The Generalized Group Lasso

In this paper the Generalized Lasso model of R. Tibshirani is extended to consider multidimensional features (or groups of features) à la Group Lasso, by substituting the ℓ₁ norm of the regularizer by the ℓ_2,1 norm. The resultant model is called Generalized Group Lasso (GenGL), and it contains as particular cases the already known Group Lasso and Group Fused Lasso (GFL), but also new models as the Graph-Guided Group Fused Lasso, or the trend filtering for multidimensional features. We show how to solve them efficiently combining FISTA iterations with the Proximal Operator of the corresponding regularizer, which we compute using a dual formulation. Moreover, GenGL makes possible to introduce a new approach to Group Total Variation, the regularizer of GFL, that results in a training much faster than that of previous methods.