Concentration for Poisson U-statistics: Subgraph counts in random geometric graphs

Bachmann, Sascha; Reitzner, Matthias

doi:10.1016/j.spa.2017.11.001

Cited by 21 publications

(27 citation statements)

References 31 publications

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“…The precise form of I 1 (u) is rather complicated and to deduce the dependence on t and δ t seems to be demanding. In a follow-up paper to our investigations presented here and the paper by Bachmann and Peccati, Bachmann and Reitzner [4] are generalizing both approaches to subgraph counts of the Gilbert graph.…”

Section: Concentration Inequalitiesmentioning

confidence: 91%

Limit theory for the Gilbert graph

Reitzner

Schulte

Thäle

2017

Advances in Applied Mathematics

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For a given homogeneous Poisson point process in R d two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random geometric graph, is investigated as the intensity of the Poisson point process is increased and the distance parameter goes to zero. The asymptotic expectation and covariance structure of a class of length-power functionals are computed. Distributional limit theorems are derived that have a Gaussian, a stable or a compound Poisson limiting distribution. Finally, concentration inequalities are provided using a concentration inequality for the convex distance.

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Section: Concentration Inequalitiesmentioning

confidence: 91%

Limit theory for the Gilbert graph

Reitzner

Schulte

Thäle

2017

Advances in Applied Mathematics

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“…Apart from minor modifications, the two upcoming proofs are very similar to the proofs of [25, Proposition 3.1 and Proposition 3.3] as well as [3,Theorem 4.2 (i)]. We will therefore present these proofs very briefly just for the sake of completeness.…”

Section: Proofs For the Asymptotic Behavior Of The Expectationmentioning

confidence: 82%

“…The geometric graph model that will be considered in the present work was particularly investigated in [3,16,20,21] and slightly generalizes the classical model of random geometric graphs. The latter model has been investigated by many authors and is extensively described in Penrose's book [25].…”

Section: Frameworkmentioning

confidence: 99%

“…First applications for these techniques are worked out in [2] and also in [3], where concentration bounds for certain Poisson U-statistics with positive kernels are established. A crucial property that was exploited in the latter investigations is that adding a point to the Poisson process cannot decrease the value of the considered functionals.…”

Section: Introductionmentioning

confidence: 99%

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Concentration for Poisson functionals: Component counts in random geometric graphs

Bachmann

2016

Stochastic Processes and their Applications

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Upper bounds for the probabilities P(F ≥ EF + r) and P(F ≤ EF − r) are proved, where F is a certain component count associated with a random geometric graph built over a Poisson point process on R d . The bounds for the upper tail decay exponentially, and the lower tail estimates even have a Gaussian decay.For the proof of the concentration inequalities, recently developed methods based on logarithmic Sobolev inequalities are used and enhanced. A particular advantage of this approach is that the resulting inequalities even apply in settings where the underlying Poisson process has infinite intensity measure.MSC: primary 60D05; secondary 05C80, 60C05

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“…The main result regarding to generalization bounds in this setting is manifested in next section, the proof of this result relies on the techniques of U -statistics and its applications which can be found in Fuchs et al [22], Bouzebda and Nemouchi [23], Fuglsby et al [24], Privault and Serafin [25], Bachmann and Reitzner [26], and Garg and Dewan [27], and we skip the details here.…”

Section: B Multi-dividing Ontology Algorithm By Maximizing Auc Measurementioning

confidence: 99%

Theoretical Perspective of Multi-Dividing Ontology Learning Trick in Two-Sample Setting

Zhu

Hua

2020

IEEE Access

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The multi-dividing ontology learning framework has been proven to have a higher efficiency for tree-structured ontology learning, and in this work, we consider a special setting of this learning framework in which ontology sample set for each rate is divided into two groups. This setting can be regarded as the classic two-sample learning problem associated with multi-dividing ontology framework. In this work, we mainly focus on the theoretical analysis of multi-dividing two-sample ontology learning algorithm, whose ontology objective function is proposed, and the generalization bounds in this setting is obtained in terms of U-statistics technique. The theoretical result given is of potential guiding significance in the field of ontology engineering applications. INDEX TERMS small ontology, multi-dividing ontology learning, similarity measure I. INTRODUCE OF ONTOLOGY

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Concentration for Poisson U-statistics: Subgraph counts in random geometric graphs

Cited by 21 publications

References 31 publications

Limit theory for the Gilbert graph

Limit theory for the Gilbert graph

Concentration for Poisson functionals: Component counts in random geometric graphs

Theoretical Perspective of Multi-Dividing Ontology Learning Trick in Two-Sample Setting

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