Estimation and Testing Under Sparsity: École d'Été de Probabilités de Saint-Flour XLV – 2015
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.
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12 Some WorkedOut Examples
13 Brouwers 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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&local ˇ ˇ ˇ active set allowed set apply approximation error arg min assume candidate oracle Chap chapter coefficients compatibility constant Consider convex cone convex conjugate convex function Corollary Define Definition dual norm inequality effective sparsity eigenvalue empirical process example exponential family expſ—t follows Gaussian Geer graphical Lasso Hence high-dimensional International Publishing Switzerland KKT-conditions least squares least squares loss Lecture Notes Let X1 linear model loss function lower bound Mathematics 2159 minimizer Moreover non-zero notation Notes in Mathematics nuclear norm penalty point inequality point margin condition precision matrix probability at least Problem Proof of Lemma Proof of Theorem Publishing Switzerland 2016 random variables regression result Sect sharp oracle inequality sigma-finite measure sparse sparsity estimator Springer International Publishing square-root Lasso structured sparsity sub-Gaussian Suppose symmetric Testing Under Sparsity Theorem 7.2 triangle inequality triangle property tuning parameter vector