## Group Sparse Optimization via lp,q Regularization

*Yaohua Hu, Chong Li, Kaiwen Meng, Jing Qin, Xiaoqi Yang*; 18(30):1−52, 2017.

### 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 an oracle property and a global recovery bound of
order $\mathcal{O}(\lambda^\frac{2}{2-q})$ 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
$\mathcal{O}(\lambda^2)$ 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 a local linear convergence rate of the
proximal gradient method for solving the $\ell_{1,q}$
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 $\ell_{q}$ 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.

[abs][pdf][bib]