EXTENDED L-ENSEMBLES: A NEW REPRESENTATION FOR DETERMINANTAL POINT PROCESSES

• 成果类型: Article
• 署名作者: Tremblay, Nicolas; Barthelme, Simon; Usevich, Konstantin; Amblard, Pierre-Olivier
• 署名单位: Communaute Universite Grenoble Alpes; Institut National Polytechnique de Grenoble; Universite Grenoble Alpes (UGA); Centre National de la Recherche Scientifique (CNRS); Universite de Lorraine; Universite de Lorraine; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1824
• 发表日期: 2023
• 页码: 613-640
• 关键词: monte-carlo; matrices

摘要

Determinantal point processes (DPPs) are a class of repulsive point pro-cesses, 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 patholo-gies 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 ma-trices, and we prove an extension of the Cauchy-Binet theorem for such matrices that may be of independent interest.