We deal with the problem of variable selection when variables must be selected group-wise, with possibly overlapping groups defined a priori. In particular we propose a new optimization procedure for solving the regularized algorithm presented in Jacob et al. 09, where the group lasso penalty is generalized to overlapping groups of variables. While in Jacob et al. 09 the proposed implementation requires explicit replication of the variables belonging to more than one group, our iterative procedure is based on a combination of proximal methods in the primal space and constrained Newton method in a reduced dual space, corresponding to the active groups. This procedure provides a scalable alternative with no need for data duplication, and allows to deal with high dimensional problems without pre-processing to reduce the dimensionality of the data. The computational advantages of our scheme with respect to state-of-the-art algorithms using data duplication are shown empirically with numerical simulations.

In this paper we investigate the regularization property of Kernel Principal Component Analysis (KPCA), by studying its application as a preprocessing step to supervised learning problems. We show that performing KPCA and then ordinary least squares on the projected data, a procedure known as kernel principal component regression (KPCR), is equivalent to spectral cut-off regularization, the regularization parameter being exactly the number of principal components to keep. Using probabilistic estimates for integral operators we can prove error estimates for KPCR and propose a parameter choice procedure allowing to prove consistency of the algorithm.