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：本文采用离散方法处理函数与动力系统，将Forman离散Morse理论中的组合梯度与持续同调相结合，构建了统一的函数简化框架。在此过程中，两个关键要素为：牺牲胞腔几何特性以聚焦同调结构的Lefschetz复形，以及可兼作离散Morse理论向量的浅对（出生-死亡对）。核心创新概念是建立在出生-死亡对上的深度偏序集，该结构刻画了通过消去浅对所实现的所有简化过程。持续性的序关系是其一类线性扩展。