The hard thresholding technique plays a vital role in the development of algorithms for sparse signal recovery. By merging this technique and heavy-ball acceleration method which is a multi-step extension of the traditional gradient descent method, we propose the so-called heavy-ball-based hard thresholding (HBHT) and heavy-ball-based hard thresholding pursuit (HBHTP) algorithms for signal recovery. It turns out that the HBHT and HBHTP can successfully recover a $k$-sparse signal if the restricted isometry constant of the measurement matrix satisfies $\delta_{3k}<0.618 $ and $\delta_{3k}<0.577,$ respectively. The guaranteed success of HBHT and HBHTP is also shown under the conditions $\delta_{2k}<0.356$ and $\delta_{2k}<0.377,$ respectively. Moreover, the finite convergence and stability of the two algorithms are also established in this paper. Simulations on random problem instances are performed to compare the performance of the proposed algorithms and several existing ones. Empirical results indicate that the HBHTP performs very comparably to a few existing algorithms and it takes less average time to achieve the signal recovery than these existing methods.

翻译：硬阈值技术在发展稀有信号恢复的算法方面发挥着关键作用。 通过合并这一技术和重球加速法,这是传统梯度下降法的多步延伸,我们建议采用所谓的重球硬阈值和重球硬阈值算法,以恢复信号。结果显示,如果测量矩阵的限定的偏差常数分别满足了$\delta ⁇ 3k ⁇ 0.618美元和$\delta ⁇ 3 ⁇ 3k ⁇ 3k ⁇ 0.577美元,那么,硬球技术在开发信号恢复算法方面发挥着至关重要的作用。我们建议采用所谓的重球硬阈值和重球硬阈值算法,用于保证成功。此外,HBHT和HBTP的有限趋同和稳定性在本文中也能够成功地恢复。对随机问题实例进行了模拟,以比较拟议的算法和现有的若干现行算法的绩效。Empricalalalal结果显示,HBT和HTP的现有平均恢复方法比其现有平均恢复率要低。