[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84514-en":3,"doc-seo-84514-105":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},84514,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Exact Nonnegative Matrix Factorization via Cone-Ray Witnesses","Exact nonnegative matrix factorization (NMF) for small exact-rank matrices is analyzed through a cone-ray pipeline that unifies truncated SVD, a polyhedral cone of nonnegative preimages, Double Description Method enumeration (via cddlib), and an alternating linear program for slack minimisation. Under a uniform-support restriction, the joint factorisation constraint reduces to entrywise nonnegativity of an r×r witness matrix MT=R−T1(R)−1. Feasibility is characterized and supports are ranked by geometric obtuseness, revealing saturation behavior and structural failure modes at larger problem sizes. A hybrid augmentation resolves the scaling ceiling and attains near machine-precision reconstruction error when successful.","arXiv :2606 .22451v1 [math .NA] 21 Jun 2026  \nEXACT NONNEGATIVE MATRIX FACTORIZATION VIA CONE-RAY WITNESSES: OBTUSENESS RANKING, SATURATION CURVES,  \nAND AN AUGMENTED ALT-LP BREAKTHROUGH∗  \nMITHIL RAMTEKE†  \nAbstract. We study exact nonnegative matrix factorization (NMF) of small exact-rank-r matrices through a cone-ray pipeline that combines the truncated singular value decomposition (SVD), the polyhedral cone of nonnegative preimages, the Double Description Method (DDM, realised via Fukuda’s cddlib), and an alternating linear program (alt-LP) for slack minimisation. Each column of the change-of-basis matrices Q, P is parameterised as a nonnegative combination of r cone rays. Under the uniform-support restriction the joint factorisation constraint QP⊤ = Ir reduces to the entrywise nonnegativity of an r × r witness matrix MT = R−T1 (R)−1 for an r-subset pair (T, K) of cone-ray indices. This closed-form witness recovers an exact NMF in microseconds whenever it is feasible.  \nWe characterise when it is feasible. We rank candidate r-subsets by their geometric nearorthogonality ( obtuseness, |det(RT) |/ Qi ∥RTi ∥) and walk the top of each list. A 100-trial Monte Carlo at m = n = 10 sweeps the candidate budget through maxTries ∈ {5 , 200 , 400} and exposes a clean saturation curve: the heuristic recovers 44/32/8 of 100 at the top-5 tightening, 79/85/58 at top-200, and only 79/87/59 at top-400 for r = 4 , 5 , 6. Beyond top-200 the residual failures are not search-budget-limited but structural: the cones simply do not contain a near-orthogonal r-subset on at least one side. Increasing m, n at fixed r hurts rather than helps—a parallel 100-trial sweep at m = n = 15 collapses to 37/7/0/0/0 out of 100 at r = 4 , . . . , 8, and doubling the budget thereto top-400 recovers zero additional successes. On Olivetti faces (400 × 4096) the DDM step itself becomes intractable: even at r = 8 the n = 4096 side times out at a one-hour cddlib budget.  \nOur principal new contribution is a hybrid that breaks through this ceiling. At each pair (T, K) visited by the obtuseness walk we first attempt the closed-form MT . When MT is infeasible we augment both supports by k = 2 additional rays maximally angularly-separated from the existing ones, producing r × (r + k) supports, and solve for µ, ν ∈ R ≥(rk)× r with RTµν⊤ R = Ir via a shortslack-LP alternation on the restricted bases. On the same m = n = 10 Monte Carlo this hybrid pushes success from 79/85/58 to 99/95/75 of 100 at r = 4 , 5 , 6—gains of +20, +10, +17 over the Mt-only top-200 baseline. Cone reconstruction error stays at or near machine precision whenever the heuristic succeeds. We close with the four scaling walls the cone-ray pipeline faces (DDM ray-count blow-up, uniform-support restriction, numerical conditioning, and slack-LP plateau on the fully-non-uniform variant) and concrete next-step suggestions.  \nKey words. nonnegative matrix factorization, polyhedral cones, double description method, obtuseness, alternating linear programming, exact factorization  \nAMS subject classifications. 15A23, 15A48, 90C05, 90C26, 52B55  \n1. Introduction. Nonnegative matrix factorisation seeks W ∈ Rr and H ∈ R×0r such that A ≈ WH⊤ for a given A ∈ Rn of nonnegative rank at most r. Two distinct problems hide inside this template. The approximate problem, used in topic modelling, image decomposition and recommendation systems, is typically attacked by multiplicative updates [13], hierarchical alternating least squares [5], or block coordinate descent [10], and is NP-hard in the worst case [19] . The exact problem assumes the data A admits an r-factorisation with zero residual and asks for a certified (W, H) recovering it; structural conditions such as separability [7, 2, 9, 18] make this tractable. This paper is firmly on the exact side, but without imposing separability.  \n∗ Software drafts and prose for this manuscript were prepared with the assistance of Anthropic’s Claude, an AI coding assistant. The autho","cbCaipmA5Wj29c7W","https://ap.wps.com/l/cbCaipmA5Wj29c7W","pdf",406075,1,10,"English","en",105,"# Introduction\n## Pipeline Toolkit and Certification\n## Contributions\n## Closed-Form Witness under Uniform Support\n## Obtuseness-Ranked Support Search and Saturation\n## Scaling Walls","[{\"question\":\"What role does the cone-ray pipeline play in exact NMF?\",\"answer\":\"It combines truncated SVD, polyhedral cone representation of nonnegative preimages (via DDM and cddlib), and an alternating linear program for slack minimisation to produce a certified exact factorization when feasible.\"},{\"question\":\"How does the uniform-support restriction simplify the factorization constraint?\",\"answer\":\"With uniform support, the constraint QP⊤=Ir collapses to an entrywise nonnegativity test of an r×r witness matrix MT=R−T1(R)−1 for a chosen r-subset pair (T, K).\"},{\"question\":\"Why does the success rate saturate or decline beyond certain search budgets and sizes?\",\"answer\":\"Beyond top-200 in the reported Monte Carlo, residual failures are structural (the cones do not contain a near-orthogonal r-subset on at least one side). Increasing m and n can further reduce success, and for Olivetti the DDM step becomes intractable under the stated time budget.\"}]",1784196245,25,{"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},"exact-nonnegative-matrix-factorization-via-cone-ray-witnesses","",{"@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/exact-nonnegative-matrix-factorization-via-cone-ray-witnesses/84514/",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","2026-07-22","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 role does the cone-ray pipeline play in exact NMF?","Question",{"text":75,"@type":76},"It combines truncated SVD, polyhedral cone representation of nonnegative preimages (via DDM and cddlib), and an alternating linear program for slack minimisation to produce a certified exact factorization when feasible.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the uniform-support restriction simplify the factorization constraint?",{"text":80,"@type":76},"With uniform support, the constraint QP⊤=Ir collapses to an entrywise nonnegativity test of an r×r witness matrix MT=R−T1(R)−1 for a chosen r-subset pair (T, K).",{"name":82,"@type":73,"acceptedAnswer":83},"Why does the success rate saturate or decline beyond certain search budgets and sizes?",{"text":84,"@type":76},"Beyond top-200 in the reported Monte Carlo, residual failures are structural (the cones do not contain a near-orthogonal r-subset on at least one side). Increasing m and n can further reduce success, and for Olivetti the DDM step becomes intractable under the stated time budget.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":21,"slug":133},"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]