Estimation and Testing Under Sparsity

Taking the Lasso method as its starting point, this book describes the main ingredients needed to study general loss functions and sparsity-inducing regularizers. It also provides a semi-parametric approach to establishing confidence intervals and tests. Sparsity-inducing methods have proven to be very useful in the analysis of high-dimensional data. Examples include the Lasso and group Lasso methods, and the least squares method with other norm-penalties, such as the nuclear norm. The illustrations provided include generalized linear models, density estimation, matrix completion and sparse principal components. Each chapter ends with a problem section. The book can be used as a textbook for a graduate or PhD course.

1 Introduction. - The Lasso. - 3 The square-root Lasso. - 4 The bias of the Lasso and worst possible sub-directions. - 5 Confidence intervals using the Lasso. - 6 Structured sparsity. - 7 General loss with norm-penalty. - 8 Empirical process theory for dual norms. - 9 Probability inequalities for matrices. - 10 Inequalities for the centred empirical risk and its derivative. - 11 The margin condition. - 12 Some worked-out examples. - 13 Brouwer s fixed point theorem and sparsity. - 14 Asymptotically linear estimators of the precision matrix. - 15 Lower bounds for sparse quadratic forms. - 16 Symmetrization, contraction and concentration. - 17 Chaining including concentration. - 18 Metric structure of convex hulls.