{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/extended-l-ensembles-a-new-representation-for","title":"Extended L-ensembles: a new representation for Determinantal Point Processes","arxiv_id":"2107.06345","date":"2021-07-12","proceeding":null,"authors":["Nicolas Tremblay","Simon Barthelmé","Konstantin Usevich","Pierre-Olivier Amblard"],"abstract":"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.","url_abs":"https://arxiv.org/abs/2107.06345v2","url_pdf":"https://arxiv.org/pdf/2107.06345v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"extended-l-ensembles-a-new-representation-for","repo_url":"https://github.com/dahtah/DPP.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"LGPL-3.0"}},{"paper_slug":"extended-l-ensembles-a-new-representation-for","repo_url":"https://github.com/dahtah/determinantal.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"LGPL-3.0"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}