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Acyclic Partial Matchings for Multidimensional Persistence: Algorithm and Combinatorial Interpretation

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Abstract

Given a simplicial complex and a vector-valued function on its vertices, we present an algorithmic construction of an acyclic partial matching on the cells of the complex compatible with the given function. This implies the construction can be used to build a reduced filtered complex with the same multidimensional persistent homology as of the original one filtered by the sublevel sets of the function. The correctness of the algorithm is proved, and its complexity is analyzed. A combinatorial interpretation of our algorithm based on the concept of a multidimensional discrete Morse function is introduced for the first time in this paper. Numerical experiments show a substantial rate of reduction in the number of cells achieved by the algorithm.

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Authors and Affiliations

  1. Department of Computer Science, Bishop’s University, Sherbrooke, QC, J1M 1Z7, Canada

    Madjid Allili

  2. Département de mathématiques, Université de Sherbrooke, Sherbrooke, QC, J1K 2R1, Canada

    Tomasz Kaczynski

  3. Dipartimento di Scienze e Metodi dell’Ingegneria, Università di Modena e Reggio Emilia, Reggio, Emilia, Italy

    Claudia Landi & Filippo Masoni

Authors
  1. Madjid Allili
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  2. Tomasz Kaczynski
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  3. Claudia Landi
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  4. Filippo Masoni
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Correspondence to Madjid Allili.

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Tomasz Kaczynski was partially supported by NSERC Canada Discovery Grant. Claudia Landi was supported by INdAM-GNSAGA.

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Allili, M., Kaczynski, T., Landi, C. et al. Acyclic Partial Matchings for Multidimensional Persistence: Algorithm and Combinatorial Interpretation. J Math Imaging Vis 61, 174–192 (2019). https://doi.org/10.1007/s10851-018-0843-8

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  • DOI: https://doi.org/10.1007/s10851-018-0843-8

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