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We consider a linear regression problem in a high dimensional setting wherethe number of covariates $p$ can be much larger than the sample size $n$. Insuch a situation, one often assumes sparsity of the regression vector, \textiti.e., the regression vector contains many zero components. We propose aLasso-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 theregression coefficients used to exploit the sparsity of the regression as doneby the Lasso estimator, whereas the second is a quadratic penalty termintroduced to capture some additional information on the setting of theproblem. We detail two special cases: the Elastic-Net $\hat{\beta}^{EN}$, whichdeals with sparse problems where correlations between variables may exist; andthe Smooth-Lasso $\hat{\beta}^{SL}$, which responds to sparse problems wheresuccessive regression coefficients are known to vary slowly (in somesituations, this can also be interpreted in terms of correlations betweensuccessive variables). From a theoretical point of view, we establish variableselection consistency results and show that $\hat{\beta}^{Quad}$ achieves aSparsity Inequality, \textit i.e., a bound in terms of the number of non-zerocomponents of the 'true' regression vector. These results are provided under aweaker assumption on the Gram matrix than the one used by the Lasso. In somesituations 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, theElastic-Net $\hat{\beta}^{EN}$, and the Fused-Lasso with respect to theestimation accuracy. This is especially the case when the regression vector is'smooth', \textit i.e., when the variations between successive coefficients ofthe unknown parameter of the regression are small. The study also reveals thatthe theoretical calibration of the tuning parameters and the one based on 10fold cross validation imply two S-Lasso solutions with close performance.

The Smooth-Lasso and other $\ell_1+\ell_2$-penalized methods