Topics in Harmonic Analysis, Sparse Representations, and Data Analysis

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Classical harmonic analysis has traditionally focused on linear and invertible transformations. Motivated by modern applications, there is a growing interest in non-linear analysis and synthesis operators. This thesis encompasses applications of computational harmonic analysis, with a strong emphasis on time-frequency methods, to modern problems arising in deep learning, data analysis, imaging, and signal processing.

The first focus of this thesis deals with scattering transforms, which are particular realizations of convolutional neural networks. While the latter uses trained convolution kernels, scattering transforms use fixed ones, and this simplification allows mathematicians to develop a model of deep learning. Mallat originally introduced a wavelet scattering transform, but we study a complementary Fourier based version. We prove that the Fourier scattering transform enjoys properties that make it an effective feature extractor for classification, and we also construct a rotationally invariant modification of this transform. We provide experimental evidence that shows its effectiveness at representing complicated spectral data.

The second focus of this thesis pertains to the mathematical foundations of super-resolution, which is concerned with the recovery of fine details from low-resolution observations. This imaging model can be mathematically formulated as an ill-posed inverse problem in the space of bounded complex measures. While the current theory primarily deals with the recovery of discrete measures with minimum separation greater than the Rayleigh length, we present alternative approaches. One direction exploits Beurling's results on minimal extrapolation to obtain a general theory that is pertinent to a wide class of measures, including those with geometric structure. Another approach is information theoretic and studies the min-max error for robust super-resolution of discrete measures below the Rayleigh length.