[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81911-en":3,"doc-seo-81911-105":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},81911,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","On the Complexity of Low-Rank Matrix Signing and Entrywise Power Matrix Factorization","Entrywise power matrix factorization (EPMF) decomposes a nonnegative matrix X into a low-rank component Xr followed by an entrywise power, targeting exact equality X=|Xr|◦p or Frobenius-norm approximation X≈|Xr|◦p. The work unifies special cases such as the modulus model (p=1) and componentwise squaring (p=2), tied to square-root rank. It resolves computational complexity: EPMF decision and Frobenius approximation become NP-hard in the exact and approximate settings, while fixed ranks yield polynomial-time solvability and fixed-parameter tractability via rank r.","arXiv :2607 .04875v2 [ cs .CC] 9 Jul 2026  \nOn the Complexity of Low-Rank Matrix Signing and Entrywise Power Matrix Factorization  \nNicolas Gillis∗ Subhayan Saha∗ Stefano Sicilia∗,† Arnaud Vandaele∗  \nUniversity of Mons, Mons, Belgium  \nAbstract  \nGiven a nonnegative matrix X, a factorization rank r and a positive integer p, entrywise power matrix factorization (EPMF) looks for a low-rank matrix Xr such that X = |Xr | ◦p (exact case) or X ≈ |Xr | ◦p (approximate case), where (·)◦p denotes the componentwise exponent. EPMF includes the modulus model (p = 1) and componentwise square factorization (p = 2) as special cases, the latter being closely related to the square root rank. We analyze the computational complexity of the exact decision problem and the Frobenius-norm approximation problem, and establish a complete complexity landscape. In the exact case, we show that EPMF is equivalent to the combinatorial problem of flipping the signs of the entries of a given matrix X to obtain a rank-r matrix, which we refer to as the low-rank matrix signing (LRMS) problem. We first show that LRMS, and hence exact EPMF, is strongly NP-hard, improving a weak NP-hardness result for the square-root-rank (Math. Prog., 2015) . We then show that LRMS can be solved in polynomial time when r is fixed. Moreover, when the rank r is part of the input, we show that for generic matrices the algorithm is fixed-parameter tractable (FPT) in the parameter r; in fact, the running time is fixed-parameter linear in the number of entries of the input matrix. In the approximate case using the Frobenius norm as an error measure, we show that EPMF is NP-hard, already when r = 2, the smallest nontrivial case.  \nKeywords: nonlinear matrix decompositions, entrywise power matrix factorization, signless rank, low-rank matrix signing, fixed-parameter tractable, strong NP-hardness.  \n1 Introduction  \nLow-rank matrix approximation is a central tool in numerical linear algebra, data analysis, machine learning, and signal processing. Given a data matrix X ∈ Rm ×n and an integer r ≤ min(m, n), the classical problem is to approximate X by a matrix of rank at most r, or, equivalently, to find factors W ∈ Rm ×r and H ∈ Rr ×n such that X ≈ WH. This is equivalent to linear dimensionality reduction: each column of X is approximated by a linear combination of the columns of W. The best known example is the truncated singular value decomposition (SVD), which gives an optimal rank-r approximation in the Frobenius norm, and any unitary invariant norm, by the Eckart– Young theorem [13,25,43] . Many variants, including robust PCA, sparse PCA, weighted low-rank approximation, and matrix completion, modify the loss function or impose additional constraints in order to reflect the structure of the data; see, e.g.,[44] and the references therein. The computational  \n∗ Department of Mathematics and Operational Research. We acknowledge the support by the European Union (ERC consolidator, eLinoR, no 101085607). [Emails: firstname.lastname@umons.ac.be](Emails: firstname.lastname@umons.ac.be).  \n†SS is a member of the Gruppo Nazionale Calcolo Scientifico-Istituto Nazionale di Alta Matematica (GNCSINdAM) .  \ncomplexity of such models has been studied in depth. Computing the (truncated) SVD is a classical problem in numerical linear algebra, and can be done in polynomial time in m, n, r and log(1/ε), where ε is the desired precision; see, e.g., [25,43] . The SVD can be used to compute the best rankr approximation in any unitary invariant norm, including the matrix ℓ 2 norm and the Frobenius norm. However, as soon as different norms are used, finding the best rank-r approximation becomes NP-hard, already when r = 1; this is true for the componentwise ℓ 1 [24] and ℓ∞ norms [23], and weighted norms or when data is missing [20] . Moreover, by writing X r = WH, where W has r columns and H has r rows, it is often useful in practice to add constraints on the factors, W and H, typically to improve the int","cbCaipjWCI9y94bQ","https://ap.wps.com/l/cbCaipjWCI9y94bQ","pdf",491723,6,1,28,"English","en",105,"# Introduction\n## Low-rank matrix approximation and complexity background\n## Entrywise power matrix factorization (EPMF)\n## Relationship to low-rank matrix signing (LRMS)\n## Exact and approximate complexity results","[{\"question\":\"What does entrywise power matrix factorization (EPMF) aim to compute?\",\"answer\":\"EPMF seeks a low-rank matrix Xr such that X equals |Xr|◦p exactly or is closely approximated by |Xr|◦p in Frobenius norm, where ◦p applies componentwise exponentiation.\"},{\"question\":\"How is the exact EPMF problem related to low-rank matrix signing (LRMS)?\",\"answer\":\"In the exact setting, the paper shows EPMF is equivalent to flipping signs of entries of a given matrix X to obtain a rank-r matrix, called the low-rank matrix signing (LRMS) problem.\"},{\"question\":\"What complexity results are proved for EPMF when the rank r is fixed versus part of the input?\",\"answer\":\"LRMS (hence exact EPMF) is strongly NP-hard. When r is fixed, LRMS can be solved in polynomial time; when r is part of the input, generic matrices admit fixed-parameter tractable algorithms with running time fixed-parameter linear in the number of entries.\"}]","On the Complexity of Low-Rank Matrix Signing and Entrywise Power Matrix Factorization | PDF",1784177010,71,{"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},"on-the-complexity-of-low-rank-matrix-signing-and-entrywise-power-matrix-factorization","",{"@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/on-the-complexity-of-low-rank-matrix-signing-and-entrywise-power-matrix-factorization/81911/",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","2026-08-04","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 does entrywise power matrix factorization (EPMF) aim to compute?","Question",{"text":77,"@type":78},"EPMF seeks a low-rank matrix Xr such that X equals |Xr|◦p exactly or is closely approximated by |Xr|◦p in Frobenius norm, where ◦p applies componentwise exponentiation.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How is the exact EPMF problem related to low-rank matrix signing (LRMS)?",{"text":82,"@type":78},"In the exact setting, the paper shows EPMF is equivalent to flipping signs of entries of a given matrix X to obtain a rank-r matrix, called the low-rank matrix signing (LRMS) problem.",{"name":84,"@type":75,"acceptedAnswer":85},"What complexity results are proved for EPMF when the rank r is fixed versus part of the input?",{"text":86,"@type":78},"LRMS (hence exact EPMF) is strongly NP-hard. When r is fixed, LRMS can be solved in polynomial time; when r is part of the input, generic matrices admit fixed-parameter tractable algorithms with running time fixed-parameter linear in the number of entries.","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},[95,99,103,107,112,116,121,124,129,132,136],{"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":108,"doc_module":4,"doc_module_name":47,"category_name":109,"show_sort_weight":110,"slug":111},5,"Comic",60,"comic",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},"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":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":108,"slug":139},19,"General","general"]