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High-dimensional regression problems with special structure

High-dimensional variable selection has received tremendous attention in the last decade. Sparse estimators like the Lasso (Tibshirani, 1996) have proven to be powerful for high-dimensional settings, both from a theoretical and a practical point of view. In this thesis we propose several extensions of the Lasso (and of variants thereof) when either additional structure is available in the data or when more flexibility with respect to sparsity or functional modeling is of interest. First, we consider the Group Lasso (Yuan and Lin, 2006). The Group Lasso is an extension of the Lasso to do variable selection on (predefined) groups of variables in linear regression models. The estimates have the attractive property of being invariant under groupwise orthogonal reparametrizations. We extend the Group Lasso to logistic regression models and present an efficient algorithm, especially suitable for high-dimensional problems, which can also be applied to generalized linear models to solve the corresponding convex optimization problem. The Group Lasso estimator for logistic regression is shown to be statistically consistent even if the number of predictors is much larger than sample size but with sparse true underlying structure. We further use a two-stage procedure which aims for sparser models than the Group Lasso, leading to improved prediction performance for some cases. Moreover, due to the two-stage nature, the estimates can be constructed to be hierarchical. The methods are used on simulated and real datasets about splice site detection in DNA sequences. When a series of (related) linear models has to be estimated it is often appropriate to combine the different data-sets to construct more efficient estimators. We use l1-penalized estimators like the Lasso or