Sparse signal recovery or compressed sensing can be formulated as certain sparse optimization problems. The classic optimization theory indicates that the Newton-like method often has a numerical advantage over the classic gradient method for nonlinear optimization problems. In this paper, we propose the so-called Newton-step-based iterative hard thresholding (NSIHT) and the Newton-step-based hard thresholding pursuit (NSHTP) algorithms for sparse signal recovery. Different from the traditional iterative hard thresholding (IHT) and hard thresholding pursuit (HTP), the proposed algorithms adopt the Newton-like search direction instead of the steepest descent direction. A theoretical analysis for the proposed algorithms is carried out, and sufficient conditions for the guaranteed success of sparse signal recovery via these algorithms are established in terms of the restricted isometry property of a sensing matrix which is one of the standard assumptions used in the field of compressed sensing and signal approximation. The empirical results from synthetic signal recovery indicate that the performance of proposed algorithms are comparable to that of several existing algorithms. The numerical behavior of our algorithms with respect to the residual reduction and parameter changes is also investigated through simulations.

中文翻译：

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用于稀疏信号恢复的基于牛顿步长的硬阈值算法

稀疏信号恢复或压缩感知可以表述为某些稀疏优化问题。经典优化理论表明，对于非线性优化问题，类牛顿法往往比经典梯度法具有数值优势。在本文中，我们提出了用于稀疏信号恢复的所谓基于牛顿步的迭代硬阈值 (NSIHT) 和基于牛顿步的硬阈值追踪 (NSHTP) 算法。与传统的迭代硬阈值（IHT）和硬阈值追踪（HTP）不同，所提出的算法采用类牛顿搜索方向而不是最速下降方向。对所提出的算法进行了理论分析，根据传感矩阵的受限等距特性，建立了通过这些算法保证稀疏信号恢复成功的充分条件，该特性是压缩传感和信号近似领域中使用的标准假设之一。合成信号恢复的经验结果表明，所提出算法的性能与几种现有算法的性能相当。我们的算法在残差减少和参数变化方面的数值行为也通过模拟进行了研究。合成信号恢复的经验结果表明，所提出算法的性能与几种现有算法的性能相当。我们的算法在残差减少和参数变化方面的数值行为也通过模拟进行了研究。合成信号恢复的经验结果表明，所提出算法的性能与几种现有算法的性能相当。我们的算法在残差减少和参数变化方面的数值行为也通过模拟进行了研究。