A lifted ℓ1 framework for sparse recovery

Abstract

We introduce a lifted $\ell _1$ (LL1) regularization framework for the recovery of sparse signals. The proposed LL1 regularization is a generalization of several popular regularization methods in the field and is motivated by recent advancements in re-weighted $\ell _1$ approaches for sparse recovery. Through a comprehensive analysis of the relationships between existing methods, we identify two distinct types of lifting functions that guarantee equivalence to the $\ell _0$ minimization problem, which is a key objective in sparse signal recovery. To solve the LL1 regularization problem, we propose an algorithm based on the alternating direction method of multipliers and provide proof of convergence for the unconstrained formulation. Our experiments demonstrate the improved performance of the LL1 regularization compared with state-of-the-art methods, confirming the effectiveness of our proposed framework. In conclusion, the LL1 regularization presents a promising and flexible approach to sparse signal recovery and invites further research in this area.