A remark on the convergence of Betti numbers in the thermodynamic regime

Trinh, Khanh Duy

doi:10.1186/s40736-017-0029-0

Cited by 7 publications

(13 citation statements)

References 8 publications

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“…Divide both sides by n. It follows from Proposition 3.10 and Lemma 4.2 that in the thermodynamic regime, almost surely, Now what remains is to prove Lemmas 4.2 and 4.3. To obtain the required results, the following implications of the coupling property of Poisson point processes, taken from [23,26], are needed. Since n/α N n → 1 as n → ∞ in the thermodynamic regime, choose Λ > 0 such that for all n and x ∈ W n,i , f n (x) ≤ Λ.…”

Section: Betti Numbers In a Compact Regionmentioning

confidence: 99%

“…Now in the Euclidean setting, the following strong law of large numbers for β k (C(X n , r n )) and β k (C(P n , r n )) in the thermodynamic regime holds, i.e., as n → ∞ with n 1/N r n → r ∈ (0, ∞), β k (C(X n , r n )) n resp. β k (C(P n , r n )) n → R Nβ (N ) k (f (x), r)dx a.s., provided that the probability density function f (x) is Riemann integrable, has convex compact support and is bounded both below and above on the support [23,26]. Although in stochastic geometry, weak and strong laws of large numbers have been established for a general class of local functionals [19,20], Betti numbers do not belong to that class.…”

Section: Introductionmentioning

confidence: 99%

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Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime

Goel¹,

Trinh²,

Tsunoda³

2018

J Stat Phys

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We establish the strong law of large numbers for Betti numbers of random Cech complexes built on R N -valued binomial point processes and related Poisson point processes in the thermodynamic regime. Here we consider both the case where the underlying distribution of the point processes is absolutely continuous with respect to the Lebesgue measure on R N and the case where it is supported on a C 1 compact manifold of dimension strictly less than N . The strong law is proved under very mild assumption which only requires that the common probability density function belongs to L p spaces, for all 1 ≤ p < ∞.

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Section: Betti Numbers In a Compact Regionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime

Goel¹,

Trinh²,

Tsunoda³

2018

J Stat Phys

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“…Specifically, [22] shows the vague convergence of the rescaled diagram n −1 dgm q [K(S n )] to some Radon measure µ q . The two recent papers [21,35] prove that a similar convergence actually holds for S n a binomial sample on a manifold. However, vague convergence deals with continuous functions φ with compact support, whereas we are interested in functions of the type pers α , which are not even bounded.…”

mentioning

confidence: 77%

“…The latter is extended to non-homogeneous processes in [35], and to processes on manifolds in [21]. Note that our results constitute a natural extension of [35]. In [33], higher dimensional analogs of minimum spanning trees, called minimal spanning acycles, were introduced.…”

mentioning

confidence: 95%

“…Our contributions to the matter consists in proving, for samples on the cube [0, 1] d , a stronger convergence, allowing test functions to have non-compact support and polynomial growth. As a gentle introduction to the formalism used later, we first recall some known results from geometric probability on the study of Betti numbers, and we also detail relevant results of [21,22,35].…”

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On the choice of weight functions for linear representations of persistence diagrams

Divol

Polonik

2019

J Appl. and Comput. Topology

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A. Persistence diagrams are efficient descriptors of the topology of a point cloud. As they do not naturally belong to a Hilbert space, standard statistical methods cannot be directly applied to them. Instead, feature maps (or representations) are commonly used for the analysis. A large class of feature maps, which we call linear, depends on some weight functions, the choice of which is a critical issue. An important criterion to choose a weight function is to ensure stability of the feature maps with respect to Wasserstein distances on diagrams. We improve known results on the stability of such maps, and extend it to general weight functions. We also address the choice of the weight function by considering an asymptotic setting; assume that X n is an i.i.d. sample from a density on [0, 1] d . For the Čech and Rips filtrations, we characterize the weight functions for which the corresponding feature maps converge as n approaches infinity, and by doing so, we prove laws of large numbers for the total persistences of such diagrams. Those two approaches (stability and convergence) lead to the same simple heuristic for tuning weight functions: if the data lies near a d-dimensional manifold, then a sensible choice of weight function is the persistence to the power α with α ≥ d.

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Limit theorems for process-level Betti numbers for sparse and critical regimes

Owada

Thomas

2020

Adv. Appl. Probab.

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The objective of this study is to examine the asymptotic behavior of Betti numbers of Čech complexes treated as stochastic processes and formed from random points in the d-dimensional Euclidean space ${\mathbb{R}}^d$ . We consider the case where the points of the Čech complex are generated by a Poisson process with intensity nf for a probability density f. We look at the cases where the behavior of the connectivity radius of the Čech complex causes simplices of dimension greater than $k+1$ to vanish in probability, the so-called sparse regime, as well when the connectivity radius is of the order of $n^{-1/d}$ , the critical regime. We establish limit theorems in the aforementioned regimes: central limit theorems for the sparse and critical regimes, and a Poisson limit theorem for the sparse regime. When the connectivity radius of the Čech complex is $o(n^{-1/d})$ , i.e. the sparse regime, we can decompose the limiting processes into a time-changed Brownian motion or a time-changed homogeneous Poisson process respectively. In the critical regime, the limiting process is a centered Gaussian process but has a much more complicated representation, because the Čech complex becomes highly connected with many topological holes of any dimension.

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A remark on the convergence of Betti numbers in the thermodynamic regime

Abstract: The convergence of the expectations of Betti numbers ofČech complexes built on binomial point processes in the thermodynamic regime is established.

Cited by 7 publications

References 8 publications

Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime

Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime

On the choice of weight functions for linear representations of persistence diagrams

Limit theorems for process-level Betti numbers for sparse and critical regimes

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