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