Zigzag persistent homology and real-valued functions
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.
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- Authors
- Carlsson, G., de Silva, V., & Morozov, D.
- Affiliations
- Stanford University (US)
- Pomona College, Claremont, CA, USA
- Venue
- Proceedings of the twenty-fifth annual symposium on Computational geometry
- Publisher
- ACM
- Published
- 2009-06-08
- Pages
- 247-256
- OpenAlex
- W2158407067
- Cited by
- 272
- References
- 17
- Citations by year
- 2026: 3 · 2025: 9 · 2024: 8 · 2023: 10 · 2022: 7 · 2021: 19 · 2020: 18 · 2019: 14 · 2018: 15 · 2017: 16 · 2016: 10 · 2015: 93 · 2014: 14 · 2013: 13 · 2012: 9
- FWCI
- 6.33
- Type
- conference paper
- Language
- en
- Open access
- closed
- Locations
- Annual Symposium on Computational Geometry (SoCG) (publishedVersion)
- Funders
- Pomona College
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
- Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Keywords
- zigzag persistence, persistent homology, level sets, real functions, homology groups, computational topology, parallel algorithms, memory-efficient algorithms
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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