Analysis of the ratio of ℓ1 and ℓ2 norms for signal recovery with partial support information

Abstract

The ratio of $\ell _{1}$ and $\ell _{2}$ norms, denoted as $\ell _{1}/\ell _{2}$, has presented prominent performance in promoting sparsity. By adding partial support information to the standard $\ell _{1}/\ell _{2}$ minimization, in this paper, we introduce a novel model, i.e. the weighted $\ell _{1}/\ell _{2}$ minimization, to recover sparse signals from the linear measurements. The restricted isometry property based conditions for sparse signal recovery in both noiseless and noisy cases through the weighted $\ell _{1}/\ell _{2}$ minimization are established. And we show that the proposed conditions are weaker than the analogous conditions for standard $\ell _{1}/\ell _{2}$ minimization when the accuracy of the partial support information is at least $50\%$. Moreover, we develop effective algorithms and illustrate our results via extensive numerical experiments on synthetic data in both noiseless and noisy cases.