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 $(q + 1)$-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 $(n-1)$-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.