• Corpus ID: 207870987

# 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}
}
• Published 5 November 2019
• Mathematics, Computer Science
• arXiv: Representation Theory
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, $\delta^{\ast}(M… ## Tables from this paper Hyperplane Restrictions of Indecomposable$n$-Dimensional Persistence Modules Understanding the structure of indecomposable$n$-dimensional persistence modules is a difficult problem, yet is foundational for studying multipersistence. To this end, Buchet and Escolar showed Generalized persistence diagrams for persistence modules over posets • Mathematics J. Appl. Comput. Topol. • 2021 The barcode of any interval decomposable persistence modules of finite dimensional vector spaces can be extracted from the rank invariant by the principle of inclusion-exclusion, leading to a promotion of Patel's semicontinuity theorem about type$\mathcal{A}$persistence diagram to Lipschitz continuity theorem for the category of sets. Computing Generalized Rank invariant for 2-Parameter Persistence Modules via Zigzag Persistence and its Applications • Computer Science, Mathematics ArXiv • 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. 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. Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions and Rank-Exact Resolutions • Mathematics, Computer Science ArXiv • 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 • Mathematics ArXiv • 2022 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 • Mathematics SIAM J. Appl. Algebra Geom. • 2021 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 Science Journal 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 Science ArXiv • 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 • Mathematics SoCG • 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$number of staircase-like blocks in the plane. ## References SHOWING 1-10 OF 32 REFERENCES Generalized persistence diagrams • A. Patel • Mathematics J. Appl. Comput. Topol. • 2018 The persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer is generalized to the setting of constructible persistence modules valued in a symmetric monoidal category and a second type of persistence diagram is defined, which enjoys a stronger stability theorem. On Interval Decomposability of 2D Persistence Modules • Mathematics Computational Geometry • 2022 Generalized persistence diagrams for persistence modules over posets • Mathematics J. Appl. Comput. Topol. • 2021 The barcode of any interval decomposable persistence modules of finite dimensional vector spaces can be extracted from the rank invariant by the principle of inclusion-exclusion, leading to a promotion of Patel's semicontinuity theorem about type$\mathcal{A}\$ persistence diagram to Lipschitz continuity theorem for the category of sets.
Persistence Modules on Commutative Ladders of Finite Type
• Mathematics
Discret. Comput. Geom.
• 2016
It is proved that the commutative ladders of length less than 5 are representation-finite and explicitly show their Auslander–Reiten quivers.
The theory of multidimensional persistence
• Mathematics
SCG '07
• 2007
This paper proposes the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and proves its completeness in one dimension.
Proximity of persistence modules and their diagrams
• Mathematics, Computer Science
SCG '09
• 2009
This paper presents new stability results that do not suffer from the restrictions of existing stability results, and makes it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence.
Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension
• Mathematics
SoCG
• 2018
This work proposes a simple algebraic construction to illustrate the existence of infinite families of indecomposable persistence modules over regular grids of sufficient size and provides realizations by topological spaces and Vietoris-Rips filtrations, showing that they can actually appear in real data and are not the product of degeneracies.
Decomposition theory of modules: the case of Kronecker algebra
• Mathematics
• 2017
Let A be a finite-dimensional algebra over an algebraically closed field $$\Bbbk$$k. For any finite-dimensional A-module M we give a general formula that computes the indecomposable decomposition of
On the representation type of tensor product algebras
The representation type of tensor product algebras of finite-dimensional algebras is considered. The characterization of algebras A, B such that A⊗B is of tame representation type is given in terms