Related papers: On interval decomposition of persistence modules
We show that a pointwise finite-dimensional persistence module indexed over a small category decomposes into a direct sum of indecomposables with local endomorphism rings. As an application of this result we give new, short proofs of…
Using a result by Chazal, Crawley-Boevey and de Silva concerning radicals of persistence modules, we show that every lower semi-continuous q-tame persistence module can be decomposed as a direct sum of interval modules and that every upper…
We show that a persistence module (for a totally ordered indexing set) consisting of finite-dimensional vector spaces is a direct sum of interval modules. The result extends to persistence modules with the descending chain condition on…
The study of persistent homology has contributed new insights and perspectives into a variety of interesting problems in science and engineering. Work in this domain relies on the result that any finitely-indexed persistence module of…
We show that every infinite zigzag persistence module decomposes into a direct sum of interval persistence modules.
Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules,…
The bedrock of persistence theory over a single parameter is decomposition of persistence modules into intervals. In [HLM24], the authors leveraged interval decomposition to produce a cell decomposition of the minimal model of a simply…
Persistence modules that decompose into interval modules are important in topological data analysis because we can interpret such intervals as the lifetime of topological features in the data. We can classify the settings in which…
For any persistence module $M$ over a finite poset $\mathbf{P}$, and any interval $I$ of $\mathbf{P}$, we give a formula for the multiplicity $d_M(V_I)$ of the interval module $V_I$ in the indecomposable decomposition of $M$ in terms of the…
The algebraic stability theorem for $\mathbb{R}$-persistence modules is a fundamental result in topological data analysis. We present a stability theorem for $n$-dimensional rectangle decomposable persistence modules up to a constant…
We develop a unifying framework for the treatment of various persistent homology architectures using the notion of correspondence modules. In this formulation, morphisms between vector spaces are given by partial linear relations, as…
We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified…
In persistent homology analysis, interval modules play a central role in describing the birth and death of topological features across a filtration. In this work, we extend this setting, and propose the use of bipath persistent homology,…
In the persistent homology of filtrations, the indecomposable decompositions provide the persistence diagrams. However, in almost all cases of multidimensional persistence, the classification of all indecomposable modules is known to be a…
Persistence modules serve as the algebraic foundation for topological data analysis, typically studied as representations of posets over a field. This article extends the structural and decomposition theory of persistence modules to the…
Multiparameter persistence modules can be uniquely decomposed into indecomposable summands. Among these indecomposables, intervals stand out for their simplicity, making them preferable for their ease of interpretation in practical…
It is shown the construction of a module structure [2] with universe over a set of a particular kind of mathematical proofs, the base ring of this module will be built on a maximal consistent extension of a set of propositions, this…
The literature in persistent homology often refers to a "structure theorem for finitely generated graded modules over a graded principal ideal domain". We clarify the nature of this structure theorem in this context.
A persistence module is a functor $f: \mathbf{I} \to \mathsf{E}$, where $\mathbf{I}$ is the poset category of a totally ordered set. This work introduces saecular decomposition: a categorically natural method to decompose $f$ into simple…
We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of…