Persistence stability for geometric complexes

Chazal, Frédéric; Silva, Vin de; Oudot, Steve

doi:10.1007/s10711-013-9937-z

Cited by 214 publications

(254 citation statements)

References 18 publications

(34 reference statements)

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“…We delay the proofs of Proposition 7.5 and Theorem 7.6 until Section 8. Note that Theorems 7.4 and 7.6 together provide a complete description of the homotopy types of VR(S 1 ; r) for arbitrary r. They also give the persistent homology of VR(S 1 ; r), where we refer the reader to [9] for information on the persistent homology of Vietoris-Rips complexes. Remark 7.8.…”

Section: Moreover Ifmentioning

confidence: 99%

The Vietoris–Rips complexes of a circle

2017

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Abstract. Given a metric space X and a distance threshold r > 0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of JeanClaude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applications using persistent homology. We show that as r increases, the VietorisRips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, . . . , until finally it is contractible. As our main tool we introduce a directed graph invariant, the winding fraction, which in some sense is dual to the circular chromatic number. Using the winding fraction we classify the homotopy types of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset of the circle, and we study the expected homotopy type of the Vietoris-Rips complex of a uniformly random sample from the circle. Moreover, we show that as the distance parameter increases, the ambientČech complex of the circle (i.e. the nerve complex of the covering of a circle by all arcs of a fixed length) also obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, . . . , until finally it is contractible.

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Section: Moreover Ifmentioning

confidence: 99%

The Vietoris–Rips complexes of a circle

2017

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“…Theorem 11.31 is a particular case of a result proven in [43] and it has found various applications in shape classification [37] and in statistical analysis of data -see, e.g., [79,78,10,44].…”

Section: Persistent Homology Of a Filtrationmentioning

confidence: 99%

Geometric and Topological Inference

Boissonnat

Chazal

Yvinec

2018

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“…Given a k-simplex σ with vertices from L and a points w ∈ W we say that w is an α-witness of σ if the vertices of σ are all within d k (w) + α of w, where d k (w) is the distance from w and its (k + 1)th nearest neighbour in L. We write W α (L, W ) for this simplicial complex. Witness complexes have been further studied by Chazal and Oudot [20] and by Chazal, de Silva, and Oudot [19].…”

Section: Foundations In Usementioning

confidence: 99%

“…Theorem 20 (Proposition 5.1 of [19]). If (X, d X ) is a precompact metric space, then theČech and VietorisRips persistent homology modules are q-tame.…”

Section: Order Module View Of Interleavingmentioning

confidence: 99%

Sketches of a platypus: a survey of persistent homology and its algebraic foundations

Vejdemo-Johansson¹

2014

Algebraic Topology: Applications and New Directions

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Abstract. The subject of persistent homology has vitalized applications of algebraic topology to point cloud data and to application fields far outside the realm of pure mathematics. The area has seen several fundamentally important results that are rooted in choosing a particular algebraic foundational theory to describe persistent homology, and applying results from that theory to prove useful and important results.In this survey paper, we shall examine the various choices in use, and what they allow us to prove. We shall also discuss the inherent differences between the choices people use, and speculate on potential directions of research to resolve these differences.Johnstone [51] named his book on topos theory "Sketches of an elephant", referencing a joke: three blind wise men encounter an elephant. They each try to describe it to each other. The wise man who caught hold of the elephant's trunk says "An elephant is like a snake."; the wise man holding the ear says "An elephant is like a palm leaf."; and the wise man holding its leg says "An elephant is like a tree.".The joke is highly relevant to topos theory; which has its roots in logic, in geometry, and in topology, with the three perspectives being fundamentally different and enriching each other in surprising ways.The title of this paper is similar, but different. The platypus is well-known to be a hybrid of an animal: sharing traits both with the phylum of birds and with the phylum of mammals. The field of persistent homology is in a similar situation to the platypus: there are at least two different viewpoints of what persistent homology should be, and they interact in sometimes unexpected ways.

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Persistence stability for geometric complexes

Cited by 214 publications

References 18 publications

The Vietoris–Rips complexes of a circle

The Vietoris–Rips complexes of a circle

Geometric and Topological Inference

Sketches of a platypus: a survey of persistent homology and its algebraic foundations

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