  In regression problems where covariates can be naturally grouped, the group
Lasso is an attractive method for variable selection since it respects the
grouping structure in the data. We study the selection and estimation
properties of the group Lasso in high-dimensional settings when the number of
groups exceeds the sample size. We provide sufficient conditions under which
the group Lasso selects a model whose dimension is comparable with the
underlying model with high probability and is estimation consistent. However,
the group Lasso is, in general, not selection consistent and also tends to
select groups that are not important in the model. To improve the selection
results, we propose an adaptive group Lasso method which is a generalization of
the adaptive Lasso and requires an initial estimator. We show that the adaptive
group Lasso is consistent in group selection under certain conditions if the
group Lasso is used as the initial estimator.
