arXiv subjects

Madjid Allili

Publications and source records attributed to Madjid Allili.

3 recordsLinked to original sources

arXiv2024

Multiparameter Discrete Morse Theory

The main objective of this paper is to extend Morse-Forman theory to vector-valued functions. This is mostly motivated by the need to develop new tools and methods to compute multiparameter persistence. To generalize the theory, in addition to adapting the main definitions and results of Forman to this vectorial setting, we use concepts of combinatorial topological dynamics studied in recent years. This approach proves to be successful in the following ways. First, we establish a result which is more general than that of Forman regarding the sublevel sets of a multidimensional discrete Morse function. Second, we find a way to induce a Morse decomposition in critical components from the critical points of such a function. Finally, we deduce a set of Morse equation and inequalities specific to the multiparameter setting.

math.GT↗

arXiv2017

A New Matching Algorithm for Multidimensional Persistence

An algorithm is presented that constructs an acyclic partial matching on the cells of a given simplicial complex from a vector-valued function defined on the vertices and extended to each simplex by taking the least common upper bound of the values on its vertices. The resulting acyclic partial matching may be used to construct a reduced filtered complex with the same multidimensional persistent homology as the original simplicial complex filtered by the sublevel sets of the function. Numerical tests show that in practical cases the rate of reduction in the number of cells achieved by the algorithm is substantial. This promises to be useful for the computation of multidimensional persistent homology of simplicial complexes filtered by sublevel sets of vector-valued functions.

cs.CG↗

arXiv2015

Reducing complexes in multidimensional persistent homology theory

The Discrete Morse Theory of Forman 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. 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.

cs.CG↗