Theoretical analysis for ℓ1-ℓ2 minimization with partial support information

Abstract

We investigate the recovery of k-sparse signals using the ℓ₁-ℓ₂ minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume k-sparse signals x with the prior support T which is composed of g true indices and b wrong indices, i.e., ∣T∣ = g+b ⩽ k. First, we derive a new condition based on RIP of order 2α (α = k − g) to guarantee signal recovery via ℓ₁-ℓ₂ minimization with partial support information. Second, we also derive the high order RIP with tα for some t ⩾ 3 to guarantee signal recovery via ℓ₁-ℓ₂ minimization with partial support information.