = {{ACM}},
urldate = {2018-04-20},
date = {2009},
pages = {247--256},
keywords = {algorithms,extended persistence,levelset zigzag,Mayer-Vietoris pyramid,zigzag persistent homology},
author = {Carlsson, Gunnar and family=Silva, given=Vin, prefix=de, useprefix=true and Morozov, Dmitriy},
file = {/home/dimitri/Nextcloud/Zotero/storage/WNIUXA7Y/Carlsson et al. - 2009 - Zigzag Persistent Homology and Real-valued Functio.pdf}
}
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