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dc.contributor.authorFawzi, Hamza
dc.contributor.authorParrilo, Pablo A.
dc.date.accessioned2016-07-15T21:47:49Z
dc.date.available2016-07-15T21:47:49Z
dc.date.issued2015-08
dc.identifier.issn0025-5610
dc.identifier.issn1436-4646
dc.identifier.urihttp://hdl.handle.net/1721.1/103632
dc.description.abstractThe nonnegative rank of a matrix A is the smallest integer r such that A can be written as the sum of r rank-one nonnegative matrices. The nonnegative rank has received a lot of attention recently due to its application in optimization, probability and communication complexity. In this paper we study a class of atomic rank functions defined on a convex cone which generalize several notions of “positive” ranks such as nonnegative rank or cp-rank (for completely positive matrices). The main contribution of the paper is a new method to obtain lower bounds for such ranks. Additionally the bounds we propose can be computed by semidefinite programming using sum-of-squares relaxations. The idea of the lower bound relies on an atomic norm approach where the atoms are self-scaled according to the vector (or matrix, in the case of nonnegative rank) of interest. This results in a lower bound that is invariant under scaling and that enjoys other interesting structural properties. For the case of the nonnegative rank we show that our bound has an appealing connection with existing combinatorial bounds and other norm-based bounds. For example we show that our lower bound is a non-combinatorial version of the fractional rectangle cover number, while the sum-of-squares relaxation is closely related to the Lovász [bar over ϑ] number of the rectangle graph of the matrix. We also prove that the lower bound is always greater than or equal to the hyperplane separation bound (and other similar “norm-based” bounds). We also discuss the case of the tensor nonnegative rank as well as the cp-rank, and compare our bound with existing results.en_US
dc.publisherSpringer Berlin Heidelbergen_US
dc.relation.isversionofhttp://dx.doi.org/10.1007/s10107-015-0937-7en_US
dc.rightsCreative Commons Attribution-Noncommercial-Share Alikeen_US
dc.rights.urihttp://creativecommons.org/licenses/by-nc-sa/4.0/en_US
dc.sourceSpringer Berlin Heidelbergen_US
dc.titleSelf-scaled bounds for atomic cone ranks: applications to nonnegative rank and cp-ranken_US
dc.typeArticleen_US
dc.identifier.citationFawzi, Hamza, and Pablo A. Parrilo. “Self-Scaled Bounds for Atomic Cone Ranks: Applications to Nonnegative Rank and Cp-Rank.” Mathematical Programming 158.1–2 (2016): 417–465.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Scienceen_US
dc.contributor.departmentMassachusetts Institute of Technology. Laboratory for Information and Decision Systemsen_US
dc.contributor.mitauthorParrilo, Pablo A.en_US
dc.contributor.mitauthorFawzi, Hamzaen_US
dc.relation.journalMathematical Programmingen_US
dc.eprint.versionAuthor's final manuscripten_US
dc.type.urihttp://purl.org/eprint/type/JournalArticleen_US
eprint.statushttp://purl.org/eprint/status/PeerRevieweden_US
dc.date.updated2016-06-30T12:07:25Z
dc.language.rfc3066en
dc.rights.holderSpringer-Verlag Berlin Heidelberg and Mathematical Optimization Society
dspace.orderedauthorsFawzi, Hamza; Parrilo, Pablo A.en_US
dspace.embargo.termsNen
dc.identifier.orcidhttps://orcid.org/0000-0001-6026-4102
dc.identifier.orcidhttps://orcid.org/0000-0003-1132-8477
mit.licenseOPEN_ACCESS_POLICYen_US
mit.metadata.statusComplete


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