Estimation consistency of the group lasso and its applications

We extend the ℓ ₂-consistency result of (Meinshausen and Yu 2008) from the Lasso to the group Lasso. Our main theorem shows that the group Lasso achieves estimation consistency under a mild condition and an asymptotic upper bound on the number of selected variables can be obtained. As a result, we can apply the nonnegative garrote procedure to the group Lasso result to obtain an estimator which is simultaneously estimation and variable selection consistent. In particular, our setting allows both the number of groups and the number of variables per group increase and thus is applicable to high-dimensional problems. We also provide estimation consistency analysis for a version of the sparse additive models with increasing dimensions. Some finite-sample results are also reported.