Time: Tuesday, April 3, 2018, 2:00pm @ MTH1311
Speaker: Weilin Li (UMD)
Title: Topics in harmonic analysis, sparse representations, and data analysis
Abstract: 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, while we study a complementary Fourier based
version. We prove that the Fourier scattering transform enjoys properties that makeit 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 lowresolution
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.
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