Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology
The goal of the paper under review is to develop techniques to ease multi-parameter persistence computations. Inspired by the success of discrete Morse theory as a tool to preprocess in the classical persistence, the authors introduce the notion of a relative-perfect discrete gradient vector field. The authors prove that a relative-perfect discrete gradient vector field has the least number of critical cells for multi-parameter persistence, the natural extension of the defining property of perfect discrete Morse functions. The authors further show a generalization of the classical discrete Morse inequalities by considering a filtration structure. This yields a pair of inequalities by using the number of critical cells to bound the Betti tables of persistence modules.
scientific article; zbMATH DE number 7603271
scientific article