On the ℓ₁-ℓ_q Regularized Regression

In this paper we consider the problem of grouped variable selection in high-dimensional regression using ℓ₁-ℓ_q regularization (1 ≤ q ≤ ∞), which can be viewed as a natural generalization of the ℓ₁-ℓ₂ 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 >> 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 ℓ₁-regularization (Lasso) naturally carry on to the ℓ₁-ℓ_q cases (1 ≤ q ≤ ∞), 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 = ∞ (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).