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dc.contributor.authorDaskalakis, Konstantinos
dc.contributor.authorKamath, Gautam Chetan
dc.contributor.authorTzamos, Christos
dc.date.accessioned2017-07-25T17:58:54Z
dc.date.available2017-07-25T17:58:54Z
dc.date.issued2015-10
dc.identifier.isbn978-1-4673-8191-8
dc.identifier.urihttp://hdl.handle.net/1721.1/110840
dc.description.abstractAn (n, k)-Poisson Multinomial Distribution (PMD) is the distribution of the sum of n independent random vectors supported on the set Bk={e1,...,ek} of standard basis vectors in Rk. We prove a structural characterization of these distributions, showing that, for all ε > 0, any (n, k)-Poisson multinomial random vector is ε-close, in total variation distance, to the sum of a discretized multidimensional Gaussian and an independent (poly(k/ε), k)-Poisson multinomial random vector. Our structural characterization extends the multi-dimensional CLT of Valiant and Valiant, by simultaneously applying to all approximation requirements ε. In particular, it overcomes factors depending on log n and, importantly, the minimum Eigen value of the PMD's covariance matrix. We use our structural characterization to obtain an ε-cover, in total variation distance, of the set of all (n, k)-PMDs, significantly improving the cover size of Daskalakis and Papadimitriou, and obtaining the same qualitative dependence of the cover size on n and ε as the k=2 cover of Daskalakis and Papadimitriou. We further exploit this structure to show that (n, k)-PMDs can be learned to within ε in total variation distance from Õk(1/ε) samples, which is near-optimal in terms of dependence on ε and independent of n. In particular, our result generalizes the single-dimensional result of Daskalakis, Diakonikolas and Servedio for Poisson binomials to arbitrary dimension. Finally, as a corollary of our results on PMDs, we give a Õk(1/ε2) sample algorithm for learning (n, k)-sums of independent integer random variables (SIIRVs), which is near-optimal for constant k.en_US
dc.description.sponsorshipAlfred P. Sloan Foundation (Fellowship)en_US
dc.description.sponsorshipMicrosoft Research Faculty Fellowshipen_US
dc.description.sponsorshipNational Science Foundation (U.S.) (Award CCF-0953960 (CAREER))en_US
dc.description.sponsorshipNational Science Foundation (U.S.). Division of Computing and Communication Foundations (CCF-1101491)en_US
dc.description.sponsorshipSimons Award for Graduate Students in Theoretical Computer Scienceen_US
dc.language.isoen_US
dc.publisherInstitute of Electrical and Electronics Engineers (IEEE)en_US
dc.relation.isversionofhttp://dx.doi.org/10.1109/FOCS.2015.77en_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.titleOn the Structure, Covering, and Learning of Poisson Multinomial Distributionsen_US
dc.typeArticleen_US
dc.identifier.citationDaskalakis, Constantinos, Gautam Kamath, and Christos Tzamos. “On the Structure, Covering, and Learning of Poisson Multinomial Distributions.” 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (October 2015).en_US
dc.contributor.departmentMassachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratoryen_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Scienceen_US
dc.contributor.mitauthorDaskalakis, Konstantinos
dc.contributor.mitauthorKamath, Gautam Chetan
dc.contributor.mitauthorTzamos, Christos
dc.relation.journal2015 IEEE 56th Annual Symposium on Foundations of Computer Scienceen_US
dc.eprint.versionAuthor's final manuscripten_US
dc.type.urihttp://purl.org/eprint/type/ConferencePaperen_US
eprint.statushttp://purl.org/eprint/status/NonPeerRevieweden_US
dspace.orderedauthorsDaskalakis, Constantinos; Kamath, Gautam; Tzamos, Christosen_US
dspace.embargo.termsNen_US
dc.identifier.orcidhttps://orcid.org/0000-0002-5451-0490
dc.identifier.orcidhttps://orcid.org/0000-0003-0048-2559
dc.identifier.orcidhttps://orcid.org/0000-0002-7560-5069
mit.licenseOPEN_ACCESS_POLICYen_US


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