High-Dimensional Statistics: A Non-Asymptotic Viewpoint (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 48)

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. Read more... 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 Cover......Page 1 Frontmatter......Page 2 List of chapters......Page 8 Contents......Page 10 Illustrations......Page 16 Acknowledgements......Page 18 1- introduction......Page 20 2 - Basic tail and concentration bounds......Page 40 3 - Concentration of measure......Page 77 4 - Uniform laws of large numbers......Page 117 5 - Metric entropy and its uses......Page 140 6 - Random matrices and covariance estimation......Page 178 7 - Sparse linear models in high dimensions......Page 213 8 - Principal component analysis in high dimensions......Page 255 9 - Decomposability and restricted strong convexity......Page 278 10 - Matrix estimation with rank constraints......Page 331 11 - Graphical models for high-dimensional data......Page 366 12 - Reproducing kernel Hilbert spaces......Page 402 13 - Nonparametric least squares......Page 435 14 - Localization and uniform laws......Page 472 15 - Minimax lower bounds......Page 504 Subject index......Page 559 reference......Page 543 Author index......Page 567 Recent years have seen an explosion in the volume and variety of data collected in scientific disciplines from astronomy to genetics and industrial settings ranging from Amazon to Uber. This graduate text equips readers in statistics, machine learning, and related fields to understand, apply, and adapt modern methods suited to large-scale data.