License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2016.103
URN: urn:nbn:de:0030-drops-62389
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6238/
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Chistikov, Dmitry ; Kiefer, Stefan ; Marusic, Ines ; Shirmohammadi, Mahsa ; Worrell, James

On Restricted Nonnegative Matrix Factorization

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Abstract

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.

BibTeX - Entry

@InProceedings{chistikov_et_al:LIPIcs:2016:6238,
  author =	{Dmitry Chistikov and Stefan Kiefer and Ines Marusic and Mahsa Shirmohammadi and James Worrell},
  title =	{{On Restricted Nonnegative Matrix Factorization}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{103:1--103:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6238},
  URN =		{urn:nbn:de:0030-drops-62389},
  doi =		{10.4230/LIPIcs.ICALP.2016.103},
  annote =	{Keywords: nonnegative matrix factorization, nonnegative rank, probabilistic automata, labelled Markov chains, minimization}
}

Keywords: nonnegative matrix factorization, nonnegative rank, probabilistic automata, labelled Markov chains, minimization
Collection: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 23.08.2016

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