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Penalized selection criteria like AIC or BIC are among the most popularmethods for variable selection. Their theoretical properties have been studiedintensively and are well understood, but making use of them in case ofhigh-dimensional data is difficult due to the non-convex optimization probleminduced by L0 penalties. An elegant solution to this problem is provided by themulti-step adaptive lasso, where iteratively weighted lasso problems aresolved, whose weights are updated in such a way that the procedure convergestowards selection with L0 penalties. In this paper we introduce an adaptiveridge procedure (AR) which mimics the adaptive lasso, but is based on weightedRidge problems. After introducing AR its theoretical properties are studied inthe particular case of orthogonal linear regression. For the non-orthogonalcase extensive simulations are performed to assess the performance of AR. Incase of Poisson regression and logistic regression it is illustrated how theiterative procedure of AR can be combined with iterative maximizationprocedures. The paper ends with an efficient implementation of AR in thecontext of least-squares segmentation.

An adaptive Ridge procedure for L0 regularization