[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85099-en":3,"doc-seo-85099-105":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},85099,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Locally Approximating the Top Eigenvector of Bounded Entry Matrices","The paper presents a local computation algorithm that approximates the top eigenvector x∈Rn of a symmetric matrix A∈Rn×n whose entries lie in [−1,1]. Building on prior work for eigenvalue estimation, it achieves preprocessing ~O(1/ε^4) and per-coordinate query complexity O(1/ε^2) for additive-εn accuracy under a spectral balance condition between |λ̃min(A)| and λmax(A). When the gap is large, the cost degrades to ~O(1/ε^6.6) preprocessing and O(1/ε^3.3) per query. A lower bound of Ω(n/ε^2) queries is shown, and the method is applied to dense-graph sparsest-cut and max-cut via local Cheeger-type and Trevisan-style algorithms.","arXiv :2607 .08556v 1 [ cs .DS] 9 Jul 2026  \nLocally Approximating the Top Eigenvector of Bounded Entry Matrices  \nNicolas Menand∗ Erik Waingarten†  \nAbstract  \nWe provide a local computation algorithm to approximate the top eigenvector x ∈ Rn of a symmetric matrix A ∈ Rn × n with entries between −1 and 1, building on the work of Swartwor˜th and  \nWoodruff [SW25] who show how to approximate the eigenvalues up to additive-εn error using O(1/ε4 )  \nqueries.  \nOur local comput˜ation algorithm has a preprocessing complexity of ˜O(1/ε4 ) and per-coordinate query complexity of O(1/ε2 ) for an additive-εn approximation whenever | λ˜min (A)  | = O (λmax (A)) . When λ˜min (A)  greatly exceeds λmax (A), our complexity degrades to at most O(1/ε6.6 ) in preprocessing and  \nO(1/ε3.3 ) per query. Furthermore, we show a lower bound of Ω(n/ε2 ) on the total number of queries needed to output an approximately top eigenvector (implying that the per-coordinate query complexity of Ω(1/ε2 ) is necessary) .  \nAs an application, we use our algorithm to provide local computation algorithms for the sparsest-cut and max-cut problems in the dense graph model of Goldreich, Goldwasser, Ron [GGR98] . By accessing the top eigenvectors (of an approximate normalized adjacency), we implement local versions of Cheeger’sinequality and Trevisan’s algorithm [Tre12] to obtain “square-root-opt” approximations in polynomial time (as opposed to exponential-in-poly(1/ε) time which is incurred in [GGR98]) .  \n∗ University of Pennsylvania. Supported by the National Science Foundation (NSF) under Grant No. CCF-2337993, as well [as No. CCF-2045128.](as No. CCF-2045128. nmenand@cis.upenn.edu)[ nmenand@cis.upenn.edu](as No. CCF-2045128. nmenand@cis.upenn.edu).  \n†University of Pennsylvania. Supported by the National Science Foundation (NSF) under Grant No. CCF-2337993 . [ewaingar@seas.upenn.edu](ewaingar@seas.upenn.edu).  \nContents  \n1 Introduction 3  \n1.1 Our Contributions .......................................... 4  \n1.2 Applications to Graph Partitioning in Dense Graphs ....................... 5  \n1.3 Technical Overview .......................................... 6  \n2 Preliminaries 9  \n2.1 Local Computation Algorithms ................................... 9  \n2.2 Randomized Linear Algebra ..................................... 9  \n3 Local Computation Algorithm for the Top Eigenvector 10  \n4 Analysis of the Algorithm 12  \n4.1 Theorem 6 assuming Lemma 4.4 , 4.2 and 4.3 ........................... 12  \n4.2 Proof of Lemma 4.4 ......................................... 14  \n4.3 Spectral Decompositions ....................................... 15  \n4.4 The Lift ................................................ 17  \n4.5 Proof of Lemma 4.2 ......................................... 19  \n4.6 Proof of Lemma 4.3 ......................................... 21  \n5 Lower Bound 23  \n5.1 Equivalent Method for Sampling from Hard Distribution ..................... 25  \n5.2 Using the Deterministic Algorithm A to solve the Coin Problem ................ 