The degrees of freedom of the Group Lasso for a General Design

In this paper, we are concerned with regression problems where covariates can be grouped in nonoverlapping blocks, and where only a few of them are assumed to be active. In such a situation, the group Lasso is an attractive method for variable selection since it promotes sparsity of the groups. We study the sensitivity of any group Lasso solution to the observations and provide its precise local parameterization. When the noise is Gaussian, this allows us to derive an unbiased estimator of the degrees of freedom of the group Lasso. This result holds true for any fixed design, no matter whether it is underor overdetermined. With these results at hand, various model selection criteria, such as the Stein Unbiased Risk Estimator (SURE), are readily available which can provide an objectively guided choice of the optimal group Lasso fit.