[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83393-en":3,"doc-seo-83393-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83393,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","High-Dimensional Procrustes Matching via Tree Counts","High-Dimensional Procrustes matching considers two sets of n correlated Gaussian vectors in Rd whose correspondence is hidden by an unknown permutation of indices and an unknown orthogonal rotation. The goal is exact recovery of the permutation aligning the two datasets. Prior low-dimensional results fail to extend to d ≫ log n, where earlier guarantees required nearly perfect correlation. A polynomial-time algorithm based on weighted comparisons of specially chosen wide trees achieves exact recovery at constant correlation, for d ≥ polylog(n).","arXiv :2607 .08538v1 [ stat .ML] 9 Jul 2026  \nHigh-Dimensional Procrustes Matching via Tree Counts  \nXiaochun (Nora) Niu Tselil Schramm Jiaming Xu ∗  \nAbstract  \nSuppose we observe two sets of n Gaussian vectors in Rd , with the promise that, after applying a permutation of [n] and a rotation of Rd , the two sets are ρ-correlated. The Procrustes matching problem asks us to recover the unknown permutation of [n] that aligns the two sets. The problem is well-studied in the low-dimensional regime d = O (log n), but the high-dimensional regime d ≫ log n has remained largely uncharted: prior matching guarantees require nearly perfect correlation ρ = 1 − o(1), even for information-theoretic recovery.  \nOur main result is a polynomial-time algorithm for exact recovery at constant correlation. The algorithm works by computing and comparing weighted counts of a specially chosen family of “wide” trees. So long as d ≥ polylog(n), the algorithm succeeds with high probability for any ρ2 > √α , where α ≈ 0.338 is Otter’s tree-counting constant.  \nWe complement this algorithmic result with an improved information-theoretic guarantee, showing that exact recovery is possible when ρ2 ≳ max{log n/d, plog n/n} . We also carry out a low-degree advantage calculation, which suggests that the condition ρ2 > √α is necessary for any tree-counting algorithm.  \n∗X. Niu and J. Xu are with The Fuqua School of Business, Duke University, Durham NC, USA,{xiaochun.niu,[jx77}@duke.edu](jx77}@duke.edu. T. Schramm is with)[. T. Schramm is with](jx77}@duke.edu. T. Schramm is with) the Department of Statistics, Stanford University, Stanford, CA, USA, [tselil@stanford.edu](tselil@stanford.edu).  \nContents  \n1 Introduction 3  \n1.1 Key Challenges and New Ideas .............................. 4  \n1.2 Notation and Organization ................................. 6  \n2 Main Results 6  \n2.1 Matching Algorithm .................................... 6  \n2.2 Wide Trees ......................................... 7  \n2.3 Statistical Properties of Our Similarity Scores ...................... 8  \n2.4 Performance Guarantees .................................. 11  \n2.5 Information-Theoretic Thresholds and Computational Gaps .............. 12  \n3 Statistical Guarantees 13  \n4 Joint Moments Calculation 15  \n4.1 Joint Moments of Gaussian Random Variables ...................... 15  \n4.2 Weingarten Calculus for Orthogonal Groups........................ 17  \n4.3 Joint Moments Calculation via Alternating Circuit Decompositions .......... 19  \n5 Mean Calculation 21  \n5.1 True Pairs .......................................... 21  \n5.2 False Pairs .......................................... 25  \n6 Variance Calculation 25  \n6.1 Decomposition of Decorated Union Graphs ....................... 27  \n6.2 Proof of Proposition 3.2 .................................. 30  \n6.3 Bounding the Grafted Part ................................ 31  \n6.3.1 Counting the Grafted Part ............................. 33  \n6.4 Bounding the Fully Overlapping Part ........................... 37  \n7 Polynomial-Time Approximation via Color Coding 39  \n8 From Almost Exact to Exact Recovery 43  \n9 Quadratic Assignment Estimator 48  \n9.1 Signal Lower Bound .................................... 49  \n9.2 Noise Upper Bound ..................................... 53  \n10 Limits of Tree Polynomials via Low-Degree Analysis 54  \n10.1 Graph Polynomials ..................................... 55  \n10.2 Proof of Theorem 10.1 ................................... 56  \n1 Introduction  \nAligning high-dimensional datasets is a fundamental problem in modern statistics and machine learning. It arises in tasks such as aligning word embeddings across languages [LCR+18], matching entities across networks [FLM+20], integrating biological datasets across experiments [SBH+19], and registering point clouds or shapes in computer vision [ ZSN+17] . In these settings, two datasets may encode the same underlying objects, but their correspondence is obscured by measure","cbCaitFCSq8aucAD","https://ap.wps.com/l/cbCaitFCSq8aucAD","pdf",1005540,2,1,64,"English","en",105,"# Contents\n## Introduction\n## Main Results\n## Statistical Guarantees\n## Joint Moments Calculation\n## Mean Calculation\n## Variance Calculation\n## Polynomial-Time Approximation via Color Coding\n## From Almost Exact to Exact Recovery\n## Quadratic Assignment Estimator\n## Limits of Tree Polynomials via Low-Degree Analysis","[{\"question\":\"What problem does the document study?\",\"answer\":\"It studies the Procrustes matching problem, where two correlated Gaussian datasets are related by an unknown permutation and an unknown orthogonal transformation, and the task is to recover the hidden permutation that aligns the datasets.\"},{\"question\":\"Why is the high-dimensional regime difficult?\",\"answer\":\"When d is much larger than log n, earlier matching guarantees require correlation ρ extremely close to 1, leaving much of the high-dimensional setting largely unresolved.\"},{\"question\":\"What is the core idea behind the proposed algorithm?\",\"answer\":\"The polynomial-time method computes weighted counts of a specially chosen family of “wide” trees and compares these similarity scores to recover the permutation.\"}]",1784187199,161,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"high-dimensional-procrustes-matching-via-tree-counts","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/high-dimensional-procrustes-matching-via-tree-counts/83393/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","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},"What problem does the document study?","Question",{"text":75,"@type":76},"It studies the Procrustes matching problem, where two correlated Gaussian datasets are related by an unknown permutation and an unknown orthogonal transformation, and the task is to recover the hidden permutation that aligns the datasets.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why is the high-dimensional regime difficult?",{"text":80,"@type":76},"When d is much larger than log n, earlier matching guarantees require correlation ρ extremely close to 1, leaving much of the high-dimensional setting largely unresolved.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the core idea behind the proposed algorithm?",{"text":84,"@type":76},"The polynomial-time method computes weighted counts of a specially chosen family of “wide” trees and compares these similarity scores to recover the 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