Distance functions, critical points, and the topology of random Čech complexes

Bobrowski, Omer; Adler, Robert J.

doi:10.4310/hha.2014.v16.n2.a18

Cited by 43 publications

(92 citation statements)

References 22 publications

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“…Alternatively we could only provide separate limit theorems for each individual region. Similar phenomena have been pointed out in a series of works [8,11,17,24,35], in which various limit theorems for topological invariants in different regimes were derived (though they are not directly related to the layered structure described above).…”

Section: Introductionsupporting

confidence: 73%

Convergence of persistence diagrams for topological crackle

Owada¹,

Bobrowski²

2020

Bernoulli

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In this paper we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction of homological cycles (generalizations of loops or holes) in different dimensions. Topological crackle is a term that refers to homological cycles generated by "noisy" samples where the support is unbounded. We aim to establish weak convergence results for persistence diagramsa point process representation for persistent homology, where each homological cycle is represented by its (birth, death) coordinates. In this work we treat persistence diagrams as random closed sets, so that the resulting weak convergence is defined in terms of the Fell topology. In this framework we show that the limiting persistence diagrams can be divided into two parts. The first part is a deterministic limit containing a densely-growing number of persistence pairs with a short lifespan. The second part is a two-dimensional Poisson process, representing persistence pairs with a longer lifespan.2010 Mathematics Subject Classification. Primary 60G70. Secondary 55U10, 60F05, 60G55.

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Section: Introductionsupporting

confidence: 73%

Convergence of persistence diagrams for topological crackle

Owada¹,

Bobrowski²

2020

Bernoulli

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“…Proof of Corollary 3.2 For every point p ∈ M, we have from (3.2) that √ det(g ij ) = 1 − Ric ij 3 x i x j + O(|x| 3 ). Thus, if we take (p) = |Ric(p)| + for some fixed > 0, then we can find r (p) > 0 such that Lemma 3.3 holds.…”

Section: Discussionmentioning

confidence: 94%

“…Similarly to the Euclidean case (cf. ), critical points of index k are generated by subsets

scriptY \subset scriptP

containing k + 1 points, and are located at the “center” of the set. While in the Euclidean case the center of

scriptY

is simply taken to be the center of the unique ( k − 1)‐sphere that contains

scriptY

, in the general Riemannian case we need to carefully define the notion of a center.…”

Section: Morse Theory For the Distance Functionmentioning

confidence: 99%

“…We will refer to these generalizations as random combinatorial complexes (see [26] for a survey). It turns out that the Erdős -Rényi threshold for connectivity can be generalized to that of "homological connectivity," where the higher homology groups H k become trivial.In parallel to the study of combinatorial complexes, a line of research was established for random geometric complexes [3,5,24,42,43]. This type of complexes generalizes the model of the random geometric graph G(n, r) (introduced by Gilbert [18]), where vertices are placed at random in a metric-measure space, and edges are included based on proximity [38].…”

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confidence: 99%

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Random Čech complexes on Riemannian manifolds

Bobrowski

Oliveira

2018

Random Struct Algorithms

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In this paper we study the homology of a random Čech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M. In particular, we focus on the phase transition for “homological connectivity” where the homology of the complex becomes isomorphic to that of M. The results presented in this paper are an important generalization of , from the flat torus to general compact Riemannian manifolds. In addition to proving the statements related to homological connectivity, the methods we develop in this paper can be used as a framework for translating results for random geometric graphs and complexes from the Euclidean setting into the more general Riemannian one.

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“…(i) Theorem 5.2 of [74] establishes expectation asymptotics for n −1 N k (P n , r) for stationary input, though without a rate of convergence and [15] 2.3.3. Statistics of germ-grain models.…”

Section: Remarksmentioning

confidence: 99%

Limit theory for geometric statistics of point processes having fast decay of correlations

Błaszczyszyn

Yogeshwaran²,

Yukich³

2019

Ann. Probab.

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Let P be a simple, stationary point process on R d having fast decay of correlations, i.e., its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let Pn := P ∩ Wn be its restriction to windows Wn := [− 1 2 n 1/d , 1 2 n 1/d ] d ⊂ R d . We consider the statistic H ξ n := x∈Pn ξ(x, Pn) where ξ(x, Pn) denotes a score function representing the interaction of x with respect to Pn. When ξ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and central limit theorems for H ξ n and, more generally, for statistics of the re-scaled, possibly signed, ξ-weighted point measures µ ξ n := x∈Pn ξ(x, Pn)δ n −1/d x , as Wn ↑ R d . This gives the limit theory for non-linear geometric statistics (such as clique counts, the number of Morse critical points, intrinsic volumes of the Boolean model, and total edge length of the k-nearest neighbors graph) of α-determinantal point processes (for −1/α ∈ N) having fast decreasing kernels, including the β-Ginibre ensembles, extending the Gaussian fluctuation results of Soshnikov [72] to non-linear statistics. It also gives the limit theory for geometric U-statistics of α-permanental point processes (for 1/α ∈ N) as well as the zero set of Gaussian entire functions, extending the central limit theorems of Nazarov and Sodin [53] and Shirai and Takahashi [71], which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in [12,13] to show the fast decay of the correlations of ξ-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing and consequently yields the asymptotic normality of µ ξ n via an extension of the cumulant method.

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Distance functions, critical points, and the topology of random Čech complexes

Cited by 43 publications

References 22 publications

Convergence of persistence diagrams for topological crackle

Convergence of persistence diagrams for topological crackle

Random Čech complexes on Riemannian manifolds

Limit theory for geometric statistics of point processes having fast decay of correlations

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