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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 theoriginal sparse solution, there are a lot of papers designing alternative modelfor $l_0$-minimization. As one of the most popular choice, the logarithmicalternative model is widely used in many applications. In this paper, wepresent an unified theoretical analysis of this alternative model designed for$l_0$-minimization. By the theoretical analysis, we prove the equivalencerelationship between the alternative model and the original $l_0$-minimization.Furthermore, the main contribution of this paper is to give an unified recoverycondition and stable result of this model. By presenting the local optimalcondition, this paper also designs an unified algorithm and presents thecorresponding convergence result. Finally, we use this new algorithm to solvethe multiple source location problem.

The unified theoretical frame of logarithmic alternative model for  sparse information recovery and its application in multiple source location  problem by TDOA