Abstract

In this dissertation, we consider an application of overcomplete dictionaries to the solution of general ill-posed linear inverse problems. In the context of regression problems, there has been an enormous amount of effort to recover an unknown function using such dictionaries. While some research on the subject has been already carried out, there are still many gaps to address. In particular, one of the most popular methods, lasso, and its variants, is based on minimizing the empirical likelihood and unfortunately, requires stringent assumptions on the dictionary, the so-called, compatibility conditions. Though compatibility conditions are hard to satisfy, it is well known that this can be accomplished by using random dictionaries. In the first part of the dissertation, we show how one can apply random dictionaries to the solution of ill-posed linear inverse problems with Gaussian noise. We put a theoretical foundation under the suggested methodology and study its performance via simulations and real-data example. In the second part of the dissertation, we investigate the application of lasso to the linear ill-posed problems with non-Gaussian noise. We have developed a theoretical background for the application of lasso to such problems and studied its performance via simulations.

Notes

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Graduation Date

2019

Semester

Fall

Advisor

Pensky, Marianna

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Mathematics

Degree Program

Mathematics

Format

application/pdf

Identifier

CFE0007811

URL

http://purl.fcla.edu/fcla/etd/CFE0007811

Language

English

Release Date

December 2019

Length of Campus-only Access

None

Access Status

Doctoral Dissertation (Open Access)

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Included in

Mathematics Commons

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