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 \ L), we define operations -- reductions, perforations, coreductions, and coperforations -- for efficient computation. 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, as it is common in topological data analysis. We show that we retrieve existing methods when the vertex map is injective; in this case, the complex partitions into lower stars, facilitating parallel processing. Thus, this paper proposes simple, flexible, and computationally efficient approaches to obtain Morse sequences from arbitrary stack functions, allowing to generalize previous approaches dedicated to computing gradient vector fields from injective vertex maps.