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dc.contributor.authorWang, Tengyao
dc.date.accessioned2016-10-19T14:07:35Z
dc.date.available2016-10-19T14:07:35Z
dc.date.issued2016-10-04
dc.identifier.otherPhD.40166
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/260825
dc.description.abstractSpectral methods have become increasingly popular in designing fast algorithms for modern highdimensional datasets. This thesis looks at several problems in which spectral methods play a central role. In some cases, we also show that such procedures have essentially the best performance among all randomised polynomial time algorithms by exhibiting statistical and computational trade-offs in those problems. In the first chapter, we prove a useful variant of the well-known Davis{Kahan theorem, which is a spectral perturbation result that allows us to bound of the distance between population eigenspaces and their sample versions. We then propose a semi-definite programming algorithm for the sparse principal component analysis (PCA) problem, and analyse its theoretical performance using the perturbation bounds we derived earlier. It turns out that the parameter regime in which our estimator is consistent is strictly smaller than the consistency regime of a minimax optimal (yet computationally intractable) estimator. We show through reduction from a well-known hard problem in computational complexity theory that the difference in consistency regimes is unavoidable for any randomised polynomial time estimator, hence revealing subtle statistical and computational trade-offs in this problem. Such computational trade-offs also exist in the problem of restricted isometry certification. Certifiers for restricted isometry properties can be used to construct design matrices for sparse linear regression problems. Similar to the sparse PCA problem, we show that there is also an intrinsic gap between the class of matrices certifiable using unrestricted algorithms and using polynomial time algorithms. Finally, we consider the problem of high-dimensional changepoint estimation, where we estimate the time of change in the mean of a high-dimensional time series with piecewise constant mean structure. Motivated by real world applications, we assume that changes only occur in a sparse subset of all coordinates. We apply a variant of the semi-definite programming algorithm in sparse PCA to aggregate the signals across different coordinates in a near optimal way so as to estimate the changepoint location as accurately as possible. Our statistical procedure shows superior performance compared to existing methods in this problem.en
dc.description.sponsorshipSt John's College and Cambridge Overseas Trust
dc.language.isoenen
dc.publisherDepartment of Pure Mathematics and Mathematical Statistics, University of Cambridgeen
dc.rightsAll Rights Reserveden
dc.rights.urihttps://www.rioxx.net/licenses/all-rights-reserved/en
dc.subjectResearch Subject Categories::MATHEMATICS::Applied mathematics::Mathematical statisticsen
dc.subjectspectral methodsen
dc.subjectDavis-Kahan theoremen
dc.subjectprincipal component analysisen
dc.subjectPCAen
dc.subjectrestricted isometryen
dc.subjecthigh-dimensional changepoint estimationen
dc.subjectsemi-definite programmingen
dc.titleSpectral methods and computational trade-offs in high-dimensional statistical inferenceen
dc.typeThesisen
dc.type.qualificationlevelDoctoral
dc.type.qualificationnameDoctor of Philosophy (PhD)
dc.publisher.institutionUniversity of Cambridgeen
dc.publisher.departmentDepartment of Pure Mathematics and Mathematical Statisticsen
dc.publisher.departmentFaculty of Mathematicsen
dc.publisher.departmentSt John's Collegeen
dc.identifier.doi10.17863/CAM.913
cam.supervisorSamworth, Richard
cam.supervisor.orcidSamworth, Richard [0000-0003-2426-4679]en


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