Group sparse optimization via ℓp,q regularization

Abstract

In this paper, we investigate a group sparse optimization problem via ℓp,q regularization in three aspects: theory, algorithm and application. In the theoretical aspect, by introducing a notion of group restricted eigenvalue condition, we establish an oracle property and a global recovery bound of order O(λ22−q) for any point in a level set of the ℓp,q regularization problem, and by virtue of modern variational analysis techniques, we also provide a local analysis of recovery bound of order O(λ2) for a path of local minima. In the algorithmic aspect, we apply the well-known proximal gradient method to solve the ℓp,q regularization problems, either by analytically solving some specific ℓp,q regularization subproblems, or by using the Newton method to solve general ℓp,q regularization subproblems. In particular, we establish a local linear convergence rate of the proximal gradient method for solving the ℓ1,q regularization problem under some mild conditions and by first proving a second-order growth condition. As a consequence, the local linear convergence rate of proximal gradient method for solving the usual ℓq regularization problem (0<q<1) is obtained. Finally in the aspect of application, we present some numerical results on both the simulated data and the real data in gene transcriptional regulation.