Results 1 to 10 of about 5,905 (108)

Approximation by interval-decomposables and interval resolutions of persistence modules [PDF]

Journal of Pure and Applied Algebra, Vol 227, no.10 (2023): 107397, 2022
In topological data analysis, two-parameter persistence can be studied using the representation theory of the 2d commutative grid, the tensor product of two Dynkin quivers of type A. In a previous work, we defined interval approximations using restrictions to essential vertices of intervals together with Mobius inversion.
arxiv   +1 more source

Matrix Method for Persistence Modules on Commutative Ladders of Finite Type [PDF]

Asashiba, H., Escolar, E.G., Hiraoka, Y., Takeuchi, H. "Matrix Method for Persistence Modules on Commutative Ladders of Finite Type" Japan J. Indust. Appl. Math. (2018). https://doi.org/10.1007/s13160-018-0331-y, 2017
The theory of persistence modules on the commutative ladders $CL_n(\tau)$ provides an extension of persistent homology. However, an efficient algorithm to compute the generalized persistence diagrams is still lacking. In this work, we view a persistence module $M$ on $CL_n(\tau)$ as a morphism between zigzag modules, which can be expressed in a block ...
arxiv   +1 more source

Interleavings and Matchings as Representations [PDF]

arXiv, 2020
In order to better understand and to compare interleavings between persistence modules, we elaborate on the algebraic structure of interleavings in general settings. In particular, we provide a representation-theoretic framework for interleavings, showing that the category of interleavings under a fixed translation is isomorphic to the representation ...
arxiv  

Hierarchical structures of amorphous solids characterized by persistent homology [PDF]

Proceedings of the National Academy of Sciences (2016) vol. 113 no. 26, 7035-7040, 2015
This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the PDs is given by an atomic configuration and the output is expressed as 2D histograms.
arxiv   +1 more source

The Whole in the Parts: Putting $n$D Persistence Modules Inside Indecomposable $(n + 1)$D Ones [PDF]

arXiv, 2020
Multidimensional persistence has been proposed to study the persistence of topological features in data indexed by multiple parameters. In this work, we further explore its algebraic complications from the point of view of higher dimensional indecomposable persistence modules containing lower dimensional ones as hyperplane restrictions.
arxiv  

Summand-injectivity of interval covers and monotonicity of interval resolution global dimensions [PDF]

arXiv, 2023
Recently, there is growing interest in the use of relative homology algebra to develop invariants using interval covers and interval resolutions (i.e., right minimal approximations and resolutions relative to interval-decomposable modules) for multi-parameter persistence modules. In this paper, the set of all interval modules over a given poset plays a
arxiv  

Composición de relaciones y $τ$-factorizaciones [PDF]

Revista SICES, 2da. Ed. (2019), 2020
The theory of $\tau$-factorizations on integral domains was developed by Anderson and Frazier. This theory characterized all the known factorizations and opened the opportunity to create new ones. It can be visualized as a restriction to the structure's multiplicative operation, by considering a symmetric relation $\tau$ on the set of non-zero non-unit
arxiv  

Hyperplane Restrictions of Indecomposable $n$-Dimensional Persistence Modules [PDF]

arXiv, 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 that any finitely presented rectangular $(n-1)$-dimensional persistence module with finite support is a hyperplane restriction of an $n$-dimensional ...
arxiv  

Persistence Modules on Commutative Ladders of Finite Type [PDF]

arXiv, 2014
We study persistence modules defined on commutative ladders. This class of persistence modules frequently appears in topological data analysis, and the theory and algorithm proposed in this paper can be applied to these practical problems. A new algebraic framework deals with persistence modules as representations on associative algebras and the ...
arxiv  

Every 1D Persistence Module is a Restriction of Some Indecomposable 2D Persistence Module [PDF]

arXiv, 2019
A recent work by Lesnick and Wright proposed a visualisation of $2$D persistence modules by using their restrictions onto lines, giving a family of $1$D persistence modules. We give a constructive proof that any $1$D persistence module with finite support can be found as a restriction of some indecomposable $2$D persistence module with finite support ...
arxiv