[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-156004-105":59,"doc-detail-156004-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","hardness-of-maximum-likelihood-learning-ofdpps","Hardness of Maximum Likelihood Learning ofDPPs","","Determinantal Point Processes (DPPs) model negatively correlated sets and are widely used to select diverse yet representative subsets in machine learning. Maximum-likelihood learning fits DPP parameters to data, but existing methods either restrict to limited DPP families or rely on local heuristics without optimality guarantees. This work proves hardness for maximum-likelihood DPP learning: computing even strong approximation of the maximum log-likelihood is NP-complete. It also provides the first polynomial-time algorithm with nontrivial worst-case approximation bounds via reductions to a hypergraph vector-coloring gap instance.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/hardness-of-maximum-likelihood-learning-ofdpps/156004/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/hardness-of-maximum-likelihood-learning-ofdpps/156004.png","ImageObject",300,407,{"name":92,"@type":93},"Quinn","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-08-28",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",14,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What problem does the document address for Determinantal Point Processes (DPPs)?","Question",{"text":113,"@type":114},"It studies maximum likelihood learning of DPP models from data, focusing on computational efficiency and the difficulty of finding optimal parameters.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What hardness result is proved for maximum-likelihood DPP learning?",{"text":118,"@type":114},"The work proves Kulesza’s conjecture, showing strong NP-completeness for approximating the maximum log-likelihood of a DPP on a ground set.",{"name":120,"@type":111,"acceptedAnswer":121},"What key technique is used to obtain the results?",{"text":122,"@type":114},"The document reduces the approximation task to solving a gap instance of a vector-coloring problem on a hypergraph built from bounded-degree graph constructions and enhanced using strong expanders.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},156004,1787942585,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":46,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":31,"preview_page_count":4},962075114765,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Hardness of Maximum Likelihood Learning ofDPPs*  \nElena Grigorescu ELENA-G@PURDUE . EDU  \nPurdue University  \nBrendan Juba BJUBA @WUSTL . EDU  \nWashington University in St. Louis  \nKarl Wimmer WIMMERK @DUQ. EDU  \nDuquesne University  \nNing Xie NXIE @CIS . FIU . EDU  \nFlorida International University  \nEditors: Po-Ling Loh and Maxim Raginsky  \nAbstract  \nDeterminantal Point Processes (DPPs) are a widely used probabilistic model for negatively correlated sets. DPPs have been successfully employed in Machine Learning applications to select a diverse, yet representative subset of data. In these applications, the parameters of the DPP need tobe fitted to match the data; typically, we seek a set of parameters that maximize the likelihood of the data. The algorithms used for this task to date either optimize over a limited family of DPPs, or use local improvement heuristics that do not provide theoretical guarantees of optimality.  \nIt is natural to ask if there exist efficient algorithms for finding a maximum likelihood DPP model for a given data set. In seminal work on DPPs in Machine Learning, Kulesza conjecturedin his PhD Thesis (2012) that the problem is NP-complete. The lack of a formal proof prompted Brunel, Moitra, Rigollet and Urschel (2017a) to conjecture that, in opposition to Kulesza's conjecture, there exists a polynomial-time algorithm for computing a maximum-likelihood DPP. They also presented some preliminary evidence supporting their conjecture.  \nIn this work we prove Kulesza's conjecture. In fact, we prove the following stronger hardness of approximation result: even computing a 1 􀀀 poly~~ 1~~log~~ ~~N-approximation to the maximum loglikelihood of a DPP on a ground set of N elements is NP-complete. At the same time, we also obtain the first polynomial-time algorithm that achieves a nontrivial worst-case approximation to the optimal log-likelihood: the approximation factor is (1+o(1~~1~~))~~ ~~log~~ ~~m unconditionally (for data sets that consist of m subsets), and can be improved to 1 􀀀 1l+oog(1N) ifall N elements appear in a O(1=N ) -fraction of the subsets.  \nIn terms of techniques, we reduce approximating the maximum log-likelihood of DPPs on a data set to solving a gap instance of a “vector coloring” problem on a hypergraph. Such a hypergraph is built on a bounded-degree graph construction of Bogdanov, Obata and Trevisan (2002), and is further enhanced by the strong expanders of Alon and Capalbo (2007) to serve our purposes.  \n* Please see the associated technical report for complete proofs: [http://arxiv.org/abs/2205.12377](http://arxiv.org/abs/2205.12377)  \n© 2022 E. Grigorescu, B. Juba, K. Wimmer & N. Xie.  \nGRIGORESCU JUBA WIMMER XIE  \n1. Introduction  \nDeterminantal Point Processes (DPPs) are a family of probability distributions on sets that feature repulsion among elements in the ground set. Roughly speaking, a DPP is a distribution over all 2N subsets of f1; : : : ; Ng defined by a positive semidefinite (PSD) N 􀀂 N matrix K (called a marginal kernel or correlation kernel) whose eigenvalues all lie in [0 ; 1], such that, for any S 􀀒 f1; : : : ; Ng, random subsets X drawn according to the distribution satisfy Pr[S 􀀒 X] = det(KS), where KSis the principal submatrix of K indexed by S.  \nDPPs were first proposed in quantum statistical physics to model fermion systems in thermal equilibrium (Macchi, 1975), but they also arise naturally in diverse fields such as random matrix theory, probability theory, number theory, random spanning trees and non-intersecting paths (Dyson, 1962 ; Burton and Pemantle, 1993 ; Rudnick and Sarnak, 1996 ; Soshnikov, 2000) . After the seminal work of Kulesza and Taskar (2012), DPPs have attracted a flurry of attention from the machine learning community due to their computational tractability and excellent capability at producing diverse but representative subsets, and subsequently fast algorithms have been developed for sampling from DPPs (Hough et al., 2006 ; Kulesza and Taskar, 2010 ; Re","cbCaiqG3cMpKgiIU","https://ap.wps.com/l/cbCaiqG3cMpKgiIU","pdf",331772,"English","# Introduction\n## Determinantal Point Processes and repulsion\n## Maximum likelihood estimation for DPP parameters","[{\"question\":\"What problem does the document address for Determinantal Point Processes (DPPs)?\",\"answer\":\"It studies maximum likelihood learning of DPP models from data, focusing on computational efficiency and the difficulty of finding optimal parameters.\"},{\"question\":\"What hardness result is proved for maximum-likelihood DPP learning?\",\"answer\":\"The work proves Kulesza’s conjecture, showing strong NP-completeness for approximating the maximum log-likelihood of a DPP on a ground set.\"},{\"question\":\"What key technique is used to obtain the results?\",\"answer\":\"The document reduces the approximation task to solving a gap instance of a vector-coloring problem on a hypergraph built from bounded-degree graph constructions and enhanced using strong expanders.\"}]","Hardness of Maximum Likelihood Learning ofDPPs | PDF"]