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Computing Persistent Homology [PDF]
Discrete and Computational Geometry, 2004 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Afra Zomorodian +2 more
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Stratifying Multiparameter Persistent Homology [PDF]
SIAM Journal on Applied Algebra and Geometry, 2019 Minor improvements throughout. In particular: we extended the introduction, added Table 1, which gives a dictionary between terms used in PH and commutative algebra; we streamlined Section 3; we added Proposition 4.49 about the information captured by the cp-rank; we moved the code from the appendix to github.
Heather A Harrington +2 more
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Geometric Approaches to Persistent Homology
SIAM Journal on Applied Algebra and Geometry, 2022 We introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric descriptions of the "life span" or "persistence" of a homology class. As a case study, we analyze the power filtration on
Henry Adams, Baris Coskunuzer
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Relative Persistent Homology [PDF]
Discrete & Computational Geometry, 2022 AbstractThe alpha complex efficiently computes persistent homology of a point cloud $$X$$ X in Euclidean space when the dimension $$d$$ d is low. Given a subset $$A$$ A of $$X$$ X , relative Čech persistent homology
Nello Blaser, Morten Brun
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Weighted persistent homology [PDF]
Rocky Mountain Journal of Mathematics, 2018 In this paper we develop the theory of weighted persistent homology. In 1990, Robert J. Dawson was the first to study in depth the homology of weighted simplicial complexes. We generalize the definitions of weighted simplicial complex and the homology of weighted simplicial complex to allow weights in an integral domain $R$. Then we study the resulting
Chengyuan Wu, Jie Wu
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Homological properties of persistent homology
Journal of Algebra and Its ApplicationsIn this paper, we investigate to what extent persistent homology benefits from the properties of a homology theory. We show that persistent homology benefits from a Mayer–Vietoris sequence and a long exact sequence for a pair if one works with graded persistence modules.
Hani̇fe Varlı +2 more
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The persistent homology of genealogical networks. [PDF]
Appl Netw Sci, 2023 AbstractGenealogical networks (i.e. family trees) are of growing interest, with the largest known data sets now including well over one billion individuals. Interest in family history also supports an 8.5 billion dollar industry whose size is projected to double within 7 years [FutureWise report HC-1137]. Yet little mathematical attention has been paid
Boyd ZM +6 more
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G-invariant persistent homology [PDF]
Mathematical Methods in the Applied Sciences, 2014 Classical persistent homology is not tailored to study the action of transformation groups different from the group Homeo(X) of all self-homeomorphisms of a topological space X. In order to obtain better lower bounds for the natural pseudo-distance d_G associated with a subgroup G of Homeo(X), we need to adapt persistent homology and consider G ...
Frosini, Patrizio, Patrizio Frosini
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SIAM Journal on Applied Algebra and Geometry, 2022 Final ...
Saugata Basu, Nathanael Cox
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Fast computation of persistent homology representatives with involuted persistent homology
Foundations of Data Science, 2023 Persistent homology is typically computed through persistent cohomology. While this generally improves the running time significantly, it does not facilitate extraction of homology representatives. The mentioned representatives are geometric manifestations of the corresponding holes and often carry desirable information.
Ziga Virk
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