Ladder decomposition for morphisms of persistence modules

The output of persistent homology is an algebraic object called a persistence module. This object admits a decomposition into a direct sum of interval persistence modules described entirely by the barcode invariant. In this paper we investigate when a morphism  \(\Phi :V \rightarrow W\) $\mathrm{\Phi }:V\to W$ of persistence modules admits an analogous direct sum decomposition. Jacquard et al. (J Appl Comput Topol 66:1–30, 2022) showed that a ladder decomposition can be obtained whenever the barcodes of V and W do not have any strictly nested bars. We refine this result and show that even in the presence of nested bars, a ladder decomposition exists when the morphism is sufficiently close to being invertible relative to the scale of the nested bars.