A Sufficient Condition of Non-Convex ${\ell }_{p} -\beta {\ell }_{q}$ Minimization for Sparse Recovery

To recover sparse signals via <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\ell }_{p}-\beta {\ell }_{q}$</tex-math></inline-formula> minimization with parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$0 &lt; p\, \leq \, 1,\, 1\, \leq \, q\, \leq \, 2\,\ (p\,\ne q)$</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">$0 \leq \beta \leq 1$</tex-math></inline-formula>, this paper employs the Restricted Isometry Property (RIP) and Restricted Orthogonality Property (ROP) to investigate sparse signal recovery in the noise setting. Based on these analyses, a sufficient condition is proposed, which generalizes and improves state-of-the-art results. In Section III, several key remarks are presented. Our results also demonstrate that the derived condition outperforms the existing ones.