  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 $O(\epsilon^2)$ for the $\ell_q$ minimization
method and the oracle inequality and $\ell_2$ recovery bound
$O(\lambda^{\frac{2}{2-q}}s)$ 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.

Machine Learning;Machine Learning;Optimization and Control