High-Dimensional Statistics

Lecture Notes

Complete Lecture Notes (PDF 1.3MB)

Introduction (PDF)

• Regression Analysis and Prediction Risk
• Models and Methods

Chapter 1: Sub-Gaussian Random Variables (PDF)

• Gaussian tails and MGF
• Sub-Gaussian Random Variables and Chernoff Bounds
• Sub-Exponential Random Variables
• Maximal Inequalities

Chapter 2: Linear Regression Model (PDF)

• Fixed Design Linear Regression
• Least-Squares Estimation
• The Gaussian Sequence Model
• High-Dimensional Linear Regression

Chapter 3: Misspecified Linear Models (PDF)

• Oracle Inequalities
• Nonparametric Regression

Chapter 4: Matrix Estimation (PDF)

• Basic Facts About Matrices
• Multivariate Regression
• Covariance Matrix Estimation
• Principal Component Analysis

Chapter 5: Minimax Lower Bounds (PDF)

• Optimality in a Minmax Sense
• Reductions to Finite Hypothesis Testing
• Lower Bounds Based on Two Hypotheses
• Lower Bounds Based on Many Hypotheses
• Application to the Gaussian Sequence Model