Computing gradient vector fields with Morse sequences

Abstract

We rely on the framework of Morse sequences to enable the direct computation of gradient vector fields on simplicial complexes. A Morse sequence is a filtration from a subcomplex $L$ to a complex $K$ via elementary expansions and fillings, naturally encoding critical and regular simplexes. Maximal increasing and minimal decreasing schemes allow constructing these sequences, and are linked to algorithms like Random Discrete Morse and Coreduction. Extending the approach to cosimplicial complexes ($S=K\setminus L$) allows for efficient computation using reductions, perforations, coreductions, and coperforations. We further generalize to $F$-sequences, which are Morse sequences weighted by an arbitrary stack function $F$, and provide algorithms to compute maximal and minimal sequences. A particular case is when the stack function is given through a vertex map, common in topological data analysis. For injective maps, the complex decomposes into lower stars, recovering established methods and enabling parallel computation; for non-injective maps, our approach applies directly without requiring perturbations. Thus, the paper adopts Morse sequences as a framework that simplifies and connects some important existing propagation-based methods, while also introducing new schemes that extend their scope and practical applicability.