Matrix method for persistence modules on commutative ladders of finite type

@article{Asashiba2018MatrixMF,
title={Matrix method for persistence modules on commutative ladders of finite type},
author={Hideto Asashiba and Emerson G. Escolar and Yasuaki Hiraoka and Hiroshi Takeuchi},
journal={Japan Journal of Industrial and Applied Mathematics},
year={2018},
volume={36},
pages={97-130}
}
• Published 30 June 2017
• Mathematics
• Japan Journal of Industrial and Applied Mathematics
The theory of persistence modules on the commutative ladders $$CL_n(\tau )$$CLn(τ) 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 $$CL_n(\tau )$$CLn(τ) as a morphism between zigzag modules, which can be expressed in a block matrix form. For the representation finite case ($$n\le 4$$n≤4), we provide an algorithm that uses certain permissible row and…
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References

SHOWING 1-10 OF 15 REFERENCES
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.
Zigzag persistent homology and real-valued functions
• Mathematics
SCG '09
• 2009
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.
Constructing Homomorphism Spaces and Endomorphism Rings
• Mathematics, Computer Science
J. Symb. Comput.
• 2001
A new deterministic algorithm for constructing homomorphism spaces and endomorphism rings of finite-dimensional modules of right Grobner bases is presented, implemented in the computer algebra system GAP and included in Hopf, a computational package for noncommutative algebra.
Computing Persistent Homology
• Mathematics
• 2005
Abstract We show that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis
Elements of the Representation Theory of Associative Algebras: Techniques of Representation Theory
• Mathematics
• 2006
This first part of a two-volume set offers a modern account of the representation theory of finite dimensional associative algebras over an algebraically closed field. The authors present this topic
The Theory of Multidimensional Persistence
• Mathematics
Discret. Comput. Geom.
• 2009
The rank invariant is proposed, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and it is shown that no similar complete discrete invariant exists for multidimensional persistence.
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.
Computing Decompositions of Modules over Finite-Dimensional Algebras
• Mathematics, Computer Science
Exp. Math.
• 2007
An algorithm to compute decompositions of modules of finite-dimensional algebras over finite fields is described and implemented in the C-Meat-Axe.
Polynomial time algorithms for modules over finite dimensional algebras
• Mathematics
ISSAC
• 1997
We present polynomial time algorithms for some fundamental tasks from representation theory of finite dimensional algebras. These involve testing (and constructing) isomorphisms of modules as well as
Unzerlegbare Darstellungen I
LetK be the structure got by forgetting the composition law of morphisms in a given category. A linear representation ofK is given by a map V associating with any morphism ϕ: a→e ofK a linear vector