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A minimising principle for interval covers of zigzag persistence modules
Abstract
Zigzag persistence modules serve as an essential tool in topological data analysis. The fundamental theorem of persistent homology states that any zigzag persistence module decomposes into a direct sum of simple building blocks called interval modules. We consider an interval cover, a possibly redundant family of interval submodules spanning every vertex space of a finite zigzag persistence module. We give an elementary descent argument that replaces a redundant cover with a minimal one that is an interval decomposition. This provides an optimisation perspective on persistence modules and their interval decomposition. As an application, we give an estimate of the generalised rank invariant in terms of a cover.
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BibTeX
@misc{chu2026minimising,
title = {{A minimising principle for interval covers of zigzag persistence modules}},
author = {Junyan Chu and Shizuo Kaji and Meiyan Kang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36843},
eprint = {2610.36843},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36843},
}
RIS
TY - UNPB
TI - A minimising principle for interval covers of zigzag persistence modules
AU - Junyan Chu
AU - Shizuo Kaji
AU - Meiyan Kang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36843
UR - https://arxiv.org/abs/2610.36843
AB - Zigzag persistence modules serve as an essential tool in topological data analysis. The fundamental theorem of persistent homology states that any zigzag persistence module decomposes into a direct sum of simple building blocks called interval modules. We consider an interval cover, a possibly redundant family of interval submodules spanning every vertex space of a finite zigzag persistence module. We give an elementary descent argument that replaces a redundant cover with a minimal one that is an interval decomposition. This provides an optimisation perspective on persistence modules and their interval decomposition. As an application, we give an estimate of the generalised rank invariant in terms of a cover.
ER -