26  \n5.3 Putting Everything Together: Proof of Theorem 2 ........................ 28  \n6 Local Computation Algorithm for Sparsest Cut 29  \n6.1 Reduction to (α,β)-Dense Graphs, for α,β = poly(ε): Proof of Theorem 3 ........... 29  \n6.2 Sparsest Cut on (α,β)-dense Graphs with α,β = poly(ε): Proof of Theorem 9 ......... 33  \n7 Local Computation Algorithm for Max-Cut 37  \n7.1 Reduction to (α,β)-Dense Graphs, for α,β = poly(ε): Proof of Theorem 4 ........... 37  \n7.2 Max Cut on (α,β)-dense Graphs with α,β = poly(ε): Proof of Theorem 12 .......... 41  \nA Spectral Norm Decay 44  \nB Negative Eigenvalue Interference 48  \n1 Introduction  \nComputing the top eigenvector of a symmetric matrix is a fundamental algorithmic primitive and often the starting point of spectral algorithms (see [KV09] for a survey on spectral algorithms) . A canonical such algorithm receives as input a matrix and utilizes a polynomial-time algorithm to find the top (or top-few) eigenvector","cbCainSSozc7ouDV","https://ap.wps.com/l/cbCainSSozc7ouDV","pdf",790302,2,1,52,"English","en",105,"# Introduction\n## Our Contributions\n## Applications to Graph Partitioning in Dense Graphs\n## Technical Overview\n# Preliminaries\n## Local Computation Algorithms\n## Randomized Linear Algebra\n# Local Computation Algorithm for the Top Eigenvector\n# Analysis of the Algorithm\n## Theorem and Lemma Structure\n## Spectral Decompositions\n## Lift and Proofs\n# Lower Bound\n## Sampling from Hard Distributions\n## Deterministic Reduction\n# Local Computation Algorithm for Sparsest Cut\n# Local Computation Algorithm for Max-Cut\n# Appendix: Spectral Norm Decay\n# Appendix: Negative Eigenvalue Interference","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It studies how to output an approximately correct top eigenvector of a bounded-entry symmetric matrix using local computation and sublinear query access to matrix entries.\"},{\"question\":\"How does the algorithm’s query complexity depend on ε?\",\"answer\":\"For additive-εn accuracy under a spectral condition, it uses preprocessing ~O(1/ε^4) and per-coordinate query complexity O(1/ε^2). If the smallest relevant eigenvalue magnitude greatly exceeds the largest, the complexities degrade to ~O(1/ε^6.6) preprocessing and O(1/ε^3.3) per query.\"},{\"question\":\"What lower bound is proved and why does it matter?\",\"answer\":\"The paper proves an Ω(n/ε^2) lower bound on the total number of queries needed to output an approximately top eigenvector, showing that a per-coordinate Ω(1/ε^2) dependence is necessary and near-optimal up to constants.\"}]",1784201101,131,{"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},"locally-approximating-the-top-eigenvector-of-bounded-entry-matrices","",{"@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/locally-approximating-the-top-eigenvector-of-bounded-entry-matrices/85099/",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","2026-07-21","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 paper address?","Question",{"text":75,"@type":76},"It studies how to output an approximately correct top eigenvector of a bounded-entry symmetric matrix using local computation and sublinear query access to matrix entries.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the algorithm’s query complexity depend on ε?",{"text":80,"@type":76},"For additive-εn accuracy under a spectral condition, it uses preprocessing ~O(1/ε^4) and per-coordinate query complexity O(1/ε^2). If the smallest relevant eigenvalue magnitude greatly exceeds the largest, the complexities degrade to ~O(1/ε^6.6) preprocessing and O(1/ε^3.3) per query.",{"name":82,"@type":73,"acceptedAnswer":83},"What lower bound is proved and why does it matter?",{"text":84,"@type":76},"The paper proves an Ω(n/ε^2) lower bound on the total number of queries needed to output an approximately top eigenvector, showing that a per-coordinate Ω(1/ε^2) dependence is necessary and near-optimal up to constants.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]