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

In this paper, we consider the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell_{1}-\alpha\ell_{2}$</tex-math></inline-formula> minimization method for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha \in (0, 2]$</tex-math></inline-formula>. Specifically, we present sparse signal recovery conditions under two types of noise: <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell_{2}$</tex-math></inline-formula>-bounded noise (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\|\mathbf{v}\|_{2}\leq\epsilon$</tex-math></inline-formula>) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell_{\infty}$</tex-math></inline-formula>-bounded noise (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\|\mathbf{A}^{T}\mathbf{v}\|_{\infty}\leq\epsilon$</tex-math></inline-formula>). First, based on the RIP framework, we provide a new theoretical guarantee for the successful recovery of sparse signal in the presence of noise when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha\in(0,1]$</tex-math></inline-formula>, and show that our results are superior to existing ones. Second, we extend the range of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(1,2]$</tex-math></inline-formula> and provide a new theoretical guarantee for sparse signal recovery under the RIP framework. Finally, numerical experiments demonstrate that the recovery success rate for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha\in(1,2]$</tex-math></inline-formula> is higher than that for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha\in(0,1]$</tex-math></inline-formula>.