The recently proposed iterative sparsification projection (ISP), a fast and robust sparse signal recovery algorithm framework, can be classified as smooth-ISP and nonsmooth-ISP. However, no convergence analysis has been established for the nonsmooth-ISP in the previous works. Motivated by this absence, the present paper provides a convergence analysis for ISP with soft-thresholding (ISP-soft) which is an instance of the nonsmooth-ISP. In our analysis, the composite operator of soft-thresholding and proximal projection is viewed as a fixed point mapping, whose nonexpansiveness plays a key role. Specifically, our convergence analysis for the sequence generated by ISP-soft can be summarized as follows: 1) For each inner loop, we prove that the sequence has a unique accumulation point which is a fixed point, and show that it is a Cauchy sequence; 2) for the last inner loop, we prove that the accumulation point of the sequence is a critical point of the objective function if the final value of the threshold satisfies a condition, and show that the corresponding objective values are monotonically nonincreasing. A numerical experiment is given to validate some of our results and intuitively present the convergence.

最近提出的迭代稀疏投影（ISP）是一种快速且健壮的稀疏信号恢复算法框架，可以分为平滑ISP和非平滑ISP。但是，在以前的工作中，尚未为非平滑ISP建立收敛分析。出于这种缺席的动机，本文提供了具有软阈值（ISP软）的ISP的收敛性分析，这是非平滑ISP的一个实例。在我们的分析中，软阈值和近端投影的复合算子被视为不动点映射，其不扩展性起着关键作用。具体来说，我们对ISP-soft生成的序列的收敛性分析可总结如下：1）对于每个内部循环，我们证明该序列具有唯一的累加点（即固定点），并证明它是柯西序列; 2）对于最后一个内部循环，我们证明了如果阈值的最终值满足条件，则序列的累加点是目标函数的临界点，并证明相应的目标值是单调递增的。进行了数值实验，以验证我们的一些结果并直观地给出收敛性。