文章基本信息

标题：On Restricted Nonnegative Matrix Factorization
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作者：Dmitry Chistikov ; Stefan Kiefer ; Ines Marusic 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2016
卷号：55
页码：103:1-103:14
DOI：10.4230/LIPIcs.ICALP.2016.103
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative n*m matrix M into a product of a nonnegative n*d matrix W and a nonnegative d*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 negatively, thus falsifying a positive answer claimed in 1974. Furthermore, we investigate whether a rational matrix M always has a restricted NMF of minimal inner dimension whose factors W and H are also rational. We show that this holds for matrices M of rank at most 3 and we exhibit a rank-4 matrix for which W and H require irrational entries.
关键词：nonnegative matrix factorization; nonnegative rank; probabilistic automata; labelled Markov chains; minimization