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dc.contributor.authorKennedy, Christopher
dc.contributor.authorSinger, Amit
dc.contributor.authorSousa Bandeira, Afonso Jose
dc.date.accessioned2016-10-06T21:02:34Z
dc.date.available2017-03-01T16:14:48Z
dc.date.issued2016-03
dc.date.submitted2014-02
dc.identifier.issn0025-5610
dc.identifier.issn1436-4646
dc.identifier.urihttp://hdl.handle.net/1721.1/104663
dc.description.abstractThe little Grothendieck problem consists of maximizing ∑[subscript ij]C[subscript ij]x[subscript i]x[subscript j] for a positive semidefinite matrix C, over binary variables x[subscript i]∈{±1}. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C∈R[superscript dn×dn] a positive semidefinite matrix, the objective is to maximize ∑[subscript ij]tr(C[superscript T][subscript ij]O[subscript i]O[superscript T][subscript j] restricting O[subscript i] to take values in the group of orthogonal matrices O[subscript d], where C[subscript ij] denotes the (ij)-th d×d block of C. We propose an approximation algorithm, which we refer to as Orthogonal-Cut, to solve the little Grothendieck problem over the group of orthogonal matrices O[subscript d] and show a constant approximation ratio. Our method is based on semidefinite programming. For a given d≥1, we show a constant approximation ratio of α[subscript R](d)[superscript 2], where α[subscript R](d) is the expected average singular value of a d×d matrix with random Gaussian N(0,1/d) i.i.d. entries. For d=1 we recover the known α[subscript R](1)[superscript 2]=2/π approximation guarantee for the classical little Grothendieck problem. Our algorithm and analysis naturally extends to the complex valued case also providing a constant approximation ratio for the analogous little Grothendieck problem over the Unitary Group U[subscript d]. Orthogonal-Cut also serves as an approximation algorithm for several applications, including the Procrustes problem where it improves over the best previously known approximation ratio of 1/2√2 . The little Grothendieck problem falls under the larger class of problems approximated by a recent algorithm proposed in the context of the non-commutative Grothendieck inequality. Nonetheless, our approach is simpler and provides better approximation with matching integrality gaps. Finally, we also provide an improved approximation algorithm for the more general little Grothendieck problem over the orthogonal (or unitary) group with rank constraints, recovering, when d=1, the sharp, known ratios.en_US
dc.description.sponsorshipUnited States. Air Force Office of Scientific Research (Grant FA9550-12-1-0317)en_US
dc.publisherSpringer Berlin Heidelbergen_US
dc.relation.isversionofhttp://dx.doi.org/10.1007/s10107-016-0993-7en_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.sourceSpringer Berlin Heidelbergen_US
dc.titleApproximating the little Grothendieck problem over the orthogonal and unitary groupsen_US
dc.typeArticleen_US
dc.identifier.citationBandeira, Afonso S., Christopher Kennedy, and Amit Singer. “Approximating the Little Grothendieck Problem over the Orthogonal and Unitary Groups.” Mathematical Programming (2016): n. pag.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Mathematicsen_US
dc.contributor.mitauthorSousa Bandeira, Afonso Jose
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.updated2016-08-18T15:36:18Z
dc.language.rfc3066en
dc.rights.holderSpringer-Verlag Berlin Heidelberg and Mathematical Optimization Society
dspace.orderedauthorsBandeira, Afonso S.; Kennedy, Christopher; Singer, Amiten_US
dspace.embargo.termsNen
dc.identifier.orcidhttps://orcid.org/0000-0002-7331-7557
mit.licensePUBLISHER_POLICYen_US


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