  In this paper we study nonconvex penalization using Bernstein functions.
Since the Bernstein function is concave and nonsmooth at the origin, it can
induce a class of nonconvex functions for high-dimensional sparse estimation
problems. We derive a threshold function based on the Bernstein penalty and
give its mathematical properties in sparsity modeling. We show that a
coordinate descent algorithm is especially appropriate for penalized regression
problems with the Bernstein penalty. Additionally, we prove that the Bernstein
function can be defined as the concave conjugate of a $\varphi$-divergence and
develop a conjugate maximization algorithm for finding the sparse solution.
Finally, we particularly exemplify a family of Bernstein nonconvex penalties
based on a generalized Gamma measure and conduct empirical analysis for this
family.
