The relaxed optimal $k$-thresholding pursuit (ROTP) is a recent algorithm for linear inverse problems. This algorithm is based on the optimal $k$-thresholding technique which performs vector thresholding and error metric reduction simultaneously. Although ROTP can be used to solve small to medium-sized linear inverse problems, the computational cost of this algorithm is high when solving large-scale problems. By merging the optimal $k$-thresholding technique and iterative method with memory as well as optimization with sparse search directions, we propose the so-called dynamic thresholding algorithm with memory (DTAM), which iteratively and dynamically selects vector bases to construct the problem solution. At every step, the algorithm uses more than one or all iterates generated so far to construct a new search direction, and solves only the small-sized quadratic subproblems at every iteration. Thus the computational complexity of DTAM is remarkably lower than that of ROTP-type methods. It turns out that DTAM can locate the solution of linear inverse problems if the matrix involved satisfies the restricted isometry property. Experiments on synthetic data, audio signal reconstruction and image denoising demonstrate that the proposed algorithm performs comparably to several mainstream thresholding and greedy algorithms, and it works much faster than the ROTP-type algorithms especially when the sparsity level of signal is relatively low.

翻译：松弛最优k阈值追踪算法是近期提出的一种求解线性逆问题的算法。该算法基于最优k阈值技术，该技术可同时执行向量阈值化和误差度量缩减。尽管松弛最优k阈值追踪算法可用于求解中小规模线性逆问题，但在处理大规模问题时其计算成本较高。通过融合最优k阈值技术、带记忆的迭代方法以及稀疏搜索方向优化，我们提出了所谓的带记忆动态阈值算法。该算法通过迭代且动态地选择向量基来构建问题解。在每一步中，算法利用当前已生成的一个以上（或全部）迭代点来构造新的搜索方向，且每次迭代仅需求解小规模二次子问题。因此，带记忆动态阈值算法的计算复杂度显著低于松弛最优k阈值追踪类方法。结果表明，当相关矩阵满足约束等距性质时，带记忆动态阈值算法能够定位线性逆问题的解。在合成数据、音频信号重建和图像去噪上的实验表明，所提算法与多种主流阈值算法和贪婪算法性能相当，且其运行速度远快于松弛最优k阈值追踪类算法，尤其在信号稀疏度相对较低时更为显著。