High-Dimensional Structured Regression Using Convex Optimization

Abstract

While the term "Big Data" can have multiple meanings, we consider the type of data in which the number of features can be much greater than the number of observations (also known as high-dimensional data). High-dimensional data is abundant in contemporary scientific research due to the rapid advances in new data-measurement technologies and computing power. Recent advances in statistics have witnessed great development in the field of high-dimensional data analysis. This dissertation proposes three methods that study three different components of a general framework of the high-dimensional structured regression problem. A general theme of the proposed methods is that they cast a certain structured regression as a convex optimization problem. In so doing, the theoretical properties of each method can be well studied, and efficient computation are facilitated. Each method is accompanied by thorough theoretical analysis of its performance, and also by an R package containing its practical implementation. We show that the proposed methods perform favorably (both theoretically and practically) compared with pre-existing methods.