Discover New Methods for Dealing with High-Dimensional Data
A sparse statistical model has only a small number of nonzero parameters or weights; therefore, it is much easier to estimate and interpret than a dense model. Statistical Learning with Sparsity: The Lasso and Generalizations presents methods that exploit sparsity to help recover the underlying signal in a set of data.
Top experts in this rapidly evolving field, the authors describe the lasso for linear regression and a simple coordinate descent algorithm for its computation. They discuss the application of ℓ1 penalties to generalized linear models and support vector machines, cover generalized penalties such as the elastic net and group lasso, and review numerical methods for optimization. They also present statistical inference methods for fitted (lasso) models, including the bootstrap, Bayesian methods, and recently developed approaches. In addition, the book examines matrix decomposition, sparse multivariate analysis, graphical models, and compressed sensing. It concludes with a survey of theoretical results for the lasso.
In this age of big data, the number of features measured on a person or object can be large and might be larger than the number of observations. This book shows how the sparsity assumption allows us to tackle these problems and extract useful and reproducible patterns from big datasets. Data analysts, computer scientists, and theorists will appreciate this thorough and up-to-date treatment of sparse statistical modeling.
Table of Contents
The Lasso for Linear Models
The Lasso Estimator
Cross-Validation and Inference
Computation of the Lasso Solution
Degrees of Freedom
Uniqueness of the Lasso Solutions
A Glimpse at the Theory
The Nonnegative Garrote
ℓq Penalties and Bayes Estimates
Generalized Linear Models
Multiclass Logistic Regression
Log-Linear Models and the Poisson GLM
Cox Proportional Hazards Models
Support Vector Machines
Computational Details and glmnet
Generalizations of the Lasso Penalty
The Elastic Net
The Group Lasso
Sparse Additive Models and the Group Lasso
The Fused Lasso
Convex Optimality Conditions
A Simulation Study
Least Angle Regression
Alternating Direction Method of Multipliers
Biconvexity and Alternating Minimization
The Bayesian Lasso
Post-Selection Inference for the Lasso
Inference via a Debiased Lasso
Other Proposals for Post-Selection Inference
Matrix Decompositions, Approximations, and Completion
The Singular Value Decomposition
Missing Data and Matrix Completion
A General Matrix Regression Framework
Penalized Matrix Decomposition
Additive Matrix Decomposition
Sparse Multivariate Methods
Sparse Principal Components Analysis
Sparse Canonical Correlation Analysis
Sparse Linear Discriminant Analysis
Graphs and Model Selection
Basics of Graphical Models
Graph Selection via Penalized Likelihood
Graph Selection via Conditional Inference
Graphical Models with Hidden Variables
Signal Approximation and Compressed Sensing
Signals and Sparse Representations
Random Projection and Approximation
Equivalence between ℓ0 and ℓ1 Recovery
Theoretical Results for the Lasso
Bounds on Lasso ℓ2-error
Bounds on Prediction Error
Support Recovery in Linear Regression
Beyond the Basic Lasso
Bibliographic Notes and Exercises appear at the end of each chapter.
Trevor Hastie is the John A. Overdeck Professor of Statistics at Stanford University. Prior to joining Stanford University, Professor Hastie worked at AT&T Bell Laboratories, where he helped develop the statistical modeling environment popular in the R computing system. Professor Hastie is known for his research in applied statistics, particularly in the fields of data mining, bioinformatics, and machine learning. He has published five books and over 180 research articles in these areas. In 2014, he received the Emanuel and Carol Parzen Prize for Statistical Innovation. He earned a PhD from Stanford University.
Robert Tibshirani is a professor in the Departments of Statistics and Health Research and Policy at Stanford University. He has authored five books, co-authored three books, and published over 200 research articles. He has made important contributions to the analysis of complex datasets, including the lasso and significance analysis of microarrays (SAM). He also co-authored the first study that linked cell phone usage with car accidents, a widely cited article that has played a role in the introduction of legislation that restricts the use of phones while driving. Professor Tibshirani was a recipient of the prestigious COPSS Presidents’ Award in 1996 and was elected to the National Academy of Sciences in 2012.
Martin Wainwright is a professor in the Department of Statistics and the Department of Electrical Engineering and Computer Sciences at the University of California, Berkeley. Professor Wainwright is known for theoretical and methodological research at the interface between statistics and computation, with particular emphasis on high-dimensional statistics, machine learning, graphical models, and information theory. He has published over 80 papers and one book in these areas, received the COPSS Presidents’ Award in 2014, and was a section lecturer at the International Congress of Mathematicians in 2014. He received PhD in EECS from the Massachusetts Institute of Technology (MIT).
"The authors study and analyze methods using the sparsity property of some statistical models in order to recover the underlying signal in a dataset. They focus on the Lasso technique as an alternative to the standard least-squares method."
—Zentralblatt MATH 1319
"The book includes all the major branches of statistical learning. For each topic, the authors first give a concise introduction of the basic problem, evaluate conventional methods, pointing out their deficiencies, and then introduce a method based on sparsity. Thus, the book has the potential to be the standard textbook on the topic."
—Anand Panangadan, California State University, Fullerton
"It always first discusses regularized models based on equations, followed by example applications, before ending with a bibliography section detailing the historical development of the given method. Software recommendations (mostly open source R packages) are typically provided either in the main part or bibliography section of each chapter. And each chapter concludes with a set of selected exercises meant to deepen the gained knowledge on the given subject, which of course is of great help for teachers of statistics. For these reasons, we congratulate the authors of Statistical Learning with Sparsity and recommend the book to all statistically-inclined readers from intermediate to expert levels. In addition, it is worth pointing out that even for non-statisticians, the book is able to demonstrate,based on numerous real-world examples, the power of regularization."—Ivan Kondofersky and Fabian J. Theis, Institute for Computational Biology