Standard and robust intensity parameter estimation for stationary determinantal point processes

Coeurjolly, Jean‐François; Biscio, Christophe Ange Napoléon

doi:10.1016/j.spasta.2016.04.007

Cited by 5 publications

(3 citation statements)

References 22 publications

(40 reference statements)

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“…The other approach is due to Bolthausen () who considered stationary random fields and whose proof was later generalised to nonstationary random fields by Guyon () and Karácsony (). This approach is, for example, used in Waagepetersen and Guan (), Coeurjolly and Møller (), Biscio and Coeurjolly (), Coeurjolly (), and Poinas, Delyon, and Lavancier (). Regarding the point process references mentioned above, essentially the same central limit theorems are (re)invented again and again for each specific setting and statistic considered.…”

Section: Introductionmentioning

confidence: 99%

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

Biscio

Waagepetersen

2019

Scandinavian J Statistics

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We establish a central limit theorem for multivariate summary statistics of nonstationary α‐mixing spatial point processes and a subsampling estimator of the covariance matrix of such statistics. The central limit theorem is crucial for establishing asymptotic properties of estimators in statistics for spatial point processes. The covariance matrix subsampling estimator is flexible and model free. It is needed, for example, to construct confidence intervals and ellipsoids based on asymptotic normality of estimators. We also provide a simulation study investigating an application of our results to estimating functions.

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Section: Introductionmentioning

confidence: 99%

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

Biscio

Waagepetersen

2019

Scandinavian J Statistics

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“…Such problems are pervasive to a variety of applications such as document or timeline summarization [LB12, YFZ `16], image search [KT11,AFAT14], audio signal processing [XO16], image segmentation [LCYO16], bioinformatics [BQK `14], neuroscience [SZA13] and wireless or cellular networks modelization [MS14, TL14, LBDA15, DZH15]. DPPs have also been employed as methodological tools in Bayesian and spatial statistics [KK16,BC16], survey sampling [LM15,CJM16] and Monte Carlo methods [BH16].…”

Section: Introductionmentioning

confidence: 99%

Maximum likelihood estimation of determinantal point processes

Brunel¹,

Moitra²,

Rigollet³

et al. 2017

Preprint

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Determinantal point processes (DPPs) have wide-ranging applications in machine learning, where they are used to enforce the notion of diversity in subset selection problems. Many estimators have been proposed, but surprisingly the basic properties of the maximum likelihood estimator (MLE) have received little attention. The difficulty is that it is a non-concave maximization problem, and such functions are notoriously difficult to understand in high dimensions, despite their importance in modern machine learning. Here we study both the local and global geometry of the expected log-likelihood function. We prove several rates of convergence for the MLE and give a complete characterization of the case where these are parametric. We also exhibit a potential curse of dimensionality where the asymptotic variance of the MLE scales exponentially with the dimension of the problem. Moreover, we exhibit an exponential number of saddle points, and give evidence that these may be the only critical points.

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“…Such problems are pervasive in a variety of applications such as document or timeline summarization ([LB12, YFZ `16]), image search ([KT11, AFAT14]), audio signal processing ( [XO16]), image segmentation ( [LCYO16]), bioinformatics ([BQK `14]), neuroscience ( [SZA13]) and wireless or cellular networks modelization ([MS14, TL14, LBDA15, DZH15]). DPPs have also been employed as methodological tools in Bayesian and spatial statistics ( [KK16,BC16]), survey sampling ( [LM15,CJM16]) and Monte Carlo methods ( [BH16]).…”

Section: Introductionmentioning

confidence: 99%

Rates of estimation for determinantal point processes

Brunel,

Moitra,

Rigollet

et al. 2017

Preprint

View full text Add to dashboard Cite

Determinantal point processes (DPPs) have wide-ranging applications in machine learning, where they are used to enforce the notion of diversity in subset selection problems. Many estimators have been proposed, but surprisingly the basic properties of the maximum likelihood estimator (MLE) have received little attention. In this paper, we study the local geometry of the expected log-likelihood function to prove several rates of convergence for the MLE. We also give a complete characterization of the case where the MLE converges at a parametric rate. Even in the latter case, we also exhibit a potential curse of dimensionality where the asymptotic variance of the MLE is exponentially large in the dimension of the problem.

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Standard and robust intensity parameter estimation for stationary determinantal point processes

Cited by 5 publications

References 22 publications

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

Maximum likelihood estimation of determinantal point processes

Rates of estimation for determinantal point processes

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