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Persistent and Zigzag Homology: A Matrix Factorization Viewpoint
[article]
2021
arXiv
pre-print
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Two fundamental computations in this field are persistent homology and zigzag homology. ...
In this paper, we show how these computations in the most general case reduce to finding a canonical form of a matrix associated with a type A quiver representation, which in turn can be computed using ...
Acknowledgements We thank Jonathan Taylor for extensive comments and suggestions.
Conflict of interest The authors declare that they have no conflict of interest.
References ...
arXiv:1911.10693v2
fatcat:b4lv6kt6xnfkzoiohwpxjbq2uy
The persistent homology of a sampled map: From a viewpoint of quiver representations
[article]
2019
arXiv
pre-print
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Our definition of homology induced maps is given by most persistent direct summands of representations, and the direct summands uniquely determine a persistent homology. ...
This paper aims to introduce a filtration analysis of sampled maps based on persistent homology, providing a new method for reconstructing the underlying maps. ...
Escolar for important questions about the well-definedness of the persistence analysis. I would like to thank Zin Arai for his constructive suggestions and comments on the paper. ...
arXiv:1810.11774v2
fatcat:idk3niz2wrdovctam3zmxy6gqu
The persistent homology of a sampled map: from a viewpoint of quiver representations
2021
Journal of Applied and Computational Topology
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Our definition of homology induced maps is given by most persistent direct summands of representations. The direct summands uniquely determine a persistent homology. ...
AbstractThis paper is intended to introduce a filtration analysis of sampled maps based on persistent homology, providing a new method for reconstructing the underlying maps. ...
Decomposition of the zigzag persistence module as a representation yields a persistence diagram again, where each interval captures the persistence of a homology generator in the deformations of spaces ...
doi:10.1007/s41468-021-00065-3
fatcat:szyc3q2ofvawpgota7djmoifru
Spatio-temporal Persistent Homology for Dynamic Metric Spaces
[article]
2019
arXiv
pre-print
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Applying the homology functor to this filtration gives rise to multidimensional persistence module derived from the DMS. ...
Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere. ...
In [46, 47] , the thread of ideas in [11] is blended with ideas in zigzag persistence theory [14] . ...
arXiv:1812.00949v4
fatcat:sfscv666u5hsbdib2dlalv2ywm
Persistent homology and Floer–Novikov theory
2016
Geometry and Topology
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2015 pre-print arXiv:1502.07928v2 fatcat:eb5s4xhh5jcuxcewv5nreglbbuOther Versions
We moreover prove a continuity result which is a natural analogue both of the classical bottleneck stability theorem in persistent homology and of standard continuity results for spectral invariants, and ...
In the case of classical Morse theory these coincide with the barcodes familiar from persistent homology. ...
These works are based around the notion of the (zigzag) persistent homology of level sets of the function; this is a rather different viewpoint from ours, as in order to obtain insight into Floer theory ...
doi:10.2140/gt.2016.20.3333
fatcat:repiha7a7fhinkmal3coevd5xe
Parallel decomposition of persistence modules through interval bases
[article]
2021
arXiv
pre-print
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We subsequently provide a parallel algorithm to build a persistent homology module over ℝ by leveraging the Hodge decomposition, thus providing new motivation to explore the interplay between TDA and the ...
This construction works for general persistence modules on a field 𝔽, not necessarily deriving from persistent homology. ...
Acknowledgements The authors acknowledge the support from the Italian MIUR Award "Dipartimento di Eccellenza 2018-2022" -CUP: E11G18000350001 and the SmartData@PoliTO center for Big Data and Machine Learning ...
arXiv:2106.11884v1
fatcat:343bjbaanna4holjhsbjhj55qu
Generalized Persistence Diagrams for Persistence Modules over Posets
[article]
2020
arXiv
pre-print
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By specializing our idea to zigzag persistence modules, we also show that the barcode of a Reeb graph can be obtained in a purely set-theoretic setting without passing to the category of vector spaces. ...
This leads to a promotion of Patel's semicontinuity theorem about type 𝒜 persistence diagram to Lipschitz continuity theorem for the category of sets. ...
Acknowledgement The idea of studying the map from the limit to the colimit of a given diagram stems from work by Amit Patel and Robert MacPherson circa 2012. ...
arXiv:1810.11517v5
fatcat:g6hm5dglwnfjjidcgtnsd7qlue
An Introduction to Multiparameter Persistence
[article]
2022
arXiv
pre-print
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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 filtered space often does not adequately capture the structure of interest in the data, and one is led to consider multiparameter persistence, which associates to the data a space equipped ...
