[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82399-en":3,"doc-seo-82399-105":30,"detail-sidebar-cat-0-en-105":83},{"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},82399,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Graph-Regularized Low-Rank Matrix Completion by Variable Projection","Graph-Regularized RTRMC (GRRTRMC) addresses low-rank matrix completion by adding graph regularization to the Riemannian Trust-Region Matrix Completion (RTRMC) framework. The method leverages low-rank geometry to reformulate completion as unconstrained optimization on a single Grassmann manifold. GRRTRMC exploits relationships between rows and columns, aiming to enhance accuracy and robustness when missing entries arise from data with strong row/column correlations.","arXiv :2607 .09546v 1 [ cs .LG] 10 Jul 2026  \nGraph-Regularized Low-Rank Matrix Completion by Variable Projection  \nBenoˆıt Loucheur 1*, P.-A. Absil 1 and Michel Journ´ee2  \n1* ICTEAM Institute, UCLouvain, 1348 Louvain-la-Neuve, Belgium.  \n2 Department of Climatology, Royal Meteorological Institute of Belgium, 1180 Uccle, Belgium.  \n*Corresponding author(s) . E-mail(s): benoit.loucheur@uclouvain.be; Contributing authors: [pa.absil@uclouvain.be](pa.absil@uclouvain.be) ; [michel.journee@meteo.be](michel.journee@meteo.be) ;  \nAbstract  \nWe address the low-rank matrix completion problem by incorporating graph regularization into the existing Riemannian Trust-Region Matrix Completion (RTRMC) framework. The latter uses the geometry of the low-rank constraint to remodel the problem as an unconstrained optimization problem on a single Grassmann manifold. Our approach, named Graph-Regularized RTRMC (GRRTRMC), exploits the inherent relationships between rows and columns of the matrix. By using these relationships, we aim to improve the accuracy and robustness of matrix completion, particularly in scenarios where the underlying data exhibits strong correlations between rows or columns.  \nKeywords: low-rank matrix completion, graph regularization, Riemannian  \noptimization, missing data imputation of weather data  \nMSC Classification: 15A83 , 65F55 , 90C35  \n1 Introduction  \nIn many application domains, such as recommendation systems [1], weather forecasting [2], and network analysis [3, 4], the available data is often incomplete due to missing measurements, transmission failures, or other technical limitations [5] . These gaps in the data make it challenging to analyze and utilize the information effectively, requiring robust approaches to estimate the missing values.  \n1  \nLow-rank matrix completion has emerged as a widely used method to address this issue. It is based on the assumption that the observed data can be represented as a lowrank matrix, enabling the reconstruction of missing entries from a partially observed subset.  \nThroughout this paper, we denote by M ∈ Rm ×n the complete target matrix that we aim to recover. In practice, we only observe a subset of its entries at indices Ω ⊆ {1,..., m} × {1,..., n} .  \n1.1 Related work  \nLow-rank matrix completion is classically formulated as the problem of finding a lowrank matrix that satisfies constraints on the observed entries:  \nmin rank(X) s.t. PΩ (X) = PΩ (M) . (1)  \nX∈Rm ×n  \nHere, PΩ is the projection operator that restricts a matrix to its observed entries:  \n[PΩ (X)]ij = (X0ij if (i, j) ∈ Ω  \n(2)  \notherwise.  \nDespite its intuitive formulation, this problem is NP-hard [6] due to the non-convexity of the rank function, making it computationally infeasible for large-scale matrices.  \nTo circumvent the challenges posed by rank minimization, a convex relaxation replaces the rank function with the nuclear norm, defined as the sum of the singular values of the matrix. The relaxed problem is expressed as [6]:  \nmin ∥X∥∗ s.t. PΩ (X) = PΩ (M), (3)  \nX∈Rm ×n  \nwhere ∥X∥∗ serves as a convex surrogate for the rank. While effective for small matrices, nuclear norm minimization scales poorly with the matrix dimensions due to its reliance on repeated singular value decomposition (SVD) during optimization.  \nTo improve computational efficiency, many methods adopt a factorized representation. These methods approximate the matrix as X = UW, where U ∈ Rm ×r and W ∈ Rr×n. This reformulation enforces a rank constraint implicitly, transforming the problem into [7]:  \nmin ∥PΩ (UW) − PΩ (M)∥2F + λ(∥U∥2F + ∥W∥2F) , (4)  \nU∈Rm ×r  \nW∈Rr ×n  \nwhere λ is a regularization parameter that controls the trade-off between fitting the observed entries and penalizing the complexity of U and W. This factorized formulation significantly reduces the computational cost, making it suitable for large-scale problems [8] .  \nHowever, the standard factorized approach treats all rows and columns independently, ignoring","cbCaie2GzGdpcGRi","https://ap.wps.com/l/cbCaie2GzGdpcGRi","pdf",1769091,2,1,24,"English","en",105,"# Introduction\n## Related work","[{\"question\":\"Why is graph regularization used in matrix completion?\",\"answer\":\"Graph regularization enforces smoothness over graph structures by penalizing large differences between connected nodes, so related rows or columns learn similar latent representations.\"}]",1784180130,60,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":28},"graph-regularized-low-rank-matrix-completion-by-variable-projection","",{"@graph":36,"@context":77},[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/graph-regularized-low-rank-matrix-completion-by-variable-projection/82399/",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","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"Why is graph regularization used in matrix completion?","Question",{"text":75,"@type":76},"Graph regularization enforces smoothness over graph structures by penalizing large differences between connected nodes, so related rows or columns learn similar latent representations.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":84},[85,89,93,97,101,106,111,114,119,122,126],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":29,"slug":100},5,"Comic","comic",{"id":102,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},6,"Technology",50,"technology",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":112,"slug":113},30,"research-report",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":98,"slug":129},19,"General","general"]