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Multidimensional Discrete Morse Function for Persistent Homology Computation Kaczynski, Tomasz


Our primary motivation for persistent homology is in its applications to shape similarity measures. Multidimensional or multiparameter persistence comes into play in that context when two objects are to be simultaneously compared according to several features. The ideas go back to early 1900s when Paretoâ s optimal points of multiple functions were studied with applications to economy on mind. In our previous work, we developed an algorithm that produces an acyclic partial matching (A, B, C) on the cells of a given simplicial complex, in the way that it is compatible with a vector-valued function given on its vertices. 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. Until now, any simplex added to C by our algorithm has been defined as critical. It was legitimate to do so, because an application- driven extension of Formanâ s discrete Morse theory to multi-parameter functions has not been carried out yet. In particular, no definition of a general combinatorial critical cell has been given in this context. We now propose new definitions of a multidimensional discrete Morse function (for short, mdm function), of its gradient field, its regular and critical cells. We next show that the function f used as input for our algorithm gives rise to an mdm function g with the same order of sublevel sets and the same partial matching as the one produced by our algorithm. This is a joint work with Madjid Allili, Claudia Landi, and Filippo Masoni.

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