Forman's discrete Morse theory appeared to be useful for providing filtration-preserving reductions of complexes in the study of persistent homology. So far, the algorithms computing discrete Morse matchings have only been used for one-dimensional filtrations. This paper is perhaps the first attempt in the direction of extending such algorithms to multidimensional filtrations. An initial framework related to Morse matchings for the multidimensional setting is proposed, and a matching algorithm given by King, Knudson, and Mramor is extended in this direction. The correctness of the algorithm is proved, and its complexity analyzed. The algorithm is used for establishing a reduction of a simplicial complex to a smaller but not necessarily optimal cellular complex. First experiments with filtrations of triangular meshes are presented.

Reducing complexes in multidimensional persistent homology theory / Allili, Madjid; Kaczynski, Tomasz; Landi, Claudia. - In: JOURNAL OF SYMBOLIC COMPUTATION. - ISSN 0747-7171. - 78:(2017), pp. 61-75. [10.1016/j.jsc.2015.11.020]

Reducing complexes in multidimensional persistent homology theory

LANDI, Claudia
2017

Abstract

Forman's discrete Morse theory appeared to be useful for providing filtration-preserving reductions of complexes in the study of persistent homology. So far, the algorithms computing discrete Morse matchings have only been used for one-dimensional filtrations. This paper is perhaps the first attempt in the direction of extending such algorithms to multidimensional filtrations. An initial framework related to Morse matchings for the multidimensional setting is proposed, and a matching algorithm given by King, Knudson, and Mramor is extended in this direction. The correctness of the algorithm is proved, and its complexity analyzed. The algorithm is used for establishing a reduction of a simplicial complex to a smaller but not necessarily optimal cellular complex. First experiments with filtrations of triangular meshes are presented.
78
61
75
Reducing complexes in multidimensional persistent homology theory / Allili, Madjid; Kaczynski, Tomasz; Landi, Claudia. - In: JOURNAL OF SYMBOLIC COMPUTATION. - ISSN 0747-7171. - 78:(2017), pp. 61-75. [10.1016/j.jsc.2015.11.020]
Allili, Madjid; Kaczynski, Tomasz; Landi, Claudia
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1123249
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