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dc.contributor.authorHan, Andi
dc.date.accessioned2023-08-22T06:21:39Z
dc.date.available2023-08-22T06:21:39Z
dc.date.issued2023en_AU
dc.identifier.urihttps://hdl.handle.net/2123/31584
dc.description.abstractLearning over smooth nonlinear spaces has found wide applications. A principled approach for addressing such problems is to endow the search space with a Riemannian manifold geometry and numerical optimization can be performed intrinsically. Recent years have seen a surge of interest in leveraging Riemannian optimization for nonlinearly-constrained problems. This thesis investigates and improves on the existing algorithms for Riemannian optimization, with a focus on unified analysis frameworks and generic strategies. To this end, the first chapter systematically studies the choice of Riemannian geometries and their impacts on algorithmic convergence, on the manifold of positive definite matrices. The second chapter considers stochastic optimization on manifolds and proposes a unified framework for analyzing and improving the convergence of Riemannian variance reduction methods for nonconvex functions. The third chapter introduces a generic acceleration scheme based on the idea of extrapolation, which achieves optimal convergence rate asymptotically while being empirically efficient.en_AU
dc.language.isoenen_AU
dc.subjectRiemannian geometryen_AU
dc.subjectRiemannian optimizationen_AU
dc.subjectPositive definite matricesen_AU
dc.subjectVariance reductionen_AU
dc.subjectAccelerated gradient methodsen_AU
dc.titleOptimization and Learning over Riemannian Manifoldsen_AU
dc.typeThesis
dc.type.thesisDoctor of Philosophyen_AU
dc.rights.otherThe author retains copyright of this thesis. It may only be used for the purposes of research and study. It must not be used for any other purposes and may not be transmitted or shared with others without prior permission.en_AU
usyd.facultySeS faculties schools::The University of Sydney Business School::Discipline of Business Analyticsen_AU
usyd.degreeDoctor of Philosophy Ph.D.en_AU
usyd.awardinginstThe University of Sydneyen_AU
usyd.advisorGao, Junbin


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