The log-sum function as a penalty has always been drawing widespread attention in the field of sparse problems. However, it brings a non-convex, non-smooth and non-Lipschitz optimization problem that is difficult to tackle. To overcome the problem, an iterative threshold algorithm for the sparse optimization problems with log-sum function is proposed in this paper. For brevity, the sparse optimization problems with log-sum function are named log-sum regularization. Firstly, by introducing an intermediate function to construct another new function, a property theorem about solution for log-sum regularization is established. Secondly, based on the above theorem, the optimal setting rules of the compromising parameters are elaborated, and an iterative log-sum threshold algorithm is proposed. Thirdly, under the situation that the compromising parameters of log-sum regularization are relatively small, it can be proven that the proposed algorithm converges to a local minimizer of log-sum regularization. Finally, a series of simulations are implemented to examine performance of the proposed algorithm, and the results exhibit that the proposed algorithm outperforms the state-of-the-art algorithms.