{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:39:37Z","timestamp":1787236777912,"version":"build-2736575974"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["AVS-ISS (648701)"],"award-info":[{"award-number":["AVS-ISS (648701)"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Algebra Geometry"],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative $n \\times m$ matrix $M$ into a product of a nonnegative $n \\times d$ matrix $W$ and a nonnegative $d \\times m$ matrix $H$. A longstanding open question, posed by Cohen and Rothblum in 1993, is whether a rational matrix $M$ always has an NMF of minimal inner dimension $d$ whose factors $W$ and $H$ are also rational. We answer this question negatively, by exhibiting a matrix for which $W$ and $H$ require irrational entries.<\/jats:p>","DOI":"10.1137\/16m1078835","type":"journal-article","created":{"date-parts":[[2017,6,29]],"date-time":"2017-06-29T10:18:09Z","timestamp":1498731489000},"page":"285-307","source":"Crossref","is-referenced-by-count":5,"title":["Nonnegative Matrix Factorization Requires Irrationality"],"prefix":"10.1137","volume":"1","author":[{"given":"Dmitry","family":"Chistikov","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stefan","family":"Kiefer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ines","family":"Maru\u0161i\u0107","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mahsa","family":"Shirmohammadi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"James","family":"Worrell","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,6,29]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/0890-5401(89)90049-7"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/130913869"},{"key":"atypb3","first-page":"1","author":"Arora S.","year":"2012","journal-title":"NJ"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/110825455"},{"key":"atypb5","first-page":"460","author":"Canny J.","year":"1988","journal-title":"New York"},{"key":"atypb6","unstructured":"D. Chistikov, S. Kiefer, I. Marus\u030cic\u0301, M. Shirmohammadi, and J. Worrell,\n                      Nonnegative Matrix Factorization Requires Irrationality\n                      , preprint,https:\/\/arxiv.org\/abs\/1605.06848, 2016."},{"key":"atypb7","first-page":"1","author":"Chistikov D.","year":"2016","journal-title":"Germany"},{"key":"atypb8","first-page":"1290","author":"Chistikov D.","year":"2017","journal-title":"Philadelphia"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(93)90224-C"},{"key":"atypb10","first-page":"74","author":"Das G.","year":"1995","journal-title":"Heidelberg"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2016.06.036"},{"key":"atypb12","first-page":"3349","volume":"13","author":"Gillis N.","year":"2012","journal-title":"J. Mach. Learn. Res."},{"key":"atypb13","first-page":"257","author":"Gillis N.","year":"2014","journal-title":"Boca Raton, FL"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2012.06.038"},{"key":"atypb15","first-page":"223","volume":"8","author":"Gregory D. A.","year":"1983","journal-title":"J. Combin. Inform. System Sci."},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1214\/14-AOS1282"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1155\/2008\/764206"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1080\/00401706.1971.10488823"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/140990139"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2003.1150"},{"key":"atypb21","unstructured":"B. Poonen,\n                      Hilbert's Tenth Problem over Rings of Number-Theoretic Interest\n                      , Note from the lecture at the Arizona Winter School on \u201cNumber Theory and Logic,\" 2003,http:\/\/math.mit.edu\/ poonen\/papers\/aws2003.pdf."},{"key":"atypb22","unstructured":"Y. Shitov,\n                      Nonnegative Rank Depends on the Field\n                      , preprint,https:\/\/arxiv.org\/abs\/1505.01893, 2015."},{"key":"atypb23","unstructured":"Y. Shitov,\n                      Nonnegative Rank Depends on the Field\n                      II, preprint,https:\/\/arxiv.org\/abs\/1605.07173, 2016."},{"key":"atypb24","unstructured":"Y. Shitov,\n                      A Universality Theorem for Nonnegative Matrix Factorizations\n                      , preprint,https:\/\/arxiv.org\/abs\/1606.09068, 2016."},{"key":"atypb25","unstructured":"D. Simanek,\n                      How to View 3D Without Glasses\n                      ,https:\/\/www.lhup.edu\/ dsimanek\/3d\/view3d.htm. (Accessed May 19, 2017.)"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/1016064"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1137\/070709967"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1145\/2447712.2447732"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0000(91)90024-Y"},{"key":"atypb30","unstructured":"S. Kiefer,\n                      NMF Computations,\n                      http:\/\/www.cs.ox.ac.uk\/people\/stefan.kiefer\/NMFcomputations.html(accessed May 19, 2017)."}],"container-title":["SIAM Journal on Applied Algebra and Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1078835","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:02:18Z","timestamp":1787234538000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1078835"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":30,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/16M1078835"],"URL":"https:\/\/doi.org\/10.1137\/16m1078835","relation":{},"ISSN":["2470-6566"],"issn-type":[{"value":"2470-6566","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}