  We consider a linear regression problem in a high dimensional setting where
the number of covariates $p$ can be much larger than the sample size $n$. In
such a situation, one often assumes sparsity of the regression vector, \textit
i.e., the regression vector contains many zero components. We propose a
Lasso-type estimator $\hat{\beta}^{Quad}$ (where '$Quad$' stands for quadratic)
which is based on two penalty terms. The first one is the $\ell_1$ norm of the
regression coefficients used to exploit the sparsity of the regression as done
by the Lasso estimator, whereas the second is a quadratic penalty term
introduced to capture some additional information on the setting of the
problem. We detail two special cases: the Elastic-Net $\hat{\beta}^{EN}$, which
deals with sparse problems where correlations between variables may exist; and
the Smooth-Lasso $\hat{\beta}^{SL}$, which responds to sparse problems where
successive regression coefficients are known to vary slowly (in some
situations, this can also be interpreted in terms of correlations between
successive variables). From a theoretical point of view, we establish variable
selection consistency results and show that $\hat{\beta}^{Quad}$ achieves a
Sparsity Inequality, \textit i.e., a bound in terms of the number of non-zero
components of the 'true' regression vector. These results are provided under a
weaker assumption on the Gram matrix than the one used by the Lasso. In some
situations this guarantees a significant improvement over the Lasso.
Furthermore, a simulation study is conducted and shows that the S-Lasso
$\hat{\beta}^{SL}$ performs better than known methods as the Lasso, the
Elastic-Net $\hat{\beta}^{EN}$, and the Fused-Lasso with respect to the
estimation accuracy. This is especially the case when the regression vector is
'smooth', \textit i.e., when the variations between successive coefficients of
the unknown parameter of the regression are small. The study also reveals that
the theoretical calibration of the tuning parameters and the one based on 10
fold cross validation imply two S-Lasso solutions with close performance.
