  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.
