  In this paper we consider the problem of grouped variable selection in
high-dimensional regression using $\ell_1-\ell_q$ regularization ($1\leq q \leq
\infty$), which can be viewed as a natural generalization of the
$\ell_1-\ell_2$ regularization (the group Lasso). The key condition is that the
dimensionality $p_n$ can increase much faster than the sample size $n$, i.e.
$p_n \gg n$ (in our case $p_n$ is the number of groups), but the number of
relevant groups is small. The main conclusion is that many good properties from
$\ell_1-$regularization (Lasso) naturally carry on to the $\ell_1-\ell_q$ cases
($1 \leq q \leq \infty$), even if the number of variables within each group
also increases with the sample size. With fixed design, we show that the whole
family of estimators are both estimation consistent and variable selection
consistent under different conditions. We also show the persistency result with
random design under a much weaker condition. These results provide a unified
treatment for the whole family of estimators ranging from $q=1$ (Lasso) to
$q=\infty$ (iCAP), with $q=2$ (group Lasso)as a special case. When there is no
group structure available, all the analysis reduces to the current results of
the Lasso estimator ($q=1$).
