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The output of persistent homology is an algebraic object called a persistencemodule. This object admits a decomposition into a direct sum of intervalpersistence modules described entirely by the barcode invariant. In this paperwe investigate when a morphism $\Phi \colon V \to W$ of persistence modulesadmits an analogous direct sum decomposition. Jacquard et al. showed that aladder decomposition can be obtained whenever the barcodes of $V$ and $W$ donot have any strictly nested bars. We refine this result and show that even inthe presence of nested bars, a ladder decomposition exists when the morphism issufficiently close to being invertible relative to the scale of the nestedbars.

Ladder Decomposition for Morphisms of Persistence Modules