A functorial Dowker theorem and persistent homology of asymmetric networks

Chowdhury, S.; Mémoli, Facundo

doi:10.48550/arxiv.1608.05432

Cited by 7 publications

(24 citation statements)

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“…Approximate homological versions were obtained in [14] and [5]. On the other hand, a functorial version of the Dowker duality was proved in [8]. The functorial versions of the Nerve Lemma (Lemma 5.1) and of the Dowker duality (Theorem 5.2) presented in this paper are more general than the previously known versions.…”

Section: Functorial Dowker-nerve Diagrammentioning

confidence: 64%

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Rips complexes as nerves and a Functorial Dowker-Nerve Diagram

Virk

2019

Preprint

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Using ideas of the Dowker duality we prove that the Rips complex at scale r is homotopy equivalent to the nerve of a cover consisting of sets of prescribed diameter. We then develop a functorial version of the Nerve theorem coupled with the Dowker duality, which is presented as a Functorial Dowker-Nerve Diagram. These results are incorporated into a systematic theory of filtrations arising from covers. As a result we provide a general framework for reconstruction of spaces by Rips complexes, a short proof of the reconstruction result of Hausmann, and completely classify reconstruction scales for metric graphs. Furthermore we introduce a new extraction method for homology of a space based on nested Rips complexes at a single scale, which requires no conditions on neighboring scales nor the Euclidean structure of the ambient space.

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Section: Functorial Dowker-nerve Diagrammentioning

confidence: 64%

“…Functorial versions of these results have been considered before. A functorial version of the Nerve Lemma appears in [7] and later in [8] for pairs of finite good open covers of paracompact spaces. Approximate homological versions were obtained in [14] and [5].…”

Section: Functorial Dowker-nerve Diagrammentioning

confidence: 99%

Rips complexes as nerves and a Functorial Dowker-Nerve Diagram

Virk

2019

Preprint

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“…1] can be applied to Čech complexes. We reason by the functorial Dowker theorem of Chowdhury and Mémoli [4,Thm. 3].…”

Section: Dowker's Theorem For čEch Complexesmentioning

confidence: 99%

“…where the same elements are related, but they have changed position. The functorial Dowker theorem [4,Thm. 3] states that this homotopy equivalence gives diagrams that commute up to homotopy when looking at inclusions of relations R ′ ⊆ R. Fixing a positive number α > 0, we can consider any Čech complex of the form C Y (X, α) as the Dowker complex of the relation R α = {(x, y) | d(x, y) < α} ⊆ X × Y where two elements are related if the distance between them is less than α.…”

Section: Dowker's Theorem For čEch Complexesmentioning

confidence: 99%

“…If we let α, ǫ, ǫ ′ > − → d H (X, A) such that α+ǫ+ǫ ′ ≤ τ , in particular − → d H (X, A) < τ /3, then by Theorem 6.5 we get a homotopy equivalence |C X (A, r)| ≃ X for r = α, r = α + ǫ and r = α + ǫ + ǫ ′ . The functorial Nerve theorem [17,Thm 5,4] gives homotopy equivalences between the three Čech complexes for the given r's. Going to the level of homology and using the second interleaving of Corollary 5.4, we get the following commutative diagram…”

Section: Determining Homologymentioning

confidence: 99%

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Determining homology of an unknown space from a sample

Brun¹,

Pascual²,

Salbu³

2021

Preprint

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The homology of an unknown subspace of Euclidean space can be determined from the intrinsic Čech complex of a sample of points in the subspace, without reference to the ambient Euclidean space. More precisely, given a subspace X of Euclidean space and a sample A of points in X, we give conditions for the homology of X to be isomorphic to a certain persistent homology group of the intrinsic Čech complex.

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Sparse Nerves in Practice

Blaser

Brun

2019

Lecture Notes in Computer Science

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Topological data analysis combines machine learning with methods from algebraic topology. Persistent homology, a method to characterize topological features occurring in data at multiple scales is of particular interest. A major obstacle to the wide-spread use of persistent homology is its computational complexity. In order to be able to calculate persistent homology of large datasets, a number of approximations can be applied in order to reduce its complexity. We propose algorithms for calculation of approximate sparse nerves for classes of Dowker dissimilarities including all finite Dowker dissimilarities and Dowker dissimilarities whose homology isČech persistent homology. All other sparsification methods and software packages that we are aware of calculate persistent homology with either an additive or a multiplicative interleaving. In dowker homology, we allow for any non-decreasing interleaving function α. We analyze the computational complexity of the algorithms and present some benchmarks. For Euclidean data in dimensions larger than three, the sizes of simplicial complexes we create are in general smaller than the ones created by SimBa. Especially when calculating persistent homology in higher homology dimensions, the differences can become substantial.

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A functorial Dowker theorem and persistent homology of asymmetric networks

Cited by 7 publications

References 0 publications

Rips complexes as nerves and a Functorial Dowker-Nerve Diagram

Rips complexes as nerves and a Functorial Dowker-Nerve Diagram

Determining homology of an unknown space from a sample

Sparse Nerves in Practice

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