Taking a discrete approach to functions and dynamical systems, this paper integrates the combinatorial gradients in Forman's discrete Morse theory with persistent homology to forge a unified approach to function simplification. The two crucial ingredients in this effort are the Lefschetz complex, which focuses on the homology at the expense of the geometry of the cells, and the shallow pairs, which are birth-death pairs that can double as vectors in discrete Morse theory. The main new concept is the depth poset on the birth-death pairs, which captures all simplifications achieved through canceling shallow pairs. One of its linear extensions is the ordering by persistence.

翻译：采用函数与动力系统的离散化方法，本文将福尔曼离散莫尔斯理论中的组合梯度与持续同调相结合，构建了函数简化的统一框架。该工作的两个关键要素是：莱夫谢茨复形（以牺牲胞腔几何为代价聚焦同调结构）与浅对（可作为离散莫尔斯理论中向量的出生-死亡对）。核心新概念是建立在出生-死亡对上的深度偏序集，它捕捉了通过消去浅对所实现的所有简化过程。该偏序集的一种线性延拓即为按持续性排序。