Concave 1-norm group selection.

Grouping structures arise naturally in many high-dimensional problems. Incorporation of such information can improve model fitting and variable selection. Existing group selection methods, such as the group Lasso, require correct membership. However, in practice it can be difficult to correctly specify group membership of all variables. Thus, it is important to develop group selection methods that are robust against group mis-specification. Also, it is desirable to select groups as well as individual variables in many applications. We propose a class of concave [Formula: see text]-norm group penalties that is robust to grouping structure and can perform bi-level selection. A coordinate descent algorithm is developed to calculate solutions of the proposed group selection method. Theoretical convergence of the algorithm is proved under certain regularity conditions. Comparison with other methods suggests the proposed method is the most robust approach under membership mis-specification. Simulation studies and real data application indicate that the [Formula: see text]-norm concave group selection approach achieves better control of false discovery rates. An R package grppenalty implementing the proposed method is available at CRAN.