## 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 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 \BbbR ^{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.

Original language | English |
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Pages (from-to) | 508-534 |

Number of pages | 27 |

Journal | SIAM Journal on Applied Algebra and Geometry |

Volume | 2 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2018 |

Externally published | Yes |

## Keywords

- Algebraic topology
- Optimization
- Persistent homology

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology
- Applied Mathematics