Linear inverse problems arise in diverse engineering fields especially in signal and image reconstruction. The development of computational methods for linear inverse problems with sparsity tool is one of the recent trends in this area. The so-called optimal $k$-thresholding is a newly introduced method for sparse optimization and linear inverse problems. Compared to other sparsity-aware algorithms, the advantage of optimal $k$-thresholding method lies in that it performs thresholding and error metric reduction simultaneously and thus works stably and robustly for solving medium-sized linear inverse problems. However, the runtime of this method remains high when the problem size is relatively large. The purpose of this paper is to propose an acceleration strategy for this method. Specifically, we propose a heavy-ball-based optimal $k$-thresholding (HBOT) algorithm and its relaxed variants for sparse linear inverse problems. The convergence of these algorithms is shown under the restricted isometry property. In addition, the numerical performance of the heavy-ball-based relaxed optimal $k$-thresholding pursuit (HBROTP) has been studied, and simulations indicate that HBROTP admits robust capability for signal and image reconstruction even in noisy environments.

翻译：线性反问题广泛出现在各类工程领域，尤其在信号与图像重构中。利用稀疏性工具求解线性反问题的计算方法已成为该领域近年来的研究热点。所谓最优$k$阈值法，是一种新近提出的稀疏优化与线性反问题求解方法。与其他稀疏感知算法相比，最优$k$阈值法的优势在于能够同步执行阈值处理与误差度量缩减，从而在求解中等规模线性反问题时保持稳定且鲁棒的性能。然而，当问题规模较大时，该方法的运行时间仍较高。本文旨在提出该方法的加速策略：具体而言，针对稀疏线性反问题，我们提出了基于重球的最优$k$阈值(HBOT)算法及其松弛变体。在约束等距性条件下，证明了这些算法的收敛性。此外，对基于重球的松弛最优阈值追踪(HBROTP)算法进行了数值性能评估，仿真结果表明，即使在噪声环境下，HBROTP仍具备鲁棒的信号与图像重构能力。