Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology

Abstract

The main objective of this paper is to introduce and study a notion of perfectness for discrete gradient vector fields with respect to (multi-parameter) persistent homology. As a natural generalization of usual perfectness in Morse theory for homology, persistence-perfectness entails having the least number of critical cells relevant for persistent homology. The first result about a persistence-perfect gradient vectorfield is that the number of its critical cells yield inequalities bounding the Betti tables of persistence modules, as a sort of Morse inequalities for multi-parameter persistence. The second result is that, at least in low dimensions, persistence-perfect gradient vector-fields not only exist but can be constructed by an algorithm based on local homotopy expansions. These results show a link between multi-parameter persistence and discrete Morse theory that can be leveraged for a better understanding of the former.

Critical Sets of PL and Discrete Morse Theory: a Correspondence

“…In particular, Robins et al in [15] show that such optimality is achievable for cubical complexes up to dimension 3. In the present paper, we rephrase this kind of optimality in terms of relative homology, hence calling it relative perfecteness, and prove that it is also achievable for simplicial complexes up to dimension 3, improving [10].…”