  In sparse information recovery, the core problem is to solve the
$l_0$-minimization which is NP-hard. On one hand, in order to recover the
original sparse solution, there are a lot of papers designing alternative model
for $l_0$-minimization. As one of the most popular choice, the logarithmic
alternative model is widely used in many applications. In this paper, we
present an unified theoretical analysis of this alternative model designed for
$l_0$-minimization. By the theoretical analysis, we prove the equivalence
relationship between the alternative model and the original $l_0$-minimization.
Furthermore, the main contribution of this paper is to give an unified recovery
condition and stable result of this model. By presenting the local optimal
condition, this paper also designs an unified algorithm and presents the
corresponding convergence result. Finally, we use this new algorithm to solve
the multiple source location problem.
