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dc.contributor.authorRoot, Jonathan
dc.date.accessioned2016-12-20T19:58:47Z
dc.date.available2016-12-20T19:58:47Z
dc.date.issued2016
dc.identifier.urihttps://hdl.handle.net/2144/19741
dc.description.abstractIn this thesis we consider concentration inequalities and the concentration of measure phenomenon from a variety of angles. Sharp tail bounds on the deviation of Lipschitz functions of independent random variables about their mean are well known. We consider variations on this theme for dependent variables on the Boolean cube. In recent years negatively associated probability distributions have been studied as potential generalizations of independent random variables. Results on this class of distributions have been sparse at best, even when restricting to the Boolean cube. We consider the class of negatively associated distributions topologically, as a subset of the general class of probability measures. Both the weak (distributional) topology and the total variation topology are considered, and the simpler notion of negative correlation is investigated. The concentration of measure phenomenon began with Milman's proof of Dvoretzky's theorem, and is therefore intimately connected to the field of high-dimensional convex geometry. Recently this field has found application in the area of compressed sensing. We consider these applications and in particular analyze the use of Gordon's min-max inequality in various compressed sensing frameworks, including the Dantzig selector and the matrix uncertainty selector. Finally we consider the use of concentration inequalities in developing a theoretically sound anomaly detection algorithm. Our method uses a ranking procedure based on KNN graphs of given data. We develop a max-margin learning-to-rank framework to train limited complexity models to imitate these KNN scores. The resulting anomaly detector is shown to be asymptotically optimal in that for any false alarm rate α, its decision region converges to the α-percentile minimum volume level set of the unknown underlying density.en_US
dc.language.isoen_USen_US
dc.subjectMathematicsen_US
dc.subjectAnomaly detectionen_US
dc.subjectCompressed sensingen_US
dc.subjectConcentration inequalitiesen_US
dc.subjectConcentration of measureen_US
dc.subjectMachine learningen_US
dc.subjectNegative associationen_US
dc.titleConcentration of measure, negative association, and machine learningen_US
dc.typeThesis/Dissertationen_US
dc.date.updated2016-12-07T02:08:05Z
etd.degree.nameDoctor of Philosophyen_US
etd.degree.leveldoctoralen_US
etd.degree.disciplineMathematics & Statisticsen_US
etd.degree.grantorBoston Universityen_US


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