Big Data Analysis
Description
This book covers the methods and theory of high dimensional probability, statistics, large-scale optimization, and inference. We aim to quickly bring readers to the frontier and interdisciplinary areas of statistics, optimization, probability, and machine learning. This book covers topics in:
High dimensional probability, Concentration inequality, Sub-Gaussian random variables, Chernoff bounds, Hoeffding's inequality, Maximal inequalities, High dimensional linear regression, Ordinary least square, Compressed sensing, Lasso, Variations of Lasso including group lasso, fused lasso, adaptive lasso, etc., General high dimensional M- estimators, Variable selection consistency, High dimensional Optimization, Convex geometry, Lagrange duality, Gradient descent, Proximal gradient descent, LARS, ADMM, Mirror descent, Stochastic optimization, Large-Scale Inference, Linear model hypothesis testing, high dimensional inference, Chi-square test, maximal test, and Higher criticism, False discovery rate control.
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Person
Junwei Lu is an Assistant Professor in Harvard T.H. Chan School of Public Health. His research focuses on the intersection of statistical machine learning and clinical studies, revealing scientific associations among clinical treatment strategies and patient phenotyping, especially focusing on precision medicine leveraging real-world clinical data such as electronic health records data for risk prediction and clinical optimization.
Content
Part I Foundations of Big Data Analysis.- Chapter 1 Introduction.- Chapter 2 Preliminaries in Probability.- Chapter 3 Preliminaries in Linear Algebra.- Part II High-Dimensional Probability.- Chapter 4 Concentration Inequalities.- Chapter 5 Sub-Exponential Random Variables.- Chapter 6 Maximal Inequality.- Part III High-Dimensional Statistics.- Chapter 7 Ordinary Least Squares.- Chapter 8 Compressive Sensing.- Chapter 9 Restricted Isometry Property.- Chapter 10 Statistical Properties of Lasso.- Chapter 11 Variations of Lasso.- Part IV High-Dimensional Optimization.- Chapter 12 Convexity and Subgradient.- Chapter 13 Gradient Descent.- Chapter 14 Proximal Gradient Descent.- Chapter 15 Mirror Descent and Nesterov's Smoothing.- Chapter 16 Duality and ADMM.- Part V High-Dimensional Inference.- Chapter 17 High Dimensional Inference.- Chapter 18 Debiased Lasso.- Chapter 19 Multiple Hypotheses.- Chapter 20 False Discovery Rate.- Chapter 21 Knock-Off.- References.
