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Geometrical aspects of statistical learning theory [Elektronische Ressource] / vorgelegt von Matthias Hein

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Geometrical Aspects of StatisticalLearning TheoryVom Fachbereich Informatikder Technischen Universitat Darmstadt¨genehmigteDissertationzur Erlangung des akademischen GradesDoctor rerum naturalium (Dr. rer. nat.)vorgelegt vonDipl.-Phys.Matthias Heinaus Esslingen am NeckarPrufungskommission:¨Vorsitzender: Prof. Dr. B. SchieleErstreferent: Prof. Dr. T. HofmannKorreferent : Prof. Dr. B. Scholkopf¨Tag der Einreichung: 30.9.2005Tag der Disputation: 9.11.2005Darmstadt, 2005Hochschulkennziffer: D17AbstractGeometry plays an important role in modern statistical learning theory, and manydifferent aspects of geometry can be found in this fast developing field. This thesisaddresses some of these aspects. A large part of this work will be concerned withso called manifold methods, which have recently attracted a lot of interest. The keypoint is that for a lot of real-world data sets it is natural to assume that the datalies on a low-dimensional submanifold of a potentially high-dimensional Euclideanspace.Wedeveloparigorousandquitegeneralframeworkfortheestimationandap-proximation of some geometric structures and other quantities of this submanifold,using certain corresponding structures on neighborhood graphs built from randomsamples of that submanifold. Another part of this thesis deals with the generalizati-on of the maximal margin principle to arbitrary metric spaces.



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Published 01 January 2006
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