Supervisor's Foreword
Acknowledgements
Parts of This Thesis Have Been Published in the Following Articles
Contents
List of Algorithms
List of Figures
List of Tables
Notations
1 Introduction
1.1 Contributions
1.2 Thesis Outline
References
2 Preliminaries
2.1 Sparse Linear Regression and Compressed Sensing
2.2 Nonlinear Inference Problems
2.2.1 Generalized Linear Models
2.2.2 1-Bit Compressed Sensing
2.2.3 Phase Retrieval
References
3 Sparsity-Constrained Optimization
3.1 Background
3.2 Convex Methods and Their Required Conditions
3.3 Problem Formulation and the GraSP Algorithm
3.3.1 Algorithm Description
3.3.1.1 Variants
3.3.2 Sparse Reconstruction Conditions
3.3.3 Main Theorems
3.4 Example: Sparse Minimization of L2-Regularized Logistic Regression
3.4.1 Verifying SRH for 2-Regularized Logistic Loss
3.4.2 Bounding the Approximation Error
3.5 Simulations
3.5.1 Synthetic Data
3.5.2 Real Data
3.6 Summary and Discussion
References
4 1-Bit Compressed Sensing
4.1 Background
4.2 Problem Formulation
4.3 Algorithm
4.4 Accuracy Guarantees
4.5 Simulations
4.6 Summary
References
5 Estimation Under Model-Based Sparsity
5.1 Background
5.2 Problem Statement and Algorithm
5.3 Theoretical Analysis
5.3.1 Stable Model-Restricted Hessian
5.3.2 Accuracy Guarantee
5.4 Example: Generalized Linear Models
5.4.1 Verifying SMRH for GLMs
5.4.2 Approximation Error for GLMs
5.5 Summary
References
6 Projected Gradient Descent for Lp-Constrained Least Squares
6.1 Background
6.2 Projected Gradient Descent for Lp-Constrained Least Squares
6.3 Discussion
References
7 Conclusion and Future Work
Appendix
A Proofs of Chap.3
A.1 Iteration Analysis For Smooth Cost Functions
A.2 Iteration Analysis For Non-smooth Cost Functions
Appendix
B Proofs of Chap.4
B.1 On Non-convex Formulation of BM:PlanRobust2013
Reference
Appendix
C Proofs of Chap.5
Appendix
D Proofs of Chap.6
D.1 Proof of Theorem 6.1
D.2 Lemmas for Characterization of a Projection onto Lp-Balls
References