ORCID Identifier(s)

0000-0002-8510-8234

Graduation Semester and Year

2020

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Ren-Cang Li

Second Advisor

Li Wang

Abstract

Sparse reconstruction occurs frequently in science and engineering and real-world applications, including statistics, finance, imaging, biological system, compressed sensing, and, today more than ever, machine learning and data science in general. Mathematically, they are often modeled as l1-minimization problems. There are a number of existing numerical methods that can efficiently solve such l1-minimization problems, such as Alternating Direction Methods of Multipliers (ADMM), Fast Iterative Shrinkage Thresholding Algorithm (FISTA), and Homotopy algorithm. In this dissertation, we will introduce a special type of l1-minimization problem called the Sylvester Least Absolute Shrinkage and Selection Operator (SLASSO) problem. In theory, an SLASSO problem can be converted to a standard LASSO problem and then solved by any existing numerical method, but the converted LASSO problem is too large scale to be practical even if the SLASSO problem is modest. The first contribution of this dissertation is a novel method to solve an SLASSO problem without conversion, making it practical to solve a fairly large sized SLASSO problem. Our second contribution is a new structured Electroencephalogram (EEG)/Magnetoencephalogram (MEG) Source Imaging (ESI) model that groups the time-varying signals of a similar structure and uses the mixed norm estimation for accurate results. The model is then solved alternatingly. Numerical simulations compare favorably with the state-of-the-art ESI methods, demonstrating the effectiveness of the model and efficient numerical treatment.

Keywords

Brain Source Imaging, Convex optimization, l1 optimization

Disciplines

Mathematics | Physical Sciences and Mathematics

Comments

Degree granted by The University of Texas at Arlington

Included in

Mathematics Commons

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