Maximum likelihood estimation of determinantal point processes

Brunel, Victor-Emmanuel; Moitra, Ankur; Rigollet, Philippe; Urschel, John D.

doi:10.48550/arxiv.1701.06501

Cited by 4 publications

(4 citation statements)

References 14 publications

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“…First, our final objective function is nonconvex, and our algorithm is only guaranteed to increase its objective function. Experimental evidence suggests that our approach recovers the synthetic kernel, but more work is needed to study the maximizers of the likelihood, in the spirit of Brunel et al [2017a] for finite DPPs, and the properties of our fixed point algorithm. Second, the estimated integral kernel does not have any explicit structure, other than being implicitly forced to be low-rank because of the trace penalty.…”

Section: Discussionmentioning

confidence: 99%

Nonparametric estimation of continuous DPPs with kernel methods

Fanuel¹,

Bardenet²

2021

Preprint

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Determinantal Point Process (DPPs) are statistical models for repulsive point patterns. Both sampling and inference are tractable for DPPs, a rare feature among models with negative dependence that explains their popularity in machine learning and spatial statistics. Parametric and nonparametric inference methods have been proposed in the finite case, i.e. when the point patterns live in a finite ground set. In the continuous case, only parametric methods have been investigated, while nonparametric maximum likelihood for DPPs -an optimization problem over trace-class operators -has remained an open question. In this paper, we show that a restricted version of this maximum likelihood (MLE) problem falls within the scope of a recent representer theorem for nonnegative functions in an RKHS. This leads to a finite-dimensional problem, with strong statistical ties to the original MLE. Moreover, we propose, analyze, and demonstrate a fixed point algorithm to solve this finite-dimensional problem. Finally, we also provide a controlled estimate of the correlation kernel of the DPP, thus providing more interpretability.Preprint. Under review.

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Section: Discussionmentioning

confidence: 99%

Nonparametric estimation of continuous DPPs with kernel methods

Fanuel¹,

Bardenet²

2021

Preprint

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“…Likelihood estimation in this setting, based on the observation of n i.i.d. discrete DPPs, has been studied in Brunel et al (2017), who investigate asymptotic properties when n tends to infinity. In the continuous case, likelihood estimation based on n i.i.d.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic approximation of the likelihood of stationary determinantal point processes

Poinas

Lavancier

2022

Scandinavian J Statistics

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Continuous determinantal point processes (DPPs) are a class of repulsive point processes on ℝd$$ {\mathbb{R}}^d $$ with many statistical applications. Although an explicit expression of their density is known, it is too complicated to be used directly for maximum likelihood estimation. In the stationary case, an approximation using Fourier series has been suggested, but it is limited to rectangular observation windows and no theoretical results support it. In this contribution, we investigate a different way to approximate the likelihood by looking at its asymptotic behavior when the observation window grows toward ℝd$$ {\mathbb{R}}^d $$. This new approximation is not limited to rectangular windows, is faster to compute than the previous one, does not require any tuning parameter, and some theoretical justifications are provided. It moreover provides an explicit formula for estimating the asymptotic variance of the associated estimator. The performances are assessed in a simulation study on standard parametric models on ℝd$$ {\mathbb{R}}^d $$ and compare favorably to common alternative estimation methods for continuous DPPs.

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“…Likelihood estimation in this setting, based on the observation of n i.i.d. discrete DPPs, has been studied in [10], who investigate asymptotic properties when n tends to infinity. In the continuous case, likelihood estimation based on n i.i.d.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic approximation of the likelihood of stationary determinantal point processes

Poinas,

Lavancier

2021

Preprint

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Continuous determinantal point processes (DPPs) are a class of repulsive point processes on R d with many statistical applications. Although an explicit expression of their density is known, this expression is too complicated to be used directly for maximum likelihood estimation. In the stationary case, an approximation using Fourier series has been suggested, but it is limited to rectangular observation windows and no theoretical results support it. In this contribution, we investigate a different way to approximate the likelihood by looking at its asymptotic behaviour when the observation window grows towards R d . This new approximation is not limited to rectangular windows, is faster to compute than the previous one, does not require any tuning parameter, and some theoretical justifications are provided. The performances of the associated estimator are assessed in a simulation study on standard parametric models on R d and compare favourably to common alternative estimation methods for continuous DPPs.

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Maximum likelihood estimation of determinantal point processes

Cited by 4 publications

References 14 publications

Nonparametric estimation of continuous DPPs with kernel methods

Nonparametric estimation of continuous DPPs with kernel methods

Asymptotic approximation of the likelihood of stationary determinantal point processes

Asymptotic approximation of the likelihood of stationary determinantal point processes

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