[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83592-en":3,"doc-seo-83592-105":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},83592,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Robustifying Sparse Matrix Multiplication","Seminal sparse matrix multiplication asks for the product of two n×n sparse matrices, where the input and/or output nonzero counts are far below n². This work generalizes the task to (approximately) computing the k largest output entries, where approximation error depends only on smaller “noise” entries, yielding a robust sparse matrix multiplication variant. A black-box reduction achieves only polylogarithmic overhead, converting any conventional sparse multiplication into a robust recovery scheme. Using a state-of-the-art SODA’24 algorithm, improved bounds follow.","arXiv :2607 .0 1427v 1 [ cs .DS] 1 Jul 2026  \nRobustifying Sparse Matrix Multiplication  \nKarl Bringmann \\# ETH Zurich  \nNick Fischer \\#  \nMax Planck Institute for Informatics  \nVasileios Nakos \\#  \nNational and Kapodistrian University of Athens  \n~~ Abstract ~~  \nIn the seminal sparse matrix multiplication problem the goal is to compute the product of two n × n matrices when the matrices are sparse, i.e., when the number of nonzeros in the input matrices min and/or the number of nonzeros in the output matrix mout are much smaller than n2 . In this paper, we explore the generalized problem of (approximately) computing the k largest output entries, with an approximation error dependent solely on the smaller entries—from the viewpoint of sparse recovery, this can be seen as a robust variant of sparse matrix multiplication. Despite the substantial research dedicated to sparse matrix multiplication, almost no existing algorithms are robust in this sewnhseeth. eTrhoetohnereaelxceptiongorithmsis Pcanaghbe’ssialgorithm inmilarly madteimerobt(min + nk) [ITCS ’12], and it remained open  \nOur principal contribution is a black-box reduction from robust sparse matrix multiplication to conventional sparse matrix multiplication with only polylogarithmic overhead. Specifically, we show that any sparse matrix multiplication algorithm with running time T (n, min , m out ) can be transfan extoermed insive tnotoolkaitrobusfromtsaplgoarsriethmrecovreunry,nianngd iinnttriimgueil(Ty,(l,sininv, )lv) .esThsoilsvirnegductioa knanplseveack-ragtypese  \nproblem.  \nBy plugging in the state-of-the-art algorithm for sparse matrix multiplication by Abboud, Bringmann, Fischer, and Künnemann [SODA’24], we achieve significantly improved bounds such as O((min + k)1 .346 ) . Notably, in the regime where k ≥ m1in.762 , our reduction culminates in an almost-optimal k 1+o(1)-time algorithm.  \n 1  Introduction  \nFew problems have sparked as much theoretical and practical research in computer science asthe quest for efficient matrix multiplication algorithms. Motivated by countless applications, several communities have contributed to the development of Strassen-like algebraic algorithms (see [2] for the current record ω \u003C 2.3714 on the matrix multiplication exponent), lower bounds [8, 58 , 62 , 47], and specialized practically fast algorithms [5, 28] . An especial focus lies on sparse matrix multiplication, where the goal is to achieve faster algorithms when either the input-sparsity min (the number of nonzero entries in A and B) or the output-sparsity m out (the number of nonzero entries in AB) is much smaller than n2 . This problem has been studied extensively [32, 71 , 4 , 51 , 56 , 46 , 68 , 38 , 31 , 29 , 45 , 60 , 20 , 1 , 6 , 7], and many faster algorithms in terms of the three parameters n, min , mout have been proposed.  \nIn this paper we consider the generalization of sparse matrix multiplication where the output matrix AB is not necessarily sparse, but consists of k ≪ n2 large, significant entries plus up to n2 small, insignificant entries which we regard as noise—our goal is to recover the k significant entries (with some small approximation error depending on the noise) . This problem is arguably very natural and well-motivated for various applications, for instance ina setting where the output matrix is sparse up to small measurement errors, or whenever the output matrix represents some numerical scores out of which only k scores are expected tobe significant; see e.g. [16, 24, 19] .  \n2 Robustifying Sparse Matrix Multiplication  \nA concrete application in the context of information retrieval and pattern recognition is to search a database of documents (e.g., searching for a phrase on wikipedia) . This task is commonly modelled as follows; see e.g. [24] for more details.1 We fix a set of t terms (i.e. , important phrases in a search query) and store a database of n documents by a term-bydocument matrix A ∈ Rn ×t. Here each column is associated to a term","cbCaiccDidshsE9G","https://ap.wps.com/l/cbCaiccDidshsE9G","pdf",761881,4,1,31,"English","en",105,"# Introduction\n# Robustifying Sparse Matrix Multiplication\n## Robustification via Sparse Recovery","[{\"question\":\"What problem does the paper generalize beyond standard sparse matrix multiplication?\",\"answer\":\"It generalizes exact sparse matrix multiplication by focusing on approximately computing the k largest output entries, treating the remaining small entries as noise.\"},{\"question\":\"How is the robustness notion formalized in the paper?\",\"answer\":\"Robustness is expressed through sparse recovery-style guarantees, where the approximation error depends only on the insignificant part of the output (e.g., ℓ2/ℓ2 and stronger ℓ∞/ℓ2 guarantees).\"},{\"question\":\"What is the main technical contribution?\",\"answer\":\"The paper gives a black-box reduction from robust sparse matrix multiplication to conventional sparse matrix multiplication with only polylogarithmic overhead, enabling improved runtime bounds when combined with a modern sparse multiplication algorithm.\"}]",1784189064,78,{"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},"robustifying-sparse-matrix-multiplication","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/robustifying-sparse-matrix-multiplication/83592/",{"url":52,"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","2026-07-25","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 generalize beyond standard sparse matrix multiplication?","Question",{"text":75,"@type":76},"It generalizes exact sparse matrix multiplication by focusing on approximately computing the k largest output entries, treating the remaining small entries as noise.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the robustness notion formalized in the paper?",{"text":80,"@type":76},"Robustness is expressed through sparse recovery-style guarantees, where the approximation error depends only on the insignificant part of the output (e.g., ℓ2/ℓ2 and stronger ℓ∞/ℓ2 guarantees).",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main technical contribution?",{"text":84,"@type":76},"The paper gives a black-box reduction from robust sparse matrix multiplication to conventional sparse matrix multiplication with only polylogarithmic overhead, enabling improved runtime bounds when combined with a modern sparse multiplication 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