Sparsity oracle inequalities for the Lasso. (English) Zbl 1146.62028

Summary: This paper studies oracle properties of \(\ell_1\)-penalized least squares in a 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.


62G08 Nonparametric regression and quantile regression
60E15 Inequalities; stochastic orderings
Full Text: DOI arXiv