[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81839-en":3,"doc-seo-81839-105":31,"detail-sidebar-cat-0-en-105":93},{"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":28,"seo_description":14,"update_tm":29,"read_time":30},81839,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Partition Rank and Algebraic Circuit Lower Bounds","Strassen’s theory links bilinear computation complexity to tensor rank, but this link fails for higher multilinearity degrees. This work establishes a new connection using a generalized tensor rank from Naslund’s partition ranks, showing how partition ranks bound multiplicative (and hence arithmetic) complexity from below for any constant degree. The bilinear case recovers Strassen’s characterization. Applications are outlined for fine-grained complexity problems such as the hyperclique conjecture, and relationships are given to tensor slice rank and its symmetric variant, including a simple NP-hardness result for the symmetric case.","arXiv :2607 .0224 1v 1 [ cs .CC] 2 Jul 2026  \nPARTITION RANK AND ALGEBRAIC CIRCUIT LOWER BOUNDS  \nCORNELIUS BRAND, PETTERI KASKI, JIAHENG WANG  \nAbstract . Strassen’s theory ofbilinear complexity provides a mathematical characterization of the arithmetic complexity of primitives such as matrix multiplication via the rank of tensors. However, the connection to tensor rank is known to break down in higher degrees of multilinearity.  \nIn this work, we highlight an unexplored connection between a generalized notion of tensor rank, which can be defined in Naslund’s framework of partition ranks (JCTA 2020), and multiplicative complexity. These partition ranks allow us to control the multiplicative complexity, and thus arithmetic complexity, in any constant degree of multilinearity from below, while recovering Strassen’s seminal characterization in the bilinear case. This enables novel potential applications of the rankbased approaches to problems in fine-grained algorithms and complexity, such as the hyperclique conjecture of Lincoln-Williams-Vassilevska Williams (SODA 2018) . Moreover, we exhibit connections to established notions of rank, such as tensor slice rank (in the sense of Tao and Sawin), as well as its symmetric variant. For computing the latter symmetric variant, we point out a simple NP-hardness proof, contrasting the rather involved NP-hardness proof for ordinary, non-symmetric tensor slice rank by Bläser et al. (SODA 2021) .  \n1. Introduction  \nStrassen’s 1969 breakthrough on algorithms for matrix multiplication [18] has had substantial impact both within and outside of algebraic complexity theory. Besides this algorithmic result, Strassen also laid out a structural theory around lower bounds on the so-called bilinear complexity of computation in algebraic structures such as matrices and polynomials. One of the cornerstones of this theory is the result that the number of multiplications between inputs from a field F needed to perform such operations using any possible computation scheme (their multiplicative complexity L) is characterized up to a small and explicit factor by the rank of an associated tensor [19, Korollar 5]; see the discussion of [6, Eq. (14.8)] for an account in English. For example, Strassen’s theory reduces the fast evaluation of a matrix product W = UV for given square matrices U = (Ui,j : i,j ∈ [n]) and V = (Vi,j : i, j ∈ [n]) into the task of finding low-rank decompositions for the associated 3-tensor, viewed as a polynomial  \nMMn = X Ui,jVj,kWi,k (1)  \ni,j,k∈[n]  \nthat is linear in each of the three sets of indeterminates, Ui,j , Vi,j , and Wi,j for i, j ∈ [n] := {1,..., n} . To state Strassen’s main result in the language of polynomials, we define the multiplicative complexity L (f) of f ∈ F [X1 ,..., Xq] as the smallest ℓ such that f is computed by any computation scheme that performs no more than ℓ essential multiplications (formal definitions are given later) .  \n(Cornelius Brand) University of Regensburg .  \n(Petteri Kaski) Aalto University.  \n(Jiaheng Wang) University of Regensburg, and University of Helsinki.  \nThis project is funded by the European Union (ERC, CountHom, 101077083) . Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. JW has received financial support from a postdoctoral fellowship funded by the Helsinki Institute of Information Technology (HIIT) .  \nStrassen’s main result now connects, up to small constants, the multiplicative complexity of abilinear map, viewed as a 3-tensor or the associated polynomial f, with the tensor rank R (f) of the tensor. Adapting the constants to the present formalism, this reads:  \nTheorem 1 (Strassen [19]) . For every 3-tensor f ∈ (Fm )⊗3, we have  \n196 · R(f) ≤ L (f) ≤ 2 · R(f) . (2)  \nThe power of Strassen’s approach is exemplified by matrix multiplication algorithms: the exponent of matrix multiplication ω encode","cbCaiipU9BJg8o8a","https://ap.wps.com/l/cbCaiipU9BJg8o8a","pdf",903391,5,1,12,"English","en",105,"# Introduction\n## Strassen’s bilinear complexity and tensor rank\n## Failure for higher multilinearity and motivation\n## Hyperclique conjecture and lower bounds\n## Main results and λ-rank framework","[{\"question\":\"What problem does the document address regarding Strassen’s theory?\",\"answer\":\"It addresses that Strassen’s connection between tensor rank and multiplicative complexity is known to break down for tensors of multilinearity degree d≥4.\"},{\"question\":\"What alternative parameter is proposed to obtain lower bounds?\",\"answer\":\"The document uses a symmetric instance of Naslund’s partition rank, called λ-rank, to derive lower bounds on multiplicative complexity for any d-tensor with d≥3.\"},{\"question\":\"How do the results relate to the hyperclique conjecture?\",\"answer\":\"The work notes an application pathway: rank-based lower bounds for relevant multilinear polynomials can be used to tackle fine-grained complexity questions, including the hyperclique conjecture.\"}]","Partition Rank and Algebraic Circuit Lower Bounds | PDF",1784176561,30,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"partition-rank-and-algebraic-circuit-lower-bounds","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/partition-rank-and-algebraic-circuit-lower-bounds/81839/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What problem does the document address regarding Strassen’s theory?","Question",{"text":77,"@type":78},"It addresses that Strassen’s connection between tensor rank and multiplicative complexity is known to break down for tensors of multilinearity degree d≥4.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"What alternative parameter is proposed to obtain lower bounds?",{"text":82,"@type":78},"The document uses a symmetric instance of Naslund’s partition rank, called λ-rank, to derive lower bounds on multiplicative complexity for any d-tensor with d≥3.",{"name":84,"@type":75,"acceptedAnswer":85},"How do the results relate to the hyperclique conjecture?",{"text":86,"@type":78},"The work notes an application pathway: rank-based lower bounds for relevant multilinear polynomials can be used to tackle fine-grained complexity questions, including the hyperclique conjecture.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,111,116,121,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":30,"slug":122},"research-report",{"id":124,"doc_module":4,"doc_module_name":47,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":47,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":47,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":47,"category_name":137,"show_sort_weight":20,"slug":138},19,"General","general"]