Theses and Dissertations from UMD
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Item 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.Item Uniqueness for continuous superresolution by means of Choquet theory and geometric measure theory(2021) Cinoman, Ryan M; Benedetto, John J; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The problem of superresolution is to recover an element of a vector space from data much smaller than the dimension of the space, using a prior assumption of sparsity. A famous example is compressive sensing, where the elements are images with a large finite resolution. On the other hand, we focus on a continuous form of superresolution. Given a measure $\mu$ on a continuous domain such as the two dimensional torus, can we recover $\mu$ from knowledge of only a finite number of its Fourier coefficients using a total variation minimization method? We will see that the answer depends on certain properties of $\mu$. Namely, a necessary condition is that $\mu$ be discrete.We use methods from geometric analysis to investigate the continuous superresolution problem. Tools from measure theory relate properties of the support of a measure, such as Hausdorff dimension, to properties of its Fourier transform. We also use measure theory to investigate the possibility of alternatives to total variation that may allow us to recover surface measures defined on space curves. There is a theorem of Choquet concerning representations of points in convex sets as sums of their extreme points. As it turns out, we can apply this to the superresolution problem because the extreme points of the set of measures with total variation $1$ are precisely the set of delta measures. We consider superresolution as a convex optimization problem, where the goal is to find representations of the initial data as sums of delta measures. Choquet theory provides tools to investigate the previously unresolved problem of uniqueness. We use this to give a novel sufficient condition for a measure to be uniquely superresolved, given data on a known finite set of frequencies.Item Expedition in Data and Harmonic Analysis on Graphs(2016) Begué, Matthew Joseph; Okoudjou, Kasso A; Benedetto, John J; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The graph Laplacian operator is widely studied in spectral graph theory largely due to its importance in modern data analysis. Recently, the Fourier transform and other time-frequency operators have been defined on graphs using Laplacian eigenvalues and eigenvectors. We extend these results and prove that the translation operator to the i’th node is invertible if and only if all eigenvectors are nonzero on the i’th node. Because of this dependency on the support of eigenvectors we study the characteristic set of Laplacian eigenvectors. We prove that the Fiedler vector of a planar graph cannot vanish on large neighborhoods and then explicitly construct a family of non-planar graphs that do exhibit this property. We then prove original results in modern analysis on graphs. We extend results on spectral graph wavelets to create vertex-dyanamic spectral graph wavelets whose support depends on both scale and translation parameters. We prove that Spielman’s Twice-Ramanujan graph sparsifying algorithm cannot outperform his conjectured optimal sparsification constant. Finally, we present numerical results on graph conditioning, in which edges of a graph are rescaled to best approximate the complete graph and reduce average commute time.Item Anisotropic Harmonic Analysis and Integration of Remotely Sensed Data(2015) Murphy, James Michael; Czaja, Wojciech; Benedetto, John J.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This thesis develops the theory of discrete directional Gabor frames and several algorithms for the analysis of remotely sensed image data, based on constructions of harmonic analysis. The problems of image registration, image superresolution, and image fusion are separate but interconnected; a general approach using transform methods is the focus of this thesis. The methods of geometric multiresolution analysis are explored, particularly those related to the shearlet transform. Using shearlets, a novel method of image registration is developed that aligns images based on their shearlet features. Additionally, the anisotropic nature of the shearlet transform is deployed to smoothly superrsolve remotely-sensed image with edge features. Wavelet packets, a generalization of wavelets, are utilized for a flexible image fusion algorithm. The interplay between theoretical guarantees for these mathematical constructions, and their effectiveness for image processing is explored throughout.Item Frame Multiplication Theory for Vector-valued Harmonic Analysis(2014) Andrews, Travis David; Benedetto, John J; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)A tight frame is a sequence in a separable Hilbert space satisfying the frame inequality with equal upper and lower bounds and possessing a simple reconstruction formula. We define and study the theory of frame multiplication in finite dimensions. A frame multiplication for a frame is a binary operation on the frame elements that extends to a bilinear vector product on the entire Hilbert space. This is made possible, in part, by the reconstruction property of frames. The motivation for this work is the desire to define meaningful vector-valued versions of the discrete Fourier transform and the discrete ambiguity function. We make these definitions and prove several familiar harmonic analysis results in this context. These definitions beget the questions we answer through developing frame multiplication theory. For certain binary operations, those with the Latin square property, we give a characterization of the frames, in terms of their Gramians, that have these frame multiplications. Combining finite dimensional representation theory and Naimark's theorem, we show frames possessing a group frame multiplication are the projections of an orthonormal basis onto the isotypic components of the regular representations. In particular, for a finite group G, we prove there are only finitely many inequivalent frames possessing the group operation of G as a frame multiplication, and we give an explicit formula for the dimensions in which these frames exist. Finally, we connect our theory to a recently studied class of frames; we prove that frames possessing a group frame multiplication are the central G-frames, a class of frames generated by groups of operators on a Hilbert space.Item The Cohomological Equation for Horocycle Maps and Quantitative Equidistribution(2011) Tanis, James Holloway; Forni, Giovanni; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)There are infinitely many distributional obstructions to the existence of smooth solutions for the cohomological equation u o φ1 - u = f in each irreducible component of L2(Γ\PSL(2,R)), where φ1 is the time-one map of the horocycle flow. We study the regularity of these obstructions, determine which ones also obstruct the existence of L2 solutions and prove a Sobolev estimate of the solution in terms of f. As an application, we estimate the rate of equidistribution of horocycle maps on compact, finite volume manifolds Γ\PSL(2,R)) using an auxiliary result from Flaminio-Forni (2003) and one from Venkatesh (2010) concerning the horocycle flow and the twisted horocycle flow, respectively.