Big Data Analysis

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.