Sparsity oracle inequalities for the Lasso

Details Sparsity oracle inequalities for the Lasso Sparsity oracle inequalities for the Lasso

2007-05-23

arxiv.org/abs/0705.3308v1

Electronic Journal of Statistics 2007, Vol. 1, 169-194

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/07-EJS008 in the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by t

Scientific paper

10.1214/07-EJS008

This paper studies oracle properties of $\ell_1$-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.

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