Analyzing Multifiltering Functions Using Multiparameter Discrete Morse Theory

TL;DR

This paper extends discrete Morse theory to analyze multiparameter filtrations, providing algorithms to approximate multifiltering functions and linking Pareto sets to critical simplices, with applications to triangular meshes.

Contribution

It introduces multiparameter discrete Morse theory, proving approximation algorithms for multifiltering functions and connecting Pareto sets to critical simplices.

Findings

Any multifiltering function can be approximated by a compatible MDM function.

The Pareto set of a multifiltering function relates directly to critical simplices.

Experimental validation on triangular meshes demonstrates practical applicability.

Abstract

A multiparameter filtration, or a multifiltration, may in many cases be seen as the collection of sublevel sets of a vector function, which we call a multifiltering function. The main objective of this paper is to obtain a better understanding of such functions through multiparameter discrete Morse (MDM) theory, which is an extension of Morse-Forman theory to vector-valued functions. Notably, we prove algorithmically that any multifiltering function defined on a simplicial complex can always be approximated by a compatible MDM function. Moreover, we define the Pareto set of a discrete multifiltering function and show that the concept links directly to that of critical simplices of a MDM function. Finally, we experiment with these notions using triangular meshes.