Book Review: High-dimensional statistics: A non-asymptotic viewpoint

Abstract

Recent years have witnessed an explosion in the volume and variety of data collected in all scientific disciplines and industrial settings. Such massive data sets present a number of challenges to researchers in statistics and machine learning. This book provides a self-contained introduction to the area of high-dimensional statistics, aimed at the first-year graduate level. It includes chapters that are focused on core methodology and theory - including tail bounds, concentration inequalities, uniform laws and empirical process, and random matrices - as well as chapters devoted to in-depth exploration of particular model classes - including sparse linear models, matrix models with rank constraints, graphical models, and various types of non-parametric models. With hundreds of worked examples and exercises, this text is intended both for courses and for self-study by graduate students and researchers in statistics, machine learning, and related fields who must understand, apply, and adapt modern statistical methods suited to large-scale data. Introduction -- Basic tail and concentration bounds -- Concentration of measure -- Uniform laws of large numbers -- Metric entropy and its uses -- Random matrices and covariance estimation -- Sparse linear models in high dimensions -- Principal component analysis in high dimensions -- Decomposability and restricted strong convexity -- Matrix estimation with rank constraints -- Graphical models for high-dimensional data -- Reproducing kernel Hilbert spaces -- Nonparametric least squares -- Localization and uniform laws -- Minimax lower bounds.