[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84377-en":3,"doc-seo-84377-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},84377,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","On the Convergence of Belief Propagation for Multipath Data Association in Target Tracking","Belief propagation (BP) is widely used for data association in target tracking, but prior convergence analyses typically cover only two-way target–measurement correspondences. Multipath data association (MPDA) permits a single target to generate multiple measurements through distinct propagation paths, creating a three-way relationship among targets, paths, and measurements. This work establishes a complete convergence proof for BP message updates in MPDA, proving convergence to a unique fixed point. Simulations confirm convergence behavior and show an accuracy–efficiency advantage over single-scan and two-scan variants of the multipledetection multiple-hypothesis tracker.","On the Convergence of Belief Propagation for Multipath Data Association in Target Tracking  \nKuilong Yang, Zengfu Wang*, Hua Lan, Jing Fu  \narXiv :2607 .0852 1v 1 [ cs .IT] 9 Jul 2026  \nAbstract—Belief propagation (BP) is widely used for data association (DA) in target tracking. Existing convergence analyses of BP for DA address only the two-way correspondence between targets and measurements, where each target generates at most one measurement per scan. Multipath DA (MPDA) allows a single target to produce multiple measurements via distinct propagation paths, creating a three-way correspondence among targets, paths, and measurements, for which a complete convergence proof has not yet been provided. We provide such a proof for the BP updates in MPDA, establishing convergence to a unique fixed point. Simulations illustrate the convergence behavior of BP in MPDA and demonstrate a favorable accuracy–efficiency trade-off relative to both single-scan and two-scan variants of the multipledetection multiple-hypothesis tracker.  \nIndex Terms—Belief propagation, data association, multipath, convergence, target tracking.  \nI. INTRODUCTION  \nMultipath data association (MPDA) is a critical challenge in multipath detection systems (MDS), including skywave over-the-horizon radar (OTHR) [1], [2], [3], passive radars [4], urban radar networks [5], [6], and wireless simultaneous localization and mapping [7],[8] . In MDS, a single point target may generate multiple measurements via different propagation paths. The unknown correspondence between targets, propagation paths, and measurements has to be resolved. Traditional data association (DA) methods, including multipath probabilistic data association (PDA) [2], multiple-detection joint PDA (MD-JPDA) [9], and multiple-detection multiple hypothesis tracking (MD-MHT) [10], may suffer from combinatorial explosion in joint target-measurement-path correspondences, or information loss from probabilistic approximations.  \nBelief propagation (BP) provides an efficient and scalable inference framework for MPDA via factor graphs [7],[11], [12] . Since BP is an iterative algorithm, establishing its convergence is important for reliable deployment. The prior convergence of BP for two-way DA—where each target generates at most one measurement per scan—was proved in [13] using the contraction mapping theorem. While [11] observed a convergence argument for BP in MPDA by treating each (target, path) pair as a pseudo-target within the framework of [13], a complete convergence proof tailored to MPDA has not yet been demonstrated; a detailed discussion is given in Section III-A.  \nKuilong Yang, Zengfu Wang, Hua Lan are with the School of Automation, Northwestern Polytechnical University, Xi’an 710072, China. Jing Fu is with the School of Engineering, RMIT University, Melbourne, VIC, 3000, Australia. This work was supported in part by the National Natural Science Foundation of China (Grants No. 62473317, U21B2008, 62371398) . (Corresponding author: Zengfu Wang.)  \nIn this correspondence, we prove that, for the MPDA formulation of [11], every execution of the sum-product BP algorithm converges to a unique fixed point. The proof buildson the contraction lemmas of [13], which apply to the MPDA message updates after a straightforward algebraic reformulation, and establishes convergence via the Banach fixed-point theorem [14] . Simulation results demonstrate the convergence behavior of BP and its favorable tracking accuracy–efficiency trade-off relative to MD-MHT [10] .  \nII. BP ALGORITHM FOR MPDA  \nA. Problem Description  \nWe follow the description of the MPDA problem in the context of multiple target tracking (MTT) in [11] . For any positive integer N , let [N] represent the set {1, 2 ,..., N} . Let xi,k ∈ Rnx be the kinematic state of target i at time k. The discrete-time dynamics for Xk independent targets are given by xi,k+1 = fi,k (xi,k ) + ui,k , i ∈ [Xk], where fi,k (·) is the known state transition function and","cbCaipuVZA0bhb2r","https://ap.wps.com/l/cbCaipuVZA0bhb2r","pdf",932765,1,9,"English","en",105,"# Introduction\n# BP Algorithm for MPDA\n## Problem Description","[{\"question\":\"What limitation do existing BP convergence analyses have for data association in target tracking?\",\"answer\":\"They consider only two-way correspondence between targets and measurements, assuming each target produces at most one measurement per scan. Multipath settings with multiple measurements per target are not fully covered by existing proofs.\"},{\"question\":\"How does multipath data association (MPDA) change the correspondence structure?\",\"answer\":\"MPDA allows a single target to generate multiple measurements via distinct propagation paths. This introduces a three-way correspondence among targets, paths, and measurements.\"},{\"question\":\"What does the document prove about belief propagation (BP) in MPDA?\",\"answer\":\"Every execution of the sum-product BP algorithm for the MPDA formulation converges to a unique fixed point. The proof uses contraction lemmas and applies the Banach fixed-point theorem to the MPDA message updates.\"}]",1784195186,23,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"on-the-convergence-of-belief-propagation-for-multipath-data-association-in-target-tracking","",{"@graph":35,"@context":84},[36,53,67],{"@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/on-the-convergence-of-belief-propagation-for-multipath-data-association-in-target-tracking/84377/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What limitation do existing BP convergence analyses have for data association in target tracking?","Question",{"text":74,"@type":75},"They consider only two-way correspondence between targets and measurements, assuming each target produces at most one measurement per scan. Multipath settings with multiple measurements per target are not fully covered by existing proofs.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does multipath data association (MPDA) change the correspondence structure?",{"text":79,"@type":75},"MPDA allows a single target to generate multiple measurements via distinct propagation paths. This introduces a three-way correspondence among targets, paths, and measurements.",{"name":81,"@type":72,"acceptedAnswer":82},"What does the document prove about belief propagation (BP) in MPDA?",{"text":83,"@type":75},"Every execution of the sum-product BP algorithm for the MPDA formulation converges to a unique fixed point. 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