Volume-Optimal Cycle: Tightest Representative Cycle of a Generator in Persistent Homology
Abstract
The present paper shows a mathematical formalization of---as well as algorithms and software for computing---volume-optimal cycles. Volume-optimal cycles are useful for understanding geometric features appearing in a persistence diagram. Volume-optimal cycles provide concrete and optimal homologous structures, such as rings or cavities, on a given dataset. The key idea is the optimality on a $(q + 1)$-chain complex for a $q$th homology generator. This optimality formalization is suitable for persistent homology. We can solve the optimization problem using linear programming. For an alpha filtration on $\mathbb{R}^n$, volume-optimal cycles on an $(n-1)$st persistence diagram are more efficiently computable using a merge-tree algorithm. The merge-tree algorithm also provides a tree structure on the diagram containing richer information than volume-optimal cycles. The key mathematical idea used here is Alexander duality.
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Information & Authors
Information
Published In
SIAM Journal on Applied Algebra and Geometry
Pages: 508 - 534
DOI: 10.1137/17M1159439
ISSN (online): 2470-6566
Copyright
© 2018, Society for Industrial and Applied Mathematics. This work is licensed under a Creative Commons Attribution 4.0 International License
History
Submitted: 14 December 2017
Accepted: 22 August 2018
Published online: 23 October 2018
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Funding Information
Innovative Structural Materials Association https://doi.org/10.13039/100010757
Core Research for Evolutional Science and Technology https://doi.org/10.13039/501100003382 : 15656429
Japan Society for the Promotion of Science https://doi.org/10.13039/501100001691 : JP 16K17638
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