图书简介
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.
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