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1 Introduction 1.1 The framework 1.2 The possibilities and challenges 1.3 About the book 1.3.1 Organization of the book 1.4 Some examples 1.4.1 Prediction and biomarker discovery in genomics2 Lasso for linear models 2.1 Organization of the chapter 2.2 Introduction and preliminaries 2.2.1 The Lasso estimator 2.3 Orthonormal design 2.4 Prediction 2.4.1 Practical aspects about the Lasso for prediction 2.4.2 Some results from asymptotic theory 2.5 Variable screening and -norms 2.5.1 Tuning parameter selection for variable screening 2.5.2 Motif regression for DNA binding sites 2.6 Variable selection 2.6.1 Neighborhood stability and irrepresentable condition 2.7 Key properties and corresponding assumptions: a summary 2.8 The adaptive Lasso: a two-stage procedure 2.8.1 An illustration: simulated data and motif regression 2.8.2 Orthonormal design 2.8.3 The adaptive Lasso: variable selection under weak conditions 2.8.4 Computation 2.8.5 Multi-step adaptive Lasso 2.8.6 Non-convex penalty functions 2.9 Thresholding the Lasso 2.10 The relaxed Lasso 2.11 Degrees of freedom of the Lasso 2.12 Path-following algorithms 2.12.1 Coordinatewise optimization and shooting algorithms 2.13 Elastic net: an extension Problems3 Generalized linear models and the Lasso 3.1 Organization of the chapter 3.2 Introduction and preliminaries 3.2.1 The Lasso estimator: penalizing the negative log-likelihood. 3.3 Important examples of generalized linear models 3.3.1 Binary response variable and logistic regression 3.3.2 Poisson regression 3.3.3 Multi-category response variable and multinomial distribution Problems4 The group Lasso 4.1 Organization of the chapter 4.2 Introduction and preliminaries 4.2.1 The group Lasso penalty 4.3 Factor variables as covariates 4.3.1 Prediction of splice sites in DNA sequences 4.4 Properties of the group Lasso for generalized linear models 4.5 The generalized group Lasso penalty 4.5.1 Groupwise prediction penalty and parametrization invariance 4.6 The adaptive group Lasso 4.7 Algorithms for the group Lasso 4.7.1 Block coordinate descent 4.7.2 Block coordinate gradient descent Problems5 Additive models and many smooth univariate functions 5.1 Organization of the chapter 5.2 Introduction and preliminaries 5.2.1 Penalized maximum likelihood for additive models 5.3 The sparsity-smoothness penalty 5.3.1 Orthogonal basis and diagonal smoothing matrices 5.3.2 Natural cubic splines and Sobolev spaces 5.3.3 Computation 5.4 A sparsity-smoothness penalty of group Lasso type 5.4.1 Computational algorithm 5.4.2 Alternative approaches 5.5 Numerical examples 5.5.1 Simulated example¡­¡­ 6 Theory for the lasso7 Variable selection with the lasso8 Theory for -penalty procedures9 Non-convex loss functions and -regularization10 Stable solutions11 P-values for linear models and beyond12 Boosting and greedy algorithms14 Probability and moment inequalitiesAuthor indexIndexReferences