The Bernstein Function: A Unifying Framework of Nonconvex Penalization in Sparse Estimation
Year: 2013
Type: preprint
Abstract: 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 $φ$-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.
Source: arXiv (Cornell University)
Author: Zhang, Zhihua
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W342175662