Persistent homology of the sum metric

Carlsson, Gunnar; Filippenko, Benjamin

doi:10.1016/j.jpaa.2019.106244

Cited by 12 publications

(13 citation statements)

References 7 publications

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“…It would thus be interesting to see if such a Künneth formula could provide an alternative proof of Theorem 1. We remark that Künneth formulae in persistent homology have been studied in [34], [35]. We now proceed to the proof of Theorem 1.…”

Section: Path Homology Of Mlpsmentioning

confidence: 89%

Path Homologies of Deep Feedforward Networks

Chowdhury

Gebhart

Huntsman

et al. 2019

2019 18th IEEE International Conference on Machine Learning and Applications (ICMLA)

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We provide a characterization of two types of directed homology for fully-connected, feedforward neural network architectures. These exact characterizations of the directed homology structure of a neural network architecture are the first of their kind. We show that the directed flag homology of deep networks reduces to computing the simplicial homology of the underlying undirected graph, which is explicitly given by Euler characteristic computations. We also show that the path homology of these networks is non-trivial in higher dimensions and depends on the number and size of the layers within the network. These results provide a foundation for investigating homological differences between neural network architectures and their realized structure as implied by their parameters.

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Section: Path Homology Of Mlpsmentioning

confidence: 89%

Path Homologies of Deep Feedforward Networks

Chowdhury

Gebhart

Huntsman

et al. 2019

2019 18th IEEE International Conference on Machine Learning and Applications (ICMLA)

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“…The study of Vietoris-Rips complexes is finding connections to many different subareas of geometry and topology, including combinatorial topology, metric geometry, equivariant topology, polytope theory, and quantitative topology, among others. Our paper directly relates to the study of independence complexes of Kneser graphs [10], and to Künneth formulas for persistent homology [16]; we hope that the study of Vietoris-Rips complexes of hypercube graphs will find further connections to other areas of mathematics.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Motivated by this Künneth formula, [16,Proposition 4.10] gives the persistent homology of VR(Q n ; r) up to homological dimension two. The 2-dimensional homology is always zero, and all 1-dimensional homology appears prior to scale parameter r < 2.…”

Section: Introductionmentioning

confidence: 99%

“…A better understanding of the homotopy types of VR(Q n ; r) could relate to stronger versions of the Künneth formula for persistent homology of Vietoris-Rips complexes. Indeed, [16] considers a Künneth formula for persistent homology in which the metric on the product X × Y is given by the sum d X + d Y . Motivated by this Künneth formula, [16,Proposition 4.10] gives the persistent homology of VR(Q n ; r) up to homological dimension two.…”

Section: Introductionmentioning

confidence: 99%

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On Vietoris–Rips complexes of ellipses

2019

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Abstract. For X a metric space and r > 0 a scale parameter, the Vietoris-Rips simplicial complex VR<(X; r) (resp. VR ≤ (X; r)) has X as its vertex set, and a finite subset σ ⊆ X as a simplex whenever the diameter of σ is less than r (resp. at most r). Though Vietoris-Rips complexes have been studied at small choices of scale by Hausmann and Latschev [13,16], they are not well-understood at larger scale parameters. In this paper we investigate the homotopy types of Vietoris-Rips complexes of ellipses Y = {(x, y) ∈ R 2 | (x/a) 2 + y 2 = 1} of small eccentricity, meaning 1 < a ≤ √ 2. Indeed, we show there are constants r 1 < r 2 such that for all r 1 < r < r 2 , we have VR<(Y ; r) S 2 and VR ≤ (Y ; r) 5 S 2 , though only one of the two-spheres in VR ≤ (Y ; r) is persistent. Furthermore, we show that for any scale parameter r 1 < r < r 2 , there are arbitrarily dense subsets of the ellipse such that the Vietoris-Rips complex of the subset is not homotopy equivalent to the Vietoris-Rips complex of the entire ellipse. As our main tool we link these homotopy types to the structure of infinite cyclic graphs.

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Homological Algebra for Persistence Modules

Bubenik

Milićević

2021

Found Comput Math

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We useČech closure spaces, also known as pretopological spaces, to develop a uniform framework that encompasses the discrete homology of metric spaces, the singular homology of topological spaces, and the homology of (directed) clique complexes, along with their respective homotopy theories. We obtain six homology and homotopy theories of closure spaces. 1 Closure spacesIn this section we provide background on EduardČech's closure spaces [9,25].2.1. Elementary definitions. We start with the elementary definitions we need to work with closure spaces. Definition 2.1. Let X be a set and let P(X) denote the collection of subsets of X. A function c : P(X) → P(X) is called a closure operation (or just closure) for X if the following axioms are satisfied:for all A, B ⊆ X. An ordered pair (X, c) where X is a set and c a closure for X is called a (Čech) closure space. Elements of X are called points.Lemma 2.2. Let (X, c) be a closure space. If A ⊆ B are subsets of X then c(A) ⊆ c(B). Proof. Note thatExample 2.3. Let X be a set. Then the identity map 1 P(X) : P(X) → P(X) is a closure operation for X. It is called the discrete closure for X. The closure operation defined by the map A → X for A = ∅ and ∅ → ∅ is called the indiscrete closure for X.Definition 2.5. Given a set X and a closure c for X we can associate to it an interior operation for X, int c : P(X) → P(X) given by int c (A) := X − c(X − A).The interior operation satisfies the following:1) int c (X) = X, 2) For all A ⊆ X, int c (A) ⊆ A, 3) For all A, B ⊆ X, int c (A ∩ B) = int c (A) ∩ int c (B). If int : P(X) → P(X) is a function satisfying the 3 conditions in Definition 2.5 and one defines c int : P(X) → P(X) by c int (A) := X − int(X − A), then c int is a closure operation for X.Proposition 2.6. [9, 14.A.12] A subset A of a closure space (X, c) is open if and only if int c (A) = A.Definition 2.7. [9, Definition 14.B.1] Let (X, c) be a closure space. Let A ⊆ X. We say B ⊆ X is a neighborhood of A if A ⊆ X − c(X − B) = int c (B). In the case that A = {x}, we say B is a neighborhood of x. The neighborhood system of A is the collection of all neighborhoods of A. Definition 2.8. [9, Definition 16.A.1] Let (X, c X ) and (Y, c Y ) be closure spaces. A map). If f is continuous at every point of X we say f is continuous. Equivalently, f is continuous if for every A ⊆ X, f (c X (A)) ⊆ c Y (f (A)). A continuous map f is called a homeomorphism if f is a bijection with a continuous inverse.

show abstract

Persistent homology of the sum metric

Cited by 12 publications

References 7 publications

Path Homologies of Deep Feedforward Networks

Path Homologies of Deep Feedforward Networks

On Vietoris–Rips complexes of ellipses

Homological Algebra for Persistence Modules

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