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dc.contributor.advisorDavenport, Mark
dc.contributor.authorMcrae, Andrew Duncan
dc.date.accessioned2022-05-18T19:33:59Z
dc.date.available2022-05-18T19:33:59Z
dc.date.created2022-05
dc.date.issued2022-04-28
dc.date.submittedMay 2022
dc.identifier.urihttp://hdl.handle.net/1853/66587
dc.description.abstractThis thesis shows how we can exploit low-dimensional structure in high-dimensional statistics and machine learning problems via optimization. We show several settings where, with an appropriate choice of optimization algorithm, we can perform useful estimation with a complexity that scales not with the original problem dimension but with a much smaller intrinsic dimension. In the low-rank matrix completion and denoising problems, we can exploit low-rank structure to recover a large matrix from noisy observations of some or all of its entries. We prove state-of-the-art results for this problem in the case of Poisson noise and show that these results are minimax-optimal. Next, we study the problem of recovering a sparse vector from nonlinear measurements. We present a lifted matrix framework for the sparse phase retrieval and sparse PCA problems that includes a novel atomic norm regularizer. We prove that solving certain convex optimization problems in this framework yields estimators with near-optimal performance. Although we do not know how to compute these estimators efficiently and exactly, we derive a principled heuristic algorithm for sparse phase retrieval that matches existing state-of-the-art algorithms. Third, we show how we can exploit low-dimensional manifold structure in supervised learning. In a reproducing kernel Hilbert space framework, we show that smooth functions on a manifold can be estimated with a complexity scaling with the manifold dimension rather than a larger embedding space dimension. Finally, we study the interaction between high ambient dimension and a lower intrinsic dimension in the harmless interpolation phenomenon (where learned functions generalize well despite interpolating noisy data). We present a general framework for this phenomenon in linear and reproducing kernel Hilbert space settings, proving that it occurs in many situations that previous work has not covered.
dc.format.mimetypeapplication/pdf
dc.language.isoen_US
dc.publisherGeorgia Institute of Technology
dc.subjectHigh-dimensional statistics
dc.subjectStructured estimation
dc.subjectOptimization
dc.subjectMachine learning
dc.titleStructured Statistical Estimation via Optimization
dc.typeDissertation
dc.description.degreePh.D.
dc.contributor.departmentElectrical and Computer Engineering
thesis.degree.levelDoctoral
dc.contributor.committeeMemberRomberg, Justin
dc.contributor.committeeMemberKoltchinskii, Vladimir
dc.contributor.committeeMemberMuthukumar, Vidya
dc.contributor.committeeMemberNemirovski, Arkadi
dc.date.updated2022-05-18T19:34:00Z


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