Statistical learning with sparsity. The Lasso and generalizations. (English) Zbl 1319.68003

Hastie, Trevor; Tibshirani, Robert; Wainwright, Martin

Monographs on Statistics and Applied Probability 143. Boca Raton, FL: CRC Press (ISBN 978-1-4987-1216-3/hbk; 978-1-4987-1217-0/ebook). xv, 351 p. (2015).

This monograph deals with statistical learning with sparsity. 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 [R. Tibshirani, J. R. Stat. Soc., Ser. B 58, No. 1, 267–288 (1996; Zbl 0850.62538)], as an alternative to the standard least-squares method. The Lasso estimator is applied to linear models and generalized linear models, and some generalizations of the Lasso penalty are then considered. Within the framework of the optimization methods, the particular relevance of the Lasso is underlined. In addition, statistical inference, matrix decomposition, sparse multivariate methods, and graph selection are also reviewed. The sparsity in signal representation and approximation is approached, together with the utilization of \(l_1\)-methods for capitalizing the sparsity in solving problems regarding different aspects of signals. The book ends with some theoretical results regarding the behavior of the Lasso, being intended to those working with big data wanting to use the sparsity assumption for extracting useful information from big datasets.

Reviewer: Florin Gorunescu (Craiova)

MSC:

68-02	Research exposition (monographs, survey articles) pertaining to computer science
62-02	Research exposition (monographs, survey articles) pertaining to statistics
62H12	Estimation in multivariate analysis
62J05	Linear regression; mixed models
62J07	Ridge regression; shrinkage estimators (Lasso)
68T05	Learning and adaptive systems in artificial intelligence

Keywords:

Lasso method; statistical learning; sparsity; linear models, generalized linear models; optimization methods; matrix decomposition; sparse multivariate methods; graph selection; signal representation; signal approximation; big data

Citations:

Zbl 0850.62538