[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84334-en":3,"doc-seo-84334-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84334,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Structure Learning on Clustered Data","Recent algorithmic advances enable scalable directed acyclic graph (DAG) structure learning for causal discovery, but most methods require a fully homogeneous population, limiting use on clustered data. A new approach estimates a global causal structure while accounting for local cluster-level variations. It extends fixed- and random-effects mixed modeling to graph learning, using a differentiable coupling mechanism that preserves acyclicity of the combined fixed and random effects. The work provides a convergent first-order method, identifiability, and asymptotic recovery, with experiments showing improved dependency detection on real and synthetic clustered datasets.","arXiv :2607 .08238v 1 [ cs .LG] 9 Jul 2026  \nStructure Learning on Clustered Data  \nRyan Thompson [ryan.thompson-1@uts.edu.au](ryan.thompson-1@uts.edu.au)  \nSchool of Mathematical and Physical Sciences University of Technology Sydney  \nUltimo, NSW 2007, Australia  \nMatt P. Wand [matt.wand@uts.edu.au](matt.wand@uts.edu.au)  \nSchool of Mathematical and Physical Sciences University of Technology Sydney  \nUltimo, NSW 2007, Australia  \nVeerabhadran Baladandayuthapani [veerab@umich.edu](veerab@umich.edu)  \nDepartment of Biostatistics University of Michigan Ann Arbor, MI 48105, USA  \nAbstract  \nRecent algorithmic advances have made directed acyclic graph (DAG) structure learning scalable for causal discovery. Yet, the currently available techniques assume a completely homogeneous population, precluding their application to clustered data where cluster-specific variations (e.g., patient-specific effects) are common. We address this issue by introducing a new approach that estimates a global structure while accounting for local cluster-level effects. The key idea is to extend the fixed-and random-effects framework of classical mixed models to the structure learning setting. Towards this end, we present a differentiable graph coupling mechanism that guarantees the union of the fixed- and random-effects graphs remains acyclic. Computationally, we provide a provably convergent first-order method and leverage efficient batched updates across clusters. Statistically, we establish identifiability of the model and show that our approach recovers the true structure asymptotically. In experiments on real and synthetic data, our proposal detects dependencies missed by alternative estimators, underscoring its value for structure learning in clustered settings.  \nKeywords: causal discovery, continuous acyclicity, directed acyclic graphs, heterogeneous data, mixed effects  \n1 Introduction  \nDirected acyclic graphs (DAGs)—graphs with directed edges and no cycles—are a fundamental tool for probabilistic modeling and causal inference (Spirtes et al. , 2000 ; Pearl, 2009) . Their capacity to compactly encode conditional independence relations makes them useful in a wide range of domains, e.g., psychology (Foster, 2010), economics (Imbens, 2020), and epidemiology (Tennant et al. , 2021) . The first formal treatments in statistics and machine learning appeared in the late 1980s (Lauritzen and Spiegelhalter, 1988 ; Pearl, 1988) . Three decades later, Zheng et al. (2018) made a pivotal advance that transformed structure learning from a notoriously difficult combinatorial optimization problem to a tractable continuous optimization problem, opening the way to scalably learn graphs with hundreds of nodes using first-order algorithms. Surveys by Vowels et al. (2022) and Kitson et al. (2023) provide comprehensive overviews of this shift and its impact on modern structure learning.  \n©2026 Ryan Thompson, Matt P. Wand, and Veerabhadran Baladandayuthapani. License: CC-BY 4.0, see [https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/) .  \nThompson, Wand, and Baladandayuthapani  \nFigure 1: Schematic of mixed DAGs. The fixed-effects graph B, shown with solid black arrows, is added to cluster-specific random-effect deviations U (i), shown with colored dashed arrows, to form W (i) = B + U(i) . Dotted colored arrows indicate edges with both fixed-and random-effect contributions. Different colors correspond to different clusters.  \nWhile recent advances have made structure learning scalable, most methods are confined to the iid setting, where the population is assumed to follow a single homogeneous causal model. This assumption is convenient but fails when causal effects vary within the population. Several strands of work have begun to relax homogeneity, including mixtures of DAGs (Saeed et al. , 2020), DAGs adapted to distributional shifts (Huang et al. , 2020), and DAGs that vary with modifying covariates (Thompson et al. , 2024) .","cbCaipVVTyhx5SBb","https://ap.wps.com/l/cbCaipVVTyhx5SBb","pdf",951617,1,46,"English","en",105,"# Introduction\n## Directed acyclic graphs for causal inference\n## Limitations of homogeneous iid assumptions\n## Clustered data and heterogeneity\n## Mixed-effects framework for DAG learning","[{\"question\":\"Why do existing DAG structure learning methods struggle with clustered data?\",\"answer\":\"Most methods assume a homogeneous population with a single causal model, so they cannot represent cluster-specific variations in effect strength or polarity common in clustered settings.\"},{\"question\":\"How does the proposed method model DAGs on clustered data?\",\"answer\":\"It decomposes each cluster’s weighted adjacency matrix into fixed (global) effects and random (cluster-specific) deviations, yielding a structural equation model with W(i)=B+U(i).\"},{\"question\":\"How is acyclicity guaranteed when combining fixed and random effects graphs?\",\"answer\":\"A differentiable graph coupling mechanism is introduced to guarantee that the union of the fixed-effects and random-effects graphs remains acyclic.\"}]",1784194894,116,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"structure-learning-on-clustered-data","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/structure-learning-on-clustered-data/84334/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why do existing DAG structure learning methods struggle with clustered data?","Question",{"text":75,"@type":76},"Most methods assume a homogeneous population with a single causal model, so they cannot represent cluster-specific variations in effect strength or polarity common in clustered settings.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method model DAGs on clustered data?",{"text":80,"@type":76},"It decomposes each cluster’s weighted adjacency matrix into fixed (global) effects and random (cluster-specific) deviations, yielding a structural equation model with W(i)=B+U(i).",{"name":82,"@type":73,"acceptedAnswer":83},"How is acyclicity guaranteed when combining fixed and random effects graphs?",{"text":84,"@type":76},"A differentiable graph coupling mechanism is introduced to guarantee that the union of the fixed-effects and random-effects graphs remains 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