The incoherence condition in additive models

We extend the idea of regularization using the Lasso, to the case of an additive model with p components, p being larger than the sample size n. Our method has a group Lasso type structure, and penalizes non-smoothness of the components in the additive model. To arrive at a sparsity oracle in- equality, we need an incoherence condition which generalizes the incoherence conditions used for the Lasso. Bickel et al. (2008) impose the “restricted eigenvalue assumption”, which is closely related to the “compatibility con- dition” in van de Geer (2007), which we simply call “Condition C”. We will formulate a version of such a “Condition C” for additive models. To verify it, we discuss the case of random design. We prove new results for weighted empirical processes, which make the transition from random to fixed design possible, and which only requires a population version of “Condition C”. A consequence is that the sparsity oracle property of our procedure holds when the variables are independent,andthatalsovariousdependencystructuresare allowed.