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dc.contributor.authorMoitra, Ankur
dc.date.accessioned2017-05-04T17:00:24Z
dc.date.available2017-05-04T17:00:24Z
dc.date.issued2016-02
dc.date.submitted2014-10
dc.identifier.issn0097-5397
dc.identifier.issn1095-7111
dc.identifier.urihttp://hdl.handle.net/1721.1/108665
dc.description.abstractHere, we give an algorithm for deciding if the nonnegative rank of a matrix M of dimension m \times n$ is at most r which runs in time (nm)[superscript O(r2)]. This is the first exact algorithm that runs in time singly exponential in r. This algorithm (and earlier algorithms) are built on methods for finding a solution to a system of polynomial inequalities (if one exists). Notably, the best algorithms for this task run in time exponential in the number of variables but polynomial in all of the other parameters (the number of inequalities and the maximum degree). Hence, these algorithms motivate natural algebraic questions whose solution have immediate algorithmic implications: How many variables do we need to represent the decision problem, and does M have nonnegative rank at most r? A naive formulation uses nr + mr variables and yields an algorithm that is exponential in n and m even for constant r. Arora et al. [Proceedings of STOC, 2012, pp. 145--162] recently reduced the number of variables to 2r[superscript 2] 2[superscript r], and here we exponentially reduce the number of variables to 2r[superscript 2] and this yields our main algorithm. In fact, the algorithm that we obtain is nearly optimal (under the exponential time hypothesis) since an algorithm that runs in time (nm)[superscript o(r)] would yield a subexponential algorithm for 3-SAT [Proceedings of STOC, 2012, pp. 145--162]. Our main result is based on establishing a normal form for nonnegative matrix factorization---which in turn allows us to exploit algebraic dependence among a large collection of linear transformations with variable entries. Additionally, we also demonstrate that nonnegative rank cannot be certified by even a very large submatrix of M, and this property also follows from the intuition gained from viewing nonnegative rank through the lens of systems of polynomial inequalities.en_US
dc.description.sponsorshipNational Science Foundation (U.S.) (Computing and Innovation Fellowship)en_US
dc.description.sponsorshipNational Science Foundation (U.S.) (grant DMS-0835373)en_US
dc.language.isoen_US
dc.publisherSociety for Industrial and Applied Mathematicsen_US
dc.relation.isversionofhttp://dx.doi.org/10.1137/140990139en_US
dc.rightsArticle is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.en_US
dc.sourceSociety for Industrial and Applied Mathematicsen_US
dc.titleAn Almost Optimal Algorithm for Computing Nonnegative Ranken_US
dc.typeArticleen_US
dc.identifier.citationMoitra, Ankur. “An Almost Optimal Algorithm for Computing Nonnegative Rank.” SIAM Journal on Computing 45.1 (2016): 156–173. c 2016 Society for Industrial and Applied Mathematicsen_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Mathematicsen_US
dc.contributor.mitauthorMoitra, Ankur
dc.relation.journalSIAM Journal on Computingen_US
dc.eprint.versionFinal published versionen_US
dc.type.urihttp://purl.org/eprint/type/JournalArticleen_US
eprint.statushttp://purl.org/eprint/status/PeerRevieweden_US
dspace.orderedauthorsMoitra, Ankuren_US
dspace.embargo.termsNen_US
dc.identifier.orcidhttps://orcid.org/0000-0001-7047-0495
mit.licensePUBLISHER_POLICYen_US


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