  High-dimensional data pose challenges in statistical learning and modeling.
Sometimes the predictors can be naturally grouped where pursuing the
between-group sparsity is desired. Collinearity may occur in real-world
high-dimensional applications where the popular $l_1$ technique suffers from
both selection inconsistency and prediction inaccuracy. Moreover, the problems
of interest often go beyond Gaussian models. To meet these challenges,
nonconvex penalized generalized linear models with grouped predictors are
investigated and a simple-to-implement algorithm is proposed for computation. A
rigorous theoretical result guarantees its convergence and provides tight
preliminary scaling. This framework allows for grouped predictors and nonconvex
penalties, including the discrete $l_0$ and the `$l_0+l_2$' type penalties.
Penalty design and parameter tuning for nonconvex penalties are examined.
Applications of super-resolution spectrum estimation in signal processing and
cancer classification with joint gene selection in bioinformatics show the
performance improvement by nonconvex penalized estimation.
