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

The ratio of ℓ₁ and ℓ₂ norms, denoted as ℓ₁/ℓ₂, has presented prominent performance in promoting sparsity. By adding partial support information to the standard ℓ₁/ℓ₂ minimization, in this paper, we introduce a novel model, i.e. the weighted ℓ₁/ℓ₂ 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 ℓ₁/ℓ₂ minimization are established. And we show that the proposed conditions are weaker than the analogous conditions for standard ℓ₁/ℓ₂ 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.