Discrete frames for high-dimensional data: constructions on regular and irregular domains
Date
2020
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Publisher
University of Delaware
Abstract
The theory of discrete frames was introduced in the 1950s by Duffin and Schaeffer, when they initiated a systematic study of dictionaries for the efficient and robust representation of data. This work was generalized by Daubechies, Grossmann, and Meyer in the late 1980s. As the representations in a Hilbert space provided by a basis often fail to provide sufficient flexibility, their work led to an explosion of interest in the development of discrete frame theory, persisting to this day. Applications range from the analysis, compression, and transmission of data to the study of function spaces. The primary results for high-dimensional spaces are existence theorems and generalizing known constructions of frames to other data sets. ☐ The main theme of this thesis is the systematic construction of discrete frame systems in two distinct settings. In the first part, we develop Gabor-type frames for functions defined on the vertex set of a graph. We propose a general framework which captures many of the existing discrete frame constructions in this setting. We also provide necessary and sufficient conditions on the choices of translations and localizing function such that they lead to Gabor-type frames. Sharp frame bounds are calculated for several existing cases, and a general formula for finding them is provided. We conclude this part with a study of frames for Cayley graphs, where tools from representation theory for finite (not necessarily abelian) groups can be applied to simplify many operations. ☐ In the second part of this thesis, we develop wavelet frames through sampling continuous reproducing formulas for functions defined on the regular domain of M_n(R). The work in this context relies heavily on the theory of square-integrable representations of locally compact groups, but all necessary background is provided. In particular, we generalize a previous construction of Ghandehari, Syzdykova, and Taylor for L^2(M_2(R)) to L^2(M_n(R)) for any dimension n. We also significantly improve the frame bounds for these constructions, improving the numerical stability of the frame design. We conclude this part with the specific details for the case n=2 to show how the general theory can be applied to construct discrete frames.
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Keywords
Discrete Frames, Discretization Problem, Graph Signal Processing, Graph Theory, Representation Theory