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

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.