Model Free Techniques for Reduction of High-Dimensional Dynamics




Berry, Tyrus

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There 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.



Mathematics, Diffusion maps, Dynamical systems, Geometry of data, Kalman filtering, Local kernels, Manifold learning