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 is one of the recent trends in this field. 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 is generally high when the size of the problem is 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 evaluated, and simulations indicate that HBROTP admits robustness for signal and image reconstruction even in noisy environments.

翻译：线性逆问题广泛出现于各类工程领域，尤其在信号与图像重建中。针对具有稀疏性的线性逆问题开发计算方法，是当前该领域的研究趋势之一。所谓最优 $k$-阈值法是近年来提出的稀疏优化与线性逆问题求解新方法。与其他稀疏感知算法相比，最优 $k$-阈值法的优势在于能同时执行阈值化操作与误差度量最小化，从而在求解中等规模线性逆问题时表现出稳定性和鲁棒性。然而，当问题规模较大时，该方法的运行时间通常较高。本文旨在提出该方法的加速策略。具体而言，我们针对稀疏线性逆问题提出了基于重球优化的最优 $k$-阈值法算法及其松弛变体。在受限等距性质条件下，证明了这些算法的收敛性。此外，我们评估了基于重球优化的松弛最优 $k$-阈值追踪算法的数值性能，仿真结果表明，即使在噪声环境下，该算法对信号与图像重建仍具有鲁棒性。