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dc.contributor.authorFawzi, Hamza
dc.contributor.authorSaunderson, James
dc.contributor.authorParrilo, Pablo A.
dc.date.accessioned2019-07-09T14:20:22Z
dc.date.available2019-07-09T14:20:22Z
dc.date.issued2018-03-21
dc.identifier.issn1615-3375
dc.identifier.issn1615-3383
dc.identifier.urihttps://hdl.handle.net/1721.1/121533
dc.description.abstractThe matrix logarithm, when applied to Hermitian positive definite matrices, is concave with respect to the positive semidefinite order. This operator concavity property leads to numerous concavity and convexity results for other matrix functions, many of which are of importance in quantum information theory. In this paper we show how to approximate the matrix logarithm with functions that preserve operator concavity and can be described using the feasible regions of semidefinite optimization problems of fairly small size. Such approximations allow us to use off-the-shelf semidefinite optimization solvers for convex optimization problems involving the matrix logarithm and related functions, such as the quantum relative entropy. The basic ingredients of our approach apply, beyond the matrix logarithm, to functions that are operator concave and operator monotone. As such, we introduce strategies for constructing semidefinite approximations that we expect will be useful, more generally, for studying the approximation power of functions with small semidefinite representations. Keywords: Convex optimization, Matrix concavity, Quantum relative entropyen_US
dc.language.isoen
dc.publisherSpringer Natureen_US
dc.relation.isversionof10.1007/s10208-018-9385-0en_US
dc.rightsCreative Commons Attribution-Noncommercial-Share Alikeen_US
dc.rights.urihttp://creativecommons.org/licenses/by-nc-sa/4.0/en_US
dc.sourcearXiven_US
dc.titleSemidefinite Approximations of the Matrix Logarithmen_US
dc.typeArticleen_US
dc.identifier.citationFawzi, Hamza, et al. “Semidefinite Approximations of the Matrix Logarithm.” Foundations of Computational Mathematics 19, no. 2 (April 2019): 259–96.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Laboratory for Information and Decision Systemsen_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Scienceen_US
dc.relation.journalFoundations of Computational Mathematics Foundations of Computational Mathematicsen_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.updated2019-06-28T18:50:33Z
dspace.date.submission2019-06-28T18:50:34Z


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