Group sparse optimization via ${\ell }_{p,q}$ regularization

Abstract

In this paper, we investigate a group sparse optimization problem via ${\ell }_{p,q}$ regularization in three aspects: theory, algorithm and application. In the theoretical aspect, by introducing a notion of group restricted eigenvalue condition, we establish some oracle property and a global recovery bound of order $O({\lambda }^{\frac{2}{2-q}})$ for any point in a level set of the ${\ell }_{p,q}$ regularization problem, and by virtue of modern variational analysis techniques, we also provide a local analysis of recovery bound of order $O({\lambda }^2)$ for a path of local minima. In the algorithmic aspect, we apply the well-known proximal gradient method to solve the ${\ell }_{p,q}$ regularization problems, either by analytically solving some specific ${\ell }_{p,q}$ regularization subproblems, or by using the Newton method to solve general ${\ell }_{p,q}$ regularization subproblems. In particular, we establish the linear convergence rate of the proximal gradient method for solving the ${\ell }_{1,q}$ regularization problem under some mild conditions. As a consequence, the linear convergence rate of proximal gradient method for solving the usual ${\ell }_q$ 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.

arXiv comment48 pages, 7 figures

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