{"paper":{"title":"On Restricted Nonnegative Matrix Factorization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.LG"],"primary_cat":"cs.FL","authors_text":"Dmitry Chistikov, Ines Maru\\v{s}i\\'c, James Worrell, Mahsa Shirmohammadi, Stefan Kiefer","submitted_at":"2016-05-23T15:26:26Z","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 $H$. Restricted NMF requires in addition that the column spaces of $M$ and $W$ coincide. Finding the minimal inner dimension $d$ is known to be NP-hard, both for NMF and restricted NMF. We show that restricted NMF is closely related to a question about the nature of minimal probabilistic automata, posed by Paz in his seminal 1971 textbook. We use this connection to answer Paz's question neg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.07061","kind":"arxiv","version":1},"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"}