Local Characterizations for Decomposability of 2-Parameter Persistence Modules

@article{Botnan2020LocalCF,
  title={Local Characterizations for Decomposability of 2-Parameter Persistence Modules},
  author={Magnus Bakke Botnan and Vadim Lebovici and Steve Oudot},
  journal={Algebras and Representation Theory},
  year={2020}
}
We investigate the existence of sufficient local conditions under which representations of a given poset will be guaranteed to decompose as direct sums of indecomposables from a given class. Our indecomposables of interest belong to the so-called interval modules, which by definition are indicator representations of intervals in the poset. In contexts where the poset is the product of two totally ordered sets (which corresponds to the setting of 2-parameter persistence in topological data… 
[PDF] Semantic Reader
2 Citations
Background Citations
1
Methods Citations
1

Figures and Tables from this paper

2 Citations

On interval decomposability of 2D persistence modules

Multiparameter persistence modules in the large scale

A persistence module with $m$ discrete parameters is a diagram of vector spaces indexed by the poset $\mathbb{N}^m$. If we are only interested in the large scale behavior of such a diagram, then we…

Structure and Interleavings of Relative Interlevel Set Cohomology

The relative interlevel set cohomology (RISC) is an invariant of real-valued continuous functions closely related to the Mayer–Vietoris pyramid introduced by Carlsson, de Silva, and Morozov and it is stable in the sense that any space with two functions that areδ-close induces a δ-interleaving of the corresponding relative inter level set cohomeologies.

Efficient Approximation of Multiparameter Persistence Modules

This article presents the first approximation scheme, which is based on bered barcodes and exact matchings, two constructions that stem from the theory of single-parameter persistence, for computing and decomposing general multi-parameters persistence modules.

An Introduction to Multiparameter Persistence

In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology. However, a single…

24 References

On Rectangle-Decomposable 2-Parameter Persistence Modules

Algorithms to check whether a module induced in homology by a bifiltration is rectangle-decomposable, and to decompose it in the affirmative, with a better complexity than state-of-the-art decomposition methods for general 2-parameter persistence modules.

Homological approximations in persistence theory

It is shown that both the dimension vector and rank invariant are equivalent to homological invariants taking values in groups generated by spread modules, and that the free abelian group generated by the “single-source” spread modules gives rise to a new invariant which is finer than the rank invariants.

Decomposition of Exact pfd Persistence Bimodules

We characterize the class of persistence modules indexed over $$\mathbb {R}^2$$ R 2 that are decomposable into summands whose supports have the shape of a block —i.e. a horizontal band, a vertical…

Algebraic Stability of Zigzag Persistence Modules

This paper functorially extends each zigzag persistence module to a two-dimensional persistence module, and establishes an algebraic stability theorem for these extensions, which yields a stability result for free two- dimensional persistence modules.

∞-Categories for the Working Mathematician

homotopy theory C.1. Lifting properties, weak factorization systems, and Leibniz closure C.1.1. Lemma. Any class of maps characterized by a right lifting property is closed under composition,…

Structure and Interleavings of Relative Interlevel Set Cohomology

The relative interlevel set cohomology (RISC) is an invariant of real-valued continuous functions closely related to the Mayer–Vietoris pyramid introduced by Carlsson, de Silva, and Morozov and it is stable in the sense that any space with two functions that areδ-close induces a δ-interleaving of the corresponding relative inter level set cohomeologies.

Zigzag persistent homology and real-valued functions

The algorithmic results provide a way to compute zigzag persistence for any sequence of homology groups, but combined with the structural results give a novel algorithm for computing extended persistence that is easily parallelizable and uses (asymptotically) less memory.

Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions and Rank-Exact Resolutions

The signed barcode is introduced, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or of a poset representation, and the theory behind these rank decompositions is developed.

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…

The indecomposable representations of the dihedral 2-groups

Let K be a field. We will give a complete list of the normal forms of pairs a, b of endomorphisms of a K-vector space such that a 2 b 2 = 0. Thus, we determine the modules over the ring R = K ( X ,…