Volume Optimal Cycle: Tightest representative cycle of a generator on persistent homology
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Abstract (arXiv)
This paper shows a mathematical formalization, algorithms and computation software of volume optimal cycles, which are useful to understand geometric features shown in a persistence diagram. Volume optimal cycles give us concrete and optimal homologous structures, such as rings or cavities, on a given data. The key idea is the optimality on (q + 1)-chain complex for a qth homology generator. This optimality formalization is suitable for persistent homology. We can solve the optimization problem using linear programming. For an alpha filtration on ℝn, volume optimal cycles on an (n-1)-th persistence diagram is more efficiently computable using merge-tree algorithm. The merge-tree algorithm also gives us a tree structure on the diagram and the structure has richer information. The key mathematical idea is Alexander duality.
Key facts. Cited by 12 papers in the map (data through 7 Oct 2026). First posted on arXiv 14 Dec 2017.
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Citing papers per year: 2021: 2; 2022: 2; 2023: 4; 2024: 3; 2025: 1.
12 papers.
- 2021The decomposition of the higher-order homology embedding constructed from the k-Laplaciancited by 8
- 2021Minimal Cycle Representatives in Persistent Homology using Linear Programming: an Empirical Study with User's Guidecited by 6
- 2023Topological Point Cloud Clusteringcited by 5
- 2022Reduction Algorithms for Persistence Diagrams of Networks: CoralTDA and PrunITcited by 2
- 2023Predicting and Interpreting Energy Barriers of Metallic Glasses with Graph Neural Networkscited by 2
- 2022Persistent homology-based descriptor for machine-learning potential of amorphous structurescited by 1
- 2023Visualizing Topological Importance: A Class-Driven Approachcited by 1
- 2024Train-Free Segmentation in MRI with Cubical Persistent Homologycited by 1
- 2024A topological analysis of the space of recipescited by 1
- 2023Notes on Applicability of Explainable AI Methods to Machine Learning Models Using Features Extracted by Persistent Homologycited by 0
- 2024Geometric Localization of Homology Cyclescited by 0
- 2025Cycles Communities from the Perspective of Dendrograms and Gradient Samplingcited by 0
- 2025Cycles Communities from the Perspective of Dendrograms and Gradient Sampling15 Dec 2025
- 2024A topological analysis of the space of recipes12 Jun 2024
- 2024Geometric Localization of Homology Cycles5 Jun 2024
- 2024Train-Free Segmentation in MRI with Cubical Persistent Homology2 Jan 2024
- 2023Predicting and Interpreting Energy Barriers of Metallic Glasses with Graph Neural Networks8 Dec 2023
- 2023Notes on Applicability of Explainable AI Methods to Machine Learning Models Using Features Extracted by Persistent Homology15 Oct 2023
- 2023Visualizing Topological Importance: A Class-Driven Approach22 Sep 2023
- 2023Topological Point Cloud Clustering29 Mar 2023
- 2022Reduction Algorithms for Persistence Diagrams of Networks: CoralTDA and PrunIT24 Nov 2022
- 2022Persistent homology-based descriptor for machine-learning potential of amorphous structures28 Jun 2022
- 2021The decomposition of the higher-order homology embedding constructed from the k-Laplacian23 Jul 2021
- 2021Minimal Cycle Representatives in Persistent Homology using Linear Programming: an Empirical Study with User's Guide14 May 2021