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DOI: 10.4230/LIPIcs.ICALP.2016.103
URN: urn:nbn:de:0030-drops-62389
URL: https://drops.dagstuhl.de/opus/volltexte/2016/6238/
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 | |
| Seminar: | 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016) | |
| Issue date: | 2016 | |
| Date of publication: | 23.08.2016 |
