🤖 AI Summary
Topological modeling of multiparameter filtration functions remains challenging due to the absence of rigorous approximation frameworks. Method: This paper introduces a theoretically sound approximation framework based on Multiparameter Discrete Morse (MDM) theory. We first prove that any vector-valued multiparameter filtration function can be algorithmically approximated by an MDM-compatible function. We introduce the notion of the Pareto set for discrete multiparameter filtrations and establish a bijective correspondence between this set and MDM critical simplices. Our approach integrates simplicial topology, vector-valued function analysis, and Pareto optimality theory. Contribution/Results: We establish a topological approximation theory for multiparameter filtrations. Experiments on triangular meshes empirically validate the geometric consistency between the Pareto set and the MDM critical structure. The framework provides both a novel tool and a computationally tractable foundation for interpretable topological analysis of high-dimensional data.
📝 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.