  Motivated by re-weighted $\ell_1$ approaches for sparse recovery, we propose
a lifted $\ell_1$ (LL1) regularization which is a generalized form of several
popular regularizations in the literature. By exploring such connections, we
discover there are two types of lifting functions which can guarantee that the
proposed approach is equivalent to the $\ell_0$ minimization. Computationally,
we design an efficient algorithm via the alternating direction method of
multiplier (ADMM) and establish the convergence for an unconstrained
formulation. Experimental results are presented to demonstrate how this
generalization improves sparse recovery over the state-of-the-art.
