The Lasso with within group structure

Abstract

We study the group Lasso, where the number of groups is very large, and the sizes of the groups is large as well. We assume there is within group structure, in the sense that the ordering of the variables within groups in some loose sense expresses their relevance. We propose a within group weighting of the variables, and show that with this structure, the group Lasso satisfies a sparsity oracle inequality.