The DOI matches its Crossref record; 17 of the 17 references it cites were found in a registry or an index.
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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Citations
Cites
A Global Geometric Framework for Nonlinear Dimensionality Reduction
Tenenbaum, J. B., Silva, V. D., & Langford, J. C. (2000) · Science · doi:10.1126/science.290.5500.2319 · Cited by 13,585
(1975) · CERN Document Server (European Organization for Nuclear Research) · Cited by 5,958
(2001) · Cited by 5,374
Carlsson, G. (2009) · Bulletin of the American Mathematical Society · doi:10.1090/s0273-0979-09-01249-x · Cited by 2,302 · cited 1×
“We claim that diagrams of other shapes should also be useful in understanding point cloud data [2, 3].”
BackgroundSelf-citationTopological Persistence and Simplification
Edelsbrunner, Letscher, & Zomorodian (2002) · Discrete & Computational Geometry · doi:10.1007/s00454-002-2885-2 · Cited by 1,779
Zomorodian, A., & Carlsson, G. (2005) · Discrete & Computational Geometry · doi:10.1007/s00454-004-1146-y · Cited by 1,591
Self-citationStability of Persistence Diagrams
Cohen-Steiner, D., Edelsbrunner, H., & Harer, J. (2007) · Discrete & Computational Geometry · doi:10.1007/s00454-006-1276-5 · Cited by 1,280
Gabriel, P. (1972) · manuscripta mathematica · doi:10.1007/bf01298413 · Cited by 771
Proximity of persistence modules and their diagrams
Chazal, F., Cohen‐Steiner, D., Glisse, M., Guibas, L. J., & Oudot, S. (2009) · Annual Symposium on Computational Geometry (SoCG) · doi:10.1145/1542362.1542407 · Cited by 411
Carlsson, G., & de Silva, V. (2010) · Foundations of Computational Mathematics · doi:10.1007/s10208-010-9066-0 · Cited by 263
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Research works printed but not linked: 2
- Abdi Herve, 2007, Sage
- Carlsson Gunnar, 2008, Stanford University, 0812
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Metadata
- Authors
- Carlsson, G., de Silva, V., & Morozov, D.
- Affiliations
- Stanford University (US)
- Pomona College (US)
- 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
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