Model Free Techniques for Reduction of High-Dimensional Dynamics

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dc.contributor.advisorSauer, Timothy
dc.contributor.authorBerry, Tyrus
dc.creatorBerry, Tyrus
dc.date.accessioned2013-08-09T15:39:28Z
dc.date.available2013-08-09T15:39:28Z
dc.date.issued2013
dc.description.abstractThere is a growing need in science and engineering to extract information about complex phenomena from large data sets. A rapidly developing approach to building a model from data is manifold learning, and analysis of such a model may allow isolation of the desired features of the data. By introducing an additional geometric structure, the techniques of differential geometry become available for analyzing the model. In this dissertation we extend previous methods of analyzing the geometry of data. Our key contribution is the theory of local kernels, which generalizes previous nonparametric techniques such as Laplacian eigenmaps and diffusion maps. We show that every geometry can be represented by a local kernel in the limit of large data. Moreover, using the discrete exterior calculus (DEC) we show that a local kernel can be used to introduce a discrete Hodge star operator on a data set. This shows that local kernels introduce a discrete geometry on a data set without the need for an explicit simplicial complex.
dc.format.extent150 pages
dc.identifier.urihttps://hdl.handle.net/1920/8261
dc.language.isoen
dc.rightsCopyright 2013 Tyrus Berry
dc.subjectMathematics
dc.subjectDiffusion maps
dc.subjectDynamical systems
dc.subjectGeometry of data
dc.subjectKalman filtering
dc.subjectLocal kernels
dc.subjectManifold learning
dc.titleModel Free Techniques for Reduction of High-Dimensional Dynamics
dc.typeDissertation
thesis.degree.disciplineMathematics
thesis.degree.grantorGeorge Mason University
thesis.degree.levelDoctoral

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