Yaohua Hu

Xin Li, Yaohua Hu, Chong Li et al.

In this paper, we discuss the statistical properties of the $\ell_q$ optimization methods $(0<q\leq 1)$, including the $\ell_q$ minimization method and the $\ell_q$ regularization method, for estimating a sparse parameter from noisy observations in high-dimensional linear regression with either a deterministic or random design. For this purpose, we introduce a general $q$-restricted eigenvalue condition (REC) and provide its sufficient conditions in terms of several widely-used regularity conditions such as sparse eigenvalue condition, restricted isometry property, and mutual incoherence property. By virtue of the $q$-REC, we exhibit the stable recovery property of the $\ell_q$ optimization methods for either deterministic or random designs by showing that the $\ell_2$ recovery bound $O(ε^2)$ for the $\ell_q$ minimization method and the oracle inequality and $\ell_2$ recovery bound $O(λ^{\frac{2}{2-q}}s)$ for the $\ell_q$ regularization method hold respectively with high probability. The results in this paper are nonasymptotic and only assume the weak $q$-REC. The preliminary numerical results verify the established statistical property and demonstrate the advantages of the $\ell_q$ regularization method over some existing sparse optimization methods.

Hao Wang, Fan Zhang, Yuanming Shi et al.

We propose a general formulation of nonconvex and nonsmooth sparse optimization problems with convex set constraint, which can take into account most existing types of nonconvex sparsity-inducing terms, bringing strong applicability to a wide range of applications. We design a general algorithmic framework of iteratively reweighted algorithms for solving the proposed nonconvex and nonsmooth sparse optimization problems, which solves a sequence of weighted convex regularization problems with adaptively updated weights. First-order optimality condition is derived and global convergence results are provided under loose assumptions, making our theoretical results a practical tool for analyzing a family of various reweighted algorithms. The effectiveness and efficiency of our proposed formulation and the algorithms are demonstrated in numerical experiments on various sparse optimization problems.