The smooth-lasso and other ℓ1 + ℓ2-penalized methods

Abstract

We consider a linear regression problem in a high dimensional setting where the number of covariates p can be much larger than the sam- ple size n. In such a situation, one often assumes sparsity of the regression vector, i.e., the regression vector contains many zero components. We pro- pose a Lasso-type estimator βQuad (where 'Quad' stands for quadratic) which is based on two penalty terms. The first one is the ℓ1 norm of the re- gression coefficients used to exploit the sparsity of the regression as done by the Lasso estimator, whereas the second is a quadratic penalty term intro- duced to capture some additional information on the setting of the problem. We detail two special cases: the Elastic-Net βEN introduced in. [42], which deals with sparse problems where correlations between variables may exist; and the Smooth-Lasso† βSL, 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 βQuad achieves a Sparsity Inequality, 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 βSL performs better than known methods as the Lasso, the Elastic-Net βEN, and the Fused-Lasso (introduced in. [30]) with respect to the estimation accuracy. This is especially the case when the regression vector is 'smooth', 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.