[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82154-en":3,"doc-seo-82154-105":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},82154,1099514067438,"River Wang","https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542",8,"Research & Report","Geometric Planted Matchings in High Dimensions: The Power of Multiple Views","Studies recovering an unknown correspondence between n points in Rd and a noisy permuted version under a Gaussian planted matching model. In high dimensions d = ω(log n) with noise level σ^2 = d/(b log n), it proves an all-or-nothing phase transition: for any fixed b \u003C 2, no estimator attains positive overlap with the true permutation, and even Euclidean reconstruction of the matched point cloud is asymptotically as uninformative as ignoring the correspondence. With K independently permuted noisy views, a polynomial-time method achieves efficient recovery of all relative matchings up to n errors whenever b > K/(K − 1), showing multiple views can overcome the b = 2 barrier.","arXiv :2607 .09026v1 [math . ST] 10 Jul 2026  \nGeometric planted matchings in high dimensions: The power of  \nmultiple views  \nTimothy L. H. Wee∗ Kaylee Y. Yang† Zhou Fan‡ Cheng Mao §  \nAbstract  \nWe study the problem of recovering the correspondence between a collection of n points in Rd and a noisy, permuted version of those points. In the high-dimensional regime d = ω(log n), under a Gaussian model with noise variance σ 2 = d/ (blog n), prior work identifies b = 2 as the threshold for almost exact recovery. We prove that this threshold is all-or-nothing: for every fixed b \u003C 2, no estimator recovers a positive fraction of the matching, and even estimating the matched point cloud in Euclidean distance is asymptotically no better than ignoring the correspondence. On the other hand, we consider a multi-view generalization of the problem where K noisy, independently permuted copies of the same latent point cloud are observed. Here we show that a simple polynomial-time procedure recovers all relative matchings up too (n) errors whenever b > K/ (K − 1) . Thus multiple views can break the impossibility barrier b = 2 for the original matching problem: in particular, for 3/2 \u003C b \u003C 2, the two-view model has no nontrivial recovery, but a third view makes all latent correspondences efficiently recoverable.  \nContents  \n1 Introduction 2  \n1.1 Related work ........................................ 3  \n2 Main results 4  \n2.1 Two-view geometric planted matching .......................... 4  \n2.2 Multi-view geometric planted matching ......................... 6  \n2.3 Gaussian weighted matching ................................ 8  \n3 Proof overview of Theorems 2.1 and 2.5 9  \nA Proofs for the two-view negative result 12  \nA.1 Upper bound on restricted free energy .......................... 15  \nA.2 Asymptotics of the full free energy ............................ 18  \nA.2.1 High probability event ............................... 21  \nA.2.2 First moment computations ............................ 28  \nA.2.3 Second moment ................................... 29  \nA.2.4 Proof of Theorem A.3 ............................... 34  \nA.3 Proof of Corollary 2.2 ................................... 36  \n∗ School of Mathematics, Georgia Institute of Technology, [timothy.wee@gatech.edu](timothy.wee@gatech.edu)[ ](timothy.wee@gatech.edu)†Department of Statistics and Data Science, Yale University, [yingxi.yang@yale.edu](yingxi.yang@yale.edu)[ ](yingxi.yang@yale.edu)‡Department of Statistics and Data Science, Yale University, [zhou.fan@yale.edu](zhou.fan@yale.edu)  \n§School of Mathematics, Georgia Institute of Technology, [cheng.mao@math.gatech.edu](cheng.mao@math.gatech.edu)  \nB Proofs for the multi-view positive result 37  \n1 Introduction  \nLet X1 , ... , Xn be a point cloud consisting of independent standard Gaussian vectors in Rd. In the geometric planted matching model, one observes the original point cloud as well as a noisy permuted copy  \nYi = X Π∗(i) + σZi, i = 1 , . . . , n, (1)  \nwhere the noise Z1 , ... , Zn are independent standard Gaussian vectors in Rd , σ is the noise scaling, and Π∗ is an unknown uniformly random permutation, viewed as a bijection on [n] . The goal is to recover Π∗ .  \nThis model captures the fundamental task of identifying unknown correspondences between unordered objects from noisy features. Such matching problems arise across science and engineering, including in single-cell data integration and multi-omics alignment [HLMM18, DSS+22 , CJM+22], particle tracking [CKK+10], image matching [MJF+21], record linkage [SBSBS16], and network alignment [BGG+09] . Across these settings, the same statistical and algorithmic question recurs: what level of noise can the observations tolerate so that the latent correspondence remains recoverable, and when can this be done efficiently?  \nOur results and techniques pertain to the high-dimensional regime where d = ω(log n) and where the noise scales as  \nσ 2 =  d  blog n ,  \nwhere b > 0 is a fixed ","cbCaiia0l2BxKQuw","https://ap.wps.com/l/cbCaiia0l2BxKQuw","pdf",577660,1,43,"English","en",105,"# Introduction\n## Related work\n# Main results\n## Two-view geometric planted matching\n## Multi-view geometric planted matching\n## Gaussian weighted matching\n# Proof overview of Theorems 2.1 and 2.5\n# Proofs for the two-view negative result\n# Proofs for the multi-view positive result","[{\"question\":\"What problem does geometric planted matching address in this work?\",\"answer\":\"It studies recovering the unknown permutation correspondence between a latent point cloud and a noisy, permuted observation. The objective is to reconstruct the true bijection between points despite Gaussian noise and random relabeling.\"},{\"question\":\"What is the main recovery threshold in the two-view (single-copy) setting?\",\"answer\":\"In the high-dimensional regime d = ω(log n), previous work identified b = 2 as the threshold for almost exact recovery. This paper proves an all-or-nothing behavior: for any fixed b \\u003c 2, no estimator can achieve nonvanishing overlap.\"},{\"question\":\"How do multiple views change the recoverability barrier?\",\"answer\":\"With K independently permuted noisy copies of the same latent point cloud, a polynomial-time procedure recovers all relative matchings up to n errors when b \\u003e K/(K − 1). This shows that additional views can break the impossibility barrier at b = 2.\"}]",1784178480,108,{"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},"geometric-planted-matchings-in-high-dimensions-the-power-of-multiple-views","",{"@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/geometric-planted-matchings-in-high-dimensions-the-power-of-multiple-views/82154/",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 problem does geometric planted matching address in this work?","Question",{"text":74,"@type":75},"It studies recovering the unknown permutation correspondence between a latent point cloud and a noisy, permuted observation. The objective is to reconstruct the true bijection between points despite Gaussian noise and random relabeling.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What is the main recovery threshold in the two-view (single-copy) setting?",{"text":79,"@type":75},"In the high-dimensional regime d = ω(log n), previous work identified b = 2 as the threshold for almost exact recovery. This paper proves an all-or-nothing behavior: for any fixed b \u003C 2, no estimator can achieve nonvanishing overlap.",{"name":81,"@type":72,"acceptedAnswer":82},"How do multiple views change the recoverability barrier?",{"text":83,"@type":75},"With K independently permuted noisy copies of the same latent point cloud, a polynomial-time procedure recovers all relative matchings up to n errors when b > K/(K − 1). 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