1. Introduction; 2. Basic tail and concentration bounds; 3. Concentration of measure; 4. Uniform laws of large numbers; 5. Metric entropy and its uses; 6. Random matrices and covariance estimation; 7. Sparse linear models in high dimensions; 8. Principal component analysis in high dimensions; 9. Decomposability and restricted strong convexity; 10. Matrix estimation with rank constraints; 11. Graphical models for high-dimensional data; 12. Reproducing kernel Hilbert spaces; 13. Nonparametric least squares; 14. Localization and uniform laws; 15. Minimax lower bounds; References; Author index; Subject index.
A coherent introductory text from a groundbreaking researcher, focusing on clarity and motivation to build intuition and understanding.
Martin J. Wainwright is a Chancellor's Professor at the University of California, Berkeley, with a joint appointment between the Department of Statistics and the Department of Electrical Engineering and Computer Sciences. His research lies at the nexus of statistics, machine learning, optimization, and information theory, and he has published widely in all of these disciplines. He has written two other books, one on graphical models together with Michael I. Jordan, and one on sparse learning together with Trevor Hastie and Robert Tibshirani. Among other awards, he has received the COPSS Presdients' Award, has been a Medallion Lecturer and Blackwell Lecturer for the Institute of Mathematical Statistics, and has received Best Paper Awards from the Neural Information Processing Systems (NIPS), the International Conference on Machine Learning (ICML), and the Uncertainty in Artificial Intelligence (UAI) conferences, as well as from the Institute of Electrical and Electronics Engineers (IEEE) Information Theory Society.
'Non-asymptotic, high-dimensional theory is critical for modern
statistics and machine learning. This book is unique in providing a
crystal clear, complete and unified treatment of the area. With
topics ranging from concentration of measure to graphical models,
the author weaves together probability theory and its applications
to statistics. Ideal for graduate students and researchers. This
will surely be the standard reference on the topic for many years.'
Larry Wasserman, Carnegie Mellon University, Pennsylvania
'Martin J. Wainwright brings his large box of analytical power
tools to bear on the problems of the day - the analysis of models
for wide data. A broad knowledge of this new area combines with his
powerful analytical skills to deliver this impressive and
intimidating work - bound to be an essential reference for all the
brave souls that try their hand.' Trevor Hastie, Stanford
University, California
'This book provides an excellent treatment of perhaps the fastest
growing area within high-dimensional theoretical statistics -
non-asymptotic theory that seeks to provide probabilistic bounds on
estimators as a function of sample size and dimension. It offers
the most thorough, clear, and engaging coverage of this area to
date, and is thus poised to become the definitive reference and
textbook on this topic.' Genevera Allen, William Marsh Rice
University, Texas
'Statistical theory and practice have undergone a renaissance in
the past two decades, with intensive study of high-dimensional data
analysis. No researcher has deepened our understanding of
high-dimensional statistics more than Martin Wainwright. This book
brings the signature clarity and incisiveness of his published
research into book form. It will be a fantastic resource for
both beginning students and seasoned researchers, as the field
continues to make exciting breakthroughs.' John Lafferty, Yale
University, Connecticut
'This is an outstanding book on high-dimensional statistics,
written by a creative and celebrated researcher in the field. It
gives comprehensive treatments on many important topics in
statistical machine learning and, furthermore, is self-contained,
from introductory materials to most updated results on various
research frontiers. This book is a must-read for those who wish to
learn and to develop modern statistical machine theory, methods and
algorithms.' Jianqing Fan, Princeton University, New Jersey
'This book provides an in-depth mathematical treatment and
methodological intuition of high-dimensional statistics. The main
technical tools from probability theory are carefully developed and
the construction and analysis of statistical methods and algorithms
for high-dimensional problems is presented in an outstandingly
clear way. Martin J. Wainwright has written a truly exceptional,
inspiring and beautiful masterpiece!' Peter Bühlmann,
Eidgenössische Technische Hochschule Zürich
'This new book by Martin J. Wainwright covers modern topics in
high-dimensional statistical inference, and focuses primarily on
explicit non-asymptotic results related to sparsity and
non-parametric estimation. This is a must-read for all graduate
students in mathematical statistics and theoretical machine
learning, both for the breadth of recent advances it covers and the
depth of results which are presented. The exposition is
outstandingly clear, starting from the first introductory chapters
on the necessary probabilistic tools. Then, the book covers
state-of-the-art advances in high-dimensional statistics, with
always a clever choice of results which have the perfect mix of
significance and mathematical depth.' Francis Bach, INRIA Paris
'Wainwright's book on those parts of probability theory and
mathematical statistics critical to understanding of the new
phenomena encountered in high dimensions is marked by the clarity
of its presentation and the depth to which it travels. In every
chapter he starts with intuitive examples and simulations which are
systematically developed either into powerful mathematical tools or
complete answers to fundamental questions of inference. It is not
easy, but elegant and rewarding whether read systematically or
dipped into as a reference.' Peter Bickel, University of
California, Berkeley
'… this is a very valuable book, covering a variety of important
topics, self-contained and nicely written.' Fabio Mainardi, MAA
Reviews
'This is an excellent book. It provides a lucid, accessible and
in-depth treatment of nonasymptotic high-dimensional statistical
theory, which is critical as the underpinning of modern statistics
and machine learning. It succeeds brilliantly in providing a
self-contained overview of high-dimensional statistics, suitable
for use in formal courses or for self-study by graduate-level
students or researchers. The treatment is outstandingly clear and
engaging, and the production is first-rate. It will quickly become
essential reading and the key reference text in the field.' G.
Alastair Young, International Statistical Review
'Martin Wainwright takes great care to polish every sentence of
each part of the book. He introduces state-of-the-art theory in
every chapter, as should probably be expected from an acknowledged
specialist of the field. But it is certainly an enormous amount of
work to organize all these results in a complete, coherent,
rigorous yet easy-to-follow theory. I am simply amazed by the
quality of the writing. The explanations on the motivations
(Chapter 1) and on the core of the theory are extremely
pedagogical. The proofs of the main results are rigorous and
complete, but most of them are also built in a way that makes them
seem easier to the reader than they actually are. This is the kind
of magic only a few authors are capable of.' Pierre Alquier,
MatSciNet
'... provides a masterful exposition of various mathematical tools
that are becoming increasingly common in the analysis of
contemporary statistical problems. In addition to providing a
rigorous and comprehensive overview of these tools, the author
delves into the details of many illustrative examples to provide a
convincing case for the general usefulness of the methods that are
introduced.' Po-Ling Lo, Bulletin of the American Mathematical
Society
'An excellent statistical masterpiece is in the hands of the
reader, which is a must read book for all graduate students in both
mathematical statistics and mathematical machine learning.' Rózsa
Horváth-Bokor, ZB Math Reviews
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