Volume-Optimal Cycle: Tightest Representative Cycle of a Generator in Persistent Homology

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.