Ladder Decomposition for Morphisms of Persistence Modules

TL;DR

This paper explores conditions under which morphisms between persistence modules can be decomposed into simpler components, extending previous results to cases with nested bars by considering near-invertibility.

Contribution

It refines existing ladder decomposition results by establishing when such decompositions exist even with nested bars, based on the morphism's near-invertibility.

Findings

Ladder decomposition exists when morphisms are close to invertible despite nested bars.

Refinement of conditions for ladder decomposition in persistence module morphisms.

Extension of previous results to more complex barcode configurations.

Abstract

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 $\Phi \colon V \to W$ of persistence modules admits an analogous direct sum decomposition. Jacquard et al. 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.