[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82180-en":3,"doc-seo-82180-105":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},82180,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Rank Independent Spectral Hypergraph Sparsification via Global Dictionary Chaining","Every weighted hypergraph on n vertices admits a spectral ε-sparsifier with O(n log n / ε^2) hyperedges. The result strengthens prior STOC 2023 independent work by eliminating the rank dependence and resolving whether the rank-related loss is inherent. The approach uses global-dictionary chaining: clique edge weights are chosen with balanced effective resistances so each hyperedge seminorm becomes Lipschitz under a common global-dictionary norm built from normalized vertex-pair directions, replacing local rank complexity with Gaussian width.","arXiv :2607 .09074v 1 [ cs .DS] 10 Jul 2026  \nRank-Independent Spectral Hypergraph Sparsification via Global-Dictionary Chaining  \nChenghua Liu* 1,2 and Yuxin Zhang†3  \n1 Institute of Software, Chinese Academy of Sciences, Beijing, China  \n2 University of Chinese Academy of Sciences, Beijing, China  \n3 University of Illinois Urbana-Champaign, IL, USA  \nAbstract  \nWe show that every weighted hypergraph on n vertices admits a spectral ε-sparsifier with O (n log n /ε2 ) hyperedges, strengthening the independent STOC 2023 works of Lee and Jambulapati–Liu–Sidford by removing their rank dependence and answering Lee’s open question on whether this loss is inherent. The key idea is global-dictionary chaining: after choosing clique edge weights with balanced effective resistances, every hyperedge seminorm is Lipschitz with respect to the same global-dictionary norm generated by normalized vertex-pair directions; the local rank complexity is thereby replaced by the Gaussian width of this common dictionary. Since these STOC 2023 works have become standard analytic primitives across a broad subsequent literature on spectral hypergraph sparsification and its variants, our rank-independent theorem sharpens many later guarantees that inherit their sampling bounds.  \n1 Introduction  \nSpectral sparsification is a central paradigm for compressing large weighted graphs while preserving the associated Laplacian quadratic form uniformly over all vertex potentials. For graphs, this paradigm admits sparsifiers with O (n /ε2 ) edges [BSS12], building on a long line of work [ST11, SS11] . In hypergraphs, the corresponding object is a natural nonlinear Laplacian energy. For a weighted hypergraph H = (V, E, w ) whose hyperedges have size at least two, the spectral energy is  \nQH (x) = e we {mu,va}e (xu − xv)2, x ∈ RV . (1)  \nWhen every hyperedge has size two, this is exactly the graph Laplacian energy. For general hypergraphs, the contribution of a hyperedge is the squared diameter of the vertex potential restricted to that hyperedge, so QH is a weighted sum of squared range seminorms.  \nThiA ws gueagatteeissubhypergraalgorithmi|pQcaHy)a(VQniHn, ()l| )() E ise hysxerctralRVaphn-sparsergyfieimroulftaHneusly ca(2p)-tures several primitives. On indicator vectors it is exactly the hypergraph cut function, which has  \n* [liuch.russell@gmail.com](liuch.russell@gmail.com)[ ](liuch.russell@gmail.com)†[yuxinz17@illinois.edu](yuxinz17@illinois.edu)  \na separate near-linear-size sparsification theory [CKN20], while on arbitrary vertex potentials it is the nonlinear Laplacian energy underlying spectral methods for hypergraphs. Thus a spectral sparsifier gives a sparse subhypergraph that preserves both cut values and nonlinear-Laplacian quantities, including eigenvalue-type parameters and Laplacian-based semi-supervised learning objectives [SY19] . These motivations have led to substantial recent activity, including spectral hypergraph sparsification in online, streaming, sliding-window, linear-sketching, fully dynamic, directed, and quantum models [STY26, CAWXZ26, KLP25, GM25, OST23, KPS24b, FGM26, LGJY25] .  \nSoma and Yoshida [SY19] initiated spectral hypergraph sparsification and proved that every rtexisanthgobtre th[i,rra]aphimdonkcha-siip,,ai- )(tineo1n((pnO1n)(r)syr.[nKe,SvkKKTYdsaa1,, dence or lost the optimal ε −2 dependence. The independent STOC 2023 works of Lee [Lee23] and Jambulapati–Liu–Sidford [JLS23] introduced chaining methods and showed that every nvertex rank-r hypergraph has a spectral sparsifier with O (n log n log r/ε2 ) hyperedges. Here r = max e∈E |e| is the rank of the hypergraph. Their analyses brought the size to nearly linear for all ranks, but left an additional logarithmic dependence on the rank. This left open a question highlighted by Lee [Lee23]:  \nCan the logarithmic dependence on the rank be removed?  \nIn this work, we study this question and answer it affirmatively.  \nTheorem 1 (Main theorem). For every n-vertex weighted hypergraph","cbCail5L1Q7fAZBB","https://ap.wps.com/l/cbCail5L1Q7fAZBB","pdf",254529,1,19,"English","en",105,"# Abstract\n# Introduction\n## Spectral sparsification and hypergraph Laplacian energy\n## Motivation from prior STOC 2023 work\n## Main theorem and rank factor source","[{\"question\":\"What does the main theorem guarantee for weighted hypergraphs?\",\"answer\":\"For every n-vertex weighted hypergraph and ε in (0,1), the theorem guarantees a spectral ε-sparsifier with O(n log n / ε^2) hyperedges, preserving the relevant spectral quantities.\"},{\"question\":\"How does this work differ from the STOC 2023 chaining results?\",\"answer\":\"It removes the additional logarithmic dependence on the hypergraph rank present in earlier bounds, answering Lee’s open question about whether that rank loss is inherent.\"},{\"question\":\"What is the core technique called global-dictionary chaining?\",\"answer\":\"After choosing clique edge weights with balanced effective resistances, hyperedge seminorms become Lipschitz with respect to a shared global-dictionary norm generated by normalized vertex-pair directions, so local rank complexity is replaced by Gaussian width.\"}]",1784178618,48,{"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},"rank-independent-spectral-hypergraph-sparsification-via-global-dictionary-chaining","",{"@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/rank-independent-spectral-hypergraph-sparsification-via-global-dictionary-chaining/82180/",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},"What does the main theorem guarantee for weighted hypergraphs?","Question",{"text":75,"@type":76},"For every n-vertex weighted hypergraph and ε in (0,1), the theorem guarantees a spectral ε-sparsifier with O(n log n / ε^2) hyperedges, preserving the relevant spectral quantities.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does this work differ from the STOC 2023 chaining results?",{"text":80,"@type":76},"It removes the additional logarithmic dependence on the hypergraph rank present in earlier bounds, answering Lee’s open question about whether that rank loss is inherent.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the core technique called global-dictionary chaining?",{"text":84,"@type":76},"After choosing clique edge weights with balanced effective resistances, hyperedge seminorms become Lipschitz with respect to a shared global-dictionary norm generated by normalized vertex-pair directions, so local rank complexity is replaced by Gaussian 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