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.

翻译：多参数滤波（或称多重滤波）在许多情况下可视为向量函数子水平集的集合，我们称此类函数为多滤波函数。本文的主要目标是通过多参数离散Morse（MDM）理论深化对此类函数的理解，该理论是将Morse-Forman理论扩展至向量值函数的框架。特别地，我们通过算法证明：定义在单纯复形上的任意多滤波函数总可由相容的MDM函数逼近。此外，我们定义了离散多滤波函数的帕累托集，并证明该概念与MDM函数临界单纯形的概念直接关联。最后，我们通过三角网格对这些概念进行了实验验证。