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

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 mergetree algorithm also gives us a tree structure on the diagram and the structure has richer information. The key mathematical idea is Alexander duality.