We especially thank our past and current collaborators in this area, who have had a major influence on this article. ...
arXiv:2203.14289v1
fatcat:ylrfxfdljrcidondwsiiqytea4
The Jones Polynomial – Diagrams and Categories
[article]
2022
arXiv
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In each case the Jones invariant appears as a key example for patterns and connections of these mathematical and physical contexts. ...
This paper is a memory of the work and influence of Vaughan Jones. It is an exposition of the remarkable breakthroughs in knot theory and low dimensional topology that were catalyzed by his work. ...
a slightly different viewpoint [44] . ...
arXiv:2204.12104v1
fatcat:7m7tcxeshbczdov3uycoqztfum
Persistence Modules on Commutative Ladders of Finite Type
[article]
2015
arXiv
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A new algebraic framework deals with persistence modules as representations on associative algebras and the Auslander-Reiten theory is applied to develop the theoretical and algorithmic foundations. ...
Furthermore, a generalization of persistence diagrams is introduced by using Auslander-Reiten quivers. ...
Acknowledgments The authors would like to thank Hideto Asashiba, Hiroyuki Ochiai, and Dai Tamaki for valuable discussions and comments. ...
arXiv:1404.7588v2
fatcat:k2lrlz4ekzafrh3roegfore5ka
Comparing Graphs via Persistence Distortion
[article]
2017
arXiv
pre-print
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This topological perspective along with the metric space viewpoint provide a new angle to the graph matching problem. ...
In this paper, we propose a new distance between two finite metric graphs, called the persistence-distortion distance, which draws upon a topological idea. ...
Below we only provide an intuitive and informal description of the persistent homology induced by a function under our simple setting. ...
arXiv:1503.07414v4
fatcat:5fzvzkot3ndnjkklgmzeiw2fpi
Hair curvature: a natural dialectic and review
2014
Biological Reviews
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Zigzag hair formation has been specifically related to the expression of Dickkopf homolog 4 (Dkk4), an EDAR target gene and a Wnt inhibitor (Sharov et al., 2006; Cui et al., 2010; see also Section V.25 ...
Gene products of KRT-IA (K31, K33a, b and K34) and KRT-IIA (K81, K83 and K86) are highly homologous
and are expressed in the hair cortex. ...
doi:10.1111/brv.12081
pmid:24617997
fatcat:qiyajpaynjhv3mhrsbhlaz5jle
Carbon Nanostructures by Macromolecular Design – From Branched Polyphenylenes to Nanographenes and Graphene Nanoribbons
2020
Faraday discussions
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Nanographenes (NGs) and graphene nanoribbons (GNRs) are unique connectors between the domains of 1D-conjugated polymers and 2D-graphenes. ...
Acknowledgements The authors extend cordial thanks to all their collaborators as well as current and previous co-workers in their group who contributed to some of the results described in this article. ...
uorescence quenching, a typical problem for large aromatic compounds, was observed, solid-state emission and emission were achieved by using a polymer matrix. ...
doi:10.1039/d0fd00023j
pmid:33290471
fatcat:x376hqed2zhqlkjgu3gxspqrbi
Networks beyond pairwise interactions: Structure and dynamics
2020
Physics reports
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We conclude with a summary of empirical applications, and an outlook on current modeling and conceptual frontiers. ...
However, a mounting body of evidence is showing that taking the higher-order structure of these systems into account can enhance our modeling capacities and help us understand and predict their dynamical ...
Other measures of shape in simplicial complexes Homology in all its variants (persistence, zigzag, multiparameter) is a powerful tool to classify structure according to key mesoscale features. ...
doi:10.1016/j.physrep.2020.05.004
fatcat:pkugz4i5obainmcrwod6i4od7q
Analytic combinatorics in several variables
2014
ChoiceReviews
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The following example may seem a natural candidate for the transfer matrix method, but it is simpler to analyze it as from the viewpoint of compositions. ...
9.5.13 from the viewpoint of coalsecing saddles. ...
Checking where H and H z have a common factor, we find an algebraic condition on u and v that appears to have no positive solutions. ...
doi:10.5860/choice.51-3299
fatcat:phaza5ah5vbmxemh2mmtchrr4u
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