Multi-Parameter Persistence Modules are Generically Indecomposable

Abstract

Algebraic persistence studies persistence modules (typically, linear representations of the poset ${\mathbf{R}}^n$ with $n\ge 1$) and the algebraic relationships between persistence modules that are interleaved. The notion of $\epsilon$-interleaving between persistence modules is a generalization of the notion of isomorphism (recovering isomorphism when $\epsilon =0$), which can be used to quantify how far any two persistence modules are from being isomorphic. An emblematic example of this kind of study is the algebraic stability theorem, which strengthens the Krull--Schmidt property of one-parameter persistence modules (representations of $\mathbf{R}$) by generalizing isomorphism to interleaving: If a pair of one-parameter persistence modules is $\epsilon$-interleaved, then there exists a partial matching between the indecomposable summands of the two modules such that matched indecomposables are $\epsilon$-interleaved and unmatched indecomposables are $\epsilon$-interleaved with the zero module. Our first main result implies that the obvious extension of the algebraic stability theorem to the case of multi-parameter persistence modules (representations of ${\mathbf{R}}^n$ with $n\ge 2$) fails spectacularly: Any finitely presentable multi-parameter persistence module can be approximated arbitrarily well by an indecomposable module. Our second main result states that modules that are sufficiently close to an indecomposable decompose as a direct sum of an indecomposable and a nearly trivial module. We derive from these two results several consequences about the interplay between the algebraic and the topological properties of multi-parameter persistence modules. These results provide strong motivation for approaching multi-parameter persistence in a way that does not rely on directly decomposing modules by indecomposables.

29 pages, 3 figures; v3: improvements to exposition

References in corpus (20)

• Accelerated Hierarchical Density Clustering
• The Theory of the Interleaving Distance on Multidimensional Persistence Modules
• Interactive Visualization of 2-D Persistence Modules
• Categorification of persistent homology
• Algebraic Stability of Zigzag Persistence Modules
• Persistent homology detects curvature
• Generalized Persistence Diagrams for Persistence Modules over Posets
• Induced Matchings and the Algebraic Stability of Persistence Barcodes
• Edit Distance and Persistence Diagrams Over Lattices
• Multidimensional Persistence and Noise
• Spatio-temporal Persistent Homology for Dynamic Metric Spaces
• Ephemeral persistence modules and distance comparison
• Topological spaces of persistence modules and their properties
• Homological approximations in persistence theory
• On the stability of multigraded Betti numbers and Hilbert functions
• Approximation by interval-decomposables and interval resolutions of persistence modules
• On the bottleneck stability of rank decompositions of multi-parameter persistence modules
• Homological algebra of modules over posets
• Exact weights, path metrics, and algebraic Wasserstein distances
• On the Universality of Random Persistence Diagrams

Cited by in corpus (1)

• Generalized Rank via Minimal Subposet