{"paper":{"title":"Nonnegative Matrix Factorization Requires Irrationality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.NA"],"primary_cat":"cs.CC","authors_text":"Dmitry Chistikov, Ines Maru\\v{s}i\\'c, James Worrell, Mahsa Shirmohammadi, Stefan Kiefer","submitted_at":"2016-05-22T20:17:24Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06848","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}