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.
Highlights
The persistent homology has been intensely developed in the last decade as a tool for studying problems of two kinds
The 0–dimensional persistent homology case, where the topological invariants are based on the number of connected components, where known under the name of the size function theory since the paper by Frosini [17]
If a reduction by Morse pairings can be performed in a filtration–preserving way, that leads to a faster persistent homology computation
Summary
The persistent homology has been intensely developed in the last decade as a tool for studying problems of two kinds. If a reduction by Morse pairings can be performed in a filtration–preserving way, that leads to a faster persistent homology computation This goal motivated the contributions of King, Knudson, and Mramor [21], Mischaikow and Nanda [23], Robins et al [25], and Dłotko and Wagner [13]. As pointed out in [23] for the one dimensional case, the complexity of computing multidimensional persistence homology of a filtration is essentially determined by the sizes of its complexes. These experiments show a fair rate of reduction but not an optimal one in the sense that the remaining cells are not all relevant in the computation of persistence homology. An improvement of our methods towards the optimality is a research in progress
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