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

@article{Asashiba2019OnAO, title={On Approximation of \$2\$D Persistence Modules by Interval-decomposables}, author={Hideto Asashiba and Emerson G. Escolar and Ken Nakashima and Michio Yoshiwaki}, journal={arXiv: Representation Theory}, year={2019} }

In this work, we propose a new invariant for $2$D persistence modules called the compressed multiplicity and show that it generalizes the notions of the dimension vector and the rank invariant. In addition, we propose an "interval-decomposable approximation" $\delta^{\ast}(M)$ of a $2$D persistence module $M$. In the case that $M$ is interval-decomposable, we show that $\delta^{\ast}(M) = M$. Furthermore, even for representations $M$ not necessarily interval-decomposable, $\delta^{\ast}(M…

## 10 Citations

Hyperplane Restrictions of Indecomposable $n$-Dimensional Persistence Modules

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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.

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

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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.

The Generalized Persistence Diagram Encodes the Bigraded Betti Numbers

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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.

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

- Mathematics, Computer ScienceArXiv
- 2021

The signed barcode is introduced, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or, more generally, of a poset representation that has its roots in algebra.

An Introduction to Multiparameter Persistence

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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…

Elder-Rule-Staircodes for Augmented Metric Spaces

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A barcode-like summary, called the elder-rule-staircode, is proposed as a way to encode H0(K), a 2-parameter persistent homology in an augmented metric space, and an efficient algorithm is developed to compute it in O(n2 log n) time.

Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces

- Mathematics, Computer ScienceJournal of Applied and Computational Topology
- 2022

It is shown that the 1-Wasserstein distance extends to virtual persistence diagrams and to signed measures, and the Cauchy completion of persistence diagrams with respect to the Wasserstein distances is characterized.

On the stability of multigraded Betti numbers and Hilbert functions

- Mathematics, Computer ScienceArXiv
- 2021

A stability result for multigraded Betti numbers is proved, using an efﬁciently computable bottleneck-type dissimilarity function, and the notion of matching is inspired by recent work on signed barcodes, and allows matching bars of the same module in homological degrees of different parity.

Elder-rule-staircodes for Augmented Metric Spaces

- MathematicsSoCG
- 2020

It is shown that if $\mathrm{H}_0(\mathcal{K})$ is interval decomposable, then the barcode of $H_0(n)$ is equal to the elder-rule-staircode, which is equivalent to $n$ number of staircase-like blocks in the plane.

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