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We study the interplay between algebraic and metric properties ofmulti-parameter persistence modules (linear representations of the poset$\mathbf{R}^n$ with $n \geq 2$). In the context of topological data analysisand inference, a fundamental property of one-parameter persistence modules(linear representations of $\mathbf{R}$) is that the operation of decomposingthem into their indecomposable summands is Lipschitz continuous, under suitablechoices of metrics on modules and on collections of indecomposables. This isfar from true of multi-parameter persistence modules: We show that any finitelypresentable multi-parameter persistence module can be approximated arbitrarilywell by an indecomposable module, and that modules that are sufficiently closeto an indecomposable decompose as a direct sum of an indecomposable and anearly trivial module. These results have several consequences of interest,which provide strong motivation for approaching multi-parameter persistence ina way that does not rely on directly decomposing modules by indecomposables.

Generic Multi-Parameter Persistence Modules are Nearly Indecomposable