Sparse estimation via $\ell_q$ optimization method in high-dimensional linear regression
This provides theoretical guarantees for sparse estimation techniques, which is important for statisticians and data scientists working with high-dimensional data, but it is incremental as it extends existing ℓ_1 methods to ℓ_q frameworks.
The paper tackles the problem of sparse parameter estimation in high-dimensional linear regression using ℓ_q optimization methods, showing that these methods achieve stable recovery with ℓ_2 bounds of O(ε^2) for minimization and O(λ^{2/(2-q)}s) for regularization under weak conditions.
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.