February 2023 Extended L-ensembles: A new representation for determinantal point processes
Nicolas Tremblay, Simon Barthelmé, Konstantin Usevich, Pierre-Olivier Amblard
Author Affiliations +
Ann. Appl. Probab. 33(1): 613-640 (February 2023). DOI: 10.1214/22-AAP1824


Determinantal point processes (DPPs) are a class of repulsive point processes, popular for their relative simplicity. They are traditionally defined via their marginal distributions, but a subset of DPPs called “L-ensembles” have tractable likelihoods and are thus particularly easy to work with. Indeed, in many applications, DPPs are more naturally defined based on the L-ensemble formulation rather than through the marginal kernel.

The fact that not all DPPs are L-ensembles is unfortunate, but there is a unifying description. We introduce here extended L-ensembles, and show that all DPPs are extended L-ensembles (and vice versa). Extended L-ensembles have very simple likelihood functions, contain L-ensembles and projection DPPs as special cases. From a theoretical standpoint, they fix some pathologies in the usual formalism of DPPs, for instance, the fact that projection DPPs are not L-ensembles. From a practical standpoint, they extend the set of kernel functions that may be used to define DPPs: we show that conditional positive definite kernels are good candidates for defining DPPs, including DPPs that need no spatial scale parameter.

Finally, extended L-ensembles are based on so-called “saddle-point matrices”, and we prove an extension of the Cauchy–Binet theorem for such matrices that may be of independent interest.

Funding Statement

This work was supported by ANR project GenGP (ANR-16-CE23-0008), ANR project LeaFleT (ANR-19-CE23-0021-01), ANR project GRANOLA (ANR JCJC ANR-21-CE48-0009), LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01), Grenoble Data Institute (ANR-15-IDEX- 02), LIA CNRS/Melbourne Univ Geodesic, and partial funding from the IRS (Initiatives de Recherche Stratégiques) of the IDEX Université Grenoble Alpes.


We thank Guillaume Gautier for helpful comments on preliminary versions of this manuscript.


Download Citation

Nicolas Tremblay. Simon Barthelmé. Konstantin Usevich. Pierre-Olivier Amblard. "Extended L-ensembles: A new representation for determinantal point processes." Ann. Appl. Probab. 33 (1) 613 - 640, February 2023. https://doi.org/10.1214/22-AAP1824


Received: 1 July 2021; Revised: 1 February 2022; Published: February 2023
First available in Project Euclid: 21 February 2023

MathSciNet: MR4551559
zbMATH: 07692270
Digital Object Identifier: 10.1214/22-AAP1824

Primary: 60G55

Keywords: determinantal point process , L-ensemble

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 1 • February 2023
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