The Lasso has been widely studied and used in many applications over the last decade. It has also been extended in various directions in particular to ensure asymptotic oracle properties through adaptive weights (Zou, 2006). Another direction has been to incorporate additional knowledge within the penalty to account for some structure among features. Among such strategies the Fused-Lasso (Tibshirani et al., 2005) has recently been extended to penalize differences of coefficients corresponding to features organized along a network, through the Generalized Fused-Lasso. In this work we investigate the theoretical and empirical properties of the Adaptive Generalized Fused-Lasso in the context of Generalized Linear Models, with emphasis on Logistic Regression. More precisely, we establish its asymptotic oracle properties and propose an extensive simulation study to explore its empirical properties. We especially show that it compares favorably with other strategies. We also propose an adaptation of the Relaxed Lasso (Meinshausen, 2007). Finally we present an original application of the Generalized Fused-Lasso to the Joint Modeling framework where the design itself suggests the graph to be used in the penalty; an illustration is provided on road safety data.
