Volume Optimal Cycle: Tightest representative cycle of a generator on persistent homology
Abstract
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 -chain complex for a 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 , volume optimal cycles on an -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.
Keywords
Cite
@article{arxiv.1712.05103,
title = {Volume Optimal Cycle: Tightest representative cycle of a generator on persistent homology},
author = {Ippei Obayashi},
journal= {arXiv preprint arXiv:1712.05103},
year = {2017}
}