# Volume Optimal Cycle: Tightest representative cycle of a generator on persistent homology

arXiv: https://arxiv.org/abs/1712.05103 · HTML: https://mlsanity.com/paper/1712.05103/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Dec 2017.

## Built on by (most cited first, 12 of 12)

- [The decomposition of the higher-order homology embedding constructed from the k-Laplacian](https://mlsanity.com/paper/2107.10970/) (2021)
- [Minimal Cycle Representatives in Persistent Homology using Linear Programming: an Empirical Study with User's Guide](https://mlsanity.com/paper/2105.07025/) (2021)
- [Topological Point Cloud Clustering](https://mlsanity.com/paper/2303.16716/) (2023)
- [Reduction Algorithms for Persistence Diagrams of Networks: CoralTDA and PrunIT](https://mlsanity.com/paper/2211.13708/) (2022)
- [Predicting and Interpreting Energy Barriers of Metallic Glasses with Graph Neural Networks](https://mlsanity.com/paper/2401.08627/) (2023)
- [Persistent homology-based descriptor for machine-learning potential of amorphous structures](https://mlsanity.com/paper/2206.13727/) (2022)
- [Visualizing Topological Importance: A Class-Driven Approach](https://mlsanity.com/paper/2309.13185/) (2023)
- [Train-Free Segmentation in MRI with Cubical Persistent Homology](https://mlsanity.com/paper/2401.01160/) (2024)
- [A topological analysis of the space of recipes](https://mlsanity.com/paper/2406.09445/) (2024)
- [Notes on Applicability of Explainable AI Methods to Machine Learning Models Using Features Extracted by Persistent Homology](https://mlsanity.com/paper/2310.09780/) (2023)
- [Geometric Localization of Homology Cycles](https://mlsanity.com/paper/2406.03183/) (2024)
- [Cycles Communities from the Perspective of Dendrograms and Gradient Sampling](https://mlsanity.com/paper/2512.12974/) (2025)

## 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.