New Restricted Isometry Property Analysis for $\ell_{1}$-$\alpha\ell_{2}$ Minimization

Abstract

In this paper, we consider the $\ell_{1}$-$\alpha\ell_{2}$ minimization method for $\alpha\in(0,2]$. Specifically, we present sparse signal recovery conditions under two types of noise: $\ell_{2}$-bounded noise ($\|\mathbf{v}\|_{2}\,{\boldsymbol\leq}\,\epsilon$ for some constant $\epsilon$) and $\ell_{\boldsymbol\infty}$-bounded noise ($\|\mathbf{A}^{T}\mathbf{v}\|_{\boldsymbol\infty}\,{\boldsymbol\leq}\,\epsilon$ for some constant $\epsilon$). First, based on the RIP framework, we provide a new theoretical guarantee for the successful recovery of sparse signals in the presence of noise when $\alpha\,{\boldsymbol\in}\,(0,1]$, and show that our results are superior to existing ones. Second, we extend the range of $\alpha$ to $(1,2]$ and provide a new theoretical guarantee for sparse signal recovery under the RIP framework. Finally, numerical experiments demonstrate that the recovery success rate for $\alpha\,{\boldsymbol\in}\,(1,2]$ is higher than that for $\alpha\,{\boldsymbol\in}\,(0,1]$.