Electronic Journal of Statistics

Sparsity oracle inequalities for the Lasso

Florentina Bunea, Alexandre Tsybakov, and Marten Wegkamp

Full-text: Open access

Abstract

This paper studies oracle properties of 1-penalized least squares in nonparametric regression setting with random design. We show that the penalized least squares estimator satisfies sparsity oracle inequalities, i.e., bounds in terms of the number of non-zero components of the oracle vector. The results are valid even when the dimension of the model is (much) larger than the sample size and the regression matrix is not positive definite. They can be applied to high-dimensional linear regression, to nonparametric adaptive regression estimation and to the problem of aggregation of arbitrary estimators.

Article information

Source
Electron. J. Statist., Volume 1 (2007), 169-194.

Dates
First available in Project Euclid: 21 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1179759718

Digital Object Identifier
doi:10.1214/07-EJS008

Mathematical Reviews number (MathSciNet)
MR2312149

Zentralblatt MATH identifier
1146.62028

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62C20: Minimax procedures 62G05: Estimation 62G20: Asymptotic properties

Keywords
sparsity oracle inequalities Lasso penalized least squares nonparametric regression dimension reduction aggregation mutual coherence adaptive estimation

Citation

Bunea, Florentina; Tsybakov, Alexandre; Wegkamp, Marten. Sparsity oracle inequalities for the Lasso. Electron. J. Statist. 1 (2007), 169--194. doi:10.1214/07-EJS008. https://projecteuclid.org/euclid.ejs/1179759718


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