Regularized Risk Minimization

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Table of contents

1 Introduction and Related Work 
1.1 Summary of the Contributions of the Thesis
1.2 Notation
1.3 Variable Selection and Sparsity-Inducing Regularizations
1.4 Dictionary Learning with Structured Sparsity-Inducing Penalties
1.5 Some Elements of Convex Analysis and Convex Optimization for Sparse Methods
1.6 Some Ingredients to Study Statistical Properties of Sparse Methods
2 Understanding the Properties of Structured Sparsity-Inducing Norms 
2.1 Introduction
2.2 Regularized Risk Minimization
2.3 Groups and Sparsity Patterns
2.4 Optimization and Active Set Algorithm
2.5 Pattern Consistency
2.6 Experiments
2.7 Conclusion
3 Structured Sparse Principal Component Analysis 
3.1 Introduction
3.2 Problem Statement
3.3 Optimization
3.4 Experiments
3.5 Conclusions
4 Proximal Methods for Structured Sparsity-Inducing Norms 
4.1 Introduction
4.2 Problem Statement and Related Work
4.3 Optimization
4.4 Application to Dictionary Learning
4.5 Experiments
4.6 Discussion
4.7 Extension: General Overlapping Groups and `1/`1-norms
5 Application of Structured Sparsity to Neuroimaging 
5.1 Multi-scale Mining of fMRI Data with Hierarchical Structured Sparsity
5.2 Sparse Structured Dictionary Learning for Brain Resting-State Activity Modeling
6 Local Analysis of Sparse Coding in Presence of Noise 
6.1 Introduction
6.2 Problem statement
6.3 Main result and structure of the proof
6.4 Some experimental validations
6.5 Conclusion
6.6 Proofs of the main results
6.7 Control of the Taylor expansion
6.8 Computation of the Taylor expansion
6.9 Technical lemmas
7 Conclusion 
A Proofs
A.1 Proofs and Technical Elements of Chapter 1
A.2 Proofs and Technical Elements of Chapter 2
Bibliography