The algebraic stability theorem for persistence modules is a central result in the theory of stability for persistent homology. We introduce a new proof technique which we use to prove a stability theorem for n-dimensional rectangle decomposable persistence modules up to a constant \(2n-1\) that generalizes the algebraic stability theorem, and give an example showing that the bound cannot be improved for \(n=2\). We then apply the technique to prove stability for block decomposable modules, from which novel results for zigzag modules and Reeb graphs follow. These results are improvements on weaker bounds in previous work, and the bounds we obtain are optimal.

持久模块的代数稳定性定理是持久同源性稳定性理论的中心结果。我们引入了一种新的证明技术，用于证明n维矩形可分解的持久性模块的一个稳定性定理，直到一个常数\（2n-1 \），该定理推广了代数稳定性定理，并给出了一个示例，证明了边界不能为针对\（n = 2 \）进行了改进。然后，我们应用该技术来证明块可分解模块的稳定性，由此产生之字形模块和Reeb图的新颖结果。这些结果是对先前工作中较弱界限的改进，并且我们获得的界限是最佳的。