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Solution of linear ill-posed problems using overcomplete dictionaries

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Date Issued:
2019
Abstract/Description:
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.
Title: Solution of linear ill-posed problems using overcomplete dictionaries.
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Name(s): Gupta, Pawan, Author
Pensky, Marianna, Committee Chair
Swanson, Jason, Committee Member
Zhang, Teng, Committee Member
Foroosh, Hassan, Committee Member
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2019
Publisher: University of Central Florida
Language(s): English
Abstract/Description: 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.
Identifier: CFE0007811 (IID), ucf:52345 (fedora)
Note(s): 2019-12-01
Ph.D.
Sciences,
Doctoral
This record was generated from author submitted information.
Subject(s): Inverse Problem -- Lasso -- Random Matrices -- Overcomplete Dictionary
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0007811
Restrictions on Access: public 2019-12-15
Host Institution: UCF

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