Reducing complexes in multidimensional persistent homology theory. (January 2017)

• Record Type: Journal Article
• Authors: Allili, Madjid; Kaczynski, Tomasz; Landi, Claudia
• Is Part Of: Journal of symbolic computation. Volume 78(2017)
• Page Start: 61
• Page End: 75
• Publication Date: 2017-01
• Subjects: Multidimensional persistent homology -- Discrete Morse theory -- Acyclic partial matchings -- Matching algorithm; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2015.11.020
• Languages: English
• ISSNs: 0747-7171

Abstract

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.