# On Interval Decomposability of 2D Persistence Modules

@article{Asashiba2022OnID, title={On Interval Decomposability of 2D Persistence Modules}, author={Hideto Asashiba and Micka{\"e}l Buchet and Emerson G. Escolar and Ken Nakashima and Michio Yoshiwaki}, journal={Computational Geometry}, year={2022} }

## Figures from this paper

## 15 Citations

On Approximation of $2$D Persistence Modules by Interval-decomposables

- Mathematics, Computer Science
- 2019

A new invariant for $2$D persistence modules called the compressed multiplicity is proposed and it is shown that it generalizes the notions of the dimension vector and the rank invariant.

Every 1D persistence module is a restriction of some indecomposable 2D persistence module

- MathematicsJ. Appl. Comput. Topol.
- 2020

It is given a constructive proof that any 1D persistence module with finite support can be found as a restriction of some indecomposable 2D persistence Module with finiteSupport as line restrictions.

Hyperplane Restrictions of Indecomposable $n$-Dimensional Persistence Modules

- Mathematics
- 2020

Understanding the structure of indecomposable $n$-dimensional persistence modules is a difficult problem, yet is foundational for studying multipersistence. To this end, Buchet and Escolar showed…

On the complexity of zero-dimensional multiparameter persistence

- Mathematics
- 2020

Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full…

Generalized Persistence Algorithm for Decomposing Multi-parameter Persistence Modules

- Computer ScienceJournal of Applied and Computational Topology
- 2022

A generalization of the persistence algorithm based on a generalized matrix reduction technique that runs in $O(n^{2\omega})$ time where $\omega<2.373$ is the exponent for matrix multiplication.

The Generalized Persistence Diagram Encodes the Bigraded Betti Numbers

- Computer Science, Mathematics
- 2021

The generalized persistence diagram by Kim and Mémoli encodes the bigraded Betti numbers of finite 2-parameter persistence modules and implies that all invariants of 2- parameter persistence module that are computed by the software RIVET are encoded in the generalized persistence diagrams.

Computing Generalized Rank invariant for 2-Parameter Persistence Modules via Zigzag Persistence and its Applications

- Computer Science, MathematicsArXiv
- 2021

The generalized rank over a ﬁnite interval I of a Z 2 -indexed persistence module M is equal to the generalized rank of the zigzag module that is induced on a certain path in I trac-ing mostly its boundary.

1 5 M ay 2 01 9 Every 1 D Persistence Module is a Restriction of Some Indecomposable 2 D Persistence Module

- Mathematics
- 2019

A recent work by Lesnick and Wright proposed a visualisation of 2D persistence modules by using their restrictions onto lines, giving a family of 1D persistence modules. We give a constructive proof…

Decomposition of Certain Representations Into A Direct Sum of Indecomposable Representations

- Mathematics
- 2019

Representations of quivers and posets are produced by persistent homology. It is possible to decompose such representations into direct sums of indecomposable representations. The indecomposable…

Generalized persistence diagrams for persistence modules over posets

- MathematicsJ. Appl. Comput. Topol.
- 2021

The barcode of any interval decomposable persistence modules of finite dimensional vector spaces can be extracted from the rank invariant by the principle of inclusion-exclusion, leading to a promotion of Patel's semicontinuity theorem about type $\mathcal{A}$ persistence diagram to Lipschitz continuity theorem for the category of sets.

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