Abstract
In this work, we discuss the problem of finding optimal cycles for homology groups of simplicial complexes and for persistent homology of filtrations. We review the linear programming formulation of the optimal homologous cycle problem and its extension to allow for multiple cycles. By inserting these linear programming problems into the persistent homology algorithm, we are able to compute an optimal cycle, that has been optimized at birth, for every persistent interval in the persistent diagram.
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Acknowledgments
The authors would like to thank Genki Kusano and Hiroshi Takeuchi for interesting discussions, especially from the side of applications, and Prof. Hayato Waki for the chance to give this talk in the workshop “Optimization in the Real World—Towards Solving Real-World Optimization Problems”.
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Escolar, E.G., Hiraoka, Y. (2016). Optimal Cycles for Persistent Homology Via Linear Programming. In: Fujisawa, K., Shinano, Y., Waki, H. (eds) Optimization in the Real World. Mathematics for Industry, vol 13. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55420-2_5
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DOI: https://doi.org/10.1007/978-4-431-55420-2_5
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