On the $\ell_1-\ell_q$ Regularized Regression

Understand

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.