UMD Theses and Dissertations

Permanent URI for this collectionhttp://hdl.handle.net/1903/3

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    U(R) PHASE RETRIEVAL, LOCAL NORMALIZING FLOWS, AND HIGHER ORDER FOURIER TRANSFORMS
    (2022) Dock, Christopher Barton; Balan, Radu; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    The classical phase retrieval problem arises in contexts ranging from speech recognition to x-ray crystallography and quantum state tomography. The generalization to matrix frames is natural in the sense that it corresponds to quantum tomography of impure states. Chapter 1 provides computable global stability bounds for the quasi-linear analysis map $\beta$ and a path forward for understanding related problems in terms of the differential geometry of key spaces. In particular, Chapter 1 manifests a Whitney stratification of the positive semidefinite matrices of low rank which allows us to ``stratify'' the computation of the global stability bound. We show that for the impure state case no such global stability bounds can be obtained for the non-linear analysis map $\alpha$ with respect to certain natural distance metrics. Finally, our computation of the global lower Lipschitz constant for the $\beta$ analysis map provides novel conditions for a frame to be generalized phase retrievable. In Chapter 2 we develop the concept of local normalizing flows. Normalizing flows provide an elegant approach to generative modeling that allows for efficient sampling and exact density evaluation of unknown data distributions. However, current techniques have significant limitations in their expressivity when the data distribution is supported on a low-dimensional manifold or has a non-trivial topology. We introduce a novel statistical framework for learning a mixture of local normalizing flows as ``chart maps'' over the data manifold. Our framework augments the expressivity of recent approaches while preserving the signature property of normalizing flows, that they admit exact density evaluation. We learn a suitable atlas of charts for the data manifold via a vector quantized auto-encoder (VQ-AE) and the distributions over them using a conditional flow. We validate experimentally that our probabilistic framework enables existing approaches to better model data distributions over complex manifolds. In Chapter 3 we examine higher order Fourier transforms in both discrete and continuous contexts. We demonstrate a connection to a matrix time variant of the free Schr\"{o}dinger equation, as well as a potential application to magnetic resonance imaging. In the discrete case we show that the reconstruction properties of higher order Fourier frames are intricately related to quadratic Gauss sums.